Thin-Walled Structures 49 (2011) 682–690

Contents lists available at ScienceDirect

Thin-Walled Structures

0263-82

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/tws

Web crippling behaviour of thin-walled lipped channel beams

M. Macdonald a, M.A. Heiyantuduwa Don a, M. Kote"ko b,n, J. Rhodes c

a School of Engineering & Computing, Glasgow Caledonian University, Glasgow, UKb Department of Strength of Materials & Structures, Technical University of Lodz, Stefanowskiego 1/15, 90-924 Łodz, Polandc Department of Mechanical Engineering, University of Strathclyde, Glasgow, UK

a r t i c l e i n f o

Available online 6 October 2010

Keywords:

Web-crippling

Load-capacity

Code predictions

FE analysis

Plastic mechanism

Experiment

31/$ - see front matter & 2010 Elsevier Ltd. A

016/j.tws.2010.09.010

esponding author.

ail address: Maria.Kotelko@p.lodz.pl (M. Kote

a b s t r a c t

This paper presents the results of an investigation into web crippling behaviour—conducted on cold-formed

thin-walled steel lipped channel beams subjected to Interior-One-Flange (IOF), Interior-Two-Flange (ITF),

End-One-Flange (EOF) and End-Two-Flange (ETF) loading conditions as defined by the American Iron and

Steel Institute (AISI). An experimental program was designed to obtain the load-deformation characteristics

of beam members with varying cross-sectional and loading parameters under the three web crippling

loading conditions. The results obtained from the experiments comprised of the ultimate web crippling

strength values and displacements of the thirty-six beam specimens tested. Nonlinear finite element

models were developed to simulate web crippling failure of the two loading conditions considered in the

experimental program. Also, a combination of elastic analysis with a plastic mechanism approach was

employed to investigate the load-deformation characteristics of lipped channel members subjected to the

IOF loading condition. The comparison of experimental, finite element and plastic mechanism approach

results revealed that the nonlinear finite element models were best capable of closely simulating the web

crippling failure behaviour observed in the experiments for all ranges of displacement. Web crippling

strength predicted from the Eurocode 3, Part 1.3 [1], and the Polish PN-B-0327 [2] design specifications

were also compared with the experimental results and the comparisons indicated considerable

underestimations for the range of specimens under EOF and ETF loading conditions.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Web crippling failure may occur at places where thin-walledflexural members are subjected to high concentrated loadings orsupport reaction forces. Fig. 1 illustrates web crippling failure at aloading point. Web crippling deformation is defined as the decreaseof cross-section height below the load-bearing plate [3].

Four different loading conditions, where web crippling maytake place, have been defined by the AISI based on the number ofloadings involved and the location of failure initiated, namely,Interior-One-Flange (IOF), Interior-Two-Flange (ITF), End-One-Flange (EOF) and End-Two-Flange (ETF) loading conditions[4]—Fig. 2(a)–(d), respectively.

A considerable amount of research has been carried out onweb crippling over many years by numerous researchers,particularly to validate various design rules for web crippling,and the majority were based on experimental investigations. Theearly research work conducted by Winter and Pian [5], Ratliff [6],Hetrakul and Yu [7], etc. provided the basis for web cripplingdesign rules that appeared in the early versions of the AISI

ll rights reserved.

"ko).

Specification and is consequently adopted by the other majordesign codes including Eurocode 3, Part 1.3. In the recent past,number of investigations were carried out by Young and Hancock,Prabakaran and Schuster, and Shaojie, Yu and LaBoube, and theseresulted in a more unified form of design rule which was adoptedby the AISI Specification, 2001 edition.

Web crippling (crushing behaviour) of hat section beams wasinvestigated by Hofmeyer [3] and Hofmeyer et al. [8], whoimplemented the yield-line analysis (plastic mechanism analysis)to the investigation of the crushing behaviour of top hat-sectionbeams subject to three-point bending—a similar approach wasapplied by Bakker and Stark [9].

A research program was initiated to investigate web cripplingbehaviour of cold-formed thin-walled lipped channel beamsunder the four loading conditions. The results of the preliminaryexperimental investigations and the finite element analysis oflipped channel beams under IOF and ITF loading conditions werereported in previous publications [10,11]. The aim of this paper isto present the results of experimental investigations and finiteelement analysis carried out to investigate the web cripplingbehaviour of lipped channel sections under IOF, EOF and ETFloading conditions. The experimental results were also comparedwith the web crippling strength predictions from Eurocode 3-1.3and with the Polish code PN-B-0327, which adopted the Eurocode

M. Macdonald et al. / Thin-Walled Structures 49 (2011) 682–690 683

recommendations. The results for the IOF loading condition werealso compared with the analytical solution of the plasticmechanism problem approach [12,13], leading to an evaluationof the unloading path.

2. Experimental investigations

Experimental investigations were designed to examine theinfluence of various cross-sectional and loading parameters onweb crippling strength.

Experimental tests for IOF loading conditions (Fig. 2a) wereperformed by Heiyantuduwa and others and described in detail in[12,14]. Two separate series of tests were performed consideringEOF and ETF loading conditions. The test specimens were fixed onto load bearing plates during both series of tests to prevent flangerotations and possible lateral movements of specimens during

Fig. 1. Web crippling at loading point.

IOF Loading.

EOF Loading.

Fig. 2. (a) IOF loading; (b) ITF loading; (

loading. Each series comprised of eighteen test specimensmanufactured from 0.78 mm thickness carbon steel sheets. Thetest specimens were designed to have three different corner radiiand two different web heights and were loaded with threedifferent sizes of load bearing plate. Fig. 3 illustrates the cross-sectional and loading parameters used in the specimen design.

A separate series of tensile tests was carried out prior tospecimen manufacture in order to obtain the material propertiesof the individual steel sheets.

During the web crippling tests, applied load, displacement atthe loading point and the displacement at a number of othercritical points were measured. Results of the experimentalinvestigations were used to validate the finite element models,to check the validity of web crippling strength predictionsobtained from design codes and to compare with theoreticalunloading paths obtained from a plastic mechanism analysis.Detailed results of material tests are given in [10,14].

2.1. EOF loading tests

EOF loading tests were performed as three-point bendingtests; however, the failure was intended to occur at the end of thebeam (at supports) and the loading was applied to the mid-pointof the beam.

The load bearing plate was fully fixed at the mid-point in orderto prevent failure around this area. The test rig used in the EOFloading tests is shown in Fig. 4.

2.2. ETF loading tests

ETF loading tests were performed by applying a load which wasdirectly above the support. Hence, the failure was initiated at theend of the beam due to the heavy loading and the support reactionforce. The test rig used in ETF loading tests is shown in Fig. 5.

ITF Loading

ETF Loading.

c) EOF loading; and (d) ETF loading.

Fig. 3. Cross-sectional and loading parameters.

Fig. 4. Test rig for EOF loading tests.

Fig. 5. Test rig for ETF loading tests.

M. Macdonald et al. / Thin-Walled Structures 49 (2011) 682–690684

3. Finite element (FE) models

Finite element models were developed to simulate the testsconducted in the experimental investigations. The finite elementanalysis package ANSYSs was employed for the modelling andanalysis procedure. Nonlinear characteristics such as materialnonlinearity, geometric nonlinearity and contact situations wereconsidered to accurately represent web crippling failure. Twodifferent finite element models were developed to represent EOFand ETF loading tests described in the experimental investigations.

FE model for IOF loading conditions was developed byHeiyantuduwa and others and described in detail in [12,14].

3.1. FE models for EOF loading condition (EOF-FE models)

EOF-FE models were developed to simulate the EOF loadingtests carried out in the experimental investigations. The geo-metric model for the EOF-FE models was similar to the test setupused in the EOF loading tests. However, the advantage of vertical

Support Block (Solid Elements)

Fig. 6. Finite element mesh for EOF-FE model.

Fig. 7. Element mesh for EOF-FE model: boundary conditions.

M. Macdonald et al. / Thin-Walled Structures 49 (2011) 682–690 685

symmetry was used to create a half-model in this case.The geometry was initially created using the solid modellingtechniques within ANSYS. Fig. 6 shows the element meshgenerated for the EOF-FE models. In this case, web cripplingfailure was expected to occur at the support reaction point. Thus,the mesh was controlled to have relatively finer elements closerto the support area and coarser elements further away from thesupport area.

The support reaction force was applied using a support blockmodelled with solid elements and appropriate boundary condi-tions were employed to simulate the actual supports used in theexperiments. Contact elements were employed in betweenthe support block and the lipped channel beam to represent theactual loading situation. Furthermore, the flange-fixed conditionwas represented using a set of nodes with coupled degrees-of-freedom. The loading was applied with displacement control on toa set of nodes selected along the bottom centre line of the supportblock. The rotation about the Z-axis was restrained along thecentre line to represent the actual support conditions in the testsetup. Fig. 7 shows the boundary conditions used for the EOF-FEmodels. A set of nodes around the mid-span of the beam was fullyrestrained against translations and rotations in all directions.

3.2. FE models for ETF loading condition (ETF-FE models)

ETF-FE models were developed to simulate the carried out ETFloading tests. In this case, web crippling failure was expected tooccur at the end of the beam under two opposite forces in line

with each other. The geometric model for the ETF-FE models wassimilar to the test setup used in the ETF loading tests. The ETFloading setup was symmetrical about the horizontal plane passingthrough the centre line of the beam. Therefore, only one-half ofthe setup was modelled to use the advantage of symmetry. Fig. 8shows the element mesh generated for the ETF-FE models. Webcrippling failure was identified to occur around the central area ofthe web under the load bearing plates. Thus, the mesh wascreated to have relatively small elements around the central partof the web and larger, coarser elements further away from thefailure region.

The loading was applied through a load bearing plate using thedisplacement control method. Contact elements were usedbetween the load bearing plate and the top flange of thebeam. Fig. 9 shows the boundary conditions used in the ETF-FEmodels.

4. Post-failure behaviour—modelling and test results

A very competitive and time-saving method (in comparisonwith FEM), which enables the determination of the ultimate loadand the post-failure behaviour of a thin-walled profile, is thecompilation of the post-buckling analysis (performed using FS,effective width approach or other analytical method) with theplastic mechanism solution. The advantage of using this approachis that it provides not only a very simple algorithm of the ultimateload determination but also a very quick answer of the questionabout the post-failure behaviour of the structure. On the otherhand, the main drawback may be a relatively high errordepending on the level of approximation of the real plasticmechanism of failure. The yield-line analysis leading to thedetermination of the plastic mechanism of failure was performedby Bakker and Hofmeyer, who used this approach to investigatethe web crippling of hat section beams [3,8,9].

On the basis of experimental results obtained for IOF loadingconditions [12,14], the theoretical model of the plastic mechan-ism was developed (Fig. 10a), close to the failure pattern observedin the experiment (Fig. 10b).

Post-failure behaviour of beams subject to EOF and ITF loadingconditions was substantially different (Fig. 11). Thus, results oftheoretical yield-line analysis can be compared with the testresults for IOF conditions only.

The geometry of ‘‘W-shaped’’ plastic mechanism and details ofthe theoretical solution based on the energy method are given in[12]. After performing a numerical derivation of the energy ofplastic deformation with respect to the angle of rotation atthe global plastic hinge, the load vs. deflection curve is obtained.

Beam (Shell Elements – Shell 181)

Load Bearing Plate (Solid Elements)

Fig. 8. Finite element mesh for ETF-FE model.

Fig. 9. Element mesh for ETF-FE model: boundary conditions.

Fig. 10. Web crippling post-failure modes: (a) ‘‘W-shaped’’ theoretical plastic mechanism model and (b) post-failure mode for IOF test specimen.

Fig. 11. Web crippling post-failure modes for: (a) EOF and (b) ITF loading conditions.

M. Macdonald et al. / Thin-Walled Structures 49 (2011) 682–690686

An intersection point of the latter with the elastic (pre- or post-buckling) path is generally considered as an upper-boundestimation of the ultimate load.

Comparisons of load vs. deflection curves obtained from plasticmechanism analysis, FE simulation and test results for IOF loadingconditions are shown in Fig. 12.

2,000

1,800

1,600

1,400

1,200

1,000

0,800

8,000

0,600

6,000

0,400

4,000

0,200

2,0000,000

0,000

Load

[kN

]

-2,000 10,000 12,000 14,000Deflection [mm]

ExperimentalFEAMechanism

Fig. 12. Comparison of load vs. displacement curve from plastic mechanism

approach with experimental and finite element analysis results for a beam

member with 100 mm web height, corner radius 1.25 mm and loaded with

100 mm wide bearing plate.

Displacment of Load

K

Loa

d

Curve-ACurve-B

Curve-C

Fig. 13. Typical load vs. deflection curves observed in web crippling under IOF

loading conditions.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0 10 20 30 40 50Displacement (mm)

Load

(kN

)

Collapse Curve- Test 43

FE Curve -Test 43

Fig. 14. Comparison of collapse curves obtained from plastic mechanism approach

and a nonlinear finite element analysis

M. Macdonald et al. / Thin-Walled Structures 49 (2011) 682–690 687

Experimental results indicated (Fig. 13) that the load–dis-placement curves for IOF loading conditions can have variousshapes depending on the section parameters and the load bearinglength. The type of failure shown in Curve-A is exhibited insections with small corner radii and loaded with wide loadbearing plates. In this case, the load–displacement curve shows arelatively sharp transition from elastic to plastic failure due to thesmall top corner and the web is subjected to more direct loading.In the literature, such a failure mode is referred to as a ‘‘brittlefailure’’. Load–displacement curves similar to Curve-C wereobserved in members with large corner radii and small bearinglengths. In this case, the top corner and the top portion of the webboth undergo high stress levels due to the bending produced bythe offset of loading. This effect is represented in the gradualchange from elastic to plastic collapse through the elasto-plastictransition of the load–displacement curves. This failure mode istermed by some researchers as a ‘‘ductile failure’’. Curve-B istypical for intermediate values of corner radii and bearing lengths.

The theoretical model of ‘‘W-shaped’’ mechanism coincideswith failure mode displayed by sections with small corner radiiand loaded with wide bearing plate (Curve-A).

Fig. 14 shows the comparison of FE simulation and the failurecurve obtained from the plastic mechanism solution for a beamwith larger corner radius (2.5 mm) and length of bearing plate(100 mm). In this case the plastic mechanism curve is onlycapable of representing the post-failure behaviour well beyondthe initial collapse stage.

5. Eurocode and Polish code (PN) web crippling strengthpredictions, test results and comparisons

The nominal web crippling strength of thirty-six specimensunder EOF and ETF loading conditions was determined usingEurocode 3, Part 1.3, and the equivalent Polish code. It should behighlighted that the Polish Code incorporates the Eurocoderecommendations entirely, and that the Eurocode providesindividual design equations to predict the nominal web cripplingstrength of each loading condition. However, these equations canbe rearranged in terms of the factors corresponding to thecontribution of individual parameters such as corner radii, webheight and load bearing length. The local transverse resistance ofa web may be determined from:

For EOF loading:

Rw,Rd ¼ k1k2k3½9:04�ðhw=tÞ=60�½1þ0:01ðss=tÞ�t2fyb=gM1 ð1Þ

For ETF loading:

Rw,Rd ¼ k1k2k3½6:66�ðhw=tÞ=64�½1þ0:01ðss=tÞ�t2fyb=gM1 ð2Þ

All terms are as defined in the Eurocode 3, Part 1.3.The load–displacement graphs obtained from the tests and

finite element analysis were used to determine the ultimate webcrippling strength of the specimens under EOF and ETF loadingconditions. Tables 1 and 2 show the web crippling strength resultsobtained from the tests (Pexp:ult.), finite element analysis (PFE:ult.)and Eurocode (PEuro) for these loading conditions.

Figs. 15 and 16 illustrate sample load–displacement graphsobtained from tests and finite element analysis along with thecorresponding Eurocode web crippling strength predictions. Thefinite element strength and the nominal web crippling strengthpredicted from Eurocode were compared with the experimentalweb crippling strength results.

Table 3 shows the mean and standard deviation of ratiosbetween finite element strength and experimental strength(PFE:ult./Pexp:ult.), as well as Eurocode predictions and experimentalstrength (PEuro/Pexp:ult.).

Figs. 17 and 18 show graphs comparing PEuro/Pexp:ult. withvarying lipped channel specimen corner radii for the EOF loadingcondition, with web depths of 75 and 100 mm, and load bearingplate lengths varying from 25 to 100 mm. Figs. 19 and 20 showgraphs comparing PEuro/Pexp:ult. with varying lipped channelspecimen corner radii for the ETF loading condition, with webdepths of 75 and 100 mm, and load bearing plate lengths varyingfrom 25 to 100 mm.

Table 1Web crippling strength results for EOF loading condition.

Test No. h (mm) ri (mm) N (mm) t (mm) Span Length—

Ls (mm)

0.2% Prf stress

(MPa)

Pexp:ult (kN) PFE:ult. (kN) PEuro (kN)

EOF-1 95.2 4.0 25 0.78 600 220 1.18 1.14 0.46EOF-2 95.5 2.6 25 0.78 600 220 1.24 1.19 0.80EOF-3 97.3 1.2 25 0.78 600 220 1.46 1.41 1.13EOF-4 95.2 4.0 100 0.78 600 220 1.74 1.61 0.80EOF-5 95.5 2.6 100 0.78 600 220 2.00 1.92 1.38EOF-6 97.3 1.2 100 0.78 600 220 2.25 2.39 1.96EOF-7 95.5 2.6 50 0.78 600 220 1.43 1.52 0.99EOF-8 97.3 1.2 50 0.78 600 220 1.70 1.93 1.41EOF-9 95.2 4.0 50 0.78 600 220 1.34 1.34 0.58EOF-10 73.3 1.2 50 0.78 600 220 1.80 2.09 1.51EOF-11 70.0 2.6 50 0.78 600 220 1.44 1.57 1.07EOF-12 69.2 4.0 50 0.78 600 220 1.32 1.40 0.78EOF-13 73.3 1.2 25 0.78 600 220 1.44 1.43 1.22EOF-14 70.0 2.6 25 0.78 600 220 1.12 1.23 0.86EOF-15 69.2 4.0 25 0.78 600 220 1.10 1.20 0.50EOF-16 73.3 1.2 100 0.78 600 220 2.35 2.61 2.10EOF-17 70.0 2.6 100 0.78 600 220 1.90 2.17 1.49EOF-18 69.2 4.0 100 0.78 600 220 1.62 1.75 0.87

Table 2Web crippling strength results for ETF loading condition.

Test No. h (mm) ri (mm) N (mm) t (mm) Span length—

Ls (mm)

0.2% Prf stress

(MPa)

Pexp:ult (kN) PFE:ult. (kN) PEuro (kN)

ETF-1 73.0 1.6 25 0.78 400 220 0.87 0.91 0.77

ETF-2 73.4 2.4 25 0.78 400 220 0.81 0.95 0.62

ETF-3 65.2 5.0 25 0.78 400 220 0.76 0.82 0.17

ETF-4 98.2 1.6 25 0.78 400 220 1.25 0.98 0.69

ETF-5 96.2 2.4 25 0.78 400 220 0.98 0.92 0.57

ETF-6 89.8 5.0 25 0.78 400 220 0.80 0.81 0.16

ETF-7 73.0 1.6 50 0.78 400 220 1.38 1.29 0.96

ETF-8 73.4 2.4 50 0.78 400 220 0.92 1.18 0.78

ETF-9 65.2 5.0 50 0.78 400 220 1.14 0.95 0.21

ETF-10 98.2 1.6 50 0.78 400 220 1.24 1.20 0.86

ETF-11 96.2 2.4 50 0.78 400 220 1.22 1.12 0.71

ETF-12 89.8 5.0 50 0.78 400 220 0.98 0.96 0.19

ETF-13 73.0 1.6 100 0.78 400 220 1.84 1.90 1.33

ETF-14 73.4 2.4 100 0.78 400 220 1.76 1.72 1.08

ETF-15 65.2 5.0 100 0.78 400 220 1.58 1.34 0.23

ETF-16 98.2 1.6 100 0.78 400 220 1.72 1.64 1.20

ETF-17 96.2 2.4 100 0.78 400 220 1.56 1.52 0.98

ETF-18 89.8 5.0 100 0.78 400 220 1.28 1.32 0.27

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Displacement (mm)

Load

(kN

)

Experimental FEA Eurocode

0.0 5.0 10.0 15.0 20.0 25.0 30.0

Fig. 15. Experimental, FE load–displacement graphs compared with Eurocode web crippling strength prediction—EOF-1.

M. Macdonald et al. / Thin-Walled Structures 49 (2011) 682–690688

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 5.0 10.0 15.0 20.0 25.0 30.0

Displacement (mm)

Load

(kN

)

Experimental FEA Eurocode

Fig. 16. Experimental and FE load–displacement graphs compared with Eurocode web crippling strength prediction—ETF-5.

Table 3Summary of comparisons.

Loading conditionMean of strength ratios Standard deviation of strength ratios

PFE:ult./Pexp:ult. PEuro/Pexp:ult. PFE:ult./Pexp:ult. PEuro/Pexp:ult.

EOF 1.01 0.66 0.13 0.16

ETF 0.98 0.52 0.11 0.26

0.0

0.2

0.4

0.6

0.8

1.0

1.2

P Eur

o/P e

xp.u

lt.

n=25 mm n=50 mm n=100 mm

0.0 1.0 2.0 3.0 4.0 5.0Corner Radii, ri (mm)

Fig. 17. Comparison of Eurocode web crippling strength predictions with

experimental ultimate strength values for specimens of 75 mm web height under

EOF loading condition (tests: EOF-10 to EOF-18).

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 1.0 2.0 3.0 4.0 5.0Corner Radii, ri (mm)

P Eur

o/P

exp.

ult.

n=25 mm n=50 mm n=100 mm

Fig. 18. Comparison of Eurocode web crippling strength predictions with

experimental ultimate strength values for specimens of 100 mm web height

under EOF loading condition (tests: EOF-1 to EOF-9).

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 1.0 2.0 3.0 4.0 5.0 6.0Corner Radii, ri (mm)

P Eur

o/P e

xp.u

lt.

n=25 mm n=50 mm n=100 mm

Fig. 19. Comparison of Eurocode web crippling strength predictions with

experimental ultimate strength values for specimens of 75 mm web height under

ETF loading condition (tests: ETF-1 to ETF-9).

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 1.0 2.0 3.0 4.0 5.0 6.0Corner Radii, ri (mm)

P Eur

o/ P e

xp. u

lt.

n=25 mm n=50 mm n=100 mm

Fig. 20. Comparison of Eurocode web crippling strength predictions with

experimental ultimate strength values for specimens of 100 mm web height

under ETF loading condition (tests: ETF-10 to ETF-18).

M. Macdonald et al. / Thin-Walled Structures 49 (2011) 682–690 689

M. Macdonald et al. / Thin-Walled Structures 49 (2011) 682–690690

6. Conclusions

The results showed that the nonlinear finite elements modelsdeveloped were capable of closely representing the web cripplingfailure of the specimens considered in this research. An averagedeviation of 72% of finite element strength from experimentalresults was observed.

The nominal web crippling strength of thirty-six specimenssubject to EOF and ETF loading conditions predicted usingEurocode 3, Part 1.3 (equivalent to the Polish code), werecompared with the experimental results. The comparisonsindicated averages of 34% and 48% underestimations of Eurocodeweb crippling strength predictions for the EOF and ETF loadingconditions, respectively.

It was also observed that the length of the load bearing plate,along with the value of corner radii and web depth, all had aneffect on the web crippling strength of the lipped channelstested—particularly noted for the IOF and EOF loading conditions.However, no definite trends could be observed for the ETF loadingcondition.

Unloading paths for beams under IOF loading conditionsobtained from the plastic mechanism analysis represented thesame character of ‘‘brittle failure’’ as experimental results forspecimens of small corner radii. Quantitative agreement oftheoretical and experimental results was rather unsatisfactory.Theoretical failure curves used in this case in compilation withelastic paths to the upper-bound estimation of the beam ultimateload would give an over-estimation of about 40% in comparisonwith FE and experimental results. One of the reasons fordiscrepancies was that the analysis did not account for any strainhardening effect.

For sections with large corner radii, failure curves obtainedfrom the plastic mechanism analysis were only capable ofrepresenting the global collapse mechanism, which normallyoccurs well beyond the initial collapse stage.

References

[1] ENV 1993-1-3, Eurocode 3: design of steel structures; Part 1.3: supplemen-tary rules for cold formed thin gauge members and sheeting, Brussels,Belgium; February 1996.

[2] PN-B-0327: steel structures—structures of cold formed thin gauge membersand sheeting—design and execution.

[3] Hofmeyer H. Cross-section crushing behaviour of hat sections (Part I:analytical modelling). International Journal of Thin-Walled Structures2005;43:1155–65.

[4] Rhodes J. Design of cold formed steel members. England: Elsevier AppliedScience; 1991.

[5] Winter G, Pian RHJ. Crushing strength of thin steel webs. EngineeringExperiment Station, Cornell University, Bulletin No. 35, Part 1; 1946.

[6] Ratliff GD. Interaction of concentrated loads and bending in C-shaped beams.In: Proceedings of the 3rd international specialty conference on cold-formedsteel structures, St. Louis, Missouri, U.S.A., November 1975.

[7] Hetrakul N, Yu WW. Cold-formed steel I-beams subjected to combinedbending and web crippling. In: Proceedings of the international conference onthin-walled structures. Glasgow: University of Strathclyde; April 1979.

[8] Hofmeyer H, et al. New prediction model for failure of steel sheeting subjectto concentrated load (web crippling and bending moment). InternationalJournal of Thin-Walled Structures 2001;39:773–96.

[9] Bakker MCM, Stark JWB. Theoretical and experimental research on webcrippling of cold-formed flexural steel members. International Journal ofThin-Walled Structures 1994;18:261–350.

[10] Heiyantuduwa MA, Macdonald, M, Rhodes J. Investigation of web cripplingbehaviour of thin-walled lipped channel beam members. In: Proceedings ofthe 3rd international conference on structural engineering, mechanics andcomputation, Cape Town, South Africa; September 2007. p. 375–76.

[11] Macdonald M, Heiyantuduwa MA, Rhodes, J. Finite element analysis of webcrippling behaviour of cold-formed steel flexural members. In: Proceedings ofthe 18th specialty conference on cold-formed steel structures, Orlando,Florida, USA; October 2006.

[12] Heiyantuduwa MA, Macdonald M, Kote"ko, M., Rhodes, J. Plastic mechanismapproach to web crippling behaviour of cold formed channel section beams.In: Proceedings of the 6th international conference on steel and aluminiumstructures (ICSAS’2007). Oxford Brookes University; July 2007.

[13] Heiyantuduwa MA, Macdonald M, Rhodes J, Kote"ko M. Theoreticalinvestigation of web crippling behaviour of thin-walled lipped channelbeams. In: Proceedings of the 5th international conference on coupledinstabilities in metal structures (CIMS 2008) vol. 1. Australia: University ofSydney; June 2008.

[14] Heiyantuduwa MA. Web crippling behaviour of cold formed thin-walled steellipped channel beams. PhD Thesis. Glasgow Caledonian University; 2008.