Structural Design of Steel Bins and Silos
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The Structural Design of Steel Bins and Silos … August, 01
1
INTRODUCTION
1.1
General
The storage of granular solids in bulk represents an important stage in the production of many substances derived in raw material form and requiring subsequent processing for final use. These include materials obtained by mining, such as metal ores and coal; agricultural products, such as wheat, maize and other grains; and materials derived from quarrying or excavation processes, for example sand and stone. All need to be held in storage after their initial derivation, and most need further processing to yield semi or fullyprocessed products such as coke, cement, flour, concrete aggregates, lime, phosphates and sugar. During this processing stage further periods of storage are necessary. In the Southern African region, with its vast raw material resources, the storage of bulk solids plays an essential part in many industries, including coal and ore mining, generation of electricity, manufacture of chemicals, agriculture, and food processing. The means of storage of these materials is generally provided by large storage vessels or bins, built in steel or reinforced concrete, located at or above ground level.
1.2
Design
The functional planning and structural design of such containers represent specialised skills provided by the engineering profession. Unfortunately there is a lack of comprehensive literature, covering all aspects of bin design, available to the practising engineer. It is the purpose of this publication to present the necessary guidelines to enable the design function to be carried out efficiently and safely, as related to the wide range of typical small, medium and fairly large storage containers or bins built in steel. In the past the design of bins was based on static pressures derived from simple assumptions regarding the forces exerted by the stored material on the walls of the bin, with no allowance for increased pressures imposed during filling or emptying. In the present text, advantage has been taken of a large amount of research work that has been carried out during recent decades in various countries, especially the United States and Australia. It is hoped that the application of the better understanding of flow loads and the analysis of their effects will lead to the design of safer bins and the avoidance of serious and costly failures such as have occurred in the past.
1.3
Terminology
Regarding descriptive terminology applicable to containment vessels, it should be noted that the word "bin" as used in this text is intended to apply in general to all such containers, whatever their shape, ie whether circular, square or rectangular in plan, whether at or above ground level, whatever their height to width ratio, or whether or not they have a hopper bottom. More specific terms, related to particular shapes or proportions, are given below, but even here it must be noted that the definitions are not necessarily precise.
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The Structural Design of Steel Bins and Silos … August, 01
a)
A bin may be squat or tall, depending upon the height to width ratio, Hm D, where Hm is the height of the stored material from the hopper transition level up to the surcharged material at its level of intersection with the bin wall, with the bin full, and where D is the plan width or diameter of a square or circular bin or the lesser plan width of a rectangular bin. Where Hm D is equal to or less than 1,0 the bin is defined as squat, and when greater as tall.
b)
A silo is a tall bin, having either a flat or a hopper bottom.
c)
The hopper transition level of a bin is the level of the transition between the vertical side and the sloping hopper bottom.
d)
A bunker is a container square or rectangular in plan and having a flat or hopper bottom.
e)
A hopper, where provided, is the lower part of a bin, designed to facilitate flow during emptying. It may have an inverted cone or pyramid shape or a wedge shape; the wedge hopper extends for the full length of the bin and may have a continuous outlet or several discrete outlets.
f)
A multicell bin or bunker is one that is divided, in plan view, into two or more separate cells or compartments, each able to store part of the material independently of the others. The outlets may be individual pyramidal hoppers (ie one per cell) or may be a continuous wedge hopper with a separate outlet for each cell.
g)
A groundmounted bin is one having a flat bottom, supported at ground level.
h)
An elevated bin or bunker is one supported above ground level on columns, beams or skirt plates and usually having a hopper bottom.
1.4
Design procedure
The full design procedure for a typical steel bin would comprise a series of activities as described in the ensuing text, but which can be summarised as follows: a)
Assessment of material properties This involves an examination of the stored material with a view to determining its properties as affecting both the functional and the structural design of the bin. The properties include the density of the material, its compressibility, and its angle of internal friction, angle of repose and angle of wall friction. For the majority of stored materials such as ores, coal, grain, etc these properties can be obtained from the tables given in Chapter 2, but for unusual materials or very large silos the properties should be determined from laboratory tests or by reference to specialist materials handling technologists.
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The Structural Design of Steel Bins and Silos … August, 01
b)
Assessment of flow characteristics Based on the material properties mentioned above, it is necessary to determine the flow characteristics of the material and thus determine the optimum shape or geometry of the bin to ensure satisfactory emptying and the prevention of hangups such as arching or bridging. It should be noted that there are three main flow patterns when a bin is being emptied, viz mass flow, funnel flow and expanded flow. These are discussed later, but the particular type of flow applicable to a bin depends both on the geometry of the bin and the flow characteristics of the material. Specialists should be consulted in the case of uncommon or suspect materials.
c)
Functional design of bin The design of the bin from a functional or operating point of view, based on the material characteristics described above, is usually undertaken by material flow technologists. This will involve the selection of the required depth, width and height to accommodate the specified volume of material, the slope of the hopper bottom, location of hopper hip, size and location of outlets, etc. Some guidance is given in chapter 3.
d)
Determination of pressures and forces The normal and frictional forces exerted by the material on the inner surfaces or walls of the bin are determined, considering the dynamic effects during filling, the static effects during storage and the dynamic effects during emptying, plus effects due to temperature, expansion of contents, etc, when present. The magnitude and distribution of the wall forces will depend on the applicable flow mode, the effects of switch pressure in bins with hopper bottoms, and the effects of eccentric discharge where applicable. Pressure diagrams showing the magnitude and distribution of pressure and frictional force are prepared for each inner surface of the bin for the filling and emptying phases, for use in the structural design of the bin.
e)
Structural design The structural design of the bin, including all of its components, can now be carried out, for the various loads and load combinations applicable. Methods are given in the text for the analysis of rectangular and circular bins, bunkers, hoppers and silos, using conventional design practice or more recently developed methods.
1.5
Flow chart
A flow chart depicting the activities described above is given in Fig 1.1 for easy reference. The four main phases, viz (a) assessment of material characteristics, (b) functional design of bin, (c) determination of design loading, and (d) structural design, are clearly identified. The first two activities, may be undertaken by the client or by a – 1.3 –
The Structural Design of Steel Bins and Silos … August, 01
specialist retained by him. The third and fourth activities would be the responsibility of the structural design engineer. (a) MATERIAL FLOW TESTS
(b) FUNCTIONAL DESIGN OF BIN
EXPANDED FLOW
FUNNEL FLOW
MASS FLOW
(c) DESIGN LOADING
FILLING CONDITIONS
EMPTYING CONDITIONS
ECCENTRIC DISCHARGE CONDITIONS
(d) STRUCTURAL DESIGN OF BINS
RECTANGULAR BINS Plating, stiffeners, hoppers, support beams and columns
CIRCULAR BINS Plating, stiffeners, ring beams, columns, hoppers, skirt plates
Fig.1.1 – Flow chart of bin design activities
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The Structural Design of Steel Bins and Silos … August, 01
1.6
Scope of text
The contents of this publication are intended to serve as guidelines for the design of the various types of containment vessel built in steel for the storage of bulk solids, including bins, bunkers, hoppers and silos. The subject matter presented covers the large majority of such vessels of small, medium and fairly large size and of conventional shape, containing materials with known or predictable properties and flow characteristics. It will thus be of assistance in the typical engineering design office and will enable the structural design of bins to be carried out efficiently and safely. As implied in the title of the publication, and as stated above, the text concentrates on the structural aspects of bin design, on the assumption that the functional or operating aspects have been dealt with by a specialist materials flow technologist. It must be emphasized that the text does not cover all aspects of bin design, because of the wide range of variables that may apply in the case of nonstandard material types, bin geometries, etc. Such variables would include eccentric filling and emptying points, asymmetric bin geometry, stored materials having unusual properties, etc. Where any of these unusual circumstances are present, reference should be made to the publications or papers dealing with the particular topic, as quoted in the text. Alternatively advice may be obtained from specialist sources locally, as mentioned in Chapter 8. Finally, it must be stated that the structural design of the bin must be undertaken by persons suitably experienced in this class of work, and especially in the interpretation of the theories and methods employed. The overall responsibility for the structural design must be taken by a registered Professional Engineer.
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The Structural Design of Steel Bins and Silos ... August, 01
2
PROPERTIES OF STORED MATERIALS
2.1
Introduction
Materials stored in bins have their own material flow characteristics which have to be taken into account in the design of the bins and silos. These flow characteristics govern the flow pattern during discharge and the loads on the vertical and hopper walls are governed by the flow pattern. Not taking account of the flow characteristics can lead to improper functioning of the bin, and assumptions of loading conditions which are not concurrent with the flow pattern occurring in the bin during discharge can lead to serious problems. The recommended procedure is to test the material for its flow characteristics, perform the functional or geometrical design, ie establish the desired flow pattern in the bin during discharge conditions, and only then establish all design loads for the structural design. Chapter 4 gives all of the equations necessary to determine the forces on the vertical walls and hopper walls for mass flow and funnel flow conditions, as well as filling (or initial) and emptying (or flow) conditions.
2.2
Material flow tests
In order to establish the flow characteristics of a stored material, a sample of the material is tested by means of specially designed test equipment. In most countries of the world equipment designed by Jenike and Johanson is used, and tests are performed in accordance with the procedures and recommendations developed by them. The test procedures used are outlined in the publications Storage and Flow of Solids, by Dr Andrew W Jenike, Bulletin No 123 of the UTAH Engineering Experiment Station of the University of Utah, Salt Lake City, Utah.
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The Structural Design of Steel Bins and Silos ... August, 01
The following information is obtained from the tests: •
Bulk density, γ;
•
Angle of internal friction, φ;
•
Effective angle of internal friction, δ;
•
Angle of friction between the solid and the wall or liner material, φw.
All of the above values are obtained by test under varying pressures. Additional results may be derived from the tests, but these are not relevant to this guideline because they are mainly used for the functional or geometrical design of a bin or silo. (some guidance is given in chapter 3) A report, reflecting all minimum requirements for continuous gravity flow conditions derived from the test results, can be obtained from bulk solids flow consultants. This report is used for the final geometrical or functional design of the bin, and the chosen geometrical design governs flow patterns and subsequent loading conditions.
2.3
Tables of material properties
Although it is advisable to test materials in order to establish their flow characteristics, tables reflecting typical flow properties of various materials with different moisture contents are provided at the end of this chapter. These tables have been developed from averagedout results derived from numerous tests, and it should be noted that some of these material characteristics show large variances. The data provided should only be used for the loading assessment of small bins with capacities not exceeding about 100 t. In order to eliminate arching, piping and other related flow problems, the functional or geometrical design, ie the design required for proper functioning of the bin, should always be based on test results. For storage facilities with capacities in excess of 100 t, it is highly recommended that the stored material be tested for its flow characteristics prior to the design of the geometrical arrangement or the determination of the loading on vertical and hopper walls.
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The Structural Design of Steel Bins and Silos ... August, 01
2.4
Flow patterns
Bins may be classified into three different types, each type having its relevant vertical and hopper wall loads.
2.4.1 Mass flow bins (Type 1) Mass flow bins are bins in which all of the stored material is in motion during discharge. These bins are especially recommended for cohesive materials, materials which degrade in time, fine powders, and material where segregation causes problems. The smooth, steep hopper wall allows the material to flow along its face and this will give a firstin, firstout pattern for the material. When material is charged into a bin it will segregate, with coarse material located at the wall face and fines in the middle of the bin. When material is discharged from a bin, it will remix in the hopper and segregation is minimised. Fine powders have sufficient time to deaerate and so flooding and flushing of material will be eliminated. Pressures in a mass flow bins are relatively uniform across any horizontal cross section of the hopper. The bins should not have any ledges, sudden hopper transitions, inflowing valleys, and particular care should be taken in assuring flow through the entire discharge opening.
2.4.2 Funnel flow bins or silos (Type 2) A funnel flow bin is a bin in which part of the stored material is in motion during discharge while the rest is stagnant. These bins are suitable for coarse, free flowing, slightly cohesive, nondegrading materials and where segregation is not a problem. The hoppers of these bins are not steep enough to allow material to flow along their face. Material will flow through a central core and this will give a firstin, lastout flow pattern for the material. Flow out of these bins can be erratic, and fine powders can aerate and fluidize. If not properly designed the nonflowing solids might consolidate and a pipe will form through which the material will flow while the rest will remain stagnant.
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The Structural Design of Steel Bins and Silos ... August, 01
2.4.3 Expanded flow bins (Type 3) An expanded flow bin is a combination of a mass flow and a funnel flow bin. The lower part, eg the hopper, forms the mass flow section and the upper part, ie the vertical walled section, represents the funnel flow section. These bins are used especially for large storage capacities and where multiple outlets are required. The flow patterns of the three types of bin are illustrated in Figure 2.1.
Type 1 Mass Flow
Type 2 Funnel Flow
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Type 3 Expanded Flow
The Structural Design of Steel Bins and Silos … August, 01
3.
ASSESSMENT OF FLOW CHARACTERISITCS AND FUNCTIONAL DESIGN
3.1
Introduction
The design of the bin from a functional or operating point of view, based on the material characteristics described in chapter 2, is usually undertaken by material flow technologists. This involves the selection of the required depth, width and height to accommodate the specified volume of material, the slope of the hopper bottom, location of the hopper hip, size and location of the outlets. The engineer should never take responsibility for the functional design of the bin unless he/she is qualified to do so. It is better to pass this responsibility back to the client who will employ a material flow technologist, or employ a material flow technologist himself after discussion with the client.
3.2
Typical flow problems
There are a number of flow problems of which the designer should be aware. These are summarised as follows: No Flow condition A stable arch forms over the discharge opening or a pipe (rathole) forms within the bulk solid above the hopper. This is caused by either the cohesive strength of the material or by the mechanical interlocking of the larger particles. Erratic flow Momentary arch formation/collapse within the bulk solid or partial/total collapse of a rathole. Flushing Mainly a problem with powders which in funnel flow conditions aerate, fluidise and flush resulting in spillage, no control at the feeder and quality problems down the line due to irregular feed. Inadequate capacity Due to rathole formation or hangups in poorly designed hoppers a large proportion of the material remains dead in the silo, reducing the live capacity to a fraction of the total volume and requiring severe hammering, prodding or mechanical vibration to restore flow of the material in the dead regions.
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The Structural Design of Steel Bins and Silos … August, 01
Segregation The different particle sizes within the bulk solid tend to sift through eachother causing accumulation of fine particles in the centre of the storage facility and coarse particles around it. This problem causes serious effects on product quality and plant operation for certain process applications Degradation Spoilage, caking, or oxidation may occur within bulk solids during handling and when kept in a silo for too long a period. In firstinlastout flow conditions through a silo (Funnel flow), some material may be trapped within the silo for extended periods and will only come out when the silo is completely emptied. Spontaneous combustion Certain combustible bulk solids (coal, grains, sponge iron etc) subject to firstinlastout flow conditions, where pockets of material are trapped for extended periods, may be subject to spontaneous combustion with disastrous consequences. Vibrations Vibrations caused by solids flow can lead to serious structural problems. Structural failure Drag forces on silo walls can exceed the buckling strength of the silo walls. This is covered in more detail in chapter 5.
3.3
Variables affecting solids flowability
Before geometrical design of a silo commences, it is essential that the flow characteristics of the bulk solid have been established and the conditions the material will be subjected to inside the silo under operating conditions are adequately defined. Variables affecting the flow of bulk solids include: Consolidating Pressure The magnitude of surcharge loads exerted by the material inside the silo has a significant effect on the flowability of the material because it increases mechanical interlocking and cohesive arch formation. Moisture Content The flow of bulk solids is generally affected by the surface moisture content up to 20% of the saturation point. Temperature Some bulk solids are affected by temperature or variation in temperature, such as thermoplastic powders or pellets. Chemical composition Chemical reaction of materials stored in a silo may change the flow characteristics of the material – 3.2 –
The Structural Design of Steel Bins and Silos … August, 01
Relative humidity Hygroscopic materials are particularly sensitive to conditions of high relative humidity with significant effect on flowability of the material, e.g. burnt lime, fertiliser, sugar etc. Time under consolidation Materials subject to consolidation pressure for extended periods of time may compact with a resulting decrease in flowability. Strain rate Bulk solids with a viscous component need to be testes at various strain rates to determine the effect on flow properties. ( Carnallite harvested from dead sea brines). The majority of bulk solids are however not strain rate sensitive. Gradation Particle size distribution and in particular fines content in many bulk solids can have a significant effect on flowability of the material particularly if moisture is present Effect of liner materials Friction angles of the material against the liner change from one type of liner to another.
3.4
Flow Testing
In addition to the testing of basic material properties such as bulk density, angle of wall friction etc , specific tests can be done to determine the flowability of a material. These tests are beyond the scope of this guideline. Facilities for flowability testing of bulk solids and the expertise for analysis and interpretation of the results are available at Bulk Solids Flow S.A .
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The Structural Design of Steel Bins and Silos … August, 01
3.5
Determination of Mass and Funnel flow
The following curves have been taken from the Institution of Engineers Australia “Guidelines for the Assessment of Loads on Bulk Solids Containers” Please note that they are to be used as a guide and do not provide absolute values.
Figure 3.1
The boundaries between mass flow and funnel flow (Coefficient of wall friction vs Half hopper angle)
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The Structural Design of Steel Bins and Silos ... August, 01
4
LOADING
4.1
Introduction
This chapter deals with the various live loads to which a typical bin structure is subject. These may be summarised as follows: •
Loads from stored materials: filling or initial loads; emptying or flow loads.
•
Loads due to eccentric discharge conditions.
•
Loads from plant and equipment.
•
Loads from platforms and bin roofs.
•
Internal pressure suction
•
Wind loads.
•
Effects of solar radiation
•
Settlement of supports
4.2
Classification of bins — Squat or tall
Regarding the loads imposed by the stored material, bins may be classified as squat or tall, depending on their ratio of height to diameter or width. In the material loading equations given later a distinction is made between the load intensities applicable to squat bins and tall bins respectively. A squat bin is defined as one in which the height from the hopper transition to the level of intersection of the stored material with the wall of the bin is less than or equal to the diameter of a circular bin, or the width of a square bin, or the lesser plan dimension of a rectangular bin. A tall bin is one in which this height is greater than the above limit. This is illustrated in Figure 4.1. 4.3
Loads from stored materials
The loadings applied by the stored material to the inner surfaces of a bin are based on various theories, applicable to the initial and flow conditions and relating to the walls of squat and tall bins and the hoppers, respectively. This is indicated in the following sections.
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4.1 —
The Structural Design of Steel Bins and Silos ... August, 01
D
Hm
Hm
D
(a) Squat bin Hm ≤ D
(b)Tall bin Hm ≤ D
(c) Plan Shapes
Fig 4.1: Bin classification – Squat or tall In all cases the pressures normal to the surfaces are obtained from the calculated vertical pressures by use of a factor K, which is the ratio of horizontal to vertical pressure. This factor is dependent on the effective angle of internal friction δ, and since the latter has upper and lower limits for each type of stored material, K also has maximum and minimum values. The wall loads are furthermore dependent on the coefficient of friction µ between the material and the vertical wall and hopper of the bin. This value also has upper and lower limits for each type of stored material and type of bin wall or lining material. 4.3.1
Loads on vertical walls of squat bins
The method used for determining the loads during the filling or initial condition is based on the Rankine theory. The maximum K and µ values derived from the lower limits for δ and .φ are used. The minimum K and µ values are used to obtain maximum loads on the hopper walls and in cases where internal columns are used, to obtain extreme maximum and minimum loads on these structural members. For the emptying or flow condition the maximum K and µvalues derived from the upper limits for δ and .φ are used.
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4.2 —
The Structural Design of Steel Bins and Silos ... August, 01
4.3.2
Loads on vertical walls of tall bins
For the filling or initial condition, the Janssen theory is used for load assessment. The maximum K and µ values, derived from the lower limits for δ and .φ’, apply. For the emptying or flow condition, the Jenike method, based on strain energy, is used. The wall loads depend on the flow pattern, viz mass or funnel flow (see section 2.4). For this condition the maximum K and µ values, derived from the upper limits for δ and .φ, apply. 4.3.3
Loads on walls of mass flow hoppers
Walker's theory is used in determining loads during the filling or initial stage. Maximum K and µ values, derived from the lower limits for δ and .φ’, apply. For the emptying or flow condition, the Jenike method is used, with maximum values of K and µ. derived from the upper limits for δ and .φ’, apply. During flow an overpressure occurs on the hopper wall just below the transition, which has a peak value at the transition level and extends downwards in a diminishing triangular pattern for a distance of about 0,3 times the top width of the hopper (see section 3.6.2). This localised pressure intensity is also referred to as 'switch pressure'. 4.3.4
Loads on walls of funnel flow hoppers
The methods used here, including the K and µ values, are the same as for mass flow hoppers, except that no overpressure occurs. 4.3.5
Examples of bin shapes and types of flow
Examples of various combinations of bin shape and type of flow are illustrated in Figure 3.2. The bins are shown as either squat or tall, and the hopper wall slopes are either steep (for mass flow of the contents during emptying) or not so steep (for funnel flow). Also shown are bins having flat bottoms with hoppers having plan shapes occupying less than the plan area of the bin (examples 4, 5 and 6); these shapes apply mainly to concrete bins with slab bottoms, with either steel or concrete hoppers. In all cases the design of the bin and hopper walls would require consideration of the initial or filling condition and the flow or emptying condition, the latter being either the mass flow or the funnel flow condition.
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The Structural Design of Steel Bins and Silos ... August, 01
Remarks
H
Shape
_ _
H D >1 The hopper is steep enough to allow material to flow along its face
D
This is a MASS FLOW SILO Hopper and vertical wall to be designed for mass flow conditions.
H
_ _
H D 1 Hopper top diameter smaller than the silo diameter
H D >1 The hopper valley angles are steep enough to allow materil to flow along its face. Both hoppers are operational at the same time
This is a MASS FLOW SILO Hopper and vertical wall to be designed for mass flow conditions.
Fig4.2a: Examples of bin shapes and types of flow
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The Structural Design of Steel Bins and Silos ... August, 01
Shape
Remarks
H
_
H D >1 Hopper one is not steep enough to allow material to flow along its face
_
D
_
Hopper two is steep enough to allow flow along its face.
This is an EXPANDED FLOW BIN Vertical wall to be designed for funnel flow Hopper 1 to be designed for funnel flow Hopper 2 to be designed for mass flow. H B >1 Hopper valley angles are steep enough to allow material to flow along the faces
H

B
This is an EXPANDED FLOW SILO The hoppers to be designed for mass flow, and vertical wall to be funnel flow conditions.

The hoppers are steep enough to allow material to flow along their faces Both hoppers are operational at the same time. (This is to prevent stable rat holing or piping in the stockpile.)

This is an EXPANDED FLOW SYSTEM The hoppers shall be designed for mass flow conditions.  The hopper is steep enough to allow material to flow along its face. This is an EXPANDED FLOW SYSTEM The hopper shall be designed for mass flow conditions.
Fig 4.2b: Examples of bin shapes and types of flow
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The Structural Design of Steel Bins and Silos ... August, 01
4.4
Equations for loading on walls – Introduction
Equations for the determination of the forces acting on the inner surfaces of the vertical walls and hopper walls of bins are given in parts 3.5 and 3.6 of this chapter. The sequence of the clauses and subclauses is summarised in the following table, for easy reference. 4.5 Loads on vertical walls
4.6
4.5.1
Initial loading
4.5.2 4.5.3 4.5.4
Flow loading Mass flow loading Funnel flow loading
Squat bins Tall bins Squat bins Tall bins Tall bins
Loads on hopper walls 4.6.1 4.6.2 4.6.3
Initial loading Tall bins Mass flow loading Tall bins Funnel flow loading Tall bins
Squat bins Squat bins Squat bins
The symbols used in the equations are defined in the list given at the beginning of the book. The dimensional symbols are illustrated in the figure following the list. Values of the hydraulic radius R for hoppers of different shapes and types are given in Table 4.1.
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4.6 —
The Structural Design of Steel Bins and Silos ... August, 01
Table 4.1: Values of hydraulic radius R for hoppers (For surcharge calculations)
Hopper
Silo
Silo
Silo
Type A
Type B
Type C
Type D
Di 4
Di 4
Da 4
L x Ba 2(L +Ba )
Mass flow
Di 4 = Dc 4
Dc 4
Dc 4
Dc 4
Funnel flow
Di 4 = Dc 4
Dc 4
Dc 4
Dc 4
Da 4
Da 4
Di 4
L x Ba 2(L +Ba )
Mass flow
Da 4 = Db 4
Db 2 4
Db 2 4
Db 2 4
Funnel flow
Da 4 = Db 4
Db 2 4
Db 2 4
Db 2 4
L x Ba 2(L +Ba )
Da 4
Di 4
L x Ba 2(L +Ba )
0,25 La2 + Bb2
0,25 La2 + Bb2
0,25 La2 + Bb2
0,25 La2 + Bb2
0,25 La2 + Bb2
0,25 La2 + Bb2
Condition Initial
Conical
Initial Square
Initial
Rectangular
Mass flow
L x Ba = 2(L +Ba ) La x Bb 2(L a +Bb )
Funnel flow
L x Ba = 2(L +Ba ) La x Bb 2(L a +Bb )
Silo
The characteristic hopper dimensions Db, Dc, Bb and La are illustrated in Figure 4.3. Note: For silo Type B, C and D material is flowing through a channel with a diameter equal to the top diameter of a conical hopper or the diagonal of square or rectangular hopper.
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The Structural Design of Steel Bins and Silos ... August, 01
4.5 Loads on vertical walls
4.5.1
Initial loading
Squat bins Ph = γ1 h K2
(4.5.1)
where K2 is the greatere of:
a) 0,400 b)
c)
1 − sin δ2 1 + sin δ2 1 − sin2 δ2 1 + sin2 δ2 Sv = µ2 Ph
(4.5.2)
—
4.8 —
Ha
Ha
hi
hi
hi
hi
Ha
H Ha
The Structural Design of Steel Bins and Silos ... August, 01
TYPE B
TYPE C
TYPE D
Dc
Dc
Dc
Dc
Db
Db
Db
Db
Bb
Bb
Bb Bb
La
La
La
La
RECTANGULAR HOPPERS SQUARE HOPPERS CONICAL HOPPERS
TYPE A
Fig 4.3: Characteristic hopper dimensions for different bin shapes and hopper types
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The Structural Design of Steel Bins and Silos ... August, 01
Tall bins Ph =
(
γ1R 1 − e − µ 2K 2 h R µ2
)
(4.5.3)
Di for circular bins 4 D = a for square bins 4 L Ba for rectangular bins = 2(L + Ba )
where R =
Sv = µ2 Ph
(4.5.4)
4.5.2 Flow loading, squat bins Ph = γ 1 h K 1
(4.5.5)
where K1 is the greater of: a) 0,400 b) c)
1 − sin δ1 1 + sin δ1
1 − sin2 δ1 1 + sin2 δ1 Sv = µ1 Ph
(4.5.6)
4.5.3 Mass flow loading, tall bins
For horizontal pressure Ph: M =
2(1 − ν )
(4.5.7)
where ν = 0,3 for axisymmetric flow
N
= 0,2 for plane flow 2ν = µ1 M 2(1−m )
(4.5.8)
where m = 0 for plane flow = 1 for axisymmetric flow Kh = x
=
ν 1− ν µ1 Mm R
(4.5.9)
( H − h)
(4.5.10)
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The Structural Design of Steel Bins and Silos ... August, 01
(
)
So =
1 1 − e− µ1K1 h R µ1 K1
A
=
− K h Mm − 1 (So − N) e− x + Mm µ1−1 − K h N K h Mm + 1 e x − K h Mm − 1 e− x
B
= So − N − A
Ph =
(
(
)
)
(4.5.11)
( )
(
)
(4.5.12) (4.5.13)
γ1R µ 1 − ( A − B ) 1m µ1 M
(4.5.14)
In calculating the horizontal pressure Ph from the top of the vertical wall down wards, a maximum value will be reached somewhat below mid point of the vertica wall. This value shall be used for the remaining part of the vertical wall. For frictional force U kN per linear m circumference: M = N
=
2(1 − ν )
(4.5.7)
2ν µ1 M 2(1− m )
(4.5.8)
ν 1− ν µ1 H = m M R
Kh = x
(
(4.5.15)
)
(
− K h M m − 1 (− N )e − x + M m µ1 − K h N K h M m + 1 e x − K h M m − 1 e −x
(
A
=
B
= –A–N
U
=
=
(4.5.9)
γ 1D2 H
− 4 D
γ 1 LBa
)
(
−1
)
)
(4.5.16) (4.5.17)
Ae x + Be − x + N for circular and square bins 4
(
LBa H − Ae x + Be − x + N 2(L + Ba ) 2(L + Ba )
) for rectangular bins
= Di = diameter of circular bin = Da = width of square bin Ba = width of rectangular bin L = length of rectangular bin
where D
— 4.11 —
(4.5.17a)
(3.5.17b)
The Structural Design of Steel Bins and Silos ... August, 01
4.5.4 Funnel flow loading, tall bins For horizontal pressure Ph: θ
D for circular and square bins = tan−1 2(H1 − h)
(4.5.19a)
Ba for rectangular bins = tan−1 2(H1 − h)
(4.5.19b)
where D = Di = diameter of circular bin = Da = width of square bin
(
)
β
= 0,5 φ w′ 1 + sin −1 (cos φ w′ 1 )
(4.5.20)
x
2 m sinδ 1 sin (2β + θ ) + 1 = 1 − sinδ 1 sinθ
(4.5.21)
y
=
(2 {1 − cos(β + θ)})m
(β + θ)1−m sin θ + sin β ⋅ sin1+m(β + θ) (1 − sin δ1) sin2+m (β + θ)
where (β + θ )
1− m
q
=
Ka =
(4.5.22)
is in radians
2 y(tanθ +sinδ 1 ) − 1 24sinθ ( x − 1)sinθ
(4.5.23)
(24 tan θ + π q) (1 − sin δ1 tan θ) 16 (sin δ1 + tan θ)
(4.5.24)
π
Ph = K a
(
γ1 R 1 − e−µ1 K1h R µ1 K1
)
(4.5.25)
(
(γ x R ) The minimum pressure at the outlet, Ph = 1 i − eµ1K1H R µ 1
) .
In calculating the horixontal pressures Ph from the top downwards, a maximum value will be reached. For the pressure calcultions, a straight line pressure diagram can be adopted from the maximum achieved pressure downwards to the minimum pressure at the outlet. For frictional force U (kN per linear metre circumference) (As for mass flow loading in 3.5.3): M =
2(1 − ν )
(4.5.7)
—
4.12 —
The Structural Design of Steel Bins and Silos ... August, 01
N
=
2ν
(4.5.8)
µ1M2(1−m )
ν 1− ν µ1 H = m M R
Kh = x
(
(4.5.9) (4.5.15)
)
(
=
B
= –A–N
U
=
=
(
)
− K h Mm − 1 (− N)e − x + Mm µ1−1− K h N K h Mm + 1 e x − K h Mm − 1 e− x
A
)
(4.5.16) (4.5.17)
γ 1 D2 H
− 4 D
γ1LBa 2(L + Ba )
) (
for circular and square bins
Ae x + Be − x + N 4
(
(4.5.18a)
)
LBa H − Ae x +Be− x + N for rectangular bins (4.5.18b) 2(L + Ba )
where D = Di = diameter of circular bin = Da = width of square bin Ba = width of rectangular bin L = length of rectangular bin
4.6
Loads on hopper walls
Note: In sections 3.6.1, 3.6.2 and 3.6.3 below, α is the half hopper angle, ie the inclination of the hopper wall to the vertical (for rectangular hoppers, α = inclination of wall under consideration, ie either side wall or end wall of hopper).
4.6.1 Initial loading For normal pressure Pn: K
= K min = the greater of
n
=
α
= half hopper angle
tan α tan φ′h2 + tan α
(m + 1) K min 1 + tanφ h 2 − 1 ′
tan α
or 0,400
(4.6.1) (4.6.2)
where m = 0 for plane flow = 1 for axisymmetric flow
— 4.13 —
The Structural Design of Steel Bins and Silos ... August, 01
n h − z ho ho − z o Pn = γ1K min + hc − n − 1 ho n −1 1 Qc where hc = γ 1 Ac
(4.6.3)
(4.6.4)
ho = based on section
Qc = γ 1 Ha for squat bins Ac γ1R = 1 − e − µ 2 K 2 H R for tall bins µ2 K 2
)
(
(4.6.5a) (4.6.5b)
For values of R for hoppers see Table 3.1 For shear force Sh: Sh = µh2 Pn
(4.6.6)
4.6.2 Mass flow loading For normal pressures n t and n tr: β
sin φ h′1 = 0,5 φ h′1 + sin −1 sin δ 1
x
=
y
=
2 m sin δ 1 sin(2 β + α ) + 1 1 − sin δ 1 sin α
(4.6.8)
(2 {1 − cos (β + α)})m (β + α)1−m sin α + sin β sin1+m (β + α) (1 − sin δ1) sin2 +m (β + α)
where (β + α )
1− m
n tr =
(4.6.7)
(4.6.9)
is in radians
y 1 + sinδ 1 cos2β x − 1 2sin α
γ 1 D
(4.6.10)
where D = Dc, Db, Bb or La, as applicable; see Fig 3.3. For rectangular hoppers, Bb is used when considering the long sides of the hopper and La when considering the ends. q
m 1 2n tr π (tanα + tanφ′h1) − 1 = 0,25 1+ m 3 tanα γ1D
—
4.14 —
(4.6.11)
The Structural Design of Steel Bins and Silos ... August, 01
m Q 4 3,3 c − qγ1D Ac π n t = n tr + (sinα +cosα tanφ′h1) (2 − 0,4sinα )m
where
Qc Ac
(4.6.12)
= surcharge at top of hopper = γ 1 Ha for squat bins
(4.6.13a)
=
γ1R (1 − e −µ K H R ) for tall bins Type A µ1 K 1
=
for tall bins γ1R 1 − e − tan φ1K1Ha R Types B, C and D tanφ1K1
1
1
(
(4.6.13b)
)
(4.6.13c) For values of R see Table 3.1 For distribution of pressures see figure at right. Note: For bins of Types B, C and D the material flows through a channel with diameter Dc. For square and rectangular bins it flows through a channel with a diameter equal to the diagonal of the top shape of the hopper. For shear forces Sh: Sh = µh1 n tr
(4.6.14)
Sh = µh1 n t
(4.6.15)
4.6.3 Funnel flow loading For normal pressure Pn:
Ph = γ 1 K1 (H a + h1 ) for squat bins =
γ1R ( 1 − e −µ K ( H µ1
=
γ1R ( 1 − e − tanφ K ( H tanφ1
1
1
1
a + h1
1
)
R
) for tall bins Type A
a + h1
)
R
) for tall bins Types B, C and D
sin 2α 4r Pn = Ph +cos 2α + µ h1sinα cosα D K1
(4.6.16a) (4.6.16b) (4.6.16c)
(4.6.17)
where r = horizontal distance from centre of hopper to point on hopper wall where pressure Pn applies (see below), and
D = Dc, Db, Bb or La, as applicable; see Figure 3.3
For rectangular hoppers, Bb is used when considering the long sides of the hopper and La when considering the ends.
— 4.15 —
The Structural Design of Steel Bins and Silos ... August, 01
For values of R see Table 3.1. For shear force Sh: 1 2r Sh = Ph − 1sinα cosα + µ h1 (cos 2α − sin 2α ) (4.6.18) D K1
Switch Pressures Switch pressures are only occurring where mass flow hopper meets with the vertical wall of an overall man flow silo, so where a mass flow hopper is a part of an expanded flow design, there are no switch pressures occurring. Some judgement in the calculation and use of switch pressure should also be taken in account as with very steep hoppers, the switch pressures tend to be very high. The judgement should be based on a vertical wall design approach, taking account of the hopper loads with a modified switch pressure.
— 4.16 —
The Structural Design of Steel Bins and Silos ... August, 01
4.7
Eccentric discharge
When the discharge opening at the bottom of a circular bin is displaced laterally in plan from the vertical centroidal axis of the bin, eccentric discharge conditions are introduced. The material flows through an eccentric channel as shown in Figure 4.4. The ratio of the horizontal pressure in the flow channel to the horizontal pressure in the rest of the bin is in direct proportion to that of the radii of the flow channel and the bin respectively, ie Po Ph = r R (Ref ...A W Jenike). Using Jenike's moment equations, the moment per unit length due to eccentric discharge is M = K R2 P where K =
(4.7.1)
sin 2θ tanθ sinθ 1 − π cos(φ w′ 1 − θ )
(4.7.2)
R
= radius of bin θ = eccentricity angle θ′w1 = maximum angle of friction between material and wall P
= normal pressure
The value of θ recommended for use in the above equation is 21º, although larger values may occur. Because of the large difference between the pressures Po and P, deformation of the cylindrical shell in plan tends to occur, and strengthening of the shell becomes necessary. For this reason, eccentric discharge outlets should be avoided if at all possible in circular bins.
— 4.17 —
The Structural Design of Steel Bins and Silos ... August, 01
Fig4.4: Eccentric discharge of circular bins 4.8
Corrugated steel sheet bins
Circular bins or silos made from corrugated steel sheets (with the crests and valleys of the corrugations running circumferentially) are usually mounted on flat concrete bases, and so are subject to funnel flow during emptying. The vertical friction forces at the walls are not generated by the sliding of the contents against the walls, but by the sliding of the contents against the static material trapped in the corrugations. The coefficient of friction is therefore not µ but tan δ, where δ is the effective angle of internal friction of the material. Thus in calculating lateral pressures Ph and frictional forces Sv and U for the vertical walls under initial and emptying conditions, equations (4.5.2), (4.5.3), (4.5.4), (4.5.6), (4.5.18) and (4.5.25) may be used, but with the effective angle of internal friction δ substituted for φ′w , and the tangent of this angle substituted for µ.
— 4.18 —
The Structural Design of Steel Bins and Silos ... August, 01
4.9
Wind loading
The wind loading on bin structures can be assessed by reference to SABS 0160 (Ref ...), where force and pressure coefficients are given for structures of square, rectangular and circular shape in plan, for various height to width ratios. Since wind loading is usually only significant in tall bins, and as such bins are often located in unprotected sites, it is recommended that the terrain be assumed as category 2. Wind loading on square or rectangular bins is usually not critical (but must of course be allowed for), because the bin shape is inherently stable and stiff, and has properly stiffened plate elements. Circular bins, on the other hand, are very sensitive to wind loading because of the varying pressure/suction distribution of the wind loading around the circumference, and the lack of stiffness of the shell in resisting this loading. The required thickness of plate in the upper strakes of a circular bin is often determined by the wind loading. Wind buckling is characterised by the formation of one or more buckles on the windward face of the shell. Wind also produces an overturning moment on a tall bin, which induces a vertical compressive stress in the leeward face; this reached a maximum at the base of the bin, where the shell needs to be checked against buckling. The distribution of pressure around a cylindrical structure is given in Table 14 of SABS 0160, in terms of external pressure coefficients Cpe. Force coefficients, for calculating the total wind force on the bin, are given in Table 1 of the code for circular structures and in Figure 6 for square and rectangular structures. The great majority of circular bins exposed to the weather are furnished with covers or roofs, which serve the dual purpose of protecting the interior of the bin and of maintaining the circular shape of the top of the shell. In the case of a bin exposed to wind loading and having an open top, however, internal suction forces are generated that aggravate the nonuniform loading pattern referred to above. Such bins are much more subject deformation, and require special consideration to cater for this severe form of loading. What has been stated above applies to single or isolated bins. Where a row or group of closelyspaced circular bins is located across the wind direction the wind resistance per bin is much higher than if the bins were widely spaced because the free flow of air around each bin is inhibited. Where a single row of bins is located
— 4.19 —
The Structural Design of Steel Bins and Silos ... August, 01
parallel to the wind direction the windward bin would probably be subject to wind loadings as determined above, but the downwind bin or bins would be largely shielded by the windward one. It is not possible to suggest actual load factors for these conditions because of the number of variables involved and advice should be sough from wind loading specialists if wind loading is thought to be critical.
4.10 Loading from plant and equipment Items such as pumps, blowers, filters, conveyor head pulleys and drive units, etc, are often mounted on the roofs of storage bins. The loading imposed can usually be catered for quite simply in the design of the roof support beams, but there are certain aspects of conveyor loading that need special attention. If the conveyor belt tensions at the head pulley are to be resisted by the bin (ie if the tensions are not carried back into the conveyor stringers), then the bin roof structure will need to be proportioned to resist this extra loading and the bin as a whole be checked for the overturning effects. Likewise if the conveyor is housed in a gantry and the head end of the gantry is supported on the top of the bin, the bin structure should be designed to cater for all of the conveyor loading components, including side wind on the gantry. A situation to be specially allowed for is where the gantry (or series of gantries) is anchored at its lower end and is not provided with a sliding bearing at its support on the bin roof. Here, differential thermal expansion of the bin caused by solar radiation on one side of the bin will result in horizontal displacement of the top, which in turn will induce a compressive or tensile force in the gantry structure, with a corresponding horizontal reaction at the top of the bin. Tall circular bins are particularly sensitive to these effects. A suitable means of avoiding the above situation is to have the gantry head end supported on sliding bearings and for the conveyor belt tensions to be transmitted back into the gantry; in this way only vertical loading will be applied to the bin.
4.11 Effects of solar radiation All bins in exposed situations are subject to the effects of solar radiation as described above, even where conveyor loading is not present. If it is necessary to investigate this aspect, it is suggested that the temperature of the wall exposed to the sun be taken as 40ºC above the ambient shade temperature.
— 4.20 —
The Structural Design of Steel Bins and Silos ... August, 01
4.12 Live loads on roofs and platforms Where the top cover of a bin serves simply as a roof and not as a platform (ie where it is nontrafficable), the live loading may be taken as specified for roofs in SABS 0160, Clause 5.4.3.3, ie a distributed load varying from 0,3 kPa to 0,5 kPa depending on the loaded area, or a point load of 0,9 kN, whichever is more severe. For trafficable roofs the loading may be taken as given in Clause 5.4.3.2, ie a distributed load of 2,0 kPa or a point load of 2,0 kN. If material spillage or excessive dust collection is a possibility it should be allowed for in addition to the above loading. The live loading on access platforms and stairways in industrial structures is not specified in the code, but it would be good practice to allow for a distributed load of 3,0 kPa or a point load of 3,0 kN.
4.13 Internal pressure suction In the case of bins having pneumatic discharge systems, positive internal pressures are generated by the blowers, but as safety vents are usually provided the full blower pressure is not likely to be realised. The maximum pressure exerted should be obtained from the supplier of the system and the pressure acting on localised areas of the bin wall be taken as say 80% of the specified pressure. Rapid discharge of bulk solids having low permeability to gases can cause negative air pressure in a bin. Circular bins, and especially their upper parts (including the roof), are particularly sensitive to this effect. Safety vents may be installed to limit the negative pressure, but in any case the pressure required to open the vent should be ascertained.
4.14 Settlement of supports Most bin structures, especially cylindrical ones, are very stiff in the vertical direction because of their great depth and fullyplated construction. Consequently, settlement of one support point — whether a beam, a column or a foundation — may induce high stresses in the shell structure and also cause a redistribution of load on the remaining supports. In the extreme case of the complete failure of one column — say due to vehicle impact — under a bin supported on four columns, the load on each of the two remaining loadbearing columns is doubled.
— 4.21 —
The Structural Design of Steel Bins and Silos ... August, 01
Even relatively small settlements of foundations can cause significant redistribution of load at the remaining supports, and it would therefore be prudent to introduce an overload factor for these.
4.15 Load combinations When designing bin structures by the limitstate method, the partial load and load combination factors as laid down in SABS 0160, Table 2, should be used, but certain variations as mentioned below may be advisable. Since the bulk density of the stored material is usually welldefined, and in any case its upper limit value is used in design, a partial load factor γi of 1,3, as specified for stored fluids, would seem reasonable for this material when at rest, eg in the design of the support. But since the maximum material loading may well be present when other live loads are active, the load combination factor Ψi should be taken as 1,0. Thus where the effects of initial material loading and wind loading, for example, are cumulative, the partial load and load combination factors would be taken as 1,3 and 1,0 respectively, for both the material load and the wind load. For the emptying or flow condition, however, a Ψi factor of 1,6 on the material loading would be advisable. On the other hand, where the effects are not cumulative, the material load or the wind load combination factor would be taken as zero, as applicable. Suggested values of partial load and load combination factors for the various types of load are given in Table 4.2.
— 4.22 —
The Structural Design of Steel Bins and Silos ... August, 01
Table 4.2: Partial load and load combination factors, ultimate limit state. Partial load factor γi
Load combination factor Ψi
Maximum, acting in isolation
1,5
—
Maximum, acting in combination
1,2
1,0
Minimum
0,9
1,0
Gravity (material at rest)
1,3
1,0
Initial (filling) condition
1,6
1,0
Flow (emptying) condition
1,6
1,0
1,5
1,0
Dead load
1,5
1,0
Live load
1,6
1,
Loads from internal external pressure in bin
1,6
1,0
Wind load
1,3
0
Loads from vehicle impact
1,3
0
Loads from differential settlement of supports
1,3
0
Type of load
Loads from selfweight of structure
Loads from stored material:
Dead loads from plant and equipment Loads from conveyors:
— 4.23 —
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