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Helsinki University of Technology Laboratory of Steel Structures Publications 15 Teknillisen korkeakoulun teräsrakennetekniikan laboratorion julkaisuja 15 Espoo 2000
Seminar on Steel Structures: DESIGN OF COLD-FORMED STEEL STRUCTURES
Jyri Outinen, Henri Perttola, Risto Hara, Karri Kupari and Olli Kaitila
TKK-TER-15
Helsinki University of Technology Laboratory of Steel Structures Publications 15 Teknillisen korkeakoulun teräsrakennetekniikan laboratorion julkaisuja 15 Espoo 2000
Seminar on Steel Structures: DESIGN OF COLD-FORMED STEEL STRUCTURES
Jyri Outinen, Henri Perttola, Risto Hara, Karri Kupari and Olli Kaitila
Helsinki University of Technology Department of Civil and Environmental Engineering Laboratory of Steel Structures
Teknillinen korkeakoulu Rakennus- ja ympäristötekniikan osasto Teräsrakennetekniikan laboratorio
TKK-TER-15
Distribution: Helsinki University of Technology Laboratory of Steel Structures P.O. Box 2100 FIN-02015 HUT Tel. +358-9-451 3701 Fax. +358-9-451 3826 E-mail:
[email protected]
Teknillinen korkeakoulu ISBN 951-22-5200-7 ISSN 1456-4327 Otamedia Oy Espoo 2000
FOREWORD This report collects the papers contributed for the Seminar on Steel Structures (Rak83.140 and Rak-83.J) in spring semester 2000. This time the Seminar was realized as a joint seminar for graduate and postgraduate students. The subject of the Seminar was chosen as Design of Cold-Formed Steel Structures. The seminar was succesfully completed with clearness in presentations and expert knowledge in discussions. I will thank in this connection all the participants for their intensive and enthusiastic contribution to this Report.
Pentti Mäkeläinen Professor, D.Sc.(Tech.) Head of the Laboratory of Steel Structures
DESIGN OF COLD-FORMED STEEL STRUCTURES
CONTENTS 1
Profiled Steel Sheeting…...…………………………………………………………1 J. Outinen
2
Design of Cold Formed Thin Gauge Members………………….……………….14 R. Hara
3
Design Charts of Single-Span Thin-Walled Sandwich Elements……….………34 K.Kupari
4
Numerical Analysis for Thin-Walled Structures……….………………………..45 H Perttola
5 Cold-Formed Steel Structures in Fire Conditions…………………………….….65 O. Kaitila
1
PROFILED STEEL SHEETING Jyri Outinen Researcher, M.Sc.(Tech) Laboratory of Steel Structures Helsinki University of Technology P.O. Box 2100, FIN-02015 HUT - Finland Email:
[email protected] (http://www.hut.fi/~joutinen/)
ABSTRACT The ligthness of cold-formed thin-walled structures was formerly their most important feature and therefore they were used mostly in products where the weight saving was of great importance, This kind of products were naturally needed in especially transportation industries e.g. aircrafts and motor industry. A wide range of research work during many decades has been conducted all over the world to improve the knowledge about the manufacturing, corrosion protection, materials and codes of practise of thin-walled steel structures. This has led to a constantly increasing use of cold-formed thin-walled structures. Profiled steel sheeting is used in various kind of structures nowadays. In this paper, a short overview of the manufacturing, products, materials and structural design of profiled steel sheeting is given. Also a short overview of some current research projects is given.
KEYWORDS Profiled steel sheeting, sheet steel, cold-formed, thin-wall, corrugated, steel, structural design, steel materials, cladding, roof structures, wall structures, floor structures.
2 INTRODUCTION
There is a wide range of manufacturers making different kind of profiled steel sheeting products. The manufacturing processes have beensignificantly developed and different shapes of sheeting profile are easy to produce. Steel sheeting is also easy to bend to different shapes e.g. curved roof structures., cylindrical products e.g. culvers etc. The products are delivered with a huge range of possible coatings. Normally the coating is done by the manufacturer and so the products are ready to be used when delivered. Cold-formed steel sheeting can be used to satisfy both structural and functional requirements. In this paper, the structural use is more thoroughly considered. Profiled steel sheeting is widely used in roof, wall and floor structures. In these structures, the profiled steel sheeting actually satisfies both the structural and functional requirements. In floor structures the steel sheeting is often used as part of a composite structure with concrete. In northern countries the roof and wall structures are almost always built with thermal insulation. The sound insulation and the fire insulation have also to be considered, when designing structures. There are several codes for the design of profiled steel sheeting. Almost every country has a national code for this purpose, e.g. DIN-code in Germany, AISI-code in USA, etc. The structural design of profiled steel sheeting in Europe has to be carried out using the Eurocode 3: part 1.3, though there are several national application documents (NAD), where the national requirements are considered with the EC3. An extensive amount of tests has been carried out and analyzed to gather together the existing design codes, and there are numerous formulae in these codes that are based partly on theory and partly on experimental test results. Some of the important aspects of structural design of coldformed profiled steel sheeting is presented in this paper. Numerous different kind of fastening techniques are developed suitable for thin-walled structures. Suitable fasteners are bolts with nuts, blind rivets, self tapping screws, selfdrilling screws and some other kinds of fasteners. The materials used in cold-formed thin-wall members have to satisfy certain criteria to be suitable for cold-forming and usually also for galvanising. The yield strength is normally in the range of 220…350 N/mm2 , but also some high-strength sheet steels with yield strength of over 500 N/mm2 are used in some cases. The practical reasons i.e. transportation, handling etc., limit the range of thickness of the material used in profiled sheeting. A lot of interesting research projects have been carried out concerning the behaviour of profiled steel sheeting all over the world. Some of the current researches are shortly described in this paper. In different parts of the world the focus of the research is naturally on the regional problems. An example of this is Australia, where the main research area of cold-formed steel structures is concentrated on the problems caused by high-wind and storm loads.
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DEVELOPMENT OF THE PROFILED SHEETING TYPES
The profiled sheeting types have been developed significantly since the first profiled steel sheets. The first plates were very simple and the stiffness of these was not very high. The manufacturing process and the materials limited the shape of the profiles to simply folded or corrugated shapes. The height of the profile was roughly in between 15 and 100 mm. Two types of typical simply profiled steel sheet forms are illustrated in figure 1.
Figure 1: Simple forms of profiled steel sheeting
From the early 1970's the shape of the profiling in steel sheeting developed considerably. This naturally meant possibilities for their widerange usage especially in structural uses. The stiffeners were added to flanges of the profile and this improved notably the bending resistance. The maximum height of the profile was normally still under 100mm. In Figure 2 a profile with stiffeners in flanges is illustrated.
Figure 2: More advanced form of profiled steel sheeting. Stifferners in flanges.
From the mid 1970's, the development of the shapes of sheeting profiles and also better materials and manufacturing technologies lead to possibilities to provide more complex profiles. This improved substancially the load-bearing capacities of the developed new profiled steel sheets. In figure 3 is shown an example of this kind of more complex profile.
4
Figure 3: Modern form of profiled steel sheeting. Stiffeners in flanges and webs. A huge range of profile types are available nowadays used for structural and other kind of purposes. The thin-walled steel structures and profiled steel sheeting is an area of fast growth. In the next chapter, a few typical examples where cold-formed profiled steel sheeting is used are presented.
USE OF PROFILED STEEL SHEETING IN BUILDING Cold-formed profiled sheeting is able to give adequate load bearing resistance and also to satisfy the functional requirements of the design. This aspect is considered in this chapter briefly in relation to the common usage of cold-formed sheeting in floor, wall and roof structures. Floor structures Profiled steel sheeting in floor structures have sheeting, e.g. trapezoidal or cassettes, as load bearing part, either alone or in composite action with other materials such as different kind of board, plywood decking or cast in-situ concrete. In the first case, the composite action is provided by adhesives, and mechanical fasteners, in the second by means of indentation and/or special shear studs. The bending moment resistance is the main requirement, and so the profiles used for flooring purposes are similar to those for roof decking.
Figure 4: A Steel-concrete composite floor slab with profiled steel sheeting
5 Wall structures In wall structures, the structure is comprised of an outer layer, the facade sheeting that is usually built with relatively small span, and a substructure which transmits the wind loading to the main building structure. The substructure can be a system of wall rails or horizontal deep profiles, or cassettes with integrated insulation. Another solution combines the load-bearing and protecting function in a sandwich panel built up by metal profiles of various shapes and a core of polyurethane or mineral wool.
Figure 5: A facade made with profiled steel sheeting
Roof structures The roof structures using steel sheeting can be built as cold or warm roofs A cold roof has an outer waterproof skin with internal insulation if required. The main requirement of preventing the rain water or the melting snow leads to shallow profiles with a sequence of wide and narrow flanges. Sheets fixed using fasteners applied to the crests or the valleys of the corrugations.
Figure 6: A roof structure made with profiled steel sheeting of a subway station in Finland
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The use of few points of fastening means that the forces are relatively high and therefore the spans are usually quite small. A wide range of special fasteners have been developed to avoid the failure of the fasteners or the sheeting e.g. pull-through failure at that point. This is a problem in especially high-wind areas, e.g. Australia. Warm roof includes insulation and water proofing and it is built up using a load-bearing profile, insulation and an outer layer e.g. metal skin, as mentioned before. The loadbearing profiled sheeting in this type of roof normally has the wider flanges turned up in order to provide sufficient support for the insulation. Fasteners are placed in the bottom of the narrow troughs. In this case, the tendency is towards longer spans, using more complex profiles of various shapes and a core of polyurethane.
Other applications The highly developed forming tecnology makes it possible to manufacture quite freely products made of profiled steel sheets with various shapes. Such are for example curved roof structures., cylindrical products e.g. culvers etc. There are not too much limitations anymore concerning the shape of the product. In Figure X. a few examples of this are presented.
Figure 7: Profiled sheet steel products in different shapes
7 MANUFACTURING
Cold-formed steel members can be manufactured e.g. by folding, press-braking or coldrolling. Profiled steel sheets are manufactured practically always using cold-forming. Also the cylindrical products are manufactured by cold cold rolling from steel strips. In figure 8, a steel culvert and a profiled steel sheet is manufactured by cold-rolling.
Figure 8: Cold-rolling process of profiled steel products
Cold-rolling technique gives good opportunities to vary the shape of the profile and therefore it is easy to manufacture optimal profiles that have adequate load bearing properties for the product. The stiffeners to flanges and webs are easily produced. During the cold-forming process varying stretching forces can induce residual stresses. These can significantly change the load-bearing resistance of a section. Favourable effects can be observed if residual stresses are induced in parts of the section which act in compression and, at the same time, are susceptible to local bucling. Cold-forming has significant strain-hardening effects on ductility of structural steel. Yield strength, ultimate strength and the ductility are all locally influenced by an amount which depends on the bending radius, the thickness of the sheet, the type of steel and the forming process. The average yield strength of the section depends on the number of corners and the width of the flat elements. The principle of the effect of cold-forming on yield strength is illustrated in Figure 9.
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Figure 9: Effect of cold forming on the yield stress of a steel profile
STRUCTURAL DESIGN OF PROFILED STEEL SHEETING
Because of the many types of sheeting available and the diverse functional requirements and loading conditions that apply, design is generally based on experimental investigations. This experimental approach is generally acceptable for mass produced products, where optimization of the shape of the profiles is a competitive need. The product development during about four decades has been based more on experience of the functional behaviour of the behaviour of the products than on analytical methods. The initial "design by testing" and subsequent growing understanding of the structural behaviour allowed analytical design methods to be developed. Theoretical or semiempirical design formulae were created based on the evaluation of test results. This type of interaction of analytical and experimental results occurs whenever special phenomena are responsible for uncertainties in the prediction of design resistance (ulimate limit state) or deformations (serviceability limit state). At the moment there are several codes for the structural design of cold-formed steel members. In Europe, Eurocode 3: Part 1.3 is the latest design code which can be used in all european countries. Still, almost every country has a national application document (NAD), in which the former national code of practice is taken into account. In Other parts of the world e.g. in USA (AISI-specifications), Australia, (AS) there are several different codes for the design. All the design codes seem to have the same principles, but the design practices vary depending on the code.
9 The design can basicly be divided in two parts: 1.) Structural modelling and analysis which is normally quite a simple procedure and 2.) Checking the resistances of the sheeting. The values that are needed in the design are: moment resistance, point load resistance and the effective second moments of area Ieff corresponding to the moment resistances. The deflections have to be also considered. The deflections during construction e.g. in steel-concrete composite floors are often the limiting factor to the structure. The load-bearing properties, i.e. moment resistance, point load resistance etc., are almost always given by the manufacturer. Profiled sheeting has basically the following structural functions: 1. To transfer the surface loads (wind, snow,etc.) to the substructure. 2. To stabilise the substructure and the components of it. 3. Optionally to transfer the in-plane loads (e.g. wind load in roofs to the end cables) "Stressed skin design" One important weak point of profiled steel sheeting is the low resistance against transverse point loads as mentioned earlier. The reason is that the load is transmitted to the webs as point loads that create high stress peaks to it. The web is then very vulnerable to lose the local stability at these points. All the manufactures have recommendations for the minimum support width, which has a notable effect on the previous phenomenon. The fire design of cold formed structures is basicly quite simple using the existing codes, but the methods are under new consideration in various research projects, from which a short description is given in chapter "Current research projects". The design for dynamic loading cases is constantly under development in countries, where the wind and storm loads are of high importance. For example in Australia, a large amount of experimental research has been carried out on this subject. Most of this research is concentrated on the connections. Different types of fasteners have been developed to avoid the pull-through, pull-over or pull-out phenomena under dynamic high-wind loading.
Figure 10: Examples of pull-through failures under dynamic loading. Local pull-through by splitting and fatigue pull-through (high-strength steel).
10 MATERIALS
The most common steel material that is used in profiled steel sheets is hot dip zinc coated cold-formed structural steel. The nominal yield strength Reh (See Fig. 4) is typically 220…550N/mm2 . The ultimate tensile strength is 300…560 N/mm2 . The modulus of elasticity is normally 210 000 N/mm2 . The mechanical properties of low-carbon coldformed structural steels have to be in accordance with the requirements of the European standard SFS-EN 10 147. The mechanical properties are dependent on the rolling direction so that yield strength is higher transversally to rolling direction.. In the inspection certificate that is normally delivered with the material, the test results are for transversal tensile test pieces. In Figure 4, typical stress-strain curves of cold-formed structural sheet steel with nominal yield strength of 350 N/mm2 at room temperature both longitunidally and transversally to rolling direction are shown. The difference between the test results for the specimens taken longitudinally and transversally to rolling direction can clearly be seen. The results are also shown in Tables 1 and 2. 450 Transversally to rolling direction
400 350
Longitudinally to rolling direction
Stress σ [N/mm 2]
300 250 200 150 100 50 0 0
0.2
0.4
0.6
0.8
1 1.2 Strain ε [%]
1.4
1.6
1.8
2
Figure 11: Stress-strain curves of structural steel S350GD+Z at room temperature. Tensile tests longitudinally and transversally to rolling direction
11 TABLE 1 MECHANICAL PROPERTIES OF THE TEST MATERIAL S350GD+Z AT ROOM TEMPERATURE. TEST PIECES LONGITUDINALLY TO ROLLING DIRECTION Measured property Modulus of elasticity E Yield stress R p0.2 Ultimate stress Rm
Mean value (N/mm2 ) 210 120 354.6 452.6
Standard deviation (N/mm2 ) 13100 1.5 2.3
Number of tests (pcs) 5 5 5
TABLE 2 MECHANICAL PROPERTIES OF THE TEST MATERIAL S350GD+Z AT ROOM TEMPERATURE. TEST PIECES TRANSVERSALLY TO ROLLING DIRECTION Measured property Modulus of elasticity E Yield stress Rp0.2 Ultimate stress Rm
Mean value (N/mm2 ) 209400 387.5 452.5
Standard deviation (N/mm2 ) 8800 1.3 1.9
Number of tests (pcs) 4 4 4
The thickness of the base material that is formed to profiled steel sheets is normally 0.5…2.5 mm. The thickness can't normally be less than 0.5 mm. If the material is thinner than that, the damages to the steel sheets during transportation, assembly and handling are almost impossible to avoid. The thickness of the sheet material is not normally over 2.5 mm because of the limitations of the roll-forming tools. The base material coils are normally 1000…1500 mm wide and that limits the width of profiled steel sheets normally to 600…1200 mm. Steel is naturally not the only material that profiled sheeting is made of. Other materials, such as stainless steel, aluminium and composite (plastic) materials are also widely used. Stainless steel products are under development all the time and the major problem seems to be the hardness of the material, i.e. there are problems in roll-forming, cutting and drilling. On the other hand, excellent corrosion resistance and also fire resistance give it big advances. Aluminium profiles are easy to roll-form and cut because of the softness of the material. On the other hand, the ductility is quite restricted, especially at fire conditions. The composite (plastic) materials are also widely used e.g. in transparent roofs, but not usually in structural use.
12 CURRENT RESEARCH WORK A wide range of different kind of research activities concerning profiled steel sheeting is going on in several countries. Most of the studies are based on both experimental test results and usually also modelling results produced with some finite element modelling programs. Usually the aim is to increase the load-bearing capacity of the studied product. Also the materials, coatings and the manufacturing technology are developed constantly. In Finland, there are a lot of small resarch projects concerning the steel-concrete composite slabs with profiled steel sheeting. In these projects, which are mainly carried out in Finnish universities, e.g. Helsinki University of Technology, and in the Technical Research Centre of Finland, the aim is simply to increase the load bearing capacity. This is studied using different profiles and stud connectors. The experiments are normally bending tests, but also some shear tests for the connection between steel sheeting and concrete with push-out tests have ben carried out. During the next few years, several research projects are starting in Finland concerning the design of lightweight steel structures. In these projects, the fire design part is of great importance. In Australia, e.g. in Queensland University of Technology, and also in several other universities, there are numerous on-going research projects concerning mainly the behaviour of the connections of steel sheeting under wind-storm loads e.g. "Development of design and test methods for profiled steel roof and wall claddings under wind uplift and racking loads" and "Design methods for screwed connections in claddings." are recently completed projects. The current situation can be found on their www-site (given in next chapter: References). In these projects a significant amount of small-scale and also large scale tests have been conducted. The small-scale tests are usually carried out to study the pull-out or pull-over phenomena of screwed connections. The large-scale tests aim to study the behaviour of the profiled steel sheeting in wall and roof structures under high-wind loading cases. The research work that is carried out concerning cold-formed steel in USA can be found from the American Iron and Steel Institutes www-site (given in next chapter: References). In this paper, just a few examples of the research work that is currently going on were mentioned. Different research programs concerning the cold-formed profiled steel sheeting are going on in Europe and other parts of the world. A major conference, "The International Specialty Conference on Recent Research and Developments in ColdFormed Steel Design and Construction", where the latest research projects are presented regurarly, is held in St. Louis, Missouri.
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REFERENCES Eurocode 3, CEN ENV 1993-1-3 Design of Steel Structures- Supplementary rules for Cold Formed Thin Gauge Members and Sheeting, Brussels, 1996 Standard SFS-EN 10 147 (1992): Continuously hot-dip zinc coated structural steel sheet and strip. Technical delivery conditions. (in Finnish), Helsinki Outinen, J. & Mäkeläinen, P.: Behaviour of a Structural Sheet Steel at Fire Temperatures. Light-Weight Steel and Aluminium Structures (Eds. P. Mäkeläinen and P. Hassinen) ICSAS'99. Elsevier Science Ltd., Oxford, UK 1999, pp. 771-778. Kaitila O., Post-graduate seminar work on "Cold Formed Steel Structures in Fire Conditions", Helsinki University of Technology, 2000. Helenius, A., Lecture in short course: "Behaviour and design of light-weight steel structures" , at Helsinki University, 1999 Tang, L.,Mahendran, M., Pull-over Strength of Trapezoidal Steel Claddings, . LightWeight Steel and Aluminium Structures (Eds. P. Mäkeläinen and P. Hassinen) ICSAS'99. Elsevier Science Ltd., Oxford, UK 1999, pp. 743-750.
ESDEP Working Group 9 Internet-sites concerning cold-formed steel: http://www.rannila.fi http://www.rumtec.fi http://www.civl.bee.qut.edu.au/pic/steelstructures.html http://www.steel.org/construction/design/research/ongoing.htm
14
DESIGN OF COLD FORMED THIN GAUGE MEMBERS
Risto Hara M.Sc.(Tech.) PI-Consulting Oyj Liesikuja 5, P.O. BOX 31, FIN-01601 VANTAA, FINLAND http://www.pigroup.fi/
15 INTRODUCTION In this presentation, cold formed thin gauge members (for simplicity: ‘thin-walled members’) refer to profiles, which the design code Eurocode 3 Part 1.3 (ENV 1993-1-3) is intended for. These profiles are usually cold rolled or brake pressed from hot or cold rolled steel strips. Due to the manufacturing process, sections of cold formed structural shapes are usually open, singly-, point- or non-symmetric. Most common cross-section types of thin-walled members (U, C, Z, L and hat) are shown in Figure 1.1, see ref. (Salmi, P. & Talja, A.). Other forms of sections i.e. special single- and built-up sections are shown e.g. in ENV 1993-1-3, Figure 1.1.
Figure 1.1 Typical cross-section types of thin-walled members. Thin-walled structural members have been increasingly used in construction industry during the last 100 years. They are advantageous in light-weight constructions, where they can carry tension, compression and bending forces. The structural properties and type of loading of thin-walled members cause the typical static behaviour of these structures: the local or global loss of stability in form of different buckling phenomena. To have control of them in analysis and design, sophisticated tools (FEA) and design codes (ENV 1993-1-3, AISI 1996, etc.) may have to be used. Unfortunately, the complexity of these methods can easily limit the use of thin-walled structural members or lead to excessive conservatism in design. However, some simplified design expressions have been developed, see refs. (Salmi, P. & Talja, A.), (Roivio, P.). The main features of the design rules of thin-walled members are described in this paper. The present Finnish design codes B6 (1989) and B7 (1988) are entirely omitted as inadequate for the design of cold formed steel structures. However, the viewpoint is ‘Finnish-European’, i.e. the main reference is the appropriate Eurocode 3 (ENV 1993-1-3) with the Finnish translation (SFS-ENV 1993-1-3) and National Application Document (NAD). The paper concentrates on the analytical design of members omitting chapters 8-10 of the code (ENV 1993-1-3) entirely. Reference is made also to a seminar publication (TEMPUS 4502), where theory and practice for the design of thin-walled members is presented in a comprehensive way. The reference contains also a summary of Eurocode 3 – Part 1.3.
ABOUT THE STRUCTURAL BEHAVIOUR OF THIN-WALLED MEMBERS The cross-sections of thin-walled members consist usually of relatively slender parts, i.e. of flat plate fields and edge stiffeners. Instead of failure through material yielding, compressed parts tend to loose their stability. In the local buckling mode, flat plate fields buckle causing
16 displacements only perpendicular to plane elements and redistribution of stresses. In this mode the shape of the section is only slightly distorted, because only rotations at plane element junctures are involved. In the actual distortional buckling mode, the displacements of the cross-section parts are largely due to buckling of e.g. flange stiffeners. In both buckling modes, the stiffness properties of the cross-section may be changed, but the member probably still has some post-buckling capacity since translation and/or rotation of the entire crosssection is not involved. In the global buckling mode, displacements of the entire cross-section are large, leading to over-all loss of stability of the member. Global buckling modes depend primarily on the shape of the cross-section. Flexural buckling usually in the direction of minimum flexural stiffness is common also for cold formed members. Low torsional stiffness is typical for open thin-walled members,so buckling modes associated with torsion may be critical. Pure torsional buckling is possible for example in the case of a point symmetric crosssection (e.g. Z-section), where the centre of the cross-section and the shear centre coincide. In torsional buckling, the cross-section rotates around the shear centre. A mixed flexuraltorsional buckling mode, where the cross-section also translates in plane, is possible in the case of single symmetric cross-sections (e.g. U, C and hat). Due to the low torsional stiffness of open thin-walled cross-sections, lateral buckling is a very probable failure mode of beams. Analogy with flexural buckling of the compressed flange is valid in many cases, but does not work well with low profiles bent about the axis of symmetry or with open profiles bent in the plane of symmetry, when the folded edges are compressed (e.g. wide hats). Naturally, plastic or elastic-plastic static behaviour of compressed or bended members are possible when loaded to failure, but with normal structural geometry and loading, stability is critical in the design of thin-walled members. Structural stability phenomena are described in more detail e.g. by (Salmi, P. & Talja, A). BASIS OF DESIGN In cold formed steel design, the convention for member axes has to be completed compared with Structural Eurocodes. According to ENV 1993-1-3, the x-axis is still along the member, but for single symmetric cross-sections y-axis is the axis of symmetry and z-axis is the other principal axis of the cross-section. For other cross-sections, y-axis is the major axis and z-axis is the minor axis, see also Figure 1.1. According to the ENV code, also u-axis (perpendicular to the height) and v-axis (parallel to the height) can be used ”where necessary”. Depending on the type of contribution to the structural strength and stability, a thin-walled member belongs to one of two construction classes. In Class I the member is a part of the overall stiffening system of the structure. In Class II the member contributes only to the individual structural strength of the element. The Class III is reserved for secondary sheeting structures only. However, this classification for differentiating levels of reliability seems not to have any influence in design. In ultimate limit states (defined in ENV 1993-1-1), the value of partial safety factors (γM0 and γM1) needed in member design are always equal to 1.1. Factor γM0 is for calculation of cross-section resistance caused by yielding and factor γM1 is for calculation of member resistance caused by buckling. The serviceability limit states are defined in form of principles and application rules in ENV 1993-1-1 and completed in ENV 1993-1-3 with the associated Finnish NAD. The partial factor in both classes γMser has a value equal to 1.0.
17 The design of adequate durability of cold formed components seems to require qualitative guide lines according to base code ENV 1993-1-3, but also much more exact specifications according to the NAD. The structural steel to be used for thin-walled members shall be suitable for cold forming, welding and usually also for galvanising. In ENV 1993-1-3, Table 3.1 lists steel types, which can be used in cold formed steel design according to the code. Other structural steels can also be used, if the appropriate conditions in Part 1.3 and NAD are satisfied. In ENV 1993-1-3 Ch. 3.1.2, exact conditions have been specified about when the increased yield strength fya due to cold forming could be utilised in load bearing capacity. Fortunately for the designer, Ch. 3.1.2 has been simplified in the NAD: nominal values of basic yield strength fyb shall be applied everywhere as yield strength (hence in this paper fyb is replaced in all formulas by fy ). This can be justified, because on the average, the ratio fya /fyb ≈ 1.05 only. Normally yield strengths fyb used in thin-walled members lay in the range 200-400 N/mm2 , but the trend is to even stronger steels. TABLE 3.1 TYPICAL STRUCTURAL STEELS USED IN COLD FORMED STEEL STRUCTURES .
18 Obviously, other material properties relevant in cold formed steel design are familiar to designers: e.g. modulus of elasticity E = 210 000 N/mm2 , shear modulus G = E/2(1+ν) N/mm2 = 81 000 N/mm2 (Poisson’s ratio ν = 0.3), coefficient of linear thermal elongation α = 12 × 10-6 1/K and unit mass ρ = 7850 kg/m3 . The draft code ENV 1993-1-3 is applicable only for members with a nominal core thickness of 1.0 < tcor < 8.0 mm. In the Finnish NAD, however, the material thickness condition is changed: 0.9 < tcor < 12.0 mm. Up to 12.5 mm core thickness is reached in roll-forming process in Finland by Rautaruukki Oy. The nominal core thickness can normally be taken as tcor = tnom – tzin where tnom is the nominal sheet thickness and tzin is the zinc coating thickness (for common coating Z275 tzin = 0.04 mm).
Figure 3.1: Determination of notional widths. Section properties shall be calculated according to normal ‘good practice’. Due to the complex shape of the cross-sections, approximations are required in most cases. Specified nominal dimensions of the shape and large openings determine the properties of the gross crosssection. The net area is reached from gross area by deducting other openings and all fastener holes according to special rules listed in Ch. 3.3.3 of the Eurocode. Due to cold forming, the corners of thin-walled members are rounded. According to the design code, the influence of rounded corners with internal radius r ≤ 5 t and r ≤ 0.15 bp on section properties may be neglected, i.e. round corners can be replaced with sharp corners. The notional flat width bp is defined by applying the corner geometry shown in Figure 3.1, extracted from the code. If the above limits are exceeded, the influence of rounded corners on section properties ‘should be allowed for’. Sufficient accuracy is reached by reducing section properties of equivalent cross-section with sharp corners (subscript ‘sh’) according to the formulas: Ag ≈ Ag,sh (1-δ)
(3.1a)
Ig ≈ Ig,sh (1-2δ)
(3.1b)
Iw ≈ I w,sh (1-4δ),
(3.1c)
19 Where Ag is the area of the gross cross-section, Ig is the second moment area of the gross cross-section and I w is the warping constant of the gross cross-section. Term δ is a factor depending on the number of the plane elements (m), on the number of the curved elements (n), on the internal radius of curved elements (rj) and notional flat widths bpi according to the formula: n
δ = 0.43
m
∑ rj / ∑ bpi , j=1
(3.2)
i=1
This approximation can be applied also in the calculation of effective cross-section properties. Due to the chosen limits, typical round corners can usually be handled as sharp corners. In order to apply the design code ENV 1993-1-3 in design by calculation, the width-thickness ratios of different cross-section parts shall not exceed limits listed in Table 3.2. In conclusion, they represent such slender flat plate fields that the designer has rather ‘free hands’ in the construction of the shape of the cross-section. However, to provide sufficient stiffness and to avoid primary buckling of the stiffener itself, the conditions 0.2 ≤ c/b ≤ 0.6 and 0.1 ≤ d/b ≤ 0.3 for the edge stiffener geometry shall be satisfied. TABLE 3.2 M AXIMUM WIDTH- TO-THICKNESS RATIOS OF PLATE FIELDS .
20 LOCAL BUCKLING One of the most essential features in the design of thin-walled members is the local buckling of the cross-section. The effects of local buckling shall be taken into account in the determination of the design strength and stiffness of the members. Using the concept of effective width and effective thickness of individual elements prone to local buckling, the effective cross-sectional properties can be calculated. The calculation method depends on e.g. stresslevels and -distribution of different elements. The code ENV 1993-1-3 Cl. 4.1. (4-6) states that in ultimate resistance calculations, yield stress fy ‘should’ be used (on the ‘safe side’) and only in serviceability verifications, actual stress-levels due to serviceability limit state loading ‘should’ be used. Thus the basic formulas for effective width calculations of flat plane element without stiffeners in compression could be presented in the general form, in accordance with the complex alternative rules of ENV code, compare to (Salmi, P. & Talja, A.): ρ = 1, when λp ≤ 0.673
(4.1a)
ρ = (λp – 0.22) / λp 2 , when λp > 0.673
(4.1b)
λp = √ (σc / σel) = 1.052 (bp / t) √ (σc / E / kσ)
(4.1c)
σel = kσ π2 E / 12 / (1 - ν 2 ) / (bp / t)2 ,
(4.1d)
where ρ is the reduction factor of the width, λp relative slenderness, bp width, σc maximum compressive stress of the element and k σ buckling factor. For compressed members σc is usually the design stress (χfy ) based on overall buckling (flexural or flexural-torsional). For bent members, in an analogical way, σc is usually the design stress for lateral buckling (χfy ). In special cases, σc really can have the value fy in compression or bending. Obviously, the safe simplification σ c = fy may always be used and to avoid iterations, it is even recommended. The reduction factor ρ shall be determined according to Table 4.1 for internal and Table 4.2 for external compression elements, respectively. The design of stiffened elements is based on the assumption that the stiffener itself works as a beam on elastic foundation. The elasticity of the foundation is simulated with springs, whose stiffness depends on the bending stiffness of adjacent parts of plane elements and the boundary conditions of the element. A spring system for basic types of plate fields needed in analysis is shown in Table 4.3. The determination of spring stiffness in two simple cases is presented in Figure 4.1. For example, in the case of an edge stiffener, the spring stiffness K of the foundation per unit length is determined from: K = u / δ, where δ is the deflection of the stiffener due to the unit load u:
(4.2)
21 δ = Θ bp + u bp 3 / 3 ⋅12 (1 - ν 2 ) / (E t3 ),
(4.3)
Typically for complex tasks, it is not shown in the code how to calculate exactly the rotational spring constant C Θ required in the formula Θ = u bp 3 / CΘ. The spring stiffness K can be used to calculate the critical elastic buckling stress σcrS: σcrS = 2 √ (K E Is) / As,
(4.4)
where Is is the effective second moment of area of the stiffener taken as that of its effective area As. In the simplified method of (Salmi, P. & Talja, A.), Is and As have been replaced by their full-cross sectional dimensions in consistence of general principles in calculation of elastic buckling forces. The general iterative as well as simplified procedures according to the code to determine the effective thickness of the stiffener teff are in their complexity hard to apply in practical design. Hence, only the simplified, conservative method of (Salmi, P. & Talja, A.) is presented here: teff = χS t,
(4.5)
where χ S is the reduction factor for the buckling of a beam on an elastic foundation. The factor is determined according to the buckling curve a0 (α = 0.13, see also Figure 6.1) from the equations: χS = 1, when λs ≤ 0.2
(4.6a)
χS = 1 / ( φ + √ (φ 2 - λs2 )), when λs > 0.2
(4.6b)
λs = √ (σc / σcrS )
(4.6c)
φ = 0.5 [1 + α (λ - 0.2) + λs2 ],
(4.6d)
In this study, distortional buckling is considered as a local stability effect. This buckling mode is included in clause 6 of ENV 1993-1-3, where design rules for global buckling are introduced. Distortional buckling is handled only qualitatively in the design code, without any equations. Implicitly it may mean, that FEA is required to be used to analyse this buckling mode in design. However, if in the case of a section with edge or intermediate stiffeners the stiffener is reduced according to the code, no further allowance for distortional buckling is required. Fortunately, distortional buckling mode should not be very probable in thin-walled members with ‘normal’ dimensions.
22 TABLE 4.1 DETERMINATION OF EFFECTIVE WIDTH FOR INTERNAL P LATE FIELDS.
23
24 TABLE 4.3 MODELLING OF ELEMENTS OF A CROSS-SECTION.
Figure 4.1 Determination of spring stiffness in two simple cases.
25 LOCAL RESISTANCE OF CROSS-SECTIONS Axial tension The design value of tension Nsd shall not exceed the corresponding resistance of the crosssection NtRd : Nsd ≤ NtRd = fy Ag / γM0 ≤ FnRd,
(5.1)
where FnRd is the net-section resistance taking into account mechanical fasteners. Axial compression The design value of compression Nsd shall not exceed the corresponding resistance of the cross-section NcRd : Nsd ≤ NcRd = fy Ag / γM0, when Aeff = Ag
(5.2a)
Nsd ≤ NcRd = fy Aeff / γM1, when Aeff < Ag
(5.2b)
In the equations Aeff is the effective area of the cross-section according to section 4 by assuming a uniform compressive stress equal to fy / γM1. If the centroid of the effective crosssection does not coincide with the centroid of the gross cross-section, the additional moments (N sd ⋅ eN ) due to shifts eN of the centroidal axes shall be taken into account in combined compression and bending. However, according to many references this influence can usually be considered negligible. Bending moment The design value of bending moment Msd shall not exceed the corresponding resistance of the cross-section McRd : Msd ≤ McRd = fy Wel / γM0, when Weff = Wel
(5.3a)
Msd ≤ McRd = fy Weff / γM1, when Weff < Wel
(5.3b)
In the equations Weff is the effective section modulus of the cross-section based on pure bending moment about the relevant principal axis yielding a maximum stress equal to fy / γM1. Allowance for the effects of shear lag to the effective width shall be made, if ‘relevant’ (normally not). The distribution of the bending stresses shall be linear, if the partial yielding of the cross-section can not be allowed. In case of mono-axial bending plastic reserves in the tension zone can generally be utilised without strain limits. The utilisation of plastic reserves in the compression zone is normally more difficult because of several conditions to be met. The
26 procedures to handle cross-sections in bending have been explained e.g. in the code ENV 1993-1-3 and in the paper (Salmi, P. & Talja, A.). For biaxial bending, the following criterion shall be satisfied: MySd / McyRd + MzSd / MczRd ≤ 1,
(5.4)
where MySd and MzSd are the applied bending moments about the major y and minor z axes. McyRd and MczRd are the resistances of the cross-section if subject only to moments about the major or minor axes. Combined tension or compression and bending Cross-sections subject to combined axial tension Nsd and bending moments MySd and MzSd shall meet the condition: Nsd / (fy Ag / γM) + MySd / (fy Weffyten / γM) + MzSd / (fy Weffzten / γM) ≤ 1,
(5.5)
where γM = γM0 or = γM1 depending on Weff is equal to Wel or not for each axis about which a bending moment acts. Weffyten and Weffzten are the effective section moduli for maximum tensile stress if subject only to moments about y- and z-axes. In the ENV code there is also an additional criterion to be satisfied, if the corresponding section moduli for maximum compressive stress Weffycom ≥ Weffyten or Weffzcom ≥ Weffzten. The criterion is associated with vectorial effects based on ENV 1993-1-1. Cross-sections subject to combined axial compression Nsd and bending moments MySd and MzSd shall meet the condition: Nsd / (fy Aeff / γM) + MySd / (fy Weffycom / γM) + MzSd / (fy Weffzcom / γM) ≤ 1,
(5.6)
where the factor γM = γM0 if Aeff = Ag, otherwise γM = γM1. In the case Weffycom ≥ Weffyten or Weffzcom ≥ Weffzten, an additional criterion has again to be satisfied. In this occasion, reference is also made to the basic steel code ENV 1993-1-1 for the concept of vectorial effects. For simplicity, in the expression above the bending moments include the additional moments due to potential shifts of the centroidal axes. Torsional moment In good design practice of thin-walled open members, torsional effects should be avoided as far as practicable, e.g. by means of restraints or ideal cross-sectional shape. If the loads are applied eccentrically to the shear centre of the cross-section, the effects of torsion “shall be taken into account”. The effective cross-section derived from the bending moment defines the
27 centroid as well as the shear centre of the cross-section. Probably, design problems will be expected, because the following criteria have to be satisfied: σtot = σN + σMy + σMz + σw ≤ fy / γM
(5.7a)
τtot = τVy + τVz + τt + τw ≤ (fy / √3) / γM0
(5.7b)
√ (σtot2 + 3 τtot2 ) ≤ 1.1 fy / γM,
(5.7c)
where σtot is the total direct stress having design stress components σN due to the axial force, σMy and σMz due to the bending moments about y- and z-axes and σw due to warping. The stress τtot is the total shear stress consisting of design stress components τ Vy and τ Vz due to the shear forces along y- and z-axes, τt due to uniform (St. Venant) torsion and τw due to warping. The factor γM = γM0 if Weff = Wel, otherwise γM = γM1. To be taken on note that only the direct stress components due to resultants NSd, MySd and MzSd should be based on the respective effective cross-sections and all other stress components i.e. shear stresses due to transverse shear force, uniform (St. Venant) torsion and warping as well as direct stress due to warping, should be based on the gross cross-sectional properties. Shear force The design value of shear Vsd shall not exceed the corresponding shear resistance of the crosssection, which shall be taken as the lesser of the shear buckling resistance VbRd or the plastic shear resistance VplRd. The latter should be checked in the case λw ≤ 0.83 (fvb / fv ) (γM0 / γM1) = 0.83 (according to NAD) using the formula: VplRd = (hw / sinφ) t (fy / √3) / γM0,
(5.8)
where hw is the web height between the midlines of the flanges and φ is the slope of the web relative to the flanges, see Figure 3.1. The shear buckling resistance VbRd shall be determined from: VbRd = (hw / sinφ) t fbv / γM1,
(5.9)
where fbv is the shear buckling strength, which depends on the relative web slenderness λ w and stiffening at the support according to the Table 5.2 in ENV 1993-1-3. The relative web slenderness λw is e.g. for webs without longitudinal stiffeners: λw = 0.346 (hw / sinφ) / t √ (fy / E)
28 Local transverse forces To avoid crushing, crippling or buckling in a web subject to a support reaction or other local transverse force (for simplicity: ‘concentrated load’) applied through the flange, the point load Fsd shall satisfy: Fsd ≤ RwRd,
(5.10)
where RwRd is the local transverse resistance of the web. If the concentrated load is applied through a cleat, which is designed to resist this load and to prevent the distortion of the web, the resistance for concentrated load needs not to be checked. Thin-walled members normally used can be designed for concentrated load according to ENV 1993-1-3 Cl. 5.9.2. The resistance formula to be used in the case of single unstiffened web depends on the number (one or two), the location and the bearing lengths of the concentrated loads. In addition, the resistance depends on the geometry (hw, t, r and φ) and material of the web (fy / γM1). In the case of two unstiffened webs, the approach is totally different, although the same parameters affect the point load resistance. As a result, only one formula with supplementary parameters is needed. The equations for stiffened webs enforces more the impression that the background of the point load resistance evaluations is rather empirical. Combined forces A cross-section subject to combined bending moment Msd and shear force Vsd shall be checked for the condition: ( Msd / McRd )2 + ( Vsd / VwRd )2 ≤ 1,
(5.11)
where McRd is the moment resistance of the cross-section and VwRd is the shear resistance of the web, both defined previously. A cross-section subject to combined bending moment Msd and point load Fsd shall be checked for the cond itions: Msd / McRd ≤ 1
(5.12a)
Fsd / RwRd ≤ 1
(5.12b)
Msd / McRd + Fsd / RwRd ≤ 1.25,
(5.12c)
where RwRd is the appropriate value of the resistance for concentrated load of the web, described previously.
29 GLOBAL BUCKLING RESISTANCE OF MEMBERS Axial compression A member is subject to concentric compression if the point of loading coincides with the centroid of the effective cross-section based on uniform compression. The design value of compression Nsd shall not exceed the design buckling resistance for axial compression NbRd : Nsd ≤ NbRd = χ Aeff fy / γM1,
(6.1)
Where, according to ENV 1993-1-3, the effective area of the cross-section Aeff is based conservatively on uniform compressive stress equal to fy / γM1. The χ-factor is the appropriate value of the reduction factor for buckling resistance: χ = min ( χy , χz, χT ,χTF ),
(6.2)
where the subscripts y, z, T and TF denote to different buckling forms i.e. to flexural buckling of the member about relevant y- and z-axes, torsional and torsional-flexural buckling. The calculation of factor χ according ENV 1993-1-3 Cl. 6.2.1 is formulated in (Salmi, P. & Talja, A.): χ = 1, when λ ≤ 0.2
(6.3a)
χ = 1 / ( φ + √ (φ 2 - λ2 )), when λ > 0.2
(6.3b)
λ = √ ( fy / σcr )
(6.3c)
φ = 0.5 [ 1 + α ( λ - 0.2 ) + λ2 ],
(6.3d)
where α is an imperfection factor, depending on the appropriate buckling curve and λ is the relative slenderness for the relevant buckling mode.
30
Figure 6.1 Different buckling curves and corresponding imperfection factors. In Figure 6.1 is shown the χ-λ-relationship for different buckling curves and corresponding values of α. The buckling curve shall be obtained using ENV 1993-1-3 Table 6.2. The selection of cross-section types in Table 6.2 is very limited. However, the correct buckling curve for any cross-section may be obtained from the table “by analogy” (how?). As a conclusion from the tables (Salmi, P. & Talja, A.), in the case of typical C- and hat profiles European buckling curve b (α = 0.34) for flexural buckling about both principal axes shall be chosen. In the case of other profiles buckling curve c (α = 0.49) shall be used. Regardless of the open cross-section form, the buckling curve b shall be chosen in the case of torsional and flexuraltorsional buckling modes. The critical buckling stress in any mode shall be determined in a traditional way, using equations e.g. from the code ENV 1993-1-3 or reference (Salmi, P. & Talja, A.). These equations for critical buckling stresses are more suitable for everyday design, especially because the cross-sectional properties (iy , iz, It , Iw etc.) can be calculated for gross cross-section. Naturally, in the case of complex cross-sections or support conditions, handbooks or more advanced methods are required. One problem in design may be the determination of buckling length in torsion taking into account the degree of torsional and warping restraint at each end of the member. Lateral-torsional buckling of members subject to bending The design value of bending moment Msd shall not exceed the design lateral-torsional buckling resistance moment MbRd of a member: Msd ≤ MbRd = χLT Weff fy / γM1,
(6.4)
31 where Weff is the effective section modulus based on bending only about the relevant axis, calculated by the stress fy / γM1 according to code ENV 1993-1-3 or e.g. χLT fy (Salmi, P. & Talja A.). Analogically to compressive loading, the reduction factor χ LT for lateral buckling is calculated by means of buckling curve a (α LT = 0.21): χ LT = 1, when λLT ≤ 0.4
(6.5a)
χ LT = 1 / ( φ LT + √ ( φ LT 2 - λLT 2 )), when λLT > 0.4
(6.6b)
λLT = √ ( fy / σcr )
(6.6c)
φ LT = 0.5 [ 1 + αLT ( λLT - 0.2 ) + λLT ], 2
(6.6d)
where the relative slenderness λLT is calculated using elastic buckling stress σcr . This stress is the ratio of the ideal lateral buckling moment Mcr and section modulus of gross cross-section. The elastic critical moment Mcr is also determined for the unreduced cross-section. The fo rmula for critical moment Mcry for singly symmetric sections is normal buckling description, but determination of critical moment Mcrz as well as handling of complex sections yields problems for sure. Bending and axial compression In addition to that each design force component shall not exceed the corresponding design resistance, conditions for the combined forces shall be met. In the case of global stability, the interaction criteria introduced in the code ENV 1993-1-3 are extraordinarily complex. For practical design purposes, a more familiar approach for combined bending and axial compression represented by (Salmi, P. & Talja, A) is more practical: Nsd / NbRd + Mysd / MyRd / (1 - Nsd / NEy ) + Mzsd / MzRd / (1 - Nsd / NEz ) ≤ 1.0,
(6.7)
where the meanings of the symbols have been described previously, except the elastic flexural buckling forces NEy and NEz corresponding to the normal Euler flexural buckling formula. In accordance with the code the effective cross-sectional properties can be calculated separately. Naturally, the resistance value shall be taken as smallest if several failure modes are possible. Here again, the additional moments due to potential shifts of neutral axes should be added to the bending moments. For simplicity and for the fact that they usually can be omitted, no additional moments are shown in the formula. Interaction between bending and axial compression are considered thoroughly in Cl. 6.5 of the code, but without any explanations of the backgrounds.
SERVICEABILITY LIMIT STATES In the design code ENV 1993-1-3, serviceability limit states have been considered on one page only. The deformations in the elastic as well as in the plastic state shall be derived by
32 means of a characteristic rare load combination. The influence of local buckling shall be taken into account in form of effective cross-sectional properties. However, the effective second moment of area Ieff can be taken constant along the span, corresponding the maximum span moment due to serviceability loading. In the Finnish NAD a more accurate approach is presented, where the effective second moment of area may be determined from the equation: Ie = ( 2 Iek + Iet ) / 3,
(7.1)
where effective second moments Iek and Iet are to be calculated in the location of maximum span moment and maximum support moment, respectively. On the safe side, ultimate limit state moments may be used. Plastic deformations have to be considered, if theory of plasticity is used for ultimate limit state in global analysis of the structure. The deflections shall be calculated assuming linear elastic behaviour. In stead of strange limit value (L/180) for deflection in the ENV draft code the NAD has defined reasonable limits for different thin gauge structure types. For example, the maximum deflection in the serviceability limit state for roof purlins is L/200 and for wall purlins L/150.
CONCLUSIONS In this paper, the main design principles of cold formed thin gauge members (‘thin-walled members’) have been considered. The manufacturing process results in typical features of thin-walled members: quite slender parts in very different open cross-sections and consequently many local or global failure modes. The desired properties (usually strength to weight ratio) of the members can be reached by optimising cross-sections, but as a by-product, the design procedures can be extremely complicated. The total lack of design codes seems to have been tranformed into a situation, in which some guidelines are available, but they are hard to adapt in practical design. The theoretical background for analytical design should be rather well known, but according to comparative tests, the accuracy of predicted resistance values is still often very poor - sometimes the deviation can even be on the ‘unsafe side’. However, taking into account several parameters affecting to analytical and test results, this inaccuracy can be expected and kept in mind in every day design. Complex structural behaviour of thinwalled members has produced inevitably complex design codes (e.g. ENV 1993-1-3). Hence all efforts to derive simplified design methods are naturally welcome. Because all manual methods are probably still to laborious, FEA is too heavy a tool and some design programs ‘already’ available may not guarantee sufficient results in practice, the biggest contribution at the moment should be made to reliable calculation programs, which are as simple as possible to use. This challenging task should preferably be carried out by the same institutions, which produce these comprehensive design codes.
33 REFERENCES ENV 1993-1-3. 1996. Eurocode 3: Design of steel structures. Part 1.3: General rules. Supplementary rules for cold formed thin gauge members and sheeting. European Committee for Standardisation CEN. Brussels. SFS-ENV 1993-1-3. 1996. Eurocode 3: Teräsrakenteiden suunnittelu. Osa 1-3: Yleiset säännöt. Lisäsäännöt kylmämuovaamalla valmistetuille ohutlevysauvoille ja muotolevyille. Vahvistettu esistandardi. Suomen Standardisoimisliitto SFS ry. Helsinki. 1997. NAD. 1999. National Application Document. Prestandard SFS-ENV 1993-1-3. 1996. Design of steel structures. Part 1.3: General rules. Supplementary rules for cold formed thin gauge members and sheeting. Ministry of Environment. Helsinki. ENV 1993-1-1. 1992. Eurocode 3: Design of steel structures. Part 1.1: General rules and rules for buildings. European Committee for Standardisation CEN. Brussels. TEMPUS 4502. Cold formed gauge members and sheeting. Seminar on Eurocode 3 – Part 1.3. Edited by Dan Dubina and Ioannis Vayas. Timisoara, Romania. 1995. Salmi, P. & Talja, A. 1994. Simplified design expressions for cold-formed channel sections. Technical Research Centre of Finland. Espoo. Roivio, P. 1993. Kylmämuovattujen teräsavoprofiilien ohjelmoitu mitoitus (Programmed design of cold-formed thin gauge steel members). Thesis for the degree of M.Sc.(Tech.), Helsinki University of Technology. Espoo.
34
DESIGN CHARTS OF A SINGLE-SPAN THIN-WALLED SANDWICH ELEMENTS Karri Kupari Laboratory of Structural Mechanics Helsinki University of Technology, P.O.Box 2100, FIN-02015 HUT, Finland
ABSTRACT There are four different criteria, which must be determined in order to design a capacity chart for a single-span thin-faced sandwich panel. These criteria are bending moment, shear force, deflection and positive or negative support reaction. The normal stress due to bending moment must not exceed the capacity in compression of the face layer. The shearing stress due to shear force must not exceed the shearing capacity of the core layer. The maximum deflection can be at the most one percent of the span and the reaction force from external loads has to remain smaller than the reaction capacity. This paper presents some details of an investigation using full-scale experiments to determine the estimated level of characteristic strength and resistance of the sandwich panel.
KEYWORDS Thin-walled structures, metal sheets, mineral wool core, shear modulus, deflection, normal (Gaussian) distribution, flexural wrinkling, shear failure of the core.
INTRODUCTION A typical thin-faced sandwich panel consists of three layers. The top and the bottom surface are usually 0.5 … 0.8 mm thick metal sheets and covered with a coat of zinc and preliminary paint. The outer surface is coated with plastic. The most commonly used core layers are polyurethane and mineral wool.
35 surface layer, metal sheet
core layer
h = 100…150 mm
b = 1200 mm
Figure 1: The cross-section of a typical sandwich panel. Sandwich panels are usually designed to bear only the surface load, which causes the bending moment and the shearing force. The bending moment causes normal stress to the top surface. The core layer must bear the shearing stress and the compression stress from the reaction force. STRUCTURAL FORMULAS AND DEFINITIONS The surface layer is presumed to be a membranous part and its moment of inertia insignificant compared with the moment of inertia for the whole sandwich panel. This gives us the simplification that the compression and tension stresses are uniformly distributed across the surface layer. The value of the modulus of elasticity for the surface layer is more than ten thousand times larger than the value of the modulus of elasticity for the core layer. The influence of the normal stresses across the core layer equals zero when considering the behavior of the whole sandwich panel. The normal stress of the surface layer is ó 1, 2 = ±
M eA f (1, 2 )
(1)
and the shearing stress of the core layer is ôs = M Q e b Af(1,2)
Q eb
= bending moment = shear force = the distance between the surface layers center of gravity = the width of the sandwich panel = the area of the surface layers cross section
(2)
36 σ1
τs e
σ2
Figure 2: The approximation of normal and shear stresses. When calculating the deflection in the mid-span of a simply supported sandwich panel we concentrate on two different load cases: Load case A is uniformly distributed transverse loading (Eq. 3 and Fig. 4.) and load case B consist of two symmetrically placed line loads (Eq. 4 and Fig. 5.).
()
(3)
()
(4)
5 qL4 1 gL2 wL = + 2 384 B 8 Geb 23 FL3 FL wL = + 2 1296 B 6Geb DEFINING THE SHEAR MODULUS
At the beginning of the testing procedure we can determine the shear modulus. Assuming that the loaddeflection curve is linear and using the Hooke´s law we can write F = kw + C. After differentiation we get ∂F =k ∂w
(5)
load [q]
where k equals the slope of the regression line.
k deflection [w] Figure 3: The load-deflection curve.
37 The experimentally defined parameter k leads to the formula that gives us the shear modulus for load case A 1 5L2 G = 8eb 2 − 384B kL
−1
(6)
and respectively for load case B 1 23L2 G = 6eb − kL 1296 B
−1
(7)
where B = ½EAf e2 is the bending stiffness. e is the distance between the centers of the surface layers as shown in the Fig. 2. The value of the modulus of elasticity is E = 210 000 N/mm2 and the area of the surface layer Af = 0.56 ∗ 1230 mm2. The width of the core layer is 1200 mm.
q
L
Figure 4: Load case A. Uniformly distributed transverse loading. F 2
L 3
F 2
L 3
L 3
Figure 5: Load case B. Two symmetrically placed line loads.
38 FULL SCALE EXPERIMENTS A vacuum chamber was used to produce a uniformly distributed transverse loading of the panels, enabling flexural wrinkling failures to occur in bending. All these experiments were done at the Technical Research Center in Otaniemi, Espoo. Once the panels were positioned in the chamber, the measuring devices for force and deflection were set to zero. A polyethylene sheet was placed over the panel and sealed to the sides of the timber casing. The compression force was produced by using a vacuum pump to decrease the air pressure in the chamber. A total of twelve panels were used in this experiment. This procedure models the distributed load caused by wind. The results of these tests give us the capacity in compression of the surface layer.
Polyethene sheet
Sandwich Panel
Vacuum Chamber Timber Casing
The measuring devices
Supports
= Force
= Deflection
Figure 6: Experimental Set-up and the positioning of the measuring devices (Vacuum Chamber). For the load case B, two symmetrically placed line loads, all experiments were made at the Helsinki University of Technology in the Department of Civil and Environmental Engineering. From the results of these tests we can calculate both the shearing and reaction capacity. Altogether 28 panels were used in this part. The loading was produced by two hydraulic jacks with deflection controlled speed of 2 mm/min. The testing continued until the sandwich panels lost their load bearing capacity.
Force
39
Fu (ultimate force)
0.4 F u
0.2 F u Time Figure 7: The loading history of load case B.
THE CHARACTERISTIC STRENGTHS Defining the characteristic strengths is based on the instructions from “European Convention for Constructional Steelwork: The Testing of Profiled Metal Sheets, 1978”. It is assumed that all testing results obey the Gaussian distribution The Formulas used in defining the characteristic strengths The value of characteristic strength MK can be calculated from the equation M K = M m (1 − cä ) where
(8)
Mm = average of the test results c = factor related to the number of test results (From Table 1) δ = variation factor TABLE 1 The relation between factor c and the number of test results n
n c
3 2.92
4 2.35
5 2.13
The square of the variation factor is
6 2.02
8 1.90
10 1.83
12 1.80
20 1.73
∞ 1.65
40 2 Mi 1 n M i − ∑ M i n ∑ M m i =1 2 ä = i =1 n −1 n
where
2
(9)
n = the number of test results Mi = the value of test number i Mm = average of the test results
The characteristic strengths are calculated based on the test results. The factor related to aging and defining the factor related to temperature The mineral wool core material was tested in three different temperatures. First test was made in normal room temperature +20 oC with the relative humidity RH of 45-50 %. Second test was made after the material was kept for 36 hours in a +70 oC temperature with the relative RH of 100 %. The final part included 36 hours of storage in a +80 oC temperature before testing. The factor related to aging, degradation factors dft and dfc can be calculated from the formulas ó ó df t = t 70 and df c = c70 ó t 20 ó c20 where
σt20 σt70 σc20 σc70
(10)
= tensile strength at +20 oC temperature, average value = tensile strength at +70 oC temperature, average value = compression strength at +20 oC temperature, average value = compression strength at +70 oC temperature, average value
The factors dft and dfc are divided into two groups ≥ 0.7 ( I) df t , df c 〈0.7 ( II )
(11)
For the case (I) test results of characteristic strengths for the capacity in compression of the surface layer and the shearing and reaction capacity of the core layer are valid. For the case (II) test result must be multiplied by the following reduction factors Φ ttd = df t + 0.3 and Φ tcd = df c + 0.3.
(12)
The factor related to temperature can be calculated from Φ Tt =
E t80 E and Φ Tc = c80 E t 20 E c20
(13)
41
where
Et80 Et20 Ec80 Ec20
= Modulus of elasticity in tension at +80 oC temperature, average value = Modulus of elasticity in tension at +20 oC temperature, average value = Modulus of elasticity in compression at +80 oC temperature, average value = Modulus of elasticity in compression at +20 oC temperature, average value
The connection between bending moment and capacity in compression The connection can be given as ã k (ó fw + 0.5 ∗ ó f ∆T ) ≤ where
Φ ttd ∗ f fcK ãm
(14)
γk = the partial safety factor of external load σfw = the normal stress caused by external load σf∆T = the normal stress caused by the temperature difference between inner and outer surface layers Φ ttd = the reduction factor related to aging ffcK = the characteristic strength of the face layers capacity in compression γm = the partial safety factor of material
In case of a single span, statically determined structure, the term σf∆T = 0. The normal stress caused by external load can be calculated from the formula ó fw = where
e b t
qL2 8ebt
(15)
= the distance between the surface layers' centres of gravity = 1 [m] = the thickness of the surface layer
The connection between shear force and shearing capacity The connection can be given as ã k (ôCw + 0.5 ∗ ôC∆T ) ≤ where
Φ ttd ∗ f CvK ãm
(16)
γk = the partial safety factor of external load τCw = the shearing stress caused by external load τC∆T = the shearing stress caused by the temperature difference between inner and outer surface layers
42 Φ ttd = the reduction factor related to aging fCvK = the characteristic strength of the face layers shearing capacity γm = the partial safety factor of material In case of a single span, statically determined structure, the term τC∆T = 0. The shearing stress caused by external load can be calculated from the formula ôCw = where
e b
qL 2eb
(17)
= the distance between the surface layers' centres of gravity = 1 [m]
The connection between reaction force and reaction capacity The connection can be given as
(
)
ã k R wp + 0.5 ∗ R ∆T ≤ where
Φ tcd ∗ R K ãm
(18)
γk = the partial safety factor of external load Rwp = the reaction force caused by external load R∆T = the reaction force caused by the temperature difference between inner and outer surface layers Φ tcd = the reduction factor related to aging RK = the characteristic strength of the reaction capacity γm = the partial safety factor of material
In case of a single span, statically determined structure, the term R∆T = 0. The reaction force caused by external load can be calculated from the formula R wp = 1 qL
(19)
2
The boundary conditions concerning deflection The maximum deflection must remain less than one percent of the span. From external load and temperature difference between inner and outer surface we get two equations:
(
)
L 100
(20)
(
)
L 100
(21)
ã k w q + 0.5 ∗ w ∆T ≤
ã k 0.5 ∗ w q + w ∆T ≤
43
where
γk = the partial safety factor of external load in serviceability limit state (=1.0) wq = the deflection caused by external load w∆T = the deflection caused by the temperature difference between inner and outer surface layers
The deflection caused by external load is mentioned in Eq. (3) and Eq. (4). The deflection caused by the temperature difference between inner and outer surface layer is w ∆T =
where
α ∆T
á ∗ ∆T ∗ L2 8e
(22)
[ ]
−1 = coefficient of linear thermal expansion for surface layer material, 12 ∗ 10 − 6 o C = temperature difference between inner and outer surface layers, 60 oC
From equations (20) and (21) we choose the one that gives the larger deflection.
DESIGN CHARTS From the four criteria we can construct the design chart by drawing four curves from the equations (14), (16), (18) and (20)&(21). The X-axis represents the span L [m] and the Y-axis represents the external load q [kN/m2]. The area located under all four curves represents the permissible combination of external load and span.
44 Design Chart of a single-span thin-walled sandwich element (example)
4
External load q [kN/m2]
3.5 3 2.5 2 1.5 1 0.5
12.00
11.00
10.00
9.00
8.00
7.00
6.00
5.00
4.00
3.00
2.00
1.00
0.00
0
Span L [m] Deflection
Bending moment
Reaction force
Shear force
REFERENCES European Convention for Constructional Steelwork, The testing of Profiled Metal Sheets, 1978. CIB Report, Publication 148, 1983. Rakentajain kalenteri (in Finnish), 1985. McAndrew D., Mahendran M., Flexural Wrinkling Failure of Sandwich Panels with Foam Joints, Fourth International Conference on Steel and Aluminium Structures, Finland, Proceedings book: Light-Weight Steel and Aluminium Structures, edited by Mäkeläinen and Hassinen, pp. 301-308, Elsevier Science Ltd, 1999. Martikainen L., Sandwich-elementin käyttäytyminen välituella, Masters Thesis (in Finnish), 1993.
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