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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
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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
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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
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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
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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: [email protected](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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17
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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18
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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19
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.
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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:
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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.
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22
TABLE 4.1
DETERMINATION OF EFFECTIVE WIDTH FOR INTERNAL PLATE FIELDS.
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23
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24
TABLE 4.3MODELLING OF ELEMENTS OF A CROSS-SECTION.
Figure 4.1 Determination of spring stiffness in two simple
cases.
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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
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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
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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)
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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.
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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.
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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)
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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
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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.
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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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