VESSEL DESIGN
I
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d
LLOYD E. BROWNELL Professor
of
Chemical
and
Nuclear
University
Engineering of
Michigan
EDWIN H. YOUNG Associate of
Reg.
pu’o...@.
1
. . ACC.
No..
Chemical
and
Metallurgical University
357
Professor Engineering
of
Michigan
Lib. Asstt . . . . . . . . . . . . . . . r/C . . . . . . . . . .
JOHN WILEY & SONS New York
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Chichester Brisbane Toronto l
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Singapore
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19
18
17
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14 (r
C o p y r i g h t @ 1959 by John Wiley
.411
8 Sons, Inc.
rights reserved.
Reproduction or translation of any part of thi\ work beyond that permitted by Sections 107 or 108 of the 1976 Unlted States Copyrtght Act without the permisbton of the copyright owner is unlawful. Requests for perm!sston or further tnformatton should be addressed to the Permissions Department, John Wiley&Sons, Inc.
l i b r a r y o f C o n g r e s s C a t a l o g C a r d N u m b e r : 5?--5882 Printed
in
the
United
States
of
America
ISBN 0 4 7 1 1 1 3 1 9 0
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PREFACE
This book was prepared primarily for se~io-and students in engineering. The needs of design engineers and consultants as well as those of students were considered in selecting the topics and methods of presentation. The book is based upon our experiences gained in industrial design offices and in 16 years of teaching courses in equipment design at the University of Michigan. We both have supervised research and development of process equipment, and have acted as consultants in this field. The book was originally prepared as class notes, which have been used for about ten years in teaching courses in process equipment design at the senior and graduate levels in the Chemical and Metallurgical Engineering Department of the University of Michigan. Typical problems have involved the design of fraction@ingt o w e r s , trm vacu>m cry&all&m, condensers, heat exch-rs, high-pres?re reactors, and other types of process equipment. The design of process equipment requires a thorough knowledge of the functional process, the materials ___- ._.... involved, and the methods of fabrication. The design factors to be considered are many and varied and, in most cases, so interwoven that exact methods of attack are often impossible to formulate. Compromises are necessary and the design engineer often has only experience in similar or related fields to guide him in his choice. Thus, the engineer must realize that considerable engineering judgment is required in applying all recommended specific methods of design. One purpose of this book is to consolidate the basic concepts, industrial practices, and theoretical relationships useful in the design of processing equipment. Many of these considerations and much of this vital information are widely scattered throughout the technical literature, industrial bulletins, appropriate codes, and handbooks. It is not intended that this book should cover all the ramifications of design problems, but it will serve as a guide to the student and the practicing engineer for efficient and economical design of equipment for the processing industries. vii
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... VIII
Preface
The organization is based on the premise t)hut t,he vessel is the basic part of most types of processing equipment. For example, a heat exchanger or evaporator is a vessel with tube bundles and a fractionating tower is a vessel with trays. The first 12 chapters are concerned in part with the development of fundamental relationships on which many of the code specifications are based. Chapter 13 is concertled entirely with code practice and covers selected code specifications not covered in the earlier chapters. Chapters 14 and 15 are concerned with the design of vessels beyond the scope of the ASME code. The sequence of chapters was selected to permit the introduction of a briet review of elementary theories of mechanics and strength of materials early in the book. More advanced theory is developed as needed in subsequent chapters. The integration of theory with practice in design eliminates the necessity of a separate section on erigineeritlg mechanics. The sequence of presentation allows for an orderly development of theoretical relatiotlships when the book is being used as a textbook in teaching design. The material presented covers the rauge from simple vessels for low-pressure service to thick-walled vessels for highpressure applications. Tl~tf rxperierlced designer will find the book useful as a reference in a design office. In all but a few cases derivations of equations and the method of analysis have been given so that the etlgirleer will utlderstaltd the assumptions and limitations involved. Also, example calculations and designs have been included to illustrate the use of the relationships and recommended procedures. We wish to acknowledge the assistance given by a large number of individuals and companies in providing subject material and illustrations on process equipment, design and in making reviews and suggestions. We are particularly iudebted to the following: C. E. Freese, Mechanical Consultant, and B. B. Kuist, The Fluor Corporation; W. H. Burrows, Chief Engineer, Manufacturing Department, Standard Oil Company of Indiana; A. E. Pickford, Department Head, .dpparatus Design, C. F. Braun and Company; H. B. Boardman, Director of ltesearch, L. P. Zick, Research Engineer, and E. N. Zimmerman, Chicago Bridge and Iron Company; W. T. Gur m and Walter Samans, American Petroleurn Institute; J. M. Evans, Chief Engineer, and F. L. Maker, Standard Oil Company of California; R. S. Justice, Chief Engineer, Gulf Oil Corporation; F. L. Plummer, Director of Engineering, Hamrnond Iron Works; W. D. Kinsell, Manager, Construction, Engineering Department, The Pure Oil Company; G. E. Fratcher, Director of Engineering, A. 0. Smith Company; F. E. Wolosewick, Sargeut and Lundy Engineers; P. E. Franks, Chief Engineer, Sinclair Refining Cornpany; D. W. Carswell and H. B. Peters, Chief Engineer, The Texas Company; W. T. Brown, Manager, Mechanical Division, and Harry Wearne, Construction Manager, Shell Oil Cornpany ; F. J. Feeley, Jr., Assistant Director, Engineering Design Division, Esso Hesearch aud Engineering Company; J. H. Faupel, E. I. du Pont de Nemours and Company; W. H. Funk, Lukens Steel Company; and the following additional companies and organizat3ons: Horton Steel Works, Ltd.; BlawKnox Company; Graver Tank and Manufacturing Company; American Cyanamid Cornpany; Inland Steel Company; Ryerson Steel Company; Taylor Forge and Pipe Works; Aluminum Company of America; M. W. Kellogg Company; American Standard Association, Inc.; The Girdler Company, Inc.; Baldwiu-LimaHamilton Corporation; Bethlehem Steel Compally, Inc.; ITnited States Department of Interior, Bureau of Miues; Great Lakes Steel Corporation; McGraw-Hill
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Preface
ix
Book Company, Inc.; I‘uivrrsal-Cyclops Steel Corporatiorr: at~d the United States Steel Corporation. We also wish to express our appreciation to the Amrricau Society of Mechanical Engineers and the American Petroleum Institute for permissiou to use selected material from the 1956 edition of the Unfired Pressure L‘essel Code and the API Specification for Welded Oil Storage Tanks and Production Tanks, respectivei). We are also indebted to Dr. J. McKetta, Mr. F. L. Standiford, Dr. H. H. Yang, and Dr. M. D. S. Lay, who assisted in the preparation of the course notes while enrolled in the Graduate School of the University of Michigan, and to Professor Donald L. Katz, Chairman, Department of Chemical and Metallurgical Engineering, University of Michigan, for encouragement and advice in the preparation of this book. Many individuals have given valuable suggestions, comments, and assistance in the preparation of this book and ally omissions irr ackuowledgment are not iutended. LLOYD E. BROWNELL EDMIK i-z. YOUNG Ann Arbor, Michigm! April, 1959
CONTENTS
.. c
Chapter 1
Factors
Influencing
the
Design
2
Criteria
in
Vessel
Design
3
Design
of
Shells
4
Design of Bottoms and Roofs for Flat-Bottomed Cylindrical
for
of
Vessels
1 19
Flat-Bottomed
Cylindrical
Vessels
36
Vessels
5
58
Proportioning Formed
6
Stress
Stress cal,
Head
for
in
Considerations and
the
Cylindrical in
Cylindrical
of
under
External
Pressure
9
Design
of
Vertical
10
Design
of
11
Design
of
12
De&n
of Flanges
13
Design
of
14
High-Pressure
15
Multilayer
Tall
Supports
B.
with
Pressure
of
Flat-Plate
and
Conical
98 of
Closures with
Elliptical, for
Formed
Torispheri-
Cylindrical Closures
Vessels
120
Operating
141 Vessels
155
Vertical
Vessels
Vessels
with
Saddle
183 Supports
203 219
Vessels
Monobloc
to
Code
Vessels
Vessels
Specifications
249 268 296
Design Welding
Conventions
323
Conventions
327 xi
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Vessels
317
Appendix A.
I
Cylindrical
Selection
Vessels
for
Horizontal
Selection
Dished
References
r
for
Vessels the
Hemispherical
Design
8
Selection
76
Considerations
Closures
7
and
Closures
\I /
xii
Contents C.
Pricing of Steel Plate
330
D.
Allowable Stresses
335
E.
Typical
F.
Shell
G.
Properties
H.
Values of Constant C of Eq. 13.27
I.
Tank
Sizes
and
Capacities
346
Accessories of
349
Selected
Rolled
Structural
Members
353 362
Charts for Determining Shell Thickness of Cylindrical and Spherical
Vessels
J.
Properties
of
Various
K.
Properties
of
Pipe
1.
Strength
of
under
Materials
External
Sections
and
Pressure Beam
364 Formulas
381 386 392
Author Index
395
Subject Index
399
C H A P T E R
0 1
FACTORS INFLUENCING THE DESIGN OF VESSELS
1.1 SELECTION OF THE TYPE OF VESSEL
e
hemical engineering involves the application of the sciences to the process industries which”‘&e primarily concerned with the conversion of one material into another dy chemical or physical means. These processes require the handling and storing of large quantities of materials in containers of varied construction, depending upon the existing state of the material, its physical and chemical properties, and the required operations which are to be performed. For handling such liquids and gases a container, or “vessel,” is used. Thzyessel is the basic part of most types of processing equipment. Most process equipment units may be considered to be vessels with various modifications necessary to e ble the units to perform certain required functions. x or example, an a_utoclave may be considered to be a high-pressure vessel equipped with agitation and heating sources; a distillation or absorption column may be considered to be a vessel containing a series of vapor-liquid contactors; a heat exchanger may be considered to be a vessel d-containing a suitable provision for the transfer of heat through tube walls; and an evaporator may be considered to be a vessel containing a heat exchanger in combination with a vapor-disengaging space. Regardless of the nature of the application of the vessel, a number of factors usually must be considered in designing the unit. The most important consideration often is the selection of the type of vessel that performs the required service in the most satisfactory manner. In developing the design a number of other criteria must be considered, such as the properties of the material used, the induced stresses, the elastic stability, and the aesthetic appearance of the unit. The cost of the fabricated vessel is also important in relation to its service and useful life.
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Usually the first step in the design of any vessel is the selection of the type best suited for the particular service ipquestio~.- The primary factors influencing this choice are: the function and location of the vessel, the nature of the fluid, the operating temperature and’pressure, and the necessary volume for storage or capacity for processing. Vessels may be classified according to service, temperature and pressure service, matkrials of construction, or geometry of the vessel. The most common types of vessels may be classified according to their geometry as: ‘1. Open tanks. . Flat-bottomed, vertical cylindrical tanks. 23. Vertical cylindrical and horizontal vessels with formed ends. 4. Spherical or modified spherical vessels.
1
Vessels in each of these classifications are widely used as s~o%& vessels and as processing vessels for fluids. The range of service for the various types of vessels overlaps, and it is difficult to make distinct classifications for all applications. It is possible to indicate some generalities in the existing uses of lhe common types of vessels. Large volumes of nonhazardous liquids, such as brine and other aqueous solutions, may be stored in ponds if of very low value, or in open steel, wooden, or concrete tanks if of greater value. If the fluid is toxic, combustible, or gaseous in the storage condition, or if the pressure is greater than atmospheric, a closed system is required. For storage of fluids at atmospheric pressure, cylindrical tanks with flat bottoms and conical roofs are commonly used. Spheres or spheroids are
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Factors Influencing the Design of Vessels
I-
lo’-0”
Inside diameter
I
:,t
lo’-0” UnAred Pressure Vessel 150 Ib.sq Fig. 1.1.
in. at 85O’F
Example of a cylindrical vessel with formed ends designed to the original API-AWE code. __----
employed for pressure storage where the volume required is large. For smaller volumes under pressure, cylindrical tanks with formed heads are ,more economical. l.la Open Vessels. Open vessels are commonly used as surge tanks between operations, as vats for batch operations where materials may be mixed and blended, as settling tanks, decanters, chemical reactors, reservoirs, and so on. Obviously, this type of vessel is cheaper than covered or closed vessels of the same capacity and construction. The decision as to whether or not open vessels may be used depends upon the fluid to be handled and the operation. Very large quantities of aqueous liquids of low value may be stored in ponds. It is doubtful if ponds may be correctly referred to as vessels. They are, however, the simplest
(Courtesy of Amer. Pet. Inst.!
containers made from the cheapest of materials, rolled earth. Not all types of earth can be used for storage ponds; a clay which will form an almost watertight bottom is essential. An example of the use of ponds of rolled earth is found in the process whereby salt is crystallized from sea water by solar evaporation (1). When more valuable fluids are handled, more reliable but more expensive containers are required. Large circular tanks of steel (2) or reinforced (or prestressed) concrete (3), (4) are often used for settling ponds in which a slowly rotating rake removes sediment from a slightly inclined conical bottom. Vessels of this type, as exemplified by the Dorr classifier, may have diameters ranging from 100 to 200 ft and a depth of several feet. Smaller open vessels are usually of a circular shape and
Selection of the Type of Vessel
ardize design for purposes of safety and economy. Tanks used for the storage of crude oils and petroleum products are generally designed and constructed in accordance with API Standard 12 C, API Specification for Welded OilStorage Tanks. This is the standard reference used in designing tanks for the petroleum industry, but it is also a useful guide for other applications. CYLINDRICAL VESSELS WITH FLAT BOTTOMS AND CONICAL OR DOMED ROOFS. The most economical design for a closed vessel operating at atmospheric pressure is the vertical cylindrical tank with a conical roof and a flat bottom resting directly on the bearing soil of a foundation comIn cases where it is posed of sand, gravel, or crushed rock. desirable to use a gravity feed, the tank is raised above the ground, and the flat bottom may be supported by columns and wooden joists or steel beams. Cylindrical, flat-bottomed, cone-roofed tanks are provided with “breathers” or vents which permit expansion and contraction of the fluids as a result of temperature and volume fluctuations. Tanks up to 24 ft in diameter may be covered with a self-supporting roof; tanks with larger diameters, up to 48 ft, usually require at least one central column for support. Tanks larger than 48 ft in diameter are frequently designed with multiplecolumn supports or with a floating or pontoon roof which rises and falls with the level of liquid in the vessel. In general, tanks with conical roofs are limited to essentially atmospheric pressure. If domed roofs are used, pressures from 244 to 15 lb per sq in. gage may be permitted. These vessels are normally smaller in diameter and of greater height for a given capacity than tanks with conical roofs (8, 9).
are constructed of mild carbon steel, concrete, and sometimes of wood (5). Other materials find limited use where serious corrosion or contamination problems are encountered. However, in the process industries in general, the major portion of existing vessels are constructed of steel because of its low initial cost and ease of fabrication. In many cases such vessels are lined with lead, rubber, glass, or plastic to improve resistance to corrosion. In the food industry fir is commonly used for pickle and kraut tanks, whereas quarter-sawed white oak is employed for wine and spirits. Redwood or Cyprus tanks are often employed for water storage reservoirs. Wood is also used in place of steel for handling dilute solutions of hydrochloric, lactic, and acetic acids and salt solutions and is indispensable as a lowcost tank in the tanning, brewing, and pickling industries (6). In the food and pharmaceutical industries it is often necessary to add materials to open vessels in the preparation of mixtures. Small open tanks or kettles are usually employed for such purposes. Glass-lined steel, copper, Monel, and stainless steel tanks are widely used in these applications to resist corrosion and prevent contamination of the process materials. 1 .l b Closed Vessels. Combustible fluids, fluids emitting toxic or obnoxious fumes, and gases must be stored in closed vessels (7). Dangerous chemicals, such as acid or caustic, are less hazardous if stored in closed vessels. The combustible nature of petroleum and its products necessitates the use of closed vessels and tanks throughout the petroleum and petrochemical industries. The extensive use of tanks in this field has resulted in considerable effort on the part of the American Petroleum Institute to stand-
Fig. 1.2. Oil refinery installation.
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(Courtesy of C. F. Braun & Company.)
/
4
Factors Influencing the Design of Vessels CY L I N D R I C A L
V ESSELS
one type or another. Figure 1.2 shows a wide variety of such items in a petroleum refinery. Note that nearly all of the processing equipment shown consists of cylindrical vessels with formed ends. SPHERWAL AND M ODIFIED S PHERICAL V ESSELS. Storage containers for large volumes under moderate pressure are usually fabricated in the shape of a sphere or spheroid. Capacities and pressures used in this type of vessel vary greatly. Capacity ranges from 1000 t.o 25,000 bbl, and pressures range from 10 lb per sq in. gage for the larger vessels to 200 lb per sq in. gage for the smaller ones. Figure 1.3 shows a battery of horizont.al cylindrical vessels and spherical vessels for storing petroleum products at pressures up to 100 lb per sq in. gage.
F ORMED E N D S . Closed
WITH
cylindrical vessels with formed heads on both ends are used where the vapor pressure of the stored liquid may dictate a stronger design. Codes have been developed through the efforts of the American Petroleum Institute (10) and the American Society of Mechanical Engineers (11) to govern the design of such vessels. These vessels are usu$ly less than 12 ft in diameter if they are to be shipped by rail. However, field-erected vessels may exceed 35 ft in diameter and 200 ft in length. If a large quantity of liquid is to be stored, a batt,ery of vessels may be used. A variety of formed heads arca used for closing I hta ends of cylindrical vessels. The formed heads include the hemispherical, elliptical-dished, torispherical, standard-dished,
Fig. 1.3.
Spherical and horizonial
storage tanks at Crown Central Petroleum Plant near Houston, Texas.
conical, and toriconical shapes. For special purposes flat plates are used to close a vessel opening. However, flat he;ids are rarely ustd for large vessels. For pressures uot covered by the ASME code, the vessels are often equipped with standard dished heads, whereas vessels that require code construction are usually equipped with either the ASME-dished or elliptical-dished heads. The most common shape for the closure of “pressure vessels” is the elliptical dish. Figure 1.1 shows a drawing of a vertical cylindrical vessel with formed ends designed to the original API-ASME code. Most chemical and petrochemical processing equipment such as distilling columns, desorbers, absorbers, scrubbers, heat exchangers, pressure-surge tanks, and separators are essentially cylindrical closed vessels with formed ends of
-_
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--_-
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7- _
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..-,
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r-.‘
(Courtesy of Hammond Iron Works.)
Where a given mass of gas is to be stored under pressure, it is obvious that the required storage volume will be inversely proportional to the storage pressure. In general, for a given mass the spherical type of tank is more economi cal for large-volume, low-pressure storage operation. At higher storage pressures, the volume of gas is reduced, and therefore the cylindrical type of storage vessel becomes more economical. If allowance is made for the cost of compression and cooling of the gas, some of this apparent saving is lost. When handling small masses of gas, there is an advantage in the use of cylindrical storage vessels because the cost of fabrication becomes the controlling factor and small cylindrical vessels are more economical than small spherical vessels. Further economy can sometimes be realized by using
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Methods
Fig. 1.4. gage.
h
Two multispheres for storage of nitrogen under 400 lb per sq in. (Courtesy of Chicago Bridge
& Iron Company.)
modified spherical vessels such as the two multispheres shown in Fig. 1.4. These storage vessels were designed to handle nitrogen at 400 lb per sq in. gage working pressure. Modified spherical vessels are also used for storage of large Large ellipsoidal vesvolumes under moderate pressures. sels have been built to hold 55,000 bbl at a pressure of 75 lb per sq in. gage. The largest vessels for storage under pressure are the semi-ellipsoidal tanks, which have been made to hold as much as 120,000 bbl at a pressure of 235 lb per sq in. gage. As the capacity of an individual vessel is increased, the pressure that the vessel can safely maintain (wit,hout very heavy construction) c!ecreases. A hemispheroid with a capacit,y of 20,000 hhl of natural gasoline at a working pressure of 2 35 lb per sq in. gage is shown in Fig. 1.5.
of
Fabrication
5
and greater reliability as compared with cast iron, it i s m o r e suitable for high-pressure service where metal porosity is not a problem. The vessel diameter is still limiting because of problems in casting. Alloy cast-steel vessels can be used for high-temperature and high-pressure installations. Form is a method of shaping metal that is commonly usedfor certain vessel parts such as closures, flanges, and fittings. Vessels with wall thicknesses greater than 4 in. are often forged. Ot$her special methods of shaping metal, such as pressing, spinning, and rolling of plates, are used for forming closures for vessel shells and are discussed later in the text. Sheet-metal forming is similar to pressing in that metal is shaped by means of presses and dies, but this method is limited to relatively thin stock. The process of sheet-metal forming as a method of vessel fabrication finds its greatest application in the field of nonferrous metals such as copper, Monel, and stainless steel, where cost considerations often preclude the use of heavier stock. Riveting was widely used, prior to the improvement of modernwelding techniques, for many different kinds of vessels, such as storage tanks, boilers, and a variety of pressure vessels (12). It is still used for fabrication of nonferrous vessels such as copper and aluminum. However, welding techniques have become so advanced that even these materials are often welded today. Because of the trends away from riveted construction, the designs based upon riveting as a method of fabrication will not be discussed in this text. M_achining is the only method other than cold forming that can be used to secure exact tolerances. Close tolerances are required for the mating parts of equipment.. Flange --.- faces, bushings, and bearing surfaces are usually machined in order to provide satisfactory alignment. Laboratory and pilot plant equipment for very-high-pressure service is sometimes machined from solid stock, pierced ingots, and forgings. Multilayer vessels for high-pressure
1.2 METHGCS OF FABRICATION
,
Process equipment _____ - . .is^_- fabricatel b”y a _ num$er.o&r& estam’methods such as fusion welding, casting, forging, machining, brazing and soldering, and sheet-metal forming. __. . Each method has certain advantages for particular types of equipment. However, fusion welding is the most important method. The size, shape, service, and material properties of t.he equipment all may influence the selection ‘of the fabrication method. GLet-irqn castings have been widely used for the mass production of small pipe fittings and are used to a considerable extent for larger items such as cast-iron pipe, heatexchanger shells, and evaporator bodies because of the superior corrosion resistance of cast iron as compared with steel. Large-diameter vessels cannot be easily cast, and the strength of gray iron is not reliable for pressure-vessel service. C& steel may be used for small-diameter thickwalled vessels. Furthermore, because of its higher strength
Fig. 1.5.
A
20,030-bbl
diameter by 35 ft high. pressure.
hemispheroid
gasoline-storage
Designed for 2% lb per
(Courtesy of Chicago Bridge
tank
$9 in. gage
64
ft
in
woridng
& Iron Company.)
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----
6
Factors Influencing the Design of Vessels
Fig. 1 . 6 . W e l d i n g e x t e r n a l
circum-
fere ntiol seam of shell of large vessel with automatic
welder.
(Courtesy
of
C. F.Braun 8 Company.)
services may be fabricated by machining a series of concentric shells and shrink fitting for producing desirable prestress conditions. This method of vessel fabrication is discussed in a later section of the text. In general, machining is an expensive operation and is limited to small vessels and parts in which the cost can be justified. 1.2a Fusion Wejam. Fusion welding is the most widely used method of fabrication for the construction of steel vessels (12). This method of construction is virtually unlimited with regard to size and is extensively used for the fabrication and erection of large-size process equipment in the field. Often such equipment is fabricated by the method of subassembly. In this process, sections of the unit are shop welded and then assembled in the field. Equipment having a size sufficiently small to permit transportation by trucks, rail, or barge is usually completely shop welded beta se of the lower cost and greater control of the welding p cedure in the shop. L/J There are two tvpes of fusion welding t-hat are exte.nsively used for the fabrrqation of vessels. These are: (1) the gas welding ‘&ocess, in which a combustible mixture of acetylene and oxygen supply the necessary heat for fusion, and (2) the electric-arc welding process, in which the heat of fusion is ‘supplied by an electric current (13, 14, 15, 16). Arc welding is the preferred process because of the reduction of heat in the material being welded, the reduction of oxidation, and better control of the deposited weld metal. A wide range of arc-welding equipment is available, from the small portable welding units to the large automatic welding machines. Small arc-welding machines are widely used in welding shops that fabricate small equipment whereas the
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automatic machines are better suited for the welding of heavy sections involving the deposition of a large quantity of weld metal. Figure 1.6 illustrates the use of an automatic welding machine in fabricating a large-diameter vessel. Gas welding is the preferred type of welding for light gages of metal (20 gage or less), which are difficult to weld by the arc-welding process. Gas welding equipment is extremely useful in flame cutting either in the field or in the shop. One of the most recent and successful developments in the field of arc welding of vessels is the submerged-arc welding process (17). This process was virtually unknown at the beginning of World War II. The necessity of expediting production of welded equipment during the war years resulted in the realization of the advantages of this technique. The process involves submerging of the arc beneath a blanket of granulated mineral flux. The, arc beneath the blanket generates heat to melt the electrode and deposits weld metal. A portion of the granulated flux melts, forming a protective layer on the weld metal, and solidifies with the weld metal. In addition to completely protecting the weld metal from the atmosphere, this process makes the weld metal virtually free of hydrogen. As the arc is covered, there is no arc flash, and also a lesser quantity of smoke and obnoxious fumes is produced as compared with the earlier welding processes. As the weld can not be observed by the operator, mechanical attachments are used to control the dimensions of the weld. Several inches of weld metal can be deposited in one pass, a fact which greatly decreases the welding time involved. However, the greatest advantage
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-..
--P -
Methods
1
<
of the submerged-arc process is t.he elimination of the operatyble. ‘w, 1.2b Welding Standards. The success of fabrication by welding is dependent upon the control of the welding variables such as exFrience and training of the welder, the An inexuse of proper materials, and welding procedures. perienced welder or a welder using inferior materials or incorrect procedures can fabricate a vessel that has good appearance but has unsound joints which may fail in service. Thus it is absolutely essential that the welding variables be controlled in order to produce sound joints in the equipm e n t . A number of codes and standards have been established for this purpose. Some of these standards are: “ASME Code Welding Qualifications” (Section IX of the ASME Boiler Code) ASA Code for Pressure Piping (B 131.1, Section 6 and’ Appendices I and II) Standard Qualijication Procedures of the American Welding Society API Standard 12 C, API Speci$cation for Welded Oil Storage Tanks (Sections 7 and 8) The American Welding Society (AWS) established the basic standards for qualifying operators and procedures. These standards of qualification form the basis for most of the standards in the various codes. For practical purposes, therefore the rules for qualifying welders and welding procedures are essentially the same in the various codes and standards. Regardless of whether or not the welded vessel is intended to meet one of the codes or standards, it is advisable that the welding conform to one of the minimum standards. Each fabrication shop should establish welding procedures best suited to its need and its equipment. To meet the welding standards previously mentioned, it is not necessary that welding procedures be the same in all shops. But it is necessary that, regardless of the procedures used, the welded joints pass the qualification tests for welding procedures and that the welding operators be qualified in using these ,wrne procedures. To meet welding standards, welds made by the shop proced&es must be tested to determine tensile strength, ductility, and soundness of the welded joints. The req&ed tests for the welding procedures specified by APmndard 12 C involve the following: A. For groove welds: 1. Reduced-section-tension test (for tensile strength). 2. Free-bend test (for ductility). 3. Root-bend test (for soundness). 4. Face-bend test (for soundness). 5. Side-bend test (for soundness). B. For fillet welds: 1. Transverse-shear test (for sbear strength). 2. Free-bend test (for ductility). 3. Fillet-weld-soundness test.
,-
,
The minimum results required by tests such as those listed above are described in detail in the various codes. A few representative requirements are: 1. The tensile strength in the reduced-section-tension test
Double-welded butt joint (V-type groove)
T
Single-welded butt joint with backing strip
Double
Fig. 1.7.
full-fillet
lap
Fabrication
7
Double-welded butt joint (U-type groove) rover Y
Single-welded butt joint with backing strip (may be V-or U-type groove)
of
joint
Examples of welded joints.
Q In.
Smgle-welded butt joint wlthout backing strip (may be V-or U-type groove)
wlthout backing strip
Single full-fillet lap joint with plug welds
(Note: The two types of lap welds
shown may be used only for circumferential joints and for shell plates not over 36 in. thick, and for attachment of nozzles and reinforcements without thickness limitation.)
(From the API-ASME code [lo].)
shall not be less than 95 y0 of the minimum tensile strength of the material being welded. 2. The minimum permissible elongation in the free-bend test is 20%. 3. The shearing strength of the welds in the transverseshear test shall not be less than 87 y0 of the minimum tensile strength of the material being welded. 4. In the various soundness tests, the convex surface of the specimen is examined for the appearance of cracks or other defects. If any crack exceeds $6 in. in any direction, the joint is considered to have failed. The individual welders, as well as the shop procedures, must meet certain standard qualifications. The individual welders must qualify under the established procedure according to the test previously described. This is important because a welder may qualify when using one procedure but may be unable to qualify when using another procedure. For example, an operator of an automatic welding machine may produce satisfactory welds with that machine but may not qualify when using manual equipment. 1.2~ Types of Welded Joints. A variety oi types of welded joints are used in the fabrication of vessels. The selection of the type of joint depends upon the service, the thickness of the metal, fabrication procedures, and code requirements. Figure 1.7 is a diagram from the API-ASME code for unfired pressure vessels which illustrates some of the types of welded joints used in the welding of steel plates for the fabrication of pressure vessels. Other types of weld joints and details for the preparation of such joints are given in Appendix B. Instead of drawing weld details to specify the type of weld desired, most engineering of&es now use standard symbols for ,welding conventions (16). Typical welding symbols are shown in Fig. 1.8.
8
Factors Influencing the Design of Vessels Type of weld Fillet
Bead
Plug and
Groove
Feld weld
Weld all around
Flush
Location of welds Arrow (or near) side of joint
1. The side of the joint to which the arrow points is the arrow side, and the opposite side of the joint is the other side.
with a definite break toward the member which-is to be chamfered.
2. Arrow-side and other-side welds are some size unless otherwise shown. 3. Symbols apply between abrupt changes in the direction of welding, or
4.
6. When a bevel- or J-groove weld symbol is used, the arrow shall point (In cases where the member to be chamfered is obvious, the break in the orrow may be omitted.)
to the extent of hatching or dimension lines, except where the all-
7. Dimensions of weld sizes, increment lengths, and spacing, in inches.
around symbol is used.
8. For more detailed instruction in the use of these symbols refer to Stondord
All
welds
are
continuous
and
of
user’s
standard
proportions
unless
Welding Symbols, published by American Welding Society.
otherwise shown. 5. Tail of arrow used for specification process or other reference.
(Tail
may be omitted when reference not used.) Fig. 1.8.
I.3
Welding symbols recommended by API Standard 12 C.
TYPES OF CRITERIA IN VESSEL DESIGN
The selection of the. type of vessel is based primarily upon the ?&&Z&l-s&vice required of the vessel. The funcI &al requirements impose certain operating conditions in respect to such things as temperature, pressure, dimensional If the vessel is not designed limitations, and various loads. properly, so as t,o accommodate these requirements, the ve el may fail in service. / Failure may occur in one or more manners, such as by plasticformation resulting from excessive stress, by rupture without plastic deformation, or by elastic instabilit,y. I:aiJuz-may also result from corrosion, wear, or fat,igue. Design of the vessel to prot,ect against such failures involves the considerat,ion of these factors and the physical properI ies of the mat.eriuls. Various types of possible vessel failure and criteria in vessel design are discussed in the following chapter. 1.4
ECONOMIC
CONSIDERATIONS
.ilthough the chemical-process requirements generally limit the choice of materials of fabrication, the final selection is frequently dictated by economic considerations. Fo1 purposes of comparison the relative costs of lO,OOO-gal tanks fabricated from various materials are tabulated in Table 1.1 (w&h steel as the unit reference) (18). An examination of t.his ._. table indicates t,hat the cheapest const&&~materials, provided they can be used, are wood, concrete, and steel. These materials can frequently be lined with a thin protective layer this eliminates the necessity of fablicat.ing the
I 1.
I--
(Courtesy of American Petroleum Institute.)
vessels from more expensive metals or alloys. As the size of the tank is increased to handle larger volumes, the relative costs of using alloys and nonferrous metals increases. Prestressed or reinforced concrete may sometimes be used to advantage for the construction of large vessels. 1.4a Steel Pricing. The bulk of chemical and petrochemical process equipment is fabricated from plain carbon steel. -1 knowledge of the method of pricing steel is essenTable 1.1.
Relative Costs of Materials of Construction for /
J WOOd Concret y’(reinforced) St,eel’ Lithcotr-lined steel Rubber-lined steel Lead-lined steel coppe:L Aluminum Glass-lined st.eel Xi-Clad steel Stain-clad steel Stainless steel, type 304 ,Mon&clad steel Incouel-clad steel Stainless steel, type 316 &lone1 metal Silver-lined steel
Tanks
Cost Relative to Steel ~.~______-. 10,000 gal 100,000 gal 0.4 0.6 0:; 1.0 1.0 1.2 1.2 1.8 2.0 1.8 2 .o 2.0 2.6 2.4 3.0 2.7 3.0 2.7 3.0 2.7 3.0 3 4 3.5 3.5 3.4 3.4 3.5 4.4 4.8 4.4 4.8 12.8
.,_
Economic Table
1.2.
Grade
Classification
Cold-rolled
Carbon
by
Size
of
Flat,
Steel
(Courtesy of Great Lakes Steel Corporation, Division of National Steel Corporation, Detroit, Michigan) Thickness. inches 0.250 o r 0.249.9 to Width, inches t,hicker 0.0142 up to 12 Bar Strip (1) Over 12 to 24 Strip (2) Strip (2) Over 12 to 24 Sheet (3) Sheet (3) Over 24 to 32 Sheet Sheet Sheet Sheet Over 32
:(
Notes: (1) Up to 35 in. wide and less than 0.225 in. in thickness, and not to exceed 0.05 sq in. in cross section, having rolled or prepared edges is “flat-wire.” (2) If special edge, finish, or definite t,emper, as defined by ASTM Specification A-109. (3) If no special edge, finish, or temper is specified or required. tial in order to arrive at economical designs for equipmenL fabricated of steel. Steel may be purchased from t,wo sources-a steel mill or a st,eel warehouse. The prices paid for the steel from the two sources are very different, the warehouse prices being appreciably higher. The reason for these price differences is found in the methods used by steel mills to obtain maximum-volume production in order to minimize unit costs. It is the present practice of the steel mills to accumulate orders until they have sufficient tonnage to permit economical rolling. Therefore, the steel mills usually serve customers who require material in reasonably large quantit,ies and- who can anticipate their requirements well in advance. It is apparent that this mode of operation is not conducive to quick delivery; three or four months, or more, depending upon the rolling schedule, may elapse before delivery. This situation makes necessary another means of furnishing steel to customers who require material quickly and in quantities too small for mill production schedules. The steel warehouse fills this distribution need, supplying steel immediately from large warehouse stocks. The steel warehouse secures steels from many rolling mills, in a full range of qualities, finishes, shapes, and sizes, and stores these steels. Thus fabricators using steel may purchase any particular product immediat,ely from stock or combine orders for various pr0durt.s and buy all at one time from one convenient source. Obviously the warehouse must. be paid an increase over Table
1.3.
Grade
Classification Carbon
of
Flat,
Considerations
9
mill price to compensate for t.he handling, storage, and delivery of the steel st.ock. Therefore, the difference between warehouse prices and mill prices is essentially :I service charge. The relative amount of “millproduction” that was shipped to warehouses for warehouse distribution for the ten-yea1 period 1944-1954 is indicat.ed in Fig. 1.9. This figure indicates that for the seven-year period 1945-1952 about 18yc of the total steel-mill production on the average was shipped to the warehouses. For the year 1951 the stezl mills produced 78,928,950 tons of steel products and shipped 14,399,432 tons (18.50’%) to the warehouses. In the design of equipment for large process plants, it is not unusual to place vessel orders with the vessel fabricator from 6 to 12 months before the required shipping dates I,O enable t)he vessel manufacturer t.o order t,he steel plate from the mill rather than from a warehouse. MILL PRICING. In general, steel is purchased from the mill or warehouse in the “hot-rolled” or “cold-rolled” condition. The steel is further classified as sheet, strip, plates, or bars. Alloy steels and &ructural steels are classified separately. “Hot-rolled, plate steel,” or “cold-rolled strip steel,” or “alloy steel bars,” and so on are combined classificat,ions of types of available steel. Table 1.2 shows the grade classification, by size, of flat, cold-rolled carbon steel by a typical steel mill. Table 1.3 shows t.he corresponding grade classification by size of flat, hot-rolled carbon steel. The steel mills and the warehouses quote “base prices” for each class of steel product. Table 1.4 shows a section of a typical mill base-price list as of January, 1956. The prices are all F.O.B. cars or trucks at the mill works (Indiana Harbor, Indiana.) The prices quoted in Table 1.4 apply to an order of 10,000 lb or more of the size ordered at one time (one thickness and one width is considered one size), of one grade or analysis, released for shipment to one destination at one time. For weights of less than 10,000 lb, “itemquantity extras” apply. Item la, of Appendix C lists the quantity extras charged by a typical steel mill (Inland St.eel Company, as of May 13, 1953) for carbon-steel plates.
Hot-rolled
Steel
(Courtesy of Great Lakes Steel Corporation) Width, inches Thickness, To 3% Over Over Over inches incl. 355 to 6 6 to 12 12 to 48 0.2300 and thicker Bar Bar Plate Plate Bar Strip Sheet 0.2299 to 0.2031 Bar Strip 0.2030 to 0.1800 Strip Strip Sheet 0.1799 to 0.0568 Strip Strip Strip Sheet
.,_ /
_
.-.-- _._~____~__ I
-
-.-~\
Over 48 Plate Plate Plate Sheet
-i---‘
-
\I
Year Fig. 1.9.
Percentage of total millproduction of steal products shipped to
warehouses.
..__--_ 7
_
4.=-s _ .T-
10
Factors Influencing the Design of Vessels Table 1.4. Mill Price List
of Inland Steel Company, Chicago, Illinois, January, 1956) Base Price per 100 lb $4.325 Hot-rolled sheets (18-gage and heavier) 4.325 Hot-rolled strip 5.325 Cold-rolled sheets 4.65 Hot-rolled carbon-steel bars (merchant quality) 5.575 Hot-rolled alloy-steel bars 4.65 Reinforcing bars 4.50 Carbon-steel plates 4.60 Carbon-steel structural shapes (Courtesy
Carbon-steel plates fall into three classifications: (1) those furnished to chemical requirements, (2) those furnished to physical requirements, and (3) those furnished to both chemical and physical requirements. Item lb of Appendix C lists the “classification extras.” Other mill-price extras of primary interest are given in item 1, Appendix C and are classified as: quality extras, length extras, width-and-thickness extras and killed-steel extras. Circular- and sketch-plate extras are involved when items such as blanks for formed heads are purchased. As these plates are usually flame cut, gas-cutting extras also apply. Gas-cutting extras are also charged for rectangular plates when the thickness limits for shearing are exceeded. Item 2 of Appendix C lists the gas-cutting extras per linear foot of cutting. The previously mentioned extras such as quality, thick-
Fig. 1 .lO.
_ __- _ I_- - -_
Base price of steel plates in Pittsburgh.
i
--T - - -- - - - 7 7
ness, and width are calculated on a dollar-per-loo-lb basis, whereas the extras for circular and sketch plates are calculated on a percentage basis, as listed in item 3 of Appendix C. The percentage is calculated on the net-per-loo-lb price of the smallest rectangular plates from which each circular or sketch plate is obtained exclusive of freight and extras for gas cutting a quantity. The outside dimension of each circular or sketch plate determines the size of the smallest rectangular plate from which the circular or sketch plate is obtained. A wide variety of other “mill extras” are quoted by the various steel mills. The reader is referred to company price lists for complete quotations on these other extras, among which are: 1. Heat-treatment extras 2. Surface-finish extras 3. Testing extras 4. Chemical-requirements extras 5. Specification extras 6. Special-requirements extras 7. Dimensional and workmanship extras 8. Extras for special-shipment requirements 9. Special-marking-of-plates extras 10. Loading extras 11. Bundling-of-plates extras. Each extra is usually separate and distinct. The individual items are combined to form a “full extra” applicable to the order. The steel-mill base prices given in Table 1.4 and the steel-mill extras given in Appendix C are quoted as of January 31, 1956.. It must be emphasized that these prices are representative of the prices quoted by steel mills at that time. As economic conditions vary, prices charged for manufactured products fluctuate, and the base and extra prices are subject to change. Figure 1.10 illustrates the changes in the base price of steel plate in Pittsburgh from July, 1938 to January, 1956 (19). The horizontal line to April, 1945 is for the period during which government controls were maintained on steel prices because of the national emergency of World War II. The curve indicates that the price of steel plate at the mill doubled between 1945’ and 1956. Reference should always be made to the most-recent available price lists for estimation purposes. W AREHOUSE P R I C I N G. Steel warehouses are strategically located throughout the country to provide a convenient source of supply for steel products. Whereas the steel mills produce steel products of standard length and width, the warehouse will supply steel cut to the customer’s requirements. Typical operations in the warehouse include shearing, sawing, slitting, and flame cutting. Some warehouses will supply steel plates rolled to cylindrical shapes and bar shapes, bar stock rolled to rings or bent to other shapes, and plates with drilled or punched holes. Figure 1.11 shows typical stocks of steel in a warehouse. Prices vary somewhat from warehouse to warehouse, depending upon the location of the warehouse, the distance from the mill, and the service performed. Item 4 of Appendix C gives typical warehouse prices from one warehouse
(20).
-I-‘
.--_c___-_
.--
.---
~~
!
Economic Considerations 11
Fig.
>’
.
\ b
I I I
i
i
1.11.
Interior view of warehouse showing typical
1.4b Fabrication Costs. The direct costs of producing a piece of process equipment include the cost of materials and the cost of labor. Material costs consist of the shop material used in the fabrication plus the parts purchased from an outside source. The cost of steel plate, which has been discussed in the previous section, usually comprises a major portion of the material costs for vessels. The labor costs involved in the actual fabrication of the equipment are often difficult to estimate accurately in advance. How has reported methods of short-cut estimations of welded process vessels (21). F ABRICATION P R O C E D U R E. One of the first steps in the fabrication of the vessel is usually the preparation of the shell for rolling. The edges of the individual plates for the shell require machining to true the edges and, in the case of code welding, to prepare the edge for welding. Figure 1.12 shows a 40-ft planer machining a double “U” edge on a lx-in. plate 29 ft long for a vessel shell. The next step is usually crimping the edges of the plate which will be joined by a longitudinal weld. The crimping step is required because the rolls cannot be used to form the two ends to the desired curvature. Figure 1.13 shows a 350-ton hydraulic press in the foreground, crimping the edge of a plate before rolling. In the background plates are shown being rolled into a cylindrical shape on pyramid rolls. M AN -HOURS AND M ATERIALS. After the shell has been given edge preparation and rolled into shape, the vessel components must be fitted and assembled by welding. Figure 1.14 gives curves according to How (21) for estimating the man-hours involved in the various stages through assembly of the shell and closures. The upper curve of
stocks of steel.
(Courtesy of Joseph T. Ryerson 8 Son, Inc.)
Fig. 1.14~ gives the cutting time in hours per linear foot for flame cutting the shell plate as a function of plate thickness. This curve may be used when the shell is cut from standard plate kept in stock, such as mill plate. If the plate is purchased from a warehouse, it may be obtained, cut to size, and the cutting cost included in the purchase price. In addition to the man-hours involved in flame cutting, a machine rate burden which includes the cost of machine time and gas consumed is also involved in flame cutting. The curve for cutting-machine rate burden is shown in the lower part of Fig. 1.14~~. The number of man-hours involved in edge preparation prior to crimping and rolling are given in Fig. 1.14d. The combined number of man-hours involved in crimping the longitudinal seam ends and rolling the plate into a cylindrical form are given as parameters in Fig. 1.14b; the man-hours are treated as a function of plate lengths and thicknesses. The parameters given in Fig. 1.14b are based upon the rolling and crimping of a few plates; the figure therefore gives a liberal allowance for these operations when more than a few plates are rolled and crimped at one time. The man-hours required for the fitting and assembling of the shell and closures for the vessel are given by the solid lines in Fig. 1.14~ as a function of the plate thickness and with three parameters for different degrees of complexity of the vessel. Also included in this figure are three curves having nearly the same shape as the parameters and indicated by the dotted lines that may be used as a rough check on the total man-hours involved in fabrication. These latter three curves are intended to be used only as a check to disclose any gross errors in the total estimation. In addi-
Factors Influencing the Design of Vessels
Fig.
1.12.
Machining a double U edge on a plate 1
yh in. thick and 29 ft long for
a vessel shell by means of o 40-ft
planer.
(Court+sy o f C, F. Broun
Company.)
Fig. 1.13.
A 350-ton
hydraulic press crimping the end of
o plate before rolling.
(Courtesy of C. F.
Bran & Company.)
g
Economic Considerations 13
%
4
k
i%il 1 I%12 I3 % % 1% 1% 2%
4
5 6
Plate thickness, in. (4 Plate thickness.
in.
(b)
3.5 9
I
14
/+I
I
I
I
I\
I
I
I
I
I
I
I
8
0.6
01111111/1(( % b % % % % 1 I?$ 1% 1% 1%
Plate
Average steel thickness, in. Cd
i
thickness, in. (4
(a) Cutting time and machine rote burden for tlgme c u t t i n g w i t h a u t o m a t i c Fig. 1.14. C u r v e s o f H o w (21) fsr e s t i m a t i n g s h o p t i m e f o r v e s s e l f a b r i c a t i o n . (d F i t t i n g a n d a s s e m b l y , a n d total f a b r i c a t i o n t i m e ( f o r machines. (b) T i m e f o r r o l l i n g plates 60 to 72 in. in width and of various lengths and thicknesses. rough check), for steel tanks and weldments. (d) W e l d i n g a n d e d g e - p r e p a r a t i o n t i m e a n d w e l d i n g - r o d w e i g h t f o r c o d e b u t t w e l d s i n c a r b o n s t e e l . (Cowtesy of McGraw-Hill Publishing Co.)
t ion to giving the number of man-hours for edge preparation, Fig. 1.14d also gives the welding time and quantity of welding rod per linear foot, involved in assembly of the vessel ends and shell. Most vessels contain two or more nozzles for charging and discharging operations. The man-hours and weldingrod requirements for attaching different types of nozzles are given in Fig. 1.15.
t
1
I
\
Formed closures such as dished heads can be purchased from fabricators with the edges beveled for welding. Costs for this preparation are given in a later section describing formed heads. However, if it is practical to machine the heads in the shop that fabricates the vessel, the man-hours required for this operation may be estimated from Fig. 1.16~ and b. Bolting flanges for nozzles may be shop fabricated from flat plates. The machining time for t.his opera.
\--I---
-
14
Factors Influencing the Design of Vessels
1.6 -
1.61
1.4 -
1.4
I -
0.5 -
4.5
1.0 -
4.0
l.2t-,
1.2 B l.Oe $ CL p 0.8 -
o-
2 1.0 m Tii .E $0.8
I
2 1 . 5 - p 3.5 # =lo e x $ 2.0 - .E 3.0
I
f$ 12 $ 2.5 - “r 2.5 &cl % 3 3.0 - 2 2.0
// / / / / / I/ I/ I
I
I
/
Labor. hr
2
a g 0.6 - LO.6
Weld rod. I
0.4 0.2 I
O-
I
1
1.5
4.0 -
1.0
4.5 -
(1.5
I
0 %
3.5 -
1% 2 2% 3 1% Nominal nozzle size, in.
3%
I 4
2 2%
3
3354 5 6 Nominal nozzle size, in.
8
(b)
(4
oJ
!
slip-on
30
5 -
flange
-
25
25 5
2
5 6 8 10 12 14 16 18 2%33%4 Nominal nozzle size, in.
I
30 0
2
2%33%4 5 6 8 10 12 14 16 18 Nominal nozzle size, in. UJ
(d Fig. 1.15.
Curves of How (21) for estimating welding-time in hours and welding-rod requirements for nozzle attachments to vessels.
welding rod for installing XH steel couplings in nozzles. 300-psi
unfired pressure vessels.
(c) Welding time and welding rod for fabricating r,ozzles in untired pressure vessels.
(a)
Welding time and
(b) Welding time and welding rod for installing long-welding-neck forged-steel
150-psi nozzles in
unfired pressure vessels.
(Courtesy of McGraw-Hill Publishing Co.)
(d)
Welding time and welding rod for fabricating
Economic
Considerations
15
6 t E 0 E 2 m 4 I I uAZz k
5 4
2 ..8
3
..6 ..4
2 1.8 ’ 1.6 E -0 1.4
I- 0.8
n
’ ,+’
1.2 0.9 0.8
/ 15
I
/
20 25 30 35 40 50 60 70 80 Outside diameter of head, in. (4
100 120
15
20 25 30 35 40 50 Outside diameter of head, in. (6)
6.5 6.0 5.0
8 ; 3.0 8 i 2.5 2 8 2.0 5 1.8
0.6
k 1.6 L ; 1.4 8 1.2 1.1 1.0 0.9 0.8
c
Fig.
1.16.
(a)
flanges up to 6 ips inclusiv
1 2 3 4 5 6 8 10 12 14 16 18 20 22 24 26 Nominal flange size, in. (cl
%
k
% t Size of weld, in. (4
J6
1
1%
Curves of How (21) for estimating man-hours of machining time and welding-rod requirements for miscellaneous operations in vessel fabrication, (b) Machining time, flanged and dished heads.
Machining time, flanged and dished heads, grooved face, carbon-and-nickel- or stainless-clad steel.
beveled face, carbon-and-nickel- or stainless-clad steel.
(c) Machining time for carbon-steel plate flanges, 1 to 26 ipr,
time and weight of welding rod for fillet welds in carbon steel.
r’ -
4
I
\7- -
(Courtesy of McGraw-Hill Publishing Co.)
\I/.
-
-
>/4 to 2% in. thick.
(d)
Welding
16
Factors lnftuencing the Design of Vessels Table
1.5.
Engineering News-Record Cost Index
Construction
Twenty-City Average of Skilled Labor
Hourly
Rates
for
(From Engineering News-Record) (Courtesy of McGraw-Hill Publishing Co.)
(Courtesy of McGraw-Hill Publishing Co.) Index Year Index Year 100 1946 346 1913 1947 413 1915 94 1920 235 1948 461 206 1949 477 1925 208 1950 510 1926 1930 202 1951 543 1932 157 1952 569 1953 600 1935 195 628 1940 242 1954 1945 308 1955 660 (July)
Year 1926 1932 1939 1945 1946 1949
i.ion is given in Fig. 1.16~. Various attachments such as skirts, saddles, and lugs may be added to the vessel, usually by fillet welding. The man-hours per foot and the welding required for fillet welding is given in Fig. 1.16d. Additional curves for some alloy and rumferrous metals are giver1 Iry How (21). 1.5 ESTIMATING CURRENT COSTS COST INDICES. Because of the constant change of costs for material, labor, taxes, and plant overhead, available COSI data rapidly become obsolete. Thus some method of bringing cost data up to date is required. The procedure normally followed is t,he application of available “cost. indices.” The cost indices are relative numbers giving the variation in a group of costs with reference to a base year. To use a cost index the estimator simply multiplies t.he known cost at a given date by the ratio of the current index value to the index applicable a.t the date of the known cost.
Index A Cost A = Cost B ___ Index B
Table 1.7.
(1.1)
A number of indices are in wide use; they differ somewhat because of the basis used in their preparation and the reference year. Three widely used indices are the Engineering ‘Vews-Record (ENR) construction-cost index (22), the
Rate, dollars/hr 1.27 1.03 1.44 1.66 1.80 2.41
Year 1950 1952 1953 1954 1 9 5 5 (July)
Rate, dollars/hr 2.52 2.84 3.01 3.14 3.25
Marshall and Stevens eyuipment-cost index (23), and the Nelson refinery index (24). The ENR construction-cost, index (22) reflects labor-wagerate and material-price trends. The index consists of the cost of a hypothetical block of construction requiring 6 bbl of cement, 1.088 M fbm of lumber, 2500 lb of steel, and 200 hours of common labor. This cost was $100 in the year 1913, which is taken as the reference year. Although this index is intended to reflect average construction costs and has no particular relation to the cost of equipment, it has proved extremely useful in estimating changes in costs for complete plant,s. Because of its wide use it has often been the basis of estimating changes in equipment costs. Table 1.5 lists some values of this index as a function of time. The Marshall and Stevens equipment-cost index reflects the comparative costs of equipment (23). It is based upon t.he costs of machinery and major equipment, installation labor, plant furniture and fixtures, tools and minor equipment, and oflice furniture. These costs are estimated quarterly for 47 different industries, with a separate formula for each industry and with the year 1926 as a reference of 100. The petroleum-industry index contains the following component percentages: process machinery, 25; installation labor, 19; power, 12; maintenance equipment, 2; and administration, 6. Other process industries for which indices are prepared are: the cement industry, the chemical industry, the clay-products industry, the glass industry, the paint
Table
1.8.
Average
Boilermaker
Wages
in
July,
1954, as a Function of Locale Table 1.6.
Marshall and Stevens Equipment-Cost Index
(Average for All Industries) (Courtesy of McGraw-Hill Publishing Co. [235]) lndex Year Year Index 57.9 1947 150.6 1913 55.9 1948 162.8 1915 1920 153.3 1949 161.2 105.3 1950 167.9 1925 100 .o 1951 180.3 1926 1930 87.0 1952 180.5 1932 66.1 1953 182.5 1935 78.0 1954 184.6 1940 86.1 1955 190.6 1945 103.4 1956 208 .a 1946 123.2 1957 (June) 224.1
(U. S. Bureau of Labor Statistics) U. S. Average New England (Me., Vt.., Mass., Conn., R. I., N. H.) Mid-Atlantic (N. Y., Pa., N. J.) Border States (Del., Md., Ky., W. Va., Va.) Southeast (Tenn., S. C., N. C., Ala., Ga., Miss., Fla.) Great Lakes (Minn., Wis., Mich., Ill., Ind., Ohio) Midwest (N. Dak., S. Dak., Kans., Nebr., MO., Iowa) Southwest (Tex., Okla., La.) Mountain (Mont., Idaho, Wyo., Utah, Ariz., N. Mex., Colo.) Pacific (Wash., Oreg., Calif., Nev.)
dollars+ 3.11 3.00 3.44 3.01 2.90 3.13 2.96 2.90 3.01 3.05
Typical Procurement Procedure for Vessels
industry, the paper industry, and the rubber industry. A weighted average for the process industries is also reported, which contains, in percentages: cement, 2; chemicals, 48; clay products, 2; glass, 3; paint, 5; paper, 10; petroleum, 22; and rubber, 8. Also, an average for all 47 industries is published, which has differed only by about 1 y0 to 2% from the average for the process industries alone. As this index is based primarily upon industrial-equipment costs, it is considered more reliable for estimating changes in equipment costs than the ENR index. Values of the Marshall and Stevens index for the average of all industries are given in Table 1.6. The Nelson refinery index (24) is a construction-cost index somewhat similar to the ENR construction-cost index but based upon the cost of materials and labor for the construction of petroleum refineries. Although price indices are extremely valuable in estimating costs, it should be mentioned that they are based on national averages and may be inconsistent with price changes for a particular locale. Also, price indices are based on wage rates and material costs but make no allowance for such factors as: availability of materials, productivity of labor, competitive conditions, influence of new techniques, business optimism, relation of demand to production capacity, and other intangibles. L ABOR - C O S T V ARIATIONS . The cost of labor varies from year to year and from area to area. These costs have risen rapidly since World War II, and at the end of 1955 the hourly rates in dollars were at an all-time high. Table 1.7 gives the average hourly rate for skilled labor for 20 cities since 1926 as reported by t,he Engineering News-Record (22). Skilled laborers such as machinists, welders, and boilermakers are required in the fabrication of vessels. The hourly rates, as a funct,ion of locale, that were paid boilermakers in July, 1953, as reported by the Bureau of Labor Statistics (22), are given in Table 1.8. S HOP O VERHEAD . In addition to the direct costs involved for materials and labor, all fabricators must add an indirect cost often termed the “shop overhead” or “burden.” This overhead includes a variety of items, such as the cost of supervision, administration, engineering, sales, utilities, maintenance, depreciation, taxes, and other fixed and indirect costs. These costs vary from shop to shop, area to area, and year to year, and are established by the conditions for a particular shop and by the accounting practice followed. This overhead usually ranges from 100 % to 200 % of the total cost for labor and materials. P ROFIT . The profit of the fabricator is estimated on the total cost to the fabricator, including materials, labor, and overhead. The profit usually ranges from 5% to 205~ of this total cost but may be higher if the state of competit,ion permits. SOURCES OF P RICE I NFORMATION. In recent years a considerable attempt has been made to collect, group, and correlate price information. A series of post World W-ar II articles was published in Chemical and Metallurgical Engineering in 1946, and additions to them were made in Chemical Engineering from 1947 through 1955. These articles have been collected and reprinted in three booklets (25). Furthermore, rather recently two texts on the subject of
17
chemical-engineering costs have been published (26, 27). In October, 1954, Weaver listed a bibliography of 351 articles dealing with equipment costs, operating costs, and estimating methods (28). SCALING EQUIPMENT COST WITH SIZE . Frequently a piece of process equipment having a size different from that for which the cost is known is desired. A comprehensive study of the cost of a variety of process equipment as a funct,ion of size and capacity was made by Chilton (18). III H subsequent article (229) Chilton analyzed these data and similar data by Williams (230) and concluded that the “six-tenths factor” rule is useful as a short-cut method for approximating the cost of a similar piece of equipment of a different size. This rule states that the cost of a second size is equal to the cost of the first size times the ratio of the sizes (or capacities) raised to the six-tenth power, or Cost A = Cost B
o.6
(1.2)
The general validity of this rule has been well established, but some discretion should be exercised by limiting it to less than a tenfold range unless cost data are available for two or more units over a range of sizes. 1.6
NPICAL PROCUREMENT
PROCEDURE
FOR
VESSELS
The typical procedure followed in the procurement of vessels for a process application will be discussed briefly to give a perspective of the sequence of steps involved. Normally a process-design group develops flow sheets for the process involved. The flow sheets include information relative to the operating temperat.ure and pressure, capacities, heat duties, and any particular information concerning corrosion. The equipment-design group prepares detailed sketches of the various items of equipment, specifying the materials of construction, shell and closure thickness, type of closure, and code stamping. The nozzle and manhole types and their ratings; corrosion allowances; stress relieving, radiographing, and hydrostatic or air-testing requirements are also specified by the design group. In addition, shipping limitations; the weight, of the vessel empty, with internal attachments, and filled with water; and the operating weight are usually estimated by this group. From the above information specification sheets for each item of equipment are prepared. The procurement group sends copies of the detailed sketches and the specification sheets to various vessel fabricators for quotations on prices and delivery dates. In t bc meantime the plant-design group are also sent copies of f.he sketches and specifications. This group prepares a layoul. of the plant design. This includes specifications for roads, utilities, sewers, fire protection, structural foundations, pumps, piping, and detail design of the various components involved. On the basis of the original est,imates received from the fabricators, one or more fabricat.ors are selected and final drawings are furnished for a rigid price quotation on the various vessels and other items of equipment to be purchased. The fabricator is then granted permission to purchase material and prepare shop drawings. These shop drawings are usually submitted to the purchaser for
18
Factors Influencing the Design of Vessels
approval. On approval of these drawings the fabricator proceeds with construction of the vessels and other items of equipment being supplied. Where major items of equipment are involved, it is customary for the purchaser to set up an inspecting-and-expediting group at the fabricator’s plant. The inspectors for the purchaser normally follow
every step in fabrication, from initial inspection of plate and heads through testing and shipping. This text covers the design problems of the engineering offices of both the purchaser and the fabricator. No attempt has heen made to separate these problems, which often overlap.
PROBLEMS
1. A vertical vessel designed as shown in Fig. 1.1 and 32 ft from head junction to head junction is in use. If an identical vessel were to be fabricated of SA-285, Grade C steel today, what would be the estimated cost of: a. The two circular blanks for the two elliptical dished heads at the warehouse (no edge preparation) ? b. Four shell plates cut to length only at the warehouse? (Note: see Chapter 5, section 5, Common Types of Formed Heads and Their Selection, for diameters of head blanks.) 2. Estimate the cost of labor today for the fabrication of the vessel shown in Fig. 1.1. How much weld metal will be required?
i
C H A P T E R
m 2
CRITERIA IN VESSEL DESIGN
A
For axial compression:
A
unit of process equipment may fail in service for a variety of reasons. Consideration of the types of failure which may occur is one of the criteria which should be used in equipment design. Failure may result from excessive elastic or plastic deformation or from creep. As a result of such deformation, the equipment may fail to perform its specified function without rupture or may fail catastrophically with rupture. Failure can usually be classified in one of the following catagories: excessive elastic deformation, elastic instability, plastic instability, brittle rupture, creep, or corrosion. 2.1
EXCESSIVE
ELASTIC
(2.2) when j = induced axial stress, pounds per square inch P = load, pounds a = cross-sectional area, square inches Stresses resulting from bending and torsion are more complex, and a large number of texts have been written on the subject of the evaluation of such stresses (29, 30,31, 32). Induced stresses result in corresponding induced elastic deformations. The deformations may interfere with the functional operation of the equipment. A common example of this is found in the use of excessively thin flanges for a bolted closure with a gasket at the interface of the flanges. Tightening of the flange bolts in an attempt to seat the gasket in such a way that it will contain the internal pressure may result in excessive elastic bending of the flange between the bolts without transfer of the bolt load to the gasket. Another example is the excessive deflection of a tray in a distillation column under the tray load, a condition which produces a nonuniform liquid seal on the bubble caps and possible instability in tray operation. 2.1 b Modulus of Elasticity. In order to avoid such situations as described in the previous examples, sufficient rigidity must be incorporated into the design of the part to restrict the amount of deformation to a permissible value. The deformation which can be tolerated is determined by the function of the part. Parts in simple tension or compression, such as exist in axial loading, deform in the elastic region in direct proportion to the induced stress and in indirect proportion to the modulus of elasticity of the material of construction. Thus the proportionality con-
DEFORMATION
Elastic deformation is induced by a load such that when the load is removed, the part resumes its original shape. A typical example is the steel spring in a watch. Under service conditions the various parts of the equipment will be subjected to a variety of induced stresses. A stress is defined as the force per unit area in the member under consideration. Various types of stresses are induced, depending upon the loading condition, and are classified as: tensile, compressive, shear, bending, and torsion. These stresses may be the result simply of the weight of the material of construction or may be caused by loads resulting from fluid pressure, forces, wind moments, and so on. Parts under axial-compressive or tensile forces have induced stresses which may be computed by the simple relations: 2.1 a Induced Stresses.
For axial tension:
j=i
(2.1) 19
\-\
\I
/
- - -___-__ - - --_ - -~.---~
__-
20
Criteria
in
-r--l
Vessel
Design
I
I
I
I
Gray cast iron (ASTM-A-276-No. 60) Cold-rolled mild steel (ASTM-A-374)
60,000
Hot-rolled mild stee (ASTM - A - 283,
loaded beam having uniform cross section and freely su ported at the ends as indicated in Fig. 2.2. Consider an element, dx, of a beam having a uniform cr( section, supporting a distributed load of w pounds per in of length of the beam as indicated in Fig. 2.2. The toi load acting on the element is w (dx). If w is consider’ positive when the load acts downward and if dx is positi\ the differential shear force, dV, must be negative. 1 summation of vertical forces: Vz - VI + w (dx) = 0, or Vz - VI = -w (dx)
and
dV = -w (dx) or dV dx
U’
(2.
Taking a summation of bending moments about point gives :
0
Ib io ;o 4b
w (dx)2 . Since ---2 is negligible,
Percentage of elongation
dM = MS - Ml= Vldx
Fig. 2.1. Typical stress-strain curves for various metals.
stant between stress and strain (under axial loads) is the modulus of elasticity. Typical stress-strain curves for a few selected materials are shown in Fig. 2.1. (Note that two scales are used on the abscissa in order to enlarge the elastic region of the curves.) The elastic portion of the total strain is represented by the straight-line segments of the curves. The slope of these straight-line segments, when the strain is expressed in inches per inch, is the modulus of elasticity of the material, E, or: f-=E E
dM - = 1’ dx
Any beam under a load def1ect.s. A particular radius curvature exists for the portion of the beam under consider tion. Thus the loaded beam has a radius of curvature P, a dist,ance 2 from the perpendicular to the neutral axi The bending of the beam will result, in a deformation Ax the fiber at any distance ?/ from the neutral axis, as indicate in Fig. 2.3. The corresponding strain or unit deformat.ic eZ is equal to Ax/x, and by similar triangles Ax ?/ ez=--= x P
(2.3)
where j = axial stress, pounds per square inch e = unit strain, inches per inch E = modulus of elasticity, pounds per square inch 2.1~ Elastic Bending. The deflection of a part subjected to forces which produce bending is a more complex phenomenon. In such cases the amount of deflection is inversely proportional to the modulus of elasticity and the moment of inertia of the member. The use of relationships developed in the field of theoretical mechanics are required for evaluation of the deflections. General procedures for such calculations are presented in a number of texts on the subject of strength of materials (29, 30, 31, 32), and on the subject of the theory of elasticity (33, 34, 35, 36, 37). Selected procedures for particular calculations involved in vessel design are presented in later chapters of this text. The basic relationships for such calculations may be developed by considering the shear and bending in a uniformly
(2.
(2.1
As given by Eq. 2.3, the ratio of stress to strain for elast deformat,ion is equal to the modulus of elast,icity, E, or ji = EEL
(b) Fig.
2.2.
Forces on an element of a uniformly loaded beam. (a) UI
formly loaded
beam. (b) Detail o f e l e m e n t d x .
Excessive Elastic Deformation
21
/dA
Section A-A
Fig. 2.3.
Stress and strain in on elemental strip of
of itat is.
of
a curved plate or beam.
By definition the radius of curvat.ure r is defined (38) as:
By substitution of Eq. 2.6
.f, = E Y I
2
1.- = r, [l
(2.6a)
M=/&
Also, by definition (29) the moment of inertia is: +a 1--c
By Eq. 2.5:
d M = Vak .v2 dA
M = !?!? r
t,herefore (2.9:
) Uni-
(shear force)
dx3
(2.15)
.And hy Eq. 2.4
!!lL~~d!L
dx
Gr for the outermost fiber where y = c, (2.10)
C
dx
! tmefore
jpf = &! ?
I v-here t = --) section modulus, inches3
dM = EI 9 = 1 -
d V = -wdx
By combination of Eq. 2.9 with Eq. 2.6a,
M2
(2.1‘1)
dx2
(2.7)
I =
(2.13)
Substituting Eq. 2.13 into Eq. 2.9 gives:
on
Therefore
(2.12)
1-Nd2y r- dx2
By summation of moments
3tic
d2yidx2 -+ (dy/d~)~]‘~J
For small deflections the quaniit ?; rl?, kc is small compared unity; therefore
By summation of forces
ed
8.6)
Stress diagram
(2.11)
I = moment of inertia of the cross section, inches’ c = distance from neutral fiber to outermost fiber
J
dx4
vu>
;ioad)
Anot,her important relationship, the equation of defleclioa curve, is obtained from EC{. 2.14:
(2.16) I
he
These relationships for i)thiulls may be applied to plates and shells under certain corldilions, {~a y0 per 1000 hours (0.00001 y0 per hour) or on 60 ‘% of the average stress or 80 y0 of the minimum stress to produce rupture at the end of 100,000 hours, whichever is the lower value. The stress to produce rupture after a specified length of time varies with the material. Figure 2.17 compares the stress to produce rupture in 1000 hours as a function of temperature for a variety of materials (86). The superalloys shown in Fig. 2.17 may be rather expensive for use in
Creep-rupture test
Fig.
2.15.
Correlation
test
and
creep-rupture-test
for
18-8
of
creepdata
molybdenum (type 316)
steel (05). icourtesy negie-!li’nois
o
f
Car-
Steel Corp.)
Minimum rate of extension, per cent per hour
b
_
r
\
-
\
\r-T-
-
32
Criteria
in
Vessel
Design
1’
Fig. 2.16. for
Stress vs. rupture time
18-8
molybdenum
(type
316)
steel at 1100, 1300, and 1500” F (85).
(Courtesy
of
Carnegie-
Illinois Steel Corp.)
10
100 Rupture time, hours
1000
vessel construction; they find their principal use in such items as turbine blades, parts for jet engines, and similar applications, but may be used for vessels for extreme service. It should be noted that creep- and stress-rupture-test data are obtained under atmospheric exposure conditions in laboratories and under uniaxial loading. The stress condition existing in a vessel part under field service conditions usually comprises stresses in three directions, a fact which complicates the application of the experimental data. In addition, the vessel material may be exposed to a corrosive atmosphere and be subject to scaling, hydrogen embrittlement, intergranular corrosion, and strain hardening. 2.6
CORROSION
The extent to which corrosion will occur in process equipment depends upon the nature of the films that form on the
10,000
100,000
surface of the parts. The excellent corrosion resistance of copper and its alloys, for example, is the result of their ability to form thin, protective films on their surfaces. These films may be the result of simple oxidation or may be To be protective the coat,inp composed of insoluble salts. should be thin, adherent, continuous, and relatively insoluble. Equipment operating under conditions that allow the formation of a uniform protective film generally corrodes slowly and may last for a great many years. Under severe corrosive conditions, however, rapid corroBye sion occurs, resulting in costly delays and replacements. judicious selection of material and by careful improving of operating conditions, corrosion can be reduced 01’ retarded, and substant,ial savings in operating and maintenance costs may be realized. Thus an appreciation of the factors which contribute to corrosion is of value in the design of equip-
60
Fig. 2.17. Stress to rupture in
1000
hours
vs.
temperature
for
various materials (86). (Courtesy Corp.
of
Universal-Cyclops
Institute.)
0
400
600
800
1000 1200 1400 Temperature, degrees F
1600
1800
2
Steel
a n d Battelle M e m o r i a l
Corrosion
ment. A thorough treatment of the theory of corrosion and its control is presented in the Corrosion Handbook (87). 2.6a Uniform Corrosion. When corrosion occurs at the surface of equipment by the formation of soluble salts, a uniform thinning of the wall occurs. The rate of corrosion depends upon the corroding medium, the velocity of fluid flow, the temperature, and other factors. This type of corrosion is encountered in acid solutions (particularly those containing oxygen), in waters carrying a high oxygen or carbon dioxide content (such as mine water), and in solutions havmg a solvent action on the corrosion products (such as those containing ammonium hydroxide, which dissolve the corrosion products from copper alloys). Attempts have been made to reduce uniform corrosion by the application of an external electric current to provide cathodic proThe limited success attained by this method has I.ertion. been attributed more to the formation of protective films than to cathodic protection. 2.6b Impingement Attack. Under normal operating conditions certain localized areas of the parts may be exposed to t,he destructive forces of a relatively high-velocity circuCorrosion under such conditions is la ling medium. described as “impingement attack.” Turbulence of the fluids causes a rapid and repeated destruction of t.he protective film with subsequent corrosion of the exposed metal. This condition is considerably aggravated when the circula1 ing medium carries in suspension an abrasive material. 2.6~ Concentration-cell Attack. C o r r o s i o n m a y b e caused by differential aeration wit,h the formation of concentration cells at the metal surfaces under certain operating conditions. Cracks, crevices, porous coatings, and breaks in protective films are sources of trouble since they trap liquid and set up differences in concentration of salts, ions, or gases in the circulation medium. As a result of an electrochemical type of concentration-cell action, severe pit,ting of the metal surface and localized perforation of the material occur. An example of this type of corrosion is the “rusting” of plain carbon steel. This type of corrosion can be materially reduced by the observation of the following suggeslions (88) : I. Specify butt joinls and tsttll)llasizr the necessity for complete penetration of tlrt. weld tllat,rrial to guard against minute crevices. 2. Avoid the use of lap j&Its, or completely seal them with weld metals, solder, or a suitable caulking compound. :%. Avoid sharp corners and stagnant areas or other sites favorable to the accumulation of precipitates and other solids. .1. Endeavor to provide uniform flow of liquid with a minimum of turbulence and air entrainment. 3. Provide suitable strainers in lines to prevent local obstructiou within the equipment which may start deposit attacks or result in impingement attack. 2.6d Deposit Attack. When small particles settle out or lodge on the wall of t,he equipment, part of the metal wall bcxcomes protected by the deposit, and a special type of concentration-cell action may take place. Usually the shielded area becomes anodic and intense pitting results. Filtering of the circulating medium and frequent cleaning will minimize the occurrence of deposit a (t ack .
r
r
\ - \
__-~ .-- . . -
Table
2.2.
33
Galvanic Series in Sea Water (236)
(Courtesy of American Society of Testing Materials) CORRODED
END (ANODIC
OR
LEAST
NOBLE)
Magnesium, magnesium alloys Zinc, galvanized steel, or galvanized wrought iron Aluminum (52 SH, 4S, 3S, 2S, 53S-T in this order) Alclad, cadmium Aluminum, (A 17S-T, 17S-T, 24S-T in this order) Mild steel, wrought iron, cast iron Ni-resist 13 %-chromium stainless steel, type 410 (active) 50-50 lead-tin solder 18-8 stainless steel, type 304 (active), 18-8-3 stainless steei, type 316 (active) Lead, tin Muntz metal, manganese bronze, naval brass Nickel (active), Inconel (active) Yellow brass, admiralty brass, aluminum bronze, red brass, copper, silicon bronze, Ambrac, 70-30 copper-nickel, camp. G-bronze, camp. M-bronze Nickel (passive), Inconel (passive) Monel 18-8 stainless steel, type 304 (passive) ; 18-8-3 stainless steel, type 316 (passive) PROTECTED
END
(CATHODIC
OR
MOST
NOBLE)
2.6e Galvanic-cell Attack. When dissimilar metals and alloys are in contact with each other in a conducting medium, a galvanic action is set up which results in the dissolution of the less-noble or anodic metal. From the electromotive or galvanic series it is possible to predict the tendencies of metals and alloys to form galvanic cells and to predict the probable direction of the galvanic action. Table 2.2 gives the galvanic series as determined for sea water (89, 236). When use is made of such a series determined for the fluid under consideration, one may be relatively safe in choosing metals from the same group; however, if the metals are distant from each other in the list, the metal higher in the list will corrode rapidly. When one is using metals that produce a galvanic action, the relative areas of the two materials have a very important bearing on the extent of corrosion. Usually the extent of galvanic action will be proportional to the ratio of the area of the metal lower in the series to the area of the metal higher in the list. Thus it is wise to avoid galvanic couples where the exposed area of the cathodic metal is much greater than that of the anodic metal. (For example, a steel part in a copper vessel would rapidly corrode, but a copper part in a steel vessel would be relatively safe). 2.6f Stress Corrosion. As a result of the simultaneous action of stresses and certain corrosive conditions, parts may fail by cracking. When the stress is applied externally, the break often is called a “stress-corrosion crack.” When residual internal stresses are involved, the resulting break often is called a “season crack.” Annealing to relieve residual stresses greatly reduces season cracking. W h e n the part is subjected to repeated cyclic stresses during service, failure by “fatigue cracking” may occur, as dis-
\ P T---
34
Criteria in Vessel Desigd
cussed previously. Such failures are characterized by their suddenness and by the absence of plastic deformation in the failing section. Stress-corrosion data on systems free of corrosion in the unstressed state indicate that there is little or no attack on material that is either completely elastically deformed or completely plasticly deformed. But such parts subjected
to both plastic and elastic deformation in the same section may undergo severe stress corrosion. Thus stress corrosion is caused by strain hardening in combination with high elastic stresses rather than by plastic deformation (45). Such strain hardening followed by corrosion has been observed in boilers and pressure vessels where local stress concentrations exist without severe plastic deformation.
PROBLEMS
1. A full-floating-head horizontal condenser has 400 >6-in-diameter No. 16 BWG admiraltymetal tubes. The floating tube sheet and head assembiy weigh 1000 lb when the head is filled with water. If the tube-support plate is located 18 in. from the center of load of the floating head assembly and the overhanging tubes and head assembly are assumed to behave as a simple cantilever beam, a. What bending stress is developed in the tubes as a result of this load? b. What is the floating-head deflection from the horizontal at the center of load? e. Is the design satisfactory if the allowable stress is 6000 psi? 3
For a simple cantilever, y = g n4 max
For a tube, Imctanpular =
= Pl
s(do4 - di4) 64
For a g-in. No. 16 BWG tube, OD = 0.625 in. ID = 0.495 in. Modulus of elasticity of admirality metal = 15 X lo6 psi 2. A horizontal stiffener supporting a bubble-cap tray in a fractionating column may be considered to act as a uniformly loaded, simply supported beam. The deflection equation for such a beam is: -
-
-
-
WZ3X
24 I
where y = dellection’at point z, inches z = distance from end of stiffener, inches 2 = total length of stiffener, inches E = modulus of elasticity, pounds per square inch Z = moment of inertia, inches4 ta = uniform load, pounds per linear inch The stiffener has a moment of inertia of 1.785 in4 and a section modulus of 1.02 in.3 and is of steel (E = 30 X lo6 psi). The stiffener has a length, 1, of 100 in. and carries a uniform load, w, of 2.4 lb per linear in. Determine: a. The maximum deflection at the center of the span, z = 50 in. b. The maximum stress at the center of the span. c. The stress 20 in. in from the support, z = 20 in. d. The shear load, pounds, at x = 20 in. 3. A copper fractionating column has vertical tray-support rods between trays. The trays are 20 in. apart and the rods are spaced so as to support 100 lb each, under column action. What is the diameter of the rods required to limit the column loading to one half of the critica column load if (a) the rods are free to pivot at both ends or (b) the rods are fixed at both ends?
k = i: E = 15 X lo6 psi
Problems
35
4. A vessel is to be fabricated of type 316 stainless steel (see Figs. 2.15 and 2.16). The The creep vessel is intended to be used at 1300” F for an expected service life of five years. deformation permitted during the service life of the unit is not to exceed 5%. Determine the allowable stress if the allowable stress is not to exceed either (a) two thirds of the stress to proAlso estimate the total creep duce the creep rate or (b) 50% of the stress to produce rupture. and rupture life if the allowable design stress as determined in this manner is used in the design of the vessel.
r
\ - ‘i -
\I
1
e
_-
--
C H A P T E R
DESIGN OF SHELLS FOR FLAT-BOTTOMED CYLINDRICAL VESSELS
u
storage of other fluids. oil-storage tank.
hroughout the chemical and petrochemical industries, gases, liquids, and solids are stored, accumulated, or processed in vessels of various shapes and sizes. Such a large number of storage vessels or tanks are used by these industries that the design, fabrication, and erection of these vessels have become a specialty of a number of companies. As a result of economic considerations, only a few companies iu the process industries now design storage vessels having large volumetric capacity, and the usual practice is to contract for this equipment with companies specia!izing in t.his field. However, the design of this equipment involves basic principles which are fundamental to the design of other types of more specialized equipment. This and the following chapter cover these fundamentals. Before the extensive use of welding as a means of fabrication, vessels fabricated of metal were assembled either hy bobing or riveting (90). At the present time most fluids held at atmospheric or low pressure are contained in cylindrical tanks of welded construction. Because of the large quantities of petroleum and its products stored in this manner, the American Petroleum Institute has established specifications governing the design of welded, vertical, cylindrical tanks for oil storage above ground. This specification code, API Standard 12 C (2), is intended to provide the oil industry with design specifications for tanks of adequate safety and reasonable economy in a wide variety of capacities. It is recommended for use by the oil indust.ry in all sections of the United States except those areas which are subject to local regulat.ions that conflict with it. Although these specifications were devised primarily for the design of storage vessels for petroleum and its products, they are useful guides in the design of a variety of vessels for
Figure 3.1 sholvs a typical welded
3.1 MATERIAL SPECIFICATIONS The materials used in the construction of storage vessels are usually metals, alloys, clad-metals, or materials with linings which are suitable for containing t.he fluid. W h e r e no appreciable corrosion problem exists, the cheapest. and most easily fabricated construction material is usually hotrolled mild (low-carbon) steel plate. The particular types of steel plate specified by API Standard 12 C are SA-7 (open-hearth or electric-furnace processes only), SA-283, Grade C for all thicknesses greater than l>i in., or SA-283, Grade D for thicknesses less t,han I>4 in. Copper-bearing steel, which is not specified by this code, has some advantages in resisting atmospheric corrosion. SA-7 is specified for structural steel shapes such as angles, channels, and I-beams, and ASTM-A-27 grade 60-30 fully annealed is specified for steel castings. The physical properties and chemical composition of these plain carbon steels are listed in Table 3.1. Low-carbon steels are rather soft and ductile and are easily sheared, rolled, and formed into the various shapes used in fabricating vessels. These steels are also easily welded to give joints of uniform strength relatively free from localized stresses. The ultimate tensile strength is usually between 55,000 and 65,000 psi and the carbon content between 0.15% and 0.25%. However, both the ultimate strength and the carbon content cannot be specilied since one is a function of the other. In Table 3.1 the ultimate strength is specified but not the carbon ccntent. Low-alloy, high-strength steels are a specific class of lowcarbon steels which are made stronger by the addition of a 36
Material
Fig. 3.1.
Welded
oil-storage
tank,
Humble
Oil
and
Refining
Company, Baytown,
Texas.
Diameter,
120
ft;
height,
40
ft;
Specifications
nominal
capacity,
80,000
37
bbl.
(Courtesy of Bethlehem Steel Company.)
small amount of alloying elements. Such steels have yield points that are considerably higher than plain carbon steel of the same carbon content and also have higher ultimate tensile strengths. The chief disadvantage of some of the high-strength alloy steels of moderate alloy content is that they are more difficult to fabricate because they have low ductility. Some have an increasing tendency to air-harden, which may result in localized stresses in welded joints. However, these disadvantages may be avoided if the alloy content and the carbon content are both kept relatively low, as they are in low-alloy, high-strength steels. Many steel manufacturers do not produce these steels to standard specifications as they do plain carbon steels but use a variety of formulas marketed under various trade names.
Table 3.1.
0)
Properties of Plain Carbon Steels, Recommended by API Standard 12 C
(Courtesy of American Petroleum Institute)
(2) Steel Specilications
J?lat,es
Structural Shapes Steel Castings
Most manufactures claim t,heir low-alloy, high-strength steels t.o be readily “weldable.” However, this is not a precisely defined characteristic, and it should be understood that such “weldability” refers to conventional metal-arc welding processes usually employed for plain carbon steel. Table 3.2 gives properties of low-alloy, high-strength steels recommended by API Standard 12 C, 1955. In selecting material for large-diameter vertical vessels, consideration should be given to the types of failure that have occurred in the past. In 1952 two unusual tank failures occurred in England while the tanks were being hydrostatically tested wit,h water at 40” F (91). Figure 3.2 shows the catastrophic nature of the failure. One tank was a crude-oil storage tank 140 ft X 54 ft with a floating roof,
SA-283, SA-7 Gradeand D SA-283, Grade C
(3) (4) Phosphorus, * Sulfur,* max, max, per cent per cent
(5) Tensile Strength, Psi
(6)
(7)
(8)
Elongat.iou, min, per cent Yield Point, min, psi In 2 in. In 8 in.
Basic 0.04 Acid 0.06
0.05
60,000-72,000
33,000
22
Acid 0.06 Basic 0.04
0.05
55,000-65,000
30,000
24
1,500,000 Tensile st.rengtht (
1,500,000 Tensile strength? - ., Requirements are same as given above for A-7 plates. 0.05 0.06 60,000 min 30,000 24
SA-7 ASTM-A-27, Grade 60-30, fully annealed “’ From ladle analysis made by manufacturer; check analysis from finished material by purchaser may show 25 y0 more. t See exceptions listed in particular specification. Copper, when copper-steel plates or structural shapes are specilied. miuimum percentage, 0.20.
- _ ~--- - I
- - - - -
\-F---
\
I
7---------
38
Design of Shells for Flat-bottomed Cylindrical Vess
and the other was a gas-oil storage tank 150 ft X 48 ft with a cone roof. A fairly large number of similar tanks have failed in this country. These failures and those encountered in welded “Liberty” ships and T-2 tankers (92, 93) have stimulated studies on the reason for such failures (53, 94, 95, 96). The failures of the two British tanks were traced to flaws in the welds in the shell. In the case of the crude-oil storage tank, the flaw initiating failure was incomplete penetration in a weld-probe replacement located across the first horizontal welded joint. The crack progressed vertically in both directions with brittle fracture up to the Gfth course; the fracture of the upper four courses was by shear. It is significant that the rupture occurred entirely across the plates and not along any of the vertical welded joints in the nine courses. The gas-oil storage tank failed in a similar manner. In this case the flaw was in a partially repaired crack in the top 10 in. of a vertical joint in the top of the first course. As in the case of the crude-oil tank, this crack progressed vertically in both directions with brittle fracture in the first four courses and with shear fracture in the remaining upper courses. The fracture occurred chiefly across the plates as in the other failure except for the fifth course, where it traveled through a vertical welded joint. These two ruptures appear to be examples of failure by notch brittleness as a result of a crack existing in the plate at a location of high stress and having a length greater than the critical crack length for crack propagation. See Chapter 2 for discussion of notch brittleness and critical crack lengths. A complete investigation was made of samples of steel from the two tanks which failed. The samples were impact tested at various temperatures and the results were compared with corresponding data for other steels used for vessel construction (53, 91, 94, 95). The steel used in the construction of these tanks was identified as BS-13, a British steel similar to SA-283, Grade C. Some of the
Table
3.2.
Principal Provisions of Specifications
results of these tests are given in Fig. 2.9 (53), in the previous chapter. The conclusion made was that the British steel had properties within the specification given and was as good as similar American steels of equivalent quality. The steel at the time of hydrostatic test was at a temperature (40” F) within the transition-temperature range (see Fig. 2.9). If the flaws had been smaller in size, or if a more ductile steel had been used, or if the hydrostatic test had been carried out at a higher temperature, the vessel probably would not have failed. These failures demonstrate the necessity of sound welding procedures and thorough inspection of welded joints. Even if cracks cannot be avoided, a high degree of protection against brittle fracture can be obtained with low-carbon steels if the steel has a minimum specification of 15 ft-lb in the Charpy V-notch test at the lowest service temperature (47, 96). When specifying fine-grained steels or semikilled steels of low carbon with manganese, a 30 ft-lb Charpy V-notch value would give similar protection (96). The majority of the steels presently used for storage-tank construction do not meet this specification at temperatures below 50’ F. It is recommended that for service temperatures below 50” F, steels meeting the above specification be used, such as ABS, class C or SA-201, Grade B (67) and that all weld joints in the shell be radiographed. 3.2 ESTIMATED COST OF TANKS
Storage tanks require considerable capital investment. If a limited quantity of fluid is to be handled, a single tank may suffice, in which case the magnitude of the proportions is controlled by the volume of fluid to be stored. Where a large number of tanks are required, it is generally true that larger tanks give a lower cost per unit volume of storage than smaller tanks. This is indicated in Fig. 3.3, which shows that the total installed cost of a l,OOO,OOO-gal coneroofed tank is approximately $32,000 whereas the corresponding cost of a lO,OOO-gal tank is about $3000, a hundred-
on low-alloy, High-strength Steels, Recommended by API Standard 12 C
(Courtesy of American Petroleum Institute) (1) (2) (3) (4) (5) Chemical Requirements, max, per cent Carbon ASTM-A-242 Ladle Analysis Check Analysis Thicknesses : 4is to N in., incl. Over W to 135 in., incl. Over 136 to 4 in., incl. SAE 950 Thicknesses: 0.0710 to 0.2299 in., incl. 0.2300 in., incl. Over >s to to $6 1 in., incl. Over 1 to 2 in., incl.
0.22 0.26
0.20
M a n g a - Phosnese phorus 1.25 1.30
1.25
... ...
0.150
Sulfur
Silicon
0.050 0.063
... ...
0.050
0.90
(7) (8) (9) Tensile Requirements, min Tensile Yield Elongation, min, per cent Strength, Point, psi psi In 2 in. In 8 in.
(f-9
70,000
50,000
18
67,000 63,000
46,000 42,000
24
70,000 70,000 67,000 65,000
50,000 50,000 47,000 45,000
22 22 22 22
19 19
... 1,500,000 Tensile strength
-
Optimum
Fig.
3.2.
Failure of British storage tank at Fawley, England (91).
Tank
Proportions
39
(Courtesy of Esso Research and Engineering Company and Petroleum Publishing Co.)
fold increase in volume for approximately 11 times the cost of the smaller tank. However, the large tanks are not always selected because of the greater flexibility permitted in storing a variety of fluids in a battery of smaller tanks. For such reasons no general rule can be made for the selection of an optimum number of tanks. The total cost of various types of large-sized storage tanks is given in Fig. 3.4 (97). The cost does not include the cost of the foundation. The cost of tanks fabricated from various materials (18), field-erected steel tanks (98), and small tanks under 1000 gal (99) has been reported in the literature. (See also 26, 27, and 28.) Figure 3.5 shows the weight of steel and the cost in dollars per ton for large-diameter tanks (97).
” P z %i.E
= 103 5 l-
J 3.3
OPTIMUM
TANK
PROPORTIONS
Before a storage tank can be designed, the proportion of height to diameter must be established. The diameters of standard steel tanks for storage at atmospheric pressure usually range from 10 to 220 ft, and the heights vary from 6 to 64 ft. Typical tank dimensions are listed in Appendix E. No general rule can be given for the selection of the height-to-diameter ratio because t.his ratio is often a function of the processing requirements, available land area, and height limitations. Figure 3.6 shows a group of oil-storage tanks of various height-to-diameter ratios. The volume of a single tank, which may be one of a
1
0 103
2 106
104
2 10s
Volumetric capacity in gallons Fig.
3.3.
Estimated
cost
of
small
and
medium-sized
tlat-bottomed
cone-
roofed storage tanks. Assumption,: Steel base cost
j- extras -(- freight =
Total cost = total weight X
(6~
6c/lb
+ fabrication cost
+ erection cost)
Marshall and Stevens “Process industries Average” cost index = 180
-.. -
- _~
--~~-..-.
_-
_..
\
-
\
7 -
\
I
- _--_-- ---. -
-
- -
40
Design of Shells for Flat-bottomed Cylindrical Vessels
k
B 5 2 5 .G % 8 z P
80
Fig.
3.4. Estimated
costs
70
of
insblleP
l a r g e - d i a m e t e r tanks
197). ICourtesy o f P e t r o l e u m Processing,
60
McGraw-Hill
Pub-
lishing Co.)
50 40 30
0’.
10I
20’
30’
’ 40
50’
60’
70’
80’
’
’
:J
’
’ ’ ’ ’ ’ ’ 90’ 100 110 120 130 140 150 160 170 180 i90 200
Capacity in thousands of barrels
battery of tanks, may be determined by the process requirements and other considerations such as production flexihility and seasonal variations in storage requirements. The optimum proportion of the tank diameter, D, to height, H, varies between two limits. The lower limit for the optimum ratio of D/H occurs when the shell, bottom, and roof costs per unit area are independent of D and H. This condition
exists with tanks of small volume, in which elastic stability and corrosion control the thicknesses. The upper limit for the optimum ratio of D/H occurs when the shell thicknesses is a function of both D and H, and the unit area costs of the bottom and roof are independent of D and H. This condition exists with tanks of large volume. The optimum proportions of a mnk are influenced by the
700 650 600 550 500 450 400 Fig. 3.5.
350
ton
of
(Courtesy
300
Weight and cost pev
large-diameter of
tanks
Petroleum
(97).
Procesr-
ing, McGraw-Hill Publishing Co j
250 200 150 100 50 n -0
10
20
30
40
50
60
70
80
90 100 110 120 130 140 150 160 170 180 190 ZOO
,.”
Capacity in thousands of barrels
ii; \
I
\
\
-
\
‘I /
,,/
--
Optimum
Fig. 3.6.
Oil-storage
tanks
of
various
Tank
Proportions
tanks 120 ft in diameter by 48 ft high with three fire walls 172 height-to-diameter ratios. Three 96,000-bbl (Courtesy of Hammond Iron Works.)
41
R in
diameter by 24 ft high appear in the background.
c4 = annual cost of the installed foundation under the vessel, dollars per square foot of tankbottom area cs = annual cost of the land in the tank area chargeable to the tank, dollars per square foot of tank-bottom area C = total annual cost, of the vessel, dollars per year
cost of the foundations and the cost of the land in the tank area that is chargeable to the tank as well as by the cost of the bottom, shell, and roof, if required. The value of t.he land chargeable to the tank may be expressed in terms of an For purposes of tank proportionannual cost per unit area. ing, it is also convenient to express the cost of the foundation, bottom, shell, and roof, if used, in terms of cost per unit area as follows. --I
then
Let D = diameter of the vessel, feet H = height of t.he vessel, feet
C =
+
\
-
\
+
CQ
+ ~4 +
~5)
(3.2)
T’
(c2
+
c3 + ~4 +
~5)
Simplifying the equation by substituting for H in terms of D, we obtain:
(3.1)
When A1 = arca of the shell, square feet = TDH As = area of the vessel bottom or the projected area of the roof, square feet TD2 =4 c1 = annual cost of fabricated shell, dollars per square foot c2 = annual cost of fabricated bottom, dollars per square foot c3 = annual cost of the fabricated roof, dollars per square foot of projected area
I
Az(c2
C = TDHCI + $
or n-
+
Substituting for the areas -41 and 42, we obtain:
aD’(H) V = volume of the vessel, cubic feet = ___ 4
“2$
(Am)
c =
7 +
F (c2
+
c3 +
c4 +
c5)
(3.3)
To determine the optimum tank proportions by using Eq. 3.3, it is necessary to determine which of the cost terms are variables prior to differentiation. 3.30 Tanks Having Shell Thickness Independent of D and H. The stress in the shell is a function of both the diameter and the height of the tank, as indicated in Eqs. 3.18 and 3.19. For reasons of elastic stability, the minimum shell thickness is limited to N s in. for 45-ft tanks and smaller and to x in. for tanks of larger diameters. Therefore,
,
..-T--
-
.-_
_
42
Design of Shells for Flat-bottomed Cylindrical Vessels
tanks having shell thicknesses of >/4 in. or less may be considered to have a shell thickness independent of H and D. Substituting )i-in. shell thickness into Eqs. 3.18 and 3.19, results in the following: 0.25 = 0.0001456 (H - 1)D
proportioning may be assumed to be directly proportional to the shell thickness as follows: cl = cg(H - l)D
(3.7)
Or
(3.8)
Or
and
D(H - 1) = 1720 for butt-welded shells
(3.4)
D(H - 1) = 1515 for lap-welded shells
(3.5)
Substituting Eq. 3.7 for cl in Eq. 3.3 gives: c = 4VMH - l)D] D
Thus, all tanks with butt-welded or lap-welded shells having a value of D(H - 1) equal to or less than 1720 or 1515 respectively fall into this category. Tanks of small volume falling into the category of Eq. 3.4 or Eq. 3.5 have shell, bottom, and roof costs per unit area which may be considered independent of D and H. Using Eq. 3.3 and differentiating the total cost, C, with respect to the diameter of the vessel, D, and considering the volume, V, to be known and the cost factors, cl, y, cz, c4, and cs, as constant known factors for the volume considered, we obtain : dC -= dD
- 4+ + r$ (C2 +
C 3
+ C4 +
- % +
y (c2 +
cg +
c4 +
c5) =
Solving for D, we obtain: 03 = !!! n-
Cl C2 +
C3 +
C 4
+ C5
Substituting for the volume, V, we obtain: Cl
or D = 2H 3.3b
Cl C 2
+ C3 + C4 +
C5
(3.6)
Tanks Having Shell Thicknesses Dependent upon
D and H. Tanks having heights and diameters such that the quantity D(H - 1) exceeds 1720 for butt-welded shells or 1515 for lap-welded shells have shell thicknesses which are dependent upon D and H. The cost of the shell per unit area, cl, is a function of D and H and for purposes of
\
\-
- 4vC6 +
\I/- -
+
C3 +
c4 f ca)
T$ (C2 +
C3 +
C 4
+ C5)
Differentiating and setting equal to zero gives: dC dD
-32csV2 =
~03
- 0+
$ (c2 +
0
uD2 4_ (C2
Expanding and substituting for H by Eq. 3.1 gives:
cs).
Actually, the individual unit costs, cl, ~2, cg, and ~4, of the various tank components will vary somewhat with tank proportions and with tank volume and other factors such as design considerations. Many correlations for estimating tank costs have been presented in the literature. These correlations indicate that for purposes of estimating installed tank costs, the cost is primarily a function of the total volume and that total installed costs do not vary appreciably with the unit costs of component parts. Therefore, the consideration of unit-cost factors as constants is a reasonable approximation. For minimum cost the slope of the curve of cost versus the diameter of the vessel to contain the fixed volume, V, must be equal to zero (dC/dD = 0). Therefore
+
y cc2 +
C3 +
c4 +
cg +
c5) =
c4 +
es) = 0
s
Substituting Eq. 3.1 for V and Eq. 3.8 for cg gives: *D
-y (Cz + C3 + C4 +
c 5 )
=
((H “;;&) te)’
Or
=+ cc2 +
c 3 +;c4 +
c5) =
((H T;u3r+)
Since H - 1 for large tanks is approximately equal to I-r, it will be assumed that (H - 1) = H. Therefore D(c2 +
c3 +
c4 +
c5)
= 4c1H
Or
D = 4H
Cl C2 +
C3 +
c4 +
c5 >
(3.9)
3.3~ Estimation of Cost Factors. In most designs the cost factors, cl, ~2, and so on, cannot be accurately evaluated until the design of the tank is known, a requirement which necessitates successive approximations in determining the optimum proportions. The costs of the tank components, shell, bottom, and roof, are a function of the plate t.hickness, grade of steel, cost of forming, cost of field welding, and so forth, and include the cost of the accessories such as nozzles, manholes, pumps, ladders, platforms, and so forth which are attached to the various components. These factors are all interrelated, and in order to make an estimate it is usually more convenient to express them in terms of the cost of the fabricated component parts of the vessel per pound of fabricated material because information concerning cost per pound is more readily available. 3.3d S i m p l i f i e d C a s e s o f O p t i m u m P r o p o r t i o n s . To demonstrate the use of Eq. 3.6, three simplified cases will be presented. For the first case, consider a small open tank and disregard the costs for land and foundation. For small tanks
-
- - -
-
43
Shell Design of Large Storage Tanks
the shell thickness is often the same as the bottom thickness. If the cost per square foot of shell, cl, is taken as equal to the cost per square foot of bottom, c2, then cl = c2; and if ~3, ~4, cg = 0, then, by Eq. 3.6, D = 2H I
Cl
= 2H
(3.10)
c1+0+0+0
However, the shell is generally more expensive than the bottom partly because of the additional cost of rolling the shell plates. For the second case, consider a small closed vessel having the same cost per unit area for shell, roof, and bottom and disregard the cost of foundations and land.
k
Cl
=
cp
=
c3
And if c4 and cg = 0, by Eq. 3.6,
r
cl
D = 2H-
i
Cl +
1
= H
Cl
+
(3.11)
0
For closed vessels also the cost of the shell per unit area is generally greater than the cost of the bottoms per unit area for the reasons previously stated. Furthermore, the roof costs are generally greater than the bottom costs per unit area because of the structural steel required for the roofs of all but small vessels. For the third case, consider a large closed tank in which the roof and shell costs are twice the cost of the bottom per unit area. Cl = 2cz = c3 And if c4 and cg = 0, 2c2
D zz 4H c2 +
1
2c2 +
= #H
0
Introducing the actual values of foundation and land costs, c4 and cg, into the equations has the obvious effect of increasing the height-to-diameter ratio. It is apparent that in areas where land is cheap and the tanks can be easily supported on the soil without expensive foundations, Table 3.3.
, i
Y
(3.12)
Nominal Capacity, bbl 90 200 210 300 400 H-500 750 L-500 II-1000 1500 L-1000 2000 3000
---
Approximate Working Capacity, bbl 72 166 200 266 366 479 746 407 923 1438 784 1774 2764
economical designs result in tanks having low heigbt-todiameter ratios. 3.4
SHELL DESIGN OF SMALL AND VESSELS (PRODUCTION TANKS)
MEDIUM-SIZED
Small and medium-sized vertical vessels with flat bottoms, called production vessels in contradistinction to storage vessels) are usually fabricated from steel plate of a single thickness. Their optimum proportions are similar to those discussed previously in case two (D = H). The design of such vessels is simple and has been standardized for the oil industry (100, 101) as described in Fig. 3.7 and Table 3.3. The shells of such vessels are usually fabricated of either Ns-in. or sd-in. plate with plate widths preferably not less than 60 in. A mild steel meeting SA-7, SA-283, Grade C or D (open-hearth or electric steel only) is specilied. The shell plates are usually either double-welded butt joints with complete penetration of weld metal or single-welded butt joints with backing strips with complete penetration of weld metal. The design of the bottom and roof (deck) is covered in Chapter 4. 3.5
SHELL
DESIGN
OF
LARGE
STORAGE
TANKS
The majority of tanks and vessels are cylindrical because a cylinder has great structural strength and is easy to fabricate. Several types of stresses may occur in a cylindrical shell. These may be recognized as: 1. Longitudinal stress resulting from pressure within the vessel; 2. Circumferential stress resulting from pressure within the vessel ; 3. Residual weld stresses resulting from localized heating: 4. Stresses resulting from
[email protected] such as wind, snow, and ice, auxiliary equipment, and impact loads; 5. Stresses resulting from thermal differences; 6. Others, such as may be encountered in practice. 3.5a S t r e s s e s i n T h i n S h e l l s B a s e d o n M e m b r a n e Theory. Simple equations may be derived to determine the
minimum wall thickness of a thin-walled cylindrical vessel
Typical Dimensions for Production Tanks (100, 101)
(Courtesy of American Petroleum Institute) Height of Height of Outside Overflow Walkway Diameter, Height, Connection, Lugs, ft in. ft in. ft ft in. 7-11, 10 9-6 7-7 10 9-6 7-7 12- 0 15 14-6 12-7 lo- 0 14-6 12-7 12- 0 15 17-7 12- 0 20 19-6 13-7 15- 6 16 15-6 23-6 21-7 15- 6 24 7-6 5-7 21- 6 8 13-7 21- 6 16 15-6 24 23-6 21-7 21- 6 8 7-6 5-7 29- 9 15-6 13-7 29~ 9 16 21-7 29- 9 24 23-6
-
--
I---\I-/o
__ .-
-____ .--
Location of Fill-Line Connection, in. 14 14 14 14 14 14 14 14 14 14 14 14 14
Size of Connections,. in. 3 4 4 4 4 4 4 4 4 i 4 4
-_-.
--i-
-
\
-
\
44
Design
of
Shells
for
Drain-line connection-
Flat-bottomed
Cylindrical
Vessels
&*24* x 36” cleanout lank
’
Detail, thief-hatch outlet
Plan
,- 20”-diam dome
]j[
I-T 40”
-= 7
Detail of dome
‘14” min
Walkway bracket lugs
I +--I_.-& I
D
- - & i t - 1Yd T A-
Shell - plate
24” x 36”
Detail, walkway bracket lugs
i c -. -
-
II %4-
I
-----.
4
Elevation Fig. 3.7.
Standard
design for sm-all and medium-sized ’
vesseh (production tanks) as
recommended
by API Standard 12 D.
(Courtesy of American Petroleum Institute.)
L
\
\
\I
/
___- __...
- . . - --- -
- -.
(See Table 3.3 fo. dimensions.)
Shell Design of large Storage Tanks
A comparison of Eq. 3.13 with Eq. 3.14 indicates that for a specific allowable stress, fixed diameter, and given pressure, the thickness required to contain the pressure for the condition of Eq. 3.14 is twice that required for the condition of Eq. 3.13. Therefore, the thickness as determined by Eq. 3.14 is “controlling” and is the commonly used “thinwalled equation” referred to in the various codes for vessels. This equation makes no allowance for corrosion and does not recognize the fact that welded seams or joints may cause weakness. J O I N T E F F I C I E N C I E S A N D C O R R O S I O N A L L O W A N C E . In vessels for atmospheric storage the welded joints are seldom stress relieved or radiographed. The welded seam may not be as strong as the adjoining rolled-steel plate of the shell. It has been found from experience that an allowance may be made for such weakness by introducing a “joint efficiency factor,” E into Eqs. 3.13 and 3.14. This factor is always less than unity and is specified for a given type of welded construction in the various codes. The thickness of metal, c, allowed for any anticipated corrosion is then added to the calculated required thickness, and the final thickness value rounded off to the nearest nominal plate size of equal or greater thickness. Equation 3.13 becomes:
Fig. 3.8. Longitudinal forcer acting on thin cylinder under internal pWSS”r.3.
I b
with an internal pressure. Figure 3.8 shows a diagram representing a thin-walled cylindrical vessel in which a uniform stress,f, may be assumed to occur in the wall as a result of internal pressure. The nomenclature used in Figs. 3.8 and 3.9 is: 1 = length, inches d = inside diameter, inches t = thickness of shell, inches p = internal pressure, pounds per square inch gage
+
hngiludinat Stress. If one limits the analysis to pressure stresses only, the longitudinal force, P, resulting from an internal pressure, p, acting on a thin cylinder of thickness t, 1enpt.h I, and diamet,er d is:
a = area of met,al resisting longit~udinal rupture = tnd
t= FE + c
(3.16)
M ODIFICATION OF EQUATIONS. For storage vessels the maximum allowable working stress is considered approximately one third of the ult,imate tensile strength of the steel; that is, a safety factor of 3 is employed, which is common for static structural loads on steel. The stresses
l‘hert~fore P p7rd”i-l j = stress = - = --- --- = 2Pd= induced stress, pounds a hd per square inch or
t = pd ?f
(3.15)
where t = thickness of shell, inches p = internal pressure, pounds per square inch d = inside diameter, inches j = allowable working stress, pounds per square inch E = joint efficiency, dimensionless (see Table 13.2) c = corrosion allowance, inches
Inrd2 z-4
1
t= and Eq. 3.14 becomes:
P = force tending to rupt.ure vessel longitudinally
and
45
(3.13)
CIRCUMFERENTIAL S TRESS. If oue refers to Fig. 3.9 and -,onsiders the circumferential s~ressrs caused by internal ,:ressure only, the following analysis rrriiy be developed :
I’ = force tending to rapture vcssrl circumferentially = pdl a = area of metal resisting force = 2t1 P pdl pd f =I sf,ress = -. = --.. = _
a
2t1
2t
Fig. 3.9.
(3.14)
1 = PC” ?f
pWS*“W.
_ _ - l._
- . - . - \ \
-
_ . . -
-
-
- . ~
I / . -
Circumferential forces acting on thin cylinder under
intern;s
46
Design of Shells for Flat-bottomed Cylindrical Vessels
are computed on the assumption that the tank is tilled level with water at 60” F (density 62.37 lb per cubic ft) and that the tension in each ring is calculated at. a point 12 in. above the center line of the lower horizontal joint of the horizontal row of welded plates being considered. The hydrostat,ic pressure in cylindrical storage tanks varies from a minimum at the top of the upper most course to a maximum at the bottom of the lowest course. In determining the plate thickness for a particular course, a design based upon the pressure at the bottom of the course results in overdesign for the rest of the plate and therefore does not represent maximum economy. A design based upon the pressure at the top of the course would result in underdesign, which would not be good engineering practice. However, some consideration should be given to the additional restraint offered by the plates adjoining a particular course. In the lowest course, the plates of the vessel bottom offer considerable restraint to the bottom shell course. This additional restraint of the bottom edge is effective for an appreciable distance or height from the bottom of the lowest course. In an intermediate course with a course of heavier plates below, the top of the heavier plates will be understressed; this will tend to cancel any overstressing of the bottom of the course in question. Therefore, a design based upon the pressure at a height of 1 ft from the bottom of the course may be considered conservative. The following equations may be derived if one assumes that the density of the fluid will not exceed that of water, which is used for the hydrostatic test of the tank.
p = p (ff - 1) 144
(3.17)
where p = density of water at 60” F = 62.37 lb per cubic ft H = height, in feet, from the bottom of the course under consideration to the top of the top angle or to the bottom of any overflow which limits the tank’s filling height p = internal pressure, pounds per square inch For double-welded butt-joint construction, the above ,definition of p may be substituted into Eq. 3.16. When one uses an allowable design stress of 21,000 psi for SA-7 plate and a joint efficiency of 0.85 for doubled-welded butt.joint construction, the following substitution results:
therefore
t = O.O001456(H - l)D + c
(3.18)
where t = shell thickness, inches H = height as defined in Eq. 3.17, feet D = inside diameter, feet c = corrosion allowance, inches If double-full-billet lap-joint construction is assumed, the corresponding joint efficiency, E, is 0.75. Then Eq. 3.18 becomes: t = O.O001650(H - l)D + c (3.19) If low-alloy high-strength steel is used, the maximum allowable stress is taken as 60% of the minimum yield
’ -------r -- _ -----------
point. When one uses double-welded butt-joint and lowalloy high-strength steel construction Eq. 3.18 becomes: 5.096(D)(W - 1) + c t=.fY.P.
(3.20)
For lap-welded construction, t = 5.775(D)(H - 1) P .rS.P.
(3.21)
wheref,,,. = minimum specified yield point of steel plate, pounds per square inch Steels of different composition should not be mixed in any course of a vessel, with the exception that it is permissible to use a different steel for providing reinforcing area for shell openings. In case two different steels are used in any part of an area of reinforcement the stress corresponding to the weaker steel should be used to determine the thickness of shell metal to be reinforced. Equations 3.18 and 3.19 are given in API Standard 12C and apply only to the steel materials approved by the code. When other materials of construction are used, the constants of these equations must be recalculated through the use of the proper value for the allowable stress. 3.5b Practical It Dimensions.
Considerations
in
Selecting
Shell-plate
should be emphasized that Eqs. 3.15, 3.16, 3.18, 3.19 are useful only for predicting the thickness of metal required to withstand the int,ernal pressure. Other factors such as structural stability, live loads, wind, ice and snow loads, and fabrication procedures must be considered. Minimum-thickness specifications for tanks by API Standard 12 C include some of these other considerations; they are listed in the tables of Appendix E. To summarize the important points of these tables, consider first the thickness of the tank shell. It should not be less than >/4 in. for tanks 50 to 120 ft in diameter, >is in. for tanks 120 to 200 ft, and 34 in. for tanks over 200 ft; or less than xs in. for tanks smaller than 50 ft in diameter. These minima are derived from practical considerations of stiffness, corrosion allowance, wind loads, and so on. Tanks having shell thicknesses greater than these minima may use decreasing thicknesses for the upper courses. The thickness for the upper courses necessary to contain the hydrostatic pressure may be determined by substituting the appropriate depth of liquid into Eq. 3.18 or 3.19. However, the thickness should not be less than the minimum. It should be noted that these minima are expressed in fractions which correspond to mill plates of standard thickness. In general it is more economical to fabricate the smaller vessels from mill plate of standard thickness than to order plate rolled to a specified thickness. However, for the large vessels the shell must be thicker to withstand the hydrostatic pressure. With this greater shell thickness, rigidity and corrosion are no longer the controlling factors. Some reduction in cost may result from ordering these plates rolled to thickness, especially if the required plate thickness lies about midway between standard plate sizes. In specifying shell-plate widths a compromise must be made between the costs of the material and the costs of field welding including plate preparation such as edge
\ .-p - - -.--- - - -~
46
Design
of
Shells
for
Flat-bottomed
Cylindrical
Vessels
are computed on the assumption that the tank is tilled level with water at 60” F (density 62.37 lb per cubic ft) and that the tension in each ring is calculated at a point 12 in. above t.he center line of the lower horizontal joint of the horizuntal row of welded plates being considered. The hydrostatic pressure in cylindrical storage tanks varies from a minimum at the top of the upper most course to a maximum at the bottom of the lowest course. In determining the plate thickness for a particular course, a design based upon the pressure at the bottom of the course results in overdesign for the rest of the plate and therefore does not represent maximum economy. A design based upon the pressure at the top of the course would result in underdesign, which would not be good engineering practice. However, some consideration should be given to the additional restraint offered by the plates adjoining a particular course. In the lowest course, the plates of the vessel bottom offer considerable restraint to the bottom shell course. This additional restraint of the bottom edge is effective for an appreciable distance or height from the bottom of the lowest course. In an intermediate course with a course of heavier plates below, the top of the heavier plates will be understressed; this will tend to cancel any overstressing of the bottom of the course in question. Therefore, a design based upon the pressure at a height of 1 ft from the bottom of the course may be considered conservative. The following equations may be derived if one assumes that the density of the fluid will not exceed that of water, which is used for the hydrostatic test of the tank.
p = p (H - 1) 144
(3.17)
where p = density of water at 60” F = 62.37 lb per cubic ft H = height, in feet, from the bottom of the course under consideration to the top of the top angle or to the bottom of any overflow which limits the tank’s filling height p = internal pressure, pounds per square inch For double-welded butt-joint construction, the above ,detinition of p may be substituted into Eq. 3.16. When one uses an allowable design stress of 21,000 psi for SA-7 plate and a joint efficiency of 0.85 for doubled-welded buttjoint construction, the following substitution results: t = 62.37(H - 1)(12D) + c 2(21,000)(0.85)(144) therefore
t = O.O001456(H - l)D + c
(3.18)
where t = shell thickness, inches H = height as defined in Eq. 3.17, feet D = inside diameter, feet c = corrosion allowance, inches If double-full-fillet lap-joint construction is assumed, the corresponding joint efficiency, E, is 0.75. Then Eq. 3.18 becomes: t = O.O001650(H - l)D + c (3.19) If low-alloy high-strength steel is used, the maximum allowable stress is taken as 60% of the minimum yield
point. When one uses double-welded butt-joint and lowalloy high-strength steel construction Eq. 3.18 becomes:
t = 5z096(D)W - 1) + c
(3.20)
dY.P.
For lap-welded construction,
t = 5.775(D)(H - 1) E JY.P.
(3.21)
wheref,.n. = minimum specified yield point of steel plate, pounds per square inch Steels of different composition should not be mixed in any course of a vessel, with the exception that it is permissible to use a different steel for providing reinforcing area for shell openings. In case two different steels are used in any part of an area of reinforcement the stress corresponding to the weaker steel should be used to determine the thickness of shell metal to be reinforced. Equations 3.18 and 3.19 are given in API Standard 12C and apply only to the steel materials approved by the code. When other materials of construction are used, the constants of these equations must be recalculated through the use of the proper value for the allowable stress. 3.5b Practical Considerations in Selecting Dimensions. It should be emphasized that Eqs.
Shell-plate
3.15, 3.16, 3.18, 3.19 are useful only for predicting the thickness of metal required to withstand the int,ernal pressure. Other factors such as structural stability, live loads, wind, ice and snow loads, and fabrication procedures must be considered. Minimum-thickness specifications for tanks by API Standard 12 C include some of these other considerations; they are listed in the tables of Appendix E. To summarize the important points of these tables, consider first the thickness of the tank shell. It should not be less than s/4 in. for tanks 50 to 120 ft in diameter, >i6 in. for tanks 120 to 200 ft, and 35 in. for tanks over 200 ft; or less than xs in. for tanks smaller than 50 ft in diameter. These minima are derived from practical considerations of stiffness, corrosion allowance, wind loads, and so on. Tanks having shell thicknesses greater than these minima may use decreasing thicknesses for the upper courses. The thickness for the upper courses necessary to contain the hydrostatic pressure may be determined by substituting the appropriate depth of liquid into Eq. 3.18 or 3.19. However, the thickness should not be less than the minimum. It should be noted that these minima are expressed in fractions which correspond to mill plates of standard thickness. In general it is more economical to fabricate the smaller vessels from mill plate of standard thickness than to order plate rolled to a specified thickness. However, for the large vessels the shell must be thicker to withstand the hydrostatic pressure. With this greater shell thickness, rigidity and corrosion are no longer the controlling factors. Some reduction in cost may result from ordering these plates rolled to thickness, especially if the required plate thickness lies about midway between standard plate sizes. In specifying shell-plate widths a compromise must be made between the costs of the material and the costs of field welding including plate preparation such as edge
Shell
Design
of
Large
Single-V double-welded butt joint
Square-groove double-welded butt joint tpatiial penetration)
I 4
47
Single- bevel double-welded butt joint
fh)
Double-welded lap joint
Double-welded full-fillet lap joint
Vertical
Fig. 3.10. Typical shell joints recommended by API Standard 12 C.
working for butt welding. Plates 80 to 90 in. wide may be purchased at base cost without the inclusion of a price extra for width. Plate widths over 90 in. carry “width extra” charge which increases appreciably with increase ig width. Therefore, it is advantageous to use the widest e that does not iqvolve an excessive extra cost. plates havmg a wld$h of 96 m. are me& extensiv&y M&i. In specifying plate lengths, theire are no price extras for leng&s between 8 and 50 ft when miM plates are purchased. Therefore, the longest plates which can be readily handled and shipped are specified. Thus plate lengths of approximately 20 to 30 ft, are selected since longer plates are difficult to handle. The exact length of the plates is determined by dividing the circumference by the number of shell plates, with proper allowance made for the vertical weld joints. 3.5~ Butt-welding versus Lap-welding. The plates of the shell may be butt- or lap-welded depending upon the design and economic considerations. However, x in. is the maximum plate thickness for lap-welded horizontal joints and N in. is the maximum plate thickness for lapwelded vertical joints. Butt-welded joints may be used for shell plates for all thicknesses up to and including 134 in. for plain carbon-steel-plates and up to and including lgs in. for low-alloy high-strength steel plat,es. The plates for butt welded joints must be squared. Squaring of the plates for lap-welded joints is not necessary. For this reason, plates for lap-welded joints are less expensive; however, erection by butt welding is somewhat faster. Because of the present high labor costs most tanks are now fabricated
Double-U double-welded butt joint
Square-groove double-welded butt joint
Horizontal Joints
I
Tanks
Single-U double-welded butt joint
double-welded butt joint
Double-bevel double-welded butt joint (Park4 penetration)
Storage
Joints
(Courtesy of American Petroleum Institute.)
by butt welding. Each course of the tank must be inside the course beneath it when the horizontal joints are lap welded. Vertical seams should not be in alignment for any of three consecutive courses. This is a precaution against localized conditions of stress at welds and aids in assuring the distribution of the stresses uniformly throughout the vessel. The requirement of a minimum distance of 2 ft between vertical joints in adjacent courses is an additional safety measure. In the butt welding of the shell, the joints should preferably be doubled welded with complete penetration and fusion. A single-welded butt joint with backing may be used instead, with the same joint efficiency. It is particularly important that the vertical butt joints have complete penetration and fusion because these joints are under the full tensile stress in the shell. The horizontal joints are not under this tensile stress. However, for structural strength against wind loads, and so on, and for prevention of failure by notch brittleness, all horizontal single-beveled joints should have complete penetration, as shown in Fig. 3.10b. With squared plates (square-groove) and doublebeveled butt plates for horizontal joints, as shown in Fig. 3.10~ and c respectively, incomplete penetration may be used for the sake of economy. However, with partial penetration the thickness of the unwelded portion should not exceed one-third the thickness of the thinner plate, and the unwelded portion should be located at approximately the center of the thinner plate. If a horizontal butt joint is offset because of different plate thicknesses, the inside surfaces should be flush.
40
Fig.
Design of Shells for Flat-bottomed Cylindrical Vessels
3.11.
Circumferential joints for tank shells.
(Courtesy of Hammond
iron Works.)
Lap joints should have an overlap of at least 5 X t inches, and in no case should the overlap be less than 1 in. Vertical lap joints should have continuous full-fillet welds both inside and out. On horizontal lap joints, as shown in Fig. 3.11, the fillet should have a size not less than one third t,he thickness of the thinner plate, and in no case should it be less than x,j in. Figure 3.11 also shows a horizontal butt-welded shell of a storage vessel. Note that the heavier courses at the base are V butt-welded, whereas the upper courses are plain butt-welded. 3.5d Cold Forming of Shell Plates. Plain carbon-steel shell plates having a thickness of XG to N in. for tanks having a diameter of 40 ft or more or low-alloy high-strength steel plates for tanks having a diameter of 50 ft or more can be deflected on erection and therefore need not be cold formed by rolling to the radius of curvature of the shell. If the diameter is 60 ft or more for plain carbon st,eel or 100 ft or more for’ low-alloy hi:h-strength steel, shell plates having a thickness of g to 55 in. may be deflected on erection without cold forming. Plain carbon-steel plates of thicknesses from $4 to N in. and diameters over 120 ft need not be cold formed. However, all plain carbon-steel plates having a thickness of 36 in. and over and all lowalloy high-strength steel plates having a thickness of W in. or over must be cold formed to the shell radius regardless of the shell diameter. Figure 3.12 shows the field welding of horizontal butt-welded seams of the shell of a large tank. 3.5e S h e l l P a r t s . In addition to the shell plates a variety of other shell parts and accessories must be considered in the shell design. Figure 3.13 shows typical tank accessories including shell nozzles, manholes, ladders, and so on. SHELL NOZZLES. Pipe lines which bring the fluid to and from the tank are attached to short pipe connections welded into the tank shell. These connections are called “nozzles” arid may be fabricated of screwed pipe fittings if t,he pipe
c
r
\
\ -
\I /
size is not over 3 in. nominal pipe size. For small pipe sizes, screwed fittings are usually preferred because they are cheaper than flange fittings. However, pipe with screwed fittings having a nominal size greater than 2 in. is rather difficult to fit because such heavy pipe is so rigid that it cannot be deflected easily to aid in aligning the threads during fitting. Therefore, for practical reasons, it is recommended that any nozzle having a nominal size greater than 2 in. have flange fittings. It is usually desirable to locate nozzles for filling and discharging near the bottom of the tank to obtain the benefit of gravity in discharging and to avoid pulling a partial vacuum on fluids which are volatile. However, water and sludge may separate and collect on the bottom. To avoid pumping this sludge out of the discharge line, the discharge nozzles are usually placed on the shell a short distance above the bottom. Another nozzle with a sump is placed at the bottom to remove material accumulating below the discharge nozzle and to completely drain the vessel. Typical nozzles of both the screwed-fitting and flange-fitting type are shown in Fig. 3.14, and standard dimensions for these nozzles are given in Items 1 and 2 of Appendix F. SHELL MOLES. Manholes are necessary in closed vessels to permit inspection, cleaning, repairs, and so on. These manholes may be located on the shell or on the roar or at both locations. Manholes located on the shell have the advantage that it is somewhat easier to use a shell opening to clean or repair a vessel. Shell manholes have the disadvantage that they usually cannot be opened unless the vessel is empty and therefore are not used as often for inspection as roof manholes. Items 3.4, and 5 of Appendix
Fig. 3.12. Field welding of shell circumferential buit Hammond Iron Works.)
jotnr.
(Cow&v
04
Shell Design of Large Storage Tanks
sidered as available for reinforcement out to a distance of four times the neck-wall thickness, measured from the out.side of the shell. The metal in the neck lying within the shell-plate thickness may also be included. If the neck of the fitting extends both inward and outward as shown in the center and right of Fig. 3.15, credit may be taken fol the metal of the neck over a dist.ance of eight neck-wall thicknesses plus the shell thickness.
% gives typical dimensions for shell manholes and manholecover plates designed as shown in Fig. 3.15. REINFORCEMENT OF S HELL OPENINGS. All openings such as nozzles and manholes, made in the shell in which t.he The opening is over 2 in. in diameter should be reinforced. reinforcement prevents local overstressing of the shell around the opening. The minimum cross-sectional area of the reinforcement should not be less than the product of the vertical diameter of the hole cut in the tank shell times the shell plate thickness. The cross-sectional area of the reinforcement is measured parallel to the axis of I.he shell across the center of t,he opening.
02 Conservation
vent
49
In the case where two or more openings are located ciosc together and the toes of t.he fillet welds fixing the reinforciup plates for these openings come within t.wice t.he shell t hi&-
@ Free vent y
\
manhole
1 Sheave @
Fig.
3.13.
Usual accessories and fittings on standard cone-roof tanks.
Included as standard 4. One 20” shell manhole
(Courtesy of Hammond Iron Works.)
Included as extra 9. Sump
15. Connection for foam chamber
1. One 20” roof manhole
10. Swing line unit complete
16.
3. One 6” gauge hatch
11. Water draw-off
17. Flame arrester
4. Roof nozzle for vent
12. Conservation vent (volatile products)
18.
Antifreeze
5. Ladder (small tanks only)
13. Free vent (nonvolatile products)
1-8
Extra
6.
14.
(a)
Target-type
14.
(b)
Ground-reading-type
Spiral
(12 or 13)
Drain
stairway
7. Two shell nozzles 8.
Flange
for
water
float
gouge float
valve
units
5. (a) Inside ladder gouge
draw-off
Reinforcement metal may include any one of the four metal parts listed below or any combinat,ion of them: 1. The metal in the at,tachment. flange of an attached iitting, 2. The metal of a reinforcing plate, 3. The metal of any excess shell-plate thickness beyond that required from a calculation of the minimum plate thickness, 4. The metal in the neck of a fitting. This can be considered as part of the reinforcement area. If the fitting e.xtends only outward, the metal in the neck may be con-
ness of each other, a single reinforcing plate should be usc~l. This plate should be proportioned for the largest opening in the group. If the reinforcing plates for one or more small openings are of such a size that they lie entirely within I 1 1% area of the reinforcement plate for the largest opening. I hcb small openings can be included in the normally designc~tl reinforcement plate for the largest opening without incrrasing the size of this reinforcing plate. If, however, any opening intersects the vertical center line of another opeuing, the width of the reinforcement (along the verticalcenter line of eit.her opening) should not be less than the sum of lhe widths of the two plat,es that would normally be used.
50
Design
of
Shells
for
Flat-bottomed
Cylindrical
Vessels
r
A circular reinforcing plate may be substituted for the plate shown, for the 3- to l&in. size nozzles inclusive, provided the diameter of the circular plate is made equal to W
Permissible alternative square cut
radius of tank shell Bolt holes shall straddle the flange center lines
Reinforcfng plate
Se no
Single flange
Double flange
(4
Fig. 3.14.
(4
Special flange (a)
Shell nozzles recommended by API Standard 12 C (see items 1 and 2 of Appendix F for typical dimensions.)
(Courtesy
of
American
Petroleum
Institute.)
Finally, if the normal reinforcing plates for the smaller openings do not fall within the area limits of the reinforcement for the largest opening, the group reinforcing plate should include within its outer limits the normal reinforcing plates for all openings in the group. 3.5f Reinforcement of Top Course of Shell for Large Open Tanks. Open vessels of large diameter may not have
the necessary inherent rigidity to withstand wind loads without deforming and excessively straining the structure. Two methods of stiffening are available: shell plates may be made thicker, or suitable stiffening girders may be added to the structure. The use of thicker shell plates usually is more costly than the use of stiffening girders. W i n d girders or stiffening rings for open tanks are located at or
Shell
Design
of
Large
Gasket: 20” manhole-25%” OD x 20” ID x 4” thick 24” manhole-29%” OD x 24” ID x !$” thick Long-fiber asbestos sheet
Storage
Tanks
51
Shape manhole flange to suit curvature of tank
I 30” Increase if necessary for clearance i
finished to provide a minimum gasket-bearing width of L in. for minimum thickness at bolt circle, see Appendix F.
Alternative designs of manholes Fig. 3.15.
Shell manholes recommended by API Standard 12 C (see items 3,4, and 5 of Appendix F for typical dimensions).
(Courtesy cf A m e r i c a n P e t r o l e u m
Institute.)
I
H = height of shell including any “freeboard” provided above the maximum filling height, feet
near the top of the vessel on the top course of shell plate. The stiffening ring is placed preferably on the outside of the shell rather than the inside. The required section modulus as specified by API Standard 12 C for the stiffening ring may be computed by Eq. 3.22: z = 0.0001D2H (3.22)
The calculation of the available section modulus of the stiffening ring may include a portion of the tank shell which is considered to be effective for a distance of 16 plate thicknesses from the ring, as indicated in Fig. 3.16. W h e n curb angles are attached to the top edge of the shell ring by butt welding, this reinforcing distance sbnuld be reduced. by the width of the vertical leg of the angle. Table 3.4
:vhere z = section modulus, inches3 D = nominal diameter of the tank, feet
L-. -I-- - - - - -
_..- - -
.-.._
--\- - ~~
\I
/
-~
_.
-uorrotned
Cylindrical
.=“I
Vessels
appropriate dimension for the minimum width of the web, b, is 12 in., which corresponds to a section modb 28.1 in3 The rest of the sectional dimensions, incl wind-girder plate thickness of $6 in., are fixed by E of Fig. 3.16. A stiffening ring such as this is fabri by bending plate steel into the appropriate shape. a wide-webbed ring cannot, be easily rolled to the req
Table 3.4.
Detail B
Section Moduli of Various Stiffening-ril
Sections on Tank Shells, Recommended by API Standard 12 C
(Courtesy of American Pet,roleum tnstitute) Member Size? in.
Fig. 3.16.
Typical reinforcement for top course of shells for open vessels
secommended
by API Standard 12 C.
(Courtesy
of
American
Petroleum
Institute.)
lists the section moduli for the rings shown in Fig. 3.16 for two shell thicknesses. Stiffening rings may be made either of structural sect,ions or formed-plate sections or combinations of the two. The minimum size of the angle specified by Standard 1.2 C either alone or with a built-up section is 236 X 255 X j/, and the minimum plate thickness is >/4 in. If the stiffening rings are located more than 2 ft below the top of the shell, a minimum of a 235 X 235 X Ns top curb angle is required for Ns-in. shells. A minimum of a 3 X 3 X $/4 angle is required for M-in. shells. Other members of equivalent section modulus may be substituted. Drain holes should be provided in rings that may trap liquid. Stiffening rings are sometimes used as walkways, in which case they should provide at least 24-in. of clear walking space and should be located preferably 3 ft, b in. below the top curb angle. Any such angle having a horizontal web exceeding 16 t.imes the web thickness requires suitable vertical support. EXAMPLE D ESIGN 3.1, W IND G IRDER FOR O PEN VESSEL. h wind girder is required for an open vessel 80 ft, 0 in. in inside diameter and 40 ft high, having a top-course plate thickness of f$ in. When one uses Eq. 3.22, the minimum section modulus of Lhe wind girder is: z = 0.0001D2H = o.oool(so)y4o) z = 25.6 im3 When one uses the type of construction shown in detail F of Fig. 3.16, referring to Table 3.4, he finds that the
Shell in. _--~ Thickness, --.-___ $16 s/4 Section Moduli
Top angle: det,ail A, Fig. 3.16 234 x 21.& / x 5c 0.41 235 x 21,s x $iF, 0.51 3 x3 x 53 0.89 Curb angle: det.ail B, Fig. 3.16 254 x ZS$ x 53 1.61 235 x 2q x p1;la 1.89 3 x 3 XSk 2.32 3 x 3 xps 2.78 4 x 4 X!i 3.64 4 x 4 xyg k. 17 One angle: detail I:, Fig. 3.16 256 x 2% x $4 1.68 ajg x 255 x $f(j 1.98 4 x 3 XQ 3.50 4 x 3 x3is 4.14 5 x 3 XX6 5.53 5 x 3$$ x 3’6 6.13 5 x31,5x36 7.02 6 x4 ~9s 9.02 Two angles: detail D, Fig. 3.16 4 x3 x 546 11.27 4 x 3 x7* 13.06 5 x 3 xpi, 15.48 5 x 3 x3$z 18.17 5 x315x- 41 6 16.95 5 x335x$; 19.99 6 x 4 ~“6 27.?4
0.42 0.52 0.91 1.72 2.04 2.48 3.35 4.41 5.82 1.78 2.12 3.73 4.45 5.95 6.60 7.61 10.56 11.78 13.67 16.24 18.89 17.70 20.63 28.92
Formed plate: detail E, Fig 3.16 6, in. 10 22.3 12 28.1 14 34.3 16 40.8 18 47.7 20 54.9 22 62.4 24 70.3 26 78.5 28 87.0 30 95.9 32 105 1 31 114.7 36 124.5 38 134.7 40 145 3
Shell Design of Large Storage Tanks
diameter because of its stiffness. Therefore, it is more convenient to weld straight sections of formed plate, making a polygonal stiffening member. The inside edge will be flame cut and made smooth to form an arc having a radius equal Usually two or more to that of the shell outside diameter. sections are welded end-to-end in the shop to minimize the field welding required. Figure 3.17 shows a preliminary sketch of a wind-girder subassembly, using 20 equal sections with every two sect,ions welded end-to-end for subassembly in the shop. For determination of web dimension, 5, cos 9" = 40
ft, 0% in. + 12 in. -___-- = 0.98769 ft, O+ in. + s
40
492.25 x=-- in . - 480.25 0.98769 = 18.135
in.
in. = 1 ft. 6; in.
For determination of inside chord length of subassembly, Chord 2
= 40 ft, 0: in. (sin 18”)
Chord = 2(480.25 in.)(0.30902) = 296.814
in. = 2.4 fl, 8:# in.
For determination of out,side chord length of subassembly Chord
__ = (40 2
continuous double-wielded but 1. joint or a continuous doublewelded lap joint. SHELLS WITH SELF-SUPPOHTING ROOFS. A self-support ing roof is one which is supported only on its periphery-, without added structural support. Such roofs cause :I compressive stress in t,he roof plates, which is transferred to t.he shell as hoop tension. A stiffening angle should be added to the top shell course at the junction of the roof and shell to absorb the stress as a tensile load. The forcc,s acting On the ring are shown in Fig. 3.19. The following nomenclature will be used in the equation* explaining Fig. 3.19: a = cross-sectional area of stiffening ring, square inches I) = nominal tank diameter, feet 0 = angle of the cone element wit.h the horizontal, degrees P = roof load, pounds per square foot, (live load of 25 lh per sq ft plus dead load) 7’1 = compressive force per linear inch along an element of the cone, pounds per linear inch Tz = tensile force per linear inch in a circumferential diwction, pounds per linear inch TX = horizontal component of Tl, pounds per linear inch F = circumferential tensile force act.ing in stiffening ring, pounds
W = t.otal load on roof, pounds = r$ P f = tensile stress, pounds per square inch
ft, 0) in.) + (1 ft, 6; in.) sin 18”
Chord = 2(41 ft, 63 in.)(0.30902) = 2(498.375)(0.30902) = 308.016 = 25
53
in.
ft, 8 in.
For determination of outside chord length of section, Chord = 2(41 ft, 6g in.) sin 9” = 2(498.375)(0.15643) = 155.9216 in. = 12 ft, 11 ++ in. Figure 3.18 shows typical detail designs for a wind girder for an open tank 80 ft in diameter and 40 ft high as described in example design 3.1. 3.5g Reinforcement of Top Course of Shell for Large Closed Tanks.
S H E L L S W I T H R O O F S H A V I N G C O L U M N S U P P O R T . If tanks are closed with a roof, the roof provides additional structural rigidity to the upper course of shell plates. As a result, smaller stiffening rings are used for closed vessels. For -Q&S =&\I column supports having diameters of 35 ft or fess, Zjg x .Zf$ x ji in. is the minimum-size stiffening angle. For tanks having diameters of 35 to 60 ft; the size of the angle should be increased to 235 x 235 x 54s in.
Fig. 3.17.
design 3.1.
Preliminary sketch of wind-girder subassembly for example
54
Design of Shells for Flat-bottomed Cylindrical Vessels Chord = 25’-8”
I4
Cut back both ends. as shown, 8” shop after welding sections together. See secbon E-E
4 36” h 2’-8*,W.P.-
W.P. Cut thts edge true and smooth after shop welding sections together-
;ow
-’
Arc on edge Of PL. 25,-l%”
Section
E-E
Chord = 24’-8’x6”
Wind-girder assembly, 10 required
Section A-A Section
Detail of wind-girder section, 20 required Fig. 3.18.
Details for wind girder for example design 3.1.
In reference to Fig. 3.19a, the roof load, W, results in compressive force, Tr, in the roof plates as follows:
T1=
w
sin e(?rD) 12
B-B
also F = aj
PD zzrp
TV = af = PD cot e 60 48
48 sin 0 To solve for a,
Compressive force, Tr, has a horizontal component, Ta, or
a =
T3 = T1 cos fj = ED+Le
PD2 cot 0 8.f
f = 18,000 psi, allowable
In reference to Fig. 3.19b,
Therefore
2F = 12DT3
PDZ a = 8(18,000) tan 8
T3 = F‘
a=- PD’ 144,000 tan B
or 60
if: live load = 25 lb per sq ft -
F
(for 4 in. plates)
dead load = 11 lb per sq ft P = 36 lb per sq ft By substituting, a =
(a) Fig. 3.19. Loads on conical roofs.
(b)
F
D 36D2 144,000 tan 8 = 4000 tan e
For small angles, tan e is approximately equal to sin 8; therefore 02 a = 4000 sin e
Example Design 3.2, Complete Shell Design for a Closed Vessel
The API Standard 12 C recommends the use of Eq. 3.23 for determining the required reinforcing area. 02 a = 3000 sin 19 In applying Eq. 3.23 credit may be taken for the crosssectional area of the shell and roof plates within a distance of 16 times their thickness from the stiffening angle. In other words, the sum of these areas must be equal to or greater than (D2/3000 sin 0) for cone-roof construction. For dome or umbrella-roof construction a similar equation may be derived. (3.24) where D = diameter of tank, feet R = radius of curvature of dome, feet a = reinforcing area, square inches 3.6
EXAMPLE DESIGN 3.2, COMPLETE FOR A CLOSED VESSEL
SHELL
Since corrosion allowance = 0, tl = 0.57 in. By the use of 10 plates and with an allowance of %a in. for vertical weld joint, the center length of each plate is calculated from the circumference. L =
nd - weld length = 3.1416(1200.57) - lO(?ia) 12n 12(10)
3770.136 = ~ = 31.4178 ft 120
(or 31 ft, 5 in.)
Standard mill plates of 96-in. width will be specified and butt welding of the shell plates will be used for fabrication. Shearing of the shell plates is required to square the plates for butt welding. Therefore, the final height of the shell plates will be slightly less than 96 in. The thickness of the shell course above the bottom course can also be determined by Eq. 3.18. The proper height, however, for this calculation will be (40 ft - 8 ft) or 32 ft
DESIGN
t2 = 0.0001456(32 - 1)lOO + c
The design calculations and drawing for the steel shell only for a 55,000-bbl oil-storage tank having a cone roof are required. The cone roof is to be supported by internal columns, girders, and rafters. 3.6a Proportioning. It is estimated that the ratio of annual cost of the shell per unit area, cl, is two times the annual cost of the bottom per unit area, ~2, and that the annual cost of the roof, cg, is 1.8 times the bottom annual cost per unit area. The annual cost of the land area, cd. and preparation, cg, together is estimated at 0.40 of the cost of the tank bottom, ~2. Corrosion allowance will be negligible. When we substitute the above information into Eq. 3.9, D = 4H
55
= 0.452 in. + c Specify that t2 = 0.46 in. L = 7r0200.46) - lO(O.15625) 10 x 12
= 31 ft 4g in
Accordingly, the thickness of the third course (H = 24 ft) will be:
t3 = 0.0001456(24 - 1)lOO + c = 0.335 in. + 0
2.002 --~ = 2.5H
~2 + 1.8~2
+ 0.4~2
When we substitute the same information into Eq. 3.1,
\
H = 4(55,000 X 42/7.48) (2SH)% H3 = 62,910 cu ft
or H = 39.8 ft Since D = 2.5H, D = 2.5 X 39.8 = 99.5 ft The tank dimensions will be 100 ft inside diameter by 40 ft high. Appendix E, item 3 indicates that such a tank will have a volume of 55,950 bbl. 3.6b Design of Shell Courses. The thickness of the bottom shell course can be determined readily by Eq. 3.18 (for butt.-welded assembly).
Fig.
3.20.
Elevation view of shell for example design 3.2. A.
Cable
sheave
B. Winch C. 4” steam nozzles D. 10” shell nozzle E. 1 s” extra-heavy couplings
tl = 0.0001456(40 - l)(lOO) + c
F.
4”
water-draw-off
nozzle
G. Shell nozzle for double swing joint
= 0.568 in. + c
- --
1-- - - -- .---
H. 24” shell manhole
\
\
\ r-7--
--
._
56
Design of Shells for Flat-bottomed Cylindrical Vessels
PL +4-k” L+- 31’ -5 g,“-
#5 shell PL
P L . $3-0.34
PL. %‘-0.46’
k
circumference P L . Ql-0.57
iI+-31’-5%“-
Section through shell
+l shell PL. Section through vertical joints
Bill ~$$$?~ Mark D e s c r i p t i o n /o */ PL95~YO.57” lo x2 P/. 95 Y4”x 0.46 * IO *3 P,! 93%*x0.34” IO #4 Pl 95%“X%# .’ IO “5 Pl: 9sy6/e’x IO TA5 L 3”X3”X+L8/B”
74#
o f Moferioh Lrnqth ft. in.
31 5 I 31 4%; 31 4% 31 4% I 31 4%” I 31 4%
O IO Pl: 96”XO.46’ IO Pl 94”XO.34” O P/. 96”X$+” O Pl: 94”X;/4” IO L 3”X3”X%”
Specify that tS = 0.34 in. L = ~(1200.34) - lO(O.15625)
10 x 12
= 31
ft, 48 in.
Likewise, the thickness of the fourth course (N = 15 ft) will be :
t4 = 0.0001456(15 - 1)lOO + c = 0.204 in. L = ~(1200.25) - lO(O.125)
10 x 120
= 31
ft, 4s in.
Specify 14 and t5 = 0.25 in. since the thickness as determined from the appropriate relationship results in a thick-
31
5Y4.
31
5%
31 31 31
5?&
5%” 8
ness of less than >a in. This means simply that hydrostat iv pressure stress considerations are not controlling and thr structural stability of the thin shell is the prime consderalion. Thus the minimum shell thickness of >/4 in. for shells of this size as set by API 12 C for tanks 50 ft in diametel and larger will be specified (see Appendi:: E, item 4). The required thicknesses for the various shell courses could have been determined from Appendix E, item 4. It will be seen that the calculated and tabulated shell-course thickness agree. Only the shell plates of I he bottom course need to be cold formed to the shell diameter. Frequently, however, thinner shell plates are rolled to facilitate erection. Accordingly. the shell specifications will call for rolling of the bot,t.om three courses.
Example
Design
The minimum-size top ft, 0 in. in diameter with a roof supported on columns is 3 in. X 3 in. X s in. and will, therefore, be used. Specifications will call for butt welding of this angle to the top course. By using 10 lengths of top angle, the length of each angle section is calculated as follows : 3.6~ Design of Top Angle. angle for a tank larger than 60
3.2,
Complete
Shell
Design
for
a
Closed
Vessel
57
L = a(1200.375) - lO(O.15625) 10 x 12
L=
31
ft, 4+$ in.
Figure 3.20 shows the elevation view of tank she!!, and Fig. 3.21 shows the shell details.
PROBLEMS
1. A cone-roof tank having a filled capacity of 100,000 bbl is to be designed. Determine the optimum proportion of D/H from the following cost considerations. The fabricated shell, roof (including plates, rafters, girders and columns), and bottom are estimated to cost 18, 20, and 14 cents per lb respectively. Foundation costs are estimated to be $4000. Fixed annual charges including amortization, interest, and so on are estimated to be 40% per year based on initial installed cost. The annual charge for the land allocated to the tank area is $500. 2. A wind girder is required for an open vessel 120 ft inside diameter and 48 ft high. The top course of the shell is fabricated from x-in. plate. If the wind girder consists of 30 identical sections corresponding to detail E of Fig. 3.16, determine the minimum section modulus and the girder dimensions. 3. Determine the required cross-sectional area of the stiffening ring for a self-supporting conical. roof 30 ft 0 in. in diameter having an angle 0 of 15” with the horizontal. 4. The required shell plate thicknesses for the vessel described in problem 2 may be determined from item 2 of Appendix E. Using these plate thicknesses and 18 courses, prepare a plot of circumferential stress versus height for the conditions of the hydrostatic test in which the vessel is filled with water.
C H A P T E R
4 A
m DESIGN OF BOTTOMS AND ROOFS FOR FLAT-BOTTOMED CYLINDRICAL VESSELS
u
he bottoms and roofs of vertical cylindrical storage vessels are usually fabricated of steel plates having thicknesses less than those used in the shell. This is possible for the bottom because it is normally supported by a prepared base of sand or gravel resting on the soil. The roof load is usually limited to wind and snow load with a proper allowance made for any anticipated additional loads. 4.1 BOTTOM DESIGN The shape and design of the bottom for a storage vessel will depend upon such considerations as: the foundation used to support the vessel, the method for removal of the stored material, the degree of sedimentation of suspended
n
Intermittent weld
Shell-to-bottom
I’
joints
Single-welded full-fillet lap joint
Single-welded butt joint with backing strip Bottom-plate joints
Fig. 4.1. Typical bottom joints recommended by API Standard 12 C. [Courtesy of American P&r&urn Institute.)
58
I
\
\
solids, corrosion of the bottom, and the size of the storage vessel. If the considerations mentioned dictate the use of a flat bottom and the safe bearing capacity of the soil is at least 3000 lb per sq ft the bottom is usually placed on a sand or gravel pad directly on the soil. If the tank bottom is directly supported by the ground, flexure of the bottom is prevented, and the bottom plates are under a simple compression load. Theoretically a lightgage sheet metal, 16-gage or less would be sufficient for such a bottom. However, to provide greater ease in welding and to allow additional metal for corrosion, plates having a thickness of at least s/4 in. should be used. For many years xs-in. plate was the most common plate thickness used for tank bottoms. Bottom plates of 72 in. or more in width are preferred and plates 96 in. wide are usually specified. Plates of >d-in. thickness are usually lap welded with a lap margin of at least 1>/4 in. for all joints. The bottom plates should extend beyond the shell-plate bottom weld at least 1 in. No more than three plate laps should be located within 12 in. of each other or of the shell. Typical welded joints for shell-to-bottom and bottom plates are shown in Fig. 4.1. Figure 4.2, a and b, gives alternate methods of shaping the sketch plates under the shell ring. The sketch plates should be formed and welded in such a manner that a smooth bearing surface for the shell plates is produced. In regard to selecting the plates for the bottom, the largest-sized plates available that can be conveniently handled and that have no cost extra for size are usually the most economical. Plates 96 in. wide x 20 or 30 ft long are often used. If the bottom plates are laid symmetrically in relation to the center lines of the bottom plan, t-he number of different shaped plates will be reduced to a minimum.
Example Design 4.1, Bottom for a Tank 150 Ft, 0 In. in Diameter
59
h3ottom plate’
Fig. 4.2.
Methods of shaping sketch plates under the shell ring recommended by API Standard 12 C.
This is an advantage because the plates can then be scribed and cut in groups of four, whereas if the bottom plates are symmetrical in relation to only one center line, only two plates can be scribed and cut at one time. A bottom asymmetrical along both center lines makes a large number of plates of different sizes necessary. The simplest symmetrical layout is to arrange the corners of four plates to intersect at the center of the tank bottom. However, this layout should not be used with lap-welded construction because four plates will lap at the joint. Also, this layout is sometimes wasteful in that with some groups of dimensions considerable scrap may result from the plates at the perimeter. In such a case the bottom plates may be rearranged with one plate centered on the bottom. In this layout the center row is single, but all other rows have mates. The center row will have two perimeter plates of t.he same size, but there will be four identical perimeter plates for each succeeding row from center. The sizes of plates and the location of cuts on perimeter
Table 4.1.
Dimensions of Welded Draw-off Elbow,
(Courtesy of American Petroleum Institute.)
plates can be readily calculated by use of Eq. 4.1 and by reference to Fig. 4.3.
A2 = B(D - B) A2
(4.1)
= 4 - ,9
Means must be provided for the removal of liquid from the bottom of the vessel. A sump, shown in Fig. 4.4, may be used with a sump-pump discharge. Flat-bottomed tanks using gravity or pump discharge may discharge by means of a draw-off elbow, as illustrated in Fig. 4.5. Dimensions for a draw-off elbow are given in Table 4.1. 4.2 EXAMPLE DESIGN 4.1, BOTTOM FOR A TANK 150 FT, 0 IN. IN DIAMETER
A bottom is required for a tank 150 ft, 0 in. in inside diameter. The minimum allowable (2) plate thickness is >/a in.; however, because of the large tank size a plate thickness of x6 in. will be specified to provide additional protection against loss by corrosion. The bottom course of shell plates for this vessel is l)is in. thick, and a XS in. fillet weld will be used between the shell and the bottom plates.
Recommended by API Standard 12 C, All Dimensions in Inches-See Fig. 4.5
Nominal Pipe Size* 2 3 4 6 8
(Courtesy of American Petroleum Institute) Distance Distance from Distance from Diameter Diameter Center of from Center of of Hole of Rein- Elbow to Center of Outlet to in Tank forcing Face of Plate, Elbow to Bottom, Bottom, Outlet Shell, B C Flange, E DR DP 7% 6 3% 6% 12
8%
9%
11 13
7 7’Ns 9% 1236
4% 5% 7% 9%
* Extra-strong pipe, API Standard 5 L
7% 9% 123i 1655
0 -I IIQ 63 : ------
-
4 Q
-
t
d
--A
13 14 16 18 Fig. 4.3.
Relationship of bottom plate dimensions.
60
Design of Bottoms and Roofs for Flat-bottomed Cylindrical Vessels
Fig. 4.4.
Draw-off sump recommended by API Standard 12 C. (Courtesy of American Petroleum Institute.)
Alternative for bottom corner
The hottom plates must extend a minimum of 1 in. beyond the shell weld, or in this case, the radius must be increased il minimum of (1 + 1s.i~ + x~ in.) or 27; in. A radius of 75 ft, 3 in. will be used for the bottom. The central hottom plates will be 96 in. wide by 31 ft, 8>/4 in. in length. The bottom plates will be lap welded, and the joints will be staggered so that no more than three plates are lapped within 12 in. of each other or of the shell. Figure 4.6 shows a layout, for such a bottom. It should he noted that the layout is symmetrical with respect t,o one axis, and thus only half of the bottom is shown. It is also evident that, except for the necessary staggering of the plates, it is symmetrical w-ith respect to the other axis. To demonstrate t,he use of Eq. 4.1 in this design, dimensions A and C will be calculated for points 2 and y on the layout. At point s dimension C is equal to one half the plate length of 31 ft, 812 in., and D is 150 ft,, 6 in. c=
3% 8% in. = 15 ft, lOi in. 2-
=
(150 ft, 6 in.):! - (15 ft, 104 in.)2 k
= X62.562 - 251.024 = 5411.338 l.harrfort!
A = 73 fl, 6% iu. To obtain the width of plate S-2 at point L, 81~-in. plate ’ m u s t br subtracked widt,hs less (10 - 1) laps of 134 m. from dimension A. The plate width is 96 in. S-2 plate width at z = (73 ft, 68 in.) - 8+(8 ft., 0 iu., - 9(1+ in.) = (73 fl, 6% in.) - (67 ft, 0% in.) = 6 it, :J in.
Tank shell
Fig. 4.5. Welded draw-off elbow recommended by API Standard 12 C. (See Table 4.1 for dimensions.) (Courtesy of American Petroleum Institute.)
’ Alternative ,,‘I’ mitred pipe
Example Design 4.1, Bottom for a Tank 150 Ft, 0 In. in Diameter
61
1?7’-sy’L1?9.-‘1~“I!,,-l~%.,,~,-~
Half plan of bottom All PLS.-12.75+- 1%” laps 2940 iin fi of 5/;.” weld
1% =3+
R a d i u s = 75’4” to outside edge of bottom
Section through corner
$$'gz 47 IO 4 4 4 4 4 4 4 2 2 2 ? 4 4 4 4 2
Mark SW32 5124 5W/6 S- / S-2 5-3 5 -4 S-5 S-6 s-7 S-7-A S-8 S-8-A S-9 S-IO S-N S -I2 S-13
Dercnption
B//l of Materiot Ft.LWfh h. wt. No.
P/s-96"X/2.75" 31 84 P/r.-96'Xl2.75" 2 3 9fi P/s. -96"Xl2.75" 15 10% Pls. - SK X12.75* Fls -SK X 12.75" ,&-SK X/2.7!? P/s: -SK X 1275" P&-SK X 12.75+ Plx-SK X/275* P/x-N x/2.75* ?/s-SK Xt2.7.V' ,&-SK X/2.75* Pls -SK X /2.75* P/S.-SK X 12.75' P/s-SK X/275* Ph.-SK Xt2.75# P/s. -SK X /2.75# f'ls-SKXI2.75'
Section through lap
Length
Order
Ft.
in.
47 10 4 2 2 2 2 2 2 2
Ph.-96Y12.75" P/r.-96"X12.75y P/s.-96"X/2.75# Pls-96Vl2.75~ Pls -96112.75" Pk-96"X/2.75X Pl~-96'X12.75x Pls.-96"X12.75x Pk-96"X/2.75* PJ~.-96'X12.75~
31 23 I5 16 I4 19 13 22 28 15
8% 9% /OS 2% 6%? 0% 9% 3Q 3% S?Q
2
f/s.-96'X12.75"
21 Ilk
2 Ph.-96'Xl2.75" I3 5% 2 /'Is.-96112.75" I6 8 2 Pk96"X/2.75* 20 I& 2 P/s-96"X/2.75" 23 5% 2 Ph.-96"Xl2.75" /2 I%
Similarly, at point y dimension C is equal to 135 pla\v lengths plus 2 plate widths less 3 plate laps of l!i in., (11’ C = 1.5(31 ft, 8i in.) + 2(8 ft, 0 in.) - 3(1% in.) = (63 ft,, 6; in.) - 32 in. = 63 ft, 2; in. By
Eq. Lla,
A-2 = ; - (;2 = 5662.562 - (63 fl, 2#in.)* = 5662.562 - 3996.610 = 1665.952 A = 40 ft.. 9t-i in.
Ag. 4.6.
Typical bottom layout for a
150~ft-diameter
tank.
Fig. 4.7.
Weight of snow loads equaled or exceeded one year in ten years-pounds per square foot (137).
Minimum Design loads in Buildings and Other Structures, A58.1-1955,copyrighted by the American
(Th’IS material is reproduced from the American Stonderds Association.)
Sfondord Building Code
Requirements
for
Roof Design 63
Therefore, for the layout shown in Fig. 4.6, the dimension for S-8 at point y is: or
(40 ft, 9% in.) - (34 ft, 7 in.) + la in. 6 ft, 4& in.
4.3 R O O F D E S I G N The most comnion shape for a tank roof is a cone although dome or umbrella roofs are also used. In addition to these shape classifications, tank roofs may be classified into two types, self-supporting and nonself-supporting. Regardless of shape or method of support, tank roofs are designed to carry a minimum live load of 25 lb per sq ft in addition to the dead load. This live load is an average figure which allows for combined wind and snow loads and for the weight plant personnel who may travel across a roof to inspect the vessel or to reach a manhole and so on. Figure 4.7 shows the maximum snow loads to be anticipated in various parts of the United States (137). 4.3a Self-supporting Conical Roofs. A self-supporting roof is one which is supported only on its periphery without the aid of additional support from columns. Tank diameters for self-supporting roofs generally do not exceed 60 ft and usually are less than 40 ft. Any greater spans require such heavy rafters that it is simpler to use one or more supporting columns and thereby reduce the span. Such roofs usually consist of roof plates supported on rafters. Small and medium-sized flat bottomed cylindrical tanks having capacities of 400 and 3000 bbl or under respectively are extensively used in the petroleum industry (100, 101). Figure 3.7 of the previous chapter shows proportions for such tanks, and Table 3.3 gives typical dimensions. The roofs of these tanks are known as “decks” and are fabricated of mild steel having specifications meeting ASTM-A-7, ASTM-A-283 grade C or D (open-hearth or electric-furnace steel only). The deck plates have the same thickness as the shell plates, and a slope of 1 in. in 12 in. is used for the cone. If xs-in. plates are specified, the deck must be reinforced with structural support if it is 15>6 ft or larger in diameter but does not require additional reinforcing if it is smaller in diameter. If >a-in. plate is used, no support is needed for lS>&ft diameter tanks, but support is required for all larger-diameter tanks. The deck may be attached to the shell by one of the following methods. The deck may be flanged and welded by: a double-welded butt joint with complete penetration, a single-welded butt joint with backing strip, or a full-fillet double-welded lap joint. If the deck is not flanged, it should have full-fillet welded joints both inside and outside. For larger tanks with cone roofs the equation for stress in a cone under either an internal or external pressure can be derived as shown in Chapter 6 (see Eq. 6.139). The maximum stress will exist at the greatest diameter of the cone and will be:
d = diameter, inches t = cone shell thickness, inches 0 = angle between cone element and horizontal The stress as calculated by Eq. 4.2 will be controlling only in the case of thick cones used with pressure vessels of limited diameter. In the case of large-diameter conical roofs such as those used for storage tanks, the controlling factor is elastic instability. The theoretical critical compressive stress that causes failure of a curved plate by wrinkling is given by Eq. 2.24. (2.24) where E = modulus of elasticity of material, pounds per square inch t = thickness, inches P = radius of curvature, inches (see Fig. 4.8) fcritical = theoretical critical stress at which failure bg wrinkling occurs, pounds per square inch The safe compressive stress that can be carried without wrinkling was investigated by Wilson and Newmark (43) in a series of experimental tests. As a result of these tests and others (44), it was found that the safe compressive stress that can be imposed on a steel cylindrical shell without failure by wrinkling is one twelfth of the theoretical critical stress and can be expressed for P as follows:
f allowable = 1.5(106) 4P 7 13 yield point
Equation 2.25 can be modified for use with a conical roof
\ ‘\
I
I I
\ I 90-a
e
D = diameter of tank, feet r = radius of curvature of cone at periphery, inches _ 6 D
sin B
0 = angle of cone element with horizontal
B
T \i
f* = -+!!L 2t sin e where p = internal or external pressure pounds per square inch gage
1’
(2.25) e?
\
\
I
Fig. 4.8.
Radius of curvature of conical
roofs.
a
64
Design of Bottoms and Roofs for Flat-bottomed Cylindrical Vessels
Fig.
4.9.
Field
photograph
structural
support
(Courtesy
of
for
Aluminum
a
showing_
tank
roof.
Company
of
America.)
by referring to Fig. 4.8 and substituting for r as follows: fallow. = 1.5(106) l+
(4.3) (4.5)
01 sin e =
f~ll0w.D
(4.4)
250.000t
It is very important to recognize that the allowable compressive stress, fallow., is not the conventional allowable stress for the material but is the safe stress that can be applied without danger of failure by wrinkling. The compressive stress induced by live and dead loads on the roof must not exceed the allowable compressive stress, fallow.. Equation 4.2 can be used to calculate the compressive stress induced by the roof loads, or Eq. 4.2 can be substituted into Eq. 4.4 as follows: sh e =
P/144)(124 (2t sin @)
\
\
D
250,OOOt
\I
/
If live load = 25 lb per sq ft and dead load = 7.65 lb per sq ft (for ?ia-in. roof plates)
P = 32.65 lb per sq ft If one substitutes for P in Eq. 4.5, min sin 0 = +& d32.65/6
= ?f4301
(4.6)
Equation 4.6 is the equation specified by API Standard 12 C for self-supporting conical roofs. It should be emphasized that the derivation of the constant is based upon selected roof loads and 3is-in. roof-plate thicknesses and should be modified for other conditions. 4.3b Conical Roofs with Structural Support. When the design calls for a conical roof with structural support, a
Roof Design
slope, or pitch, of the roof of a y;i-in. rise per 12 in. is recommended. The roof plates may be ridged in order to decrease the number of rafters required. Roof plates should not be attached to the rafters. Roof plates of lap design should have a minimum lap of 1 in. when tack welded; moreover, if a continuous full-fillet weld is used on all seams, it is necessary to weld only the top side of the roof. For steel construction a minimum thickness of yi 6 in. is recommended for the roof plates. Figure 4.9 is a photograph showing assembly of structural support for a tank roof before installation of roof plates. Storage tanks and’ other large vessels with conical roofs usually are designed with no attempt made to prevent the roof plates from flexing. In such a design the rafters are spaced sufficiently close to each other to prevent overstressing of the outer fiber of the roof plales as a result of flexure. The roof plates are assumed to ad as flat continuous beams wit,h a uniform roof load. The rafters and girders are assumed to act as uniformly loaded beams with free ends. Roof design involves the consideration of bending and shear in the roof plates, rafters, and girders. Column action in the rafters of self-supporting roofs and in the columns of roofs having supports must be also considered. A brief discussion of these relationships follows. UNIFORMLY L OADED B EAMS WITH F REE E NDS. Referring t.o Fig. 4.10, consider any point, 2, between supports RI and Rz in the beam uniformly loaded with w pounds per linear inch. The forces acting on the beam to the left of point x produce a bending moment, M, which can be evaluated by a summation of the moments at x. For a uniformly loaded beam freely supported at the ends, RI = R,; therefore
By substituting x = l/2 into Eq. 4.7, we obtain: M max
=
Wl2 -
(for a uniformly loaded beam freely
8
supported)
M = E’I dx2cy wx2 - wl2x
To obtain the location of t,he maximum bending moment, let dM - 0 dx
‘I
therefore
;($-!$)=n
2
therefore
To evaluate the constant of integration Cl apply the boundary condition dy/dx = 0 where x = l/2. Therefore
By substituting in Eq. 4.7 and integrating again, we obtain:
EIy = !f!$ - !$ - !$t! + C2
(4.10)
Since y = 0 wherq r = 0,
c2 = 0 By substituting and solving Eq. 4.10 for y, we obtain: -
(4.7)
-
-
- WPX 24
The maximum deflection occurs at t,he center of the span where x = l/2; therefore 5wl” = maximum deflection ’ = 384EI
(4.11)
UNIFORMLY LOADED CONTINUOUS BEAM. A uniformly loaded continuous beam having a large number of equally spaced supports reacts the same as a simple uniformly loaded beam with fixed ends. Consider the uniformly loaded beam shown in Fig. 4.11. The beam is assumed to be a section of a continuous beam with a large number of equally spaced supports 1 distances apart. A bending moment MO exists over the supports.
OP Wl - - wx = 0 2 therefore 1 x=-2
(4.8)
To determine the deflection of the beam Eq. 2.41 is substituted into Eq. 4.7, and the equation is integrated.
R1 ,!!! 2 The force RI produces a clockwise or positive moment equal to Rlx, and the uniform load IO the left of x results in a force, wx, which produces a counterclockwise or negative moment equal to -wx(s/2),or
65
Fig. 4.10.
Uniformly loaded beam with free
end+
Design
46
of
Bottoms
and
Roofs
for
Flat-bottomed
Cylindrical
Vessels
maximum bending moment occurs over the supports, as defined by Eq. 4.14. The maximum deflection of the beam by inspection is observed to occur at x = l/2 and may be obtained by substituting Eq. 4.14 into Eq. 4.13, integrating, and evaluating the new constant of integration, Ct. Fig. 4.11.
Uniformly loaded continuous beam.
EIy = but
Taking the summation of moments at a distance z from the support RI, we obtain: M = MO + Rlx - wx ; 0
But
y=Oatx=O therefore
cz = 0 Substituting and solving for y where x = l/2, we obtain: WP __ =
’ = - 384EI
for a beam with clamped ends, and (2.14)
Substituting for M and R1, we obtain:
M=E~+fo+~-w~
(4.12)
maximum deflection
(4.17)
COLUMN ACTION. Slender structural members under axial compression tend to deflect. This deflection results in a bending stress superimposed on the compressive stress. Referring to Fig. 2.4 of Chapter 2, we find that the axial force, P, causes a deflection of y in the column of length, 1, and cross-sectional area, a. The bending moment, M (equal to the force, P, times the lever arm, y) induces a bending stress equal to MC/I, which must be added to the compressive stress, P/a, or j=yc+LP~+P a a
Integrating, we obtain: By definition Applying boundary conditions to evaluate the constant Cl, we obtain:
-= “” 0
(where r = radius of gyration)
therefore (4.18)
when x = 0
d x
c1 =
I = ar2
The column may be compared to a uniformly loaded beam freely supported, in which by Eqs. 4.12, 4.8, and 4.11,
0
Therefore
E+Mox+!$2-w~
(4.13)
Also,
Solving for the product yc for the uniformly loaded beam, we obtain:
\
-= dy 0 dx
whens=l
I.
j+; M=!$; y=?!!&
A.
= c112
therefore MO = - $
(maximum moment for uniformly loaded beam with fixed ends)
(4.14)
Substituting into Eq. 4.12, we obtain:
(4.19)
where Cr = constant. For a column the product yc is assumed to vary as 12, as in the case of a beam: yc = CZP
M = EI ‘2 = _ w; + w+ - w$
(4.15)
Substituting the quantity Cs12 for yc in Eq. 4.19 and solving for P/a, we obtain:
The bending moment at the center of the span where 2 = l/2 is:
(4.20)
(4.16) Comparison of Eq. 4.14 with Eq. 4.16 indicates that the
\
\
\I
I
where Cs is a constant depending upon the material, the method of loading, and the method of support. No method is known for calculating from theory the value of
Roof Design 67
the constant Cs. The constant is usually determined by experiment. Rankine experimentally evaluated the constants for round- and square-ended columns and found Cs to be l/18,000 and l/36,000, respectively (29). Since any slight displacement of a fixed-end or square-end column either laterally or axially will result in an unknown eccentric loading, columns are usually designed as round-ended members. For values of l/r between 60 and 200, the American Institute of Steel Construction (102) recommends the following equation for steel columns: 18,000 P -= a 1 + (12/18,000r2)
(4.2i)
For a rectangular beam, bt2 z=6 where b = width of beam, inches t = thickness of beam, inches Thus, for the case where b = 1.0 in.,
t2 z=6 Substituting Eq. 4.24 and Eq. 4.20 into Eq. 2.10, we obtain:
For columns having values of l/r between 0 and 60, a
i; column formula is not used, but a maximum value of
15,000 for compressive stress, P/a, for steel columns is specified. For values of l/r greater than 200, Euler’s column formula is used. (See Table 2.1, Chapter 2.) Self-supporting roofs have rafters under combined compressive load and bending load. Such rafters may be considered to act as beams under column loading. The constants f and Cs of Eq. 4.20 have been specified as given in Eq. 4.22 by the American Institute of Steel Construction (102) for steel beams under column action.
,
P 20,000 -= a 1 + (12/2000b2) where 1 = unsupported length, inches b = width of compression flange, inches
l
/
,
The application of Eq. 4.22 is limited to conditions in which 1 exceeds 15b but is less than 40b. If the member is shorter than 15b, the rafter may be designed as a beam. Lateral stiffeners may be used to maintain 1 within the upper limit of 40b. The value of 20,000 specified in the numerator of Eq. 4.22 is permitted because the bending stress is maximum only at the outer fiber, and therefore the maximum combined compressive stress exists only at the outermost fiber on the top side of the rafter. The average compressive stress across the member will be less than 20,000 psi. RAFTER SPACING. Consider a circumferential strip of roof plate 1 in. wide located at the outer periphery of the conical roof, and disregard the support offered by the shell. This strip is considered to be essentially a straight, flat, continuous, uniformly loaded beam. The controlling bending moment is equal to w12/12 and occurs over the supporting rafters. By Eq. 4.14, -w12 -pP -P(1Y2 M max = - = ___ = 12 12 12
(4.23)
where 1 = length of beam (strip) between rafters, inches P = unit load, pounds per square inch = w when width = 1.0 in. Introducing the stress resulting from flexure by Eq. 2.10, we obtain:
or (4.25)
1 = t d2j/p
For an allowable stress, j, of 18,000 psi (the maximum specified by API Standard 12 C for steel roofs) a roof-plate thickness of Ns in., and a roof load of 32.65 lb per sq ft, that is, p = 0.227 psi, a substitution into Eq. 4.25 gives:
1 = I 16
(2)WOW = 74.6 in. 0.227
It is apparent from the above calculations for a %a-in. roof plate that rafter spacing in large-diameter cone-roof tanks should not exceed 6 ft unless heavier roof plates are used. API Standard 12 C specifies a maximum rafter spacing of 27r feet or 75% in. on the outer perimeter of a ring of rafters and a maximum of 535 ft on the inner perimeter (2). The minimum number of rafters adjacent to the shell is determined by dividing the shell circumference by the maximum rafter spacing. The actual number of rafters to be specified should be a multiple of the number of sides of the polygon of girders supporting the other end of the rafters to provide a symmetrical layout; this is a further restriction. The minimum number of rafters to be used between two adjacent inner girders should be based on the perimeter of the outer polygon of girders. The length of one side of a polygon having sides of equal length is:
where L = polygon-side length, feet N = number of sides of polygon R = radius of circle circumscribing polygon, feet The minimum number of rafters, n, required will then be equal to (12NL/I) or, n = 24NR sin 360
1
2N
(4.27)
where n = minimum number of rafters 1 = maximum rafter spacing, inches The actual number of rafters to be specified should be a
68
Design of Bottoms and Roofs for Fiat-bottomed Cylindrical Vessels
multiple of the number of sides of the polygon, N, to mainlain a symmetrical layout. SELECTION O F RAFTERS AND G IRDERS . Rafters are designed as uniformly loaded beams with free ends. Each rafter is considered to support the roof plates and roof load over an area extending on either side of the rafter and bounded by the center line to the adjacent rafter. F r o m Eq. L8 it is seen that the maximum bending momeut for such a beam is equal to w12/8 and occurs at the center of thr~ bi111. The maximum fiber stress is directly proportional to ~hc square of the length of the beam, 12. Therefore, to avoid use of excessively heavy rafters the length of the rafter is usually limited to from 20 t.o 24 ft or less. This may be demonstrated by considering 20 ft, 0 in. rafttxrs spaced 6 ft. 0 in. apart at the outer side and t ft. 0 in. apart at the inner side with a roof-design Io;~d of 0.5 psi. The maximum bending moment is, by K’:(I. 1.8: M = Ef _ (0.25)(5 x 12)(20 x 12)” 8 8 = 108,000 in-lb Rewriting Eq. 2.10 as, M
z=f
for f = 18,000 psi (assumed allowable value) we obtain: 2=
108,000 in.-lb = 6 o in 3 18,000 lb per in.2 ’ ’
From item 1 of Appendix G a 7 x 2Jg in. channel weighing 9.8 lb per ft in which z = 6.0 in.3 may be select.ed to fulfill the requirements. For vessels of large diameter in which the rafter spru is reduced by the use of girders, column supports must be used for each ring of girders and also at the center of thr tank. Usually five or more straight girders art’ joined end-to-end to form a polygonal support for the ends of the rafters. The girders are designed in the same manner as the rafters. The girder load is considered to be a uniform load equal to the roof load plus the weight of the rafters. The roof area contributing this girder load is the length of the girder times the distance on either side halfway to t.hr next rafter support. The rafters form a series of concentrated loads on the girders, but for practical considerat.ions the load may be treated as uniform whenever four or more rafters are supported on one girder. Self-supporting roofs have rafters under a combined compressive load and bending load. In such designs, if the unsupported length, 1, exceeds 15b where b equals the width of the compression flange, the stress in pounds per square inch should not exceed the value calculated from Eq. 4.22. The laterally unsupported length of beams and girders should not exceed 40 times b (the width of the compression flange). The above restrictions, limiting beams to lengths with an 1 ‘b ratio not greater than 40 and to stress not greater than permitted by the formula for Z/b ratios greater than 15, do not apply to rafters which are in contact with the steel roof plating. It is assumed that under full-load conditions, friction between the roof sheets and the rafters will provide
adequate lateral support to the compression flanges of the rafters. SELECTION 0F COLUMNS. The design of beams and rafters for roofs is based on a safety factor of approximately 3, that is, one third the ultimate strength. For rolled-steel structural shapes of 55,000 psi ultimate tensile strength, an allowable tensile stress of 18,000 psi is recommended. The same value may be used for the maximum allowable compressive st,ress of rolled-steel sections if lateral deflection is prevented. In the case of columns, the lateral deflection should be considered, and the maximum allowable compressive sl.ress should not exceed 15,000 psi. This stress can be computed by Eq. 4.21. The allowable compressive stress so calculated is based on the gross sect,ion of the column (including area of weld) if the column consists of two or more sections welded together. For main compression members the l/r ratio should not exceed 180, and the ratio for bracing and secondary compression members should be limited to 200. Supporting columns for roofs may be either of standard structural shapes or of pipe, depending upon preference in design. In the installation of columns, clip angles should be used on the tank bottoms to prevent any possible lateral movement of the column bases. 4.3~ Dome and Umbrella Roofs. A dome roof is one formed to a spherical surface. At the beginning of the century, tanks with dome roofs were used for a great variety of services. Today they are seldom used for atmospheric storage as the more simple cone-roof tank is cheaper. ’ Dome roofs are still used extensively for cylindrical flatbottom storage vessels designed for low-pressure service. The umbrella roof is formed so that any horizontal section through the roof is a regular polygon wit,h as many sides as there are plates. Umbrella roofs are a compromise between cone roofs and dome roofs. Umbrella roofs have approximately the strengt,h of dome roofs but are easier to install because the roof plates are curved in only one direction. The equation for stress in a spherical thin-walled vessel can be developed in a manner similar to that used in developing Eq. 3.13 with t.he following result:
t=p_d+c 4f By comparing Eq. 4.28 with Ey. 3.14 it is apparent that for the same radius of curvature and the same shell thickness the spherical shape is twice as strong as the cylindrical shape. Thus, for equal strength, the radius of curvature of a sphere should be twice that of the cylinder. Therefore, it is customary to make the radius of curvature of a dome roof equal to twice the radius-of the shell. The API Standard 12 C recommends this proportion and permits a 20 7, variation in either direction, or R=D
(4.29)
Rmir, = 0.8D
(4.30)
or or
Rmax
=
1.20
/
69
Example Design 4.2, Design of Roof and Structural Supports for a 122-Ft Tank but
where R = radius of curvatjure of dome, feet D = radius of tank shell, feet
R=;
ELASTIC S TABILITY OF U MRRELLA R OOFS. The specifications of an umbrella roof are also determined by the roof loads and the elastic stability of the roof under load. By following the same method of derivation as that used for Eq. 4.6 for self-supported conical roofs, Eq. 4.32 can be derived for umbrella roofs as follows, start.ing with Eq. 2.25:
1.5 X 1OV = pr2 (as for conical roofs)
1.5 x 10” t2 = r2 0.227 &iXXlo”=r t 2.58 X 1000 = ; = y R = 215t
(4.32)
Equation 4.32 is recommended by API Standard 12 C for hoth umbrella roofs and dome roofs. When one applies the same fact.or of safety, 12, dome-shaped roofs have greater elastic stahilit,yt.han umbrella roofs. This may be shown as follows. ELASTIC STARILIT~ OF D OME HOOFS. For elastic stahility of a thin sphere under external pressure, (42) Peril
iwl
=
2Et” _-r2 d3(1 - p2)
(4.33)
For steel, p = 0.3, and E = 30 X 10” psi; therefore 2 x 30 x 10”t2
Introducing an elastic-stability fact,or of safety of 12.0, the same safety factor that was included in Eq. 2.25, used in derivations of cone- and umbrella-roof operations, we ohtain:
.
or
3650 ~- - 30-1 12
R = 304t
(4.34)
The constant in Eq. 4.34 differs from the constant in Eq. 4.32 by 4. Th’1s is to be expected since the constant in Eq. 4.28 differs from the constant in Eq. 3.14 by a factor of 2.0; moreover, t and r occur in the Pcriticar equation to the sfxwid power.
t,herefore p = 32.65 lb per sq ft. = .227 psia
R t
4.4 EXAMPLE DESIGN 4.2, DESIGN OF ROOF AND STRUCTURAL SUPPORTS FOR A 122 FT., 0 IN. DIAMETER STORAGE TANK 48 FT, 0 IN. HIGH SELWTION OF ROOF PLATES. A tank of t,his diameter requires column supports and rafters and girders. If the rafter lengths are to be limited to about 20-ft spans, two polygonal rings of girders plus one central column will be necessary. All roof plates will be cut from two st.andard sizes of plates carried in stock: plate size A, 72 in. wide x 25 ft, 3$/, in. long, and plate size B, 72 in. wide x 22 ft, l\$ in. long. All plates will be g{s in. thick (7.65 lb per sq ft). A study of various combinations of the above plates together with sketch-plate requirements indicates that an economical roof-plate layout will be like the one shown in Fig. 4.12. Such a layout results in a small amount of scrap in cutting out the sketch plates. R AFTER A N D G IRDER SPARING. A suitable rafter-spacing layout based on the use of two polygonal groups of girders is next determined. Girders of approximately 26-ft in length will be used because spans greater than 30 ft require excessively- heavy structural sections. Using a radius of 22 ft, 0 in. t.o circumscribe the inner polygon of girders requires a polygon of five sides. To maintain symmetry a decagon will be used for the outer polygon, as shown in Fig. .I. 13. To determine t,he rafter spacing. Eq. -t.25 may be used. The design load is 25 lb per sq ft live load plus 7.65 lb per sq ft dead load @ifi, in. plate). The allowable design st,ress for the roof plat.es will be taken as 18,000 lb per sq ft. Therefore, by Eq. 1.25, 1=&2flp=
3 16
; = d(36.3 X 106)/12p = 103 d3.03/p
= 74.6 in., maximum rafter spacing
Assuming p = 0.227, the same as for conical and umbrella roofs, we oht.ain:
Also, maximum rafter spacing = 2x ft = 75.4 in. The minimum number of rafters in the outer ring can be determined by dividing the circumference of the shell by the maximum rafter spacing and equals (2xr/t) or
P-= 10” 4X03/.227 t r ~- = 3650 t
&in
=
2 x 3.14 x 61 x 12 74.6
= 61.5 rafters, minimum number
70
Design of Bottoms and Roofs for Flat-bottomed Cylindrical Vessels W-IO”
B” 1/5’-9k” 6’~,q m 13,-Z%” C u t 2 P L S . 7 2 ” I 10.2# x 25’~X5”
‘- C u t 2 P L S . 7 2 ” x lO.f#
I 25’4%”
9’4%”
3’$!+r5’-9%;p’-‘!q
( 12’ 7 % ”
C u t 2 P L S . 7 2 ” x 1 0 . 2 # x 25’-3%”
C u t 2 P L S . 7 2 ” I I O . 2 9 I( 25’-3%” W-8%”
,
,_
l3’-6%”
diameter
NO. 30 36 4 4 4 4 4 4 4 4 4 4 4 4 4 2
C u t 2 P L S . 7 2 ” x I O 2 # x 25’-3%”
ppl 11’ 0%” C u t 2 P L S . 7 2 ” x 10 2# x 22’-1%”
C u t 2 P L S . 7 2 ” x I O 2 # x 22’- 1%’
2 4 4
Pls P/s. P/s. P/s Pls. P/s P/s P/s
Description Mark 7Z"Xl~?2"XZ5-3~" A 72"XlO.2*X22'/~2" B 72*X/0.2+X lP'7?6" C 72"X/O.2+X II-074. 0 IO2 *X Skefch 2 lO2"XSketch I2 IO2 *X Sketch 7 IO2 x X Sketch II
P/s I02 x X Sketch P/s IO2 #X Sketch P/s 102 * X Sketch P/s 102 * X Sketch Pls IO2 #X Sketch Pfs IO2 *X Sketch P/s P/s Pls P/s P/s
IO2 # IO2 x 102 e /02# IO2 +#
X X X X X
Sketch Sketch Sketch Sketch Sketch
Make From 3 0 Pf~.72"xlO.2~X25~3~' 36 P/s. 72"X10.2*X22~l~'2P/s 72"XIO2"X25'3%' 2Pls 72"X102xX22'I%' 2Pls 72"%/02 2Pls
72"X10.txX25'-3%"
8 I4
2 P/s
72"X102xX2S'-3%"
5 4 6 9
2Pls 72"XlO..2'X25'-3%"
IO I lA 3 I3
2 P/s 72"X102cX25'3%" 2Pls 72"XlO2*X22-I%' 2P/s 72"X/O2*XZ5'3%' 2Pls
Shipping Weights 30 "A"PlS. 46,410 36 B"Pls. 48,744 /2;1"Sketches 18,564 1.
10,-O”
_I!_
II’-11%”
C u t 2 P L S . 7 2 ” x 10.2+
I
22’-1%”
BB'Skefches
4 C u t 2 P L S . 7 2 ” I 10.2$ I 22’-1%
Fig. 4.12.
\I
Roof plates for a 122~ft-diameter
I
tank.
"X25-374'
10,832
72"X/OZ*X25'-3tim
Example Design 4.2, Design of Roof and Structural Supports for a 122-Ft Tank
71
As indicated in Fig. 4.13, a lo-sided polygon, or decagon, will be used to support one end of these rafters. Therefore, the minimum number of rafters required must be a multiple of 10. Thus it may be seen that the minimum number of rafters between the outer shell and the decagon of girders is equal to 6.15 rafters per girder. An integral number of 7 rafters per decagon girder will be used, giving a total of 70 rafters. To check the spacing of these rafters on the girders Eq. 4.26 may be used.
L = 2R sin 360 2N 360 = (2)(39.75) sin (2)(10) = 79.5 sin 18” = (79.5)(0.309) = 24.6 ft Average rafter spacing = 24.6 ft/7 rafters = 3.51 ft This spacing is less than the 5jd ft maximum allowable rafter spacing on the inner ring. The minimum number of rafters required between the decagon and the pentagon can be determined by using Eq. 4.27, or
24NR . 360
n = ~sm 1 2N
360 (24)(10)(39.75) sin (2)(10) n
=
74.6 = 127.8 sin 18’ = (127.8)(0.309) = 39.5 The actual number of rafters to be used must be a multiple of the number of sides in a pentagon and the number of sides in a decagon. Therefore, 40 rafters will be specified. The minimum number of rafters within the pentagon may also be determined by use of Eq. 4.27, or
$” inside of shell Notes: All holes I%.” I#Z unless other&e noted All caps to be welded to columns in shop
NO.
5 Girders 4J IO G i r d e r s L uqs IO Rafters IO 40 70 I Column Channel Channel Clip
=
74.6
= 34.7 sin 36” = (34.7)(0.588) = 20.4 The actual number to be specified should be a multiple of five. This means that 20 rafters would be a most convenient number if its use can be justified. An examination of Fig. 4.12 indicates that the maximum span between rafters is less than that determined by Eq. 4.27, which is conservative. Therefore, 20 rafters will be specified. SELECTION OF RAFTER SIZE. The rafter having the greatest span controls the size of the structural section required. The maximum rafter span is indicated as rafter R4 in Fig. 4.13 and has a length of 22 ft, 9xi6 in. The spacing between the rafters at the shell periphery is approxi-
Billo f Materio/s Mark Cl A to F incl 62 G r0 L incl RI R2 R3 R4 Cl
CIC
Cap Columns Channel Channel Clip
c2c
IO
70
11
l0&2”@ 20.7*X43 ‘7%~’ IO& 9 ° C M4~X47’N’%s” IO& 6”X6”Xs/8B”XO’-2W /OP/s. /2”X2~7.4~X/‘2” 5Pls 4”xj/ss”xl’o~u’8” fOPIs. 4UxJ/6 ” x l’O3/4/r” 486 8”@ll.S*X2’6” 32Pls10”@10.2xXl”10”
Bolt list Rafters to cap countersink Girders to cofumns Girder splices R a f t e r s t o girdem Rofters to lugs Columns to bases Columns to bases
Fig. 4.13.
56/2*@20.7xX50’10%k” 5131 9*@ 13.4eX5050”3?aa 5b 6”Xb”X %“XO’2%” 5Pls. /2”X20.4xXl’5”
c3
N C G/S G2S Cl9
40 60 60 340 140 6 4 3 2
/L 6”X6’XJ/B’XO’-272” IPL 32”( X94”
c2
Cap 5 Splices IO Splices I6 C o l u m n b a s e s Channels Gussets
lugs
Make from 5 6 l5’@33.9~X25’9%’ lBar S”XWX33’-9%” IO& 15’833.9XXZ44”6~ /Bar 5”X%‘X46’-9Y2’2’ IO t b%llS~ x II’ 7%’ /0&8%‘Il.5*X/9’6%6= do& 8’@f/.5x x2/‘- 70/l” 70688ll.5*X24’7~e” lu /2”@20.7*XS4’7~e /UlO’@ 15.30xs3’-ll~4/r’
Cop 5C o l u m n s Channel Channel CIIP
360 (24)(5)(22.0) sin o(5) n
Descrrption
Roof-support assembly for a
706 6”X4”X%“XO”4” Y--n x 2%” 74’X2” v4’X2” Y4/** x I%’ 74. x /J/4* o/4’ x /Y2# 3/4=x /Y4# 122~ft-diameter
tank
Design
72
of
Bottoms
and
Roofs
for
Flat-bottomed
Cylindrical
Vessels
Inner-row girders make 5, mark Gl
m -1
__
2%”
m
_______----------------
2” -a. --F---C” e
-----------------------
2%“+ L”
24’-6%”
_
A+ - -2” * *
Outer-row girders make 10, mark G2
Inner-girder splices make 5, mark GlS 10 PLS 5” Y ;L”x 531”
Required mark
10 PLS 5” x %“x 6&” Required mark @ 10 PLS 5” x 3j”x 7%” Required mark @ 10 PLS 5” x %j”x 8y,” Reqwed mark @
Girder lugs
10 PLS 5” x %,“x 2OPLS
8Q” Required mark @
5”X%“X4”
Required mark @
20 PLS 5”~ 4”~ 4hM
Required mark @
POPLS 5”X%“X5%”
Required mark @
20 PLS 5” x %“x 5%” Required mark @ 20 PLS 5” x s” x 51%,” Required mark @ 1OPLS
5-x
%“x 6%” Required mark @ Fig.
4.14.
Girder details
mately 5p; ft., and the rafter spacing at lhe decagon end of ! he R4 rafters is approximately 31,~ ft. The design of the rafter is based upon Lhe roof load plus the weight of the rafter. Since the rafter size and weight is unknown, a preliminary design based on the roof load only will be made and later will be checked with the weight of the rafter selected. For the preliminary calculation the raft.er load will be
\
\
\I
/
for
a 122-ft-diameter tank.
assumed to be uniform and will be taken as t.he load induced by a roof plate having an average width of 4j4 ft. The live load is taken as 25 lb per sq ft, and the x6-in. roofplate weight is 7.65 lb per sq ft; this gives a total design load of 32.65 lb per sq ft. This corresponds to a load of 0.227 psi. Following the procedure presented in the section entitled “Selection of Rafters and Girders,” we find that the uni-
-.---
Example Design 4.2, Design of Roof and Structural Supports for a formly loaded beam with this t,hin type of end support has a maximum bending moment. as given hy Eq. 4.8. M mitx
=
WP -
8
w=~57.6 + 33.9 = co.~ 12-'
= 0.227 psi X (4.5 X 12) = 12.25 lh per in.;roof l(jad 1 = 22.67 X 12 = 272 in. M ma.x
.1r,,,,* =
113,500 M 2=-= __ = 6.28 in.3 18,000 f Referring to item 1 of Appendix G, we lirld I hat the lightest, American Standard channel sect ion that can he specified is an 8-in. x 21i in. 11.5-lb-per-ft h(Ailnl having a section modulus of 8.1 in.3. The weight of rafters should he included in I he r:lfter toad. This added load amount s I 0: 11.5 lb per fl = 0.938 It) prr in. w = - ~~~- ~~ 12
M,,,,, = (!“:&!!:‘“) 8
1h
prc
in.
The minimum radius of gyratjion of the column section is a function of the length of the column under consideration. If the ratio of (I/r is not to exceed 180 a11Cl the 1enpt.h of the column is 48 ft, Ox, in. (576.5 in.) as shown in Fig. 4.15. then the minimum radius of gg.raIion is:
’ = 122,000 in-lb
Therefore, the rafter selected is satisfactory. S ELECTION OF GIRDER S IZE . Consider first the girders (G2 of Figs. 4.13 and 4.14) of the decagon, which have a span of 24 ft, 6% in. The girders wilt he assumed to art. as a uniformly loaded beams carrying the raft,er loads. Each of these girders supports one end of 11 rafters. The maximum rafter loading is 13.21 lb per tin in. over an average rafter span of (61 - 22)/2 or 19.5 ft. Assuming that half of the total load carried on each raft,er is supported by the girder, we can calculate the roof-plus-raft.er load as follows: (13.21)(19.5)(12)(11) Roof-plus-rafter load = --~ ~---
(29% (2)
= 57.6 lb per lin in. =
WP -
8
= (57.6)(295)2 8
\
\
1 576.5 r = - = -~ = 3.21 in. 180 180 Referring to item 9 of Appendix G under the heading, “properties of sections consisting of t.wo channels,” one observes that the lightest channel combination which will provide a radius of gyration of 3.21 in. or more about both the X-Z and y-y axis is the combination of a 9-in., 13.4-lb channel and a 12-in., 20.7-lb channel. This combination has a value of 3.41 in. in relation to the 2-z axis and a value of 3.62 in. in relation to the y-y axis. This combination provides a cross-sectional area of 9.92 sq in. and has a weight of 34.1 lb per ft of combined sect.ion. The allowable compressive st.ress for t,he column may he calculated by use of Ey. 4.21 as follows: 18,000 1 + (L2/18,000r2)
z = _M = 627,000 -= 34.9 in.3 18,000 f The lightest channel which will provide this section
I = 180 r
f=-
= 627,000 in-lb
c
Therefore, the girder selected is sat,isfartory. S ELECTION OF COLUMN S IZE . A total of 16 columns wilt be required: 10 for the decagon, 5 for the pentagon plus I support,ing the apex of the cone as shown in Fig. 4.13. The roof area and its corresponding lo;ld increase per column as the distance from the tank cent,er increases as a result of the roof-support layout. The total roof load supported by each column (C3 of Figs. k.13 and 4.15) of the decagon is equal to the toad peg decagon girder plus the weight of the girder itself or is 60.4 lh per tin in. of girder length from t.he previous girderdesign calculation. Therefore, P = 60.4 X 295 = 17,800 Ih
122,000 z = ~ ~ = 6.78 in.” 18,000
M max
(60.S)(2Y5)2 = 656.000 in-lb 8
. 3 2 = M - = 656,000 ~-~~~ = ,y(j.,, _ 111 18,000 f
= 113,500 in-lb
The total toad is 12.25 + .96 or 13.21 Recalculating M,,,,, we oblain:
73
(see item 1 of Appendix G) is a 15 x 3$6 in. channel weighing 33.9 lb per ft and having a section modulus of ‘41.7 in.R. Checking the girder by including the weight of t,he girder, w e find:
w
(12.25)(272)” = 8
122-Ft Tank
rrlohh~
=
18,000 (600)2 l + (Is,000)(3.41)2
18,000 = ___ ‘7 -7.7 = 6620 psi LI.‘L
\ I 7
-
.
Columns-
Inner-row rafters
Make 1, mark Cl Make 5, mark C2 Make 10, mark C3
Make 10, mark Rl Make 10. mark R2
19’-6%” 20'- 1 1’+1/,,”
Middle-row rafters Make 40. mark R3
22"7'&"w 22’-93;,”
Outer-row rafters Make 70. mark R4
6"x4"x%"L
xO>4"
Rafter lugs Make 70. mark L1 j+- 2’-8”+
i!,
3" 8”@11.5* x2’-6” 2 PLS.
[email protected]#x l’-1w -Column bases Make 16. mark CL% R
Outer-cob cap Make 10, mark C3C
Inner-cob cap Make 5, mark CZC
Center-col. cap Make 1. mark ClC
Fig. 4.15.
Rafter and column detoilr for o 122-ft-diameter tank.
Example Design 4.2, Design of Roof and Structural Supports for a 122-Ft Tank
it does not support girders. 648 in. Therefore,
The actual induced stress is:
j=;
.f=
17,800 + (50 X 34.1) 9.92
Therefore, the column combination is satisfactory, and the radius of gyration is controlling. The size of the pentagon column supports (C2 of Figs. 4.13 and 4.15) can be readily determined because it is recognized that here also the radius of gyration is controlling. These columns have a length of 50 ft, 4 in., or 604 in., as shown in Fig. 4.15. Therefore, 604 1 P = - = - = 3.36 in. 1 80 180 Thus the same column combination called for at the decagon supports will also be specified here. The central column (Cl of Figs. 4.13 and 4.15) has a greater length because of the roof pitch and the fact that
Consider a combination of a lo-in. and a 12-in. channel. This combination has a radius of gyration of 3.83 in. in relation to the x-x axis and a radius of gyration of 3.52 in. in relation to the y-y axis; therefore the average radius of gyration is 3.68 in. In view of the low induced compressive stress on this column and the average radius of gyration of 3.68 in. as compared with the 3.61 in. required, its use in this application can be justified. It is customary for many tank fabricators to use built-up structural sections for columns as is done in this example. However, it is apparent that an appreciable saving in column material could be realized by using pipe in which the induced stress would be higher. For example, lo-in. schedule-10 pipe having a radius of gyration of 3.74 in. and a cross-sectional area of 5.49 sq in. would be satisfactory. This area compares with the area of 10.50 sq in. for the lo-in., 15.3-lb and 12-in., 20.7-lb channel combination for the center column. Thus a material saving of 48’% can be realized. This gain may be partially offset by the greater cost of pipe,
PROBLEMS
1. Using the dimensions given in Fig. 4.16, determine the required section modulus, Z, for rafters RA, RB, RC, and RD. 2. Using the dimensions given in Fig. 4.16, determine the required section modulus, Z, for girders GA, GB, and GC. 3. Using the dimensions given in Fig. 4.16, determine the required radius of gyration for columns Cl, C2, C3, and c4. 4. Derive a relation comparable to Eq. 4.6 for use with aluminum roof plates. C- 1 I1 reauiredl
tank.
4.16.
Rafter and girder layout for a
Its length is 54 ft, 0t.d in. or
648 1 r = - - - = 3.61 in. 180 180
19,505 = ~ = 2270 psi 9.92
Fig.
75
150-ft-diameter
cone-rooi
.
C H A P T E R
PROPORTIONING AND HEAD SELECYION FOR CYLINDRICAL VESSELS WITH FORMED
CLOSURES
t.he shell. These vrsst~ls had the fau11 of frequent leakage around the rivet. heads. AI tempts to correct this diffrcul! y were made by means of fillet welding the plate edges and seal welding the rivet heads. These vessels often were not satisfactory unless the fillet welds were made so large that the loads were carried by the fillet. welds rather than hy the rivets. When it was realized that the welds were carry ing the loads rather than the rivets, a large number of vessels for low-pressure service (walls less than 1 in. thick) were fabricated entirely by oxyacetylene we!ding. The limitations of the welding art, at this time, in particular the brit.tleness of the bare electrode welds, made the construction of heavy-walled vessels impracticable. With the development of flux-coated electrodes ductile welds were possible. This development resulted in practical obsolrscence of riveted-fabrication techniques for pressure-vessel service. 5.1 b Use of Formed Heads. Cylindrical vessels with formed heads are used for a wide variety of applications in which cylindrical tanks with flat bottoms cannot be used. These applications can be grouped into three Gasses: (1) funct.ional use, (2) pressure consideration, and (3) size limitations. Processing equipment such as distillation columns, desorption units, packed i owers, evaporators, crystallizers, and heat exchangers are essentially cylindrical vessels having formed heads plus other required functional parts. If the working pressure of the process vessel is to be 01 her than atmospheric pressure, formed heads are usually used to close the vessel. In general, all cylindrical vessels requiring a working pressure in the vapor space of abo& 5 lb pep sq in. gage or more are fabricated with formed heads. Large-diameter
u h e real need for the use of formed c!osurrs on cyliutl rical vessels arose w-ii h the development of the power steam boiler early in the nineteenth century. As a result of the frequent occurrence of boiler explosions, the British House of Commons in 1817 made the recommendation that the heads of cylindrical boilers be hemispherical (12). Since then a wide variety of formed closures termed “heads” have been developed, standardized, and extensively used in the fabrication of process pressure vessels. The development of the thermal cracking process in the petroleum industry during the period from 1915 to 1930 resulted in the construction of thousands of pressure vessels with formed heads operating in the range of from 100 to 400 psi. The heads of these early vessels usually were of the torispherical-dish type with a small knuckle radius. The first formed heads were of a small size and were hand-forged by “bumping out” a flat plate. One of t.he early American steel producers, Lukens Steel Company, in 1885 formed a 5-ft-diameter dished head by digging a hole in the ground to the approximate radius of the dish and bumping the heated plate into the depression by the use of mauls. Since then metShods of forming heads have been highly developed by the use of dies and forging and spinning techniques. Figure 5.1 shows a photograph of the world’s largest flanging machine spinning a head with a 20 ft, 6 in. diameter. 5 . 1 G E N E R A L CONSICERATIONS 5.la Development of Welded Ccnstruction. The early thermal-cracking plarts of the prtrcleum industry used pressure vessels in which the formed htads were rivrtrd to 76
Material
Fig. 5.1.
f I(
7
World’s largest flanging machine spinning a head 20 ft, 6 in. in outside diameter.
flat-bottomed, cone-roofed storage vessels are limited to .I -working pressure in the vapor space of only a few ounces. Xowever, cylindrical vessels with flat bottoms and considerably smaller diameters may operate under allowable working pressure of several pounds per square inch if a domed or umbrella roof is used. Equipment designed to operate under less than atmospheric pressure will also require the use of formed heads. Small horizontal storage vessels supported off the ground are usually fabricated with formed heads although flat ends of heavy plate are someI imes used. 5.1~ Vertical versus Horizontal Vessels. In general, ! he functional requirements of the vessel determine whether the vessel shall be vertical or horizontal. For example, distilling columns and packed towers, which utilize the force :*f gravity for phase separation, require vertical installation. Heat exchangers and storage vessels may be either vertical or horizontal. In the case of heat exchangers, the selection is often controlled by the routing of the fluids and heattransfer considerations. In the case of storage vessels, the installation location is important. If the vessel is to be installed outdoors, the wind loads on vertical vessels may impose the necessity of heavy foundations to prevent overt.urning. For this reason, horizontal storage vessels are
(Courtesy of Lukens
Specifications
77
Steel Company.)
usually more economical. However, ot.her important considerations such as available floor space or ground area, head room, and maintenance requirlmcnts may be determining factors. 5.2
MATERIAL
SPECIFICATIONS
Vessels with formed heads are most commonly fabricated from low-carbon steel wherever corrosion and temperature considerations will permit its use because of the low cost, high strength, ease of fabrication, and general availability of mild steel. Low- and high-alloy steels and nonferrous metals are used for special services. The steels commonly used fall into two general classificetions: (1) the steels specified by the ASME code for unfired pressure vessels (ll), often referred to as “boiler-plate steels, ” of flange or firebox quality; and (2) structuralgrade steels, some of which are permitted by the above code in certain applications and which arc widely used fcr the construction of storage vessels under specifications given in API Standard 12 C (2). The design of vessels il; accordance with the ASME code for unfired pressure vessels is treated in Chapter 13, which includes a description of the materials and specifications. The discussion in this chapter will be restricted to those st,eels used in the fabrica-
78
Proportioning and Head Selection for Cylindrical Vessels with Formed Closures
ASTM-A-7, A-283, Grade C, and A-283, Grade D are the most commonly used plain carbon steels in the construction of storage vessels and are widely used for vessels with formed heads, especially the steel designated as ASTMA-283, Grade C. Steel A-283-54 is of the structural quality intended for general applications. It is available in four grades, A, B, C, and D, having minimum tensile strengths of 45,000, 50,000, 55,000 and 60,000 psi respectively, as given in Table 5.1. This steel is available in thicknesses up to and including 2 in., but its use in vessels designed to code specification is limited to thicknesses up to and including K in. Grades A and B are primarily used in severe cold-forming applications where high ductility is of prime importance and tensile strength is a minor consideration. On the other hand, Grade D does not have sufficient ductility for easy shell and head forming and is not as easily welded as Grade C. As a result, Grade C is the most widely used structural-quality plate steel for vessel construction. The major. portion of all oil-storage tanks, elevated tanks, water standpipes, and other varieties of tanks of all descriptions, involving both dishing and rolling, are constructed of ASTM-A-283, Grade C. Steel A-7 is intended for use in the construction of bridges and buildings and for general structural purposes. It has physical properties identical to A-283, Grade D. These steels are the same whether made by the open-hearth or electric-furnace processes. However, steel A-7 is also made by the acid-Bessemer process, and steel made by this process is not recommended for vessel construction. Steel A-7 is available in all standard thicknesses, and its use is permitted . in vessels designed to present code specifications and having shell thicknesses up to and including s/4 in., providing the steel has properties equivalent to A-283, Grade D. Steel ASTM-A-113-55 is a structural steel used for the construction of locomotives and railroad cars except where firebox boiler plate is required. It is made by either the open-hearth or the electric-furnace process and is available in nearly all standard thicknesses. This steel is made in 3 grades, A, B, and C. Steel A-113-55, Grade B has properties approximately midway between those of steels A-28354, Grades C and B, as shown in Table 5.1. Note that the grade specifications for tensile strength for the A-113 steels run in the reverse order of the grade specifications for A-283 steels. There is no particular advantage to using this steel in preference to A-283 steels except when it is more readily available. It may be used for vessels designed to present code specifications with the same limitations as for A-283 grade steel. Steel ASTM-A-131-55 is an improved structural steel intended primarily for use in ship-construction. Formerly, the specifications for this steel were essentially the same as for A-7 and A-283, Grade D. To improve the quality of ship-hull steels, the specification was changed in 1950 in order to include an increase in quality specifications with increasing thicknesses. This logical requirement of . increased quality with increased thickness warrants consideration of this steel as a material of construction for heavy-vessel fabrication. For this steel there is a limitation on the maximum percentage of carbon and a range of from 0.60 % to 0.90 y0 manganese for all plates thicker than G in.
tion of vessels with formed ends not requiring fabrication in accordance with these codes. 5.20 Comparison of Specifications for Structural- and Boiler-quality Steel Plates. Structural-quality steel rather
than boiler-plate-quality steel is used in the fabrication of many vessels with formed heads because of economic considerations and its availability. Both types of steel are available in the “killed ” and the “semikilled ” or rimmed quality. A “killed ” steel is one completely deoxidized by the addition of aluminum, silicon, or manganese at the time of the casting of the ingot. The purpose of killing is to minimize the interaction of carbon and oxygen and to reduce the formation of blow holes. A completely killed steel requires “hot capping,” more time in the soaking pit, and more time for the ingot heating. “Hot capping ” is, the use of an insulated mold on top of the ingot mold to hold a molten reservoir of metal for feeding the ingot as it shrinks on solidifying. A partially killed or rimmed steel is a partially deoxidized steel. An ingot of rimmed steel has a high-purity, low-carbon steel rim from which it obtains its name. Fully killed structural steels have no advantages .over boiler-plate steel because of their high cost and limited availability. One of the major differences between boiler-plate steel and structural-plate steel is the “ quality ” control dictated by the number and severity of test requirements. As far as chemical requirements are concerned, the principal difference expressed by ladle analysis is the more restrictivelimit placed on phosphorus and sulfur for boiler-plate steels. The thickness tolerances are the same for boiler-plate steels and structural steels when plates are ordered to a given thickness. The physical tests are the same for both steels except for the number of tests and the stipulated location for test specimens. Structural-quality plate steels require only two tension and two bend tests from each heat ,of metal which may contain over 100 tons. Flange-quality boiler-plate steel requires one tension and one bend test from each plate rolled. Firebox-quality boiler-plate steel requires two tension tests and one bend and one homogeneity test from each plate as rolled. There are also minor differences in the methods permitted for repairing surface defects in the slabs prior to rolling. Boiler-plate steel such as SA-285 flange quality and lirebox quality had mill quality extras of $0.40 and $0.50 per 100 lb respectively as of January 1956 (see Appendix C). Other boiler-plate steels such as SA-212 and SA-201 had mill quality extras of from $1.20 to $1.55 per 100 lb, depending upon thickness and grade. Killed steels had mill extras of $0.65 per 100 lb. The use of structural-grade steels results in the minimum of quality-extra charges, and the use of these steels is justified whenever permissible. In selecting steels for pressure-vessel fabrication to satisfy code requirements, Chapter 13 should be consulted. 5.2b Types of Structural-steel Plates. The most widely available types of plain-carbon structural-steel plates are listed (67) in ASTM-A6-54T. Those most suitable for vessel construction are A-7, A-113, A-131 and A-283. Specification ASTM-A6-54T gives the general requirements such as permissible variations in dimensions and weight, methods of testing, correcting of defects, and rejection (67).
I
---- -
---T--\-\ - - - \
- - -\I - - l - -I- r - - -
--
--_-
-- _ ~__ ~-~- - I -I
Proportioning of Vessels with Formed Heads Table 5.1.
.i
t I
i
T
Steel A-283-54 Grade A Grade B Grade C Grade D A-7-55T A-131-55* Grade A Grade B Grade C A-l 13-54 Grade A Grade B Grade C * See text for
1955 ASTM Steel Specifications (67)
Min Yield Point, psi
Max Thickness Available, in.
55,000 60,000 60,000 72,000 72,000
24,000 27,000 30,000 33,000 33,000
2 2 2 2 15
58,000 to 71,000 58,000 to 71,000 58,000 to 71,000
32,000 32,000 32,000
$5 and less 35 to 1 1 and over
60,000 to 72,000 50,000 to 62,000 48,000 to 58,000 limitations.
33,000 27,000 26,000
Tensile Strength, psi 45,000 50,000 55,000 60,000 60,000
to to to to to
79
Also, for plates having a thickness of 1 in. or more, a requirement of 0.15% to 0.30% silicon is specified. In addition, it is stipulated that this steel be manufactured to have an inherent fine-grained structure. This steel is available in a wide range of thicknesses and is of higher quality than A-7 but presently is not permitted in the construction of vessels designed to meet unfired-pressure-vessel codes. The additional quality requirements for heavier plates of this steel will increase its cost and may thereby eliminate any savings from using it instead of boiler-plate steels. Other structural-quality steels listed in ASTM designation A-6-54T are A-8, A-94, A-284, and A-242. Steel A-8 is a 3.0% to 4.0~~ nickel steel containing a maximum of 0.43~~carbon and having a tensile strength of from 90,000 to 115,000 psi. It is intended for use in main stress-carrying structural members. The nickel addition results in liner, stronger, and tougher pearlite than is found in plain carbon steel and appreciably increases the yield point, fatigue limit, and impact strength. The difficulties of welding this steel plus the cost extras for nickel addition precludes its use for vessel construction. Steel A-94 is a structural silicon steel containing a maximum of 0.40% carbon and a minimum of 0.20 y. silicon and having a tensile strength of from 80,000 to 95,000 psi and a minimum yield point of 45,000 psi. This steel may be eliminated from consideration for vessel construction on the basis of welding difficulties and the cost extras for fully killed steel. Steel A-284 is a low- and intermediate-strength carbon-silicon steel containing from 0.10% to 0.300/, silicon and having tensile strengths of from 50,000 to 60,000 psi depending upon the grade. The steel is coarse-grained and requires heat treating for grain refinement. The presence of the silicon tends to dissociate carbides to form soft graphite thereby weakening the welded joints. For these reasons and because this steel is a fully killed steel and therefore involves cost extras, it is not economical to use it for vessel construction. Steel A-242 is a low-alloy structural steel intended primarily for use as a stress-carrying material of structural members when saving in weight and atmospheric-corrosion
Min % Min To Elone.. in.. Elone.. in.. 8X. ’ 2in. ’
Max %
Max % S (ladle) (basicj
Max % C (ladle)
(1at;le) (basicj
30 28 27 24 24
no spec. no spec. no spec. no spec. no spec.
0.04 0.04 0.04 0.04 0.04
0.05 0.05 0.05 0.05 0.05
21 24 (see Ref. 67) (see Ref. 67)
no spec. 0.23 0.25
0.04 0.04 0.04
0.05 0.05 0.05
21 24 26
no spec. no spec. no spec.
0.04 0.04 0.04
0.05 0.05 0.05
27 25 23 21 21
24 38 no spec.
.
resistance are important. Thicknesses are limited to not under sis in. and not over 2 in. It contains a maximum of 1.25Cy manganese and a maximum of 0.20yo carbon. This steel has a yield point of 50,000 psi for thicknesses of from xi6 to N in., 45,000 for thicknesses of K to 134 in. and 40,000 for thicknesses of l>s to 2-in. in comparison to a yield point of 30,000 psi for A-283, Grade C. For the plates l>s-in. thick and less this represents an increase of 50% or more in yield strength. Using the same design factor of safety based on yield point results in a proportional decrease in metal thickness required to resist a given load. In designs in which stress rather than elastic stability or brittle fracture is controlling, the use of t,his steel rather than a plain carbon steel such as A-283, Grade C may result in a saving. See Table 3.2 for specifications for this steel and Chapter 3 for further discussion of its use. 5.3 PROPORTIONING OF VESSELS WITH FORMED HEADS
In general, the cost of a vessel may be considered to be proportional to the weight of the steel used in its construction. It would therefore appear that for storing a fluid under uniform pressure a vessel having the minimum surface area and thickness per unit volume would be the most economical. A spherical vessel has the minimum surface area per unit volume and the minimum shell thickness for a given pressure and volume. If the cost of fabrication were not a prime consideration, the most economical shape for a vessel would therefore appear to be a sphere. However, the fabrication costs of spherical vessels are so great that their use is limited to special applications. Cylindrical vessels are more easily fabricated, in the majority of cases are considerably simpler to erect, are readily shipped, and are therefore more widely used in the process industries. For a simple cylindrical vessel with formed heads, the optimum ratio of length to diameter, L/D, is a function of the cost per unit area of the shell and the formed heads. More complex vessels such as distillation columns, heat exchangers, and evaporators have additional parts such as
.
*od .
Proportioning
and
Head
Selection
for
Cylindrical
Vessels
with
Formed
Closures
Solving for x2 we obtain:
z2 = 4b” - ty2 = 4(b2 - s”; Differential volume, dV = A dy = rx2 dy Integrating we obtain:
Dimensions for a 2: 1 ellipsoidal dished head.
trays in distillation columns and tube bundles in heat exchangers which must also be considered in determining the optimum proportions. The proportioning of a simple vessel may be based either on the cost per pound of the material or the cost per unit area of the material. In Chapter 3 the proportioning of flat-bottomed, cylindrical, cone-roofed tanks was based on the cost per unit area because land and foundation costs, which are important for such vessels, can best be considered on a unit-area basis. In addition, the cost of coned roofs and flat bottoms are relatively constant on a unit-area basis for large-diameter tanks. However, cylindrical tanks with formed ends for various pressure services have wide variations in thickness and therefore vary in cost per unit area. The cost of land area and foundations is usually a minor consideration for such vessels. Therefore, ii is more advantageous to consider the cost of shell and heads in terms of unit weight rather than in terms of unit area. 5.30 with
Equations for Optimum Proportions of Vessels Elliptical Dished Heads.
VOLUME RELATIONSHIPS _---__.- A cylindrical vessel closed at l,ot’i;-~ds-~wi~~-~llipt,icaldished heads has a volume equal to the volume of the cylindrical section plus twice the volume contained in one of the heads. The volume contained in a head can be expressed in terms of a cylinder of equivalent volume having the same inside diameter as the cylindrical section of the head. Figure section . of c -,___._.5.2 :.. -.._is. .a. .cross .. an ellipsoidal head having a 2: 1 malor-to-mmor-axis ratio. ~~qi&ions for the volume relationships for a 2:l ellipsoidal head ( 103) are as follows. The equation of au ellipse is: (5.1) For a 2: 1 ellipsoidal dished head a = 2b Substituting we obtain:
-$+$=I
The
v-olume of an equivalent cylinder is: 1. = vra2H
where H = length of cylinder Equating we obtain:
3 Ta2H = !!!. 3
Thus the volume of IWO ellipsoidal heads having a majort,o-minor-axis ratio of 2.0 is:
I-n=(3(y),=Tg Therefore, the t,o a1 vohmte coutwiueti in the vessel is: ~~;‘_, = [($)I* +gj
where I, = length of the vessel, t.angent line t.o tangent line, between heads, feet. Solving for L, we obtain:
L+p] COST R ELATIONSHIPS. The diameter of a circular plate required for forming an ellipsoidal head is approximately 22% greater than the internal diameter of the finished vessel (103). Also, the cost of the formed heads is approximately 50% greater than the cost of the steel from which they are formed. This increase in cost results from cost extras for circular plates and the cost of forming and machining. Let c, = cost of fabricated shell, dollars per pound 1.5 c, = cost of fabricated head, dollars per pound 1 = thickness of head and shell, inches p = density of steel, pounds per cubic foot The cost of the shell section of t,he fabricat.ed vessel is:
Expanding we obtain:
.
x2 + 4y2 = 4b2 ‘, A ; ”
. ’i
* t
Selection of Optimum Plate Dimensions
For vessels fabricated from plates from 2 in. to 6 in. in thickness, the thickness extra will modify the cost per unit. weight. In this range of thickness the cost of the vessel may be estimated as varying approximately inversely with D46 (103). Or ,I cs cs = gr’l
and the cost of two elliptical dished heads is
1
2 x 1.&p ; (1.22D)Z ; [ or the total cost of the vessel is:
t 4V D2 + G=c,pr- 12 [ uD T %(1.22D)2
81
1
Substituting into Eq. 5.3, we obtain: C = c,“k
1
F + 0.782D2.75
Differentiating and equating to zero to obtain the minimum, we find that But according to Eq. 3.14,
dG 1 1.275V + 9-(0.782)0~.~~ = 0 &j = - z ~ D5/4 ,
. t = I!!! = PO w
8.60D3
24f
= 1.275V
D3 = 0.148V
Substituting we obtain: C = c,pu !@ 2885
1.275 ; + 0.782D2
= c,k[1.275V +
0.782D31
1
Substituting for V, we obtain: (5.3) D = 0.116L + 0.039D
where k = f!? 288j P R O P O R T I O N I N G . The cost of the shell is not a constant but is a function of the weight of the vessel, which in turn is a function of the pressure and diameter. For vessels having a shell plate thickness of up to 2 in., the cost of the vessel may be estimated as varying approximately inversely with D% (103). Or C8’ C8 = Dys
Substituting in Eq. 5.3, we obtain: G = c,‘k F + 0.7820%
1
Holding V constant, differentiating, and equating to zero in order to obtain the minimum, we find that dC -= -g 1.275V D,i + fi(0.782)05* = 0 dD 6.25D3 = 1.275V D3 = 0.204V . . Substltutmg for V we find that
D = 0.16OL + 0.0530
1 (5.4)
Or use L/D = 6 for vessels with plate thickness up to 2 in.
---3?+
.-
--iv-\-
0.961 L=- = 8.280 z 80 0.116 Or use L/D = 8 for vessels with plate thickness of from 2 in. to 6 in. D IAMETER AND LENGTH LIMITATIONS. The selection of the proportions of a vessel may be influenced by other factors such as the maximum diameter or length that can be shipped by railroad flatcar. In general, the maximum diameter that can be shipped on most railroad lines is 13 ft, 6 in. Larger diameters may be shipped by rail but require special routing of the shipment. If water transportation is available between the fabrication shop and the erection site, large-diameter equipment may be shipped by barge or floated to the site. Two other alternatives are: (I) shop forming and partial fabrication by welding in sections with final fabrication in the field or (2) field assembly of plates cut and formed in the shop. The length of a vessel is not as critical as the diameter with respect to railroad shipping limitations because more than one flatcar may be used. Figure 5.3 shows an oil-refinery fractionation column loaded on three flatcars, supported on the two end flatcars with no load supported on the middle car. This permits the cars to negotiate a curve with the vessel pivoting on the end cars. Other considerations such as the selection of plate widths and plate lengths to minimize the number of welded joints may influence the proportions of the vessel. (See the following section.) 5.4 SELECTION OF OPTIMUM PLATE DIMENSIONS
L = 0.947 ~ D = 5.930 rr 60 0.160
I
1
-
\I
P?ATE W IDTH . The cylindrical shells of vessels with formed heads may be fabricated by rolling and welding one or more plates together. A choice exists as to the plate
/-
-
--
-
--
- - - -.-+ . . .--..-.
Proportioning and Head Selection for Cylindrical Vessels with Formed Closures
82 “T
s*
Fig. 5.3.
Oil-refinery fractionating tower ready for shipment on three flat cars.
widths and number of plates to be used. Usually a circumferential weld and sometimes a longitudinal weld may be avoided by using a larger plate width. Plates having widths in excess of 90 in. bear a cost extra which increases with increasing width. The most economical design is often one in which a wider plate is used; providing that a welded joint is thereby eliminated and the cost saved by eliminating such a joint exceeds the extra cost of wider plates. An example of the reduction in cost that may be realized by the selection of a plate size that will eliminate a welded joint is given by W. G. Theisinger (104) in regard to a purchase order involving 20 vessels 48 in. in diameter and 20 vessels
.= : 60 g 50 : 4o aJ 30 $ f a- 20 i 10 0.1
Fig. 5.4.
0.2
0.3 0.4
0.6 0.8 1.0
C, = cost extra, dollars per 100 lb
2
3
Width extras for carbon-steel plates os of 1953.
4
(Courtesy of C. F.
Braun
& Co.)
54 in. in diameter. The shells for these vessels might have been ordered as follows: 1. Two-plate shells a. 48-in.-diameter vessels 20 plates 15’735 X 87 X 141 in. 20 plates 15735 X 85 X 134 in. b. 54-in.-diameter vessels 20 plates 1763s X 91 X llT4-in.-wide plate and $1.50 per 100 lb for the 17634-in.-wide plate and the overweight allowance totaled $9,853.00; therefore, a net saving of $7587.00 was realized by the purchaser by using singleplate shell construction. In addition, the fabricating time was reduced by 5800 man-hours, and this resulted in quicker delivery. It should be pointed out that these figures are for prices existing in 1944 and are not representative of current prices. When a vessel shell may be fabricated by one- or twopiece construction, the selection may be made by simply estimating the costs for each design and selecting the design giving the lesser cost. However, for larger vessels in which the shell must be fabricated from many plates, the above
Selection Table 5.2.
Average Extra Fabrication Cost, C, (104)
Dollars per Foot as of 1944 (Based on 133% Shop Burden) Code Welded Unclassified 3.75 2.50 3.00 4.50 5.40 3.60 6.38 4.25’ 6.98 7.50 8.10 8.74
4.65 5.00 5.40 5.83
9.30 9.83 10.43 11.10
6.26 6.55 6.95 7.40
11.78 12.60 13.43 14.25
7.85 8.40 8.95 9.50
15.15 16.13 17.25 18.53
10.10 10.75 11.50 12.35
19.50 20.55 21.53 22.80
13.00 13.70 14.35 15.20
23.85 24.90 25.88 27.00
15.90 16.60 17.25 18.00
28.28 29.10 30.15 31.35
18.85 19.40 20.10 20.90
.
i
‘E
Optimum
Plate
Dimensions
83
The total cost of all of the circumferential shell welds for N number of plates (excluding the head welds) will be: +WL.
(N - l)wDC, = s
(5.7)
where 1 = length of shell, inches D = shell diameter, feet The additional cost of using plates wider than 90 in. is given by the equation: t$) (2) = (l;f;;o) [A] (w - 9O)‘.23
(5.8)
The total extras for using plate widths wider than 90 in. plus the costs for all the circumferential joints exclusive of head joints is given by the sum of Eqs. 5.7 and 5.8 as follows:
C=?rD
’ - 1 C, + O.O0023511(w - 9O)‘.23 KW > I
G-9)
Differentiating the cost, C, with respect to plate width, w, and equating to zero to obtain the minimum, we find that
dC
-GE - TD 7 + 0.000235t1.23(~ - 90)“.23 = 0
dur-
c, w2 - (w t = 3460
- g(Qo.23
(5.10)
Solving Eq. 5.10 for w gives the optimum width of plate to give minimum fabrication cost for the shell as a function of joint fabrication cost, C,, and shell thickness, 1. This equation is plotted in Fig. 5.5 for convenience. Since 1953 the width extra has been combined with the thickness extra (see Appendix C). Therefore Fig. 5.5 is useful only for first approximations. P LATE T HICKNESS . Plates having thicknesses of from --- -.w $6 &.‘to~&~areXvailable from mills at base cost with no thickness extras. To avoid extras for plates thicker than 1 in., a higher-strength steel often may be used to advantage. ^- This is of particular importance in connection with
procedure is not so simple since a number of designs may be possible. To determine thepEt~.~~-~.ofplates, the plate width resulting in the minimum cost for the fabricated shell can be evaluated mathematically. The width extras for plain-carbon-steel plates as of 1953 are shown in Fig. 5.4, in which the cost extra in dollars per 100 lb is plotted against w - 90 where w is the plate width in inches. The equation of the line given in Fig. 5.4 is:
ce = $ (w - 90)1.23 where C, = dollars per 100 lb w = plate width, inches
i
of
The cost of circumferential welding, Cw, including the cost of preparation of the joint, is usually expressed in terms of dollars per foot of weld and is given in Table 5.2. The fabrication cost per circumferential weld will be
6 5 4 3 90 100 110 120 130 140 150 160 170 180 190 200
w = optimum plate width, inches
Fig. 5.5. Optimum plate widths for vessel-shell width extras as of 1953.
construction bared on
84
Proportioning and Head Selection for Cylindrical Vessels with Formed Closures
Fig. 5.6. F o r m i n g dished heads by “drawing” in o press. D e t a i l o: Toking c i r c u l a r b l a n k p l a t e f r o m f u r n a c e f o r f o r m i n g . thick for o vessel 60 in. in inside diameter in o 1000~ton press. ( C o u r t e s y o f C . F . Broun & C o . )
Detail
b: Forming of head 1
x 6 in.
vessels designed to meet code requirements and is conAlthough the cost of heads formed from flat plates sidered in detail in Chapter 13. The 1957 practice in involves the additional cost of forming, the use of forrne!l steel pricing combined thickness extras with width extras heads as closures is usually more economical than the use (see Appendix C). of flat plates as closures except for closures of small diamP_.._ LATE eter. This can be shown by comparing the thickness _.-- -LENGTH.. Plates having lengths between 8 and 50 ft are available from mills at no length extra. If posrequired for closures of flat plates with that of various types sible, the plate lengths selected for a vessel should be $1 of formed heads. within these limits. Warehouses usually do not stock Figure 5.7 illustrates various types of the more common plates longer than 40 ft, and this length is usually carried formed heads where only in plate thicknesses of $Q in. or less and plate widths of 72 in. or less. The plates of heavier gage (up to and “\ f = head thickness, inches including 3 in.) and greater width are usually carried in icr = inside-corner radius, inches 20-ft lengths at the warehouses. The maximum plate ;/-. i length, thickness, and width that can be handled by the sf = straight flange, inches shop fabricating the vessel may impose a limitation on the c P = radius of dish, inches size of plate that can be handled. OD = outside diameter, inches 5.5 COMMON TYPES OF FORMED HEADS AND THEIR
SELECTION
Nearly-all formed heads are fabricated from a single circular flat plate by spinning, as shown in Fig. 5.1, or by “drawing”i+itb dies in a press, as shown in Fig. 5.6. Detail a of Fig. 5.6 shows a single-blank plate being removed from a furnace for forming to a head, and detail b shows the plate in a press during the forming operation.
b = depth of dish (inside), inches a = ID/2 = inside radius, inches s = slope of cone, degrees OA = overall dimension, inches H = diameter of flat spot, inches
___ ._ -..--
Common Types of Formed Heads and Their Selection
The inside depth of dish and overall dimension, OA, may be determined by use of the dimensional relationships for flanged and dished heads given in Fig. 5.8. For purposes of welding heads to the shells of vessels, various styles of machined edges can be supplied on the formed head by the manufacturer. Standard machining styles for heads supplied by one manufacturer are shown in Fig. 5.9. It should be noted that in styles C6 and C7 the dimension t (the head thickness) must exceed dimension s (the shell thickness) by at least & in. and in styles D8, D9 and DlO the dimension t must exceed dimension s by k in. ’ “Table 5.3 gives the cost extras for the various standard machining styles and applies to all types of formed heads. Quantity differentials (1955) must be applied to t.he cost extras given in Table 5.3 as follows for types A and B only: list plus 90 *h for 1 to 4 heads, list plus 50 CA for 11 to 50 heads. All other styles, diameters, and gages are list plus 90%. 5% F l a n g e d - o n l y H e a d s . The formed head most economical to fabricate is that produced by simply forming a flange with a radius on a flat plate. This head is identified as a “flanged-only head” and is illustrated in detail a of
Fig. 5.7. The radius of a flanged-only head decreases somewhat the abruptness in change of shape at the junction of the flat head and the cylinder. The resulting gradual change in shape reduces local stresses. The flanged-only head finds its widest application in closing the ends of horizontal cylindrical storage vessels a~ atmospheric pressure. These vessels typically store I’url oil, kerosene, and miscellaneous liquids having low vab)()t pressures. Flanged-only heads may be used for the bot IWII heads of vertical cylindrical vessels that rest on concrr~c? slabs and do not have diamet,ers in excess of 20 ft. Table 5.4 gives the straight-flange length and inside-corner radius for such heads as functions of head thickness. These hwtls are fabricated on the basis of using the outside diameter as the nominal diameter. Head diameters based on the outside diameter are available in increments of 2 in. from 12 to 42 in., in increments of 6 in. from 42 to 144 in., and in increments of 12 in. from 144 to 240 in. A head wit.h a 246-in. outside diameter is also available. $\During @@i&&of the heads, thinning out of the plate I %curs at the corner radius. Therefore, for heads having an outside diameter of under 150 in., plate thicknesses must
rt
‘;T Inside depth ofI dish Fig. 5.1. mon
rt
Various types of more com-
formed head,:
(a)
flanged only,
(b) Aangad a n d s h a l l o w d i s h e d ,
(c)
tlanged
(d)
and
standard
dished,
wOD----- ’ (b)
AWE and API-AWE code flanged and dished dished (g)
(torispherical), (ellipsoidal),
fiunged
[toriconical)
and
(105).
(e)‘Jilliptical
(f)
hemispherical,
conical
dished
(Courtesy of Lukenr
Steel Co.)
+nside depth of dish /
cc-----+OD--------l (d)
85 1/’
86
Proportioning and Head Selection for Cylindrical Vessels with Formed Closures Table 5.3.
Cost Extras for Standard Machining Styles for Heads (105)
(Courtesy of Lukens Steel Company) Gage g’ 1” 134” 1>4” $3.50 $4.00 $5.00 $5.50 $8.00 $5.00 $5.50 $7.50 $6.50 $7.00 $9.00 $9.50 $8.50 $9.00 $11.50 $12.00
Outside Diameter Style A 24” and under 36” 48” 60”
>p” $2.00 $4.50 $5.50 $7.00
72” 84” 96” 108” 120”
$8.50 $10.00 $11.00 $12.00 $14.00
$9.50 $11.50 $12.50 $13.50 $15.00
$10.00 $12.00 $13.00 $15.00 $17.00
$13.00 $15.00 $17.00 $18.00 $20.00
132” 144” 160” 176” 192”
$18.00 $19.00 $21.00 $28 00 $35.00
$19.50 $20.50 $22.50 $30.00 $37,00
$20.50 $22.50 $24 50 $32.00 $39.00
Style B or C 24” and under 36” 48” 60”
$2.50 $5.00 $6.50 $8.50
$4.00 $6.00 $7.50 $10.00
72” 84” 96” 108” 120”
$9.50 $11.50 $12.50 $13.50 $15.00
132” 144” 160” 176” 192”
-
2”
&YOO $10.00 $12.50
&L50 $10.50 $13.00
$ii.ho $14.00
$12.00 $15.00
$13.50 $15.50 $17.50 $19.00 $20.50
$14.00 $16.00 $18.00 $19.50 $21.00
$14.50 $16.50 $18.50 $20.00 $22.00
$15.50 $17.50 $19.50 $21.00 $23.50
$16.50 $19.00 $21.00 $22.50 $25.00
$21.50 $23.00 $25.00 $35.00 $45.00
$22.00 $24.00 $25.50 $37.00 $48.00
$23.00 $24.50 $27.00 $39.00 $51.00
$23.50 $25.50 $28 50 $41.00 $54.00
$25.00 $28.00 $31.00 $43.00 $56.00
$26.50 $29.50 $33.00 $46.00 $60.00
$4.50 $6.50 $8.00 $10.50
$6.00 $8.50 $10.50 $13.50
$6.50 $9.00 $11.00 $14.00
$ii.bO $11.50 $15.00
sii.50
$12.50 $16.50
$ii.bo
$i+.‘oo
$11.00 $12.50 $14.00 $15.00 $17.00
$11.50 $13.50 $15.00 $17.00 $19.00
$14.50 $17.50 $19.00 $20.50 $22.50
$15.00 $18.00 $20.00 $21.50 $23 00
$16.50 $19.50 $21.50 $23.50 $25 00
$18.00 $20.50 $23.00 $25.50 $27.00
$22.00 $25.50 $28.00 $30.00 $33.00
$25.50 $29.00 $33.00 $37.00 $39.50
$20.00 $21.00 $22.50 $30.00 $39.00
$21.00 $22.50 $24 50 $32.00 $41.00
$23 00 $25.00 $27.00 $35.00 $43.00
$24.00 $25.50 $28.00 $38.00 $48 00
$24.50 $26.00 $28.50 $41.00 $53.00
$26.50 $29 00 $32.00 $45.00 $58.00
$36.00 $39.00 $44.00 $57.00 $70.00
$42.00 $44.00 $50.00 $662.00 $75.00
$4.00 $6.50 $8.00 $10.50
$5.00 $7.50 $9.50 $12.00
$5.50 $8.00 $10.00 $13.00
$7.00 $10.50 $13.00 $17.00
$7.50 $11.00 $13.50 $17.50
72” 84” 96” 108” 120”
$12.00 $13.50 $14.50 $15.50 $17.00
$13.00 $15.00 $17.00 $18.00 $20.00
$14.00 0$16.00 $17.50 $19.00 $21.00
$18.00 $20.50 $22.50 $24.00 $25.00
132” 144” 160” 176” 192”
$21.50 $22.50 \ $24.00 $35.00 $46.00
$25.00 $26.50 $28.00 $40.00 $52.00
$26.00 $27.50 $29.00 $45.00 $61.00
$27.00 $28.50 $30.00 $47.00 $65.00
- - - Style D 24” and under 36” 48” 60”
be increased by >is in. for plates up to 1 in. in thickness and 34 in. for plates 1 to 2 in. in thickness if the minimum plate thickness is to be maintained throughout the corners. 3 I The manufacturer’s catalog should be consulted for greater thicknesses and diameters and for blank weight and forming costs.
\
\
\I
I
5.5b Dished
-
256”
3 I, ...
136”
. . .
$20.50
$28.50 $32.00 $37.00 $50.00 $63.00 -___
$23.00
. .
$ii. 50 $14.00 $19.00
$15’.bO $16.00 $21.00
$1;. bo $26.00
$21.50 $29.50
$19 00 $21.50 $23.00 $24.50 $26 00
$21.00 $23.50 $25.50 $27.00 $29.00
$22.50 $25.50 $28.00 $30.00 $32.00
$29.50 $34.00 $37.00 $40.00 $43.00
$35.00 $38.00 $41.50 $45.00 $49.00
$27.50 $29 00 $31.00 $50.00 $70.00
$31 .oo $33.00 $35.50 $55.00 $75.00
$34.50 $37.00 $40.00 $60.00 $80.00
$46.50 $49.00 $52.00 $68.00 $85.00
$52.00 $55.50 $61.00 $77.00 $93.00
Flanged Heads.
Standard
Dished
and
Flanged
Shallow
The pressure rating of a flanged-only head can be increased if the flat portion is dished. Such heads, not designed to code specifications, are formed from a flat plate into a dished shape consisting of two radii: the “crown” radius or radius of dish and the inside-corner radius, some-
..--.-- -
---
87
Common Types of Formed Heads and Their Selection Table
5.4.
Dimensions
of
Standard
Flanged-only
Heads for All Diameters
(Courtesy of Lukens Steel Company) Inside-corner Gage Standard Straight Radius (in.) (Thickness) Flange (in.) t icr s.f l>h-2 Ns 3is 1>+2>5 4i s/a 135-3 l5.;6 3i6 135-3 1% 34 15’ 135-3 $5 /lG 5i6 lf+-3j$ 1% !ci lj$-3>5 1% 34 13+3jg 2% 39 1>84 2% 74 1 135-4 3 344 1% 1X-4% 1>6-4j& 3% l?L 19j4ijs 144 4% 1>+4js 4% 1% 5% 1% 1%4X 6 2 1 jq--4>5
Fig. 5.0.
Dimensional relationships for flanged and dished heads. ID a=--
2
b = r - d(BC)2
times referred to as the “knuckle” radius. If the radius of dish is greater than the shell outside diameter, the head& known as a “flanged and shallow dished head.” If the radius of dish is equai to or less than theoutside diameter, the head is known as a “flanged and standard dished head.” A flanged and shallow dished head is shown in Fig.-5.7, detail b, and a flanged and standard dished head is shown in Fig. 5.7, detail c. These heads are fabricated on the basis of using the outside diameter as the nominal diameter. Head diameters based on the outside diameter are avalable in increments of 2 in. from 12 to 42 in., in increments of 6 in. from 42 to 144 in., and in increments of 12 in. from 144 to 240 in. A 246-in.-outside-diameter head is also available. It should be emphasized that because of the high localized stresses due to the small inside-corner radius, the use of flanged and shallow dished heads and flanged and standard dished heads is not permitted in vessels which must meet pressure-code requirements. Typical applications of these heads occur in the construc-
2
1 Style A
3
----~
/
AC = d(BC)2
\I
- (ASI
tion of vertical process vessels for low pressures, of horizontal cylindrical storage tanks for volatile fluids such as naphtha, gasoline, and kerosene, and of large-diameter storage tanks in which the vapor pressure and hydrostatic pressure is too great for the practical use of flanged-oniy heads. Vessels with flanged and shallow dished heads are primarily used for horizontal storage tanks. Table 5.5 gives the dimensions of flanged and shallow dished heads. Table 5.6 gives the dimensions of flanged and standard dished heads except for the radius of dish. The radius of dish varies with thickness and diameter, and
- .~
\
2
O A = t + b + s f
Standard machining styles for heads (105).
\
r
BC = r - (icr)
5
Style I3 Fig. 5.9.
I
4
A0 =
- (Al+
ID - - (kr)
/
6
7
8 I-.
Style C
9 Style 0
I
‘ I.
,.
(Courtesy Op Lukens
10 .’ ,’ I’
Steel Company.)
Proportioning
88
Table 5.5.
and
Head
Selection
for
Cylindrical
Vessels
Dimensions of Flanged and Shallow Dished Heads in Inches (See Fig. 5.7.)
(Courtesy of Buffalo Tank Company) 66 72 76 .__ 84 90
OD Gage
icr
-
sf
120 120 120 120 120
120 120 120 120 120
108
114 120 126 132 ~~.... ~-- _ _ _
34
N
3% 338 4%
%6
74
6
?4
35
?iS
54
35
1
6
OD Gage
icr
-
sf
$4
!i
355
197
946
36
3%
197
34
94
4.34
197
746
74
6
197
6
197
$5
1
46
1%
P
r
r
120 120 120 120 120
r
197 300 197 197 300 197 197 300 197 197 300 197 197 300 197
r
r
120 120 120 120 120
r
r
197 197 197 197 197
r
300 197 300 197 300 197 300 197
r
96 r
102 r
197 197 197 197 197 197 197 197 197 197 /f--l 138 144 r
r
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
with
Formed
Closures
economical to use an elliptical hanged and dished head. These heads are used principally for vessels designed t(J meet the ASME codes for unfired pressure vessels. In general, these heads are used either for horizontal or vertical vessels for a great variety of process equipment within the pressure ranges specified above. For pressures in the range of 150 lb per sq in. gage and for higher pressures, a cost comparison should be made between the code flanged and dished heads and the code elliptical dished heads. The optimum choice based on total cost varies with pressure, diameter, thickness, and material of construction. Table 5.7 gives the inside-corner radius and radius of dish for code flanged and dished heads. Table 5.8 gives .the straight-flange length for different head thicknesses of flanged and dished heads. For purposes of cost estimation it is necessary to know the blank weight in order to obtain the cost of the steel used and the cost of forming the head at the fabrication plant. The approximate blank .diam;fter may, be determined by use of the following relatronshrps: .-.__ _.. . . diameter = OD + s + 2sf + Qicr . . ~
(for gages under 1 in.)
(5.12)
diameter = OD + g + 2sj + $icr + t the manufacturer’s catalog should be consulted for this dimension, blank weight, and forming costs. 5.5~ Flanged and Dished Heads (Torispherical) to ASME Code. The pressure rating of flanged and dished
heads can be increased by decreasing the local stresses which occur in the inside corner of the head. This may be accomplished by forming the head so that the inside-corner radius is made at least equal to three times the metal thickness; for code construction, the radius should in no case be less than 6% of the inside diameter. Also, the radius of dish may be made equal to or less than the diameter of the head. Figure 5.7, detail d shows a sketch of a cross section of a flanged and dished head meeting the ASME Code, in which is identified as a “torispherical” head. These heads are fabricated on the basis of using the outside diameter as the nominal diameter. Head diameters based on the outside diameter are available in increments of 2 in. from 12 to 42 in., in increments of 6 in. from 42 to I44 in., and in increments of 12 in. from 144 to 240 in. Heads having outside diameter of 210 in. and 246 in. are also available. The volume in cubic feet of heads having icr equal to 6% of the outside diameter (not including the straightflange portion)‘is approximately equal to: V = 0.000049d;3
\,I
(5.11)
where di = inside diameter of vessel, inches 1’ = volume of torispherical dished head to straight flange, cubic feet Heads of this type are used for pressure vessels in the general range of from 15 to about 200 lb per sq in. gage. These heads may be used for higher pressures; however, for pressures over 200 lb per sq in. gage it may be more
(for gages 1 in. and over)
(5.13)
where OD = outside diameter of dish, inches sf = straight-flange length, inches icr = inside-corner radius, inches f = gage thickness, inches
Table 5.6.
Dimensions of Flanged and Standard Dished
Heads
(Courtesy of Lukens Steel Company) Inside-corner Thickness (in.) Radius (in.)
t %6 ?4
icr 956 % l?is 1% 1x6 1%.
174
2% 2%
33js 3% 4% 435 416 5w 5% 6
G co
-
?ll’hI!! ’ +II - l I : I ’ ! I I I I
I 1I 1; ! I I 1I I ’ I
I
Z-
I
I
I
2 xi&~‘+llli1~ I+ I I I l ll$$ I
I
I I
I
/
90
Proportioning and Head Selection for Cylindrical Vessels with Formed Closures Table 5.7.
40
Dimensions of ASME Code Flanged and Dished Heads (Continued)
42
48
r
icr
r
40 k
2%
42 40 I
i I
I
I
I f 40 36 t
k I I I 1
I I G
2%
3 3% --I- - - . - -t- - - -1 --. . - - --I- - . - 1_ - - _-!-+-: - - 36 8%
40 42 _- +I-- - -
icr
icr
r
icr
r
48
3%
Y
3% a
60 k
3 3% - ----_ - --. - - t .-It: - - i- - 42 Wi-
84,
I I -I-$I-4%
481 -- -i I+ - -I-+ 48
I 9 54
k
I
"i
i 3% 3% - - . .-- - - - - - - - . - 9
JI 3% c3% - - - - _-- - - - 9
I A. -t-t__. 7 4’ 4h
-A -i54
JI 60 54 I
icr
r
r
78 78 78 72
5%
84 A
5% k
90 90
5% A
96 96 90 6
96
1
I
I /
I 1-1 - --L _- 4-72
I I 1 5% 5% - ,---. --. .-9
84 78 k
I
I 3 --I--o78
I
I I
I I t 5%
-5% - ----- -
I I I I
I I z.1 ---l--
9
,-~-~
I I
I
I I i96
r
4% A I
72 4
I
I
/
i 72 !
I -r --I-t -t z 60
i 4% -4% 3% -5% -5% -6 -6%
-7% -Wi
I
5% ~6 -------
I I
.I_ -+-
-l- v-
-!4-
_
-.
a I
102
I I
102 96
I
I
f
i
6%
1
84
_
I t 6% 6%
---A
90
9
-
-.-
I
I I
I I I t 66
9 114
r
I
66
I
icr
90
90
I
_
t
I I I $I
9
\
icr
l(
r
t
72
k I
+
icr
I i
I
4 c4%i - - - - - - T 54 9
r
I
I
I -t--. I --t- -
icr
I I I
1 v 66 60
*r
I
r
84
r
66
s
!
3
icr
I
48
I I
"i
9
78
icr
I
i
66
r
t 48 42
I
60
54
T
icr
r
108
k
6
I $
I I
6
I
I I t
108 102
I I --t - 96
-..-
k I I 0 LO2
’
-. 8
z --------*= 5c z-E+-0 0
WcQ-4 xx+-----------g
s WOCO-----
------oa
x
WC0 wf-----~-----~ x
4: A
w’- --------E
WC0 --------v----d x
i?
5-e ------*= 0
+-----------,G
15’
sir---------->~ 0 0
El
-g- -----+z x &- ---+g 0
Y
p-----------g
5
b’ 3
+---------- --,E
p- -----------z x
k-5
Proportioning and Head Selection for Cylindrical Vessels with Formed Closures
92
m 250 =5
Fig. 5.10.
Approximate 1955 cost of
forming per head for
B
ASME c o d e
flanged and dished heads if two to
I
four heads are purchased.
0
10
1 Note: Multiply costs by factor gwn in table 5.10. , I ,
20
30
40
50
: I
60
I /
70
80
90
100
I
110
120
130
140
15 '0
Head, outside diameter, inches
The weight of material may be calculated from the blank diameter, gage thickness, and density of steel (490 lb per cu ft). The cost of the steel may be obtained from Appendix C. Table 5.9 gives the cost of forming per head for heads having an outside diameter of up to 48 in. Figure 5.10 presents the cost of forming per head for larger-diameter heads. Table 5.9 and Fig. 5.10 are based on 1955 forming costs and include the quantity differential that applies when two to four heads are purchased. Table 5.10 gives the quantity-differential pricing factors for the purchase of other quantities of heads. If the straight flanges are machined for welding by the fabricator, Table 5.3 with appropriate quantity differenGals may be used for estimating the machining cost’ extras. If heads having straight flanges in excess of standard lengths
\I
/
are desired, an extra equal to 5y6 of list price fcr each additional !6-in. flange length is charged. To determine the total cost of the head at the mill, the cost for the steel blank is first determined. To this is added the forming cost, the machining cost, quantity differentials, and any extras for ordering flanges longer than standard. Elliptical Dished Heads Meeting ASME and API5.5d ASME Code Saecifications. The elliptical dished (ellip-
soidal) heads are used in preference to code flanged and dished heads formed with two radii for many vessels designed for pressures in the range of 100 psi and for most vessels designed for pressures over 200 psi. The elliptical dished heads are formed on dies in which the diametrical cross section is an ellipse. If the ratio of major to minor
-.----,_ - .---_-._
Common Typesof Formed Heods and Their Selection Table
5.8.
Typical Code
Stondord
Flanged
and
Recommended Max Straight Flange, in. 2 3 3% 4% 6 6 6 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8
Straight Dished
Flange
for
ASME
Heads
Table
5.9.
Code
Flonged
93
Typical Forming Costs as of 1955 for ASME and
Dished
Heads
Hoving
Diameters
up
to 48 Inches
Prices per Head for 2-4 Heads* Notes on Max Straight Flange 3” for 60” diam 3” for 60” diam 3” for 96” + 10” diam 3” for 126” diam 4” for 132” - 144” diam 335” for 156” diam 5” for 168” diam 3” for 180” diam and above 3” for 180” diam and above 3” for 180” diam and above 3” for 180” diam and above 3” for 180” diam and above 3” for 180” diam and above 3” for 180” diam and above 3” for 180” diam and above 3” for 180” diam and above 3” for 180” diam and above 3” for 180” diam and above 3” for 180” diam and above 3” for 180” diam and above 3” for 180” diam and above 3” for 180” diam and above 3” for 180” diam and above
Thickness s6- >i x6- 46 x6- $5 36 38: I 136 l$h Ia, 17/x 2q 2’; ;? 3 i 3
-1’4 - l!& --134 -2
12-18 9.61 10.23 11.16 12.40 15.50
20-24 10.54 10.85 12.40 13.18 17.05
23.40 35.10 54.60
30.23 40.95 60.45 87.75 117.00 136.50
Head, OD 26-30 32-36 36-42 48 .~ 11.63 12.40 15.50 17.82 12.40 13.95 1 7 . 0 5 20.15 13.95 16.28 19.38 23.25 15.50 17.05 21.70 24.80 20.15 22.48 26.35 31.00 33.15 46.80 66.30 92.63 117.00 136.50 165.75 210.60
37.05 52.65 72.15 97.50 121.88 146.25 177.45 220.35
* Zlultiply by factors in Table 5.10.
42.90 48.75 58.50 64.35 78.00 83.85 102.38 107.25 126.75 132.60 154.05 161.85 189.15 200.85 230.10 239.85 278.85
axis is 2: 1, the strength of the head is approximately equal to the strength of a seamless cylindrical shell having the corresponding inside and outside diameters. For this rea* son most manufacturers have standa d’ ed on elliptical dished heads having a 2 : 1 ratio of axis.&‘he inside depth -
i
Fig. 5.11. Brow
Horizontal storage tanks with elliptical closures on the right and a hemispherical closure on a vessel in the left foreground.
a C O.)
iCmriesyof C. F.
94
Proportioning and Head Selection for Cylindrical Vessels with Formed Closures Table
5.10.
Typical Quantity-differential Factors for Formed Heads
Table
Pricing
5.11. Standard Straight Flanges for Available ASME Code Elliptical Dished Heads
(Courtesy of Lukens Steel Company) Diameter Diameter Available Standard Gage Available Standard (in.) sf (in.) (in,) (in.) sf (in.)
Based on 1954 and 1956 Prices and on Fig. 5.10 and Table 5.9 Heads up to 48” OD and 48” ID for Gages over Heads up to ?4 “, and 48” OD and Heads over 48” ID for 48” OD and Gages up to 48” ID for 76” Inclusive All Gages Multiply by Multiply by 1954 1956 1954 1956 Quantity 1.32 2.20 1.05 2.20 1 head of a size 2 to 4 heads of a size 1.00 1.65 1.00 2.10 5 to 10 heads of a size 0.92 1.50 0.897 1.85 11 to 20 heads of a size 0.775 1.30 0.795 1 65 0.71 1.20 0.720 1.50 2 1 to 50 heads of a size 0.645 1.05 0.615 1.30 5 1 to 60 heads of a size 0.565 1.20 Over 60 heads of a size 0.58 0.95
546
5% w N N % 1
?‘iS
1% 1% 1% 1% 1% 1% 1% 2
of dish is half of the minor axis and is equal to one fourth of the inside diameter of the head.p The right side of Fig. 5.11 shows elliptical heads 1 We in. thick on the ends of butane-storage tanks 144 in. in diameter by 120 ft long. These tanks are rated at 100 lb per sq in. gage at 400’ F. Figure 5.7, detail e, shows a sketch of a cross section of an elliptical dished head. These heads are fabricated on the basis of using the inside diameter as the nominal diameter.
12- 36 1% 42 12- 66
2-2x 2-3 2-3
2% 2% %i
12-120 12-126 12-138 12-156 12-180 12-192 12-204 12-216 12-216 12-216 12-216 12-216 12-216
2-3x 2-335 2-3x 24 2-4 2-4 2-4s 2-434 241,-4x 2%-4x 2x6-4% 2%4% 2is6-4% 34%
bi 3% 3% 4 4% 4% 4% 5 5% 5% 5% 6
20-204 26-192 26-186 34-174 34-168 34-156 34-138 48-132 54-126 54-120 60-120 60-114 72- 96 72- 96 ‘i’2- 96 72- 96
2x4~ 3%-p% 4%4% 4% 4x-5 5X-5% 5%-6 6
6%-6M 6N-7 7?+-7% 7% 7%-8
w-8%
8%-9 9
Head diameters based on the inside diameter are available in increments of 2 in. from 12 to 42 in. and in increments of 6 in. from 42 to 216 in. Table 5.11 gives the standard straight flanges for various elliptical dished heads. The manufacturer’s catalog should he consulted for the maximum straight flange available at extra host. The volume
Blank diameter, inches
20
30
40
50
60
70
80 1
Fig.
5.12.
head
inside
Blank dlamltc.: diameter
for
versus various
t h i c k n e s s e s o f ASME elliptical
110 100 90 80 70 60 I
0 70 80 90 100 110 120 130 140 150 160 170 180 190 20:’ Blank diameter. inches
heads.
95
Common Types of Formed Heads and Their Selection
600
Fig.
5.13.
Approximate
1955
cost of forming par head for ASME two
coda elliptical heads if to
four
heads
are
pur-
chased.
Thickness 3’ /l-
I
I
I
Note: Multiply costs by factors in table
5.1~.
/
1
1
I i0
Head, inside diameter, inches
in cubic feet contained within the head, not including the straight-flange portion, is approximately --\ ___- - .- -., . _equal to:
V = 0.000076 di3 -----T_ - -..-. --..-.. -_I __ I
Table 5.12. ASME
Typical Forming Costs as of 1955 for
Code Elliptical Heads Having Diameters up to 48 Inches
(5.14)
where di = inside diameter of vessel, inches V = volume of elliptical dished head to straight flange, cubic feet For cost-estimating purposes the procedure presented in the previous section for code flanged and dished heads is followed; Fig. 5.12 is used for the blank diameter and Table 5.12 and Fig. 5.13 are used for the cost of forming per head for two to four heads. Table 5.10 may be used for determining other quantity diflerentials. 5.5e Hemispherical Heads. For a given thickness, hemispherical heads are the strongest of the formed heads. These heads can be used to resist approximately twice the pressure rating of an elliptical dished head or cylindrical shell of the same thickness and diameter. The degree of forming and accompanying costs are greater than for any of the heads previously described; also, the available sizes
Thickness
x N-- x6 KG-j6 x x 1
l>+lK 1%--l% 1%-1x 1x-2 2% 2% 2% 3
Prices per Head for 24 Head Lots* 48 12-18 20-24 26-30 32-36 3842
12.40 13.95 13.95 15.50 20.15 30.23 37.05 46.80 66.30 85.80
13.95 14.73 17.05 19.38 24.80 35.10 46.80 50.70 70.20
17.83 18.60 20.93 23.25 31.00 44.85 52.65 60.45 79.95
91.65
101.40
20.15 22.48 26.35 31.00 24.80 29.45 34.10 27.90 33.33 38.75 35.65 41.85 48.05 50.70 58.50 66.30 58.50 66.30 75.08 70.20 79.95 88.73 91.65 101.40 109.20
111.15
120.90
132.60
115.05 124.80 138.45 148.20 159.90 150.15 173.55 183.30 195.00 189.15 212.55 224.25 235.95 247.65 259.35 271.05
* Multiply by factors in Table 5.10.
PC
Proportioning
and
Head
Selection
for
Cylindrical
Vessels
with
Formed
Closures
the rolls on one end according to the angle of the cone. The smaller opening of the conical section must be large enough to accommodate the diameter of the bending roll, which usually is 4, 8, 10, or more inches in diameter. The greater the apex angle of the cone, the larger the smaller opening must be in order to accommodate the roll. Toriconical heads differ from simple conical heads in that they have a radius at the flanged end, as illustrated in detail g of Fig. 5.7. Toriconical heads may be formed from flat plates in the same manner as flanged and dished heads. They are more expensive t,han simple conical heads but are better suited for pressure-vessel applications because the localized stresses near the junction of the cone and the shell are more uniformly distributed in the toriconical section. Thus the concentrated localized stresses which would exist without the knuckle radius are greatly reduced. Spun toriconical heads are available from manufacturers in a variety of sizes ranging from 30 to 198 in. outside diameter (or inside diameter) in 6-in. increments for included angles of 90” and 120°, and in gages from M in. through 2 in. Included angles of 114’ can also be obtained in diameters of from 42 to 216 in. and, included angles of 140’ in diameters of from 66 to 240 in. Diameters of 240 in. are available in each of the above angles. Pressed segmental cones having included angles of 60”, 75“, and 90” are also available in a variety of sizes. 5.5g Some Oiher Types of Formed Heads. Three types of formed heads not previously described are shown in Fig. 5.15. The flanged and reverse dished head shown
Fig.
5.14.
3-tn.
straight
Flanged hemispherical head 168 in. in inside diameter with flange
weighing
16,500
lb
and
constructed
from
seven
formed sections. (Courtesy of Lukens Steel Company.)
formed from single plates are more limited. One-piece spun hemispherical heads available from one fabricator are given in Table 5.13. A great variety of hemisphericalhead diameters and gages are available in segmental form and can be field or shop welded. Figure 5.14 shows a hemispherical head of a 168-in. inside diameter fabricated from six plates dished to shape over ‘an 84-in. radius former and then welded together with a dished head at the end to complete the hemisphere. 5.5f Conical and Toriconical Heads. Conical heads are widely used as bottom heads for a variety of process equipment such as evaporators, spray driers, crystallizers, and settling tanks. The particular advantage of the use of conical bottoms lies in the accumulation and removal of solids from such equipment. Cones having an angle at the apex of 60” are commonly used for the removal of solids. Greater angles may cause accumulation of solids as a result of the frictional resistance between the solids and the inside cone surface. Another common application of conical sections is for changing the diameter of cylindrical shells; this is often necessary in some fractionating-column designs. Figure 5.3 illustrates such an application. Conical sections can be formed on the rolling equipment used to roll cylindrical shells. This is done by spreading
Table 5.13.
Dimensions of Available One-piece Hemispherical Heads
(Courtesy of Lukens Steel Company) ID (in.) Thickness sj (in.) Lightest to Heaviest 12 o-2 % 1% o-2 Wi6 1% w o-2 36 1% 14% 1534 o-2 96 1% o-2 l@%t5 N 1% o-2 17% 1% N 1835 o-2 N 1% 23 o-2 N 1% 24 1% O-2% N 04% .W% N 1% o-2 35 28% M 1% 30 ; 2 0-254 34 21,s @-wi 35 34 3 o-3 36 5% 3 o-3 38 % 3 o-3 ; 33 o-3 41% 42 o-3 94 3 o-3 47% 48 36 3 o-3 5034 % 3 o-3 54 o-3 34 3% 60 94 3 ;,4 O-3% 64% % 3% O-3% 72 0-3x % 3% 94 952 31,g 0-3x
Spun
Common Types of Formed Heads and
in detail a of Fig. 5.15 is often used on the ends of vessels intended to face against a plane surface such as the bottom of steel vessels resting on concrete slabs. In this respect this type of head has the advantage of a bearing surface in one plane and also the additional strength resulting from dishing. The radius of dish, R, is equal to .the outside diameter of the vessel, and the heads are available in Z-in. increments from 18 through 24 in. outside diameter and in 6-in. increments from 24 through 132 in. outside diameter. A head 144 in. in diameter also is available. The gages vary from x~ in. through 1 in. Dished-only heads such as shown in detail b of Fig. 5.15 can be used as the center piece of built-up hemispherical heads of large sizes andmay also beusedin specially designed equipment where flanges are undesirable. Such heads are available in a large number of sizes, from 12 through 24 in. in increments of 2 in. on the outside diameter, from 24 through 144 in. in increments of 6 in., and from 144 through 180 in. in increments of 12 in. Gages range from xs through 3 in. Flared and dished heads, shown in detail c of Fig. 5.15, may be used for cover plates of kettles and hoppers and in specially designed equipment. Such heads are available in sizes from 18 through 132 in. inside diameter in increments of 6 in. The corresponding outside diameter is equal to the inside diameter plus 6 in. The radius of dish is
d
I
equal to the inside diameter. x6 in. through 2 in.
Their
(cl flared and dished
Steel Co.)
PROBLEMS
1. A horizontal vessel 10 ft in diameter and 60 ft long is to be fabricated from ASTM-A-283, Grade C plate 54 in. thick by code welding. Determine the optimum number of plates for the shell and the plate width. (See Appendix C, item 4, section g.) 2. Heads for six vessels approximately 48 in. in diameter for 190 psi, service are required. Determine the cost of 12 elliptical dished heads 48 in. in inside diameter and 44 in. thick, and compare it with the cost of 12 torispherical heads 48 in. in outside diameter and s in. thick. (Roth heads are suitable for 190 psi service.) Steel is to be ASTM-A-285, Grade C, and heads are to be shipped with standard straight flanges.
97
Gages are available from
Fig. 5.15. Some other types of formed heads: (a) dished, (b) dished only,
Selection
(105).
flanged and reverse (Courtesy of
Lukens
C H A P T E R
STRESS CONSIDERATIONS IN THE SELECTION OF FLAT-PLATE AND CONICAL CLOSURES FOR CYLINDRICAL VESSELS
6.1 RELATIONSHIPS BASED ON THE THEORY OF ELASTICITY
0
n the fabrication of process equipment flat plates or cones may he used as closures for cylindrical vessels because such closures are easily formed with conventional shop equipment, but their use usually is limited to low-pressure service or to closures for small-diameter vessels. Flat plates are often used as closures for hand holes, manholes, and so on. The sharp discontinuity in shape existing at the junction of a cylindrical vessel and either a flat plate or a conical closure results in localized stress concentrations at the junction. In low-pressure service, where the magnitude of these stresses is low, they often may be disregarded. However, for adequate evaluation of a design, a knowledge of the magnitude of these stresses is essential. The nature of the stress concentrations is complex in that bending moments, shear, and stress reversals must be considered in addition to the membrane stresses resulting from internal pressure. The use of a flat plate as a closure often results in the plate’s being considerably thicker than the cylinder to which it is attached. The difference between the flexibility of the flat plate and of the cylinder results in the two parts of the vessel attempting to deform radially and angularly at different rates under the influence of internal pressure. This movement is prevented by the rigid juncture of the two elements and results in shear and flexural stresses which may be quite severe. Similar but less severe localized stresses result when conical heads are used as closures. In addition to the juncture stresses, bending stresses in the central portion of the flat closure may also be critical.
6.1 a Stress-Strain Relationships. By definition the modulus of elasticity, E, is the slope of the straight line or elastic portion of the “stress-strain” diagram (see Fig. 2.1) or by Eq. 2.3: EJ
e
(2.3)
where j = stress, pounds per square inch e = unit strain, inches per inch E = modulus of elasticity, pounds per square inch Therefore, if the elastic limit is not exceeded, elastic deformation occurs under induced stresses. The amount of deformation or “strain” is simply related to the induced stress by the above relation. Thus
When a specified segment of metal is loaded in one direction only, with resulting induced stress and corresponding strain, strain is also induced in a direction or directions at. right angles to the induced stress. For example, a tensiletest specimen elongates under tensile load and reduces its diameter by lateral contraction. Experiments have proved that such axial elongation is related to the corresponding lateral contraction. The ratio of these two deformations is a constant within the elastic limit and is known as Poisson’s ratio. This ratio may be 98
Relationships
EC -=p c
(6.2)
wh& eC = unit lateral contraction c3 = unit axial elongation ~1 = Poisson’s ratio, a constant depending upon the material (0.3 for structural steel) This relationship may also be used to calculate the lateral expansion resulting from axial compression of a material. In the case of a closed cylindrical vessel containing a pressure, the shell may be considered to be subjected to three forces. One force results from the pressure pushing against the heads. This load on the heads is transmitted to the shell in an axial direction and therefore results in a longitudinal tensile force and tensile stress being set up in the shell. A circular or “hoop“ stress is also induced in the shell by the contained pressure acting against the circular shell. A third stress exists in the radial direction, which may be disregarded in thin-walled vessels. The two principal tensile stresses act at right angles to each other producing a two-dimensional stress condition. The resulting elongation in one of these directions will depend not only upon the tensile stress in this direction but also upon the stress in the perpendicular direction. If one refers to one direction as z and the other as y, the unit elongation by Eq. 6.1 in the x axis direction due to the tensile stress, ji, will be: fi E 2=-
E
f E El= 2
E
the
Theory
(from Eq. 6.2)
cc=/.-.f!l
E
If both stresses ji and j, are acting simultaneously, the net unit elongation, ~~2, in the x direction will be (the subscript 2 refers to the net stress or strain for the biaxial loading condition) :
= .f,-E - p fz-E
.f,z =
Eq,2
E
(6.41
or
=
EE,Z
99
= fi - pjg
(6.4a)
(6.5)
= .f, - ~jz
(6.5aj
Substituting in Eq. 6.4, we obtain:
Expanding we obtain: or
* cz2 = I2 - pry2
- /.p f-?
E
E
cz2E = f.z - 13,2E - cc2fz
Factoring we obtain: cz2E.=.fi(l
- p2) -
peZ12E
Solving for ji, ,we obtain: fz=
jx =
+ w,zE IP2
ez2.E
(%2 + wu2w
1 - p2
(6.6)
In like manner it may be shown that Y
=
(~2
+ wzz)E
1 - /.Ls
(6.7)
Equations 6.6 and 6.7 are convenient relationships giving the stresses in two direct,ions perpendicular to each other in terms of the respective strains resulting from the stresses, Poisson’s ratio, and the modulus of elasticity of the material in question. It is assumed that. the material is homogeneous and the deformations are all within the elastic limit. 6.lb General Bending Relationships. It has previously been shown by Fig. 2.3 and by Eq. 2.6 that the unit elongation in a deflected beam is equal to:
Substituting for l/r as given by Eq. 2.13, we obtain:
d2y
c2AL&! z
fi2
Elasticity
Equation 6.4 can be combined with Eq. 6.5 to give a pair of useful equations in which the stresses ji: and jr, are functions of the strains e22 and ~2 as follows. Noting that f, E in Eq. 6.4 may be substituted for by its equivalent from f2 = Ey2 + p fi Eq. 6.5, we find that from Eq. 6.5 -. E E
j
(6.lb)
(for f J
E
of
or
or
The accompanying contraction, eC, in the x direction will equal PQ, (because t, is equal to the ey resulting from j2/) or
6 = Pqj
e2/2
(6.la)
The tensile stress in the y direction, jr,, will produce an elongation, e2/, in the y direction and a lateral contraction, 6, in the x direction, as indicated previously.
Therefore
on
And the corresponding net unit elongation, ~2, in the y direction will be:
expressed as follows:
E, = e2/
Based
%=Ys
(6.8)
An elemental strip of a plate under deflection can be compared to a simple beam, and Eq. 6.8 can be used to
Selection
100
of
Flat-Flate
and
Conical
Closures
for
Cylindrical
express the unit strain in terms of the radius of eur~aturr of the deflected plate. In reference to Fig. 2.3, the unit strain of an elemental strip in the z direction is assumed to be negligible. Rewriting Eqs. 6.4, 6.5, and 6.6 for the element shown in Fig. 2.3, we obtain:
fi E,2 = fi - - /.E E
and (6.18)
(6.9) 6.1~
(6.11) hut
Vessels
Bending
Relationships
in
a
Circular
Flat
Plate.
Referring to Fig. 2.3 and considering it to represent a strip in a circular plate, we find that the dishing of a circular flat plate under uniform pressure will result, in curvatures in both the z and z directions. In reference to Eq. 2.6 the unit strains map he written:
if
E,2
= 0
then jz+ P2
(6.12) Substituting for t.he straius in Eqs. 6.6 autl 6.i, we find that
Substituting Eq. 6.8 for E, in Eq. 6.12, we obtain:
j
(6.13) For the strip of the plate in the J: direction having thirkness, t, and unit width in the z direction, Eq. 2.7 ma! be modified to:
M=
/
+;f j-g dA
(6.14)
Substituting forji by use of Eq. 6.13, we obt.ai!::
z
=:
(w
+
wz2W
=
-3.
1 - /.l2
(6.19)
(6.20) Figure 2.3 also shows that the strain and corras~on&ng stress is zero at the midplane and is at a maximum at the outer fibers. The effect of the stress on either side of the midplane is to produce a couple which may be expressed as a bending moment. Equation 2.7 may be applieci k noth the x and z directions to determine the bending moments from the combined stresses as follows (see pa@ 41 of Peferonce 107) :
But by Eq. 2.8 I 5=
+t/2 I -tra
y2 dA
Il.st, I _ o.st~ jzy dy dx = M, d3:
:;. 8’
and for a unit skin
(6.22)
Substituting Eqs. 6.19 and 6.20 for jJ and j, in Eqs. 5.21 and 6.22, respectively, integrating, and substituting l”;;q. 6.15 (107) we obtain:
t’ ; s 12
Therefore M = _ ~-E t3 d2y 1 - p2 12 dx2 Let
D=
Et3 12(1 - p2)
(6.15)
where D = flexural rigidity; then
h uniformly loaded circular plate will dish in a spherical manner; t,herefore r, = r, = rzz, and M, = M, = MS+ Therefore, Eqs. 6.23 and 6.24 reduce to: 1 MZZ ___-= rxz W + PCL)
A comparison of Eq. 6.16 and Eq. 2.14 shows that D, the flexural rigidity of a plate, is equivalent to the quantity EZ for beams. It therefore follows that the Eqs. 2.15 and 2.16 for beams may be modified for plates by the sub-
(6.25)
Referring to Eq. 2.13 and letting the radial distanre r. to x, we obtain:
equal
_1 = 3
rx
(6.26)
Relationships Based on the Theory of Elasticity
101
Referring 1.0 Pig. 6.1, by similar triangles, we obtain: rz
dl
01 1
rz
dj P dr
(6.27)
Substituting Eqs. 6.26 and 6.27 into Eqs. 6.23 and 6.21, we obtain: (6.28)
(6.29)
/‘wIltIre Mz is the bending moment prr unit lengt,h along the
? circumferential section, and M, is the bending moment pe1 i
unit, length along the diametral section of circular plate. In reference to Fig. 6.2, a circ*umferential section of a flat rover plate having a uniformly distributed load, p, is designated as element ah-d. A summation of the forces resulting from the bending moment may be taken about the sides ad and bc. Arc length acl = r d+, and arc length hc =
(r + dr) d+ ad
Fig. 6.1.
Deflection in a dished circular plate.
If t.he small differences in shear on these two sides are disregarded, t,hesetwo forces result in a couple in the x-y plane equal to - Qr d+ d r
couple = M,r d+
(6.30)
M, + f$ d r (r + dr) dr$ I
(6.31)
M, dr d+
Summing up the couples in the x-y plane wit.11 proper regard for sign, we find that (107) M, + $< d r
The sides ab and cd have couples which are each equal to Mz dr, and they have a resultant in the z-y plane equal to
(6.34)
I
(r + dr) d+ - M,r d+ - &r d+ d r -M,
(6.32)
d r d+ = 0 (6.35)
Disregarding small high-order quantities, we find that
The symmetry of the element results in no shear on the sides a b and dc. The shear per unit length, &, times the length of the arc ad gives tbe tot.al shear on the arc and is equal to &r d+. The tobl shear on the side bc is correspondingly equal to
Subst.itut.ingEqs. 6.28 and 6.29 into Ey. 6.36, we obtain:
(6.33)
(6.37)
Fig. 6.2. circular
M, + $f r - M, - Qr = 0
Bending moments in a flat
plate.
IY L,
I
Y
View A - A
(6.36)
102
Selection
of
Flat-plate
and
Conical
Closures
for
Cylindrical
Vessels
Combining terms and factoring gives:
Equation 6.37 may be written as (107) : (6.38)
(6.48)
The shear per unit length of circumference, &, at any radial distance, P, in a uniformly loaded circular plate is:
The maximum defection will occur at the center of the plate, where r = 0.
force
pm-2
pr
Q=-===circumference 2ar 2
(6.39)
Substituting Eq. 6.39 into Eq. 6.38 gives: (6.40)
(6.49) To determine the stresses in a flat cover plate, Eqs. 6.28 and 6.29 are used with the substitution of Eq. 6.48 for y, by which
Integrating once gives:
f (1 + P) - r2(3 + P)
Id dy --(r-)=g+Co r dr dr
(6.41)
and
I
$ (1 + j.~) - r2(1 + 3~)
Multiplying both sides of the equation by r and integrating again gives:
(6.50)
(6.51)
I At the edge of the plate, where r = d/2,
(6.42)
(6.52;
Dividing by r and integrating again gives: y = f$ + 7 + Cl log r + c2
(6.53)
(6.43) At the center of the plate, where r = 0,
UNIFORMLY LOADED FLAT PLATE WITH THE EDGES CLAMPED. The constants of integration, CO, Cr, and Cs must be evaluated from the boundary conditions at the edge of the plate. For a circular plate with the edge clamped, the slope of the plate at the center and at the periphery is zero. Therefore,
and
($)i=o=o= [f&+T+tqrzo
1
(6.44)
(6.45)
c 0 =-pd2 320
(6.46)
(6.47)
y P_ pr4 d2r2 + pd4 640 1 . 2 8 0 10240
f=!$!f where z for a unit-width rectangular strip of the flat plate located on the center line is equal to th2/6. Therefore,
or (655a)
UNIFORMLY LOADED FLAT CIRCULAR PLATE SIMPLY SUPAT THE EDGE . The condition of the circular plate with a clamped edge differs from the condition of a circular plate freely supported by the bending moment at the edge of the plate as given by Eq. 6.52. If this bending moment is removed, spherical dishing will result. The radius of curvature, rZ, of a spherical dished plate may be determined by reference to Fig. 4.3 and Eq. 4.1. PORTED
Applying the condition that y = 0 at r = d/2 (for a clamped plate) and solving for C2 gives:
Substituting for C’s in Eq. 6.46 gives:
The maximum moment is at the edge, as indicated by the comparison of Eqs. 6.52, 6.53, and 6.54. The maximum stress may be determined by substituting Eq. 6.52 into Eq. 2.10.
th = d db%)(P/f)
Substituting for CO and Cr in Eq. 6.43, gives: d2r2 1280+c2
(6.54)
r=(d/2)
From Eq. 6.44, Cr = 0 and by substituting for Cl in Eq. 6.45, CO is evaluated. Therefore,
y P_ pr4 640
M, = M, = (1 +6;)Pd2
A2 = BD - B2
(4.i;
For small values of B, the term B2 may be disregarded. Therefore A2 = BD (approximately). Using the notation for circular plates, A = r, B = y, D = 2r,, gives: r2 S 2yr,
Relationships
therefore Y
d2 - 8r,
max
(6.56)
To determine the deflection at any point, r,
Based
on
the
Theory
of
Elasticity
103
CONCENTRIC L OAD ON A F LAT C IRCULAR P LATE. A concentric load on a flat circular plate creates a bending moment, Ml, and a deflection, y, which dishes the plate. Equation 6.27 gives the radius of curvature of a spherically dished cylindrical plate in terms of r. 1 dy _=
r dr
Rewriting we obtain:
Substituting for l/r, by Eq. 6.25, we obtain:
idI
(6.57)
rz
rz
*
Equation 6.58 is the deflection equation for pure bending of a circular plate with a bending moment at the edge equal and opposite to the bending moment of a uniformly loaded circular plate with clamped edges. If the deflections for these two conditions are combined by superposition, the deflection of a uniformly loaded circular plate simply supported at the edges will be obtained. Thus, Eq. 6.48 may be combined with Eq. 6.58 to give (107) :
I‘,
--p E-g Y= (A, ) [(z) (t? - r2]
(6.59)
The maximum deflection occurs at r = 0; therefore Ymax
= ~
f
(6.59a)
(1 + p) - r2(3 + p )
therefore
1
+ ‘g
(6.60)
Mz=~[C3+pl~-r2)] and Mz = 5 f (1 + IL) - r2(1 + 3~) [
therefore
: (3 + p ) - r2(1 + 3~)
1 1
+ ‘g
(6.61)
The maximum bending moment occurs at the center of the circular flat plate when r = 0; therefore M,
= M,
= ;
$
(3 + /.I)
0
(6.61a)
6M th
1 MZ -= rz Wl + PI Substituting gives: (6.64) where y = deflection due to concentric load on flat circular plate MI = moment caused by concentric load on flat circular plate
strip of the shell or radial strip of the flat-plate closure is similar to the deflection of a beam on an elastic foundation. Therefore, the relationships for such a beam will be discussed. The deflection of such a beam at any point in the beam will be proportional to the reacting force on the beam at the point in question. Such a beam under a point load will deflect immediately under the load because the supporting foundation is elastic. The stiffness of the beam will transfer a portion of the load to either side of the force; this will resul in a smaller elastic deflection which is a function of the resiskante of the foundation and the distance from the point of load application. Let 5 = horizontal distance along the beam y = deflection of the beam w = resistance of the foundation per unit length at any point cl = constant depending upon the stiffness of the beam and the resistance of the foundation E = modulus of elasticity of the beam Z = moment of inertia at the beam M = bending moment at point z For a beam on an elastic foundation, the deflections of the unloaded portion of the beam at any distance, 2, from the point of load application is given by:
The corresponding maximum stress is: = 7=
But by Eq. 6.25,
6.ld Bending Relationships for a Beam on an Elastic Foundation. The deflection of an elemental longitudinal
The bending moments for a uniformly loaded flat circular plate simply supported may be obtained by adding the edge bending moment, pd2/32, to Eqs. 6.50 and 6.51 to give :
fmax(r=O)
r dr
(6.63)
(6.58)
.,.
sr
therefore
By Eq. 6.52
’ *
d/2
1 -=-
dy = /;2 rd r
6pd2(3 + P>
yz
(16)(4)th2
(6.62)
=
Cl%
(6.65)
The resistance w of the foundation per unit length at any
Selection
of Flat-plate
and Conical Cfosures for Cylindrical Vessels
Qo
the load approaches zero as the distance from the pain‘. of load application increases. This reyuires that the conTherefore stants A and B be equal to zero.
Qo
y = ewPx(C cos /3x + F sin 82)
d’
The constants C and F must be evaluated for the particular conditions of the beam under consideration. In the following section these constants are evaluated for a cylindrical shell with a flat-plate closure.
--b-=4
6.2
Fig. 6.3.
Forces and moment at junction of fiat cover plate and cylindrical
shell (108, 109).
p
where
= internal pressure, pounds per square inch gage
lh = thickness of flat cover plate, inches 1, = thickness of cylindrical shell, inches Q. = shear force at junction per unit length of circumference, pounds per inch No = tensile force at junction per unit length of circumference, pounds per inch A&, = bending moment at junction per unit length of circumterence, inch pounds per inch d = diameter of cover plate and shell, inches x = longitudinal distance along shell from junction, inches yl = deflection of shell at junction, inches ys = deflection of flat plate at junction, inches I = radial distance along flat plate, inches NI = head radial tensile force per unit length of circumference, pounds per inch MI = axial head bending moment per unit length of circumference
zr, can be expressed in I erms of (1~ (Ir, bhe slope of this deflection curve. poillt,
(2.16)
STRESSES IN CYLINDRICAL VESSELS WITH FLAT-PLATE. CLOSURES
In reference to Fig. 6.3, the internal pressure, p, acts upon the flat cover of t.hickness,lh, and upon the cylindrical shell of thickness, t,, to produce the forces QO and No and the moment MO. The force Qo is the shear force per unit length of circumference, which acts to restrain the shell from expanding and separating circumferentially from the flat cover plate. The force ‘lie is the axial tensile force per unit length of circumference resulting from the pressure load on the flat cover plate, which acts to separate the cover plate axially from the shell. The result of these forces is the bending moment MO. The relations, based upon the theory of elasticity, between QO and MO can be derived. A longitudinal strip of the shell in the neighborhood of the junction which is bent inward is selected for analysis. The force causing this deformation can be considered to be an inward radial shear force acting on the end of the strip. This force is resisted by the bending forces set up in the strip and by the compressive hoop stress opposing a tendency for the shell circumference to decrease. The total resistance to this tendency to deform inward results in radial shear forces, longitudinal bending stress, and circumferential compressive stresses. 6.20 Bending in the Shell. The deflection curve of an elemental longitudinal strip of a shell resulting from the forces and moments in Fig. 6.3 is comparable to that of a beam on an elastic foundation. The general solution of the eyuation for the deflection curve for such a beam is given by Eq. 6.69.
Substituting for u, in Eq. 2.16 by means of Eq. 6.65 and reii rranging gives :
d;y
(6.69)
_
Y
dx” - - clEI
(6.66)
y = eppZ(C cos /3x + F sin /3x)
(6.69)
The two constants C and F can be determined from the conditions at the loaded end of the elemental strip. From Ey. 6.16
lett.ing the yuantitb- l;‘Elc, = -f/Y’ gives: M” = (Mz)x:,o = --I~1
(6.70) ZCO
(6.67) :~nd from
Ey. 6.17
Equation 6.67 hds the following solution (107): .y = c@(A cos px + B sin PC) + Cpz(C cos /3r + F sin @r) (6.68) The arbitrary constants must be evaluated from the known conditions at certain points along the element. The deflection,’ y, of the beam at points greatly removed from
Qo where
=
(Qr1.m
= ($$)xzo = -Dl (>z=o
D1 =
(6.71)
El,3 12(1 - /.42)
By subst,ituting Eq. 6.69 for .v in Eqs. 6.70 and 6.71 and
Stresses
differentiat,ing,
the constants C and F are determined.
(6.73) Therefore
,
y = 2g [pMo(sin
/%z - (‘OS /3x) - Qo (‘OS pz] (6.74)
,
.
The maximum deflection due to bending nccurs at the end of the shell at the junction with the head. Therefore
(y)z=o = - & W~fo + Qo)
(6.75)
in
Cylindrical
Vessels
with
Flat-plate
Closures
105
the positive s direction taken away- from the junction. The strain due to bending (Eq. 6.75) and the strain due t,o pressure-stress considerations (Eq. 6.79) are in the same direcCon and must have the same sign. The slope of the deflection curve at CC = 0 resulting from the combined effects of bending and pressure stresses is obtained by differentiation of Eq. 6.80. This equation is identical with Eq. 6.76. To eva1uat.e t,he constant /3 in Eq. 6.80 reference is made to Fig. 6.4. Detail a of t,his figure shows a cross-sectional view of a longitudinal elemental strip of a shell of radius r. The width of the element is h inches. The curvat.ure of the shell results in the forces .f.J, on each side of t,he element react,ing less t,han 180” apart,. This results in a component, w, radially inward, which is normal t,o the surface. (See 1 detail h of Fig. 6.4.)
1
The slope of this deflection curve at. this point is equal to:
& z2 x=0 = L 2P2D1 0
@PM0
+
Qo)
(6.76)
The relationships as given in Eqs. 6.75 and 6.76 are limited to the bending resulting from the reaction at t,he junction of the shell and the closure. The shell will also deform as a result of the effect. of longit,udinal and circumferential stresses from internal pressures. These two stresses as determined by Eqs. 3.13 and 3.14 may be combined by the use Eq. 6.5 to give the radial deformation as follows:
Substituting Eqs. 3.14 and 3.13 into Eq. 6.5 gives:
pd pd Ey2 = 2t,E - Cc St,E
:
For small auples sin (0 7) = h/21+, or
For R Iongitudin~l
*trip of unit widt.h b = 1; t.herefore
where u’ = normal force. pounds per linear inch By- Eqs. h.la
Therefore
anti
6.78
’
(6.77)
(6.82)
To convert from the unit radial deformation as given by Eq. 6.77 to total deformat,ion it is necessary t.o multiply Eq. 6.77 by P.
(6.83)
(6.78)
pd2 Yp = ~ 4&E
or
F---b---l
/.vd” 8&E
:t, (6.79)
t
912
where yP = total strain due to pressure The deformation as given by Eqs. 6.75 and 6.79 are directly additive (in accordance with the principle of superposition). Therefnre ?‘oombined
1
=
(6.80)
The second term is negative because the location of the .r-y axis was originally taken at the center line of the shell wall with the positive .y direr&n radially outward and
fd Fig.
6.4.
pressure.
Circumferential
(b) stresses
in
cylindrical
shell
under
internal
106
Selection
of
Flat-plate
and
Conical
Closures
for
Substituting Eq. 6.83 into Eq. 6.81 gives: -4t,Ey W=d2
(6.84)
Substituting for w in Eq. 6.18 gives:
Cylindrical
Vessels
6.2b Bending in a Flat-plate Closure. The flat-plate closure is considered to deflect as a uniformly loaded circular plate simply supported at its edge with a superimposed concentric load from the reaction at the junction with the shell. Superimposing the deflections by means of Eqs. 6.59 and 6.64 gives Eq. 6.93
-p d” -
Substituting for D by Eq. 6.15 and rearranging gives:
$q12(yp2)] +y = 0
y2 =
(:,,,
g
)[G::)(:) -r2] +
(6.85)
(6.93)
where
Or
D2
~+4[!zL&)]y=o by Eq. 6.67 $+4/34y
= Eth3 12(1 - $)
Differentiating Eq. 6.93 with respect to P and evaluating at r = d/2 gives:
= 0
therefore (6.86) A stress intensification factor for the stress caused by the deflection y is obtained by dividing Eq. 6.82 by Eq. 3.14.
(3r=d,2 = - [ ,,,a:‘: p) - 2D2;‘: ,)]
(““)
Inspection of Fig. 6.3 shows that (dyl/dx) = (dyz/dr). Substituting for (dyz/dr) in Eq. 6.94 and multiplying through by Eq. 6.89 gives:
(6.87) Equation 6.87 can be expressed in terms of the moment and shear by substituting Eq. 6.80 for y.
. . . -pd and by 2t,3p2 Drvidmg through Eq. 6.95 by 21, ~ gives : th2
(6.96)
(6.88)
Since by Eqs. 6.15 and 6.86
(6.89)
fl4& = $?
substituting Eq. 6.89 into Eq. 6.88 results in a dimensionless relationship.
25 fh o o p
-
2QoP -
2MoP2 4-aYl P
d2p
(6.90)
P
Another dimensionless equation in terms of the bending moment, MO, and the shear, Qs, can be derived from Eq. 6.76. Multiplying Eq. 6.76 by Eq. 6.89 gives: = P4D1
hl WMo +
Therefore
Et, dyl -(->x p2d2 d
=
z=.
2PMo
+ Qo
1
(6.98)
Substituting Eq. 6.98 for Ml in Eq. 6.96 gives: p2 d2t, B24Qo z=o = - 640 + P)th + 4p(l + EL) B2tsMo
2P(l
(6.99)
+ P)th
Substituting Eq. 6.86 for /3 and rearranging gives:
(6.100)
gives:
-PMo
Qo =--pd
Ml = ‘A - M 0 2
(6.91)
2
Dividing Eq. 6.91 by (-pd/2t,)
Qo)
The shear force at the junction QO results in a radial tensile force, Nr, in the cover plate. If Ni is taken at the midplane of the cover plate, a lever arm equal to &/2 exists between these forces. This results in a bending moment, Ml, which may be evaluated by the summation of moments. (6.97) NI = -Qo and
2pd
(6.92)
(6.101)
Stresses
where br, bs, and bB are the respective coefficients in the parenthesis of Eq. 6.100. (See Table 6.1 for tabulated values.) Inspection of Fig. 6.3 shows that the radial stress in the flat plate is equal to Nl/th and the unit radial strain is equal to 6/r. By elastic theory and Eq. 6.4, 6 -=e,=
r
Cylindrical
Vessels
with
Flat-plate
107
Closures
Multiplying through Eq. 6.106 by (th/&) and rearranging gives:
(6.107)
+ [ 3d(;2; “1
(6.4)
&&
E
in
E
Ethyl
(See Reference 32, p. 121.) Substituting dy/dr and d2y/dr2 by differentiating Eq. 6.93 gives:
-= d2p
MO
a1 - + pd2
Qo
u2 - + pd
(6.108)
a 3
where al, ~22, and u3 are the respective coefficients given in the brackets of Eq. 6.107. (See Table 6.1 for tabulated values.) 6.2C Combination of Relationships in Dimensionless Groups. Watts and Lang (108) combined the relationships
for bending and shear in the shell and cover plate in the form of dimensionless groups as given by Eqs. 6.101 and 6.108. The advantage of this procedure is to give genera1 relationships independent of the system of units. The coefficients or “influence numbers” group the variables. describing the geometry of the vessel. The equations for the shell may be put in the same form as Eqs. 6.101 and 6.108. Multiplying 6.90 by (th/4ts) and rearranging gives:
But
and
therefore
~=[-yy$)]!$
Substituting for jr and jC in Eq. 6.4 gives:
a+-p)i$P
(6.102)
The strain 6 is measured at the mid-plane of the plate, and the displacement at the junction will be: (6.103) At the junction the displacement of the shell must equal the displacement of the head. Therefore y1= -61
(6.104)
= -6
By using Eq. 6.82 and substituting for yl, we obtain:
+[?(?)I$+
Ethyl
MO
[2(q)]
(6.109)
Qo pd
(6.110)
-=ur-++5-+ufj
pd2
pd2
where ~4, ~5, and a6 are the respective coefficients given in the brackets of Eq. 6.109. (See Tables 6.2 through 6.9 for tabulated values.) Equation 6.92 is multiplied by the ratio z 0 L rearranged to give:
2
and is
~(~)z=o = [-(W ($I$
,J
+ [ +“]$+O ( 6 . 1 1 1 )
(6.112) 64DI;;3+e) 2
+
)I
-Mid2D2U +
P)
(6.105)
Substituting Eq. 6.98 for M1 and Eq. 6.15 for Dz and dividing through by (pd/2t,) gives: 3tsd(l - cc) Eta1 - = --1dl - p)Qor 32tf&2 thpd2 d2p
+ x3(1 - P)QO _ x4 - PL)MO P dth2 2P dth
where bq, bg, and bs are the respective coefficients given in Eq. 6.111. (Note that be = 0.) (See Tables 6.2 through 6.9 for tabulated values.) Equation 6.110 may be equated to Eq. 6.108 to give: al $ + u2 p% + u3 = u4 p% +
a 5 p$ + a6
(6.113)
Equation 6.101 may be equated to Eq. 6.112 to give: (6.106)
b1$+b2$+b3
= bdp%+ba$+be
(6.114)
Selection
108
of
Fiat-plate
and
Conical
Closures
for
Cylindrical
Vessels Table 6.4.
Or (a3
M,, = pd2 (a4
- ad@5
- al)@5
- h2)
- h2)
- (a5 - a2)h3
- (a5 - u2)vh
--- 1
- h)
(6.115)
and Q. = pd
(~4 - adba - (~3 - ad(ba ~____~-~ (~4
- ad&
- bd
1
- bd
- (a5 - az)(h - h)
(6.116)
Table 6.1.
Coefficients for Cylinder, /h/l,
= 1.0
(Extracted from Transactions of the ASMK with Permissilm of the Publisher, the Amrrican Society of Mechanical Engineers, 29 West 39th St., New York, N. Y.) (108)
Coefficients for Flat Head (108)
d/t.
a5
a4
4.00 -6.6091 10.00 -16.5227 20.00 -33.0454 30.00 -49.5681 40.00 -66.0908 80.00 -132.1817 1 0 0 . 0 0 -165.2271 300.00 -495.6813 500.00 -826.1356
a6
-1.8178 -0.2125 -7.87-13 - 0 . 2 1 2 5 -4.0618 -0.2125 -4.9784 -0.2125 --5.7185 - 0 . 2 1 2 5 -8.1296 -0.2125 -9.0892 -0.2125 -15.7229 -0..2125 -?0.3”41 - 0 . 2 1 2 5
64
h:,
-3.6357 -5.7485 -8.1296 -9.9567 -11.4970 -16.2592 -18.1784 -31.4859 -40.6481
(Extracted from Transactions of the ASME with Permission of the Publisher, the American Society of Mechanical Engineers, 29 West 39th St., New York, N. Y.) Wlh
(1 I
,I 2
02
61
b.0 10 n o to.00 RO.00 100.00 300.00 500.00
-8.4000 -21 0000 -84.0000 - 168.0000 -2lo.nnnn -630,OOOO -1n50.0000
+I 4000 +I 4000 +I.4000 +I.4000 +1,4Onn f1.4000 +1.4nnO
+0.2625 +0.6563 f2.6250 +5 2 5 0 0 +6 5 6 2 5 i-lY.6875 +32.815
+ 5.0839 C12.7098 f50.83Yl +101.6782 f12i.0978 f381 2 9 3 3 +635.4889
hr
bs = 0
hr
-0.6355 -K158Y - 0 . 6 3 5 5 -0.3972 -0.6355 -1.5887 -0.6355 -3.1774 -0.6355 -3.9718 - 0 . 6 3 5 5 -11.9154 - 0 . 6 3 5 5 -19.8590
Table 6.5.
Coefficients for Cylinder, /h/l,Y
= 0.5
(Extracted from Transactions of the ASMlS with Permission of the Publisher, the American Society of Mechanical Engineers, 29 West 39th St., New York, N. Y.) d/t.
a4
as
a6
___-.-4 10 20 30 40
-3.30513 -8.2630 -16.5240 -24 7885 -33.0510
-0.9090 - 1 .4373 -2.0325 -2.4894 -2.8745
h
hs
-.---. -0.10625 -0.10625 -0.10625 -0.10625 -0.10625
-0.9090 -1.4373 -2.0325 -2.4894, -2.8745
-0.1250 -0.1250 -0.1250 -0 1250 -0.1250
Coefficients for Cylinder, /h/Is = 1.2
(Extracted from ‘I’ransac%ions I,f the ASMK with Permis:;ilbll of the Publisher. the Ameri(~an Society of Mechanical Engineers, 29 \\‘est 39th Si... New York, N. Y.) (108) d/t, ___.
Table 6.2.
-0.5000 -0.5000 -0.5000 -0.5000 -0.5000 -0.5000 -0.5000 -0.5000 -0.5000
4.80 12.00 24.00 36.00 48.00 96.00 120.00 360.00 600.00
(I b
a4
a6
b4
b5 -
-9.5171 -23.7927 -47.5854 -71.3781 -95.1708 -190.3416 -237.9271 -713.7812 -1189.6352
-2.3896 -3.7783 -.5.3433 -6.5442 -7.5566 -10.6867 -11.91,81 -20.69,l: -26.7167
-0.2550 -0.2550 -0.2550 -0.2550 -0.2550 -0.2550 -0.2550 -0.2550 -0.2550
-5.7351 -9.0679 -12.8240 -15.7061 -18.1359 -25.6480 -28.6754 -49.6672 -64.1200
-0.7200 -0.7200 -0.7200 -0.7200 -0.7200 -0.7200 -0.7200 -0.7200 -0.7200 be = 0
bg = 0
Table 6.3.
Coefficients for Cylinder, Ih/&
Table 6.6.
= 0.8
(Extracted from Transactions of the ASME with Permission of the Publisher, the American Society of Mechanical Engineers, 29 West 39th St., New York, N. Y.) (108) d/t.
a4
3.20 -4.2298 8.00 - 10.5745 16.00 -21.1491 24.00 -31.7236 32.00 -42.2981 64.00 -84.5963 80.00 - 105.7454 740.00 -317.2361 400.00 -528.7268
a6
a6
b4
-1.3007 -0.1700 -2.0812 -2.0567 -0.1700 -3.2906 -2.9085 -0.1700 -4.6537 -3.5622 -0.1700 -5.6996 -4.1133 -0.1700 -6.5813 -5.8171 -0.1700 -9.3073 - 6 . 5 0 3 7 - 0 . 1 7 0 0 .- 10.4059 -11.2647 -0.1700 -18.0236 -14.5427 -0.1700 -23.2684
hs
-0.3200 -0.3200 -0.3200 -0.3200 -0.3200 -0.3200 -0.3200 -0.3200 -0.3200 bG = 0
Coefficients for Cylinder, th/&
=
1.6
(Extracted from Transactions of the ASME with Permissicnl of the Publisher, the American Society of Mechanical Engineers, 29 West 39th St,.. New York, N. Y.) (108) d/t,
a4
(15
6.40 -16.9193 -3.6790 16.00 -42.2981 -5.8171 32.00 -84.5963 -8.2266 48.00 -126.8944 -10.0755 64.00 -169.1926 -11.6342 128.00 -338.3851 -16.4532 160.00 -422.9814 -18.3952 480 00 -1268.9443 -31.8615 800.00 -2114.9071 --41.1330
a6
br
bs
-0.3400 -11.7730 -1.2800 -0.3400 -18.6147 -1.2800 -0.3400 -26.3251 -1.2800 -0.3400 - 3 2 . 2 4 1 6 - 1 . 2 8 0 0 -0.3400 -37.2294 - 1.2800 -0.3400 - 5 2 . 6 5 0 3 - 1 . 2 8 0 0 -0.3400 - 5 8 . 8 6 4 8 - 1 . 2 8 0 0 -0.3400 - 1 0 1 . 9 5 6 8 - 1 . 2 8 0 0 -0.3400 - 1 3 1 . 6 2 5 7 - 1 . 2 8 0 0
,
Stresses Table 6.7.
Coefficients for Cylinder, G/t, =
2.0
(Extracted from Transactions of the ASME with Permission of the Publisher, the American Society of Mechanical Engineers, 29 West 39th St., New York, N. Y.) (108) d/L
a4
a6
a6
br
ba
in
Cylindrical
Vessels
with
Flat-plate
Closures
109
~1 and x2. The bending moment at x1 is Ml and at 22 is Ms. The induced bending stresses vary from zero at the x axis to a maximum at the outer fiber. The bending stress in the strip at any distance y from the x axis is given by Eq. 2.10.
At xl. 8.00 20.00 40.00 60.00 80.00 160.00 200.00 600.00 1000.00
-26.4363 -5.1416 -0.4250 -20.5665 -2.0000 -66.0908 -8.1296 -0.4250 -32.5185 -2.0000 -132.1817 -11.4970 -0.4250 -45.9881 -2.0000 -198.2725 -14.0809 -0.4250 -56.3237 -2.0000 -264.3634 -16.2593 -0.4250 -65.0370 -2.0000 -528.7268 -22.9941 -0 4250 -91.9762 -2.0000 -660.9085 -25.7081 -0.4250 -102.8325 -2.0000 -1982.7254 -44.5278 -0.4250 -178.1112 -2.0000 -3304.5424 -57.4851 -0.4250 -229.9406 -2.0000 -
bg = 0
At x2.
The corresponding forces acting upon the-element dA are f,.fdA.
fgA,, = F Table
6.8.
Coefficients for Cylinder, th/ls
=
6.0
At x2,
IHxtracted from Transactions of the ASME with Permission of the Publisher, the American Society of Mechanical Engineers, 29 West 39th St., New York, N. Y.) tl/l.
4
10 10 30 40 X0 100 300 M O
a4
a6
-39.6612 -99.1560 -198.288 -297.4614 -396.6120 -793.1640 -991.5372 -2973.916 -4957.268
-10.908 -17.2476 -24.2820 -29.8728 -34.4940 -48.7800 -54.5400 -94.4550 --121.9500
a6 -1.275 -1.275 -1.275 -1.275 -1.275 -1.275
-1 275 -1.275 -1.275
ba
6s
-130.8960 -206.9640 -292.6800 -358.4736 -413.928 -585.360 -654.480 -1133.460 -1463.400
-18.00 -18.00 -18.00 -18.00 -18.00 -18.00 -18.00 -18.00 -18.00
j2/AYz = 7
Coefficients for Cylinder, &/Is
= 10
‘y dA sY
Also, a shear force exists at y equal to:
Since the element is in equilibrium, the summation of the forces must be equal t,o zero. Therefore e _MAY _ _ ' y dA - M+ y dA - fshesrb dx = 0 (6.117) z J2/ s1/ Rearranging gives:
66 = 0
Table 6.9.
‘y dA J1/
fshear =
CM2 - MI)Y
Zb dx
cydA
(6.117a)
J21
But (M2 - Ml) = dM, and by Eq. 2.5 dM/dx = V. Therefore
rlxtracted from Transactions of tht= ASME with Permission of the Publisher, the American Society of Mechanical Engineers, 29 West 39th St., New York, N. Y.)
(6.118) In a rectangular section of width b and depth t,
4
10 20 30 40
80 100 300 ,500
-66.1025 -165.260 -330.480 -495.769 -661.020 -1321.940 -1652.562 -4956.526 -8262.113
-18.18 -28.745 -40.65 -49.788 -57.49 -81.30 -90.90 -157.40 -203.25
-2.1250 -2.1250 -2.1250 -2.1250 -2.1250 -2.1250 -2.1250 -2.1250 -2.1250
-363.60 -574.90 -813.00 -995.76 -1149.80 -1626.00 -1818.00 -314'8.00 -4065.00
-50.00 -50.00 -50.00 -50.00 -50.00 -50.00 -50.00 -50.00 -50.00
j
’ ‘by dy = 3 (c2 shear = z sY
(6.119)
Y2)
The value of fshear = 0 when y = c and is maximum at
be = 0
6.2d
Stresses in the Shell.
Figure 6.5 shows a hmgitudinal strip of the shell of width b under bending it&on. Consider an elemental length, &c, between points SHEAR
STRESSES
AT
THE
JUNCTION.
H
Ll J
Fig. 6.5.
i2 Bending in an element of
Section H - H
a plate.
110
Selection
of
Flat-plate
and
Conical
Closures
for
Cylindrical
the vertical axis where y = 0. Therefore the maximum shear stress is:
f
vc2 v(t2/4) 3 v shear = ?? = 2(bt3/12) = 2bt
Vessels
This stress exists at xa, the location of the maximum axial stress in other places than at the junction.
(6.120)
If the shear at the junction is expressed as shear per unit length of circumference, Q(bo = l.O),
Qo fshear -2 -2 -
6.2e
Stresses
in
SHEAR STRESSES Eq. 6.39
the
IN THE
(6.121)
faxial
oombined = $ + F = $ + s 8
(6.122)
CIRCUMFERENTIAL STRESSES AT THE JUNCTION . The combined circumferential stresses will consist of the hoop stress from internal pressure and the circumferential bending stress, plus the component of the axial bending stress. The circumferential bending stress (circum. bending) may be obtained by substituting Eq. 6.75 for y in Eq. 6.82, or fcircum. bending = F = $ “Mi2i
1
Q”)
Substituting in Eq. 6.123 gives:
fcircum.
combined
= f 8
+ Q”) s
+ F .s II
I
(6.125)
STRESSES IN THE SHELL IN O THER PLACES THAN AT THE Watts and Lang (108) have given the following JUNCTION. relationships for stresses in the shell in other places than at the junction: flshesr =
3Qo
d1 +
AXIAL S TRESSES AT THE J UNCTION. The combined axial stress in the plate at the junction is composed of the axial shear stress plus the bending stress from the action of the shell on the plate, or
2(i3;;s(~)
+
2@Mo/Qd2
(6.126)
where x, = location of maximum shear stress in the shell in other places than at the junction,
Substituting Eq. 6.98 for MI in Eq. 6.131 faxial
combined = l$l+!$-~~ ( 6 . 1 3 2 )
combined = longitudinal pressure stress + axial
= E + 3.41 fi s
(flshear)
(6.128)
+ c2 1K - M,I
(6.133)
Substituting Eq. 6.98 for MI and Eq. 6.61 for Mz evaluated at P = d/2 in Eq. 6.133 gives: fcircum. combined = t I 1
+
?-3$-6p
0
thd 2 (1 1(6.134)
STRESS IN C ENTER OF F LAT P LATE. At the center of the flat-plate closure the axial stress is equal to the circumferential stress. The combined maximum stress is equal to the sum of the shear stress and the bending stress from the action of pressure on the plate (see Eq. 6.62a).
f
(6.127)
f)axisl
(6.131)
f&al combined =
fcireum. combined = !$ I I
+ iBd@Mlo
From
(6.130)
(6.124)
f&cum. bending = pd(‘M: + Q”) s
JUNCTION .
CIRCUMFERENTIAL STRESSES AT THE JUNCTION . The combined circumferential stresses in the plate at the junction are composed of the circumferential shear stress, the circumferential bending stress from the action of the shell on the plate, and the circumferential bending stress from the action of pressure on the plate (see Eq. 6.61). The bending moments causing these stresses have opposite signs.
(See Eq. 6.89.)
= $
AT THE
The shear force per unit length in the plane at 90’ is of equal magnitude (29). Therefore by Eq. 6.121
(6.123)
but /34&
PLATE
Closure.
Q=$!
4
AXIAL SrnEssEs. The combined axial stresses in the shell at the junction will consist of the axial stress from internal pressure and the axial bending stress, or by Eqs. 3.13 and 2.10:
Flat-plate
max combined =
Ia0i th
+ ; 1%
- Mzzl
(6.135)
Substituting Eq. 6.98 for Ml and Eq. 6.62a for M,, in Eq. 6.135 gives:
f
max combined = (6.136)
111
Stresses in Cylindrical Vessels with Flat-plate Closures
1
Solution of the previous equations shows that when the thickness of the flat plate is greater than the thickness of the shell, the maximum stress is the combined axial stress located in the shell at the junction. If the flat plate has a thickness equal to or less than the shell, the maximum stress is the combined axial stress located in the flat plate at the junction. Thus, there is a ratio between the thickness of the cover plate and the thickness of the shell, th/tsr which will result in equal stress in the plate and shell. This ratio will vary from 1:0 to 1:2 depending upon the ratio of diameter to thickness of the cover plate. However, it is usually advantageous to use a ratio of th/ts greater than 1:2 in order to reduce the magnitude of the maximum stress concentration. Therefore, for such designs the maximum stress concentration is located in the cylinder at the junction with the flat cover plate and may be determined by Eq. 6.122. For purposes of comparison it is convenient to express the stress concentration at the junctionf,,,, in terms of the hoop stress in the shell for the same conditions. f Max stress ratio, I = ‘s = L%?!fhoop
I !
4_h=si=6 tp:
~ = 0.606
a2 = 2(1 - p) a3
= 1.4
b
= 3; (1 - p) ; = 1.575
b
2
- cl) = -0.635 = _ 3(I ___ P2d4
3
=-?l-~ = -0.953 16 ,B2thts
u4 = -(@d)2 th 2 4
= - 1 4 2 8 b4 = -/3d
= -785.4
as= -pd$
= -@j&j b5 = - f
= -18
s
a6= - !!!k! 4 8
(6.137)
b6 co
= -1.275
Substituting into Eqs. 6.115 and 6.116 gives: - a.s)(bs - bz) - (a5 - a2)bs - al)(bs - b2) - (a5 - a2)(b4 - bl) = 478
(a3
MO = pd2 (a4
Qo = pd
(a4 (a4
- alhi -
(a3
- al)(bs - bz) -
- as)(b4 - bl) - a2)(b4 - bl) = -426
(a5
Stresses in the shell: By Eq. 6.121, f
shear = 3_ (-426) = -2,550 psi 21 z
By Eq. 6.122, faxial c o m b i n e d = ‘“1” ,“2” + p
4
= 49,300 psi
8 6
Ratio of shell diameter to shell thickness,
d 36 = I44 -=-i4 ;i
Fig. 6.6. closures.
\I
= 30.5
P2tht.s
pd/%
-\ - \
(6.86)
b1 = 6(1 - PI
a1 = -3(1 - p) $ = -50.4
Figure 6.6 is a plot of I, the ratio of the combined axial stress in the shell at the junction to the hoop stress in the shell. Watts and Lang (108) have evaluated and tabulated the stress-intensification factors for the major stress in the shell and flat-plate closure. Figure 6.6 indicates the stress concentrations that would exist if the theory of elasticity held up to the stress levels indicated. These stresses usually are not reached because the yield point of the material used in the construction of the vessel usually is exceeded. Above the yield point, plastic deformation occurs and the theory of elasticity no longer holds. For analysis of the plastic stresses, the theory of plasticity must be applied. The present state of this theory is such that it is not possible to evaluate these stresses. Strain gages applied at anticipated points of stress concentrations make it possible to evaluate the actual stress concentration. Figure 6.6 shows that the stress concentration at the junction between the shell and the head may result in a localized bending stress in the shell many times greater than the hoop stress in the shell even when relatively thick cover plates are used. This is quite often ignored in many designs. Fortunately, plastic deformation at the point of stress concentration tends to relieve an excessive localized bending stress and thereby to prevent failure. This illustrates the definite advantage of using formed heads which tend to reduce localized stresses at the junction. 6.2f E x a m p l e D e s i g n 6 . 1 . Determine the various theoretical stresses in the shell and in the flat-plate closure of a steel condensate collector operating at 100 lb per sq in. gage. The shell of the collector is fabricated of rolled steel plate s/4 in. thick, and the flat-plate closures have a thickness of 134 in. The vessel is 6 ft long and 3 ft in diameter. Solution:
. I _ -
T
7
d/t,
Maximum stress ratio in the shell at the junction for flat-plate
.
Selection of Flat-plate
112
and Conical Closures for Cylindrical Vessels
The stress in the center of a flat plate by Eq. 6.136 is:
By Eq. 6.125,
jmax
fcircum. combined = ‘F
478) - 42611 + I y.py(36NO.606 - x -__-__ 1 a + (0.3 x 6)(478) ~ T-5 = 7,200 + 11,900 + 13,750 = 32,850
psi
Stresses in the shell in other places than at the junction: By Eq. 6.126,
f
.shear
=
(3/2)(-426) l/i.+-(2)(--0.68)-+ (2)(0.463) _.~~~._~ -~~-
W4)e
0.606x4.45
- 2 5 5 6 4 0 . 5 6 6 - (-2556)(+0.752)= . ~___- - -. =e2.i 14.9
+1288psi .
Location of this stress is given by Eq. 6.127.
1 - ~~~ -426 = -hn--’ (0.606)(478) 0.606
combined
= /-2841 + 11275 + 852 - 14,080\ = 12,137
If this vessel had been fabricated from SA-283, Grade C steel, two of the stresses in the shell, faxis combined and would be above the minimum yield fcircumferential oombinedv point of 30,000 psi for this material. This indicates that some plastic deformation will occur and that these computed theoretical stresses will not be reached since they will be relieved by plastic deformation. If this vessel was operated under cyclic loading conditions, failure might have occurred by.brittle fracture if the stress range had exceeded twice the yield point. (See Chapt.er 2.) It should be pointed out that vessels of such design are seldom used for pressures higher than 25 psi because they are impractical. Vessels of such size operating at pressures above 25 psi usually have formed closures, termed dished heads. Practical Design of Cylindrical Vessels with Flat6.29 plate Closures. As a first approximation, the thickness of a
+ 1
= 4.45 in. from junction
cylindrical shell under internal pressure may be determined by the membrane equation (Eq. 3.14). The thickness of the flat-plate closure may be determined by considering the plate as a circular plate with clamped edges (Eq. 6.55a) or
t 8 =Ei
1~~ Eq. 6.128,
flaxial = !yg
t,,
= 3600 + (3.41)(+12)(+128.8)
1
4P
0.606
psior -1660 psi
T,, = 2* - - - = 4 . 4 5 -; x __- = I.45 - I .3 = 3. I5 in. from junction Stresses in thr ,llal-plalr closure: By Eq. 6.130.
By Eq. 6.132, j&, eomk,inrr, = q!? j + j !y! - wp2, 2 H = 2 8 4 + 1275 + 8 5 2 = 2441
=
d d3p/16j
th d 43pD6.f -= tx pd
I.ocation of t.his stress is given by Eq. 6.129. 7rl
(See Eq. 3.14)
w
+ 3.41 &i/a k128.8
= 3600 _+ 3260 = +8860
psi
By Eq. 6.134, i !1
= j-2841 + 1127s + 852 - 5980' = 1137 psi
(See Eq. 6.55a) W) _- 0.866
! / ‘:
i --___--\
\
\I
/
u/j/p
(6.138)
The thickness ratio obtained by the use of Eq. 6.138 results in ratios in which the thickness of the flat-plate closure is several times the thickness of the shell for usual values of allowable stress and operating pressure. Some improvement in design proportions may be made possible by increasing the thickness of the shell to reduce the maximum theoretical combined axial stress in the shell at the junction. However, such an increase in shell thickness does not result in a corresponding decrease in the required thickness of the flat-plate closure. (The maximum theoretical combined axial stress in the shell at the junction also can be decreased by increasing the thickness of the flat cover plate.) The maximum theoretical local combined stress may exceed the yield point appreciably with imperceptible permanent strain since only the outer fibers undergo plastic deformation. (See Fig. 2.5, detail b.) If the vessel is stressed under cyclic conditions, the maximum theoretical local stress range can approach a value equal to twice the yield point without danger of causing failure by brittle fracture. (See Eq. 2.37). If the vessel is not to be stressed in a cyclic manner, a higher theoretical (elastic) local stress might be tolerated. Under such conditions the theoretical stress loses its significance, and it is suggested that strain be used as a design criterion.
: i /
I
psi
.- ~-..~ _ -
-
~--
Stresses in Cylindrical Vessels with Conical Closures 6.3
STRESSES CONICAL
IN CYLINDRICAL CLOSURES
VESSELS
A flat plate may be cut and rolled to the shape of a cone and used as a closure for a cylindrical vessel. Shops having facilities for rolling cylindrical shells usually can roll conical shapes; this makes this type of head convenient for many types of process vessels. Figure 6.7 shows an element in a conical closure defined by length dl and angle dqi For very small angles the side opposite the angle is numerically equal to the angle in radians times the length of a long side. Therefore the area, A, of the element is:
F hrn = Fhv
COS
Ct = j,,l dl tk$ (‘OR CY
By a summation of forces in the normal direction Fhrn
jhtdld+cosa
-
P, =
0
- ptllrd+ = 0
therefore (6.130)
A = di (P d+) The force, FP, on the element and uormal to the surface is equal to the internal pressure times t.he area, or F, = pA = p dl (P dq5)
,
113
By small-angle relationships, we obtain (see Fig. 7.6b and c):
WITH
This force is resisted by the normal components of the forces of the stresses induced in the element. The meridional forces, F,, are 180” apart on either side of the element and therefore have no normal component. However, the hoop forces, Fh, are not 180” apart and have a normal component, Fhrn. The normal component, FhTn, may be determined from the radial component, Fhr, as follows:
(For angle 0, see Fig. 4.8 and Eq. 1.2.) At the junction of the conical head and the cylindrical shell, a compressive force is exert.ed by the cone on the cylinder. The shell under the influence of internal pressure attempts to expand radially outward against this inward force; this results in a bending moment and shear at the junction. The inward compressive force produced by the conical head can be evaluated as follows, by referring to Fig. 6.8. Axial tension in the shell = I’ pounds per linear inch of shell circumference. p = pd
Fh = jhj dl
4
Fig. 6.7.
Hoop stress in a conical head.
(6.140)
114
Selection
of
Flat-plate
and
Conical
Closures
for
Cylindrical
Vessels
a7 =
b&o2 tan a! sin cr 2
b7 =
a8 = (Eo2B/4) - ~‘G sin cr C + .bG
2G,$02 tan a~ C + 2pG4
A/2 bg = ~ C + 2pG
a9 = 2-P - set ff - y tan cr + 3ba(l + P) set a 8 F02 3 set+ ffPCC) cb7 csc2 (y b = 60 9
E04
b8 kUIs ff 4
c
%02
where A = fo(berz’
[O beiz &J - be&’ (0 berz to)
B = (berz’ 50)’ + (beiz’ [o)2 C = Eo(berz .fo berz’ [o + beiz to beiz’ 50) G = (berz
toI2
+ (bei
(01
2
and Fig. 6.8.
.$!o = cone parameter at the base
Compressive force at junction of cone and cylinder.
= /3d 1/2(t,/t,) cot (Y csc (Y The component of axial tension in the cone = T pounds per linear inch. P cos a
pd T=--.--
4 cos ff
(6.141)
The ber and bei Bessel functions are given by Mac Lachlan (111). The calculations for the coefficients are based on the relations:
The component of axial tension in the cone expressed as compression at the junction = C pounds per linear inch. C = P tan LY = p_d tan ix 4
(6.142)
As a result of this compressive force (Eq. 6.142) it is impossible to design a conical head to eliminate moment and shear at the junction since the cone always tends to deflect inward, and the shell outward under the influence of internal pressure. 6.30 Bending in the Shell and Conical Closure at the It was shown previously, in the analysis of a shell Junction. having a flat-plate closure, that two dimensionless equations (Eqs. 6.109 and 6.111) could be derived for the section of the shell at the junction. For convenience these were rewritten to give Eqs. 6.110 and 6.112. These equations, including the coefficients, are valid for the shell at the junction of the cone. Relationships for the conical closure comparable to Eqs. 6.101 and 6.108 for the flat-plate closure may be developed as reported by Watts and Lang (110). Eta ~1 = a,$ + as ;; + as pd2 = b,p%+b.F;+bg
Figure 6.9 shows diagrammatically the location of these forces and moments. The nomenclature used in Fig. 6.9 is the same as for Fig. 6.3 in the previous section on flat cover plates.
Qo
(6.143) (6.144)
where y1 = radial deflection of shell at junction, inches tc = thickness of conical section, inches
Fig.
6.9.
Forces and moment at junction of
h e a d (1 101.
cylindrical
shell and conical
L
Stresses
Equation 6.143 may be equated to Eq. 6.110 to give:
Table 6.13.
(6.146)
Or
(a - as)(bs
M,, = pd2 (a4
- a7)(bs - b8)
(~4
Qo = pd
- ba) -
(a4
-
- (a5
- as)bs - us)(br - b7) 1 (6.147)
(a5
- a7)bg - (us - &j)(b4 - b7) a716 - b8) - (a5 - us)(br - bi) 1 (6.148)
The coefficients u4 through a9 and bd through bg are tabulated in Tables 6.2 through 6.13.
3.9231 9.8076 29.4229 49.0382 104.9320 194.0819 310.4304 498.6781
01
(19
87
ao
be
ba
-6.3685 +I.6824 +0.1020 +3.6189 -0.4913 f0.0199 -16.0642 +2.7141 +0.0360 +5.7543 -0.4957 +0.0276 -48.4194 +4.7646 -0.0999 +9.9995 -0.4980 +0.0314 -80.7915 +6.1753 -0.1941 +12.9205 -0.4986 +0.0322 -173.0657 +9.0713 -0.3879 +18.9165 -0.4991 +0.0329 -320.2775 +12.3660 -0.6085 +25.7384 -0.4994 +0.0331 -512.4285 +15.6604 -0.8291 +32.5598 -0.4995 +0.0333 -823.3524 +19.8700 -1.1111 +41.2758 -0.4996 +0.0333
ai
as
ao
87
4.0825 -6.3676 f1.5220 +0.0030 +3.8148 16.3299 -26.5094 f3.2502 -0.2297 +7.7977 40.8248 -66.8682 +5.2433 -0.5138 +12.3988 77.0489 -126.6027 +7.2665 -0.8047 +17.0707 122.4743 -201.5488 +9.2045 -1.0840 +21.5459 204.1239 -336.2996 f11.9299 -1.4770 +27.8398 308.1955 -508.0946 +14.6953 -1.6760 +34.2261 500.0751 -824.8978 +18.7620 -2.4629 +43.6180
-0.4720 -0.4913 -0.4957 -0.4972 -0.4980 -0.4986 -0.4989 -0.4992
115
+0.0447 +0.0634 to.0683 +0.0700 +0.0707 f0.0713 f0.0715 +0.0718
-0.4372 -0.4720 -0.4881 -0.4913 -0.4949 -0.4957 -0.4966 -0.4972 -0.4980
+0.1486 -CO.1809 +0.2009 +0.2052 +0.2103 +0.2114 +0.2126 +0.2135 +0.2145
combined = , $ - Bd(pM; + Q") 8 s
Note that the sign of the second term in Eq. 6.149 is negative whereas that of the second term in Eq. 6.125 is positive because the direction of bending in the shell with a conical closure is in opposite direction to the bending in the shell with a flat-plate closure. Stresses in the Shell in Other Places than at the Junction. Watts and Lang (110) have derived relationships for the maximum stresses in the shell in other places than at the junction as follows: 3Qol/l + 2 max =
(PMolQo)
2t, tan-’
b9
68
Closures
Coefficients for Cone, a = 60” (110)
Coefficients for Cone, a = 30” (110)
(Extracted from Transactions of the ASME with Permission of the Publisher, the American Society of Mechanical Engineers, 29 West 39th St., New York, N. Y.) Utn
Conical
Stresses at the Junction. The maximum shear stress at the junction is given by Eq. 6.121. The maximum combined axial stress is given by Eq. 6.122. The maximum combined circumferential stress (circum. combined) is given by Eq. 6.125 with the following modifications:
fshear Table 6.11.
with
6.3b Stresses in the Shell.
Coefficients for Cone, a = 15” (110)
(Extracted from Transactions of the ASME with Permission of the Publisher, the American Society of Mechanical Engineers, 29 West 39th St., New York, N. Y.) d/lb
Vessels
8.4853 -12.2604 +1.5114 -0.3207 +6.9846 21.2132 -33.0871 +2.6361 -0.7559 +11.4443 58.1018 -93.7150 +4.6013 -1.5823 +19.2842 84.8528 -137.7469 +5.6295 -2.0229 +23.3930 169.7056 -277.5290 +8.0911 -3.0837 +33.2351 212.1320 -347.4572 +9.0817 -3.5114 +37.1965 2 9 4 . 1 4 0 6 -482.6728 +10.7465 -4.2314 +43.8550 400.3580 -657.8465 +12.5860 -5.0271 +51.2120 6 3 6 . 3 9 5 9 -1047.2780 +15.9426 -6.4797 +64.6377
f&cum.
Table 6.10.
Cylindrical
(Extracted from Transactions of the ASME with Permission of the Publisher, the American Society of Mechanical Engineers, 29 West 39th St., New York, N. Y.) as a9 d/h a7 br ba 69
and Eq. 6.144 may be equated to Eq. 6.112 to give: b,~+bg$+b9=h4~+b5!$+bi
in
+
WMO/QO)~
(&+1)
(6 150) .
The location of this maximum shear stress in the shell is given by Eq. 6.127. The maximum axial stress is given by Eq. 6.128, and the location of this stress is given by Eq. 6.129. 6.3~ Stresses in the Conical Closure. Relationships for evaluating the stresses in the cone at the junction have been presented by Watts and Lang (110).
(6.151) Table 6.12.
Coefficients for Cone, a = 45” (110)
(Extracted from Transactions of the ASME with Permission of the Publisher, the American Society of Mechanical Engineers, 29 West 39th St., New York, N. Y.) as ao d/h a7 b7 bs bv 4.0000 10.0000 40.0000 80.0000 100.0000 300.0000 500.0000
-5.7796 +1.2340 -15.5974 +2.1524 -64.9345 +4.5965 -130.8284 +6.6064 -163.7929 +7.4152 -493.6916 +13.0171 -823.7623 +16.8714
-0.0725 -0.2654 -0.8558 -1.3547 -1.5562 -2.9547 -3.9179
+4.0325 +6.6074 +13.5059 +19.1883 +21.4754 +37.3186 +49.2200
-0.4372 -0.4720 -0.4913 -0.4949 -0.4957 -0.4980 -0.4986
+0.0741 +0.0977 +0.1163 to.1202 +0.1211 +0.1235 +0.1240
faxialoombined
= f/ y + y +6;s
c
c
p cos ff -I- 2a3 + ~ 4
I
7
I
! (6.152)
(6.153)
116
Selection of Flat-plate and Conical Closures for Cylindrical Vessels
14
v
I
I
100
I
200
0 = s apex angle of cone I
I
300.
400
I
I
dlts Fig. 6.10. Maximum stress ratio in the shell at the junction for conical chnursr.
6.3d Location of Maximum Stresses. If the thickness of the cone is equal to or greater than the thickness of the shell, the maximum stress concentration usually is the axial bending stress in the shell at the junction. This stress may he calculated by use of Eq. 6.122, which assumes elastic behavior to the stress level indicated. For purposes of comparison it is convenient to express the stress concentration as a stress ratio, as indicated in Eq. 6.138. Figure 6.10 graphically indicates the stress concentration in the shell at the junction with conical heads as a function of angle a! for head-to-shell thickness ratios (110) of 1 and 2. Conditions under which the axial stress in the shell may nor.jbe maximum are as follows:
stress concentration and determining its value, Watts ant1 Lang (110) have presented suitable tables. For cones with LY equal to or less than 45’, it is desirable to make the thickness of the cone equal to the shell thirkness. When the cone is thinner than the shell there is a large axial stress in the cone at the junction. Similarly. when the cone is thicker than the shell, there exists a 1;rrpe axial stress in the shell at the junction. If it is not practicable to make the shell and cone the same thickness in is desirable to make the cone t,hirker than the shell. When a is greater than 45”, the cone should be made thicker than the shell. All the stress c,oncentrations except the circumferent irl I stresses increase in magnitude as the angle (Y is increased. This is indicated for the shell axial-stress ratio in Fig. 6.10. At one limit, where (Y is O’, the cone becomes a cylinder having the same mean diameter as the shell, and no stress intensifications will occur if the two cylinders have the same thickness. In the ot,her limit, when QI = 90’. the cone becomes a flat cover plate, and the analysis of I he previous section on flat plates holds (see the dashed line in Fig. 6.10). This limit is shown in Fig. 6.6. Comparing the ordinate of Fig. 6.6 to that of Fig. 6.10 indicates that an approximately tenfold reduction in stress concentration exists when conical heads are used rat,her than flat cover plates. The high stress ratios shown in Figs. 6.6 and 6.10 are a direct result of sharp-corner construction. The use of :I formed “toriconical” head rather than a simple conical head will greatly reduce the stress concentrations at the shelland-cone junction. 6.3e E x p e r i m e n t a l l y D e t e r m i n e d S t r e s s e s . To verif\ the theoretical relationships, experimental tests were conducted by O’Brien et al. on conical vessels by the use (Jf strain gages (112). The experimental vessels were 48 in. in diameter wi~tr conical heads having a = 45’. The cones were slightl) thicker than the shells, the cones having a thickness of 0.755 in. and the shells a thickness of 0.633 in. Figure 6.11 shows the profiles of the junction of the cones with the shells. Vessel A-2 (Fig. 6.11) approximates a toriconical head. Strain gages were suitably placed both inside and out,side of the vessel in the vicinity of the shell-cone junction to measure the circumferential and axial strains. The stresses
1. When thickness of the cone is less than the thickness of the shell. 2. When the thickness of the cone is equal to the thickness of the shell, but the ci/& ratio is small. 3. When (Y = 45” or less, and the ratio of d/t, is small. These exceptions apply to designs of unusual proportions that rarely occur in process equipment. For exceptions 1 and 2 the maximum stress concentration will be the axial stress in the cone. For exception 3 the maximum stress concentration will be the circumferential stress in the cone at the junction. For convenience in locating the maximum
Radius 0.00 assumed 0.80 actual
Profile at cone-shell junction, vessel A-2
Fig.
6.11.
(Courtesy
Profiles of
at
American
Profile at cone-shell junction, vessel A-l cone-shell Welding
junctions
Society.)
of
experimental
vessels
(112).
Stresses
in
Cylindrical
Vessels
with
Conical
Closures
117
Cylinder
Outer circumferential stress curves
Fig. 6.12.
Measured and computed stresses in experimental vessel (1 12).
were computed from the strains determined experimentally by means of Eqs. 6.6 and 6.7.
.f, = 1”- (%a? + EIq/z) P2
(6.6)
(6.7)
(Courtesy
of
American
Welding
Society.)
where jr, f, = longitudinal and circumferential st.ress. respectivelg , pounds per square inch p = Poisson’s ratio E = modulus of elast.icity, pounds per square inch E,Z, ~~2 = longitudinal and circumferential strain, re spectively, inches per inch The inner and outer 1on;ritudinal stresses computed from strain measuremenls are compared in Fig. 6.12 wit,h the
118
Selection
of
Flat-plate
and
Conical
Closures
for
stresses as determined from theoretical considerations. The solid curves represent the theoretical stresses for the vessel A-l having the sharp junction, and the circles are the experimentally determined stresses for the same vessel. The dashed curves represent the theoretical stresses for vessel A-2 having a toriconical junction, and the solid circles are the corresponding experimentally determined stresses. The inner longitudinal (or axial) stress is a tensile stress whereas the corresponding outer stress is compressive. In Eq. 6.122 the two terms are additive for the inner surface and are of opposite sign for the outer surface because the bending moment Ma is positive in one case and negative in the other. Thus, the greatest stress occurs at the inner surface, where the stress due to bending and the longitudinal pressure stress are additive. Examination of Fig. 6.12 indicates that, except for a few local discrepancies, the general trends of the stress distribution for the head-to-shell junction in vessel A-l are the same whether determined experimentally or calculated from theory. The agreement is not quite so good for vessel A-2, however. The maximum stress in this vessel is considerably lower than that in vessel A-l; this indicates the reduction in maximum stress obtained by the use of a toriconical head rather than a conical head. Figure 6.12 indicates that the theoretical relationships satisfactorily predict the stress distribution within the elastic range. A comparison may be made between Figs. 6.10 and 6.12 by calculation of the maximum stress at the junction as follows: Diameter of vessel A-l = 48 in. Shell thickness, t, = 0.633 in. Cone thickness, t, = 0.755 in. d 48 ts - 0.633
75’7
2 t ----El2 _ 0.755 4 0.633 . Referring to Fig. 6.10 and using the line for a! = 45” and t,/t, = 1.2, we find that the maximum stress ratio is 3.8.
Cylindrical
Vessels
The hoop stress in the cylindrical shell is:
pd ~48 j = ?i = (2)(0.625) The maximum stress is: Ifp = 1 psi
fnlsx = (3.8 X 38.4~)
= 146~
f max = 146
This compares favorably with the maximum inner longitudinal stress indicated in Fig. 6.12, which is also plotted for 1.0 psi. Although the stress concentrations shown in Fig. 6.12 which exceed the yield point are seldom reached because of plastic deformation, the results of the elastic analysis indicate the location of the stress concentration and the zone in which plastic deformation can be expected to occur. Most commercial pressure vessels, including those having conical heads, are given a hydrostatic test at one and onehalf times the working pressure. The vessel deforms elastically during the early part of the test. At some pressure level the yield point of the material is exceeded at zones of local stress concentration. Some plastic deformation follows which permanently deforms the zone of stress concentration, giving the vessel a new shape. This permanent deformation may not be very great and may not be apparent. However, a system of residual stresses is set up in the plastic zone that remains after the pressure is removed. In service this vessel may deform elastically at the working pressure, but the stresses cannot be calculated because the shape and the stress-intensity pattern are not known. Thus, in many practical cases plastic flow permits a vessel to resist ccncentrated stresses without damage, provided that the material of construction has sufficient ductility. The general discussion in the last paragraph of section 6.2g on flat-plate closures also applies to conical closures. The ASME code for unfired pressure vessels (11) uses the following modified form of Eq. 6.139 for conical heads that have a half-apex angle, cy, not greater than 30’:
t=
pd 2 cos oc(jE - 0.6~)
(6.154)
Additional information on the design of conical closures is given in Chapter 13.
PROBLEMS
1. A cylindrical vessel has a shell fi in. thick and 20 in. in inside diameter. The vessel is ciosed at both ends with a flat cover plate welded to the shell and has a thickness of 135 in. I f the internal pressure is 200 psi, calculate: MO, 00, and the maximum local fiber stress in the vessel. 2. For the vessel described in problem 1, calculate: a. shear stress in the plate at the junction, b. maximum combined circumferential stress in the shell, c. maximum combined axial stress in the plate at the junction,
I - - - - - - - -__- ~ - ~ - - - T - - - - I - - - - -
= 38’4p
Problems d. maximum combined circumferential stress in the plate at the junction, e. maximum combined stress in the center of the plate. 3. For the vessel in problem 1, determine the maximum shear and maximum axial combined stresses in the shell in obher places than at the junction and the location of these stresses. 4. If one of the flat cover plates in problem 1 is replaced with a 90”cone (01 = 45’) having a thickness of >/4 in., calculate: Me, Qe, and the maximum local fiber stress at the cone end. 5. For the vessel in problem 4, calculate the maximum shear stress in the shell at the cone end in other places than at the junction, and determine its location.
119
C H A P T E R
STRESS CONSIDERATIONS IN THE SELECTION OF ELLIPTICAL, TORISPHERICAL, AND HEMISPHERICAL DISHED CLOSURES FOR CYLINDRICAL VESSELS
7.1 ELLIPTICAL DISHED CLOSURES lliptical, torispherical, and hemispherical dished closures, shown previously in Fig. 5.7, are considerably stronger than conical or flat-plate closures for cylindrical vessels. The great majority of cylindrical vessels have either elliptical or torispherical dished heads. Figure 7.1 shows the welding of an elliptical dished head to a lOZ-in.diameter shell. Although the hemispherical dished head is stronger than either the elliptical or torispherical dished head, it is not so widely used because of the excessive forming required in its fabrication. In general, this results in a higher fabricating cost and a more limited range of available sizes. However, hemispherical heads in limited sizes are now extensively used as closures for propane and butane horizont.al storage vessels and for this service are *more economical than elliptical dished heads. See Fig. 5.11. The principal advantage of dished heads over flat cover plates or cones as closures is the large reduction in the discontinuity in shape at the junction between the cylindrical vessel and the closure, resulting in a-_reduction of discontinuity stresses at or near the junction. The hoop and meridional stresses in the central portion of these three types of dished heads are relatively easy to evaluate. For a given vessel these stresses will be at a maximum in the torispherical head and at a minimum in the hemispherical head. However, the discontinuity stresses at or near the junction are difficult to evaluate. As a result of the difficulty of evaluating these stresses, an extensive literature has developed in this field (1133125). A history of the design of heads for pressure vessels is included in a recent article entitled “Report on the Design of Pressure Vessel Heads” prepared by the Design Division of the Pressure Vessel Research Committee (12).
The excessively high stress concentrations existing in the sharp-corner junctions of vessels with flat cover plates and with conical heads have been discussed in the previous chapter. The use of a knuckle radius on a cone to form a toriconical head reduces the stress concentration at the junction. An additional improvement in design is obtained if the torus ring is retained and the conical element is replaced with a spherical dished element. Such a head is referred to as a torispherical head and is widely used in t,he fabrication of process equipment. A further improvement in design resulting in a greater reduction of stresses in a formed head is obtained from the use of elliptica! dished heads. The stresses in such heads have been analyzed by two methods: (1) by strain measurements on heads of vessels subjected to internal pressure and then by m;t t hematical computation of the stresses from the strain measurements, and (2) by mathematical analysis ui the head geometry. 7.2 STRESS ANALYSIS FROM STRAIN MEASUREMENTS An early study of strain measurements on elliptical dished heads was reported by E. Hijhn (113) a translatiou of which appeared in Mechanical Engineering (114). In this study an elliptical head having a 1.97 to 1.00 major-tominor-axis ratio and having an outside diameter of 31.7 in., a head-plate thickness of 0.47 in., and a shell plate thickness of 0.313 in. was subjected to internal pressures of 8, 16, and 24 atm. Before and after application of the pressure to the vessel, HShn made tomplates of the contour of t,be elliptical dished head and also templates of torispherical dished heads that were included in the investigation. He found that the 120
\- \
\I
I
Stress
Analysis
from
Strain
Measurements
torispherical dished heads deformed considerably, approaching the shape of an ellipse with a major-to-minor-axis ratio of about 2.0, whereas the elliptical head retained its original contour. This tends to indicate that the natural shape for a disllzc: head on a cylindrical vessel under internal pressure is an ellipse having a 2 : 1 major-to-minor-axis ratio. Strain gages were placed about the vessel in such a manner that strains were measured in the radial and longitudinal directions at 18 different positions along the shell and around the head. The strains measured were the result of elastic deformation in three directions. In thin-walled vessels radial stresses are of minor importance, and the two principal stresses in the head act at right angles to each other and are identified as hoop stress, fiL, and meridional stress fm. At the junction of the head with the shell, these stresses become the hoop (or circumferential) stress of the shell and the longitudinal stress of the shell, respectively. If both of these stresses are acting simultaneously, they may be calculated from strain measurements by use of Eqs. 6.6 and 6.7. fh
=
(~h2
+
wm2)E
1 - $ and f = m
(~n2 + /.e2)E
(6.7)
1 - /ls
If these stresses are divided by the shell hoop stress, stressintensification factors are obtained which are convenient for comparing the two principal head stresses with the corresponding shell hoop stress existing outside of the discontinuity at the junction. Figures 7.2 and 7.3 are plots
1.6,
Fig. 7.2.
Hoop-rtrers-intenriflco-
tion factor for vessels with elliptical
closures-from
mental
data
(114).
Hijhn’s
experi-
,
Fig.
7.1.
Welding
elliptical head
%
in.
thick
to
shell
with
automatic
welder--vessel is 102 in. in diameter and designed for 50 lb per sq gage, 200” F service.
,
,
,
,
,
,
,
,
I 0.2
I 0.3
I 0.4
I 0.5
I 0.6
I 0.7
I 0.8
I 0.9
in.
(Courtesy of C. F. Braun & Co.)
s 0.6 .5 2 0.4 ‘ci 5 0.2 .E I D 0 s co 8 -0.2 2 2 -0.4 II
z
-0.6 -0.8 -10~J 0 0.1
1 5 0.1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0.0 i !_H e a d radial d i s t a n c e t o m e r i d i a n -- Shell axial distance from junction -.Shell radius Shell radius
122
Selection of Elliptical, Torispherical, and Hemispherical Dished Closures
the central section of the head thinned out in forming thus accounting for the peculiar hump in the head-stress curve near the center of the head. Figure 7.3 shows the corresponding curve for the meridional-stress-intensification factor in the head and the longitudinal-stress-intensification factor in the shell. Note that similar reversals also occur.
of these stress-intensification factors as calculated from Hijhn’s data. Figure 7.2 shows the hoop-stress-intensification factor, Ih = f&h shell, in the head and in part of the shell of the vessel. Note that Ih in the head from the center of the head to the point of tangency of shell and head is plotted on the left-hand portion of the figure whdpeas Ih in the shell is shown in the right-hand portion of the figure. Therefore, @he horizontal axis to the left of the point of tangency of ‘.head and shell represents fractional radial distances from the axis of the vessel to the point under consideration. The horizontal axis to the right of the tangency represents distances along the longitudinal axis of the shell measured from the point of tangency in terms of fractions of the radius. The vertical-a% intensification factor represents tensile stress above the reference line of zero and compressive stress below the reference line. Figure 7.2 shows that
7.3
HUGGENBERGER’S THEORETICAL MEMBRANE STRESSES
ANALYSIS
FOR
The work of Hijhn previously described in which stresses were determined from strain measurements indicated that two critical stress groups exist : (1) the stresses in an element of an elliptical dished head resulting from internal pressure and the geometry of the head and (2) the stress concentrations in the neighborhood of the junction of t,he head and the shell. Huggenberger (115, 116) developed an analysis of the stresses in an element of an elliptical dished head
F i g . 7 . 3 . Meridional-~hass-intensification factor for vessels with elliptical closure&rom
mental
,O
L
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Head radial distance to meridian Shell radius
0.8
0.9
1,0 _
the central portion of the head is under tensile stress and that a reversal to a compressive stress occurs at 0.86 radius units from the axis. A maximum compressive stress is reached at about 0.95 radius units from the axis, after which point the compressive stress decreases to zero at the point of tangency. The curve further indicates the influence of the head on the hoop stress in the shell. The hoop-stress intensification factor in the shell increases from zero at the point of tangency, reaching a maximum at about 0.3 radius units from the point of tangency; it then drops to a lesser value, after which drop it levels off at about 0.7 radius units from the tangent line to a constant value of 1.0, the normal hoop-stress-intensification factor in the shell. The dotted line on Fig. 7.2 from the center of the head to the experimental curve indicates the probable curve that would have resulted if the elliptical dished head used by HShn had been of uniform thickness throughout. Actually,
0.1
0.2
0.3
0.4
0.5
0.6
data
Hiihn’s
experi-
(114).
L
Shell axial distance from junction Shell radius
from mathematical considerations. In Huggenberger’s analysis the bending moments at the junction of the head and shell were disregarded. The equations of Huggenberger are developed in the following section. Figure 7.4 is a three-dimensional sketch of a differential element of an elliptical dished head. Arc m-n is a meridian formed by passing a plane through point 0 and through the w-w axis. Arc O-O’ is formed by passing a plane through point 0 perpendicular to the w-w axis. The stress jm is the stress on the element in the meridional direction and is called the meridional stress. Stress fh is the stress acting circumferentially on the element and is termed the hoop stress. These are principal stresses, and no shearing stresses exist on the sides of the element. Figure 7.5 is a view of the meridional plane through the element. Because of the symmetry of the head about the w-w axis, the location of this meridian is not significant.
Huggenberger’s
Theoretical
Analysis
for
Membrane
Stresses
123
Similarly, the circumferential or hoop force acting on opposite sides is: Fh = fhtrl d4 As shown in Fig. 7.6, detail b, this force has a resultant, force in the radial direction which is perpendicular to the w-w axis and is: de 2
F h r = 2Fh - = f&l
dc# de
As shown in Fig. 7.6, detail c, this resultant force has a component in the direction normal to the plane of the element, which is: Fh,.,, = Fh de sin $I = fhtrl sin 4 d4 d0 Summing the forces normal to the plane of the element results in: Fmn i- Fhrn - FP = 0
or Fig.7.4.
Stresses acting on differential element in an elliptical dished head
de d4 -I- fhtrl sin 4 d4 de - prlro d$ de = 0
f&o
Combining gives : The location of the hoop plane is defined by angle do made by the normal to the surface of the element and the W--W axis. The radius, ~1 defines the curvature of the element in the plane of the meridian; the radius r2 defines the curvature of the element about the W-W axis, about which axis it generates a cone; and PO is the radial distance of the element from the W-W axis. The length of both sides of the element in the meridional plane is equal to rr d+, and the length of the upper side of the element in the circumferential plane through point 0 is rg d0. This follows from the fact that for very small angles expressed in radians, the side opposite the angle is numerically equal to the angle in radians times the length of a long side. The surface area of the element is equal to the product of the two sides, or A = rlro d+ de
f&0
+ fhtrl sin 4 = prm
Dividing by rlrot gives: fh sin 4 f”+-=rl
r0
P t
From Fig. 7.5 it follows that rg = r2 sin 4; therefore, fm
fh p ,+,=i
(7.1)
The internal pressure acting on this area produces a normal force having the value of: FP = pA = prlro dq5 de
d
(7.2)
where p = internal pressure, pounds per square inch This force is resisted by the components of the principal forces, the components being taken in the direction of the normal to the plane of the element. The principal force in the meridional direction acting on opposite sides is: F, = f&o de
r1 = meridional radius of curvature at point P. inches
(7.3)
r2 = radius of curvature of the section perpendicular to the meridian at point I? inches
Figure 7.6, detail a shows that this force has a component in the direction normal to the plane of the element because the forces on opposite faces are not 180’ apart. For a finite element, F mn = F, d+ = f&o de d+ This follows because of small angle relationships.
Fig. 7.5.
Meridional section of element of an elliptical dished head.
Selection
124
of
Elliptical,
Torispherical,
and
Hemispherical
Dished
Closures
(a) Fig. 7.6.
WJ
Components of forces in meridional and circumferential planes.
(a) Components of forces in a meridional plane.
(b) Components of forces in a
c i r c u m f e r e n t i a l p l a n e . (c) N ormol component of forces from o circumferential plane.
Therefore the distance I in Fig. 7.5 is given by.
Tire axial force (along the w-w axis of Fig. 7.5) equals meridional force, F,,,, times sin 4. Axial force = m”“p = P,, sin #J Force = (SI ress) (area)
b2 ro2 a2
F,, = .fN1 (2mt) 7rrg’p = f,,L(2ar,,l)
2
Let k = u/b. Therefore
sin 4
ro2 z = k2z
i,r
(7.7)
From Fig. 7.5 and by Ey. 7.7, The equation of an ellipse is : (7.6) where rg, 2, a, and b have the significance of 5, y. a, and b, respectively, as indicated in Fig. 5.2. Figure 7.5 indicates that fill is perpendicular to rp and is tangent 1.0 the ellipse at the point P. The slope of jm is c&&ted as follows:
Substituting for Z2 its equivalent, as given by Eq. 7.6 and making t: = a/b, we obtain: (7.8)
Z 2 = -$ (a2 - ro2)
Therefore
lu slopr = - dro
Dilfereutiating
Eq. 7.6 gives: r0
sin C#I \lultiplyinp dZ dr,,
\
\
ro2(1 - k2)]‘y
[(ak)2
\I
/
-
-~---
(7.9)
both sides of Eq. 7.8 by (p/2t) gives:
rg -b” -. a2 z
=
- [(ak)” +
-
+ ro”(l - k 2
)I 51
Huggenberger’s lht jm
=
_‘op.L
(from Eq. 7.5)
sin 4 2t
Therefore P [(ak)Z + r&l - P)]” jm = 2t
(7.10)
Theoretical
Analysis
Membrane
Stresses
125
dished head. These equations were first given by Huggenherger (115) in 1925. Examining these equations for the extreme conditions when r = a and b = m, we find that the stresses at the junction of the head and the attached cylinder for this condition reduce to the longitudinal and circumferenlial stresses in a cylinder, .or
By mathematical definition t,he radius of curvature, rl, is: (7.11)
for
fm = z and
From Eq. 7.8,
.fh = T
Z = i (a2 - ro2)E At the crown center, when rg = 0, the two st,resses fh and jnL hecome equal and have the value:
dZ = Z’ = -; (a2 ro2)-%0 dro
(7.16)
a2 d2Z _ zjt = 2
dro2
k (U2”T
For a hemispherical head, when k = 1.0, Eq. 7.16 reduces IO the st.ress equation for a hemi,;pherical head.
3 2 l+ gp-Ir”ij 1
.fh =
kbstitufing in F-q. 7.11, we find that
r1 = -
- ; (a2
_
ro2)%
jh = j,,, = !f
Simplifying, we find that [(ak)2 + ro2(1 - k2)]“” r1 = __ a2k2
(7.12)
Rearranging Eq. 7.4, we find that .fh=r2
I
Substituting
from
fm
--; t
Ecp. 7.10 and 7.12
gives:
i [(ak)2 + ro(l - k2)]ts -P - t [(ak)’ + ro2(1 - k2)]“z _____-____ ..~_ a2k2
jh = p2
jh = 1’2 1 -[
FIYJIII
[ 1 P
a2k2 2[(ak)2 + ro2(1 -
This indicates that an elliptical dished head ,,aving a majorto-minor-axis ratio of 2: 1 should be expected to have the same stress in the crown as the hoop stress in the attached shell if bending moments at the junction are not considered. For any elliptical dished head having a value of k greater than 1.0 and less than 2.0, the maximum stress will occur in the center of the crown and will lie between t.he values of rp/t and rp/2t, as given by Eq. 7.16. Huggenberger’s equations, 7.10 and 7.15, are cumbersome because of their length. The principal advantage in t,heir use lies in the expression of the stresses in terms of the radial distance rg. These equations may be modified so as to express the stresses in terms of the radius of curvature d the closure. From Fig. 7.5 it is evident that
(7.13) r2
Fig. 7.j and Eq. 7.9, t-0 = [(ak)2 r2 = mF sm $I
rp zl
For HII elliptical dished head whrre a = 2b, Eq. 7.16 heconies:
PO2
[
.I,,, =
Substituting into Eq. 7.5 gives: + ro2(1 - k2)]s’
(7.14)
.rm = pz
Therefore 3 a2k2 E - -. 2[(ak)’ + ro2(l - k’i1 t (7.15) Equations 7.10 and 7.15 describe t.he two principal stresses, jm and .fh, in terms of the geomeky of the elliptical
This relationship ran be transferred t.o stress-intensification form (press. is pressure) by dividing by fhOop(st&) to give :
I
(7.27)
126
Selection of Elliptical, Torispherical, and Hemispherical Dished Closures
value at the junction with the shell (if bending :nc,ments at the junction are ignored). For all elliptical dished heads in which k is greater than 2.0, the maximum stress will be this compressive stress (bending moments being ignored) occurring at or near the junction with the shell. Although Huggenberger’s relationships do not take into account the bending-stress concentrations in the head near the junction with the shell, they. are useful in analyzing the stresses in the remainder of an elliptical dished head for any major-to-minor-axis ratio. 7.4
Head radial distance to meridian Shell radius F i g . 7 . 7 . Elliptical-headstress-intensi~cation theory of
Huggsnberger
f a c t o r s b a s e d on m e m b r a n e
(115, 116).
If lh = t*,
A comparison of Eqs. 7.15 and 7.14 indicates that Eq. 7.15 can be written in the following form: (7.19)
Converting Eq. 7.19 to stress-intensification form gives: .fh Ihbress.)
=
fhoop(shell)
= 21 m(press.)
k2t ’ -
COATES’S THEORETICAL ANALYSIS FOR BENDING STRESSES AT THE HEAD-SHELL JUNCTION
The relationships developed by Huggenberger are limited because no allowance was made for bending moments that exist in the head at and near the junction. The existence of this bending moment is graphically indicated by the curves determined from Hijhn’s data as shown in Figs. 7.2 and 7.3 and is further indicated in Fig. 7.7, in which hoop stresses calculated from Eqs. 7.17 and 7.20 are plotted. Comparing the hoop pressure stress for an elliptical dished head in which k = 2.0 as predicted by the appropriate curve of Fig. 7.7 with the combined hoop stress in Pig. 7.2. we find that a sharp discontinuity is indicated at the junction of the head and the shell. This discontinuity exists because the head and shell have been treated as separate, disconnected membranes with no restraints at the edges. Figure 7.8 shows the deformation of an ellipsoid or revolution having a major-to-minor-axis ratio of 2.0 when under internal pressure. A vessel of such a shape when under internal pressure will deform to become more spherical, the shell deforming outward along the minor axis and inward along the major axis. Figure 7.9 illustrates the effect of the deformation of an elliptical dished head when attached to a cylindrical vessel. Under internal pressure the elliptical dished head shown in Fig. 7.9 will tend to deform inward as was shown in Fig. 7.8. The cylindrical shell will tend to deform radially outward under the influence of internal pressure. The joining of the head to the shell restrains both of these deformations. This results in the introduction of bending moments in the head and in the shell. The effect of the junction is to bend the shell inward and to bend the head outward with respect to their unrestrained positions. As a result a compressive stress is induced in the outermost fiber of the knuckle of the
(7.20)
4zm(preL%)th2
where Zm(pre,,.) is given by Eq. 7.17. If k = a/‘b = 2, and 1, = Zh, 1 Zh(press.) = 21 m(press.) - ~ Z WL(Pr’28S.)
(7.21)
The values of Zh(press.) and Zm(press.) given by Eqs. 7.17 and 7.20 are plotted in Fig. 7.7. As shown in Fig. 7.7, the hoop stress, jh, has a maximum value in tension at the center of the crown where PO = 0. As ~0 approaches a, the hoop stress in tension decreases, passes through zero, and reaches a maximum negative
Fig. 7.8.
Deformation of an ellipsoid of
revolution under internal pressure.
Coates’s
Theoretical
Analysis
for
Bending
Stresses
at
the
Head-Shell
Junction
127
y1 = deflection of shell at junction T2 = normal hoop force Q = meridional shearing force MI = meridional bending moment Mz = hoop bending moment The following relationships summarize Coates’s derivations for the shell: 1 ~ e-“+~ [Qo cos /3gcl + /31Mr,(cos /31x, ” = - 2~3D - s i n /3rzr)] (6.74) with zero pressure
Ets3
where D = Fig. 7.9. zone.
Deformation of an elliptical dished head and shell in the junction
(Courtesy
of
F. 1.
Maker.)
head whereas a tensile stress is induced in the innermost fiber at the same location. Similarly, the bending of the shell in the opposite direction tends to increase the tensile stress in the outermost fiber of the shell in the neighborhood of the junction. at
,
7.4a the
12(1 - $+)
Coates’s Relotionships for Junction when Head and
local Bending Shell Are Not
p = Poisson’s ratio
- sin &zr)] (7.22)
Stresses Joined.
W. M. Coates (117) mathematically investigated the bending at the junction of a dished head and cylindrial shell. A longitudinal strip of the shell in the neighborhood of the junction which is bent inward under the influence of the head is selected for analysis. The force causing this deformation can be considered to be an inward radial shear force acting on the end of the strip. This force is resisted by the bending forces set up in the strip and by the compressive hoop stress opposing a tendency for the shell circumference to decrease. The total resistance to this tendency to deform inward results in radial shear forces, longitudinal bending stresses, and circumferential compressive stresses. Since the shell behaves in an elastic manner and is relatively thin in comparison to the diameter. the radial force on the strip under consideration can be considered to be proportional to the radial deflection. Therefore, the strip may be considered to act like a beam on an elastic foundation. Such a beam under a point load will deflect immediately under the load because the supporting foundation is elastic. The stiffness of the beam will transfer a portion of the load to either side of the force, and this will result in a smaller elastic deflection, which is a function of the resistance of the foundation and the distance from the point of load application. The theory for such a beam was developed in Chapter 6. The equation of the deflection curve for such a beam is given by Eq. 6.74. By use of this relationship Coates developed an analysis of the discontinuity stresses. Figure 7.10 illustrates an element of the shell under consideration where P = radius of shell & = thickness of shell x1 = longitudinal distance from junction
and
Q = D $ = -e-B1z1 [QO(COS /3lzl - sin /3rzr) 1
day1 1 MI = - D - = - e-flP1 da2 81
+ M2
[Q. sin
,&MO(COS
2/?lM0
sin /31x1] (7.23)
plzl
PIZI + sin 8141
(7.24) (7.25)
= lrMt
The corresponding relationships of Coates for the head are : (7.26)
I92
Fig.
7.10.
=
* 3(1 - UP) rz2th2
-\i.
Forces and moments acting on element of shdl
(7.27)
Selection
128
of
Elliptical,
Torispherical,
and
Hemispherical
Dished
Closures
and
MO = 0 M2 = /.LM,
Substituting these values in the equations developed for the cylinder and for the head, gives the following equations. For the shell,
6Ml t,2
=
6M2
t,2 =
f
m(bending)
=
f7+1 sin plxl (7.33)
f Qbending)
=
?--41’r1
6Ml -= th2
.f
m(bending)
-
- 3pk2a2
-_____
=
4
d33(1
ax
p2)thr2 I
22 p2 dx2 j cos
GM2 -= f h(bending) = th2
where x2 = distance measured along meridian from plane of junction, inches r2 = radius of curvature of hoop section of head, inches (See Fig. 7.5.) =
(7.35)
Functions for computing stresses based on the theory of beams
7.11.
on elastic foundations (29).
3%
(7.34)
For an elliptical dished head (r2 is a variable)
0 1 ’ ’ ! ’ !\ -0.1 , , , , / , / -0.21 11 11 11 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 Fig.
sin PlrCi
1 2p13D
(e-~SZI)[Qo
c o s (32X.2
- /3&fo(cos
pzx2 - sin /32s2)]
(7.28)
T2 = 3 yz = 2Plr r-2
2x2 - sin @2x2)] (7.29)
(7.30)
d%
Ml = D __ = p- (e-@2”2)[ -Qo sin p2x2 dxz2 r2Pl +
P~Mo(cos P2x2 +
sin P2x2)l
tan (Y dy2 M2 = /.LM~ - D(l - /L’) ~ .r2 dm
(7.31) (7.32)
where LY is the angle between the normal to the head and the plane of the junction. Figure 7.11 is shown for convenience in Pvaluating the functions of ,Blxl or &x2. 7.4b Analysis of Edge Loads and Local Stresses when Head and Shell Are Joined. The joining of the head to the
7.4~ Solution of Equations for the Shell. Equations 7.33 and 7.34 predict the major bending stresses developed in the shell as a result of the discontinuity at the junction. The terms in the right-hand side of these equations are usually constant for a given vessel and pressure with the only variable, x1, the linear axial distance (inches) from the junction measured along the shell. Use of Fig. 7.11 reduces the solution of these equations to a simple calculation. The constant @I is evaluat,ed for the shell, and suitable increments of x1 are selected (such as Ax1 = 0.05a). Values of the product plzl are tabulated, and the corresponding functions of e-alZl sin ,81x1 and e-Slzl cos plzl are determined by use of Fig. 7.11. These functions are multiplied respectively by 3pk2/4@12t,2 and by -pk2a/4t, to give the tW0 major discontinuity stresses,fm(hendiIlr) andfh(bending), in the shell as a function of x1. The StreSS-intenSifkatiOn factors, zm(bendinr) and zh(bending), may be computed by the use of the following relationships:)
I m(bending)
=
fm(bending)
Z h(bending)
=
f___ h(bending)
\
\I
I
=
e--Bl+ sin /31x1 (7.38)
If k = 2, Eqs. 7.37 and 7.38 reduce to:
Z h(bending)
\
sin /lIzI (7.37)
fhbreas.)
1. The radial displacement of the shell and of the head at the junction must be equal. 2. The slope of the total-displacement curves for the head and for the shell at the junction must also be equal.
- - -
e-@+l
fh(preas.)
Z
Q. 2k” fvl
0
(7.36)
shell imposes the following restrictions if the shell and head are of the same thickhess:
Therefore
(/-‘I P2dsz)
(7.39)
=
The functions e-sz sin px and eYflz cos @x have maximum values at bx = a/4 and /3x = s/2, respectively. As a result the maximum meridional bending stress in the shell occurs at a distance of ,81x1 = ?r/4, and the maximum hoop bending stress in the shell occurs at /?lxl = 0. This corresponds to the following.
Coates’s Theoretical Analysis for Bending Stresses at the Head-Shell Junction
The location of maximum Zm(bending
in sheu) is:
Mathematicians have proved that Eq. 7.45 can not he evaluated in finite form in terms of the elementary functions of CY. Equation 7.45 defines a function denoted by E(K, (Y) which because of its origin is termed an “elliptic integral,” or
x1 = 0.61 dat, The location of maximum Zh\l,,.nding in sherr) is: x1 = 0 The distance to damp out hending
E(K, a) = lo” msin’ (Y dol (0 < K < 1)
stresses is:
x = 2.44 dat, 7.4d
Solution of Equations for an Elliptical Dished Head.
Equations 7.35 and 7.36 predict the major bending stress in an elliptical dished head as a result of the discontinuity at the junction. The solution of these equations is more difficult than the solution of those for the shell since the radius of curvature of a hoop section of the head, r2 (see Fig. 7.5) is a variable. Therefore f12, which is dependent upon r2 (see Eq. 7.27) is also a variable. A further complication arises from the fact that ~2 is the linear distance along the meridian of the head (as measured from the junction). This distance is not a simple function of r-0 (the head radial distance to the meridian) and can only be evaluated graphically or by the use of elliptic integrals. This prevents the formal integration of the term p2 dx2 in both Eq. 7.35 and 7.36. To show a method of determining the distance 22 as a function of ro, reference is made to Fig. 7.12, which illustrates an ellipse of axis u-b circumscribed by a circle of radius a (126). The distances ro and Z may be expressed as: r0 = a sin cy And by Eq. 7.6
(7.41)
.~z = P d,2 - ro2 a Z = b cos 01
(7.42)
(7.43)
Rut
dro = a cos (Y dor dZ = -bsinardor I herefore S=
I
ca da + b2 sin2 (Y dol
(7.44)
where the upper limit of cx corresponds to the value of a! at point P. Substituting 1 - sin” (Y for cos2 (Y gives: s =
/0
u l/iie- (a2 - b2) sin2 ar dar
=a oa di - K2 sin2 a! dcu /
where
K i= 1/a2? = eccentricity of the ellipse a
(7.45) (7.46) (7.46a)
(7.47)
Values of the function E in terms of’K and cy have been determined by the use of infinite-series calculations. Tables of values of the function are given in the literature (127). To determine ~2 as a function of r-0, the value of s at the point in question is subtracted from the value of s for a = 90”. As the function E is equal to s/a (see Eqs. 7.45 and 7.46) and as it is convenient to plot stress-intensihcation factors versus the dimensionless ratio ro/a, a table may be prepared for an elliptical dished head giving corresponding values of CY, ro/a, s/a, x2/a, and rz/a. Such a tabulation is shown in Table 7.1 for an ellipse with a major-tominor-axis ratio (k) equal to 2.0 to 1. In preparing this table even angles of a were selected rather than even increments of ro/a in order to avoid the necessity of interpolation of tables of elliptic integrals (127) used to determine values of E. Selecting o( fixes ro/a because by Eq. 7.41 ro/a equals sin CL The value of rz/a (radius of curvature divided by shell radius) is determined by use of Eq. 7.14. Equations 7.35 and 7.36 may be solved by tabulating rz, fi2, and z2 as a function of rg. A summation of the increments of pz Ax2 is used to determine the integral of p2 dxz. The values of this integral may then be used to determine the corresponding functions of e-‘zz2 sin &x2 and e%% cos &xz by means of Fig. 7.11. Substitution of the values of these functions and the corresponding value of r2 into Eqs. 7.35 and 7.36 permits the determination of the major
Measuring the arc length, s, from point B, the top of the minor axis, we obtain: BP = / ds = 1 l/dro2 + dZ2
129
h’g. 7.12.
Trigonometric variables for an ellipse.
139
Selection of Elliptical, Torispherical, and Hemispherical Dished Closures Table 7.1.
If k = 2 and t, = th,
Dimensionless Ratios for an Ellipse in
a/b = 2:O (See Fig. 7.12 [127].) Which
I
ro -=
S -=
a
a
52 -=
a
sin (Y
E.2
(Ego - Ed
90” 89” 88” 87” 86” 85’ 80” 75” 70” 65=’ 60” 55O 5o” 45O 4o” 35”
1.0000 0.99985 0.99939 0.99863 0.99756 0.99619 0.98481 0.96593 0.93969 0.90631 0.86603 0.81915 0.76604 0.70711 0.64279 0.57358
1.2111
0 0.0088 0.0175 0.0263 0.0350 0.0438 0.0886 0.1352 0.1845 0.2368 0.2927 0.3523 0.4157 0.4829 0.5536 0.6278
1.2023 1.1936 1.1848 1.1761 1.1673 1.1225 1.0759 1.0266 0.9743 0.9184 0.8588 0.7954 0.7282 0.6575 0.5833
f-2 -
a 1.000 1.0015 1.002 1.004 1.008 1.012 1.044 1.096 1.164 1.239 1.322 1.410 1.496 1.581 1.660 1.734
bending stresses, fm(bendinp) and fh(bendins),in an elliptical dished head. Equations 7.35 and 7.36 may be converted into stressintensification-factor form by dividing by @d/2&), or - 3t,k2a =
Ih(bending)
7.4e C o m b i n e d S t r e s s - i n t e n s i f i c a t i o n F a c t o r s . The bending stresses can be combined with the pressure stresses to give the maximan combined stresses. For the shell,
I n+(combined)
Ih(combined)
(7.49)
Ih(bending)
Table 7.2.
Xl
a 0
F2(zl)
(4)
(5)
01x1
F,(Pl&
FZ(@lxl)
= e&r1 sin /3lzl
+ 0.5
= I h(bending)
+
= zh(bending)
+ 1-O
(See Eq. 7.37.) (7.52)
Z h(press.)
(See Eq. 7.38.) (7.53)
= In(bending)
+ Zm(press.)
1.0 0.82 0.66 0.50 0.37 0.15 0.03 -0.04 -0.06 -0.06 -0.04 -0.01 0
(7.54)
where Im(ben&ns) is given by Eq. 7.50, and Im(press.) is given by Eq. 7.17. Z h(combined) where Ih(bendins) by Eq. 7.20.
iS
= Ih(bending)
+ Ih(press.)
given by Eq. 7.51, and Ih(preSS.)
EXAMPLE CALCULATIONS METHOD OF ANALYSIS
USING
(7.55) iS given
COATES’S
A cylindrical steel vessel has elliptical dished closures with a major-to-minor-axis ratio of 2.0, The diameter of the vessel is 80 in., and the vessel is to operate under an internal pressure of 100 psi. The shell and the heads have the same thickness, 1>/4 in. Determine the maximum
Solution of Stress-intensification Factors in the Shell
(3)
--_~~- ~..~-__ 0 0 0 1 0.025 0.182 2 0.05 0.364 3 0.075 0.546 4 O.JO 0.728 6 0.15 1.092 8 0.20 1.456 10 0.25 1.820 12 0.30 2.185 14 0.35 2.558 16 0.40 2.912 18 0.45 3.276 20 0.50 3.640 where Fl(xr) = e-p1z1 cos plzl
+ Zm(press.)
I m(bendina)
Z m(combined)
‘ithr2 d3(1 - P2) (7.48)
= Im(bending) =
For the elliptical closure,
7.5
(sin [ /3s &s)
Xl
=
a
(7.47a)
I m(bending)
m(bending)
(7)
(8)
Ih(bending)
Im(combined)
(6) Im(bending)
0 0.15 0.255 0.30 0.325 0.30 0.23 0.155 0.100 0.025 0.010 0
0 0.272 0.463 0.544 0.590 0.545 0.418 0.281 0.182 0.045 0.018 0
0 0.082 0.139 0.163 0.177 0.164 0.125 0.084 0.055 0.014 0.005 0
0
0
0
0.50 0.772 0.963 1.044 1.090 1.045 0.918 0.781 0.682 0.545 0.518 0.500 0.500
(9)
zh(combined)
1.00 1.08 1.14 1.16 1.18 1.16 1.13 1.08 1.06 1.01 1.01 1.00 1.00
f--
Example Calculations Using Coates's Method of Analysis
To solve the above equations, Table 7.2 was prepared. One-inch increments of ~1 were selected up to 4 in. from the junction, and 2-in. increments from 4 to 20 in. These values were divided by a (40 in.) and tabulated in column 2, and also were multiplied by PI (0.182) and tabulated in column 3. The values of eVP121 cos @rzr and e-@lzl sin @rzr were determined from Fig. 7.11 and tabulated as Fr(zr) and Fs(sr), respectively, in columns 4 and 5. Substitution of values from COhmn 4 intoEq. 7.40 gives Ih(bending) tabulated in column 7. Substitution of values from column 5 into Eq. 7.39 gives Im(ben&ns) tabulated in column 6. Substitution of values from column 6 into Eq. 7.52 and of COhnn 7 into Eq. 7.53 gives zm(combined) and zh(oombined) tabulated in columns 8 and 9, respectively. The example calculation for the head is as follows. Since the head has a major-to-minor axis ratio of 2.0 tol,
stress-intensification factors in the shell and closures in the neighborhood of the junction. For the shell, a = 40 in., b = 20 in., k = a/b = 2.0. By Eq. 6.86 and with p = 0.3, Pl
=
4 3(1 - P2) = 4 3(1--0.32) = o 182 y’ (40)2(1.25)2 ’ d r2t 8 2
Therefore 3k2 ___ = 1.815 ( 4812t,a ) By Eq. 7.39, zm(ben&ng) = 1.815eCsl”l By Eq.
7.40,
BY Eq. 7.52,
zh(bending)
=
~m(combined)
= zm(bending) = zh(bending)
+ 0.5 + 1.0
Table 7.3.
Solution of Stress-intensification Factors in the Head
(1)
(2)
(3)
(4)
(5)
(f-9
(7)
(8)
a
r2
no/a
P2
x2 ( i n . )
Ax2
B2 Ax2
Wz Axz
90 89 88 87 86 85 80 75 70 65 60 55 50 45 40 35
40.00 40.06 40.08 40.16 40.32 40.48 41.76 43.84 46.56 49.56 52.88 56.40 59.84 63.24 66.40 69.30
0.999 0.999 0.998 0.997 0.996 0.985 0.966 0.940 0.906 0.866 0.819 0.766 0.707 0.643 0.573
0.1818 0.1817 0.1815 0.1813 0.1811 0.1808 0.1780 0.1738 0.1688 0.1634 0.1580 0.1531 0.1488 0.1447 0.1405 0.1382
0.000 0.352 0.700 1.052 1.400 1.752 3.544 5.408 7.380 9.472 11.708 14.092 16.628 19.316 22.144 25.112
0.000 0.352 0.348 0.352 0.348 0.352 1.792 1.864 1.972 2.092 2.236 2.384 2.536 2.688 2.828 2.968
0.0000 0.0640 0.0632 0.0638 0.0630 0.0636 0.3190 0.3240 0.3329 0.3418 0.3533 0.3650 0.3774 0.3890 0.3973 0.4102
0.0000 0.0640 0.1272 0.1910 0.2540 0.3176 0.6366 0.9606
(9)
FliW2 1.000
0.936 0.874 0.811 0.751 0.226 0.426 0.220 0.075 -0.012 -0.055 -0.070 -0.060 -0.044 -0.028 -0.014
__r_
S i n @iZi
-e -@?l c o s plxl
BY Eq- 7.53, ~h(combined)
1.000
1.2935
1.6353 1.9886 2.3536 2.7310 3.1200 3.5173 3.9275
(10)
Ad
F2W2~d 0 0.058 0.110
0.156 0.193 0.226 0.314 0.313 0.264 0.194 0.125 0.068 0.026 0.001 -0.011 -0.014
~-I . ..-- .-
hbend.)
0.934 0.872 0.809 0.745 0.684 0.408 0.201 0.065 -0.010 -0.042 -0.047 -0.040 -0.028 -0.017 -0.008
-\--\-
zm(bend.)
-0.001
0.012 0.015
-
\
I
zh(press.)
zm(press.)
0.500 0.501 0.502 0.503 0.505 0.506 0.523 0.548 0.583 0.620 0.661 0.705 0.748 0.790 0.830 0.865
0.000 -0.105 -0.198 -0.282 -0.347 -0.405 -0.544 -0.518 -0.411 -0.284 -0.172 -0.087 -0.032
1.000
131
7-
-----
-1.000
-0.993 -0.990 -0.986 -0.975 -0.966 -0.868 -0.729 -0.545 -0.375 -0.193 0.158 0.314 0.455 0.572
-
0.500 0.396 0.304 0.221 0.158 0.101
-0.010
--
kcomb.)
-.
-0.021 0.030 0.172 0.336 0.490 0.618 0.716 0.789 0.842 0.880
&comb.)
0.000 -0.059 -0.118 -0.177 -0.230 -0.282 -0.460 -0.528 -0.480 -0.385 -0.235 -0.057 0.118 0.286 0.438 0.564
132
Selection of Elliptical, Torispherical, and Hemispherical Dished Closures
1.2 1.(1 I = 0.8 2 5: 0.6 g 0.4 2 .$ 0.2 .5L! II / ca .E& -0.2 2 50.4 8 * -0.6
Fig. 7.13. Hoop-stress intensification based on combination o f H u g g e n b e r g e r ’ s membrane s t r e s s e s a n d Gates’s b e n d i n g stresses.
-0.8 -1.0
\ I
0.1
0.2
0.3
0.4
0.5
0.6
Head radial distance to meridian Shell radius
0.7
0.8
0.9
1.0
0.2
0.3
0.4
0.5
0.6
Shell axial distance from junction = (2)=(2)-L_ Shell radius
Table 7.1 may be correctly used in the analysis of in the head. By Eq. 7.27,
I
hr st rcsses
The value of r2 may be determined from colrmm 6 of Table 7.1 (a = 40 in.) for the values of angle a selected in column 1 of Table 7.3. The coefiicien( for Eq. 7.50 is: -3a (--3)(JO) ,~~~~~~ = r2 d~~~33) P2 d3(1 - P2)
0.1
-72.5 = pp
By Eq. 7.50,
By Eq. 7.51,
As pz varies with ~2, the above equations can not be formally integrated. Integration of /32 dx2 is performed by steps. By use of column 5 of Table 7.1, values of x2 are determined for corresponding values of LY and r2. These Differvalues of x2 are tabulated in column 4 of Table 7.3. ences are taken to give the incremental values of Ax2 listed in column 5. Values of Ax2 are multiplied by /32 and tabulated in column 6. Accumulative summations of /32 Ax2 are tabulated in column 7 and correspond to the integral from 0 to 52 of /32 dxz. Then values of Fl(x2) and Ft(x2) are determined by the use of Fig. 7.11 in the same manner as previously indicated to give Zh(hendinp) and Z,,,(l,en~~i~~e) listed in columns 10 and 11.
To det,ermine the combined stress-intensification factors, the pressure-stress-intensification factors must be added to the bending-stress-intensification factors. The meridional and hoop pressure-stress-intensification factors may be determined by Eqs. 7.17 and 7.20, respectively, and are tabulated in columns 12 and 13, respectively, of Table 7.3. The combined meridional and hoop stress-intensification factors determined by use of Eqs. 7.54 and 7.55 are tabulated in columns 14 and 15, respectively. The bending-stress-intensification factors for this vessel at and Ilear the junction, zh(bending) and zm(bending), are plotted in Figs. 7.13 and 7.14. It should be pointed OUI that these curves are specific for a vessel having the dimensions given in the problem and are not general. Referring to Fig. 7.13, we find that the computed hoop stress-intensification factor at or near the junction by use Coates’s method of computation is shown as a dashed curve labeled B. Inspection of the curve indicates that the bending stresses in both the head and the shell are rapidly damped out within a short distance from the junc*tion. Huggenberger’s theoretical curve for t.he hoop stress in an elliptical dished head having a major-to-minor-axis ratio of 2 to 1 and the theoretical hoop stress in the shell are shown in Fig. 7.13 by the dotted curves labeled A, Addition of curve A to curve B results in the combined curve shown as a solid line labeled C. This combined curve, C, gives the predicted stress-intensification factor in both the elliptical dished head and shell for the vessel described. Figure 7.14 shows the corresponding meridional stressintensification factors for the same vessel. Although the combined curves of Figs. 7.13 and 7.11 are specific for the vessel described, it is interesting to compare these curves with those of Figs. 7.2 and 7.3, which were In gencomputed from strain measurements of E. Hiihn.
Effect
era1 it may be seen that the meridional and hoop stressintensification factors have the same shapes in the corresponding rurves. 7.6 EFFECT OF MAJOR-TO-MINOR-AXIS RATIO
The American Pet.roleum Institute formed the API Committee on Untired Pressure Vessels in 1930 to formulate a code for the petroleum induslrj. An unpublished memorandum entitled “Streses in Heads of Pressure Vessels” was prepared for this committee in 1932 by F. L. Maker, then of the Standard Oil Company of California.* In this memorandum curves were presented for computed maximu111 stresses and maximum strains for elliptical heads of various depth ratios; the computations were made by using Gates’s method of calculation and a ratio of shell radius, P, lo head thickness, t, of 32 (r/t = 32). Figure 7.15 shows the ratio of maximum stress or strain in the head to the hoop stress or strain in the shell as a function of the ratio of major to minor axis k of the elliptical head. The following is a quotation from Maker’s original memorandum in reference to Fig. 7.15: “It was found that as the ratio of the axes of the ellipse varied there was a change in the location of the point at which the maximum stress occurred. For heads flatter than k = 2.5, hoop compression on the outside face of the knuckle is the governing point. For heads having the value of k between 2.5 and 1.2, the meridional tension on the inside face of the knuckle governs. For heads having the value of k between 1.2 and 1.0 (the last representing a hemispherical he’dd) hoop tension at the For the usual shape of juncture is the governing st,ress. elliptical head having a depth ratio of 2.0, the discontinuity Ptresses added to the membrane stresses give a maximum * Private communiratiou.
Fig. 1
7.14.
siflcation
Meridional-stress
‘hggenberger’s
and
intan-
based on combination of membrane stresses
Coat&s bending stresses.
of
Major-to-minor-axis
Ratio
133
of 1.09 times the stress in the cylindrical shell of the same thickness.” The upper dashed line in Fig. 7.15 shows the maximum strain ratios calculated by Coates’s relationships. The solid line from k = 3.5 to k = 2.5 and the dotdash line from k = 2.5 to k = 1.0 are also maximum stress ratios calculated from Coates’s relationships. The maximum of strain ratios is greater than the maximum of stress ratios over the entire range for k greater than 1.25. However, the maximum-stress theory is generally used as the basis of pressure-vessel design since experimental tests have come out in agreement with calculations based on this theory. Experimental tests have shown that elliptical heads having k = 2.0 are as strong or stronger than the shell. (One. point from a test by T. M. Jasper is shown in Fig. 7.15 at k = 2.0.) For this reason the proposed curve was lowered slightly to pass through 1.0 at k = 2.0. Calculations based on the maximum-strain theory failure have indicated that a hemispherical head should have a thickness of only 41 y0 of the shell thickness. Simple membrane theory without allowance for the discontinuity stresses would indicate a head thickness of 50% of the shell thickness. Tests of vessels with hemispherical heads having thicknesses of 50 y0 of the shell thickness have shown such heads to be as strong or stronger than the shell. For these reasons the proposed curve was drawn to pass through 0.5 at k = 1.0. For many years the proposed curve shown as a solid line was used as the stress-intensification factor, referred to as v, by the ASME code (11). In recent years this curve has been replaced by Eq. 7.56. l’ = $(2 + k2)
t= 2fE..@--+e - 0.2~
(7.56) (7.57)
ii 0 . 8 s -” 0 . 6 E E 0.4 s ‘C yj 0.2 z ‘Z 0 5 .E :, f$ - 0 . 2 z 1 -0.4 .-s 0 2, - 0 . 6 E -0.8
-1.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 liO 0.1 0.2 0.3 0.4 0.5 ( LH e a d
radial
i
distance to meridian Shell axial distance from junction = (;)A I (2)Shell radius Shell radius
134
Selection
Hoop
of Elliptical, Torispherical, and Hemispherical Dished Closures
strain-outside
f;Y face. of knuckle \
Proposed I ;J minimum depth for eUiptica\ heads
Fig. 7.15. Computed maximum stresses and strains in elliptical heads as ratios of stresses and strains in attached cylindrical shell. (Courtesy of F. 1. Maker.)
I-
Meridional tensioninside face of knuckle
,II
I
I
I
I
3.0
I
I
I
I
I
I
I
I
I
I
I
I
2.5 2.0 k, ratio of major to minor axis of elliptical head
where V = stress-intensification factor E = welded-joint efficiency
from Eq. 7.56 (See Chapter 13.)
c = corrosion allowance, inches a
The most widely used elliptical dished head is that having a major-to-minor-axis ratio of 2.0 and a thickness approximately equal to the thickness of the shell to which it is attached. The discussion here will be limited primarily to vessels with heads of this shape. Inspection of the hoop and meridional stress-intensification curves computed from strain measurements (see Figs. 7.2 and 7.3) and from elastic theory (see Figs. 7.13 and 7.14) indicates that in the neighborhood of the junction, the maximum tensile stress is the combined meridional bending and longitudinal stresses in the outer fibers of the outer surface of the shell near the junction. The combination of hoop discontinuity and shell-hoop stresses in the same region produces a tensile stress nearly as great. The maximum compressive stress is the combined membrane and meridional bending stresses located in the outer fiber on the outer surface of the head near the junction. The meridional bending stress produces a tensile stress on the outer surface of the shell near the junction and a compressive stress of about equal magnitude on the inner surface. The net effect is to produce a nonuniform stress distribution across the thickness of the head and of the shell. A very small amount of plastic deformation will relieve these high longitudinal stresses and cause a redistribution of the stresses which cannot be evaluated by the theory of elasticity. The tendency of the elliptical dished head to deform
I
I1
1.5
I
I
I
1.0
inwardly at the junction under internal pressure results in a radial shear force inward on the shell. This force opposes the internal pressure acting in the opposite direction on the shell at the junction. Thus the head may be considered to
act as a reinforcing ring on the end of the shell reducing the average hoop stress on the shell near the junction. Although the outer-fiber stresses resulting from bending may exceed the hoop stress in the shell proper, the net result of the bending stress, membrane stress, and shear will be a reduction in average stress in the shell c r o s s s e c t i o n in the longitudinal direction at and near the junction. The shear between the head and the shell acts radially outward on the head opposing the tendency of the head to bend inward at the junction. The net result is to reduce the compressive stress in the head near the junction. Thus the shell serves to reinforce the head. At the center of an elliptical dished head having a majorto-minor-axis ratio of 2.0, both the meridional and circumferential hoop stresses are equal to the hoop stress in the shell proper. However, this is a point condition, and the average stress over a finite area at the center of the head is less than the hoop stress in the shell proper. The statement is sometimes made that an elliptical dished head having a major-to-minor-axis ratio of 2.0 is as strong as the cylindrical shell to which it is attached. O n the basis of average stresses over any finite area of the head or the shell near the junction, such a head and the junction can be considered to be slightly stronger than the cylinder proper. This has been verified by years of experience and a multitude of tests. In 1931, T. M. Jasper, then Director of Research of A. 0. Smith Company, made the comment (124) “Elliptical heads of 2 to 1 ratio will fail a cylinder wall of the same thickness if properly made and attached.”
---
Relationships of Rhys-Stresses at Junction of Knuckle and Crown in Torispherical Closures 7.7 TORISPHERICAL DISHED CLOSURES The torispherical head, often referred to as the flangedand-dished head, has previously been shown in Fig. 5.7 (details c and d). This head is formed from a flat plate with two radii, the larger being the crown radius or radius of dish and the smaller being the knuckle radius, sometimes referred to as the inside-corner radius. Heads of this type were widely used before the development of dies for the elliptical dished head. In the early years the knuckle did not have any particular radius, and the radius was frequently only 1 or 2 in. The radius used was whatever the boilermaker or flangesmith could hammer out on hot forming blocks. The crown radius is always made equal to the diameter of the shell on the theory that this provides a stress in the spherical portion of a head equal to the stress in the cylindrical shell. Fai 1 ures in the field and a number of tests indicated the weakness of the knuckle of this type of dished closure. Because of these failures, larger knuckle radii were used, and the ASME Boiler Construction Code added the restriction that the knuckle radius be not less than 6% of the vessel diameter. This limitation is an improvement but does not define an optimum dished head. Figure 7.16 illustrates some of the various shapes of torispherical dished heads possible for a given depth of dish, b, less than the diameter of the vessel. One extreme case is shown by the uppermost curve where the crown radius is infinity and the knuckle radius, rb, is a maximum and is equal to the depth of dish. The other extreme case is shown by the lowermost curve where the knuckle radius is equal to zero and the crown radius, ~3, is a minimum. An infinite number of knuckle and crown radii combinations exists between these two extremes, one of which is illustrated by ~1 and rz. An examination of Fig. 7.16 indicates that as the crown radius increases from its minimum value of rg to infinity, the ratio of the knuckle radius to the crown radius (r1 to rc) goes from zero to a maximum value and back to zero. Hiihn (114) and Boardman (128) showed that when this ratio is maximum, the resulting torispherical dished head most closely approximates an elliptical dished head. HShn showed that the ratio of rl/r, of a torispherical head is maximum when
r1 rcmrtx
m-k
135
where r1 = knuckle radius, inches a = radius of shell, inches (90 - 4) = angle between rl and diameter In determining /3z (Eq. 7.27) Eq. 7.59 is used to evaluate r2 rather than Table 7.1, which was used for the elliptical closure. For the knuckle zone,
Equation 7.60 may be used with Eq. 7.58 to determine the values of /3szZ for the solution of Eqs. 7.50 and 7.51 for the knuckle. The limit of the knuckle in terms of the angle $I is given by: a - r1 sin 4 = ___ rc ( - rl ) where r, = radius of crown, inches The value of the angle r$ determined by the above equation may be substituted into Eq. 7.60 to determine the maximum value of 2s in the knuckle zone. Tables similar to Tables 7.2 and 7.3 may be established for the torispherical head to determine values of both I m(bending) and Ih(bendinz) in the shell and head, respectively, at and near the junction of the head and shell. 7.9 RELATIONSHIPS OF RHYS FOR STRESSES AT THE JUNCTION OF THE KNUCKLE AND CROWN IN TORISPHERICAL CLOSURES The change in the meridional radius of curvature at the junction of the knuckle and crown produces an additional discontinuity. C. 0. Rhys (124) has suggested the following equations for calculating the bending stresses at and near the junction of the knuckle and the crown. For the knuckle, (7.61)
.fh(press.) = ‘sh 2 - ; ( )
(7.58)
d(k2 + 1) - 1
where k = ” b 7.8 APPLICATION OF COATES’S RELATIONSHIPS TO TORISPHERICAL CLOSURES FOR STRESSES AT THE JUNCTION OF THE SHELL AND KNUCKLE The same procedures used for calculating hoop and meridional bending stresses in elliptical closures may be used for torispherical closures in the knuckle zone. The radius of curvature, r2, of a torispherical knuckle is different from that in an ellipse and may be calculated as follows: r2 = rl + (a - r1) set (90 - 4)
\
(7.59)
--
-
Fig. 7.16. Possible of F. 1. Maker.)
TFT- ~--- -~ - -
variations
of
radii
in
torispherical
heads.
(Courtesy
Selection of Elliptical, Torispherical, and Hemispherical Dished Closures
136
Zm(bendinr) 7.10 DISCUSSION OF EQUATIONS OF COATES AND RHYS
(4 Fig.
7.17.
(after
Deformation
Hiihn [l 141). (a)
(b) of
o
torispherical
Elastic
head
deformation.
(b)
under Plastic
internal
pressure
deformation.
(7.62)
fh(bending)
(7.63)
(7.64) Zh(press.)
Z m(press.)
zh(txnding)
=
r2ty 2ath
______
f m(bending)
4s =
In using the equations of both Coates and Rhys for torispherical closures, the different radii that are used should not be confused. The knuckle radius, rl, is a radius in a meridional plane, is a constant for a given head, and is identical to icr in Table 5.7. The knuckle has another radius of curvature, r2, in the plane perpendicular to the meridian, as shown for the case of the ellipse in Fig. 7.5. This radius is a variable, as given by Ey. 7.59, is equal to the shell radius at the junction of the knuckle and shell, and is equal to the radius of the crown, r,, at the junction of the crown and knuckle. Because pt and r2 are variables in the knuckle, integration of the functions of p dx are required. In the crown, r2 is a constant equal to r,. Therefore 62 is also constant, and &x can be determined without integration. In using the relationships of Rhys, the distance z is the positive distance from the junction of the knuckle and crown in the direction of the knuckle for the knuckle stress equations and in the direction of the crown for the crown stress equations. According to the relationships of Rhys, the maximum bending stress is the meridional bending stress in the knuckle due to the discontinuity at the crown-knuckle junction. This stress is equal to:
(7.66)
- 2ath
= [ 3;2!! (2 - l)] e-J”os=f” COS r/j* & h
r1
0
(7.67)
Jo (7.68) %or the crown, (7.69) (7.70) (7.71) (7.72) (7.73)
max =
ath
(7.75)
The relationships of Coates and Rhys for bending stresses in torispherical heads are discontinuous when combined with the pressure stresses in the head. Also, these calculated combined stresses fail to damp out in the same manner as those determined by Coates’s relationships for an elliptical dished head. A further complication exists in the fact that prior to failure a torispherical head plastically deforms. This fact invalidates the use of elastic-theory relationships. 7.11 DEFORMATION OF TORISPHERICAL HEADS UNDER INTERNAL PRESSURE Although a torispherical head having approximately the shape of an ellipse is considered to be the strongest head of this type, it is rarely used because of the availability of dies for elliptical dished heads when heads of such strength are desired. However, for reasons of economy torispherical heads having less than the maximum ratio of rl/rc are extensively used. Torispherical heads having less than the maximum ratio of rl/r, t,endto deform to the shape of an ellipse under the influence of internal pressure, as was shown by Hijhn (114). Figure 7.17a (from Hiihn) illustrates the change in shape that occurs within the elastic limit of the material, and Fig. 7.17b indicates the permanent set produced when such a head is plasticly deformed. The crown expands out-
-- -
-
-
- ---
Development
of
Stress-intensification
Factor
ward, the knuckle flattens, and the profile becomes more elliptical. As a result of this plastic deformation, the head becomes stronger with a redistribution of the stresses. Elastic t.heory does not permit the calculation of such stresses after plastic set has occurred. Therefore, it has been necessary to use experimental tests of vessels under pressure supplemented with observations of vessel failures to establish an empirical correlation for the design of torispherical heads. Hiihn reported 344 cases of failures of dished heads in Europe, most of which had ring cracks on the inside of the knuckle discovered in the course of inspections and in fifteen of which violent ruptures occurred. Hiihn pointed out that as the meridional and hoop stresses are of opposite signs at the inner surface of the knuckle, large shearing strains exist at an angle of 45” with the meridians but observed that no case of failure had occurred which could be attributed to these strains. The cracks that were observed always occurred along a ring in the knuckle. The bending stresses in the knuckle are large because of the sharp radius of the knuckle, and the bending stresses do not follow the linear rule in passing from compression to tension throughout the thickness of the knuckle. The stress-distribution curve has a hyperbolic shape, and the stress is much greater at the inner surface than it would be in the bending of a straight beam. It has not been feasible to determine the stresses on the interior surface of a torispherical dished head by strain measurements on t.he exterior surface. However, such external st.rain measurements
for
Code-designed
Torispherical
Closures
137
have been useful in developing empirical correlations for t,he design of torispherical heads. 7.12
DEVELOPMENT FACTOR FOR CLOSURES
OF STRESS-INTENSIFICATION CODE-DESIGNED TORISPHERICAL
Hiihn analyzed all of the reliable test data then available on torispherical heads and performed many tests of his own. He noted that the yield point of the head material was firs1 reached in the knuckle, and measured the pressure required to produce such yielding. He then computed the stress in the center of the crown at. this pressure by use of the equations for spherical shells based upon membrane theory. Next he determined the ratio of the yield point to the stress in t,he center of the crown and used this stress-intensification factor for correlation purposes. After trying a number of methods of correlation, he concluded that the results could best be correlated by comparing the computed stress ratios with the ratio of the knuckle radius to the crown radius h/r,. Hiihn plotted 20 points with rl/rC varying from 0.025 to 0.19 and drew a curve through these points for which he fitted an empirical equation. Hijhn’s data and equation are shown in Fig. 7.18. The equation or curve gives a factor by which the head thickness computed by means of t,he equation for spherical shells may be multiplied to give a t,orispherical head thickness having suficient strength. Figure 7.18 was originally part of a memorandum of F. L.
3.5
A Elastic tests by Siebel and Koerber 0 Elastic tests by Hahn + Test of A.O. Smith Co.
\
n -NW I 1 -Iq-1
\
a
A \-
1.
- b-----~ 1% -- .
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
+
.
-,-,
-
- 0
_
-
- J To 1.0 at r/R=
1.0
-
0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20 r/R
Ratio of minimum knuckle radius to crown radius, Pig. 7.18. Computed maximum stresses and strains in elliptical heads as American
Welding
Society.)
ratios of stresses and strains in attached cylindrical shell (12). (Courtesy of
138
Selection of Elliptical, Torispherical, and Hemispherical Dished Closures
Maker* prepared for the API-ASRIE committee and was later published (12). The data and curves of Hiihn were reproduced, and additional data from tests in this country were added. A curve was also included based on computations made by using Coates’s method for determining stresses in elliptical dished heads of various depth ratios. A comparison of this curve with the empirical curve of Hijhn indicated that the curve for torispherical dished heads should turn upward more abruptly than was specified by Hiihn. Therefore, the “proposed” curve was bent upward to the form given in Fig. 7.18 in order to give better agreement with some of the data for the low ratios of ri/rc. It should be emphasized that the curves for torispherical heads in Fig. 7.18 are based on tests in which stress-intensification factors were determined by observation of the pressure at which yielding occurred on the outside of the knuckle of the head, and are empirical. The “proposed” curve was included in the first edition of the API-ASME code, published in 1934, and has withstood the test of 20-years’ use. The 1956 edition of the ASME code (11) substituted Eq. 7.76 for evaluating the stress-intensification factor, W. w = $3 + dim
t=
preW 2fE - 0.2~
+c
(7.76) (7.77)
where W = stress-intensification factor for torispherical dished heads c = corrosion allowance, inches
al,, = -Mlbz c-ill =
a12
j,2 + M2 7-. p(J2J1 - J2i1) +
+
=
MlMz bl0 = + ~ 2 [ r(JzJ1
MI
bll =
DISHED
-
-1
J12 + J2’
-
Jzth)
+
Mz(Jk
+
Jzlbz)
p(jzJ1
- Jzi)
+
M’L(J&
+
J2j2)I
bl2 = 0 Ml = d12(1
- p2) ; 0 n
The coefficients (influence numbers) ala, all, ais, and blo, bll are given in Table 7.4. The quantities J1 and J2 are, respectively, the values at the junction of the real and imaginary parts of the hypergeometric function resulting from the theory of spherical shells. The quantities jr and is are the derivatives of J1 and J2, respectively (109). Because of the difference between the coefficients for a hemispherical head and for a flat plate, Eqs. 6.113 and 6.114 become Eqs. 7.78 and 7.79.
MO
Qo
a11 - +
a12 =
pd
Mo a4
pd2
+
u5 :; + u6
(7.78)
CLOSURES
A thin-walled spherical vessel theoretically requires only half the shell thickness of a cylindrical vessel of the same diameter to contain the same internal pressure. This is indicated by a comparison of Eq. 3.14 and 4.25, which define the theoretical hoop stress based on membrane theory. If large access openings for manways or nozzles for piping are to be cut into formed closures, the hemispherical head may be used to advantage because its greater strength will make less reinforcement required. Some consideration must be made of the thickness discontinuity that exists when a hemispherical dished head having lesser thickness than the shell is attached to a cylinder. The extent of bending and shear at the junction should also be considered. G, W. Watts and H. A. Lang (129) analyzed the stresses in a pressure vessel with a hemispherical head. The method used was the same used by these authors for flatplate closures and conical closures, described in Chapter 6. Dimensionless equations having the form of Eqs. 6.113 and 6.114 were used. However, for a hemispherical head attached to a cylindrical shell, the coefficient brs = bs = 0. For the cylinder ~4, us, u& b4, bg, and b6 are the same as for the cylinder in Eqs. 6.109, 6.110, 6.111, and 6.112 and are tabulated in Tables 6.2 through 6.6. The constants alo, ail, ar2, and blo, bll, blz for the hemispherical head were defined by Watts and Lang as follows: * Private communication.
Jzi)
JZJI - - JZJI
- 4
pd2
HEMISPHERICAL
Mz(J& +
- 0.0875
a10 - +
7.13
1
jg
blo p2 + bll$ + be = b4 p$ + bs: + bs
(7.79)
but bs = 0, b12 = 0; therefore
(a4 - alo)& -
Qo = pd
- ad(bs - bll) bd - ( a s - w)(br - blo) (7.80)
(a12
Mo = pd2
-(a12 (a4 -
ulo)(bs
- as)@4
- ho)
- bll) - (us - m)(br
- blo) (7.81)
7.14 STRESSES IN THE SHELL AT ITS JUNCTION WITH HEMISPHERICAL DISHED CLOSURES
The maximum shear stress in the shell at the junction is given by Eq. 6.121. The maximum combined axial stress
Table 7.4.
Coefficients
for
Hemispherical
Head
(129)
(Extracted from Transactions of the ASME with Permission of the Publisher, the American Society of Mechanical Engineers, 29 West 39th St., New York, N. Y.) d/h 4.00 10.00 20.00 30.00 40.00
a10
a11
a12
ho
bll
-6.9690
f1.8200
-0.0875
+3.6608 -0.5272
-16.7131
+2.8631
-0.0875
+5.7319
-33.2457 -49.7670 -66.2900
+4.0574 -0.0875 +4.9729 -0.0875 +4.7438 -0.0875
-0.5058
+8.1196 -0.5030 +9.9470 -0.5020 +11.4886 -0.5015
Location of the Maximum Stress
139
!n the shell is given by Eq. 6.122. The maximum combined circumferential stress is given by Eq. 6.125.
(7.84)
7.15 STRESSES IN THE SHELL IN OTHER PLACES THAN AT THE JUNCTION
The maximum combined axial stress is given by Eq. 6.125 with the same substitution for wall thickness, or
The maximum shear stress in the shell in other places than at the junction is given by Eq. 6.126, and its location is defined by Eq. 6.127. The maximum axial stress in the shell in other places than at the junction is given by Eq. 6.128, and its locat.ion is defined by Eq. 6.129. The maximum circumferential stress in the shell in other places than at the junction is given by: f&cum.
= $ + 0.733 s
(7.85j
The maximum combined circumferential stress is given by Eq. 7.86.
fcircum. = 5: + 1'9 /+ z r* +
(7.82)
a(fshear?
UllQO
For a hemispherical head the axial and meridional stresses are identical and are given by Eq. 7.87.
The location of this stress is given by: 1.072 ccc = za + __ P
(7.83)
where xa is from Eq. 6.129, and /3 is from Eq. 6.86.
7.18 LOCATION OF THE MAXIMUM STRESS
7.16 STRESSES IN THE HEMISPHERICAL HEAD AT THE JUNCTION The maximum shear stress in the hemispherical head is given by Eq. 6.121 with the modification that th, the thickness of the head, is used rather than t,, the thickness of the shell. or
The maximum stress in a vessel having a hemispherical closure is dependent upon the ratio of the thickness of the head to the thickness of the shell, as indicated in Fig. 7.19, in which the stresses are plotted as stress-intensification factors versus the ratio of th/ts. For ratios of head to shell thickness of from 0.6 to 2.0, the maximum stress is
1.4
L
0.4
0.6
0.8
1.0
1.2
1.4
Ratio of head thickness to shell thickness, Fig. 7.19.
(7.86)
7.17 STRESSES IN THE HEMISPHERICAL HEAD IN OTHER PLACES THAN AT THE JUNCTION
wherefshear’ = Eq. 6.126
0
)
Stress-intensification
1.6
1.8
t,,/t,
factors for vessels with hemispherical closures.
2.0
2.2
140
Selection of Elliptical, Torispherical, and Hemispherical Dished Closures
located in the shell and is the circumferential stress as defined by Eq. 7.82; it is equal to 1.037 times the hoop stress in the cylinder. For head-to-shell ratios of 0.6 and under the maximum stress is the circumferential combined stress in the head at or near the junction as defined by Eq. 7.86. The other stress-intensification factors plotted in Fig. 7.19 may be identified as follows: I aj(shell)
=
~rj(shell)
=
zaj(head)
=
I am(head)
=
7.19
maximum combined axial stress intensification in the shell at the junction, Eq. 6.122 maximum combined circumferential stress intensification in the shell at the junct,ion, Eq. 6.125 maximum combined axial stress intensification in the head at the junction, Eq. 7.85 meridional hoop stress intensification in the head in other places than at the junction, Eq. 7.87
PRACTICAL
CONSIDERATIONS
Figure 7.19 shows that the bending st.resses at or near the junction in vessels with hemispherical closures are much less significant than in vessels with flat-plate, conical, elliptical, or torispherical closures. If one judges on the basis of membrane theory, the hemispherical head has twice the strength of a cylindrical shell of the same diameter, as indicated by Eqs. 3.13, 3.14 (note that the hoop stress in a hemisphere is the same as the axial stress in a cylinder). This is shown in Fig. 7.19 where the curve I,, is equal to 1.0 when th/ts = 0.5. The maximum
combined stresses in the head and shell are equal at th/tp = 0.6. This is indicat,ed in Fig. 7.19 where the curves I em(shel1) and Z,j(l,e*d) intersect at Zh/Zs = 0.6. This, in turn, indicates that the optimum ratio of th/ts is 0.6. For such a ratio, the shell thickness may be evaluated by the thin-wall equation, and the maximum stress in the shell will be only 1.7% greater than the theoretical circumferential stress. If the head thickness is equal to 0.6 times the shell thickness, the maximum shell stress will be only 3.7% greater than the theoretical circumferential shell stress. For Zh/Zs ratios of less than 0.6, both the axial and the circumferential stresses in the head at the junction are increased rapidly as this ratio decreases. If the head is made with half the l.hickness of the shell, the combined circumferential head st.ress at the junction is 11.5 y$, greater than the theoretical hoop stress in the shell, and the combined axial stress in the head is 7.9 ‘% greater. This increase in stress is a result of t.he bending moments at and near the junction. If either of t,hese stresses exceeds the yield point of the material, plast.ir deformation will result, which will relieve the excessive st,ress. If some plastic deformation can be tolerated in high-stress conditions or in hydrostatic testing, a thickness ratio of 0.5 can be considered satisfactory. The ASME Code (I 1) gives the following relationship fc;: the thickness of hemispherical heads: Ih =
where E = joint efKrienc:y
PROBLEMS
1. A natural-gasoline stabiliziug colurnu is to be constructed of ASTM-A 285, Grade C steel. The column is to operate at 200 psi. The vapor leaves the top of the column at 160” F. The tower has a nominal diameter of 6 ft. Select a torispherical closure and au elliptical dished closure for this application if the allowable design stress is 13,750 psi and the minimum corrosion allowance is Sic in. (The jom. et iciency ff for a single-piece closure is loo%.) For the case of the elliptical closure, assuming a joint efficiency, E, of 85 y0 in the shell, drtrrmine the combined hoop stress-intensification factor curves in the shell and in the closure.
t ehe” = fEPrinaide - OJjp + c
pdi ..~___ + c IjE - 0.4p
(See Reference 11.)
2. For the conditions giver1 in problem 1, determine the combined meridional stress-intrusification factor curves in the shell and closure. 3. For the case of the torispherical closure in problem 1, assuming a joint efficiency of 8SyG in the shell, determine the meridional bending stresses in the knuckle resulting from the change in the meridional radius of curvature at the junction of the knuckle and crown. 1. For the conditions in problem 3, calculate the meridional bending stresses in the crown resulting from the change in the meridional radius of curvature at the junction of the knuckle aud crown. 5. For purposes of heat transfer it is desired to attach a steel hemispherical head !i iu. thick to a steel cylindrical vessel 1 in. thick and 10 in. in inside diameter. If the vessel is under a pressure of 1000 psi, calculate the maximum stress in the head. 6. Calculate the plot the maximum-shear-stress-intensification factor in the shell at the junction for various hemispherical closures as a function of th/ts when d/th = .40.
(7.88)
C H A P T E R
DESIGN OF CYLINDRICAL VESSELS WITH FORMED CLOSURES OPERATING UNDER EXTERNAL PRESSURE
A A
The relationships for the conditions beyond the critical length will be developed first. (See Timoshenko [42] .)
wide variety of chemical and petrochemical processes require equipment operating under partial vacuum. Examples are: vacuum condensers for evaporators and distillalion columns, vacuum columns such as lube oil columns, vacuum crystallizers, and so on. Such vessels are under external pressure from the atmosphere. Vessels are sometimes jacketed and heated by means of steam, Dowtherm, or other condensing vapors under pressure in the jacket; These this also produces an external pressure on the vessel. vessels are usually cylindrical with formed heads as closures. A cylindrical vessel under external pressure has an induced circumferential compressive stress equal to twice the longit udinal compressive stress because of external-pressure effects alone. Under such a condition the vessel is apt to collapse because of elastic instability caused by the circumferential compressive stress. The collapsing strength of such vessels may be increased by the use of uniformly :;paced, internal or external circumferential stiffening rings. From the standpoint of elastic stability such stiffeners have the effect of subdividing the length of the shell into subsections equal in length to the center-to-center spacing of the stiffeners. Long, thin cylinders without stiffeners or with stiffeners spaced beyond a “critical length” will buckle at stresses below the yield point of the material. The corresponding critical pressure at which buckling occurs is a function only of the t/d ratio and the modulus of elasticity, E, of the material. If the length of the shell with closures, I, or the distance between circumferential stiffeners, 1, as the case may be, is less than the critical length, the critical pressure at which collapse occurs is a function of the l/d ratio as well as of the t/d ratio and the modulus of elasticity, E.
8.1 ELASTIC STABILITY OF LONG, THIN CYLINDERS UNDER EXTERNAL PRESSURE A cylindrical shell under external pressure tends to deform inward as a result of the external radial pressure. The relationship between the radius of curvature, the product EI, and the bending moment producing curvature is given by Eq. 2.9.
Consider a cylindrical shell having an original radius of curvature of ro under no load. A local section of this shell under external radial pressure will have a new radius of curvature of P. The relationship between the bending moment, the two radii, and the product EI is given by:
M=EI
L-1 r) ( PO
Figure 8.1 is a diagram of a shell section before and after deformation under external pressure. The shell has an original radius of curvature of ro and at the point under consideration has a radius of curvature of P after deforming through a radial distance, w. Points a and b represent the limits of an elemental strip, ds, in the shell prior to deformation. The points a’ and b’ are the corresponding limits of the same elemental strip after deformation. The elemental strip subtends an angle of d0 before deformation and an angle of de -l- A0 after deformation. 141
142
Design
of
Cylindrical
Vessels
with
Formed
Closures
Operating
under
External
Pressure
Substituting Eqs. 8.5 and 8.7 into Eq. 8.3 for AdS and Ado, respectively, gives: 1-= r
de + $$ dS
W)
Substituting l/r0 for de/d& by Eq. 8.2a, gives:
1 1 -=1+w +$ r r> r0 (
(8.9)
Substituting Eq. 8.9 into Eq. 8.1 gives: 1 - - 1r r0
( >
Fig. 8.1.
Deformation of a cylindrical-shell section under external pressure.
= R) + @!f2 = - E;
Assuming rro = ro2, we find that !k+E2= -z
(8.10)
From Fig. 8.1 and small-angle relationships, ab = dS = ro de
(8.2)
therefore 1 de -=q,
Multiplying through by ro2 and substituting for Eq. 8.2e gives, therefore (42) : (8.11)
(8.2a)
dS
After bending 1 de + Ad0 -= P dS + Ad&’
(8.3)
Figure 8.2 shows a quadrant section of a cylindrical vessel under external pressure. The dotted lines show a possible deformation of this shell under the influence of external
where dS + Ads = length of element a’b’ If the small angle dw/dS is disregarded, for small-angle functions where the tangent of the angle equals the angle expressed in radians, de
=
dS
+ Ads
(8.4)
PO - w de (r. - w) = dS + Ads Subtracting Eq. 8.2 from Eq. 8.4 gives: A& = -w&j = -wds r0
(8.5)
Inspection of Fig. 8.1 indicates that the difference between the angles ($$ and ($$+sdS) is the same as the ,difference between the angles de and (de + Ado), or ($+$dS)-($)=(dO+AdO)-de
(8.6)
therefore (8.7)
Fig. 8.2.
Bending moments in o shell deformed by external pressure.
Elastic
Stability
pressure, p. In the deformed condition the bending Considmoment Ms and the force F will exist at point c. ering a circumferential element of unit longitudinal width, we find that the compressive force F will be equal to the pressure times the projected area, or F = p(uc) = p(ro - wo)
(8.12)
of
long,
Thin
Cylinders
under
External
(8.13)
w = A sin q0 + B cos q0 + pro2+pE;rc” (8.21) Introducing the conditions at c and g (Fig. 8.2) where the deformed shell is perpendicular to the axes, we find that 0 and
0 dw
Substituting for F in Eq. 8.13, by Eq. 8.12, we obtain: M = MO + p[(ac)(bc) - tee’]
(8.14)
Considering the two triangles abe and cbe, we find that
143
The solution to this differential equation is:
Taking a summation of moments about any arbitrary point, e, in the deformed shell gives: M = MO + F(bc) - p(ce)(he)
Pressure
de
e=*/2
=
0
Equation 8.21 may be differentiated and set equal to zero for the conditions of 0 = 0:
0
dw = A(cos qe) - B sin q0 = 0 de *=o
(ae)2 = (~b)~ + ( b e ) ’
and (ce)’ = (bc)2 + (be)2
therefore
A=0
Substituting gives:
For the condition of 0 = s/2 we have:
(ae)’ = (~b)~ + (ce)2 - (bc)2
= -B sin q0 = 0
= (ce)2 + (UC - b~)~ - (bc)2 (ae)2 = (ce)2 + (a~)~ - 2(bc)(ac)
therefore
or +(ae2) = 8(ce)2 + fan - (bc)(ac)
-Bsinq:=O
therefore (ac)(bc) - i(ce)2 = ~[(uc)~
- (ae)21
(8.15)
sin q: = 0
Substituting Eq. 8.15 into 8.14 gives: M = MO + &~[(uc)~ - (ue)2]
(8.16)
l3ut UC = r. - ~0, and ue = PO + (-w). Therefore, substituting into Eq. 8.16 gives: M = MO + +p-P[(Po2
- 2row0 + wo2) - (ro2 - 2r0w + w2)]
2 - wo = MO + p row - row0 + S-r
[
1
M
O
+
pr0(w
-
w0)
(8.17)
Substituting Eq. 8.17 into Eq. 8.11 gives: d2w M0r02 WC -__ z+ EI ~+w(1+~)~pro3w0~~0r02
(8.18)
(8.22)
These unique values of q define corresponding unique values of p in Eq. 8.19. The lowest of these values is q = 2 and defines p critical. Equation 8.22 is satisfied when q is equal to 2 or multiples of 2. Substituting this value of q in Eq. 8.19 and solving for p gives (42) Ptheoretioal =
Furthermore, the small quantity $(wo’ - w2) may be dis,*egarded. Therefore M =
and therefore
3EI 24EI -=-----ro3 do3
(8.23)
Equation 8.23 expresses the theoretical or critical load per unit circumferential length of unit width of circumference. For a strip of unit width the critical load is the pressure at which buckling theoretically occurs. If the ring is a part of a long cylindrical shell, the adjacent metal or either side of the ring will offer restraint to the longitudinal deformation of the strip. To allow for this restraint Eq. 8.23 may be divided by (1 - ~1~). (See Eqs. 6.la and 6.12.) To express the critical stress in terms of the shell thickness, t,, a substitution for I may be made for a rectangular strip.
I=!!?
Let (8.19) Therefore (42)
$+q2w = pro3wo EI- Moro2
(8.20)
12
where b = 1 for a strip of unit width. stitutions in Eq. 8.23 gives:
Making these sub-
(8.242
Design of Cylindrical Vessels with Formed Closures Operating under External Pressure
144
where 1, = critical length, inches d = diameter of shell, inches t = shell thickness, inches 8.3 COLLAPSING PRESSURE OF VESSEL SHELLS W!Tbi CIRCUMFERENTIAL STIFFENERS
For vessels in which circumferential stiffeners are spaced at less than the critical length, the coefficient of Eq. 8.25 must be modified according to the proportions of the vessel (158, 159), or Ptheoretical
=
KE
f
3
0
Applying a factor of safety of 4 gives: (8.30) dtstance between stiffeners (+I = ( diameter of vessel > Fig. 8.3.
Collapse coefficients for cylindrical shells under external pressure
(134).
where fi = roefiicient accordin g IO the proportions of the vessel, as indicated in Fig. 8.3 (134). (Noie that the minimum value of K is 2.2 as by Eq. 8.25.)
Substituting Ey. 8.29 into Eq. 3.14, (disregarding corrosion) we find that the circumferential compressive stress from external pressure at which collapse occurs is:
Substituting for Poisson’s ratio, p = 0.3, gives:
Equation 8.24 gives the theoretical “critical” pressure (external) at which a long cylindrical vessel will buckle. This equation is the generally accepted theoretical formula of Bresse (130) and Bryan (131) for long, thin tubes under external pressure. Stewart in a number of tests using commercial tubing and pipe investigated the applicability of Ey. 8.25 and found what collapse occurred at a critical pressure of 27 y0 lessI han the theoretically predicted pressure. For design of long, thin cylindrical vessels operating under external pressure, a factor of safet.y of 4 may be applied to Eq. 8.25. giving (1,%2) : t 3 Pallo\\nlIle
8.2
=
0.55E
0
(8.26)
2
CRITICAL LENGTH BETWEEN STIFFENERS
Equations 8.25 and 8.26 apply to long, thin cylinders under external pressure without circumferential stiffening rings or with the stiffening rings spaced at or beyond the “critical length.” To make allowance for the added restraint offered by stiffeners spaced at less than the critical length, the critical length may first be evaluated. The expression for the critical length was first developed by Southwell (133). Southwell’s analysis involves a 15-row determinant solution and is beyond the scope of this text. The relationship resulting from Southwell’s analysis is given by: 4~ 4% 1, = - - - - (i/i-T)(d
l/dlt,
(8.27)
Substituting, p = 0.3, gives: 1, = l.lId m
(8.28)
(8.31) Equations 8.25 and 8.31 may he plotted for convenienca of solution as shown in Fig. 8.4 (IL). The inflections in the parameters occur at the critical lengths, which correspond to the critical lengths determined by Eq. 8.28. The vertical parameters of d/t above the inflections represent the region where the spacing between stiffeners exceeds the critical length and the collapsing pressure is independent of the l/d ratio. Equation 8.25 applies in this region. The inclined parameters below the inflection represent the region where stiffeners have an effect and the collapsing pressure is a function of the l/d ratio as expressed by the coefficient K in Eq. 8.29. It is significant to note that Fig. 8.4 is general and is independent of the material of construction. To use the chart to predict the ratio of l/d at which collapse occurs, it is necessary to know the value of (f/E) for the material at the temperature under consideration. Figure 8.5 shows a group of stress-strain curves for se>,era1 materials and indicates that for mild steel at room temperature the stress-strain curve can be approximated by two straight lines (135). Unfortunately, this approximation is the exception rather than the rule and is limited to carbon and low-alloy steels at temperatures below 500” F. Carbon steels at temperatures above 500” F and other materials such as high-alloy steels and nonferrous metals have nonlinear stress-strain curves with a variable modulus of elasticity and no definite yield point. Figure 8.6 shows the variation of the modulus of elasticity, E, for plain carbon steel and austenite steel as a function of temperature. The error caused by using a constant modulus of elasticity
Collapsing Pressure of Vessel Shells with Circumferential Stiffeners
145
6
4 3
F i g . 8 . 4 . G e n e r a l c h a r t f o r cd-
lapse
of
vessels
pressure
under
showing
between
the
,d/t and
I/d
property f/E
(l/d)
external
*
relationship
dimensional
1.0 0.8
ratios
and the physical
0.6 0.4 0.3 0.2
(135).
I llllll
I /llll/ I IllIll
0.08 0.06 0.000001
2
3
0.00001 8
4 5 6
I
l/llll
2
3 4 5 6
0.0001 8
! 3 4 5 6 8 0.001 ;
2
3
456
8
2
3 4 5 6 8
0.01
0.1
t = (f/E)
80,000 p
:
70,000
I : I
Structural silicon steel
- V-
60,000
I I
27 ST aluminum alloy
Fig. 8.5. Typical stress strain curves
for
several
materials.
(Extracted
from
the
11351 with permission
ASME
Transactions
of
of the publisher, the American Society
29
of West
Mechanical 39th
St.,
Engineers, New
York,
N.Y.)
i
aluminum-annealed 0
r-- - -- --T==-‘
._-__-
-
-\
3
0.002
\
0.004
\IT
0.008 0.010 Strain, inches per inch determined on 8-in. gage length
0.006
--~-
-
0.012
0.014
0.016
-
146
Design
of
Cylindrical
Vessels
with
Formed
Closures
32 30 ‘g 28 2 26 .o 2 24 q 22 z gj 20 g, 1 8 16 14
0
200
400
600
800
loo0
Temperature, deg, F Fig. 8.6. function
Modulus of elasticity of plain carbon and oustenitic steels as o of temperature.
(Extracted from the 1956 edition of the ASME
Boiler and Pressure Vessel Code, Unfired Pressure Vessels [l 11, with permission
of
the
publisher,
the
American
Society
of
Mechanical
Engineers,
29 West 39th St., New York, N.Y.)
for materials having nonlinear stress-strain curves may be avoided by using a “tangent” modulus of elasticity, that is, the slope of the stress-strain curve at the stress and temperature under consideration. For convenience, to avoid the necessity of measuring the tangent modulus, the simultaneous value of (j/E) can be plotted versus stress in terms of pressure and dimensions of the vessel. In designing a vessel for a given value of (j/E) based upon the material of construction and operating temperatures, one bases his design upon the strain at which collapse occurs rather than upon an allowable stress. Using a design factor of safety of 4, in which the allowable pressure is considered to be one fourth of the theoretical pressure at which collapse occurs, we obtain: Ptheoretical = -
w --
20,000
10,000
8WJ cq II N” 3 JO00 II z 2 3,000 % 3 2,000
Rearranging Eq. 8.32, we obtain:
0
d W
=
Pressure
Given: A fractionating tower 14 ft in inside diameter by 21 ft in length from tangent line to tangent line of the closures. The tower contains removable trays on a 39-in. tray spacing and is to operate under vacuum at 750” F. The material of construction is SA-283, Grade B plain carbon steel, which has a yield strength of 27,000 psi (see Table 5.1.). The required thickness of the shell will be determined both without stiffeners and with stiffeners located at the tray positions.
= theoretical external pressure at which collapse occurs, pounds per square inch ~~11~~. = allowable external pressure, pounds per square inch
Ptheoretical
External
8.4 EXAMPLE DESIGN OF A SHELL
where Ptheoretical
=
under
horizontal line and corresponds to the horizontal line of Fig. 8.5. The 650” F line shows a break at a lower value of (j/2) and levels off thereafter. Figures 8.4 and 8.7 can be used to determine the safe external working pressure of an existing vessel under external pressure. The dimensional ratios l/d and d/t are first computed and the corresponding value of j/E is determined from Fig. 8.4. This value is used with Fig. 8.7 to determine the value of the quantity (panow.) (d/t) from which the value of psnoW. is directly computed. In designing a vessel the dimensional ratio Z/d is usually known, but the value of d/t is unknown as t is to be determined. The value of t must first be assumed and the calculated safe allowable working pressure checked with the desired working pressure as indicated above. As both of the curves of Figs. 8.4 and 8.7 have a common abscissa of j/E, they may be conveniently superimposed as indicated in Figs. 8.8 and 8.9 (for plain carbon steel up to 900“ F).
(8.32)
4Psllow.
d
Operating
2‘B =
or 1,000
(8.33)
Using Eq. 8.33 and appropriate stress-strain diagrams, we can determine simultaneous values of (j/E) and (j/2) and can plot them as shown in Fig. 8.7. The 100” curve in Fig. 8.7 has an inclined line of constant slope below the yield point because at 100” F and below the modulus of elasticity does not vary with stress. Above the yield point in the plastic region the stress-strain curve is nearly a
800
Strain, e = (f/E) Fig. 8.7.
Chart for plain carbon steel showing allowable pressure under
external loading at 100’ F to 900’ F. ASME
[135]
(Extracted from Transactions of the
with permission of the publisher, the American Society of
Mechanical Engineers, 29 West 39th St., New York, N.Y.)
Example Design of a Shell
2.0 1.8 1.6 1.4
0.90
0.80
I
III
I
III
I\1
I
0.60 0.50 0.40 0.35
0.06
I
I
I I -r--1 I II
0.05
’
! ’ !
! 1 ! ! !‘I MCtOr A = f/E = B
Fig. 0.0.
Combined chart for determining thickness for carbon-steel shells under external
from the 1 956 edition of the
pressure-for yield strengths of 24,000 to 30,000 psi.
(Extracted
ASME Boiler and Pressure Vessel Code, Unfired Pressure Vessels [1 11, with permission of the publisher, the American Society
Mechanic :C II Engineers, 29 West 39th St., New York, N.Y.)
of
14%
Design of Cylindrical
Vessels with Formed Closures Operating under External Pressure
40,000 35,OilO 30,000 25
25,000
,-7OOOF
7-
-
/-8OO’F .--900°F
I I, I II
I
6.0 2 5.0 08 35 4.0 1 7 3.5 gw E p 3.0 g g 2.5
r,
I1.111111 I I
I\ I
I
I
I
I I
I
I
I I I I
I
! I
I
I I
I
I
I I
I
14,000 12,000 10.000
,,,,I I
I I I 6.000 I11111
I I I II
5,000
m 2,500
5
-2 g 2.0 .f : :f 22 1.4 *:! . @ 1.2 Es 1.0 g; 0.90 0 $ 0.80 B z 0.70 $ 0.60
kt+i# 800 o--
0.50
500
0.40 0.35 0.30
400 350 300
0.25
250
0.20 0.18 0.16 0.14 0.12 0.10 0.09 0.08 0.07 0.06 0.05 0.00001
Kg. 8.9.
2
3
4
5678
0.0001
2
3
4
5678
2
0.001 Factor A = fJE = e
3
4
1
5 6 7 8
0.01
Combined chart for determining thickness for carbon-steel shells under external pressure-for yield strengths of 30,000 to 38,000 psi.
from the 1956 edition of the
(Extl-acteL
ASME Boiler and Pressure Vessel Code, Unfired Pressure Vessels [l I], with permission of the publisher, the American Society of
Mechanical Engineers, 29 West 39th St., New York, NY./
r -
Design 8.4~ Required Shell Thickness without Stiffeners. The determination of the shell thickness is a successive-approximation calculat,ion. Assume a shell thickness of x in.
1
2 1 X 12
252
= ~ = 1.49
do = (14 x 12) + 1.25
pallow. = ‘&&$ = 16.1 psi
B
(8.33)
- dolt
= 2&$c
=
8.48 psi
Since this pressure is considerably lower than the desired external pressure of 15 psi for full vacuum, the calculation must be repeated with a greater thickness assumed. Therefore assume a shell thickness of lKs in. 252 1 -YE ~= 1 6 9.63 do
1.49
Entering Fig. 8.8 with Z/d = 1.49 and moving to do/t = gives E = 0.00029 in. per in. Moving vertically to the ‘i50” F material line and horizontally to the right gives B = 3400; therefore
208.5
Pallow.
3400
= ~ 208.5 = 16.3
wt = vrdlp
1 = 39
1 do
-= ‘10 t
I
Shell
DESIGN
OF
CIRCUMFERENTIAL
STIFFENERS
In designing circumferential stiffening rings for VA&S under external pressure each stiffener is considered to resist the external load for a (1/2 distance on either side of I he ring (where E is the spacing between rings). Thus the load per unit length on the ring at collapse is equal t,o Z(ptheorrtical). We may rewrite Eq. 8.23 noting that in t,his equation iho term 1 is taken as unity. Therefore ~theoretical(~) =
‘9’
where P = load on combined shell and stiffener in pounus per inch of circumferential length. Or I = Pth.o;;;c
Substituting Eq. 3.14, f = pd/2t, and Eq. 6.1, E = *f/E, gives: (8.3.5)
The required shell thickness of 1%~ in. far the condition of no internal stiffeners can be materially reduced by the inclusion of stiffening rings at the tray locations. Assume a shell thickness of 7/i6 in. Required
This represents a saving in shell steel of (30,700 - 16,450) = 14,250 lb. However, this is offset in part by the weight. of the stiffening rings. This point is covered in the example design in the section following the stiffening-ring sectioa The weight of the rings of a satisfactory design was found to be 2700 lb. Therefore a net saving of (14,250 - 2700) or 13,100 lb of steel is realized.
x 21 x 33.15
wt = 30,700 lb 8.4b
Therefore
Multiplying by t/t and rearranging gives:
psi
Therefore a 1Ws in. plate, if available, is adequate. A shell plate of this thickness weighs 33.15 lb per sq ft. The shell weight is:
wt = 3.14 x 14
(which is adequat.e)
= 16,450 lb
p =
169.63 do -= __ = 208.5 13/16 t
Thickness
with
149
the shell weight = 3.14 X 14 X 21 X 17.85
8.5
Therefore Pallow.
Stiffeners
The weight of the shell is 17.85 lb per sq ft.
Enter Fig. 8.8 with l/d = 1.49 and move horizontally to intersect the diagonal line for d/t = 271; this gives: E = 0.0002 in. per in. Move vertically to the material line for 750’ F (interpolating between the 700” F and 800” F material lines) and then move horizontally to the right-hand side of the chart and read B = 2300. The maximum allowable external pressure for the assumed shell thickness of $4 in. is: =
Circumferential
Enter Fig. 8.8 with an l/do of 0.231 and move to a do t of 386; this gives: E = 0.0012 in. per in. Move verticall\ to the 750” F line and to the right to give B = 6200. The maximum allowable pressure for t,he assumed shell thickness is:
169.25
169.25 d-= o ~- = 271 t 0.625
Pallow.
of
Stiffeners.
in.
The moments of inertia of the stiffening ring and the shell act together to resist collapse of the vessel under extcl,rl;t: pressure. Timoshenko (42) has shown that the combiurd moment of inertia of the shell and stiffener may be cotisidered as equivalent to that of a thicker shell, or t,=~+$Lt++
(II.:iC!)
u
39
where t, = equivalent thickness of +ll, irlchcs A, = cross-sectional area c:! , ,I,? ~it,c,rlniferrnli:ll stiffener, square inches 1 = tl, = distance betwecu ( il,(‘lllilfei.(~lltii?; stiii’erle:,a, inches
39
(14 x 12) + 0.875 = i%%%
= 0231
168.875 ~ - 386 0.4375
\
\
\I /
_ -__-.-.-.T_~.~
Design of Cylindrical Vessels with Formed Closures Operating under External Pressure
150
24.1 in.4 (see Appendix G) and A, = 3.58 sq LB Substituting into Eq. 8.39 gives:
Substituting Eq. 8.36 into Eq. 8.35 gives: (8.37) where Z = required moment of inertia of stiffening ring, inches’ Equation 8.37 is the same as that specified by the 1956 ASME code (11) for stiffening rings for vessels under external pressure except that the coefficient in the denominator is 14 in the code equation rather than 12 as in Eq. 8.37. The value of 14 in the code equation may be approximated empirically. In general, the combined moment of inertia of the stiffener and the shell together varies from 30% to 7Oa/, greater than the moment of inertia of the stiffener alone (10). Using a conservative allowance of a 30y0 increase in the Z of the stiffener when combined with the shell and introducing additional safety in design by increasing the load by lo%, we obtain:
B =
Enter the right side of Fig. 8.8 with B = 4790 and move horizontally to the material line for 750” F; then move vertically to the bottom of the chart where E = c.00045. Substituting into Eq. 8.38 gives: (169)2 X 39
>
0.00045
14
Z = 18.95 in.4
As the required moment of inertia is less than tnat provided by the assumed 7-in. channel, the design is satisfactory. The weight of five such stiffening rings is: Wt of rings = 5 X 3.14 X 14 X 12.25
1.1 d21
= 2,700 lb
(1.3)(12)
The weight of the shell is: (8.38)
Wt of shell = 16,450 lb + 2700 lb
where
= 19,150 E = unit
strain
(see Figs. 8.8 and 8.9)
Equations 8.37 and 8.38 give the moment of inertia required for the stiffening for the same collapsing pressure as that of the vessel designed by use of Fig. 8.8. The allowable operating pressure is one-fourth the pressure at which collapse theoretically occurs. In order to utilize Fig. 8.8 in the design of stiffening rings, it is necessary to proceed in the opposite direction, entering the figure with the value of B. If B is expressed in terms of the equivalent shell thickness, leq., (including the contribution of the stiffening ring), then by Eq. 8.33, B = J = ~eoretic~ldo
2
ux&)
=
mlo\v.do
t + (.AIllil
(8.39)
If gaps are placed in the stiffening rings, they should be staggered between alternate rings. In this case the distance between stiffening rings should be taken as twice the ring spacing to allow for the lack of continuous support of the shell at the gaps in the stiffening rings. Permissible gaps in the stiffening rings of external-pressure vessels are specified by the ASME code (11). (See Chapter 13 on code vessels.) 8.6
0.4375 + $8
z =
I=-
Therefore
15 X 168.875 pallow.do = 4739 t + (4,/O = 0.4375 + (3.58/39)
EXAMPLE DESIGN STIFFENERS
OF
CIRCUMFERENTIAL
The design of the circumferential stiffening rings for the 14-ft-diameter fractionating tower given previously (Required Shell Thickness with Stiffeners) in this chapter will be illustrated. The required shell thickness was found to be xG in., and the stiffeners spaced at 39 in. The tower operates under full vacuum. The calculation of the required moment of inertia of the stiffening rings requires successive approximations. Assume a 7 in. channel weighing 12.25 lb per ft. 1 =
The total weight of the shell with stiffeners is 17,600 lb, compared to a total weight of 30,700 lb if a shell without stiffeners is specified; this represents a saving of l&100 lb steel. 8.7
OUT-OF-ROUNDNESS
OF
SHELLS
Any out-of-roundness after fabrication of a vessel designed for external pressure will reduce the strength of the vessel. The out-of-roundness results in increased stress concentrations, and the effect of external pressure is to aggravate the condition. Thus a shell of elliptical shape or a circular shell, either dented or with flat spots, is less strong under external pressure than a vessel having a true cylindrical shape. The following procedure may be used to determine the additional stress from elliptical out-of-roundness. In reference to Fig. 8.2, the dashed section may be COGsidered to be a deformed cylinder in which zag is the maximum eccentricity radially inward. Timoshenko (136) has shown that the initial radial displacement, w’, at any point may be assumed to follow the relationship: WI = wo cos 28
(8.40)
If a uniform external pressure, p, is superimposed upon this initially deformed cylinder, an additional radlai displacement, w, will result,. This displacement w was previously defined by Eq. 8.11. d2w s+w=+
(8.11)
Noting that D, the flexural rigidity of a plate as given by Eq. 6.15, may be substituted for EZ, we obtain: -12(1 - &M? Ets3
(8.41)
Elastic
Stability
of
Hemispherical
To determine the bending moment M in the shell, consider a strip in the circumferential direction of unit width. By Eq. 8.17 for the region A to g (see Fig. 8.2)
and
Torispherical
@mwo
and for t.he region c to A (see Fig. 8.2)
Substituting into Eq. 8.41 gives:
Et38
pro3
Ptheoretical
f
1 (8.45)
8 However, by Eq. 8.24, E
t 3
_ p2I ;
0 Substituting into Eq. 8.45 gives: d’w - d02
= -32~~ cos 28 (8.46)
=
>
151
P
M = pro(w + wo)
$+w=
Closures
For a longitudinal strip of the shell of unit width to = 1)
(8.42)
A4 = pro(w - wo)
Dished
ELASTIC STABILITY OF HEMISPHERICAL TORISPHERICAL DISHED CLOSURES
AND
Formed closures under external pressure are subject to failure by elastic instability as are shells. Equation 4.33 applies in the case of hemispherical or torispherical heads and gives the theoretical pressure at which collapse would occur because of elastic instability.
I
Timoshenko (42) has shown that by substituting Eq. 8.24 into Eq. 8.45 and applying the condition at A of Fig. 8.2 (45’ position), the following solution to the differential equation Eq. 8.46 is obtained: WOP
w =
Ptheoretical
cos 28 -
(8.47)
P
= theoretical collapsing pressure (Eq. 8.24) p = external pressure acting on shell, pounds per square inch
where Ptheoreticsl
The bending moment at point A of Fig. 8.2 is zero, whereas the maximum bending moment occurs at 0 = 0 and at 0 = g, where 123 lnax =
pro
wo +
WOP Ptheoretical
-
P> (8.48)
This bending moment resulting from external pressure acting on an out-of-round shell produces a stress, f,,,:
Ptheoretical =
2 E(t)2
(4.33)
r2 d3(1 - p2)
or
t theoretical
= r 43(1 - ~‘1
dp/2E
(8.53)
Applying an approximate design factor of safety of 4.4, that is, using a thickness 4.4 times as great as that at which buckling theoretically occurs, we obtain:
th = hit theoretical
=
4.4r dp/2E g3(1-- + c (8.54)
For steel construction where p = 0.3, +
C
= 4r dp/E + C
(8.55)
where th = thickness of head, inches p = maximum external pressure, pounds per square inch c = corrosion allowance, inches E = modulus of elasticity at operating temperature, pounds per square inch r = radius of dish for hemispherical and torispherical dished heads, equivalent head radius for elliptical dished heads, inches
152
Design of Cylindrical Vessels with Formed Closures Operating under Externol Pressure
As Ey. 8.55 contains the modulus of elasticity, E, which may decrease aL elevat,ed temperatures as a function of stress, it is convenient to use Fig. 8.8 in a manner similar to that for cylindrical shells. The dashed line labeled To use “Sphere line” of Fig. 8.8 is shown for this purpose. the same chart, the scale for the sphere line is modified in that, the vertical axis is now equal to r/100& where P is the radius of curvature (outside of head) and th is the head Figure 8.8 is used to determine thickness (both in inches). B, the same procedure being used as for shell design. The maximum allowable pressure, psllou,., is then determined by Eq. 8.56.
B Pallow.
8.9
EXAMPLE DESIGN CLOSURE
OF
=
(8.56)
-
r/th
A
HEMISPHERICAL
DISHED
A hemispherical closure will be designed for the vessel described in the section entitled “Example Design of a Shell.” The design is a successive approximation because the tangent modulus of elasticity at this temperature (750” F) is also a function of the stress, f. Radius of curvat,ure
= 9 = 84.5 in.
Assume a head thickness of ?x in.
84.5 rc ~~~~ = 2.7 100th = 100x 0.3125 Enter the left-hand side of Fig. 8.8 at a value of 2.7 and move horizontally to intersect the sphere line at f/E = 0.00044. Move vertically to the material line for 750” F (interpolating between 700 and 800” F) and then move horizontally to read B = 4800. The maximum allowable external pressure for the assumed shell thickness of x~ in. is:
B Pallow. = -
Pallow.
4800 = 17.8 2.7 x 100
(which is greater rhan
psi;1
= -~
14.7 psia) Therefore the thickness assumed is satisfactory. 8.11 ELASTIC STABILITY OF ELLIPTICAL DISHEC CLOSURES UNDER EXTERNAL PRESSURE
The radius of curvature of an elliptical dished closurc~ changes aboul the meridian of lhe head. To use the prrvious relationships for elliptical closures, an eqnivaienl radius of curvature must be used. The radius of curvat ur’e of an elliptical dished head is maximum at the center of the head and at this point is equal to twice the radius of t hc> shell for a head having a major-to-minor-axis ratio of 2.0. Design of the head based on this maximum radius of curvature would result in considerable overdesign be-ause thrk radius of curvature decreases as the point under consideration is moved away from the center toward the junrt.ion with the shell. This decrease in the radius results in an increase in rigidity and greater elastic stability. Thus, aI1 elliptical dished head has greater elastic stability than a torispherical dished head having the same diameter, thickness, and radius of curvature at the center of the head. As the radius of curvature of an elliptical dished head varies along the meridian, an average radius may be used. How-ever, the average must not be taken too far from the centr[ of the head, which is the least stable point on the head. Table 8.1 lists the equivalent radius of curvature as a funrtion of the major-to-minor-axis ratio for heads of vessels under external pressure (11). (Not same as Code.) 8.12
EXAMPLE DESIGN CLOSURE
OF
ELLIPTICAL
DISHED
An elliptical dished closure, a/b = 2.0, will be designed for the vessel described in the section entitlerl “Example Design of a Shell.” This design also involves successive approximation. From Table 8.1,
r/th
Pallow. =
rc - = 0.90 d
4800 cc 17.8 --2.7 X 100
r, =
(0.90)(169) = 152.1 in.
As the vessel is to be designed for 1 atm. (14.7 psia) the assumed head thickness is satisfactory. 8.10
EXAMPLE DESIGN CLOSURE
OF
A
TORISPHERICAL
DISHED
Table 8.1. Equivolent Radius of Curvature to Be Used for Design of Elliptical Dished Heads under External Pressure (11)
A torispherical closure will be designed for the vessel described in the section entitled “Example Design of a Shell.” This design is also a successive approximation. Radius of dish (radius of curvature) = 169 in. Assurlle a head thickness of $6 in.
(Kxtracted from the 1956 Edition of the ASME Boiler a11t1 Pressure Vessel Code l~nfirrd Pressure Vessels, with Permission of the Publisher, the American Society 3f Mechanical Engineers, “9 \\‘est 39th St., New York, N.Y.) ,\laJor-to-lninor-axIs
u/h
ratio,
hvg radms of curvature P< -’ d b~essel diametw
3.0
2.8
2.6
2.4
1.36
1.27
1.18
1.08
1.6
1.4
1.2
1.0
0.73
0 65
0.57
0.W
PC = 2.7 ~ 100th Proceeding as before, B = -1800. (See example design of shell.)
Major-to-minor-axis ratio. u/b Avg radius of curvature rC _ Vessrl d i a m e t e r ‘7
2
2
2 0
1 II
0.99
0.90
0.81
Pipes and Tubing under External Pressum Assume a head thickness of gj6 in. -~ = 152.1/(100)(0.5625) = 2.7
100th
Proceeding as before, B = 4800. (See example design of shell.)
Thereiore the assumed thickness of x/i6 in. is satisfactory. 8.13 ELASTIC STABILITY OF CONICAL CLOSURES UNDER EXTERNAL PRESSURE Conical closures under external pressure can be classed in three groups. If the apex angle is small (45O or less) the conical closure is considered to behave as a cylindrical shell having the same diameter as the large end of the cone and a length equal to the axial length of the cone, provided the cone has no stiffening rings. If circumferential stiffening rings are used, the metal thickness may be decreased in each successive section as the apex is approached. In this case, each section is designed by using the greatest diameter of the section as the equivalent shell diameter, D, and t.he axial length between stiffeners (center to center) as the equivalent shell length between stiffeners, L. For conical heads having an intermediate apex angle (45’ to 120’) the same procedure is followed except that the diameter at the large end of the cone is taken as the length of the equivalent cylinder if no circumferential stiffeners are used. If circumferential stiffeners are used, the procedure is the same as for stiffened cones with apex angles of less than 45”, described above. For flat cones having apex angles greater than 120”, the conical head is designed as a flat plate having a diameter equal tc the largest diameter of the cone.
153
8.15 PIPES AND TUBING UNDER EXTERNAL PRESSURE The relationships presented earlier for long, thin cylinders under external pressure are conservative. Tubing and pipe usually have (t/do) ratios greater than 0.02 and as a result are subject to failure by elastic-plastic buckling rather than by elastic failure. Because of this and the general uniformity of commercial tubing and pipes, more liberal values of the allowable external working pressure may be permitted than for the case of vessel shells. The ASME Special Research Committee on Vessels under External Pressure reviewed this problem with the object of increasing the value of allowable external working pressures (213). Stewart (132) developed the following empirical relationship for the collapsing pressure of a steel pipe having a yield strength of 37,000 psi at room temperature: p = 86,670 G 0do
- 1386
(8.57)
The ASME code committee revised Stewart’s formula (213) to include the effect of material having a yield point other than 37,000 psi as follows:
p = 2.344+) - 1.064.f~
f;r
0.04
0.24
(8.58)
10,000 8,000 6,000
8.14 EXAMPLE DESIGN OF A CONICAL CLOSURE A conical closure having an apex angle of 45” and without stiffeners will be designed for the vessel described in the section entitled “Example Design of a Shell.” This design is also a successive approximation. &sume a thickness of I’{6 in.
84.5 l=A!.c =-~__. tan o(
do th
0.4142
= 204 in.
169 - 246 0.6875
Entering Fig. 8.8 with l/do = 1.21, move horizontally to do/& = 246 intersecting curve at f/E = 0.00030. M o v e vertically to 750” F material line to read B = 3700. .P,lhv.
B = G _- 2s’100 46 = 15.05 psia
Therefore. the thickness of 1 ?.I6 in. is satisfactory.
E- UJUU
j
,000
800 600 ; 500 F 400 i 300 B F 200 3 E 100 80 60 50 40 30 0
0.08
0.12
0.16 0.20 t/D ratlo
0.28
0.32
Fig. 8.10. A l l o w a b l e w o r k i n g p r e s s u r e s f o r t u b i n g a n d p i p e u n d e r e x t e r n a l pressure (213).
154
Design of Cylindrical Vessels with Formed Closures Operating under External Pressure
where p = collapsing pressure, pounds per square inch t = tube thickness, inches d = tube outside diameter, inches fy = yield strength at operating temperature, pounds per square inch E = modulus of elasticity at operating temperature, pounds per square inch The code committee considered that the modified Stewart formula, given by Eq. 8.58, was conservative when used for (t/do) ratios up to 0.14. No extensive data and no theoretical relationships exist for the collapsing pressure of very thick tubes having a t/do ratio of 0.30 or greater. However the committee was of the opinion that such thick tubes fail by plastic yielding rather than by collapse. On this basis the allowable pressure for tubes having a t/do ratio of 0.30 or greater was calculated upon a modified hoop-stress relation as given by Eq. 8.59. 4.fY t p=1.05 0 d,
(8.59)
Equation 8.59 is based upon thin-wall theory (see Eq. 3.14) and the assumption that the yield strength is one half of the ultimate strength. A 5% reduction factor is included in the denominator on the basis of both theoretical considerations and experience. For the intermediate range of t/do ratios lying between 0.14 and 0.30, a gradual transition was selected by the committee (213). By selecting allowable working stresses (w.s.) equal to 40% of the yield strength for steel and allowable working pressure equal to one fifth of the collapsing pressure as determined by Eq. 8.58, Fig. 8.10 was constructed. In reference to Fig. 8.10, the transition zone was drawn as a straight line from t/do = 0.14 to t/do = 0.30 for each of the curves. A cross plot of this curve with numerous parameters of t/d is presented in Fig. 13.5 of Chapter 13 for convenient use in design applications. Although Fig. 8.10 and Fig. 13.5 were prepared for mild-steel tubes, these figures can be used for other ferrous materials when a factor of safety of either 4 or 5 is employed and for nonferrous materials when a factor of safety of 5 is employed.
PROBLEMS
1. Determine the maximum allowable vacuum that can be applied to a low-carbon-steel cylindrical vessel 10 ft in outside diameter with elliptical dished heads (k = 2.0). The bead and shell thickness are both 36 in., and the vessel is 30 ft long (tangent line to tangent line). 2. Determine the minimum number of equally spaced circumferential shell stiffeners required to permit use of full vacuum with the vessel described in problem 1. 3 . (a) Determine the minimum required moment of inertia for the stiffeners required in problem 2. (6) Select the minimum-weight channel suitable (see Appendix G) for this purpose, 4. The shell for a vacuum crude tower 30 ft in diameter is constructed of low-carbon-steel plate 1x6 in. thick. Circumferential shell stiffeners are to be located 6 ft apart. Determine the required moment of inertia for the stiffeners, and suggest a possible design for the stiffeners. A corrosion allowance of 41s in. is required. 5. A vacuum crystallizer 12 ft in diameter is to be fabricated of mild steel and is to have a 60’ cone (apex angle) at the base and a torispherical closure at the top. The distance from the junction of the cone with the shell to the point of tangency of the top closure is 18 ft. Specify the design for: (a) the torispherical head. Cb) the shell, and (c) the cone.
I
\
\
\I
/
C H A P T E R
90
I-1 DESIGN OF TALL VERTICAL VESSELS
Fig. 9.2 shows the welding of an inside longitudinal seam of a vessel shell. Figure 9.3 shows a completed welded vessel entering the oven for stress relieving.
T
u a!1 vertical vessels may or may not be designed to be self-supporting. The design of self-supporting vertical vessels is a relatively recent concept in equipment design; high structures were formerly stabilized by the use of guy wires. The self-supporting type of tower is widely used today since it has been found uneconomical to allocate valuable space for the wires of guyed towers. Furthermore, the esthetic appearance of a clean-looking plant has been recognized as having commercial value. Self-supporting columns 200 or more feet high that possess attractiveness, safety, utility, and economy of construction are in use today. The conditions under which vertical pressure vessels operate are often severe, and since the contents are quite often inflammable, structural failure is a serious matter. Simple membrane stress relationships are insuficient to predict the stresses induced by the action of wind and seismic forces.
9.2 AXIAL AND CIRCUMFERENTIAL PRESSURE STRESSES A cylindrical vessel under internal pressure tends to retain its shape in that any out-of-roundness or dents resulting from shop fabrication or erection tend to be removed when the vessel is placed under internal pressure. Thus, any deformation resulting from internal pressure tends to make an imperfect cylinder more cylindrical. However, the opposite is true for imperfect cylindrical vessels under external pressure, and any imperfection will tend to be aggravated with the result of possible collapse of the vessel. For this reason, a given vessel under external pressure in general has a pressure rating only about 60% as high as it would have under internal pressure. This is only an approximation and other considerations must be taken into account in determining the rating of vessels under external pressure.
9.1 INDIVIDUAL STRESSES IN THE SHELL The stresses in the shell of vertical vessels are essentially: (1) the axial and circumferential stresses resulting from internal pressure or vacuum in the vessel; (2) the compressive stresses resulting from dead loads including the weight of the vessel itself plus its contents and the weight of insulation and attached equipment; (3) stresses resulting from bending moments caused by wind loads acting on the vessel and its attachments; (4) stresses caused by any eccentricity resulting from irregular load distribution; (5) stresses resulting from seismic (earthquake) forces. In addition, stresses may result from fabrication procedures such as cold forming, and welding. Figure 9.1 shows the cold forming of a cylindrical shell from flat plates, and
9.20 Tensile Stresses Resulting from Internal Pressure. The axial and circumferential stresses due to internal pressure in the shell of a closed vessel were developed in Chapter 3 and are given by Eq. 3.13 and Eq. 3.14, respectively. The axial tensile stress is: Rd
fw = 4(t,
(3.13)
The circumferential tensile stress is: fcp = -EL.. 2(& - c) 155
(3.14)
Design of Tall Vertical Vessels
where W = weight of shell above point, T, pounds Do = outside diameter of shell, feel Di = inside diameter of sheli, feet X = distance from top to point under consideration, feet ps = density of shell material, pounds per cubic foot = 490 lb per cu ft for steel construction And
Wiw = $ Dins.Xtins.Pim.
(9.2)
where Dins, = mean diameter of insulation, feet W i n s . = weight of insulation Pins. = insulation density, pounds per cubic foot = 40 lb per cu ft for most insulation tin.3, = insulation thickness, inches Fig. 9.1.
Rolling vessel shell from
3-in. plate.
Since compressive stress is force per unit area, disregarding corrosion allowance, c, gives:
(Courtesy of C. F. Brow &
Co)
fdead wt s h e l l =
9.2b C o m p r e s s i v e Skesses R e s u l t i n g f r o m E x t e r n a l Pressure. External pressure acting upon a cylindrical shell
ir/4(Do2 - Di2)Xp, Xp, a/4(Do2 - Di2)144 = z (if py = 490 lb per cu fL)
= 3.4x
and its heads may result in failure of a vessel either by J-ielding or by buckling. If the vessel has a relatively thin w-all, the stress at which wrinkling or buckling begins to occur is usu3Uy below the yield strength of the material. If the vessel has a relatively thick wall, the stress at which buckling occurs is the yield point of the material under consideration at the temperature of service. Since vessels operating under external pressure must be designed in accordance with elastic-stability criteria, the design is based upon the critical pressure at which buckling occurs rather than upon an allowable stress for the material. The design procedure to be followed is given in the previous chapter. After designing the vessel for external pressure service (using procedures given in Chapter 8) we may use Eys. 3.13 and 3.14, referred to above, to evaluate the induced compressive axial and circumferential stresses.
(9.3a)
The stress due to the dead weight of insulation is:
fdead
(9.4)
wt ins. = __
144mD,(t, - c)
where D, = mean diameter of the shell, feet Dim. x DO = diameter of insulated vessel, feet t, = shell thickness, inches
9.3 COMPRESSIVE STRESSES CAUSED BY DEAD LOADS
The dead load acting on the vessel is determined by the weight and location of all the exterior and interior attachments such as: trays, overhead condensers, platforms, insulation, and so on. Those loads which act eccentrically may be reduced to vertical forces and moments acting at the central axis of the tower. This section will consider only the vertical compressive forces acting on the vessel, and a later section will cover the summation of the moments produced by eccentric loads. Stresses caused by dead loads may be considered in three groups for convenience: (1) Stress induced by shell and insulation (2) Stress induced by liquid in the vessel (3) Stress induced by attached equipment. STRESS I N D U C E D BY SH E L L AND IN S U L A T I O N . At any distance, X feet, from the top of a vessel having a constant shell thickness, Wshell =
; (Do2
- Di2)p,X
Fig.
(9.1)
9.2.
Welding
inside
longitudinal
(Courtesy of C. F. Brow 8 Co.)
-
-
-
-
---r-
-. -.--
seam
of
3-Lthick
shell
section.
Tensile and Compressive Stresses
Fig. 9.3.
Welded column entering oven for stress relieving.
Therefore,
where .h, Pins.Xfins. fd e a d WC ins. = 144Ct, _ el
(9.4a)
S TRESS INDUCED RY S UPPORTED LIQUID . I
1: liquid wt
(9.5)
S TRESS INDUCED BY A TTACHMENTS S UCH AS TRAYS, O V E R H E A D C O N D E N S E R S. TO P H E A D , FLATFORMS, A N D LADDERS.
2 weight of attachments
.,
f dead wt attach. = -__-~--~~---__lzaD,(t,
- c)
The weight of steel platforms m;ly be estimated at 35 lb per sq ft of area, and the weight of steel ladders at 25 lb per lin ft for cagec! ladders and 10 lb per lin ft for plain ladders !i Z). ‘l‘rays in distilling columns, including liquid hold-up on trays, IXIY be estimated to have a weight of 25 lb per sq ft of tray area. The total dead-load stress, fd7, ncticg along the longitudinal axis of the shell is then the sum of the above daadweight stresses I
f&z = .fdead w t she’1 f .filrrtd wt ina. -b fdead w t lir! + fdasd
st attach.
( C o u r t e s y o f C . F . Bran & C o . )
= lhr total dead-load stress acting along the longitudinal axis at point X, pounds per square inch
If the vessel does not contain internal attachments, such as trays which support liquid, but consists only of the shell insulation, the heads, and minor attachments such as manholes, nozzles, and so on, the additional load may be estimated as approximately equal to 18yo of the weight of a steel shell, or as shown by Nelson (139), j& = (1.18)(3.4)X = 4.0X
(9.6)
(9.7)
157
(9.8)
9.4 TENSILE AND COMPRESSIVE STRESSES CAUSED BY WIND LOADS IN SELF-SUPPORTIKG VESSELS The stresses produced in a self-supporting vertical veseei by the action of the wind are calculated by considering the vessel to be a verticle, uniformly loaded cantilever beam. The wind loading is a function of the wind velocity, air density, and the shape of the tower. The United States Weather Bureau (137) has correlated the above factors in the following relation: (9.9)
158
Design of Toll Vertical
Vessels
9.4. Minimum allowable resultant wind pressures (137). (Th is material is reproduced from the American Standard Building Minimum Design Loads in Buildings and Other Strucfures, A58.1-1955, copyrighted by the American Standards Association.) Fig.
where B = barometric pressure, inches, mercury P, = wind pressure on a flat surface, pounds per square foot VW = wind velocity, miles per hour F, = shape factor = 1.0 for flat plate at 90” to the wind For a barometric pressure of 30 in. of mercury Eq. 9.9 becomes: P, = 0.004Vw2F, (9.10) The shape factor, F,, for a smooth cylinder has been found to be 0.60 (137). Thus the resistance of a smooth cylinder is 60% of that of a flat surface normal to the wind and having the same projected areas as the cylinder. Projections of auxiliary equipment loaded on the tower will cause turbulence, and the use of a value of F, based on smooth cylinders is questionable. Therefore, the value of F, used by designers varies from 0.60 to 0.85, depending on the amount and shape of the projections on the vessel. If a value of 0.60 is used for the shape factor then Eq. 9.10 becomes P, = 0.0025Vw2 (9.11) The appropriate wind velocity that should be used in Eq. 9.11 is dependent upon the location in which the equip-
Code
Requiremenfr
for
ment is to be erected. In the Gulf Coast area winds up to 125 mph are experienced. Most other regions experience intermediate maximum wind velocities; therefore a figure of 100 mph is often used. Figure 9.4, published by the American Standards Association (137) is a map of the United States indicating minimum allowable resultant wind pressures at 30 ft of elevation. To obtain the design force, P,, the wind-velocity pressures should be multiplied by a shape factor and a height factor. In the use of Fig. 9.4 a shape factor of 0.6 is recommended for chimneys and clean circular towers, and a shape factor of 1.0 for rectangular buildings and structures. The height factor is 1.0 for structures having heights from 30 to 49 ft. For higher structures the height factor varies directly as the (height/30) raised to the >f power (138). In using Table 9.1 reference is made to Fig. 9.4 to determine the wind pressure at an elevation of 30 ft for the locality in question. The design pressure for the tower is obtained from Table 9.1 after one knows the height of the tower. The value obtained from Table 9.1 should be multiplied by the appropriate shape factor, F, for cylindrical towers. These design values are recommended as minimum and do not provide allowance for tornadoes. As pointed out by Bergman (140) the relationships given by the ASA (137) presented here for use with Fig. 9.4 and
Tensile and Compressive Stresses
559
where M,, = bending moment due to wind at X distance from the top, inch-pounds d eff. = effective diameter of vessel, inches This equation is subjected to the limitation that the wind acts over the total distance, X. The stress in the extreme fiber of the shell. due to the wind, is obtained by use of Eq. 2.10: (9.13) At the base of the tower, fwb = W&o)
I
,
- _ _~~...
Re Fig. 9.5. Drag coefficients for circular cyliiders
(141). (Courtesy of
McGraw-Hill Book Co.)
Table 9.1, do not consider the effect of velocity on the drag coeflicient. The drag coefficient is similar to a friction factor and varies with the Reynolds number, Re, as shown in Fig. 9.5. Figure 9.5 shows that between Re = 500 and Re = 500,000, the drag coefficient is fairly constant, with a value of about 1.1 for cylinders with L/D = A. However, at a value of Re equal to 6 X 106, the drag coeflicient drops abruptly to 0.7 for rough-surface cylinders and to 0.3 for smooth cylinders. The wind pressure determined by use of Table 9.1 is based upon the higher drag coefficient and values of Re between 5 X lo2 and 5 X 106. The value of Re is equal to 9100 DV where D is the vessel diameter in feet, and V is the wind velocity in miles per hour (140). Thus a vessel having a diameter of 8 ft in a wind having a velocity of only 10 mph would have a Reynolds number of 7.28 X 105, which is above the transition value of about 6 x 105. Therefore the use of Fig. 9.4 and Table 9.1 results in a wind-pressure safety factor of about 2 to 3, depending upon the smoothness of the vessel. The force, P,, acts over the projected area of the column, and some designers compensate for the turbulence caused by the projections by using an “effective” diameter, d,a., of the vessel and the allied equipment. This effective diameter is the diameter of the vessel plus twice the thickness of the insulation plus an allowance for the projected area of piping and attached equipment. For open-framed structures the effective area is taken as twice the projected area, and an allowance of 17 in. is made for caged ladders (139). Figure 9.6 shows a group of self-supported vertical vessels with caged ladders and platforms. Note also the external piping, which increases the effective diameter (d& to wind loads. After determining the values of the wind loading and the projected area upon which it acts, the bending moment any distance X from the top of the tower can be expressed as: Mm, = P,X ($) (y) = ~PwX2d,,.
- --\--- \
- -
(9.12)
\
where PO = outside radius of shell, inches I = rectangular moment of inertia perpendicular to and through the longitudinal axis, inches 4 fWZ = stress at extreme fiber due to wind load, pounds per square inch (compressive stress on downwind side, tensile stress on upwind side) In design calculations it is assumed that auxiliary equipment will add load to the vessel but will not aid in its support; therefore, the extreme fiber is at the outside surface of the shell. For any values of t/r that would be encountered in vessel design, this relationship can be simplified as follows. The equation of a circle is: x2 + y2 = P2 and by Eq. 2.8:
I, = j-z” dA
If the integration is performed in the first quadrant, then I, = 4 or x2 dA J Assume that the area of a thin shell is 2mt and that dA = tds. Introducing the derivative of an arc length (38) gives: ds = 41 + (dy/dx)2
dx =
Therefore dA = d&2 dx I, = 4t
Table 9.1.
XP dx T 0 2/772 s
(9.15)
ASA Recommended Wind Pressures for
Various Height Zones above Ground (137)
(Courtesy of American Standards Association) Wind pressure-map areas Height zone (lb per sq ft) et) 20 25 30 35 40 45
50
Less than 30 30 t o 49 50 t o 99 100 to 499
40 50 60 75
15 20 25 30
i--f-----L----‘
20 25 30 40’
25 30 40 45
25 35 45 55
30 40 50 60
35 45 55 70
-. w..- -
169
Design of Tall Vertical Vessels
Fig. 9.6.
S&f-supporiing
verfkol vessels with caged ladders, platforms, and external piping.
(Courfe~y
of C. F. drain
g Co.)
Guyed
Substituting x = P sin 0
9.5
GUYED
Vessels
161
VESSELS
The chief advantage of using guy wires for restraining a tall vertical vessel is the reduction in the size of the foundation and in the size and number of foundation bolts. This is offset in part by the anchorages for the guy wires and by sin 0 cos e the nuisance created by the wires. The design of guyed 2 0 [2 vessels has been discussed by Marshall (142, 143). = 4lr3 $ The wires used for guying purposes are usually of wire [I rope fabricated of high-strength-steel strands around a therefore hemp core saturated with a preservative and a lubricant. (9.16) I, = 7dt The wire rope is generally specified by two numbers, of which the first gives the number of strands per cable, and Or, including an allowanc~efor corrosion and using Ihe the second gives the number of wires per strand. Thus, a mean radius, r,, we obtain: 6 x 7 wire rope, which is commonly used for guying pur(9. I6a) Z = 7rrm3(f, - c) poses, contains 6 strands with 7 wires per strand. Commercial wire rope is available in several grades: / \vliere \? extra-high strength, plow steel, and cast steel. The approximate breaking strengths in tons for these grades are given by the following equations for 6 x 7 cable: Substituting E(4. 9.16 hit.0 E(4. 9.13 gives (lwwuse = ro): P?lL T = 41d2 (for extra-high strength) (9.22) .I (9.17) (for plow steel) T = 36d2 (9.23)
I, = 4tr3
r/2
sin’ 0 dt9
= 4tr” e _ 1
1
T/2
T = 30d2
By substitut.ing Eqs. 9.12 and 9.15 into E(4. 9.13 the general equation for the bending stress on the vessel shell due to a given wind load, P,, may be obtained. f,,,x -
PwX2&vo 27rrm3(fs - c)
But r, = ro and doi2 = r,n; therefore, Eq. 9.18 simplified to:
The approximate costs of 6 x 7 cable is (144) :
(9.19)
J”,,, =
15.89deE.X2 do2& - c)
Dollars per foot = 0.4.4d2
(for extra-high strength) (9.25)
Dollars per foot = 0:EOd’
(for plow steel)
(9.26)
Dollars per foot = 0.28d2
(for cast steel)
(9.27)
be
Simplifying Eq. 9.19 for the case where P,,; is 25 lb per sq ft gives I he following two equations. For insulated towers, (9.20)
For noninsulated t,owers, (9.21) Equations 9.20 and 9.21 are limited by the following rtasl rictions: I. The wind pressure is 25 lb per sq ft. 2. The wind acts at the above intensity over the entire length of the column. 3. There are no external attachments on the tower. 4. The moment of inertia of the shell about its transverse axis is: Z = 7rrm3(t, - c) 5. The mean radius of the shell is approximately equal to the outside radius. The general form of Eq. 9.19 incorporates assumptions 2, 4, and 5 above.
(9.24)
where T = approximate breaking strength, tons d = wire-rope diameter, inches
(9.18) rz111
(for cast steel)
.
For design purposes a factor of safety of 4 is usually applied to Eqs. 9.22, 9.23, and 9.24 to obtain the allowable cable loads. Greater factors of safety are used where sudden loads are anticipated. Usually three or four sets of guy wires are equally spaced around the vessel. For vessels up to 50 ft in height one cable is used in each position; for vessels 75 ft in height two cables are often used in each position; for vessels over 75 ft in height three or more cables are commonly used in each position. These wires are attached to a rigid collar usually located at two thirds and sometimes three quarters the height of the vessel. 9.50 Tension in Guy Wires. To avoid the danger of having a slack guy wire, an initial tension of one fourth of the allowable cable load is applied by tightening the turnbuckles on the cable. The guy wires are used to counteract the bending moment caused by the wind load, given by Eq. 9.12. Using this equation with X equal to the total height, H, and with the guy ring located at y 1.0 set 0.05 0.02/T 0.02 0.10 0.04/T 0.04 0.20 0,08/T 0.08
168
Design of Toll Vertical
Vessels 9.8
EXAMPLE
CALCULATION
9.2
For the unguyed vertical tower described in Example Calculation 9.1, calculate the seismic bending moment and resultant stress at 25 ft and 50 ft above the base of the tower. The tower weighs 1800 lb per vertical foot of height (180,000 lb total) and is to be erected in southern California. Calculation of vessel period: By Eq. 9.68
cw ~ r
H
T = 2.65 X IO-” (;>‘(,>, therefore
Fig.
9.11.
Seismic forces on a vertical vessel.
T = 0.856 set
of seismic coeficient: In reference to Fig. 9.7, southern California is in seismic zone 3, and by Table 9.3 the period of vibration 0.856 set lies between 0.4 and 1.0 sec. Therefore t,he seismic coefficient, C, is given hy Determinuiion
9.7f Shear and Bending Moment Resulting from Seismic Forces in Unguyed Vessels. The seismic forces act to pro-
duce horizontal shear in vertical unguyed vessels. This shear force in turn produces a bending moment about the base of the vessel. The shear loading will be triangular with the apex at the base, as shown in Fig. 9.11. The center of action for such a triangular loading is located at ,2.SH. The shear force at the base resulting from seismic forces is given by Eq. 9.69. The shear force, V,, (pounds) at any horizontal plane in the tower X feet down from the top is given by: v
= CWX@H - X) 82
HZ
c - 0.08
T
c = 0.08 = 0.0935 0.856 of seismic bending moments at 25 and 50 ft:
Determination
By Eq. 9.71 (9.70)
where C = seismic coefficient from previous section W = total weight of tower, pounds H = total height of tower, feet
M = 4cWX2(3H - X ) sz H”
TV = 180,000 lb, at 25 ft (X = 75 ft), therefore
The bending moment MS, (inch pounds) at plane X resulting from the shear forces above plane X is given by:
,~~, = ~f~0.0935)(180,000)(75)2(300 - 7 5 ) h.r (IO@2 ___--~-= 8.50 X lo6 in-lb
At X = 50 ft
M,, = 4.20 X lo6 in-lb The corresponding heading stress may be determined by Eq. 9.17. Therefore
Determination
By Eq. 9.72
of seismic stresses at the 25 arid 50 ft levels:
(9.72) The maximum shear and bending moment are located at the base of the tower and may be found by substituting X = H in Eqs. 9.70 and 9.71, respectively, or and
vsb = cw 2CWH(12 in. per ft) = 8(,w~~ 1 A&t, = ~~- ~~ 3
At 25 ft
fsz =
(9.73) At 50 ft (9.74)
Therefore
Substituting Eq. 9.74 into Eq. 9.72 gives the seismic hending stress at the base of the skirt of the vessel. (9.75) where r = tower radius, inches 1, = skirt thickness, inches
M,, = 8.50 X lo6 in-lb
Therefore 8.50 X lo6 = 3080 psi ~(42)~(0.5)
M,, = 4.20 X lo6 in-lb fw = ;q2=; = 1514 psi
9.9
STRESS
CAUSED
BY
ECCENTRIC
LOADINGS
In vessels such as bubble-cap columns the shell and trays are placed symmetrical about the longitudinal axis, but external attached equipment usually acts as an eccentric
r-- --
Combined Stresses in the Shell
Erection of
Fig. 9.12.
a
169
carbon
dioxide absorption tower 84 ft. 0 in. by
7 ft.
Girdler
6
in.
(Courte
sy of The
Company and the Mississippi
Chemical Corp.)
load and should be considered as such. Most external attached equipment produces a negligible moment, and engineering judgment must be used in the calculation of stresses. Equipment such as small ladders, pipes, arltl manholes may usually be disregarded, but the total combined moment of heavier equipment such as overhead or side condensers is important. The eccentricity is calculated by: ZM, e=Zw,
Eccentric loads produces a bending moment equal to zW,,(e). The additional bending stress at plane X caused hy this n~omen~ is:
.“*
--\
(9.77)
(9.7C)
where e = eccentricity, the distance from the column axis to center of reaction, inches Zkf, = summation of moments of eccentric loads, inch pounds ZW, = summation of all eccentric loads, pounds
I
Substituting ftn I hy Eq. 9.16 gives:
-~
\
-\r 7-
9.10
COMBINED
STi?ESSES
IN THE SHELL
A controlling combined tensile or compressive stress occurs as a resu!t of combinations of stresses. It is important to consider the intended construction, erection, and test schedule which is to be followed in erecting the vessel
_
?$
Design of Tall Vertical Vessels
fWz (from wind load) [J (from seismic load) fh (from total developed load)
fW (axial stress from internal pressure)
Fig.
9.13.
Stress conditions in the
ihell
of a vertical vessel.
and placing it on stream. With this in mind, the conditions must he determined which establish the controlling stresses. Figure 9.12 shows the erection of a tall selfsupporting tower on its foundation. An evaluation must be made of the effect of the combined stresses induced in the shell of a vertical pressure acted upon : by wind loads. These stresses are additive at specific points in the shell of the vessel. During the construction and subsequent use of the vessel, the total combined stresses will vary according to the forces acting on the vessel at any given time. The stressed condition of the vessel may be divided into the following possible cases: Case 1: vessel under construction a. empty shell erected b. shell and auxiliary equipment such as trays or packing but no insulation Case 2: vessel completed but shut down Case 3: vessel under test conditions a. hydrostatic test b. air test Case 4: vessel in operation In the consideration of wind and earthquake loads it is assumed that the possibility that the most adverse wind and earthquake load will occur simultaneously is remote and the possibility that these lateral forces will occur in the same direction is even more remote. Therefore, the resulting stresses for wind loads and earthquake loads are computed separately, and the most adverse loading condition used in the design. In analyzing the combined stresses calculations are usually made beginning at the top of the vessel. The mini-
mum shell thickness in the upper portion of the tower is usually controlled by the circumferential stress resulting from internal pressure or vacuum. The shell plate thickness at the top of the column can usually be specified on this basis by using a standard plate thickness slightly larger than the minimum. At lower sections of the vessel, where compressive dead loads and wind loads become significant, the shell thickness must be increased in order to resist these additional loads. The distance down the vessel for which the initial shell plate thickness may be used without exceeding the allowable stress is determined by calculating the combined stresses again, remembering that the specified distance should be reduced to some multiple of standard plate widths to avoid cutting a plate unnecessarily. In designing the shell it isnot necessary to allow for the compressive stress resulting from the weight of the liquid in the hydrostatic test since the bottom head of the shell transfers this load directly to the skirt. It is essential to check the lower sections for wrinkling failure before the tinal specifications are fixed. Therefore, in the case where thermal, eccentric, and liveload stresses are negligible, and for positive pressures, the following equations for the maximum combined stress can be applied in the calculation of plate thickness. It must be noted, however, that if the stress due to eccentricity is appreciable, it should be included in the dead-weight-stress and bending-stress terms. Figure 9.13 is a diagram indicating the stress conditions in a vertical vessel resulting from the loads indicated. By reference to this figure it is apparent that the maximum tensile stress exists on the upwind side of the vessel and the maximum compressive stress exists on the downwind side of the vessel. Maximum tensile stress (upwind side) at point X with an unguyed vessel under internal pressure and in the absence of eccentric loads is: flmnx
= UWZ
or f& + fap
- fd~
(9.78)
For external pressure the equation becomes: ftmax = (fwz or fsz) - fap - .fdz
(9.79)
Maximum compressive stress (downwind side) at point X with an unguyed vessel under internal pressure and in the absence of eccentGc loads is: fcmax = (fwz or fsz) + fdz - fap
(9.80)
For external pressure the equation becomes: fcmax = (fwz or .fsz) + fdz + fap
(9.81)
9.11 DETERMINATION OF THICKNESSES OF SHELL PLATES FROM TENSILE STRESSES The diameter and height of the vertical vessel are determined by the process requirements. In the case of a distillation column the diameter is determined by the maximum vapor load and allowable vapor velocity. The height of the tower is determined by the number of trays required to effect the desired separation and the selected tray spacing. The material of construction is determined by corrosion
Checking Shell Compressive Stresses for Elastic Stability
requirements, temperature and pressure levels, economic considerations, and availabilitv. The minimum required shell thickness usually occurs in the top course of the vessel where cumulative stresses from wind loads and dead weight are small. A trial thickness for the top course can be selected on the basis of pressure considerations by using Eq. 13.1. However, seismic loads may be significant in the upper portion of the vessel if the vessel is to be erected in seismic zone 3, in which case the selected thickness must be checked for the combination of pressure stresses and stresses from seismic forces by using Eq. 9.72 in Eq. 9.78 or 9.81, whichever is appropriate. If the selected thickness is satisfactory, it is next necessary to determine how many courses of this thickness may be used in the vessel. The limiting value of X for the initial plate thickness selected may be determined by substituting into Eq. 9.78, using Eq. 9.7 forjdz, Eq. 9.17 forjwz, Eq. 9.72 forjSz, and Eq. 3.13 for jap. This may be done by trial and error bv assuming the number of courses down from the top and checking the combined stresses at this level. An alternative procedure is to solve directly for the distance X down from the ton of the vessel at which the maximum induced stress is equal to the allowable stress by means of a quadratic relationship. For unguyed vessels under internal pressure fabricated of steel and for the case in which the maximum lateral loads are determined by wind loads rather than seismic forces for operating conditions, the following.Cquation may be used:
I
2P,X2 pd + 4(t, t*X = ?rd(t, - c)
f
awft)x - d(t,,- c)
where P, is given by Eq. 9.11. This equation may be modified to lit erection and other To deterconditions not covered by the limitations above. mine X, the distance at which the induced tensile stress is equal to the allowable stress, jrtllOw. may be substituted for ft max and the equation arranged in the form of a quadratic.
+
pd ~~ - fdhv. 4(k T c)
Equation 9.82 has the following form:
1
,
where the coefllcients a, b, and c are defined by the quantities in the respective brackets. The solution of the binomial equation is:
x=
i ‘4‘
in thickness in order to withstand the increase in tensile stress resulting from wind load. In this case the calculation using Eq. 9.83 is repeated, with the larger plate thickness substituted. The bending moment due to wind loads increases with X2. Therefore, the plate thickness required will increase more rapidly with respect to X near the bottom of the tower than in the upper region. In checking the tensile load conditions the various cases of stress conditions should be investigated individually to determine the controlling condition. If the vessel is designed for high-pressure service, the limiting condition will usually exist when the vessel is operating under pressure and under a wind load. As the vessel would not be tested with air pressure under high-wind conditions, this case is not considered to be controlling. Vessels designed for low-pressure service may encounter maximum-stress conditions when the erected empty vessel is exposed to a high wind load. Any additional compressive loads such as those induced by trays with liquid inside a distillation column or overhead external condensers will relieve the tensile load. 9.12
CHECKING SHELL COMPRESSIVE ELASTIC STABILITY
STRESSES
FOR
9.12a Critical Compressive Stress. After determining the shell plate thicknesses in order to satisfy tensile stress requirements, the design should be checked for compressive stresses on the downwind side. This analysis is more complex because elastic stability must be considered. Thinwalled columns stressed along their longitudinal axis may fail in two ways: by Euler’s buckling or by wrinkling. Failure due to Euler’s buckling involves bending of the shell as a whole and is seldom controlling in vertical vessels with cylindrical shells. Wrinkling is local in nature and depends upon the combined compressive stresses at the point under consideration. It is necessary to determine the stress under which this phenomenon occurs. The allowable critical compressive stress at which wrinkling does not occur when a steel cylinder is under axial compression was given in Chapter 2 by Eq. 2.25.
= 0 (9.82)
aX2+bX+C=0
,/
171
-b + 1/b2 - 4ac 2a
After determining the value of X by means of Eq. 9.83, it may be desirable to adjust the plate thickness, t,, for the top courses so that the height of the section, X, will be a multiple of the plate width used. Usually the plate thickness originally selected is satisfactory for a number of courses. Plates below distance X must have an increase
fcallow.
= 1.5 x
10s c 5 Q y.p. r
(2.25)
9.12b The Influence of Stiffeners. Vertical vessels may be stiffened by the addition of internal or external members attached to the shell. The members may be attached in either the longitudinal or circumferential direction and in some cases in both directions. Timoshenko has shown how allowance may be made for the stiffening effect if the members are uniformly spaced (42). Equation 2.25 may be modified as follows to allow for the stiffening effect:
f callow. =
1.5 x lo6 f
%qt,t,/t2)
= 1.5 x 10s fi 6 35 y.p. P
(9.84)
172
Design of Tall Vertical Vessels
where t,=t+$ Y t,=t+$ z
(the equivalent thickness of the (9.86~) shell in the circumferent.ial direclion) (lhe equivalent thickness of the (9.84h) shell in t.he longitudinal direr tion)
where .12/ = cross-sectional area of one ~ircumferenlial sliffener, square inches d, = distance hetween circm~lferential stiffeners, inches A, = cross-sectional area of one longitudinal sbiffener, square inches (1, = distance between longitudinal stiffeners, inches It is significant that any additional area added to t.he shell as stiffeners increases the buckling strength of the vessel in proportion to the square root whereas the additional metal area added uniformly to the shell by increasing its thickness will increase its strength in direct proportion. The obvious conclusion is that it is more economical to stiffen a shell by increasing the shell thickness t.han hy adding stiffeners. 9.13
EXAMPLE DESIGN, SHELL A TALL VERTICAL VESSEL
CALCULATIONS
FOR
Therefore use ?is in. SeWion of head: A preliminary calculation indicates that the required ellipt,ical-head thickness will be 516 in. Since elliptical dished heads 7 ft, 0 in. in diameter are not made this t,hin, a t.orispherical dished head will he used. Also, torispherical heads are nominal on the out.side diameter. Calculation: By Eq. 13.12 t = !):885pr, .fE - O.lp
= -----___ ,12,6;;;;;::“~~&oj
= 0. &OO in. = &-in. t.hick head By Eq. 3.12 Diameter = 011 + tz + 2sf + Qicr From Table 5.7, sj= 3l$in.
A close fractionation tower is to he fabricated and insMled in the west-central area of Texas. The vessel has the following specifications: Shell outside diameter = 7 ft, 0 in. Shell length, tangent line to tangent line = 150 ft., 0 in. Operating pressure = 40 lb per sq in. gage Operating temperature = 300” F Shell material = SA-283, Grade C Shell, double welded butt joints stress relieved but not radiographed Skirt height = 10 ft, 0 in. Tray spacing = 24 in. (71 trays) Top disengaging space = 4 ft, 0 in. Bottom separator space = 6 ft, 0 in. Tray loading including liquid = 25 lb per sq ft Tray-support rings = 236 in. x 2>$ in. x N in. angles Corrosion allowance, c = $6 in. Overhead vapor line = 12 in., outside diameter Insulation (ins.) = 3 in. on column and vapor lines Accessories = one caged ladder Allowable stress (see Chapter 13 on code vessels): SA-283, Grade C stress relieved but not radiographed has an allowable stress of 12,650 psi (see Table 13.1), and a welded-joint efficiency of 0.85 is specified by the ASME code (11) (see Table 13.2). Calculation of minimum shell thickness: t = -J-Q + c SE - 0.4p (12,650) (0.85) - (0.4) (40)
\
in., and from Table 5.8,
= 84 + 3.5 -+ 3 + 3.4 = 93.9 in. rrd2t p Weight. of head = --- _~ 4 1728 - 7r(93.9)?&) ~-.___-~ 4
#&
= 860 lh Calculation of axial sfrrss in shell: d< z do; therefore 11s~ = 84 in. By Eq. 3.13 jap = .A?-. = (-to) CS4) = .W80 psi 4(& - c) (4)(0.1875) Calculation of dead weights: j-dead
wt shell =
3.4x
(by Eq. 9.3a)
Note: Eq. 9.3a applies only t,o shells of constant thicknesses and may he used for t.he top section where the thickness is constant.
(13.1) I)
+-fi
\I
Pins.Xtins. w t i n s . = 1~4(t, _ ej
(by Eq. 9.4a)
40x3 = 4.44x = (144)(0.1875)
= 0.156 + 0.125 = 0.281
\
icr = 514
/
(5. L2j
Diameter = 84 + $$ + 2(1+) + g(5;)
Jdead
_ (40)(42)
+ c = 0.275 + 0.125
-.
Example Design, Shell Calculations for a Tall Vertical Vessel \\
I
of top head =
Calculation of combined stresses under operating conditions: Upwind side: By Eq. 9.78
860 lb 25 lb per ft
\\ t of ladder = WI. of 12-in. schedule 30 pipe (from Appendix K) = \\‘ 1, of pipe insulation zz
.f t (max)
43.8 lb per ft
; (1.52 - 1.02)40 =
.fwz +
= 1.297X2
.fap - .fdr
- 19.67X + 4.178
For an allowable stress of 12,650 psi and a joint efficiency of 0.85.
(860 + 108.1X)lh
W = 860 + 108.1X #‘deed wt attachments = (not including trays)
=
= 1.297X2 + 4480 - 19.76X + 2.05
39.3 lb per fl Total
ZW nd(t, -
173
(12,650)(0.85) psi = 1.297X2
- 19.76X + 4478
01 X2 - 15.25X - 4830 = 0
e)
860 + 108.tA = 17.4 + 2.19X = (3.14)(84)(0.1875)
Solving for X gives:
The weight of trays plus liquid (below X = 4) is calculated as follows. Lhwnwind side: By Eq. 9.80
fc(nrax) .fdrad wt (liquid + trays) =
=
fw, - jap +
.fdz
= 1.297X2
- 4480 + 19.76X - 2.05
= 1.297X2
+ 19.76X - 4478
From elastic stability, by Eq. 2.25, = (9.73X - 19.45)
jc = 1.5 x 106 4 (= +y.p. 0P
jclw = 3.4X + 4.44X + 2.19X + 17.4 + 9.73X - 19.45 = (19.76X - 2.05)
is:
Calculation of stress due to wiud loads: The wind pressure HS obt.ained from Fig. 9.4 and Table 9.1
= 6690 psi Therefore 1.297X2 + 19.76X - 4478 = 6690
From Pig. 9.4, wind pressure at 30 ft = 25 psf From Table 9.1, the corresponding wind pressure above 100 ft = 40 psf
x2 + 15.3x - 8640 = 0 x = --lJ5 4 d(15.25)2 + (4)(1)(8630) 2 = 85.5 ft
If a shape factor of 0.65 is applied, the effective wind pressure above 100 ft will be 26 psf. Table 9.1 shows that wind pressure corrected for shape will be about 20 psf below 100 ft elevation. Therefore, a design wind pressure of 25 psf will be used in the design calculations. This permits the use of Eq. 9.20. (To minimize wind load place ladder 90” to vapor line.) deff. = (insulated tower + vapor line) = (84 + 6) + (12 + 6) = 108 By Eq. 9.20
If credit is taken for the stiffening effect of tray support rings, a higher allowable compressive stress will result. Therefore
t, = t, + 9II
in.
whew I, = equivalent thickness of sheli, inches ill, = cross-sectional area of one rircun!ferential stiffner d, = distance between circumferrntial stiffeners, inches t, = t, (since no longitudinal stiffener are used)
jTOZ = 15.89deR.X” ____ = 15.89(108X2) (8-4)2(0.1875) d&t, - c) 4
Therefore fwt = 1.297X2
\
-
\
\I
(see Eq. 9.8.t;l)
/
-
v-
174
Design
of
Tall
Vertical
Vessels
The tray-support rmgs are 235 x 21’ ,s x s in. angles. Therefore A2/ = 1.73 sq. in. dg = 24 in.
Calculation of combined stresses for condition of partial erection: Upwind side: jtcmax) = 0.727X2 - 4.234X - 10.42
(tray spacing)
= (12,650)(0.85)
t, = 0.1875 + g = 0.1875 + 0.072
x = +5.81 + d(5.81)2 + (4)(1)(14,800) 2
By Eq. 9.84
.fc =
1.5 x 10s dg 5 6 Y.P. P
(see Table 5.1)
30,000 = =-$) 4(0.26)(0.1875) 5 g y.p. 5 3
= 125 ft Downwind side: (No credit is taken for stiffening rings.) fc(msx) = fwz + fdz = 6690
= 7880 < 10,000 1.297X’ + 19.76X - 4478 = 7880 X2 + 15.25X - 9.540 = 0 Therefore
x=
X2 - 5.81X - 14,800 = 0
Therefore
t, = 0.26 in.
= 10,750
-15.25 + 2/(15.25)2 + (4)(9540) 2
= 90.0 ft
X2 + 5.81X - 9,200 = 0
Therefore x =
-5.81 + l/(5.81)2 + (4)(1)(9,200) 2
= 93.1 ft
This credit for stiffness by the tray-support rings results in a X value of 90.0 compared to the previous computed value of 85.5 ft. The next step is to check the shell for empty condition, no trays, no insulation, no pressure, vapor line in place, only wind load acting. Credit may be taken for corrosion allowance under erection conditions. Calculation of stresses: Upwind side: Calculation of dead weight: fdead wt shell =
3.4X
Other dead weights are: Wt of top head =
Total
Thus the controlling stress conditions exist under operating load with a superimposed wind. The 72-ft distance will be acceptable since the upwind condition under operating load is controlling. For this reason, specify nine courses of 8-ft-wide > 10,000 fc(allow.) = 10,000
= 0.3125 + 0.072
Calculation
= 0.383 in.
combined compressive stresses:
of
By Eq. 9.84
fc(allow.)
fc(msx) =
1.5 x 10” 42 d(O.383)(0.3125)
and
= fwz + fdw = 10,000
5 10,000
0.519X2 + 4.0X - 63.06 = 10,000
= 12,350 > 10,000 Therefore
6 gy.,,
x2 + 7.7x - 19,400 = 0 7.7 f 2/(7.7)2 + (4)(19,400) 2 = 135.5 ft
fc = 10,000
x=
0.776X2 + 13.2X - 2779 = 10,000
Therefore the erection condition is not controlling, and four courses of plate 8 ft, 0 in. wide and xs in. thick will be used for the second section. At the end of the Ws-in. plate section X = 72 + 32 = 104 ft. Calculation of third section: It is apparent that since the wind load stress varies with
X2 + 17.00X - 16,450 = 0 Therefore - 1 7 0 0 + 1/(17.00)2 + (4)(16,450) 2 = 120.0 ft
x=
\
~\-
.--
------T-r
I
176
Design of Tall Vertical Vessels
X”, the plate thickness must increase at a more rapid rate. Therefore ?s-in. plate will he used for the third section. Calculation oj third (lower) tower section consisting of si-in. plate: Operating cortdilions: Calculation of axial slress:
pd (J) (40) (84) fw = ;&I
= ~~~ = 1680 (4)(0.501
0.486X2 + 9.53 - 1680 - 195 = 10,000 X" + 19.6X - 24,400
= 0
x = -J(J,6 + d(19.6)2 + (4)(24,400) 0
psi
= 146.7
ft
Use five courses of plates 8 ft, 0 in. wide and s in. thick. X at the end of the 36-in.-plate section = 72 + 32 + 40 = 1’41 ft. The bottom course will be made of one plate 6 ft, 0 in. wide and 3/4 in. thick. (This s-in. course must be specified as ASTM-A-285, Grade C to meet code requirementssee Table 13.1, footnote.) The calculation of stress at the bottom tangent line is X = 150 ft for yd-in. plate. Calculation of axial stress:
(Ialcuiatiorr of dead wrights:
.fd e a d
therefore
40X(3) wt i n s . = (144)(o.50j= 1.67x
860 + 108.1 = 6.52 + 0.82X attaohmmts = a(84)(0.50)
fdw
fdead
wt (trays + liquid)
=
Calculation of dead weights:
(X/2 - l)W)U) (48)(0.50)
5 .fdlrr(shell)
= 3.64X - 7.28 .f,jx = 3.4.x - l94.2 + 1.67X + 6.52 + 0.82X + 3.64X
.fdw(ins.)
7.28
= [(3.4)(150) - 194.2]&
TiT
(40)(150)(3) = (r44)(o.5625) =
= 9.53x - 195
fw.c
=
15.89(108)X2 ~84)2(o.50)
Calculation of combined Upwind side: f t(max)
=
0.486X2
.tkc(trayti
stresses under operating conditions:
(3.14)(84)(0.5625)
17,060 = -__ = 149
115 pi
-____
+ liquid)
f&,,(t&,)
psi
222 Psi
860 + (108.1)(150) .fdr+h-llments) =
Calculation of stress due to wind loads:
= 283.4
= - (48)(o.5625)
= 283.4 + 222 + 115 + 460 = 1080
psi
Calculation of stress due to wind loads:
= .fw.r +.fa, - fdz
= 0.,486X" + 1680 - 9.53X
j
+ 195
UIT
For an allowable sl rta.++ 0 f 1’3,750 psi 10,750 = 0.186X” - 9.53X + 1680 + 195 X2 - 19.6X18,250 = 0 Therefore
= (~5.WUWWW2 (84)2(0.5625) = 9700
psi
Upwind side: ft(max)
= .fwT + .fa,> - .fd., = 9700 + 1495 - 1080
x = fl9.6 I!Z fi.6)’ + (4)(18,250) 2
= I 1.5.3 ft
psi
Downwind side: jcknax) = .fwz - .fu, +
Downwind side: ?CilllHXl
= 10,105
fd3.
= 9700 - 1495 + 1080
= .f,l,., - .f,r ,, + .fd, = 0.486X" .fcka,\\-.)
- 1680 + 9.53.Y - I95 = 4 y.p. = IO,000
= 9285
Therefore the design is satisfactory with regard to loading conditions in which the wind load rather t,han the seismic load is controlling. Check of stresses due to seismic loads:
Oiher Methods of Analysis Referring to Fig. 9.7, we find that West. Central Texas is located in seismic zone 2. The weight of the tower plus attachments, liquids, and so on at the bottom tangent line may be calculated t)\ multiplying the total compressive stress due to dead weights by the cross-sectional area of the tower at this position, or ~W(z=150)
- e)
= (13.2X - 98.8)(7r)(84)(0.3l25) = 105,500 lb lherefore
= 2,783,OOO
W = 196,000 lb W(avn)
= fd.d(ts
M,,(u=lo?, = (~1~)(.04)(103,OO.i)(lO &)'(A80-- 104) , I -(160)'
= .fdw(total)~4
= (1080)1r(84)(0.6875) or
ZW(x404)
177
in-Il)
The corresponding stress is:
196,000 = 1305 lb per ft = ~~150
The period of vibration may be calculated by use of Eq. 9.68.
= 1550 psi The stress due to wind load at this pcGul is 0.776X2.
T = 2.65 x LO-” ($2 (T!y
fu.(X=,04 = (0.776)(10!)" = 8380 The determination of the period of vibration of a column having thickness variations is somewhat involved. It is sufficient for design purposes to approximate the period by using a single shell thickness. The most conservative design will result when the greatest thickness is used. Therefore T = 2.65 x lo-” (~)‘((~~~:~))11
Therefore the wind load is controlling itI this level of I.hv column. The evaluation of seismic stresses at X = 72 ft where the shell thickness is 5~~ in. is: w(X=72)
= f&?rd(&
- e )
= (19.76X - 2.0.')(~)(8~)(0.1875)
= 1.59 set
= 70,300 lb Referring to Table 9.3, we find that the seismic coefficient C, for zone 2 and for a period greater than 1.0 set is equal to:
h&(X=72)
c = 0.04
82
= sWX2(3H - x) H2
The seismic moment at the bottom tangent line is: Ms(x=160)
= (4)(0.04)(196,000)(150)2[(3)(160) (160)2
(4)(0.04)(70,300)(72)2(C80 (160)"
-~72)
= 930,000 in-lh
The moment due to seismic forces may be calculated by Eq. 9.71. M
=
- 1501
= 9,090,OOO in-lb The corresponding stress is given by Ey. 9.72.
9,090,000 = ~(42)~(0.5626) = 2910 psi h comparison of t.he seismic stress of 2910 psi with the wind load stress of 9700 psi indicates that the wind stresses are controlling at the bottom tangent line. Is the next upper course occurs only 6 ft above t,he bottom tangent line, no calculation of the seismic load will be made at lhis point, but it will be evaluated at X = 104 ft where t, = x6 in.
fs(x=72)
930,000 = ----~~-~(42),~(0.1875)
fs(x=72)
= 893 psi
The stress due to wind load at this point is 1.297x”.
fw(x=72) =
(1.297)(72)' = 6700 psi
Therefore t.he wind stresses rather than seismic stresses we controlling over the column height.. This is due to a long period of vibration and corresponding low seismic coefflcien~. However, if this vessel had been located on the lower West Coast in seismic zone 3 rather t,han in zone 2, the seismic* stresses would have been doubled, and the wind stressc~s reduced. The design of the skirt, bolt.ing riug, and foundations f’o~ this tower is covered in Chapter 10. Figure 9.14 is a sketch of the tower designed in this example. 9.14 OTHER METHODS OF ANALYSIS In the preceding sections of this chapter essentially one method of combining t,he stresses has been employed. This method is known as the maxirliulll-i)riIlrip;ll-stress t,heorband is the method most widely used. Other theories may be used to det,ermine the t,hickness of lhe shell required. T h e m:~xiIriurn-prirlcipal-strrss I heorg autl t h r e e other
178
Design of Tall Vertical Vessels
Torispherical head KC’ ihick, 7’-0”
9 courses 8’4’ ea. -of %C plate -3” insulation on shell 12’ OD vapor line with 3’ insulati=
-71 trays (24” spacing)
-4 courses 8’-0” ea. of %.C plate
Some structural materials have physical properties such that the maximum-principal-stress theory is not the best method of analysis. High-tensile-strength materials in which the tensile strength to shear strength ratio is higher than usual are more likely to fail in shear than in tension. Other evidence of failure by shear has been observed in simple tension tests when there is slipping of planes inclined to the tensile-load axis. This would indicate the need of considering more than just the principal stress. 9.14b Maximum-shear Theory. The maximum-shear theory, developed by J. J. Guest (147) assumes that yielding starts when the maximum shearing stress becomes equal to the maximum shearing stress at the yield point in a simple tensile test. The relationships involved can be developed with reference to Fig. 9.15, which shows the stress condition in an element under uniaxial tensile load. Consider any plane, m-n, at an angle, 0, to the axis of the element. The crosssectional area at right angles to the axis of the element is taken as a. The area of the element lying in the plane m-n is:
a %7&n = cos e 5 courses V-0” ea. -of Y plate
1 course 6’-0” -of %s” plate
The tensile stress, jt, in the axial direction measured over the area in the m-n plane is:
P P jt = - = Gn a
(COS
e) = j*ri*l
CO9
e
The component of jt normal to the plane m-n is: jn = jt COS 0 = j*xial COS* 0
The tangential component called the shearing stress, j,, is: Fig. 9.14.
Sketch of a 160-ft, O-in. tower for the example design. j8
methods of analysis are listed below and compared in the following sections. 1. 2. 3. 4.
= jt sin u = faxiar sin 0 cos
v
j, will be maximum when 0 = 45”, or
Maximum-stress theory Maximum-shear theory Maximum-strain theory Modified-strain-energy theory
9.140 Maximum-stress Theory. The maximum-stress theory, sometimes known as Robinson’s theory, is the oldest and simplest theory for designing any section subjected to stresses in three directions. It simply assumes that the maximum of the three stresses ji, j,, and jz controls the design because yielding is taken to occur when the maximum stress reaches the yield point of the material. This theory makes no allowance for the effect of the components of the two minor &resses on the principal stress. This simple theory may’result in over or underdesign of the section under copsideration. The theory is most applicable for brittle materials. However, vessels with designs based on relationship3 developed from this theory and fabricated from structural steels are widely used and have proved satisfactory in service. It is assumed that ji > j, > ji.
e
fe(mar)
Fig. 9.15.
=
faxis1 __
2
Stresses in an element under uniaxial tensile lu&
Other
In the case of two-dimensional stress such as exists in the shell of a thin-wall cylindrical vessel, the two principal tensile stresses at right angles to each other are: 1. The longitudinal stress, jr (y-axis direction).
fz = g
23 f-=--= fw
E
23 f=
E
2. The hoop stress, fh (x-axis direction). (3.14)
Consider the plane m-n with its normal at an angle 0 to the axis of the vessel as shown in Fig. 9.16 noting that jh = Zjr. The shearing component of jr in the plane m-n will be obtained by letting ji in the y axis = jaxiar.
cz3
of
Analysis
= -$fz - PL(fu +fJ1
f2/3 = fyP = Ey3 = 1 [ji - p( j, + j*)] E E E
(3.13)
fh = $
E
Methods
fz = cz3
E
= 1 E[fz - P(fi + &)I
fi3
= pE (1 + Pco(1 -
2Pcr)
(%3 +
q/3 +
ez3) +
The components fsh and jsl are in opposite directions; therefore the resulting combined shear stress in two-dimensional analysis (f& is: 20
The value of j8s will be maximum when 8 is 90”. fore
fs2max
There-
= 8(fZ - fh)
In a similar manner it can be shown (29) that in the case of three-dimensional-stress analysis the maximum shear stress is defined by one half the algebraic difference between the maximum and minimum of these stresses, or fs3
= kfi - fi) = +fy.p.
(9.85)
This theory is in good agreement with experimental results obtained on ductile materials. For purposes of comparison with other theories the maximum actual shear stress and the allowable shear stress will be doubled by removing the 35 from each side of the equation. Therefore
fi
- fi = fmax
(9.89)
E- CT3 l+P
fv3 =
P*E
(l+co(l -2p) (%3 +
eY3+ tzr)
+
E
(l+p) %3 (9.91)
E PE fi3 = (1 +~)(l -2r) (c,3+ ‘U3+ e,3> + (1 +~j Ez3
Sin
(9.88)
(9.90)
The shearing component of fh in the plane m-n will be obtained by letting fh in the x axis = faxisr.
= +(fi - fh)
(9.87)
where the subscripts x3, y3, and 23 refer to the 2, y, and z directions in a three-dimensional system. Equations 9.87, 9.88, and 9.89 can he transformed to express the stresses in terms of strains as follows:
j81 = 6ji sin 28
fs2
179
(9.92’
In the case of thin-walled vessels in which the radial stresses can be disregarded, ji may be taken as equal to zero. This simplification results in Eqs. 6.4a and 6.5a. There is limited evidence against the use of this theory. In the case of a simple plate subjected to tension in two perpendicular directions, the elongation in each direction will be reduced by the tension in the perpendicular direction. This would indicate that a plate loaded in this manner would have a greater yield point than a plate loaded in simple tension in one direction. This conclusion is not supported by limited experimental tests. 9.14d M o d i f i e d - s t r a i n - e n e r g y T h e o r y . There are a number of modifications of the theory based upon the premise that yielding begins when a given quantity of strain energy is accumulated in a given volume of material. Equation 9.35, which gives the strain energy in a deflected
(9.86)
It should be noted that in Eqs. 9.85 and 9.86 both stresses ji and ji are assumed to have t,he same sign, that is, both are compressive stresses or tensile stresses. Designs based on relationships developed from this theory are more conservative than designs based on the principal-stress theory. 9.14~ Maximum-strain Theory. The maximum-strain theory developed by Saint-Venant (148) assumes that yielding of a ductile material begins when the maximum strain becomes equal to the strain at the tensile-test yield point. This theory permits the combining of stress-strain relationship in three dimensions. Equations 6.4 and 6.5 give the strains, and Eqs. 6.4a and 6.5a give the stresses from two-dimensional stressstrain analysis. Equations 6.4 and 6.5 can be modified for three-dimensional analysis as follows:
Fig. stress.
9.16.
Stresses in an element of a vessel shell under two-dimensional
180
Design of Tall Vertical Vessels Substituting into Eq. 9.94 for the condition at the yield point gives: h 1 UZ(Y.P.) = jj fi dfi s0 therefore (9.99) Setting Eq. 9.97 equal to Eq. 9.99 gives:
f.r2 + f,’ + fz2 - a4fifU + fifi + f,fi) =
fy.,."
(9.JW
SeLting p equal to the theoretical maximum value of 0.3 gives a simplified equation
4Kf.z - f,j2 + Cfi/ - fd2 + (f* - fz121 = Fig.
9.17.
Graphical comparison of four theories of failure.
beam, can be modified for the strain energy of an element strained in IWO directions as follows: UY = ~ft,h
(9.93)
ux =
(9.94)
Jo’=fz de,
The total strain energy stored in the element is equal to the sum: (9.95) uxy = uy + ux Equations 6.4 and 6.5 give the strains in the x and y directions resulting from the stresses fi and f,
&c2. = fi - - Pfl/ E
E
f, Pfz %2 = E - E Substituting into Eq. 9.95 gives:
therefore
uz, = f (fg2 +fz2 - 2Pfz/fz)
(9.96)
For an element strained in three directions it can be shown in a similar manner that
crz,z =
& Kfi” +fg2 +fz2) - 2PLfzfz/ +fifi +fufi)l (9.97)
Equation 9.97 is due to Haigh (149) and has been called t,he strain-energy function. Yielditig is assumed to occur when the total strain energy is equal to that obtained in simple tension at the yield point: therefore
fY.P.2
(9.101)
Equation 9.101 is identical to that developed by Hubel (150), Hencky (151), and Von Mises (152) to bring the theory into agreement with the fact that materials can undergo large hydrostatic pressures without yielding. Equation 9.100 for two-dimensional strain reduces to:
fi2 +fg” - %-Jfzf, =
fy.,."
(9.102)
9.15 GRAPHICAL COMPARISON OF THE FOUR THEORIES Figure 9.17 graphically shows a comparison of the foul methods of analysis for a two-dimensional stress system (fi = 0). The upper right quadrant represents tension fol both fi and f,. The upper left quadrant represents fi, in tension and fi in compression, and the lower right quadrant represents fi in tension and f2, in compression. The solid lines in the figure represent the locus of the conditions at which yield is assumed to begin according to the four theories. The square u-b--c-d represents the maximum stress theory. Point a represents equal tension in both the x and y perpendicular directions, both of which are considered to be equal to the yield-point stress obtained from a simple tensile test. According to the maximum-stress theory, yielding does not occur inside the square. As f, is decreased and fz is held constant, fi is controlling from point 1 to point 2, at. which points fyy.p. in compression is taken as equal to f,u,.,. in tension. The irregular hexagonal figure l-a-2-3-& represents the maximum-shear theory and is the same as the maximumstress theory in the upper right quadrant, l-u-2, and the lower left quadrant, 3-c+ but is more conservative in the other two quadrants where the stresses are of opposite signs. This can be explained by Eq. 9.85, which is for the three dimensional theory. If f2/ and fi are both positive and fi is equal to zero, the maximum minus the minimum stress will be equal to either fi minus zero or f, minus zero. However, iffi: is negative and f, is positive and f2 is zero, the maximum minus the minimum will be f, - fi, which will give the diagonal line l-4. The rhombus A-B-C-D represents the maximum-strain theory. By referring to Eqs. 9.87 and 9.88, in which fi is taken as equal to zero for two-dimensional stress, it may hc seen that if the stresses fi and f, have the same sign, then
Example of a Vessel in Which the Four Theories are Compared
i
either jZ or j, can appreciably exceed the yield point of the material. This is true because the strain in the y direction reduces the strain in the z direction and vice versa when the stresses have the same sign, as is the case at point A or point C. The strain-energy theory is represented by the ellipse and can be evaluated by means of Eq. 9.102.
I
9.16 EXAMPLE OF A VESSEL IN WHICH THE FOUR THEORIES ARE COMPARED
i !
1 1 I
/ :
Because a tall fractionating tower has inherent stresses resulting from dead weight, pressure stresses, and superimposed loads such as wind or seismic forces, it is ideal for comparing the four theories discussed in the previous sections. Brummerstedt (153) in 1943 presented an example design of a tower comparing the four theories. His problem concerned an analysis of stresses in a 70-tray fractionating tower which had a 10 ft inside diameter and a height of 160 ft. The tower was to withstand an internal pressure of 200 psi and a superimposed seismic force equal to 20% of its operating weight. This force was assumed to apply at the center of gravity of the tower. The tower was to be designed to a maximum allowable stress of 13,750 psi for the material of construction. The welded-joint efficiency was to be 82 c&. This resulted in an allowable stress is 13,750 X 0.82 or 11,300 psi. (These values came from the API-ASME code restrictions in effect in 1943). The operating weight including steel plate, attachments such as piping platforms, ladders, and operating liquid was estimated to be 620,000 lb. The seismic force of 20”/0 resulted in a horizontal force of 124,000 lb. This force induced a moment, of 9,600,OOO ft-lb on the column. The circumferential- and longitudinal-pressure stresses resulting from internal pressure were computed by Brummerstedt (153) and combined with the dead weight and seismic stresses under the assumption that the radial stress in the tower shell was zero. These stresses were combined by the four theories, and the results tabulated for comparison. Tables 9.4 and 9.5 summarize the results. The example tower design is such that the percentage difference in total weights as indicated in Table 9.5 is not very great. This is due to the comparatively high design pressure of 200 psi. A similar comparative analysis made of such a tower operating under a low pressure of under 50 psi often results in a considerably greater variation. 111 such a case Lhe method of analysis becomes of greater
Table
9.4.
Summation of Combined Stresses in a Tall
Tower (According to Brummerstedt) (153 i
Resulting Stress, psi Ratio Theory (based on l>+in. shell) (basis of max stress) Maximum-stress 9,620 1 .OOO:l Maximum-strain 9,204 0.957:1 Maximum-shear 11,780 1.225:1 Strain-energy 10,450 1.088:1
importance. However, as indicat.ed in an earlier section. :I t.all tower operating under a low desigu pressure may fail because of elastic instability. To design for such a condiLion t,he appropriate relationships presented must be applied. Shell and head thicknesses based only upon membranestress equations provide no allowance for superimpc#setl loads on vertical vessels. However, the shell thicknesses obtained by such equations provide a convenient stari.ing point for evaluating the thicknesses for vertical vessels since the thicknesses thus obtained may be modified to sat,isfy structural requirements. In the case of vessels operating under internal pressures of 30 lb per sq in. gage or more it is usually convenient to first check the cumulaG\-e tensile stresses from pressure, wind bending moments, and/or seismic moments. The design of tall vessels fol operation at low internal pressures or the design of ‘wn!vessel under external pressure is controlled by the cumul:~tive compressive forces. The design of such vessels can be determined most rapidly by beginning t.he calculations with the cumulative compressive stresses rat.her than with the t,ensile stresses.
Table 9.5.
Summation of Thicknesses and Weights Required by the Four Theories (153) Prr-
Theory Maximumstress Maxirnurnstrain
Maxilnumshear
Sl.rainrwrgy
Bottom 20 ft
Next 20 ft
1316 in. 1M in. 1>/4 in.
t
Next
20
ft
Weight crntof av Ol Shell Top Plates MHX
100
ft
(lb)
st>rrss
136 in. 136 in. SO,000 100.0
I ss in. 136 in. llg in. 285,000 98.2 .
1x6 in. 136 in. I%/4 in. 1’6 in. 300,000 103. 5 1756 in. 154s in. I:j’16
PROELEMS
I
181
1. An insulated steel fractionating column located at Oakland, California, is 6 ft, 0 in. in inside diameter and 160 ft, 0 in. from tangent to tangent between heads. The heads project 1 ft, 6 in. beyond the point of tangency. The skirt is 8 ft, 0 in. from the base to the shell junction at the point of tangency with the bottom head. The vessel is designed to operate at 100 lb per sq in. gage. It is const,ructed of SA 285 grade C steel (13,750 psi maximum allowable tensile stress). The effective wind area of external attachments i s e&mated to be 10% of the
in.
1'6 in. 295,000 IQ! .I!
182
Design of Tall Vertical Vessels
The area of the uninsulated column. The insulation is 3 in. thick and weighs 40 lb per cu ft. tray spacing is 18 in., and there are 102 trays with an estimated weight of 25 lb per sq ft of Calculate the column cross section. The top tray is 4 ft, 0 in. below the top tangent line. minium shell thickness at the bottom tangent line resulting from wind moment. 2. For the vessel in problem 1, calculate the maximum stress at the bottom tangent line resulting from the seismic moment. 3. If the shell of the vessel in problem 1 is fabricated from 20 plates 96 in. wide, specify the thickness for each course allowing a minimum of xs in. for corrosion. 4. Redesign the vessel in the example design for full vacuum operation. 5. Redesign the vessel in the example design for 160 lb per sq in. gage operating pressure. 6. Redesign the vessel in the example design for the same conditions but on the basis of the maximum-shear theory. 7. Redesign the vessel in the example design for the same conditions but on the basis of the maximum-strain theory. 8. Redesign the vessel in the example design for the same conditions but on the basis of the maximum-strain-energy theory. Preliminary 9. A fractionating tower is required to separate styrene from a dilute feed. calculations indicate that 70 trays will be required with the feed entering on tray 32 (from the The reboiler will be separate from the column. Ninety-five per cent recovery of bottom). the styrene in the feed is desired. Annual production of styrene from the columns is to be 8000 tons. Stream Comnosition. Weight Percentages Feed Mol. Wt (saturated liquid) Bottoms Styrene Ethylbenzene Toluene Benzene Temperatures
104.14 106.16 92.13 78.11
Top of column Bottom of column Pressures Top of column Bottom of column Reflux ratio, L/D = 7 (mol ratio)
37.0 61.1 1.1 0.8 54” 90” 30 310
99.7 0.3
c c mm Hg mm Hg
The tower is self-sustaining (no guy wires) and is to have a IO-ft skirt extending from the top of the foundation to the tangent line of the bottom dished head. The erected tower is to be located in the Houston, Texas, area. The overhead condenser is to rest on the ground, and the reflux is to be pumped back. The client specifies that the bubble caps are not to be larger than 5 in. but may be smaller if desired. The tower is to be designed for full-vacuum service. A tray layout and tower design excluding skirt, foundation bolts, nozzles, and bubble-cap details are required. REFERENCES FOR PROBLEM 9 Bolles, W. L., “Optimum Bubble Cap Tray Design,” Part I, Petroleum Processing, Vol. 11, No. 2 (1956); Part II, No. 3; Part III, No. 4; Part IV, No. 5. Boundy, Ray, Styrene ACS Monograph No. 115, Reinhold Publishing Company, New York, 1952. Davies, J. A., “Bubble Trays-Design and Layout-Part I,” Petroleum Refiner, Vol. 29, No. 8 (1950); Part II, ibid., No. 9.
C H A P T E R
DESIGN OF SUPPORTS FOR VERTICAL VESSELS
ertical vessels are normally supported by means of a suitable structure resting on a reinforced-concrete foundation. This support structure between the vessel and the foundation may consist of a cylindrical steel shell termed a “skirt.” An alternate design may involve the use of lugs or brackets attached to the vessel and resting on columns or beams. These more common designs for supporting vertical vessels will be described. 10.1 SKIRT SUPPORTS FOR VERTICAL VESSELS 10.1 Skirt Thickness. Tall vertical vessels are usually supported by skirts. Because cylindrical shells have all the metal area located at the maximum distance (for a given diameter) from the neutral axis, the section modulus, 2, is maximum, and the induced stress minimum for the metal involved. Thus the cylindrical skirt is an economical design for a support for a tall vertical vessel. The skirt is usually welded directly to the vessel. Because the skirt is not required to withstand the pressure in the vessel, the selection of material is not limited to the steels permitted by the pressure-vessel codes, and structural steels with corresponding allowable stresses may be used with some economy. (The steels used in the design of flat-bottomed cylindrical storage tanks (se&hapter 3) are suitable for the skirts of vertical vessels 9 For structural loads a factor of safety of 3 based on the ultimate tensile strength is usually used, whereas@ factor of safety of 4 is used with pressure vessels.) Thus t&e, allowable stress in the skirt is usually 3334 ‘% higher than that in the shell of a pressure vessel when the steels in each case have the same ultimate tensile strength. The skirt may be welded directly to the bottom dished
head, flush with the shell, or to the outside of the shell. Ib the skirt is welded flush with the shell, the weight of the vessel in the absence of wind and seismic loads places the weld in compression. On the other hand, if the skirt is welded to the outside of the vessel, the weld joint is in shear; therefore this method is not so satisfactory, but it is an easy method of erection and is often used for small vessels. There will be no stress from internal or external pressure for the skirt, unlike for the shell of the vessel, but the stresses from dead weight and from the wind or seismic bending moments will be a maximum. The same procedure may be used for designing the skirt as for designing the shell, which was described in Chapter 9. Note: Subscript b refers to thebase of the skirt. Wind-load stress = fwt, =
15.89deR.H2 d 2t 0
(9.20) (9.17)
8CWH Seismic-load stress = fsb = 7 Dead-weight stress = f& = E T dt Max permissible compressive stress = fcattoW. = 1.5 x 106 l/t,t, s 6 y.p. P Max tensile stress = ftmax = (fwb orf&) - fdb Max compressive stress = foaX = (f& or fsb) i- fdb
(9.75)
(9.6)
(9.841 (9.78) (9.80)
Design of Supports for Vertical Vessels
184
and
Substituting gives:
But because of the bond, E, = e,s,
fs
I -,” (b) Fig. 10.1.
Sketch of loading of anchor bolts.
After the skirt and bearing plat,e have been designed, the skirt design should be checked for the reaction of the bolting chairs or ring. (See section lO.lg.) 10.1 b Skirt-bearing-plate and Anchor-bolt Design.
The bottom of the skirt of the vessel must be securely anchored to the concrete foundation by means of anchor bolts embedded in the concrete to prevent overturning from the bending moments induced by wind or seismic loads. The concrete foundation is poured with adequate reinforcing steel to carry tensile loads (143, 154, 155). The anchor bolts may be formed from steel rounds threaded at one end and usually with a curved or hooked end embedded in the concrete. The bolting material should be clean and free of oil so that the cement in the concrete will bond to the embedded surface of the steel. When either a compressive or tensile load is applied to the anchor bolts, the load is transferred from the steel through the bond to the concrete. Surface irregularities, bends, and hooks aid in transferring loads from steel to concrete. As the steel and concrete are bonded, the resulting strain is the same for both the,steel and concrete at the bond. The modulus of elasticity of steel, E,, is about 30 X lo6 psi while that of concrete, EC, varies from about 2 X lo6 to 4 X lo6 psi depending upon the mix employed. The ratio of these moduli is: ES n=-
FJC rewrit.ing gives: Ecn = E, But
(10.1)
‘10.2)
induced = %fc induced
Table 10.1 gives the value of n as a function of tY,e compressive strength of the concrete, which in turn is a fmction of the mix used for the concrete. The bending moment and weight of the vertical vessel result in a loading condition on the concrete foundation somewhat similar to that in a reinforced-concrete beam. Figure 10.1 is a sketch representing the loading condition of the anchor bolts in the concrete foundation. Figure 10.1, detail a is a sketch showing the bearing plate at the base of a skirt for a vertical vessel. In the calculations it is assumed that the bolt circle is in the center of the bearing plate. Sometimes the bolt circle is made larger than the mean diameter of the bearing plate but should be taken equal to it for simplicity of calculation since the error is small and is on the safe side. “,The wind l.oad and the dead-weight load of the vessel result in a tensile load on the upwind anchor bolts and a compressive load on the downwind anchor bolts.’ If je is the compressive stress in the concrete, the induced compressive stress in the steel bolts in the concrete is given by Eq. 10.2. Thus njc !s the induced compressive stress in the steel bolts on the downwind side, and js is the maximum tensile stress on the upwind side. As the stress is directly proport,ional to the distance from the neutral axis, a straight line may be drawn from js to nfc, as shown in detail b of Fig. 10.1. The neutral axis is located a distance kd from the downwind side of the bearing plate and a distance (d - kd) from the upwind side. By similar triangles, we obtain:
fs
~-
7&
(d - kd) = kd therefore 1 k = nfc nfc = ~_ + fs 1 + U../n.fJ
Table 10.1.
(10.3)
Average Values of Properties nf Three Concrete
Mixes
Water Content fc’ U.S. Gallons 28-day Ultimate 3. per 94-lb Sack Compressive of Cement Strength, psi 7% 2000 6% 2500 6 3000 5 3750
n x IO6 Ec 15 12 10 n
.“c Allowable Compressive St.rength,psi 800 1000 1200 1400
Skirt Supports for Vertical Vessels
where ,‘; = maximum induced tensile stress in steel at bolt-circle center line on upwind side, pounds per square inch. fe = maximum induced compressive stress in concrete at bolt-circle center line on downwind side, pounds per square inch / E ‘n=2 x7
where Ct is the term in the brackets and is a constant for a given value of k. To determine the distance 11 consider the element which is located a distance of r(cos a + cos 0) from the neutral axis. The moment of the force on this element times this lever arm is: dMt = dFt r(cos a + cos 13)
If the maximum induced tensile stress in the volts, f8, and t.he maximum induced compressive stress in the concrete, fc, at the center line of the bolt circle-are known, k may be determined by use of Eq. 10.3. Taylor, Thompson, and. Smulski (156) have expressed the area of bolting steel% terms of an equivalent shell of steel of thickness tl having the same total cross-sectional tIrea of steel as shown in Fig. 10.2. Referring to Fig. 10.2, we find that the location of the neutral axis may be defined in terms of angle (Y (156). d/2 - kd --cl--k ‘OS a = ~ 42
= fd1r
Mf = j,tlr22 = 2fstlr2
(1 + cos a)
* Ja
r(cos e + cos CX)
r(cos 8 + cos a!)2 (1 + cos a)
1
1
de
de
a + cos epde (1 + cos a)
(COS
(7r - a) cos2 a + Q(sin (Y cos (11) + &(7r -- a~) 1 + cos (11 [ I (10.10)
Dividing Mt by FL gives 11. 11 =
(10.5)
The distance of this element from the neutfal axis is
(T - a) cos2 a + &sin (Y cos (Y) + +(7r-). r (7r - a) cos a + sin LY [ I (10.11)
(Note that 11 is a constant for a given value of k.)
r(cos a + co9 e>
R E L A T I O N S H I P S FOR T H E C O M P R E S S I O N S I D E . On the compression side a similar procedure is used. A differential element of concrete and steel is considered having an area of: dA, = tzr de (10.12)
The maximum distance from the neutral axis for such an element is: r(1 + cos (Y) The stress in the element, fs’, is directly proportional to the distance from the neutral axis, and if the maximum stress is fs, p tcos a + cos e) =.A; (1 + cos a)
ccos a + cos e) ,
By integration,
In the same figure consider an element of the bolting steel measured by angle d0. The area of this element is given by:
A
[
=fdlr [
(10.4)
dA, = tlr d0
185
where t2 = concrete width (exclusive of bolting steel, under the bearing plate, inches. The distance of this element from the neutral axis is:
(10.6)
r(cos
e
tl)
- cos a)
Multiplying the stress by the elemental area gives the elemental force in tension, dF,. dFt = fstlr
a -I- cam e) de (1 + cos a)
(co9
(10.7)
The summation of the elemental forces on the bolting steel in tension can be represented by tensile force Ft located at the center of tension and distance 11 from the neutral axis. Similarly the summation of the compressive forces on the concrete in compression can be represented by a compressive force F, located at distance l2 from the neutral axis. RELATIONSHIPS FOR THE T ENSION S IDE. By integration of Eq. 10.7 for the upper and lower halves on both sides of the center line, we obtain: (co9 a! + toss) de (1 + cos cy)
Ft = f,tlr2
= f..rlr [
J
r
-
-
l
=jstlrct
-
-
-
-
L- ((T - n) cos cx + sin a)] (10.8) 1 + cos a (10.9)
-
_
‘..-
\
\
-\I
Fig.
T
10.2.
Plan view of loading on bolting steel and bearing plate.
Design of Supports for Vertical Vessels
186 Table
10.2.
Values of Constants C,, C,, Z, and j as a Function of k (156)
k
CC
Ct
z
j
0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 0.550 0.600
0.600 0.852 1.049 1.218 1.370 1.510 1.640 1.765 1.884 2.000 2.113 2.224
3.008 2.887 2.772 2.661 2.551 2.442 2.333 2.224 2.113 2.000 1.884 1.765
0.490 0.480 0.469 0.459 0.448 0.438 0.427 0.416 0.404 0.393 0.381 0.369
0.760 0.766 0.771 0.776 0.779 0.781 0.783 0.784 0.785 0.786 0.785 0.784
element times the lever arm is: dM, = dFc r(cos tl - cos a) = (t2 + ntl)r2fc “: ~mc~~ap)2 de By integration, MC = (tz + ntdfcr22 =
(t2 +
The stress fc’ in the element is directly proportional to the distance from the neutral axis, and if the maximum induced stress is fc, r(cos e - cos a) fc’ = fc r(1 - cos a)
The corresponding compressive (camp.) stress, Sg(comp.), in the steel on the compression side (see Eq. 10.2) is:
Ss(com*.) = nfc ‘y y;oy;)a)
(10.14)
The corresponding compressive forces in the element are obtained by multiplying the elemental stresses by the elemenlal areas.
dFc (concrete) = fc’ dA, = f&r dFe(steel)
ntdfcr22
de
cos2 a - ?&sin a! cos a) + & 1 - cos 4 -1 (10.19)
a cos2 a - +(sin (Y cos (Y) + +a sin LY - (Y cos Q! [
I1
I
ld (T - a) cos2 a + +(7r - a) + $ sin a cos a =2 (7r - a) cm a + sin a [ +cu-$sinarcoscu+a~os~,i (10.21) ++ sin (Y - ff cos cr i
[
Referring to Fig. 10.2, we find that the distance from the neutral axis to the center line of the vessel is (d/2)(cos a) and distance zd is equal to: d zd = l2 + - cos a 2
(10.22) +a - 3 sin a cos LY + a! cos2 LY sin 01 - (Y cos (Y
= nfc’ dA, = nf,tlr
dFc(totau
=
(t2
+ nh)rfc
F, = (t2 + ntl)rfc2 F, = (tz + ntdrfc F, = 02 + ntl)rfJA
>I
(10.23)
The quantities Ct, C,, j, and z are given in Table 10.2 as a function of k. B OLTING AREA AND B EARING- PLATE W IDTH . Taking a s&&&nof moments about F, (see Fig. 10.2) we obtain:
“O;’ I3 cos - cos CY aI de
M w i n d - W&,zd
By integration,
(r) ( 1 0 . 2 0 )
+ 12
2 = ; cos a + [ (
The total compressive force on the element is equal to the sum of the above two equations, or
1
(Note that 12 is a constant for a given value of k.) The total distance between the forces Ft and F,: is equal to II + 12. This distance divided by d gives the dimensionless ratio, j. j=
J10.13)
s0
e - cos a]2
1 - cos LI1
Dividing MC by F, gives 12.
12 = The maximum distance from the neutral axis for such an element is: r(1 - cos f-I)
p (COS
- Ftjd = 0
therefore /0
Ft = Mwind - Wdwzd jd
p ““1” I3 -cos cosa a de
2(sincu - (YCOS(Y) 1 - cos (Y I
(10.17)
'(10.24)
Substituting for Ft by Eq. 10.9 we obt.a!‘a: (10.25)
(10.18)
where C, = the term in the bracket and is a constant for a given value of k. To determine the distance 12 the same procedure is used as for the tension side. The moment of the force on the
And A, = 2mtl; therefore A, E zT
Mwind - Wd,,,.zd Gfdd
I
(10.26)
Skirt
Referring to Fig. 10.2 and taking a summation of vertical force? we obtain: i
(10.27)
Ft + wdw - F, = 0
Substituting for Ft by Eq. 10.9 and F, by Eq. 10.18, we obtain: jstlrct
+
wdw - ct.2 +
ntl)rfccc =
0
Solving for tr, we obtain: 1
2
= wdw
+ (GA - CJ,n)rli WJ
(10.28)
The total width of the bearing plate will be tl + tz (Eq. 10.25 plus Eq. 10.28). Therefore Width of bearing plate, t3 = tl + t2
(10.29)
Nomographs for the solution of anchor-bolt problems by the method of Taylor, Thompson, and Smulski have been presented by Gartner (233). An alternate procedure has been presented by Jorgensen (234). D ETERMINATION OF B EARING-PLATE THICKNESS. The thickness of the bearing plate is determined by the compression load on the downwind side of the vertical vessel. The minimum required width of the bearing plate was previously determined by use of Eq. 10.29. The maximum compressive stress between the bearing plate and the COITCrete occurs at the outer periphery of the bearing plate. The induced compressive stress at the bolt-circle center line was determined by successive approximation in calculating the required width of bearing plate (see Eq. 10.29). Equation 10.30 gives the relationship between the maximum induced compressive stress at the outer periphery and the corresponding stress at the bolt circle. fc(max induced) = (fc(bolt
2kd + t3 2kd >
circle induced)) ~
(
(10.30)
Although the compressive stress varies from the maximum given in Eq. 10.30 to a lesser value at the junction of the skirt and bearing plate, the value at the bolt circle may be used for simplicity of calculation in determining the required thickness of the bearing plate. BEARING PLATES WITHOUT G USSETS. A bearing plate without gussets may be assumed to be a uniformly loaded cantilever beam with je(max induced) the uniform load. The maximum bending moment for such a beam occurs at the junction of the skirt and bearing plate for unit circumferential length (b = 1 in.) and is equal to: MC ,,,a=, = jo ,,,a= bl f 0
= -ff
(for b = 1) (10.31)
where 1 = outer radius of bearing plate minus outer radius of skirt, inches The maximum stress in an elemental strip of unit width is given by: .f( max)
6M (maw 3.flY In** l2 = ~ = btd2 t42
(for b = :)
where t4 = bearing-plate thickness, inches
Supports
for
Vertical
Vessels
187
Letting jcmax) = f@no&,le) and solving for t4 gives us: I (10.32a) tr = 11/ 3.fc msx/f(allow.) The thickness of the bearing plate, t4, as calculated by Eq. 10.32~ is usually rounded off to the next larger standard thickness of plate. ‘I B E A R I N G P LATES WITH G U S S E T S . If gussets are used to stiffen the bearing plates, the loading condition on the section of the plate between two gussets may be considered to act similarly to that of a rectangular uniformly loaded plate with two opposite edges simply supported by the gussets, the third edge joined to the shell, and the fourth and outer edge free. Timoshenko (107) has tabulated ,the deflections and bending moments for this case as shown in Table 10.3. Note in Table 10.3 that for the case where l/b = 0 (no gussets or gusset spacing, b = 00) the bending moment reduces to Eq. 10.31, and the thickness of the flange is determined by Eq. 10.32. Also note that when l/b is equal to or less than %, the maximum bending moment occurs at the junction with the shell because of cantilever action. If l/b is greater than 4, the maximum bending moment occurs at the middle of the free edge. To determine the bearing-plate thickness from the bending moments, Eq. 10.33 may be used. (10.32b) I
D ESIGN PROCEDURE FOR BOLTING CALCULATIONS AND SIZING OF BEARL~G PLATE . The location& the neutral axiyiae&ined by&e Rio of induced stresses, as indicated by Eq. 10.3. Thus the determination of minimum bolting and minimum width of bearing plate requires successive-approximation calculations. The value of k determines the constants Ct, C,, j, and z, which in turn determine the values of Ft and F, and their locations. As a first approximation in the determination of k, js may be taken as the maximum allowable stress in the bolting steel, but je should not be taken as the maximum allowable compressive stress in the concrete since the maximum com-
Table
10.3.
Maximum Bending Moments in a Bearing Plate with Gussets (107)
(Courtesy of McGraw-Hill Book Co.)
0
8. - 0.5OOjJ2 45 0078jcb2 - 0 428jJ2 0 0293j,b2 -0.319j~P w I% 0. 0558f,b2 - 0 . 227jJ2 1 0. 0972f,b2 -0.119j012 % 0. 123f,b2 - 0 . 124f,12 2 0 131f,b2 - 0 . 125f,12 3 0. 133f,b2 - 0. 125f,12 00 0. 133f,b2 - 0 . 125f,12 b = gusset spacing (5 direction) inches. 1 = bearing-plate outside radius minus skirt outside radius (y direction) inches.
188
Design
of
Supports
for
Vertical
Vessels Table
Bolt
Size d
Standard Thread No. of Root Area Threads 0.126 13 11 0.202 10 0.302 9 8
0.419 0.551
10.4.
Bolt
Data
(157)
(Courtesy of Taylor Forge & Pipe Works) Minimum Bolt Spacing* 8-thread Series Radial ~.-~ No. of Root Minimum PreDistance ferred R Threads Area B, INO. 8
thread series below 1”
1 Q”
3 I,
11~5 1%
3 3 3 3
2’16 2%
Edge Distance
E 5/‘8” N
‘x6” ‘346
1?,4 1% 1%
Nl.lt Dimem:i on (across flats)
Maximum Fillet Radius P
w
M )’
1!16
%6
‘%S
l?G
w
‘546
13iS
N
1%
746
I1946
x6
2
746
2x6
%6
2%
36
8
0.551
0.693 0.890 1.054 1.294
8 8 8 8
0.728 0.929 1.155 1.405
5% 5 5 4 %
1.515 1.744 2.049 2.300
8 8 8 8
1.680 1.980 2.304 2.652
2
4 % 4 4 4
3.020 3.715 4.618 5.621
8 8 8 8
3.423 4.292 5.259 6.324
2% 2% 2% 2%
1’4 6
2%6
2%
1%
i
w6
36
3 %
‘Y6
3% 3 % 4% 4 %
‘416 ‘946
N ‘546
* B, = center-t.o-cent,er distance het.ween bolts, inches .
pressive stress in the concrete is developed at the outer periphery of the bearing plate rather than at the center line of the bolt circle. After evaluating k by Eq. 10.3, the minimum area of bolting steel required may be determined by Ey. 10.26 and the preceding relationships. This permits the selection of the number and size of bolts having sufficient root area to equal or slightly exceed the minimum required bolting area. Table 10.4 gives the necessary information for t.his selection. Usually the number of bolts selected is a multiple of four to permit greater ease in bolt layout.
A s(act.) = ~VAH 2 ,Js(min)
(10.33)
where N = number of bolts Ae = root area of bolt, square inches (Table 10.3) A s(min) = minimum area of bolting steel, square inches (Eq. 10.26) A a(act.) = actual area of bolting steel used, square inches The width of the bearing plate may be evaluated by use of Eqs. 10.25, 10.28, and 10.29. In the first trial the area of bolting steel is select.ed and the width of the bearing plate determined by use of t,he original evaluation of k. The values of the induced tensile
and compressive stresses in the stee! and concrete based on this area and this width are next determined. An induced tensile stress in the steel based upon the first calculation of k may be determined by Eq. 10.34. .fs (induced) = .fs(first estimate) e
s(act.)
(10.34)
An induced maximum compressive stress in the concrete also based upon the first calculation of k may be determined by use of Eq. 10.30. These values of f8 and fc may be used to obtain a more correct value of k by Eq. 10.3. If the new value of k differs appreciably from that originally calculated, a complete recalculation should be made using the constants Ct, C,, j, and z based upon the new value of k. After consistent values of fs, fe, and k have been determined, the bolting design and bearing-plate width are established. The thickness of the bearing plate may then be established by use of Eq. 10.31. To complete the design of -the skirt and bolting ring the compression ring or bolting “chairs” should also be considered. 10.1~
Example
Calculation
10.1,
Bearing-plate
Design.
A proposed fractionation column is 10 ft in diameter and 150 ft high and rests on a foundation of 3000-psi-strength
189
Skirt Supports for Vertical Vessels
concre$e. The proposed bearing plate under the skirt has a 9 ft, 8 in. inside diameter and an outside diameter of 11 ft, 8 in. The bolt circle is 11 ft, 0 in. in diameter and contains 24 steel bolts 215 in. in diameter (from Table 10.4, the area per bolt = 3.72 sq in). Assume that under operating conditions the dead weight of the tower is 600.000 lb. A high wind velocity develops a wind moment of 8,OOO.OOO ft-lb. A continuous compression ring is used. From Table 10.1, n = Es/EC = 10; js(atlon.) for the structuralsteel skirt is 20,000 psi. Determine the maximum induced stress in tension in the bolts and the maximum induced stress in compression in the bolts. Also determine I he maximum compression stress in the concrete at the outermost edge of the bearing plate on the downwind side and the width and thickness of the bearing plate. For first trial assume js E 20,000. From Table 10.1, jccmax) = 1200.
The compressive load may be cakulated by a summation of vertical forces using Eq. 10.27. Ft + Wa,,. - F, = 0 600,000 + 598.000
k (spprox.)
=
1 1+f”
-
de
=
1 = 0.333 1+-20,000 (10)(1000)
By rearranging Eq. 10.30 and solving for jc+,l+ rirclp) we* obtain :
fc(bolt
circle)
(2)(0.333)(11)(12) (2)(0.333)(11)(12) + 12_1 = 1055
psi
To evaluate the induced stresses the constants are read from ‘I’able 10.2. For k = 0.333 Cc = 1.588 Ct = 2.376 t = 0.431
F, = (tz + rrt,)rj-J:, But
IS = tB - t, = 12 - 0.215 = 11.785 in.
1.198.000 = [(11.785) + (10 X 0.21~)](5.5)(12)(1.588)j~ jc = 818 psi Rechecking k by Eq. 10.3 gives: k =
~~ = Mwind
8 X lo6 - 6 X 10S(0.431)(11) - Wdw.zd =(0.782)(11) .id
= 8 ' lo6 - 2'85 ' lo" = 598 000 lb 8.6
The induced stress in t,he steel js based upon k = 0.333 may be evaluated by Eg. 10.9.
tl = A = (24)(3*72) = 0.215 in. c-w-. ._ ,& 411) (12)
598,000 = j,(O.215)(5.5)(12)(2.376) fs = 17,700 psi
1 = ~~= 0.317 1 + 2.16
Rechecking const,ant,sand stresses gives the following. From Table 10.2 for k = 0.31i, c, = 1.554 Cf = 2.405 i = 0.434 j = 0.782
2
F
=
8 X lo6 - 10”(6)(0.-4,3~4)(11) = 396000 (0.782)(11)
596,000 .f* = co-q-( 12)(2.
iOj) = 17AjO Pi
F,. = 600,000 + 396.000 = I, 196,000 lb
.fc =
1,196,OOO
-- = 835
(11.785 + 2.15)(5.5)(12)(1.55.4)
psi
I -- = 0.321 1 + 2.08
1
k =
17,400
’ + (10)(835) k = 0.32, approximalely , by interpolation The rechecks (Jf fS, .f,, and k are in sufbcienl agreement for design purposes. To check maximum compressive stress in bolts and concrete, E(t. 10.2 is used. fs(comp.) = gafc = 10jc = 8350 psi By Eq. 10.30
fr(msx
,c
1 17,700
l +
(10)(X18)
j = 0.782
The tensile load may be calculated by Eq. 10.24.
lb
The induced stress in the concrete al the bolt circle. jC based upon k = 0.333 may he evaluated by Eq. 10.18.
t(p posed) = (11 ft, 8 in.) - (9 ft. 8 in.) = 12 in. ro 2 By Eq. 10.3, estimating jr(t)olt circle) = 1000,
= F, = 1,198.OOO
induced) =
(.fr(tmlt
= 835
circle induced))
(2)(0.3w)(w
+ 12
(2)(0.32)(11)(12)
= 835(-$) = 965psi
>
Design of Supports for Vertical Vessels
190
By Eq. 10.32b ,-Full-fillet weld
Ski,rt-
t4 = ~6(11,700/20,000) = 1.875 Therefore, use lid-in. plate. 10.1
d
Practical
Considerations
in
Designing
Bearing
Plates.
Fig. 10.3.
Rolled-angle bearing plate.
Determination of bearing-plate thickness by Eq. 10.32 is as follows:
1 =
t4
11 8 in. - 10 ft, 0 in. = 1 0 in. ft, 2
= 10
(‘)(“5) 20,000
-=.
3
81
in
*
(without gussets)
As this thickness is considered to be excessive, the bearing plate will be stiffened with 24 gussets equally spaced and straddling the bolts. The gusset spacing, b, is b = dll)W = 17.3 in.
24
1 10 -z--z 0.58 b
11.3
Interpolating between l/b = 34 and l/b = 45 from Table 10.3 gives: M max = M, = -0.26fJ2 = (-0.26)(965)(100) = -25,100 in-lb
ROLLED-ANGLE BEARING PLATE. If the vertical vessel is not very high and a skirt is used to support the vessel rather than legs, lugs, or columns, a simple design may suffice for the bearing plate. If the calculated thickness of the bearing plate is 36 in. or less, a steel angle rolled tti fit the outside of the skirt may be lap welded as shown in Fig. 10.3. SINGLE-RING BEARING PLATE. If the required bearingplate thickness is 35 in. to N in., a design using a single-ring bearing plate may be employed, as shown in Fig. 10.4. If the bearing-plate width is less than 5 in. and the thickness less than $4 in., the rolled-angle design (Fig: 10.3) will probably be more economical. CENTERED CHAIRS. If the required bearing-plate thickness is N in. or greater for the design shown in Fig. 10.4, a bolting “chair” can be used to advantage. Figure 10.5 shows a typical design for a centered anchor-bolt chair. Although the number and size of bolts required should be checked for each design, Table 10.5 gives some typical values of the maximum number of chairs usually inserted in a vessel or skirt of a given diameter. In checking the bearing-plate thickness for a -entered chair the plate inside the stiffeners may be considered to act as a concentrated loaded beam with fixed ends. The concentrated load, P, is produced by the bolt and is equal to maximum bolting stress times the bolting area, or P = fax&, (10.35) wheref, = maximum induced stress in bolting steel Ab = root area of anchor bolt, square inches (The values of f8 and & are those determined in an earlier section.) P = maximum bolt load, pounds The maximum bending moment in the bearing plate inside the chair occurs at upwind dead center and is located
By Eq. 10.32b t4 = 1/6(25,100/20,000) = 2.75 in. Further reduction in bearing-plate thickness could be realized if the gusset spacing were decreased by using 4 8 gussets. For 48 gussets, gusset spacing, b is: b
=
Skir+
I
7411lW) = 8.65 in. 48
10 1-=-= 1.255
b
8.65
Interpolating again from Table 10.3 gives: j-- 5” minimum _I_t(
M msx = M, = -0.121fJ2 = -11,700 in-lb
I
\
- \
Fig. 10.4.
\I
/
Single ring beam, plate
with gusrats.
Skirt Supports for Vertical Vessels Table
10.5.
191
Maximum Number of Centered Chairs in Various-sized
Vessel
Skirt diameter, ft 3 4 5 6 7 8 9 10
Skirts
No. of Chairs 4 8 8 12 16 16 20 24
width of the plate. With this consideration the required bearing-plate thickness inside the chair may be calculated by Eq. 10.37.
Plan View B-B
t4 =
f5Mmx J (t3 - bWfan,w.
(10.37)-
where t3 = bearing-plate width, inches t4 = bearing-plate thickness, inches bhd = bolt-hole diameter in bearing plate, inches f&llow. = allowable stress, pounds per square inch
Washer Bolt size + 1 Section A-A
The bending moment in the bearing plate outside the stiffeners (between chairs) may be controlling and can be determined by use of Table 10.3. The thickness can be determined by use of Eq. 10.32b. E MPIRICAL DIMENSIONS FOR E XTERNAL CHAIRS. If the n&be;-of bolts required exceeds the number given in Table 10.5, external bolting chairs may be used, as shown in Fig. 10.6. The proportions for the chair may be determined empiricaliy by the relationships given in detail b of Fig. 10.6. Note that thehole -in- the - - bearing - _ _ plate _ _ is--.made - -
LWasher (thickness equal to bolt diameter) Bolt size + 9” Bolt size + %’ , , .T
Elevation Fig. 10.5.
Centered
anchor-bolt
chair.
at or near tl bolt where the cross-sectional area is minimum. T .roment is given by Eq. 10.36.
Bolt size + y” min
where M,,, = maximum bending moment, inch-pounds b = spacing inside chairs, inches (usually 8 in.) The hole in the bearing plate reduces the effective beam
Fig. 10.6.
External
bolting
chair.
192
Design of Supports for Vertical Vessels
plate
Fig. 10.7.
Vessel skirt with external bolting chairs.
larger than the hole in the__top - plate - for ease in erection of the vessel. CALCULATION OF C O M P R E S S I O N - P L A T E T H I C K N E S S . The zximum load on the compression plate at the top of an external chair occurs on the upwind side of the vertical vessel where the reaction of the bolts produces a compression load. The compression plate may be considered to act as a rectangular plate bounded by the two gusset plates, t,he skirt, and the outside of the plates. The bolt load may be considered to be a uniformly distributed load acting over a circufar area cquaf to the bolt area. The f a c t t h a t t h e compression plate is welded to the skirt and gusset plates as indicated in Fig. 10.7 provides additional rigidity on these sides, which tends to compensate for the lack of support on the fourth side. As an approximation the plate will be considered to act as a plate freely supported on four sides. Timoshenko (107) has developed the relationships for a rectangular plate freely supported on four sides with a concentrated load acting as a uniformly distributed load over a circular area of radius e. In reference to the “Plan” view of Fig. 10.6 with y in the radial direction and r in the circumferential direction, the maximum bending moments M, and M, are given by Eqs. 10.38 and 10.39, respectively.
where M, = maximum bending moment along radisi axis, inch-pounds M, = maximum bending moment along circumferential axis, inch-pounds P = maximum bolt load on upwind side (see Eq. 10.35) pounds cc= Poisson’s ratio (0.30 for steel) In = natural logarithm a = radial distance from outside of skirt to bolt circle, inches I= radial distance from outside of skirt to outer edge of compression plate, inches b = gusset spacing, inches e = radius, of action of concentrated load, inches sz one-half distance across flats of bolting nut, inches 71, Y2 = constant,s from Table 10.6 A comparison of Eqs. 10.38 and 10.39 using the constants in Table 10.6 indicates that for (b/Z) = unity, M, = M,, and that for all cases in which (b/l) is greater than unity, M, is greater than M, and M, is therefore controlling. After the determination of the size of the bolt and the width of the bearing plate and after the selection of the bolt-circle diameter and gusset spacing, the dimensions a,b,e, and 1 are fixed. The constants yi and ys may be evaluated by use of Table 10.6, and the maximum bending moments in the radial and circumferential directions may be computed by use of Eqs. 10.38 and 10.39. For the case in which a is selected to be l/2 and M, is controlling, Eq. 10.38 reduces to: My = 0
(1O.N~)
To determine the maximum stress in the compression ring a strip of unit width is considered. For this case,
fmax
Table 10.6.
6Mll = 2 t5
Constants for Moment Calculation in Compression Ring (107)
b/l Yl Yz (10.38)
(1 + p) 111: + (1 - -YI)
(Courtesy of McGraw-Hill Book Co.) 1.0 1.2 1.4 1.6 1.8 2.0 .-
=
0.350 0.211 0.125 0.073 0.042 0 0.565 0.115 0.085 0.057 0.037 0.023 0 0.135 Note: for a b/l less than 1.0 invert b/l and rotate axes 90”.
Skirt Supports for Vertical Vessels
Or. if fi is assumed to be follow. (10.41)
i5 = dWK,/f,n0w.)
where t5 = thickness of compression plate, inches .fsuoW. = allowable working stress, pounds per square inch lO.le
Example Calculation 10.2, External-chair Design.
An external chair will be designed for a column 8 ft, 0 in. in diameter having 12 bolts 114 in. in diameter with a calculated induced stress of 17,500 psi. The bolt-circle diameter is 8 ft, 6 in., and the outside diameter of the bearing p1at.e is 9 ft, 0 in. The gusset height, h, is 12 in. By Fig. 10.6, t6 = (Sg)(lys In.) = 0.515 in.
(Use 34-in. plate.)
A = 9 in. + (lj$ in.) = 1014 in. b = 8 m. + (l?,< m.) = 936 m. By Table 10.4, root, area of bolt, Ab = 1.405 sq in. The bolt load by Eq. 10.35 is: P = fs&, = (17,500)( 1.405) = 24,600 lb By Fig. 10.6, (8 ft, 6 in.) - (8 ft. 0 in.) = 3 iI a=--2
as the upper plate of the bolting ring. Such a continuous ring is preferred when the spacing of external chairs becomes so small that the compression plates approach a continuous ring. As in the case of the compression plate the maximum load on a continuous compression ring occurs on the upwind side of the vertical vessel where the reaction of the bolts produces a compression load on the ring. This load produces a bending stress in the compresson ring. As in the case of external chairs the vertical gusset plates transfer this compression load to the bearing plate. In determining the thickness of the continuous compression ring the assumption is made that each section of the ring between gussets acts as a rectangular plate bounded by the two gusset plates, the shell, and the outer ring. The bolt load will be considered to be a uniformly distributed load acting over the area of the bolt. Therefore the method used in determining the thickness of the compression plates for external bolting chairs is applicable. This method involves the use of Eq. 10.38, 10.39, or 10.40 and of Table 10.6 plus Eq. 10.41. CALCULATION OF GUSSET-PLATE THICKNESS FOR COMPRESSION RINGS. If the gussets are evenly spaced alternately between bolts, the gusset plate may be considered to react as a vertical column. Normally the gusset is welded to the shell, but no credit is taken for the stiffening effect produced by the shell. The moment of inertia of the gusset about the axis having the least radius of gyration is given in Appendix J, item 1 as: .
1 = (9 ft, 0 in.) - (8 ft, 0 in.) - = 6 in. 2 From Table 10.4, nut’dimension across flats 2.375 e = --~~~~ = ~ = 1.188 in. 2 2 The ccmpression-plate thickness is:
b 9.5 1 .J-8 -=-= 1
6
Interpczlating from Table 10.6 gives: ‘,
y1 = 0.134 Substituting in Eq. 10.40 gives:
,
1
21 (I + p)h-~en- + 1 - y1
( o ( 6 ) ) + 1 - 0.134 7r1.188 I
I!
I I
I
Compression
/ ring
= +8200 in-lb Substituting into Eq. 10.41 withf,llow. = 17,500 psi gives: i,=g;.=dT=1.672 Therefore use lx-in. plate for compression plate. 10.1 f Continuous-compression-ring Thickness. Figure 10.8 shows a sketch of a continuous compression ring used
193
Fig.
10.8.
Skirt with continuous compression ring and strap.
194
Design of Supports for Vertical Vessels
OF
r2 = ta2 12
(10.42)
force, Qc (see (Fig. 6.3) produced in shells with closures, and the calculation may be treated accordingly. In Chapter 6 the following relationships were derived:
where a = area of cross section, square inches is = radius of gyration, inches t6 = gusset-plate thickness, inches 1 = width of gusset, inches
(YLO = &1 @MO + Qo> (6.76)
Equation 4.21 may be used to express the relationship for steel columns in which the value of (h/r) is from 60 to 200. 18,000 1 + (h2/18,000r2)
(4.21)
In detail a of Fig. 10.6 y is the horizontal deflection of the skirt corresponding to yr in Fig. 6.3 and varies with distance above the compression ring in accordance with Eq. 6.65. Applying the boundary condition that the slope of the deflection curve dy/dx is equal to zero at the top of the ring where x = 0, we obtain:
where h = height of gusset, inches
dy
Substituting Eq. 10.42 into this relationship gives. 18,000 f anow~ = 1 + (h2/15001sZ)
(10.43)
=
Bolt load P
0s
=a
its
@PM0
(10.44),
00)
(10.48)
Qo>
(10.49)
Solving Eqs. 10.48 and 10.49 for MO and Qo gives:
Qo
(10.50)
= 4P3&y
MO = -2#S2Dly
(10.51)
P= J4 3(1 r2t2- p2)
(6.86)
where
or h2(bolt load) = 0 (10.45) 1500
Examination of Eq. 10.45 indicates that when the gusset height, h, is small, the third term in the equation may be disregarded. In this case Eq. 10.45 reduces to the relationship for straight compression without column action or (10.46) Equations 10.45 and 10.46 are based on the asumption that the compression plate is sufficiently thick for the bolt load to be transferred to the gusset plates without introduction of eccentric action. The stiffening resulting from the welding of the compression plates and gussets to the shell introduces a margin of safety which justifies the above assumption. If the gussets are not evenly spaced, an eccentric loading will result in an induced bending moment. The thickness ef such gussets may be proportioned empirically, as in the xase of gussets for external chairs.
ts = gts
+
1
18,dbO = 1 + (hz/1500teq
18,000Z1r,3 ‘- (bolt load)ts2 -
W2D1
(YLO = & @MO +
Substituting in Eq. IO.43 gives: Bolt load
L
Noting that y1 of Fig. 6.3 is taken as equal to -y in Fig. 10.6, detail a, we obtain:
The allowable stress, fsuoW., in Eq. 10.43 must be:
fallow.
=(I=
dz z=o 0
(10.47)
lO.lg Reaction of External Bolting Chairs and Compression Rings. The use of external bolting chairs or a compres-
sion ring results in a loading condition that produces a reaction in the skirt. This reaction, R, is similar to the shear
(6.15) By Eq. 6.84 tE W -=r2 Y
(10.52)
where w = load, pounds per linear inch t = skirt thickness, inches P = radius of curvature, inches Substituting Eqs. 6.15 and 10.52 into Eqs. 10.50 and into Eq. 10.51 gives: (10.53) MO = -p2t2wr2 6(1 ,- p2)
_ -w 2f12
(10.54)
Taking a summation of moments about the junction of the skirt with the bearing plate gives: P(a) = Q&m where Qs is the force per linear inch on the skirt and is assumed to act over an arc distance of m, or (10.55)
Skirt Supports for Vertical Vessels
This thickness can’ be reduced by increasing the gusset height. Assuming a gusset height of 18 in. rather than 12 in. will reduce the skirt thickness to:
Substltutine Eq. IO.53 into 10.55 gives: W WC-
mh
1 = 1.214(+$)1a = 0.926 in. or 1.0 in.
Substituting Eq. 10.56 into Eq. 10.54 for u) gives: Mo
=
_
P3t2Par2
(10.57)
6(1 - p2)mh For a strip of unit width under flexure
fallow.
=
6Mo -cm t2
p3Par2 (1 - p2)mh
(10.58)
Substituting for /3 by Eq. 6.86 and solving for t gives:
[1
t = [3(I - P2)P p” ( 1 - p2)?’ m h j
% $6
For steel in which p = 0.3,
t = l.76(&yrts
(10.59)
where t = skirt thickness required to resist reaction of external chairs or compression ring, inches P = radius of skirt, inches m = 2A (see Fig. 10.6) or bs (bolt spacing) P = maximum bolt load, pounds a = radial distance from outside of skirt to bolt circle, inches h = gusset height, inches lO.lh Ring.
Example Calculation 10.3, Reaction of a Bolting
The tower described in Example Calculation 10.1 is to be modified so that it has a bearing plate extending 635 in. out from the skirt with a bolt circle 3>/4 in. outside the skirt. Twenty-four bolts 235 in. in diameter are to be used with n continuous compression ring, and the gusset height is to be 12 in. Determine the required thickness of the skirt to resist the reaction of the bolting ring. The maximum induced stress in the bolts is 17,450 psi, and the maximum allowable stress in the skirt is 20,000 psi. a = 3>/,-in. ~(126.5) m = ~ = 16.6 in.
24
,
fallow.
= 2%000
Psi
h = 12 in.
P = 60 in. By Eq. 10.59
Substituting gives:
(17,450)(3.72)WW (16.6)(12)(20,000) = 1.214
in. or 134 in.
lO.li Thermal Stresses in the Skirt. For the case in which the vessel is operated at a temperature considerably different from atmospheric temperature, a thermal stress may be induced in the skirt as a result of the temperature gradient near the junction of the skirt and the vessel. T EMPERATURE GRADIENT I N S KIRT . To minimize the temperature gradient in the skirt, the skirt may be insulated both inside and out. Skirts of vessels are insulated inside and out for fire protection when manways are cut into the skirt. The modulus of elasticity decreases rapidly with increasing temperature above 600” F with resulting loss in elastic stability. F. E. Wolosewick (160) has given an approximate equation for the skirts of vessels with 2 to 435 in. of insulation both inside and out.
T, = (TV - 50°) - 6.037x - 0.2892’ + 0.009~~ - 0.00007~~ (10.60) Differentiating with respect to z gives: dTZ -6.037 - 0.578~ + 0.027~~ - 0.00028~~ da:=
1
j6 (60)s
(10.61)
where T, = temperature of skirt at z distance below junction of skirt and shell, degrees Fahrenheit TV = temperature of fluid in vessel bottom, degrees Fahrenheit 5 = distance below junction of skirt and shell, inches T HERMAL EXPANSION. As an insulated vessel is brought ‘up to operating temperature, it will undergo thermal expansion. If there is no restraint to this expansion, no stress will be induced. The metal both in the skirt and in the shell at the junction will have the same temperature. From the junction to the foundation a temperature gradient will exist, which will tend to produce a varying thermal expansion. At any given point in the skirt the radial thermal expansion, y, is proportional to the coefficient of thermal expansion, LY, the radius of the skirt, F, and the temperature difference T’, or
y = cw(Tv - Tz) = MT’
p = 17,450 X 3.72 sq in. per bolt
195
(10.62)
where y = radial thermal expansion, inches (Y = coeflicient of thermal expansion inches per inch per degree Fahrenheit r = radius of skirt, inches T’ = (TV - T,) = temperature of vessel bottom minus skirt temperature, degrees Fahrenheit Differentiating Eq. 10.62 with respect to 2, the distance along the skirt from its junction with the vessel, gives: ar dT’ dy -=dx
dx
196 but
Design of Supports for Vertical Vessels
dT’ = d(T, - T,) = -dT,;
and
therefore
dy -= - ar dT,dx dx STRESS
FROM
BENDING
SHEAR.
AND
(YLO = - 2i&l (@MO + Qo) dy = $n, (Wfo + Qo) 0iii z=o
The
(6.75)
(YP dT’ _dx
= dr -2 0.dx
-!2P3D,
s=o =
ij;D
(@MO
~IRW~FERENTIAL THERMAL STRESS, SHELL. (See Eq. 6.125.)
(6.76)
frt
=
/#(PMo
WITH
fCt,
+
Qo) -~
(10.73) I
But bj Eq. 10.71 /3Mo = -Qo; therefore fct = I !!$p SHEAR
+ Qo)
THERMAL
STRESS,~~~,
(10.74) JUNCTION
AT
SHELL.
WITH
(See Eq. 6.121.)
Clearing fractions on I he right side of these equationsgives: arT’(2P3D,) = -@MO
(10.75)
- Qo lO.li
a=$ (2/3*&) = 2PMo + Qo
(10.61)
Adding the two equations gives: arT1(2P3D1)
+ zd$: (2p2D1)
= PM,
+g)
Example Calculation 10.4,
Thermal Stresses.
Consider a vessel having a diameter of 130 in. and a skirt thickness of, 3% in. insulated inside and out with the skirt supporting a shell in which the bottom temperature is 700” F. Assume that the temperature distribution in the skirt is given by Eq. 10.60. Calculate the thermal stresses a1 the junction. a = 7.6 X 10e6 deg F
therefore MO = 2c@D~(T’/3
SHELL.
JUNCTION
AT
+‘$y
1
+ Qo)
GVMo 1
(10.71)
PMo = -Qo ,Y AXIAL IHEAMAL STRESS, f,t, AT JUNCTION (See Eq. 6.122.)
WITH
Substituting Eq. 10.62 into Eq. 6.75 and Eq. IO.63 inlo Eq, 6.76 gives: = yz,o = -
(10.70)
A comparison of Eqs. 10.69 and 10.70 indicates that.
MOMENTR
term dyjdz represent the slope of the skirt.from t.he vertical as a result of thermal deformation. This deflection can be compared to the deflection dyl/dx for a cylindrical shell joined to a flat-plate closure, shown in Fig. 6.3. ht the junction, where 2 = 0,
crrT’
Q. = a dp2 D1 ‘2
(10.63)
(10.63)
E = 25.5 X 106 psi
(from Fig. 8.6)
/.L = 0.27 By Eq. 6.86
P=
Substituting into Eq. 10.64 gives: =$ (2/3201) = 4M2h(T’B+~)+Qo
- P2) = 4 J r2t2
By Eq. 6.15 x 10”(0.5)3 = 28-7 )’ 1o4 D, = --t”_- = 25.5 ______
therefore
12(1 - jL2)
Q o = 2&12D1 (ZT’/l - z)
(10.66)
At the junction of the skirt and bottom dished head ,T’ = TV - T,, and T,, = T,; therefore T’ = 0. And, t.herefore, dT’ Mo==ad@D,dx Qo = -ad/3zD,dg
(10.67)
12(1 - 0.272)
’
’
The temperature gradient at the junction, 1’ = 0, by Eq. 10.61 is: x=0
= -6.037
Axial thermal stress: Substituting into Eq. 10.69 gives:
(10.68)
(10.69)
= -(7.6 x 10-6)(130)(0.227)(28.7 = 388 in-lb per in.
X lo’)\ -6.037)
Lug Supports for Vertical Vessels
By Eq. 10.72
~‘ircumjerential
set up on the windward side when the vessel is empty because in this case the dead load is subtracted from the wind load. Therefore the stresses on the leeward side are the determining factor for design of the supports. The maximum total compression load in P pounds in the most remote column is (164): are
= 9320 psi
By Eq. 10.70
197
thermal stress:
p = 4P,(H - L) I ZW n n&e = (7.6 X 10-6)(130)(0.227)2(28.i
X
104)(-6.037)
= -88.2 lb per in. By Eq. 10.7.4 jet = /!!!y!
= [(0.27)(9320)] = 2115 psi
Shear thermal slress:
By Eq. 10.75
10.2 LUG SUPPORTS FOR VERTICAL VESSELS
The choice of the type of supports for a vertical pressure vessel depends on the available floor space, the convenience of location of the vessel according to operating variables, the size of the vessel, the operating temperature and pressure, and the materials of construction. (Brackets or lugs offer many advantages over other types of supports. They are inexpensive, can absorb diametrical expansions 4 y sliding over greased or bronze plates, are easily attached to the vessel by minimum amounts of welding, and are easily leveled or shimmed in the field. As a result of the eccentricity of this type of support, compressive, tensile, and shear stresses are induced in the wall of the vessel. The tensile and compressive forces cause indeterminate flexural stresses which must be combined with pressure stresses circumferentially and longitudinally. The shear forces act in a direction parallel to the longitudinal axis of the vessel, and the shear stress induced by these forces is relatively so small that they are often disregarded. Lug supports are ideal for thick-walled vessels since the thick wall has a considerable moment of inertia and is therefore capable of absorbing the flexural stresses due to the eccentricity of the loads. In thin-walled vessels, however, this type of support is not convenient unless the proper reinforcements are used or many lugs are welded to the vessel. / If a vessel with lug supports is located out of doors the w-ind load, as well as the dead-weight load should be considered in the calculation of P. However, as lug-supported vessels are usually of much smaller height than skirt-supported vessels, the wind loads may be a minor consideration. The wind load tends to overturn the vessel, particularly when the vessel is empty. The weight of the vessel when filled with liquid tends to stabilize it. The highest compressive stresses in the supports occur on the leeward side when the vessel is full because dead load and wind load are additive. The highest tensile st.resses
(10.76)
where P,, = total wind load on exposed surface, pounds H = height of vessel above foundation, feet I, = vessel clearance from foundation to vessel bottom, feet DbC = diameter of anchor-bolt circle, feet n = number of supports ZB’ = weight of empty vessel plus weight of liquid and other dead load, pounds 10.20 Lugs with Horizontal Plates. Figure 10.9 shows a sketch of a vessel supported on four lugs, each lug having two horizontal-plate stiffeners. Such lugs are of essentially the same design as that shown in Fig. 10.6 for external chairs, and the same design procedure may be used. This type of lug uses to advantage the axial stiffness and strength of the cylindrical shell to absorb the bending stresses produced by the concentrated loads of the supports. Both the top and bottom plates should have continuous weids as the maximum compressive and tensile stress occurs in these two plates, respectively. These welds and the intermit.tent welds of the vertical gussets to the shell carry the vertical shear load. The load, P, on the column has a lever arm, a, measured to the center line of the shell plate. This moment
Fig.
10.9.
stiffeners.
Sketch
of
vessel
on
four-lug
supports
with
horizontal-plate
Design of Supports for Vertical Vessels
198
The following assumptions are made in the development of the bending-moment equations:
u
f R
7 1
51
1E
R
P
Front view
Fig. 10.10.
Side view
Sketch of lug in which only the vertical gussets ore welded to
shell. (Courtesy of Petroleum Refiner.1
is resisted by the couple, Qah, acting at the center lines of t,he top and bottom horizontal stiffening plates. As in the case of bolting loads on external chairs, discussed previously, a reaction is established in the shell producing axial and circumferential bending stresses which should be considered. The axial stress is defined by: faxial
6Mo
- 03Par2
= 12 = t1 _ p2jmh
(10.58)
where m = effective arc width, taken as equal to 2A. This method of analysis is recognized as being approximate because it is based upon an assumed effective arc width and an assumed uniformly distributed moment over this effective arc width. This method does not provide for the calculation of the bending moments and stresses in the circumferential direction between lug supports. If the horizontal plates are not used or if the bottom plate is not welded to the shell, this method of analysis is not applicable. 10.2b L u g s w i t h o u t H o r i z o n t a l P l a t e s . Wolosewick (161) has considered the case of lugs fabricated of two vertical gussets welded to the shell without horizontal plates welded to the shell. This analysis is based upon the stresses resulting from bending moments induced in a shell ring section. No allowance is made for the axial stiffening effect of the shell. This method of analysis was developed in the Boulder Canyon Project and was presented in a report of the project, Bulletin No. 5, Penstock Analysis and Stiflener Design (162). Figure 10.10 shows a sketch of a lug without a top stiffening plate in which the two vertical gussets but not the bottom plate are welded to the shell. D E R I V A T I O N O F E Q U A T I O N S. In Wolosewick’s analysis of a lug of the type shown in Fig. 10.10, the stiffening effect of the heads of the vessel as well as the restraint of longitudinal fibers of the shell is ignored. Therefore the calculated stresses based on the assumed ring action of the shell will be larger than the actual stresses induced by the loads; this places this approximate analysis on the conservative side.
1. The stress in the ring is a linear function across the thickness of the shell. 2. The modulus of elasticity in tension and the modulus of elasticity in compression are the same. 3. Elementary circumferential fibers behave freely and are not restrained by longitudinal fibers of the vessel. (This assumption may be modified by multiplying the stresses by (1 - P2)). 4. There is no warping of the vessel cross section, and a section that is plane section before bending remains a plane section after bending. For a ring on which a set of loads symmetrically located are acting, such as the one shown in Fig. 10.11, the only unknown to be determined is the moment MO acting on the center line of the cut section. Shear forces for this model of symmetrically arranged forces are considered to act as follows: The shear force at A will be opposed by the shear force at B, which acts in the opposite direction; and, therefore, the net shear force on the ring above is zero since the magnitude of the shear force at A is the same as that of the shear force at B. Figure 10.12 is a diagram showing the angles involved in the derivation of the relationships. The angle 6 is the angle between the y axis and R and is equal to one half the angle bounded by the two vertical gusset plates of a lug. The angle $I is the angle from the y axis to any point s. The moment MO is the internal moment in the shell at the center line, and the moment M, is the internal moment in the shell at any point, s, between axes. The strain energy stored in a structural element resulting from the action of bending moments is given by: lJ=L M2ds 2EI .I
(9.39)
For small-angle relationships in reference to Fig. 10.12, ds = rd4 Therefore lJ = &
R
s
(10.77)
M2r d+
R
4-l
- yMo+--R u ‘A
R R
&4-e e
\ OB -N$MO \
Fig. 10.11.
--R
Loading condition on shell ring resulting from action of
lugs (161). (Courtesy of Petroleum Refiner.)
four
I
Lug Supports for Vertical Vessels Table
10.7.
Bending Moments for Four Uniformly
Spaced Double-Gusset Lugs with Various Angle Spacing between Gussets (161)
8”
r#l=o
+=tJ
f$ = 45O
0
0.137 0.091 0.055 0.027 0.002 -0.018 -0.033 -0.043 -0.049 -0.050
0.137 0.095 0.065 0.045 0.033 0.030 0.035 0.048 0.068 0.095
-0.071 -0.068 -0.060 -0.050 -0.033 -0.013 +0.010 0.038 0.067 0.095
5 10 15 20 25 30 35 40 45
In an analogous manner, the following generalized equation can be obtained: M
Ma/M
(10.78)
Equation 10.78 represents the value of the slope of the ring at the center line, where MO is acting and is equal to zero since no rotations are possible. Therefore au ---CO aM0
(10.79)
MS = MO + Rr(1 - cos 4)
(10.80)
From 4 = 0 to 4 = 13
From & = 6 to 4 = (w/2 - 0) M, = MO + Rr(1 - cos 4) - Rr(sin 4 - sin 0)
1
sin e 1 (COS e + 0 sin e) -__ - + cot; 2 2 7r
0
(10.85)
where rz = an even number of lug supports symmetrically spaced
Castigliano’s first theorem states that the partial derivative of the total strain energy with respect to the load is equal to the momement of the load in the line of action of the load. In the case of couples the partial derivative with respect to the moment is equal to the rotation of the beam at the axis of the couple (30). au I &,f aw -=s--r@ aM,, EZ J aM0
199
Using the above equations Wolosewick (161) has presented: graphical solutions for the cases of two, four, and eight-lug supports; graphical solutions for the conditions of two and four uniform loads; equations for two, four, and eight lugs with shell-reinforcing plates. The value of the moments along the circumference can be determined from Eqs. 10.80, 10.81, and 10.82 if the value of MO from Eq. 10.85 and the direct forces (represented by R in Fig. 10.12) are available. Table 10.7 gives values of the dimensionless group (M,/Rd) at 4 = 0, 4 = 0, and 4 = 45” for various values of 0. (Note: at 9 = 0, M, = MO). As shown by Table 10.7, a decrease in B will increase M,. The reduction in moment by an increase in the angle 0 of the gussets is useful in design. If the stresses induced in the vessel by a lug exceed the allowable value, the angle of spread of the gussets can be increased to reduce the stresses. EFFECTIVE W IDTH OF S HELL. In the developmentofthe bending-moment equations it is assumed that only a portion of the shell withstands the flexural stresses caused by the reaction between the lugs and the vessel. This assumption disregards the reinforcement effect of the remainder of the shell as well as of the heads of the vessel. Thus the computed bending moments for the shell are conservative. A rigorous determination of the effective width of shell that resists the flexural stresses produced by the eccentric loading imposed by the lugs requires a laborious mathematical analysis. Wolosewick assumed that for this type of loading the effective width of shell equals the width of shell adjacent to the upper end of the hypothetical column
(10.81)
From + = (w/2 - 0) to 4 = s/2 MS = MO + Rr(1 - cos 4) - Rr(sin 4 - cos 4)
f’ R
(10.82)
Therefore, MO [MO
(*/Q--B t
+
/9
+ RrU - cos +)I dd
[MO + Rr(1 - cos 4) - Rr(sin 4 - sin 0)] d+
+ .I - 8 s/2
(r/2)
[MO + Rr(1 - cos 4)
1
- Rr(sin 9 - cos $J)] d4 = 0 (10.83) Formal integration of Eq. 10.83 gives:
2 i cot i 7r (cos 8 + fl sin 0) - sin 0 2
1
(10.84)
Fig. 10.12.
One quadrant of a shell with four
lugs as supports.
2oC
Design of Supports for Vertical Vessels
width of t,he base plate mult,ipliedby the cosine of the angie 9. III reference t.o Fig. 10. IO, the bending moment Pa is resisted by a couple in the shell, R(3@), for each gusset or for the case of two gussets as shown in Fig. 10.10.
Fig. 10.13.
DESIGN OF W ELDED L UGS. If the two vertical gussets shown in Fig. 10.10 are welded to the shell by four vertical welds, the welds will be subjected to both a vertical shear and tensile and compressive stresses produced by the bend.. ing moment. The vertical shear in t,he welds expressed in pounds per linear inch will be equal to the vertical force P divided by the length of the welds, or
Lug with single vertical gusset (continuous weld all around).
cf the gusset plus 0.78 V% inches on each side, OI
V=;
b = h + (2)(0.78) k’?t = h + 1.56 V5i
(10.86)
utlere b = effective width of shell, inches h = width of shell adjacent t.o hypothetical colu~~lr~ gusset, inches (See Fig. 10.10.) r = mean radius of shell, inches t = thickness of shell, inches
of
The effective width extends beyond the value given by Eq. 10.86. Therefore, the value of the computed stress will be higher than the value of the actual localized stress, and the design will be conservative. However, the value of the stress as computed is not exceedingly high, and the size of the designed lugs is practical. C O M P U T A T I O N OF B E N D I N G S T R E S S E S I N S H E L L . The calculation of the bending stress in the shell follows once the values of the moments are obtained. If the dimensions of t,he lugs are assumed, the value of b, the effective width of shell, can be computed, and the stress calculated from Eq. 2.10. (2.10)
or for a plate Z = (bt”/6) (10.87) wheref = bending stress, pounds per square inch M = bending moment, inch-pounds t = thickness of shell, inches b = effective width of shell, inches = h + 1.56 V% (See Fig. 10.10.) Several methods are used for the design of lugs for vertical pressure vessels. The assumptions involved in any method are diversified, and the procedure used often depends on the judgme-,; and experience of the design engineer. One method frequently used is to consider the gussets as columns fixed by the bearing plate at one end and by the wall at the other. The contact plane with the remainder of the lug area adds additional restraint to the column. Figure lO.l(! illustrates the strip of the rib that is assumed to resist the load. The effective width, b, of the column is controlled by Ihe
‘f
\
\I
/
The maximum tensile force produced in the welds expressed in pounds per linear inch will exist in the lowermost fiber of the vertical welds and will be equal to: T - MU2
6Pa
13/12
= 12
(10.90)
The resultant combined force (shear plus tension) in the welds is: F=+f2+T2 (10.91) The usual allowable stress for weld metal in shear is 13,600 psi (102). If .15” fillets are used, the weld will have a minimum cross-sec:tional area of 0.707w sq. in. per in. of weld length. Therefore Fallo,r. is: F anow, = (13,600)(0.707)~ = 9,600~ lb per lin in.
(10.92)
where w = width of one leg of fillet, inches A single vertical gusset plate with a horizontal base plate and with a continuous weld all around is sometimes used ae a lug, as shown in Fig. 10.13. In such a case the welds are no longer symmetrical with respect to the neutral axis. The location of the neutral axis (center of gravity or centroid) and the moment of inertia of the welds about this axis must be evaluated. This may be accomplished by determining the distance, b, from the base to the neutral axis. The welds are treated as lines, and the summation of the moments (the length of the line times the distance from the centroid of the line to the reference axis) is divided by the I otal length of the lines, or (164) 11 + l2 + 2dh b=2(1$l+2d)
(10.93)
The moment of inertia I, of the welds about the neutral axis at the center of gravit,y, e.g., is given by: I I’.“. = z[sy2 dA + Az2] 2
(10.94)
+ s + 2 b 2” ’ + & + tc2 0 + (2d + t)b”
= b3&!f + f + tc2 + lb2 + ; + 2d(g2 + b2)
(10.95’
Lug Supports for Vertical Vessels
The shear force on the weld expressed in pounds per linear inch is v = ----!I!--
as a beam fixed at one end but guided al the ot.her end wit.h a concentrated load af the guided end (30).
(10.96)
2 + 4d + 2t
fbend. =
The maximum tensile force on the weld (in pounds per linear inch) produced by t.he bending moment occurs a I Ihe bottom horizontal weld and is given by: (10.97)
The resultant force is given hl Eq. 10.91, and the allowable force by 10.92. 10.2~
Column Supports and Column Bearing Plates.
S UPPORTS FOR LUGS. If the column is attached in such a manner that it can be considered as a column under concentric axial load, the allowable fiber st,ress is given by Eq. 4.18. COLUMN
c
fc =
18,000 1 + (1’/18,000r2)
(4.21)
where 1 = unbraced length of column inches P = least radius of gyration of column, inches fc = allowable compressive st.ress in column, pounds per square inch The maximum permissible l/r ratio is 120, and t,he maximum fiber stress = 15,000 lb per sy in. The required cross-sectional area, A, of the column for axial compression is, by Eq. 2.2:
cw c “=P
where p = maximum compression load per column, pounds fc = allowable compression stress from Eq. 4.21 above Usually the columns are attached to the vessel with a distance, a, between the center line of the column and the center line of the shell, as shown in Figs. 10.9 and 10.10. This produces an eccentric loading and an additional stress in the column supports. This stress is given as:
fee = 4
201
(10.98)
where 2 = section modulus of columrl, inches3 The column supports may also be subjected to bending as a result of a wind load. The moment produced by the wind load was considered by Siemon (164) to be equal to PL/2; this is comparable to considering the column
(PW/ fl)l, 2
(10.99)
2
where Pw = wind load on vessel, pounds R = number of colu~nn supports I = column height, inches Z = column se&ion modulus, cubic inches When columns are subjected both to direct loads and to bending produced by eccentric loads, the American lnstitute of Steel Construction specification (102) states that, the sum of the axial compressive stresses divided by the allonable column stress, plus the bending stress divided by the allowable flexural stress shall not exceed unity, or
Z) + ” (Puh3nZ) y! + (Pa ~--~ .-..pm’-m 5 1
fc
.ft
(10.100)
C OLUMN BEARING P L A T E . The procedure used for designing bearing plates for column supports is similar IO that used for designing a bearing ring for skirts of vertical towers discussed in the earlier part of this chapter. The column load is usually transferred t.o a concrete foundat.ion. The allowable compressive stress for concrete foundations may be selected from Table 10.1. The maximum column load, P, divided by t.he allowable compressive stress, fc, gives the minimum required area for t,he bearing plate. The column is usually located in the center of the plate, and the base plate is usually proportioned to give approximately equal overhang on all sides. If the load is assumed to be uniformly distributed, a uniform load, fc, is applied to the underside of the overhang of the p1at.e. This overhang may be considered to act as a uniformly loaded cantilever beam which has a maximum bending moment of (fcP/2) (see Eq. 10.31) at the junction with the column. The required thickness is given by Eq. 10.32 where 1 is t.he greatest overhang distance. A comprehensive analysis of stresses from local loadings in cylindrical pressure vessels was reported by Bijlaard (163). The method of analysis involves the development of loads and displacements into double Fourier series. This method may be used for: (1) a load uniformly distributed within a rectangle, (2) a point load, (3) a moment in the longitudinal direction uniformly distributed over a short distance in the circumferential direction, and (4) a moment in the circumferential direction uniformly distribut,ed over a short distance in the longitudinal direction. For the t,angentiti loading condition eyuations for displacement, bending moment, and membrane forces are covered.
PROBLEMS
1. An external chair is to be designed for a column 6 ft, 0 in. in diameter with eight anchor bolts 1 in. in diameter and with a calculated induced stress of 18,000 psi. The bolt-circle diam-
~____ \
\
\I
/
-~--I
-
. .-m
. 202
Design of Supports for Vertical Vessels
-I
eter is 6 ft, 6 in., and the outside diameter of the bearing’plate is 7 ft, 0 in. iq 8 in. Determine the required thickness of the compression ring.
The gusset height
2. The concrete foundation supporting the tower base shown in Figure 10.14 is a 2500-psi 28-day-ultimate-strength concrete. Assuming a loo-mph wind velocity, determine the required size of anchor bolts if the allowable stress is 18,000 psi for the bolting steel.
~24 holes equally spaced
L
I
Gussets straddle bolts
Tower height: 150 ft (from top of concrete slob to top of tower) Uninsulated tower diameter: 8 ft, 0 in. Insulation: 3 in. Effective projected diameter for wind action: IO f t Estimated tower weight, empty = 175,000 lb Number of trays = 85 at 18-in.
spacing (top vapor-disengaging.zone
3 ft) Normal operating load = 30 lb par sq ft of Aoor troy orea
=
(includes shell
and insulation) Skirt height = 11 ft, 0 in. (from top of concrete slab to bottom tangent line) The 24 gussets are equally spaced. Fig. 10.14.
Sketch of distillation-tower base for problems.
3. For the tower base shown in Fig. 10.14 and the conditions given in problem 2, determine the required thickness, z, of the compression ring if the allowable stress in the ring is 18,000 psi. 4. If the concrete foundation supporting the tower base shown in Figure 10.14 is a 3000-psi 28-day-ultimate-strength concrete, the compression-ring thickness, z, is 1% in., the 24 bolts are 2 in. in diameter (eight-thread series), and the maximum wind pressure is 30 lb per sq ft, determine the stress in the skirt resulting from the interaction between the compression-ring assembly and the skirt. 5. For the tower base shown in Fig. 10.14 and the conditions given in problem 4, determine the maximum stress in the bearing plate. 6. If the tower described in problem 2 were located in seismic zone 3, determine the size of bolts required to withstand the seismic loads, assuming the wind moment is not controlling.
/
C H A P T E R
DESIGN OF HORIZONTAL VESSELS WITH SADDLE SUPPORTS
u
he selection of the type of support for a pressure vessel 1s dependent on several variables, such as: the size of the vessel, its wall thickness, the floor space available, the elevation of the vessel in relation to the ground or floor, the materials of construction, and the operating temperature. Horizontal cylindrical pressure vessels are commonly supported by saddle supports or cradles. If the underside of the vessel is to be located only a short distance above the grade line, steel saddles resting on the top of concrete piers may be used. When vessels are elevated, a structural-steel-~ __II) ,?> I. frame may be usea to sup@~&?l~s?~ccradles. If ~~~znpport~t?YGd~the‘ load resulting fr%iiiie welght‘Pifflb’~ei;essle.~~~~~~.contei;ts will be equally d&ded though ..,one than-. the --_ support may settle- more ‘~~71r”l. -. _ .._ - even . ..- ____._._ other. .- Smce the loads may not be equally drvrded after .,_ ..-” tG supports settle -if m--~;-~~~~~~~r~~~~~~~~~, ,---_. _“. ..-..“-x.,-- ,. i th~two-sul$ort system has an advantage over --a system ,.-----.----employing a larger number o~sup~oir&$ Figure 11.1 shows a group of horizontal butane- and gasoline-storage tanks each 12 ft in diameter by 120 ft long supported on two saddles. Horizontal vessels when resting on saddle supports such as shown in Fig. 11.2 behave as beams. An analysis of the stresses induced in the shell by the supports was reported by Zick (165) who developed equations for the stresses. Zick’s relationships contain empirical constants determined experimentally.* By using this method of analysis following stresses can be evaluated: 1. The maximum longitudinal stress. 2. The tangential shear stress. 3. The circumferential stress at the horn of the saddle. 4. The additional stress in the head used as a stiffener. * For steel vessels.
The maximum unstiffened length of the vessel between the heads, the ring compression in the shell over the saddle, the stresses on the ring stiffeners, and the total horizontal force acting against the horns of the saddle may also be determined. advantage of the stiffening effect of the head. Dimension A=-.-“.---is often selected* __..so .that A = 0.4R. Dimension A should _Ic_____, ‘~-~- ..F never exceed 20% of dlmensron L; otherwise the stresses resulting from cantilever action will be excessive. A cylindrical vessel with dished closures at the ends may be treated as an equivalent cylinder having a length equal to (L + $iH) where L is the distance between the tangent lines of the vessel and H is the depth of a dished closure. *his approximation assumes that the weight of the head and the fluid contained in the head is equal to two-thirds of the weight of a cylinder of length,H and the fluid contained in it. This approximation ,is valid for hemispherical heads and elliptical dished heads and can be demonstrated by use of Eq. 5.14 for an elliptical closure for a lOO-in.-diameter vessel. V = (0.000076)(100)3 = 76 cu ft The depth of dish, from Fig. 5.7, is ID/4 = 25 in. The volume of a cylinder 100 in. in diameter and 25 in. deep is 114 cu ft. The ratio of the volume of the head to the volumhe cylinder is 76/114 or N. The weight of the fluid and the vessel may be considered to be a uniform load equal to the total weight divided by the equivalent length, or
2Q W=LH
J”
where w = uniform load, lb per ft
\/
204
Design of Horizontal Vessels with Saddle Supports
Fig.
11.1.
Butane and gasoline horizontal storage tanks 12 ft in diameter by 120 ft long supported on two saddles.
In the loaded condition the shell, over the distance L, The load of the heads behaves as a uniformly loaded beam. introduces a shear load at the junction of the heads and the cylinder equal to ~~HuJ. This load produces a vertical couple acting at a distance of 3gjH from t,he point oTtsg: gency and a horizontal couple acting with a lever arm of R/4 where R is the radius of the vessel in feet.
L -H-l
I
11.1
LONGITUDINAL
BENDING
(Courtesy of C. F. Braun & Co.)
STRESSES
As in the case of an overhanging beam with two supports, two maximum bending moments exist in the longitudinal direction of the vessel. One maximum occurs over the saddle supports, and the other maximum occurs in the center of the vessel span. The shell acts as a beam over the two supports under the uniform load of the vessel and its contents, as shown in Fig. 11.3. -aThe maximum moment over the supports, M,, may be determined by referring to Fig. 11.3 and by taking bending moments about the center of reaction, Q, over the distance
H+A: Vertical shear moment = sHw(A)
counterclockwise
Vertical couple = $Hw($H) counterclockwise
A
Overhanging-shell moment = WA z
counterclockwise
0
clockwise Fig.
11.2.
Sketch of horizontal vessel with two saddle supports.
A = distance from tangent line to saddle, feet ’ L = length of vessel,
tangep!
to tangent, feet
Therefore
H = depth of heod,%et Q = total load per saddle, pounds =
total weight divided by two
R = radius of vessel, feet b = width of saddle (or width of concrete for formed concrete saddles, inches r = radius of vessel, inches I
t = shell thickness, inches v B = total included angle, degrees
! I
w = load per unit length,
pounds/ft
Substituting Eq. 11.1 for w, we obtain:
..-. 1 .I
(11.2)
Longitudinal
Bending
205
Stresses
The ceptroid of the shell included in angle 24 is located a distance of r&n A/A) from the ~-2 axis (where A is measured in radians). Therefore the moment of inertia of the arc of the shell about its own centroid (cent.) is:
I cent. = I, - Ad2 = 2tr3 =
tr3
A
sinA?A] _ (~s~A)‘~~~~(~)
ii+ sin2A A + sin A cos A - 2 ~ A
1
(11.4)
The section modulus, 2, for the side in tension at the saddle is: z = J = r3t[A + sin A cos A - 2(sin2 A/A)] C r(sin A/A) - r cos A - or Bending-moment diagram in Fig. 11.3.
z = r2t
ft-lb
Cylindrical shell acting os beam over supports, according
to Zick (165).
(Courtesy of American Welding Society.)
The maximum bending moment at the center of the span is determined by taking the summation of the bending moments about the saddle over t.he distance H + L/2. In addition to the moments over the distance H + A is the moment :
A-+--sin A cos A - 2(sin2 A/A) .’ (sin A/A) - cos A
(11.5)
The stress f, at t.he saddle will be (from Eqs. 11.2 and 11.5) :
fl = 12QA
1 -
R2 - H2l-;+2A 1+g
=
W
Taking a summation of these moments with due regard for signs giyesghe moment MC at the center of the span, or
-Mc=w
(L - 2A)2 8
1
)I
sin2A A + sin A cos A - 2 ~ A To-det.ermine the stress the moment of inertia of the shell _.. must be evaluated. Above ,each saddle support circumferential bending moments are produced which permit the unstiffened upper portion of the shell to deform. This deformation makes this portion of the shell ineffective as a beam and reduces the effective cross section in the same manner as if a horizontal section were cut from the vessel some distance above the saddle. The arc A measured from both sides of the center line of the saddle up to these fictitious “cuts” defines the effective cross section of the vessel, shown in Fig. 11.4. By Eq. 9.15 the moment of inertia, I,, of the arc of the shell in the lower tw~@iZIi%its &luded by angle 2A is:
13. = 2tr3 = 2tr3
or
f1= k3S
,,’ sin2 A dA
(11.6)
Section A-A
/
1-I
Fig. 11.4.
Sketch of effective area of shell under boom
(Courtesy of American Welding Society.)
action (165).
206
Design of Horizontal Vessels with Saddle Supports Values of A/L when R = H
I
/
/
I
/
I
/
I
/
/
/
//
’ 1.6
/I
1.0 K, and Kg 0.8
0.8
0.6
0.6
I lo
H/L=0 --!? H/L = 0.05
\\\\\\\\\\\\
H/L=O.lO 0
Values of A/L when R = 2H Fig. 11.5.
Plot of longitudinal bending moment constants KI
where
(11.7)
+ sin A cos A - 2 ~ In a similar fashion, using Eqs. 11.2 and 11.5, we find that the stress at the mid-span, ji, will be:
and KZ (165).
(Courtesy
of
American
Society.)
plotted for the condition of H = R when jr governs, and Kg for the condition of H = 0 when js governs. These approximations simplify the calculations and give conservative designs. It should be noted that Eq. 11.9 was obtained by dividing the maximum bending moment by the corresponding section modulus. The stress SO obtained will be the maximum axial stress in pounds per square inch in the shell due to bending as a beam. This maximum bending stress may be either tension or compression. The tensile stress as obtained by Eq. 11.6 or 11.8 when combined with the axial stress due to internal pressure should not exceed the allowable tensile stress of the material times the efficiency of the girth joints. According to Zick (165) the compressive stress as determined by Eq. 11.6 or 11.8 when combined with axial pressure stress should not exceed one half of the compression yield point of the material or the value given by: fallow.
(11.8)
Welding
=
XY
(11.10)
where Y = 1 for 4 7 60 r
where
Y =
21,600 for 4 > 60 18,000 + (L/r)2 r
X=(l,OO0,OOO~)(2-~lOO~)forf~O.015
Values of K1 and K2 for different design proportions can be obtained from Fig. 11.5 (165). In Fig. 11.5, K1 is
X = 15,000 fort 2 0.015 r Equation 11.10 is applicable when t 2 j/4 in. (166).
-----
T-7-----
Tangential
Shear
Stress
207
then v = Q (L - 2.4 - 0.7H) ~ Q L + 1.3H
L - 2A L + H
H
>
(11.11)
Consider a section of shell of length dx, as shown in Fig. 11.7. From Eq. 2.10
By Eq. 2.5 Fig. 11.6 tesy of
Shear diagram for shell stiffened American Welding Society.)
with
ring
(165)
(Cour-
(11.12)
z = 7rr3t It should he noted that the reduction in compression stress as a result of elastic instability is not a factor in a vessel which is designed for pressure or in which t/r 2 0.005. Consideration must be given to the stress due to bending moment before adding the stress due to internal or external pressure. This is especially important when the combined stress is less than the bending stress before internal or external pressure is applied. 11.2
TANGENTIAL
SHEAR
STRESS
1.2a S h e l l S t i f f e n e d b y R i n g i n P l a n e o f S a d d l e . d
When the shell is held to a cylindrical shape, the tangential shear stress varies as the sine of the central angle, 4, measured from the vertical. The maximum shear stress occurs at the equator. In this case the analytical solution is simple. Let V = shear force as shown in Fig. 11.6. Then between supports V = Q - w(A + H + m) where w = 2Q/(L + $H) lb per ft, or V=Q-
2Q
(3L + 4H)/3
(A + ff + m)
At the saddle, where m = 0, V=Q-6QsH
(see Eq. 9.16)
y = r cos f#J
(11.13) (11.14)
dA = tdl = trd4
(11.15)
dP = df dA
(11.16)
On section ABDC the moment at AB is M, and at CD is (M + dM). If the element WA0 on the ring from -4 to +$J is isolated, flexural forces will exist on the ends, and longitudinal shear forces on the radial planes at Wand 0, as shown in Fig. 11.8. By a static balance of forces, shown in Fig. 11.8, ZF, = 0
or
zfdA+zdfdA-zfdA+2u=O where u is the total longitudinal shear force on the section W and 0. Substituting Eq. 11.16 into the above equation and canceling terms, we have: IdP = j’df dA = -2a
(11.17)
Substituting Eqs. 11.12, 11.13, and 11.14 into Eq. 11.16 gives : AP=
s s dP=
++ Vcos+d4
-*
=
=$ ( 1 1 . 1 8 )
7rP
dM
Fig. 11.7. Shear and moment diagram for shell stiffened by ring (165). (Courtesy of Weldi;.g
Society.)
American
Design of Horizontal Vessels with Saddle Supports
208
0.319. The value for K3 is independent of 13, the angle c-f contact with the support saddle. For design purposes the value of j3 should not exceed the allowable tensile stress of the material times 0.8, or
fa + dfdA/’
j3 = 0.8 X allowable tensile stress of material 11.2b Fig.
11.8.
Shear on side of element shown in Fig. 11.7.
where AP is the change in the longitudinal force on the portion WA0 per unit length of the ring. 4P is balanced by the longitudinal shear on a unit length of the radial sections W and 0. Subst.it,uting Eq. 11.18 into Eq. 11.17 gives: 2V sin 4 -2~ = (2t) (unit shear) = Bp OL‘
Unit shear = -I’ si?rrtn ?
(11.19)
If a shearing stress occurs at a given point on a plane in a stressed body, there must exist a shearing stress of equal magnitude at that point on a second plane at right angles to the first plane (231). Since the shear has the same intensity on adjacent edges of the rectangular element, the unit shear on the ends of the free body WA0 at the points W and 0 also equals (V sin +/d), and it,s direction is normal to the radial planes and is, therefore, tangent to the shell, or V sin C#J or=n-r
flt(unstiffened) ‘JtWffened)
~t(unstiffened)
(11.20)
1=
sin $I cos 4 * 2 0
2
=
V sin I$ sin a! cos a)
r(7r - a +
The shear stress j4 will then be:
where rt = transverse tangential shear per unit length of arc The shear force CL is tangent to the shell at all points and varies from zero at the top to a maximum at mid-point, and back to zero at the bottom. The summation of the vertical components of the transverse tangential shears on both sides of the stiffener gives Q. For this case the term C in Eq. 11.20 is replaced by Q and the vertical component. is P sin I$ times the shear.
Unstiffened Shell with Saddles Awoy from Head.
When the shell of the vessel is free to deform above the saddle, the tangential shear stresses act on a reduced effective cross section, and the maximum stress occurs at the horn of the saddle. Here the shears are proportional to sin $I but act only on twice the arc given by (e/2) + (b/20) or * - cy. This angle is the assumed position at which maximum tangential shears occur on a shell which is free to deform above the saddle and beyond the influence of the head. Zick reported (165) that this assumption was verified very closely by strain-gauge experiments. Fig. 11.9 represents a section taken in the plane of the saddle for a shell with supports away from the head. If a portion of the shell is noneffective, as shown in Fig. 11.4, the shear (it is increased in the effective portion. Since the summation of the vertical components must still equal the vertical load Q the shears will be increased in inverse proportion to the integral of the function, or
f4
=
Q sin 4 rt(7r - a sin (Y cos (Y)
or j4 L_
Q& rt
H
L-H-2A] LfH j
~4 L + fZ
where K4 =
1
sin $I 7r - ff + sin a! cos a!
(11.24)
(11.25)
Q (11.21)
The tangential transverse shear stress at any point on a section on both sides of the stiffener is: V sin C#J j3 = srt = St [ !*$!Isin #I
Location of - assumed pomt of maxtmum shear
or (11.22) where KS = YiL4 *
(11.23)
For the maximum value of js, sin #J = 1, and Ka = l,/?r =
Fig. 11.9. (165).
Location of assumed point of maximum shear in unstiffened shell
(Courtesy
of
American
Welding
Society.)
209
Circumferential Stress at Horn of Saddle
For design, f4 $ 0.8 X the allowable tensile stress of the material. The maximum shear stress occurs at the point of maximum shear, or where $I = cy,
A-7
sin (Y K4 = ___.-.-7r - a + sin (Y cos (Y Values of Ka are given in Fig. 11.10. 11.2~ Shell Stiffened by Head. When the saddle supports are located near the head, the tangential shear stresses are first carried from the saddle to the head. Then the load is transferred back to the head side of the saddle by :aniential shear stresses which act on an arc of angle larger than the angle of contact of the saddle. Here the shears vary as the sine of 42. The angle 42 varies from (r - a) t.0 n-. Above angle (Y these shears are directed downward and vary as the sine of & from 0 to (Y. Below angle (Y they are directed upward, on the head side of t,he saddle. This can be represented as shown in Fig. 11.11. In order to have static balance at the left of section A-A of Fig. 11.11, the downward forces must balance the upward forces. a Q sin2 41 forces down = 2 ___ I- d+, TP c .I0
2Q =-
41
- sin 41 cos 41 a
n-
2
2
[
ou sin’ 42 d& / T sin’ +J d& Ja
forces up = 2
=,~Qs~~& a-sinpcosa [ 2Q =lr
Fig.
11.11.
Welding
A
Shear in shell stiffened by head (165).
(Courtesy of American
Society.)
The shear stress in the head is:
2Q sin $12
f5 =
2th(rr)
a - sin a cos (Y
’
7r - a + s i n (Y c o s (Y 1
or
-f5A!c!
(11.27)
rth
5
I
a
(11.26)
170 160 150 g 140 $4 1 3 0 .E = 120 I
(11.28)
__ n-
a - sin a! cos a! --__7r [ - a + sin (Y cos 01
The maximum stress occurs at 42 = cr.
1
(11.29)
Then
Values of Kg as a function of 13 are given in Fig. 11.10. 11.3
CIRCUMFERENTIAL
STRESS
AT
HORN
OF
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 K, and KS Values of Kd and KS as a function of saddle angle 0.
SADDLE
The theoretical analysis leading to t,he determination of the circumferential stress at the horn of the saddle has not been carried out successfully. The maximum stress occurs at the point of maximum bending moment due to tangent.ial shear. When a stiffening ring is used to restrain the shell from deforming above the saddle, the mathematical analysis of the bending moment due to tangential shear can be solved. The one-half arc of Fig. 11.12 is in equilibrium under the action of the forces shown. From symmetry the vertical shear is zero both at point A and at point C. At any point, U, the shear ut is: Q sin I,!J ut = ~ m
iO0
Fig. 11.; 0.
-
Shear diagram when saddle is near head
K = sin 42
*
[ 2
\
ki w A
a - sin (Y cos ff -~ --___K5 =Sma * a F 7r - a + sin (Y cos (YI
- - + sin +2 co9 $I~
110
I
where
T]d42
I
+- sin ~11 cos cy)
,
fs = shear stress in shell = Qh-6 T 8
1 r dh
?r - a + sm (Y cos a! a - sin OL cos a! 7r - a + sin (Y cos ff 42
Shell stiffened by head
Similarly,
-0 I
sin (Y cos a)
I
per unit length of arc, or for a length of arc dl, is:
7rr
,210
Design of Horizontal Vessels with Saddle Supports Roark has shown (see Reference 166, p. 147) that the horizontal deflection of an element in a curved beam may be expressed as follows: x=
= maximum shear on both sides of ring per unit length of arc
I
$dl I
Likewise, from symmetry about the vertical axis of tha shell, the horizontal movements of A and C are both zero. or
FY-A!M
Gravity
C
c Mbmd15=Oor
EI
c A
c
M+mdl=
0
A
But dl = r d+; therefore c
IQ c A Fig.
11.12.
Forces acting on one-half arc of shell stiffened by ring in plane
of saddle (165).
(Courtesy
of
American
Welding
Society.)
C
M+mr
dqb = 0 or
c
M4m
d$ = 0
A
Likewise, m dl = r d+ (r cos + -- r cos 0).
Therefore
C
M# d$ (cos qi - cos ,8)] = 0
But dl = P d#; therefore
c
A
c
= g sin $J d# a
c A
The 2 component of this shear = Q= = (Q/r) sin Ic, cos J/ d#. The moment arm of uZ with respect to N = P cos # - r cos 4. The y component of this shear = uU = (Q/r) sin2 # d$. The moment arm of uy with respect to N = r sin $I - r sin #. Therefore the moment of the tangential shear about N is: ji&,=/dM=[;
sin # cos #(rxos # - r cos 4) dfi + Q sin2 # ___ (r sin $J - r sin #) d# n/0
Or by integration M, = p .L - cos f.j - z sin 4 7r [
1
(11.30)
As stated before, from symmetry, the vertical shear is zero at both A and C. There are, therefore, only three unknowns acting on the free body, Pt, MA. and MC. Timoshenko has shown (see Reference 29, Part II, p. 68) that for a thin curved beam the small angle of rotation, A dq5. between two neighboring cross sections may be expressed as follows:
M, d2 A d + = EI From symmetry about the vertical axis of the shell, the rotations of A and C are both zero, or c c A
C
M+ cos I$ d4 M+d4 = 0 cos p c A
c
But
2 A
M+ d4 = 0; therefore
(11.31)
c c
Mmcos+d+=O
(11.32)
A
From a static balance the total moment is: ZM=O M$ = -Ptr(l - cos 4) + MA + $ (2 - 2 cos Cp - C$ sin 4)
(11.33)
Substituting Eq. 11.33 into Eq. 11.31 and integrating from 0 to p gives: -P&P - sin /3) + /~MA = $ [3 sin /3 - /3 cos /3 - 2/3] (11.34) Subst,ituting Eq. 11.33 into Eq. 11.32 and integrating from 0 to p gives: -Ptr[sin @ - +@ - t sin 2p] + MA sin ,f3 = 2 [Q sin 2/3 - k/3 cos 2/3 + 0 - 2 sin 01
(11.35)
Simultaneous solution of Eqs. 11.34 and 11.35 gives:
C
Mg dl -=Oor M.+dl=O EI c A
(11.36)
Circumferential Stress at Horn of Saddle
0.24 0.10
0.20 q 0.16 &$ 0.12
0.08$ 0.06 $ b
z 0.08
0.04 tYf 0.02
0.04 0
0 20 40 60 80 100 120 140 160 lsoo Values of fi in degrees
Fig. 11 .13.
Constants Klo and K1l as a function of angle 8.
times the radius of the shell or one half the length of the vessel, whichever is smaller. Therefore, the use of the value-of’the hypothetical moment MB given by Eq. 11.40 wi render calculated stresses in accord with actual stresses. JI”11.3b S h e l l S t i f f e n e d b y H e a d . When the shell is stiffened by the head, the shear stresses are carried across the saddle to the head, and then the load is transferred back to the saddle, as previously shown. As in the case of the unstiffened shell, the shears tend to concentrate near the horn of the saddle. Since the stiff members are relatively short, this transfer reduces the circumferential bending moment still more; that is, the circumferential bending moment is smaller in the shell stiffened by the head than in the unstiffened shell. This effect is introduced when the circumferential bending moment is defined as: (11.41)
MB = KTQr MA = SK [sin2 /?(l - + cos /3 + ip sin /3 - Q2) - $0 sin /3 + $@ cos j3(2/3 + sin 2/3 - 5 sin /3 + /3 cos fi)]
(11.37)
where K, = KG for values of A/R greater than 1. For ’ values of A/R less than 0.5, K7 = >/4K,+ For design purposes the following equations are recommended:
Substituting Eqs. 11.36 and 11.37 into Eq. 11.33 gives: f7 =
M+ = & [cos +(sin2 /3 - Q@ sin Z/3 + +B2 cos 2fi) + 4 sin q5(+fi2 + + sin 2/3 - sin2 /3) + $3 cos PGV + sin 33) - sin /3(&b + 4 sin 2@ + %p cos 2p)]
f7
4t(b
(11.38) (11.39)
note that p = 180 - i the maxi( > mum moment occurs at 4 = /3. Therefore As seen in Fig. 11.14
ifLh 8R
(11.42)
2t2
1 2 K7QR -___ if L < 8R ,1 (11.43) + 1.56 6) Lt2
f, =
It should be noted that K and the quantities in parentheses in Eqs. 11.36, 11.37, and 11.38 are functions of p and will have the same value for all values of 4 for a given saddle support. Values of P,/Q and MA/@ computed by the use of Eqs. 11.36 and 11.37 are given by the diagrams of Fig. 11.13, and values of M,/Qr for various values of 4 and p computed by the use of Eq. 11.38 are given in Fig. 11.14.
MB = KeQr
-__
Q
= -
where K = sin2 p - $32 - $3 sin 2/3
3K7Q
Q
-
4t(b + 1.56 6)
where
211
maximum circumferential combined compressive stress at the horn of the saddle t= vessel-shell thickness for unattached wear plate t= combined thickness of shell and wear plate when wear plate extends r/10 inches above horn of saddle and saddle is located near head (A/R 5 0.5). (Otherwise, t equals thickness of shell only). b = width of saddle, inches
0.08
0
0.06 ~fl
(11.40)
where MB = maximum circumferential bending moment in inch-pounds Values of KG were determined from Fig. 11.14 and plotted in Fig. 11.15. The use of Ks in the design of shells with ring stiffeners in the plane of the saddle is treated in section 11.6. J11.30 Unstiffened Shell. When the saddles are located away from the head so that the shell is free to deform, the shears tend to accumulate near the horn of the saddle so that the actual maximum circumferential moment in the shell is less than the value obtained for M from Eq. 11.40. Zick reports (165) that this has been confirmed by straingauge measurements, which show that the effective length on top of the saddle that resists the moment is about four
I
0.04
I
I
I
I
60
80
I-
i%t
100
120
t’= 180’2 *
0 Fig. 11 .14.
I
20
40
Value of 4 in degrees Variation of circumferential moment around shell.
212
Design of Horizontal Vessels with Saddle Supports
The total load due to the horizontal shears will be (see Fig. 11.11):
“0 sin 41 cos +lr d+l /0 rr -
sL1! “0In-
[
a - sin cy cos a! C#~Z ~0s & ~ ~ 7r - ff -1 s i n cy cos a!
s i n
1 r &a
=p([sq!!!]y[ ?r ~.~~F25~%!%] [ ?q!!q:) = 120
sin’ o( Q =-4 2 [ 3r--cu+sinacosa:
110 100
The maximum cross-sectional area of the disk will be 2rtb..
901 ’ ’ ’ ’ ’ ’ 0 0.01 0.02 0.03 0.04 0.05 0.060.07 0.08 0.09 0.1 0.2
fs = g x 1.5
K a and K,, Fig. 11.15.
1
Values of Ka and K13 as Al function of the saddle angle 0.
(sin’ a)/(?r - a + sin cy cos a) ~~-_ 1 . 5 2 [-2rth 01
Zick suggests t.hat for multiple supports one should use L equal to twice the length of the load carried by the saddle. If L 2 8R, use the first formula. Equation 11.42 takes into account the assumed value of the effective length of shell that resists the bending moment, as outlined previously. It also takes into consideration the fact that the change in shear distribution reduces the direct load at the horns of the saddle. This reduced direct load is assumed to be equal to Q/4 for shells without ring stiffeners. Equation 11.42 also takes into account the fact that the effective length of the shell resisting this reduced load is limited by that portion which is stiffened by the contact saddle. The assumed value for this effective length is 0.72 drt on each side of the saddle plus the portion directly above the saddle. Values of K, can be obtained from Fig. 11.16; they were derived from Fig. 11.15 through use of the assumptions listed above. For design purposes Zick recommends:
fs=QKs
(11.44)
rth
where o( Ks = 8 7r~~ - a +sin’ sin a! cos a
1
(11.45)
The value of Ks as a function of saddle angle 0 are given in Fig. 11.17. The stressfs is a tensile stress in the head and should be. combined with the stress induced by internal pressure.
f7 s 1.25 X allowable tensile stress of material LNote: when rings are placed in the plane of the saddle, a longitudinal bending stress occurs at the edge of the ring. This local stress would be 1.8 times the design ring stress. 11.4
ADDITIONAL STRESS AS STIFFENER
IN
HEAD
USED
The stiffness of the head is often utilized by locating the saddles near the heads. In the derivation of Eq. 11.20 it was shown that shears have both tangential and horizontal components, as illustrated in Fig. 11.8. When the saddle is close to the head, the horizontal components will cause tension across the entire height of the head as if the head were a flat disk. The following analysis is based on the assumption that the head is a flat disk and that the maximum stress fs induced in the head by the horizontal components of the tangential shears is 1.5 times the average.
0' 0
Fig. 11.16.
I
0.5
I
I
1.5 Ratio of A/R
1.0
I
J
2.0
Plot of circumferential bending moment constant
2.5
K:.
213
Design of Ring Stiffeners
on each side of the saddle plus the width of the wear plate, if used. I
I
I
nl
I I I I
I
I
I
I
i
fg l
o-~._p _cOs _ + a! t(b + 1.56 dz) ( 7r - a + sin (Y cos a! )
QKY ~_ l(b + 1.56 h) where
110
b = wear-plate widt,h,inches 1 + cos a ____--~Ky = ~ 7r - a + sin LY cos cy
100 0
fig.
0.50 0.75 KS and K,
0.25
11.17.
1.00
1.25
Valves of KS and Kp as o function of the saddle angle 0.
For design purposes the combined stress on the head should he permitted to he 257, great,er than the allowable tensile stress of the material. For cases involving negative pressures the head stress is compressive and the combined stress equals (+fs - head stress). This stress can usually be disregarded. 11.5 WEAR PLATES-RING COMPRESSION IN SHELL. OVER SADDLE There are forces acting on the shell hand directly over the saddle causing ring compression in the shell hand. The tangential shear forces act over the arc from a! to ?r and are directed toward the center, 0, because the saddle reactions are radial. Fig. 11.18 shows these reactions with t,he assumption that there is frictionless contact between the surfaces of the shell and the saddle. Taking moments about point 0 indicates that the ring compression at any point, A, is given by the summation of the tangential shears between cr and 0, or
“(2 sin dwd41 = _ 4 TP
=
Q
-
(ii.47i
The values of Kg as a function of saddle angle 8 are pi\-cbn in Fig. 11.17. The stress fs is especially important when concretn saddles are used. It should also be checked for large-diameter vessels. For design purposes fs = 0.5 X compression yield point of material. The ring compression stress may be reduced by attaching a wear plate somewhat larger than the surface of the saddle to the shell directly over the saddle. The thickness, t, ma> be taken as the combined thickness of the shell and I he wear plate in the formula for fg provided the width of lhe plate equals (b + 1.56 &). Note: when the wear-plate thickness is added to the shell thickness as stated above, the thickness, t, in Eq. 11.42 can also be taken as the combined thickness of the shell and the wear plate if the wear-plate width equals (b + 1.56 fi) and if the plate extends r/10 inches above the horn of a saddle near the head. 11.6 DESIGN OF RING STIFFENERS In the case of thin-walled vessels or the case of saddles located away from the head (A/R > 34) the shell alone may not resist the circumferential bending moment. Ring stiffeners are then attached to the shell to alleviate the ioar on the shell. The length, I, of the shell that will act with each stiffener can be assumed (162) to be equal to 0.78 fi Fig. 11.19 shows two recommended types of internal ring
Q sin 4; d& (T - LY + sin O( cos 01)
This summation of shears w-ill become a maximum when cp = n-. The above expression w-ill then become: lOad
(11.46)
$(
’ +:!I!=
7r - ct + sin Q! cos Q!
Fig.
The width of shell that will resist. this load will be 1.56 42
11.18.
Welding
loads and reactions on saddles (I 65).
Society.)
(Courtesy of American
214
Design
of
Horizontal
Vessels
with
Supports
‘lxc
l--l Vassel
Saddle
s h e l l -. -7 =+Composite section A-A
P-I‘
MB = KsQr
t+-l~C
-7 Alternate composite section A-A Fig. 11 .19.
Values of MJQr (equal to Krr) are plotted in Fig. 11.13 for several values of the angle /3. The maximum circumferential bending moment occurs at the point at which #I = /l, as shown previously. Then by Eq. 11.40 The value of KG is plotted in Fig. 11.15 as a fun&on of the angle of contact of the saddle, 8. The moment due to the tangential shear at any point is given by Eq. 11.30, or
Example of internal stiffening rings.
1 - cos 4 - 2” sin 4 >
M, = 9 n-
stiffeners, and Fig. 11.20 shows corresponding types of external ring stiffeners. An inside ring stiffener is most desirable from the strength standpoint because the maximum stress is compression in the shell, which is reduced by the internal operating or test pressure. An external ring stiffener is not very desirable from the appearance standpoint and is even less desirable from the strength standpoint because the maximum stress may be either compression in the outer flange or tension in the vessel shell due to load Q. The value of load Pt on the top of the ring can be developed by a procedure identical to the one followed to obtain Eq. 11.36. If the radius of the ring is taken equal as to r, then
The moment due to the tangential shear at the horn of the saddle will be:
Pt = $K [3 sin2 B - zp sin 2/l - B2 + */3’ cos 261
or
n-
(Mb)~ = 2 (1 - cos /3 - $/3 sin /3) n-
or U%)B = K12Qr
Table 11.1 gives values of Kl2 for different values of p. Consider now the section of ring from the vertical to the horn of the saddle (at which the circumferential bending moment is a maximum) under the action of the forces shown in Fig. 11.21. The load on the ring at the horn of the saddle can be determined by taking moments about the center, 0.
Pt + Pdr + MB = MA + (M,)B
(See Eq. 11.36.)
PB = 1 [
or
Pt
r
= KloQ
(11.43)
where
(11.49)
in which
K = sin2 /3 - ;/3’ - i sin 29
(11.39)
Values of P,/Q (equal to Klo) are plotted in Fig. 11.13 for several values of the angle p. Likewise, the circumferential bending moment at the top will be:
M
A
+
(MC)0 - MB] - Pt
(11.52)
Substituting Eqs. 11.50, 11.51, 11.40, and 11.48 into Eq. 11.52 gives: PB =
Klo = GK (3 sin2 a - -# sin 2b - p2 + &3’ cos 2@)
(11.51)
(Ku +
or
K12 - Kc -
KIO)Q (11.53)
PB = KnQ
Table 11.1 gives values for K13 for several values of fi. The stress on the ring will be the sum of the stresses due to the load Pp plus the stress due to the circumferential
Bcl
MA = SK [sin2 /I(1 - t cos 0 + +3 sin /? - $12) - $fi sin p + &l cos /3(2/3 + sin 2@ - 5 sin p + /? cos /3)]
Composite section
B-B
or
MA = KllQr
(11.50)
where K11 = bK [sin2 /3(1 - : cos p + +p sin p - $3’) - &? sin p + $B cos 8(28 + sin 2p - 5 sin /3 + fi cos fl)]
Alternate composite section Fig. 11.20. Example of external stiffening rings.
B-B
Zick’s Table 11.1.
P 135 120
105 90
0 90 120
150 180
Approximate Values of Constants for the Evaluation of Ring Stresses Kll
KlZ
Km
Ks
K13
0.03 0.02 0.012 0.006
0.286 0.189 0.119 0.0684
0.132
0.082 0.0528 0.0316 0.017
0.102 0.056 0.021 0.0004
0.100
0.078 0.057
moment MB, or (11.54)
where A, = cross-sectional area of the ring stiffener, square inches Z
- = section modulus of the ring
where K14
=
1 + cos p - + sin2 p 7r - p + sin /3 cos @
1 + cos 120” - i sin2 120” K14
=
A
- (120/180) + sin 120” cos 120’
K14 = 0.260 (11.55)
When the ring is attached to the outside surface of the shell adjacent to the saddle or to the inside surface of the shell directly over the saddle, the maximum combined stress is a compressive stress in the shell, fro being negative. For design Zick recommends that the maximum combined compressive stress resulting from liquid load and pressure should not exceed one half the compression yield point of the material (165). The maximum combined tensile stress resulting from liquid load and pressure should not exceed t k allowable tensile stress of the material. DESIGN
OF
SADDLES
The saddle must be capable of resisting the loads imposed by the vessel. Fig. 11.18 indicates the radial-load condition acting on a saddle. To resist the horizontal components of these radial loads, the saddle must be designed to prevent separation of the horns of the saddle when the vessel is carrying a full liquid load. Therefore, at the lowest point of either a steel or concrete saddle a minimum cross-sectional area must exist sufficient to resist the horizontal components of the reactions. A summation of the horizontal components on one half of the saddle is given by: F =
- cos Cj J + cos p 7r - /3 + sin fl cos fl
P sin (z - 4) d+
=
- cos #I + cos p 7r - p + sin fl cos /3
sin C#J dqt
11.8 ZICK’S NOMOGRAPH FOR AID IN THE DESIGN OF VESSEL SUPPORTS
As an aid in the design of supports for horizontal vessels, Zick (165) has presented the nomograph shown in Fig. 11.22, which indicates the most economical locations and types of supports for vessels on two supports. The nomograph is based on a liquid density of 42 lb per cu ft. If liquids of different densities are involved or different materials of construction are to be used, a preliminary design may be obtained by use of the figure. Large-diameter vessels constructed of thin-wall material should be supported near the closures provided that the shell can support the load between the saddles. The closures must be stiff enough to transfer the load to the saddles. Thick-walled horizontal vessels are sometimes too long to act as simple beams. According to Zick such vessels should be supported where the maximum longitudinal bending stress in the shell at the saddles is about equal to the maximum longitudinal bending stress at the mid-span. The shell must be stiff enough to resist this bending and to transfer the load to the saddles. If the shell is unable
-+ sin2 4. - cos 4 cos p T ?r - p + sin /3 cos /3 1 p [ = Q 1 + cos p - & sin2 /3 [ 7r - /3 + sin ,f3 cos p =
Q
1
= K14Q
(11.57)
According to Zick the effective section of the saddle resisting this horizontal force should be limited to a distance of r/3 below the shell at the lowest point of the saddle. This same restriction should also apply to the reinforcing steel cross section in a concrete saddle. The average design stress should be limited to two thirds of the allowable tensile stress of the saddle material. For a saddle where 0 = 120”, /3 = 120”, and
When n rings are used,*
11.7
215
Similarly, for a saddle where 0 = 150”,
C
$
Nomograph
(11.56)
* When two circumferential stiffening rings per saddle are attached to the shell (one on each side of a saddle) the minimum spacing between the rings should be 1.56 & inches, and the maximum spacing, R feet.
Fig. 11.21.
Forces on ring stiffener,
1.
216
Design of Horizontal Vessels with Saddle Supports
Shell thickness, t, inches
AHI+-----L------~HH
Basis of design A-285 Grade C carbbn steel liquid wt = 42 lb per cu tf
Example shown by arrows Use 120” saddles
Fig.
11.22.
to provide the necessary stiffness, ring stiffeners should be added near the saddles. 11.9
(Courtesy
Location and type of support for horizontal pressure vessels on two supports by L. P. Zick (165).
EXAMPLE
CALCULATION
OF
STRESSES
Estimate the stresses induced by the supports in a vessel designed for storing lube oil and having the following design data: Lube-oil API gravity Working pressure Design pressure Design temperature Material Allowable working stress Joint efficiency Corrosion allowance Shell diameter (ID) Shell thickness (including corrosion allowance) Head thickness (including corrosion allowance) J Tangent length Bearing-plate width, b Heads
16.5 75 psi 9$ psi 500’ F SA-285, Grade C 13,750 psi
80%
36 in. 10 ft
American
Welding
Society.)
To analyze the type of support to be employed for supporting the vessel, use is made of Fig. 11.22. By entering the figure with a shell-thickness value of s in. (with allowance for corrosion) and with a tangent length of 68 ft, it is found that the resulting zone indicates that A/R S 0.5 with 0 equal to 120” and that the head-plate thickness should he checked. For this vessel R = 5 ft; therefore A will be taken as 234 ft (W R) in order to take advantage of the stiffening.effect of the head. A sketch of the vessel with 120” saddle supports is shown in Fig. 11.23. Following is the calculation of the weight of one head of the vessel. Fig. 5.12 the required blank diameter for 3ia20-in-diameter elliptical dished head -t-i52 iQ The weight of the plate is: ~(152)‘(0.75)(490; -.- = 3850 lb per head (4) (1728)~
N in. K in. 68 ft.“\ 10 in. elliptical dished, 2:l ratio employed
of
Y,. 1$I;
R q -.
The weight of two heads is 7700 lb. The shell weight is:
410) (68) (0-W (490’) /
\
12
-~I
:
1 $
= 65
5oo ,
lb
’
’
Example Calculation of Stresses
The volume of one head is given by Eq. 5.14 ‘-G.= 0.000076 Di3_ \-_\ = 0.000076(120)3 = 131 cu ft per head
217
From Fig. 11.10 and with 0 equal to 120” K5 = 0.88 therefore
J
fs = (204,000)(0.88) = 4780 psi (60)(0.625)
The total volume of the two heads is 262 cu ft. The volume of the shell is: n-y(68) = 5340 cu ft
The tangential shear stress in the shell is given by Eq. 11.28; and as &hell equals thead, the shear stress in the head equals the shear stress in the shell. fs = fi = 4780 psi P
The total volume is: 262 + 5340 = 5600 cu ft The densit’y of the fluid is 59.7 lb per cu ft. The total weight of the fluid (vessel full) = (59.7)(5600)
Circumferential stress at horn of saddle: Since the shell is stiffened by the head and since L > 8R, Eq. 11.42 gives the circumferential stress at t.he horn of the saddle.
= 334,000 lb The weight of the vessel and its contents = 334,000 + 65,500 + 7700 = 407,200 lb
From Fig. 11.16, A/R = 0.5, and 0 = 120°; therefore K, = 0.013
Therefore Q (load per saddle) = 204,000 lb
and
-=H A 2.5 = 0 0368 L L=68 .
f2 =
From Fig. 11.5 therefore
-204,000
f, =
Maximum longitudinal bending stress: The saddles are located close to the heads of the vessel. The maximum longitudinal bending stress exists at the center of the span between the saddles and is given by Eq. 11.8: v
For the condition in which no credit for t is taken for the wear-plate thickness,
+=zQL 7rr2(t - c)
(4)(0.625)[10 + 1.56 d(60)(0.625)] _ (3)(0.013)(204,000) (2) (0.625) 2 = -4180 - 10,180 = -14,360 psi
The maximum permissible stress equals: 12,650 X 1.25 = 15,800 psi As the stress f7 is less than the allowable stress, it is not necessary to take credit for t,he wear plate.
K2 = 0.82 .
= +4920 psi Since t/r = 0.625/60 = 0.0104 > 0.005, the compression stress is not a f&&r in the design. f, = longitudinal pressure stress
L ----
I
By Eq. 3.15 (4) (0.80) (0.625)
z- - - - - - -
rT-r
5400 psi
4 fz +f, = 10,320 < 0.8faaoW. = 11,000 psi Tangential shear stress: the tangential shear stress in the head is given by Eq. 11.127. fs _ Q! 4th - c)
ji _
Fig.
11.23.
Sketch of vessel in example calculation.
218 ‘.’
Design of Horizontal Vessels wtth Saddle Supports
Additional stress in head used as stiffener: The additional stress induced in the head when it is used as a stiffener is given by Eq. 11.44. js zz --QL
From Fig. 11.17 and with ~9 equal to 120’ Kg = 0.40 j = (204,000)(0.40) s (60)(0.625) = 2180 psi
For an elliptical dished head (K = 2.0) the maximum pressure stress may be taken as equal to the circumferential hoop stress in the shell (see Chapter 7 and Eq. 7.57). From Eq. 7.57 p[Vd + 0.2(t - c)] fP = 2E(t - c) By Eq. 7.56 v = i(2 + k2) = 1.0 Using a one-piece head, we find that E = 1.0. j
P
2180 + 8650 = 10,830 psi The maximum allowable stress in the head is: (12,650)(1.25) = 15,800 psi
4th - c)
therefore
The maximum combined stress in the head equals:
= 90[(1)(120) + (0.2)(0.625)1 = 8650 psi (2)(l) (0.625)
Ring compression in shell over saddle: The stress in the shell band directly over the saddle is given by Eq. 11.46. jg
K9 = 0.76 (204,000) (0.76) jg = (0.625)[10 + 1.56 1/(60) (0.625)] = 12,700 psi The allowable stress equals the yield point divided by two. According to reference 67 the yield point of SA-285, Grade C steel equals 30,000 psi. Therefore 30,000 allowable stress = __ = 15,000 psi 2
1. Recalculate the stresses in the vessel described in the example calculation (see Fig. 11.23) shell and head thickness is r+{a in. rather than K in. 3. In reference to Fig. 11.1, the horizontal storage tank is 12 ft, 0 in. in inside diameter x 120
if the
ft, 0 in. long. The vessel is used to store butane at 100 psi and 400” F and has elliptical dished heads. The heads and shell have a thickness of rgs in. with a >Q-in. corrosion allowance. Assume the saddles are located 80 ft apart, 8 equals 120”, wear-plate width equals 10 in., wearplate thickness equals N in., and the wear-plate extends 6 in. above the horn of the saddle. Assume the vessel is fabricated of ASTMA-285, Grade C steel with a joint efficiency of 80%, and calculate the stresses in the shell and head for the case in which internal stiffeners are not used. 3. Redesign the storage vessel described in problem 2 for gasoline storage at 50 psi and 400” F using internal stiffeners in the plane of the saddle (see Fig. 11.19), ASTMA-283, Grade C steel, a joint efficiency of 80%, and no corrosion allowance. 4. Design the concrete saddles for the vessel described in problem 2, and specify the area of reinforcing steel if a 1:2>4:3js concrete mix is used (see Table 10.1).
’
-
--.
-
\
\
\I /
QKg
+ 1.56 &)
From Fig. 11.17 and with 8 equal to 120”
PROBLEMS
-- -
=
t(b
C H A P T E R
DESIGN OF FLANGES
A
J 1.
Welding-neck. 2. Slip-on. 3. Screwed. 4. Lap-joint. 5. Blind.
A variety “”
of attachments and accessories are essential to vessels. These items include flanges for closures, nozzles, manholes, and handholes; and flanges for two-piece vessels. Supports, platforms, and ladders are examples of other typical accessories. Flanges may be used on the shell of a vessel to permit disassembly and removal or cleaning of internal parts. Flanges are also used for making connections for piping and for nozzle attachments of openings larger than lj+in. nominal pipe size. Threaded-pipe connections such as couplings and half couplings are used for openings smaller than l$d-in. pipe size. Figure 12.1 shows a sectional view of an autoclave with a shell flange on the vertical vessel and nozzle flanges on the discharge and feed attachments.
Other standard types such as reducing flanges, socketwelding flanges, orifice flanges, and nonstandard flanges are also available for certain ratings. 12.la Welding-neck Flanges. Figure 12.2 shows a sectional view of one and lists the dimensions of several standard X0-lb welding-neck flanges from 55 to 24-in. nominal pipe size. Welding-neck flanges differ from other types in that they have a long, tapered hub between the flange ring and the weld joint. This hub provides a more gradual transition from the flange-ring thickness to the pipe-wall thickness, thereby decreasing the discontinuity stresses and consequently increasing the strength of the flange. This type of flange is preferred for extreme service conditions such as: repeated bending from line expansion or other forces, wide fluctuations in pressure or temperature, high pressure, high temperature, and subzero temperature. These flanges are recommended for the handling of costly, flammable, or explosive fluids, where failure or leakage of a flange joint might bring disastrous consequences. 12.1 b Slip-on Flanges. Figure 12.3 shows a sketch of a standard 150-lb slip-on flange and gives dimensions for such flanges of $4 to 24-in. nominal pipe sizes. The slip-on type of flange is widely used because of its greater ease of alignment in welding assembly and because of its low initial cost. The strength of this flange as calculated from internalpressure considerations is approximately two-thirds that of a corresponding welding-neck type of flange. The use of
12.1 SELECTION OF STANDARD FLANGES A great variety of types and sizes of “standard” flanges are available for various pressure services. The flanges designated as “American Standards Association (ASA) B16.5-1953” are used for most steel pipe lines over l>S-in. nominal pipe size; therefore these flanges are extensively used for nozzles and other attachments to vessels (187). These flanges are normally forged from ASTM A-181 and ASTM A-105 carbon steels. Forged-alloy-steel flanges are also available. These flanges are called “companion flanges” because they are almost always used in pairs. Although they are usually manufactured from forged steel, cast-iron companion flanges may be used for low-pressure service. Forged-steel flanges are manufactured in the following standard types for all pressure ratings: 219
...-
--;-.:
__. ‘-\- __..r-- -- -
~--, _-._-_ _____
n-7
--
--
220
Design of Flanges
B 0
Adapter block with 1 gage 0 to 18 00 lb 1 rupture-disc assembly for 1150 lb at 300°F 1 gas-inlet valve.
Thermometer well
*
-Coil
;P
If/la 2% B
5.66 6.72 8.72 10 88 12.88
8 8 8 12 12
1 1
74, ‘/b ?/d
3,i ?‘a $‘r 74 7/6
8Yz 941 11% 14$1 17
15 19 30 43 64
14 16 18 20 24
21 23% 25 27Si 32
1% l?,io l%a l’$(e 1%
16% 18ji 21 23 27ti
2% 241 2’4ie 2% 3%
14.19 16.19 18.19 20.19 24.19
12 16 16 20 20
1% 1% 1% 1H 1%
1 1 1% 1% 1%
18% 21y1 23% 25 2955
85 93 120 155 210
15% 18 19% 22 26411
AND
width equal to the width of the raised face whereas flat metal gaskets may be used having a width equal to that used with the large tongue-and-groove type of face. The male-and-female facings have the advantage of confining the gaskets thereby minimizing the possibility of blowout of the gaskets. They have the disadvantage that the two mating flanges are not identical. For this reason these flanges are not as widely used on pipe-line connections as are the raised-face flanges. They are used extensively on heat exchangers, and sometimes for manholes and as end flanges. Male-and-female facings have another disadvantage compared to tongue-and-groove flanges in that they offer no protection against forcing the gasket into the vessel. The male-and-female facings are standardized with a x6-in-deep recess on the female face and a j/4.-in.-high raised face on the male part. The gasket surfaces are usually smooth finished as the outer diameter of the female face serves to locate and retain the gasket. The width of the face is so large in the case of large male-and-female gasket-contact surfaces that full-face metal gaskets cannot
Fig. 12.5.
F i g . 1 2 . 3 . S t a n d a r d 150-lb f o r g e d s l i p - o n f l a n g e s ( 1 6 8 ) . ( C o u r t e s y o f
Sectional view of a screwed flange.
Taylor Forge and Pipe Works.)
6’
A MERICAN STANDARD~ASA OutDiamFig.
12.4.
Thickof
Sectional view of steel lop-joint flange ond lap-joint stub.
B16~-1939 FORGED AND ROLLED STEEL- ASTM A 181 outside Drilling Template Approx. et.er __ N o . Diam. Diam. Weight Bolt E&l, Circle Pounds
52.3 STANDARD FLANGE FACINGS Standard flanges are available with a variety of machined faces, as shown in Fig. 12.7 (corresponding dimensions are given in Table 12.2). Steel flanges with a raised face are extensively used because of the simplicity of the design and because they have been proved adequate for average service conditions. For severe service involving high pressure, high temperature, thermal shock, or cyclic operations, this type of flange facing may not be satisfactory. Flanges with ratings of 150 and 300 lb have a >is-in.-high raised face, and flanges having higher ratings have >i-in.-high raised faces. The raised face is machined with spiral or concentric grooves approximately 364 in. deep with about >s2-in. spacing. The edges of these grooves serve to deform and hold the gasket. Flat-ring composition gaskets normally are used having a
4 7 9 13 17 5 6 8 10 12
10 11 13% 16 19
15;s 1 1% 1x5 13;
7X6 asi lO>i; 123; 15
14 I6 18 20 24
21 2336 25
1% IN6 1916 l’Me 1%
163; 18%
27%
32
Fig. 12.6. Standard
21 23 27% 150-lb
Forge and Pipe Works.)
8 a
12 12
20 26 45 70 110
12 16 16 20 20
311 170 209 272 411
8
blind flange (168). (Courtesy of Taylor
Nonstandard Table
12.1.
Pressure
Carbon-steel
Ratings
Flanges
Fluid
Wd‘3-. St-; or Oil
Oil OdY
12.4 150
300
400
600
900
1500
2506
Maximum Hydrostatic-shell-test Pressures* ( l b p e r sq in.) 350
900
1200
1800
2700
4500
7500
Service Temperatures (dee F) 100 150 200 250
Maximum Nonshock Service-pressure Ratings (lb per sq in.) 600 800 1200 1800 3000 5000 590 785 1180 1770 2950 4915 580 770 1160 1740 2900 4830 570 760 1140 1710 2850 4750
190 180 170 160
560 550 540 525
740 725 710 700
1120 1095 1075 1050
1680 1645 1615 1580
2800 2740 2690 2630
4660 4565 4475 4380
i 500 550 600 650
15ot 140 130 120
500 475 445 415
665 630 590 550
1000 950 890 830
1500 1420 1330 1240
2500 2370 2220 2070
4165 3950 3700 3450
700 750 800 850
110 100 92 82
380 340 300t 245
500 450 4oot 330
1
300 350 400 450
230 220 210 200
900 70 210 280 { 950 55 165 220 1000 40 120 160 * Temperature of test water shall not exceed t Primary service-pressure ratings.
223
are below the flange face, the gasket-contact faces are protected from damage. The main disadvantage of this type of facing is the high cost of manufacturing it; this is the most expensive gasket face.
for
(168)
With Standard Facings (Other than Ring Joints) for Water, Steam, and Oil Service Primary Servicepressure Ratings (lb p e r sq in.)
Flanges
NONSTANDARD
FLANGES
Nonstandard flanges are available in sizes from 26 in. to 96 in. These flanges are fabricated by rolling a hot annular blank, as shown in Fig. 12.8. Nonstandard flanges are available in a variety of ratings. Typical ratings for flanges supplied by one manufacturer are 50, 125, and 250 psi. Figure 12.9 shows a sectional view of a large-diameter welding-neck flange rated at 50 psi (at 100” F service temperature) and, gives the dimensions for such flanges.
760 1140 1900 3160 680 1020 1700 2830 600t 9 0 0 t 15OOt 25OOt 490 740 1230 2050 420 330 240 125O F .
630 495 360
1050 825 600
1750 1375 1000
be used because of the excessive tightening loads required to seat the gasket. The tongue-and-groove type of gasket face has advantages and disadvantages similar to those of the male-and-female type of gasket face. The presence of retaining metal on either side of the gasket gives protection against deforming soft gaskets into the interior of the vessel; this is an advantage over the male-and-female type of face. Also, the gasket is less subject to erosive or corrosive contact with the fluid in the vessel. In service the tongue is more likely to be damaged than the groove; therefore the tongue should be placed on the part that can be most ‘easily removed from the vessel. This type of facing is standardized for both small and large flanges. The small area of the tongue-andgroove surface provides the minimum area that it is advisable to use with flat gaskets. Therefore, this type of facing provides the minimum bolting load for compressing a flat gasket. One advantage of the ring-joint type of facing is that it offers the greatest protection under severe service conditions or with the use of hazardous fluids. This type of flange is widely used in petroleum, petrochemical, and high-pressure service. Close tolerances and high standards of machining are required, and as a result, this type of flange is seldom used in nominal sizes larger than 36 in. Another advantage lies in the fact that the internal pressure acts on the ring to increase the sealing force on the joint. The fact that mating flanges are identical reduces the problems of stocking and of assembly. Also, because the gasket-contact surfaces
Raised face
Lap joint
Large male and female
Small male and female
Large tongue and groove
Small tongue and groove
Ring-type joint Fig. 12.7. American Standard Range facings (168). (Courtesy of Taylor Forge and Pipe Works.)
Design of Flanges
224
Table 12.2. American Standard Flange Facings (168j For 150, 300, 400, 600, 900, 1500, and 2500-lb Flanges
Outside Diameter 3 Raised Face, Inside Lap Outside Diameter 3 DiameJoint, Large ter of Large Male, Large Female and and Nominal and Small Large Small Small Large Small Small Pipe Tongue5 Malea.5 Tongue5 Tongue3*5 G r o o v e 5 Female4f5 Groove5 Size, i; iJ W X Y Inches R s 1 1x16 I2%2 1x6 1% 2x2 1% ?4 1% 1% l’.YiS 'X6 11!46 1x6 N 1 2 1416 1% 1% 2x6 1% I'%6 1% 2%6 1x6 2x6 1% 255 1% 2%
Height Inside Raised Face, DiameLarge and ter of Kaised Small Male Large Face, and Tongue, and 250 and 400, 600, Depth of Small 300-lb 900, 1500, Groove Groove3v5 Stand- and 2500-lb or 2 ardsl Standards2 $4 '%6 1% s/4 17i6 54 113,ic ,ki
1%
2%
w6
I'%6
2%G
23/16
l/i 6
3% 3%
%S
2x6
3x6
21936
%6
2%
2% 21xfi
4%6
2%
3%6
3x6
%6
3
3x6
4%
5x6
3%
41x6
4Ke
'46
1% 2
8 10 12 14
10%
12% 15 16>/4
wi6
5%
5x6
wi6
6%
3% 4%
411d,j 5%
14 6
4x6
5%
61946
7%
5x6
6%
6%
8
f4 6
8946
me
7x6
'4 6
8% 1ojg
10 12 14?4 15%
12% 13%
9% 11>/4 13% 14%
?4 %i K M
'% 6
101$i6
8x6
9x6
'/i' 6
Ns
12%6
10x6
12x6
11x6
%s
946
w6
w6
14x6
13x6
%6
ws
16546
13%6
l5%6
14l146
%6
%6
lo?46
16 17l3i6 16% 18% 15% 17% 18756 wi6 1@?46 %6 ?'i6 21 17% 2034 17lg,j 18 19% %iS 20x6 19Ns x6 946 23 21 20 19% 22 23x6 19%6 =?iS wi6 ?46 %6 25M 24 26>i 23% 27x6 29x6 26x6 25x6 27% X6 %6 1 Regular facing for 150 and 300-lb steel flanged fittings and flange standards is a x s-in. raised face included in the minimum flange thickness. A x6-in. raised face is also permitted on the 400, 600, 900, 1500, and 2500-lb flange standards, but it must be added to the minimum flange thickness. * Regular facing for 400, 600, 900, 1500, and 2500-lb flange standards is a x-in. raised face not included in minimum-flanpethickness dimensions. 3 A tolerance of plus or minus 0.016 in. (364 in.) is allowed on the inside and outside diameters of all facings. 4 Care should be taken in the use of joints of these dimensions (they apply particularly on lines where the joint is made on the end of pipe) to insure that pipe used is thick enough to permit sufficient bearing surface to prevent crushing the gasket. Threaded companion flanges are furnished with plain face and are threaded with American Standard Locknut Thread. 6 Gaskets for male-female and tongue-groove joints shall cover the bottom of the recess with minimum clearances taking nto account the tolerances prescribed in note 3.
12.5 GASKETS AND THEIR SELECTION Leakproof metal-to-metal surfaces in which gaskets are not used are difficult to fabricate even by use of very accurate machined surfaces. Irregularities in clearances of only a few millionths of an inch will permit the escape of a fluid under pressure. The function of a gasket is to interpose a semiplastic material between the flange facings, which material through deformation under load seals the minute surface irregularities to prevent leakage of the fluid. The amount of flow of the gasket material that is required to produce a tight seal is dependent upon the roughness of the surface. The amount of force that must be applied to the
gasket to cause the gasket to flow and seal the surface irregularities is known as the “yield” or “seating” force. This force is usually expressed as a unit stress in pounds per square inch and is independent of the pressure in the vessel. Thus, this yield stress represents the minimum load that must be applied to the gasket to seat it even though very low pressures are used in the vessel. Usually the gasket is seated by tightening the bolt load on the flanges prior to the application of the internal pressure in the vessel. Upon the application of the internal pressure in the vessel, an end force tends to separate the flanges and to decrease the unit stress on the gasket. Figure. 12.10 shows the three major forces acting on the gasket.
Gaskets and Their Selection
Leakage will occur under pressure if the hydrostatic end force is sufficiently great that the difference between it and the bolt-load force reduces the gasket load below a critical value. Also, it may be possible with too low a contact pressure on the gasket for the gasket to be blown out by the internal pressure. The ratio of the gasket stress, when the vessel is under pressure, to the internal pressure is termed the “gasket factor.” The gasket factor is a property of the gasket rnxdthe construction and is independent of the internal pressure over a wide range of pressures. In selecting the proper gasket for an existing closure, one of the first steps should involve the determination of the total amount of force necessary to make the gasket yield and to maintain a tight seal under operating conditions. Figure 12.11 shows sectional views of some common types of gaskets and lists the gasket factor, m, and the minimum design seating stress, y, for each type of gasket (11). Figure 12.11 also indicates the recommended facings for the various types of gaskets. The effective width of the gasket, b, for various types of facings is shown in Fig. 12.12 (11). Flat-ring gaskets are widely used wherever service conditions permit because of the ease with which they may be cut from flat sheets and installed. They are commonly fabricated from such materials as rubber, paper, cloth, asbestos, plastics, copper, lead, aluminum, nickel, monel, and soft iron. The gaskets are usually made in thicknesses of from 5d4 to $4 in. Paper, cloth, and rubber gaskets are not recommended for use above 250” _.. , . ..-.c F. Asbestos-composition gaskets may be used up to 650” F or slightly higher, and ferrous and nickel-alloy metal gaskets may be used up to the maximum temperature rating of the flanges. Laminated gaskets are fabricated with a metal jacket and a soft filler, usually of asbestos. Such gaskets can be used up to temperatures of about 750” F to 850” F and require less bolt load to seat and keep tight than solid-metal flatring gaskets. Serrated metal gaskets are fabric;lted of solid metal and have concentric grooves machined into the faces. This greatly reduces the contact area on initial tightening, thereby reducing the bolt load. As the gasket is deformed,
Fig. 12.8.
Rolling of a large
flange
(168).
J-m--~\,
225
the contact surface area increases. Serrated gaskets are useful where soft gaskets or laminated gaskets are unsatisfactory and the bolt load is excessive with a flat-ring metal gasket. Smooth-finished flange faces should be used with serrated gaskets. Corrugated gaskets with asbestos filling are similar to laminated gaskets except that the surface is rigid with concentric rings as in the case of serrated gaskets. Corrugated gaskets require less seating force than laminated or serrated gaskets and are extensively used in low-pressure liquid and gas service. Corrugated metal gaskets without asbestos may be used to higher temperatures than those with asbestos filling and are extensively used in sealing water, steam, gas, oil, and acid and other chemicals. Two standard types of ring-joint gaskets are available for high-pressure service. One type has an oval cross section, and the other has an octagonal cross section. These rings are fabricated of solid metal, usually soft iron, soft steel, monel, 4-6% chrome, and stainless steels. The alloy-steel rings should be heat treated to soften them. For lowtemperature service plastic rings may be used for corrosion resistance and as a means of electrically insulating the flange joint. There is a considerable possible choice of gasket material in many applications. The decision as to which gasket material is to be selected is often based upon the required gasket width. If the gasket is made too narrow, the unit pressure on it may be excessive. If the gasket is made too wide, the bolt load will be unnecessarily increased. A relationship for making a preliminary estimate of the proportions of the gasket may be derived as follows: (Gasket seating force) - (Hydrostatic pressure force) = (Residual gasket force) The residual gasket force can not be less than that required to prevent leakage of the internal fluid under operating pressure. Therefore
(Courtesy of Taylor Forge and Pipe
r
-
Works.)
-
226
Design of Flanges
FORGED
AND
Size Inches 26 28 30 32
ASTM A 181 50 Lb Pressure at 100” F-Welding-neck Type Outside Outside Diameter Thickness Diameter CompressedLength Diameter Drilling Template Approx. of of Taper-Hub through of of~~~~d AsbestosNo. Diam. Weight Flange Flange Diameters Hub Bore of Gasket of Bolt Each, A T R Size E K L B Holes 1Holes Circle Pounds 3 26 32 31% 1% 2846 273C x 28% 2734 26% 98 29% 1 13-i 3036 29% x 30% 2934 2 8 % 3 28 36 105 33% 31% 1 3556 31% x 32% 31% 30% 3 30 36 1 3394 112 1% 32% 35 33% x 35 33% 32% 3% 32 36 140 1% 38% 1% 36% ROLLED
STEEL
34 36 42 48
Wi
1% 1% 1% 1%
37 39 45% 51%
54 60 66 72
61?4
67% 74 80
1% 1% 1% 2%
57% 634i 7036 7636
26 28 30 32
33 35 37 39%
1% 1% 1% 1%
30 32 34 36%
34 36 42 48
41% 43% 50 56
15-i 1% 1% 1%
38% 40% 46% 52%
3736 x 38% 36% 36% 39%x40% 38% 38% 45% ~46% 44% 44% 51%x52% 50% 50%
54 60 66 72
6235
1% 1% 1%
59 65 71% 77%
57% x 59 63% x 65 69Xx71x 75Xx77X
42x 49 55
68% 75%
2
8134 Fig.
12.9.
Nonstandard
35% x 37 37% x 39 44%x45% 50% x 5154 56% ~57% 6234 x 63% 68Xx 7036 74%~ 76Ji 50 Lb Pressure 29% x 30 31% x 32 33% x 34 3534 x 36%
large-diameter
flanges
where y = yield stress, pounds per square inch (see
Fig. 12.11)
m = gasket factor (see Fig. 12.11) p = internal pressure, pounds per square inch d,, = outside diameter of gasket, inches 4 = inside diameter of gasket, inches
In Eq. 12.1 it is assumed that the hydrostatic force extends to the outer diameter, 4, of the gasket and that all the hydrostatic force is utilized in relieving the gasket load that existed prior to application of the internal pressure. These assumptions disregard elastic deformation of the bolts, gasket, and flanges, but the relationship is a useful one for the initial proportioning of the gasket. For convenience
35% 37% 43% 49vh
34% 36% 42% 483/4
3% 3% 335 3%
56 54% 4 62 60~$ 4% 4% 68 67 5% 74 73 at 100’ F-Slip-on Type 28% 28 2% 2% 30% 30 32% 32 2% 34% 3444 2%
57% 56% 63% 62% 69% 68% 75% 74% (168).
(Courtesy
2% 2%
3-i 2%
3% 3% 4 4% of
Taylor
34 \ 36 42 48
40 40 48 52
1% 1% 1% 1%
38% 40% 46% 52%
149 157 209 241
54 60 66 72
-64 72 72 80
1% 1M 1% 1%
59 65 71% 77%
312 398 556 705
%l?da 28%
32 36 36 40
1 1 1 1%
31 33 35 37%
122 140 148 171
40 44 48 56
1% 1% 1% 1%
39% 41% 47% 53%
181 191 234 269
%-72 80
1% 1% 1% 1%
60%
335 451 591 728
3w6 3wiS
34l%6 361X,5 42l9i6
48% 55%6 61316
67% 734iS Forge
and
Pipe
661,/, 73 79
Works.)
Eq. 12.1 may be rewritten as follows: do &=
-_-Y-pm - P(m + 19
(12.2)
In the case where it is desirable to retain the gasket material selected and to decrease the gasket width, a gasket seating stress greater than y may be used with certain reservations. If the seating stress greatly exceeds y. the gasket may be crushed, or a ductile, unrestrained gasket may be squeezed out between the &urge faces. In general the use of seating stresses exceeding y should be limited to solid-metal gaskets in tongue-and-groove joints.
Design of Special Flanges
227
12.6 OPTIMUM SELECTION OF BOLTS FOR SPECIAL FLANGES
4
,’
.-
1 \
’ \
The maximum bolt load will be the greater of the two following forces: the force required to seat the gasket and the force required to withstand the internal pressure and maintain the gasket-factor pressure (mp) at the same time. After the greater of these two forces has been determined, the required bolting area of bolting steel may be determined by dividing the maximum force by the allowable bolting stress. A number of combinations are possible in providing the required bolting area. In general a larger number of smaller-sized bolts will provide the same bolting area as a lesser number of larger-sized bolts. The minimum bolt spacing based on wrench clearances limits the number of bolts that can be placed in a given bolt circle. The maximum bolt spacing is limited by the permissible deflection that would exist between flanges. If this deflection is excessive, the gasket joint will leak. Taylor Forge recommends (188) the following empirical relationship for maximum bolt spacing: Bs(max) = 2d + 5
(12.3)
where Ba(max) = maximum bolt spacing for a tight joint, inches d = bolt diameter, inches t = flange thickness, inches (see Fig. 12.11) m = gasket factor
. r
1,
1 I ,
I
Before the bolting calculations can be completed, the diameter of the vessel, B, and the value of gr (hub thickness) must be known. The wrench clearances limit the minimum bolt distance from the hub. ~%n general it is desirable to use a minimumdiameter bolt circle and an even number of bolts, preferably a multiple of four. The minimum diameter of a bolt circle may be determined by setting up a table in which the number of bolts required, the root area, the preferred bolt spacing, B,, and the radial spacing, R, are tabulated as functions of bolt size (see Table 10.4 for root area, B,, and R). The minimum bolt-circle diameter will be either the diameter necessary to satisfy the radial clearances [d = B + s(gl + R)] or the diameter necessary to satisfy the bolt-spacing requirement [d = (NBS/r)], whichever is greater. The optimum design is usually obtained when these two controlling diameters are approximately equal. The following example demonstrates the procedure recommended.
: ‘,
Table 12.3. Selection of Optimum Bolt Size Min NB, Eolt. Root No. of Actual Size Area Bolts No. (N) B, R n- B+%l+R) N 0.302 73.7 76 3 l>Q 72.5 36% ‘A 0.419 53.3 56 3 1% 53.4 37 1 0.551 40.4 44 3 1% 42.0 37% 1% 0.728 30.6 32 3 1% 30.5 37% lx 0.929 24.0 24 3 1% 22.9 38
Fig.
12.10.
The three major forces acting on a gasket (169).
Example of selection of optimum size and number of bolts. Given : Inside diameter, B = 32 in. Hub thickness, g1 = 12 in. (see Fig. 12.14) Allowable bolt stress = 20,000 psi .: Maximum bolt load = 446.000 lb The minimum bolting area, AB(min) is given by: W A~(min)
=
- =
fallow.
446,000 - = 22.3 sq in. 20,000
.
Root area, B, preferred, and radial distance, R, are obtained from Table 12.3. Inspection of Table 12.3 indicates that the minimum bolt circle will result from the use of 32 bolts 136 in. in diameter. The size of the bolt circle is 3734 in. 12.7 DESIGN OF SPECIAL FLANGES Process vessels are often of such large size that standard pipe flanges are not available in the sizes required. In such cases special flanges must be designed. Large-size flanges may be rolled from an annular ring (see Fig. 12.8) or may be rolled from bar stock and welded. If a slip-on flange without a hub is to be used, the ring for the flange may be flame cut from flat steel plate. The earliest method of designing flanges was the so-called “locomotive method” of Risteen (170). Cracker and Stanford (171) developed a method of flange design in which the flange was considered to behave as a beam. Den Hartog (172) compared the “locomotive” and Cracker Stanford methods by vector analysis and showed them to be the same although the deriGations were different. Waters and Taylor (171) developed a method of analysis
220
Design of Flanges
Gasket Factors (m) for Operating Condit,ions and JIinimum Design Seating Stress (y) Note: This table gives a list of many commonly used gasket materials and contact facings with sug gested design values of m and y that have generAy pro~rd satisfactory in actual service when using elfective gasket, seating w-idth? h, given in Fig. 12.13. hlin design Sketche.< ~IIICI 5 seating Gaskel ~u:~trrial notes stress. y :~
-_Rubber without f;rbric or ;I high ~wrcc~l~~rge Relow 75, Show I Wometer 75 or higher, Short, I )uromet w
~.
Refer to Fig. 12.12 Facing imitations
Use col.
-
uf ushtos lilwr: 0 50 I
00
2 0 0 2.
i3
3 *so
0 ‘00
Lisp 19 4, 6 OIll\-
1600 3700 fl300
Ilubber with iIst)f~st~I~-f;~l)r~i(~ wire reiriforcemfv~ 1
::-[‘I! 2-ply I-d\
insertion. wilh or willlout
., -. ‘ ? -3-
-
2 4- 0 7.i.i I .;5
1 LOO
Sl)irill-wOlJIlfl
Ill~‘lill.
:ls~~f~stoc;
2900 I300
lilltd
-
Corrugated nlr*t ;11, asbt‘st.os inst~rtt~tl or Corrugated melal. j:~cl,c~~~d, nsl~r~os filled
sot’{ il~llIIIiJlUIlI Ydt CO~~pt~J
Tron
o r tbl?lS?;
01‘ s o f t stwl
MOlld
2.7-i
3700
2 3 0 - 2 ,a
2900 3700 4500 .X00 h.iOO
3 0 0
3.25
01’ ,I-hC;) chlY~lllf
- 3.30
Slainless slr& S o f t alufiiiiluiti
2.73 3.00 3 25 3.50 3.i.5
3700 ‘1500 5500 6500 7600
3 "3 3.50 3.75 3.50 3.75 3.75
5300 6500 7600 8000 9000 9000
Grooved iron )I metal jackel,
3.25 3.50 3.75 4 06 I 23
5500 6500 7600 8800 10100
Solifl fkdt metal
I.00 ‘k.i5 .i 50 0.00 h .50
8800 13000 18000 "1800 26000
3.50 6.00 6.50
18000 21800 26000
Soft CC,,,I”“’
Corrugated mrtal
or I,rans
Iron or soft slwl \lo~irl o r l-h’,‘; (*liroliit Stainloss steels Soft :iluriiiriurii Soft ccbr)wr
Flat metal, jacketed, asbestos filled
or l)razs lron or’sht stcv~l Alone1
Iron or ~ofl. slrrl Kin:: joint,
Yvlorlf~l 0,’ I - h ’ ;, f~hrolllc
St:iinlea$ stfds
* The surface of a gasket. llil\miIlg 21 lap should Fig. 12.1’,
Gasket materials and contact facings.
1~1:
ag:hst ltic
s1~1001l1
snrtiw of the hciri,u and
(Extracted from the 1956 edition of the
with permission of the publisher, the American Society of Mechanical Engineers [I 11.)
ASME
Use la only
Use lit, 2* only-
Use I. 2, 3 011ly
h-me
JIOI
ap;linst the nubbin.
Boiler and Pressure Vessel Code, Unfired Pressure
Versel~,
229
Design of Special Flanges
combining the theory for a beam on an elastic foundation with the theory for a flat plate which made possible the calculation of the stresses in the radial, tangential, and axial directions. The Taylor-Waters method was extended by \Vaters, Rossheim, Wesstrom, and Williams (173, 174). This method of flange design has been the basis of the MIME-code (11) procedures for flange design. A compdrison of the theoretical st,ressea with stresses determined from strain measurements has been reported (175) for flanged joints of vessels and piping in low-pressure service.
Facing Sketch Exaggerated
Basic Gasket S Column ,I
Loose ring flange
Loose hubbed flange
Riveted flange
Fusion lap-welded ring flange
ltin W i d t h . b,j CoIumu I I
Forged integral flange
W+3N 8 Fig.
!!+!Y; (y ruin)
3N a
7N 16
N 4
3N T-
12.13.
Fusion thru-welded ring flange
Various types of flanges
Waters, et al.
Fusion butt-welded hubbed flange
subject to the method of analysis of
(Extracted from Transocfions of
the ASME with permission
of the publisher, the American Society of Mechanical Engineers [174].)
The following sections describe a method of flange design based upon the procedures developed by Waters, Rossheim, IVesstrom, and Williams. The method is general and applies to circular flanges of bolted joints under pressure and free to deflect under the action of the bolt load. This includes all types of flange facing in which the gasket or cont.acting flange surfaces are entirely within the bolt circle and excludes all types in which there is any contact outside the bolt. circle. Various types of flanges to which this method applies are shown in Fig. 12.13.
W-I-‘-J l IIT-
Ith, b b = b,
Fusion lap-welded hubbed flange
w h e n b. 5 >;”
I
VT” b = 2 w h e n b, > fL”
rz (radius)
Location of Gasket-load i ?actiou
Note: The gasket factors listed only apply to tlanged joints in which the gasket is contained entirely within the inner edges of the holt holes.
The design values and other details given we suggested only end mandatory.
ure not
Shell .’
Fig.
12.12.
Hub
Ring
r, (radius) -
Effective gasket width and location of gasket load reaction.
(Extracted from the 1956 edition of the ASME Boiler and Pressure Vessel
Fig.
Code, Unfired Pressure Vessels, with permission of the publisher, the Ameri-
(Extracted from Tronsocfions
12.14.
Analysis
of
forces
ond
moments
in o
con Society of Mechanical Engineers [l 11.)
the American Society of Mechanical Engineers, [174].)
tapered hub flange.
of fhe ASME with permission of the publisher,
230
Design of Flanges
The bending moment is given by Eq. 6.16.
M=DS Fig. 12.15.
Type-l loading of tapered hub Range
(6.16‘,
where D is given by Eq. 6.15. The shear is given by Eq. 6.17.
(173).
Q=D$
(6.17)
The assumptions made in the derivation are: 1. Creep and plastic yielding do not occur. 2. The bolt load has been determined. 3. The lever arm of the bolt load has been computed. 4. The effect of the external moment applied to the flange, equal to the product of the bolt load and the lever arm, is independent of the location of the bolt-loading circle and of the forces balancing the bolt load. The tapered hub flange is analyzed by dividing the flange into three parts, as shown in Fig. 12.14, and considering each part as an independent unit. The loading is assumed to consist of: (1) a moment acting on the ring, so distributed that it may be replaced by an equivalent couple produced by the force Wr at the inside diameter and outside diameter of the ring, as shown in Fig. 12.15; (2) internal hydrostatic pressure acting radially on the base of the flange and axially through an assumed closure, as shown in Fig. 12.16. The effects of each loading on the flange are analyzed independently and are assumed to be linearly related in such a way that the complete solution may be obtained by superposition, as shown in Fig. 12.17. 12.7a Analysis of a Shell Connected to a Flange. A shell connected to a flange is considered to act as a beam on an elastic foundation. The deflection equation for this condition is given by Eq. 6.69, which may be written as follows: y = gz(cl sin &r + c2 cos @z)
(12.4)
Equation 12.4 has the opposite sign on the exponent /3x from that of Eq. 6.69 because of a difference in sign convention used by Waters, Rossheim, Wesstrom, and Williams (173,174). In Eq. 6.69 the distance x was taken as positive away from the junction along the shell whereas Waters et al. take x as positive in the direction of the ring. The deflection y is taken as positive in the radial direction outward in Eq. 12.4 whereas in Eq. 6.69 y was taken as positive in the radial direction inward. . The constant p has the same value as it had in Eq. 6.86. The constants cl and c2 may be evaluated by considering two of the boundary conditions at the junction.
The load is given by Eq. 6.18. d4Y u,=Ddx4 Successively differentiating Eq. 12.4 gives: 2 = Pt+[cr(sin px + cos /3x) + cz(cos px - sin /3x)] (12.5) 2 = 2P2$“[c1 co9 px -
Type-II loading of tapered hub
flange
(173).
sin Px]
c2
2 = 2P3t@[cr(sin /3x - cos @x) + cs(sin /?x + cos Bs)] $ = 4/34eSz[cr sin 82 + cs cos /3x] Therefore M, Q, and w are: M = Et3p22z[cl cos 82 - c2
6(1
sin Pxl
(12.6)
- ~‘1
Q = Et3B3eBZ[cr(sin px - cos /3x) + cs(sin ox + cos &)I
6(1 - p2) (12.7) w=
Et3/342z[cl sin /3x + cs cos @xl
(12.8)
3(1 - P2)
In Eq. 12.8, p4 is given by Eq. 6.86.
Substituting Eqs. 12.4 and 6.86 into Eq. 12.8 in terms of the shell radius, rsr gives: Et3y WCP s 2t * 2
27rr1 Fig. 12.16.
(6.18)
Fig. 12.17.
p
27r-r#
(12.9)
27r+
Combined loading of tapered hub flange
(173).
Design of Special Flanges
231
(Q+dQ)(r+dr)dB
To evaluate the constants cl and c2 the boundary conditions at the junction of the shell with the hub are applied. At z = 0 (by Eq. 12.4) (12.10)
cz = Yo
+dM,)(r+dr)dB
At z = 0 (by Eq. 12.6) M
=
0
Et=P%
6U
- ~‘1
(12.11)
cl = MO [6(;;B:1)]
The above equations for the shell cannot be solved until the corresponding relationships have been developed for the ring and the hub. A relationship 12.7b Analysis of the Ring of a Flange. giving the shape of the deflection curve of a flat plate having radial symmetry in terms of the shear, Q, and the flexural rigidity, D, is given by Eq. 6.38 with z as the axial direction instead of y.
Fig.
12.18.
Shear and moments on on element of ring (173).
of Fig. 6.2 with Fig. 12.18 indicates that these relationships must be written in the following forms: M,=D($+;$)
(12.19)
Mt= D(;$+$)
(12.20)
(6.38) or for Q = 0 (see Reference 107, p. 63) Q=-!%+“;”
rf[3$(r$)] = 0
Substituting Eqs. 12.16 and 12.17 into Eqs. 12.19 and 12.20 gives:
Differentiating with respect to r gives:
z {r$$(r$)] = 0
(12.12) M, =
D
2c5(1 + P) ln r + (3 + dc5 + 2(1 + P)C6
By dividing Eq. 12.12 by P and rearranging it, we can write it as:
(~+~~)($+~~)I=0
(12.13)
Waters, Rossheim, Wesstrom, and Williams (173, 174) have shown that Eq. 12.12, derived for the specific case of Q = 0, can also be derived for the general case of Q f 0. (See Appendix A of References 173.) By four successive integrations of Eq. 12.12, we obtain:
- (1 - /J) $ Mt = D 2c5(l + cc) h r + (1 +
z=Klnr-fr2+ 4 8
(12.14)
By successive differentiation we obtain: dz - = 2cgr In r + (c5 + 2cg)r dr d2z dr2 /
,
= 2C5
ln r +
&2 2~5 2c3 -= -p + 3 dr3
3C5
+ 2~6 -
(12.16)
dMr -=
2c5(l + cc) +x(1
r
(12.17)
2
(12.18)
and shear relationships for a circular flat e given by Eq. 6.28, 6.29 and 6.36. A comparison
(12.22) Phi
1
(12.23)
- c0c3
r3
1
(12.24)
Substituting Eqs. 12.22, 12.23, and 12.24 into Eq. 12.21 gives: (12.25)
But
therefore
Q = (total load)W _ - W circumference 27rr Wl c5 = go
c3
2(1 +
Q+!
(12.15)
1
Differentiating Eq. 12.22 with respect to r gives:
- 2c) Letting f = ~5, and @co 8 = ~6, we obtain:
2 = cgr2 In r + c+jr2 + c3 In r + c4
3P)cs +
+ (1 - /J) s
dr
2
,
(12.21)
(12.26)
where WI = equivalent bolt load or total force applied at the outside diameter of the ring, and (oppositely) at the inside diameter of the ring, which multiplied by the radial breadth of the ring equals the total moment loading on the ring, pounds
The constant c4 is obtained from Eq. 12.15 by noting that z = 0 at r = r1, and therefore c4
= -c5r12
In
rl - c6q2
- c3
In
r1
Substituting the values of ~3, ~5, and cg given by Eqs. 12.30, 12.26, and 12.29, respectively, into Eqs. 12.22 and 12.23 for the case r = r 1 gives: Fig.
12.19.
MT,
Segment of a tapered hub (173).
Q = shear on o unit sector of hub, at any point, pounds per inch.
K2 - 1
= -
Sub-
scripts 0 and 1 some os for M,,. T = Hoop tension, pounds per linear inch of axial hub length.
(12.31)
Mn = moment on o unit sector of hub, at ony point, inch pounds per inch. Subscripts 0 and 1 refer to this moment at the small and large ends
Mt, =
of the hub, respectively.
In reference to Fig. 12.18, where t.he angle B is the slope of the middle surface at the inner edge of the ring, the deflection, slope, and moment at any point may be expressed in terms of ~9, the loading, the ring dimensions, and the elastic constants. For a small angle of rotation (9 = dz dr
(12.27)
Substitution of Eqs. 12.26 and 12.16 int,o Eq. 12.27 gives: ~=~(2lnr~+l)+2ryr+~( 1 2 . 2 8 ) Solving for cg gives:
Examination of Fig. 12.14 indicates that the monlrut on the ring at ~2 is equal to zero. Substituting Eq. 12.26 for (‘5 and Eq. 12.29 for c6 in Eq. 12.22 for the condition of P = r2 gives: MC9 = 0 = iyil [2(1 + p) In 7i-
r z + (3 +
p)] + 2(1
K-2 + I
(12.32) 12.7~ Analysis of the Hub of a Flange. In the analysis of the hub the assumption is made that the stresses and deformations are the same as those for a beam with a varying section and on an elastic foundation. In this case the beam is represented by a longitudinal strip of the hub of unit width. The unit dimesion is taken at the inner surface where r = rl. Consider a segment of a transverse section of the hub like the one in Fig. 12.19. A summation of the forces in the radial plane must equal zero for equilibrium to exist; therefore
or rldQ
-
Force r d2 = hoop stress = E 2 .\rea = g~rl dx
- p)
dV 41 -=.-7 =dx
r1
(12.33)
sdx = 0
r12
(see Eq. 6.82)
Y
Taking a summation of moments in the radial plane gives: Regrouping gives:
(.ZIf,
+ t/,Vh)(r1)
dr#~
- Mh(rl)
d4 + Qrl(dz)
d4 = 0
0 = iyh 2(1 + p) In ‘? + 2 n- [ rl 1
(12.35) For a beam of infinite breadth (see Eq. 6.16) (6.16)
r-2’ (12.30) I[--~(1 + PW’---~+ (1 + PI 1 where K = ‘2 r1
Difl’erentiatiug Eq. 6.16 once and substituting Eq. 12.35, differentiating again and substituting Eq. 12.34 gives: d2
>j
(12.36)
233
Design of Special Flanges
Equation 12.36 can be written in dimensionless form.
1
d2 (1 + CYj)3 $ + $(l + Cxj)w = 0 dj2
The moments at either end and the shear at the shell end are given by:
(12.37)
2 where j = - = dimensionless axial dist.ance along hub
h
CY= taper factor for hub = @-ILqd go w = dimensionless radial displacement of hub or shell
ZIh 0 = Ego3rlA o 12(1 - p2)h2
(12.41)
,Mh = &0~(1 + a13rlAl I 12(1 - p2)h2
(12.42)
Q o = -
M
go = shell thickness, inches
91 = maximum hub thickness, inches 9= intermediate hub thickness, inches h = hub length, inches
(12.41a)
hfh = (1 - + d3-Qoh2A1 .--__ L rlfi
(12.42a)
Here only the first boundary condition is known (zero) and the rest are unknown. Let
1
d2w define the curvature factor at the large A1 = _ dj2 I 1 end of the hub when z = h and j = 1 d3w define the shear factor at the small end B. = 7 dJ3 I o of the hub when x = 0 and j = 0
h,
Q. = _ .%oWaAo + Bo) rl+
1. The radial displacement at the large end. 2. The moment at the large end. 3. The moment at the small end. 4. The shear at the small end.
define d2w the curvature factor at the small 0dj2 a end of the hub when x = 0 and j = 0
= Egoh2Ao
~-4
9=
Equation 12.37 may be solved in one of three ways: (1) by an exact solution of the differential equation, with g as a variable, which will give a solution in terms of Bessel functions; (2) by writing the total energy of the system as a function of the deflections and minimizing the total energy; (3) by writing the strain energy as a function of the loads and determining the deflection at any point. Waters, Rossheim, Wesstrom, and Williams (173) using the strainenergy method derived an approximate solution. In the strain-energy method three parameters, al, ~2, and us, are selected and so related that if al is used alone, a first approximation is obtained with all boundary conditions satisfied. Similarly, if al and us are used together, or al, us, and us are used together, second and third approximations, respectively, are obtained with all boundary conditions satisfied. There are four boundary conditions to be satisfied; therefore the solution involves a fourth-order equation. The four boundary conditions for the hub that are to be specified are :
(12.38)
(12.39)
(12.40)
(12.43)
12(1 - /.@I3
or in terms of the huh modulus, $, by:
at any point, r rl
A
Erlgo3@aAo + Bo)
(12.43a)
Waters et al. (173) have shown that w can be written as a polynomial in powers of j with al, us, and us, and Ao, AI, and B. appearing in the coefficients as follows: w = (1 - j)ur + (j - $j4 + $j5)u2 + (j” - Sj’ + +j6)a3
- (&j - ij-j” + &j4)Ao - (hzj - &$)A1 - (&j - ijj” + &j4)B0 ( 1 2 . 4 4 ) The known boundary condition is applied to Eq. 12.44. If j = 1, we obtain: w = 0. This satisfies this condition. Also, if Eq. 12.44 is differentiated and j = 0 and j = 1 are substituted, the three boundary conditions given by Eqs. 12.38, 12.39, and 12.40 are satisfied. The parametersu2and u3 represent successive approximations and may be dropped without affecting the validity of Eq. 12.44 with regard to boundary conditions. In solving Eq. 12.44 use was made of the total energy of the hub, that is, the sum of the energy of bending, Ur; the energy of stretching, U2; the external energy of rotation, Us; and the external energy of translation, Ud. The sum of these energies after deformation, lJtotsl = U1 + lJ2 + U3 + U4, must be a minimum at equilibrium or (dUtot,l)/(du,) = 0. This condition approximately s&&es Eq. 12.37. This step permits a solution of Eq. 12.37 in terms of three unknown constants of integration, the fourth being zero (wr = 0). Three equations result. ~11~1
+ ~12~2
+ ~13~3
= ~14.40
+ c15A1
+ c16Bo
c21al
+ c22a2
+ c23u3
= c24AO
+ C25Al
+ c26BO
C31a1
+ c32a2
+ C33a3
= c34Ao
+ c35A1
+ C36BO
Waters et al. (173) have tabulated the solutions for the constants in the above equations and have presented curves for the determination of al, us, and us in terms of Ao, Al, and Bo. With the boundary conditions fixed, the values for al, us, and us may now be computed, and the quantities Ao, A:, and Bo determined. These determinations in turn permit the determination of the deflection, slope, and moment of
234
Design
of
Flanges
any point of the hub. If the hub is free at the small end (loose flange) A0 = Bo = 0.
An expression for Qr may be obtained by integrating Eq. 12.34:
12.7d Relationships for the Hub, Shell, and Ring When Combined. The relationships for each part of the flange
(shell, ring, and hub) have been developed independently with undetermined boundary conditions. With the parts assembled, the adjacent parts have common boundary conditions, and in addition certain conditions are known at the free boundaries. According to the theory used in developing the hub relationships, two of the constants of integration disappear for the shell condition of constant thickness, and Eq. 12.4 may be solved with: (See Eq. 12.37 for definitions.)
(12.45)
As stated above, the displacements, slopes, moments, and shears are identical on either side of the shell-hub interface, and four equations exist by which cl, ~2, Ao, and Bo may be expressed in terms of AI. With these relationships the slope 0, the moment Mh,, and the shear Qr at the large end of the hub can be expressed in terms of Al. For the shelland-hub junction, from Eq. 12.44 and its derivatives: at x = 0 and j = 0, the deflection is: (12.46)
yo = c2 = wrl = alrl
Bo 1 (-
the slope is: dy
-
z z=o = =
12
P(Cl
+
- al +
a2
- T%AO
- &AI
Pl
h
(12.47)
c2)
w dj + QO Qo is given by Eq. 12.43a. Q1 z !&oh
Therefore
' (1 + aj)w dj _ 3aAo++ BO
r1
1
Substituting Eq. 12.44 for w and integrating with substitution of the limits as indicated gives:
QI
=~[(;+++(;+$2+($j+&)as
-(&+;+y)“o-(;+;)A1
- & + ; + ; Bo >
(12.52)
There now exist four equations, 12.31, 12.50, 12.51, and 12.52, containing the four unknowns 0, Mr,, Al, and Qr. The solution of these four equations is the key to the design of the flange. Inspection of Eq. 12.52 for Qr and Eq. 12.50 for 0 indicates that both Qr and 0 are defined in terms of known dimensions, the loading factors A,,, Bs, and Al, and the parameters al, ~2, and u3. The loading factors and the parameters can be expressed in terms of cr and #. Therefore, if Eq. 12.52 and 12.50 are divided by Al and substitutions are made for parameters, the loading factors are obtained.
the shear is: - F (3aAo + B,,) = 6(fg$) (cl - ~2) (12.48). the moment is:
-Go3P2c1 W - p2)
(12.49)
The parameters al, a~, and a3 and constants ~5, cg, Ao, and Bo can all be expressed in terms of A 1 at the junction of the hub and ring. On either side of this junction the slopes and moments are equal, respectively, and the displacement is assumed to be negligible. The ring acts on the hub, producing a moment Mh,, with the additional moment of ->$Qrt resulting from the shear Qr. Therefore the deflection and the moment with respect to the intersection of the inner surface of the ring and its midpoint are: Slope = 0 = (--al - &22 - 5~13 + 4 A0 + +A1 + &BO) ; (12.50) Moment = Mr, = Mh, - fQlt Al - +Q,t (12.51) >
The two factors F and V of Eqs. 12.53 and 12.54 are functions solely of cx and #. By definition #=
12(1 - p2)h4 r12g02
therefore ti ____ = 3(1 - pZ)
(12.55)
Therefore F and V of Eq. 12.53 and 12.54 may be plotted with the group 2h/&go as a parameter. Also, F and V are functions of CZ’, by definition cy = (gr - ge)/go, which is a function of gr/go. Figure 12.20 shows plots of F and V as functions of the groups indicated above. The next step in the solution is the determination of the value of Al and the evaluation of the three remaining variables by substitution. To derive the expression for Al, Eqs. 12.51 and 12.31 are divided by Al, and M,,/Al is eliminated from the resulting equations. Substituting for
Design of Special Flanges
0.6 I
I
/IIIIII h Gl
0.5
1.1 D
1.5
2.0
2.5
3.0
3.5
2.5
3.0
3.5
4.0
4.5
5.0
g, kl 0.6
0.5
0.4
Y
0.3
0.2
0.1
0 1.0
1.5
2.0
4.0
g1’go Fig.
12.20.
Values of F and V for integral flanges (188).
(Courtesy of Taylor Forge and Pipe Works.)
4.5
5.0
236
Design 4
D and
J
of
Flanges
Et3 ti their respective equivalents, 12(1 - $) 30 - P2)
This equation may- be rewritten in t,he following dimensionless form :
and __G&, gives : Also 9 = 90 + E (sl (12.56)
=
90)
.90U
+ 4
(12.63)
From Eqs. 12.62, 6.16, and 12.63:
d2w E900 jh = -. L--(1 + a.;) 2(1 - p2)h2 dj2 7
To obtain the maximum stress, set djh/dj equal to 0. Therefore
This equation may be written as:
d3w ,lja
d2w dj2
(1 + Cxj) -- + Ly - = 0
where
M = !% B
(12.58)
1
x= f
(12.64,
(12.65)
By substituting the second and third derivatives of Eq. 12.44 into Eq. 12.65. a fourt.h-degree equation is obtained.
&Aj“ + (A + &B)j3 + (B + &C)j2 + (C + 2aD)j + (D + aE) = 0
(12.66)
(12.59) where A = 96 3
AI
B = 90Tl - 1082 C =
-60:+24?-2$+2-22 3
D = B. U=
(12.61)
ET2 Here again all the constants can be evaluated in terms ot
The above equations are too cumbersome for use in design. Therefore they were reduced by Waters, Rossheim, Wesstrom, and Williams to those for the three critical stresses (173). A study of the stress distribution shows that for the tapered hub flange (Fig. 12.14) the critical stresses are: (1) the radial and hoop stresses at the inside diameter of the ring and (2) the axial hub stress at the outer surface of the hub and at the large end of the hub for a hub with little taper or at the small end of the hub for a hub with a large taper. In the second case the critical stress may be displaced into the shell for a very rigid hub. 12.7e locating the Critical Section in the Hub. For a hub whose section modulus is variable,
jh=(~)(;)=(~)@)
(12.62)
Al, which is in turn a function of the hub quantities a! and
#. The roots of Eq. 12.66, therefore, represent values of j or X at which the axial stress reaches a maximum in the hub. The corresponding maximum stress could be evaluated by substituting the roots into Eq. 12.64; obviously~ roots outside of the region 0 _< j 5 1 are meaningless and should be disregarded. From the above operation it is found that the maximum axial st,ress usually occurs eit,her at, one end of the hub or at the other. In solving Eq. 12.42 for Mh,, Eq. 12.57 may be substitut.etl for A1 to give: M
_ g h, -
6
(12.6i)
Therefore the axial hub stress, jh,, at the inner surface of the large end is given by (12.68)
from Eq. 6.1.6
d2y
Mdl - p2)
d;c2
=
EI
The maximum axial stress, Jo at the outer surface of the hub or shell can be related to the corresponding stress,
Design of Special Flanges
237
25
J
I
Values
Fig.
12.21.
tion
factor, f ’
Taylor
of stress con
(188).
(Courtesy
Forg e a n d P i p e Works.)
3 2.5
f’ = 1 (minimum)
f’ = 1 for hubs of uniform thickness (gl/go = 1) f’ = 1 for loose hubbed
I
fr‘,, at the inner surface by the factory, or
fH = j’? fhj(max) where f’ = fH - or __ fh,
/ 1
fh,
(12.69)
Similarly, the maximum tangential stress at the inside diameter of the ring is determined from Ey. 12.57, 12.53, 12.32, and 12.32, giving:
1
(12.71)
(12.72)
fT = F - ZjR
(whichever is greater) (12.70)
In the case wheref’ = 1, the maximum stress isfH and is kated at the junction of hub and ring. Values off’ were determined by Waters, Rossheim, Wesstrom, and Williams (173) and are shown in Fig. 12.21 as functions of the hub quantities (h/ho) and (gl/go). The maximum radial stress occurs at the inside diameter :,f the ring. The moment MT1 is determined by substitution of Eqs. 12.57 and 12.53 into Eq. 12.51. The extreme fiber 41 ressis calculated from the moment M,, and is added to the radial stress due to Q1, giving:
flanges
where X = X in Eq. 12.59 M = M in Eq. 12.58 F = F in Fig. 12.20 (1 - p) + 6 (1 + F)$!~~~ In K
a
y=KZ+l J
K2 - 1
1
(12.73) (12.74)
Values of T, U, Y, and Z are presented in Fig. 12.22 for relationships in which p = 0.3. 12.7f Reduction to the Other Types of Flanges. The previous relationships apply to the general case of an integral
238
Design of Flanges
10 9 8 7 6
2
K=$
1 1.02
1.03
1.04 Fig. 12.22.
1.06
1.08 1.10
1.20
1.30
Valuer of T, U, Y, and Z when p = 0.3 (188). T=
U=
K2(1 + 8.55246 loglo
3.GO
4.00 5.00
(Courtesy of Taylor Forge and Pipe Works.)
K) - 1
(1.04720 + 1.9448K2)(K K*(l
1.80 2.00
- 1)
+ 8.55246 log lo K) - 1 1.36136(K2 1
Y = __ K - 1
- l)(K - 1) K2 log,0 K
0.66845 + 5.71690 ~K2 - I
1
z=K2f K2 - 1
Poisson’s ratio assumed = 0.3
flange composed of a ring, a tapered hub, and a shell of uniform thickness extending indefinitely beyond the hub. These relationships can be modified for special cases, such as the case of loose flanges. The ring and hub of a loose flange (see Fig. 12.13) are independent of the shell and are not subject to internal hydrostatic pressure. The curvature factor, Ao, and the shear factor, Be (see Eqs. 12.38 and 12.40) are both equal to zero, and a different
set of F and V values must be used. These values, denoted as FL and VL, are shown in Fig. 12.23. Strain measurements on hubs of loose flanges have indicated that the maximum bending stress always occurs at the large end of the hub. ThereforeS’ = 1.0 for loose flanges. The critical-stress equations for loose flanges (based on FL and VL) are: fH =
x$
(12.75)
239
Design of Special Flanges
1 + ,.,,*a > t2
j~=xM
By substituting Eq. 12.81 and the relationships M = MO/B and e = F/ho into Eqs. 12.69, 12.71, and 12.72, and into Eqs. 12.75 and 12.76 for the loose flange, the following equations giving the desired stresses are obtained. For integral-type jlanges and all hubbed jlanges:
(12.76) (12.77)
jT = F - ZjR
Longitudinal hub stress, jH = =$$
For the extreme case of a loose flange without a hub, the stress equations become: fff = 0
(12.78)
fR = 0
(12.79)
fT
= y
(12.82)
Radial flange stress, jn = csteL;2;‘Mo
(12.83)
Tangential flange stress, jr = YMo t26 - ZjR
(12.84)
(12.80) For ring jlanges of the loose type:
12.7g
Flange-Stress Code. In the
Equations
Used
in
the
Pressure-
vessel flange equations previously developed, the quantity X or its reciprocal, L, may be expressed in terms of other factors. (See Eq. 12.59.) This may be performed by noting that he = d&o and by letting the term d equal (U/V)hOge2 and the term e equal F/ho. By combining these factors the quantity L may be defined as the reciprocal of X, or
L+!+Pd
fT =
z
.fR = 0
(12.863
.fH = 0
(12.87)
Typical flanges permitted by the ASME code (11) are shown in Fig. 12.24. Table 12.4 lists the nomenclature for Fig. 12.24 and the code equations for flange design.
(12.81)
100 80 60 40 30
10 8 6 4 3 v, 2
6 5 FL
4
0.:
1 l7-
0.4 0.3 0.2 0.:
0.8 0.7 0.6 0.5
0% 0.06 0.04 0.03 0.02 1.0
Fig.
12.23.
FL , VL, valuer for loose
Aanges (1881 I.
1.5
2.0
(Courtesy of Taylor Forge and Pipe Works.)
3.0
4.0
5.0
Design of Flanges
240
Loose-type flanges loadings and dimensions not shown are the same as in sketch 2
O.lt,(min) fhveted or s c r e w e d f l a n g e with or without hub.
To b e t a k e n a t m i d p o i n t o f contact between flange and lap, independent 01 gasket location. (1)
f o r h u b t a p e r s 6O o r less. use g, = g,.
(2)
(4) Integral-type flanges p h 1 ~l.%(min)
I
W h e r e h u b slope adjacant t o f l a n g e excetdr 1: 3. usa d e t a i l 6a o r 6 b .
0.25&r, b u t n o t less t h a n % i n . , t h e m i n i m u m for e i t h e r l e g . T h i s weld m a y b e m a c h i n e d t o a c o r n e r r a d i u s a s p e r m i t t e d i n s k e t c h 5. in which case g, = go.
go
(6)
(7) Optional-type flanges These may be calculated as either loose-or Integral-type. L o a d i n g s a n d d i m e n s i o n s n o t s h o w n a r e t h e s a m e a s i n 2 f o r l o o s e - t y p e f l a n g e s o r in 7 for i n t e g r a l - t y p e t
I
:,+ k” (m
f u l l p e n e t r a t i o n &d backchtp
@a)
(8) Fig.
12.24.
Vessels,
v
with
T y p e s o f f l a n g e s p e r m i t t e d b y t h e ASME C o d e . permission
of
the
publisher,
the
American
Society
@b)
(Extracted from the 1956 edition of the ASME Boiler ond Pressure Vessel Code, of
Mechanical
Two bolt loads exist: that developed by tightening up the bolts, Wm2, and that which exists under the operating conditions, W,l. The bolt load for the tightening-up condition must exert sufficient force, H,, on the gasket to cause yielding of the gasket in order to produce a tight joint. This load is equal to the effective area of the gasket times the gasket yield stress, or (12.88)
The bolt load under the operating condition cor1sist.s of the force necessary to resist the internal pressure and to The internal preskeep the gasket tight during operation. sure produces an end force, H, given by: H = ;G2p
Engineers
Unfired Pressure
[l 11.)
given by:
Code rules .for desigr~ ing flanges: Bolt loads:
Wm2 = H, = rbG 34 in., G = outside diameter of gasket contact face minus 2b, inches. = thickness of hub at small end, inches. = thickness of hub at back of flange, inches. = total hydrostatic end force, pounds, = 0.785G’p. = hydrostatic end force on area inside of flange =
0.785B2p.
hydrostatic end force on area inside of flange, pounds, = H - HO. total joint-contact-surface sea tinp load, pounds. hub length, inches. radial distance from bolt cir& to circle on which HD acts. radial distance from gasket-load react.ion to t,otI circle, inches, = (C - G)/Z. factor = z/Bgo, inches. c radial distance from bolt circle to circle on which HT acts. ratio of outside diameter of flange to inside diameter of flange = A/B. factorJe;l
f
;.
”
5
tq+
+$
H&T.
inchlpounds.
m iv
= gasket factor; obtain from Fig. 12.11. = possible contact width of gasket, inches (see Fig.
P
= maximum allowable working pressure, pounds
R
= factor: for integral-type flanges e = f
pounds.
= moment under bolting-up conditions, inch-pounds = WhG. Mmax = maximum moment, greater of MO or Maj~o/j~a,
M,
for loose-type flanges d = $ hogo
e
241
fb ff “fH fR .fT
T t u V
12.12).
per square inch. = radial distance from bolt circle to point of intersection of hub and back of flange, inches (integral and hubbed flanges) (see Table 10.4). = maximum allowable bolt stress at atmospheric temperature, pounds per square inch. = maximum allowable bolt stress at operating temperature, pounds per square inch. = maximum allowable design stress for flange material or nozzle neck, pounds per square inch. = longitudinal stress in hub, pounds per square inch. = radial stress in flange, pounds per square inch. = tangential stress in flange, pounds per square inch. = factor involving K; obtain from Fig. 12.22. = flange thickness, inches. = factor involving K; obtain from Fig. 12.22. = factor for integral-type flanges; obtain from Fig. 12.20.
VL
W Wrnl
w,, Y Y 2
= factor for loose-type flanges; obtain from Fig. 12.23. = flange-design bolt load, pounds. operating or work= required bolt load for maxirnuru ing conditions, pounds (see Eq. 12.91). = required initial bolt load at, atmospheric-tempt+ ature conditions without. internal pressure, po~r~tds (see Eq. 12.88). = fact.or involving Zi; &lain from Fig. 12.22. = gasket or joint-contact-surface unit seating l(jiid. pounds per square inch (see Fig. 12.11). = factor involving K; obtain l’rom Fig. 12.22.
’
242
Design of Flanges
The procedure for determining the actual bolt size was presented in section 12.6. The actual bolt area provided usually exceeds the minimum required bolt area because an integral number, usually a multiple of four, is used. The excess bolting area may result in overstressing of the flange in the bolting-up operation. To provide a margin of safety against such overstressing, the code specifies that the design load, W, for the bolting-up condition be based on the average of the minimum and the actual bolting areas, or
For the operating condition w =
W,l
(See Eq. 12.91.)
(12.95)
Flange moments: The various axial forces on the flange produce bending moments. The summation of moments is taken about the bolting axis. For flanges classified as the integral type, the total moment must be at least equal to the sum of the moments acting upon the flange, or: Lever Arms =
Flange Loads X Ho = 0.785B2p
hD=R+e 2
Moments MD = H D X
ho
(12.96)
HT = H - HD
h
T
=R+g,+h, 2
MT = H T X
hT
(12.97)
Ho=W-H
,, =C-G -
G
2
Mo = Ho X ho (12.98)
and total moment,
MO = MD + MT + Mo
(12.99)
In the case of loose-type flanges in which the flange bears directly on the gasket, the force HD is considered to act on the inside diameter of the flange and on the gasket load at the center line of the gasket face. The lever arms for the momeuts are:
responding to these types are designed as integral flanges if any of the following values are exceeded: g0
> Q in.
B
-. > 300
90 p > 300 psi Operating temperature > 700” F If the operating conditions and proportions are such that none of the above limits are exceeded, flanges corresponding to sketches 8 through 9 of Fig. 12.24 may be designed as loose-type flanges. In this case the minimum flange thickness may be calculated by use of Eq. 12.85 rewritten in the following form:
t = 2/O’M,ax)/UB)
(12.85)
To illustrate the design procedure, a ring-type flange with a plain face for a heat-exchanger shell will be designed to the following specifications: Design pressure Design temperature Flange material Bolting steel Gasket material Shell outside diameter Shell thickness Shell inside diameter Allowable stress of flange material Allowable stress of bolting material Flange type
= 150 psi = 300” F = ASTM A-201, grade B = ASTM A-193, grade B-7 = asbestos composition = 31 in. = B = x in. = 30>/4 in. = 15,000 = 20,000
See sketch 8 of Fig. 12.24.
Calculation of gasket width-by Eq. 12.2: do z=
y-pm Y
- P(m + 1)
(12.100)
Assuming a gasket thickness of ~1~ in., from Fig. 12.11 we find : y = 3700
(12.101)
therefore
m = 2.75 h G=C-G 2
(12.102)
The lever arms given in Eqs. 12.100, 12.101, and 12.102 apply to the flanges shown in sketches 2, 3, and 4 of Fig. 12.24 and also to those in sketches 8 through 9 when these flanges are calculated as loose-type flanges. In the case of the lap-joint flange of sketch 1 of Fig. 12.24, the lever arm ha is given by Eq. 12.100, and lever arms ho and hT are ,identical and are given by Eq. 12.101. ’
12.7h
Design Procedure for Ring Flanges (LooseType).
Ring flanges are widely used because of their simplicity and their ease of fabrication. Some typical ring flanges are shown in sketches 7 through 9 of Fig. 12.24. Flanges cor-
d o
3 7 0 0 - (150)(2.75)
= 1 021
z=
3700 - (150)(3.75)
.
Assume that di of the gasket equals 32% in.; then do = (1.021)(32.75) = 33.5 in. Minimum gasket width =
33.5 ; 32.75)
= !
/
8
Therefore, use a >4 MO 60,000 (14) . . Pipes and Tubes Seamlese Carbon steela 48,000 (4)(6) 12,000 SA-53 A ... SA-53 B ... 60,000 (4)(6) 15,000 . . . (4)(6) 11.750 SA-83 A 48.000 :.: 12.000 SA-106 A ... SA-106 B ... 60.000 _.. 15.000 SA-192 . . . ... . . 11:750 . . . 60,000 15,000 SA-210 . . . . 55.000 (4)(6) 13,750 SA-333 C 1 55,000 (4)(6) 13,750 SA-334 C Seamlees Low-allov steeLn C-+GMo 3 55,000 13.750 SA-209 Tl C+6Mo 3 60,000 15,000 SA-209 Tla C--WMo 3 53.000 13.250 SA-209 Tlb L4g Cr-0.7 MO 60;OO0 15.000 SA-213 T3 5Cr-$5 MO 60,000 (14) SA-213 T5 7 Cr--$4 Ma 60,000 ( 1 4 ) SA-213 T7 9Cr-1 MO 60,000 (1s ,.. SA-213 T9 l)i (k-35 Mo-Si 60.000 . 15,000 SA-213 Tll 1 Cr--45 MO 60:000 15,000 SA-213 T12 5Cr--)4 Mo-Si 60,000 SA-213 T5b (14) . . 2 Cr-54 MO 60,000 15.000 SA-213 T3b 50-36 Mo-Ti 60,000 (14) :. SA-213 T5c lCr-v 60,000 (5) 15,000 SA-213 TL7 60.000 15.000 SA-213 T21 3 Cr-0.9 MO 2$i Cr-L MO 60,000 . . 15.000 SA-213 T22 65,000 16.250 SA-333 3 ;S&Ni 9 65,000 16,250 SA-333 5 65,000 SA-334 3 9 . 16.250 ;S&Ni 9 65.000 16.250 SA-334 5 C-?JMo SA-335 Pl SA-335 P2 34 G--W MO 15;ooo SA-335 P3 1% Cr-0.7 Mo 2 CT-j.4 MO 4 60,000 15,000 SA-335 P3b SCr--54 MO 5 60.000 . SA-335 P5 60;OO0 SA-335 P5b 50-g Mo-Si 5 Cr--)d Mo-Ti 60,000 SA-335 P5c or Cb 60.000 15.000 SA-335 PLL L>i 0-36 Mo-Si 15.000 LCr+B MO 60,000 SA-335 P12 134 Si-$6 MO 60,000 15;000 SA-335 PL5 60,000 SA-335 P7 7 Cr--fh MO (14) SA-335 P9 9Cr-LMo 60.000 (14) 3 0-0.9 MO 6O;OOO 15.000 SA-335 P2L
For Metal Temperatures Not Exceeding Deg F 700
750
800
850
13,250 13,250 11.650 .
12,050 12.050 10.700
10,200 10,200 9300
8350 8350 7900
900
...
950
,
6500 6500
.
.
13,250 14.350 15;soo 16,600
12.050 12.950 13,850 14,750
... ... . .
... ... .
11,000 12.100 13,250 17.700
10.250 11,150 12,050 15.650 .
10,200 10,800 11,400 12,000 . ... ... . 9000 9600 10.200 12,600 .
8350 8650 8950 9250 . . ... ... . 7750 8050 8350 9550 . . 9550 9900 8950 9250 9550 14,400 15,000 15,900
6500 6500 6500 6500
6500 6500 6500 6500
12.600 12,800 11,400 12.000 12:600 15,650 16.900 18.000 .
13,400
13,100
12.800
11,650 14,350 11,450 11.650 14,350 11,450 14,350
10.700 12.950 10:550 10,740 12,950 10,550 12,950
9300 10,800 9200 9300 10.800 d200 10.800
7900 8650 7850 7900 8650 7850 8650 . .
6500 6500 6500 6500 6500 6500 6500
13,750 15,000 13.250 15,000 13.400 13;400 13.400 15,000 15,000 13,400 15,000 13,400
13,750 15,000 13.250 15,000 13,100 13.100 13,100 15,000 15,000 13,100 15,000 13.100
13,450 14.400 15;ooo 15.000 12,800 12,500 12,800 15.000 14,750 12.800 14;700 12.800
13,150 13,750 12,750 14,400 12,400 11,500 12.500 14.400 14.200 12.400 14,000 12,400
12.500 12,500 12.500 13.100 11.500 9500 12.000 13,100 13.100 10,900 12.500 11,500
14,800 15,000 ... ... ... _..
14,500 15,000 ... ... ...
13,900 15,000
13.200 14,400
... ...
. . .
15;000 14.700 12,800 12.800 12,800
13,150 13,150 14,400 14,000 12,400 12,400 12,400
12,000 13,100 ... . ... ... 12,500 12.500 13.100 12,500 11,500 10,900 11,500
15,000 15,000 15.000 15.000 15.000 14.750 L5;ooo 15;ooo 14;400 13,400 13,100 12,500 13.400 13,100 12.800 14,800 14,500 13,900
14,400 14,200 13.750 11,500 12.500 13,200
13,100 11,000 13.100 11.000 12;500 10;000 9500 7000 Li.000 10,800 12,000 9000
15;000 15,000 13,400 13,400 13,400
251
L5;ooo 15,000 13.100 13.100 13,100
6500 6500 6500 6500 6500 12,500 12,750 13.000 ...
... ... . . . ...
2500 r... 2500 . . . 2500 . . . 2500 . . . 2500 . . . 6250 ... 6250 ... 6250 . . . ... . ... 6250 .._ 7500 5000 6250 ,.. 6250 ._,
14,400 12,500 14.200 13.100 L$9QO 13;ooo 16.800 13,250 ... 12,400 11.500
...
... ... . . ... . . ...
... ... ... ... ... ... . . ... ... ... ... ... . ... 2500 . . . ...
15.650 17,700 13;850 14.750 15:650 16.250 17.500 18;750 17.500 18:750 16,250 15,000 18,750 20,000
15,650 14.750 18;OO0 19,100
1050
2500 2500 2500 2500
17,700 19,800 15,500 16.600 17;700 16.250 17,500 18,750 17,500 18,750 16,250 15.000 18;750 20,000
.
1000
1100
.
.
.
.
.
.
.
.
.
.
.
I
.
.
.
.
,
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
_.. 2800 ... .._
7300 5200 3300
......... ......... .........
1150
1200
... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...... 1550 1000 ... ... ... ... ... ... 2200 1500
2 5 0 0 ...... 2500 ...... 2500 ...... 2500 ...... ......... .........
... ... ... ... ... ... ... ... ...
... ... ... ... ... ... ... ... ...
10,000 10.000 10.000 11,000 10,000 7000 10.800 11,000 11,000 9000 10,000 10,000
6250 6250 6250 . 7800 7300 5000 8500 7800 7500 5500 6200 7300
...... ...... . . ... 5500 4000 5200 3300 3500 2500 5500 3300 5500 4000 5000 2800 3500 2500 4200 2750 4800 2800
... ... ... 2500 2200 1800 2200 2500 1550 1800 1750 1800
... ...
9000 11,000 ...
7000 7800
5500 5800
2 7 0 0 1500 3000 2000
4500 4500 4500 4500
. .
... 10.000 10,000 11,000 10,000 10,000 9000 10.000
. . . .
4000 4200
. ... . . . .
.
. ,.. ... ,.. . .
1200 1500 1200 1500 1200 1000 1200 1200 1200
.
.
.
.
.
.
.
I
.
.
6250 6250 7800 6200 7300 5500 7300
5500 4200 5200 3500 4800
4000 2 5 0 0 2750 1750 3300 2200 2 5 0 0 1800 2800 1800
7800 7500 6250 5000 8500 7000
5500 5000 ... 3500 5500 5500
4000 2500 1200 2800 1550 1000 ,,_ ,.. 2500 1800 1200 3300 2200 1500 4000 2700 1500
.
1200 1200 1500 1200 1200
Table From the 1956
ASME
Material end Spa% NOIIliId tkation Number Grade Composition SA-335 P22 2$i C r - 1 MO S A - 3 6 9 FPl C-44 MO ja G-w MO SA-369 FP2 SA-369 FP3b 2 CT--Ji MO S A - 3 6 9 FPll lj/, Cr-36 M o - S i SA-369 FP12 1 Cr--)d MO SA-369 FP21 3 Cr-1 MO S A - 3 6 9 F P 2 2 2 % C r - 1 MO SA-369 FP5 5.cr-$( M O 7 Cr-35 MO S A - 3 6 9 FP7 SA 369 FPY 9 Cr-1 MO F0rghgs Carbon Steels SA-105 I SA-105 II SA-181 I SA-181 II SA-266 I SA-266 II SA-266 III ... S A - 3 5 0 LFl ... Low-alloy Steels S A - 1 8 2 Fl C-x M O SA-182 F12 1 Cr-$6 MO SA-182 F5 5 Cr-$6 MO S A - 1 8 2 Fll l)C Cr--35 MO SA-182 F9 9 Cr-1 MO 2>/4 C r - 1 MO SA-182 F22 S A - 3 3 6 Fl C-X MO SA-336 F2 1 Cr.-$6 MO S A - 3 3 6 F5a 5 Cr-x MO SA-336 F22 Zjl, C r - 1 MO SA-350 LF3 3% Ni Castings Carbon Steels SA-95 SA-216 WCA SA-216 WCB SA-352 LCB ... Low-alloy Steels C--j6 M O SA-217 WC1 SA-217 WC4 Ni-Cr-% MO SA-217 WC5 Ni-Cr--1 MO SA-217 WC6 l$i Cr-$6 MO SA-217 WC9 2j/, C r - 1 MO SA-217 C5 5 Cr-$4 MO SA-217 Cl2 9 Cr-1 MO S A - 3 5 2 LCl ... SA-352 LC2 ... SA-352 LC3 Bolting Carbon Steels S A - 2 6 1 BO SA-307 B S A - 3 2 5 Low-ailoy Steels SA-193 B5 5 Cr--fh MO 1 Cr-0.2 MO SA-193 B7 S A - 1 9 3 B7a 1 CT-0.6 MO SA-193 B14 1 Cr-0.3 MO-V SA-193 B16 I c-15 MO-V SA-320 L7. LY ilo, 'L43 SA-354 BB SA-354 BC SA-354 BD Bars Carbon Steels SA-306 50 SA-306 55 SA-306 60
13.1.
(Continued)
Unfired-Pressure-Vessel Code with Permission of the American Society of Mechanical Engineera Spec For Metal Temperatures Not Exceeding Deg F PMin N u m - Ten-20t0 her ‘de Notes 650 700 750 800 850 900 950 1000 1050 5 mnnn 15.000 15.000 15.000 15.000 14.400 13.100 11.000 7800 5800 55.000 1 3 . 7 5 0 i3:750 1 3 . 7 5 0 i3:450 13:150 121500 1O:OOO 13,750 13;750 13,450 13,150 12,500 10,000 6250 5 5 ; o o o 13;750 60,000 15,000 15,000 15,000 14,700 14.000 12,500 10.000 6200 4200 15.000 15,000 15,000 15,000 14,400 13,100 11;000 7800 5500 60.000 15,000 15,000 14,750 14,200 13,100 11,000 7500 5000 60,000 15,000 14.800 14.500 13.900 13.200 12.000 9000 7000 5500 60,000 15,000 60,000 15,ono 15;ooo 15;ooo 15;ooo 14,400 13;100 11,000 7800 5800 13,400 13,100 12.800 12.408 11,580 10,000 7300 5200 60,000 (14) 11.500 9500 7000 60,000 (14) 13,400 13,100 12,500 5000 3500 60.000 (14) 13,400 13,100 12,800 12.500 12,000 10,800 8500 5500
i
--.---
1100 4200
1150 3000
2750 4000 2800 4000 4200 3300 2500 3300
2500 1550 2700 3000 2200 1800 2200
5000 5200 5500 5500 5800
2800 3300 4000 3300 4200
1559 2200 2500 2200 3000
1500 1200 1500 2000
5000 5200 5800
2800 3300 4200
1550 2200 3000
1000 1500 2000
5500 5800 5200 5500
4000 4200 3300 3300
.:. -~’ ... . _.. _..
_..
60.000 70,000 60,000 70,000 60,000 70,000 75,000 60.000
9
.
1 1 3
... ...
70,000 70,000 YO,OOO 70,000 100,000 70,000 70,000 70,000 80,000 80,000 70,000
. . ..
... ...
1 1 1
.:. (14) (14) (14) (14) (14) (14) .._
15,000 17,500 15.000 17.500 15,000 17,500 18,750 15,000
14.350 12,950 10,800 16.600 14.750 12.000 14;350 12:950 lo;800 16,600 14,750 12,000 14,350 12,950 10,800 16,600 14,750 12,000 17.700 15,650 12,600 . . ... ...
17,500 ... ... .
17,500 16,150 17,500 16,150 21,200 17,500 17,500 16,150 16,500 20,000
17,500 15,500 16,000 15,500 20,000 17,500 17,500 15,500 15,500 20,000
16.900 14,850 14,500 15.000 17,700 17,500 16.900 14,850 14,500 18,000
17,500 17,500 20,000 17,500
70,000 (7) (18) 6n.000 (7) 7o;ooo (7j 65,000 (7) (16)
17,500 15.000 17:500 16,250
16,600 14,350 16,600
14,750 12,950 14,750 ...
12,000 10,800 12,000 ...
65,000 70,000 70,000 70,000 70,000 90,000 90.000 65,000 65,000 65,000
16,250 17,500 17,500 17,500 17,500 22,500 22,500 16,250 16,250 16.250
16,250 17,500 17,500 17,500 17,500 21,600 22,000
16,250 17,500 17,500 17,500 17,500 20,400 21,000
15,650 17,000 17,000 17,000 17,000 19,000 19.400
(7)(8) (7)(8) (7)(8) (V(8) (7)(8) (7)(a) (7)(8) (7)(16) (7)(16) (7)(16)
8650 9250 8650 9250 8650 9250 9550
6500 6500 6500 6500 6500 6500 6500
2500 2500 2500 2500 2500 2500 2500
15,000 12,750 14.200 13.100 13,000 11,500 14.400 13.100 15;400 13;100 16,000 14,000 15,000 12,750 14,200 13,100 13,000 11,500 16,000 i4,,oon
6250 7500 7300 7800 8500 7800 6250 7500 7300 7800
9250 8650 9250 ... 14,400 15,800 15,800 15,800 15,800 17,000 17,300
12,500
17,200
15,650
.
...
(9)(10) 20,000 (9)(10) 20,000 (9)(10) 20,000 (9) (10) 20,000 (9) (10) 20,OOb (9)(15)
20,000 20,000 20,000 20,000 20,000 . .
20,000 20,000 20,000 20,000 20,000
20,000 20,000 20,000 20,000 20,000 ...
17,250 16,250 17,250 18,750 18,750 ...
(9)(10) 18,750 (9)(10) 20,000 (9)(10) 20,000
17,200 18,400 18,400
15,650 16,750 16,750
... . ...
50,000 55,000 60,000
. _..
12,500 13,750 15,000
12.500 14,000 14,000 14,000 14,000 13.600 15,000
4500 4500 4500 10,000 10,000 11,000 11,000 11,000 10,000 11,750
. 14,950
. .
6500 6500 6500
.
100,000 (Y)(lO) 16,250 55,000 (11) (9)(10) 18,750
.
. . ... ...
(4) (4) (4) (4)
10,000
. .
6900
. ... 13,750 12,500 13,750 16,650 16,650
2500 2500 2500 6250 6250 7800 7800 7800 7300 8500
... ... ...
... ...
... ... ...
... ... ...
... . 10,300 8500 10,300 14,250 14,250
4800 2750 . . . iii0 i;io : : : 6250 2750 6250’ 2750 : .: ...
... ... ... ... ...
1200 200%
... ... ...
... ... ...
. .
F r o m t h e 1 9 5 6 ASMEPntired-PressureVessel CZde w i t h P e r m i s s i o n o f t h e A m e r i c a n S o c i e t y o f M e c h a n i c a l E n g i n e e r s Notes: T h e stress virlues i n t h i s table m a y h e i n t e r p o l a t e d t o d e t e r m i n e v a l u e s f o r i n t e r m e d i a t e t e m p e r a t u r e s . . All stress values in shear are 0.80 times the values in the above table. A l l s t r e s s valwes i n b e a r i n g a r e 1 . 6 0 t i m e s t h e v a l u e s i n t h e a b o v e t a b l e . (1) See section 13.5, first paragraph. (2) Flange quality in this specification not permitted over 850° F. (3) These stress values are one fourth the specified minimum tensile strength multiplied by a quality factor of 0.92, except for SA-283, Grade D, and SA-7. (4) F o r s e r v i c e t e m p e r a t u r e s a b o v e 850° F i t i s r e c o m m e n d e d t h a t k i l l e d s t e e l s c o n t a i n i n g n o t l e s s t h a n 0 . 1 0 % residual silicon be used. Killed steels wbicb have b e e n d e o x i d i z e d w i t h l a r g e a m o u n t s o f a l u m i n u m a n d r i m m e d s t e e l s m a y h a v e c r e e p a n d s t r e s s - r u p t u r e p r o p e r t i e s i n t h e t e m p e r a t u r e r a n g e a b o v e 850’ F which are somewhat less than those on which the values in the above table are based. (5) Between temperatures of 650 and 1000” F, inclusive. the stress values for specification SA-201, Grade B. may be used until high-temperature test data become available. (6) Only killed ateel (silicon) shall be used above 900’ F. 252
Welded-Joint
EtSciencies
253
Table 13.1. (Continued) (7) T o t h e s e stress v a l u e s a q u a l i t y f a c t o r s h a l l b e a p p l i e d , a s s p e c i f i e d i n t h e c o d e . (8) These stress values apply to normalized and drawn material only. (9) These stresss values are established from a consideration of strength only and will be satinfactory for average service. F o r b o l t e d j o i n t s , w h e r e f r e e d o m from leakage over a long period of time without retightening is required, lower stress vabms may be necaasary, determined from the rnllttive flexibility of t.he flange a n d b o l t s a n d c o r r e s p o n d i n g r e l a x a t i o n p r o p e r t i e s . (10) Between temperatures of -20 and 400’ F, st,resa values equal to the lower of the following will be pnrmit.tnd: 20 ‘7 of the specified tensile strength, or 25 % of the specified yield strength. (11) Not permitted above 450~ F; allowable stress value, 7000 psi. (12) Between t,emperatures of 750 and 1000° F, inclusive, the stress values for specification S4-212, Grade R, may he used until high-twnperature test datit become available. (13) The stress values to be used for tempnraturtis helow -20” F when steels ate made to conform with specification SA-300 shall be those that are given in the column for -20 to 650” F. (14) Maximum allowable stress values for kmperetures below 700° F are giver1 in the following table: For Metal Temperatures Not Exceeding Deg F Specification Number
PNumber
- 2 0 to 400
For Metal Temoeraturea Not Exceeding Deg F Specificatioo Numhnr
PNumber
- 2 0 to 400
500 600 650 tirade ~~___.~ SA-213 T5 5 15,000 14,500 14,000 13,700 SA-355 . 15,000 SA-213 T7 5 15,000 14,500 14,000 13,700 SA-369 FP5 5 15,000 SA-213 T9 5 15,000 14.500 14,000 13,700 SA-369 FP7 5 15,000 SA-213 T5b 5 15.000 14,500 14.000 13,700 SA-369 FP9 z 15,000 SA-213 T5c 5 15,000 14.500 14.000 13,700 SA-335 P7 15,000 SA-335 P5 5 15.000 14.500 14,000 13,700 SA-335 P9 5 15,000 SA-335 P5b 5 15,000 14.500 14.000 13,700 SA-182 F12 4 17.500 SA-335 P5c 5 15,000 14;500 14,000 13,700 SA-182 F5a 5 22;500 SA-336 F2 4 17.500 17.500 17.500 16.800 SA-182 Fll 4 17.500 F5 20;ooo 19;200 18;OOO 17;300 SA-182 F9 5 25;OO0 SA-336 5 (15) For temperatures below 400’ F, stress values equal to 20 % of the specified minimum I.ensile strengt,h will be permitted. (16) See par. UCS-67(d) of Reference 11. (17) See special ruling in regard to welding with electrodes having a tensile strength less t h a n t.he b a s e m e t a l . (18) This material not suitable for welding. Grade
tion but does list a number of mandatory requirements and a number of nonmandatory suggestions for practice in design. It is expected that the designer and the fabricators will use accepted engineering practices and good judgment when dealing with those parts of the vessel not covered by code specifications. 13.5
MATERIAL
SPECIFICATIONS
Plain-carbon- and low-alloy-steel plates are usually used where service conditions permit because of the lesser costs and greater availability of these steels. Such steels may be fabricated by fusion welding and oxygen cutting if the carbon content does not exceed 0.35 %. Vessels may be fabricated of plate steels meeting the specifications of SA-7, SA-113, Grade C, and SA-283, Grades A, B, C, and D provided that (1) the vessel does not contain lethal liquids or gases, (2) the operating temperature is between --20 and 650” F, (3) the plate thickness does not exceed 46 in., (4) the steel is manufactured by the electric furnace or openhearth furnace, and (5) the material is not used for unfired steam boilers. The allowable stresses for these and other plate steels together with those for steels used for pipes, forgings, castings, and boltings are given in Table 13.1. One of the most widely used steels for general purposes in the const,ruction of pressure vessels is SA-283, Grade C. This steel has good ductility and forms, welds, and machines easily. It is also one of the most economical steels suitable for pressure vessels. However, its use is limited to vessels with plate thicknesses not exceeding 46 in. For vessels having shells of a greater thickness, SA-285, Grade C is most widely used in moderate-pressure applications. In the case of high pressures or large-diamet.cr vessels a higherstrength steel may be used t.o advantage to reduce the wall thickness. SA-212, Grade B is well suited for such applica-
500
600
650
14,500 14,500 14,500 14.560
14,000 14,000 14,000 14,000 14,000 14,000 17,500 20,100 17.500 22.700
13,700 13,700 13,700 13,700 13,700 13,700 16,800 19,000 16$00 22,nno
la;sbo
14,500 17,500 21,600 17,500 24.000
tions and requires a shell thickness of only 79’~ of that required by SA-285, Grade C. This steel also is easily fabricated but is more expensive than the other steels. The SA-283 steels cannot be used in applications with temperat.ures over 650” F; the SA-285 steels cannot be used for services with temperatures exceeding 900” F; and the SA-212 steels cannot be used at temperatures over 1000” F. However, both the SA-285 and the SA-212 steels have very low allowable stresses at the higher temperatures. Therefore, for temperatures between 650 and 1000” F, steel SA-204, which contains 0.4 to 0.6% molybdenum, is satisfactory and has good creep qualities. For low-temperature service ( - 50 to - 150” F) a nickel steel such as SA-203 may be used. The allowable stress for this steel is not specified for temperatures below -20” F. Normally, the fabricator must run impact tests to determine the applicability of the steel and its freedom from brittle fracture for low-temperature service. 13.6
WELDED-JOINT
EFFICIENCIES
The use of a welded joint may result. in a reduction in the strength of the part at or near the weld. This may be the result of metallurgical discontinuities and residual stresses. The code rules make allowance for these factors by specifying joint efficiencies for various types of welds with and without stress relief and radiographing. The designer is permitted some option in the selection of the kind of welded joint to be used and in whether or not the vessel and its parts must be stress relieved and whether or not the welded joints must be radiographed. The general thickness limitations for various types of joints are given in Table 13.2. Additional t,hicltness limitat.ions for various types of st.eels are given in the previous sect,ion, entitled Mat,erial Specifications. All vessel shells having a thickness greater,_~ than l$/, in.
254
Design of Pressure Vessels to Code Specifications Table
13.2.
Maximum
Allowable
Efficiencies
for
Arc- and Gas-welded Joints (11)
From the 1956 ASME Unfired-Pressure-Vessel Code with Permission of the American Society of Mechanical Engineers
TYPO of Joint
Limitations
MaxilIl”lIl Basic Joint The- J o i n t Effimally Efficiency, Stress ciency, R a d i o RePer Per cent graphed lieved cent
NOIll3 No No 80 Double-welded butt joint Single-welded Longitudinal joints not 80 No Yes 85 , butt joint over 1M i n . t h i c k . N o Yes No 90 thickness limitation on with backing Yes Ye8 95 strip circumferential joints. Sindewelded Circumferential iointa 70 No No 70 butt joint only, not 0~~~56 in. Yes 75 NO without backthick. ing strip Double full-fillet Longitudinal joint. not 65 No No 65 lap joint over Q$ in. thick. CirNO Yes 70 cumferential joints not over J$ in. thick. Single full-fillet Circumferential joints 60 No No 60 lap joint with only. not over FQ in. No Yea 65 plug welds thick; attachment of heads not over 24 in. in outaide diameter to shells not over 56 in. thick. Single full-fillet Only for attachment of 50 No No 50 lap joint withheads convex to presNO Yes 55 sure to shells not over out plug welds 5$ in. thick, and for attachment of heads ConCeYe to pressure not over 24 in. in outside diameter to shells not over x in. thick.
WeGhan (d + 50)/120 (where d = inside diameter or 20 in., whichever is greater) must be thermally stress relieved. Vessels of any thickness fabricated from the following low-alloy steels must be stress relieved; SA-301, Grade B; SA-302; SA-217, Grades WC4 and WC5; SA-357; SA-387, Grades B, C, D, and E; and chrome-molybdenum steel having a chrome content greater than 0.7%. Also, vessels having a shell thickness greater than 0.58 in. must be thermally stress relieved if they are fabricated of the following steels: SA-202, SA-203, SA-204, SA-225, SA-299, SA-301, Grade A, SA-387, Grade A, and any steel having a specified molybdenum content of 0.4 to 0.65 y0 and a chrome content not greater than 0.7a/ Also, steels greater than 1 in. in thickness must be stress relieved if they meet the specifications of the following: SA-212, SA-105, Grade II, SA-181, Grade II, SA-266, Grade II, SA-95, and SA-216, Grade WCB. If high-alloy steels are used, stress relieving is not required in the case of austenitic chromium-nickel stainless steels. The increase in joint efficiency may be used if these steels are heat treated at over 900” F. If the vessels are constructed of ferritic chromium stainless steels, stress relieving is required in all vessel thicknesses except in the case of type 405 welded with electrodes, a process producing the austenitic weld. The code gives the temperatures and describes the procedures to be used in thermal stress relieving. Allowable stresses as specified by the code for highalloy steels are given in Appendix D.
Radiographic examination is required for double-welded butt joints if the plate thickness is greater than 134 in. If the plate thickness is greater than 1 in., complete radiographing of each welded joint is required if the vessel is fabricated of SA-202, SA-203, SA-212, SA-225, SA-294, SA-299, SA-301, or SA-302. Vessels of all thicknesses that are fabricated of SA-353, SA-357, or SA-387 must be radiographed. Also, vessels constructed of high-alloy steels such as type 405 welded with straight chromium electrodes and types 410 and 430 welded with any electrodes must be radiographed in all thicknesses except when the carbon content does not exceed 0.08 %, the plate thickness does not exceed 155 in., and austenitic welds are used. 13.7
DESIGN OF CYLINDRICAL INTERNAL PRESSURE
SHELLS
UNDER
The equations for determining the thickness of cylindrical shells of vessels under internal pressure are based upon a modified membrane-theory equation. The development of this equation is described in the following chapter (see Eq. 14.34). The modification empirically shifts the thinwall equation (see Eq. 3.14) to approximate the “Lame” equation for thick-walled vessels (see Fig. 14.5). The equation may be written in either of the following forms:
t=
fE 2.6~1 = fE ::.4p
(13.1)
or
p,*tdf!L t-0 - 0.4t i *
(13.2)
where t = minimum required thickness of the shell exclusive of corrosion allowance, inches p = design pressure, or maximum allowable working pressure, pounds per square inch E = welded-joint efficiency (see Table 13.2) f = maximum allowable stress, pounds per square inch (see Table 13.1 or Appendix D). ri = inside radius of the shell, inches PO = outside radius of the shell, inches If the thickness of the shell exceeds 50% of the inside radius, or when the pressure exceeds 0.385fE, the Lame equation should be used to calculate the vessel-shell thickness (see Chapter 14). The following forms of the Lame equation are given by the code (11). With the pressure p known, t = r@!:! - 1 ) where
-1 = p.23__ zw
( )
(13.3)
&fE+p fE - P
(13.4)
z=(?~)‘+y$!LJ
(13.6)
When t is known,
where
255
Design of Cylindrical Shells Under External Pressure 13.8 DESIGN OF CYCLINDRICAL EXTERNAL PRESSURE
SHELLS UNDER
I, =
The design of cylindrical vessels under external pressure is based upon consideration of the elastic stability of the shell, as described in Chapter 8. The calculation is made by successive approximation by using the following equation (see Eq. 8.33): B p = do/t
(13.7)
B-here p = allowable working pressure, pounds per square inch da = external diameter of shell, inches t = minimum thickness of shell exclusive of corrosion allowance, inches B = factor from Fig. 8.8 for carbon steel (see Appendix I for charts for other steels and alloys) Considerable reduction in vessel-shell thickness is often obtained by the use of stiffening rings, in which case Eq. 13.7 may be modified as follows (see Eq. 8.39): B =
No t + (ML)
(13.8)
where A, = cross-sectional area of the stiffening ring, square inches L = design length of a vessel section, as shown in Fig. 13.1 The required moment of inertia of the stiffening ring may be calculated by use of Eq. 13.9 (see Eq. 8.38).
(13.9)
14
where A = the factor given in Fig. 8.8 Stiffening rings should extend completely around the circumference of the vessel. If it is necessary to include joints between the ends of sections of such rings, as shown in details C, D, F, and G of Fig. 13.2, the moment of inertia of the ring must be maintained bv the addition of metal. The internal stiffening ring may be replaced in part by the external stiffener, as shown in detail H, so that the moment of inertia of the ring is maintained. In designing such stiffeners the moment of inertia of each section is taken about its own neutral axis. Gaps in the stiffening ring such as those shown in details A and E should not exceed the permissible length of arc given in Fig. 13.3 unless the additional reinforcement shown in detail H is provided. Some exceptions to this limitation are permitted by the code if the arcs of the stiffening rings are staggered 180”. Stiffening rings are attached to the shell by either continuous or intermittent welding. In the case where intermittent welding is used on each side of the stiffening ring, the total length of weld should be at least equal to one-half of the outside circumference of the vessel for external rings, and at least equal to one-third of the circumference for internal rings. Intermittent welds may be spaced a maximum distance of 8t in. apart. Any out-of-roundness of the shells of vessels subjected to external pressure reduces the strength of the vessel. This problem was discussed in section 8.7 of Chapter 8. The maximum permissible deviation, e, from a circular form permitted by the code for vessels under external pressure is given by Fig. 13.4.
Moment axis of ring
-...G h = depth of head
h = depth of head
Fig.
13.1.
Vesw!s,
Design length of vessel section
L for Eq. 13.8.
(Extracted from the 1956 edition of the
with permission of the publisher, the American Society of Mechanical Engineers [l 11.)
ASME Boiler
and
Pressure Vessel Code, Unfirad Pressure
256
Design of Pressure Vessels to Code Specifications
is found to be 0.138dc. The chord length for the template is twice this value, or 2(0.138)(169.3) = 46.7 in. Therefore, in a chord length of 46.7 in. the maximum plus or minus deviation from circular form must not exceed 0.65 in. 13.9 DESIGN OF PIPES AND TUBES UNDER EXTERNAL PRESSURE
moment of ikn.. required for ring
Length of any gap in unsupported shell not to exceed l e n g t h of arc
Unstiffened
cylinder
When tubes or pipes are subjected to external pressure, an increase in allowable pressure is permitted over that determined for shells by use of Fig. 8.8 (which is based upon elastic-stability considerations). Figure 13.5, from the 1956 code, gives the allowable design pressure for pipes or tubes subjected to external pressure as a function of the allowable stress of the material of construction and the ratio of t/do. When corrosion or erosion is expected, additional metal must be supplied. If the pipe or tubes are threaded, additional metal equal to (0.8/n) inches, where n is the number of threads per inch, must be provided. 13.10
than
This section &all have moment of inertia required for ring
H Fig. 13.2. Various arrangements of stiffening rings for unfired cylindrical vessels subjected to external pressure.
(Extracted from the 1956 edition
FORMED
CLOSURES
UNDER
INTERNAL
PRESSURE
The most common types of closures for vessels under internal pressure are the elliptical dished head (ellipsoidal head) with a major-to-minor-axis ratio equal to 2.0 : 1.0 and the torispherical head in which the knuckle radius is equal to 6% or more of the inside crown radius (ASME standard dished head). 13.10a E l l i p t i c a l D i s h e d H e a d s . In the case of the two-to-one elliptical dished head the following relationships apply (see Eqs. 7.56 and 7.57):
of the ASME Boiler and Pressure Vessel Code, Unfired Pressure Vessels, with permission of the publisher, the American Society of Mechanical Engineers
(13.10)
[ill.)
In determining the maximum out-of-roundness, e, a segmental circular template is used having the inside or the outside radius specified by the design (depending upon whether the measurements are made inside or outside). The length of the chord of the template should equal twice the maximum permissible unsupported arc length as determined by Fig. 13.3. In the case of vessels with butt joints, the value oft used in Fig. 13.4 is the nominal plate thickness less the corrosion allowance. If longitudinal lap joints are used, t is equal to the nominal plate thickness, and the permissible deviation is equal to (t + e). EXAMPLE CALCULATION. The vessel considered in section 8.4 of Chapter 8 will be used to illustrate the method of determining the permissible out-of-roundness of cylindrical shells. The shell is 14 ft in diameter and r3
where y
2fE -
In reference to Table 13.3 50 ,$ = 13,750(0.80) = o’0045
(13.14)
A = 26.5” As (o = 30’) exceeds a A of 26.5’, a compression ring is required. By Eq. 13.16
= 2w + P)
or
P
A = 0.0045
where
13.10d Conical Dished Closures. For conical closures or conical shell sections in which half the apex angle, a! (see Fig. 6.8) is not greater than 30”, Eq. 6.154 is used. The discontinuity stresses at the junction of the conical closure with the shell, described in Chapter 6, may cause excessive deformation. This may be prevented by the addition of a compression ring at the junction. When LY exceeds A, determined from Table 13.3, conical heads without a knuckle will require a compression ring at the section where the cone joins the shell.
;
0.001
0.002
0.003
0.004
0.005
0.006
13
18
22
25
28
31
A, deg
The required cross-sectional area of the compression ring in square inches is given by (11) : .4~~~+)(lzg)
(13.16)
where A = c&ical value from Table 13.3 When the thickness of the head or the shell exceeds the required thickness, exclusive of corrosion allowance, for the design pressure, credit may be taken for the excess thickness. The area included in a distance of eight plate thicknesses on each side of the joint times the excess thickness may be credited towards the required compression-ring area, A. EXAMPLE-DESIGN CALCULATION. A vessel having an inside diameter of 200 in. is to operate under an internal pressure of 50 psi. A conical closure is to be used with an apex angle of 30”. The material has an allowable stress of 13,750 psi and a joint efficiency of 80%. By Eq. 6.154
t= ;=
2 cos
[ 1 23
13.10e Toriconical Closures. Conical closures or conical shell sections in which half the apex angle, cr is greater than 30’ must be connected to the shell by means of a torus ring section at the junction to reduce junction stresses. Also, a toriconical head may be used when the angle a! is less than 30” but greater than A (see Table 13.3) in order to avoid the use of compression rings. The minimum knuckle radius must be equal to the greater of either 6% of the outside diameter of the head skirt or three times the knuckle thickness. The required thickness of the knuckle is determined by use of modified forms of Eqs. 7.76 and 7.77 with L substituted for r, where
and Value of A for Conical Closures (11)
(200)2;.577)]
= 1.52 sq in.
(13.15)
Table 13.3.
L=A.!2 CO8 a
(13.17)
dl = inside diameter of the conical portion of the toriconical head at the point of tangency with the knuckle, measured perpendicular to the axis of the cone, inches (see Fig. 13.6) The thickness of the cone is determined by use of Eq. 6.154, in which dl is substituted for d. EXAMPLE-DESIGN CALCULATION. It is desired to use a toriconical closure having a knuckle radius of 20 in. for the vessel in the previous section in order to avoid the use of a compression ring. Determine the thickness of ihe knuckle and the cone.
pd
cu(fE - 0.6~)
50(200) 2(0.866)(13,750 x 0.80 - 0.6 x 50)
= 0.526 in.
259
Fig. 13.6. Toriconical closure (11).
Design of Pressure Vessels to Code
260
Specifications
Integral-flange type
Loose-flange type (4
Ring gasket -I shown
‘,
Ring gasket shown
’
\ ‘\
Fig. 13.7.
Spherically
dished-steal-plate
coven
with
bolting
flanges.
(Extracted from the 1956 edition
of the
ASME
Boiler and
Pressure Vessel Code, Unfired
Pressure Vessels, with permission of the publisher, the American Society of Mechanical Engineers [ 111.)
The inside diameter of the cone, di, at the point of tangency to the knuckle is: dl = 200 - 2(2O)(J - 0.866) = 194.64
in.
The thickness of the knuckle is de&mined by means of Eqs. 7.76 and 7.77 modified by use of L in place of P,. where I,. is given by Eq. 13.17. I = ___194.64 1 ti2 in. i 2(0.866)
13.12 SPHERICAL DISHED COVERS
By Eq. 7.76
_ _ w = P(3 + dJr1) = t(3 + &12/20) = 1.3-L
By Eq. 7.77 t(knuckle)
=
50(112)(1.34) = 0.342 in. 2(13,750)(0.8) - 0.2(50)
The thickness of the cone by Eq. 6.154 with di substituted for d is:
t
= ~~~ (50)(194.64) 2(0.866)[(13,750)(0.8) -i
miiijj =
design pressure and a joint efficiency of 1.0 is used. The minimum thickness as determined by the above method must be compared with the thickness computed for the same closure by means of the procedure for external pressure vessels given in Chapter 8, section 8.8 or 8.11 as the case may be. Hemispherical and conical closures are designed in accordance with the procedures outlined in Chapter 8.
o'512 ln'
13.11 FORMED CLOSURES UNDER EXTERNAL PRESSURE In designing elliptical or torispherical dished heads for external pressure (pressure on the convex side) the thickness must be at least equal to that computed by use of Eq. 7.57 for elliptical closures and Eq. 7.77 for torispherical closures. In using either of these equations the pressure on the concave side is taken as equal to 1.67 times the external
A torispherical dished closure or a spherical dished flat plate may be combined with a bolting ring to produce a dished cover, as shown in Fig. 13.7. COVERS WITH PRESSURE ON THE CONCAVE SIDE . The thickness of the covers shown in detail a of Fig. 13.7 is determined by the method used for torispherical covers. The thickness of the spherical dished covers shown in details b, c, and d of Fig. 13.7 is determined by use of the membrane equation for spherical shells under internal pressure with an empirical factor of N to allow for the discontinuities at the junction with the ring. The minimum thickness of the dished covers in details b, c, and d of Fig. 13.7 is given by (11): (13.18) where p = internal design pressure, pounds per square in.ch L = radius of crown, inches f = allowable stress If the bolting rings are within the range of the American
Nozzles, Openings, and Reinforcements Slandards (ASA) B16.5.1953. the flange dimensions and facing details should conform to these standards. Bolting rings larger than these standards allow and corresponding to detail a of Fig. 13.7 are designed as ring-plate flanges, as described in Chapter 12. If the bolting ring corresponds to that shown in detail b of Fig. 13.7, the following equations may be used to determine the flange thickness (11). For a ring gasket, the flange thickness is: (13.19) For a full-face gasket, the flange thickness is: B(A + B)(C - B) (13.20) .4 - B I \vhrre A = outside diameter of bolting ring, inches B = inside diameter of bolting ring, inches C = diameter of bolt circle, inches MO = total moment as determined in Chapter 12 for ring-plate flanges f = allowable stress, pounds per square inch p = internal pressure, pounds per square inch t = 0.6
If the cover plate corresponds to detail c of Fig. 13.7, the following equations may be used to determine the flange thickness (11). For a ring gasket, the flange thickness is: t=Q[l+Jl+~]
(13.21)
For a full-face gasket, t.he flange thickness is:
where
If the bolting ring corresponds to detail d of Fig. 13.7, the following relat.ionship applies for determining the flange t,hickness(11) : Flange thickness, t = F[l + 1/l + (J/F2)] M here
F = PB
-(.frLo = - a - -$ = p. - (.fr)r=ri
2 I
[
Subtracting Eq. 14.8 from Eq. 14.9 gives: pi-p*,b2 ri2 roz
For a given material stressed within the elastic limit under a given internal pressure, the quantities eas, ja, and E are all constants, or
b = ro2ri2h - po) 2 r. - Pi2
or
b = do2di2(pi - PA
(14.lOa)
4(do2 - di2) where a is a constant. Substituting for jt by means of Eq. 14.3 gives:
For the usual condition, in which p. = 0, b =
-2jv-r$=2a T
d 2d.2 4(&l -” di2) pi
Substituting Eq. 14.10a into Eq. 14.8 and solving for a gives :
Rearranging gives: - 2d,. dfr -=.fr+a r
ro2ri2(pi [
- PO)
(ro2 - ri2)ro2
therefore
Integrating gives: In (jr + a) = -In r2 + Cl
1
-a=p,
a = [ri~o~~r~)]
where Cr is a constant of integration. Taking the antilogarithms where antilogarithm Cr = b gives: b .&+a=r2 or &.=$-a
(14.10bj
(14.6a)
where b and a are constants If the convention of positive values for tensile stresses and negative values for compressive stresses is adopted, the equation for jr becomes (radial stresses are compressive under the influence of internal pressure) :
-po
By expanding, Eq. 14.1 is obtained as follows: a = Piri2 - poro2 = p&i2 - podo2 = do2 - di2 r. 2 - ri2
fa
(14.11a)
For the case where p. = 0, (14.11b) Substituting the constants into Eqs. 14.6b and 14.7 gives:
The mathematical stress relationships-given above were
Lam&
Theory of Stress Analysis for Thick-walled Cylinders
271
The constant b is given by Eq. 14.10b. d 2d.2
b = \
i-0 ---
3 \
Substituting gives: do = 23+78 in.
r. -‘-
.--, Y3!-
4 fb)
z Pa 4(d12 - di2)
di = 12 in. pi = 20,000 psi
therefore
b = 975,800 The constant a is given by Eq. 14.1Ib. a _ pidi2 do2 - di2
Fig. 14.3. Tangential stress distribution in thick-walled cylinders.
Substituting gives: L
originally stated by Lamk (189), and the results are applicable within the elastic region. From the expressions obtained for the axial, radial, and tangential stresses, it is seen that for given conditions fa is constant and ft and f,. are inversely proportional to the square of the radius. It is then possible to give a graphical illustration of the stress distribution within the cylinder wall. 14.2~
i
I
,
Stress Distribution in a Cylindrical Shell.
T ANGENTIAL S TRESS DISTRIBUTION. To obtain a graphical presentation of the variation of the tangential stress with the radius, the limits of Eq. 14.7 may be considered. As r approaches zero, the tangential stress approaches plus infinity; and as r approaches infinity, the tangential stress approaches a, as indicated in detail a of Fig. 14.3. Detail b of Fig. 14.3 is a representation of the tangential stress variations in the wall of a thick-walled vessel. As the internal pressure is usually greater than the external pressure, the tangential stress is positive and has a maximum value at the inside surface of the cylinder. RADIAL S TRESS DISTRIBUTION. To obtain a graphical presentation of the variation of the radial stress with the radius, the limits of Eq. 14.6b may be considered. As r approaches zero, fr approaches minus infinity; and fr is equal to zero where r = &&, as indicated in detail a of Fig. 14.4, for the general case where p. # 0. If p. = 0, then r, = a. See detail b of Fig. 14.4. Examination of Fig. 14.4 indicates that the radial stress is compressive and has a maximum value at the inside surface of the cylinder.
pi = 20,000 psi di = 12 in.
a = 7106 Therefore, Lamb’s equation for the tangential stress variations in the wall of this vessel becomes:
ft = ‘F + 7106 The tangential stresses at various points in the shell are:
ftcr=s in.) = 34,212 psi ft(r
ft=s+a
=7.5 in.)
(inside surface)
= 24,454 psi
ftcr=9xis in.) = 18,666 psi ft+11.719
in.) =
14,212 psi
(outer surface)
It is apparent that in this vessel the hoop stress at the outer surface is only 41.5% of the maximum stress which exists at the inner surface.
:
i
r
14.2d E x a m p l e C a l c u l a t i o n 1 4 . 1 , B a s e d u p o n Lami Theory. A thick-walled alloy-steel vessel of monobloc con-
struction having an inside diameter of 12 in. and an outside diameter of 23xs in. is subjected to an internal pressure of 20,000 psi. The tangential and radial stress variations in the wall are desired. The Lam& relationship for tangential stresses in a thickwalled vessel is given by Eq. 14.7:
do = 23~‘~ in.
therefore
r0 ----ri .--
--
----
_/
r
‘0 1’ -m-s-- ri -
/k
/ ,’ --) &-CTi
0
Fig.
14.4.
Radial stress distribution in thick-walled cylinders.
272
High-pressure Monobloc
Vessels
The Lam& relationship for the radial stresses in a thickwalled vessel is given by Eq. 14.6h.
&=a-;
and for d = di
This relat.ionship for the vessel under consideration reduces to: fr = 7106 - y
= -20,000 psi
frcT = 7.5 in,) = - 10,240 fr(T=g4ie
.f+=
in,) = -4450
fi(d=d.)* = AK2 + l) K2 - 1 Therefore .ft(*nax)
The radial stresses at various points in the shell are: f+.=~j~.)
Therefore, for d = do,
psi
psi
= fy.p.
= P i g [ I
(14.14a)
Equation 14.14a gives the stress at failure according to this theory. For design purposes a factor of safety, X, may be introduced so that the induced stress will be less than the elastic limit, jy.r., of the material in question. For design purposes Eq. 14.14a would be written: (14.14h)
11.719 i n . ) = 0 psi
14.3 CRITERIA FOR SHELL FAILURE BASED ON THEORY OF ELASTICITY
According to the maximum-principal-stress theory, failure is considered to occur when any one of the three principal stresses, Jo, jf, or f,., reaches the st,ress equal to the elastic limit taken as the yield point (jY.p.) of the material. By Eq. 14.1
where X = factor of safety Equation 14.14b may be solved for K to give:
14.30 Maximum-principal-stress Theory.
I By
Eq. 14.12
By
Eq. 14.13
A comparison of the above three equations indicates thatft is the largest and therefore is the limiting factor to be taken into considerat,ion for design. At failure the following equality is assumed to hold: fi(max) = f,.,. Inspection of Eq. 14.12 shows that. the maximum value offt is obtained at the inside wall, that is, where d = di and p0 = 0 gage (atmospheric pressure). Rewriting Eq. 14.12 for p. = 0 gives:
ft -Pidi2
do2 - di2
Let do K -= 4 theu
K=
(f,.dx&) + ’ J (fy.,.hPi) - 1
(14.14c)
Equation 14.14a is the so-called maximum-principalstress equation and is also known as the Lame equation. It may be noted that Poisson’s ratio does not appear in this equation. In many cases it has been found that the equations resulting from the application of the maximum-strain and maximum-strain-energy theories are in much better agreement with the experimental results than the so-called Lame equation. These equations contain Poisson’s ratio, and many authors have come to the conclusion that the Lame equation is not theoretically correct because it does not include this ratio. On the contrary, it has been clearly shown that this equation follows from a rigorous mathematical stress analysis in which the lateral contraction, of which Poisson’s ratio is a measure, has been taken into account. 14.3b Maximum-shear-stress Theory. This second criterion postulates that failure should occur when the maximum shear stress equals the shear stress set up in the material at the elastic limit (taken as the yield point, fY.r.). Disregarding the effect of the axial stress, ja, we lind that the radial and tangential stresses form a two-dimensional system. Knowing that the maximum shear stress is equal to the algebraic difference between the stresses considered, we find that the corresponding maximum shear stress is given by Eq. 9.85. fs(n,ax) = at - “fr)n,ax = &fy.*. The shear stress at radius r is: fw = at - .&jr Substituting for ft by Eq. 14.7 and for j,. by Eq. 14.6b gives :
!
Criteria for Shell Failure Based on Theory of Elasticity
therefore
273
again with the proper regard for sign convention, gives:
f87 = Ji Substituting for b by Eq. U.lOb for the (‘ase in which the external pressure is zero, and noting that js is a maximum when r = di/2, we find by Eq. 9.85 that. do2Pi f Max) = do2 _ dig = ify.,,.
(14.16)
Let
therefore
Examination of Eq. 14.18 indicates that the st.rain is maximum when P is equal to di/2. Therefore
5 = I( 4 then fy.,. = i&y 1 Pi = ?fd*nax)
(14.17a)
Substituting for u by Eq. 14.1lb and for b by Eq. 14. IOb gives: %(rn%X)
The term js(max) in Eq. 14.17a is the induced shear stress that exists at failure in accordance with this theory and is numerically equal to one half of the tensile stress in simple tension at the elastic limit. This theory may be used for design if a factor of safety, X, is included. Therefore, for design purposes Eq. 14.17a may be written as: (14.17b) In discussing the strength of cylinders under high pressure, Manning (237) states: “Overstrain of a cylindrical wall occurs when the maximum shear stress reaches a value l/d times the tensile upper yield stress. . . . The most satisfactory basis of design is arranging for the maximum shear stress . . . to equal (when raised if required by an appropriate safety factor) the tensile yield stress divided by &.” Equation 14.16 may be modified (equated to &./fi instead of jY.P./2) to give: (14.17c)
pi = -
(1 - /.+A2 +
K =
_ fY.P.
Jfy.1,.
-_
(14.17d)
- XPi v5
Equations 14.17~ and 14.17d are based on the shear-strainenergy hypothesis (237). 14.3~ M a x i m u m - s t r a i n T h e o r y . In this third case, rupture is considered to occur whenever the strain set up in the material reaches the strain at, the elastic limit. In the discussion of the maximum-principal-stress theory it was shown that the tangential stress, jt, was the maximum or limit,ing stress. Consequently, within the elastic range, et is the limiting strain to consider in design. By Eq. 6.4
Substituting for jt by Eq. 14.7 and for jr by Eq. 14.6b,
1
ddo2
(14.19)
To satisfy the criterion that. failure occurs when tl,(mnx) equals the st,rain at the elastic limit we let
et2
therefore
= ey.p.
fy.,. = E+.p. = &n(n,nx, Subst.ituting for d,/di = K gives:
Equation 14.20a states the conditions at. which failure is assumed to occur according to this theory. For design purposes a safety factor, X, is introduced into this equat.ion for proportioning the vessel. Thus, for design purposes Eq. 14.20a can be written as:
Equation 14.20b may be solved for K to give: ...~___
K = Fo: purposes of design Eq. 14.17~ may be solved for K as follows:
(1 +
E
(fy.p.lxPij
t (l - Cc)
(fY.P./~Pi)
- (1 + 4
(14.20~)
14.3d M a x i m u m - s t r a i n - e n e r g y T h e o r y . St.rain energy refers to the mechanical energy absorbed by a body stressed within the elastic range. According t,o this fourth criterion, the strain energy accumulated in the material when it is stressed to its elastic limit is the det.ermining factor for rupture. Tn Chapter 2, section 2.4a, the strain energy was shown to be equal to the work done on the material (see Eq. 2.26). The work done in a one-dimensional stress system is given by:
(2.26) For a two-dimensional stress system it has been shoWn that the strain in either direction is the algebraic sum of 11le two components, as given by Eq. 6.4. E.z2
= f [.f, - /4-t/1
(6.A)
274
High-pressure
Monobloc
Vessels
Substituting (l/E)[f, - pfV] for (f,/E) in Eq. 2.26 gives:
criterion, we let u Ir(max)
(14.21)
ua! = & u-z2 - kf&/l
f . .2
(14.28:
= $!j-
It then follows that
Likewise. for the y direction,
wu - PM4 + (1 + /4d04
f,.,. =
(14.29)
(d,c-- di2)2
therefore
uz, = & LL2 +.f,” - w.&1
(14.23)
In an analogous manner it can be shown that the strain energy for a three-dimensional system is given by:
Letting do/di = K, substituting p = 0.25 for steel, and multiplying the numerator and denominator by 2 gives: 46 + 10K4 fY.P.
U q/z = & lfz2 +.h2 +.fz2 - P(fi/fi +.fi.f, + fifi)l (14.24) This last expression has been called the strain-energy function. If Eq. 14.23 is applied to the two-dimensional stress system formed by ft and fv in the case of the cylinder wall, the strain energy is given by:
ut,
1
=
pi
2(K2
(14.30aj
- 1)
Equation 14.30a gives the conditions at which failure is assumed to occur by the maximum-strain-energy theory. For design purposes a factor of safety, X, is included tc proportion a vessel. Equation 14.30a can then be written as:
fY.P. =
APi 46 + 10K4 2(K2 - 1)
(14.30b)
14.3e Comparison of the Four Theories of Failure with Experimental-test Results. Newitt (191) reported a num-
=[A2 +A2 - a-dtfrl 2E
By making the substitutions for ft and f,., Ut, may be expressed as a function of the radius of the cylinder, sign convention being used,
ut~=~[(u+~)2+(u-~) -++$)(a--$)] (14.26) Carrying out the algebraic operations and simplifying, we obtain:
ber of experimental tests on mild steel in which the elastic strain of a cylinder under pressure was measured and plotted as a function of internal pressure. He found that the strain was proportional to the pressure until the elastic limit was reached. This procedure gave values of stress, strain, and internal pressure at the elastic limit. These tests were made on a number of vessels having different K ratios (from K = 1:35 to K = 3:65). The experimental value of p/fy.p. was then compared with the values predicted by the four different theories. Table 14.1 summarizes the results obtained with mild steel.
ut.=g2+;-p(u2-;)] = ;
[
(1 - p)a2 + (1 + cc) 9
1
Table
streas at
The strain energy is a function of the fourth power of the radius and has its maximum value at the inside wall where P = dJ2. Therefore U Ir(Irax)
; = -
[
(1 - /da2
+ (1 + cc) $ z
1
By making the substitutions for the constants a and b by Eqs. 14.11b and 14.10b, the expression becomes: E (1 - /.tL) (d ry;,2)2 + (1 + cl) lilr(max, = J 0 z [
U tr(max)
(1 - PL)di4 + (1 f PIdo (do” di2)2
1
(do2
14.1.
pi2d04
- di2)2
1
(14.27)
The strain energy of the material at the elastic limit is given by (f,.,. 2/2E) (see Eq. 2.26). To satisfy the fourth
Results
of
Tests
Cylinders
(191)
Ratio of Yield in Yield ExperiExternal Simple Pressure mental t o I n t e r - T e n s i o n i n Crlin- V a l u e nal Dider -(p) , of (fy.p.)* ameter Ih/sq in. lb/w in. p/fy.p. 35,300 9,700 0.275 1:35 1:53 35,300 12,000 0.340 1:58 35,300 12,500 0.354 1:58 35,300 12,500 0.354 14,700 0.416 1:74 35,300 1:77 35.300 14.400 0.407 1:79 35;300 15;400 0.436 1:79 35,300 15,200 0.430 1:79 35.300 15.400 0.436 1:79 35;300 14;600 0.413 1:86 34,000 13,600 0.400 1:97 34,000 14.100 0.415 2:19 36,860 18,090 0.490 2~19 36,860 18,090 0.490 2:45 36,860 18,740 0.508 2~66 36,860 20,150 0.546 36,860 20.300 0.550 2:88 3:05 36,860 20:200 0.547 36,860 21,700 0.588 3~26 3:65 36,860 21,800 0.591
on
Mild-steel
P/&..~.
Calculated Valuea of According to MaxMUMaxM&Xprin.prin.shearstreinstress strain stress energy Theory Theory Theory Theory 0.291 0.402 0.430 0.430 0.506 0.515 0.525 0.525 0.525 0.525 0.554 0.590 0.655 0.655 0.713 0.752 0.784 0.806 0.827 0.860
0.295 0.393 0.415 0.415 0.475 0.483 0.490 0.490 0.490 0.490 0.511 0.539 0.583 0.583 0.625 0.649 0.672 0.684 0.697 0.718
0.225 0.287 0.300 0.300 0.336 0.340 0.344 0.344 0.344 0.344 0.356 0.372 0.395 0.395 0.416 0.429 0.439 0.446 0.452 0.467
0.262 0.344 0.363 0.363 0.41: 0.417 0.422 0.422 0.422 0.422 0.449 0.460 0.494 0.494 0.522 0.539 0.553 0.562 0.571 0.583
Criteria for Shell Failure Based on Theory of Elasticity Table 14.2.
Results of Tests on High-tensile-steel Cylinders (192)
Tensile Elastic Limit, Class of Steel Nickel steel Nickel steel Nickel steel Nickel steel Nickel-chromium steel Nickel-chromium-molybdenum Nickel-chromium-molybdenum Nickel-chromium-molybdenum
I i
(fs.,.),
steel steel steel
K tons/sq in. 2.15 28.08 2.15 28.94 2.50 21.31 2.50 28.80 2.00 29.57 2.00 37.62 2.00 33.44 2.00 32.45
A comparison of the experimental values of p/fy.*. and the theoretical predictions shows that the closest agreement, between theory and experimental value was obtained with the strain-energy equation. Similar tests were also made on a variety of high-tensilesteel cylinders by Macrae (192). The results of these tests are given in Table 14.2. A comparison of the experimental values of ~/fy.~. and those predicted by the various theories shows that in the case of high-tensile steels the best agreement is obtained by use of the maximum-shear-stress equation. This might ‘nave been anticipated because these materials have shear strengths which are quite low in comparison with their tensile strengths. Cook and Robertson (193) reported similar information on cast-iron cylinders. However, in their study the pressure was increased until the cylinders ruptured because Results cast iron does not have a well-defined yield point. cf these tests are given in Table 14.3. In Table 14.3 only the comparison between the experimental value of p/j and that predicted by the maximumprincipal-stress theory is given. The fact that the agreement is good indicates that this material follows this theory. 14.3f Comparison of the Lam6 Theory with the Membrane Theory. The membrane equation for hoop stress is
Yield Pressure in Cylinder (p), tons/sq in. 11.48 11.50 9.00 12.10 11.93 14.10 12.68 12.10
p/j,,,,,. Calculated According to MaxMaxMaxMaxnrin.nrin.shearstrainstress strain stress energy Theory Theory T h e o r y T h e o r y P/f,.,,. 1
0.409 0.397 0.422 0.420 0.404 0.374 0.379 0.373
.ft = & + 0.6 -
(14.33)
Pi
or if welded-joint efficiency included (ll), it is:
and corrosion allowance are
jtE yO.Sp’+ ’
(14.34)
Graphical Comparisons of the Various Theories.
(14.31) Table 14.3.
outside radius of shell, inches ri = inside radius of shell, inches
CK - 1)
(See Eq. 1.4.14.)
The ratio of jt/pi may be conveniently plotted against K as shown by Maccary and Fey (194) and as indicated in Fig. 14.5. The determination of shell thickness using the Lame equation involves calculation by successive approximation. The same calculation using the membrane equation is more convenient, being a direct calculation, but is limited in its application to vessels in which t/o$ is equal to or less than 0.10. The range of the membrane equation has been extended by the empirical modification of adding the constant 0.6. This new equation is known as the ASME modified membrane equation and is in much closer agreement with the Lam& equation. At a t/d+ value of 0.25 the ASME modified membrane equation agrees with the Lame equation within 1% (194). The ASME modified membrane equation is:
r,, =
ft = Pi L
0.488 0.488 0.527 0.527 0.466 0.466 0.466 0.466
A graphical comparison of the membrane theory, the ASME modified membrane theory, the principal-stress thenry, the
Results of Tests on Cast-iron Cylinders
If the ratio of r,,/‘ri equals K, then Eq. 14.31 indicates t.hat the hoop stress determined by the membrane equation becomes: D
0.392 0.392 0.420 0.420 0.375 0.375 0.375 0.375
(K2 - 1)
t=
Rewriting in terms of jt with t = (rO - PJ gives:
where
0.577 0.577 0.631 0.631 0.546 0.546 0.546 0.546
ft = pi (K2 + 1)
14.39
I
0.644 0.644 0.724 0.724 0.600 0.600 0.600 0.600
the Lame equation becomes:
given in by Eq. 3.14 as:
ft = &
275
(14.32)
and Eq. 14.14 indicates that the hoop stress determined by
Exter- I n t e r nal ml Dinm- D i a m eter, eter, in. in. 1.133 1.420 1.390 1.710 1.561 1.475 1.516 1.870
0.873 0.923 0.755 0.922 0.793 0.750 0.635 0.630
(193) calcu-
K 1.30 1.54 1.83 1.85 1.97 1.97 2.40 2.96
Bursting Tensile lated Strength Pressure Bursting CalcuPressure, Observed lated (f), (PI 9 Ib/aqin. Ib/sqio. lb/aqin. P/f P/f 18,600 5,060 4,760 0.272 0.256 0.388 Il.406 24,500 9,520 9,950 23,550 13,000 12,710 0.552 0.540 1 4 , 5 5 0 1 4 , 8 0 0 0.540 0.55n 26,900 15,100 14,300 0.623 0.590 24,200 0.665 0.590 24,750 16,460 14,600 26,700 19,250 18,800 0.720 0.704 1 7 , 4 1 0 1 7 , 3 0 0 0.802 0.796 21,700
276
High-pressure
\
Monobloc
I
I
1 0 \;\ \\ ; \?, C-IQ. 6 II I\\ 4
Vessels
I
I
I
I
I
I
I
I
I
I
I I
I
I
I
I
Fig.
1 4 . 5 . A c o m p a r i s o n of t h e
Lam&tangential brane-theory (Courtesy
of
stress tangential
with
mem-
stress
(194).
McGraw-Hill
Publishing
Co.)
0.6 1
I
I
I
1.2
1.4
1.6
I
I
I
I
I
1
0.4 0.3 1.0
1.8 2.2 2.0 K = Outside diameter Inside diameter
maximum-shear-stress theory, the maximum-strain theory, and the maximum-strain-energy theory is shown in Fig. 14.6. 14.3h Example Calculation 14.2, Comparing Theories of Failure. A vessel is to be designed
the
2.4
2.6
2.8
Maximum-shear-dress By Eq. 14.17d
Theory:
Four
to withstand an internal pressure of 20,000 psi. An internal diameter of 12 in. is specified, and a steel having a yield point of 70,000 psi has been selected. Calculate the wall thickness required by the various theories with a factor of safety of X = 1.5. Maximum-principal-stress Theory: By Eq. 14.14~
K = =
fY.P. df,.,. - APi fi 70,000 70,000 - 1.5 x 20,000 x 1.732 J---
= 1.97 d, = (1.97)(12) = 23.6 in. therefore t = 5.8 in. Using Fig. 14.6 for X~i/fy,~. = 0.428 and reading from ~hr maximum-shear-stress curve (fsmex = j,.,./ti), we obi.ain:
therefore
T&L20 t .
K = 1.58 a!,, = (1.58)(12) = 19 in.
therefore
therefore t = 3.5
t = g = 6.0 in.
in.
Using Fig. 14.6 for Ap/f = (1.5 X 20,000)/70,000 = 0.428 and reading from the principal-stress curve, we obtain:
L-34 t . therefore t = +4 = g = 3.53 in.
Maximum-strain Theory: By Eq. 14.20~ K =
(fy.p./APi)
+ (1 - PI>
(fy.,,./~Pi)
- (1 + Pcl;
Criteria for Shell Failure Based on Theory of Elasticity
therefore
t = g = 4.14
K = 1.69 d, = (1.69)(12) = 20.3 in. , I
therefore t = 4.15
277
Maximum-strain-energy By Eq. 14.30b
in.
Theory:
Referring to Fig. 14.6 for X&f,,,,, = 0.428, we obtain: di = 2.9 I
1.00
-
I--
0.95 0.90
0.75
-
0.70
-
0.85 0.80
-
0.65
-
-
0.60 0.55
-
==G
-
0.50 --
0.45
-
0.40
-
0.35
-
0.30
-
0.25
-
xl! f
Fig. 14.6. Comparison of various theories for shell design.
tlm-t
0.20
I
0.15
I
>
0.3
IIll.
I
-
1 Thin-wall equation 2 ASME modified equation 3 Maximum-principal-stress equation (Lam&) 4 Maximum-strain equation (M = 0.25) 5 Maximum-strain-energy equation 6 Maximum-shear-stress 7
0.10
I
-
equation
Cfs max = fY,P, I43
equation
Cfs milx = fY,P. /2)
Maximum-shear-stress
0.4 0.5 0.6 0.70.8
1.0
1.5 di t
2
3
4
5 6 7 8910
15
\ ~
.?(
278
High-pressure Monobloc Table 14.4.
K
1.168 1.167 1.500
2.000 3.000 4.000
Pi h-9
7,560 7,820 15,680 21,950 26,620 28,220
Vessels
Stress Conditions at Onset of Overstrain (237)
Max Tangential Stress Max Shear Stress (psi) (psi)
49,280 51,070 40,760 36,470 33,350 32,000
28,670 29,610 28,220 29,280 29,970 30,000
Substituting with pi = 20,000 psi and again using fv.r, = 70,000, we obtain:
70,000 46 +lOK* = = 2.33 1.5 x 20,000 2(K2 - 1) 6 + lOK* = (4.66)2(K2 - 1)2 6 + lOK* = 21.7K4 - 43.4K2 +
21.7
0 = 11.7K4 - 43.4K2 = 15.7 Let K2 = CT. 11.722 - 43.42 + 15.7 = 0 5=
-b& db2-4ac 43.4 f 33.9 2a = (2)(11.7)
For a maximum value z = 3.3; therefore K = dg = 1.82 d, = (12)(1.82) = 21.8 in.
14.4 CRITERIA FOR SHELL FAILURE BASED ON THE THEORY OF PLASTICITY
t = 4.9 in. From Fig. 14.6, reading from the maximum-strain-energy curve at Xp/f,.,, = 0.428, we find that
di = 2.4
t = 12 = SOin 2.4
'
'
Discussion: The required thicknesses of the vessel shell under consideration according t,o the equations for the four theories are as follows: Theory Maximum-shear Maximum-strain-energy Maximum-strain Maximum-principal-stress
is obtained. The pressure measured at the beginning of permanent set can be compared with that predicted by the various theories, and the most appropriate equation selected. In the case of the example calculations presented above a steel with a tensile yield point of 70,000 psi was specified. This would be classified as a high-strength tensile steel and would be expected to fail by shear as indicated by Macrae (192) (also see Table 14.2). Manning (237) concludes that steels with high tensile strength would be expected to fail by shear. Table 14.4 lists the maximum shear stress and the maximum tensile stress at which overstrain was observed to occur for vessels of various K ratios fabricated of hightensile-strength steel. Table 14.4 shows that onset of overstrain occurred when the maximum shear stress reached a mean value of 29,300 psi (+ about 3 %), whereas the maximum tensile stress varies widely. Furthermore, Manning states that judging on the criterion of onset of overstrain when the shear stress equals the tensile yield stress divided by 43, one would predict onset of overstrain when the shear stress reached 30,200 psi, which is less than 235 ‘% higher than the mean experimental value. In the case of a vessel of monobloc construction the term K2/(K2 - 1) in Eq. 14.17~ approaches unity at high values of K, and the maximum internal pressure, pi, becomes equal to jY.JX & as a limiting condition. For the previous problem, in which j,.,. was 70,000, this limiting pressure would be 27,000 psi. This is a serious restriction but may be circumvented by other procedures such as prestressing and multilayer construction.
Thickness (in.)
5.8 4.9 4.15 3.5
It is obvious that the various theories give widely different answers for the required thickness. Experimental results have agreed with one theory when work was being done with one material and with another theory when work was being done with another material. When working with a new material, it is important to make a preliminary test with a vessel constructed of the material. The experimental vessel may be hydrostatically tested until a permanent set
The theory of the plastic behavior of materials is in general beyond the scope of this text. A number of excellent texts are available on this subject (195-200). However, some of the relationships developed from the theory of plasticity appear to explain the yield and bursting characteristics of thick-walled cylinders better than any of the relationships based upon the theory of elasticity. Therefore, a limited discussion on the plastic failure of vessels is considered appropriate. On subjecting a thick-walled vessel to increasing internal pressure, stresses are induced in the shell which are maximum at the bore, as predicted by the various criteria for failure based upon the theory of elasticity. Contrary to what might be expected, failure of a shell of ducti metal usually does not begin at the fibers along the bore but at the fibers along the outside surface of the shell (217). O n stressing beyond the yield point, most metals pass through a region of plastic flow in which elongation progresses without an increase in resisting stress. This condition is first reached in the inner part of the cylinder. However, the st,rain of the inner zone is limited by the outer zone, which is not strained beyond the yield point; thus the inner fibers are incapable of rupture. The inner fibers of an overstressed vessel often show evidence of slip where failure began but halted because of the restraint offered by the outer fibers. The inner fibers therefore are prevented from failing, provided that the outer fibers offer suflicient
279
Criteria for Shell Failure Based on the Theory of Plasticity
restraint. There is no such protection of the outer fibers by the inner fibers. Manning has discussed the rupture of thick-walled cylinders of ductile metal (203). The pressure necessary for the yield point for the metal fibers in the bore to be reached is known as the “elastic-breakdown” pressure. At this pressure the maximum fiber stress is the tangential stress at the inner surface. The radial stress also has its maximum value at the bore, and this stress is equal to the internal pressure. As the pressure is raised, the region of plastic flow, termed “overstram, ” moves radially outward and causes the tangential stress to decrease in the inner layers and to increase rapidly in the outer layers. Progressive increase in pressure moves the elastic-plastic interface radially outward until the interface reaches the outer radius and no elastic zone remains. In this situation the maximum hoop stress is at the outside surface. Manning has reported (203) that for the beginning of overstrain for a vessel in which the outside diameter to inside diameter ratio was 2: 1 and the pressure was 12,750 psi, the tangential stress at the inner radius was 21,000 psi, and at the outer radius 8000 psi. For the same vessel with 100 y0 overstrain (plastic-elastic interface at rO) at a pressure of 27,620 psi, the tangential stress at the bore was 16,000 psi, and at the outer surface 34,000 psi. Thus it is apparent that the tangential-stress distribution is totally different in the 100 y0 plastic state than in the completely elastic state. On the other hand, the radial-stress patterns have similar shapes for the completely elastic and completely plastic states. When failure occurs in the shell of a ductile metal as the result of progressive increase in stress, it usually follows the path of a continuous helix from the outer surface inward, as shown in Fig. 14.7. Prager and Hodge (195) have defined the internal pressure in a cylindrical vessel that is required to place the elastic-plastic interface on the outside surface of the vessel. This is the pressure required to place all of the vessel wall beyond the yield point. In deriving this relationship it is necessary to establish the condition under which plastic flow is initiated. A widely used yield criterion is that of Von Mises (205). This criterion can be expressed by the following relationship:
fy.,. = f&Y. Ah
ft = f?. + I’:
(14.37)
Substituting for ft in Eq. 14.37 by Eq. 14.36 gives:
dfr = 2fy.p. r dr v3 r Integration of Eq. 14.38 gives:
P=
2fy.p. In 2 4 Pi
(14.39)
where p = internal pressure required to stress the outer surface to the yield point, pounds per square inch f Y.P. = yield point of the shell material in single tension, pounds per square inch r, = outside radius of the vessel ri = inside radius of the vessel Equation 14.39 was derived for an ideal plastic solid, that is, a material that has a stress-strain diagram illustrated by the “idealized” curve of Fig. 2.11. For this ideal condition the yield strength and tensile strength (t.s.) have the same value; therefore, the bursting strength would he that predicted by either Eq. 14.39 or Eq. 14.40. pS.%ln~ where
ft.*.
(14.40)
= ultimate tensile strength, pounds per square inch
(14.35)
where f,.,. = yield-point stress of the material in simple tension, pounds per square inch fs.v. = yield limit in simple shear, pounds per square inch It was shown previously, by Eq. 9.85, that the maximum shear stress in a three-component system is:
Combining the above relationships with Eq. 14.35 gives: ft - fr (
2
_ fY.P. Ah >
(14.36)
By making allowance for sign convention with the compressive stress, fr, negative, the Eq. 14.2b may be written as:
Fig.
14.7.
Typical failure as a result of overstraining of thick-walled cylin-
der of ductile metal.
(Courtesy of J. H. Faupel
[201].)
(Extracted
from
Transactions of fhe ASMF with permission of the publisher, the American Society
of
Mechanical
Engineers.)
High-pressure
280
Monobloc
Vessels
30 10 - 2oY E.B.
E.B. = Elastic breakdown at bore O.S. = Overstrain through the wall B. = Bursting (x = bursting for experimental curve) - Calculated ---- Experimental
E.B. (
10 0 0, 0 Jt4000L2000~ Fig.
14.8.
0
External hoop strain, microin./in.
Pressure-versus-strain curves for thick-walled cylinders under internal pressure (215).
(Extracted from Transactions of
the ASME
with permission
ot the publisher, the American Society of Mechanical Engineers.) Treatment and Properties of Cylinders Properties in Transverse Direction at Yield Strength, Cylinder No.
Material and Treatment
Bore
of Cylinder
-
Elongo-
Reduction in
psi
Ultimate,
tion in
Wall
0 . 0 1 % 0.2%
Strength,
1 in.,
area,
Ratio
offset
offset
psi
per cent
per cent
Quenched-and-tempered 2.74
74,300
79,500
105,400
24
50
9
Annealed SAE 1035
2.75
39,950
38,600
73,100
25
10
Annealed SAE 3320
2.75
58,100
68,500
112,600
20
32 41
11
Annealed Cr-Ni-MO-V
2.75
45,100
45,300
83,300
25
37
3
and quenched SAE 3320
In the actual case for ordinary metals the ultimate tensile strength is appreciably higher than the yield strength, and the stress at bursting will lie between the yield and ultimate strengths. Faupel (201) has proposed that Eqs. 14.39 and 14.40 be modified as follows: p=~(22$$31n~)+(I-~)($$1n~)(14.41)
basis (201). The vessels were tested at temperatures from ambient to 660” F. Figure 14.8 shows some of the data reported by Faupel and Furbeck (215) on elastic-breakdown pressures and observed and calculated bursting pressures. Table 14.5 presents selected test results of Faupel (201) and of Crossland and Bones (202) and includes a comparison of five theoretical methods of predicting bursting pressure, and the observed data.
Equation 14.41 reduces to: p=%[ln??][2-k]
(14.42)
Equations 14.41 and 14.42 proportionally weight the stress values to their ratios; thus, when the ratiofY.,./ft.s. is equal to 0.25, Eq. 14.39 contributes 75% and Eq. 14.40 contributes 25y0 to the bursting pressure (201). Faupel reported the tests on the rupture of nearly 100 thick-walled cylinders fabricated from a variety of high-strength steeIs and showed that Eq. 14.42 was reliable within k 15 y0 for predicting the observed rupture pressure on a 90 O/,-certainty
1
I
a
(K - 1) P = ?L (K + 1) In K
4
5 Manning method (see below and section 14.11j
Monobloc
I
Vessels
at
Elevated
Temperatures
281
I comparison of the observed rupture pressures in Table 14.5 with those obtained by the five methods of predicting the rupture pressure from theory shows that the best agreement is obtained in the case of Faupel’s data with either the Faupel or the Manning method. The Faupel met,hod makes use of Eq. 14.42, and the Manning method makes use of torsion-test data. The Manning mrt,hod involves the graphical integration of the stressstrain curve in a torsion test and cannot be expressed in a single equation (see section 14.11) (203). In the absence of torsion-test data Eq. 14.42 is recommended for the prediction of the bursting strength of thick-walled cylinders.
prior to rupture is dependent upon the time-stress history of the vessel. Voorhees, Sliepcevich, and Freeman (204) have presented a procedure for calculating the time of rupture from creep and stress-rupture data normally available to a designer. Prior to the work of Voorhees the design of thick-walled vessels at high pressures and elevated temperatures was usually based upon the maximum principal stress and an allowable stress determined from creep and stress-rupture test data. This is the current method recommended by the ASME code (11) for vessels operating at pressures up to 3000 psi.
14.5
Voorhees’ analysis assumes that the creep-rupture life of a vessel under complex stressing is controlled by an equivalent stress, 7, termed the “shear-stress invariant.” This average stress is also known as the octahedral shear stress, the effective stress, the intensity of stress, and the quadratic invariant. The theory for the biaxial-stress condition was developed by Von Mises (205), and this theory was further developed to apply to the triaxial-stress condition independently by Hencky (206, 207, 208) and by Huber (209). A derivation of the relationship between the equivalent stress, j, and the three principal stresses, fl, fi, and frc where fl > fi > f3 was given by Eichinger (210). The relationship between these stresses is:
14.50
MONOBLOC VESSELS AT ELEVATED TEMPERATURES
\I’hen pressure vessels are used at elevated temperatures with induced stresses within the creep range, the phenomenon of creep must be taken into consideration. A discussion of creep and the use of stress-rupture curves was presented in Chapter 2, section 2.5. . Thick-walled vessels for high-pressure service have steep stress gradients. These stress gradients change under the influence of creep at elevated temperatures. This redistribution of the stress gradients under the influence of creep is known as “stress leveling.” The vessel geometry and the creep- and rupture-strength properties of the material of construction influence the degree of redistribution of stress under the action of creep. The life of the vessel
Table
14.5.
Bursting
Equivalent
7” =
Pressure
of
Stress
+rc.f1 - f2) 2 +
Cylinders
(Shear-stress
(fl - j-d2 +
Invariant).
u2 - f3)2!
(14.43)
(201)
Calculated pb (psi) A Faupel Test No. 7 13 14 30 40 11 $2 44 45 46 47
R 2.49 2.43 2.44 2.75 2.76 2.75 2.74 1.75 2.75 3.69 4.71
psi 90,480 68,350 91,550 89,700 80,100 74,900 104,650 105,650 137,850 105,500 106,700
Observed P b (Psi) 79,000 57,000 83,000 63,000 55,000 67,500 98,500 59,000 143,000 168,000 192,500
Crossland and Hones (202) ‘rest No.
I I 4
2 3 4 5 6 7 8 9 10 11 12 13
I I.- --I-/
1.57 1.33 1.99 2.29 2.66 1.78 2.90 1.88 2.48 3.18 2.13 3.60 3.72
66,000 66,000 66,000 66,000 66,000 66,000 66,000 66,000 66,000 66,000 66,000 66,000 66,000
31,000 18,64,0 44,600 54,000 60,100 38,400 65,300 40,200 57,400 70,000 47,800 76,000 79,000
Lamb1 Method 65,300 48,500 65,000 68,700 61,500 57,300 79,750 53,600 105,500 91,200 97,600
c
B MeanDev. diameter2 (%) Method -17 77,500 -15 57,000 -22 77,000 -
++; -15 -19 -9 -26 -45 -49
F _ _ 27,800 -10 18,200 -2 39,400 -12 45,000 -17 49,500 -18 34,200 -11 52,000 -20 36,800 -8 47,500 -17 54,000 -23 42,000 -12 56.500 -26 57;ooo -28
81,000 75,000 74,200 97,500 57,500 128,500 121,000 138,500
_
_
G _
29,300 18,700 43,700 52,000 60,000 37,000 64,500 40,400 56,100 68,700 47,700 74,500 76,000
Dev. (%) -2 0 -7 +29 f36 +10 -10 -3 -10 -28 -28
Classical3 Method 95,000 70,500 94,500 115,000 106,500 96,000 131,000 68,300 178,000 160,000 191,000
-6 0 -2 -4 0 -4 -1 0 -2 -2 0 -2 -4
H 34,400 21,700 52,500 63,200 74,600 44,000 81,000 48,200 69,500 88,500 57,800 93,200 100,400
-
I
----. - ~..-~\ Ye---
\ r 7------ --
D Dev. (%o)
+20 +24 +14
+83 +94
+42 +43
+16 +24 -5 -1
+11
+16 +I8 +17
f24 +15 $24
+20 +21 f26 +17
+23 +27
R
Faupeld Method 85,700 54,000 84,700 67,400 63,000 66,500 111,300 63,000 158,500 152,000 184,000
Dev. Manning5 D e v . Method (% ) (o/o) 85,000 +9 59,500 :: r; 89,000 70,000 +Z +:;7 69,000 f25 70,000 f4 +Yf 105,000 +: 64,500 +9 +f i: -10 191,000 150,000 $1 -4 157,000 -18
J 25,800 16,300 39,400 47,400 56,000 32,700 60,600 36,000 52,000 66,100 43,200 73,000 74,800
K 29,000 18,000 43,500 53,000 62,000 38,000 67,000 41,000 57,000 72,000 47,000 78,000 81,000
-~-
-17 -13 -12 -12 -7 -15 -7 -10 -9 -6 -10 -4 -5
-6 0 -2 -2 +3 -1 +3 f2 -1 +3 -2 +3 3-3
282
High-pressure
Monobloc
Vessels
For a cylindrical pressure vessel under internal pressure the maximum principal stress, fl, is equal to ft as given by Eq. 14.7. The intermediate principal stress, fi, is equal to la as given by Ey. 14.11b. The minimum of the three principal stresses, fs, will be the radial stress, fr, as given by Eq. 14.6b. By comparing these equations it may be observed that fa is equal to the arithmetic average of ft and f,., orfa = (f,. + ft)/2. Substituting into Eq. 14.43 gives:
but
3” = Brut --.a2 + (ft -.M2 + cfa -fr121 4
therefore (14.44)
3 = d5 (ft - fa) At the inside surface of a thick-walled vessel
Substituting for b by Eq. 14.10b gives:
J(r=TI) = di do2pi
(do2 - di2)
(14.45)
Or in terms of the outside diameter, do, and thickness, t,
3cr=rij = d (do/02Pi
(14.46)
4(4/O - 4 Voorhees analyzed the experimental data obtained on notched-bar samples tested at elevated temperatures (211, 214) and on pressure vessels tested at elevated temperatures and high pressures (204, 211). He concluded that the equivalent stress, 3 (shear-stress invariant) was more useful in correlating these experimental data than the maximum principal stress or the maximum shear stress. Voorhees also reviewed the work of other investigators in this field and concluded that these studies also indicated the usefulness of the equivalent stress (211). 14.5b Effect of Creep in High-pressure Vessels. The initial stress distribution at the time of first application of load may be determined by means of Lamk’s analysis. Under the influence of elevated temperature and highpressure stress gradients the shell material creeps causing a redistribution of the stresses, as mentioned previously. The most rapid stress redistribution will occur in the region of greatest stress. This region is located near the inner surface of the shell, and the maximum principal stress is the tangential stress, ft. For a given material, given dimensions, and given operating conditions, a vessel at elevated temperature will have a definite life prior to rupture termed the “rupture life.” Creep-rate curves and stress-rupture curves for various materials of construction are available for design purposes. Typical examples of such curves are shown in Fig. 2.15 and Fig. 2.16, respectively. The rupture life of a vessel at elevated temperatures is dependent upon the history of the stress conditions and creep phenomena. Thus, if a vessel is held at a given temperature under a high stress, it will have a shorter life than if it is held at the same stress level at a lower temperature. Voorhees verified Robinson’s theory (212) that the
fraction of the total vessel life dissipated at any stress is equal to the ratio: Actual Time at Given Stress Level Rupture Life at That Stress in a Conventional Constant-load Test ) ( To test the validity of this relationship Voorhees raised and lowered the stress levels on 18 conventional unnotched bars. On averaging the results for the 18 tests, he found that the additions of the fractions of rupture life checked with the experimental observations within 1 y0 (204). Voorhees points out that this rule of addibility of rupture-life fractions can not reasonably be expected to hold true if appreciable structural alterations occur. To analyze the effects of stress leveling in the shell of a thick-walled vessel, Voorhees arbitrarily subdivided the cross section of the shell into a number of concentric rings or shells in such a way that the conditions at the centroid of a particular ring were representative of that ring. His study indicated that a subdivision into a minimum of six concentric shells each with twice the circular cross section of the adjacent inner shell gave a satisfactory coverage of the total range of stresses across the wall. In applying this analysis the procedure consists of replacing the continuous change in stress pattern with an equivalent series of time intervals over each of which the creep rate and stress in a given ring may be considered nearly constant. The fraction of rupture life expended during each interval is calculated for each ring; and when the accumulative fraction for any ring reaches unity, rupture should occur at that location, and failure of the entire vessel is imminent. 14.5~
Stress
Redistribution
by
Creep
Relaxation.
Consider two adjacent rings in the shell of a thick-walled vessel with the inner ring having a principal stress of f2 and a creep rate of C2 and the outer ring having a principal stress of fl and a creep of Cr. If f2 > fr, then C2 > Cr with both rings at the same temperature. After some creep has occurred, the plastic strain in the inner ring will exceed the plastic strain in the outer ring, but as both rings are joined by the fibers of the material, this difference in plastic strain must be absorbed by elastic strains in each of the two rings. These elastic strains result in an increase in the stress in the outer ring and a decrease in the stress in the inner ring. On the basis of the assumption of plastic incompressibility the changes in principal stress are equal to 2G times the corresponding principal elastic strains. A similar relationship holds for the elastic changes involving the equivalent stress and strain (204). The relationship between the modulus of elasticity, E, Poisson’s ratio, p, and the factor 2G is as follows (29) : E 2G = __ 1+/J
(14.47)
where E = modulus of elasticity in tension, pounds per square inch G = modulus of elasticity in shear, pounds per square inch p = Poisson’s ratio at operating temperature = 0.32 for most high-strength steels at lOOO1300” F
Monobloc
In the case of plain carbon steels and alloys with high creep rates, the effect of stress redistribution by creep is a stress equalization in a short period of time with 1% creep or less. This is followed by a progressive thinning of the shell, an increase in the stress, and a reduction of the rupture life of the vessel. Steels with slow creep rates behave differently. In this case the creep rate at the operating condition may not be rapid enough to produce stress equalization before an appreciable fraction of the rupture life of the vessel is consumed. Thus, the steels with iow creep rates may be used at higher temperatures and greater pressures, and the phenomenon of stress equalization may not occur. 14.5d Example Calculation 14.3, Illustration of the Voorhees Method. To illustrate this procedure, a vessel
operatmg at 1050” F and 5500-psi internal pressure will be considered. The vessel is fabricated of annealed carbon steel iraving a modulus of elasticity of 24,000,OOO psi at 1050” F. The vessel has an outside diameter of 12 in. and an inside diameter of 6 in. Creep-rate data for this steel are given in Fig. 14.9, and the stress-rupture curve is given in Fig. 14.10 (211). For purposes of calculation the cross section of the shell is arbitrarily divided into six rings each having an area equal to twice the area of the adjacent inner ring. Thus the area ratios will be 1, 2, 4, 8, 16, and 32; and the innermost ring will contain j&a of the total area, the second ring 34a, and so on. Table 14.6 presents a summary of the preliminary calculations for these six rings. In reference to Table 14.6, the six ring divisions are numbered from the outside diameter inwards, as indicated in column 1. Column 2 gives the fraction of the total cross-sectional area contained in each ring. Column 3 gives the ratio of the centroid radius of each ring to the outside radius of the vessel. Columns 4, 5, 6, and 7 give the ratios of various calculated stresses to the internal pressure that exists at the centroid of each ring (and also at the inner and outer surfaces). The tangential stress, ft, in column 4 was calculated by use of Eq. 14.12. The
Vessels
at
Elevated
Temperatures
283
axial stress, fa, in column 5 was calculated by use of Eq. 14.1; and the radial stress, fr, in column 6, by Eq. 14.13. The equivalent stress (shear-stress invariant), f, o f c o l u m n 7 was determined by use of Eq. 14.43. The next step in the calculation involves the determination of the “rate of redistribution of the initial stress gradients,” given in Table 14.7, as controlled by creep relaxation. Data on creep rate versus stress at 1050” F for the annealed carbon steel under consideration are required in this step. Such data are given in Fig. 14.9. In reference to Fig. 14.9, the dashed line is taken as the average of the test data. The creep rates corresponding to the values of stress f, used for column 7 of Table 14.6, are taken from the dashed line of Fig. 14.9 and tabulated in Table 14.7. The differential strain rate at each interface between successive pairs of rings is determined by subtraction. This is illustrated for the innermost three shells as follows: Shell No.
Effective Creep Rate
6 5 6-5
0.040 -0.025 0,015 differential rate
5 4 5-4
0.025 -0.011 0.014 differential rate
For a short time interval the creep rate may be considered constant, and the creep-rate differential may be converted to a stress change by multiplying the differential creep rate by the assumed time interval and by the shear modulus of elasticity. In the case of the interaction between rings number six and five and between rings number live and four, the stress changes are 546 psi and 509 psi, respectively, calculated as follows. For rings 5 and 6,
.I
0.015In. (in.) (hr)
(0.002 hr) (24 X lo6 psi)
= 546 psi
6
0 to 1 o- 10 to 45 (minimum . - - - - so
. - -
Creep Fig. 14.9.
rate, in.lin.lhr
Creep rate versus stress at 1050’ F. for annealed carbon steei (204).
II 0.01
I
I
rate petlod)
I I
(Courtesy of the American Chemical Society.)
0.1
284
High-pressure
t 5-
Monobloc
Sampling Code direction 0
0 . 0 Ll
Vessels
Specimen diameter
Longitudinal Longitudinal Longitudinal Tangential Radial I
0.160 0.350 0.505 0.160 0.160 I
I
I I I
I
I
I
1
10
Fig.
14.10.
Rupture life, hr
Stress verse rupture life for three conditions (204).
For rings 4 and 5, (0.002 hr) (24 X IO” psi) & = 509 psi (. ) The calculated results are summarized in Table 14.8. The stress interaction results in the relaxation of the stress in the inner shell and a transfer of the stress to the adjacent outer shell. The transfer of stress is distributed inversely as the respective areas are distributed. As the outer shell of a pair has twice the area of the inner adjacent shell because of the selection of the area sequence, the inner shell will receive two thirds of the total stress change. By following the procedure illustrated in Table 14.8, the stress changes for all six rings are determined, and the net change for each ring is established. After the initial time interval of 0.002 hours under load, creep will reduce the equivalent stress in ring number 6 to 12,430 - 364 = 12,066 psi. In ring number 5 the net change results in an equiva-
Table 14.6.
Position
and
Stress
Pattern
at
Centroids
of Each of Six Rings in Vessel with OD = 21D (204)
1
2 Fraction of Shell Total Cross NO. Section __. ..ID
3 PIP” (at centro’d,
5 4 3 2
I943 3
0.506 0.523 0.555 0.617 0.723
O’D
3%3. .
0,900
6
lax
“6 3 -%i a
96 3
Illlll
4
5
6
7
j&e 1.67 1.64 I.55 1.41 1.21 0.97
fo/P 0.33 0.33 0.33 0.33 0.33 0.33
f/P l/P -1.00 2.30 -0.97 2.26 -0.88 2.10 -0.74 1.86 -0.54 1.52 -0.30 I .lO
0.74 0.67
0.33 0.33
-0.08 0.00
0.71 0.58
100
1000
(Courtesy of the American Chemical Society.)
lent stress at the end 0.002 hours of 11,570 + 182 - 339 = 11,413 psi. The fraction of the rupture life expended in each ring during the time interval of 0.002 hours may he determined by use of the rupture-life data shown in Fig. 14.10. For ring number 6 the rupture life of the vessel at the initial stress of 12,430 psi at 1050” F is estimated from Fig. 14.10 to be 12 hours. The reduction in stress caused by creep during 0.002 hours has increased the life of the vessel to 13 hours. Using an average life of 12!5 hours, we find that the fraction of the life used up during the interval of 0.002 hours is equal to (O.O02/12>Q(lOO%) = 0.016%. The calculation sequence is repeated, starting with the stress distribution existing at the end of the first time interval. The fraction of the rupture life consumed in this second interval is then determined. When the summation of the life fractions equals unity, failure is imminent and the total of the accumulated time intervals gives the anticipated rupture life. As successive creep occurs with relaxation of the higher stresses, the stress variations across the wall tend to level out with a decrease in creep rate. Consequently, longer time intervals may be used in each successive calculation. In the above example the stresses are essentially uniform across the wall with a value of 6000 psi at the end of 1.5 hours. This corresponds to an expenditure of about, only 0.8% of the rupture life of ring number 6. The reduction of the maximum stress from 12,430 psi to 5800 psi appears to increase the rupture life of the vessel from 12 hours to about 840 hours, as indicated by Fig. 14.10. However, the continuation of creep tends to increase the stress level and accelerate the creep rate, and this shortens t)he life of the vessel. The stress rise is proportional to the rate of change of the
Prestressed Monobloc
ratio of the diameter to the wall thickness. In the absence of localized bulging the st,ress rise at the end of a period of creep will equal the initial stress at the beginning of the period times the ratio:
Shell No. 6
1 + Creep Strain during the Interval _---.1 ( - Creep Strain during the Interval )
5 5 4
The creep rate at a stress of 6000 psi is 0.00048 in. per in. per hr (see Fig. 14.9). A period of 8.5 hours at this condition would result in a total creep of (8.5 hr)(0.00048) = 0.0041 in. per in. Therefore, at the end of a total elapsed time of 10 hours the uniform equivalent stress should be:
(. > 1.0041
6000
,
psi ox
14.6 PRACTICAL CONSIDERATIONS
In the case of plain carbon steels and alloys with high creep rates the amount of creep required to cause incipient failure as a result of out-of-roundness, eccentricity, or extraneous stresses is very small. After stress equalization occurs, the vessel continues to deform plastically, and any irregularity or out-of-roundness is accentuated; thus instability is produced, followed by rupture. Voorhees, Sliepcevich, and Freeman (204) recommend that for the case of plain carbon steels and alloys with high creep rates, the selected design stress be equal to 0.8 times the value of the stress producing a 2% creep during the anticipated life of the vessel. As the initial stress is rapidly equalized, the mean integrated stress will be reached in short time and may be calculated by formal integration rather than by the stepwise procedure previously described. This integration is: (14.48)
Table
14.7.
Initial
Shell No.
7, psi
6 5 4 3 2
12,430 11,570
i
10,240
8,340 6,050 3,910
Conditions
for
an
Vessel at 5500 psi (204)
Effective Creep Rate, in./in./hr 0.040 0.025
0.011 0.0027 0.005 0.0008
14.8.
Stress Changes Resulting from Creep (211) Total Stress Interaction, psi 546
Stress Change in Each Ring, psi -g ( 5 4 6 ) = +$$ ( 5 4 6 ) = - g (509) = +g ( 5 0 9 ) =
509
-36k +182 -339 +170
But bs Eqs. 14.44, 14.7, and 14.11b,
Substituting the above equation into Eq. t-L.47 forjgives:
= 6045 psi
Although the stress rise is only 45 psi, it r,esults in a decrease in the rupture life of the shell from 840 to 780 hours. The first 1.5 hours consume 0.8%, and the next 8.5 hours 1.5% of the rupture life of ring number 6. A continuation of these calculations show that the rupt,ure life of the vessel would be reached after 300 hours if no local deformation or out-of-roundness developed. However, small irregularities in geometry are augmented by creep, and this shortens the rupture life of the vessel. Voorhees states that an experimental vessel with similar loading failed after 55 hours (204).
4nnealed-carbon-steel
Table
285
Vessels
(14.49)
Equation 14.49 may be set. equal to 0.8 of the st.ress giving 2 o/o creep during the anticipated life, and the prc )portions of the vessel determined. This proposed design procedure is not recognized by the code but should be satisfact,ory for most of the more ductile high-temperature alloys which show a marked degree of strengthening in a notched-bar rupture test. In t.he case of alloys for high-temperature service having high creep strength, the design should be based on stressrupture data rather than creep rate. The recommended stress for such a design is 0.8 of the stress for rupture in t,he anticipated service life of the vessel (204). In order for this design criterion to be used, the operating conditions of temperature and pressure must be carefully controlled, and reliable data on the alloy employed must be available. Normally the maximum equivalent, stress, J, will be the initial equivalent stress at the inner radius, pi. For ideal conditions the shell material at the inner wall should have a high creep rate in order to quickly relax t.he high initial stresses. On the other hand, a low creep rate is desirable at the outer surface in order to restrain the shell of the vessel from thinning and to maintain stability-. These two criteria are incompatible for a single material, but both can he realized if two shell constructions are used (204). The inner shell material should be chosen to have moderate creep strength and to be capable of withstanding extended creep without rupture. The material for the outer shell should have a low creep rate and a high creep strength. Such a design may be used with loose fits since the inner shell will creep until contact is made wit.h the outer shell. 14.7 PRESTRESSED MONOBLOC
VESSELS
14.7a The Advantage of Prestressing. The advantage of prestressing a vessel is either the reduction of the maximum stress existing under operating conditions or the reduction of the required shell thickness when a specified maximum allowable design stress is used. Regardless of the theory employed to calculate the maximum stress in t h e
286
High-pressure Monobloc
t ” p 5. G
Vessels
tp.s.
P
P
‘i
‘i 4
3
8
‘0
7
‘0 ET
1
2
64
(a) Fig. 14.11.
wall of a thick-walled vessel, a nonuniform stress distribution under pressure will be found to exist in a monobloc vessel that has not been prestressed. The principle of prestressing is to induce a permanent residual compressive stress existing under zero pressure at the point in the shell where the maximum tensile stress is induced under pressure. 14.7b Ideal Stress Distribution. The ideal stress distribution in a prestressed shell was discussed by Maccary and Fey (194) and was shown to depend upon the theory being used for design. When the maximum-principal-stress theory is the theory used, Fig. 14.11, detail a, shows the stress distribution under load for a nonprestressed shell whereas detail b shows the stress distribution under load for an ideal prestressed shell. In reference to detail b, by summation of forces Zpiril= 2i:‘fdrl
or = area l-2-3-4
(14.50)
where n.p.s. = nonprestressed 1
favg n.p.s. = & = PiK-l fallow. n.p.8. =
(14.51)
Pi $$
In reference to detail b, by use of the same maximum stress (p.s. stands for prestressed)
Also, by summation of forces
piri = (favn ,.s.)(~I,.s.) = area 5-6-7-8
(14.52)
Equating Eq. 14.50 to Ey. 14.52 (for the same pressure and inside diameter) gives: Pi(z)
tw. = Pi ($q)
tn.,,.,.
Shell thickness, prestressed t,.,. Shell thickness, nonprestressed = c. = ($2) (K+)
(14.54)
The concept of ideal prestressing is useful for realizing the possibilities of prestressing. However, the idealprestress condition can only be approximated by the method of autofrettage of monobloc vessels.
Stress distributions in shells under load.
piri = /: f dr = (fayg n.p,s.)(tn.p.s.)
K = difauow. + p)/(fallow. - P)
# 5 6
(b)
CF=
Thus, employing the maximum-principal-stress theory, in which the maximum stress is determined by ILamB analysis, and employing a given maximum allowable stress, one finds that the use of an ideal-prestressed vessel instead of a nonprestressed vessel will result in a reduction in shell thickness of (K + 1)/(K2 + 1). To calculate this reduction in required thickness the design pressure and maximum allowable working stress must be known. Equation 14.14b can be rearranged to solve for K.
= $$ (14.53)
14.8
THERMAL
PRESTRESSING
The work of Voorhees (211) on the creep of vessels subjected simultaneously to high-temperature- and highpressure-service conditions suggests the use of creep to thermally prestress a thick-walled vessel. Voorhees reports that a creep of 1 y0 is sufficient to produce stress equalization under the operating pressure. If a vessel has the pressure on it raised to the operating pressure and then is heated slowly and uniformly until a 1% strain from creep has resulted, the stress distribution across the vessel wall should become nearly uniform, approaching the ideal-prestressed condition shown in detail b of Fig. 14.11. If the pressure is released and the vessel cooled with sufficient rapidity to prevent additional creep, residual stresses will result in the shell. Upon placing the vessel in service at the same pressure as that used in prestressing but at a temperature below that producing creep, the ideal-prestressed condition will be approached for the loaded condition. Although this method of prestressing appears to be very promising for monobloc vessels, there is no known report of the use of this method. The most widely used method of prestressing monobloc vessels is that of “autofrettage.” 14.9
AUTOFRETTAGE
PRESTRESSING
The process of “autofrettage” is the oldest method of prestressing monobloc cylinders. Toward the end of the nineteenth century some monobloc cylinders for gun barrels were prestressed by a special casting technique using a chilled core, but this process has not been used in modern practice. The word “autofrettage” comes from the French language, and a literal translation is “self-hooping” from the similarity of prestressing by means of shrink fitting successive shells to the prestressing of barrels by hooping. The process consists simply of stressing all, or more usually, part, of the monobloc shell beyond the yield point by means of hydrostatic pressure. This produces a greater unit strain in the inner portion of the shell than in the outer portion. On release of the overstressing pressure the difference in unit elongation results in a residual compressive stress in the inner and a residual tensile stress in the outer portion of the shell. Usually the permanent set as measured on the inside diameter is limited to between 2.5 and 6.07& Although it might appear that the maximum strengthening be produced by overstressing the entire shell beyond the elastic limit,
Autofrettage Prestressing
this is not. the case. It is usually desirable to limit the overstressing to keep an outer layer of the shell within the elastic region. However, Manning (203) suggests that there are advantages to 100% overstrain and states that “it is doubtful whether they (other workers) have obtained the fullest possible advantage from the peculiar redistribution of stress which overstrain brings about.” 14.9a
,
Stress Relationships for Autofrettage Pressure.
The theory of the autofrettage process has been studied by a number of investigators. One of the original investigators was Macrae (192). His work was later reviewed by Newitt (191). Other reviews and discussions of the subject have been presented by Hill (197), Comings (218), Prager and Hodge (195), Faupel (201, 216), and Faupel and Furbeck (215). Hill considers the autofrettaging of spherical shells as well as cylindrical shells. Faupel and Furbeck report experimental data on autofrettaged vessels tested to bursting and compare the theory with experimental results. The procedure used in the analysis of autofrettaged shells consists of the following: 1. The determination of the autofrettage pressure which will locate the elastic-plastic interface at the desired position in the wall of a vessel having a specified inside and outside diameter. 2 . The determination of the autofrettage-pressure stresses. 3. The determination of the changes in pressure stresses resulting from the unloading process. 4. The combination of residual stresses with operatingpressure stresses by the method of superposition. 5. The calculation of elastic breakdown and bursting pressures. The derivation of the relationships used in the procedure outlined above is presented by Prager and Hodge (195) and will be only summarized here. Prager and Hodge (195) have shown that integration of Eq. 14.38 leads to a relationship for the autofrettage pressure required to locate the elastic-plastic interface at a radial distance of r, (between ri and ro), or
P = .fs.y.
1-k~-22ni
(
r.
PC >
(14.55)
where js,Y. = yield limit in simple shear, pounds per square inch (see Eg. 14.35) r, = radial distance of elastic-plastic interface, inches ri = inside radius of shell, inches r,, = outside radius of shell, inches p = autofrettage pressure, pounds per square inch The two limits p* and p ** of Eq. 14.55 are reached when rc = ri or when rc = r,,, respectively. p* =f*.Y. [I - ($1 P **
= 2ja.Y. In 5 ri
(14.56)
where p* = autofrettage pressure that will plastic interface at ri, pounds P ** = autofrettage pressure that will plastic interface at r,, pounds
287
piace the elasticper square inch place the elasticper square inch
The elastic-plastic interface located at radius P, divides the shell into two zones, a plastic zone and an elastic zone. Separate relationships must be used for determining the autofrettage-pressure stresses for each zone. For the plastic zone, where ri 5 r < rc, jr = -js.y.
1 - $Y - 2 In i
ft = jg.y. 1 + $ + 2 In i ( >
>
(14.583 (14.59)
.fa =6.,.(!$+2lnk) where jY = autofrettage radial stress, pounds per square inch jt = autofrettage tangential stress, pounds per square inch ja = autofrettage longitudinal stress, pounds per square inch For the elastic zone, where r, 5 r 5 r,,
jr = p” 1 - 5 ( > jt = p” 1 + $ (
P
>
,, _ .fs.y.rcz r.
2
14.9b Changes in Stresses as a Result of Unloading the Autofrettage Pressure. V ESSELS WITH U NLOADING S TRESSES WITHIN THE ELASTIC REGION. Prager and Hodge (195) have shown that the
method of computing the changes in stresses resulting from unloading the autofrettage pressure will depend upon whether or not the unloading stresses exceed the compressive yield strength of the material. Vessels will have unloading stresses within the elastic region if
and or if and
K = 2 5 2.22 pi
(14.64)
p* _< Ap < p**
(14.65)
K > 2.22 AP 5 ZP*
(.?4.66)
where Ap = the unloading pressure, pounds per square inch For vessels satisfying the above conditions, the following relationships give the changes in autofrettage-pressure,
288
High-pressure
Monobloc
Vessels
Also,
stresses for both the elastic and plastic regions:
U-t) = ft + Aft
Afa = -Ap’
(14.69) (14.70)
where Ap = (pl - pz) pl = autofrettage pressure, pounds per square inch p2 = unloading pressure, pounds per square inch (normally zero) V ESSELS WITH U NLOADING S TRESSES REGION. Vessels will have unloading
BEYOND
THE
ELASTIC
stresses beyond the
compressive yield strength if (14.71) and
2p* 5 Ap < p**
(14.72)
2js.y.
1 - :: - 2 ln z 3>
(14.73)
where rj = radial distance of elastic-plastic interface, inches The changes in autofrettage-pressure stresses upon unloading for the plastic zone are: i-i 5 r 5 rj (14.74).
I
where (ji) = residual tangential stress. pounds per square inch jt = jt from Eqs. 14.59 and 14.62 Aft = Aft from Eq. 14.68 or from Eqs. 14.75 and 14.78 And (14.82) Ma) = fa + 4-a where (ja) = residual axial stress, pounds per square inch ja = ja from Eqs. 14.60 and 14.63 Aja = Aja from Eq. 14.69 or from Eqs. 14.76 and 14.79 14.9~
Combined Stresses under Operating Pressure.
The combined stresses under operating pressure are determined by the method of superposition. Lame’s equations for j,., jt, and ja (Eqs. 14.6b, 14.7, and 14.1, respectively) are combined with the appropriate residual stresses of Eqs. 14.80, 14.81, and 14.82. The unit radial strain in both t,he elastic and plastic regions under autofrettage pressure is given by:
For the conditions above, Eq. 14.55 must be modified as follows to give the elastic-plast,ic interface after unloading: Ap =
(14.81 j
(14.83) I
E where G = ___ 20 + d and
(See Eq. 14.47.)
E = modulus of elasticity of material, pounds per square inch p = Poisson’s ratio
Upon unloading of the autofrettage pressure a change in unit radial strain, A+, occurs. For conditions satisfying Eqs. 14.64, 14.65, and 14.66 (14.84)
(14.76)
I
where Ap’ is given by Eq. 14.70. For conditions satisfying Eqs. 14.71, 14.72, and 14.73
For the elastic zone, (14.85) (14.77) (14.78)
The residual autofrettage stresses are determined as follows: (14.80) (fv) = fr + Ah where (jr) = residual radial stress, pounds per square inch jr = j,. from Eqs. 14.58 and 14.61 Aj,. = Aj,. from Eq. 14.67 or from Eqs. 14.74 and 14.77
14.9d Example Calculation 14.4, Determination of Optimum Autofrettage Pressure. An autofrettage vessel is to
be designed to contain a fluid at 20,000 psi internal pressure. An internal diameter of 12 in. is specified. A high-strength steel having a 70,000-psi yield point is to be used. Using a factor of safety, X equal to 2.0, the required shell thickness, according to the maximum-principal-stress theory, is 5.5 in. (for a similar vessel shell). Therefore, K = do, d; = 1.916. The shell thickness for the ideal-prestressed condition is, by Eq. 14.53:
therefore
Ktp.s. + 1 2.916 o 623 _---=-= tn.p.L% K2 + 1 4.67 ’ t
p.s. = (5.5)(0.623) = 3.43 in.
i /
Autofrettage
f7 = -.f8.y.
ft = f8.Y. therefore
therefore
40,300 psi
ro >
(
6.2111; .
>
For r = 6, 7, 8, and 9 in.,fa = -3,030, +9,300, +20,900, and +29,700 psi, respectively. For the elastic region: By Eq. 14.63 P
The upper limit is:
(40,300) (81) ,, _--= .fs.y.rcz = 29,700 psi 2 110.3 r.
By Eq. 14.61
p** = 2fs.y. In 2 ri I11
10.5 __ 6
fT = p” 1 - r!I ( >
therefore
= 45,200 psi
J.=29,700(1-y)
The stresses for r, = 6, 7, 8, 9, and 10.5 in. will be determined. Only the calculations for rC = 9 in. will be presented here. The autofrettage pressure required to locate the elasticplastic interface at r, = 9 in. is determined by Eq. 14.55. 1 - $1 - 2 In 2
For r = 9, 9x, and 1035 in., jr = -10,750, -4760, and 0 psi, respectively. By Eq. 14.62
I - & - 2 In +
jt = 29,700(1 + y) >
For r = 9, 9x, and 1056 in., ft = +69,000, +64,000, +59,400 psi, respectively. By Eq. 14.63
Under autofrettage pressure the radial, tangential, and longitudinal stresses for r, = 9 in. are as follows.
\
\
ft = p” 1 + 5 ( >
therefore
>
= 43,500 psi
I
(
fa = 40,300
= 27,150 psi
P
2
r 1 + ” + 2 111 r. PC >
therefore
= 40,300(, -i&)
(
>
For r = 6, 7, 8, and 9 in.,ft = +37,400, +49,800, +61,100, and +69,000 psi, respectively. By Eq. 14.60
P * = f8.y. 1 - <
= 40,300
1 - & - 2 In $
ft = 40,300 1 + & +21n$ ( >
To determine the optimum autofrettage pressure a number of values of r, must be selected, and the corresponding stresses evaluated. The upper and lower limits of the autofrettage pressure are determined by Eqs. 14.56 and 14.57. The lower limit is:
(
(
For r = 6,7, 8, and 9 in., jr = -43,500, -31,000, -19,400. and - 10,700 psi, respectively. By Eq. 14.59
f,.,. = 70,000 psi
p = f*,,,
1 - $ - 2 In k)
jr = -40,300
By Eq. 14.35
= (2)(40,300)
(
therefore
r, = 10.5 in.
(
289
For the plastic region: By Eq. 14.58
As the ideal-prestressed condit,ion cannot be obtained by autofrettage, an intermediate value between the idealprestressed and the nonprestressed value will be selected. Determine the autofrettage pressure which will give the minimum combined hoop stress if a shell thickness of 41s in. is used. Solution: Data: /ti = 6.0 in.
js.&= %I =
Prestressing
\I
fa = p" = +29,700 psi
/
-
-
-
and
290
High-pressure
Monobloc
Vessels
For r = 6, 7, 8, 9, 9x, and IO>5 in., Ajr = 43,700, 26,500, 15,400, 7640, 2390, and 0 psi, respectively. Substituting into Eq. 14.68 gives:
80 70
t--CR4
60 50 40
Ajt = - Ap’ therefore Aft = -21,200(1
+ y)
For r = 6, 7, 8, 9, 95
-50
8
therefore Afr = -21,200(1
-y)
Fig.
14.13.
9 r (inches)
10
Residual stresses for rC = 9 in.
11
Results
RESULTS FROM EXPERIMENTAL OF AUTOFRETTAGE
291
Autofrettoge
-5 -10 -15 -20 -25 -30 -35 -40
(14.87) Fig.
(14.88)
Faupel and Furbeck (215) determined experimental values of residual stresses in autofrettaged vessels by means of the method developed by Sachs (219). In this procedure, prestressed thick-walled cylinders were aligned in a lathe, and concentric layers of metal machined from the bore in successive steps. The axial and circumferential strains were recorded for each successive cut (usually about 0.05 in. measured on the diameter). Also, after each cut the bore diameter was measured to within 0.0001 in. by use of flush-cooling and temperature-compensating gages. These data on the strains and bore diameters were substituted into the following equations developed by Sachs (219) to give values of residual stresses based upon these measurements.
(14.89)
ft’ = + P2 [ (Ao-An)-$-(+&ye](14.90)
1
of
STUDIES
(14.86)
EL2
Studies
25
Faupel and Furbeck (215) experimentally studied residual stresses in autofrettaged vessels constructed of plain carbon steel and various alloys. The theoretical residual stresses were computed from relationships which may be derived from equations presented earlier in this chapter. In these derivations Eq. 14.70 is substituted into Eqs. 14.67, 14.68, and 14.69, and these last three equations added to Eqs. 14.58, 14.59, and 14.60, respectively, to give the residual stresses:
fa’ = i-“- [ ( A o - A ) $ - ct
Experimental
60
operating hoop stress calculated by Lam&‘s relationship is also shown. This operating-pressure stress is combined with the residual stresses to give a series of combined stresses for various values of rc. The curve at the top of the figure shows the locus of the maximum combined stress and indicates a minimum value where r, is about 7 in. If the vessel had not been autofrettaged, the maximum stress would have been 39,600 psi. If the vessel had been autofrettaged so that r, was equal to 7 in., this stress would have been reduced to 35,000 psi. This amounts to about an 11 y0 reduction in the maximum stress at operating pressure. Examination of Fig. 14.14 indicates that the vessel could be overautofrettaged with the result that the combined stress would be greater than that in the same vessel without autofrettage. 14.10
from
(14.91)
14.14.
6
7
8
9 r (inches)
10
11
Plot of residual hoop stresses and combined operating
hoop-
pressure stress for various values of rC.
where A0 = initial cross-sectional area of cylinder including the bore, square inches A= cross-sectional area of the bore following a machine cut, square inches CC= (E, + pE,), inches per inch O= (EC + pE,), inches per inch E, = axial strain, inches per inch EC = circumferential strain, inches per inch E = modulus of elasticity, pounds per square inch /J= Poisson’s ratio In one test a cylinder with an outside diameter to inside diameter ratio of 2.74 fabricated of Cr-Ni-MO-V steel was heat-treated to a hardness of 42 Rockwell C to remove residual stresses. It was then autofrettaged at a pressure of 200,000 psi for 4.5 hours. This procedure was followed The by a stabilizing heat-treatment at 650” F for 3 hours. cylinder was checked for stability and elastic behavior up to the autofrettage pressure by repeating the loading to 200,000 psi through four cycles of operation, and then the residual stresses were determined by the Sachs method. The results are shown in Fig. 14.15.
292
High-pressure
Monobloc
Vessels
According to Manning, two assumptions are involved: (1) the cross-sectional area remains constant under strain, (2) the relation between maximum shear stress and maximum shear strain is the same in a cylinder as in a torsion test. The first assumption permits the calculation of the shear strain at any point in the wall of the vessel and the second permits the calculation of shear stress from torsion data. Equation 14.2b (with proper allowance for sign convention) is a statement of the first assumption, or:
2.757’ -_I
P
40
ff -jr = P $ The corresponding maximum shear stress at radius P is given by Eq. 9.85: (9.85)
fs = at - .fr) If Eq. 9.85 is substituted into Ey. 14.2b we have:
In Ey. 14.92 the value of P must be that of the strained condition. This can be expressed as follows for the int.egrated form : -200 .f7 = - p i +
2
~~~~
d(r + Ar)
(14.93)
-240
where Ar = shift of point at radius r as a result of strain induced by pressure pi Ari = shift of inside radius, r-i, as a result of strain induced by pressure pi pi = internal autofrettage pressure
-280
I
6
Fig.
14.15.
I
,
6
Theoretical and experimental residual-stress distributions in an
outofrettaged with
I
4 2 0 2 4 Cross-sectional area, square inches
permission
cylinder (215). (Extracted from Transactions of of
the
publisher,
the
American
Society
of
the ASME
In order to integrate Eq. 14.93 the shear st.ress, fS, as a function of strain must be known. For this purpose Manning uses the results from a torsion test and the following eyuation given by Nadai (241):
Mechanical
(14.91)
Engineers.)
In Fig. 14.15 the theoretical stresses predicted by Eys. 14.86, 14.87, and 14.88 show good agreement with the experimental values. 14.11
MANNING’S
If the cross-sectional area is constant, the following relation is true:
METHOD
Table 14.5 compares experimental values of bursting with various theories of failure and shows good correlation between the experimental values and those predicted by the method of Manning (203). Manning’s hypothesis is that since vessels for high-pressure service are usually fabricated of high-tensile steel, which usually fails by shear, an experimentally determined shear stress should be used in the design of such vessels rather than tensile-test data. He proposes that a simple torsion test be used to determine the shear stress-strain relationships and has presented methods of using these data in the design of high-pressure vessels (203, 237-240).
I
where T = toryue 8 = unit twist d = diameter of shaft used in torque test
--\--
-
~~-
or
m2 - vi2 = r(r + 2r
Ar)” - r(ri + Ari)’
Ar + Ar’ = 2ri Ari + Ari’
(14.95)
Using Eqs. 14.94 and 14.95 and experimental data for torsion tests, Manning integrates the integral group of Ey. 14.93 and plots twice the integral as a function of radius ratio r/pi. If twice the integral is combined with -pi as indicated in Eq. 14.93, the radial stress is obtained under autofrettage conditions. This stress curve is plotted in Fig. 14.16 and is indicated by the line labeled “Manning’s value.” The curve is plotted over a range of r/pi from 1.0 to 10.0 but the integration has been made for a vessel of infinite external radius.
Manning’s
Method
293
240,000
180,000 '2160,000 3 '3 140,000 2 E 120,000 8 t g 100,000 I ex 80,000 60,000
0 1
2
3
4
5
6
7
8
9
Radius ratio Fig. 14.16.
Comparison of Manning’s method with Eq. 14.58.
For a vessel of infinite external radius the maximum value of the integral for the example given (see page 511 of Reference 239) is 185,800 psi. In the example quoted the elastic-plastic interface was placed at a radius ratio of 3.0. The dashed line was obtained by use of Eq. 14.58, using values of ri = 1 in., r, = 3 in., r, = 00, and js.Y. = 57,800 psi. The value of js.Y. = 57,800 was obtained from Manning’s plot on page 511 of Reference 239. Note that the formal integration given by Eq. 14.58 gives essentiaily the same curve as obtained by Manning, who used the rather tedious procedure of stepwise integration. The problem lies more in the determination of the correct shear yield stress than in the method of integration. The reason for using the ratio of r/ri on a logarithmic s&e is for convenience in vessel design. Manning points out that for an isotropic material the same stress and strain curves versus radius ratio (r/rJ will apply regardless of the absolute dimensions of the vessel. Jasper (242,243)) on the other hand, states that for “very thick-walled vessels the ratio of thickness to diameter is not as good a criterion as delinite thickness.” It must be recognized that vessels may not behave ideally and their characteristics may be influenced by the nonisotropic nature of the material of construction. Such deviations can only be determined by lrsts with vessels of full size. In the absence of such data correlations based on dimensional similarity are the only method of analysis. The abscissa of Fig. 14.16 is labeled “Radius Ratio.” I‘ltk ratio is equal to r,, ‘P; for the case in which r, = m.
For use in the analysis of actual vessels in which r, is less than infinity this ratio becomes 3r/r, and a correction must be made to the values of stress determined from the figure since the limits of the integration are no longer from one to infinity. The use of the figure in this manner assumes the criterion of dimensional similarity mentioned earlier. To illustrate the use of the stress curve examples will be given. Example 1. ri = 3 in., PC = 9 in., and r, = 30 in. Therefore, for r = ri = 3 in., 3r 3(3) _ 1 -= 9 rc From Fig. 14.16, j,.(uncorrected) = - 185,800 psi. For r = r,, 3’=3 i-c From Pig. 1.1.16, jrc ,,,, corlected, For r = ro,
= -57,800 psi.
3r -= (3)(30) -~ = 9 rc From
Fig. 14.16,
fr(nncorrected)
1o
= -5200 psi.
Since the radial pressure at I-,, is zero (for a vessel with zero external pressure) and since the shape of the radial
--
-. .
High-pressure Monobloc Vessels
294
X
From the figure, fr(UncOrre&d) = -20,800 psi. To correct these values so that jr, = 0, -20,800 psi is subtracted from the values given above to give: f,.< = -104,600 - (-20,800) = -83,800 = autofrettage pressure fr, = -57,800 - (-20,800) = 37,000 psi and
Change in outside diameter F i g . 1 4 . 1 7 . H y s t e r e s i s i n a u t o f r e t t a g e d v e s s e l ( 2 3 7 ) . ( C o u r t e s y of American
Chemical
Society.)
stress curve depends upon the radius ratio (assuming dimensional similarity), the curve has only to be shifted downward until the radial stress is zero at 3r/re = 10. The same result may be obtained by subtracting the right-hand intercept of 5200 psi from the uncorrected stress values obtained from the figure. Therefore the corrected stresses become: fri = - 185,800 - (-5200) = -180,600 psi = autofrettage pressure frc = -57,800 - (-5200) = -52,600 psi and fr. = 0 The radial stress at any other point in the wall of the vessel can be determined in a similar manner. Example 2 . For P = ri,
Pi = 8 in., r, = 1 2 in., r, = 2 0 in. 3r -zr (3)(8) _ 2 PC 12
From Fig. 14.16, fr(uncOrrected) For r = rc,
= - 104,600 psi.
-= 3 3r rc From the figure, fr(Unc,,rrected) For r = r,,
= - 57,800 psi.
3r (3)(20) = 5 -= __ 12 PC
It should be mentioned that Fig. 14.16 could have been prepared using r,/ri = 4 or any other convenient quantity rather than 3. If such a plot were prepared with rJri = 4, then the radius ratio would become 4r/r, and would change correspondingly for any other value. Manning recommends that the elastic-plastic interface rc be located approximately at r, = 1/z for optimum prestressing with autofrettage. 14.12 PRACTICAL CONSIDERATIONS IN AUTOFRETTAGE In the process of autofrettage it is the usual practice to provide additional metal on both the inside and outside surface of the wall. The excess metal on the inner surface is necessary in order to allow for permanent radial strain and yet maintain the desired internal diameter. Also, some metal must be provided both on the inner and the outer surfaces for machining off the scale and for truing subsequent to the heat-treating operation which should follow autofrettage. The desirability of heat treatment after autofrettage can be explained by reference to Fig. 14.17. Figure 14.17 was used by Manning (237) to illustrate the problem of hysteresis in an autofrettaged vessel. If in the original overstraining the internal pressure is carried to point B in Fig. 14.17 and released, the outside diameter will have a permanent strain as indicated by point C. Decrease in diameter to C’ may result if the vessel is allowed to remain in the unloaded state. On reloading the vessel to a pressure of B’ the strain of the outside diameter will follow the path between C’ and B’, producing a hysteresis loop. However, if the vessel is given a low-temperature anneal (575”-600” F for mild steel), the hysteresis loop will be replaced by a straight line characteristic of elastic strain. Furthermore, the yield point will rise from B’ to E. This phenomenon was first investigated by Macrae (192), who termed the rise in yield from B’ to E the “elastic gain.” Thus a low-temperature heat treatment has the advantage of producing greater dimensional stability in the final vessel. One suggested procedure of design consists of determining the thickness of a shell that would burst at twice the service pressure by means of Eq. 14.42. This vessel can then be autofrettaged with a pressure sufficient to place the elasticplastic interface at the geometric-mean radius. This should be followed by a low-temperature heat treatment and machining to desired dimensions. The vessel will have a factor of safety of 2 based on the working pressure.
Practical
Considerations
P!?OBLEMS
1. A thick-walled pressure vessel having an inside diameter of 8 in. and an outside diameter Determine the maximum induced of 16 in. is subjected to an internal pressure of 15,000 psi. stress according to the maximum-principal-stress theory, the maximum-shear-stress theory, the maximum-strain theory, and the maximum-strain-energy theory. 2. Prepare a plot of the tangential stress, ft, and the radial stress, f,., for the vessel in problem 1. 3. Determine the bursting strength of the vessel described in problem 1 if the vessel is fabricated of SA-302 grade B steel (see Chapter 12 for ft.,. and Fig. 2.6 for f,.,.). 4. Calculate the stress distribution across the vessel shell for the vessel described in section 14.5d (Illustration of the Voorhees Method) after 1% strain from creep has occurred in the shell. 5. A vessel with an inside diameter of 20 in. and an outside diameter of 32 in. is to be operated at 1100” F. What is the maximum allowable working pressure if the vessel is fabricated of 18-8 type 316 steel (see Fig. 2.16) and its anticipated life is 20,000 hours? (See Eq. 14.49). 6. For the vessel described in problem 1 calculate the optimum autofrettage pressure that will give the minimum peak stress under operating conditions. 7. A reactor is needed in our plant for service at 400” F with an internal pressure of 25,000 lb per sq in. gage. We have elected to use an inside diameter of 16 in. The thickness of the vessel is to be selected so that the vessel will rupture at twice the service pressure by use of the equation of Faupel, which is based on experimental-test data. The vessel is to be autofrettaged with a pressure sufficient to place the elastic-plastic interface at the geometric-mean radius, that is, at l/r&. The physical properties of the steel are: Tensile strength = 105,000 psi Yield strength = 90,000 psi Using the above suggested design procedure, determine: u. The required thickness of the shell. b. The autofrettage gage pressure required for locating the elastic-plastic interface at the geometric-mean radius. e. The radial compressive stress under autofrettage pressure at: 1. the inside surface 2. the elastic-plastic interface 3. the outer surface.
in
Autofrettagn
295
C H A P T E R
15 m a MULTILAYER VESSELS
a certain identical value. It is assumed that both the internal and external diameters are known and that the number of shells is to be a minimum. It is also assumed that the combined cylinder is fabricated by shrinking-on each successive shell from the inside outwards and that after each shell is shrunk-on, the outside diameter is machined to size before the next cylinder is shrunk-on to the inner shell or shells. Therefore the designer must determine the number of shells and their radii plus the interference (the amount by which the outside diameter exceeds the inside diameter of the next shell). The relationships for designing such a vessel may be derived as follows: Consider a cylinder fabricated from n shells like the one in Fig. 13.1 where:
I
n the previous chapter the Lamk relationships based upon the theory of elasticity were presented for tangential, axial, and radial stress distribution in a thick-walled vessel. Also, the relationships of Voorhees and Faupel based upon the theory of plasticity were discussed. These relationships make possible the determination of the required thickuess of thick-walled vessels for high-pressure applications and the determination of the stress variations according to the theory of “plane strain.” The possibility of reducing the required wall thickness by using multiwall construction in which the concentric shells are “shrink-fitted” together will now be considered. The Lame hoop-stress equation indicates that the maximum stress occurs at the inner surface of the vessel. By shrink-fitting concentric shells together the inner shells are placed in residual compression so that the initial compressive loop stress must be relieved by the internal pressure before hoop tensile stresses are developed. Therefore the maximum hoop tensile stress as determined by- Lame’s relationship is appreciably reduced with the result that t,here is a reduction of the total wall thickness required to contain the pressure when the vessel-wall thirkness is designed with a sprcifred allowable stress.
4, dl, . . . d,-l = diamet,er
of successive intershell surfaces, inches pi = internal pressure, pounds per square inch p,, = external pressure, pounds per square inch PI, 1J.L. ’ . p,,-1 = successive interface pressures with pi and p. acting f, = hoop stress set up at the inside of each shell with pi and p0 acting, pounds per square inch (to be the same for all shells) PI’, pr’, . * pn’ = interface shell pressure that exists when pi = p,,
15.1 MULTILAYER VESSELS WITH SHRINK-FITTED SHELLS Tbe relationships which follow were first presented by H. I,. Cox in 1936 (220). The t.beory as developed is based upon the assumption that the maximum combined stresses (hoop-pressure stress plus hoop-shrinkage stress) cxistiug :rt the inner surface of each of the several shells will attain
According to the Lamk thick-walled-vessel theory, considering the rth + 1 shell and using the sign convention as giver1 by Ey. 14.6b gives: 296
Multilayer
Vessels
with
Shrink-fitted
Shells
297
Q multilayer shell showing notation used
in dcri-
vo hit
f7 = -Pr
therefore (15.1) and 46 b pr+l = -jr+1 = - a + __ = - a - i - -~ dr+12 rv+12
(15.2)
f*=h=-$+u=f2+u
(15.3)
And by Eq. 14.7
n
where u and b are defined by Eqs. 14.llb and 14.lOa, respectively. Therefore _ pr
,
-j,
(15.4)
= --2a
Subtracting Eq. 15.3 from Eq. 15.2 gives: pr+l &+I - f&r2 = --a(dr+1’ + dr2) Dividing through by dr2 gives:
Pr+l dr+12 _ j* = _ u(“$ + d,.2
Let
Fig.
I)
d~ r+l = K rtl 4
vations.
(15.6j
Equation 15.12 then becomes: FP, - pi = (1 - Of,
Substitutiug Eq. 15.6 into Eq. 15.5 gives: pr+l Kr+12 -f, = -a(K,+12 f 1)
.P~+I &+I’ ---Lq Kr+12 + 1 Pr - fq = - 2 C
%+I~ ‘+I = 1 + K,+12
(15.7)
c+1
Nowifr = 1, I
If r = 2,
,
If r = 3,
Pr+1 -
c TPT
Pr = (1 - C+df*
- Pr-1 =
(1 - Gl.fq
ClPl - p i =
(1 - Cllfn
czpz - p1 = C3P3
If r = n,
C nP0
-
-
p2
(1 -
= (1 -
(15.8)
(15.9)
(15.10) (15.11)
C,)jq
C,)p,
=
CIC&3
*
(1 -
The smallest value of j, for a given pressure difference - po) obviously exists when F has a maximum value. The method of Lagrange multipliers may be used for determining such a constrained maximum (34). This method indicates that the maximum value of F exists when (pi
K1 cz K2 = K3 , . . = K, = K Substituting Eq. 15.6 into Eq. 15.16 gives: $k!+$+ . !, 12 1 d,-1 z Cl zz c2 = c3 = . . . c, = c
(:3).f9
(15.12)
Let
- - C, = F
(15.17)
(15.18)
SubstituCng Eq. 15.18 into Eq. 15.13 gives: F = C1C3C3
ClC2 . . Glfp
(15.16)
A comparison of Eq. 15.16 with Eq. 15.9 indicates that
P n - 1 = (1 - Wfp
- pi
(15.15)
f, = -p. + p* ( )
K=
This series can be represented by:
(C1C2C.( * * *
(pi - PO) - VP, - PJ F - l therefore
It then follows from Eq. 15.8 that and similarly
(15.14)
Solving for j, gives:
Dividing Eq. 15.7 by Eq. L5./4 gives:
Let!
Diagram of
15.1.
(15.5)
(15.13)
* . . c, = C”
(15.19j
SubstituGng Eq. 15.19 into Eq. 15.15 gives: f,
=
-po + (
p* >
(15.20)
298
Multilayer Vessels
where p. = external pressure on vessel, pounds per square inch pi = internal pressure in vessel, pounds per square inch n = total number of shells in vessel wall with thicknesses satisfying Eq. 15.17 f, = combined stress at interface of each shell (pressure stress superimposed on shrinkage stress) C = constant as defined by Eqs. 15.9 and 15.6
If this vessel had been fabricated of two shells instead of three, then in order to satisfy Eq. 15.16 the interface diameter would have had to be 16.8 in. For this vessei n=2 and K - ‘1”;” - ‘“;i3i5 _ 1.4, K2 = 1.96 Substituting into Eq. 15.25 gives:
A comparison of Eqs. 15.16 and 15.18 indicates that Eq. 15.9 may be written as:
C=2K2
(15.21)
1 + K2
where K=(2)=@)=($)= . ..(-+) (15.22) Therefore it follows that .-In o2n
‘” =
(15.23)
(l;‘;i2)”
Substituting Eq. 15.23 into Eq. 15.20 gives:
f
Q
(1 + K2)“h -ho) _ p - (1 + K2)n I ’
(15.24)
2nK2"
Equation 15.24 is the general relationship for determining the maximum combined stresses at the interfaces of the concentric shells and at the inside surface of the innermost shell. The combined stresses (hoop) at these lor.ations all have the same numerical value. This equation for the usual case where the external pressure is zero (gage), p0 = 0, reduces to: pdl + K2jn .f, = 2nK2n - (1 + K2)%
(15.25)
15.la Example Calculation 15.1. To illustrate the application of Eq. 15.25, consider a multilayer vessel having an inside diameter of 12 in. and an outside diameter of 23x6 in. which has been formed by shrink-fitting. The vessel is to operate under an internal pressure of 20,000 psi and is constructed of three shells. The interface diameters are 15 in. and 183i in., respectively. Determinef,. 23.4375 K3 = _____ =
K=Kl+K2+=
18.250
=
pi = 20,000
fq(F
therefore
+ pi
(by Eq. 15.19)
f,(c- - 1) = - c”-‘po + pr
Substituting for f, by Eq. 15.20 and rearranging gives: (15.263
Pr = [‘:I:] (Pi-P01 +Po
15.lc Example Calculation 15.2. Determine the interface pressures of Example Calculation 15.1. The combined pressure existing at the two interfaces (shrinkage stress with a superimposed 20,000-psi pressure stress) is given by Eq. 15.26. P r = [c~~~-ll]
( P i -PO)
+Po
but and
PO = 0 2(1.5625)
C=2K2
~ =
1 + K2 = 1.5626
(pr)r=2
fq = 24,582 psi (compressive stress)
\
- 1) = -Fpo
1.222
- 1 (20 ooo) -1 1 ’ Therefore when n = 3 and P = 2 (second interface), substituting for (n - r) = 3 - 2 = 1.0 gives:
20,000(1 + 1.5625)3 “67 = 23(1.25)6 - (1 + 1.5625)3
\
psi
pi = pr and F = C”-’
Pr =
n=3
:I i. -~~ ~~ I+--,
26,530
+ 1.96)2
- (1 + 1.96)2
At the fth shell
therefore
K = 1.25, K2 = 1.5625
20,000(1 22(1.4)4
If the shell had been of monobloc construction, the maximum stress would have been 34,212 psi (see Example Calculation 14.1). A reduction of 2235 y0 was achieved by two-shell construction whereas a reduction of only 28>/4% was realized by three-shell construction. It is therefore apparent that from the economic consideration of shrink-lit fabrication the use of more than two shells may not be justified. 15.1 b Determination of Interface Pressures. To determine the variations in hoop stress throughout the wall of any shell by using the Lam& relationships it is necessary to determine the combined pressure at any interface (pressure resulting from internal pressure plus shrinkage stresses). Eq. 15.15 may be rewritten as follows:
1.25
Because K1 = Kz = KS, this vessel satisfies Eq. 15.16; therefore Eq. 15.25 is applicable and may be used to calculate f,. Substituting in Eq. 15.25, we find that:
therefore
f-2 =
=
1.222cn+
[
1.222n
Cl.222
-
1)w4000)
1.2223 - 1
\I /
=
5350
----- --
psi
--
Multilayer
Also, for n = 3 and P = 1 (the first interface), (p
) TT
=
=
Iu.22a2
-
11(2o~ow = 11,920 psi
1.2223 - 1
1
Vessels
15.1 d Surfaces
Determination of the Hoop Stresses at the Outer The individual shells may of Multilayer Vessels.
be treated in accordance with Lame’s theory if the pressures at the interfaces are computed by use of Eq. 15.26. The hoop stresses may then be computed by use of Eq. 14.12 as an alternative to the use of Eq. 15.25. 15.le Example Calculation 15.3. By using the vessel under consideration in Example Calculation 15.1 and the results obtained in Example Calculation 15.2, the hoopstress variation in each of the three shells may be readily determined. The Lame relationship for hoop stresses as given in Eq. 14.12 is: d.2d 2 jt = Pdi’ - do2 +” d2 do2 - di2
[ 1 Pi - PO ___ do2 - di2
Each of the individual shells can be treated as a monobloc vessel as follows. For the inner shell di = 12 in. do = 15 in. pi = 20,000 psi p. = 11,920 psi Substituting into Eq. 14.12 gives: (jt)d=di = 24,970 psi The value of 24,970 psi is within slide-rule agreement of the value of 24,582 psi obtained by use of Eq. 15.25 in Example Calculation 15.1. (ft)d=d,
= 16,870 psi
For the middle shell di = 15 in. do = 18.75 in. pi = 11,920 psi
Shrink-fitted
Shells
(ft)d=d, = 17,820 psi For the outer shell di = 18.75 in. do = 23.4375 in. pi = 5350 psi PO = 0 Substituting into Eq. 14.12 gives: (ft)d=di = 24,400
(in comparison to 24,582 psi obtained by use of Eq. 15.25)
and (ft)d=do = 19,040 psi To summarize the stresses existing in the three shells (combined shrinkage and pressure stress), 1. For the inner shell ji = 24,970 psi
(24,582)
j. = 16,870 psi 2. For the middle shell ji = 24,270 psi
(24,582)
j. = 17,820 psi 3. For the outer shell ji = 24,400 psi
(24,582)
j. = 19,040 15.1 f Determination of Shrinkage Stresses. The shrinkage stresses may be readily determined by the method of superposition. The stress variation, assuming the vessel is a monobloc shell having the same inside diameter and outside diameter as the multilayer vessel, may be determined in each by use of LamB’s equations. The stress variation in each of the shells can be determined by the methods presented in the previous section. The shrinkage stresses are obtained by subtraction with proper allowance for signs. 15.1 g Example Calculation 15.4. The vessel under consideration in Example Calculations 15.1, 15.2, and 15.3 will be used for demonstrating the method of obtaining shrinkage stresses. In Example Calculation 14.1 a monobloc vessel was considered which had the same proportions as the multilayer vessel in Example Calculations 15.1, 15.2, and 15.3. The shrinkage (shrink.) stresses are obtained by subtracting the individual combined stresses in each of the shells of the multilayer vessel from the monobloc-vessel stresses as follows.
Inner surface of inner shell (P = 6):
p. = 5350 psi
jt = 34,212 psi
(for monobloc)
Substituting into Eq. 14.12 gives:
jt = 24,970 psi
(for multilayer)
(j&& = 24,270 psi
(in comparison to 24,582 psi obtained by use of Eq. 15.25)
299
and
To summarize the pressure-stress considerations: 1. The pressure at the interface of the inner shell is equal to the operating pressure of 20,000 psi. 2. The pressure at the interface of the inner shell and the middle shell is 11,920 psi. 3. The pressure at the interface of the middle shell and the outer shell is 5350 psi. 4. The pressure at the outer surface of the outer shell is 0 lb per sq in. gage.
with
34,212 + ft(shrink.) = 24,979 Therefore fr(.&ink,)= - 9242 psi (compression).
Vessels
Multilayer
300
40,000
1
I
I
I
I
I
I
I
I
I
I
I
I
I
I
Lamb monobloc
Fig.
15.2.
Graphical
representation
of the stress distribution in vessel of Example Calculation 15.4.
- _______-----_-O- -------._--------184 -846
6
7
8
9
10 11 Radial distance, inches
Outer surface of inner shell (P = 7.5): jt = 24,454 psi
(for monobloc)
jt = 16,870 psi
(for multilayer)
24,454 + ft(shrink.)
= 16,870
Therefore shrinkage jt = -7584 psi (compression). inner surface of middle shell (r = 7.5): jt = 24,454 psi
(for monobloc)
jt = 24,270 psi
(for multilayer)
24,454
+ ft(shrink.)
= 24,270
Therefore ft(8hrink.j = - 184 psi (CompreSSiOn). Outer surface of middle shell (r = 93ie): jt = 1 8 , 6 6 6
psi
jt = 17,820 psi
(for monobloc) (for multilayer)
18,666 + ft(shrink.)
=
12
15.1 h Determination of Interferences Required in Shrink-fitted Vessels. The necessary total difference in
diameters (interference) may he calculated by the relationships developed by Cox (220). The general relationship is: 1 EU, -= [c’r--r(K2r (C” - l)(P’ - 11 4
jt = 24,400 psi
(for multilayer)
18,666 + ft(shrink.)
jt = 14,212 psi
(for monobloc)
jt = 17,820 psi
(for multilayer)
Therefore ft(shrink,)
= +5608 psi (tension).
Figure 15.2 is a graphical representation of the superimposed Lamk pressure-stress curve for a monobloc vessel
- PO) (15.27)
C2!!5
(see Eq. 15.21) 1 + K2 n = total number of shells r = interface number, numbering outward
K = E ratio (see Eq. 15.22) ID pi = internal pressure, pounds per square inch gage pO = external pressure, pounds per square inch gage
= 24,400
Therefore ft(&i&.) = +5734 psi (tension). Outer surface of outer shell (r = 11.718):
+ 1 ) - 2P](&
where U, = difference in diameters, inches d, = outside diameter of the rlh shell, inches E = modulus of elasticity of shell material, pounds per square inch
Therefore fi(shrink.) = -846 psi (compression). Inner surface of outer shell (r = 9x6): (for monobloc)
13
having the same dimensions as the multilayer vessel, the shrinkage stresses of Ihe multilayer vessel, and the combined operating-pressure stresses of the multilayer vessel. A reduction of 9630 psi in the maximum hoop stress at t,he inner surface is obtained as a result of shrink-fitting fabrication.
17,820
jt = 18,666 psi
nQ .g E gz
Substituting for C in Eq. 15.27 by means of Eq. l5.21 gives:
EU, -= 4
2n--rK2n-2r[(h-2 + ,)r(K2’ + 1) - 2’+1K2’](pi - pO) (K2r
- 1)[2nK2”
For a multilayer shrink-fit,ted
- (K2 + l)“] (15.28)
vessel fabricated from two
Multilayer
shells. n = 2 and r = 1.
Therefore (15.29)
For a multilayer shrink-fitted vessel fabricated from three shells, n = 3, and r = 1 and r = 2. For r = 1 dl
4K4 --’ 1K4++1
For P = 2
1
(Pi - PO)
(15.30)
2K2(K4 + 4K2 + 1) ~- (Pi - PO) (15.31) (K2 + 1)(7K4 + 4K2 + I) I For a multilayer vessel fabricated from four shells R = 4, ztud r = 1, 2, and 3. For r = 1
EU1
=
with
Shrink-fitted
Shells
301
For the second interface, where n = 3 and r = 2, Eq. 15.31 applies.
EUl dl
EUl -=
Vessels
.-m-t!!?--
15K6 + llK4 + 5K2 + 1
(pi - PA 05.32)
For r = 2 4K4(K4 + 4K2 + 1) - ~~- (Pi - PO) (K2 $ 1)(15K6 + 11K” + 5K-2 + 1) (15.33) For r = 3
Kc-2 d2
2~2(#8 + X6 + UK4 + 5K2 -t l)(~i - &) EU, _- = (5~4 + Ks + 1)(15K6 jZ1K455K2 + 1) i c3 (15.34) The interferences for 1 he multilayer vessel described in Example Calculation 15.1 will be determined. The nominal dimensions of the threelayer vessel from Example Calculation 15.1 iIre: 15.li Example Calculation 15.5.
1. Inner-shell inside diameter, 12 in.; outside diameter, 13 in. 2. Middle-shell inside diameter, 15 in.; outside diameter, 183% in. 3. Outer-shell inside diameter, 18% in.; outside diameter, 23x6 in. Also, K = 1.25, pi = 20,000 psi, and p. = 0. The modulus of elasticity will be taken as 30,000,OOO psi. For the first interface, n = 3 and r = 1, Eq. 15.30 :rpplies.
EUI _ - 7K4 +4f2+ i (Pi - PO) dl Substituting and solving for U1 gives: (4)(1.25)4 (20,000) (7)(1.25)4 + (4)(1.25)2 + 1> therefore ZJ1 = 0.0040 in, The outside diameter of the inner shell must exceed the Sde diameter of the middle shell by 0.0040 in. before shrink-Wing.
2K2(h’ + AK’ + 1) EU2 -= (K2 + l)(7h-4 + 4sK2 + 1) (” - “) 4 Substituting and solving for ZJ2 gives:
u2=(&g (2)(1.25)2[(1.25)4 + (4)(1.25)2 + l] [(1.25)2 + 1][(7)(1.25)4 + (4)(1.25)2 + l] therefore
1
(20’ooo)
U, = 0.0061 The outside diameter of the subassembly consisting of the inner and middle shells must be machined to provide an interference on the diameter of 0.0061 in. with the outer shell before shrink-fitting of the outer shell. 15.1 i Simplified Relationships. The general relationships for multilayer-vessel construction have been presented in the previous sections according to the method developed by Cox (220). The use of these relationships is somewhat involved; also, for practical reasons most shrink-fitted multilayer vessels consist of two shells. Therefore it is desirable to work with simplified relationships for the case of twoshell construction in which the external pressure is zero. For this condition the combined shrinkage stress and pressure stress at the inner surface of each shell (f,) is given by Eq. 15.25. pi(l + K2jn
fq = ___ 2”K2n
- (1 + K2)n
This equation may be rewritten as: /l + K2\” (15.35)
Eq. 15.21 C is defined as:
Substituting into Eq. 15.35 gives: (15.36) Solving this equation for C gives: C=Wi
(15.37)
This relationship gives the value of C for II number of shells with pi (internal working pressure) and f, (allowable stress) specified. The corresponding value of K is given by Eq. 15.21 rewritten as follows: K = 2/C/(2 - C)
(15.38)
An examination of Eqs. 15.37 and 15.38 indicates that when n = 2, pi = 3fq, C = 2, and K = m. Thus, when
302
Multilayer
Vessels
the working pressure is three times the allowable stress, the theoretical required thickness approaches infinity. This situation may be compared with that of the Lam& equation for a monobloc shell, in which the theoretical required shell thickness approaches infinity when the working pressure approaches the allowable stress of the material. Thus, the use of multilayer-vessel theory theoretically permits a threefold extension of the design range, two-shell construction being assumed. A greater extension may be obtained by using more than two shells. The practical limits of two-shell construction occur in the range in which the working pressure is from one half to twice the allowable working stress. 15.lk Example Calculation 15.6, a Two-shell Shrinkfitted Vessel. A multilayer vessel having an ID of 12 in.
and consisting of two shells which operate under a working pressure of 20,000 psi with an allowable working stress of 24,582 psi (same as the maximum stress in the three-shell vessel of Example Calculation 15.1) will be designed. By Eq. 15.37 C= .;/(pi/jq) + 1 = 1/(20,000/24,582) + 1 = 1.348 By Eq. 15.38 K = X4(2
- C) = 41.348/(2 - 1.348) = 1.44
dl = (12)(1.44) = 17.28
where Ad = increase in shell diameter, inches d = inside diameter of shell, inches At = temperature of heating minus room tempersture, degrees Fahrenheit (Y = coefficient of thermal expansicn = 0.0000067 in. per in. per “F (for steel) Example Calculation 15.7, Determination of Pre15.lm heat Temperature for Shrink-fitting. In reference to Exam-
ple Calculation 15.5, the required interferences for a threeshell shrink-fitted vessel were found to be 0.0040 in. and 0.0061 in., for the inner and outer interfaces, respectively. The preheat temperature necessary to provide the required thermal expansions without clearances and also with an additional clearance of 0.050 in. is to be determined. The interface diameters are 15 in. and 18% in. The inside diameter is 12 in. and the outside diameter is 23Tf6 in. By use of Eq. 15.39 the temperature to which the she& must be preheated may be determined. Ad = a(At)d where LY = 0.0000067” F At = (1 - 70” F) Ad = interference + clearance For the case of no clearance: For the middle shell
dz = (17.28)(1.44) = 24.85 in.
Ad = 0.0040 in.
Thus, for the same maximum-stress limitations, the twoshell vessel has an outside diameter of 24.85 in. and a threeshell vessel has an outside diameter of 23% 6 in. (see ‘Example Calculation 15.1). If this vessel had been of monobloc construction and the maximum stress had been limited to 24,582 psi, the following would have been true. By Eq. 14.14a j K2 + 1 24,582 --= K2 + 1 -=-=-K2 1 20,000 K2 - 1 P
d = 15 in. therefore
0.0040 ’ - 7o = (15)(0.0000067)
39.9” F
t = 110°F For the out,er shell Ad = 0.0061 in.
1,229
d = 18% in.
therefore
therefore K = 3.12
t-70=
OD = (12)(3.12) = 37.4 in. Thus, times as vessel of times as
the monobloc shell would be 37.4/23.44 = 1.59 thick as the vessel of three-shell construction. The two-shell construction would be 24.85/23.44 = 1.06 thick as the vessel of three-shell construction. 15.11 Thermal Expansion for Shrink-fitting. In order to shrink-fit successive shells upon one another, the outer shell must be expanded by heating. It may then be slipped over the inner shell or shells and allowed to cool. The expansion must be sufficient to overcome the required interference afiar cooling and also to provide some clearance for easy assembly. The diametral enlargement of the outer shell resulting from heating is: Ad = a(At)d
(15.39)
0.0061 (18.75)(0.0000067)
= 48.5” F
t = 118.5” F For the practical case in which a clearance of 0.050 in. i; provided: For the middle shell Ad = 0.0040 + 0.050 = 0.054 in. d = 15 in. therefore 0.0540 = 538” i? ’ - 7o = (15)(0.0000067)
t = 608” F
Multilayer Construction Using Weld Shrinkage For the outer shell
Ad = 0.0061 + 0.050 = 0.561 in. d = lSyd in. lherefore 0.0651 t - 7o = (1a75)@.000a.@47) = 4470 F
303
method of prestressing would take advantage of the shrinkage s&ess,es in the longitudinal welded joints of shells. The A. 0. Smith Company has developed this technique (221) and haa successfully fabricated a considerable number of law pressure vessels operating at a pressure above 5000 psi. The shrinkages of longikieinal we&& have been correlated by Spraragen and Ettinger (222). For@ansv@se shrinkage of butt welds the following relations i
t = 517’ F Thus, a shrinkage resulting from a tempera&& difference of only 40” to 50’ is sulllcient, but to insure erraapf assembly the two shells age heated to about 500 to B&Y F. (Note that after the middle shell is shrunk on the inner shell, this partial assembly is cooled and machined to accurate size before the third shell is added.) 15.2 MULTILAYER CONSTRUCTION USING WELD SHRINKAGE The previous section on the thermal expansion for shrinkfitting demonstrated that the necessary interference for prestressing could be obtained with a very small increase in the temperature of the outer ring (40 to 50” F), uniform heating of the ring and no clearance for assembly being assumed. It follows that if a narrow longitudinal hand of from one tenth to one twentieth of the circumference of the outer shell were heated to 10 to 20 times this temperature difference, the same effect would be produced by cooling. In the cooling of a longitudinal welded joint such an effect can be produced. Therefore a possible convenient
Fig. 15.3.
Arc-welding of multilayer vessels.
s = 0.1716; + 4&H&+
(15.40)
where s = transserse~sh :&Al%S u = cr.--sectional area, &weld, square inches t = plate thickness,, &&es w = average width of weld, inches Substituting into Eq. 15.40 the dimensions for a single-V butt weld of a J/4-in. plate gives shrinkage values in the order of magnitude of >Q in. This shrinkage is greater than necessary to prestress successive shells to the desired amount. Therefore some of the shrinkage must be absorbed by the provision of a clearance between successive shells at the time of welding. As this procedure depends upon shop technique, it is not possible to compute the final stresses that exist upon completion of the fabrication. Peening after welding may also be used to reduce the shrinkage stresses. In the technique used by the A. 0. Smith Company an inner shell having a thickness usually greater than 3/4 in. and often $5 in. is first fabricated. This shell is not perfo-
(Courtesy of A. 0. Smith Corporation.)
Multilayer Vessels
Fig.
15.4.
Multilayer vessel tested to destruction, d/t = 9.3 (245).
rated and serves to contain the fluid to be held under pressure in the vessel. Subsequent shells usually >/4 in. in thickness are progressively wrapped around the inner shell, tightened mechanically, and welded longitudinally. These subsequent shells are perforated with small holes for venting. Cylindrical rings are inserted at both ends of the inner shell during these operations to maintain a true cylindrical shape. The welds are staggered around the circumference to minimize localization of any excessive stresses at or near the welded joints and are ground flush prior to the adding of subsequent layers. After the vessel shell has been built up to the desired thickness with successive layers, the ends of the built-up shell are machined for the welding groove for the attachment of a formed head. In the final vessel
0
Fig.
15.5.
4
12 16 20 24 Number of layers welded
Increase in compressive stress in inner layer or
number of wrapped layers fCowtesy
8
(di = 48 in., ti =
of A. 0. Smith Corporation.)
28
32
o fraction of
35 in., t, = 34 in. (245).
(Courtesy of A. 0. Smith Corporation.)
12 or more successive layers are often used. Figure 15.3 shows the arc welding of multilayer shells. A number of these vessels were tested to destruction. The test data obtained have provided some valuable information concerning this method of fabrication. Whereas monobloc vessels which fail under high-pressure service often fragment, the multilayer vessels do not fail in such a manner. Considerable deformation occurs prior to failure, and when a leak develops in the inner shell, fluid escapes through the perforations in the outer shells. This provides a system of venting which gives warning of possible rupture. In the tests to destruction some of the vessels were stress relieved and others were not. One vessel which was not stress relieved withstood 8y0 greater stress than the corresponding vessel which was stress relieved. This indicates the desirability of not stress relieving, in order to retain the compressive stresses developed in fabrication. Figure 15.4 shows a multilayer vessel tested to destruction. 15.2a Analysis of Test Data. Figure 15.5 shows a plot of the induced compressive hoop stresses at the inside surface of the inner shell as determined by circumferential strain measurements during the progressive wrapping of successive shells. The vessel used in this test had an inside diameter of 48 in. The vessel wall consisted of an inner shell 34 in. thick wrapped with 32 concentric layers each 34 in. in thickness. The curve gives an indication of the cumulative effect of the shrinkage of successive layers on the inner shell. One method of analyzing these data involves the prediction of the experimental curve by use of simplified theoretical relationships. Consider the inner shell and the first wrapped layer as an inner shell under external pressure and an outer shell under internal pressure, respectively. The interface pressure is common to both shells. A summation of forces about a diametrical plane can be made
Multilayer
according to the membrane t,heory. shells are equal and opposite, or
The forces in the t.wo
-Fi = --f&l
(for inner core)
+Fi = +fdd
(for first wrapped layer)
Construction
f2 = 6600 psi E2
(fi)n=l = -fl -- = incremental stress induced in inner t; core by first w-rapped layer but = O
therefore
- (.fi)n=o
= -.fi!
1
Assuming that there is uniform stress addition across the shell and considering that the inner core and the first layer are a unit, we may treat the second wrapped layer in like manner. The incremental stress induced in the inner core plus the first layer by a second wrapped layer is:
therefore
A(.fi)n=3 = .f3 (c~+~mi;)
(15.41)
For uniform t,hickness of successive shells, tl = tz = t3 = t,. Therefore
A(.fi)n = -fn t1,
l)& ~---
f2
- = 0.0002 in. per in.
=
E
(15.51a)
Total c~ircurrlf~relltial
strain = ~ntl
(15.4$)
= 0.0002(x) 19.25 = 0.031 ic. Apparently 0.03 I /0.045 or about 70’ ; of Ihe weld shrinkage is converted to induced stress. The total “free” weld shrinkage is a funct,ion of waldjoint dimensions. If the same plate thickness and weld joint is used in each successive weld, the free weld shrinkage will be a constant. Part of this free weld shrinkage is used in overcoming clearance between layers, and part in comThe major portion of pressing the inner layers elastically. the free weld shrinkage develops elastic strain and resultant stress in the layer itself. Assuming for purposes of simplification that each clearance in the welding operation absorbs the same amount of the free weld shrinkage, we find that the total circumferential strain contributing to stress development will remain the same. The unit strain will then be proportional to the diameter. or by Eq. 15.44 E n=
total strain 4,
0.031
= ~ = 0.0098'1 in. per in. 4
f, = & = 10.00987)r30
296,000
x 106) _
d< + 1 + 4 - 0.25 (15.45)
For the vessel data in Fig. 15.5, where ti = !,i in. and h . . . t32 = +/4 in .,
AWn = -.fn
305
Shrinkage
therefore
t1
AA = (.fi)n=~
Weld
By Eq. 15.42
Equating and solving for fi gives:
(fi)n=O
Using
Substituting Eq. 15.45 into Eq. 15.42 for jn, we obtain:
(15.42)
The stress in the rzth wrapped shell is produced by inducing strain resulting from weld shrinkage. Weld shrinkage is described by Eq. 15.40, where s = 0 . 1 7 1 6 ; + 0.0121W 0
(15.43)
OF
(-296,000)(2) + 99..5
A(& = -122 + 100.5n Or as an approximation,
For $i-in. plate with -15” bevel welds and fi6-in. clearance, a = 0.06 sq in. t = 0.25 w = 0.30 s = 0.041 + 0.004 = 0.045 in.
(circumferential strain)
A calculation from the data given in Fig. 15.5 shows that all of this shrinkage was not used to develop induced stress. In reference to Fig. 15.5, the point given for the first wrapped layer which is most consistent with the other data occurs at fi = 2000 psi.
592,000
Nfi),
=112 + lOOn + 100
(15.46)
Equation 15.46 represents the incremental increase in the induced compressive stress in the inner core. Since 1 he increments are small, s/4 in., and the number of shells is large, Eq. P5.46 will be written in differential form and will be formally integrated, or
dfi _ 4,
5.92 X 10” n2 + 1OOn
+ 100
306
Multilayer Vessels 22.000
20,000 18,000 16,000
Fig. 15.6. Comparison of predicted stresses with experimental data of Fig. 15.5.
I
I
0 Values f r o m F i g . 1 5 . 5 I I
0
2
4
6
8
10
12
14
16
18
20 22
24 26
28
30
32
Number of layers, n
Substituting for K = do/di gives:
therefore
j;: =
-5.92 x 105
ln
2/(100)2 - 4(100) fi =
-5.92 X lo5 In 98
2n + 1 0 0 - 41002 - 4(100) ( 2n + 100 + d1002 -4(100) >
2n + 100 - 9 8 2n + 100 + 9 8
fi = - 6 0 3 0 l n 2n+2 2n + 198
(15.47)
=
- %,do2
do2 - di2
(15.48)
Let. K = do/di, then
= (-2poK2)/(K2 - 1)
f~(membrane)
fl(membrane)
= __
2t
(15.50)
Substituting for t = (do - di)/Z gives:
ft(membrane)
---Pod, =do - di
(15.51)
(-poK)/(K
= +& (15.53)
- 1)
or ft(LamB)
=
.tl(membrane)
- 2(97 + n)96 - (15.54) 96(97 + n + 96) = 193 + n
Multiplying Eq. 15.47 by the ratio given in Eq. 15.54 corrects the stress based on membrane theory to the stress based on Lam& theory for thick walls, or .fi(corrected)
pod,
(15.52)
Rewriting Eq. 15.53 in terms of n number of wrapped layers for the particular vessel under consideration, we obtain: 97 + n K _ 48 + 1 + 5% - 1,: z48 96
(15.49)
The thin-wall hoop-stress equation, 3.14, can be written for external pressure as:
-poK K - l
=-
The ratio of the maximum fiber stress at the inside surface as given by Lamb’s analysis, by Eq. 15.49, to the corresponding stress based on membrane theory, by Eq. 15.52, is: fi(Lam9
Equation 15.47 gives the change in the inside-wall hoop compressive stress resulting from n number of shells. The analysis of the stresses was based on thin-wall theory. As the number of shells increases, the wall becomes progressively thicker, and the thin-wall analysis should be adjusted for this factor. This can be accomplished by application of Lame’s analysis for thick-wall vessels. The wall is under an external pressure because of the weld shrinkage. By Eq. 14.12, and for --pi = 0 and d = di, f 1(La1nb)
ft(membrane)
= [y;3'+;][- 6030h;$8] -27,200 (15.55)
Table 15.1 lists calculations for the vessel under consideration. Inspection of Table 15.1 indicates that the rate of increase in the compressive hoop stress in the inner core decreases rapidly as the number of outer shells is increased. Figure 15.6 shows a plot of the experimental data from Fig. 15.5 compared with the predicted values based on thh-
RibbonTable 15.1.
Calculation
of
Stresses
in
a
and
Wire-wound
Vessels
307
Multilayer
Vessel C& = 48
il 0
in., ti = 35 in., t, = x in.)
ln(g&) -66030 -4.501
1
-3.922 -3.527 -3.034 -2.293 -1.734 -1.378 -1.075
2 4
10
20 32 50
27,200 23,700 21,300 18,300 13,850 10,480 8,320 6,480
ln
$&$
-3,500 -5,900 -8,900 -13,350 -16,720 -18,880 -20,520
~~3~~ (:r rected) 1.01
1.02 1.03 1.06
1.10 1.15
1.21
-3,530 -6,010 -9,150 -14,100 -18,400
-21,700 -24,800
wall analysis (column 4 in Table 15.1) shown in the dashed line and the predicted values based on thick-wall analysis (column 6 in Table 15.1) shown by the solid line. 15.2b Discussion. The stress distribution across the shell of the completed vessel may be predicted by use of the relationships given. To do this the initial stress in the given layer as it is wrapped on is determined by Eq. 15.45. The contribution to this value of each subsequent layer must be determined by integration with the correct constants for the layer under consideration. The relationships given apply only to the vessel described and are limited by the assumptions made in the analysis. Variations in manufacturing methods will alter the analysis. It should also be pointed out that certain assumptions were made concerning the shrinkage of the layers. These assumptions were approximations and may or may not have accurately represented the shrinkages that were used in the fabrication of the vessel. The final relationship given by Eq. 15.45 fits the experimental data in a satisfactory manner (see Fig. 15.6). Seely and Smith (231) have presented an analysis of the stresses in a hollow cylinder made of thinwalled shells having various wrapping pressures. 15.3
RIBBON-
AND
WIRE-WOUND
VESSELS
The technique of wire winding cylindrical shells subjected to high internal pressure is old and has long been used for reinforcing gun barrels. More recently this technique has been extended to include the use of flat and interlocking ribbons for prestressing shells. 15.3a Wire and Flat-ribbon Windings. Wire and flatribbon windings are used only for absorbing hoop and
Fig.
15.7.
Typical section of a
windings (194).
prestressad
Welds 4
vessel with wire or flat-ribbon
TCore tube
‘L Profile roller Fig. 15.8.
Detail of Wickelofen wrapping (225). (Courtesy American
Society of Mechanical Engineers.)
radial stresses and offer no restraint to axial load. An inner monobloc shell must be used having a minimum thickness sufficient to absorb the axial internal-pressure load. In the process of winding, the inner shell may be considered to behave as a vessel under an external pressure induced by the winding. This pressure at the interface between the shell and wire windings also acts as an internal pressure on the wire windings. An internal pressure in the vessel induces hoop-tension stresses in both the inner monobloc shell and the outer windings. Thus, under the operating conditions the inner monobloc shell may be considered to have both internal and external pressures, and the windings to have induced stresses resulting from winding tension and internal pressure. A typical section of a flat-ribbon- or wire-wound vessel is shown in Fig. 15.7. 15.3b Interlocking-ribbon Winding. A method of constructing prestressed vessels with spiral-wound interlocking ribbons of steel was developed in Germany by Schierenbeck (223) and described by Holroyd (224) and by Donovan, Josenhans, and Markovits (225). The principle involved in this design consists of interlocking the winding by means of grooved profiles to permit the winding to carry a portion of the axial load. Figures 15.8, 15.9, and 15.10 illustrate the method of interlocking-ribbon construction, which is also known as “Wickelofen” wrapping. In prestressing Wickelofen-wound vessels, shrinkage, is obtained by preheating the ribbon to from 1100 to 1550” F (prior to winding), as indicated in Fig. 15.9. The wound
308
Multilayer
Vessels ,-Wickelofen
drum
Profileroller
Coupling-/ Fig. 15.9. Wickelofen-wrapping lathe (225). (Courtesy of American Society
of
Mechanical
Engineers.)
ribbon is cooled first by air jets and then by water jets. After each layer of tape is spirally wound around the length of the vessel, the end is welded to the previous wrapped layer. Winding speeds can be as high as 15 fpm (224). The ribbon may vary in thickness from s/4 in. to N in., and the width is usually about 10 times the thickness. Special lathes are required for wrapping. These lathes can be of the same type used in machining monobloc-vessel
Shrunk flange Double-cone \ gasket
Fig. 15.1 1.
Flanged end of a Wickelofen vessel (225). (Courtesy of
Bureau of Mines, United States Deportment of the Interior and American Society
of
Mechanical
Engineers.)
shells except that the tool carriage must be modified to accommodate the reel of tape, the tape heating and cooling equipment, and a profile back-up roll. Figure 15.11 shows the flanged end of a Wickelofenwound vessel. Figure 15.12 shows a sectional drawing of a design for a high-pressure vessel for coal hydrogenation reported by Donovan, Josenhans, and Markovits (225). This vessel was designed to be fabricated of a low-chrome vanadium steel similar to SAE 6115 suitable for being heat treated during wrapping. This steel has a yield strength of about 90,000 psi and an ultimate strength of about 130,000 psi. Donovan recommends that the yield strength be between 60 y0 and 75 ‘% of the ultimate strength and that the elongation not be less than 17%. In the designing of the vessel the strip was stressed to 70,000 psi on the outer layer; this resulted in a compressive prestress of 50,000 psi in the core. Twenty-two layers 5is in. thick were used over a core 135 in. thick (see Fig. 15.12). 15.4 THEORY OF RIBBON AND WIRE WINDING
Wickelofin flange Fig. :5.10. Mechanical
Wickelofen Engineers.)
flanges
(225).
(Courtesy
of
American
Society
of
Ribbon- and wire-wound vessels may be fabricated with either a thin or a thick shell (inner core plus windings). Thick shells with windings are used obviously because of the greater strength resulting from wound construction in comparison with monobloc construction. The reason for the use of wound thin shells is not so apparent. One example of the use of such shells is the case in which the inner core must be fabricated of a noncorrosive or ductile material that does not have sufficient strength to resist the tensile (hoop) loads produced by the internal pressure. 15.4a Thin Shells Wound at Constant Tension. The simplest application of ribbon or wire winding involves the use of a thin shell onto which is wound flat ribbon or wire of the same material of construction as the shell with constant tension in the wire during winding. HS the ribbon or wire is wound onto the shell, it causes a cncumferential compressive stress to develop in the shell. In such an application an internal shell must be designed
Theory of Ribbon and Wire Winding
o f sufficient thickness to resist the axial load resulting from internal pressure. The axial stress is only one half of the hoop stress according to membrane theory; therefore the inner shell needs to he about half as thick as the corresponding monobloc shell. The necessary additional shell material required to absorb the hoop stress is made up of the wire winding. The inner core may be considered to behave as a thin-walled vessel if the operating pressure is moderate. The compressive stress induced in the inner shell may be determined by taking a summation of forces about a diametral plane with no internal pressure in the vessel. At this plane the total tensile force in the wires is equal to and opposite in sign to the compressive forces in the shell, or
(Afhz = -fn ji + (nt?l _ l)&
(See Eq. 15.41a.)
Carrying out the operations indicated for the derivation for the ribbon-wound condition gives: (fi)n = s2 ln
7rru(rz - 1) + 2ti - ?rr,
2ti
(15.60)
I Under the operating condition of internal pressure the combined-stress, fi, total in the shell will be (f& plus the pressure stress, (fi)p. Assuming uniform distribution of the pressure stress and using the method of superposition, we obtain: w
(fi) total = mn + WP
(15.61)
But for ribbon and square-wire windings,
(fi)p = !g = 2t, p;tnt a z
(15.62)
and for round wire, Pd (.fi)p
The following equations are for a ribbon-wound vessel where n = number of layers of winding = thickness of ribbon, inches W ,, = width of ribbon, inches fn = stress in ribbon, pounds per square inch ti = thickness of inner shell, inches fi = induced compressive stress in inner shell, pounds per square inch (Afi& = induced-stress increment in inner shell from the winding on of n layers of ribbon, pounds per square inch T = tension in ribbon, pounds
309
t,
=
2ti + mm,
(15.63)
8’1lD, liquid in8’ ID, liquid out
Vickers-Anderson joint J-sections
12%-chrome core tube
and (15.56)
.fn = $ R 18 Substituting Eq. 15.56 into Eq. 15.41a
Threaded neck flange
gives: n
1
i insulation Shrink ring
(15.57)
For the case where the ribbon thickness is small compared to the shell thickness,
dji = LT s n 0
W7L
1 4 +
(n
SAE 6115
22
layer
dn -
Detail B
I)&2
Integrating and substituting limits gives the following equations. For ribbon windings, Shrink
(15.58)
For square-wire windings where wn = t,,
(fi)% = L$l ln lncnt ,-ml; + ti It
2
R
ring
Vickers-Anderso joint
(15.59)
For circular-wire windings where wn = 2r, = t,, and wire area = nrW2, Eq. 15.57 may be modified to: Fig.
15.12.
principle
Bureau of Miner converter design based on the Wickelofea
(225).
(Courtesy of Bureau of Mines, United States Department
of the Interior and American Society of Mechanical Engineers.)
310
Multilayer Vessels
Combining Eq. 15.62 with Eq. 15.58 and Eq. 15.60 with Eq. 15.63 gives the following equations. For ribbon windings, - T
(fihd = tw
n n
ln
t,(n - 1) + ii + Pi4 2ti + 2nt, ti - t,
(15.64)
Eqs. 15.58, 15.59, and 15.60, apply. However, in the case of a shell that has a modulus of elasticity which differs from that of the windings, Eqs. 15.62, 15.63, 15.64, 15.65, 15.66, and 15.67 must be modified. A summation of forces may be taken at a diametral plane as follows: 2~’ = Shell + Winding - Fprescwre = 0
For round wire,
= ji2til + j,2nt,l pi4 2ti + mr,
Wn=l
=
(15.5833)
,;
nz
= fiti + j,nt, = ‘+
(15.65)
Equations 15.58, 15.59, and 15.60 become indeterminate as t, approaches ti. This is also true in Eqs. 15.60 and 15.65 when rW approaches (2/r)&. This case is seldom encountered except when n = 1. If n = 1, the equations mentioned above may be modified as follows. For ribbon,
(G)p ri -=(6dp rn
But
(fi,, & (15.59b)
= fi” nz
For round wire,
= (4p
and
Jl?.dE = (QJp E,
therefore
riEi
op (.fJp
(15.60b) For ribbon,
(15.68)
Under the influence of internal pressure, the shell and the winding expand together. Clark (31) assumes that the unit strain in the shell and the winding is proportional to the radial distance of the point under consideration; or
For square wire, (fi)n=l
- pidil = 0
r,E,
Substituting for the mean value of ri and r,, for ribbon winding, gives: (15.64b)
+(di + ti)Ei
(15.69)
= &(di + 2ti + nt,)E,
For round wire, LfJtotal = 2, + 2ti
wz
y:,,
w
(15.6513)
For calculations in which n > 1 but t, or 2r, approaches 0, successive calculations using the above equations may be made for each value of n, and the stresses combined by the method superposition. The stress in the winding will be a maximum at the outer layer when the vessel is under internal pressure and will equal the sum of the tensile stress from the tension during winding plus the pressure stress. For ribbon winding, 12
(15.66)
For round wire,
(f?Anax = s2 +
Pi4 2ti + mr,
(15.67)
T5.4b Thin Shells Wound at Constant Tension with Winding and Shell of Dissimilar Metals. Corrosive fluids
under pressure may be contained in vessels having an inner shell of corrosion-resistant material dissimilar to the winding. The relationships derived for the induced compressive stress in the shell (ji) for n layers of winding, including
Equations 15.68 and 15.69 contain an unknown ji and an unknown j,, which may be determined by simultaneous solution.
Pi4 (4 +
(.fn)p = 2ntn + 2ti
ti)Ei
(4 + 24 + nt,)E,
and (.fi)p
1
(15.70)
Pi4
= 2nt
12
(4 +
(15.71)
24 +
&)E, +
(4 + ti)Ei
I
2t,
’
To calculate the stress in the shell under the influence of internal pressure, Eq. 15.71 may be substituted for the second term in Eq. 15.65. To calculate the maximum stress in the winding under the influence of pressure, Eq. 15.70 may be substituted for the second term in Eq. 15.66. Equations for square- and round-wire windings on a shell of dissimilar metal corresponding to Eqs. 15.70 and 15.71 may be derived. 15.4~ Winding
Example Calculation 15.8, Wound Thin Shell with and Shell of Dissimilar Metals. A copper vessel
having an internal diameter of 6 in. and a wall thickness of ~5 in. is to be used under a 2000-psi internal pressure. The copper cylinder is to be wire-wound with two layers
Theory of Ribbon and Wire Winding
of square steel wire 0.05 x 0.05 in. Calculate (1) the compressive stress in the copper cylinder before application of internal pressure, (2) the tension to be used in winding the wire if the maximum tensile stress in the copper is to be 6000 psi, and (3) the maximum tensile stress in the winding under operating pressure. E copper = 16 X lo6 psi, &tee, = 30 X lo6 psi Solution: 1. Determination of compressive skess. Solving Eq. 15.71 for (jJP, we obtain: ZOOO(6)
(fi)P = 4(. 05) .
[
(65 1 + 0.10)30 x lo6 + 2(0.5) 6.5(16 X 106) I
12,000 =-0.20(2.05) + 1.0 = 8510 psi But since the maximum tensile stress in the copper is 6000 psi, the copper must be prestressed with a compressive stress of 8310 - 6000 = 2510 psi. 2. Detcrmmation of winding tension. Substituting ( ji)P for the second term of Eq. 15.64, we obtain the following equations.
To design such a vessel it is necessary to be able to predict the residual-stress distribution throughout t.he inner monobloc core and throughout the windings prior to application of internal pressure. The pressure stresses may be superimposed upon the residual stresses to obtain the combined stresses under operating conditions. To obtain the residualstress distribution in the inner core (when pi = 0) it is necessary to predict the radial pressure at the junction (interface) between the core and the windings. The core may then be treated as a shell under external pressure, and the stress distribution computed. It is also necessary t,o predict the residual stresses in the windings after all the windings have been added. The induced hoop tensile stresses and radial compression stresses resulting from an external pressure can be related to each other by use of the relationships based on Lamb’s analysis. These relat,ionships, given by Eqs. 14.12 and 14.13 in terms of diameter, can be rewritt,en in terms of radii for the case of no internal pressure as follows. For circumferential stresses,
PJO
2
ftc = __~ pi2 - r02
r2 + ri2 r2
l-----l
For radial stresses,
The above two equations can be c*omt)ined relationship for no int,ernal pressure.
T = 2510(0.0025) 0.203 (tension in wire during winding)
26.3 = 12,360 psi jrn = (0.05)2 3. Determination of maximum wire stress under operating pressure. Substituting into Eq. 15.70, we obtain:
(f&l =
(2000) (6) 4(0.05) + 2(0.5)
6.5(16 X 106) (6 + 1 + 0.10)30 X lo6
1
12,000 12,000 ~~ = 17,800 psi = 0.20 + 0.474 = G/4 Therefore Maximum combined &tress in outer winding layer under pressure = 10,500 + 17,800 = 28,300 psi 15.4d
Thick Tension.
Shells
with
Winding
Applied
under
Con-
Windings of wire or ribbon may be applied under various degrees of tension. The simplest method of fabrication is to apply the winding under constant tension. stant
to give ;,
(15.72)
- T -2510 = - ln 0.55 0.0025 0.45
= 30.9 lb
311
Equation 15.72 gives the hoop st,ress, jtr, in terms of the radial stress, jic, and the inside radius of the vessel, ri, at any radius, r. This relationship is useful in predicting the stresses in the wire windings and in the core which result from the compression of subsequent windings. It should be noted that both jtc and jVc of Eq. 15.72 are compression stresses resulting from external pressure. Let fw = stress in wire during winding of layer under consideration (initial stress in wire) After n layers have been wound, the stress fw* in I hp inner layer becomes : f*=(f w w -f)=f tc w
-f rc
LT+ri2 ___ [ r2 - ri2 I
(15.73)
where fw * = total stress in the winding under considera tiolk for pi = 0 jte = stress induced in the winding under consideration by the external windings for pi = 0 The total hoop stress in an inner winding may also be expressed in terms of the radial stress, jr, and the radius, r, in differential form by means of Eq. 14.3; or (15.74)
312
Multilayer Vessels
Substituting for C in Eq. 15.77 gives.
Equations 15.73 and 15.74 both define the stress distribution in the windings. Equating 15.73 to 15.74 gives:
r2fTc
--p 4-m
r2 - ri2
dr Equation 15.75 is written in differential form and must he integrated to give the cumulative radial stress, jrc, at any radius, r, when a given tensile winding stress, fW, is used. The equation can be rearranged as follows to permit integration:
= II) ln (r2 - ri2) + ‘f In (ro2 - ri2)
.L (ro2 - ri2) = P- (r2 - ri2) OP
Substituting Eq. 15.78 for jrc in Eq. 15.73 gives:
--p dfrc dr
or therefore
fw* =fw [l - f*)ln$$)!j fw -= dfre r
I dr
2freri2 r(r2 - ri2)
Multiplying through by -r2/(r2 - ri2) gives :
-fur2
r2 &
2jrcri2r2 r(r2 - ri2) = dr(r2 - ri2) - r(r2 - ri2)2
(15.76)
The right-hand side of Eq. 15.76 can be shown to be equal to the following differential:
Differentiating the above expression gives: (r2 - ri2) (r2 ‘2 + 2&r) - 2r3fTc (r2 - ri2)2 Expanding the above and regrouping results in the righthand side of Eq. 15.76. Substituting the differential for the right-hand side of Eq. 15.76 gives:
-fur (r2 - ri2)
or
r2TfTc = -jw s ~
r2 - Pi2 therefore
r2fTc r2 - ri2
f,.c
r dr (r2 - ri2)
= II) ln (r2 - ri2) + C
(15.77)
To evaluate the constant of integration, it is noted that = 0 at the outer surface of the windings, where r = r,. 0 = 2 In (ro2 - ri2) + C
therefore C = $ In (ro2 - ri2)
(15.79)
It should be noted that the variable r under consideration lies between rj and r, where rj is the radius at the junction (interface). Equation 15.78 gives the radial stress at the outer surface of the monobloc core due to the windings (prior to the application of internal pressure) if rj is substituted for r. This radial stress is numerically equal to the external pressure on the core and, together with the dimensions of the monobloc, permits the computation of the residual-stress distribution. Equation 15.79 gives the hoop-stress distribution in the windings (without internal pressure, pi = 0). The combined-stress distribution, including the effect of internal pressure, may be determined by superposition of the internal-pressure stresses upon the residual stresses. The application and design conditions normally fix the inside diameter of the vessel, the operating pressure, the operating temperature, and the material of construction. The axial load induced by the operating pressure fixes the minimum wall thickness of the inner core. The material of construction and operating temperature fix the allowable stress of the inner core. The remaining variables are the total thickness of the windings, the winding tension, and the maximum stress induced in the windings under operating conditions. The maximum hoop stress in the core for the operating conditions, the winding tension.. and the maximum combined stress in the windings are a function of the thickness of the windings. As all of the variables for the windings cannot be fixed, one may be selected, and the other two calculated. In the example which follows, r, will be fixed, and the tension end combined stress calculated. 15.4e Constant
Example Calculation 15.9, Thick Shell Wound at Tension. Consider a vessel having the same inside
diameter (12 in.) and operating at the same internal pressure (20,000 psi) as the one in Example Calculation 14.1 but fabricated by wire winding with high-tensile-strength wire with the result that its outside diameter is 20 in. The stress in the inner core is not to exceed 25,000 psi. Determine the inner-core thickness, the winding tension required,
Theory of Ribbon and Wire Winding and the stress distributions with and without internal pressure. Calculation of inner-core thickness: By Eq. 14.1 f
dj2 - di2
pj at vessel ends = 0. 25
ooo
=
-64.8pj -17,500 = ___64.8 - 36 + 64.8 ik",6]
therefore
Therefore
17,500 Pj = ___ = -3890 psi -4.50
2wJw12)2 dj2 - (12)2
The residual-stress distribution in the inner-core wd may now be calculated by using pi = 3890; or
dj2 = s (144) + 144 5 259
Therefore
t = 2.05 in.
+ (64.8)(36) -3890 r2 28.8
therefore
rj = 6 in. + 2.05 in. = 8.05 in. r, = 10 in.
[ 1
f = -64.8(3890) t 28.8
dj = 16.1
or
-Pipi I $2 I ZPj 2 1 - ri2
Pj2
= 25,000 psi
= Pdi’ - Pjdj2 n
ft = -17,500 -
313
ft = -8760 25!!!!
(given)
Pi = 6 in. Induced hoop stresses in the core and winding due to internal pressure.. By Eq. 14.12 with p0 = 0 and pi = 20,000 psi, r, = 10 in., and ri = 6 in.,
For r = 8.05 in., ft = -8760 - 4870 = -13,630 psi For r = 7 in., ft = -8760 - 6440 = -15,200 psi For r = 6.5 in., ft = -8760 - 7460 = -16,220 psi Calculation of required winding stress: By Eq. 15.78 with r = rj, and frc = pj at rj,
= 20,000(36) + (100)(36) 100 - 36 7 [l::'ooo36] therefore
ft =
11,250 + 1'12f;ooo fw = 29,100 psi
For r = r, = 10 in., ft = 11,250 + 11,250 = 22,500 psi For r = 9 in., ft = 11,250 + 13,900 = 25,150 psi For r = 8.5 in., ft = 11,250 + 15,600 = 26,850 psi For r = 8.05 in. ft = 11,250 + 17,500 = 28,750 psi Forr = ?in., ft = 11,250 + 22,950 = 34,200 psi For r = 6 in., ft = 11,250 + 31,250 = 42,500 psi
Calculation of residual hoop stresses in windings: Equation 15.79 gives the residual hoop stress in the windings at any radial distance.
When r = r,, fw* = fw = 29,100 psi When r = 9 in.,
The necessary residual hoop stress at the inside surface of the inner core is obtained by subtracting the 42,500-psi hoop pressure stress from the allowable value of 25,000 psi to give: .ft(residual)
=
25,000 - 42,500 = -17,500
psi
The required external pressure on the inner core (with pi = 0) may be calculated by use of Eq. 14.12 with p0 = pi, P = ri, and r,, = r+
= 21,700 When r = 8.5 in.,
(by inspectionj
psi
ft0* = 29,100 [l - (z)ln(-$J] = 16,650 psi When r = ri = 8.05 in.,
.-
fw* = 29,100 [l - (s)In($),I = 11,080 psi
314
Multilayer Vessels
i
Wire
winding ‘0 ‘_I
high-strength colddrawn steel wire is available that has ultimate strengths in the range of 200,000 psi. 15.4f Thick Shells with Windings Applied under Variable Tension to Produce Constant Tension under Operating Conditions. It is apparent from the previous
example, Example Calculation 15.9, that the maximum combined stresses in the windings can be reduced if the windings are applied under variable tension so that an ideal prestressed condition results. Derivations of relationships for this condition have been presented by Comstock (226). If the windings are to possess constant combined stress under operating conditions, the thin-wall equation is applicable for determining the pressure at the junction, rj; or by Eq. 3.14,
25
where jtU, = tension in wire under operating conditions, pounds per square inch
6
7
The value of pj required to reduce the induced pressure stress at the inner surface of the inner core to the allowable level may be calculated by use of Eq. 14.12 after the thickness of the inner core has been established. This thickness is established by use of Eq. 14.1 and is equal to (rj - r-i). After the value of pj has been determined, a value for either r, or jte may be selected, and the value for the other determined from Eq. 15.80. If Eq. 15.80 is substituted for p,, in Eq. 14.12, rj substituted for r,,, and the equation solved for the condition r = ri, Eq. 15.81 results. 6
(jt)r=,,
= Pi(rj2
+ ri2) - 2rjftdr,
E
Fig. 15.13. Hoop-stress distribution in wire-wound vessel (Example Calculation 15.9).
.
jtw
The stress distributions are plotted in Fig. 15.13. The maximum induced hoop stress from internal pressure of 42,500 psi has been reduced to 25,000 psi by the residual stress from the outer windings. The maximum stress in the vessel occurs under operating conditions, exists in the outermost winding, and has a value of 51,600 psi. Although this appears to be a high stress, it is not excessive since
(15.81)
If (jt),.=,.i IS taken as ihe maximum allowable stress in the core, Eq. 15.81 may be rearranged to give jtw. =
Pi(rj2
+ ri2 1
Combined stresses in the windings and core: For P = 6 in., jt = 42,500 - 17,500 = 25,000 psi For P = 7 in., jt = 34,200 - 15,200 = 19,000 psi For r = 8.05 in. (in core), ft = 28,750 - 13,630 = 15,120 psi For r = 8.05 in. (in winding), jtm = 28,750 + 11,080 = 39,830 For r = 8.5 in., ftw = 26,850 + 16,650 = 43,500 psi For r = 9.0 in., jt, = 25,150 + 21,700 = 46,850 psi For r = 10 in., jt,,, = 22,500 + 29,100 = 51,600 psi
- rj)
Pi2 - ri2
- .ft(rtli0w.)(rj2
2rj(r,
-
Pj)
- ri2) --~ (15.82)
If it is desirable to fix both the combined stress in the wire, ftw, and jt(altow.), then r, can be calculated directly by either of the above equations. When the internal pressure becomes zero, the residual stress in the wire is equal to the combined stress, jtlu, minus the pressure stress, jt; or ft(residua1) = ftw -
-ftul
& r. - ri2
2 piri - 2
r. - ri2
+ r2c~~~2r~~,2j] - 2
(15.83)
The hoop stress corresponding to a radial st,ress is given by Eq. 15.72.
Comstock has shown (226) that the radial stress, jr, produced by the wire windings beyond radius r (without internal pressure) is :
fr = - !cY2s jtw _ r
(15.84)
Theory of Ribbon ond Wire Winding
Substituting Eq. 15.84 forf,. in Eq. 15.72 gives the change in hoop stress, Aft, in the wire at radius r as a result of the windings beyond r (with no internal pressure). Aft =
existing at r (without internal pressure). Thus the required winding tension is obtained by subtracting Eq. 15.85 from Eq. 15.83 to give:
fw = ftw
( 1 +q (1+$) (15.85)
The required winding tension, fW, at radius r is obtained by subtracting the change in stress, Aft, due to windings beyond radius I from the residual hoop stress, ft(residud)
315
ro(r2 + r12) - 2rri2 r(r2 - n2)
1
2piri2 - ~ (15.86) r2 - ri2
Comstock (226) has presented an illustrative design demonstrating the use of Eq. 15.86 in a wire-wound vessel to operate at 15,000 psi with a constant tension of 40,000 psi in the wire winding under operating conditions.
PROBLEMS
I. A new high-tensile-strength, high-yield-strength, low-alloy plate steel with a minimum ultimate strength of 105,000 psi and a minimum yield strength of 70,000 psi and with a minimum elongation in 2 in. of 22% has been described in the literature (228). (The chemical composition is given as: a maximum of 0.25% carbon, 1.50% manganese, and 0.35~~ as well as a minimum of 0.10% vanadium and a nickel range of from 0.40 to 0.70%.)
Design the shell of a two-shell shrink-assembled multilayer vessel using this steel.
silicon
The
vessel is to have an inside diameter of 24 in. and is to operate at a pressure of 20,000 psi. The service temperature is to be under 600” F, and an allowable tensile stress of 30,000 psi is to be used. 2. Redesign the vessel described in problem 1 as a three-shell shrink-assembled multilayer vessel. 3. What are the interface pressures for the vessel described in problem Z? 4. What metal interferences are required for the vessel described in problem 2 3 5. A 12-in.-inside-diameter pressure vessel is fabricated of an inner shell of copper 1 in. thick and an outer shell of steel 35 in. thick in such a manner that the interface pressure is zero and the two shells are in contact with each other. The reactor is 4 ft long from tangent line to tangent line with ellipsoidal heads (also of double layer). If end effects are ignored, determine the hoop stress in both of the shells if the vessel contains an internal pressure of 2500 psi and if E of copper = 15 X lo6 psi and E of steel = 30 X lo6 psi. (Suggestion: assume that thin-wall theory is valid.) 6. A copper vessel with an internal diameter of 10 in. and a wall thickness of f$ in. is to be operated at a lOOO-psi internal pressure. The copper shell is to be wire-wound with square steel wire having a cross section of 0.1 in. by 0.1 in. If the wire is wound at a constant tension of 20,000 psi, what is the minimum number of layers of winding required to keep the hoop tension in the copper core below 6000 psi under operating conditions? What is the axial stress in the copper shell under operating conditions? What is the residual compressive hoop stress irr the copper shell when the internal pressure is removed? 7 . Redesign the shell described in problem 6 using the same material and same inside diameter but using wire winding at a constant tension of 25,000 psi. 8. Redesign the shell described in problem 6 using the same material and same inside diameter but using a variable-tension winding to produce a constant tension of 35,000 psi in the winding under operating pressure.
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210. Eichinger, A., Statement in: Proceedings, Second International Congress of Applied Mechanics, Ziirich, 1926, p. 325. 211. Voorhees, H. R., “The Creep-Rupture Life of Engineering Structures with an Initial Stress Gradient,” Ph.D. Thesis, University of Michigan, 1956. 212. Robinson, E. L., “Effect of Temperature Variation on the Long-Time Rupture Strength of Steels,” Trans. Am. Sot. Mech. Engrs., 74 (1952), pp. 777-781. 213. Higgins, M. B., “Allowable Working Pressure for Long Tubes Subject to External Pressure,” Paper No. 4,8-A-123, ASME Meeting, December, 1948 (abstracted in Me& Eng., 71 (1949), p. 169). 214. Freeman, J. W., and Voorhees, H. R., “Selection of Alloys for Service Requirements,” Znd. Eng. Chem., 48, No. 5, p. 861. 215. Faupel, J. H., and Furbeck, A. R., “Influence of Residual Stress on Behavior of Thick-walled Closed-end Cylinders,” Trans. Am. Sot. Mech. Engrs., 75 (1953), pp. 345-354,. 216. Faupel, J. H., “Residual Stresses in Heavy-wall Cylinders,” J. Franklin Inst., 259 (January-June, 1955), pp405-419. 217. Bridgman, P. W., The Physics of High Pressure, G. Bell and Sons, London, 1949. 218. Comings, E. W., High Pressure Technology, McGraw-Hill, New York, 1956. 219. Sachs, G., “Residual Stresses, Their Measurement and Their Effects on Structural Parts,” Symposium on the Failure of Metals by Fatigue, Melbourne University Press, Melbourne, Australia, 1947, pp. 237-247. 220. Cox, H. L., “The Design of Built-up Cylinders,” Engineer, 162 (1936), p. 179. 221. Jasper, T. M., and Scudder, C. M., “Multilayer Construction of Thick-wall Pressure Vessels,” Trans. Am. Inst. Chem. Engrs., 37 (1941), p. 885. 222. Spraragen, W., and Ettinger, W. G., “Shrinkage Distortion in Welding,” Welding J. (N. Y.), Res. Suppl., 29 (1950), p. 323-S. 223. Schierenbeck, J., Jr., U. S. Patent 2,326,176, August, 1943. 224. Holroyd, R., “Report of Investigations by Fuels and Lubricants Teams,” U. S. Bur. Mines Inform. Circ. 7375, 1946. 225. Donovan, J. T., Josenhans, M., and Markovits, J. A., “High Pressure Vessels in Coal Hydrogenation Service,” Trans. Am. Sot. Mech. Engrs., 72 (1950), p. 357. 226. Cornstock, C. W., “Some Considerations in the Design of Thick-wall Pressure Cylinders,” Trans. Am. Inst. Chem. Engrs., 30 (1943), pp. 299-318. 227. “Synopsis of Boiler and Pressure Vessel Laws, Rules and Regulations: by States, Provinces and Cities (United States and Canada),” National Bureau of Casualty Underwriters, 1955. 228. Fratcher, G. E., “New Alloys for Multi-layer Vessels,” Petroleum Refiner, 33, No. 11 (1954), pp. 137-141. 229. Chilton, C. H., “Six-tenths Factor Applies to Complete Plant Costs,” Chem. Eng., 57, April (1950). p. 112. 230. Williams, Rodger, Jr., “Six-tenths Factor Aids in Approximating Costs,” Chem. Eng., 54, December (1947), p. 124. 231. Seely, F. B., and Smith, J. O., Advanced Mechanics of Materials, Wiley, New York, 2nd ed., 1952. 232. Taylor, C. P., Glenday, C., and Faber, O., Engineering, 85 (1908), p. 325. 233. Gartner, Abraham I., “Nomograms for the Solution of Anchor Bolt Problems,” Petroleum Rejtner, 36, No. 7 (1951), pp. 101-106.
--
322
References
234. Jorgensen, S. M., “Anchor Bolt Calculations,” Petroleum Refiner, 25, No. 5 (1946), pp. 211-213. 2 3 5 . “Equipment Cost Rise Accelerates,” Chem. Eng., 64 (March, 1957), pp. 266-267. 236. La Que, F. L., and Cox, G. L., “Some Observations of the Potentials of Metals and Alloys in Sea Water,” Am. Sot. Testing Materials, Proc. 40 (1940), pp. 670-687. 237. Manning, W. R. D., “Strength of Cylinders,” Ind. Eng. Chem., 49, No. 12 (1957), p. 1969. 238. Manning, W. R. D., “The Design of Compound Cylinders for High Pressure Service,” Engineering, 163 (1947). pp. 349-352. 239. Manning, W. R. D., “The Design of Cylinders by Autofrettage,” Engineering, 169 (1950), pp. 479, 509, 56?
240. 241. 242. 243. 244.
245.
Manning, W. R. D., “Residual Contact Stresses in Built-Up Cylinders,” Engineering, 170 (1950), p. 464. Nadai and Wahl, Plasticity, McGraw-Hill Book Company, 1931. Jasper, T. M., letter to the editor of Engineering, Engineering, 160 (1947), p. 16. Jasper, T. M., letter to the editor of Engiaeering, Engineering, 160 (1947), p. 160. Langenberg, F. G., “Effect of Cold Working on the Strength of Hollow Cylinders,” Trams. Am. Sot. Steel Treating, 8 (1925), p. 447. “Multi Layer Engineering for Safety,” A. 0. Smith Corp. Bulletin No. V-53, Milwaukee, Wisconsin, 1950.
A P P E N D I X
DRAWING
Item 1.
CONVENTIONS
DRAWING
SIZES
AND
SCALES
Item 2.
The Title Block followiilg information:
All drawmgs should be made on vellum tracmg paper unless otherw,se speclfled.
I. 2. 3. 4. 5. 6. 7.
The drawng paper or cloth to be used should be the standard trimmed sheets of such wes that they wll fold to the letter size of E+“x 11’: Sheet A B C cl E The foltowng
TITLE BLOCK AND BILL
SIX 8f”x 11” 11” x 17” 17” x 22” 22” x 34” All over D sue
identifies
OF MATERIALS
the drawing and should contain
the
Name of manufacturer an-address Name of equipment or part drawn Name of purchaser and address Date of completion of drawing. Scale Names of draftsman. checker. and tracer Drawng number
Tracing paper is usually purchased wth the border. title block, manufacturer’s name and address,etc. printed on the standard sizes of tracing paper. The remainder of the title block is lettered free hand to supply the necessary addItIonat information. When drawings are made on blank tracing paper. the enbre btle block and border must be drawn. The title block should be placed on the lower right hand corner of the paper. The revision block should always accompany the title block in the manner shown.
drawing scales are to be used: 1:l (full size) 1:2 (half size) 1:4 1:s 1:12 1:16 1:32
A bill 01 materials. listing all the material required in the form in which it is purchased or cut for fabrication should be included above the title block with two exceptions. In the case of simple details compising only one or two parts. the material, size, etc. may be indicated by notes on!he drawing. In the case of large assemblies comprising many parts.the materials are listed on separate sheets accompanying the drawings.
The scale should be selected to show a clear picture of the part being drawn. Detail drawings are usually made with scales Of l:l, 1:2, or 1:4, whereas assembly drawings are more often made wth scales of 1:4, 1:8, or 1:16. The scale used should always be indicated in the title block. If more than one scale is used on the same drawing sheet, each Scale is to be indicated under the title of that respective section or wew.
;:’ 1 PART NAME
) SIZE O F
S T O C K /MATERIALI
QUANTITY
CHEMCO.
323
INC.
I NOTEs
Drawing
324
Conventions DRAWING
Item 3.
Item 4.
CONVENTIONS
LETTERING
I
Alphabet of Lines Lettermg SUppIleS mformatlon whxh cannot be gwen by lanes alOne. and therefore It 1s One of the most Important elements of a good drawng.
The line is the basis of the englneenng drawng. The welght of an indiwdual line is used to slgnlfy the function of that lone. Three weights of linesare used,namely: heavy, medium. and light, and are used in the following manner. Heavy
Madfom
-
-
-
-
-
----------------
-
-
-
Outline of parts Cutting plane lmes Short break lines
-
Hidden
-
-
Material
Castwon
2.
Lettering may be either verbcal or inchned (between 60 and 70 degrees) but must be cowstent on each drawmg. Example:
4
All letters shall be of umform sue wth the exception that the ftrst letter Of each word Or group of words may be approxtmately 50 percent larger for emphasis. but the form of lettermg must be cofwstent. Example:
Electric windings
Electric insulation
Sound or heat insulation
Fire brick refractories
Concrete
Common brick
The material of which a part IS constructed IS designated, in a section view. by crosshatching. Accepted symbols of the most common engmeermg materials are shown above. Light parallel All slant hnes are directnons and in Of 60’ or 3O’may
lines should be used for most cross-sectlomng. 45”. All adlacent parts are cross-sectioned in opposite the case of three or more adlacent parts. an angle be used on the addItIonat parts.
ALL
CONTACT
S URFACES
Spacmg between knes of lettermg should be 2/3 of the height of the letter.
6
A permanent gude hne may be used l/16” below the letters of a word or group Of words. These gwde lines may be extended to form a leader. Double hnes may be used for tdles and for emphasis.
babbitt
Wood
FINISH
5.
Example: Aluminum
VERTICAL OR /NCLfNED /60’- 70’)
Only upper case letters are to be used.
Symbols
and
stroke commercial gothic style.
3 Cross-section lines Long break lines
Bronze. brass, and copper
Standards
All lettermg must be I” the smgle
lines
-
“I
I”
edge
Lettering
1
DETAIL 7
BUTT WELO
BUT1 WEfD
The following letter heights
TITLES HEADINGS
AND AND
OF
A RM
should be used
%s’
DRAWING NUMBERS~ 3" PROMINENT NOTES-++
BILL OF MATERIALS, DIMENSIONS, GENERAL NOTES, ETC.
Drawing DIMENSIONS
Item 5.
Dlmensmns are placed on by reading the drawmg The the necessary lnformabon as obwous wthout scaling the
I
a drawmg so that a part may be made dlmensmns must be so complete that to the we and locabon of parts 1s drawng or maklng computations.
km 5. cont.
D~mensmn lines are full. hght knes. broke” only where the dtmenslon 1s Inserted and are parallel to the object or hne bemg dlmensKwed. Dlmenslon knes are used for two purposes: for Speclfylng and for showng size. Example:
--g&J%&
Dimension
Example: 2.
by arrows at the surface of the
+------ 2’I6”-----+
Extenwn lmes or wetness lmes tndwate the distance measured when the dlmensmn tine 1s placed outsidethe object. They are full. hght lanes startmg l/16” f&n the oblect and extendmg l/8” beyond the dimension hne.
Standards
2. Dimensions for machine shop work. where tolerances are less than,tO.Ol”,should be given in inches and decimals only. Long shafts may be dlmensioned in feet and Inches. Example: 3.
4 . 5 0 3 ‘$88’0
Feet are designated by a single quotabon mark 0. Inches are designated by a double quotation mark (“). Feet and Inches are separated by a dash. Example:
Dlmenslon lanes are termmated part or at the extension knes
I
assembly drawmgs. etc, where tolerances are greater than +1/16”.ishould be given in inches and fractions of inches up to. but not includmg. 72”. Any distance of 72” or more should be given in feet and inches.
IocabOn
rIro”
DIMENSIONS
1. Dimensions for structural steel work. welded parts, castings,
DEFINITIONS I
Conventions
7’85”
4. In all cases where feet are given, the inches must also be indicated. In case of even feet, a (-0”) must be added. Example:
8’01’
1 0 ’ 0 ”
5. Dimensions for location should be made from a reference lme such as a center line or a base lme and not from an edge. This is especially true if the edge is to be machined or 8s the edge of a casting.
Example: 3
Leaders are hght. stratght knes which lead from a note or dlmensmn and whtch are terminated by an arrowhead touching the part to which attenbon 1s directed. If two or more are used in one drawmg. they should be kept parallel If powble. Example:
4
‘7 DRILL $20 N . C . T A P
Center tmes are fine lines composed 01 alternate long and short dashes whxh are used to represent axes of symmetrical parts and which serve as extension hnes I” the location of holes or other Slmilac features. Important dtimensions should not be referred to a center kne that has no finished hole on it or has no finished surface comcidmg with it
6. Horizontal and sloping hnes should read from left to right while vertical lines should read from bottom to top. Example:
7. Dimensions should not be added or repeated unnecessarily.
325
326
Drawing
Conventions Item 5. cont.
DIMENSIONS
I
8
Dlmenslons are placed OutwJe of a wew or oblect unless it wll add clearness or slmplwty to the dra’wlng 11 placed nnslde They should “ever be placed I” cut (?&owed) surfacesunless absolutely necessary. at which bme.ttw.secbo,nlng IS omltted around the numbers and hnes
9
Overall dlmenslons
should be given on all views
IO
Dlmenslons
should not be crowded If space IS small.one
I1
When dlmenstons are placed on an angle, they should be placed horuontally on the arc as on a dlmenston lhne For large angles, place the dlmenslon un hne with the arc.
of
12 For equally spaced holes in a circular flange or disc, give the dram&r of the bolt hole wth the number and sue of the holes Example
I
I
I
I
S’DRILL 6 HOLES EVENLY SPACED ON It’ 6.C
13. Circular sections should always be dlmensloned
from center
to center and never from the edge of a part.
I
14. An outhne of a drawng preferably should not be used as an extenston Ilne. and a hldden edge hne should never be used as an extension lhne I
-
-
-
-L
A P P E N D I X
WELDING
CONVENTIONS
WELDING
Item 1.
SPECIFICATIONS
ttem 2.
WELDING
INSTRUCTIONS
20 Gage and Less, Use acetylene welding or seam weldmg. weldang of light gages 1s very dlfflcult.
Arc
18 Gage and Heawer: Use arc welding or seam-weldmg. Arc welding IS cheaper than gas welding and seam-welding IS cheaper than arc welding Edges to be Welded. Sheared edges f5/8” manmum) are sufficient tar ordinary welds, heawer plates tlame cut and scale removed. Code Weldmg Is requmng edges machined or flame cut and ground.
SIX
Beveltng 30°bevel can be sheared-5/S” maximum-all other angles and larger wed plates are flame cut and ground over or machined. If both plates are beveled 30’ IS suffuent. If one plate only IS beveled 55’bevelmg should be used. Up to l/4” plates no bevel is required for ordinary weld-up to 3/4” plates smgle bevel is sufficient-heavier plates reqwre double bevel or single U weld if one side only is accessible for welding and double U welds 11 both sides are accessible. “U” grooves can be made only by machining.
of Welds
Butt and plug welds, and all welds wth grooves (V and Llf do not teqwre dlmenslons of bead The gap between the sheets or grooves has to be at feast completely talled wth weld Other welds are measured as follows
Jf& $“j$j>&~ r* Fillet Weld
=lb
b’iiir”il
Corner Weld
Smgle ‘-rs bevel
Edge Weld
TV////////// Single be=%
Double bevel
Welds made on the rotary seam welder (called seam welds) also do not reqwre any cross-sectional dunenslon as no matellal 1s deposwd Contlnulty A conhnuous weld 1s one which IS continued over the lull length of seam as shown III the followng figure
Double U
C
Spectfy on dIdwIng
.\\\\\\\\\\\\\\ Double J
Corner Welds: No beveling IS requwd, weld outside.Size of fillet is not to be specdled-fillet a ltttle heavier than plate is understood. For heavy strams weld one bead iwde in addition to outside weld. See following examples:
C Weld
7 Corner welds
327
===I!
Welding
328 km
2.
ml.
Conventions WELDING INSTRUCTIONS
WELDING INSTRUCTIONS
tern2. cont.
T Welds II beveled no speclfacataon of size of weld requred If not beveled gw sue of Illlet-specify smgle or double fillet-continuous. lntermlttent or staggered-sue of bevel and grind asrequlred.
Butt Welds. 100 percent penetrabon IS understood-no size of bead SpeClllCatlOn 1s requred-speclly single or double butt weld. Grmd,flush and bevel as reqwred.
lntermlttent and conbnuous smgle IIlkt welds are used for very hght loads only.
Smgle and double plarn butt welds are for kght loads only, single up to l/8” plate and double to l/4”-gap for smgle equal to plate thickness. for double l/2 plate thickness.
For medwm hght loads a staggered weld hawng the ttrst and last weldadouble weld should be used. For all heavier loads a double mtermittent or a double contmuous weld IS to be preferred as no bendlng I” the weld IS present Fillet weld normally l-l/Z bmes the plate thickness. For kght loads the Idlet IS same as the plate thtckness and for heawer loadstwce tfw plate thwkness. Heavy loads use smgle or double V weld wth 55Oangle and 1/16”lor gap and toe. Single V up to 5/8” plate and double up to 1” plate. Heawer welds and all code welds use single or double U welds. For all U welds gap and toe 3/32”.Use the dlmenslons gwen under code weldmg. Examples
Heawer loads “se single and double V welds-up to 5/g” single. Up to 1” double-gap and toe l/16” for all welds. Extra heavy loads and all code welds use single or double U-welds, For all U welds gap and toe 3/32”.use groove dimensions g,ven under code welding. Examples: mmSingle butt No bevel
Smgle V butt 30° bevel
Single U butt Groove code sto.
=a==-Double butt Double V butt Double U butt No bevel 30” bevel Groove code std. zzIziL&A Smgle IlM Smgle v Smgk u T weld T weld
zzii%n Oouble fillet
Double V
72JiBL Double U
Plug Weld: Use only where pantof weldmg IS not otherwiSe accessible. Ommeter 01 hole at least l-l/Z bmes the thxkness of the plate and countersunk. Holes three bmes the thickness of plate but not less than l/2” teqwres no countersmkmg. Specaly as required: Hole diameter. Csk. locabon and distance of welds, and grind.
mSmgle U butt Single V butt Welded both sides Welded both sides
Lap Welds: Single lap welds use very hght loads only-for all heaviec loads use double lap weld and make Illlets l-l/Z ttmes the plate. thickness to avadconcentrabon of stress in the weld. Specify: single or double-we of flllet-contlnuous-staggeredmtermlttent-double IntermIttent. Examples: I% TIMES PLATE THICKNESS 7 dm
Examples:
Single lap weld -e= DIA. Plug weld
=TfE Plug weld Grand flnlsh
Double lap weld
Welding j_ 1 1
WELDING INSTRUCTIONS
1
,,
Item 3. I
Conventions
CODE WELDING GROOVES
329
I
Edge Welds: Do not use for loads Edge welds are only used to make tight. I” many cases seam weld can be used Instead Up to 3/16 plates no bevel 1s reqwred. l/4” to 7/16” plates “se a 3O’bevel Over full thickness of the plates. l/Z” and heavier 30”bevel IS used over 3,‘4 of plate thickness Specify: SIX of weld (the SIX IS equal to the wdthof the weld)-conttnuous-IntermIttent-grind as required Examples.
I
Edge weld ‘Up to 3/16” plates
Edge weld l/4” to 7,‘16” plates
Edge weld l/Z” plate and
up
Single butt weld
Single tee weld
WELDING INSTRUCTIONS
em 2. cont.
I
Strength of Welds (non code): Permwbleumt stress for fillet welds made wthcoated welding rod IS 14,000 Ibs per square Inch. tak,“g Into conslderabon that practlcallyeveryf~llet weld is subfect to shear. The folIowang table gwes Safe workmg values for fillet welds: Sue Of fillet l/S” 3/16” l/4” 5/16” 3/6” l/2” 5/8” 3/4”
Load I” Ibs. per Itneal Inch
. . . . . . .._____................................ 1250 . . ..._._.___................................. 1 8 7 5 ..,......._._................................ 2 5 0 0 . . . . . . . . ..___............................... . 3125 ............................................. 3750 ..___........................................ 5 0 0 0 ............................................. 6250 _____........................................ 7 5 0 0
Double butt weld
Double tee weld
Calculated length of weld should be ancreased l/4” for startmg and stoppmg of the arc. For welds with no shear
present
the above values can be increased to:
16,000 Ibs. per square inch for tension, and 18.700 Ibs. per square Inch for compression. Butt welds and all welds employing a groove and having 100 percent PenetrattOn are ConsIdered 01 the same size as the plate even if weld IS built up higher smce the addtbonal material is not adding to the strength of the weld.
b
I
\
\
\I
I
-.7
A
A P P E N D I X
PRICING OF STEEL PLATE
Drawing quality 0.35 *Specified grain size (McQuaid-Ehn test) 0.90 *Forging quality 0.80 * Includes extras for killed steel or for any minimum specified silicon up to 0.15 %, inclusive, or any maximum specified silicon over 0.10 to 0.30’%, inclusive. d. Specification Extras The following extras, applicable to specifications listed under this caption or to equivalent specifications, include the classification extras Quality and Chemical Requirement, but no other extras, unless otherwise specified.
1. Mill Carbon-steel-plate Extras (in Dollars per 100 Pounds) (Courtesy
Inland Steel Company-Jan. 1956)
a. Item Quantity Extras 10,000 lb or over Under 10,000 lb to 6000 lb, incl. Under 6000 lb to 4000 lb Under 4000 lb to 2000 lb Under 2000 lb ta 1000 lb Under 1000 lb b. Classification Extras
$/lo0 lb None 0.10 0.20 0.50 1.00 1.50
ASTM Specification A-201, ASME SA-201, or Equivalent
$/loo lb
(1) Carbon-steel plates subject to chemicalcomposition limits or ranges None (2) Carbon-steel plates subject tu physical 0.05 requirements (melt test only) (3) Carbon-steel plates subject to ladle chemistry (carbon and/or manganese, phosphorus, and sulfur) and to physical requiremenls in qualities lower than “flange grade” 0.10 c. Quality Extras The following quality extras contain the applicable classification extras listed in item 6 above. $1100 lb Hot pressing steel (not boiler llange steel) 0.20 Cold pressing or cold flanging steel (not 0.25 boiler flange steel) Flange steel, ASTM h-285 or equivalent 0.40 Ordinary firebox steel, ASTM A-285 or o.so equivalent 0.55 Locomotive flange steel Locomotive firebox steel 0.65
Thickness 1%” and under Over 1%” to 2”, incl. *Over 2” to 3” 9 incl .
Grade B - Grade A FireFire-. Flange box Flange box 1.20 1.25 1.20 1.25 1.35 1.40 1.35 1.40 1.45 ... 1.45
ASTM Specification A-212, ASME SA-212, or Equivalent Grade A Grade B FireFireThickness Flange box Flange box 1 s” and under 1.20 1.25 1.20 1.25 Over 1%” to 2”, incl. 1.35 1.40 1.35 1.40 *Over 2” to 3”, incl. . 1 . 4 5 1.55 * Includes extra for heat treating test specimens. When material is required to be normalized or annealed, the extra shown under heat-treatment and surface330
Pricing of Steel Plate
finish extras shall apply in addition to the specification extra, which in the case of plates over 2” in thickness shall be reduced by $0.05 per 100 lb. e. Length Extras-All Plates, Rectangular or Otherwise $/lo0 lb 8’0” or over up to published limit of length, None but not over 50’0” 0.10 Under 8’0” to 5’0” incl. Under 5’0” to 3’0”’ incl. 0.20 Under 3’0” to 2’0”’ incl. 0.30 0.50 Under 2’0” to 1’0”’ incl 2.00 Under 1’0” to 6” ’ * 0.10 Over 50’0” to 60’0”, incl. 0.30 Over 60’0” to 80’0” incl. f. Width and Thickneis Extras, Dollars per 100 Pounds The size extras in Tables I and II also apply to plates ordered to weight per square foot on the basis of equivalent thicknesses in the ranges listed below. When plates are specified with the greater dimension at right angles to the direction of rolling, this dimension shall be considered as the width, and the extra figured accordingly. Table
I.
Sheared,
Gas-cut, Edge
and
Universal
Mill
Plates Thickness, Inches
?, / 4 to y; It0 J/i 5 46 to Under 3/i6, to 44, 3’2, to 1, 134, Width, Inches !/a Excl. Excl. Excl. Excl. Incl. %Over 8to12,incl. 1.55 1.35 1.20 1.10 0.95 0.95 ‘Over12 to24,incl. 1.50 1.30 1.15 1.05 0.90 0.90 1.40 1.20 1.05 0.90 0.75 0.75 Over24to30.incl. Over 30to36;incl. 1.25 1.05 0.90 0.75 0.60 0.65 Over36ta48,incl. 1.20 1.00 0.80 0.65 0.50 0.55 1.05 0.85 0.65 0.50 0.30 0.40 Over48 to60.incl. Over 60 to 80; incl. 1.00 0.65 0.45 0.30 0.10 0.25 Over80to90,incl. 1.00 0.60 0.40 0.25 None 0.15 Over 90 to 96, incl. 0.75 0.50 0.35 0.15 0.25 1 Add 0.65 per 100 lb to all extras shown above for plates OVRP thick unless killed steel is specified or implied. 2 Maximum leneth limits for aide shearine in these widths: O v e r 8” to 12”, &cl., but not over 45” thLk, incl.--120” Over 12” to 24”, excl., hut not over )-i“ thick, incI.-240” Over 12” to 24”. excl., and over x” to %” thick, incl.-180”
1OYPT 1:; to3, Incl. 1.00 0.95 0.80 0.70 0.60 0.45 0.30 0.20 0.30 1 ?.$I”
All sheared-edge plates longer than the foregoing shearing limits or over x” thick must be gas rut to size at the listed gas-cutting extras. Table II. Mill-edge Plates
Over 3 6 to 4 8 , incl. Over 4 8 to 6 0 , incl. Over 6 0 to 7 2 , incl.
1.10 1.05 1.00
0.90 0.85 0.65
0.70 0.65 0.45
Extras
per
Linear
Foot
0.55 0.50 0.30
0.40 0.30 0.20
g. Killed-steel Extra* Silicon-killed steel, aluminum-killed steel, or steel killed by any deoxidizing agent, specified or implied $0.65 per 100 lb * Extra does not apply to forging quality or specified-grain-size quality (McQuaid-Ehn test).
of
Cutting
(Court.esy Inland Steel Comp;\nl--Xfa! 1953) Thickness, Inches 1 or under 1% 1% 1% 1% 1% 1% 1% 2
2% wi 23/ 2>-;
2% 2% 2% 3 356
Extras, $ per Linear Foot of Cut 0.37 0.38 0.39 0. .40 0:41 0.42 0.43 0.44 0.45 0.46 0.17 0.48 0.49 0.51 0.53 0.55 0.57 0.59
Extras, r% per Lineal Foot of Cut 0.61 0.63 0.65 0.67 0.69 0.71 0.73 0.75 0.7; 0.79 0.81 0.85 0.80 0.93 0.97 0.99 1.03 1.0;
Plates ordered to a maximum of carbon exceeding 0.39% or a minimum of 0.25% carbon together with a manganese content of over 1.00% require edge tra;rlment. The extra for such edge treatment is $1.50 per 100 lb to be assessed in addition to the foregoing gascutting rates. Mill
Circular-
and
Sketch-plate
Extras
(Courtesy Inland Steel Company-May 1953) Circular plates Semicircular plates Sketch plates furnished to a radius Regular sketch plates with not more t,han four straight edges Irregular sketch plat.es wit,h more than four straight edges Warehouse
Base
Prices
and
35’j& 35Y, 35 76 25 ‘i;i 40 ‘;,
Extras
(Courtesy J. T. Ryerson and Son, Inc., Chicago, Ill.-April 14, 1955) a . Base Price of Hot-rolled Carbon Steels
Thickness, Inches +/a to %6 %to Under xs, to 5% Pi 1 ‘i. ;x,c; ;y; ;x,c; oT;5
- OverWidth, 3 0 to Inches 3 6 , incl.
Mill-Gas-cutting
331
$/lOO ItI Structural shapes 5.99 Junior beams 6.64 Stair channels 6.64 Bars and bar shapes 5.81 Hot-rolled strip 5.92 Plates, hot-rolled 5.82 Sheets, hot-rolled 5.68 b. Base Price of Cold-finished Carbon Bars (Rounds, Squares, Flats, Hexagons) $/lOO lb Chicago 7.25
332 C.
cl.
e.
Pricing of Steel Plate
Hot-rolled Carbon Steel $/IO0 lb Quantity Extras Base 30,000 lb and over 0.20 20,000 to 29,999 lb 0.40 10,000 to 19,999 lb 0.60 5,000 to 9,999 lb 0.70 2,000 to 4,999 lb 1.00 1,000 to 1,999 lb 400 to 999 lb 1.95 100 to 399 lb 3.70 Under 100 lb 6.70 Cold-finished Carbon Bars $/loo lb Quantity Extras Base 2000 lb and over 0.50 1000 lb to 1999 lb 1.50 500 lb to 999 lb 3.35 300 lb to 499 lb Item under 300 lb 4.00 7.00 299 to 150 lb (total order) 12.00 149 to 75 lb (total order) 17.00 Under 75 lb (total order) (Total order applies when the total weight of coldfinished carbon bars purchased in one day for shipment at one time to one destination falls within one of the quantity ranges indicated.) Cutting Extras in Dollars per 100 Pounds
Rectangles
Universal Mill Plates 1” thick and lighter 11”’ 18 thick and heavier, cut under 5’0” 1ong 136” thick and heavier, cut 5’0” and longer
Thickness, Inches 346 % .%6
N 746
JS X6 w 34 Td
1 1% 1% 1% 1% 1% 1% 2
5'0" to 12'0"
6%
0.60
0.40
0.30
0.25
Sketch Plates For simple sketches with straight sides and no re-entrant cuts, add 0.10 cwt to above width and length extras. (A simple sketch is a rectangle modified by only one additional cut.)
See structural schedule No charge for cutting
f. Plate Flame-cutting Charges
To Length Only To 8’ 5’ to Length and under Under Under Under Under Only over 8’ 5’to3’ 3’to2’ 2’tol’ 1 ’ S6 tol” 0 . 0 0 0 . 2 0 0 . 3 0 0 . 5 0 0 . 7 5 1.55 thick* To Width and Length, xs” to 1” Thick* To Width 6” Wide Over 6” Over 10” and and to lo”, to 24”, Over 24” Length+ under Incl. Incl. Wide Over 12’0” 1.50 0.75 0.60 0.45 Under 5’ to 3’, incl. 0.70 0.45 0.35 0.30 Under 3’ to 2’, incl. 0.85 0.55 0.50 0.50 Under 2’ to 1.20 0.80 0.75 0.75 l’, incl. Under 1’0” 2.80 1.85 1.80 1.75 * Over 1” thick-see flame-cutting extras, but on high-carbon and abrasion-resisting use flame-cutting extras on items over 35” thick. t This schedule also covers stock-length plates sheared to width only.
Use shearing schedule above
2% 2% 2% 3 3?d 3 % 3 % 4 4 % 5 5 % 6 7 7 % 8 9
10
Extras per Linear Foot Addl. Footage. First 100 Ft over 100 Ft of Any Item of Any Item $0.19 $0.10 0.19 0.10 0.11 0.20 0.12 0.21 0.22 0.13 0.23 0.14 0.24 0.15 0.25 0.16 0.26 0.17 0.35 0.37 0.38 0.39 0.40 0.41 0.42 0.43 0.45 0.47 0.49 0.53 0.57 0.61 0.65 0.69 0.73 0.81 0.89 0.99 1.07 1.16 1.24 1.33 1.46 1.73 2.01
0.26 0.28 0.29 0.30 0.31 0.32 0.33 0.34 0.36 0.38 0.40 0.44 0.48 0.52 0.56 0.60 0.64 0.72 0.80 0.90 0.98 1.07 1.15 1.24 1.37 1.64 1.92
Charges are based on linear feet for each item. Fractions of an inch are charged at the next full inch. The minimum charge is $1.50 net per item. Note: All other mill extras apply to warehouse stocks.
Pricing of Steel Plate
333
d. Flange-quality and Firebox-quality Steel Plates (Courtesy J. T. Ryerson and Sons, Inc., 1954-55 stock list and reference book) Hot Rolled-Open Hearth Tensile Strength 55,000 to 65,000 psi Conforms to ASME SA-285, grade C ASTM A-285 (latest) grade C Size, In. 1/4x 30 36 42 48 54 60 72 84 96 120 5/16x 30 36 42 48 54 60 72 84 96 120 3/8x 30 36 42 48 54 60 72 84 96 120 ?/16x 30 36 42 48 54 60 72 84 96 120 1/2x 30 36 42 48 54 60 72 84 96 120
Wt
per Ft Lb
Size and Quality Extra (S/100 lb) Flange Firebox
Stock Lengths, Ft
25.50 30.60 35.70 40.80 45.90 51.00 61.20 71.40 81.60 102.00
1.35 1.20 1.20 1.20 1.05 1.05 0.90 0.90 0.95 1.30
1.60 1.45 1.45 1.45 1.30 1.30 1.15 1.15 1.20 1.55
S-30 S-30 S-30 S-30 S-30 S-30 S-30 S-30 S-30 S-30
31.88 38.25 44.63 51.00 57.38 63.75 76.50 89.25 102.00 127.50
1.20 1.05 1.05 1.05 0.90 0.90 0.75 0.75 0.80 1.15
1.45 1.30 1.30 1.30 1.15 1.15 1.00 1.00 1.05 1.40
S-30 S-30 S-30 S-30 S-30 S-30 S-30 S-30 S-30 S-30
38.25 45.90 53.55 61.20 68.85 76.50 91.80 107.10 122.40 153.00
1.10 0.95 0.95 0.95 0.80 0.80 0.65 0.65 0.70 1.05
1.35 1.20 1.20 1.20 1.05 1.05 0.90 0.90 0.95 1.30
S-30 S-30 S-30 S-30 S-30 S-30 S-30 S-30 S-30 S-30
44.63 53.55 62.48 71.40 so.33 89.25 107.10 124.95 142 SO 178.50
1.10 0.95 0.95 0.95 0.80 0.80 0.65 0.65 0.70 1.05
1.35 1.20 -’ 1.20 1.20 1.05 1.05 0.90 0.90 0.95 1.30
S-30 S-30 S-30 S-30 S-30 S-30 S-30 S-30 S-30 S-30
51.00 61.20 71.40 81.60 91.80 102.00 122.40 1442.SO 163.20 204.00
0.95 0.80 0.80 0.80 0.65 0.65 0.50 0.50 0.55 0.90
1.20 1.05 1.05 1.05 0.90 0.90 0.75 0.75 0.80 1.15
S-30 S-30 S-30 S-30 S-30 S-30 S-30 S-30 S-30 S-30
1
Size, In. 9/16x 30 36 42 48 54 60 72 84 96 5/8x 30 36 42 48 60 72 84 96 120 3/4x 30 36 42 48 60 72 84 96 120 7/8x 30 36 48 60 72 120 lx 30 36 4s 60 72 84 96 120 11/8x 72 11/4x 72 11/2x 72 13/4x 72
Wt
per Ft, Lb
Size and Quality Extra (S/l00 lb) Flange Firebox
Stock Lengths, Ft
57.38 68.85 SO.33 91.80 103.30 114.80 137.70 160.65 183.60
..* ... ... ... ... ... ... ... ...
1.20 1.05 1.05 1.05 0.90 0.90 0.75 0.75 0.80
S-30 S-30 S-30 S-30 S-30 S-30 S-30 S-30 S-30
63.75 76.50 89.25 102.00 127.50 153.00 178 50 204.00 255.00
0.95 0.80 0.80 0.80 0.65 0.50 0.50 0.55 .
1.20 1.05 1.05 1.05 0.90 0.75 0.75 0.80 1.15
S-30 S-30 S-30 S-30 S-30 S-30 S-30 S-30 S-30
76.50 91.80 107.10 122.40 153.00 183.60 214.20 244. SO 306.00
0.95 0.80 0.80 0.80 0.65 0.50 0.50 0.55
1.20 1.05 1.05 1.05 0.90 0.75 0.75 l.15
S-30 S-30 S-30 S-30 S-30 S-30 S-30 S-30 S-30
89.25 107.10 142 SO 178.50 214.20 357.00
0.95 0.80
1.20 1.05 1.05 0.90 0.75 1.15
S-30 S-30 S-30 S-30 S-30 S-30
102,oo 122.40 163 20 204.00 244, SO 285 60 326.40 408.00
0.95 0.80
1.20 1.05 1.05 0.90 0.75 1.‘15
S-30 S-30 S-30 S-30 S-30 S-30 S-30 S-30
275.40
0.50
...
30
306.00
0.50
...
30
367.20
0.50
...
30
428.40
1.50
...
30
d.65 0.50
0’. 65 0.50 0.50 0.55 ...
A P P E N D I X
ALLOWABLE
Item 1.
STRESSES
Maximum Allowable Stress Values in Tension for Carbon and low-alloy Pipe and Tubes of Welded Manufucture, in Pounds per Square Inch
(Extracted from the 1956 Edition of the ASME Boiler and Pressure Vessel Code, Unfired Pressure Vessels, with Permission of the Publisher, the American Society of Mechanical Engineers) (Joint efficiencies used for preparing this table are: electric-resistance welded-85 %, lap welded-80 70, but,t welded-60s.) SpeciFor Met,al Temperatures Not Exceeding Deg F Specifified -~__--. cation Nominal Min -2oto Number Grade Composition Weld Notes Tensile 650 700 ..~______ 750 800 850 ..-__900 950 1000.-SA-53 Carbon steel (1) 45,000 9000 8800 8200 ...bo ::: Lap SA-53 ‘A’ Carbon steel Resist. (l)(2) 48,000 10,200 9000 9100 i9bo 5500 . . . SA-53 B Carbon steel Resist. (l)(2) 60,000 12,750 12,200 11,000 9200 7350 5500 . . . ... 40,000 8000 SA-72 . . Wrought iron Lap . 7800 7300 . . . ,.. . . . . . . . SA-72 Wrought iron Butt 40,000 6000 5850 5500 SA-135 *A‘ Carbon steel Resist. (i)‘(i) 48,000 10,200 9900 9100 i4bo 6jbo 5500 ” “’ . . : 1: SA-135 Resist. (2)(3) 60,000 12,750 12,200 11,000 B Carbon steel 9200 7350 5500 . . . 9700 SA-178 A Low-carbon steel Resist. (2)(3) . . 10,000 8950 7800 6650 5500 3800 iii0 SA-178 B 0. H. iron Resist. 8500 8300 7750 SA-178 C Medium-carbon steel Resist. (i)‘(i) 6d,bbO 12,750 12,200 11,000 bib0 i350 Go i&i0 2ido SA-226 Low-carbon steel Resist. (2) (3) 10,000 9700 8950 7800 6650 5500 3800 2100 S A - 2 5 0 Tl Carbon-+molybdenum R e s i s t . 5i,bbO 11,700 11,700 11,700 11,450 11,200 10,650 8500 5300 SA-250 Tla Carbon-timmolybdenum R e s i s t . 60,000 12,750 12,750 12,750 12,250 11,700 10,650 8500 5300 S A - 2 5 0 Tlb Carbon-54 molybdenum Resist. 53,000 11,250 11,250 11,250 11,050 10,850 10,650 8500 5300 SA-333 C Carbon steel 11,700 _.. . . . . . . ,.. _., .‘. . . . R e s i s t . 55,000 SA-333 3 316 nickel Resist. 65,000 23,800 _.. ... . . ... ,., . . . . . . SA-333 5 3 nickel Resist. 65,000 13,800 . . ... ... ... . . . . . . _.. SA-334 C Carbon steel Besist. . 55,000 11,700 ... ... . . . .,. . . . s-334 3 335 nickel R e s i s t . 65,000 13,800 . . . ... ... ... . . . .,. . . . SA-334 5 5 nickel Resist. 65,000 13,800 ,__ . . . . . . rliotes: The st.ress values in this table may be interpolated to determine values for intermediate tempera&es. ’ ’’ . . (1) These stress values permitted for open-hearth and electric-furnace steels only. (2) For service temperatures above 850 F it is recommended that killed steels containing not less than 0.10% residual silicon be used. Killed steels which have been deoxidized with large amounts of aluminum and rimmed steels may have creep and stress-rupture properties in the temperature range above 850 F, which are somewhat less than those on which the values in the above table are based. (3) Only (silicon) killed steel shall be used above 900 F. 335
336
Allowable Stresses Item 2.
Maximum Allowable Stress Values in Tension for Nonferrous Metals, in Pounds per Square Inch
Aluminum and Aluminum-alloy Products (Extracted from the 1956 Edition of the ASME Boiler and Pressure Vessel Code, Unfired Pressure Vessels, with Permission of the Publisher. the American Society of Mechanical Engineers) Specification Number
Alloy
Temper
Specified Tensile Strength,
pi
Minimum Yield Strength, psi Notes
For Metal Temperatures Not Exceeding Deg F 100
150
200
250
300
350
4800
1650 2500 2750 3000 2350 3000 3500 4000 3350 3600 4250 5000 3000 3600 4000 4800 5650 5750 7000 8000 5300 5500 6800 7800 4000 5000 5500 6250
1650 2150 2550 3000 2350 2800 3400 3900 3150 3250 4000 4850 2900 3200 3800 4600 5650 5750 7000 8000 5300 5500 6800 7800 4000 5000 5500 6250
1600 1950 2350 2900 2300 2550 3150 3650 2900 3000 3800 4700 2700 3000 3600 4400 5650 5750 7000 8000 5300 5500 6800 7700 4000 5000 5500 6200
1450 1700 2100 2700 2100 2250 2900 3300 2700 2800 3600 4400 2500 2800 3400 4200 5500 5500 6550 7400 5200 5200 6300 7200 4000 4900 5350 6050
1250 1500 1900 2350 1850 2000 2650 3000 2400 2500 3300 4000 2200 2500 3100 3800 4650 4650 5800 6550 4400 4400 5600 6300 4000 4500 4800 5400
1200 1300 1600 2000 1600 1700 2400 2700 2100 2200 3000 3500 2000 2200 2800 3400 3850 3850 5050 5600 3700 3700 4900 5400 3350 3700 3800 3950
1050 1100 1400 1600 1300 1400 2100 2200 1800 1900 2650 3100 1700 1900 2500 2900 3150 3150 4300 4700 3000 3000 4100 4600 2800 2800 2800 2800
6250
6250
6200
6000
5400 4650 3900
7750 8500 7500 10,500 6000 6800 9500 6000
7750 8500 7200 10,200 5900 6500 9200 5900
7650 8400 7000 9900 5700 6200 9000 5700
7100 7700 6700 9400 5400 6000 8500 5400
6400 6900 6400 7900 5000 5800 7200 5000
5600 6100 5600 6200 4200 5100 5600 4200
4800 5300 4000 4400 3200 3600 4000 3200
8400 10,000 13,000 9500 6000 15,000
8100 9700 12,200 9200 5900 14,300
7700 9400 11,600 9000 5700 13,700
7100 9000 10,400 8500 5400 12,000
6000 7800 7200 7200 5000 9100
4800 6200 4400 5600 4200 5700
3400 4600 3000 4000 3200 3950
8400 8200 7900 7500 4800 4700 4600 4400 i7j 10,000 9700 9 4 0 0 9 0 0 0 (7) 13,000 12,200 11,600 10,400
6300 4000 7800 7200
4900 3400 6200 4400
3300 2600 4600 .300G
Sheet and Plate SB-178
SB-178
SB-178
SB-178
SB-178
996A
990A
MlA
Clad MlA
MGllA
SB-178
Clad MGllA
SB-178
GlA
SB-178
GRZOA
SB-178 SB-178
GSllA Clad GSllA
0 H112 H12 H14 0 H112 H12 H14 0 H112 H12 H14 0 H112 H12 H14 0 H112 H32 H34 0 H112 H32 H34 0 H112 H32 H34 0 H112 1 H32 H34 T4 T6 T6 welded T4 T6 T6 welded
9500 10,000 11,000 12,000 11,000 12,000 14,000 16,000 14,000 14,500 17,000 20,000 13,000 14,500 16,000 19,000 23,000 23,000 28,000 32,000 22,000 22,000 27,000 31,000 18,000 20,000 22,000 25,000
2500 4000 9000 10,000 3500 5000 11,000 14,000 5000 6000 12,000 17,000 4500 6000 11,000 16,000 8500 9000 21,000 25,000 8000 8500 20,000 24,000 6000 8000 16,000 20,000
25,000
9500
31,000 34,000 30,000 42,000 24,000 27,000 38,000 24,000
23,000 26,000 16,000 35,000
T6 T4 T6 T6 T6 welded T4
42,000 62,000 65,000 38,000 24,000* 60,000
35,000 40,000 55,000 35,000
T6 T6 welded T4 T6
42,000 24,000* 62,000 65,000
35,000
14,bbo 32,000 ...
(l&h (1) $p (1) (ii& (1) (1) i($v (1) iij (1) (1)
iii (1) (1) ii; (1) .. (1) is,’ (5) I$ ...
Bars, Rods, and Shapes SB-211 SB-211 SB-211 SB-273
GSllA CG42A CS41A GSllA
SB-273
CG42A
4u,ouo
Bolting Materials / I
SB-211
GSllA
SB-211 S!B-211
CG42A CS41A
\
\
4d,bbo 55,000
\I
/
(5)
Allowable Stresses Item 2.
Specification Number
Maximum Allowable Stress Values in Tension for Nonferrous Metals, in Pounds per Square Inch (Continued)
Aluminum and Aluminum-alloy Products (Continued) Specified Minimum Tensile Yield For Metal Temperatures Not Exceeding Deg F Strength, Strength, 400 100 150 200 250 300 350 Temper psi psi Notes
Alloy
Pipe and Tube SB-274 SB-274 SB-234 SB-274 SB-274 SB-274 GSlOA SB-274 SB-234 SB-274 >
337
GSllA
H112
14,000 14,500
H14
20,000
H18 T42 T5 T6 T4 T6 T6 welded
27,000 17,000 22,000 32,000 26,000 38,000 24,000*
0
Forgings SB-247 SB-247
MlA CS41A
F T4
SB-247
GSllA
T6 T6
SB-247
GSllB
14,000
55,000 65,000 38,000
T6 welded 24,000* T6 36,000
5000 6000
iij
3350 3600
17,000 24,000
(1) (1)
10,000 15000
25,000 16,000 35,000
...
5000 30,000 55,000 35,000 sd,bbo
(5) ii;
(5) (5)
...
iij (5) (5)
iij
2700 2800
2400 2500
2100
2200
1800 1900
4700
4400
4000
3500
3100
6050 4200 4900 7200 6000 9000 5700
5700 4150 4600 6550 5800 8500 5400
5250 4050 4150 4800 5600 7200 5000
4400 3300 3300 3300 4900 5600 4200
3500 2100 2100 2100 3500 4000 3200
2700
2400 10,200 11,300 7200 5000 6100
3150 3250
2900 3000
5000
4850
6750 4250 5500 8000 6500 9500 6000
6400 4200 5100 7600 6200 9200 5900
3350 3150 13,800 12,800 16,200 15,200 9500 9200 6000 5900 9000 8400
2900 12,000 14,400 9000 5700 7900
11,000 14,000
8500 5400 7300
2100 1800 5750 3900 5750 3900 5600 4000 4200 3200 4700 3200
* Strength of full-section tensile specimen required to qualify welding procedures.
Notes:
(1) For welded construction, stress values for 0 material shall be used. (2) For nominal thicknesses not greater than 0.500 in., the stress values for H14 material may be used; for nominal thicknesses of 0.501 to 1.000 in., the values for H12 material may be used; for thicker material the values listed shall be used. (3) For nominal thicknesses not greater than 2.000 in.; for thicker material the stress values for 0 material shall be used. (4) For nominal thicknesses not greater than 0.500 in., the stress values for H12 material may be used; for thicker material the values listed shall be used. (5) The stress values given for this material are not applicable when either welding or thermal cutting are employed. (6) For nominal thicknesses not less than 0.25 in.
Item 2.
Maximum
Allowable
Stress
Values
in
Tension
for
Nonferrous
Metals,
in
Pounds
per Square
Inch
(Continued)
Copper and Copper Alloys Material and Specification NUUlhW
Coudit,iorl
Copper SB-11 SB-12 SB-13 SB-42 88-42 SB-42 SB-75 SB-75 SB-75 SB-111 58-111 SB-152
Plates Rods Seamless boiler tubes Pipe Pipe Pipe Seamless tubes Seamless tubes S e a m l e s s tubes Seamless condenser tubes Seamless condenser tubes Plate steel, at,rip and har
Red brass SB-43 SB-111
Pipe Annealed Seamless condenser tubes Annealed
Annealed Annealed Annealed Annealed Light drawn Hard drawn Annealed Light drawn Hard drawn Light drawn Hard drawn Annealed phosphorus, deoxidized
Size.
iu.
Specified Minimum Tensile Yield Strength, Strength. n.; .?a:
---____150 100 200
For Metal Temperatures Not Exceeding Deg F ~~ ____.. 300 350 400 500 450
250
550
10,000* I o,ooo* 9000* 9000* :lo~ooo* 40,000* 9000* 30;000* 40,000* 30,000* 40,000* in,ooo*
6700 6700 6000 6000 9000 11,300 6000 9000 11,300 9000 11,300 6700
6700 6700 6000 6000 9000 11,300 6000 9000 11.300 9000 11.300 6700
b500 6500 5900 5900 8700 11,000 5900 8700 11,000 8700 11,000 6500
6300 6300 5800 5800 8300 10,500 5800 8300 10,500 8300 10,500 6300
5000 5000 5000 5000 8000 8000 5000 8000 8000 8000 8000 5000
3800 3800
. . . .
30,ooo 30,000 30,000* 30,000* 36,000* 45,000’ 30,000* 36.000* 45.000* 36,000* 45,000* 30,000
38OIJ 3800 5000 5000 3800 5000 5000 5000 5000 3800
2500 2500 2500 2500 2500 2500 2500 2500 2500 2500 2500 2500
... ... . . . .
. ...
4o,olJo* 4n.nno*
12,000* 12,non*
8000 8000
8000 8000
8000 8000
8000 9000
8000 8000
6000 6000
3000 3000
2000 2000
... ...
... ...
600
650
.
. .
.
Admiralty, A. B. C. D SB-111 Seamless condenser tubes SB-171 Tube plates
Annealed Annealed
... ...
45,000* 45.000
15,000* 15,000
10,000 10.000
10,000 10.000
10,000 10,000
10,000 10,000
10,000 10,000
8000 8000
5000 5000
3000 3000
... ...
... ...
Aluminum brass. B. C. D SB-111 Seamless condenser t.uhea
Annealed
. .
50,000*
18,000
12.000
12,000
12,000
12,000
12,000
7500
3000
2000
.
. .
Naval brass SB-171 Tube platea
Anuesled
. .
50,000
20,000
12.500
12,500
12,000
11,200
10,500
7500
2000
.
. .
Annealed Annealed
... ...
50.000* 50.000
2O,UOO* 1 2 , 5 0 0 1 2 , 5 0 0 1 2 , 0 0 0 11,200 10,500 20,000 12,500 12,500 12,000 11.200 10.500
7500 7500
2000 2000
...
Copper-nickel, 70-30 SB-111 Seamless condenser tubes SB-171 Tube plates
Annealed Annealed
... ...
52,000 50,000
18.000 20.000
12.000 12,500
11,600 12,500
11,300 12,500
11,000 12,500
10,800 12,200
10,600 12,000
10,300 11,700
10,100 11,300
9900 11,000
9800 10,500
9600 10,000
9500 9500
9400 9000
Copper-nickel, 80-20 SB-111 Seamless condenser tubes
A nun&d
45.000
16,000
10,700
10,600
10,500
10,400
10,300
10,100
9900
9600
9300
8900
8400
7700
7000
40,000*
15.000*
10,000
10,000
9800
9500
9300
9000
8700
8300
7500
6700
6000
. .
...
. 10.500
7500
6000
Muntx metal S&l11 Seamless condeuser SB-171 Tube plates
tubes
Copper-nickel, 90-10 SB-111 Seamless condenser tubes Anuealed Aluminum bronze SB-111 Seamless condenser t,ubes SB-171 Tube plates
Annealed Annealed
.
50,000* 90,000
19,000 36.000
12,500 22,500
12,400 12.200 22,500 21,000
11,900 19,500
11.600 18.000
10,000 16.500
6000 15.000
4000 13.500
2000 12,000
Aluminum bronze D SB-169 Plate. sheet
A nnealnd
...
70,000
30,000
17,500
17.500
16.800
16,000
15,500
15.000
14.500
12,000
10.000
Copper-silicon, A, C SB-96 Plate, sheet (1) SB-98 Rods (1) SB-98 Rods (1)
Annealed Soft Half hard
... ... . .
50,000 52,000 70,000
18,000 15,000 38,000
12,000 10,000 14,000
12.000 11.900 10,000 10,000 14,000 14,000
11,700 10,000 14,000
10,000 10,000 14,000
5000 5000 10,000
... ... ...
... ... ...
... ...
. . ... . .
Copper-silicon B SB-98 Rods (1) SB-98 Rods (1)
Soft Helf hard
... ...
40,000 55,000
12,000 20,000
8000 11,000
8000 8000 11,000 11,000
8000 11,000
7000 10.000
5000 8000
. . . . . . . . . . . .
... ...
... ...
Soft
...
30,000
10,000*
2100
2000
. .
,..
BOLTING MATERIALS Copper SB-12 Rod
. .
.
9000
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
..
...
...
...
.
(3) 2500
2500
2500
2400
2200
.
.
......
:
.
Item 2.
Maximum Allowable Stress Values in Tension for Nonferrous Metals, in Pounds per Square Inch (Continued)
Copper and Copper Alloys (Continued) Material and Specification Number Copper-silicon, A, D SB-98 Rod (1)
Condition Soft Quarter bard Half hard
Copper-silicom B SB-98 Rod (1)
Soft Bolt temper
Aluminum bronze SB-150 Alloy No. 1
SB-150 Alloy No. 2
m-150
\llcry N o .
CASTING MATERIAL SB-61(2) SB-62(2)
3
Size, in.
For Metal Temperatures Not Exceeding Deg F 100
150
200
250
300
350
400
450
500
3800 6000 9500
3800 5900 9300
3800 5800 9000
3800 5600 8800
3500 5500 8500
...
...
...
3000 13,100 10,800 9600
2900 12,700 10,500 9400
2800 12,300
...
...
. .
95ou 8900 8300
9400 8800 8200
9000 8600 8000
7500 7500 7500
6000 6000 6000
4500 4500 4500
10,500 10,500 10,000
9000 9000 9000
7500 7500 7500
6000 6000 6000
... ...
52,000 55,000 70,000
15,000 24.000 38,000
3800 6000 9500
All Up to )i, incl. Over $5 to 1. incl. Over 1 to 19;. incl.
40,000 85,000 75,000 75,000
12,000 55,000 45,000 40,000
13,800 11,200 10,000
U p t o 36, id. Over 56 to 1. incl. Over 1
80.000 75,000 72,000
40,000 37,500 35.000
10,000 9400 8800
10,000
9400 8800
10,000 9300 8700
9900 9200 8600
9800 9200 8600
98oO 9100 8500
100,000 90,000 85,000
50,000 45,000 42,500
12,500 11,200 10,600
12,500 11,200 10,600
12,500 11,200 10,600
12,500 11.200 10,600
12,400 11,200 10,600
12,400 11,200 10.500
12,300 11,100 10,500
12,200 11.000 10;400
12,000 10.800 10;200
to,000 35,000 32,000
10,000 8800 8000
10,000 8800 soon
10,000 8800 7900
9900 8700 7800
9800 8600 7800
9700 8600 7600
9600 8400 7600
Y500 8300 7400
7000 7000 7000
6800 6000
6800 6000
6800 5800
6800 5500
6500 5000
6000 4500
5500 3500
...
36 to 1, incl. Over 1 to 2, incl. O v e r 2 to 4, incl.
...
Up to 56, incl. Over 36 to 1, incl. Over 1 to 2. iocl.
... ...
Specified Minimum Tensile Yield Strength, Strength, osi psi
90,000 75,000 7P,OOO 34,000 30,000
3000 13,800 11,200
10.000
10.000
550
9000 9700 9000 8400
4000
600
650
. . . . .
... ... . .
. . .. ..
700
... ... ... . . ... . .
. ... ... 3300
... ...
... ,..
... ...
_. * E x p e c t e d v a l u e , not i n c l u d e d i n s p e c i f i c a t i o n s . (1) C o p p e r - s i l i c o n a l l o y s a r e n o t always s u i t a b l e w h e n e x p o s e d t o c e r t a i n m e d i a a n d h i g h t e m p e r a t u r e s , p a r t i c u l a r l y s t e a m a b o v e 2 1 2 F . The user should satisfy himself that the alloy selected is satisfactory for the service for which it is to be used. (2) In the absence of evidence that the casting ia of high quality throughout, valuea not in excew of 80 x of those given in the table shall be used. This is not intended to apply to valves and fittingE made .,o r e c o g n i z e d s t a n d a r d s .
Item 2 . M a x i m u m A l l o w a b l e Stress Values in Tension for Nonferrous Metals, in Pounds per Square Inch , Nickel and High-nickel Alloys Material Form, and Specific&n Number Nicks el B a rs, rods, and shapea SB-160 SB-160 Bolting SB-160 SB-160 Pipe or tubing SB-161 :::::: Co;&..;~ tubing SR.163 Pla #a, sheet, or strip SB-162 SB-162 (plate only) Low-carbon Nickel Bagti-y6$ and shapes SB-160 Bolting SB-160 Pipe or tubing E??? ,u-I”I odenser tubing 3B-163 3B-163 PI:ke, sheet. or strip I 3B-162 I jB-162 (plate only) N i c k e l - c c ,p lx Bars, r‘0 Bs, and shapes
SB-164 (class
SB-164 (class Bolting SB-164 (class A and B) !y;;s6”A (class A only) Class B Pipe or tubing SB-165 SB-165 -am ILC >“-I”.3
odenser tubing 3B-163 3B-163 te, sheet, or strip 3B-127 jB-127 (plate only)
1
-- ___
PiPSegyi6ybing
I
Co;cl~f;~
tubing
Pl;tt&$et, or strip SB-168 (plate only)
Notes:
Condition
Yield Specified Strength Tensile (0.2 % Strength. Offset), psi N o t e s 100
Hot or cold worked-annealed Hot rolled or forged-hot finished
55.000 60.000
15,000 15.000
Hot or cold worked--annealed Hot rolled or forged-hot finished (3)
55,000 60,000
15,000 15,000
S~*IId~SS--a~Il~*led Senmless-hard. stress relieved Seamless--stress equalized
55,000 65,000 70,000
15,000 40,000 50,000
10,000 10.000 3700 3700 (1) (1)
10,000 16.200
16.200
For 200
10,000 10.000
300
400
10,000 10,000
3700 3700
3700 3700
10,000 ;?;Wz ,
10.000 ;;,o”W; ,
10,000 10,000 3700 3700 :s”,:i 15:ooo
Metal Temperaturea 500
600
10.000 9500
10,000 8300
3700 2700
3700 3400
2% 14:500
10,000 : ::
w g
(Continued)
Not Exceedin* Deg 800 700
F 900
1000
... .
. .
. . ... . . ...
.
... ... ... ... ... ... ... ... ... ... ...
...
...
. . .
. . .
1 1 0 0 12002
... ... ... ... ... ... ... ... ... ... ...
i5z ? % f
.
... ... . ..
Seamless--annealed Seamless-stress relieved
Sl,OOO 65,000
15.000 40.000
10,000 16,200
10.000 15,300
10,000 15,000
10,000 15.000
10,000 14,500
10.000
Hot or cold rolled-annealed Hot rolled-as rolled
55,000 55.000
2%M)O
15,000
10,000 13,300
10,000 13,300
10,000 13.300
10,000 13,300
10,000 12,500
10,000 11.500
Hot or cold worked-annealed Hot rolled or forged-hot finished
50,000 50,000
10.000 10,000
6700 6700
6400 6400
6300 6300
6200 6200
6200 6200
6200 6200
6200 6200
5900 5900
4500 4500
3000 3000
2000 2000
1200 1200
Hot worked or annealed
50,000
10.000
2500
2408
2300
2300
2300
2300
2300
2200
2100
2000
1800
1200
4500
3000 ...
2000 .
1200
.
...
8000 15,000
7700 14,200
7500 13.800
7500 13.500
7500 13,500
7500 .*.
7400
7200
12,000 30,000
8000 15.000
7700 14.200
7500 13.800
7500 13.500
7500 13.500
7500 13,000
7400 12,000
7200 11,000
4500 10,000
3000 ...
2000
1200
50,000 50.000
12,000 12,000
8000 8000
7708 7700
7500 7500
;i::
7500 7500
7400 7400
7200 7200
4500 4500
3000 3000
2000 2000
1200 1200
Hot or cold worked-annealed Hot rolled or forged-hot finished
70.000 80.000
25,000 40,000
16.600 20,000
14,600 18,900
13.600 18,400
13,200 18.200
32 ,
::::z
8000 4000
Hot or cold worked--annealed Hot rolled or forged-hot finished (3) Cold drawn--as drawn (4) (5) Cold drawn--as draw, (4) (5)
70,000 80.000 90,000 85,000
25,000 40,000 70,000 50,000
6100 10.000 17.400 12.400
5700 9600 16,900 12,000
5200 9400 16,200 11,500
5000 9000 15,500 11,100
i%: 15,400 11,100
4900 8300
4700 4000
Seamless-annealed Seamless-hard. strew relieved Seamless-stress equalized
70,000 85.000 85,000
28,000 55,000 65,000
17,500 21,200 21.200
16,500 20,200 20.200
15,500 19,500 19,500
14,800 19.200 19,200
14.700 19,200 19,200
Seamless--annealed Seamless-stress relieved
70,000 85,000
28,000 55.000
17,500 21,200
Hot or cold rolled-annealed Hot rolled--as rolled
70,000 75.000
28,000 40,000
... ... ... ... ... ... ... ... ... ... ... ... ...
... ... ... ... ... ... ... ... ... ... ... ... ...
Hot or cold worked--annealed Hot rolled or forged-hot finished
80,000 85.000
30,000 35,000
7000 14,500
3000 7200
2000 5500
Hot or cold worked--annealed Hot rolled or forged-hot 8ni&ed
80,000 85.000
30,000 35,000
6000 7300
3000 7200
2000 5500
Seamless-annealed
80,000
30,000
Seamless--annealed Seamless-hard, stress relieved
50,000 60,000
12,000 30,000
Seamless--annealed Seamless-stress relieved
50,000 60,000
Hot or cold rolled--annealed Hot rolled--as rolled
Seamless--annealed Hot or cold rolled-annealed Hot rolled--as rolled
80.000 80,000 85,000
30,000 30,000 35,000
(1)
(1) (1)
13.100 18,200 4900 8500
14,700
13,100 17,600 4900 8500
14,700
14,500
8000
8000 8000 4000
.
.
.
16,500 20,200
15.500 19,500
14,800 19.200
14.700 19,200
14,700 19,200
14,700 18,500
14,500 15,000
17,500 18.700
:z~ (
15.500 17.000
14,800 17,000
14,700 17.000
14.700 17.000
14,700 16,500
14.500 14,500
20,000 21,200
18.600 20,200
18,000 20,000
18,000 20,000
18,000 20,000
18,000 20,000
17,500 20,000
17,000 20,000
2%
6800 7900
%8
%:
18.000
18,000
17,500
17.000
16,000
7000
3000
2000
17.500
17,000
16,000
7000
3000
2000
17.500 20,000
17,000 20,000
16,000 19.500
7000 14,500
3000 7200
2000 5500
7300 8700 20,000 20,000 20,000 21.200
6900 8500 18,600 18,600 18,600 20,200
6800 8200 18,000 18,000 18.000 20,000
18,000 18,000 20.000
Allowable working stress on stresses established at 500 %.
18,000 18,000 20,000
6800 7900 18,000 18.000 18,000 20,000
for this material is based
upon Se-
for the hot-rolled or forged hot-finished temper 0
(2)
16.000 19.500 6300 7400
163, stress-relieved
condenser tubing. The
dess other specific data are available.
Tbr
Allowable
Stresses
341
Item 3. Typical Physical Properties of Materials (Extracted from the 1.956 Edition of the ASME Boiler and Pressure Vessel Code, Unfired Pressure Vessels, with Permission of the Publisher, the American Society of Mechanical Engineers) Thermal Approx. Conductivity, Thermal Specific Weight, Melting 32 F-212 F, Expansion, Heat, Btu/ Range, 1OV in./in., ASME lb per Btu/sq ft/hr, lb/OF cu in. “F Material Spec. No. “Fjin. “F at 212F SB-178 0.098 1195-1220 1660 13 2(a) Aluminum alloy 996A 0.23 990A SB-178 0.098 1190-1215 1540 13.1(a) 0.23 MlA SB-178 0.099 1190-1210 1340 0.23 12.9(a) MGllA SB-178 0.098 1165-1205 1130 13.3(a) 0.23 GR20A SB-178 0.097 1100-1200 960 13,2(a) 0.23 GSllA SB-178 0.098 1080-1205 1190 13.1(a) 0.23 Copper, deoxidized SB-11 and 111 0.323 1980 2352 9.8(b) 0.09 Red brass SB-111 and 43 0.316 1810-1880 1104 10 4(b) 0.09 1650-1720 Admiralty SB-111 and 171 0.308 768 11.2(b) 0.09 Aluminum brass SB-111 0.301 1710-1780 696 10,3(b) 0.09 Naval brass SB-171 0.304 1630-1650 804 11,8(b) 0.09 Muntz metal SB-171 0.303 1650-1660 852 11.6(b) 0.09 30 y0 Cupronickel SB-111 and 171 0.323 2140-2260 204 9.1(b) 0.09 20 y0 Cupronickel SB-111 and 171 0.323 2100-2200 240 0.09 9.3(b) 10 y0 Cupronickel SB-111 0.323 2020-2100 324 9.5(b) 0.09 Copper-silicon (A, C, D) SB-96 and 98 0.308 1780-1880 252 10.0(b) 0.09 Copper-silicon (B) SB-98 0.316 1890-1940 396 9’. 9(b) 0.09 Aluminum bronze (D) SB-171 0.281 1850-1900 552 0.09 9.0 Aluminum bronze (E) SB-171 0.274 1900-1930 264 0.09 Nickel and low-carbon nickel SB-162 0.321 2615-2635 420 7.2(c) 0.13 Nickel-copper SB-127 0.319 2370-2460 180 7.8(c) 0.13 2540-2600 Nickel-chromium-iron SB-168 0.300 104 6.4(c) 8.11 Steel SA-30 0.279 460 6.7(c) (a) Thermal expansion per degree F from 68 F to 212 F. (b) Thermal expansion per degree F from 68 F to 572 F. (c) Thermal expansion per degree F from 32 F to 212 F.
J
342
Allowable Stresses (Extracted from the 1956 Edition of the
Material and Specificatiofl
Namber
Item 4. Maximum Allowable Stress Values in Tension ASME Boiler and Pressure Vessel Code, Urlfired PresFor Metal Temperatures
Grade
Plate Steels 3 SA-167 SA-167 3 SA-167 5 SA-167 6 SA-167 a SA-167 10 SA-167 10 SA-167 11 SA-240 A SA-240 B SA-240 c SA-240 D SA-?9?. f SA-240 SA-240 s SA-240 s SA-240 T E’ipes and Tubes Seamless SA-213 TP304 TP304 SA-213 SA-213 TP321 SA-213 TP347 SA-213 TP316 SA-213 TP310 TP310 SA-213 SA-268 TP405 TP410 SA-268 TP430 SA-268 TP304 SA-312 TP304 SA-312 SA-213 TP309 TP310 SA-312 SA-312 TP310 SA-312 TP321 SA-312 TP347 SA-312 TP316 SA-312 TP317 SA-376 TP304 SA-376 TP304 TP321 SA-376 SA-376 TP347 TP316 SA-376 Welded TP304 SA-249 TP304 SA-249 SA-249 TP310 SA-249 TP310 SA-249 TP321 SA-249 TP347 SA-249 TP316 SA-249 TP317 TP405 SA-268 SA-268 TP410 SA-268 TP430 TP304 SA-312 SA-312 TP304 SA-312 TP309 SA-312 TP310 SA-312 TP310 SA-312 TP321 SA-312 TP347 SA-312 TP316 SA-312 TP317
Nominal Composition
Type 304 304 321 347 309 310 310 316 410
18 18 18 18 25 25 25 18 13 15 34; 18 430 17
kin Tensile
Cr-8 Ni Cr-8 Ni Cr-8 Ni-Ti Cr-8 Ni-Cb Cr--12 Ni Cr-20 Ni Cr-20 Ni Cr-IO N i - 2 M O Cr Cr Cr-8 Ni-Cb Cr
304 18 Cr-8 Ni 304 18 Cr-8 Ni 321 18 Cr-8 Ni-Ti
l
18 Cr-8 Ni . , , 18 Cr-8 Ni . 18 Cr-10 Ni-Ti 18 Cr-10 Ni-Cb . 16 Cr-13 Ni-3 . 2 5 G-20 Ni 25 Cr-20 Ni 12 Cr-Al 1 3 C r 1 6 C r . 1 8 G-8 Ni . 18 Cr-8 Ni . 25 G-12 Ni . 25 Cr-20 Ni . 25 Cr-20 Ni .. 18 Cr-10 Ni-Ti .. 18 Cr-10 Ni-Cb . . 16 Cr-13 Ni-3 18 Cr-13 Ni-t . 18 Cr-8 Ni 18 Cr-8 Ni 18 Cr-10 Ni-Ti 18 Cr-10 Ni-Cb . . . 1 6 G-13 N i - 3 . . ...
. . . . . .
...
. . .
... . . ... . ... ...
.
.
...
18 Cr-8 Ni 18 Cr-8 Ni 25 G-20 Ni 25 Cr-20 Ni 18 G-10 Ni-Ti 18 G-10 Ni-Cb 16 Cr-13 Ni-3 18 Cr-13 Ni-4 12 G-Al 13 Cr 16 Cr 18 Cr--8 Ni 18 G-8 Ni 25 Cr-12 Ni 25 G-20 Ni 25 Cr-20 Ni 18 Cr-10 Ni-Ti 18 Cr-10 Ni-Cb 16 G--i3 Ni-3 18 Cr-13 Ni-4
MO
MO MO
MO
MO MO
MO Mo
75,000 75,000 75.000 75;ooo 75,000 75,000 75,000 75,000 65.000 70;ooo 75,000 70,000 ..75,QO& 60,000 75,000 75,000 75,000
-20 to Notes
:.
(1) : . . I:; :.
.’
ibj :
iij
75,000 75,000 75,000 75,000 75,000 75,000 75,000 60,000 60,000 60,000 75,000 75,000 75,000 75,000 75,000 75,000 75,000 75,090 75,000 75,000 75,000 75,000 75,000 75;ooo
(1)
... . . . .
75,000 75,000
y)
75,000 75,000
(2)(4) (3)(4)
75,000 75,000 75,000 60.000 60;OO0 60,000 75,000
ii; (4) (4) (4) id (4) (9) (1) (4)
75,000 75,000 75,000 75,000 75.000 75;ooo 75,000
ii; (2) (4) (3) (4) (4 (4) f;j
iij (3)
Gj (1) iit j (3)
100
iij
300
400
500
600
650
18,750 17,000 16,000 15,450 15,100 14,900 18,750 16,650 15,000 13,650 12,500 11,600 18.750 18,750 17,000 15,800 15,200 14,900 18;750 18,750 17,000 15,800 15,200 14,900 18,750 18,750 17,300 16,700 16,600 16,500 18,750 18,750 18,500 18,200 17,700 17,200 18,750 18,750 18,500 18,200 17,700 17,200 18,750 18,750 17,900 17,500 17,200 17,100 16,250 15,600 15,100 14,600 14,150 13,850 17,500 17,500 16,300 15,650 15,100 14,600 15,800 18,750 18,750 17,000 15,200 14,900 17,500 17,500 16,300 15,650 15,100 14,600 1 8 , 7 5 0 18,750. 17,9(?0 1 7 , 5 0 0 l.7,200 1 7 , 1 0 0 15.000 15.000 i4:yioo 14.400 13.950 13.400 is;750 171000 161000 15:450 15;100 141900 18,750 16,650 15,000 13,650 12,500 11,600 18,750 18,750 17,000 15,800 15,200 14,900
14,850 11,200 14,850 14,850 16,450 16,900 16,900 17,050 13,700 14,300 14,850 14,300 .17,950
18,750 18,750 18,750 18,750 18,750 18,750 18,750 15,000 15,000 15,000 18,750 18,750 18,750 18,750 18,750 18,750
13.000.--14;850 11,200 14,850
18,750 18.750 181750 18,750 18.750 18;750 18,750
17,000 16,650 18,750 18,750 18,750 18,750 18,750 15,000 14,450 15,000 17,000 16,650 18,750 18,750 18,750 18,750 18,750 18,750 18,750 17,000 16,650 18,750 18,750 18,750
16,000 15,000 17,000 17,000 17,900 18,500 18,500 14,700 14,000 14,100 16,000 15,000 17,300 18,500 18,500 17,000 17,000 17,900 17,900 16,000 15,000 17,000 17,000 17,900
15,450 13,650 15,800 15,800 17,500 18,200 18,200 14,400 13,500 13,400 15,450 13,650 16,700 18,200 18,200 15,800 15,800 17,500 17,500 15,450 13,650 15.800 15;SOO 17,500
15,100 12,500 15,200 15,200 17,200 17,700 17,700 13,950 13,100 13,000 15,100 12,500 16,600 17,700 17,700 15,200 15,200 17,200 17,200 15,100 12,500 15.200 IS;200 17,200
14,900 11,600 14,900 14,900 17,100 17,200 17,200 ‘13,400 12,850 12,500 14,900 11,600 16,500 17,200 17,200 14,900 14,900 17,100 17,100 14,900 11,600 14,900 14,900 17,100
14,850 11,200 14,850 14,850 17,050 16,900 16,900 13,000 12,700 12,200 14,850 11,200 16,450 16,900 16,900 14,850 14,850 17,050 17,050 14,850 11,200 14,850 14,850 17,050
16,000 16,000 16.000 16;OO0 16,000 16,000 16,000 16,000 12,750 12,750 12,750 16.000 16;OO0 16,000 16,000 16,000 16,000 16.000 161000 16,000
14,450 14,150 16,000 16,000 16,000 16,000 16,000 16,000 12.750 12;300 12,750 14,450 14,150 16,000 16.000 16;OOO 16,000 16,000 16,000 16,000
13,600 12,750 15,750 15,750 14,450 14.450 15;200 15,200 12.500 11;900 12,000 13.600 12;750 14,750 15.750 15;750 14,450 14,450 15,200 15,200
13,150 11,600 15,500 15,500 13,400 13,400 14,900 14,900 12,250 11,500 11,400 13,15011,600 14,200 15,500 15,500 13,400 13,400 14,900 14,900
12,800 10,600 15,050 15,050 12,900 12,900 14,600 14,600 11,900 11,150 11,050 12,800 10,600 14,100 15,050 15,050 12,900 12,900 14,600 14,600
12,700 9850 14,600 14,600 12,700 12,700 14,550 14,550 11,400 10,900 10,650 12,700 9850 14,050 14,600 14,600 12,700 12,700 14,550 14,550
12,650 9500 14,400 14,400 12,650 12,650 14,500 14,500 11,050 10,800 10,400 12,650 9500 l$,OOO 14,400 14,400 12,650 12,hSO 14,500 14,500
18,750 .
200
Allowable for High-alloy Steel, in Pounds per Square Inch sure Vessels, with Permission of the Publisher, the
Stresses
343
American Society of Mechan [ical Elngineerrs)
Not Exceeding Dee F 700
750
800
850
900
950
1000
1050
1100
1150
1200
1250
1300 1350 1400 1450
1500
14,800 10,800 14,800 14,800 16,400 16,600 16,600 17,000 13,400 13,900 14,800 13,900 17,000 12,450 14,800 10,800 14,800
14,700 10,400 14,700 14,700 16,200 16,250 16,250 16,900 13,100 13,500 14,700 13,500 16,900 11,800 14,700 10,400 14,700
14,550 10,000 14,550 14,550 15,700 15,700 15,700 16,750 12,750 13,100 14,550 13,100 16,750 11,000 14,550 10,000 14,500
14,300 9700 14,300 14,300 14,900 14,900 14,900 16,500 12,100 12,500 14,300 12,500 16,500 10,100 14,300 9700 14,300
14,000 9400 14,100 14,100 13,800 13,800 13,800 16,000 11,000 11,700 14,100 11,700 16,000 9100 14,000 9400 14,100
13,400 9100 13,850 13,850 12,500 12,500 12,500 15,100 8800 9200 13,850 9200 15,100 8000 13,400 9100 13,850
12,500 8800 13,500 13,500 10,500 11,000 11,000 14,000 6400 6500 13,500 6500 14,000 4000 12,500 8800 13,500
10,000 8500 13,100 13,100 8500 9750 7100 12,200 4400 4500 13,100 4500 12,200
7500 7500 12,500 12,500 6500 8500 5000 10,400 2900 3200 12,500 3200 10,400
5750 5750 8000 8000 5000 7250 3600 8500 1750 2400 8000 2400 8500
4500 4500
3250 3250 3600 3600 2900 4750 1450 5300 ...
2450 2450 2700 2700 2300 3500 750 4000 . .
1800 1800 2000 2000 1750 2350 450 3000 .
1400 1400 1550 1550 1300 1600 350 2350 ...
1000 1000 1200 1200 900 1100 250 1850 . .
750 750 1000 1000 750 750 200 1500 . .
3600
2;oo
2b00
i5sb
1200
1bbo
5300
4bo.o
3iOfl
2350
1850
ld,bbO 8500 13,100
%I0 7500 12,500
5?sb 5750 8000
3250 3250 3600
i450 2450 2700
i&o 1800 2000
1.4.60 1400 1550
i&i0 1000 1200
lso0 .,. . 750 750 1000
14,800 14,700 14,550 10,800 10,400 10,000 14,800 14,700 14,550 14,800 14,700 14,550 17,000 16,900 16,750 16,600 16,250 15,700 16,600 16,250 15,700 12,450 11,800 11,000 12,500 12,250 11,950 11,850 11,500 11,100 14,800 14,700 14,550 10,800 10,400 10,000 16,400 16,200 15,700 16,600 16,250 15,700 16,600 16,250 15,700 14,800 14,700 14,550 14,800 14,700 14,550 17,000 16,900 16,750 17,000 16,900 16,750 14,800 14,700 14,550 10,800 10,400 10,000 14,800 14,700 14,550 14,800 14,700 14,550 17,000 16,900 16,750
14,300 9700 14,300 14,300 16,500 14,900 14,900 10,100 11,600 10,600 14,300 9700 14,900 14,900 14,900 14,300 14,300 16,500 16,500 14,300 9700 14,300 14,300 16,500
14,000 9400 14,100 14,100 16,000 13,800 13,800 9100 11,000 10,000 14,000 9400 13,800 13,800 13,800 14,100 14,100 16,000 16,000 14,000 9400 14,100 14,100 16,000
13,400 9100 13,850 13,850 15,100 12,500 12,500 8000 8800 9200 13,400 9100 12,500 12,500 12,500 13,850 13,850 15,100 15,100 13,400 9100 13,850 13,850 15,100
12,500 8800 13,500 13,500 14,000 11,000 11,000 4000 6400 6500 12,500 8800 10,500 11,000 11,000 13,500 13,500 14,000 14,000 12,500 8800 13,500 13,500 14,000
10,000 8500 13,100 13,100 12,200 9750 7100
7500 5750 7500 5750 12,500 8000 12,500 8000 10,400 8500 8500 7250 5000 3600
4500 4500 5000 5000 6800 6000 2500
29bO 3200 7500 7500 6500 8500 5000 12,500 12,500 10,400 10,400 7500 7500 12,500 12,500 10,400
1750 2400 5750 5750 5000 7250 3600 8000 8000 8500 8500 5750 5750 8000 8000 8500
loo0 1750 4500 4500 3800 6000 2500 5000 5000
3250 3250 3600 3600 5300 4750 1450 ... .
2450 2450 2700 2700 4000 3500 750 ... ...
1800 1800 2000 2000 3000 2350 450 ... ...
1400 1400 1550 1550 2350 1600 350 ... ...
1000 1000 1200 1200 1850 1100 250 ... ...
750 750 1000 1000 1500 750 200 ... ...
6800 4500 4500 5000 5000 6800
3250 3250 2900 4750 1450 3600 3600 5300 5300 3250 3250 3600 3600 5300
2.45b 2450 2300 3500 750 2700 2700 4000 4000 2450 2450 2700 2700 4000
1800 1800 1750 2350 450 2000 2000 3000 3000 1800 1800 2000 2000 3000
i4tio 1400 1300 1600 350 1550 1550 2350 2350 1400 1400 1550 1550 2350
iGo 1000 900 1100 250 1200 1200 1850 1850 1000 1000 1200 1200 1850
750 750 750 750 200 1000 1000 1500 1500 750 750 1000 1000 1500
12,400 8500 13,350 13,350 12,350 12,350 14,250 14,250 9350 10,150 9450 12,400 8500 13,350 13,350 13,350 12,350 12,350 14,250 14,250
12,150 8250 12,700 12,iO0 12,150 12,150 14,000 14,000 8600 9850 9000 12,150 8250 12,700 12,700 12,700 12,150 12,150 14,000 14,000
6400 6400 7200 4250 10,600 10,600 8850 8850
4900 4900 6150 3050 6800 6800 7200 7200
3800 3800 5100 2100 4250 4250 5800 5800
iii0 2700 6400 6400 5500 7200 4250 10,600 10,600 8850 8850
l&lb 2050 4900 4I)oo 4250 6150 3050 6800 6800 7200 7200
2750 2750 4050 1250 3050 3050 4500 4500 . . .
2100 2100 3000 650 2300 2300 3400 3400 . . ...
1550 1.550 2000 400 1700 1700 2550 2550 ... ...
1200 1200 1350 300 1300 1300 2000 2000 ... ...
850 850 950 200 1000 1000 1550 1550 ... ...
650 650 650 150 850 850 1300 1300 . . ...
bib 1500 3800 2750 2ibb lssb Go ii0 '850 3800 2750 2100 1550 1200 850 650 3250 2450 1950 1500 1100 750 650 5100 4050 3000 2000 1350 950 650 2100 1250 650 400 300 200 150 4250 3050 2300 1700 1300 1000 850 4250 3050 2300 1700 1300 1000 850 5800 4500 3400 2550 2000 1550 1300 5800 4500 3400 2550 2000 1550 1300
12,600 9200 14,100 14,100 12,600 12,600 14,450 14,450 10,600 10,650 10,100 12,600 9200 13,950 14,100 14,100 12,600 12,600 14,450 14,450
12,500 8850 13,800 13.800 12,500 12,500 14,350 14,350 10,000 10,400 9800 12,500 8850 13,800 13,800 13,800 12,500 12,500 14,350 14,350
iibo 4500 10,000 8500 8500 9750 7100 13,100 13,100 12,200 12,200 10,000 8500 13,100 13,100 12,200
11,900 11,400 10,600 8500 8000 7750 7500 7200 11,700 10,600 9350 8300 11,700 10,600 9350 6000 12,000 11,800 11,500 11,100 12,000 11,800 11,500 11,100 13,600 12,800 11,900 10,400 13,600 12,800 11,900 10,400 7750 6800 3400 9350 7500 5450 ii%0 8500 7900 5500 3800 11,900 11,400 10,600 8500 8000 7750 7500 7200 11,700 10,600 8900 7200 11,700 10,600 9350 8300 11,700 10,600 9350 6000 12,000 11,800 11,500 11,100 12,000 11,800 .l1,500 11,100 13,600 12,800 11,900 10,400 13,600 12,800 11,900 10,400
5000 3800 2500 6800
344
Allowable Stresses Item 4.
Material and Specification Number
Grade
‘be
Nominal Composition
Spec. Min Tensile
Maximum
Allowable
Stress Valutis in Tension For Meta! Temperatures
Notes
- 2 0 to 100
200
300
400
500
600
650
Forgings 410 13 Cr 21,250 20,400 19,750 19,000 18,500 18,100 17,900 SA-182 F6 304 18 C r - 8 Ni iii 18,750 17,000 16,000 15,450 15,100 14,900 14,850 SA-182 F304 SA-182 F304 304 18 C r - 8 Ni 18,750 16,650 15,000 13,650 12,500 11,600 11,200 ... F321 321 18 C r - 8 Ni-Ti 18,750 18,750 17,000 15,800 15,200 14,900 14,850 SA-182 ... SA-182 F347 347 18 C r - 8 Ni-Cb 18,750 18,750 17,000 15,800 15,200 14,900 14,850 ... F316 3 1 6 1 8 C r - 8 N i - 3 MO 18,750 18,750 17,900 17,500 17,200 17,100 17,050 SA-182 . . . SA-182 F310 310 25 Cr-20 Ni 23,750 23,750 2 3 , 7 5 0 2 3 , 2 0 0 22,400 21,500 20,850 ... SA-336 F6 410 13 Cr . . . 181750 18,100 17,500 16,900 16,400 16,000 15,700 SA-336 F8 304 18 Cr-8 Ni (1) 18,750 17,000 16,000 15,450 15,100 14,900 14,850 SA-336 F8 304 18 Cr-8 Ni ... 18,750 16,650 15,000 13,650 12,500 11,600 11,200 SA-336 Fat 321 18 Cr-8 Ni-Ti ... 18,750 18,750 17,000 15,800 15,200 14,900 14,850 SA-336 F8c 347 18 Cr-8 Ni-Cb ... 18,750 18,750 17,000 15,800 15,200 14,900 14,850 SA-336 F8m 316 18 C r - 8 N i - 3 M O ... 18,750 18,750 17,900 17,500 17,200 17,100 17,050 SA-336 F25 310 25 Cr-20 Ni ... 23,750 23,750 2 3 , 7 5 0 23,200 22,400 21,500 20,850 Castings SA-351 CA15 ... 13 Cr--fs M O (6) 22,500 22,500 22,500 22,500 22,500 22,000 2 1 , 6 0 0 SA-351 CF8 . . . 18 Cr-8 Ni (1) (6) 17,500 16,500 15,600 15,000 14,600 14,350 1 4 , 2 0 0 SA-351 CF8 . . . 18 Cr-8 Ni 70,000 17,500 15,700 14,250 13,100 12,200 11,700 11,500 17,500 SA-351 CF8M . . . 1 8 C r - 9 Ni-2)s M O 70,000 16.900 16,500 16,400 16,350 16,300 16,250 SA-351 CF8M ... 1 8 C r - 9 Ni--235 M O 70;ooo 17,500 16;900 16;500 16,300 15;900 15,350 15,000 70,000 17,500 17,100 SA-351 CF8C ... 18 Cr-9 Ni-Cb 16,600 16,100 15,500 14,700 14,200 70,000 17.000 SA-351 CF8C . . . 18 Cr-9 Ni-Cb 17,500 15.600 14.200 13.000 12.200 11,900 CH20 . . . 25 Cr-13 Ni 70,000 17;500 16;lOO 15;150 SA-35 1 141600 14;550 14;450 14,400 SA-351 CK20 . . . 25 Cr-20 Ni 65,000 16,250 15,300 14,900 14,600 14,550 14,450 14,400 Boltings 416 13 Cr 20,000 SA-193 B6 19,300 18,700 18,300 17,850 17,000 16,500 SA-193 Bat 321 18 Cr-8 Ni-Ti 75,bbo 15,000 15,000 13,600 12,650 12,200 11,900 11,850 SA-193 B8c 347 18 Cr-8 Ni-Cb v 75,000 15.000 15,000 13,600 12,650 12,200 11,900 11,850 SA-193 B8 304 18 Cr-8 Ni 75,000 15;ooo 13,300 12,000 10,900 10,000 9300 8950 SA-193 B8F 303 18 Cr-8 Ni 75,000 15,000 . . SA-320 (8 grades) Notes: The stress values in this table may be interpolated to determine values for intermediate temperatures. All stress values in shear are 0.80 times the values in the above table. All stress values in bearing. are+60 times the values in the above table. ‘(1) At temperatures of from 200 F through 1050 F these stress values meet all criteria specified for establishing stress values, except that they exceed 62;s ‘$ZO but do not exceed 90 y0 of the yield strength at temprature. They may be used where slightly greater deformation is not objectionable. (2) These stress values at temperatures of 1050 F and above should be used only when assurance is provided that the steel has a predominant grain size not finer than ASTM No. 6. (3) These stress values shall be considered basic values to be used when no effort is made to control or check the grain size of the steel
Allowable
Stresses
345
for High-alloy Steel, in Pounds per Square Inch (Continued) L
Not Exceeding Deg F 700 17,500 14,800 10,800 14,800 14,800 17,000 20,000 15,400 14,800 10,800 14,800 14,800 17,000 20,000
750
800
850
900
950
1000
1050
1100
1150
1200
1250
1300
8800 6400 4400 2900 1750 1000 17,050 16,300 14,000 11,000 7500 5750 4500 3250 2450 14,700 14,550 14,300 14,000 13,400 12,500 10,000 9700 9400 9100 8800 8500 7500 5750 4500 3250 2450 10,400 10,000 14,700 14,550 14,300 14,100 13,850 13,500 13,100 12,500 8000 5000 3600 2700 14,700 14,550 14,300 14,100 13,850 13,500 13,100 12,500 8000 5000 3600 2700 1 6 , 9 0 0 1 6 , 7 5 0 1 6 , 5 0 0 1 6 , 0 0 0 1 5 , 1 0 0 1 4 , 0 0 0 1 2 , 2 0 0 1 0 , 4 0 0 8 5 0 0 6 8 0 0 5 3 0 0 .4000 18,500 17,000 15,500 14,000 12,500 11,000 9750 8500 7250 6000 4750 3500 2900 1750 1000 15,100 14,650 14,000 11,000 8800 6400 4400 14,700 14,550 14,300 14,000 13,400 12,500 10,000 7500 5750 4500 3250 2450 9100 8800 8500 7500 5750 4500 3250 2450 10,400 10,000 9700 9400 14,700 14,500 14,300 14,100 13,850 13,500 13,100 12,500 8000 5000 3600 2700 14,700 14,550 14,300 14,100 13,850 13,500 13,100 12,500 8000 5000 4600 2700 1 6 , 9 0 0 1 6 , 7 5 0 1 6 , 5 0 0 1 6 , 0 0 0 1 5 , 1 0 0 14,,000 1 2 , 2 0 0 1 0 , 4 0 0 8 5 0 0 6 8 0 0 5 3 0 0 4 0 0 0 9750 8500 7250 6000 4750 3500 18,500 17,000 15,500 14,000 12,500 11,000
20,700 19,600 18,300 16,000 14,050 13,850 13,600 13,350 11,300 11,100 10,900 10,650 16,200 16,100 15,900 15,500 14#,700 14,350 14,000 13,500 13,700 13,300 12,900 12,600 11,700 11,600 11,500 11,350 14,350 14:300 14,150 13,900 14,350 14,300 14,150 13,900
11,000 13,000 10,400 15,000 13,000 12,300 11,200 13,500 13,500
7600 5000 12,600 12,100 10,100 9850 13,500 12,000 12,350 11,700 11,900 11,600 11,100 11,000 12,500 10,500 12,500 11,000
15,750 14,900 13,800 12,500 11,000 1 1 , 8 0 0 1 1 , 7 5 0 1 1 , 6 5 0 1 1 , 4 5 0 1 1 , 3 0 0 l;,ibO 10,k‘OO 11,800 11,750 11,650 11,450 11,300 11,100 10,800 8300 8000 7750 7500 7250 7050 8650
3300 9600 9600 10,600 10,600 11,200 10,900 8500 9750
2200 1500 7500 5750 7500 5750 9400 8000 9400 8000 10,800 8000 10,800 8000 6500 5000 8500 7250
1000 4500 4500 6800 6800 5000 5000 3800 6000
3250 3250 5300 5300 3600 3600 2900 4750
2450 2450 4000 4000 2700 2700 2300 3500
1 0 , 5 0 0 1 0 , 0 0 0 8000 SO00 3 6 0 0 2700 10,500 10,000 8000 5000 3600 2700 6800 6300 5750 4500 3250 24,50
1350
l-E00
1450
1500
750 1800 lb00 1000 750 1800 1400 1000 2000 1550 1200 1000 2000 1550 1200 1000 3000 2350 1850 1500 2350 1600 1100 750 1800 1800 2000 2000 3000 2350 18bb 1800 3000 3000 2000 2000 1750 2350
‘750 1400 1000 1400 1000 750 1550 1200 1000 1550 1200 1000 2350 1850 1500 1600 1100 750 14;O 1400 2350 2350 1550 1550 1300 1600
1000 750 7.50 1000 1850 1500 1850 1500 1200 1000 1200 1000 900 750 1100 7.50
sion 1550 1 2 0 0 1 0 0 0 2000 1550 1200 1000 i50 1800 1100 1000 . . ...
. . ‘(4) These stress values are the basic values multiplied by a joint-efficiency factor of 0.85. (5) These stress values are established from a consideration of strength only and will be satisfactory for average service. For bolted joints where freedom from leakage over a long period of time without retightening is required, lower stress values may be necessary as determined from the flexibility of the flange and bolts and corresponding relaxation properties. (6) To these stress values a quality factor shall be applied (see ASME code). (7) These stress values permitted for material that has been carbide-solution treated. (8) For temperatures below 100 F, stress values equal to 20% of the specified minimum tensile strength will be permitted. (9) This steel may be expected to develop embcittlement at room temperature after service at temperatures above 800 F: conse quently, its use at higher temperatures is not recommended unless due caution is observed.
,
A P P E N D I X
m E
‘, TYPICAL TANK SIZES AND CAPACITIES
Item 1.
Typical
Sizes
and
Corresponding
Approximate
Capacities
for
Tanks with
Recommended by API Standard 12 C
72-in.
Butt-welded Courses
(Courtesy of American Petroleum Institute) 1 Tank Diameter (ft) 10 15 20 25 30 35 40 45 50 60 70 80 90 100 120 140 160 180 200 220
2 Approx. Capacity per Foot of Height (bbl) 14.0 31.5 56.0 87.4 126
3 12 2 170 380 670 1,050 1,510
171 224 283 350 504
2,060
685
8,230 10,740 13,600 16,790 . .
895
1133 1399 2014 2742 3581 4532 5595
6770
2,690
3,400 4,200 6,030
... . . ... ... *..
4 18
5 24
250 565 1,010 1,570 2,270
4 335 755 1,340 2,100 3,020
3,080 4,030 5,100
4,110 5,370 6,800
6,290 9,060
8,390
s n
12,340 16,120 20,390
25,180 36,260 49,350 .
12,909 16,450 21,490 27,190 33,570 48,340 65,800 ...
...
...
...
...
. . .
. . .
6
7
8
9
Tank Height (ft) 30 36 42 48 Number of Courses in Comnleted Tank r3 6 420 505 1,130 945 1,680 2,010 21350 21690 2,620 3,150 3,670 4,200 3,780 4,530 5,290 6,040 5,140 6,710 8,500 10,490 15,110
6,170 8,060 10,200 12,590 18,130
11,900 14,690 21,150
20,560 26,860
24,680 32,230
28,790 37,600
33,990
40,790
47,590
41,970 60,430
82,250 107,400 136,000 167,900 203,100
50,360 72,510 98,700
128,900 163,200 201,400 243,700
7,200 9,400
58,750 84,600
115,100 150,400 190,400 235,000 284,400
8,230 10,740 13,600 16,790 24,170 32,900 42,970 54,380 67,140 96,690
131,600 171,900 217,500 268,600 322,300 D = 219
10
11
54
60
9
10
4j20 6,800
51250 7,550
9,250
10,280 13,430 17,000
12,090 15,300 18,880 27,190 37,010 48,350 61,180 75,540 108,800 148,000 193,400 244,800 284,500 D = 194
20,980
30,220
41,130 53,720 67,980 83,930
120,900 164,500 214,900 254,300 D = 174
The approximate capacities shown are based on the formula: Capacity (42-gal bbl) = 0.14D2H, where D = listed tank diameter and H = listed tank height. Capacities and diameters below the heavy lines (~01s. 9-11) are maximum for the tank heights shown, on the basis of the lx-in. maximum permissible thickness of shell plates and the maximum allowable design stresses. 346
347
Typical Tank Sizes and Capacities Item 2.
Shell Plate Thicknesses for Typical Sizes of Tanks with 72-in. Butt-welded Courses
Recommended by API Standard 12 C (Courtesy of American Petroleum Institute) 1
2
3
4
5
6
7
8
9
10
11
Tank Height (ft)
6 Tank Diam
(ft) 10 15
-.
12
18
24
30
36
42
48
5-E
60
8
9
10
0'.20 0.24
0'.22 0.26
Number of Courses in Completed Tank n
z
”.s
4
5
6
7
Shell Plate Thickness (In.) 346
946
%S
%6
336
%6
%6
12 Maximum Allowable Height for Diameters Listed (ft)
20 25 30
946
346 %
336
3/i
%S
356
%6
x6
3’i6
x6
346
346
946
0.19
94 6 0.19 0.21
35 40 45 50 60
3’i6
%6
%6
%6
346
946
>/a ?4
0.19 0.19 s/a 0.26
0.19 0.21 0.23 0.26 0.31
0.21 0.24 0.27 0.30 0.36
0.24 0.28 0.31 0.35 0.41
0.27 0.31 0.35 0.39 0.47
0.30 0.35 0.39 0.43 0.52
.
346
% 0.25 0.30
0.25 0.27 0.31 0.34 0.41
0.30 0.34 0.38 0.43 0.51
0.36 0.41 0.46 0.51 0.62
0.42 0.48 0.54 0.60 0.72
0.48 0.55 0.62 0.69 0.83
0.54 0.62 0.70 0.78 0.93
0.61 0.69 0.78 0.86 1.03
. . . . . .
0.35 0.40 0.45 0.50 0.55
0.47 0.54 0.61 0.67 0.74
0.60 0.68 0.76 0.85 0.93
0.72 0.82 0.92 1.02
0.84 0.96 1.08 1.20 1.32
0.96
1.08
1.10 1.24 1.37
1.24 1.39
1.21 1.38 . .
70 80 90 100 120 140 160 180
200 220
x6
!i +/a si >I
6
1.13
6
...
. . .
6i.3
58.2 52.5 47.8
Plate thicknesses shown in item 2 in fractions are thicker than those required for hydrostatic loading but for practical reasons have been fixed at the values given; therefore, plates for these courses may he ordered on a weight basis. Plate thicknesses shown in item 2 in decimals are based on maximum allowable stresses, and therefore plates for these courses must be ordered on a thickness basis. In deriving the plate-thickness values shown, it was assumed, on the basis of average mill practice, that the edge thickness of plates 72-in. wide and ordered on the weight basis would underrun the nominal thickness by 0.03 in. The actual thickness may underrun a calculated or specified thickness by 0.01 in.; consequently, fractional thickness values are shown only when the fractional value exceeds the calculated thickness of the course in question by more than 0.02 in. The maximum allowable height for diameters listed in feet is based on the I!,$-in. maximum permissible thickness of shell plates and the maximum allowable design stresses.
348
Typical Tank Sizes and Capacities
Item 3.
Typical
Capacities
1
for
Sizes
and
Tanks
with
Corresponding 96-in.
Approximate
Butt-welded
Item 4.
Courses
Shell
Plate
Thicknesses
Tanks with 96-in.
for
Typical
Sizes
Recommended by API Standard 12 C
Recommended by API Standard 12 C
(Courtesy of American Petroleum Institute)
(Courtesy of American Petroleum Institute)
2 APPrOX.
3
d&30-
ity per ~~~ Tank Foot of 1 6 Diam Height w Wl) 2 10 14.0 225 15 31.5 505 20 56.0 900 25 87.4 1 , 4 0 0 30 126 2,020
4
5
6
7
8
Tank Height (ft) 24 32 40 48 Number of Courses in Completed 3 T5 6 335 450 755 1,010 1,260 1,340 1,790 2,240 2,690 2,100 2,800 3,500 4,200 3.020 4,030 5,040 6,040
9
64
a ... ::: 5,600 8,060
2,740 3,580 4,530 5,600 8,060
4,110 5,370 6,800 a.390 12,090
5,480 7,160 9,060 11,190 16,120
6,850 a.950 11,330 13.990 20,140
8,230 10,740 13,600 16,790 24,170
9,600 12,530 15,860 19,580 28,200
10,960 14,320 18,130 22,380 32,230
70 80 90 100 120
685 895 1133 1399 2014
10,960 14,320 18,130 22,380 ...
16,450 21,490 27,190 33,570 58,340
21,930 28.650 36;260 44,760 64,460
27,420 35.810 45;320 55,950 80,580
32,900 42,970 54,390 67,140 96,690
38,380 50.130 63;450 78,340 112,800
43,870 57.300 72;520 89,530 128,900
140 160 180 200 220
2742 3581 4532
. . . . . .
65,800 . .
87,740 114,600 145,000
109,700 143,200 181.300
131,600 171,900 217,500
153,500 200,500 253,800
175,500 229,200 238,100
179,100 216,700
223,800 270,800
268,600 274,200 D = 163 322,300 D = la7 D = 219
...
Tank Diam 1 (W 10 4is 1 5 Ks
The capacities in item 3 are based on the formula: Capacity (42-gal bbl) = 0.14DzH. where D = listed tank diameter and H = listed tank height,. Capacities and diameters below the heavy lines (~01s. 7-9) are maximum for the tank heights shown, on the basis of the l$&in. maximum permissible thickness of shell plates and the maximum allowable design stresses.
3 4 5 6 7 a _ _ _ Tank Height (ft) ___~~ -. 16 24 32 40 48 56 Number of Courses in Completed Tank 2 3 4 5 6 7 Shell Plate Thickwas (in.) ..- -___ 3f6 3;s . 2;; K6 vi6 Hi/is
9
.
34 6
"46
s/is
He %6
se 348
Hn ?f6
3i I3 0.19 n.19 0 . 2 1
0.20 0.24
0.23 0.2a
3 5 %e 4 0 Ne 4 5 X6 50 % 60%
%a vi.5 416 % %
%a 0.19 % 5’4
0.19 0.19 0.21 0.25 0.27
u.20 0 . 2 4 0.23 0.28 0.26 0.31 0.29 0.35 0.34 0.41
0.28 0.32 0.36 0.40 0.48
0.33 0.37 0.42 0.46 0.55
70
s/a
?‘f6
2 5 3 0
346 %a
346
3fe
10
__ Maximum 6 4 AllowRbln Height for a Diameters Linted (h)
k6
20
::: 4,900 7,050
171 224 283 350 504
. .
2
a
56 Tank 7
35 40 45 50 60
5595 6770
1
of
Butt-welded Courses
_.. .,, .,_
3i
0.25
0.32
0.2i
0.3i
90 % 100 4/a 1 2 0 )/+
% 0.25 0.27
0.31 0.34 0.41
0.41 0.46 0.5s
0.40 0.48 0.46 0.55 0.52 0.62 0.57 0.69 0.69 0.83
0.56 0.64 0.72 0.80 0.97
0.65 0.74 0.83 0.92 1.10
. . ... _.. . ..,
1 4 0
0.31 0.35 0.40 0.44 0 . 4 8
0.47 0.54 0.61 0.67
U.64 0.73 0.82 0.91 1.00
0.80 0.91 1.03 1.14 1.25
1.13 !.29 1.45 ... ...
1.29 1.47 ... ... ...
6 5 3 58.2 52.5 47.8
80
160
%
5-i
%
Pi
ia0 )/a
2 0 0 >i 220 0 . 2 5
O.i4
0.96 1.10 1.24 1.37 . .
A P P E N D I X
SHELL ACCESSORIES
Item 1.
Shell Nozzle Dimensions in Inches as a Function of Nozzle Size, Recommended by API Standard 12 CUse with Item 2 and Fig. 3.14
(Courtesy of American Petroleum Instit.ute) 1 Size of .A ozzle 20 18 16 14 12 IO 8 6 1 3 1‘7 * J “i*
St 3 716
2 OD of Pipe 20 18 16 14
w4 10%
8% 6% 4% 3%
-wfl
1.90
3 Flanged Nozzle. Minimum Pipe-wall Thickness1 I1 See item 2, WI. 2
0.50 0.50 0.50 0.432 0.337 0.300 0.218 0.200
4 3 Diameter Length of Side of Hole in of Reinforcing Heinforcine Plate Plate . I. DR 2036 43 1836 39 1636 35 1434 31 2819 1x4 241,~ 10% 2O’a 8% 16’i 634 12 G4 10 3%
2% 2
6 Width of Reinforcing Plate
W 52jg 47%
4mi 38 35 3ojg 25 20>/4 1536
12%
7 8 Distance, Distance, Shell to Shell to Flange Flange Face, Face, Outside, ./ Inside, hIO 8 10 8 10 8 10 8 10 8 10 8 8 6 8 6 6 6 6 6 6 6 6 6
9 10 Distance from Rottom of Tank to Center of Nozzle Regular Low Type H Type C 24 2135 22 19% 20 1734 18 1535 17 14% 15 12>/4 13 1036 11 8% 9 6 8 5 7 3% 6 3
Coupling 10 12% Coupling . . . Coupling . . . . . . . . . 1 Coupling . . . . . . . . . Coupling . . . . . . N *\ and 1 j ,\n. in diameter do not require reinforcing plates. DR will be the diameter of the hole in the * Flanged nozzles 2 m. shell plate, and weld A will be&given in Appendix E, item 3, col. 7. Reinforcing plates may be used if desired. t Screwed nozzles 3 in. in diameter require a reinforcing plate, the details for which shall be the same as shown for 3-in. flanged nozzles. Reinforcing plates may be used on the smaller fittings if desired. $ Extra-strong pipe, API Std. 5L., made from formed plate, electrically butt welded, may be substituted. 4.000 2.875 2.200 1.576 1.313
349
350
Shell Item 2.
Accessories Shell Nozzle Dimensions in Inches as a Function of Shell Plate Thickness, Recommended by API Standard 12C-Item
1 and Fig. 3.14
(Courtesy of American Petroleum Institute)
Shell Thickness (t) and Reinhcingplate Thickness CT)*
20” 18” 16” and 14” Flanged Nozzle, Minimum Pipe-wall Thickness (n) t 35 ?d ?4
Diameter of Hole in Shell Plate, D,, Equals OD of Pipe nlus the Following Values For Max D, For Min D, Add to OD Add to OD (Weld A in (Weld A in Field) Shop) 3.6 94 I% N .% N
Size of Fillet Weld B Ji
%6
%6
‘3’i6
%6
%6
‘546
?‘i6
746 %6
7’i6
‘3’i6
‘>iS
l?‘i6,
N
lx6
‘946
?4
l3i6
1% l?i Isi 1% 1% 1% 1% 1% 1% 1%
‘346
1 ‘i 6 1% 1346
1 r/4
1x6
1%
‘%6
1x6
N
1%
5%
Size of Fillet FVeld A for Nozzles 2”, l>h”, l”, and x”
946
34 w
N N K
Size of Fillet Weld A for Nozzles Larger than 2”
K N
N K N
1
‘346
13’i6
136 1x6
I%6
1%
‘946
1%6
‘?‘f6
1%
‘?‘iS
1x6
‘%6
1%
* If a thicker shell plate is used than is required for the hydrostatic loading, the excess shell-plate thickness may be considered as reinforcement, and the thickness of reinforcing plate (T) decreased accordingly. t Based on API Std. 5L for pipe of >s-in. wall thickness; for pipe of over >$in. wall thickness, use ASTM A-53, A-135, or A-139 of latest issue. Pipe made from formed plate, electrically butt welded, may be substituted for any of the abovementioned pipe sections.
Item 3.
Shell-manhole
Cover-plate
Thickness
and BoltiAg-
flange Thickness, Recommended by API Standard 12 CSee Item 4 and Fig. 3.15
(Courtesy of American Petroleum Institute) EquivaMax lent Tank PresHeight, sure* (ft) (psi) 20 8.7 35 15.2 54 23.4 79 34.2
Bolting-flange Cover-plate Thickness after Thickness, Finishing, Min (in.) Min (in.) 20-in. 24-in. ZO-in. 24-in. Manhole Manhole Manhole Manhole 946 N 34 si M Pi w N w %i w 36 s/a N
* Based on water loading.
Shell Accessories Item 4.
351
Dimensions in Inches for 20-in. Shell Manhole, Recommended by API Standard 12 C-See Item 3 and Fig. 3.15
(Courtesy of American Petroleum Institute) Shell Thickness and Manhole Attachment Flange Thickness t, T X6 %i 316
N X6 w Ns N ‘416
94 ‘346
76 1
‘546
l5i6
1% 1946
1% 1516
1% 1x6
1%
Size of Fillet Weld A Weld B
Approx. Radius Length (Applies to of Formed Type Side Only) R L 4534 45% 45% 45 34 45 45 45 45 44 44 44 44 4335 43 34 43 34 43 43 4235 4235 42 34 42% 4235
Width of Reinforcing Plate
W 54jq 54% 54 54 5335 5334 53% 53% 52 51% 51% 5135 51 59% 50% 50 50 49 49 49 49 49
Max Diam of Hole in Shell* DP 2455 2434 24% 2435 24% 24% 25 25jq 25>/, 2535 25 34 26 26 26>i 2635 2635 2635
26% 27 27 27 27
Inside Diam of Manhole Frame Max ID Min ID 20 w4 20 2235 20 22% 20 22ji 20 2236 20 22 20 2176 20 21% 20 2194 20 2135 20 21% 20 21>/4 20 21>6 20 20 20 .w6 20 w4 20 20% 20 20)s 20 2% 20 20% 20 2036 20 20
* Hole in shell may be oval, with horizontal major diameter of 29 in., where necessary for removal of rigid scaffold brackets.
352 Item 5.
Shell Accessories Dimensions in Inches for 24-in. Shell Manhole, Recommended by API Standard 12 C-See Item 3 and Fig. 3.15
(Courtesy of American Petroleum Shell Thickness and Manhole Attachment Flange Thickness t, T
Size of.__ Fillet Weld A Weld B
Approx. Radius Lengt.h (Applies to of Formed Type Side Only) R 1,
I
Inst,itute)
Max Width of Diam Reinforcof Hole ing Plate in Shell W D, 64$i 28pi 64 2835 64 28;s 64 28 Js 63 2834 63 28% 63 29>i 629, 2 9 X/4 62 ?,$ 2934 62$,, 29N 62 29% 3 0 5i 61% 61 30’1 61 30f’4 60 30>$ 60 3054 60 30 “5 58yd 3o,a/, 58% 30% 58 30% 58 30% 58 31
Inside
of Mauhole Frame Min ID Max ID24 26% 24 2635 24 26% 24 26$/, 24 2616 24 26 24 25% 24 25% 24 25% 24 2536 24 25% 24 25>c 24 2 5 $6 24 25 24 2474 24 24N 24 24% 24 24% 24 2436 24 245.i 24 2456 24 24 Diirrn
Diam of Diam of Cover
Plate DC
32% 32%
32% 32% 32% 32% 32%
32% 32% 32%
32% 32% 32% 32% 32% 32% 32% 32%
32%
32% 32% 32%
The dimensions shown in col. 1 of items 4 and 5, are based on heaving the thickness of the attachment flange, T, and also that of the neck for a distance of at least 4T extending outward from the connecting face of the attachment flange, equal to the thickness of the tank shell, t. If the manhole att,aches to a thicker shell plate than is required for the hydrostatic loading, the excess shell-plate thickness 111ayhe considered as reinforcement, and the thickness, T, of the manhole attachment flange may be decreased accordingly. The entire neck or portions of the neck may he thinner than the attachment flange, provided there is sufficient reinforcement furnished otherwise; however, the neck should not be thinner than the thickness of the shell plate or the allowable minimum finished thickness of the bolting flange, whichever is the smaller. The finished manhole holtingflange thickness shall not be less than that of the cover plate less 36 in., with a minimum of s/4 in.
A P P E N D I X
I
I
PROPERTIES OF SELECTED ROLLED STRUCTURAL MEMBERS
Item 1.
Channels,
American
Standard,
Properties
of
Sections 7.
lt
Section Index and Nominal Size C 60 18 x 4
R = 0.625 Cl 15x3%
R = 0.50 c 20 13 x 4
R = 0.48
Weight per Foot (lb) 58.0 51.9 45.8 42.7
Area Depth of of Section Channel (in.2) (in.) 16.98 15.18 18 13.38 12.48
50.0 40.0 33.9
14.64 11.70 9.90
50.0 40.0 35.0 31.8
11.71 10.24
t-1 -4;
AVg
Widt,h F l a n g e W e b T h i c k - T h i c k - ~~~ of Flange ness ness I (in.) (in.) (in.) (in.4) 4.200 0.625 0.700 670.7 0.625 622.1 4.100 0.600 4.000 0.625 0.500 573.5 3.950 0.625 0.450 549.2 -I
Axisi-1
Axis 2-2
(in. 3, 74.5 69.1 63.7 61.0
.” (in.) 6.29 6.40 6.55 6.64
(in.4) 18.5
Ii.1 15.8 15.0
s
P
2’
(in.3) 56 5.3 5.1 1.9
(in.) 1.04 1.06 1.09 1.10
(in.) 0.88 0.8; 0.80 0.90
3.716 3.520 3.400
0.650 0.650 0.650
0.716 0.520 0.400
401.4 346.3 312.6
53.6 46.2 41.7
5.24 5.44 5.62
11.2
15
9.3 8.2
38 3.4 3.2
0.87 0.89 0.91
0.80 0.78 0 79
13
4.412 4.185 4.072 ,I. 000
0.610 0.610 0.610 0.610
0.787 0.560 0.447 0.375
312.9 271.4 250.7 237.5
48.1 41.7 38.6 36.5
4.62 4.82 4.95 5.05
16.i 13.9 12.5 11.6
4.9 4.3 4.0 3.9
1.07 1.09 1.10 1.11
0.98 0.97 0.99 1.01
3.170
3.047 2.940
0.501 0.501 0.501
0.510
0.387 0.280
161.2 143.5 128.1
26.9 23.9 21.4
4.28 4.43 4.61
3.2 4 5 3.9
2.1 1.9 1.7
0.77 0.79 0.81
0 68 0.68 0.70
0.436 0.436 0.436 0.436
0.673 0.526 0.379 0.240
103.0 90.7 78.5 66.9
20.6 18.1 15.7 13.4
3.42 3.52 3.66 3.87
C.0 3.4
2.8 2.3
1.7 1.5 1.3 1.2
0.67 0.68 0.70 0.72
0.65 0.62 0.61 0.64
14.66 9.30
c 2 12 x 3 R = 0.38
30.0 25.0 20.7
8.79 7.32 6.03
c3 10x2%
30.0 25.0 20.0 15.3
8.80 7.33 5.86 4.47
10
3.033 2.886 2.739 2.600
20.0 15.0 13.4
5.86 4.39 3.83
9
2.648 2.485 2.430
0.413 0.413 0.413
0.448 0.285 0.230
60.6 50.7 47.3
13.5 11.3 10.5
3.22 3.40 3.49
2.4 1.9 1 .8
1.2 1.0 0.97
0.65 0.67 0.67
ti.59 0.59 0.61
18.75 13.75 11.50
5.49 4.02 3.36
8
2.527 2.343 2.260
0.390 0.390 0.390
0.487 0.303 0.220
43.7 35.8 32.3
10.9 9.0 8.1
2.82 2.99 3.10
2.00 1.50 1.30
I .oo 0.86 0.79
0.60 0.62 0.63
0.57 0.56 0.58
R = 0.34 c 4 9~2%
R = 0.33 c5 8 x 2>/4
R = 0.32 y,.-
12
__YV__)__YYU__Y__IN
353
354
Properties
of
Selected Item 1.
Section Index and Nominal Size
Weight per Foot (lb) 14.75 12.25 9.80
Rolled
Structural
Channels,
Members
American
Standard,
Properties
Area Depth Width Flange Web of of of Thick- ThickSection Channel Flange ness ness (in. 2, (in.) (in.) (in.) (in.)
of
Sections
(Continued)
Axis l-l
I
Axis 2-2
s
(in. 4,
(in. “)
r
I
(in .)
(in.4)
s
I-
(in.3)
(in.)
Y
4.32 3.58 2.85
7
2.299 2.194 2.090
0.366 0.366 0.366
0.419 0.314 0.2.0
27.1 24.1 21.1
7.7 6.9 6.0
2.51 2.59 2.72
1.40 1.20 0.98
0.79 0.71 0.63
0.57 0.58 0.59
(in.) 0.53 0.53 0.55
3.81 3.07 2.39
6
R = 0.30
13.00 10.50 8.20
2.157 2.034 1.920
0.343 0.343 0.343
0.437 0.314 0.200
17.3 15.1 13.0
5.8 2.13 5.0 2.22 2.34 4.3
1.10 0.87 0.70
0.65 0.57 0.50
0.53 0.53 0.54
0.52 0.50 0.52
C8 5X1X
9.00 6.70
2.63 1.95
5
1.885 1.750
0.320 0.320
0.325 0.190
8.8 7.4
3.5 3.0
1.83 1.95
0.64 0.48
0.45 0.38
0.49 0.50
0.48 0.49
7.25 5.40
2.12 1.56
1.720 1.580
0.296 0.296
0.320 0.180
4.5 3.8
2.3 1.9
1.47 1.56
0.44 0.32
0.35 0.29
0.46 0.45
0.46 0.46
6.00 5.00 4.10
1.75 1.46 1.19
1.596 1.498 1.410
0.273 0.273 0.273
0.356 0.258
2.1 1.8 1.6
1.4 1.2 1.1
1.08 1.12 1.17
0.31 0.25 0.20
0.27 0.24 0.21
0.42 0.41 0.41
0.46 0.44 0.44
C6 7 x 2JQ
R = 0.31 c7 6x2
R = 0.29 c9 4x1s
R = 0.28 c 10 3x1s
R = 0.27
4
3
Item 2.
Beams, American Standard, Properties of Sections 2 l- , -1 1 2
/ /
Section l’ndex and Nominal Size 24” I B 18 24, x 716
Avg
Weight per Foot WI
Area Depth Width of of of Section Beam Flange (in. 2, (in.) (in.)
Flange Thickness (in.)
Web Thickness (in.)
(in. 4,
(inT3)
(in.)
120.0 105.9
35.13 30.98
24
8.048 7.875
1,102 1.102
0.798 0.625
3010.8 2811.5
250.9 234.3
9.26 9.53
84.9 78.9
21.1 20.0
1.56 1.60
100.0 90.0 79.9
29.25 26.30 23.33
24
7.247 7.124 7.000
0.871 0.871 0.871
0.747 0.624 0.500
2371.8 2230.1 2087.2
197.6 185.8 173.9
9.05 9.21 9.46
58.4 45.5 42.9
13.4 12.8 12.2
1.29 1.32 1.36
95.0 85.0
27.74 24.80
20
7.200 7.053
0.916 0.916
0.800 0.653
1599.7 1501.7
160.0 150.2
7.59 7.78
50.5 47.Q
14.0 13.3
1.35 1.38
75.0 65.4
21.90 19.08
20
6.391 6.250
0.789 0.779
0.641 0.500
1263.5 1169.5
126.3 116.9
7.60 7.83
30.1 27.9
9.4 8.9
1.17 1.21
70.0 54.7
20.46 15.94
18
6.251 6.000
0.691 0.691
0.711 0.460
917.5 795.5
101.9 88.4
6.70 7.07
24.5 21.2
1.09 1.15
50.0 42.9
14.59 12.49
15
5.640 5.500
0.622 0.622
0.550 0.410
481.1 441.8
64.2 58.9
5.74 5.95
16.0 14.6
1.05 1.08
Axis l-l
I
Axis 2-2 r
Z
(in. 4,
S (in. 3,
P
(in.)
R = 0.60 24” I Bl 24 x 7
I i’
R = 0.60 20” I B2 20 x 7
R = 0.70 1
20” I B3 20 x 639
R = 0.60 18” I B4 18 x 6
R = 0.56 I B7 15 x 535 R = 0.51 15”
Properties of Selected Rolled Structural Members Item 2.
I
.I #
Section Index and Nominal Size 12” I B8 12 x 5jq R = 0.56
Weight per Foot (Ills)
355
Beams, American Standard, Properties of Sections (Continued)
Depth Width Area of of of Section Beam Flange (in.2, (in.) (in.)
Aw
Flange Thickness (in.)
Web Thickness (in.)
r (in.4, 1
Axis l-l
s
(in.3,
Axis 2-2 P
(in.)
I (in.4,
s
P
50.0 40.8
14.57 11.84
12
5.477 5.250
0.659 0.659
0.687 0.460
301.6 268.9
50.3 44.8
4.55 4.77
16.0 13.8
(in.3) _A/-5.8 5.3
12” I B9 12 x 5 T = 0.45
35.0 31.8
10.20 9.26
12
5.078 5.000
0.544 0.544
0.428 0.350
227.0 215.8
37.8 36.0
4.72 4.83
10.0 9.5
3.9 3.8
0.99 1.01
10” I B 10 +/ 10x495 R = 0.41
35.0 25.4
10.22 7.38
10
4.944 4.660
0.491 0.491
0.594 0.310
145.8 122.1
29.2 24.4
3.78 4.07
8.5 6.9
3.4 3.3
0.91 0.97
-> 8" I --T B 1 2 8x4 R = 0.37
23.0 18.4
6.71 5.34
8
4.171 4.000
0.425 0.425
0.441 0.270
64.2 56.9
16.0 14.2
3.09 3.26
4.4 3.8
2.1 1.9
0.81 0.84
7" I B 13 7x356 R = 0.35
20.0 15.3
5.83 4.43
7
3.860 3.660
0.392 0.392
0.450 0.250
y.9 36.2
12.0 10.4
2.68 2.86
3.1 2.7
1.6 1.5
0.74 0.78
6" I B 14 6x346 R = 0.33
17.25 12.5
5.02 3.61
6
3.565 3.330
0.359 0.359
0.465 0.230
26.0 21.8
8.7 7.3
2.28 2.46
2.3 1.8
1.3 1.1
0.68 0.72
5" I B 15 5x3 R = 0.31
14.75 10.0
4.29 2.87
5
3.284 3.000
0.326 0.326
0.494 0.210
15.0 12.1
6.0 4.8
1.87 2.05
1.7 1.2
1.0 0.82
0.63 0.65
4" I B 16 4x238 R = 0.29
9.5 7.7
2.76 2.21
4
2.796 2.660
0.293 0.293
0.326 0.190
6.7 6.0
3.3 3.0
1.56 1.64
0.91 0.77
0.65 0.58
0.58 0.59
3" I B 17 3x296 R = 0.27
7.5 5.7
2.17 1.64
3
2.509 2.330
0.260 0.260
0.349 0.170
2.9 2.5
1.9 1.7
1.15 1.23
0.59 0.46
0.47 0.40
0.52 0.53
Item 3.
(in.) 1.05 1.08
Wide-flange Light Beams, Stanchions, and Joists, Properties of Sections 2
I-
-1
IlIE
Section Index and Nominal Size
Flange Weight Area Depth of of Thickper Foot Section Section Width ness (in.) (in.) (in.2) (in.) (lbs)
Web Thickness (in.)
I (in.4)
Axis l-l s (in.3)
P (in.)
I (in.“)
155.7 130.1 105.3
25.3 21.4 17.5
4.91 4.81 4.65
4.55 3.67 2.79
Axis 2-2 s (in. 3,
(ii.,
Light Beams CBL 12 12 x 4 R = 0.30
22.0 19.0 16.5
6.47 5.62 4.86
12.31 12.16 12.00
4.030 4.010 4.000
0.424 0.349 0.269
0.260 0.240 0.230
2.26 1.83 1.39
0.84 0.81 0.76
356
Properties of Selected Rolled Item 3.
Section Index and Nominal Size CBL 10 10 x 4 R = 0.30 CBL 8 8x4
R = 0.30 CBL 6 6x4
R = 0.30
Structural
Members
Wide-flonge light Beams, Stanchions, and Joists, Properties of Sections (Continued)
Flange Depth Weight Area o f Thickof per Foot Section Section Width ness (im2) (in.) (in.) (in.) (lbs) 19.0 5.61 10.25 4.020 0.394 0.329 17.0 4.98 10.12 1.010 15.0 4.40 10.00 1.000 0.269
Web Thickness (in.)
I (in.4,
0.250 0.240 0.230
96.2 81.8 68.8
Axis l-l s (in.3, 18.8 16.2
Axis 2-2 s (in.“)
(in.)
I (in.4)
4.14
4.19
13.8
4.05 3.95
2.08 1.72 1.39
0.86 0.83 0.80
3.30
P
3.45 2.79
P
(in.)
15.0 13.0
4.43 3.83
8.00
8.12
4.015 4.000
0.314 0.254
0.245 0.230
48.0 39.5
11.8 9.88
3 29 3 21
2 62
1.65 1.31
0.86 0.83
16.0 12.0
4.72 3.53
6.25 6.00
4.030
0.404 0.279
0.260 0.230
31.7 21.7
10.1
2.59 2.48
4.32 2.89
2.14 1.44
0.96 0.90
16.8 13.4
2 69 2.66 2.56
17.1 13.3 9.69
5.6 4.4 3.2
1.52 1.50
2.16 2 13
8.89 7.51
3.54 3.00
1.28
4.000
7.24
Stanchions CBS 6 6x6
R = 0.25 CB 51 5x5
R = 0.3
25.0 20.0 15.5
7.35 5.88 4.59
6.37 6.20 6.00
6.080 6.018 6.000
0.456 0.367 0.269
0.320 0.258 0.240
53.5 41.7 30.3
18.5
5.45 4.70
5.12 5.00
5.025
0.420 0.360
0.265 0.240
25.4 21.3
16.C
3.000
10.1 9.94 8.53
1.43
I.26
Joists CBJ 12 12 x 4 R = 0.30
14.0
4.14
11.91
3.970
0.224
0.200
88.2
14.8
.4 61
2.25
1.13
0.74
CBJ 10 10 x 4 R = 0.30
11.5
3.39
9.87
3.950
0.204
0.180
51.9
10.5
3.92
2.01
1.02
0.77
10.0
2.95
7.90
3.940
0.204
0.170
30.8
7.79
3.23
1.99
1.01
0.82
8.5
2.50
5.83
3.940
0.194
0.170
14.8
5.07
2 .43
1 89
0.96
0.87
CBJ 8 8x4
R = 0.30 CBJ 6 6x4
R = 0.25
Item 4.
Section Index A l R-H
Size (in.)
8x8
Thickness (in.) ,‘lf$ 7. xi 94 % Qi6
34
Equal Angles, Properties of Sections
Weight per Foot (lb) 56.9 51.0 45.0 38.9 32.7 29.6 26.4
Area of Section (in.2) 16.73 15.00 13.23 11.44 9.61 8.68 7.75
I (in.4) 98.0 89.0 79.6 69.7 59.4 54.1 48.6
Axis l-l and Axis 2-2 .- __.s P (in.3) (in.) 17.5 2.42 15.8 14.0 12.2 10.3 9.3 8.4
2.44 2.45 2.47 2.49 2.50 2.51
(in.)
Axis 3-3 r min (in.)
2.41 2.37 2.32 2.28 2.23 2.21 2.19
I.55 J.56 1.56 1.57 1.58 1.58 1.58
2
Properties of Selected Rolled Structural Members Equal Angles, Properties of Sections (Continued)
Item 4.
1.49 1.50 1.52 1.54 1.55 1.56 1.56
1.57 1.52 1.48 1.43 1.41 1.39 1.36
0.96 0.97 0.97 0.98 0.98 0.99 0.99
2.8 2.4 2.0 1.8 1.5 1.3 1.0
1.19 1.20 1.22 1.23 1.23 1.24 1.25
1.27 1.23 1.18 1.16 1.14 1.12 1.09
0.77 0.77 0.78 0.78 0.79 0.79 0.79
1.5 1.3 1.2 0.98 0.79
1.06 1.07 1.07 1.08 1.09
1.06 1.04 1.01 0.99 0.97
0.68 0.68 0.69 0.69 0.69
5 x 5 ,
27.2 23.6 20.0 16.2 14.3 12.3 10.3
7.98 ‘6.94 5.86 4.75 4.18 3.61 3.03
17.8 15.7 13.6 11.3 10.0 8.7 7.4
5.2 4.5 3.9 3.2 2.8 2.4 2.0
4x4
18.5 15.7 12.8 11.3 9.8 8.2 6.6
5.44 4.61 3.75 3.31 2.86 2.40 1.94
7.7 6.7 5.6 5.0 4.4 3.7 3.0
3% x 3%
11.1 9.8 8.5 7.2 5.8
3.25 2.87 2.48 2.09 1.69
3.6 3.3 2.9 2.5 2.0
A2
6x6
A3
R=M
A4
R=g4
R=s
(in.) 1.86 1.82 1.78 1.73 1.71 1.68 1.66 1.64 1.61
Axis 3-3 r min (in.) 1.16 1.17 1.17 1.17 1.18 1.18 1.19 1.19 1.19
Z (in. “) 35.5 31.9 28.2 24.2 22.1 19.9 17.7 15.4 13.0
Size (in.)
R=ffl
A5
Weight per Foot (lb) 37.4 33.1 28.7 24.2 21.9 19.6 17.2 14.9 12.6
Axis l-l and Axis 2-2 f s (in. 3, (in.) 8.6 1.80 7.6 1.81 6.7 1.83 5.7 1.84 5.1 1.85 4.6 1.86 4.1 1.87 3.5 1.88 3.0 1.89
of Section (in. 2, 11.00 9.73 8.44 7.11 6.43 5.75 5.06 4.36 3.66
Section Index
357
Area
X
* Special gage. Item 5.
Equal Angles, Properties of Sections 2 3;rlrj+ ’ -j-l 1 4+ T 2 ‘3
Set tion Index
Size (in.)
A7
3x3
R = x6
Thickness (in.) 55 756 N 516 *E6 %
tA 9 R=?4
N
wix2?4
%6 ‘/i 946 w
tA 11 R=?4
2x2
%6 Y4 %6 w
Axis l-l and Axis 2-2 S r (in.“) (in.) 1.1 0.90 0.95 0.91 0.83 0.91 0.71 0.92 0.58 0.93 0.44 0.94
Weight per Foot (lb) 9.4 8.3 7.2 6.1 4.9 3.71
Area of Section (in.2) 2.75 2.43 2.11 1.78 1.44 1.09
(in.4) 2.2 2.0 1.8 1.5 1.2 0.96
7.7 5.9 5.0 4.1 3.07
2.25 1.73 1.47 1.19 0.90
1.2 0.98 0.85 0.70 0.55
0.73 0.57 0.48 0.39 0.30
4.7 3.92 3.19 2.44 1.65
1.36 1.15 0.94 0.71 0.48
0.48 0.42 0.35 0.28 0.19
0.35 0.30 0.25 0.19 0.13
Z
(in.) 0.93 0.91 0.89 0.87 0.84 0.82
Axis 3-3 r min (in.) 0.58 0.58 0.58 0.59 0.59 0.59
0.74 0.75 0.76 0.77 0.78
0.81 0.76 0.74 0.72 0.69
0.47 0.48 0.49 0.49 0.49
0.59 0.60 0.61 0.62 0.63
0.64 0.61 0.59 0.57 0.55
0.39 0.39 0.39 0.40 0.40
2
358
Properties of Selected Rolled Structural Members Item 5.
Section Index
Size (in.)
tA 12 R=M
1% x 1%
Equal Angles, Properties of Sections (Continued)
Thickness (in.) % 54.5 34 946 34
iA 15 R = x6
1fC x l>i
tA 16 R = 34
1x1
% Ns M
* Special gage. t Bar size.
Item 6.
Section Index
Size (in.)
Thickness (in.) N
A 26 R=H
A 27 R=s
Pi-
4 x 3JS
x.5 N
% 4x3
;” 546 *>i
.w AZ8 R=N
335 x 3
?46
N %6
?‘f6 RAc2i6
3% x 2%
N 746
A 32 R = 956
%S
3x2~
% 946 M
* Special gage.
Weight
per
Foot (in.) 14.7 11.9 10.6 9.1 7.7 6.2
Weight ner Foot
Axis l-l and Axis 2-2 s r (in.3, (in.) 0.26 0.51 0.23 0.52 0.19 0.53 0.14 0.54 0.10 0.55
3.99 3.39 2.77 2.12 1.44
Area of Section (in.2, 1.17 1.00 0.81 0.62 0.42
I (in.“) 0.31 0.27 0.23 0.18 0.13
2.34 1.80 1.23
0.69 0.53 0.36
0.14 0.11 0.08
0.13 0.10 0.07
1.92 1.48 1.01
0.56 0.43 0.30
0.08 0.06 0.04
1.49 1.16 0.80
0.44 0.34 0.23
0.04 0.03 0.02
(lb)
(in.)
Axis 3-3 r min (in.)
0.57 0.55 0.53 0.51 0.48
0.34 0.34 0.34 0.35 0.35
0.45 0.46 0.46
0.47 0.44 0.42
0.29 0.29 0.30
0.09 0.07 0.05
0.37 0.38 0.38
0.40 0.38 0.35
0.24 0.24 0.25
0.06 0.04 0.03
0.29 0.30 0.31
0.34 0.32 0.30
Unequal Angles, Properties of Sections
Area
Axis l-1
nf -_
Section Z (in.2) (in.“)
s (in.3)
P
X
(in.) 1.22 1.23 1.24 1.25 1.26 1.27
(in.) 1.29 1.25 1.23 1.21 1.18 1.16
(in.4)
Z
6.4 5.3 4.8 4.2 3.6 2.9
2.4 1.9 1.7 1.5 1.3
9.8 8.5 7.2 5.8
3.98 3.25 2.87 2.48 2.09 1.69
6.0 5.1 4.5 4.0 3.4 2.8
2.3 1.9 1.7 1.5 1.2
1.23 1.25 1.25 1.26 1.27 1.28
1.37 1.33 1.30 1.28 1.26 1.24
2.9 2.4 2.2 1.9 1.7 1.4
10.2 9.1 7.9 6.6 5.4
3.00 2.65 2.30 1.93 1.56
3.5 3.1 2.7 2.3 1.9
1.5 1.3 1.1
1.07 1.08 1.09 1.10 1.11
1.13 1.10 1.08 1.06 1.04
2.3 2.1 1.9 1.6 1.3
9.4 8.3 7.2 6.1 4.9
2.75 2.43 2.11 1.78 1.44
3.2 2.9 2.6 2.2 1.8
1.4 1.3
1.20 1.18 1.16 1.14 1.11
1.4 1.2
0.93 0.75
1.09 1.09 1.10 1.11 1.12
8.5 7.6 6.6 5.6 4.5
2.50 2.21 1.92 1.62 1.31
2.1 1.9 1.7 1.4 1.2
0.93 0.81 0.69 0.56
1.0
1.0
0.95 0.78
1.1
1.0
Axis
Axis 2-2
4.30 3.50 3.09 2.67 2.25 1.81
13.6 11.1
X
0.91
1.00
0.92 0.93 0.94 0.95
0.98 0.96 0.93 0.91
4.5 3.8 3.4 3.0 2.6 2.1
1.1
0.94 0.78 1.3 1.2
1.0
0.90 0.74
s
3-3
P
Y
r
min (in.)
(in.3, 1.8 1.5 1.4 1.2 1.0 0.81
(in.) 1.03 1.04 1.05 1.06 1.07 1.07
(in.) 1.04 1.00 0.98 0.96 0.93 0.91
0.72 0.72 0.72 0.73 0.73 0.73
1.4 1.1 1.0
0.85 0.86 0.87 0.88 0.89 0.90
0.87 0.83 0.80 0.78 0.76 0.74
0.64 0.64 0.64 0.64 0.65 0.65
0.98 0.85 0.72 0.59
1.1
0.88 0.89 0.90 0.90 0.91
0.88 0.85 0.83 0.81 0.79
0.62 0.62 0.62 0.63 0.63
0.76 0.68 0.59 0.50 0.41
0.70 0.71 0.72 0.73 0.74
0.70 0.68 0.66 0.64 0.61
0.53 0.54 0.54 0.54 0.54
0.74 0.66 0.58 0.49 0.40
0.72 0.73 0.74 0.74 0.75
0.75 0.73 0.71 0.68 0.66
0.52 0.52 0.52 0.53 0.53
0.87 0.73 0.60
P-
Properties Item 7.
of
Selected
Rolled
Structural
Members
359
Unequal Angles, Properties of Sections 2
Section Index
Size (in.)
A 33 R = ss
3x2
Rt”=“i
2%
Weight Area nf T h i c k - per ness Foot Section Z (in.) (lb) (in.2) (in.4) 3-5 7.7 2.25 1.9 l/i6 6.8 2.00 1.7 5% 5.9 1.73 1.5 x6 5.0 1.47 1.3 ?4 4.1 1.19 1.1 0.84 *%6 3.07 0.90 N N6 ?4
x2
tA 645 R=% tA 39 R=j/,
2 x 1% 1% x 1M
(in.) 1.08 1.06 1.04 1.02 0.99 0.97
(in. 4, 0.67 0.61 0.54 0.47 0.39 0.31
P
X
1
(in.) 0.58 0.56 0.54 0.52 0.49 0.47
Y
0.91 0.79 0.65 0.51
0.55 0.47 0.38 0.29
0.77 0.78 0.78 0.79
0.83 0.81 0.79 0.76
0.51 0.45 0.37 0.29
0.36 0.31 0.25 0.20
0.58 0.58 0.59 0.60
0.58 0.56 0.54 0.51
0.42 0.42 0.42 0.43
%6
3.92 3.19 2.44
1.15 0.94 0.72
0.71 0.59 0.46
0.44 0.36 0.28
0.79 0.79 0.80
0.90 0.88 0.85
0.19 0.16 0.13
0.17 0.14 0.11
0.41 0.41 0.42
0.40 0.38 0.35
0.32 0.32 0.33
%i 94.5 %i
2.77 2.12 1.44
0.81 0.62 0.42
0.32 0.25 0.17
0.24 0.18 0.13
0.62 0.63 0.64
0.66 0.64 0.62
0.15 0.12 0.09
0.14 0.11 0.08
0.43 0.44 0.45
0.41 0.39 0.37
0.32 0.32 0.33
M
2.55 1.96
0.75 0.57
0.30 0.23
0.23 0.18
0.63 0.64
0.71 0.69
0.09 0.07
0.10 0.08
0.34 0.35
0.33 0.31
0.27 0.27
2.34 1.80 il.23
0.69 0.53 0.36
0.20 0.16 0.11
0.18 0.14 0.09
0.54 0.55 0.56
0.60 0.58 0.56
0.09 0.07 0.05
0.10 0.08 0.05
0.35 0.36 0.37
0.35 0.33 0.31
0.27 0.27 0.27
%
R=j/
(in.) 0.92 0.93 0.94 0.95 0.95 0.97
s
Axis 2-2 P min (in.) 0.43 0.43 0.43 0.43 0.43 0.44
1.55 1.31 1.06 0.81
%6
2x134
(in.3) 1.0 0.89 0.78 0.66 0.54 0.41
Axis 2-2 S P (in.3) (in.) 0.47 0.55 0.42 0.55 0.37 0.56 0.32 0.57 0.26 0.57 0.20 0.58
5.3 4.5 3.62 2.75
%6
tA 37
Axis l-1
%6
Y4 %6
34
’
* Special gage. t Bar size. Item 8.
Tees, Equal and Unequal, Properties and Dimensions of Sections
Size Radius Thickness of Weight Area Fillet of Toe Root per Section Foot Section Flange Stem a b (iff.) (in.2) (in.) (in.) (in.) (in.) Index (lb) Equal Tees Tl T8 t: ;o tT 11 tT 13 tT 14 tT 15
I
13.5 7.8 6.4 6.7 5.5 4.1 4.3 3.62
3.97 2.27
4 3
4 3
w N
1.87 1.95 1.60 1.19 1.26 1.05
3 234 2Js
32%
%6 w
2% 2%
x6
.wi 2 2
%
2 2
\
%6
%
\
%6.
%
7i6
%6
5% x6 94
x6 Pi
hi M Pi ki
316
96 ?I6
\I
/
Axis l-l (irf.‘)
(iz3)
5.7 1.8 1.6 1.0 0.88 0.52 0.44 0.37
Axis 2-2
-
(if;.)
(ii.)
(ir!.‘)
(iz3)
(i:.)
2.0 0.86
1.20 0.90
1.18 0.88
2.8 0.90
1.4 0.60
0.84 0.63
0.59 0.74 0.50 0.32 0.31 0.26
0.90 0.74 0.74 0.66 0.59 0.59
0.86 0.76 0.74 0.65 0.61 0.59
0.52 0.75 0.44 0.25 0.23 0.18
0.42 0.50 0.35 0.22 0.23 0.18
0.62 0.53 0.52 0.46 0.43 0.42
-_
-
360
Properties of Selected Rolled Structural Item 8.
Members
Tees, Equal ond Unequal, Properties and Dimensions of Sections (Continued)
Size
Weight per Section Foot Index (lb)
-~ Radius Thickness of Fillet Toe Root. a b (if.) (in.) (in.)
Area of
Sect ion Flange (in.2) (in.)
Stem (in.)
--.
Axis l-l
(ilf.‘)
(ifs)
% %
2.7 2.4
3% N 94
6.3 2.0 1.2 0.94
Axis- 2-2 -~-~. 7
(ii.)
(i:.)
(inl.‘)
(ii”)
(ii.)
1.1 1.1
0.82 0.84
0.76 0.76
5.2 3.9
2.1 1.6
1.10
2.0 0.90 0.62 0.52
1.39 0.86 0.69 0.73
1.31 0.78 0.62 0.68
2.1 2.1 1.2 0.75
1.1 1.1 1.0
Unequal Tees 13.6 11.5
4.00 3.37
5 5
3 %
T 60
11.2 9.2 8.5 6.1
3.29 2.68 Z.-E8
4 4 4
4 % 3
1 .x
3
* $6, '332 94, 1%2
T 50
T 61
T T
62 79
3
* 346, % 7
2.5
1.6
\
1.4
\\
1.600 xx 1,400 LJ 1,200 4
\
\it i.\.\, \\ \
1.2 1.0 -.0.90
1,000 8 900 2 800 700 600 500
0.80 0.70 0.60 0.50
0.40 0.35 0.30 0.25 0.26 0.18 0.16 0.14 0.12 0.10 0.09 0.08 0.07 0.06 0.05 0.00001
400 350 300 250
2
3
4
5 6 7 8
2
0 . 0 0 0 1
3 4 5678 2 0.001
3 4 5678 0.01
2 -
200 180 160 140 120 100 90 80 70 60 50 3 4 5678 -0.1
Factor A = f/E = c Item
2.
Chart for determining shell thickness of cylindrical and spherical vessels under external pressure when constructed of aluminum alloy MIA-O or
M I A - H ’ .lZ.
_.---
(From the 1956 ASME Unfired Pressure Vessel Code with permission of the Am. Sot. Mech. Engrs.)
\
\I
/
--.-
Charts
for
Cylindrical
and
Spherical
Vessels
Under
External
Pressure
367
25,000 20,000 18,000 16,000 14,000 10 9.0 8.0 7.0 6.0
.,. f
1.6 1.4
?-?I$ !jje 3-j $j .g s ;I; 3 E b2 IA p
. 1.0 0.90 0.80 0.70 0.60
10,000 9,000 8,000 H 7,000 6,000 J-K-1 5,000 4,000 ttttttttl 3,500 3,000 $ g 2,500 2 % c: 2,000 ci 1,800 2' 1,600 2% II\1 I Ill1 , I I I I I II 1,400 1,200 4
12
1,000 $ 900 800 a 700 600
2 0.50
500
0.40 0.35 0.30 0.25
400 350 300
0.20 0.18 0.16 0.14 0.12
200 180 160 140 120
0.10 0.09 0.08 0.07 0.06 0.05
100
250
ii 70 60 50 0.001 Factor A = f/E = c
Item 3.
Chart for determining shell thickness of cylindrical and spherical vessels under external pressure when constructed of aluminum alloy MIA-HI4,
(From the 1956 ASME Unfired Pressure Vessel Code with permission of the Am. Sot. Mech. Engrs.)
368
Charts for Cylindrical and Spherical Vessels Under External Pressure 501
I
IIIYII
IIIIiIlll
I I I
III llIIIIIIII
I I I
I I I
IIIIIIIll
I
I I I
40
I
lIIIr150.ooo
35 30
40,000 35,000 30,000
25
25,000
20 18
20,000 18,000 16,000 14,000 12,000
I II I I I II
1614 12
I II
I IIll
9': 8.0 7.0 6.0
8;OO0 7,000
!5.0 4.0 3.5 3.0 2.5 2.0 1.8 1.6 1.4 1.2 1.0 0.90 0.80 0.70 0.60
0.50 0.40 0.35 0.30 0.25
400 350 300 250
0.20I -
200 180
0.09 0.08 0.07 0,006 0.05 0.001 Factor A = f/E = c hem 4 .
Chart for determining shell thickness of cylindrical and spherical vessels under external pressure when constructed of aluminum alloy MGl l A-0 01
MG; :A-HI 12.
(From the 1956 ASME Unfired Pressure Vessel Code with permission of the Am.
Sot. Mech. Engrs.)
Charts for Cylindrical and Spherical Vessels Under External Pressure I
I I I
369
I III11150.oQo 40,000 35,000 30,000 25,000 20,000 18,000 16,000 14,000 12,000 10,000
0.001
-
Factor A = f/E = c Item 5 .
Chart for determining shell thickness of cylindrical and spherical vessels under external pressure when constructed of aluminum alloy MGI l A-H34.
(From the 1956 ASME Unfired Pressure Vessel Code with permission of the Am.
Sot. Mech. Engrs.)
370
Charts
for
Cylindrical
and
Spherical
Vessels
Under
External
Pressure
40 35 30 25
25,000
20 18 16 14
20,000 18,000 16,000 14,000 12,000
12
10,000 9,000 8,000 7,000 6,000
2: 8.0 7.0 6.0 ,e 5.0
5,000 4,000 3,500 3,000 e
E 2,500 8% L zp 2,000 :;--, 1,800 5% 1,600 %% 1,400 1,200 4 1,000 8 900 2 800 700
2 5 0.60
600
2 0.50
500
0.40 0.35 0.30
400 350 300
0.25
250
0.20 0.18 0.16 0.14
200 180 160 140
0.12
120
0.10 0.09 0.08 0.07 0.06
100 90 80 70 60 50 0.001 Factor A = f/E = c
Item 6.
Chart for determining shell thickness of
cylindrical
and spherical vessels under external pressure when constructed of nickel.
Unfired Pressure Vessel Code with permission of the Am. Sot. Mech. Engrs.)
(From the
1956 ASME
Charts
for
Cylindrical
and
Spherical
Vessels
Under
External
Pressure
25
25,000
20 18 16 14
20,000 18,000 16,000 14,000 12,000
12
371
10,000 9,000 8,000 7,000 6,000
10 9.0 8.0 7.0 6.0
5,000 4,000 3,500 3,000
@J 2,500 3” .E S
2.5
2,000 5 5 1,800 s$ 1,600 gz 1,400-
2.0 1.8 1.6 1.4 1.2
1,200 ;
1.0 0.90 0.80 0.70
1,000 E 900 2 800 700
0.60
600 500 400 350 300
0.25
250
0.20 0.18 0.16 0.14 0.12
200 180 160 140
0.10 0.09 0.08 0.07
100 90 80 70
0.06
60
0.05 0.00001
Item 7.
120
2
3
4
5678
o.ooo1
2
3
4
5678
2
0.001 Factor A = f/E = 6
3
4
5678
0.01
2
3
4 5678
50
0.1
Chart for determining shell thicknest of cylindrical and spherical vessels under external pressure when constructed of annealed nickel-copper alloy.
(From the 1956 ASME Unfired Pressure Vessel Code with permission of the Am.
Sot. Mech. Engrs.)
372
Charts for Cylindrical and Spherical Vessels Under External Pressure
25 20 18 16 14 12 9!00 8.0 7.0 6.0 5 . 0 4 . 0 3.5 3.0 2.5 2.0 1.8 1.6 1.4 1.2 1.0 0.90 0.80 0.70 0.60 0.50 0.40 0.35 0.30 0.25 0.20 0.18 0.16 0.14 0.12 0.10 0.09 0.08 0.07 0.06 0.05 0.00001
Item 8 alloy.
2
3
4
5678 0.0001
2
3
4
5678
2
0.001 Factor A = ffE = c
3
4
5678
Chart for determining shell thickness of cylindricbl and spherical vessels under external pressure when constructed of annealed nickel-chromium-irorl (From the 1956 ASME Unfired Pressure Vessel Code with permission of the Am. Sot. Mach. Engrs.)
Charts for Cylindrical and Spherical Vessels Under Eiternal Pressure 50
50.000
40 35 30
40,000 35,000 30.000
25
25.000
20 18 16 14
20,000 18.000 16.000 14.000
12
12.000
10 9.0 8.0 7.0
)O.OOO 9,000 8.000 7,000
6.0
6.000
5.0
5.000
4.0 3.5 3.0
4.000 3.500
373
3qooo g f 2,500 s f E : 2,000 2 5 1,800 2s 1,600 $ g 1,400 _cj 1,200 4
2.5 2.0 1.8 1.6 1.4 1.2 1.0 0.90 0.80 0.70
900 1,000
$
800 700
’
0.60
600
0.50
500
0.40 0.35 0.30 0.25
250
0.20 0.18 0.16 0.14
200 180 160
’
’ ’ ’ ’ ” I
I
I
I\
I
Y I I\II\
h
I
\
I
\I
RI
0.12
I
I\1
\
I
\I
l\l
l\l
\slll40
0.10 0.09
0.08 0.07 0.06 2 3 4 5678 0.00001
0.000 1
0.001 Factor
ltem 9.
Chart for determining shell thickness of cylindrical
A =flE = e
and spherical vessels under external pressure when constructed of annealed copper, type
(From the 1956 ASME Unftred Pressure Vessel Code with permission of the Am. Sot. Mech. Engn.)
/I b
I
\
\
\I
/
DHP-
374
Charts
for
Cylindrical
and
Spherical
Vessels
Under
External
Presusre
30,000 25,000 20,000 18,000
9,000
8,000 7,000 6,000 5,000 4,000 3,500
700 600 500 400 350 300 250
0.09 0.08 0.07 0.06 0.05
0.00001
2
3
4
5678
0.0001 2
3
4
5 678 0.001
2
Factor A = f/E = E Item 10.
3
4
5678 0.01
2
3
4
5678
0.1
Chart for determining shell thickness of cylindrical and spherical vessels under external pressure when constructed of copper-silicon alloys
(From the 1956 ASME Unfired Pressure Vessel Code with permission of the Am. Sot. Mech. Engrs.)
Aand C.
Charts for Cylindrical and Spherical Vessels Under External Pressure 50
50,000
40 35 30
40,000 35,OOu 30,000
25
25,000
20 18 16 14 12
20,000 18,000 16,000 14,000 12,000
52
;ogoo
8.0 7.0
8:OOO 7,000
6.0
6,000
5.0
5,000
4.0 3.5
4,000 3,500
3.0
3,000
2.5
2,500
2.0 1.8 1.6 1.4 1.2
zoo0
WO 1,600 1,400 u I m
1.0 0.90 0.80 0.70
800 700
0.60I
600
0.50
500
0.40 0.35 0.30
400 350 300
I q \ I\
0.25
250
0.20 0.18 0.16 0.14
200 180 160 140
0.12
120
0.10 0.09 0.08 0.07 0.06
375
I
0.05 0.00001
I
II
I111111
I
III
IllllYl
* =#
P o-
100 90 80 70
*‘o
60 50 2
3
4
5678
2
0.0001
3
4 5678
2
0.001
3
4
5 6 7 8
0.01
2
3
4
5 6 7 8
0.1
Factor A = f/E = e :lem 11. alloy.
L.
_
I
Chart for determining shell thickness of cylindrical and spherical vessels under external pressure when constructed (From the 1956 ASME U n f i r e d P r e s s u r e V e s s e l C o d e w i t h p e r m i s s i o n o f t h e A m . Sot. Mech. Engrs.)
\
\
\I
/
of annealed 90-T 3 copper-nickel
- -
376
Charts
for Cylindrical and Spherical Vessels Under External Pressure
40 35 30 25 20 18 16 14 12 SC 8.0 7.0 6.0 5.0 4.0 3.5 3.0 2.5 2.0 1.8 1.6 1.4 1.2 1.0 0.90 0.80 0.70 0.60 0.50 0.40 0.35 0.30 0.25 0.20 0.18 0.16 0.14
I /I II
I ill’
I/ I I I
I I II,
I
0.09
0.08 0.07
I
l-
0.001 Factor A = f/E = e Item 12. alloy.
Chart for determining shell thickness of cylindrical and spherical vessels under external pressure when constructed of annealed 70-30 copper-nickel (From the 1956 ASME Untired Pressure Vessel Code with permission of the Am.
\
---i-
----
\I
/
Sot. Mech. Engrs.)
---
Charts
for
Cylindrical
and
Spherical
Vessels
Under
External
Pressure
50,000 40,000 35,000 30,000 25,000 20,000 18,000 16,OOt I 14,000 12,000 10,000 9,000 8,000 7.000 6,000 5,000 4,000 3,500 3,000 7
5nn
-,-v.,
nnn -,-_7
2.0 1.8 1.6 1.4
1,800 ” 1,600 2 x 1,400 1.200
1,000 900 800 0.60
‘d
0.50 0.40 0.35
400 350
0.30
300
0.25
250
0.20 0.18 0.16 0.14
200 180
0.12 0.10 0.09
100 90 80 70
0.06 -
60 5.
0.05 0.00001
,
4 ,b,L1 0.’
2 0001
3
4
5 6 7 8
0.001
3
4
5678
0.1
4
377
378
Charts for Cylindrical and Spherical Vessels Under External Pressure AlI
l,‘l~IlAlllI’I ^ .^‘II
‘II^
‘II.- lll1_11 a”
I
I
I
I
I
Illlll 50,OOd
& / a” lml I
40,000 35,000 30.000
IIIIIll\lI i II 18jOO0 16,000 14,000
16s 400” F
12,000 10,000 9,000 8,000
1.8s 1.61
I
\
Y
\
l\hl
‘,
Y
‘,
\
\
‘,
.\
\
\
-zoo0
$2 1 , 8 0 0 Q‘,
\ ‘1
\
\
1,600 .%,
0.80 0.70
1,000 900 800 700
0.60
600
2i 2
0.50 400 350 300 250 200 180 160 140 120 0.10 0.09 0.08 0.07 0.06 0.05 t 0.00001
2
3
4
5678
0.0001 Factor A = f/E = E
Item 14. Chart for determining shell thickness of cylindrical and spherical vessels under external pressure when constructed of austenitic steel ii 6 Cr-8 Ni, t y p e 304). (From the 1956 ASME Unfired Pressure Vessel Code with permission of the Am. Sot. Mech. Engrr.)
Charts
for
Cylindrical
and
Spherical
Vessels
‘IIIIIWI
I
I
Under
External
Pressure
329
II
40,000 35,000 30,000
\I
2,500
1,600 1,400 1,200
o.ocoo1
-
-
o.ooo1
0.001 Factor
Item 15.
-
0.01
-
0.1
A = f/E = E
Chart for determining shell thickness of cylindrical and spherical vessels under external pressure when constructed of oustenitic steels (18 Cr-8 Ni j-
MO, type 316; 18 G-8 Ni Sot. Mech. Engrs.)
+ Ti, type 321; 18 G-8 Ni
f Cb, type 347).
(From the 1956 ASME Unfired Pressure Vessel Code with permission of the Am.
380
Chari ts
for
Cylindrical
and
Spherical
Vessels
Under
External
Pressure
40,ooo 35,000 30,ooo 25,000
(
I
- ,--12,m
1.4 1.2 1.0 0.90 0.80
0.50 0.40 0.35
Factor A = f/E = e Item 16.
Chart for determining shell thickness of cylindrical and spherical vessels under external pressure when constructed of cast iron.
AWE Unfired Pressure vessel Code with permission of the Am. Sot. Mech. Engrs.)
r
.
t
\
\
\I
/
A P P E N D I X
PROPERTIES OF VARIOUS SECTIONS AND BEAM FORMULAS
f
381
382
Properties
of
Various
Sections
and
Beam
Formulas Properties
Item 1.
of
lldwin
Courtesy of the Distance from Axis to Extremities of Section g and II
Area of Section ‘4
Various
Sections
Locomotive Works) Radius of Gyration
Section Modulus Moment of Inertia I
I=
I ;i d-
1
-& = 0.2890
II=0
-$ = 0.577a
a’ ii
=-$ = 0.707n
(12
-6 = 0.118d 6&
g=t 2
ad - a,’ -12
a4 - 111’
*=0
d - (114 3
a( -a,’ 3n
6a
a*+m* - = 0.289 da2 + at’ 12 \I-
a2 - 012
y = s = 0.707a
a4 - a,’ 12
bh
h up2
bha 12
bh’
bh
u-A
bh3 3
bh=
&2 = 0.289A
-ii-
= 0.577h
-27
1 -.
u’=m
y=hcosa+bsince 2
bh
bh - bth,
y=n
b(h - hl)
r/
c ; h2 tan 30’ = 0.866h2
u=” 2
h y = ~ = 0.5i7h
2 cm 30’
u=l
2
h(b + bd 2
h(br + 2b) Y=3(bl Ub + 2bd y’ = 3(b
h(b + b,) 2
y=h
A2 cd a + b1 sin* P
s
h cos LI + b sin a
bh3 - - blhls6h
b(h3 - h,J)
b(h3
12
12
A h*(l +2cos2300) 12 [ 4 cd 30’ = 0.06h’
I
A ii
1
= 0.06hI
Z/L* tan 22;O = 0.828h*
bh
bh3 - b,h,’-
A*(1 + 2 cos* 30”) 4 CD9 30”
A k2(1 + 2 co.32 223”) 12 [ 4 cm* 22:” = 0.055h~ h3(bz
+ lbb, + br*) 36(b + bd
h3(b + 3b)n
12
d6(bZ + h’9
6db2+h2
E (hz COG P + b* sin* a)
2
bh
b2h=
6(b* + AZ)
1
#,h 2
: h* tan 30” = 0.866h* 2
b3hJ
bh
bh
A
4 cd 30” = 0.12h3
1
.4 C h(1 + 2 cm* 30”) 6 4 cos 30’ = 0.104h” A
h(l + 2 cm* 22:“)
6
4 02s 221 = O.lOSh~ hz(b2
+ 4661 + bl*) 12(bl
12
bh3 - b,h,a
lZ(bh - bihd h3 - ht3
- h13) 6h
h(l + 2 cos* 30’)
6 C
h2 co82 P + b2 sin2 a
>
+ 2b)
h*(b + 3b)l
12
li(h -hi)
1
I I
I + 2 COSZ 30’ 3 = 0.264h h
4 cos 300
1 + 2 COSZ 30” 3 = 0.264h
h _.~ 4 cos 22;”
1 + 2 cos2 225” 3 = 0.257h
& d\/l(b” + 4bbl
+ blz)
,
Properties of Various Sections and Beam Formulas Item 1.
Properties of Various Sections (Continued)
Distance from Axis to Extremities of Section y and Yl
Area of Section Sections
A
Section
bh z
s=f Y
bhs 38
bh2 24
4
$ = 0.049d’
rd3 32 = 0.09&f*
d
U’d
r(d’ = O.O49(d’ - dl’)
r(d’ - dl”) 32d
d/d’
= 0.098 d’
4
-dJ(9a-* - 64)-
d 49rz - 64
F = 0.093bh’
h i
bh’
& - 0.408h
64
i
d
y = - = 0.283d en.7
d’(w - 64) - o,oo69d~ 1152~
11, = 2 = 0.212d 2(d3 - dls)
= 0.3927(d* - dl*)
u, = 3rdfd* - dl*) - 4(dJ - d?) Br(d’ - dl’)
1152*(d* - dl*)
T = 0.735468
A ;i
rbA3 K = 0.049bAJ
r(bh - blhl)
r(bAJ - b,h,‘)
A ;i
4
= 0.7854(bh - blhl) (b = h = I) It- n-r) -
64
= O.O49(bha- bth13)
I 6
4 = 0.2146r’
I=
192(3r - 4) = 0.0238d3
rI
1-T ( > = 0.7767r
!-.TL ( 3 16 36 - 9s > = 0.0075r’
2ch* + (bl - 2c)s1* + (b - 2c)(2h - 81)’
-
r(bh3 - btA13) 32h = 0.093(bh3 - blh13) A
I L = 0.0097rJ
3
_ (b - 2c)(m - s)~ 3
l g
bAJ - blAl* bh - bdt
~o.o349c* = 0.187r
;
U I
rK1
y--rCDS4
Angle 4
K
Kl
K2
5 10 15 20 25 30 35 40 50
0.00044 0.00352 0.01180 0.02767 0.05331 0.09058 0.14102 0.20573 0.38026 0.61418 0.90034 1.22525 1.57080
0.9986 0.9908 0.9796 0.9640 0.9439 0.9200 0.8921 0.8607 0.7831 0.7051 0.6144 0.5197 0.4244
O.OOOOO 0.00000 O.OOOOO
* 0 in radian measure (1 radian = 180/r darees, and I” = 0.017453 radians). 4 About the XX axis. L\ About neatral axis of shaded area.
i
I
I
2
ti 80 90
I
bw3 + b1/13 - (bl - 2c)(r - 81)’
2A ul=h-u
rK
12r = 0.132d
.l(d4 _ &“)(dz - dl*) - 64(d3 - dlJ)P
Y=3*(d2
8
bs + 2ec+ blat
I.6
UC!!
2
r(d* - dl’)
i
-!L = 0.236h
rz
d(3r - 4)
is
t= 4IiI
bha ii-
= 0.7354(d* - d?)
.
adius of Gyration
y=A
2
r(d* - dl*)
Modulus
Moment of Inertia I
bh 5
rd’ -= 0.3927dz 8
383
0.00001 0.00003 0.00010 0.00033 0.00076 0.00337 0.01064 0.02711 0.05848 0.10976
d lii i d;5
384
Properties of Various Sections and Beam Formulas Item 2. Beam Formulas
(Courtesy of the Baldwin Locomotive Works)
-.--
Moment Moment Fsw,
It? = w
Rr = w
M- = g (32 - 2)
M, = W(z -a), [z > a]
2Wl M “an= = --ii-
MIILOZ = W b
WZ Qc = T (21 - I) Q ,,m1 = w w 1113 / = F, $.J blU.9 w 1 L = jg .z
RI =R,=;
R,=!%
R,=f[W(l-a)+W,b]
&.+
1(
f&z-
R, = RI = %’
W(2a + b) Rx = 21
Ma=+-z)
MI = RIZ. [z < a or = a] W(z - a)* M, = RIZ 2b v
M
[z ; (a + b) and > a] Mz = Rdl - d. [z > (a + b)l Qc = RI, [z < a or = a] Qz=
II = dc(l +a)/31
r = ;, f haz)
!!!p _ f (= - =),
Qz
= ; - wz
Q wl 50 /=g.gg(maZ).
Iz < (a + b) and > al Qz = Rz, [z > (a + b) and < 4
Moment / ~ +
R, = R2 =; M,=
W+f;)
WI M IPI”1 = 7’ [t = JiJ] Q,.= W(;-F+T) W Qm.. - 2. [z = 01
Qw = ; W, [z = I]
Q !>,, 1_
W
I = o.01304EIlJ (mm)
w 313 /=rI.GJ20(ma2)
[z = 0.51911
M,=w~(~~~~;;~~“)~
Qr=+z
w(L. - d. 12 > Cl Q. = R I, [z < cl Qz = RI, [z > cl W / = 0.0098 jj P haz)
5701 Q",,,.. = T f = 2; . ~5(maz).[z
= 0.42211
[z = c = 0.41411
-
I
\
\
_ __ .._ . ..-
\I
/
-
-
_ ___ .-~.
II
Properties of Various Sections and Beam Formulas Beam Formulas(Con,
385
tin ued)
yfTzg2Gq RI
+H,- S h e a r I
1-R;
w3c + Cd RI=W(T) w3c1+ 0 RI=W(I;-) M .c e&z--W=s[z c Q. = RI. fz < Cl Qz = (RI - WA [z > Cl
RI = w[(c + f)* - c1*]/21 Rr = w[(c, + 1)” - c*]/21 bfz = >bur(c - 212 M., = $Pw(c + .a)* - Rm hf.” = yzwh - zrP
R = reaction M = moment w = concentrated load clr total bad 30 = unit load W b - W,a R, = ___ r1 = Wl L b) + Wda + 0 1 bfs = Ita, 1% < (I - b)l bf=, = Itan - W(b + z, - I) [ZI > (I - b) and .
Properties Item 2. Pipe Size and Outside Diameter inches
Schedule Number and/or Weight 40ST 40s 60 X8 80s
10 D = 10.750
80 100 120 140 160 5s 10s
20 30 ST 405 D&750
4’ 60
xs 80s
80 loo 120 140 160
10 20 30 ST 40
14
-14.006
D
6om
80 loo 120 140
10 20
a0 ST
D = :&hM
4oxs 60 80 100 120 140
10 20 30 18 D = 18.000
40
60
T
I
ST X8
Wall Thickness inrhen t 0 365 ,395 ,500 .531 ,593 ,718 ,750 .843 1.000 1.062 1.125
Properties
Surface Area Outside Inside sq It sq ft. per ft per ft A0 Ai 2.81 2.62 2.81 2.61 2.81 2.55 2.81 2.54 2.81 2.50 2.81 2.44 2 81 2.42 2.81 2.37 2.81 2.29 2.81 2.26 2.81 2.23
Fifth POWU ofID in.6 d’
Inside Diameter
Design
d 10.090 9 960 9.730 9.6Bi 9.564 9.314 9.230 9.064 8.750 8.625 8.500
101 98.0 88.1 85.3 80.0 iO.l 67.7 61.2 51.3 47.7 44 4
165 :tao ,203 ,219 .238 .250 .279 .300 .330 ,344 .375 ,406 ,438 ,500 ,562 ,625 ,687 .x4:( ,872 1.000 1.125 1.219 1.312
12.420 12.390 12.344 12 312 12.274 12.250 12.192 12.150 12.090 12.062 12.000 11.938 11.874 11.750 11.626 11.500 11.376 11.064 11.000 10.750 10.500 10.313 10.126
296 292 287 283 279 276 269 265 258 255 249 242 236 224 212 201 191 166 161 144 128 117 106
3.34 3.34 3.34 3.34 3.34 3.34 3.34 3.34 3.34 3.34 3.34 3.34 3.34 3.34 3.34 3.34 3.34 3.34 3.34 3.34 3.34 3.34 3.34
3.25 3.24 3.23 3.22 3.21 3.21 3.19 3.18 3.17 3.16 3.14 3.13 3.11 3.08 3.04 3.01 2.98 2.90 2.88 2.31 2.75 2.70 2.65
,188 ,220 ,238 .250 ,312 ,375 .4ofi .43x .469 .500 ,593 ,625 ,656 .75u ,937 1.993 1.250 1.344 1.406
13.624 13.560 13 524 13.500 13.375 13.250 13.188 13.125 13.062 13 000 12 814 12.750 12.688 12.500 12.125 11.814 11.500 11.313 11.183
469 45E 452 443 428 408 399 389 380 371 345 337 329 305 262 230 201 185 175
3.67 3 6i 3.67 3.67 3.67 3.67 3.67 3.67 3.67 3.67 3.67 3.67 3.67 3.67 3.67 3.67 3.67 3.67 3.67
.188 .238 .250 .281 .312 ,344' ,375 .406 ,438 ,469 .500 .531 .656 .68X ,750 ,843 1.031 I.218 1.43x I.506 1.593
15.624 15.524 15.500 15.438 15.375 15.312 15.250 15.188 15.124 15.062 15.000 14.938 14.688 14.625 14.500 14.314 13.938 13.564 13.124 13.o!lo 12.814
931 902 895 877 859 842 825 808 791 775 759 744 684 669 641 601 526 459 389 371 345
,250 ,312 ,375 .43x .500 .562 .594 ,625 ,719 ,750
17.500 17.375 17.250 17.124 17.000 16.876 16.813 16.750 16.562 16.500
1641 1584 1527 1472 1420 1369 1344 1318 1247 1223
\
-
-
of
Pipe
Weizht of Pip+ water lb lb per ft per ft w WC 40.5 34.1 43.7 33.7 54.i 32.3 58 0 31.9 64 3 31.1 76.9 29.5 80 I 29 I 89 2 27 9 104.1 26.0 109 9 25.3 115.7 24.6
Radius of Gyration inches 7” 3.67 3.66 3.63 3.62 3.60 3.56 3.55 3.52 3.47 3.45 3.43
Moment of Inertia in.’ I 160.8 172.5 212.0 223.4 244.8 286.2 296 3 324.3 367.9 384.0 399.4
11.73 12.88 13.41 14.58 15.74 16.94 19.24 21.52 23.81 26.04 31.53 32.64 36.91 41.09 44.14 47.14
121.2 120.6 119.7 119.1 118.3 117.9 116.7 115.9 114.8 114.3 113.1 111 9 110.7 108.4 106.2 103.9 101.6 96.1 95.0 90.8 86.6 83.5 80.5
22.2 24.2 27.2 29.3 31.8 33.4 37.2 39.9 43.8 45.6 49.6 53.5 57.6 65.4 73.2 80.9 88.5 107.2 111.0 125.5 139.7 150 1 160.3
52.5 52.2 51.8 51.6 51.2 51.0 50.6 50.2 49.7 49.5 49.0 48.5 47.9 47.0 46.0 45.0 44.0 41.6 41.1 39.3 37.5 36.2 34.9
4.45 4.44 4.44 4.43 4.42 4.42 4.41 4 40 4.39 4.39 4.38 4.37 4.36 4.33 4.31 4.29 4.27 4.22 4.21 4.17 4.13 4.10 4.07
3.57 3.55 3.54 3 53 3.50 3.47 3.45 3.44 3.42 3.40 3.35 3.34 3.32 3.27 3.17 3.09 3.01 2.96 2.93
8.16 9.52 10.29 10.80 13.44 16.05 17.34 18 66 19.94 21.21 24.98 26.26 27.50 31.22 38.47 44.32 50.07 53.42 55.63
145.8 144.4 143 6 143.1 140.5 137.9 136.6 135.3 134.0 132.7 129.0 127.7 126.4 122.7 115.5 109.6 103.9 100.5 98.3
27.7 32.4 35.0 36.7 45.7 54.6 59.0 63.4 67.8 72.1 84.9 89.3 93.5 106.1 130.8 150.7 170.2 181.6 189.1
63.1 62.5 62.2 62.0 60.8 59.7 59.1 58.6 58 0 57.5 55.8 55.3 54.8 53.1 50.0 47.5 45 0 43.5 42.6
4.88 4.87 4.87 4.86 4.84 4.82 4.81 4.80 4.79 4.78 4.74 4.73 4.72 4.69 4.63 4.58 4.50 4.48
195 226 244 255 315 373 401 429 457 484 562 589 614 687 825 930 1027 1082 1117
4.19 4.19 4.19 4.19 4.19 4.19 4.19 4.19 4.19 4.19 4.19 4.19 4.19 4.19 4.19 4.19 4.19 4.19 4.19 4.19 4.19
4.09 4.06 4.06 4.04 4.02 4.01 3.99 3.98 3.96 3.94 3.93 3.91 3.85 3.83 3.80 3.75 3.65 3.55 3.44 3.40 3 35
9.34 11.78
191.7 189.3 188.7 187.2 185.7 184.1 182.7 181.2 179.6 178.2 176.7 175.3 169.4 168.0 165.1 160.9 152.6 144.5 135.3 132.J 129.0
31.8 40.1 42.1 47.2 52.4 57.5 62.6 07.8 72.8 77.8 82.8 87.7 107.6 112.4 127.5 136.5 164.8 192.3 223.7 232.3 245.1
83.0 82.0 81.7 81.1 80.4 79.7 79.1 78.4 77.8 77.2 76.5
71.5 69.7 66.1 62.6 58.6 57.5 55.8
5.59 5.57 5.57 5.56 5.55 5.54 5.53 5.52 b.bO 5.49 5.48 5.47 5.43 5.42 5.40 5.37 5.29 5.23 5.17 5.15 5.12
292 366 384 429 474 519 562 605 649 691 732 773 933 972 1047 1157 1365 1556 1761 1815 1894
4.71 4.71 4.71 4.71 4.71 4.71 4.71 4.71 4.71 4.71
4.58 4.55 4.52 4.48 4.45 4.42 4.40 4.39 4.34 4.32
240.5 237.1 233.7 230.3 227.0 22a.i 222 0 220 4 21515 213.8
47.4 59.0 70.6 82 2 93.5 101. i 110 4 116 0 132 5 138.2
104.1 102.7 101.2 99.7 98 3 96.9 96 I 95.4 93 3 92.6
6.28 6.25 6.23 6.21 6.19 6.17 6 16 6.15 6.12 6.10
549 679 807 932 1053 1171 1231 1289 1458 1515
-
-1
8.00
8.62
9.36 9.82
10.93
12 37 13.88 15.40 16.92 18.41 19.89 21.41 22.88 24.35 25.81 31.62 33.07 35.90 40.14 48.48 56.56 65.79 68.33 72.10 13.94 17.36 20.76 24.li 27.49 30.79 32 46 34.12
38.98 40.64
I ---
\I
-
Pipe
389
(Continued)
Areas and Weizhia CrawSectional Metal Flow ARti AI& sq in. sq in. A A, 11.91 78.9 12.85 77.9 16 10 74.7 17.06 73.7 18.92 71 8 22.63 68.1 23.56 67.2 26 24 64.5 30 63 60.1 32.33 58 4 34.02 56.7 6.52 7.11
of
--
75.9 73.4 72.7
4.53
Section MOdUlllE in.3 Z 29.91 32.1 39.4 41.6 45.5 53.2 55.1 60.3 68.4 71.4 74.3 20.3 22.0 24.7 26.6 28.7 30.1 33.4 35.7 39.0 40.5 43.8 47.1
129.2 140.5 157.5 169.3 183.2 191.9 212.7 227.5 246.5 253 279 300 321 362 401 439 475 562 579 642 701
50.4 56.7 62.8
68.8 74.5 88.1 90.8 100.7 109.9 116.4 122.6
742 781
27.8 32.3 34.8
36.5 45.0 53.3 57.3 61 4
65 3
69 1
86.3 84.1 87.7
98.2 117.9 132.8 146.8 154.6 159.6
36.5 45.8 48.0 53.6 59.3 64.8 70.3 75.6
81.1
86.3
91.5 96.6 116.6 121 4 130 9 144.6 170.6 194.5 220.1 226.9 236.7
61.0 75.5 89.6 103.6 117.0 130.2 136.8
143.3
162.0 168.3
-
390
Properties of Pipe Item 2.
Nomrnal Pipe Size and Outside Diameter inches
I8 D = !a.000
sdledule NU?ObZ and/or Weight 80 100 120 140 160 10 20 ST 30 xs 40
20 D = 20.000
60 80 100 120 140 160
22 D = 22.000
10
ST xs
10 20 ST 30 24 D = 24.000
xs
40 6. 80 100 120 140 160
ST xs
~PLM 10 30 D = 30.000
ST
20 xs 30
Design
Properties
Surface Area OUtside Inside sq It sq ft per ft per ft A. Ai 4.71 4.29 4.22 4.71 4.71 4.11 4.71 3.99 4.71 3.89 4.71 3.83 4.71 3.78
of
Pipe
(Continued)
Areas and Weights CraneSeetional M&l Flow AWlI Am sq in. sq in. A AI 43.87 210.6 50.23 204.2 61.17 193.3 71.81 182.7 80.66 173.8 168.0 86.48 90.75 163.7
Pipe lb per ft to 149.2 170.8 208.0 244.2 274.3 294.0 308.5
Ib per R WUJ 91.2 88.4 83.7 79.1 75.3 72.7 70.9
Radii of GYiW tion inches rP 6.08 6.04 5.97 5.90 5.84 5.80 5.77
298.6 294.8 291 .o 287.2 283.5 279.8 278.0 276.1 265.2 261.6 239.8 252.7 240.5 238 8 227.0 213 8 209.0 202.7
52.7 65.8 78.6 91.5 104.1 116 8 122.9 129 3 166 4 178.7 184 8 208.9 250.3 256.1 296.4 341.1 357.5 379.1
129.3 127.6 126.0 124.4 122.8 121.2 120.4 119.6 114.8 113.3 112.5 109.4 104.1 103.4 98.3 92.6 90.5 87.8
Weight of
Momeot
16.375 16.126 15.688 15.250 14.876 14.625 14.438
Fifth Power of ID in.’ d6 1177 1090 950 825 728 669 627
,250 .312 .375 .438 ,500 ,562 ,593 ,625 ,812 ,875 ,906 1.031 1.250 1 281 1.500 1.750 1.844 1.968
19.500 19.375 19.250 19.124 19.000 18 875 18.814 18 750 18.376 18.250 18 188 17 938 17 500 17.438 17.000 16 500 16 313 16.064
2 82 2.73 2.64 2.56 2 48 2.40 2 36 2 32 2 10 2 02 1.99 1 86 1 64 1 61 1 42 1 22 1.16 1.07
5.24 5.24 5.24 5.24 5.24 5 24 5 24 5.24 5.24 5 24 5 24 5.24 5 24 5 24 5 24 5 24 5.24 5.24
5.11 5.07 5 04 5.01 4 97 4 94 4 93 4 91 4 81 4 78 4 76 4.70 4.58 4 57 4 45 4 32 4.27 4.21
15.51 19 36 23.12 26 9 30.6 34.3 36.2 38 0 48 9 52 6 54 3 61 4 73 6 75.3 87.2 100.3 105.2 111.5
,250 ,375 ,500
21.500 21 250 21.000
4 59 4.33 4 08
5.76 5 76 5 76
5.63 5.56 5.50
17 1 25.5 33 8
363 355 346
58.1 86.6 114.8
157.2 153 6 150.0
7 69 7.65 7 60
1010 1490 1953
91.8 135.4 177.5
,250 ,312 ,375 .438 ,500 ,562 625 ,687 ,750 ,968 1.031 1.218 1.531 1.812 2.062 2 188 2.343
23 500 23.376 23.250 23.125 23 000 22.876 22.750 22.626 22 500 22.064 21.938 21.564 20.938 20.376 19.876 19.625 19.314
7.17 6.98 6 79 6 61 6 44 6 26 6.09 5.93 5.77 5.23 5.08 4 66 4 02 3.51 3.10 2.91 2.69
6.28 6.28 6.28 6.28 6 28 6.28 6.28 6.28 6 28 6.28 6.28 6.28 6.28 6.28 6.28 6.28 6.28
6.16 6.12 6.09 6.05 6.02 5.99 5.96 5 92 5.89 5.78 5.74 5.65 5.48 5.33 5.20 5.14 5.06
18.7 23.2 27.8 32.4 36.9 41.4 45.9 50 3 54.8 70.0 74.4 87.2 108.1 126.3 142.1 149.9 159.4
434 429 425 420 415 411 406 402 398 382 378 365 344 326 310 302 293
63.4 78.9 94 6 110.1 125.5 140 7 156.0 171.1 186.3 238.1 252.9 296.4 367.4 429.4 483.2 509.7 542.0
187.8 185.8 183.8 181.9 179.9 178.0 176.0 174.1 172.2 165.6 163.7 158.1 149.1 141.2 134 3 131.0 126.9
8.40 8.38 8.35 8.33 8.31 8.29 8.27 8.25 8.22 8.15 8.13 8.07 7.96 7.87 7.79 7.75 7.70
1316 1629 1943 2249 2550 2840 3140 3420 3710 4653 4920 5670 6852 7824 8630 9010 9455
109.6 135 8 161.9 187.4 212.5 237 0 261 285 309 388 410 473 571 652 719 751 788
.375 ,500
25.250 25.000
10 26 9.77
6.81 6.81
6.61 6.54
30.2 40.1
501 491
102.6 136.2
216.8 212.5
9.06 9.02
2479 3257
191 250
,312 ,375 ,438 ,500 ,562 .625
29.376 29.250 29.125 29.000 28.875 28.750
21.9 21.4 21.0 20.5 20.1 19.6
7.85 7.85 7.85 7.85 7.85 7.85
7.69 7.66 7.62 7.59 7.56 7.53
29.1 34.9 40.6 46.3 52.0 57.7
678 672 666 661 655 649
98.9 118.7 138.0 157.6 176.8 196.1
293.5 291.0 288.4 286.0 283.6 281.1
10.50 10.48 10.45 10.43 10.41 10.39
3219 3833 4434 5040 5635 6230
214 255 296 336 37% 415
Wall Tbickinches f 0.812 ,937 1.156 1.375 1.562 1.688 1.781
Inside Diameter inches d
of Inertia in.’ I 1624 1834 2180 2498 2750 2908 3020
180.5 203.8 242.2 277.6 305.6 323.1 335.6
6.98 6 96 6.94 6 92 6.90 6 88 6.86 6 85 F 79 6 77 6.76 6.72 6.64 6.63 6 56 6 48 6.45 6.41
757 938 1114 1289 1457 1624 1704 1787 2257 2409 2483 2772 3251 3316 3755 4217 4379 4586
75.7 93.8 111.4 128.9 145.7 162.4 170.4 178.7 225.7 240.9 248.3 277.2 325.1 331.6 375.5 421.7 437.9 458.6
34 D = 34.000
ST X5
,375 ,540
33.250 33.000
40.6 39.1
8.90 8.90
8.70 8.64
39.6 52.6
868 855
134.7 178.9
376.0 370.3
11.89 11.85
5599 7383
329 434
36 D = 36.000
ST XS
,375 .wo
35.250 35.ow
54.4 52.5
9.44 9.44
9.23 9.16
42.0 55.8
976 962
142.7 189.6
422.6 416.6
12.60 12.55
6659 8786
370 488
ST ,375 41.250 119.4 11.0 10.80 49.0 1336 166.7 578.7 14.72 10621 42 506 11.0 1320 221.6 571.7 B = 42.%0% X6 ,500 41.000 115.9 IO.23 65.2 14.67 14037 668 Thii table ia believed to be the mast nearly complete tabulation of the dimensional properties of commercially ‘available sizes of steel pipe ever published. It includes: the older weights of pipe (ST = standard weight, XS = extra strong, XX = double extra strong), the schedules given id ASA Standard B36.10 (10, 20. etc.). and those given in ASA Standard U3G.19 (5S, IOS, 4OS, 80s). the latter being applicaable to stainless steel only. Piping designers will find herein-all the dimensional data they may need to determine: Pipe wall thickneasea required to resist internal pressure. Bending stresses resulting from line expansion. Bending streesa caused by weight loadings. Pipe column sizes required to sustain given axial loads. Flow areas and fifth powers of the diameter, useful in pressur?edrop calculations. Surface weas for use in evaluating heat losses and insulation and coating requirements. Definition of Properties Listed in Item 2 D - outside diameter of pipe, in. d = inside diameter of pipe, in. I = nominal wall thicknm of pipe, in. A, = ff = outside pipe surface, 8q ft per ft length
A, = $= crws-wtional flow area. sq in. w = 3.4.4 = weight of pipe. lb per ft length 10~ = 0.433Ay = weight of w#ter filling, lb per It lengtn I d/Dz+ - = ~ = radius of gyration, in. A lr 4 I = Ary* = 0.0491(0’ - d’) = moment of inertia in.’
r* = Ai = % = inside pipe surface, sq It per ft length (D* - d*)r A = ~ = crawsectional 4
metal area, aq in.
z = ;= o,og*2 DE++ = section modulus. in.3
i
8 ? I
A P P E N D I X
L
I
A P P E N D I X
STRENGTH OF MATERIALS* ~
c Metals
and
Stress in Thousands of Pounds per Square Inch (Cour~~esy
Tension, IJltimate 15 24-28 30-65 20-35 -1-o-.50
Alloys
Aluminum, cast Aluminum, bars, sheets Aluminum, wire, hard Aluminum, wire, annealed Aluminum, Z-70/, Ni, Cu, Fe, etc. Aluminum Bronze, 5% to 7% v0 Al Aluminum Bronze, 10% Al Brass, Brass, Brass, Brass, Brass, Brass, Brass, Brass,
i5 85-100
8% Sn 13% Sn 20% Sn 24% Sn 30% Sn gun metal, 9% Cu, 1% Sn Manganese, cast 10% Sir Manganese, rolled 1 2% M n Phosphorus, cast 9% Sn Phosphorus, wire I1% P Silicon, cast, 3% Si Silicon, cast, 5% Si Silicon, wire Tobin, cast 38% Zn Tobin. rolled 135% su Tobin, cold-rolled I >i% Pb
28.5 29.4 33 22 5.6 25-55 60 100 50 100 55 75 108 66 80 100 ‘5 X-35 35-65 36
Metal, cast 55-60s Metal, plates W-40% Metal, bars 2- 4% Metal, wire I l- 2 %
Cu Zn Fe SK1
*Courtesy of the Baldwin Loconroti\e
17.1 17.9 6
Works)
compression, [Jltimate 12
Modulus Bending, Ultimate
i2u . 42 73 lli 30
.
23.2 22.3 26.9 39 33.5 20
16’ 19 20 2 2 5.6 10 30 80 24 . .
Shearing, Ultimate 12 .
a.2 7.6 8.6
ii:1 41.1 31 1 a-24 HO 30
Copper, cast Copper, plates, rods, bolts Copper, wire, hard Copper, wire, annealed Delta Delta Delta Delta
Elastic Limit 6.5 12-14 16-30 14 25 40 60
32.6
17% Zn 23% Zn 30% Zn 39% Zn 50% Zn cast, common wire, hard wire, annealed
Bronze, Bronze, Bronze, Bronze, Bronze. Bronze, Bronze, Br.onze, Bronze, Bronze. Bronze, Bronze, Bronze, Bronze, Bronze, Bronze,
of the Baldwin Locomot.ive
I/
. .
43.7 34.5 56.7 32 12.1 52
. . .
. .
. . . .
. .
.
36
9,doo,ooo
. . . . .
4u
. . . . . . . . . .
26.7 35.8 20.7
10,000,QOO
5.5 3.3 0.04
10,ddd,000
... . . . . .
.
.
5.0 22
1 t,ooo,ooo
i25 .
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