Helsinki University of Technology Laboratory of Steel Structures Publications 15

Teknillisen korkeakoulun teräsrakennetekniikan laboratorion julkaisuja 15

Espoo 2000 TKK-TER-15

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 Laboratory of Steel Structures Publications 15 Teknillisen

korkeakoulun teräsrakennetekniikan laboratorion julkaisuja 15

Espoo 2000 TKK-TER-15

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

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: srt.sihteerit@hut.fi

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 (Rak-83.140 and Rak-83.J) in spring semester 2000. This time the Seminar was realized as ajoint seminar for graduate and postgraduate students. The subject of the Seminar waschosen as Design of Cold-Formed Steel Structures.

The seminar was succesfully completed with clearness in presentations and expertknowledge in discussions. I will thank in this connection all the participants for theirintensive 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…...…………………………………………………………1J. Outinen

2 Design of Cold Formed Thin Gauge Members………………….……………….14R. Hara

3 Design Charts of Single-Span Thin-Walled Sandwich Elements……….………34K.Kupari

4 Numerical Analysis for Thin-Walled Structures……….………………………..45H Perttola

5 Cold-Formed Steel Structures in Fire Conditions…………………………….….65O. Kaitila

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PROFILED STEEL SHEETING

Jyri Outinen

Researcher, M.Sc.(Tech)Laboratory of Steel Structures

Helsinki University of TechnologyP.O. Box 2100, FIN-02015 HUT - Finland

Email: jyri.outinen@hut.fi(http://www.hut.fi/~joutinen/)

ABSTRACT

The ligthness of cold-formed thin-walled structures was formerly their most importantfeature and therefore they were used mostly in products where the weight saving was ofgreat importance, This kind of products were naturally needed in especially transportationindustries e.g. aircrafts and motor industry.A wide range of research work during many decades has been conducted all over theworld to improve the knowledge about the manufacturing, corrosion protection, materialsand codes of practise of thin-walled steel structures. This has led to a constantlyincreasing use of cold-formed thin-walled structures. Profiled steel sheeting is used invarious kind of structures nowadays.In this paper, a short overview of the manufacturing, products, materials and structuraldesign of profiled steel sheeting is given. Also a short overview of some current researchprojects is given.

KEYWORDS

Profiled steel sheeting, sheet steel, cold-formed, thin-wall, corrugated, steel, structuraldesign, steel materials, cladding, roof structures, wall structures, floor structures.

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INTRODUCTION

There is a wide range of manufacturers making different kind of profiled steel sheetingproducts. The manufacturing processes have beensignificantly developed and differentshapes of sheeting profile are easy to produce. Steel sheeting is also easy to bend todifferent shapes e.g. curved roof structures., cylindrical products e.g. culvers etc. Theproducts are delivered with a huge range of possible coatings. Normally the coating isdone 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 functionalrequirements. In this paper, the structural use is more thoroughly considered. Profiledsteel sheeting is widely used in roof, wall and floor structures. In these structures, theprofiled 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 withconcrete. In northern countries the roof and wall structures are almost always built withthermal insulation. The sound insulation and the fire insulation have also to beconsidered, when designing structures.

There are several codes for the design of profiled steel sheeting. Almost every countryhas 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 theEurocode 3: part 1.3, though there are several national application documents (NAD),where the national requirements are considered with the EC3. An extensive amount oftests has been carried out and analyzed to gather together the existing design codes, andthere are numerous formulae in these codes that are based partly on theory and partly onexperimental test results. Some of the important aspects of structural design of cold-formed profiled steel sheeting is presented in this paper.

Numerous different kind of fastening techniques are developed suitable for thin-walledstructures. Suitable fasteners are bolts with nuts, blind rivets, self tapping screws, self-drilling screws and some other kinds of fasteners.

The materials used in cold-formed thin-wall members have to satisfy certain criteria to besuitable for cold-forming and usually also for galvanising. The yield strength is normallyin the range of 220…350 N/mm2, but also some high-strength sheet steels with yieldstrength 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 profiledsheeting.

A lot of interesting research projects have been carried out concerning the behaviour ofprofiled steel sheeting all over the world. Some of the current researches are shortlydescribed in this paper. In different parts of the world the focus of the research isnaturally on the regional problems. An example of this is Australia, where the mainresearch area of cold-formed steel structures is concentrated on the problems caused byhigh-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 steelsheets. The first plates were very simple and the stiffness of these was not very high. Themanufacturing process and the materials limited the shape of the profiles to simply foldedor 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 bendingresistance. The maximum height of the profile was normally still under 100mm. In Figure2 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 bettermaterials and manufacturing technologies lead to possibilities to provide more complexprofiles. This improved substancially the load-bearing capacities of the developed newprofiled steel sheets. In figure 3 is shown an example of this kind of more complexprofile.

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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 ofpurposes. The thin-walled steel structures and profiled steel sheeting is an area of fastgrowth. In the next chapter, a few typical examples where cold-formed profiled steelsheeting 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 tosatisfy the functional requirements of the design. This aspect is considered in this chapterbriefly in relation to the common usage of cold-formed sheeting in floor, wall and roofstructures.

Floor structures

Profiled steel sheeting in floor structures have sheeting, e.g. trapezoidal or cassettes, asload bearing part, either alone or in composite action with other materials such asdifferent kind of board, plywood decking or cast in-situ concrete. In the first case, thecomposite action is provided by adhesives, and mechanical fasteners, in the second bymeans of indentation and/or special shear studs. The bending moment resistance is themain requirement, and so the profiles used for flooring purposes are similar to those forroof decking.

Figure 4: A Steel-concrete composite floor slab with profiled steel sheeting

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Wall structures

In wall structures, the structure is comprised of an outer layer, the facade sheeting that isusually built with relatively small span, and a substructure which transmits the windloading to the main building structure. The substructure can be a system of wall rails orhorizontal deep profiles, or cassettes with integrated insulation. Another solutioncombines the load-bearing and protecting function in a sandwich panel built up by metalprofiles 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 roofhas an outer waterproof skin with internal insulation if required. The main requirement ofpreventing the rain water or the melting snow leads to shallow profiles with a sequence ofwide and narrow flanges. Sheets fixed using fasteners applied to the crests or the valleysof the corrugations.

Figure 6: A roof structure made with profiled steel sheeting of a subway station inFinland

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The use of few points of fastening means that the forces are relatively high and thereforethe spans are usually quite small. A wide range of special fasteners have been developedto 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-bearingprofile, insulation and an outer layer e.g. metal skin, as mentioned before. The load-bearing profiled sheeting in this type of roof normally has the wider flanges turned up inorder to provide sufficient support for the insulation. Fasteners are placed in the bottomof the narrow troughs. In this case, the tendency is towards longer spans, using morecomplex profiles of various shapes and a core of polyurethane.

Other applications

The highly developed forming tecnology makes it possible to manufacture quite freelyproducts made of profiled steel sheets with various shapes. Such are for example curvedroof structures., cylindrical products e.g. culvers etc. There are not too much limitationsanymore concerning the shape of the product. In Figure X. a few examples of this arepresented.

Figure 7: Profiled sheet steel products in different shapes

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MANUFACTURING

Cold-formed steel members can be manufactured e.g. by folding, press-braking or cold-rolling. Profiled steel sheets are manufactured practically always using cold-forming.Also the cylindrical products are manufactured by cold cold rolling from steel strips. Infigure 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 andtherefore it is easy to manufacture optimal profiles that have adequate load bearingproperties 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. Favourableeffects can be observed if residual stresses are induced in parts of the section which act incompression 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 amountwhich depends on the bending radius, the thickness of the sheet, the type of steel and theforming process. The average yield strength of the section depends on the number ofcorners 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 requirementsand loading conditions that apply, design is generally based on experimentalinvestigations. This experimental approach is generally acceptable for mass producedproducts, where optimization of the shape of the profiles is a competitive need.

The product development during about four decades has been based more on experienceof the functional behaviour of the behaviour of the products than on analytical methods.The initial "design by testing" and subsequent growing understanding of the structuralbehaviour allowed analytical design methods to be developed. Theoretical or semi-empirical design formulae were created based on the evaluation of test results. This typeof interaction of analytical and experimental results occurs whenever special phenomenaare 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 steelmembers. In Europe, Eurocode 3: Part 1.3 is the latest design code which can be used inall 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 partsof the world e.g. in USA (AISI-specifications), Australia, (AS) there are several differentcodes for the design. All the design codes seem to have the same principles, but thedesign practices vary depending on the code.

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The design can basicly be divided in two parts: 1.) Structural modelling and analysiswhich is normally quite a simple procedure and 2.) Checking the resistances of thesheeting. The values that are needed in the design are: moment resistance, point loadresistance and the effective second moments of area Ieff corresponding to the momentresistances. The deflections have to be also considered. The deflections duringconstruction e.g. in steel-concrete composite floors are often the limiting factor to thestructure.

The load-bearing properties, i.e. moment resistance, point load resistance etc., are almostalways 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 againsttransverse point loads as mentioned earlier. The reason is that the load is transmitted tothe webs as point loads that create high stress peaks to it. The web is then very vulnerableto lose the local stability at these points. All the manufactures have recommendations forthe 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 ashort 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 largeamount of experimental research has been carried out on this subject. Most of thisresearch is concentrated on the connections. Different types of fasteners have beendeveloped to avoid the pull-through, pull-over or pull-out phenomena under dynamichigh-wind loading.

Figure 10: Examples of pull-through failures under dynamic loading. Local pull-throughby splitting and fatigue pull-through (high-strength steel).

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MATERIALS

The most common steel material that is used in profiled steel sheets is hot dip zinc coatedcold-formed structural steel. The nominal yield strength Reh (See Fig. 4) is typically220…550N/mm2. The ultimate tensile strength is 300…560 N/mm2. The modulus ofelasticity is normally 210 000 N/mm2. The mechanical properties of low-carbon cold-formed structural steels have to be in accordance with the requirements of the Europeanstandard SFS-EN 10 147.

The mechanical properties are dependent on the rolling direction so that yield strength ishigher transversally to rolling direction.. In the inspection certificate that is normallydelivered 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 nominalyield strength of 350 N/mm2 at room temperature both longitunidally and transversally torolling direction are shown. The difference between the test results for the specimenstaken longitudinally and transversally to rolling direction can clearly be seen. The resultsare also shown in Tables 1 and 2.

0

50

100

150

200

250

300

350

400

450

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Strain εε [%]

Str

ess

σσ [N

/mm

2 ]

Transversally to rolling direction

Longitudinally to rolling direction

Figure 11: Stress-strain curves of structural steel S350GD+Z at room temperature.Tensile tests longitudinally and transversally to rolling direction

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TABLE 1MECHANICAL PROPERTIES OF THE TEST MATERIAL S350GD+Z AT ROOM TEMPERATURE.

TEST PIECES LONGITUDINALLY TO ROLLING DIRECTION

Measured property Mean value(N/mm2)

Standard deviation(N/mm2)

Number of tests(pcs)

Modulus of elasticity E 210 120 13100 5Yield stress Rp0.2 354.6 1.5 5Ultimate stress Rm 452.6 2.3 5

TABLE 2MECHANICAL PROPERTIES OF THE TEST MATERIAL S350GD+Z AT ROOM TEMPERATURE.

TEST PIECES TRANSVERSALLY TO ROLLING DIRECTION

Measured property Mean value(N/mm2)

Standard deviation(N/mm2)

Number of tests(pcs)

Modulus of elasticity E 209400 8800 4Yield stress Rp0.2 387.5 1.3 4Ultimate stress Rm 452.5 1.9 4

The thickness of the base material that is formed to profiled steel sheets is normally0.5…2.5 mm. The thickness can't normally be less than 0.5 mm. If the material is thinnerthan that, the damages to the steel sheets during transportation, assembly and handlingare almost impossible to avoid. The thickness of the sheet material is not normally over2.5 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 ofprofiled 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 seemsto be the hardness of the material, i.e. there are problems in roll-forming, cutting anddrilling. On the other hand, excellent corrosion resistance and also fire resistance give itbig 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 notusually in structural use.

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CURRENT RESEARCH WORK

A wide range of different kind of research activities concerning profiled steel sheeting isgoing on in several countries. Most of the studies are based on both experimental testresults and usually also modelling results produced with some finite element modellingprograms. 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-concretecomposite slabs with profiled steel sheeting. In these projects, which are mainly carriedout in Finnish universities, e.g. Helsinki University of Technology, and in the TechnicalResearch Centre of Finland, the aim is simply to increase the load bearing capacity. Thisis studied using different profiles and stud connectors. The experiments are normallybending tests, but also some shear tests for the connection between steel sheeting andconcrete with push-out tests have ben carried out. During the next few years, severalresearch projects are starting in Finland concerning the design of lightweight steelstructures. In these projects, the fire design part is of great importance.

In Australia, e.g. in Queensland University of Technology, and also in several otheruniversities, there are numerous on-going research projects concerning mainly thebehaviour of the connections of steel sheeting under wind-storm loads e.g. "Developmentof design and test methods for profiled steel roof and wall claddings under wind upliftand racking loads" and "Design methods for screwed connections in claddings." arerecently completed projects. The current situation can be found on their www-site (givenin next chapter: References).

In these projects a significant amount of small-scale and also large scale tests have beenconducted. The small-scale tests are usually carried out to study the pull-out or pull-overphenomena of screwed connections. The large-scale tests aim to study the behaviour ofthe 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 foundfrom 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 werementioned. Different research programs concerning the cold-formed profiled steelsheeting are going on in Europe and other parts of the world. A major conference, "TheInternational Specialty Conference on Recent Research and Developments in Cold-Formed Steel Design and Construction", where the latest research projects are presentedregurarly, is held in St. Louis, Missouri.

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REFERENCES

Eurocode 3, CEN ENV 1993-1-3 Design of Steel Structures- Supplementary rules forCold Formed Thin Gauge Members and Sheeting, Brussels, 1996

Standard SFS-EN 10 147 (1992): Continuously hot-dip zinc coated structural steel sheetand strip. Technical delivery conditions. (in Finnish), Helsinki

Outinen, J. & Mäkeläinen, P.:Behaviour of a Structural Sheet Steel at Fire Temperatures. Light-Weight Steel andAluminium Structures (Eds. P. Mäkeläinen and P. Hassinen) ICSAS'99. Elsevier ScienceLtd., Oxford, UK 1999, pp. 771-778.

Kaitila O., Post-graduate seminar work on "Cold Formed Steel Structures in FireConditions", Helsinki University of Technology, 2000.

Helenius, A., Lecture in short course: "Behaviour and design of light-weight steelstructures" , at Helsinki University, 1999

Tang, L.,Mahendran, M., Pull-over Strength of Trapezoidal Steel Claddings, . Light-Weight 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

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DESIGN OF COLD FORMED THIN GAUGE MEMBERS

Risto HaraM.Sc.(Tech.)

PI-Consulting OyjLiesikuja 5, P.O. BOX 31,

FIN-01601 VANTAA, FINLANDhttp://www.pigroup.fi/

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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. Dueto the manufacturing process, sections of cold formed structural shapes are usually open, sin-gly-, 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 sec-tions 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 duringthe last 100 years. They are advantageous in light-weight constructions, where they can carrytension, compression and bending forces. The structural properties and type of loading ofthin-walled members cause the typical static behaviour of these structures: the local or globalloss of stability in form of different buckling phenomena. To have control of them in analysisand design, sophisticated tools (FEA) and design codes (ENV 1993-1-3, AISI 1996, etc.) mayhave to be used. Unfortunately, the complexity of these methods can easily limit the use ofthin-walled structural members or lead to excessive conservatism in design. However, somesimplified 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. Thepresent Finnish design codes B6 (1989) and B7 (1988) are entirely omitted as inadequate forthe 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 transla-tion (SFS-ENV 1993-1-3) and National Application Document (NAD). The paper concen-trates 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 andpractice for the design of thin-walled members is presented in a comprehensive way. The ref-erence 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. offlat plate fields and edge stiffeners. Instead of failure through material yielding, compressedparts tend to loose their stability. In the local buckling mode, flat plate fields buckle causing

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displacements only perpendicular to plane elements and redistribution of stresses. In thismode the shape of the section is only slightly distorted, because only rotations at plane ele-ment junctures are involved. In the actual distortional buckling mode, the displacements of thecross-section parts are largely due to buckling of e.g. flange stiffeners. In both bucklingmodes, the stiffness properties of the cross-section may be changed, but the member probablystill has some post-buckling capacity since translation and/or rotation of the entire cross-section is not involved. In the global buckling mode, displacements of the entire cross-sectionare large, leading to over-all loss of stability of the member. Global buckling modes dependprimarily on the shape of the cross-section. Flexural buckling usually in the direction ofminimum flexural stiffness is common also for cold formed members. Low torsional stiffnessis typical for open thin-walled members,so buckling modes associated with torsion may becritical. Pure torsional buckling is possible for example in the case of a point symmetric cross-section (e.g. Z-section), where the centre of the cross-section and the shear centre coincide. Intorsional buckling, the cross-section rotates around the shear centre. A mixed flexural-torsional buckling mode, where the cross-section also translates in plane, is possible in thecase of single symmetric cross-sections (e.g. U, C and hat). Due to the low torsional stiffnessof 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 notwork well with low profiles bent about the axis of symmetry or with open profiles bent in theplane of symmetry, when the folded edges are compressed (e.g. wide hats). Naturally, plasticor elastic-plastic static behaviour of compressed or bended members are possible when loadedto failure, but with normal structural geometry and loading, stability is critical in the design ofthin-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 comparedwith 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 otherprincipal axis of the cross-section. For other cross-sections, y-axis is the major axis and z-axisis the minor axis, see also Figure 1.1. According to the ENV code, also u-axis (perpendicularto 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-walledmember belongs to one of two construction classes. In Class I the member is a part of theoverall stiffening system of the structure. In Class II the member contributes only to the indi-vidual structural strength of the element. The Class III is reserved for secondary sheetingstructures only. However, this classification for differentiating levels of reliability seems notto have any influence in design. In ultimate limit states (defined in ENV 1993-1-1), the valueof partial safety factors (γM0 and γM1) needed in member design are always equal to 1.1. FactorγM0 is for calculation of cross-section resistance caused by yielding and factor γM1 is for cal-culation of member resistance caused by buckling. The serviceability limit states are definedin form of principles and application rules in ENV 1993-1-1 and completed in ENV 1993-1-3with the associated Finnish NAD. The partial factor in both classes γMser has a value equal to1.0.

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The design of adequate durability of cold formed components seems to require qualitativeguide lines according to base code ENV 1993-1-3, but also much more exact specificationsaccording 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, whichcan be used in cold formed steel design according to the code. Other structural steels can alsobe 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 tocold forming could be utilised in load bearing capacity. Fortunately for the designer, Ch. 3.1.2has been simplified in the NAD: nominal values of basic yield strength fyb shall be appliedeverywhere as yield strength (hence in this paper fyb is replaced in all formulas by fy). This canbe justified, because on the average, the ratio fya /fyb ≈ 1.05 only. Normally yield strengths fybused in thin-walled members lay in the range 200-400 N/mm2, but the trend is to evenstronger steels.

TABLE 3.1TYPICAL STRUCTURAL STEELS USED IN COLD FORMED STEEL STRUCTURES.

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Obviously, other material properties relevant in cold formed steel design are familiar to de-signers: 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 thicknessof 1.0 < tcor < 8.0 mm. In the Finnish NAD, however, the material thickness condition ischanged: 0.9 < tcor < 12.0 mm. Up to 12.5 mm core thickness is reached in roll-forming proc-ess 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 (forcommon 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 com-plex shape of the cross-sections, approximations are required in most cases. Specified nomi-nal dimensions of the shape and large openings determine the properties of the gross cross-section. The net area is reached from gross area by deducting other openings and all fastenerholes according to special rules listed in Ch. 3.3.3 of the Eurocode. Due to cold forming, thecorners of thin-walled members are rounded. According to the design code, the influence ofrounded corners with internal radius r ≤ 5 t and r ≤ 0.15 bp on section properties may be ne-glected, i.e. round corners can be replaced with sharp corners. The notional flat width bp isdefined by applying the corner geometry shown in Figure 3.1, extracted from the code. If theabove limits are exceeded, the influence of rounded corners on section properties ‘should beallowed for’. Sufficient accuracy is reached by reducing section properties of equivalentcross-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 ≈ Iw,sh (1-4δ), (3.1c)

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Where Ag is the area of the gross cross-section, Ig is the second moment area of the grosscross-section and Iw is the warping constant of the gross cross-section. Term δ is a factor de-pending 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 for-mula:

n m

δ = 0.43 ∑ rj / ∑ bpi , (3.2) j = 1 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-thicknessratios 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 con-struction of the shape of the cross-section. However, to provide sufficient stiffness and toavoid 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 3.2MAXIMUM 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 bucklingof the cross-section. The effects of local buckling shall be taken into account in the determi-nation of the design strength and stiffness of the members. Using the concept of effectivewidth and effective thickness of individual elements prone to local buckling, the effectivecross-sectional properties can be calculated. The calculation method depends on e.g. stress-levels and -distribution of different elements. The code ENV 1993-1-3 Cl. 4.1. (4-6) statesthat in ultimate resistance calculations, yield stress fy ‘should’ be used (on the ‘safe side’) andonly 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 ele-ment without stiffeners in compression could be presented in the general form, in accordancewith 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) / λp2, when λp > 0.673 (4.1b)

λp = √ (σc / σel) = 1.052 (bp / t) √ (σc / E / kσ) (4.1c)

σel = kσ π2 E / 12 / (1 - ν2) / (bp / t)2, (4.1d)

where ρ is the reduction factor of the width, λp relative slenderness, bp width, σc maximumcompressive stress of the element and kσ buckling factor. For compressed members σc is usu-ally the design stress (χfy) based on overall buckling (flexural or flexural-torsional). For bentmembers, in an analogical way, σc is usually the design stress for lateral buckling (χfy). Inspecial cases, σc really can have the value fy in compression or bending. Obviously, the safesimplification 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 4.2for external compression elements, respectively.

The design of stiffened elements is based on the assumption that the stiffener itself works as abeam on elastic foundation. The elasticity of the foundation is simulated with springs, whosestiffness depends on the bending stiffness of adjacent parts of plane elements and the bound-ary conditions of the element. A spring system for basic types of plate fields needed in analy-sis is shown in Table 4.3. The determination of spring stiffness in two simple cases is pre-sented in Figure 4.1. For example, in the case of an edge stiffener, the spring stiffness K of thefoundation per unit length is determined from:

K = u / δ, (4.2)

where δ is the deflection of the stiffener due to the unit load u:

21

δ = Θ bp + u bp3 / 3 ⋅12 (1 - ν2) / (E t3), (4.3)

Typically for complex tasks, it is not shown in the code how to calculate exactly the rotationalspring constant CΘ required in the formula Θ = u bp3 / CΘ. The spring stiffness K can be usedto 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 effectivearea As. In the simplified method of (Salmi, P. & Talja, A.), Is and As have been replaced bytheir full-cross sectional dimensions in consistence of general principles in calculation ofelastic buckling forces. The general iterative as well as simplified procedures according to thecode to determine the effective thickness of the stiffener teff are in their complexity hard toapply in practical design. Hence, only the simplified, conservative method of (Salmi, P. &Talja, A.) is presented here:

teff = χS t, (4.5)

where χS is the reduction factor for the buckling of a beam on an elastic foundation. The fac-tor is determined according to the buckling curve a0 (α = 0.13, see also Figure 6.1) from theequations:

χS = 1, when λs ≤ 0.2 (4.6a)

χS = 1 / ( φ + √ (φ2 - 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 modeis included in clause 6 of ENV 1993-1-3, where design rules for global buckling are intro-duced. Distortional buckling is handled only qualitatively in the design code, without anyequations. Implicitly it may mean, that FEA is required to be used to analyse this bucklingmode in design. However, if in the case of a section with edge or intermediate stiffeners thestiffener is reduced according to the code, no further allowance for distortional buckling isrequired. Fortunately, distortional buckling mode should not be very probable in thin-walledmembers with ‘normal’ dimensions.

22

TABLE 4.1

DETERMINATION OF EFFECTIVE WIDTH FOR INTERNAL PLATE FIELDS.

23

24

TABLE 4.3MODELLING 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 cross-section 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 thecross-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 as-suming a uniform compressive stress equal to fy / γM1. If the centroid of the effective cross-section does not coincide with the centroid of the gross cross-section, the additional moments(Nsd ⋅ eN) due to shifts eN of the centroidal axes shall be taken into account in combined com-pression and bending. However, according to many references this influence can usually beconsidered negligible.

Bending moment

The design value of bending moment Msd shall not exceed the corresponding resistance of thecross-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 purebending 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’ (nor-mally not). The distribution of the bending stresses shall be linear, if the partial yielding of thecross-section can not be allowed. In case of mono-axial bending plastic reserves in the tensionzone can generally be utilised without strain limits. The utilisation of plastic reserves in thecompression 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 ENV1993-1-3 and in the paper (Salmi, P. & Talja, A.). For biaxial bending, the following criterionshall 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 themajor or minor axes.

Combined tension or compression and bending

Cross-sections subject to combined axial tension Nsd and bending moments MySd and MzSdshall 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 abending moment acts. Weffyten and Weffzten are the effective section moduli for maximum ten-sile stress if subject only to moments about y- and z-axes. In the ENV code there is also anadditional criterion to be satisfied, if the corresponding section moduli for maximum com-pressive stress Weffycom ≥ Weffyten or Weffzcom ≥ Weffzten. The criterion is associated with vecto-rial effects based on ENV 1993-1-1.

Cross-sections subject to combined axial compression Nsd and bending moments MySd andMzSd 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 orWeffzcom ≥ Weffzten, an additional criterion has again to be satisfied. In this occasion, referenceis also made to the basic steel code ENV 1993-1-1 for the concept of vectorial effects. Forsimplicity, in the expression above the bending moments include the additional moments dueto potential shifts of the centroidal axes.

Torsional moment

In good design practice of thin-walled open members, torsional effects should be avoided asfar as practicable, e.g. by means of restraints or ideal cross-sectional shape. If the loads areapplied eccentrically to the shear centre of the cross-section, the effects of torsion “shall betaken 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 beexpected, 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. Thestress τtot is the total shear stress consisting of design stress components τVy and τVz due to theshear 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 directstress components due to resultants NSd, MySd and MzSd should be based on the respective ef-fective cross-sections and all other stress components i.e. shear stresses due to transverseshear 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 cross-section, which shall be taken as the lesser of the shear buckling resistance VbRd or the plasticshear 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 webrelative to the flanges, see Figure 3.1. The shear buckling resistance VbRd shall be determinedfrom:

VbRd = (hw / sinφ) t fbv / γM1, (5.9)

where fbv is the shear buckling strength, which depends on the relative web slenderness λwand stiffening at the support according to the Table 5.2 in ENV 1993-1-3. The relative webslenderness 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 localtransverse force (for simplicity: ‘concentrated load’) applied through the flange, the point loadFsd shall satisfy:

Fsd ≤ RwRd, (5.10)

where RwRd is the local transverse resistance of the web. If the concentrated load is appliedthrough 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 normallyused can be designed for concentrated load according to ENV 1993-1-3 Cl. 5.9.2. The resis-tance formula to be used in the case of single unstiffened web depends on the number (one ortwo), the location and the bearing lengths of the concentrated loads. In addition, the resistancedepends on the geometry (hw, t, r and φ) and material of the web (fy / γM1). In the case of twounstiffened webs, the approach is totally different, although the same parameters affect thepoint 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 thepoint load resistance evaluations is rather empirical.

Combined forces

A cross-section subject to combined bending moment Msd and shear force Vsd shall bechecked 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 ofthe web, both defined previously. A cross-section subject to combined bending moment Msdand 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, de-scribed previously.

29

GLOBAL BUCKLING RESISTANCE OF MEMBERS

Axial compression

A member is subject to concentric compression if the point of loading coincides with the cen-troid of the effective cross-section based on uniform compression. The design value of com-pression 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 con-servatively on uniform compressive stress equal to fy / γM1. The χ-factor is the appropriatevalue 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 bucklingof the member about relevant y- and z-axes, torsional and torsional-flexural buckling. Thecalculation 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 + √ (φ2 - 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 therelative 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 correspondingvalues of α. The buckling curve shall be obtained using ENV 1993-1-3 Table 6.2. The selec-tion of cross-section types in Table 6.2 is very limited. However, the correct buckling curvefor any cross-section may be obtained from the table “by analogy” (how?). As a conclusionfrom the tables (Salmi, P. & Talja, A.), in the case of typical C- and hat profiles Europeanbuckling curve b (α = 0.34) for flexural buckling about both principal axes shall be chosen. Inthe case of other profiles buckling curve c (α = 0.49) shall be used. Regardless of the opencross-section form, the buckling curve b shall be chosen in the case of torsional and flexural-torsional buckling modes. The critical buckling stress in any mode shall be determined in atraditional 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 de-sign, especially because the cross-sectional properties (iy, iz, It, Iw etc.) can be calculated forgross 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 deter-mination of buckling length in torsion taking into account the degree of torsional and warpingrestraint 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 buck-ling 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 iscalculated by means of buckling curve a (αLT = 0.21):

χ LT = 1, when λLT ≤ 0.4 (6.5a)

χ LT = 1 / ( φLT + √ ( φLT 2 - λLT 2 )), when λLT > 0.4 (6.6b)

λLT = √ ( fy / σcr ) (6.6c)

φLT = 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 isthe 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 r-mula 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 yieldsproblems for sure.

Bending and axial compression

In addition to that each design force component shall not exceed the corresponding designresistance, conditions for the combined forces shall be met. In the case of global stability, theinteraction criteria introduced in the code ENV 1993-1-3 are extraordinarily complex. Forpractical design purposes, a more familiar approach for combined bending and axial compres-sion 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 flexuralbuckling forces NEy and NEz corresponding to the normal Euler flexural buckling formula. Inaccordance 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 tothe bending moments. For simplicity and for the fact that they usually can be omitted, no ad-ditional moments are shown in the formula. Interaction between bending and axial compres-sion are considered thoroughly in Cl. 6.5 of the code, but without any explanations of thebackgrounds.

SERVICEABILITY LIMIT STATES

In the design code ENV 1993-1-3, serviceability limit states have been considered on onepage 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 takeninto account in form of effective cross-sectional properties. However, the effective secondmoment of area Ieff can be taken constant along the span, corresponding the maximum spanmoment due to serviceability loading. In the Finnish NAD a more accurate approach is pre-sented, 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 maximumspan moment and maximum support moment, respectively. On the safe side, ultimate limitstate moments may be used. Plastic deformations have to be considered, if theory of plasticityis used for ultimate limit state in global analysis of the structure. The deflections shall be cal-culated assuming linear elastic behaviour. In stead of strange limit value (L/180) for deflec-tion in the ENV draft code the NAD has defined reasonable limits for different thin gaugestructure types. For example, the maximum deflection in the serviceability limit state for roofpurlins is L/200 and for wall purlins L/150.

CONCLUSIONS

In this paper, the main design principles of cold formed thin gauge members (‘thin-walledmembers’) have been considered. The manufacturing process results in typical features ofthin-walled members: quite slender parts in very different open cross-sections and conse-quently many local or global failure modes. The desired properties (usually strength to weightratio) of the members can be reached by optimising cross-sections, but as a by-product, thedesign procedures can be extremely complicated. The total lack of design codes seems to havebeen tranformed into a situation, in which some guidelines are available, but they are hard toadapt in practical design. The theoretical background for analytical design should be ratherwell known, but according to comparative tests, the accuracy of predicted resistance values isstill 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 inaccuracycan be expected and kept in mind in every day design. Complex structural behaviour of thin-walled members has produced inevitably complex design codes (e.g. ENV 1993-1-3). Henceall efforts to derive simplified design methods are naturally welcome. Because all manualmethods 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 atthe moment should be made to reliable calculation programs, which are as simple as possibleto use. This challenging task should preferably be carried out by the same institutions, whichproduce these comprehensive design codes.

33

REFERENCES

ENV 1993-1-3. 1996. Eurocode 3: Design of steel structures. Part 1.3: General rules. Sup-plementary rules for cold formed thin gauge members and sheeting. European Committee forStandardisation CEN. Brussels.

SFS-ENV 1993-1-3. 1996. Eurocode 3: Teräsrakenteiden suunnittelu. Osa 1-3: Yleiset sään-nöt. 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. Designof steel structures. Part 1.3: General rules. Supplementary rules for cold formed thin gaugemembers and sheeting. Ministry of Environment. Helsinki.

ENV 1993-1-1. 1992. Eurocode 3: Design of steel structures. Part 1.1: General rules andrules for buildings. European Committee for Standardisation CEN. Brussels.

TEMPUS 4502. Cold formed gauge members and sheeting. Seminar on Eurocode 3 – Part1.3. 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 de-sign of cold-formed thin gauge steel members). Thesis for the degree of M.Sc.(Tech.), Hel-sinki University of Technology. Espoo.

34

DESIGN CHARTS OF A SINGLE-SPAN THIN-WALLEDSANDWICH ELEMENTS

Karri Kupari

Laboratory of Structural MechanicsHelsinki 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 asingle-span thin-faced sandwich panel. These criteria are bending moment, shear force, deflection andpositive or negative support reaction. The normal stress due to bending moment must not exceed thecapacity in compression of the face layer. The shearing stress due to shear force must not exceed theshearing capacity of the core layer. The maximum deflection can be at the most one percent of the spanand the reaction force from external loads has to remain smaller than the reaction capacity. This paperpresents some details of an investigation using full-scale experiments to determine the estimated level ofcharacteristic 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 usually0.5 … 0.8 mm thick metal sheets and covered with a coat of zinc and preliminary paint. The outersurface is coated with plastic. The most commonly used core layers are polyurethane and mineral wool.

35

core layer

b = 1200 mm

h = 100…150 mm

surface layer, metal sheet

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 momentand the shearing force. The bending moment causes normal stress to the top surface. The core layermust 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 comparedwith the moment of inertia for the whole sandwich panel. This gives us the simplification that thecompression and tension stresses are uniformly distributed across the surface layer. The value of themodulus of elasticity for the surface layer is more than ten thousand times larger than the value of themodulus of elasticity for the core layer. The influence of the normal stresses across the core layer equalszero when considering the behavior of the whole sandwich panel.

The normal stress of the surface layer is

)2,1(f2,1 eA

Mó ±= (1)

and the shearing stress of the core layer is

ebQ

ôs = (2)

M = bending momentQ = shear forcee = the distance between the surface layers center of gravityb = the width of the sandwich panelAf(1,2) = the area of the surface layers cross section

36

τs

σ2

σ1

e

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 ontwo different load cases: Load case A is uniformly distributed transverse loading (Eq. 3 and Fig. 4.) andload case B consist of two symmetrically placed line loads (Eq. 4 and Fig. 5.).

( )GebgL

81

BqL

3845

w24

2L += (3)

( )Geb6FL

BFL

129623

w3

2L += (4)

DEFINING THE SHEAR MODULUS

At the beginning of the testing procedure we can determine the shear modulus. Assuming that the load-deflection curve is linear and using the Hooke´s law we can write F = kw + C. After differentiation weget

kwF =

∂∂

(5)

where k equals the slope of the regression line.

k

deflection [w]

load

[q]

Figure 3: The load-deflection curve.

37

The experimentally defined parameter k leads to the formula that gives us the shear modulus for loadcase A

12

2 B384L5

kL

1eb8G

−

−= (6)

and respectively for load case B

12

B1296L23

kL1

eb6G

−

−= (7)

where B = ½EAf e2 is the bending stiffness. e is the distance between the centers of the surface layers asshown in the Fig. 2. The value of the modulus of elasticity is E = 210 000 N/mm2 and the area of thesurface 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.

3L

3L

3L

2F

2F

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 TechnicalResearch Center in Otaniemi, Espoo. Once the panels were positioned in the chamber, the measuringdevices for force and deflection were set to zero. A polyethylene sheet was placed over the panel andsealed to the sides of the timber casing. The compression force was produced by using a vacuum pumpto decrease the air pressure in the chamber. A total of twelve panels were used in this experiment. Thisprocedure models the distributed load caused by wind. The results of these tests give us the capacity incompression of the surface layer.

Vacuum Chamber

The measuring devices = Force = Deflection

Timber Casing Supports

Polyethene sheet Sandwich Panel

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 HelsinkiUniversity of Technology in the Department of Civil and Environmental Engineering. From the results ofthese tests we can calculate both the shearing and reaction capacity. Altogether 28 panels were used inthis part. The loading was produced by two hydraulic jacks with deflection controlled speed of 2mm/min. The testing continued until the sandwich panels lost their load bearing capacity.

39

0.2 Fu

0.4 Fu

Forc

e

Time

Fu (ultimate force)

Figure 7: The loading history of load case B.

THE CHARACTERISTIC STRENGTHS

Defining the characteristic strengths is based on the instructions from “European Convention forConstructional Steelwork: The Testing of Profiled Metal Sheets, 1978”. It is assumed that all testingresults obey the Gaussian distribution

The Formulas used in defining the characteristic strengths

The value of characteristic strength MK can be calculated from the equation

( )äc1MM mK −= (8)

where Mm = average of the test resultsc = factor related to the number of test results (From Table 1)δ = variation factor

TABLE 1The relation between factor c and the number of test results n

n 3 4 5 6 8 10 12 20 ∞c 2.92 2.35 2.13 2.02 1.90 1.83 1.80 1.73 1.65

The square of the variation factor is

40

1n

MM

n1

MM

ä

n

1i

2n

1i m

i2

i

i

2

−

−

=∑ ∑= = (9)

where n = the number of test resultsMi = the value of test number iMm = 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 normalroom temperature +20 oC with the relative humidity RH of 45-50 %. Second test was made after thematerial was kept for 36 hours in a +70 oC temperature with the relative RH of 100 %. The final partincluded 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

20c

70cc

20t

70tt ó

ódfand

óó

df == (10)

where σt20 = tensile strength at +20 oC temperature, average valueσt70 = tensile strength at +70 oC temperature, average valueσc20 = compression strength at +20 oC temperature, average valueσc70 = compression strength at +70 oC temperature, average value

The factors dft and dfc are divided into two groups

〈≥

)II(7.0

)I(7.0df,df ct (11)

For the case (I) test results of characteristic strengths for the capacity in compression of the surface layerand the shearing and reaction capacity of the core layer are valid. For the case (II) test result must bemultiplied by the following reduction factors

.3.0dfand3.0df ctcdtttd +=Φ+=Φ (12)

The factor related to temperature can be calculated from

20c

80cTc

20t

80tTt E

Eand

EE

=Φ=Φ (13)

41

where Et80 = Modulus of elasticity in tension at +80 oC temperature, average valueEt20 = Modulus of elasticity in tension at +20 oC temperature, average valueEc80 = Modulus of elasticity in compression at +80 oC temperature, average valueEc20 = 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

( )m

fcKttdTffwk ã

fó5.0óã

∗Φ≤∗+ ∆ (14)

where γk = the partial safety factor of external loadσfw = the normal stress caused by external loadσf∆T = the normal stress caused by the temperature difference between inner and outer

surface layersΦttd = the reduction factor related to agingffcK = the characteristic strength of the face layers capacity in compressionγm = the partial safety factor of material

In case of a single span, statically determined structure, the term σf∆T = 0. The normal stress caused byexternal load can be calculated from the formula

ebt8qL

ó2

fw = (15)

where e = the distance between the surface layers' centres of gravityb = 1 [m]t = the thickness of the surface layer

The connection between shear force and shearing capacity

The connection can be given as

( )m

CvKttdTCCwk ã

fô5.0ôã

∗Φ≤∗+ ∆ (16)

where γk = the partial safety factor of external loadτCw = the shearing stress caused by external loadτC∆T = the shearing stress caused by the temperature difference between inner and outer

surface layers

42

Φttd = the reduction factor related to agingfCvK = the characteristic strength of the face layers shearing capacityγm = the partial safety factor of material

In case of a single span, statically determined structure, the term τC∆T = 0. The shearing stress caused byexternal load can be calculated from the formula

eb2qL

ôCw = (17)

where e = the distance between the surface layers' centres of gravityb = 1 [m]

The connection between reaction force and reaction capacity

The connection can be given as

( )m

KtcdTwpk ã

RR5.0Rã

∗Φ≤∗+ ∆ (18)

where γk = the partial safety factor of external loadRwp = the reaction force caused by external loadR∆T = the reaction force caused by the temperature difference between inner and outer

surface layersΦtcd = the reduction factor related to agingRK = the characteristic strength of the reaction capacityγm = the partial safety factor of material

In case of a single span, statically determined structure, the term R∆T = 0. The reaction force caused byexternal load can be calculated from the formula

qLR21

wp = (19)

The boundary conditions concerning deflection

The maximum deflection must remain less than one percent of the span. From external load andtemperature difference between inner and outer surface we get two equations:

( )100L

w5.0wã Tqk ≤∗+ ∆ (20)

( )100L

ww5.0ã Tqk ≤+∗ ∆ (21)

43

where γk = the partial safety factor of external load in serviceability limit state (=1.0)wq = the deflection caused by external loadw∆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 thetemperature difference between inner and outer surface layer is

e8LTá

w2

T∗∆∗=∆ (22)

where α = coefficient of linear thermal expansion for surface layer material, [ ] 1o6 C1012 −−∗∆T = 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 externalload q [kN/m2]. The area located under all four curves represents the permissible combination ofexternal load and span.

44

Design Chart of a single-span thin-walled sandwich element (example)

0

0.5

1

1.5

2

2.5

3

3.5

4

0.0

0

1.0

0

2.0

0

3.0

0

4.0

0

5.0

0

6.0

0

7.0

0

8.0

0

9.0

0

10

.00

11

.00

12

.00

Span L [m]

Ext

ern

al lo

ad q

[kN

/m2]

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, FourthInternational Conference on Steel and Aluminium Structures, Finland, Proceedings book: Light-WeightSteel and Aluminium Structures, edited by Mäkeläinen and Hassinen, pp. 301-308, Elsevier ScienceLtd, 1999.

Martikainen L., Sandwich-elementin käyttäytyminen välituella, Masters Thesis (in Finnish), 1993.

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