inelastic behaviour of cold-formed channel sections in bending · ultimate capacity of cold-formed...

Inelastic Behaviour of Cold-Formed

Channel Sections in Bending

by

Soheila Maduliat

B.Eng

A thesis submitted in fulfilment of the requirements for the degree of

Doctor of Philosophy

Department of Civil Engineering

Monash University, Australia

October 2010

Copyright Notices Notice 1 Under the Copyright Act 1968, this thesis must be used only under the normal conditions of scholarly fair dealing. In particular no results or conclusions should be extracted from it, nor should it be copied or closely paraphrased in whole or in part without the written consent of the author. Proper written acknowledgement should be made for any assistance obtained from this thesis. Notice 2 I certify that I have made all reasonable efforts to secure copyright permissions for third-party content included in this thesis and have not knowingly added copyright content to my work without the owner's permission.

iii

Declaration

This thesis contains no material that has been accepted for the award of any other

degree or diploma in any university or other institution. The author affirms that to the

best of her knowledge this thesis contains no material previously published or written

by another person, except where due reference is made in the text of the thesis.

……………….

Soheila Maduliat

October 2010

iv

Acknowledgements

First and foremost, I would like to express my deepest gratitude to my first supervisor,

Dr Michael Bambach, for initiation, guidance and supervision of my PhD work

throughout this research. Also, special thanks to Prof Xiao-Ling Zhao for taking over

my supervision, his support and encouragement.

I gratefully acknowledge Department of Civil Engineering, Monash University for

providing me a departmental scholarship and supporting me towards this study.

I would like to express my sincere gratitude to staff members Mr. Long Goh, Mr. Alan

Taylor, Mr. Kevin Nievaart, Mr. Patrick Arias, Mr. Jeffrey Doddrell, Mr. Glenn Davis,

Mr. Peter Dunbar and Mr. Don McCarthy of the Civil Engineering Laboratory for their

assistance with the experiments. I wish to thank Mr. Godwin Vaz, Mr. Chris Powel and

Mr. Rob Alexander for their cooperation. I would also like to thank all staff and my

fellow postgraduates of the Department of Civil Engineering, Monash University, for

their friendship, encouragement and support. My gratitude also goes to Mrs. Jennifer

Manson for her great help and friendship during my whole time at Monash University.

I am indebted to my husband Dr Reza Rajabpour for his patience, sacrifice, and

understanding. I would never have completed this PhD without his companionship and

support.

I want to thank my family for their constant support and help. Thanks must go to the

two whom I admire and respect the most, my mother and father, for their endless

support and encouragement throughout the course of my life.

Last not least I would like to acknowledge my beloved daughter Tara, whose love is the

main drive in my life.

v

Summary

This thesis investigates the behaviour of cold-formed channel sections with edge

stiffener under pure bending. The primary aim of this research is to examine the

inelastic bending capacity of cold-formed channel sections and in doing so provide

design rules to account for such behaviour. Design rules are prepared for cold-formed

steel specifications (inelastic reserve capacity) NASPEC (2007) and AS/NZS4600

(2005) as well as hot-rolled steel specifications (compact, non-compact and slender

classes) AS4100 (1998).

To investigate the behaviour of cold-formed channel sections under pure bending, this

study conducts an extensive experimental and numerical analysis of 42 cold-formed

channel sections in three different geometrical categories (simple channel sections,

channel sections with simple edge stiffener and channel sections with complex edge

stiffener) to determine the effect of different edge stiffeners on the ultimate strength of

cold-formed channel sections. The sections are made from cold-formed G450 steel with

nominal thickness of 1.6mm and varying theoretical buckling stresses ranging between

elastic to seven times the yield stress.

The ultimate bending moment capacities of the sections are calculated from six

methods being the: test result ( testM ), NASPEC (2007) design rules ( NASPECM ),

AS/NZS4600 (2005) design rules ( 4600ASM ), DSM ( DSMM ), EUROCODE3 (2006)

design rules ( 3EurocodeM ) and AS4100 (1998) design rules ( 4100ASM ). The testM is then

used as a benchmark to gauge the accuracy of the NASPECM , 4600ASM , DSMM , 3EurocodeM

and 4100ASM .

The results of the test investigations showed that the existing design rules in NASPEC

(2007), AS/NZS4600 (2005), DSM and EUROCODE3 (2006) are conservative and the

sections classifications in AS4100 (1998) are inaccurate for cold-formed channel

sections. Therefore, the experimental results were used to revise the existing design

Summary vi

methods (NASPEC (2007), AS/NZS4600 (2005) and DSM) for determining the

ultimate capacity of cold-formed channel sections in bending and also defining new

slenderness limits for sections classifications in AS4100 (1998).

The yield line mechanism model is proposed and compared with the test results in order

to investigate the behaviour of cold-formed channels with edge stiffener after collapse.

Numerical (finite element) analyses is then developed and verified with the test results

and used to investigate deformation process of cold-formed channel sections under

bending that could not be monitored during the experimental program to complement

the test results.

The outcome of this study is to determine the section geometry for which a cold-formed

channel section can reach the fully plastic capacity and maintain it for sufficient

rotation, such that when employed in a structure such as a portal frame it may be

considered applicable for plastic mechanism analysis, thus allowing for increased

design capacities and more economical structural solutions.

vii

Table of Contents

Declaration ....................................................................................................................... iii Acknowledgements .......................................................................................................... iv Summary ........................................................................................................................... v Table of Contents ............................................................................................................ vii List of Figures .................................................................................................................. xi List of Tables .................................................................................................................. xv NOTATIONS AND ACRONYMS .............................................................................. xvii Chapter 1 ........................................................................................................................... 1 INTRODUCTION ............................................................................................................ 1

1.0 Background ............................................................................................................. 1

1.1 Production of Cold-Formed Sections ...................................................................... 2

1.2 Plastic Design and Inelastic Reserve Capacity ....................................................... 4

1.3 Aims of the Research .............................................................................................. 6

1.4 Outline of the Research ........................................................................................... 6

Chapter 2 ........................................................................................................................... 8 LITERATURE REVIEW ................................................................................................. 8

2.0 Chapter Synopsis ..................................................................................................... 8

2.1 Section Strength ...................................................................................................... 8

2.1.1 Local buckling .............................................................................................. 9 2.1.2 Distortional buckling .................................................................................. 11 2.1.3 Interaction effects between the elements ................................................... 15

2.2 Cold-Formed Design Rules ................................................................................... 16

2.2.1 Effective width method .............................................................................. 16 2.2.1.1 Effective width of uniformly compressed stiffened and unstiffened

elements ................................................................................................. 18 2.2.1.2 Effective width of stiffened elements with stress gradient .............. 20 2.2.1.3 Effective width of unstiffened elements with stress gradient .......... 21 2.2.1.4 Effective width of uniformly compressed elements with an edge

stiffener .................................................................................................. 24 2.2.2 Direct strength method ............................................................................... 27 2.2.3 Post yielding or inelastic reserve capacity of cold-formed steel ................ 32

2.3 Hot-Rolled Design Rules ...................................................................................... 34

2.3.1 Rotation capacity ........................................................................................ 36 2.3.2 Section classification .................................................................................. 37 2.3.3 Elastic limits for compression elements ..................................................... 38 2.3.4 Elastic limits in bending elements.............................................................. 40 2.3.5 Slenderness limits for non-compact elements ............................................ 40 2.3.6 Plastic limits for compression elements ..................................................... 42 2.3.7 Plastic limits in bending elements .............................................................. 43

2.4 Plastic Design ........................................................................................................ 47

2.5 Collapse Behaviour of a Cold-Formed Structure .................................................. 53

Table of Contents viii

2.5.1 Yield line theory......................................................................................... 53 2.5.2 Yield line mechanism model...................................................................... 54

2.6 Conclusions ........................................................................................................... 56

Chapter 3 ......................................................................................................................... 58 TEST PROCEDURES OF COLD-FORMED CHANNEL SECTIONS UNDER PURE

BENDING .................................................................................................................... 58 3.0 Chapter Synopsis ................................................................................................... 58

3.1 Material Properties ................................................................................................ 59

3.2 Mechanical Properties and Preparation of the Specimens .................................... 61

3.3 Bending Rig Set up ............................................................................................... 68

3.4 Bending Rig Modifications ................................................................................... 69

3.5 Bending Test Procedures ...................................................................................... 73

3.5.1 Curvature calculations................................................................................ 73 3.5.2 Bending moment calculations .................................................................... 75

3.6 Conclusions ........................................................................................................... 80

Chapter 4 ......................................................................................................................... 81 EXPERIMENTAL RESULTS AND DISCUSSIONS ................................................... 81

4.0 Chapter Synopsis ................................................................................................... 81

4.1 Sections Classifications ......................................................................................... 82

4.2 Slender Sections .................................................................................................... 83

4.2.1 Moment-curvature graphs of the slender sections ..................................... 87 4.3 Non-Compact Sections ......................................................................................... 91

4.3.1 Moment-curvature graphs of the non-compact sections ............................ 92 4.4 Compact Sections .................................................................................................. 95

4.4.1 Failure modes from testing a compact section (section 40) ....................... 96 4.4.2 Moment-curvature graphs of the compact section ..................................... 97

4.5 Failure Modes for Tested Sections ........................................................................ 98

4.6 Comparing the Elastic Portion of the Moment-Curvature Graphs of the Test

Results with the EWM Results ......................................................................... 105

4.7 Comparing the Test with the Design Rules Results ............................................ 117

4.7.1 Nominal member moment capacity ......................................................... 119 4.8 Conclusions ......................................................................................................... 139

Chapter 5 ....................................................................................................................... 141 REVISING EXISTING DESIGN RULES AND SLENDERNESS LIMITS .............. 141

5.0 Chapter Synopsis ................................................................................................. 141

5.1 Reliability Analysis ............................................................................................. 141

5.2 Inelastic Reserve Capacity .................................................................................. 144

Table of Contents ix

5.2.1 Proposed inelastic design model for partially stiffened compression members ...................................................................................................... 150

5.3 AS/NZS4600 Design Rules................................................................................. 156

5.3.1 A proposed revision for the AS/NZS4600 design model......................... 157 5.4 Direct Strength Method Design Rules ................................................................ 162

5.4.1 A proposed revised DSM design model .................................................. 162 5.4.2 Revised proposed methods for local buckling failure .............................. 162 5.4.3 Revised proposed methods for distortional buckling failure ................... 166

5.5 Elastic and Plastic Slenderness Limits in AS4100 (1998) .................................. 173

5.6 Conclusions ......................................................................................................... 179

Chapter 6 ....................................................................................................................... 182 YIELD LINE MECHANISM (YLM) ANALYSIS OF COLD-FORMED CHANNEL

SECTIONS UNDER BENDING ............................................................................... 182 6.0 Chapter Synopsis ................................................................................................. 182

6.1 YLM Model for Cold-Formed Channel Beams .................................................. 183

6.2 Failure Curve ....................................................................................................... 186

6.3 Estimating the Ultimate Moment Capacity ......................................................... 199

6.4 A Proposed Method for Estimating the Rotation Capacity ................................. 202

6.5 Comparison between the Test and the YLM Bending-Curvature Diagrams .. 205

6.6 Energy Absorbers ................................................................................................ 208

6.6.1 Energy absorption computation ............................................................... 209 6.7 A Simplified YLM Equation for the Cold-Formed Channel Sections ................ 213

6.7.1 Estimating the ultimate moment capacity using simplified method ........ 218 6.8 Conclusions ......................................................................................................... 220

Chapter 7 ....................................................................................................................... 222 FINITE ELEMENT METHOD (FEM) ANALYSIS OF COLD-FORMED CHANNEL

SECTIONS UNDER BENDING ............................................................................... 222 7.0 Chapter Synopsis ................................................................................................. 222

7.1 ABAQUS Models ............................................................................................... 222

7.1.1 Mesh Density…………………………………………………………....223 7.2 Material and Geometrical Nonlinearity .............................................................. 226

7.3 Results of the Simulation .................................................................................... 226

7.4 Simulation Result for Two Compact Sections .................................................... 233

7.5 Deformation of the Tested Sections Prior to Their Collapse Point ..................... 234

7.6 Conclusions ......................................................................................................... 242

Chapter 8 ....................................................................................................................... 243 CONCLUSIONS AND RECOMMENDATIONS ....................................................... 243

8.0 General ................................................................................................................ 243

Table of Contents x

8.1 Conclusions ......................................................................................................... 245

8.2 Recommendations for Future Study ................................................................... 250

REFERENCES.............................................................................................................. 251 Appendix A ................................................................................................................... 262 Appendix B ................................................................................................................... 269 Appendix C ................................................................................................................... 284 Appendix D ................................................................................................................... 299 Appendix E ................................................................................................................... 314 Appendix F .................................................................................................................... 328 Appendix G ................................................................................................................... 342 Appendix H…………………………………………………………………………...356

xi

List of Figures

Figure 1.1: (a) Hot-rolling steel (b) Rolling mill for cold-forming metal ......................... 1 Figure 1.2: Cold-formed sections used in structural framing ........................................... 2 Figure 1.3: Cold forming tools (Hancock (1988)) ............................................................ 3 Figure 1.4: Roll forming process for cold-formed hollow sections (Wilkinson (1999)) .. 3 Figure 1.5: Stress and strain distribution at first yield moment and plastic moment ........ 4

Figure 2.1: Buckled member (Bambach (2003)) .............................................................. 9 Figure 2.2: Local buckling of a plate element................................................................. 10 Figure 2.3: Values of k for calculating different theoretical buckling stress

(Timoshenko and Gere (1961)) ..................................................................... 11 Figure 2.4: Different buckling modes (Hancock (1988))................................................ 12 Figure 2.5: Finite strip analysis of flange and lip (Schafer and Pekoz (1999)) .............. 15 Figure 2.6: Effective design sections (AS/NZS4600 (2005)) ......................................... 17 Figure 2.7: Stress distribution in effective width method (Bambach (2003)) ................ 17 Figure 2.8: Stiffened and unstiffened elements .............................................................. 18 Figure 2.9: Effective width of uniformly compressed stiffened and unstiffened elements

(AS/NZS4600(2005)) .................................................................................... 19 Figure 2.10: Effective width of stiffened elements with stress gradient (AS/NZS4600

(2005)) ........................................................................................................... 20 Figure 2.11: Effective width of unstiffened elements with stress gradient (EUROCODE

(2006)) ........................................................................................................... 22 Figure 2.12: Effective width of unstiffened elements with stress gradient (EUROCODE

(2006)) ........................................................................................................... 22 Figure 2.13: Slender section in minor axis bending (Bambach (2003)) ......................... 23 Figure 2.14: Effective width of an element with edge stiffener (AS/NZS4600 (2005)) . 25 Figure 2.15: Comparison of FEA and experimental data with the DSM curve under (a)

compression (Zhu and Young (2006)) (b) bending (Zhu and Young (2009)) ....................................................................................................................... 28

Figure 2.16: Comparison of DSM with test results (Yu and Schafer (2007)) ................ 29 Figure 2.17: Compression strain factor for compression flange (Hancock (1988)) ....... 32 Figure 2.18: Stress and strain for inelastic reserve capacity (Hancock (1988)) .............. 33 Figure 2.19: Measurement of Rotation Capacity (Wilkinson (1999)) ............................ 36 Figure 2.20: Moment-curvature of different type of steel section (Elchalakani et al.

(2002b)) ......................................................................................................... 38 Figure 2.21: Classification of plate width ....................................................................... 39 Figure 2.22: Classification of plate depth ....................................................................... 40 Figure 2.23: Definition of web depth .............................................................................. 42 Figure 2.24: Width of a flange in Korol and Hudoba (1972) ......................................... 43 Figure 2.25: Allowable d/t ratios of webs of fully plastic sections for σo =33 ksi

(Haaijer and Thuerlimann (1958)) ................................................................ 44 Figure 2.26: Effect of slenderness ratio in Kemp (1996) method ................................... 45 Figure 2.27: Compact limits for Cold-formed RHS beams ............................................ 47 Figure 2.28: Position of plastic hinges in Wilkinson’s portal frames based on test results

(Wilkinson (1999)) ........................................................................................ 49

List of Figures xii

Figure 2.29: Vertical deflection of Wilkinson’s portal frames based on test results (Wilkinson(1999)) ......................................................................................... 50

Figure 2.30: Geometry of the section in Baigent’s portal frames test (not to scale) ...... 50 Figure 2.31: Test data compare to Direct Strength Method result for beams (Schafer

(2006a)) ......................................................................................................... 52 Figure 2.32: Yield line mechanism for box sections under bending (Koteko (2004)) ... 55 Figure 2.33: Basic yield line mechanism (Zhao (2003)) ................................................ 56

Figure 3.1: Tensile coupon specimen in accordance to the AS1391 (2005) ................... 59 Figure 3.2: Stress-Strain Curves ..................................................................................... 60 Figure 3.3: Typical channel sections............................................................................... 62 Figure 3.4: Sections dimensions ..................................................................................... 65 Figure 3.5: Calculating I value of section 1 .................................................................... 66 Figure 3.6: Filled sections ............................................................................................... 67 Figure 3.7: Front view of Monash pure bending rig ....................................................... 68 Figure 3.8: Schematic of the Monash pure bending rig .................................................. 69 Figure 3.9: Modified wheel ............................................................................................. 70 Figure 3.10: Installation of the restraining plates ........................................................... 71 Figure 3.11: Restraining plate ......................................................................................... 71 Figure 3.12: Buckling modes for section 22 ................................................................... 72 Figure 3.13: Determination of the curvature from measured rotation angles ................. 74 Figure 3.14: Determination of the curvature from measured strains .............................. 75 Figure 3.15: Geometry of the bending rig....................................................................... 76 Figure 3.16: Force diagram at the left support wheel of the pure bending rig ................ 77 Figure 3.17: Comparing moments from two different methods ..................................... 79

Figure 4.1: Width of the element .................................................................................... 83 Figure 4.2: Normalised moment-curvature diagram with yield moment and yield

curvature respectively for three slender sections .......................................... 87 Figure 4.3: Normalised moment curvature graphs based on test results for section 2 and

17 ................................................................................................................... 88 Figure 4.4: Normalised moment curvature graphs based on test results for section 9 and

11 ................................................................................................................... 89 Figure 4.5: Different stage of the loading for section 13 ................................................ 90 Figure 4.6: Section behaviour in the different stage of the loading ................................ 91 Figure 4.7: Normalised moment-curvature diagram with plastic moment and plastic

curvature respectively for few non-compact sections ................................... 92 Figure 4.8: Normalised moment-curvature diagram with plastic moment and plastic

curvature respectively for section 10 ............................................................ 93 Figure 4.9: Normalised moment-curvature diagram with plastic moment and plastic

curvature respectively for section 38 ............................................................ 94 Figure 4.10: Normalised moment-curvature diagram with plastic moment and plastic

curvature respectively for section 39 ............................................................ 95 Figure 4.11: Flange-web distortional failure modes for section 40 ................................ 96 Figure 4.12: Normalised moment-curvature diagram with plastic moment and plastic

curvature respectively for section 40 ............................................................ 97 Figure 4.13: Local buckling mode appearance during the bending test ......................... 98

List of Figures xiii

Figure 4.14: Deformation of the failed sections.............................................................. 99 Figure 4.15: The rotation angles due to the deformation of the compression flange and

the deformation of the web flange juncture respectively verses width to depth ratio of the tested sections ................................................................. 101

Figure 4.16: The failure modes of section 17 ............................................................... 102 Figure 4.17: The failure modes of section 24 ............................................................... 103 Figure 4.18: The failure modes of section 4 ................................................................. 104 Figure 4.19: The failure modes of section 37 ............................................................... 105 Figure 4.20: Comparison between the test result with EWM results and also distortional

buckling check ............................................................................................ 113 Figure 4.21: Comparison between test results and EWM and distortional buckling check

results graph for selected tested sections .................................................... 115 Figure 4.22: Comparison between test and existing design rules results ...................... 136 Figure 4.23: Comparison between test and AS4100 design rules results ..................... 138

Figure 5.1: Normalised moment-strain and moment-curvature diagrams of section 2 . 144 Figure 5.2: Position of neutral axis ............................................................................... 146 Figure 5.3: Elastic-plastic stress distribution (Cy>1) .................................................... 147 Figure 5.4: Elastic stress distribution (Cy<1) ................................................................ 147 Figure 5.5: Position of neutral axis based on the resultant axial force ......................... 149 Figure 5.6: Dividing a typical section into smaller elements ........................................ 150 Figure 5.7: Comparison between the proposed inelastic model and the experimental

results .......................................................................................................... 151 Figure 5.8: Slenderness limits for plastic mechanism analysis ..................................... 156 Figure 5.9: Comparison between the proposed AS/NZS4600 model for nominal member

capacity due to distortional buckling and the experimental results ............ 158 Figure 5.10: Comparison between the proposed DSM models and the experimental

results for local buckling ............................................................................. 163 Figure 5.11: Comparison between the proposed DSM models and the experimental

results for distortional buckling .................................................................. 167 Figure 5.12: Comparison between the existing and the proposed slenderness limits ... 173 Figure 5.13: Sections classification into two individual groups ................................... 174 Figure 5.14: Comparison between the proposed and the existing slenderness limits ... 174

Figure 6.1: Common observed failure mode for the tested simple channel sections .... 183 Figure 6.2: YLM model in channel-section columns and beams (Koteko (2004)) ...... 183 Figure 6.3: (a) Common observed YLM model for the edge stiffener and the flange (b)

Common observed YLM model for the web and the flange ....................... 184 Figure 6.4: YLM model for cold-formed channel sections with edge stiffener ............ 185 Figure 6.5: Longitudinal cross-section of the web YLM model ................................... 187 Figure 6.6: Angle η1 ...................................................................................................... 188 Figure 6.7: Measured a1 and a2 values .......................................................................... 191 Figure 6.8: a1 from test measurement over assumed a1 ratio verses width to depth ratio

of the tested sections ................................................................................... 193 Figure 6.9: a2 from test measurement over assumed a2 ratio verses edge stiffener to

width ratio of the tested sections ................................................................. 194

List of Figures xiv

Figure 6.10: Comparing collapse curves of sections 12, 13, 20 and 21 based on different values of a2 .................................................................................................. 195

Figure 6.11: The ratio of test result over the YLM results verses the width to depth ratio of the tested sections ................................................................................... 200

Figure 6.12: Normalised moment-curvature diagram from the test and the YLM results for a slender section .................................................................................... 206

Figure 6.13: Normalised moment-curvature diagram from the test and the YLM results for a non-compact section ........................................................................... 207

Figure 6.14: Normalised moment-curvature diagram from the test and the YLM results for a compact section .................................................................................. 207

Figure 6.15: A vehicle body structure (Lu and Yu (2003)) .......................................... 208 Figure 6.16: A W beam barrier ..................................................................................... 209 Figure 6.17: Dividing a graph based on Simpson rules ................................................ 210 Figure 6.18: Divided moment-rotation graph of section 3 based on Simpson rules ..... 211 Figure 6.19: Energy absorption from test results over the YLM results ratio verses the

width to depth ratio ..................................................................................... 213 Figure 6.20: The best curve fit with the test result........................................................ 214 Figure 6.21: The best curve fit for calculating the X factor from the ratio of the sections

slenderness over their plastic slenderness limit .......................................... 216 Figure 6.22: Normalised moment-curvature diagram from the test and the simplified

proposed method for a slender section ........................................................ 217 Figure 6.23: Normalised moment-curvature diagram from the test and the simplified

proposed method for a non-compact section .............................................. 217 Figure 6.24: Normalised moment-curvature diagram from the test and the simplified

proposed method for a compact section ...................................................... 218

Figure 7.1: Commonly used elements in ABAQUS (Hibbitt et al. (2009)) ................. 223 Figure 7.2: FEM for different mesh sizes ..................................................................... 224 Figure 7.3: Normalised moment-curvature of sections 8, 35 and 41 for different mesh

density. ........................................................................................................ 225 Figure 7.4: Comparison between the normalised moment-curvature graphs of the

section 9 based on inclinometers readings and the FEM results ................. 228 Figure 7.5: Rotation at point A ..................................................................................... 228 Figure 7.6: Rotation at point B ..................................................................................... 229 Figure 7.7: Comparison between the normalized moment-curvature graphs of the

section 9 at points A and B ......................................................................... 229 Figure 7.8: Failure position for section 9 ...................................................................... 230 Figure 7.9: Comparison between the normalised moment-curvature graphs of section 9

based on strain gauges readings and the FEM result .................................. 231 Figure 7.10: Histograms of the ratio of test results over the FEM results .................... 232 Figure 7.11: Comparison between the normalised moment-curvature graphs of the

sections 40, A and B ................................................................................... 234 Figure 7.12: Buckling modes (Rogers (1995)) ............................................................. 235 Figure 7.13: Deformation of section 9 .......................................................................... 237 Figure 7.14: Deformation of section 42 ........................................................................ 239 Figure 7.15: Deformation of section 40 ........................................................................ 241

xv

List of Tables

Table 2.1: Different standard section classification (Wilkinson (1999)) ........................ 37 Table 2.2: Web slenderness limit classification in bending (Wilkinson (1999)) ............ 41 Table 2.3: Ultimate load of the Wilkinson’s portal frames based on test results ............ 49 Table 2.4: Loading pattern on Baigent and Hancock portal frames ............................... 51

Table 3.1: Tensile coupon test results ............................................................................. 61 Table 3.2: The dimensions of each section ..................................................................... 63

Table 4.1: Sections classification based on test result .................................................... 84 Table 4.2: Sections classification based on AS4100 ....................................................... 85 Table 4.3: The rotation angle due to the deformation of the compression flange, l , and

also the rotation angle due to the deformation of the web flange juncture, d ,

for the tested sections .................................................................................. 100 Table 4.4: yC value based on test result and ultM based on test result and EWM ...... 118

Table 4.5: Ultimate moment capacities of the tested sections based on NASPEC and AS/NZS4600 design rules ........................................................................... 126

Table 4.6: Ultimate moment capacities of the tested sections based on DSM ............. 128 Table 4.7: Ultimate moment capacities of the tested sections based on EUROCODE3

and AS4100 design rules ............................................................................. 133 Table 4.8: The ratio of the ultimate moment capacity over the yield moment with the

values based on AS/NZS4600 and NASPEC due to distortional buckling mode and DSM. .......................................................................................... 137

Table 5.1: Proposed inelastic reserve capacity model data ........................................... 148 Table 5.2: Proposed AS/NZS4600 model data ............................................................. 159 Table 5.3: Proposed DSM model data for local buckling ............................................. 165 Table 5.4: Proposed DSM model data for distortional buckling .................................. 168 Table 5.5: Proposed DSM model data .......................................................................... 170 Table 5.6: Proposed AS4100 model data ...................................................................... 176 Table 5.7: Mean values, COV and reliability index of proposed and existing design

methods ....................................................................................................... 180

Table 6.1: Comparison between test and assumed values for 1a and 2a ...................... 192 Table 6.2: Comparison between test results and YLM results...................................... 201 Table 6.3: Calculated rotation capacity value ............................................................... 204 Table 6.4: t-test and Wilcoxon signed rank test results for YLM verses test results .... 205 Table 6.5: Comparison between absorbed energy for the tested sections based on test

results and the YLM results ........................................................................ 212 Table 6.6: The value of X factor ................................................................................... 215 Table 6.7: Comparison between test results and simplified method results. ................ 219

Table 7.1: t-test and Wilcoxon signed rank test results for FEM verses test results..... 227

List of Tables xvi

Table 7.2: Comparison between ultimate moment capacities of the tested sections based on the FEM and the test results ................................................................... 232

Table 7.3: Dimensions and ultimate capacities of sections A and B based on revised design rules ................................................................................................. 234

Notations and Acronyms xvii

NOTATIONS AND ACRONYMS

Notations:

A Effective section area fA Flange area

b Width of section

1b Depth of section

2b Width of section

3b and 4b Edge stiffeners widths

fb Width of a section

wb Depth of a section

eb Effective width

1eb and 2eb Effective width of stiffened element with stress gradient

yC Compression strain factors

d Depth of section

mD Mean values for dead load

E Young’s modulus of elasticity

ue Elongation

*f Design stress in the compression element

crf Theoretical buckling stress

odf Section’s theoretical distortional buckling stress

olf Element’s theoretical local buckling stress

yF Yield stress

uF Tensile strength

mF Mean ratio of actual section modulus to the nominal value

dh Distance between the top and the bottom flanges

I Second moment of area of a cross section

exI Effective second moment of area of a cross section about x axis

Notations and Acronyms xviii

xfullI Full unreduced second moment of area of a cross section

k Plate buckling coefficient

k Curvature

k Buckling factor in EUROCODE3

yk Yield curvature

pk Plastic curvature

ultk Ultimate curvature

effL Effective length

mL Mean values for live load

lL Out of plane deflection for the compression flange elements

mM Mean ratio of the yield point to the minimum specified value

oM Elastic lateral buckling moment

cM Critical moment

yM Yield moment

pM Plastic moment

beM Member moment capacity for lateral-torsional buckling

bdM Member moment capacity for distortional buckling

nalbdistortioM Member moment capacity for distortional buckling

blM Member moment capacity for local buckling

olM Elastic local buckling moment

odM Elastic buckling moment in the distortional mode

testM Ultimate moment capacity based on test result

NASM Ultimate moment capacity based on NASPEC 2007 design rules

4600ASM Ultimate moment capacity based on AS/NZS4600 2005 design rules

DSMM Ultimate moment capacity based on DSM design rules

3EurocodeM Ultimate moment capacity based on EUROCODE3 2006 design rules

4100ASM Ultimate moment capacity based on AS4100 1998 design rules

Notations and Acronyms xix

sM Section moment capacity

pm Plastic moment capacity of the steel elements

mP Mean ratio of the experimental results to the predicted results

R Resistance of the section

Rotation capacity

nR Nominal resistance

mR Mean values of the resistance

er Outside bend radius

ir Inside bend radius

S Plastic section modulus

Load effect

mS Mean values of the load effect

xS Plastic section modulus about x axis

t Plate thickness

ft Flange’s thickness

wt Web’s thickness

RV Coefficient Of Variation (COV) of the resistance

SV Coefficient Of Variation (COV) of the load effect

DV COV values for dead load

LV COV values for live load

iW Work components for each plastic hinge

w Out of plane deflection for a plate element

cY Distance from the neutral axis to the compression edge of the section

Z Elastic section modulus

eZ Effective section modulus

exZ Effective section modulus about x axis

xfullZ Full unreduced section modulus

d Rotation angle due to the deformation of the web flange juncture

Notations and Acronyms xx

l Rotation angle due to the deformation of the compression flange due to

local buckling

Reliability index

Load factor

D Dead load factor

L Live load factor

Left Rotational angle on the left side of the beam

Right Rotational angle on the right side of the beam

Rotational angle of the beam

c Compression strain

t Tension strain

ult Ultimate compressive strain

y Yield strain

Slenderness ratio

d Slenderness ration subject to distortional buckling

e Element slenderness ratio

Flange Flange’s slenderness ratio

l Slenderness ration subject to local buckling

s Section slenderness ratio

sp Plastic slenderness limit

sy Elastic slenderness limit

Poisson’s ratio

Resistance factor

Effective width factor

Stress ratio

Acronyms:

C Compact section

Notations and Acronyms xxi

DSM Direct Strength Method

EWM Effective Width Method

FEA Finite Element Analysis

FEM Finite Element Method

FOSM First Order Second Moment

FSM Finite Strip Method

NC Non-compact section

S Slender section

YLM Yield Line Mechanism

1

Chapter 1

INTRODUCTION

1.0 Background

The Australian demand for pre-fabricated metal buildings is approximately $780

million per annum. This steel is used for domestic, agricultural, industrial and

temporary structures. There are two types of steel structural members that are typically

used in the construction industry: hot-rolled and cold-formed. In hot-rolled steel

members, large pieces of metal are heated above their recrystallisation temperature and

then formed into different cross sections. However, in cold-formed members the metal

is deformed by being passed through rollers at room temperature. Figure 1.1 shows the

hot-rolling and cold-forming process.

(a) (b)

Figure 1.1: (a) Hot-rolling steel (b) Rolling mill for cold-forming metal

Applications of hot-rolled sections have been mainly in large scale commercial and

industrial structures. Cold-formed sections have been used in car bodies, highway

barriers (energy absorbers) and in secondary structural elements including roof and wall

frames. The cold-formed sections are made of steel sheets, strips or plates where their

Chapter1. Introduction 2

thickness is normally between 0.5 to 6mm. Figure 1.2 shows the most common cold-

formed open sections in the steel industry.

Figure 1.2: Cold-formed sections used in structural framing

In the last 20 years, cold-formed sections have also become popular for primary

structures, particularly temporary structures which are predominately made from cold-

formed channel sections. In addition, steel frames (which are mainly cold-formed

channel sections) are used in 10 per cent of Australia’s residential buildings. Cold-

formed steel structures are booming in Australian, North American, Europe and the UK

markets. Compared with hot-rolled sections, cold-formed steel sections are easier to

fabricate into complex shapes, more compact in packaging and have higher strength to

weight ratio.

The following two sections explain the production and design rules for cold-formed

sections.

1.1 Production of Cold-Formed Sections

Cold-formed sections are produced by roll forming or brake pressing operations at

ambient temperature. Roll forming machines consist of pairs of opposing rolls that form

strips into the final shape. Each pair of opposing rolls is called a stage (Figure 1.3).

Brake pressing equipment consists of a moving top beam and a stationary bottom bed

(Figure 1.4). For sections with several folds, the pressing operation needs to be repeated

several times with different positions of the steel plate.


Figure 1.3: Cold forming tools (Hancock (1988))

Figure 1.4 shows how cold-formed hollow sections are produced from thin steel strip in

the roll forming operation.

Figure 1.4: Roll forming process for cold-formed hollow sections (Wilkinson (1999))


1.2 Plastic Design and Inelastic Reserve Capacity

The plastic design method allows for the larger application of loads on sections due to

the redistribution of yield stress through the depth of the section (Figure 1.5). This

method can increase the capacity of a channel section by up to 20-30 per cent greater

than the first yield capacity which is calculated by the elastic design method. If a

section in beams or portal frames (structural assemblies) reaches its plastic moment, a

plastic hinge will develop at that stage. If that plastic hinge can rotate sufficiently to

redistribute the moment through the member, the additional load can be resisted by the

structural assemblies which can be up to 70 per cent greater than the first yield

capacity. Therefore, the plastic design method would be more economical compared to

the traditional elastic design method.

Figure 1.5: Stress and strain distribution at first yield moment and plastic moment

In Australia, hot-rolled sections are designed under the Australian Standard for Steel

Structures AS4100 (1998) and cold-formed sections under the Australian/New Zealand

Standards for Cold-formed Steel Structures AS/NZS4600 (2005). The Australian

Standard AS4100 (1998) allows the plastic design of hot-rolled sections. However,

AS/NZS4600 (2005) does not allow for the plastic design of cold-formed sections. This

is due to the fact that cold-formed sections do not satisfy the following plastic design

rules which are given in AS4100 (1998):

The yield stress must not exceed 450 MPa. However, cold-formed sections may

have a yield stress higher than 450 MPa;

The ratio of the ultimate tensile stress to the yield stress must not be less than 1.2.

However, ultimate tensile stress to the yield stress ratio for cold-formed steel may be

less than 1.2; and


The steel must exhibit a strain hardening capacity. However, in cold-formed steel

strain hardening commences immediately after yielding.

Wilkinson and Hancock (1998) performed tests on three portal frames manufactured

from cold-formed Rectangular Hollow Sections (RHS). They concluded that the failure

was not due to the lack of material strain hardening capacity.

The additional capacity of the member beyond the first yielding is called the inelastic

reserve capacity. The inelastic reserve capacity design method is especially suitable for

portal frame structures in agricultural and industrial buildings where deflection

limitations can be relaxed. In the Australian Standard, AS/NZS4600 (2005), the

inelastic reserve capacity design method is restricted to fully effective sections. The

method cannot be used to design slender sections or sections with complex flange or

web stiffeners. While the inelastic reserve capacity design method is not applicable for

cold-formed channels with edge stiffeners, Baigent and Hancock (1981) tested seven

portal frames manufactured from cold-formed channels with edge stiffener and

illustrated the inelastic behaviour of cold-formed channel sections with a capacity of 25

per cent to 70 per cent greater than the first yield capacity. It appears that certain cold-

formed channel sections have a bending capacity beyond the yield capacity. It is noted

that by using edge stiffeners instead of increasing a section’s thickness, a slender

channel section may be fully effective at its ultimate capacity. Currently the edge

stiffener is not included in the inelastic reserve capacity design method. Experimental

data in the literature is mostly for slender sections and there is inadequate experimental

data for sections with the ultimate capacity of equal and/or greater than its yield

capacity. Therefore, the existing design rules assumption is that the ultimate moment

capacity of cold-formed sections cannot reach beyond the yield capacity, which is

unduly conservative. The purpose of this thesis is to therefore investigate the inelastic

and post-collapse behaviour, improve the existing design methods and propose a

method to determine energy absorption capacity of cold-formed channel sections in

bending. This research is therefore designed to extend the application of increasingly

sought after and valuable cold-formed channel sections.


1.3 Aims of the Research

The main aim of this study is to determine the section geometry for which a cold-

formed channel section can reach the fully plastic capacity and maintain it for sufficient

rotation, such that when employed in a structure such as a portal frame it may be

considered applicable for plastic mechanism analysis, thus allowing for increased

design capacities and more economical structural solutions.

To address the current limitation of design standards, this research investigates the

inelastic bending capacity of cold-formed channel sections, and provides design rules to

account for such behaviour. Design rules will be prepared for cold-formed steel

specifications (inelastic reserve capacity), and hot-rolled steel specifications (compact,

non-compact and slender classes). In addition, determining energy absorption capacity

of cold-formed channel sections under bending is another aim of this research.

These aims will be achieved by conducting research with the following steps:

Performing bending tests on channel sections with and without complex edge

stiffeners;

Reviewing the existing design standards and comparing them with the test results;

Developing design rules to account for inelastic strength;

Simulating the tested beams using Yield Line Mechanism (YLM) analysis to

investigate their behaviour after collapse and determining their energy absorption

capacity;

Proposing a simplified equation for determining the behaviour of cold-formed

channel sections after collapse; and

Simulating the tested beams using finite element analysis to complement the test

results.

1.4 Outline of the Research

To achieve the research aims, the following structure has been adopted. Chapter 2

presents the literature review on cold-formed sections, the range of design standards for

designing cold-formed channel sections, the sections classifications and Yield Line


Mechanism analysis. This chapter provides a theoretical basis and an understanding of

the existing theory as valuable context for the research. Chapter 3 then describes the

experimental procedures. Forty two different beams with various theoretical buckling

stresses were tested using the Monash pure bending rig to examine their ultimate

moment capacity and their post failure performance. Chapter 4 investigates the

experimental results with the forty two tested sections being classified into slender,

non-compact and compact classes based on the experimental results. Following which,

the experimental results are compared with the existing cold-formed design method

results and it is concluded that the existing design rules are conservative. In chapter 5,

the experimental results are used to revise the existing design rules in cold-formed and

hot-rolled specifications. The revised design methods apply inelastic behaviour on cold-

formed channels. Chapter 6 then focuses on YLM analysis by describing the YLM

model of the deformed beams and plotting the collapse curve of each section using

YLM analysis, compared with the test collapse curve. The YLM analysis is then used

to determine the energy absorption of the sections, and is compared with the energy

absorption from the test result. A simplified equation is then developed for determining

the collapse curve of each section based on the test results. In chapter 7, finite element

analysis of each tested section is performed using the commercial program ABAQUS.

The aim of the finite element analysis was to describe the deformation process of the

tested sections. Finally, chapter 8 summarises the findings and suggest future directions

for further research.

8

Chapter 2

LITERATURE REVIEW

2.0 Chapter Synopsis

This chapter reviews and discusses literature on cold-formed sections. The focus is on

the treatment of cold-formed sections in North America, Australia and Europe as cold-

formed steel structures are booming in these regions markets. As this research focuses

on how cold-formed steel performs under a range of conditions, specific literature that

relate to the experiments is also reviewed. This literature is on the elastic and plastic

slenderness limit for steel sections, distortional buckling in thin wall structures, the post

yielding of cold-formed steel and finally concept of Yield Line Mechanism which is a

method to determine the load-carrying capacity and also post collapse behaviour of

cold-formed structures. This literature therefore provides valuable context to draw

conclusions and determine the need for this specific research.

2.1 Section Strength

Section strength is not only controlled by a material yielding but also by local,

distortional and lateral-torsional buckling. In local buckling, the plate element buckles

without any deformation of the web flange juncture. In distortional buckling, the shape

of the cross section is changed and the flange element rotates around the web flange

intersection. In lateral-torsional buckling, the whole section twists and bends without

any changes in the section’s shape. When a member locally buckles, a number of

buckled wavelengths along the longitudinal direction are formed. As it is shown in

Figure 2.1, since in the elastic range all the locally buckled cells behave in the same

manner, instead of the whole member, the behaviour of a single locally buckled cell can

be studied (Bambach (2003)).

Chapter 2. Literature Review 9

Figure 2.1: Buckled member (Bambach (2003))

This thesis is concerned with the section strength of cold-formed channel sections in

bending subject to local and distortional buckling. Cold-formed channel sections, as

primary structures, are always fully restrained to avoid lateral instability using wall and

roof framing. Therefore, the lateral-torsional buckling is not the concern of this thesis.

The following sections are an in-depth explanation of local and distortional buckling.

2.1.1 Local buckling

Since cold-formed sections are fabricated from thin plate elements, they are prone to

local buckling. Figure 2.2 shows a plate simply supported on all four edges under a

uniform uniaxial compression stress. To calculate the force that can cause a local

buckling on this plate, Bryan’s differential equation is used. This is based on the small

deflection theory (Bryan (1891)).

2

2

4

4

22

4

4

4

2

3

)2()1(12 x

wtf

y

w

yx

w

x

wEtx

(2.1)

where, is Poisson’s ratio, t is the plate thickness and w is the out of plane deflection

for the plate element.


Figure 2.2: Local buckling of a plate element

The critical value of an element’s theoretical local buckling stress is from AS/NZS4600

(2005):

2

2

2

)1(12

b

tEkfcr

(2.2)

where, k is the plate buckling coefficient that depends on the longitudinal edge support

and distribution of stress across the plate. The values of k for different conditions are

shown in Figure 2.3 (Timoshenko and Gere (1961)). To avoid local buckling prior to

the yielding failure, crf should reach the yield stress of yF .

It is important to note that to determine the theoretical local buckling stress of a section

due to the interaction effect between elements, the THINWALL computer program

based on Papangelis and Hancock (1995) research is used.


Figure 2.3: Values of k for calculating different theoretical buckling stress (Timoshenko and Gere (1961))

2.1.2 Distortional buckling

Three modes of buckling for lipped channels in compression and bending are shown in

Figure 2.4. As can be observed, the half-wavelength of the distortional buckling is

between the local and the lateral buckling. Schafer and Pekoz (1999) concluded that the

distortional mode is firstly more sensitive to imperfection compared to the local mode

and secondly, the distortional buckling has less post buckling capacity compared to the

local buckling. Finally, the distortional buckling can cause a section failure even if its

stress is higher than the theoretical local buckling stress. Schafer and Pekoz developed

a new hand-method to calculate critical local and distortional theoretical buckling stress

in which compared well with the experimental results.


Figure 2.4: Different buckling modes (Hancock (1988))

Lau and Hancock (1987) developed a simplified expression for predicting theoretical

distortional buckling stress of cold-formed channel sections in compression. Lau and

Hancock method has been validated by comparing with an accurate Finite Strip Method

results (Lau and Hancock (1987)). When examining Lau and Hancock’s (Lau and


Hancock (1987)) method for calculating the theoretical distortional buckling stress for

flexural cold-formed channel sections, Hancock (1997) compared Lau and Hancock’s

method with an accurate solution based on Finite Strip Method (FSM). Hancock

concluded that Lau and Hancock’s method provides a close estimation of theoretical

distortional buckling stress for sections in bending to the FSM.

Schafer et al.(2006) studied the effect of complex edge stiffeners on the distortional

buckling behaviour of thin wall members. They also compared different methods of

calculating distortional buckling together with the experimental data. A summary of

their conclusions are that:

Hancock’s method is conservative for calculating the distortional buckling of sections

with slender webs. Therefore, Hancock method works well for web hight-to-thickness

ratio of less than 200;

Edge stiffener can increase distortional stress in a compression flange. However, if

the stiffener’s length increases, it may cause local instability; and

FSM provides a more accurate result compared to Hancock and Schafer hand-

method.

The Australian Standard AS/NZS4600 (2005) has design formulas for theoretical

distortional buckling stress based on Lau and Hancock’s (1987) method. Bambach et al.

(1998) used AS/NZS4600 (2005) formulas to calculate the theoretical distortional

buckling stress of sections with complex edge stiffeners. The results were compared

with finite strip buckling analysis results based on thin-wall program (Papangelis and

Hancock (1995)). By suggesting some limitations to AS/NZS4600 (2005) formulas,

Bambach et al. (1998) concluded that AS/NZS4600 (2005) has a high degree of

accuracy for calculating theoretical distortional buckling stress. Bambach et al. (1998)

limitations for sections in bending are as follows:

applying the factor of cw

w

Yb

b

2 to the calculated stress based on AS/NZS4600

method;

for channel sections with complex edge stiffener, the flange to simple edge stiffener

ratio should not be smaller than 3.33; and


the depth of the edge stiffener should comply to AISI.

To be consistent with the Australian and American design standards, the following

formulas which are based on Lau and Hancock (1987) method (Appendix D of

AS/NZS4600 (2005)) are used to calculate the theoretical distortional buckling stress of

the tested sections with edge stiffener.

25.0

3

2

28.4

t

bbI wfx (2.3)

where fb and wb are width and depth of the section respectively.

2

(2.4)

A

IIx yx2

1 (2.5)

22

11 039.0

JbI fx (2.6)

xyfy IybI

12

2

(2.7)

22

113 xyfy IbI

(2.8)

3

22121

' 42

A

Ef od (2.9)

2244

24

3

'3

39.13192.256.12

11.11

06.046.5

2

ww

wod

w bb

b

Et

f

b

Etk

(2.10)

If fullxc ZZk .0 (2.11)

If 0k k should be calculated with 0' odf (2.12)

E

kJbI fx

1

22

11 039.0

(2.13)

22

113 xyfy IbI

(2.14)


3

22121 4

2

A

Efod (2.15)

The values of yxJIIIIA wxyyx ,,,,,,, are for the compression flange and edge stiffener.

It is important to note that the elastic buckling stresses from the equations and

numerical simulations are theoretical. When the buckling stress is below the yield stress

the section is slender, but to obtain a compact section the theoretical buckling stress

needs to be many times higher than the yield stress.

2.1.3 Interaction effects between the elements

Web and flange are considered as elements which are simply supported by either one or

both edges to calculate the theoretical local buckling stress. However in reality, because

of the connection between the flange and the web, the rotational restraint is created for

either the flange or the web. Therefore, it is conservative to assume that sections

elements have simply supported edges. However, it can not be assumed that sections

element have fixed edges either. In all design Standards such as AS4100 (1998), web

and flange slenderness limits are assumed to be independent of the restraining element.

However, using FSM, Schafer and Pekoz (1999) illustrated that the boundary condition

has a great effect on the distortional buckling coefficient (Figure 2.5).

Figure 2.5: Finite strip analysis of flange and lip (Schafer and Pekoz (1999))


Yiu and Pekoz (2000) compared experimental results of different studies with the

design recommendation for the capacity of plain channels based on the Schafer’s

numerical model. This work showed consistencies between Schafer’s program, which

is based on interaction between plate elements, and the experimental results.

By calculating the theoretical local and distortional buckling stresses, the ultimate

section moment capacity of the cold-formed sections can be calculated by using the

elastic Effective Width Method (EWM). The following sections review elastic EWM

which is the design rules in North American and Australian cold-formed specifications.

Furthermore, the European code design methods are also reviewed to calculate the

effective sections. From now on, for simplicity, EWM will be used instead of elastic

EWM.

2.2 Cold-Formed Design Rules

The design assumption for EWM is that the ultimate capacity of the cold-formed

sections should not exceed the yield capacity. However, in some conditions, inelastic

reserve capacity, which is another design rule in cold-formed standards, allows sections

ultimate capacity to exceed the yield capacity. The following sections are a critical

review of literature on EWM design methods.

2.2.1 Effective width method

In cold-formed sections; geometric shapes, thinner plate elements and imperfections are

causes of local buckling failure prior to yielding. These sections are called slender

sections and are not fully effective. Effective sections are the reduced design sections to

calculate the ultimate capacity of a structural element (Figure 2.6).

(a) Compression element


(b) Element under stress gradient

Figure 2.6: Effective design sections (AS/NZS4600 (2005))

The effective width method was first introduced by Von Karman et al. (1932).

Following a series of experiments, these researchers concluded that the ultimate loads

are independent from the width and the length of a plate. They assumed that buckled

portions of a plate do not carry any load. However, unbuckled portions can carry loads

of up to the yield point. In their method, instead of a non-uniform stress along the full

width of b , it is assumed that a uniform stress, equal to the edge stress, is distributed

along the portion of the width of ( eb ). Figure 2.7 shows the stress distribution in a

stiffened and unstiffened element.

Figure 2.7: Stress distribution in effective width method (Bambach (2003))


A stiffened element is a flat element with both edges supported longitudinally by web,

flange or lip stiffener. An unstiffened element is an element with only one edge

supported longitudinally (Figure 2.8).

Figure 2.8: Stiffened and unstiffened elements

2.2.1.1 Effective width of uniformly compressed stiffened and

unstiffened elements

The effective width method, which is used in most design standards (for example

NASPEC (2007) and AS/NZS4600 (2005)), is based on the Winter (1970) method

which is explained in the following paragraph.

The effective width of uniformly compressed elements can be calculated as follows:

bbe (2.16)

In Equation (2.16) is the effective width factor and is determined as follows:

According to NASPEC (2007) and AS/NZS4600 (2005):

For :673.0

22.01

(2.17)

For :673.0 1 (2.18)

where is slenderness ratio given by:

crf

f *

(2.19)

where, *f is the design stress in the compression element which is shown on Figure

2.9. crf is the theoretical buckling stress. For calculating theoretical local buckling


stress, the plate buckling coefficient ( k ) is taken as 4 and 0.43 for stiffened and

unstiffened elements respectively (Timoshenko and Gere (1961)).

Figure 2.9: Effective width of uniformly compressed stiffened and unstiffened elements (AS/NZS4600(2005))

According to EUROCODE3 (2006):

For :673.0p 2

)3(055.0

p

p

where 03 (2.20)

For :673.0p 1 (2.21)

kt

bp

4.28 (2.22)

yF

235 (2.23)

where k is the buckling factor and is taken as 4 and 0.43 for stiffened and unstiffened

elements respectively and b is the appropriate width.

Kalyanaraman et al. (1977) compared the test results of a cold-formed unstiffened

element in compression with Equations (2.16) to (2.19) results. For a stocky element,

where 5.1oly fF , the strength results from Equation (2.16) compare well with the

test results.


2.2.1.2 Effective width of stiffened elements with stress gradient

For stiffened elements with a stress gradient, the effective width is split into two parts

as shown in Figure 2.10. The sum of the split effective widths of ( 21 ee bb ) cannot

exceed the compression portion of the element.

Figure 2.10: Effective width of stiffened elements with stress gradient (AS/NZS4600 (2005))

The value of each split effective width depends on the stress gradient on the element,

and can be calculated by using the following equations:

According to NASPEC (2007) and AS/NZS4600 (2005):

31e

e

bb (2.24)

*1

*2 ff (2.25)

For :236.0 22e

e

bb (2.26)

For :236.0 12 eee bbb (2.27)

eb is determined by using Equations (2.16) to (2.19) with *f equal to *1f . The plate

buckling coefficient for calculating olf is given by:

)1(2)1(24 3 k (2.28)



For :10

05.1

2.8

,5

2121

k

bbbbb eeeee

(2.29)

For :0

eeee

ce

bbbb

bbb

6.0,4.0

1

21

(2.30)

For 278.929.681.7:10 k (2.31)

For 9.23:1 k (2.32)

For 2198.5:31 k (2.33)

Zhou and Young (2005) have performed bending tests on cold-formed stainless steel

tubular sections. Zhou and Young compared their test results with the North American

and Australian design rules results. They concluded that the existing design rules are

conservative for calculating the cold-formed stainless steel tubular sections ultimate

moment capacity.

2.2.1.3 Effective width of unstiffened elements with stress gradient

Based on AS/NZS4600 (2005), the effective width of unstiffened elements under the

stress gradient can be calculated in two different scenarios as follow:

According to AS/NZS4600 (2005) and NASPEC (2007):

a) Where the stress increases toward the unstiffened edge of the element (Figure 2.11):

For :0

207.021.057.0

1

22.01

k

(2.34)

For :0

207.021.057.0

1

122.01

1

k

(2.35)


Figure 2.11: Effective width of unstiffened elements with stress gradient (EUROCODE (2006))

b) Where the stress decreases toward the unstiffened edge of the element (Figure 2.12):

For :0

34.0

578.0

1

22.01

k

(2.36)

For :0

21.1757.1

1

22.01

1

k

(2.37)

Figure 2.12: Effective width of unstiffened elements with stress gradient (EUROCODE (2006))


For :748.0p 2

188.0

p

p

(2.38)

For :748.0p 1 (2.39)


For :01 bbe (2.40)

For :0

1

bbe (2.41)

a) Where the stress increases toward the unstiffened edge of the element (Figure 2.11):

207.021.057.0 k (2.42)

b) Where the stress decreases toward the unstiffened edge of the element (Figure 2.12):

For :01 34.0

578.0

k (2.43)

For :01 21.1757.1 k (2.44)

Australian, North American and European codes are based on the theory of elastic

effective width which has been explained so far. A new theory, called plastic effective

width have been introduced by Bambach and Rasmussen (2004a) for the design of an

unstiffened element under stress gradient. This is due to there being an inconsistency in

the elastic effective width design of unstiffened elements under the stress gradient. In

the elastic effective width theory, the assumption is the maximum stress and strain on

an element are the yield stress and yield strain. For an unstiffened element under stress

gradient the ultimate strain at the unsupported edge can exceed the yield strain. This

effect is shown below, in Figure 2.13.

Figure 2.13: Slender section in minor axis bending (Bambach (2003))


Bambach and Rasmussen (2004a) proposed two methods for determining the effective

width of unstiffened elements with stress gradient. They have used plate test results

from Bambach and Rasmussen (2004) and compared with the elastic and plastic

effective width methods results. By doing so, it was concluded that the plastic effective

width satisfies both the ultimate force and the ultimate moment. However, the elastic

effective width satisfies the ultimate force but underestimates the ultimate moment.

Unstiffened elements have a smaller theoretical buckling stress and ultimate strength

when compared with the stiffened elements that are of the same material and

dimension. However, by adding an edge stiffener to the free edge of an unstiffened

element, its theoretical buckling stress and ultimate strength are increased. The

following section explains the behaviour of the elements with an edge stiffener.

2.2.1.4 Effective width of uniformly compressed elements with an edge

stiffener

A compression element with an edge stiffener is called a partially stiffened element.

The buckling behaviour of a partially stiffened element, depending on the edge

stiffeners size, varies between stiffened and unstiffened elements. Therefore, the plate

buckling coefficient, k value, of a partially stiffened element varies between 0.43 to 4.

Desmond et al. (1981) conducted analytical and experimental studies on partially

stiffened elements. They have concluded that the buckling behaviour of elements with

an adequate size of stiffener, and therefore their effective width, is similar to stiffened

elements that have the same material and dimension. The outcome of Desmond et al.

(1981) research led to the design rules for calculating the buckling coefficients of

uniformly compressed partially stiffened elements in AS/NZS4600 (2005) and

NASPEC (2007).

Bambach (2009) has also illustrated that if the lip to flange ratio for an element with a

simple stiffener exceeds to 0.16, the element will behave as an stiffened element.

Bambach also concluded that the lip to flange ratio should not exceed 0.25. This is due

to the fact that a large stiffener initiates buckling itself and will reduce the theoretical

buckling stress of the whole element.


Section 2.4 in AS/NZS4600 (2005) which is similar to section B4 in NASPEC (2007)

explain how to determine the effective width of an element with edge stiffener (Figure

2.14). The EUROCODE3 (2006) consider a reduction factor due to the distortional

buckling of the edge stiffener to determine the effective width of an element with edge

stiffener. Section 5.5.3.2 in EUROCODE3 (2006) described how to calculate the

effective width of an element with edge stiffener.

Figure 2.14: Effective width of an element with edge stiffener (AS/NZS4600 (2005))

It is important to note that the NASPEC (2007) and AS/NZS4600 (2005) calculations

are based on the Winter formula (Equations of (2.16) to (2.19)). To verify the Winter

formula for partially stiffened elements, Kwon and Hancock (1992) have performed

compression tests on cold-formed channel sections with edge stiffeners. Their test

results indicated that sections without adequate edge stiffeners, and a flange buckling

coefficient of less than 4, will fail due to the distortional buckling. Therefore, the value

of crf in the Equation (2.2) should be equal to the theoretical distortional buckling

stress. Based on distortional buckling failure, Kwon and Hancock (1992) compared

their test results with the Winter formula results and concluded that the Winter’s

formula provides an un-conservative design capacity for partially stiffened elements.


Consequently, they modified the Winter’s formula for distortional buckling being as

follows:

For :561.0

6.06.0

25.01y

od

y

od

F

f

F

f (2.45)

For :561.0 1 (2.46)

where, is slenderness ratio given by:

odf

f *

(2.47)

Bambach (2009) modified the Winter equation for edge-stiffened elements. Bambach’s

modification was purely based on an empirical approach using finite element analysis

and his modified equations are as follows:

For :0.443.0 k 34

22.01

(2.48)

For 0.4k or :43.0k

22.01 (2.49)

From Kwon and Hancock (1992) and Bambach (2009) studies it can be concluded that

distortional buckling failure is not clearly addressed in EWM. However, EUROCODE3

(2006) considers a reduction factor due to the distortional buckling of the edge stiffener

to determine the effective width of an element with edge stiffener. In NASPEC and

AS/NZS4600 (2005) the member moment capacity is determined subject to the

distortional buckling. The following equations show the calculation of a member

moment capacity based on AS/NZS4600 (2005) due to distortional buckling.

ccnalbdistortio fZM (2.50)

fullx

cc Z

Mf

.

(2.51)

where cM is the critical moment and can be calculated as follows:

a) For rotation of a flange and lip about the flange/web junction case:

For fullxc ZZk .,0 (2.52)


For cZk ,0 Effective section modulus with the k value equal to four for the

compression flanges; and the design stress in the compression element, *f , is equal to

cf .

For fullxycd ZFM .,674.0 (2.53)

For

dd

fullxycd

ZFM

22.0

1,674.0 . (2.54)

b) For transverse bending of a vertical web with a lateral displacement of the

compression flange case:

cZ Effective section modulus with the design stress in the compression element, *f ,

is equal to cf .

For fullxycd ZFM .,59.0 (2.55)

For

dfullxycd ZFM

59.0

,7.159.0 . (2.56)

For

2.

1,7.1

d

fullxycd ZFM

(2.57)

od

yd f

F (2.58)

The EWM with distortional buckling check are complicated methods for calculating

cold-formed sections ultimate member capacity. To this end, Schafer and Pekoz (1998)

have introduced a less complicated method titled the Direct Strength Method (DSM).

The DSM is as an alternative design method for calculating cold-formed sections

ultimate member capacity in North American and Australian standards. The following

section is a detailed discussion on DSM.

2.2.2 Direct strength method

Based on numerical methods such as Finite Strip Method (FSM), DSM was originally

presented by Schafer and Pekoz (1998). These researchers compared the DSM results

with the test results for cold-formed open sections that were under uniaxial bending.


They concluded that DSM can be used to accurately predict the section capacity for

cold-formed open sections under uniaxial bending.

Using Finite Element Analysis (FEA), Zhu and Young (2006) and Zhu and Young

(2009) performed parametric studies on rectangular and square aluminium hollow

sections subjected to compression and bending. Zhu and Young verified their FEA

model with some experimental results. In the same studies, they have compared the

FEA and the test results together with the DSM results. They modified the DSM for

slender sections to obtain a less conservative result for the aluminium slender sections.

Figure 2.15(a) is a comparison of FEA and experimental data with the DSM curve

under compression. However, Figure 2.15(b) is a comparison of FEA and experimental

data with the DSM curve under bending. Zhu and Young concluded that their modified

DSM method results are in a good agreement with the FEA results.

Figure 2.15: Comparison of FEA and experimental data with the DSM curve under (a) compression (Zhu and Young (2006)) (b) bending (Zhu and Young (2009))

Young and Yan (2004) performed column tests on the cold-formed channel sections

with a complex edge stiffener. These were then compared with the DSM results. Young

and Yan concluded that the DSM results compare well with the test results for sections

which have slender flanges. However, when compared to the test results, these results

were conservative for the sections with less slender flanges.

Yu and Schafer (2003) tested the C and Z sections under bending. In these comparative

testing, the sections were restrained to avoid the distortional and the lateral buckling


prior to the local buckling. The test and the DSM results were also compared in terms

of local buckling. Yu and Schafer concluded that the DSM was quite conservative for

the non-slender members.

In addition to this research, Yu and Schafer (2006) also tested the C and Z sections

under bending with no restraint on the elements for distortional buckling. The test and

the DSM results were also compared for distortional buckling. They concluded that,

compared to the other standards for distortional buckling, more accurate results can be

predicted using the DSM. Furthermore, Yu and Schafer (2007) illustrated that the

moment gradient may increase the distortional buckling strength of a cold-formed C or

Z section beams. Therefore, they developed an empirical equation to predict the

distortional buckling moment due to the moment gradient. Figure 2.16 compares the

DSM with the Yu and Schafer’s test results.

Figure 2.16: Comparison of DSM with test results (Yu and Schafer (2007))

By reviewing the comparison, it is evident that DSM does not include any inelastic

reserve capacity for the cold-formed sections and assumes that the maximum moment

capacity of the cold-formed sections is the yield moment. However research by Enjily

et al. (1998) that was both experimental and theoretical, shows that the ultimate

moment capacity of some cold-formed channel sections can reach up to the plastic


moment. Therefore, it can be concluded that there is still room for improvement in

DSM.

The following equations show how to determine the ultimate moment capacity of a

beam based on DSM:

Lateral buckling:

For yo MM 56.0 : obe MM (2.59)

For yoy MMM 56.078.2 :

o

yybe M

MMM

36

101

9

10 (2.60)

For yo MM 78.2 : ybe MM (2.61)

where oM is the elastic lateral buckling moment and can be calculated using section

3.3.3.2 in AS/NZS4600 (2005).

Local buckling:

For 776.0l : bebl MM (2.62)

For 776.0l :

4.04.0

15.01be

ol

be

olbebl M

M

M

MMM (2.63)

ol

bel M

M (2.64)

olfullxol fZM . (2.65)

where olf is the theoretical local buckling stress.

Distortional buckling:

For 673.0d : ybd MM (2.66)

For 673.0d :

5.05.0

22.01y

od

y

odybd M

M

M

MMM (2.67)


od

yd M

M (2.68)

odfullxod fZM . (2.69)

where odf is the theoretical distortional buckling stress and the THINWALL program

can be used to determine the value of the theoretical distortional buckling stress.

Some advantages of the DSM over the EWM are outlined by Schafer (2003). These are

that:

“DSM includes simple design improvement: no effective width, no iteration, gross

section properties used for strength; Theoretical improvement: interaction of elements

(e.g. web/flange) is accounted for, distortional buckling is explicitly treated;

improvements in applicability and scope: rational analysis method for all sections.”

The subject of discussion in the previous sections was primarily about the cold-formed

design rules (EWM, DSM) which did not allow the ultimate moment capacity of a

member to exceed the yield capacity. However for fully effective sections, section

6.1.4.1 of EUROCODE3 (2006) allows a member moment capacity beyond the yield

moment. The inelastic design methods from the EUROCODE3 (2006) are defined as

follows:

If the effective section modulus is less than the gross elastic section modulus (non-fully

effective sections):

0, / MyeRdc FZM (2.70)

If the effective section modulus is equal to the gross elastic section modulus (fully

effective sections):

000max.., /)//14( MyxMeefullxxfullxyRdc FSZSZFM (2.71)

where γM0 is equal to one for seismic and accidental design situations. maxe is the

slenderness of the element which correspond to the largest value of 0/ ee ;

For double supported plane elements

pe and 3055.025.05.00e (2.72)

where ψ is the stress ratio.


For outstand elements pe and 673.00 e (2.73)

For stiffened element de and 65.00 e (2.74)

In addition, section 3.3.2.3 of AS/NZS4600 (2005) and also C3.1.1(b) of NASPEC

(2007) introduce another method titled Inelastic Reserve Capacity. This allows the

ultimate capacity of a section to reach beyond the yield capacity. There are also

limitations for using the Inelastic Reserve Capacity method that are discussed in the

following section.

2.2.3 Post yielding or inelastic reserve capacity of cold-formed steel

Yener and Pekoz (1985) indicated that beams with stiffened compression elements, due

to the re-distribution of yielding through the section’s depth, can carry more loads after

the initial yield stress. This is called the post yielding or inelastic reserve capacity of the

beam. Reck et al. (1975) performed bending tests on three groups of cold formed hat

sections with stiffened flanges under compression to monitor their strain capacities.

Figure 2.17 shows the ratio of the ultimate strain to the yield strain, yC , versus the

beam’s compression flange slenderness.

Figure 2.17: Compression strain factor for compression flange (Hancock (1988))


As can be seen, for the slenderness ratio of less than yF530 , the ultimate strain is

almost three times bigger than the yield strain. The ratio of the ultimate strain to the

yield strain decreases by increasing the flange slenderness ratio. Figure 2.17 is a base

for inelastic reserve capacity equations in design standards. Reck et al. (1975) showed

that the ultimate moment depends not only on the width-to-thickness ratio of the

compression flange but also the location of the neutral axis. The closer the neutral axis

of a section to the compression flange, the sooner the tension flange yields. Therefore,

the section has a bigger inelastic reserve capacity compared to a section with neutral

axis close to the tension flange. Figure 2.18 illustrates the effect of the neutral axis

location in the section’s inelastic reserve capacity.

Figure 2.18: Stress and strain for inelastic reserve capacity (Hancock (1988))

Bambach (2003) collected experimental results for the I and channel sections in minor

axis bending. The collected experimental data for the I sections was from Chick and

Rasmussen (1999) and Rusch and Lindner (2001). For the channel sections the

researchers were Beale et al. (2001) and Yiu and Pekoz (2000). These experimental

results exhibited some post-elastic behaviour for some sections. For example, Yiu and


Pekoz (2000) anticipated that plain channel sections with the flange slenderness ratio of

less than 0.859 would have post-elastic behaviour. Therefore, Bambach (2003) have

proposed the inelastic reserve capacity design rules for unstiffened elements under

stress gradient.

According to AS/NZS4600 (2005) and NASPEC (2007) the Inelastic Reserve Capacity

method is restricted to a few conditions. Furthermore, that the ultimate section moment

capacity cannot exceed either ye FZ25.1 or that causing a maximum compression strain

of yyC . yC and y are compression strain factor and yield strain respectively.

According to AS/NZS4600 (2005) and NASPEC (2007), for compression elements

with edge stiffener, yC is equal to one. Therefore, inelastic reserve capacity rules are

not applicable to compression elements with edge stiffeners. As a result, this thesis

investigates the inelastic behaviour of compression elements with edge stiffener.

The previous sections briefly reviewed the relevant literature on existing design rules

that are based on cold-formed specifications. To extend this review more specifically to

the topic of this research, the following sections discuss the design rules based on the

hot-rolled specifications such as AS4100 (1998).

2.3 Hot-Rolled Design Rules

The Australian Standard AS4100 (1998) classifies the hot-rolled sections into different

classes according to their slenderness ratio. In AS4100 (1998), the hot-rolled sections

ultimate capacity is calculated using the following equations:

eyS ZFM (2.75)

where, eZ is the effective section modulus;

For compact sections:

)5.1,( ZSMinZe (2.76)

For non-compact sections:


ZSZZ

spsy

ssye

(2.77)

For slender sections:

s

sye ZZ

for plate element in uniform compression (2.78)

2

s

sye ZZ

for plate element with maximum compression at an unsupported edge

and zero or tension at the other (2.79)

where, Z is the elastic section modulus, S is the plastic section modulus, sy is the

elastic slenderness limit and sp is the plastic slenderness limit.

Section 2.3.2 explains the sections classification (compact, non-compact and slender) in

different standards. Sections 2.3.3 to 2.3.7 reviews the literature on elastic and plastic

slenderness limits for different elements, and under different loadings. It should be

noted that the slenderness ratio, which is defined in the cold-formed specifications, is

not similar to what is defined in the hot-rolled specifications. The following two

equations define the slenderness ratio in cold-formed and hot-rolled specifications.

For cold-formed specifications: crf

f *

(2.80)

For hot-rolled specifications: 250

yF

t

b (2.81)

where b and t are width and thickness of the element respectively.

To classify a section as compact (plastic) not only a plastic hinge should be developed

at the maximum moment point but also the plastic hinge should rotate sufficiently to

redistribute the moment through the member. Therefore, the concept of rotation

capacity which is an indicative parameter for the section ductility and determines how

an internal moment can redistribute when the plastic moment is reached is explained in

the following section.


2.3.1 Rotation capacity

Some beams fail before reaching the plastic moment or even the yield moment.

However, some beams do not fail before reaching the plastic moment and a plastic

hinge develops as a result. The rotation capacity )(R is a measure of how much the

plastic hinge can rotate before the section’s failure. Rotation capacity is normally

defined as:

1 pkkR (2.82)

EIMk , EIMk pp (2.83)

Figure 2.19 demonstrates that calculation to determine the rotation capacity by

normalising the moment-curvature diagram with the plastic moment and the plastic

curvature.

Figure 2.19: Measurement of Rotation Capacity (Wilkinson (1999))

Researchers (Lukey and Adams, Korol and Hudoba, Zhao and Hancock, Hasan and

Hancock) have differing opinions on the value of the rotation capacity. The plastic

slenderness limit of an element is determined in regard to the rotation capacity. Lukey

and Adams (1969) used a rotation capacity of 2.5 to satisfy the redistribution of a

moment in plastic design. However, Korol and Hudoba (1972), Zhao and Hancock

(1991) as well as Hasan and Hancock (1988) used 4R for determining the

slenderness limit. Important to note is that the most commonly used rotation capacity

for plastic slenderness limit is between 3 and 4.


2.3.2 Section classification

Depending on the sections rotation capacity )(R and maximum moment )( maxM ,

sections are classified into the different groups. EUROCODE3, CSA S16.1 and BS

5950 classify sections into four groups. Alternatively AS4100 and AISC LRFD use

three different classes of sections. For example in the EUROCODE3 design standard,

section classifications are:

Class1 3,max RMM P

Class2 3,max RMM P

Class3 py MMM max

Class4 yMM max .

Different section classifications are set out in Table 2.1, below.

Table 2.1: Different standard section classification (Wilkinson (1999)) Specifications

Eurocode 3 Class 1 Class 2 Class 3 Class 4

BS 5950 Plastic Compact Semi-Compact Slender

CSA S16.1 Plastic or Class 1 Compact or Class 2 Non-Compact or Class 3 Slender or Class 4

AS 4100 Compact Slender

AISC LRFD Compact Slender

Non-Compact

Non-Compact

Figure 2.20 shows how to determine a section’s class using its Moment-Curvature

diagram.


Figure 2.20: Moment-curvature of different type of steel section (Elchalakani et al. (2002b))

2.3.3 Elastic limits for compression elements

Plate buckling depends on its geometry, material property and external restraint. As

noticed in Equation (2.2), critical theoretical buckling stress of a plate depends on the

width-to-thickness ratio (slenderness) tb . Lay (1965) has defined the biggest value for

the width to thickness ratio of yF

500 to avoid local buckling in elastic range for

compression flange of hot-rolled I sections. The following equation shows Lay’s

definition.

16250

5002 y

y

F

t

b

Ft

b (2.84)

where,b Represents width of the element, which is shown in Figure 2.21.


Figure 2.21: Classification of plate width

For a hot-rolled channel section, the flange can represent a simply supported plate on

one edge. Consequently from the Figure 2.3, the k value is equal to 0.425. From

Equation (2.2) in which is adopted from AS/NZS4600 (2005), the width to thickness

ratio for unstiffened and stiffened flanges should be:

For unstiffened flange: yFt

b 277 5.17

250 yF

t

b (2.85)

For stiffened flange: 0.4k : 54250

yF

t

b (2.86)

From AS4100 (1998) the width-to-thickness ratio limit for a compression element is:

For unstiffened flange: 15250

yF

t

b (2.87)

For stiffened flange: 40250

yF

t

b (2.88)

In AS4100 (1998), the slenderness limits for an unstiffened flange and stiffened flange

have been decreased from 17.5 to 15 and 54 to 40 respectively. This is due to the

residual stress, which exists in a section as a result of welding in hot-rolled sections

Ueda and Tall (1967). By ignoring the residual stresses, early yielding cannot be

avoided. In addition, members stiffness is reduced and the inelastic behaviour of the

sections may not be predicted correctly (Galambos (1968)).


2.3.4 Elastic limits in bending elements

Normally web of I or channel sections are simply supported elements on both

longitudinal edges by flanges. A web can represent an element, which is in bending and

its stress gradient varies from tension to compression. For elastic buckling design of a

web, using Equation (2.2) in which is adopted from AS/NZS4600 (2005), the

slenderness limit for a web can be as follow:

where from Fig 2.3, 9.23k ; and therefore,

131250

yF

t

d (2.89)

d is shown in Figure 2.22.

ddd

Figure 2.22: Classification of plate depth

In Australian Standard (AS4100 (1998)), elastic slenderness limit for bending element

is 115. As it is mentioned in section 2.3.3, the slenderness limit in AS4100 (1998) is

lower than the calculated value from the Equation (2.2) which is adopted from

AS/NZS4600 (2005).

2.3.5 Slenderness limits for non-compact elements

Lyse and Godfrey (1935), Craskaddan (1968) and Holtz and Kulak (1973) performed

bending tests on I section beams. Their proposed slenderness limits for non-compact

(Class3 and Class2) sections were 70, 67 and 86 respectively. The Canadian standard in

1974 used the Holtz and Kulak (1973) limit being:

87250

yF

t

d (2.90)

Wilkinson (1999) classified the slenderness ratio for webs under bending from different

studies and standards. Wilkinson’s classifications are shown in Table 2.2. The Web


slenderness limits in this classification are mostly based on bending tests for I sections

with the steel grade of 33 to 44 ksi (228 to303MPa).

Table 2.2: Web slenderness limit classification in bending (Wilkinson (1999))


d d ii d iii d iv

Figure 2.23: Definition of web depth 2.3.6 Plastic limits for compression elements

When designing hot-rolled sections in the plastic range, the tb ratio limit should be

applied (Lay (1965)). This is to make sure that local buckling is not occurring prior to

forming a plastic hinge. Lay (1965) showed that local buckling in plastic range depends

on the tb ratio, moment gradient, strain hardening and yielded region length. Lay

determined a limit for the tb ratio based on previously mentioned factors (moment

gradient, strain hardening and yielded region length) for unstiffened elements. For

example for A36 steel, 55.8tb and A441 steel 7.6tb .The yield stress for A36 and

A441 are 248 and 345 MPa respectively. Therefore, the 250yFtb for A36 and

A441 should be less than 8.51 and 8.0 respectively. AS4100 (1998) is using Lay’s limit

for plasticity design of hot-rolled sections (Lay (1965)).

For the unstiffened flange: 0.8250

yF

t

b (2.91)

Kato (1965) assumed that the local buckling of a plate element depends on the

following two factors: the slenderness of the element and the yield ratio of the material

in plastic range. Kato recommended a formula that, in general, shows results close to

the Lay’s slenderness limit (Lay (1965)).

Lukey and Adams (1969) have performed twelve bending tests on hot-rolled I sections.

These researchers proposed a flange slenderness limit which was less conservative

when compared to the slenderness limit of Lay (1965).

For an unstiffened flange: 8.10250

yF

t

b (2.92)


In terms of the stiffened element, such as a box section, Korol and Hudoba (1972)

performed bending tests on box sections. According to Korol and Hudoba (1972) the

slenderness limit for the Rectangular Hollow Section (RHS) and Square Hollow

Section (SHS) flanges, with a rotation capacity of 4, is:

yFt

b 394 or 25

250yF

t

b (2.93)

where b is shown in Figure 2.24.

Figure 2.24: Width of a flange in Korol and Hudoba (1972)

Hasan and Hancock (1988) tested eighteen RHS and SHS grade C350 cold-formed

sections in bending. The result of the flange slenderness limit for the rotation capacity

of 4R was equal to 25. Zhao and Hancock (1991) performed the same test for grade

C450 cold-formed RHS and SHS sections. The result of the flange slenderness limit for

the same rotation capacity of 4R was 22. Therefore, based on Hasan and Hancock

(1988), Zhao and Hancock (1991) the following limitations are assumed for different

steel grades:

5.29250

45022

250450,22

5.29250

35025

250350,25

yy

yy

F

t

bMpaF

t

b

F

t

bMpaF

t

b

5.29250

yF

t

b (2.94)

2.3.7 Plastic limits in bending elements

According to the AS4100 (1998), the plastic buckling limitation is defined as:


82250

yF

t

d (2.95)

Haaijer and Thuerlimann (1958) proposed a theory which was supported by the test

results. They assumed that the depth to thickness ratio of a web, in both compression

and bending, depends on the stress distribution and maximum strain of the compression

flange εm. Figure 2.25 shows the depth to thickness ratio limits for different εm/εy and

axial forces. Assuming εm/εy =4, for a section which is only in bending, depth to

thickness ratio limit are:

62250

yF

t

d (2.96)

Figure 2.25: Allowable d/t ratios of webs of fully plastic sections for σo =33 ksi (Haaijer and Thuerlimann (1958))

Sanders and Householder (1978) restricted the depth to thickness ratio of box sections

to be:


62250

yF

t

d (2.97)

The web slenderness limit of a box section under bending is not the same as an I

section. It should be noted that the existing benchmark in design standards are based on

I sections test results; and the interaction effect between the elements is ignored which

sees this theory as conservative.

Kuhlmann (1989) calculated the rotation capacity through experimental analysis for

twenty four I sections. Kuhlman pointed out that the main parameters that influence the

rotation capacity are flange slenderness and web slenderness (stiffness). For instance,

sections with the same flange slenderness yet with different web slenderness have a

different rotation capacity.

Kemp (1996) found the rotation capacity of forty four I sections in four different series

of tests. By conducting this research, Kemp showed that rotation capacity depends

primarily on the lateral slenderness ratio. It is evident in Figure 2.26 with a strong

correlation between rotation capacity and the lateral slenderness ratio. However, Kemp

(1996) included the interaction of flange and web slenderness.

Figure 2.26: Effect of slenderness ratio in Kemp (1996) method


fycidwfe rLKKK )/( (2.98)

The basic parameters used in the Kemp (1996)model are:

“1-Yield stress factor for flange or web 250yF for

Fy (in Mpa) 2-Slenderness ratio in lateral-torsional buckling fcyi rL ,

in which Li is the length from the section of maximum moment to the adjacent point of inflection, and rcy is the radius of gyration of the portion of the elastic section in compression.

3-Flange slenderness factor in local buckling 9/ff tbk

in range of 0.7 < kf < 1.5. 4-Web slenderness factor in local buckling 70/fwcw thk

in range of 0.7 < kw < 1.5. 5-Distortional restrain factor kd of concrete slab in the

negative moment region of continues composite beams as discussed subsequently (kd=1 for plain steel beams and 0.71 for composite beams).”

Kato (1989) evaluated the rotation capacity of I sections in different test series. By

doing so, Kato produced an interaction formula between the web and flanges based on

rotation capacity requirement, being:

11170181

2

2

2

2

y

w

y

f

F

t

d

F

t

b

, 4R (2.99)

Wilkinson and Hancock (1998a) illustrated that the plastic web slenderness limits in

design standards are not conservative for the cold-formed RHS beams in bending.

Wilkinson and Hancock produced an iso-rotation curve which indicates that there is an

interaction between webs and flanges in RHS beams (Figure 2.27(a)). They also

proposed a compact limit for cold-formed RHS beams with a Rotation Capacity equal

to four (Figure 2.27(b)).


(a) Iso-rotation curves (Wilkinson and Hancock (1998a))

(b) Proposed compact limits for Cold-formed RHS beams (Wilkinson and Hancock

(1998a))

Figure 2.27: Compact limits for Cold-formed RHS beams 2.4 Plastic Design

The plastic design method allows larger application of loads on sections due to

spreading the yield stress over the entire section. This method can increase the capacity

of a channel section by up to 20-30% which is calculated by the elastic design method.


One of the early works on I sections was conducted by Maier-Leibnitz who tested 4

metre long 40cm x40cm fix ended I beams. Maier-Leibnitz determined that the beam

could carry a load 1.5 times greater than the yield load (Baker et al. (1956)). Baker et

al. (1956) discussed this finding after Maier-Leibnitz proved that the capacity of an I

beam can exceed its yield capacity, with various tests performed on redundant

structures. The conclusion was that after forming the first hinge in the structure, if this

moment can be maintained for sufficient rotation then other hinges in the structure can

also develop and a plastic mechanism can develop in the structure. Therefore, the

plastic design method would increase the capacity of the structure significantly greater

than the traditional elastic design method. It can be seen then that the plastic design

method is more economical in comparison with the traditional elastic design method.

However, it does have a higher deformation when compared to the elastic method.

Therefore, the plastic design method is more suitable for portal frame structures in

agricultural and industrial applications where serviceability criteria can be relaxed. In

addition, by using wall and roof framing, the whole frame is fully braced against lateral

instability.

As discussed previously, when in accordance with AS/NZS4600 (2005) and NASPEC

(2007), the inelastic reserve capacity design method cannot apply on cold-formed

channel sections with edge stiffener. Therefore cold-formed channel sections with edge

stiffener are excluded from plastic design. Based on AS4100 (1998), plastic designs

are not applicable for cold-formed section due to the brittle failure associated with high

strength steel. However, different experimental studies show plastic design restrictions

are not founded in all cases.

Wilkinson and Hancock (1998) performed tests on three pin based portal frames

manufactured from cold-formed Rectangular Hollow Sections (150x50x4 RHS) which

did not satisfy the material ductility requirement for plastic design method. They

concluded that the failure was not due to the lack of material strain hardening capacity

and a plastic collapse mechanism was developed in the tested structures. The pin based

portal frame required only two plastic hinges to develop a plastic collapse mechanism.

The positions of the plastic hinges based on test results are shown in Figure 2.28.


Figure 2.28: Position of plastic hinges in Wilkinson’s portal frames based on test results (Wilkinson (1999))

The ultimate load can be estimated where the last plastic hinge develops in the

structure. The difference between the loads for the formation of the last and first plastic

hinge is defined as the increased capacity of the structure due to the plastic analysis. It

is evident from Table 2.3 that the Wilkinson and Hancock portal frame had the capacity

beyond the formation of the first hinge.

Table 2.3: Ultimate load of the Wilkinson’s portal frames based on test results Frame Ultimate load / First hinge load

Vertical Horizontal

Frame 1nominal 57.6 1.44 1.05

measured 68.4 1.71 1.05experimental 68.2 1.75





Load at ultimate (kN)


They also concluded that the adequate rotation capacity is four to redistribute the

moment through the structure assembly. Wilkinson and Hancock portal frame test show

that cold-formed RHS sections can have the ultimate capacity beyond their first hinge

load however their deflection exceeds the deflection limit which is defined in AS4100

for beams (Figure2.29).

Figure 2.29: Vertical deflection of Wilkinson’s portal frames based on test results (Wilkinson(1999))

Baigent and Hancock (1981) tested seven portal frames manufactured from cold-

formed channels sections with edge stiffener. The geometry of their tested section is

shown in Figure 2.30.

153

79

15

t =1.85r =10i

Figure 2.30: Geometry of the section in Baigent’s portal frames test (not to scale)

They applied three loading patterns (Dead load and Live load, Transverse wind load

and Longitudinal wind load) on the portal frames (Table 2.4).


Table 2.4: Loading pattern on Baigent and Hancock portal frames Frame Loading pattern

Frame 1 Dead load and Live loadFrame 2 Dead load and Live loadFrame 3 Transverse wind loadFrame 4 Longitudinal wind loadFrame 5 Dead load and Live loadFrame 6 Transverse wind loadFrame 7 Longitudinal wind load

Their first four frames were restrained along their outside flanges; however, the last

three frames were restrained along their inside flanges by fly bracing in addition. This

work illustrated that the ultimate capacity of the cold-formed structures were 25% to

70% greater than the first yield capacity. This means the first plastic hinges have the

capability to rotate sufficiently to redistribute the plastic moment through the structure

to form a plastic collapse mechanism. However, the cold-formed channel sections do

not satisfy the plastic design requirement in AS4100 (1998). This is due to cold-formed

sections mainly being used as a secondary structure like roof or wall framings (purlin

and girt); and not being used as a primary structures. Therefore, there is limited

research on the plastic design for cold-formed sections; and cold-formed open sections

can only be designed in structure assemblies elastically. Plastic design rules are mainly

based on test results on hot-roll I sections portal frames.

Baker et al. (1956) collected all available references regarding the plastic design rules

based on I sections test results. From 1940 to 1960 Lehigh University in USA also

investigated the plastic behaviour on hot-rolled sections. It is evident from 1940

onwards that there have been numerous studies on the plastic behaviour of hot-rolled I

sections. Conversely, Schafer (2006a) has collected the experimental data for the

bending capacity of cold-formed sections from different studies and compared these

with Direct Strength Method (DSM) results (Figure 2.31).


Figure 2.31: Test data compare to Direct Strength Method result for beams (Schafer (2006a))

Figure 2.31 shows that experimental data are mostly for slender sections and there is a

gap of knowledge for sections with the slenderness ratio of less than 0.6. This is due to

the fact that cold-formed sections, which were in the market, were primarily used for

secondary structures where the serviceability limits control the designs not the strength.

Therefore, they are made as slender sections to satisfy the serviceability limit in the

most economical manner. If cold-formed sections can carry a load greater than their

yield capacity, they could become more economical when used as a primary structure

in building assemblies. This is in addition to being easier to fabricate into complex

shapes, more compact in packaging and having higher strength to weight ratio

compared with hot-rolled sections. All of these advantages can result in using cold-

formed sections as a more economical option than hot-rolled sections.

By considering the current research, and the value of determining the collapse response

of the cold-formed sections after reaching their ultimate capacity (collapse point),

another purpose of this literature is reviewing Yield Line Mechanism which is a method

to determine the collapse behaviour of cold-formed steels.


2.5 Collapse Behaviour of a Cold-Formed Structure

There are different methods to determine the load-carrying capacity of cold-formed

structures under bending and compression. Some of those methods have been classified

by Koteko (2007) as:

Analytical methods such as Effective Width method and Direct Strength Method;

Numerical methods such as Finite Element Method and Finite Strip Method; and

Analytical-numerical or semi-empirical methods where the load-carrying capacity of

the member is the intersection point of the plastic failure curve and the post buckling

path in the elastic range.

In the elastic range, where the deformations of the elements are small, the theory of

elasticity can be used to determine the load-deformation behaviour of the structure.

When increasing the load, local yielding occurs and hinges may develop. The collapse

behaviour of the element depends on the behaviour of the plastic hinges. Failure

mechanism (Yield Line Mechanism) theory can be used to determine the load-

deformation behaviour of the structure in the post failure range.

2.5.1 Yield line theory

Davies et al. (1975) analysed a plate element under uniaxial compression and proposed

a yield line theory. They showed that the ultimate load capacity of the plate depends on

the localised yielded portion of the plate. Murray (1984) proposed a yield line theory

with ignoring the shear force and twisting moment.

Based on Murray’s formulation Enjily et al. (1998) developed two modified theories for

the inelastic deformation of the channel sections. Beale et al. (2001) tested twenty six

cold-formed channel sections under bending. They compared the test results with the

Enjily et al. (1998) theory results. The Enjily et al. (1998) theory and the test results

compared well.

Zhao and Hancock (1993a) have reviewed different theories (Davies et al. (1975),

Murray (1984) and Bakker (1990)) and proposed a new theory to determine the reduced

plastic-moment capacity of an inclined yield line under an axial force.


Zhao and Hancock (1993) performed experimental tests on plastic hinges under axial

force for different inclination angles ( ). They compared their test with the theory

results of Murray (1984) and also the Zhao and Hancock (1993a) theory results. They

concluded that Murray’s theory, due to not including the shear force and twisting

moment, miscalculates the plastic moment drop.

2.5.2 Yield line mechanism model

There are different theories to analyse the collapse behaviour of a complete structure.

However, for achieving correct results from a theory, an accurate model should be

prepared. The yield line mechanism models are based on experimental observations.

Based on laboratory tests observations, Murray and Khoo (1981) developed eight basic

mechanisms for plates and five combinations of simple mechanisms for channel

columns.

Kecman (1983) studied the bending collapse behaviour of rectangular and square

hollow sections. Kecman subsequently developed a yield line mechanism including

travelling yield lines. Kecman’s model was verified using experimental results from

fifty six bending tests on twenty seven different sections.

Koteko (1996) investigated the yield line mechanism of rectangular and trapezoidal box

section beams with a high width to depth ratio compared to Kecman’s sections.

Kotelko’s models are similar to Kecman’s model with a slight difference of the web

hinge line angles (2.32).


Figure 2.32: Yield line mechanism for box sections under bending (Koteko (2004))

As the inclination angle and the number of inclined yield lines in the local plastic

mechanism have a considerable influence on the final analysis results, Zhao (2003)

collected basic yield line mechanisms from different studies which are shown in Figure

2.33. His collection of different yield line mechanism is used as a reference in this

research for modelling the collapse shape of the tested sections.


Figure 2.33: Basic yield line mechanism (Zhao (2003))

2.6 Conclusions

By reviewing the range of literature on the study of designing cold-formed channel

sections with edge stiffener a number of conclusions are evident.

Firstly, the DSM and EWM, which are the design methods for cold-formed sections in

different standards, do not include any inelastic reserve capacity for cold-formed

channel sections with edge stiffener. The assumption in the DSM and EWM is that the

maximum moment capacity is the yield moment. There is a lack of experimental data


for fully effective sections since most studies (Schafer (2006a)) concentrate on slender

sections. Therefore, it would be valuable to conduct research on channel sections with

edge stiffeners to determine whether or not inelastic reserve capacity can be applied.

Secondly, the plastic design method is based on studies for hot-rolled steel and mainly

applicable for hot-rolled sections (Lyse and Godfrey (1935), Haaijer and Thuerlimann

(1958), Lay (1965), Craskaddan (1968), Lukey and Adams (1969), Korol and Hudoba

(1972), Holtz and Kulak (1973)). Few studies (Hasan and Hancock (1988), Zhao and

Hancock (1991) and Wilkinson (1999)) were conducted on behaviour of cold-formed

closed sections in the plastic range and concluded that the plastic method in AS4100

(1998) cannot provide an accurate result for cold-formed steel. While some

experimental data (Baigent and Hancock (1981)) demonstrate the inelastic behaviour of

cold-formed channel sections, no studies were performed on the rotation capacity. This

is the key factor for determining the slenderness limits in the plastic design.

Thirdly, Kecman and Kotelko’s Yield Line Mechanism models are in good agreement

with the test results. Therefore, these models will be used as guidance for introducing

an accurate model in this thesis to investigate the collapse behaviour of cold-formed

channel sections.

Finally, these findings and the limited experimental data and research support the topic

of this thesis. By examining the behaviour of cold-formed channel sections under

bending, this work will be an important contribution to propose a less conservative

method for calculating the ultimate moment capacity of these sections.

58

Chapter 3

TEST PROCEDURES OF COLD-FORMED

CHANNEL SECTIONS UNDER PURE BENDING


From literature review it was found that there is a lack of research on the ultimate

strength of cold-formed channel sections in the inelastic and plastic range. Therefore, in

this research experimental analysis are set to investigate the behaviour of cold-formed

channel sections with edge stiffeners under bending. This is to determine if the inelastic

reserve capacity and plastic design rules are applicable on channel sections with edge

stiffener.

This chapter describes the test procedures of forty two cold-formed channel sections in

major-axis bending using Monash bending rig. The sections are made from cold-

formed G450 steel with nominal thickness of 1.6mm and varying theoretical buckling

stresses ranging between elastic to seven times the yield stress.

The material properties of the cold-formed sections are examined to determine their E

(young’s modulus), yF (yield strength), uF (tensile strength) and ue (elongation)

values. The mechanical properties of the specimens are then calculated and their

preparations are discussed. Furthermore, the value of the Monash bending rig setup and

modifications to meet the specific purposes of this research is outlined. Finally, the test

procedures and methods for calculating curvature and bending moment are also

explained. By doing so, this chapter forms the foundation for a more detailed discussion

of the experimental analysis that leads to numerical analysis, revising existing design

methods, validating finite element simulations and semi-empirical analysis of cold-

formed channel sections under bending.

Chapter 3. Test Procedure of Cold-formed Channel Sections under Pure Bending 59

3.1 Material Properties

The purpose of this experiment is to analyse the behaviour of cold-formed channel

section under bending. To achieve this, four different steel sheets were used to fabricate

forty two cold-formed channel sections. The metal sheets are G450 cold-formed steel

with a nominal thickness of 1.6mm. From each sheet two tensile coupons were cut. To

track the coupons, the steel sheets are named as G, H, I and J. The dimensions of the

tensile coupons and the tension test procedures were in accordance with the AS1391

(2005). Figure 3.1 depicts these dimensions.

Figure 3.1: Tensile coupon specimen in accordance to the AS1391 (2005)

To determine the strain from the tests, two strain gauges were attached to the centre of

each side of the coupon. Additionally, an extensometer was used to collect the strain

after the strain gauges of the coupons were detached. The tension tests were performed

using a 500 kN capacity Baldwin Universal Testing Machine. The average values of the

strain gauges were used for plotting the stress-strain diagram of up to 0.75% strain. The

extensometer data were used for the strain of beyond 0.75% where there was a chance

of the strain gauge detaching from the coupons.

In cold-formed steels, according to the AS1391 (2005), a 0.2% of proof stress was used

as a yield stress. This is due to the cold-formed steels having a rounded stress-strain

curve around the yield point, which is not the same as the hot-rolled steels that have

upper and lower yield stress. According to the AS1391, the young modulus is the slope

of the straight line of the graph prior to the yielding. Figure 3.2 shows the full stress-

strain curves for one of the coupons. This also shows how the yield stresses and

young’s modulus were determined. The seven other tests graphs are shown in Appendix


A. These stresses were calculated according to the ratio of the measured load collected

from the machine to the original cross sectional area.

Coupon G1

0

100

200

300

400

500

600

0% 2% 4% 6% 8% 10% 12%

Strain

Str

ess

(MP

a)

(a) Full Stress-Strain curve for coupon G1

Coupon G1

0

100

200

300

400

500

600

0.0% 0.1% 0.2% 0.3% 0.4% 0.5% 0.6% 0.7% 0.8% 0.9%

Str

ess

(MP

a)

Strain

Fy=535 MPa

E=194198 MPa

(b) Determining Fy for coupon G1

Figure 3.2: Stress-Strain Curves

Furthermore, the following equations show how to calculate the uF (tensile strength)

and ue (elongation) from coupon test results.


areaOriginal

machinethefromrecordedloadPeakFu (3.1)

lengthGaugeOriginal

lengthgaugeOriginallengthgaugeFinaleu

100% (3.2)

mmlengthGaugeOriginal 50

Table 3.1 shows the calculated values of E (young’s modulus), yF (yield strength), uF

(tensile strength) and ue (elongation) for the tension tests of the eight coupons. It is to

be noted that due to the inaccurate installation of the extensometer, results were not

used for coupon H1 and H2 to plot their full stress- strain curves.

Table 3.1: Tensile coupon test results

Thickness Yield AverageYield Young's Tensile

t stress Fy stress Fy Elongation modulus E stress Fu

Coupons (mm) (MPa) for each steel sheets %eu (MPa) (MPa)

G1 1.54 535.0 11% 194198.0 561.8

G2 1.57 522.0 10% 177338.0 563.5

H1 1.53 541.0 10% 176938.0 565.4

H2 1.53 544.0 12% 187905.0 581.0

I1 1.5 557.0 12% 196506.0 584.3

I2 1.51 525.0 12% 191620.0 559.3

J1 1.49 543.0 12% 198834.0 568.4

J2 1.49 561.0 12% 197997.0 595.7

Mean 1.52 541.00 11.4% 190167.0 572.2

>450.0 <15%

528.5

542.5

541.0

552.0

From Table 3.1, it is evident that the average values of the yield stress and young’s

modulus are 541 MPa and 190167 MPa, the average percentage of the elongation is

11.4% and the average ratio of the ultimate tensile stress over the yield stress is 1.06.

These values do not satisfy some of the plastic design limitations in the AS4100 (1998).

Therefore, based on the AS4100 (1998) design rules, the tested sections’ bending

capacity cannot reach the plastic moment.

3.2 Mechanical Properties and Preparation of the Specimens

This study seeks to determine the effect of different edge stiffeners on the ultimate

strength of cold-formed channel sections. Three different typical channel sections are

therefore tested which are shown in Figure 3.3.


(a) (b) (c)

Figure 3.3: Typical channel sections (a) Typical channel section with complex edge stiffener (b) Typical channel section with simple edge stiffener

(c) Typical simple channel section

The channel sections are 1,500mm long and fabricated from steel metal sheets. The

tested section’s slenderness ratio is s which is based on hot-rolled specifications that

range between 4.22 and 56.64. The sections theoretical buckling stress varies between

170 to 4000 MPa and depends on the section’s size and effective length. The

dimensions, yield moment ( yM ), plastic moment ( pM ), the ultimate moment capacity

from bending test results ( testM ) and effective length ( effL ) for each section are shown

in Table 3.2. It is to be noted that effective length is the distance between restraining

plates (this is described in more detail in section 3.4).

63

Table 3.2: The dimensions of each section Section dimensions are shown in Figure 3.4 Thickness Length Area Yield DSM

b4 b3 b2 b1 t Leff As stress Fy Steel My Mp Mb(lateral buckling)

sections (mm) (mm) (mm) (mm) (mm) (mm) (mm2) (Mpa) Sheet kN-m kN-m kN-m

1 47.40 161.22 1.54 500.00 386.50 541.00 I 9.524 11.386 9.5242 66.45 121.68 1.57 500.00 391.70 541.00 I 8.532 9.666 8.5323 12.32 15.94 44.92 122.14 1.57 500.00 397.60 528.50 G 7.798 9.243 7.7984 14.20 14.94 62.75 79.85 1.56 500.00 387.50 552.00 J 5.754 6.616 5.7545 12.62 21.67 41.49 111.16 1.57 500.00 388.50 528.50 G 6.663 8.074 6.6636 12.51 16.29 41.27 129.03 1.57 500.00 398.60 528.50 G 8.050 9.629 8.0507 12.39 15.78 34.99 139.88 1.58 500.00 396.40 528.50 G 8.315 10.090 8.3158 11.82 17.66 48.23 110.04 1.59 500.00 397.70 528.50 G 7.154 8.449 7.1549 9.78 18.06 56.65 99.00 1.56 500.00 394.30 552.00 J 6.976 8.102 6.976

10 17.12 17.98 49.36 99.83 1.54 500.00 390.60 541.00 I 6.502 7.730 6.50211 10.85 16.19 60.10 94.21 1.54 500.00 390.20 552.00 J 6.730 7.750 6.73012 10.85 16.50 50.93 113.76 1.53 500.00 390.50 541.00 I 7.552 8.843 7.55213 9.98 14.27 58.18 102.90 1.57 500.00 396.40 541.00 I 7.288 8.376 7.28814 22.74 47.59 121.10 1.58 500.00 397.50 542.50 H 7.976 9.415 7.97615 13.34 42.49 141.02 1.58 500.00 383.10 542.50 H 8.587 10.193 8.58716 18.67 31.40 159.19 1.57 500.00 391.20 542.50 H 9.080 11.166 9.08017 12.44 37.01 161.69 1.54 500.00 385.80 542.50 H 9.314 11.293 9.31418 17.34 62.09 102.68 1.56 500.00 392.20 541.00 I 7.327 8.341 7.32719 12.45 47.50 141.42 1.55 500.00 389.40 542.50 H 8.976 10.549 8.97620 14.53 55.88 121.20 1.56 500.00 392.90 542.50 H 8.312 9.566 8.31221 12.88 65.86 103.61 1.57 500.00 393.90 541.00 I 7.582 8.541 7.582

64

Table 3.2: The dimensions of each section (continued) Section dimensions are shown in Figure 3.4 Thickness Length Area Yield DSM

b4 b3 b2 b1 t Leff As stress Fy Steel My Mp Mb(lateral buckling)

sections (mm) (mm) (mm) (mm) (mm) (mm) (mm2) (Mpa) Sheet kN-m kN-m kN-m

22 20.00 39.99 89.00 1.50 500.0 298.5 541.0 I 4.408 5.215 4.40823 19.96 45.00 89.98 1.50 500.0 314.9 541.0 I 4.827 5.652 4.82724 19.96 49.99 89.96 1.50 500.0 329.9 541.0 I 5.178 6.009 5.17825 19.97 35.00 79.80 1.55 500.0 278.4 541.0 I 3.595 4.303 3.59526 20.00 40.20 79.99 1.50 500.0 285.7 541.0 I 3.824 4.517 3.82427 19.97 45.00 79.98 1.52 500.0 303.9 541.0 I 4.175 4.883 4.17528 19.96 29.97 70.05 1.50 500.0 239.9 541.0 I 2.634 3.198 2.63429 19.95 34.99 70.10 1.55 500.0 263.3 541.0 I 3.000 3.591 3.00030 19.99 39.97 70.00 1.50 500.0 270 541.0 I 3.176 3.751 3.17631 20.00 25.00 58.90 1.50 300.0 208.4 541.0 I 1.830 2.268 1.83032 19.97 29.96 60.80 1.55 400.0 233.4 541.0 I 2.217 2.696 2.21733 19.97 35.00 60.40 1.55 500.0 248.4 541.0 I 2.438 2.920 2.43834 14.80 19.90 49.50 1.55 190.0 168.6 541.0 I 1.239 1.541 1.23935 14.96 24.99 50.10 1.50 285.0 180.1 541.0 I 1.421 1.725 1.42136 14.95 29.97 50.10 1.50 290.0 195 541.0 I 1.611 1.921 1.61137 9.75 14.78 38.20 1.55 170.0 119.6 541.0 I 0.668 0.837 0.66838 9.63 19.75 39.40 1.55 210.0 136.5 541.0 I 0.850 1.032 0.85039 9.83 24.68 38.50 1.55 240.0 151 541.0 I 0.970 1.151 0.97040 9.20 10.45 28.10 1.55 85.0 88.8 541.0 I 0.327 0.430 0.32741 9.70 14.50 29.50 1.55 155.0 105.1 541.0 I 0.445 0.564 0.44542 9.73 19.55 29.00 1.55 145.0 120.1 541.0 I 0.544 0.666 0.544


b 2

b1

b 3

b 4

t

Figure 3.4: Sections dimensions

The following equations show the calculations of the yield and plastic moments of the

tested sections which are tabulated in Table 3.2.

yfullxy FZM . (3.3)

yxp FSM (3.4)

where, fullxZ . is the calculated elastic and xS is the calculated plastic section modulus

of the section about the major axis. The elastic and plastic curvatures, yk and pk , of

the tested sections can be determined as follows:

EI

Mk y

y (3.5)

EI

Mk p

p (3.6)

The following computations are an example of calculating fullxZ . , xS , yk and pk values

for section 1:

3.0,541,194100

54.1,0,4.47,22.161,46.1 4321

MPaFMPaE

mmtbbmmbmmbmmr

y

i

mmrbdmmrbb

mmrImmrcmmru

mmt

rrmmtrr

ee

c

iie

22.1552,4.44

652.1149.0,42.1637.0,5.357.1

23.22

,3

12

33


y5

y4

y3

y2

y1L2

L4L5

L3

L1

Figure 3.5: Calculating I value of section 1

35155

32

44144

33

333

32

222

3111

652.1461.159501.3

775.812

45.1602

4.44

31160012

61.802

22.155

12

.77.0

24.44

652.1579.1501.3

mmIImmcrbymmuL

mmty

Immt

bymmbL

mmd

Immd

rymmdL

mmtb

Immt

ymmbL

mmIImmcrymmuL

ce

e

ce

The dimensions of L1 to L5 and y1 to y5 are shown in Figure 3.5.

422.

1 31419,2

mmELYIyLtIb

Y iciiifullxc

1

..

3.. 65.34,524.9,17600 mmE

EI

MkmkNFZMmm

Y

IZ

fullx

yyyfullxy

c

fullxfullx

1

.

3 64.41,386.11,21050 mmEEI

MkmkNFSMmmyYLtS

fullx

ppyxpicix


Elchalakani (2003) tested cold-formed circular hollow sections using the Monash pure

bending rigs (this is described in more detail in the following section). Elchalakani

concluded that, to avoid any failure at the two ends of the members due to the local

bearing, all sections need to be filled with plaster. In the early stages of this study a

channel section has been filled with plaster and has been tested by the Monash rig. It

was observed that, even by filling the two ends of the section using plaster, the local

instability cannot be avoided. Before bending failure at mid-span some sections failed

due to the bearing failure. Figure 3.6(a) shows cracks in one end of the first tested

section filled with plaster. To avoid cracks and bearing failure complications, all the

sections have been filled with 50MPa grout concrete from each ends. No cracks were

observed in the subsequent tests for sections filled with concrete (Figure 3.6(b)).

(a) Filled sections with plaster at each end.

(b) Filled sections with grout concrete at each end. Figure 3.6: Filled sections


3.3 Bending Rig Set up

The bending tests are performed using the Monash pure bending rig. The rig was

developed by Cimpoeru (1992) for modelling the collapse during roll-over of bus

frames consisting of square thin-walled tubes. Since development of the rig, it has been

extensively used for research purposes. For example, Zhao and Grzebieta (1999) used

the rig for testing Square Hollow Sections subjected to large deformation under cyclic

bending. Elchalakani et al. (2002a) analysed the plastic collapse behaviour of circular

tubes using the Monash pure bending rig. Furthermore, Tan (2009) performed pure

bending tests on perforated hat sections using the rig.

In four-point bending tests, the applied point loads are converted to bending moment

and possibly torsion (if the point loads are eccentric to the shear centre). With the

Monash bending rig, the concrete filling and placing of the member ends in the wheels,

then the subsequent rotation of the wheels, applies a pure bending strain to the member.

Thus, the members are loaded with pure bending moment and negligible torsion. This

was validated experimentally, where no members displayed torsional deformations.

Therefore, the superiority of using Monash pure bending rig is that, it can apply a

definitive moment to the sections’ mid-span with zero axial and shear forces. Figure 3.7

shows the front view of the bending rig during a test.

Figure 3.7: Front view of Monash pure bending rig


The Monash pure bending rig includes two load cells, which are connected to the

support wheels from one side and to the hydraulic jack from the other side. The

bending moment is applied to the specimens by a hydraulic pump connected to two

hydraulic jacks. By pumping, the wheels start rotating. Therefore, two force couples

apply to the specimen from the loading pin. The applying load on each wheel is

measured by sensors which are attached to the 50kN load cells. While one of the

wheels is stationary the other is able to move horizontally to avoid any axial force

developing during the experiment. Figure 3.8 shows schematic of the Monash bending

rig.

Figure 3.8: Schematic of the Monash pure bending rig 3.4 Bending Rig Modifications

To meet the purpose of this research, the pure bending rig was modified to avoid any

lateral buckling prior to local or distortional buckling (Figure 3.9). On each loading

wheel four restraining steel plates were installed. The steel plates were used to restrain

all sections with a depth of less than 210mm. As is the value of the pure bending rig,

one of the loading wheels can be moved horizontally and therefore, the effective length

can be adjusted for each section.


SectionA

A

250mm

Steel plates torestrain the section

Front view of the loading wheelThe rest are not shown for clarity

Bolts and nuts to alterplate lateral position

Loading wheel

View A-A

Figure 3.9: Modified wheel

In some cases, to reach a higher theoretical buckling stress, the section’s effective

length needs to be as small as possible. By widening the restraining plates this objective

can be achieved. Figure 3.10 shows the designed restraining plates which are installed

on the rig’s wheels.


Figure 3.10: Installation of the restraining plates

Restraining plate design computations are as follows:

According to the AS4100 (1998):

“5.4.3.2 Restraint against twist rotation A torsional at a cross-section may be deemed to provide effective restraint against twist rotation if it is designed to transfer a transverse force equal to 0.025 times the maximum force in the critical flange from any unrestrained flange to the lateral restraint.”

F

250mm

Front view

65 15

Side view

b

Figure 3.11: Restraining plate

yf FAF 025.0 (3.7)


where fA is the flange area.

Therefore, the restraining plate has been designed according to AS4100 (1998) and

with a necessary thickness that is greater than 14mm.

To determine the local, distortional and the minimum theoretical buckling stress of the

tested sections based on their effective length, the Thin-Wall computer program is used

(Papangelis and Hancock (1995)). By introducing the size of the section and the

loading pattern to the Thin-Wall program, the maximum stress verse buckling half-

wavelength graph is produced. Figure 3.12 depicts the maximum stress verse buckling

half-wavelength graph for the section 22. Note that the effective length of section 22 is

500mm (Table 3.2).

Figure 3.12: Buckling modes for section 22

In addition, the nominal member moment capacities of the tested sections due to the

lateral buckling are calculated based on DSM in AS/NZS4600 (2005) and NASPEC

(2007); and tabulated in Table 3.2. It is evident in Table 3.2 that all the tested sections’


nominal member capacity due to the lateral buckling is equal to their yield moment. In

other words, the tested sections are restrained against lateral buckling.

3.5 Bending Test Procedures

A series of steps were conducted during the bending tests. Specimens were mounted on

the loading wheels and positioned by four locking pins at the top and the bottom of

each wheel. Two magnetic inclinometers were then placed on the specimens to monitor

rotation of the sections during the test. The moment was applied to the specimens by a

hydraulic pump connected to hydraulic jacks on each loading wheel. By pumping, the

wheels start rotating. Therefore, two force couples were applied to the specimen by the

loading pin within the grouting length. While testing, eleven parameters were

monitored to calculate the moment and curvature of each sample. Six inclinometers

were used to show the angle changes of two hydraulic jacks, two loading wheels and

two ends of specimens. The forces applied from two jacks were recorded via a

computer. The strain in the top and the bottom flanges were then collected by

connecting strain gauges to the coaxial wires and soldering to the signal transmission.

The results were then recorded via a computer. Collecting these data is essential in

determining the curvatures and moment of the sections during the test.

3.5.1 Curvature calculations

To calculate the curvature of each specimen, under the pure bending, two different

methods were used. In the first method, the average value of the rotational angle on the

left and the right side of the beam ( Left ) and ( Right ) were used to calculate the

curvature ( k ) of the sample. These angles were measured by magnetic inclinometers

attached to the top of the specimen (Figure 3.13). The value of k was calculated using

the following equation:

Lk 5.0 (3.8)

2RightLeft

(3.9)

where L is the distance between the two inclinometers and needs to be measured prior

to each test.


Due to applying constant moment on the tested beams, the curvature is constant along

the beams. Therefore, the beam is an arc of a circle and the angle measured by magnet

inclinometers is dependent on the position of the inclinometers.

Figure 3.13: Determination of the curvature from measured rotation angles

The following calculations prove the dependency of the curvature to the position of the

inclinometers.

rk

1 (3.10)

r

LSin Left

Left (3.11)

r

LSin Right

Right (3.12)

For small value of , in Radian, Sin

Lr 5.0 (3.13)


In the second method, strains of the top ( c ) and the bottom ( t ) flanges were used to

calculate the curvature of the sample. The top and the bottom strains were measured

using the two attached strain gauges at the top and bottom flanges. The value of k was

calculated using the following equation:

c

t

c

c

YdYkk

tan (3.14)

where cY is the distance from the neutral axis to the compression edge of the section

and d is the section’s depth. c and t are the absolute values of the compression and

tension strains respectively (Figure 3.14).

M

d

c

M

t

Yc

d-Yc

Figure 3.14: Determination of the curvature from measured strains

3.5.2 Bending moment calculations

Figure 3.15 shows the geometry of the left side of the bending rig. Prior to performing

each test, the relative dimensions of the loading cell, wheel and locking pins should be

measured for both wheels. This is due to using the average dimensional values of both

wheels to calculate the applying moment to the sample.


Figure 3.15: Geometry of the bending rig

By measuring a, e and d from the bending rig and using Equations 3.15 to 3.20, three

different angles of α, β and γ can be determined. Figure 3.16 demonstrates the force


being applied through the loading cell. The following equations show the calculation of

α, β and γ.

)2()()( 222 aedaeCos (3.15)

)2()()( 222 deadeCos (3.16)

180 (3.17)

erinclinometwheelLoadingnew (3.18)

erinclinometjackLoadingnew (3.19)

newnewnew 180 (3.20)

d

a

F

FCos''

'e

"

Figure 3.16: Force diagram at the left support wheel of the pure bending rig

Also:

FSinFCos

9090

180 (3.21)

When considering the applying force through the loading cell as F, the left hand side

bending moment can be calculated as follows:

eFSinM (3.22)

By performing similar computations for the right hand side wheel, the final applied

moment on the sample was calculated as the average values of the left and right

moments.


Also, the relationship between the bending moment, curvature and flexural

rigidity )(EI can be used as an alternative method to calculate the applying moment.

From Bernoulli-Euler equation:

Mx

uEI

2

2

(3.23)

where u is the deflection of the beam and x

u

is the slope of the beam )( .

Mx

EI

(3.24)

EIkMkx

(3.25)

The calculated bending moments and curvatures from readings of the strains gauges are

accurate before any local failure. Therefore, calculated moments and curvatures based

on readings of the strain gauges were used as a benchmark to calibrate the calculated

moments and curvatures, based on the inclinometers reading. For calculating the

bending moment and curvature, Equations 3.25 and 3.14 are based on readings of the

strain gauges. In addition, Equations 3.22 and 3.8 are based on inclinometer readings.

The reason for using the calibrated inclinometers records is that the readings of the

strain gauges represents the local moment and the curvature. After forming the plastic

hinge, the curvature is concentrated at the location of the hinge. Therefore, if a gauge

position is not at the same position as the hinge, the calculated curvature using the

strain-gauges record does not represent the beam’s behaviour as a whole. Therefore, the

inclinometer readings are used to determine the rotation capacity of the sections after

local failure occurs.

Figure 3.17(a) compares moment-rotation diagrams for the two different moment

calculation methods. Figure 3.17(b) shows how the calculated moment based on the

inclinometers records were calibrated using the records of the strain gauges.


0

1

2

3

4

5

6

7

0.0 0.1 0.2 0.3 0.4 0.5 0.6θ (Rad)

(a) Moment-Rotation graph of section 8 before calibration

M (

kN-m

)

M=eFSinγ

M=EIk(Strain gauges)

0

1

2

3

4

5

6

7

8

0.0 1.0 2.0 3.0 4.0 5.0 6.0θ (Rad)

(b) Final Moment-Rotation garph of section 8

M (

kN-m

)

M(Calibrated)=eFSinγ

M=EIk(Strain gauges)

Figure 3.17: Comparing moments from two different methods


3.6 Conclusions

This chapter examined the material property of the tested sections by performing

tension tests on coupons from different metal sheets. The outcome of the tension test

shows that the material property of the tested sections are not in a range to satisfy some

of the plastic design limitations in the AS4100 (1998). This issue is to be verified in the

following chapter.

To determine tested sections capability for designing in plastic range, their rotation

capacity should be measured. Since strain gauge reading is only reliable before the

local failure, inclinometers were therefore used to monitor section behaviour after

failure. It should be noted that the inclinometer data are calibrated using strain gauges

in the early stage of each test.

Local bearing and lateral-torsional buckling failures are not the concern of this

research. The local bearing failure has been avoided by grouting of the samples at both

ends. The lateral-torsional buckling has been avoided by modifying the Monash pure

bending rig.

The following chapter builds on this research by presenting an in-depth discussion of

the test results and comparing them with the existing design rules for current steel

standards.

81

Chapter 4

EXPERIMENTAL RESULTS AND DISCUSSIONS


This chapter classifies channel sections into slender, non-compact and compact

sections, according to two different methods. The first method is based on the test

results and the second method is based on the slenderness ratio of the sections. These

two methods are compared and it is shown that sections classifications based on

AS4100 (1998) do not match with the test results.

The common observed failure modes from the tests are identified and discussed. The

observations reveal that due to the sudden collapse of the sections, deformation process

of the sections cannot be monitored during the test. To this end, a more in-depth

discussion of the failure modes based on FEM is conducted in chapter 7.

In addition, the ultimate bending moment capacities of the sections are calculated from

six methods being the:

1. test result ( testM );

2. NASPEC (2007) design rules ( NASPECM );

3. AS/NZS4600 (2005) design rules ( 4600ASM );

4. DSM ( DSMM );

5. EUROCODE3 (2006) design rules ( 3EurocodeM ); and

6. AS4100 (1998) design rules ( 4100ASM ).

testM is then used as a benchmark to gauge the accuracy of the NASPECM ,

4600ASM , DSMM , 3EurocodeM and 4100ASM .

Chapter 4. Experimental Results and Discussions 82

4.1 Sections Classifications

In terms of classifying the sections, this chapter compares the ultimate moment

capacities, from pure bending tests, of forty two tested sections with their yield and

plastic moments. In addition, the rotation capacities of the tested sections are

determined by normalising the moment-curvature diagram from the test results with the

plastic moment and the plastic curvature. By using the test results, the tested sections

are classified into compact, non-compact and slender sections as follows:

Compact: 3,max RMM P

Non-compact: 3,max RMM P or py MMM max

Slender: yMM max .

According to the AS4100 (1998) design rules, and based on the tested sections

slenderness ratio, the tested sections are classified into compact, non-compact and

slender sections as follows:

Compact: ps

Non-compact: ysp

Slender: ys

s , is the value of either the web or flange slenderness ratio with the greatest value of

eye . The slenderness ratio of each element, e , is calculated according to

AS4100 (1998).

250y

e

f

t

b

(4.1)

where t is the element thickness and b is the clear width of the element between the

face of supporting elements, as shown in Figure 4.1.


Figure 4.1: Width of the element

p and y are plastic and elastic slenderness limits. The value of p and y are

defined in AS4100 (1998).

4.2 Slender Sections

A slender section is defined as a section where its ultimate moment capacity cannot

reach the yield moment. Based on the test results, fifteen sections of the forty two tested

sections are classified as slender sections. Among these slender sections, eight are

channel sections with a simple edge stiffener, five are channel sections with complex

edge stiffener and two are simple channel sections. The slenderness ratio of the tested

sections and their classifications, based on test result and also AS4100 (1998), are

tabulated in Table 4.1 and 4.2 respectively. These tables display the range of results and

the variability depending on the methods used.


Table 4.1: Sections classification based on test result Rotation

Mtest My Mp capacity Test Mbuckling Legend

sections kN-m kN-m kN-m R=k/kp-1 Classification kN-m

1 5.03 9.52 11.39 S 2.62 S:Slender

2 4.45 8.53 9.66 S 1.35 NC:Non-Compact

3 7.90 7.80 9.24 NC 5.18 C:Compact

4 4.85 5.75 6.62 S 2.45

5 7.56 6.66 8.07 NC 6.22

6 8.17 8.05 9.63 NC 7.80

7 8.60 8.32 10.09 NC 9.08

8 7.45 7.15 8.45 NC 6.59

9 6.80 6.98 8.10 S 2.59

10 6.76 6.50 7.73 NC 4.95

11 6.09 6.73 7.75 S 2.40

12 7.48 7.55 8.84 S 5.22

13 6.60 7.29 8.38 S 1.84

14 7.97 7.98 9.42 S 2.17

15 8.76 8.59 10.19 NC 2.34

16 8.57 9.08 11.17 S 3.35

17 8.73 9.31 11.29 S 8.70

18 6.38 7.33 8.34 S 4.00

19 8.37 8.98 10.55 S 2.32

20 7.82 8.31 9.57 S 6.41

21 5.78 7.58 8.54 S 1.96

22 4.98 4.41 5.22 NC 4.69

23 4.97 4.83 5.65 NC 3.50

24 4.91 5.18 6.01 S 4.39

25 3.95 3.60 4.30 NC 2.45

26 4.26 3.82 4.52 NC 2.71

27 4.46 4.18 4.88 NC 2.54

28 3.11 2.63 3.20 NC 2.91

29 3.30 3.00 3.59 NC 2.44

30 3.40 3.18 3.75 NC 1.43

31 2.24 1.83 2.27 NC 1.81

32 2.50 2.22 2.70 NC 0.59

33 2.72 2.44 2.92 NC 2.45

34 1.58 1.24 1.54 0.65 NC 1.31

35 1.70 1.42 1.73 NC 1.41

36 1.88 1.61 1.92 NC 1.61

37 0.91 0.67 0.84 1.50 NC 0.77

38 1.07 0.85 1.03 1.30 NC 0.91

39 1.22 0.97 1.15 0.70 NC 1.15

40 0.52 0.33 0.43 4.30 C 0.46

41 0.64 0.45 0.56 2.10 NC 0.56

42 0.73 0.54 0.67 2.45 NC 0.67

85

Table 4.2: Sections classification based on AS4100 controlling

element for AS4100

sections λeFlange λeWeb λeyFlange λeyWeb λepFlange λepWeb (λe/λey)Flange (λe/λey)Web Max(λe/λey) failure λs λsy λsp Classification

1 39.55 148.27 15.00 115.00 8.00 82.00 2.64 1.29 2.64 Flange 39.55 15.00 8.00 S

2 56.64 108.39 15.00 115.00 8.00 82.00 3.78 0.94 3.78 Flange 56.64 15.00 8.00 S

3 36.06 107.61 40.00 115.00 30.00 82.00 0.90 0.94 0.94 Web 107.61 115.00 82.00 NC

4 54.06 70.34 40.00 115.00 30.00 82.00 1.35 0.61 1.35 Flange 54.06 40.00 30.00 S

5 32.88 97.43 40.00 115.00 30.00 82.00 0.82 0.85 0.85 Web 97.43 115.00 82.00 NC

6 32.68 113.99 40.00 115.00 30.00 82.00 0.82 0.99 0.99 Web 113.99 115.00 82.00 NC

7 26.69 123.26 40.00 115.00 30.00 82.00 0.67 1.07 1.07 Web 123.26 115.00 82.00 S

8 38.64 95.18 40.00 115.00 30.00 82.00 0.97 0.83 0.97 Flange 38.64 40.00 30.00 NC

9 48.25 88.58 40.00 115.00 30.00 82.00 1.21 0.77 1.21 Flange 48.25 40.00 30.00 S

10 41.42 89.63 40.00 115.00 30.00 82.00 1.04 0.78 1.04 Flange 41.42 40.00 30.00 S

11 52.20 85.11 40.00 115.00 30.00 82.00 1.31 0.74 1.31 Flange 52.20 40.00 30.00 S

12 43.20 103.61 40.00 115.00 30.00 82.00 1.08 0.90 1.08 Flange 43.20 40.00 30.00 S

13 48.89 90.79 40.00 115.00 30.00 82.00 1.22 0.79 1.22 Flange 48.89 40.00 30.00 S

14 38.78 107.31 40.00 115.00 30.00 82.00 0.97 0.93 0.97 Flange 38.78 40.00 30.00 NC

15 34.02 125.88 40.00 115.00 30.00 82.00 0.85 1.09 1.09 Web 125.88 115.00 82.00 S

16 23.83 143.73 40.00 115.00 30.00 82.00 0.60 1.25 1.25 Web 143.73 115.00 82.00 S

17 29.66 148.93 40.00 115.00 30.00 82.00 0.74 1.30 1.30 Web 148.93 115.00 82.00 S

18 52.89 91.17 40.00 115.00 30.00 82.00 1.32 0.79 1.32 Flange 52.89 40.00 30.00 S

19 39.44 128.70 40.00 115.00 30.00 82.00 0.99 1.12 1.12 Web 128.70 115.00 82.00 S

20 47.10 108.78 40.00 115.00 30.00 82.00 1.18 0.95 1.18 Flange 47.10 40.00 30.00 S

21 56.09 91.46 40.00 115.00 30.00 82.00 1.40 0.80 1.40 Flange 56.09 40.00 30.00 S

S: Slender

Legend: NC:Non-Compact

C:Compact

86

Table 4.2: Sections classification based on AS4100 (continued) controlling

element for AS4100

sections λeFlange λeWeb λeyFlange λeyWeb λepFlange λepWeb (λe/λey)Flange (λe/λey)Web Max(λe/λey) failure λs λsy λsp Classification

22 33.34 81.41 40.00 115.00 30.00 82.00 0.83 0.71 0.83 Flange 33.34 40.00 30.00 NC

23 38.25 82.37 40.00 115.00 30.00 82.00 0.96 0.72 0.96 Flange 38.25 40.00 30.00 NC

24 43.15 82.35 40.00 115.00 30.00 82.00 1.08 0.72 1.08 Flange 43.15 40.00 30.00 S

25 27.53 70.05 40.00 115.00 30.00 82.00 0.69 0.61 0.69 Flange 27.53 40.00 30.00 C

26 33.54 72.57 40.00 115.00 30.00 82.00 0.84 0.63 0.84 Flange 33.54 40.00 30.00 NC

27 37.75 71.60 40.00 115.00 30.00 82.00 0.94 0.62 0.94 Flange 37.75 40.00 30.00 NC

28 23.51 62.82 40.00 115.00 30.00 82.00 0.59 0.55 0.59 Flange 23.51 40.00 30.00 C

29 27.52 60.84 40.00 115.00 30.00 82.00 0.69 0.53 0.69 Flange 27.52 40.00 30.00 C

30 33.32 62.77 40.00 115.00 30.00 82.00 0.83 0.55 0.83 Flange 33.32 40.00 30.00 NC

31 18.64 51.88 40.00 115.00 30.00 82.00 0.47 0.45 0.47 Flange 18.64 40.00 30.00 C

32 22.74 52.01 40.00 115.00 30.00 82.00 0.57 0.45 0.57 Flange 22.74 40.00 30.00 C

33 27.53 51.63 40.00 115.00 30.00 82.00 0.69 0.45 0.69 Flange 27.53 40.00 30.00 C

34 13.19 41.29 40.00 115.00 30.00 82.00 0.33 0.36 0.36 Web 41.29 115.00 82.00 C

35 18.63 43.25 40.00 115.00 30.00 82.00 0.47 0.38 0.47 Flange 18.63 40.00 30.00 C

36 23.51 43.25 40.00 115.00 30.00 82.00 0.59 0.38 0.59 Flange 23.51 40.00 30.00 C

37 8.33 30.56 40.00 115.00 30.00 82.00 0.21 0.27 0.27 Web 30.56 115.00 82.00 C

38 13.05 31.70 40.00 115.00 30.00 82.00 0.33 0.28 0.33 Flange 13.05 40.00 30.00 C

39 17.73 30.85 40.00 115.00 30.00 82.00 0.44 0.27 0.44 Flange 17.73 40.00 30.00 C

40 4.22 20.98 40.00 115.00 30.00 82.00 0.11 0.18 0.18 Web 20.98 115.00 82.00 C

41 8.07 22.31 40.00 115.00 30.00 82.00 0.20 0.19 0.20 Flange 8.07 40.00 30.00 C

42 12.86 21.83 40.00 115.00 30.00 82.00 0.32 0.19 0.32 Flange 12.86 40.00 30.00 C

S: Slender


C:Compact


Based on AS4100 (1998) sections 7, 10 and 15 are classified as slender sections.

However, based on the test results these sections are classified as non-compact sections.

On the other hand section 14 which is classified as a non-compact section, behaved as a

slender section.

4.2.1 Moment-curvature graphs of the slender sections

The methods to calculate the bending moment and curvature from the test, and the yield

moment and curvature of tested sections were explained in chapter 2 and 3. Building on

this, Figure 4.2 details the normalised moment-curvature diagram with yield moment

and yield curvature respectively for three slender sections. All the graphed sections

were fabricated from a steel sheet of the same width. As the same product was used that

was the same width, comparisons can then be made between the ultimate moments in

terms of the efficiency for various edge stiffener configurations.

Slender Sections

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 6.0 7.0 8.0

k/ky

M/M

y

Section 9

Section 1

Section 18

Figure 4.2: Normalised moment-curvature diagram with yield moment and yield curvature respectively for three slender sections

It can be seen from Figure 4.2 that section 9, which is a channel section with a complex

edge stiffener has a higher moment capacity compared with sections 1 and 18. Section

1 is a simple channel has the least moment capacity. It can therefore be concluded that

edge stiffeners have a positive effect on increasing the ultimate moment capacity of


channel sections. Figure 4.3 shows the normalised moment-curvature graphs, based on

the test results, for sections 2 and 17. These sections are an example of the slender

sections which failed before their curvatures reached their yield curvature.

Section 2

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

k/kp

M/M

p

Test Result

M=My

Fully effective

k=ky

Section 17

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

M=My

Fully effective

k=ky

Figure 4.3: Normalised moment curvature graphs based on test results for section 2 and 17

It can be seen from Figure 4.3 that sections 2 and 17 buckled elastically where their

M/Mp values are 0.14 and 0.77 respectively. Figure 4.4 shows the normalised moment-


curvature graphs of the selected tested slender sections and that their curvature at the

failure point is greater than the yield curvature.

Section 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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

pTest Result

M=My

Fully effective

k=ky

Section 11

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

M=My

Fully effective

k=ky

Figure 4.4: Normalised moment curvature graphs based on test results for section 9 and 11

It is evident in Figures 4.3 and 4.4 that the slender tested sections buckled prior to the

yielding. However, except section 17, all the other slender tested sections showed post

buckling behaviour. The buckling load and also the post buckling capacity of all the

tested sections are tabulated in Table 4.1.


Furthermore, Figure 4.4 displays the graphs after buckling and that these are not linear.

Also, the slope of the graph before failure has changed at a few points. This is due to

the nonlinearity of the computation of the effective sections. Finally, Figure 4.5 shows

the different stages of the loading, using section 13 as an example.

Section13

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

My/Mp

Fully effective

ky/kp

Beginning of stage 1




Figure 4.5: Different stage of the loading for section 13 In relation to the behaviour at each stage, the first stage sees the section behaving as a

fully effective section due to the small stress on the compression flange (Figure 4.6(1)).

In the second stage the section starts to behave as a non-fully effective section.

Therefore, the neutral axis will shift to the tension side of the section (Figure 4.6(2)).

When increasing the maximum stress in the compression flange, the section starts to be

less effective until the strain and stress in the compression flange reach to the yield

strain and stress, which is evident in the third stage of the graph. The forth stage is

where the stress can not increase; however, the strain is increasing and is beyond the

yield strain. At this stage the yield stress will distribute through the section, as shown in

Figure 4.6(4).


(1)

f* < Fy < y

(2)

f* < Fy < y

(3)

f* = Fy = y

(4)

f* = Fy > y

Figure 4.6: Section behaviour in the different stage of the loading

It can therefore be concluded that, sections which are classified as slender Figure 4.4

behaved in-elastically. This is due to their curvature and that their strain at the failure

point is greater than the yield curvature and the yield strain. It is to be noted that the

normalised moment-curvature diagram with plastic moment and curvature for the

complete number of sections tests are shown in Appendix B.

4.3 Non-Compact Sections

A non-compact section is defined as a section that its ultimate moment capacity either

could not reach to the plastic moment or does not have enough rotation capacity to

redistribute the moment along the member (Elchalakani et al. (2002b)). Twenty six of

the tested sections are non-compact sections. Six are channel sections with complex

edge stiffeners. However, the other twenty are channel sections with simple edge

stiffeners. Based on AS4100 (1998) classifications sections 25, 28, 29, 31 to 39, 41 and

42 are grouped as compact sections (Table 4.2). However, based on the test results they

behaved as non-compact sections. By comparing sections classifications with the

AS4100 (1998) results, it can be concluded that plastic slenderness limit based on the

AS4100 (1998) is not accurate. This is due to the fact that AS4100 (1998)


classifications are based on studies for hot-rolled steel. Therefore, revised slenderness

limits which are applicable for cold-formed channel sections will be proposed in the

following chapter.

4.3.1 Moment-curvature graphs of the non-compact sections

The normalised moment-curvature graph of a steel section serves as an important basis

to determine the ultimate capacity and also rotation capacity of the section. Figure 4.7

demonstrates normalised moment-curvature diagram with plastic moment and plastic

curvature for four non-compact sections respectively. The ultimate moment capacities

of the all sections, which is an anomalous result according to AS4100 (1998) rules, in

the following graph reached the yield moment. Among these sections, section 37

exceeded the yield moment and even reached the plastic moment.

Non-compact sections

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Section 29Section 32

Section 36

Section 37

M=Mp

Figure 4.7: Normalised moment-curvature diagram with plastic moment and plastic curvature respectively for few non-compact sections

According to the existing design rules, inelastic reserve capacity design method cannot

be applied for cold-formed channel sections with edge stiffener due to the lack of

research in this field. However, some tested sections had an inelastic reserve capacity.

As shown in Table 4.1, sections 3, 5, 6, 8, 10, 15, 23, 25 to 27, 29, 30 and 32 buckled


before reaching the yield moment; however, their ultimate moment capacity from the

test results is greater than their yield moment. Figure 4.8 shows the normalised

moment-curvature graph of section 10 as a sample of this group of sections.

Section 10

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

M=My

Fully effective

k=ky

Figure 4.8: Normalised moment-curvature diagram with plastic moment and plastic curvature respectively for section 10

It is evident from Figure 4.8 that section 10 buckled where its M/Mp value is 0.64.

Section 10 did not fail when its curvature reached the yield curvature. Therefore section

10 behaves in-elastically, as shown at stage four of Figure 4.6.

As shown in Table 4.1, sections 7, 22, 31, 33 to 38 have buckled between the yielding

moment and their plastic moment. In addition, their ultimate moment capacity, as

evident from the test results, is greater than their yield moment. It should be noted that

the ultimate moment capacities for sections 34, 37 and 38 have reached their plastic

moment capacity. Finally, Figure 4.9 shows the normalised moment-curvature graph of

section 38. This serves as a sample for this group of sections.


Section 38

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

pTest Result

M=My

Fully effective

k=ky


It is shown in Figure 4.9 that section 38 has buckled where its M/Mp value is 0.88,

being after the yield moment. The slope of the graph has not changed prior to the

yielding and this means the section was fully effective in the elastic range. Therefore,

the change in the slope of the graph after the yield moment can be due to the material

non-linearity behaviour of the section as well as the buckling non-linearity.

Furthermore, Table 4.1 shows that the sections 39, 41 and 42 have buckled after their

moment reached the plastic moment; and their ultimate moment capacity, as is evident

from the test results, is greater than their plastic moment. As with previously, Figure

4.10 shows the normalised moment-curvature graph of section 39. The behaviour of

this section is a useful sample for this group of sections.


Section 39

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

M=My

Fully effective

k=ky

M=Mp


Figure 4.10 shows that section 39 has buckled where its M/Mp value is equal to one.

The slope of the graph has not changed prior to the plastic moment. While the ultimate

moment capacity of the section 39 reaches to the plastic moment and a plastic hinge

develops, the plastic hinge could not rotate sufficiently to redistribute the moment

through the member. Therefore, plastic design rules are not applicable on this section

and this section should be classified as a non-compact section.

4.4 Compact Sections

A compact section is defined as a section where its ultimate moment capacity reaches

the plastic moment and has enough rotation capacity to redistribute the moment along

the member. When using this definition, section 40 is the only section classified as a

compact section from the forty two tested sections (Table 4.1). In section 40 the

ultimate moment not only reached the plastic moment but also had a rotation capacity

of 4.2. Therefore, this section has the ability to redistribute the moment through the

member.

Slenderness limits have been defined in accordance with both cold-formed and hot-

rolled international steel specifications, below which cold-formed channel sections may


display full plastic capacity with rotational capacity greater than 3 (compact sections),

and which are currently considered acceptable for plastic design. For cold-formed steel

specifications, the flange and web slenderness values must be below 0.25 and 0.15

respectively according to the effective width method (Table 5.1), or the section

slenderness values for local and distortional buckling must both be below 0.35

according to the DSM (Table 4.8). For hot-rolled steel specifications, the flange and

web slenderness values must be below 8 and 22 respectively (Section 5.5).

4.4.1 Failure modes from testing a compact section (section 40)

During the testing of section 40, it became evident that the failed shape of the cross

section changed and the web element rotated around the web flange intersection.

Rogers (1995) named this failure as a flange-web distortional buckling failure and

described it as follows:

“Flange/web distortional buckling is evident when both corners move out of alignment,

but remain parallel to each other, and an apparent lateral buckling formation of the

web appears.”

Figure 4.11 shows the flange-web distortional failure modes for section 40. It can be

observed that both corners are parallel even though they move out of alignment.

However, deformation of compression flange or the web-compression flange juncture is

not evident.

(a) Top view (b) Front view

Figure 4.11: Flange-web distortional failure modes for section 40


4.4.2 Moment-curvature graphs of the compact section

The normalised moment-curvature graph of a steel section serves as an important basis

to determine if the section is capable of being designed using the plastic design method.

Figure 4.12 demonstrates normalised moment-curvature diagram with the plastic

moment and the plastic curvature for section 40.

Compact section

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 6.0 7.0 8.0

k/kp

M/M

p

M=MpSection 40

R=k/kp-1=4.2


According to AS4100 (1998), the tested sections do not satisfy the following plastic

design limitations:

1- The yield stress is 541 MPa which is greater than 450MPa;

2- The percentage of the elongation is 11% which is less than 15%; and

3- The ultimate tensile stress over the yield stress is 1.06 which is less than 1.2.

In addition, according to NASPEC (2007) and AS/NZS4600 (2005), the tested sections,

being cold-formed channel sections with edge stiffener, cannot behave in-elastically.

However, from normalised moment-curvature graph of section 40 it is evident that not

only a plastic hinge at the maximum moment point has been developed but also the

plastic hinge rotated sufficiently to redistribute the moment through the member


( 2.4R ). Section 40 is therefore classified as a compact section and does not fit in the

range of above mentioned design rules, showing them to be conservative.

4.5 Failure Modes for Tested Sections

The testing identified that two simple channel sections of the fifteen slender sections

were the exception where the buckling in the flanges was seen during the test and prior

to the failure of the beam (Figure 4.13).

Figure 4.13: Local buckling mode appearance during the bending test

To inform some analysis of the failure mode of each tested section, two dimensions

were measured at the end of each bending test. Firstly was the distance between the top

and the bottom flanges, where the edge stiffener are located (for channel sections with

edge stiffener) or at the free edges of the section (for simple channel sections), were

measured ( dh ). By measuring this distance the rotation angle due to the deformation of

the web-compression flange juncture, d , can be calculated. This is shown in Figure

4.14(a).

dd hbL 1 (4.2)

2b

LArcTan d

d (4.3)


h

L

d

L

l

d

d

l

(a)Deformation of the (b)Deformation of the

web-compression flange juncture compression flange element

(Distortional buckling) (Local buckling)

Figure 4.14: Deformation of the failed sections

The second measurement was the out of plane deflection for the compression flange

elements, lL , to calculate the rotation angle due to the deformation of the compression

flange due to local buckling, l .This is shown in Figure 4.14(b).

25.0 b

LArcTan l

l (4.4)

The value of d and l are tabulated in Table 4.3.


Table 4.3: The rotation angle due to the deformation of the compression flange, l , and

also the rotation angle due to the deformation of the web flange juncture, d , for the

tested sections Thickness

b4 b3 b2 b1 t Width/Depth αl αd

sections (mm) (mm) (mm) (mm) (mm) Deg Deg

1 47.40 161.22 1.54 0.29 7.20 21.60

2 66.45 121.68 1.57 0.55 8.60 18.10

3 12.32 15.94 44.92 122.14 1.57 0.37 12.60 27.30

4 14.20 14.94 62.75 79.85 1.56 0.79 9.10 8.20

5 12.62 21.67 41.49 111.16 1.57 0.37 13.60 27.90

6 12.51 16.29 41.27 129.03 1.57 0.32 13.60 /

7 12.39 15.78 34.99 139.88 1.58 0.25 15.95 /

8 11.82 17.66 48.23 110.04 1.59 0.44 11.70 20.40

9 9.78 18.06 56.65 99.00 1.56 0.57 10.00 13.90

10 17.12 17.98 49.36 99.83 1.54 0.49 11.50 20.60

11 10.85 16.19 60.10 94.21 1.54 0.64 11.30 13.40

12 10.85 16.50 50.93 113.76 1.53 0.45 11.10 23.20

13 9.98 14.27 58.18 102.90 1.57 0.57 9.80 /

14 22.74 47.59 121.10 1.58 0.39 11.90 28.70

15 13.34 42.49 141.02 1.58 0.30 13.20 /

16 18.67 31.40 159.19 1.57 0.20 12.60 25.10

17 12.44 37.01 161.69 1.54 0.23 12.20 40.80

18 17.34 62.09 102.68 1.56 0.60 10.90 14.70

19 12.45 47.50 141.42 1.55 0.34 11.90 32.50

20 14.53 55.88 121.20 1.56 0.46 / /

21 12.88 65.86 103.61 1.57 0.64 8.60 16.80

22 20.00 39.99 89.00 1.50 0.45 12.70 25.30

23 19.96 45.00 89.98 1.50 0.50 12.50 14.00

24 19.96 49.99 89.96 1.50 0.56 10.20 13.00

25 19.97 35.00 79.80 1.55 0.44 8.10 18.60

26 20.00 40.20 79.99 1.50 0.50 8.50 19.10

27 19.97 45.00 79.98 1.52 0.56 10.10 13.40

28 19.96 29.97 70.05 1.50 0.43 / /

29 19.95 34.99 70.10 1.55 0.50 8.10 19.50

30 19.99 39.97 70.00 1.50 0.57 7.10 9.80

31 20.00 25.00 58.90 1.50 0.42 / /

32 19.97 29.96 60.80 1.55 0.49 7.60 25.10

33 19.97 35.00 60.40 1.55 0.58 8.10 19.00

34 14.80 19.90 49.50 1.55 0.40 / 18.00

35 14.96 24.99 50.10 1.50 0.50 0.00 25.60

36 14.95 29.97 50.10 1.50 0.60 7.60 25.10

37 9.75 14.78 38.20 1.55 0.39 0.00 11.80

38 9.63 19.75 39.40 1.55 0.50 0.00 24.30

39 9.83 24.68 38.50 1.55 0.64 0.00 26.30

40 9.20 10.45 28.10 1.55 0.37 0.00 69.60

41 9.70 14.50 29.50 1.55 0.49 0.00 12.40

42 9.73 19.55 29.00 1.55 0.67 0.00 16.10

l and d verses width to depth ratio of the tested sections are drawn in Figure 4.15.


0.0

5.0

10.0

15.0

20.0

25.0

30.0

35.0

40.0

45.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

width/depth

angl

eαl

αd

Figure 4.15: The rotation angles due to the deformation of the compression flange and the deformation of the web flange juncture respectively verses width to depth ratio of the

tested sections

Figure 4.15 shows that the rotation angles due to the deformation of the compression

flange do not vary significantly in comparison with the rotation angles from the

deformation of the web-compression flange juncture. Furthermore, for sections where

the width to depth ratio is less than 0.5, the rotation angles due to the deformation of the

web-compression flange juncture are greater than the rotation angles due to the

deformation of the compression flange. Therefore the distortional buckling failure

mode is more pronounced when compared to the local buckling failure mode. Figure

4.16 shows the top and the front view of the failure mode of section 17 with the width

to depth ratio of 0.23. It can be seen from Figure 4.16 that the distortional buckling

failure mode is more pronounced when compared to the local buckling failure.


(a) Front view

(b) Top view

Figure 4.16: The failure modes of section 17

For the sections with the width to depth ratio from 0.5 to 0.7, the rotation angles due to

the deformation of the web-compression flange juncture are greater than the rotation

angles due to the deformation of the compression flange. However, the difference

between these two angles is not significant. Therefore, the observed failures in these

sections are most likely due to the combination of the local and the distortional

buckling failure mode. The combination of the local and the distortional buckling

failure mode were observed in the column test of cold-formed steel channels with

complex stiffeners as well (Yan and Young (2002)). Figure 4.17 shows the top and the


front view of the failure mode for section 24 with the width to depth ratio of 0.56, with

this failure mode.

(a) Front view

(b) Top view


The only section that its l value is greater than the d value is section 4 with the width

to depth ratio of 0.8. Figure 4.18 shows the failure shape of section 4. As evident from

these photos taken from the front and above, the local buckling is more noticeable than

the distortional buckling failure mode.


(a) Front view

(b) Top view


Some sections such as section 37 have an l value of zero. This means they failed due

to the rotation of the web-compression flange juncture without any deformation of the

compression flange. Figure 4.19 shows the failure shape of section 37, being a

substantial distortional buckling mode at the point of failure.



Appendix H shows the failed shape photos of all the tested sections. Overall they show

consistent failure patterns.

4.6 Comparing the Elastic Portion of the Moment-Curvature Graphs

of the Test Results with the EWM Results

According to the EWM design rules, and based on the different value of *f which is

the design stress in the compression element of the section, the section’s moment

capacity and the section’s curvature can be calculated as follows:

es ZfM * (4.5)

c

c

Yk

(4.6)

E

fc

*

(4.7)

cEY

fk

*

(4.8)

For the different value of the *f the effective section modulus ( eZ ) and the position of

neutral axis from the compression flange ( cY ) are calculated. After calculating the

section moment capacity, the member moment capacity due to the distortional buckling

are determined using the AS/NZS4600 (2005) method. The following computations are

an example of how the section moment capacity, member moment capacity and the

curvature of section 1 which is a simple channel section, have been calculated.


MPaf

541

500

393

270

150

70

21

10

*

3.0,541,194100

54.1,0,4.47,22.161,46.1 4321

MPaFMPaE

mmtbbmmbmmbmmr

y

i

mmrbdmmrbb

mmrImmrcmmru

mmt

rrmmtrr

ee

c

iie

22.1552,4.44

652.1149.0,42.1637.0,5.357.1

23.22

,3

12

33

Flange element:

MPab

tEkf

k

crb 75.90112

43.02

2

2

442.2

347.2

081.2

725.1

286.1

878.0

481.0

332.0

*

crbf

f

546.16

143.17

08.19

458.22

625.28

891.37

40.44

40.44

373.0

386.0

430.0

506.0

645.0

853.0

1

1

22.01

673.0

1673.0

bbFor

For

ef


Web element:

Assume web is fully effective:

35155

32

44144

33

333

3111

652.1461.159501.3

775.812

45.1602

4.44

31160012

61.802

22.155

652.1579.1501.3

mmIImmcrbymmuL

mmty

Immt

bymmbL

mmd

Immd

rymmdL

mmIImmcrymmuL

ce

e

ce

32

2222

270.3

388.3

771.3

438.4

657.5

488.7

775.8

775.8

1277.0

2

546.16

143.14

08.19

458.22

625.28

891.37

40.44

40.44

mmtL

Immt

ymmbL ef

MPaY

frYfmm

L

yLY

cec

i

iic

081.523

395.483

837.379

822.260

765.144

462.67

218.20

628.9

,

575.90

335.90

567.89

257.88

964.85

735.82

61.80

61.80

**

1

772.0

777.0

793.0

821.0

871.0

947.0

1

1

,

042.404

736.375

237.301

032.214

082.126

865.63

218.20

628.9

*1

*2

**

2 f

fMPa

Y

fYrdf

cce


MPad

tEkfk crd

588.322

342.324

099.330

364.340

871.359

086.391

359.414

359.414

112,

681.18

783.18

116.19

71.19

84.20

648.22

24

24

121242

2

23

273.1

221.1

073.1

875.0

634.0

415.0

221.0

152.0

*1

crdf

f

If 673.0 the web is fully effective otherwise set cY and iterate until convergence.

mmYc

952.94

538.93

567.89

257.88

964.85

735.82

61.80

61.80

048.97

348.101

024.115

754.132

22.155

22.155

22.155

22.155

625.0

653.0

741.0

855.0

1

1

1

1

22.01

673.0

1673.0

ddFor

For

ef


524.48

674.50

242.56

51.50

865.42

406.40

805.38

805.38

,236.02

,236.0

,

314.26

285.27

325.30

747.34

099.40

329.39

805.38

805.38

3 2

21

12

2

1 ef

ecefef

efefef

efef

efef d

rYdd

dddFor

ddFor

dd

33

13

1313

1518

1693

2324

3496

5373

5069

4869

4869

12

157.16

643.16

162.18

373.20

049.23

665.22

402.22

402.22

2mm

dImm

drydL efefeef

33

66

6626

9521

10840

14830

10740

6563

5497

4869

4869

12

69.70

201.68

446.61

002.63

531.64

532.62

207.61

207.61

2mm

LImm

LYydL cef

33

77

777

21100

22550

26960

28540

31440

35840

38960

38960

12

586.126

879.125

893.123

239.123

092.122

477.120

415.119

415.119

2

268.63

682.64

653.68

963.69

256.72

485.75

61.77

61.77

mmL

ImmL

rdymmrYdL eec


422

1032E3

1059E3

1143E3

1183E3

1254E3

1353E3

E31419

31419

,

852.94

538.93

567.89

257.88

964.85

735.82

61.80

61.80

mm

E

LYIyLtImmL

yLY iciiiex

i

iic

1*

*3

64.29

65.27

66.22

68.15

699.8

636.4

634.1

739.6

,

877.5

662.5

013.5

619.3

188.2

145.1

370.0

176.0

,

10860

11320

12760

13400

14590

16360

17600

17600

mm

E

E

E

E

E

E

E

E

EY

fkmkNfZMmm

Y

IZ

cexs

c

exex

Calculating My, Mp, ky and kp:

35155

32

44144

33

333

32

222

3111

652.1461.159501.3

775.812

45.1602

4.44

31160012

61.802

22.155

12

.77.0

24.44

652.1579.1501.3

mmIImmcrbymmuL

mmty

It

bymmbL

mmd

Immd

rymmdL

mmtb

Immt

ymmbL

mmIIcrymmuL

ce

e

ce

422.

1 31419,2

mmELYIyLtIb

Y iciiifullxc

1

..

3.. 65.34,524.9,17600 mmE

EI

MkmkNFZMmm

Y

IZ

fullx

yyyfullxy

c

fullxfullx

1

.

3 64.41,386.11,21050 mmEEI

MkmkNFSMmmyYLtS

fullx

ppyxpicix


71.0

67.0

55.0

38.0

22.0

11.0

03.0

02.0

,

52.0

50.0

44.0

32.0

19.0

10.0

03.0

02.0

pp

s

k

k

M

M

Distortional buckling check:

The theoretical distortional buckling stress can be calculated according to the Appendix

D of AS/NZS4600 (2005).

221 996.72,0,0,4.47,22.161 mmtbAbdmmbbmmbb fllfw

02

5.0,7.232

)2()2(2

fll

lll

fll

lflflf

bdb

dbdymm

bdb

bbbbdbx

43

706.573

mmt

dbbJ llf

43

232

2 426.1412122

mmtb

ydtbtdd

ytdytbI fll

lllfx

432

23

2 1367012212

5.0 mmtb

xb

btbbxtdtb

xbtbI llflfl

fffy

05.05.05.0 ydbxbtbxbytbydxbtdI llflfflflxy

12.749,4065.5,6.1392

8.4 21

225.0

3

2

A

IIxE

t

bbI yxwfx

922.62

,052.0039.01

222

11

xyfyfx IybIJbI

MPaA

EfIbI odxyfy 112.1374

2,357.0 3

22121

22

113

0302.139.13192.256.12

11.11

06.046.5

22244

24

3

3

E

bb

b

Et

f

b

Etk

ww

wod

w


3. 176000 mmZZk fullxc

453.0,065.0039.0 22

113

1

22

11

xyfyfx IbI

E

kJbI

MPaA

Efod 9.1734

2 32

2121

764.1

695.1

503.1

246.1

929.0

634.0

348.0

24.0

*

odd f

f

dd

fullxcd

fullxcd

ZfMFor

ZfMFor

22.0

1,674.0

,674.0

.*

.*

MPaZ

MfmkNM

fullx

ccc

489.268

636.256

184.223

442.178

259.123

70

21

10

,

726.4

518.4

929.3

141.3

17.2

232.1

37.0

176.0

.


mkNfZM ccnalbdistortio

725.4

517.4

928.3

141.3

169.2

232.1

37.0

176.0

.

41.0

40.0

34.0

28.0

19.0

11.0

03.0

02.0

,

71.0

67.0

55.0

38.0

22.0

11.0

03.0

02.0

,

52.0

50.0

44.0

32.0

19.0

10.0

03.0

02.0

p

nalbdistortio

pp

s

M

M

k

k

M

M

After calculating the moment capacity and curvature for differences in the *f value

they are normalised with the plastic moment and plastic curvature respectively. The

normalised moment verses the normalised curvature are plotted and compared with the

test results in Figure 4.20.

Section 1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

k/kp

M/M

p

Test ResultMsx/MpMbdistortional/MpM=MyFully effectivek=ky

Figure 4.20: Comparison between the test result with EWM results and also distortional buckling check


Figure 4.20 shows, based on the EWM results for section moment capacity, that section

1 is fully effective until its pMM ratio reaches to 0.20. After that the compression

flange starts to not be fully effective and at 44.0pMM the compression portion of

the web starts to not be fully effective as well. This means that section 1 started to

buckle when pMM was equal to 0.20; and the ultimate moment capacity of the

section locates where pMM was equal to 0.52. Figure 4.20 also shows the member

moment capacity due to distortional buckling. It is shown that the moment-curvature

graph slope changes in when pMM is equal to 0.20; and the ultimate moment

capacity of the section locates where pMM is equal to 0.43. By comparing the test

graph with the EWM graph it can be concluded that the EWM can predict the buckling

point accurately and the graph slopes prior to the failure are in a good agreement with

the test results. However, the ultimate moment capacity from EWM is greater than the

test result. On the other hand, the calculated ultimate moment capacity due to a

distortional buckling check is smaller than the test result. The comparison of the test

result graphs with EWM and also distortional buckling check results graphs for few

selected tested sections, as representatives among all other sections, are shown in

Figure 4.21 and the rest are shown in Appendix C.

115

Section 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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

pTest ResultMsx/MpMbdistortional/MpM=MyFully effectivek=ky

Section 21

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p


Section 4

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p


Section 25

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p


Figure 4.21: Comparison between test results and EWM and distortional buckling check results graph for selected tested sections

Section 30

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M


Section 36

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p


Section 40

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p


Figure4.21: Comparison between test results and EWM and distortional buckling check results graph for selected tested sections (continued)


The following conclusions can be drawn from Figure 4.21:

Firstly, except sections 4 and 30, the test moment-curvature diagrams for all the

sections are close to the predicted EWM with distortional buckling check diagrams.

However, the EWM with the distortional buckling check predicts a smaller value of the

ultimate moment capacities compared with the test results.

Secondly, the test moment-curvature diagrams for sections 4 and 30 do not match well

with the EWM and also distortional buckling check diagrams due to the following

reasons:

Section 4 is the only section where the two ends were filled with plaster; and the

plaster is not stiff enough to avoid the local instability of the section due to the

bearing the load which is applied to the section through the loading pin.

Regarding section 30, it is clearly evident that this section buckled in the early stages

of the loading which is not consistent with the EWM prediction. This may be due to

the geometric imperfections of the section.

Finally, it is also important to note that EWM does not allow for the curvature at failure

point to exceed the yield curvature. However, the test results show that most of the

sections failed where their curvature, and therefore their strain, is greater than the yield

curvature and yield strain.

4.7 Comparing the Test with the Design Rules Results

The compression elements for each of the tested sections are either un-stiffened or

partially stiffened with an edge stiffener. According to both the AS/NZS4600 (2005)

and NASPEC (2007) standards, the inelastic reserve capacity design method is not

applicable. Therefore, the nominal section moment capacity of SM is calculated based

on the initiation of yielding in the effective section. This means that the ultimate

compressive strain ( ult ) is equal to the yield strain ( y ). Therefore, the compression

strain factor ( yultyC ) can not be greater than one.

In the previous chapter the methods to calculate the curvature from both strain gauges

and inclinometers data was explained. Drawing on these explanations it is valuable to


calculate the ultimate compressive strain and also the compression strain factor as

follows:

ultcult kY (4.9)

E

Fyy (4.10)

yultyC (4.11)

Table 4.4 shows the yC values based on the test results. The table also compares the

ultimate moment capacity, based on the test results, with the ultimate section moment

capacity based on EWM with 1yC .

Table 4.4: yC value based on test result and ultM based on test result and EWM

EWM EWM

Ms Mtest Mtest/Ms Ms Mtest Mtest/Ms

sections Cy kN-m kN-m sections Cy kN-m kN-m

1 1.30 5.88 5.03 0.86 22 1.49 4.14 4.98 1.202 1.20 4.55 4.45 0.98 23 1.42 4.39 4.97 1.133 1.63 7.69 7.90 1.03 24 1.12 4.52 4.91 1.094 1.58 4.85 4.85 1.00 25 1.72 3.45 3.95 1.145 1.87 6.66 7.56 1.13 26 1.96 3.59 4.26 1.196 1.21 8.05 8.17 1.01 27 1.61 3.80 4.46 1.177 1.39 8.32 8.60 1.03 28 1.41 2.22 3.11 1.408 1.27 6.99 7.45 1.07 29 1.92 2.89 3.30 1.149 1.60 6.39 6.80 1.06 30 1.82 2.98 3.40 1.14

10 1.76 6.24 6.76 1.08 31 2.25 1.65 2.24 1.3611 1.59 5.88 6.09 1.04 32 2.50 1.87 2.50 1.3312 1.69 7.14 7.48 1.05 33 1.79 2.35 2.72 1.1613 1.54 6.36 6.60 1.04 34 1.89 1.21 1.58 1.3114 1.76 7.16 7.97 1.11 35 2.56 1.42 1.70 1.2015 1.35 7.42 8.76 1.18 36 2.30 1.61 1.88 1.1716 1.08 8.85 8.57 0.97 37 2.35 0.67 0.91 1.3617 1.10 8.49 8.73 1.03 38 2.10 0.85 1.07 1.2618 1.70 5.63 6.38 1.13 39 1.63 0.97 1.22 1.2619 1.11 7.38 8.37 1.13 40 2.40 0.33 0.52 1.5920 1.35 6.45 7.82 1.21 41 1.70 0.45 0.64 1.4421 1.14 5.28 5.78 1.09 42 1.51 0.54 0.73 1.35

Mean(Pm)= 1.16Cov(VP)= 0.12


Table 4.4 shows the yC value of up to 2.56 for the tested sections and also shows that

the ultimate section moment capacity based on EWM with 1yC which provides

conservative results.

4.7.1 Nominal member moment capacity

The nominal member moment capacity of bM is the minimum of the member moment

capacities subjected to the lateral, local or distortional buckling. Five different methods,

NASPEC (North American Specification for the design of Cold-Formed Steel

Structural Members), AS/NZS4600 (Australian/New Zealand standard for cold-formed

steel structure), DSM (Direct strength Method), EUROCODE3 (European standard for

design of steel structures) and AS4100 (Australian standard for steel structure) are used

to analyse the nominal member capacity of the tested sections.

NASPEC and AS/NZS4600 use the same method which is based on EWM to analyse

the nominal section moment capacity. Note that all the sections are fully restrained.

Therefore, the effect of the lateral buckling is ignored and bM due to the lateral

buckling is equal to sM with the tabulations set out in Table 4.4. In the NASPEC and

AS/NZS4600 methods, the distortional buckling failure needs to be checked in addition

to the lateral buckling failure. The method to calculate the nominal member moment

capacity due to distortional buckling is the same for both NASPED and AS/NZS4600.

However, the methods to determine the theoretical distortional buckling stress are not

the same. Therefore, in this thesis the hand method of Appendix D from AS/NZS4600

is used to determine the theoretical distortional buckling stress for calculating MAS4600.

Finite Strip Method (FSM) is used to determine the theoretical distortional buckling

stress for calculating MNASPEC. The following calculations are an example of how the

nominal section moment capacity, based on EWM and also the nominal member

moment capacity due to distortional buckling according to NASPEC and AS/NZS4600

design rules, are calculated for section 9.


EWM

3.0,552,198416

56.1,78.9,06.18,65.56,99,44.1 4321

MPaFMPaE

mmtmmbmmbmmbmmbmmr

y

i

mmrbbmmrbdmmrbdmmrbb

mmrImmrcmmru

mmt

rrmmtrr

elelee

c

iie

78.6,06.122,932,65.502

63.1149.0,414.1637.0,485.357.1

22.22

,3

4312

33

Flange element:

83.9405115

,2434328.0399,268.2428.1 42

34

1

tS

btI

tS

btI

F

ES aa

y

333.0333.0,max,83.940,min,248.04

582.0 1211 nnIIItS

bn aaa

907.0,1min,23.853

a

sIs I

IRI

37.343.0582.425.0357.0 33

n

IRb

bk

b

b

MPab

tEkfcrb 75.90

112

2

2

2

673.0981.0 crb

y

f

F

mmbbFor

For

ef 045.40791.022.0

1673.0

1673.0

Element b3:

554.412124,741.022 331

k

d

rbb e


673.0201.0,13660112

2

2

2

l

ll

crd

yd

lcrd f

FMPa

d

tEkf

mmdd llef 06.12,1

Element b4:

55.5182

,43.01

1* ye F

b

rbfk

673.0356.0,4082112

2

2

2

l

ll

crb

yb

lcrb f

FMPa

b

tEkf

mmbb llef 78.6,1

Web element:


33

666

32

555

3444

33

333

32322

32

1311

125.49

293

121.812

78.02

045.40

26.32586.197.62

171.14612

03.92

06.12

63.1474.16485.3

375.112

28.172

78.6

mmd

Immrd

ymmdL

mmtb

Immt

ymmbL

mmIImmcrymmuL

mmd

Immd

rymmdL

mmIImmrcbymmuL

mmtb

Immt

bymmbL

e

efef

ce

leflefelef

ce

leflef

38188

32

7177

26.3241.9797.62

272.1012

22.982

65.50

mmIImmrcbymmuL

mmbt

Immt

bymmbL

ce

3

2

11311111

310311010

33

9199

375.112

72.812

78.6

63.153.82485.3

17.14612

97.892

06.12

mmtb

Immt

bbymmbL

mmIImmrcbbymmuL

mmd

Immd

rbymmdL

ll

ce

llel


MPaY

frYfmm

L

yLY

cec

i

iic 93.519,633.51

**

1

912.0,33.474 *1

*2

**

2

f

fMPa

Y

fYrdf

cce

MPad

tEkfk crd 0.1101

112,811.2112124

2

2

23

673.0687.0*

1 crdf

f

If 673.0 the web is fully effective otherwise set cY and iterate until convergence.

633.51cY

mmddFor

For

ef 997.91989.022.0

1673.0

1673.0

mmd

rYdd

dddFor

ddFor

mmd

d ef

ecefef

efefef

efef

efef 118.25,236.0

2,236.0

,516.233 2

21

12

2

1

33

16

1616 0.1084

12757.14

2mm

dImm

drydL efefeef

33

1212

1212212 0.1321

12074.39

2mm

LImm

LYydL cef

33

1313

131313 7278

1282.73

2367.44 mm

LImm

LrdymmrYdL eec

422 36.584,633.51 mmELYIyLtImmL

yLY iciiiex

i

iic

mkNfZMmmY

IZ exs

c

exex 25.6,11320 *3


Calculating My, Mp:

33

666

32

555

3444

33

333

32322

32

1311

125.49

293

272.1012

78.02

65.50

26.32586.197.62

171.14612

03.92

06.12

63.1474.16485.3

375.112

28.172

78.6

mmd

Immrd

ymmdL

mmbt

Immt

ymmbL

mmIImmcrymmuL

mmb

Immd

rymmdL

mmIImmrcbymmuL

mmtb

Immt

bymmbL

e

ce

llel

ce

ll

38188

32

7177

26.3241.9797.62

272.1012

22.982

65.50

mmIImmrcbymmuL

mmbt

Immt

bymmbL

ce

3

2

11311111

310311010

33

9199

375.112

72.812

78.6

63.153.82485.3

17.14612

97.892

06.12

mmtb

Immt

bbymmbL

mmIImmrcbbymmuL

mmd

Immd

rbymmdL

ll

ce

llel

422.

1 36.625,2

mmELYIyLtIb

Y iciiifullxc

mkNFZMmmY

IZ yfullxy

c

fullxfullx 976.6,12640 .

3..

mkNFSMmmyYLtS yxpicix 102.8,14680 3

Distortional buckling check according to AS/NZS4600:

The theoretical distortional buckling stress can be calculated according to Appendix D

of AS/NZS4600 (2005).

2

4321

8.131)(

78.9,06.18,65.56,99

mmtdbbA

mmbbmmbdmmbbmmbb

llf

llfw

mmbdb

dbdymm

bdb

bbbbdbx

fll

lll

fll

lflflf 02.42

5.0,1.372

)2()2(2

43

92.1063

mmt

dbbJ llf


43

232

2 590912122

mmtb

ydtbtdd

ytdytbI fll

lllfx

432

23

2 4461012212

5.0 mmtb

xb

btbbxtdtb

xbtbI llflfl

fffy

90175.05.05.0 ydbxbtbxbytbydxbtdI llflfflflxy

1759,424.27,89.6012

8.4 21

225.0

3

2

A

IIxE

t

bbI yxwfx

279.12

,317.0039.01

222

11

xyfyfx IybIJbI

MPaA

EfIbI odxyfy 1.2964

2,275.0 3

22121

22

113

0301.239.13192.256.12

11.11

06.046.5

22244

24

3

3

E

bb

b

Et

f

b

Etk

ww

wod

w


533.0,529.0039.0 22

113

1

22

11

xyfyfx IbI

E

kJbI

86.55742 3

22121

A

Efod

995.0od

yd f

F

dd

fullxcd

fullxcd

ZfMFor

ZfMFor

22.0

1,674.0

,674.0

.*

.*

MPaZ

MfmkNM

fullx

ccc 2.432,462.5

.

mkNfZM ccnalbdistortio 463.5


Distortional buckling check according to NASPEC:

The theoretical distortional buckling stress is determined by using the thin wall

program:

)(,2.735 wallThinFSMFromMPafod

866.0od

yd f

F

dd

fullxcd

fullxcd

ZfMFor

ZfMFor

22.0

1,674.0

,674.0

.*

.*

MpaZ

MfmkNM

fullx

ccc 3.475,007.6

.


The ultimate moment capacities of the tested sections based on NASPEC and

AS/NZS4600 design rules are tabulated in Table 4.5 and are compared with the test

results. By reviewing this table, it is concluded that the ultimate moment capacities of

the tested sections based on the test results are greater than the predicted ultimate

capacity, based on NASPEC design rules. In addition, the calculated ultimate moment

capacities of the tested sections based AS/NZS4600 design rules are conservative in

comparison with the ultimate moment capacity from test results. It is also evident that

NASPEC design rules provide less conservative results compare to the AS/NZS4600

design rules. This is due to the fact that FSM predicts more accurate buckling stress

compare to the AS/NZS4600 hand method and the only difference between NASPEC

and AS/NZS4600 design rules is in calculating the theoretical distortional buckling

stress.


Table 4.5: Ultimate moment capacities of the tested sections based on NASPEC and AS/NZS4600 design rules

fod from fod from

EWM Thin-wall Min hand method Min

Ms NASPEC Ms&Mbdistortional AS/NZS4600 Ms&Mbdistortional

sections Mtest Mbdistortional MNASPEC Mtest/MNASPEC Mbdistortional MAS4600 Mtest/MAS4600

kN-m kN-m kN-m kN-m kN-m kN-m1 5.03 5.88 4.71 4.71 1.07 4.73 4.73 1.062 4.45 4.55 3.29 3.29 1.35 3.60 3.60 1.243 7.90 7.69 6.96 6.96 1.13 6.21 6.21 1.274 4.85 4.85 4.71 4.71 1.03 4.40 4.40 1.105 7.56 6.66 6.36 6.36 1.19 5.16 5.16 1.466 8.17 8.05 7.27 7.27 1.12 6.41 6.41 1.277 8.60 8.32 7.46 7.46 1.15 6.61 6.61 1.308 7.45 6.99 6.51 6.51 1.14 5.72 5.72 1.309 6.80 6.39 6.01 6.01 1.13 5.46 5.46 1.2410 6.76 6.24 6.13 6.13 1.10 5.21 5.21 1.3011 6.09 5.88 5.53 5.53 1.10 5.13 5.13 1.1912 7.48 7.14 6.53 6.53 1.15 5.90 5.90 1.2713 6.60 6.36 5.84 5.84 1.13 5.50 5.50 1.2014 7.97 7.16 6.98 6.98 1.14 6.27 6.27 1.2715 8.76 7.42 6.63 6.63 1.32 6.47 6.47 1.3516 8.57 8.85 7.65 7.65 1.12 7.16 7.16 1.2017 8.73 8.49 7.22 7.22 1.21 6.95 6.95 1.2618 6.38 5.63 5.57 5.57 1.15 5.30 5.30 1.2019 8.37 7.38 6.69 6.69 1.25 6.30 6.30 1.3320 7.82 6.45 6.18 6.18 1.27 5.77 5.77 1.3521 5.78 5.28 5.02 5.02 1.15 4.92 4.92 1.1822 4.98 4.14 4.19 4.14 1.20 3.75 3.75 1.3323 4.97 4.39 4.42 4.39 1.13 4.01 4.01 1.2424 4.91 4.52 4.53 4.52 1.09 4.17 4.17 1.1825 3.95 3.45 3.60 3.45 1.14 3.19 3.19 1.2426 4.26 3.59 3.70 3.59 1.19 3.30 3.30 1.2927 4.46 3.80 3.92 3.80 1.17 3.54 3.54 1.2628 3.11 2.22 2.63 2.22 1.40 2.35 2.22 1.4029 3.30 2.89 3.00 2.89 1.14 2.71 2.71 1.2230 3.40 2.98 3.15 2.98 1.14 2.80 2.80 1.2231 2.24 1.65 1.83 1.65 1.36 1.64 1.64 1.3632 2.50 1.87 2.22 1.87 1.33 2.04 1.87 1.3333 2.72 2.35 2.44 2.35 1.16 2.25 2.25 1.2134 1.58 1.21 1.24 1.21 1.31 1.23 1.21 1.3135 1.70 1.42 1.42 1.42 1.20 1.40 1.40 1.2136 1.88 1.61 1.61 1.61 1.17 1.57 1.57 1.2037 0.91 0.67 0.67 0.67 1.36 0.67 0.67 1.3638 1.07 0.85 0.85 0.85 1.26 0.85 0.85 1.2639 1.22 0.97 0.97 0.97 1.26 0.97 0.97 1.2640 0.52 0.33 0.33 0.33 1.59 0.33 0.33 1.5941 0.64 0.45 0.45 0.45 1.44 0.45 0.45 1.4442 0.73 0.54 0.54 0.54 1.35 0.54 0.54 1.35

Mean(Pm)= 1.21 Mean(Pm)= 1.28Cov(VP)= 0.10 Cov(VP)= 0.07

Reliability Index(β)= 2.95 Reliability Index(β)= 3.27

φ=0.9, γL=1.5, γD=1.2

DSM is another method to determine the nominal member capacity of tested sections.

In DSM the lateral, local and distortional buckling are considered to analyse the

member capacity. All sections are fully restrained. Therefore, bM due to the lateral

buckling, beM , is equal to yM . Consequently, the local and distortional buckling are


the controlling factors for determining bM . The following computations show how the

nominal member moment capacity, based on DSM for section 9, is determined.

DSM

Local buckling

The theoretical local buckling stress is determined by using the thin wall program:

mkNMM

M

M

MM

M

M

mkNMMfZMMPaf

bebe

ol

be

olbl

ol

bel

ybeolfullxolol

618.615.01776.0844.0

976.6,797.9.,2.775

4.04.0

.

Distortional buckling

The theoretical distortional buckling stress is determined by using thin wall program:

)(,2.735 wallThinFSMFromMPafod

mkNMM

M

M

MM

M

M

mkNMmkNfZMMPaf

yy

od

y

odbd

od

yd

yodfullxodod

007.622.01673.0866.0

976.6,291.9.,2.735

5.05.0

.

The ultimate moment capacities of the tested sections based on DSM due to the local

and distortional buckling failure mode are tabulated (set out in Table 4.6) and compared

with the test results. The theoretical local and distortional buckling stresses are

determined using FSM. By reviewing the data in Table 4.6, it is concluded that the

calculated ultimate moment capacities of the tested sections, based DSM design rules,

predicts conservative results when compared with the ultimate moment capacity

evident from the test results.


Table 4.6: Ultimate moment capacities of the tested sections based on DSM DSM DSM Min

Local Buckling Distortional Buckling Mbl&Mbd

sections Mtest Mbl Mbd MDSM Mtest/MDSM

kN-m kN-m kN-m kN-m1 5.03 5.46 4.71 4.71 1.072 4.45 3.97 3.29 3.29 1.353 7.90 7.68 6.96 6.96 1.134 4.85 5.16 4.71 4.71 1.035 7.56 6.66 6.36 6.36 1.196 8.17 7.76 7.27 7.27 1.127 8.60 7.74 7.46 7.46 1.158 7.45 7.15 6.51 6.51 1.149 6.80 6.62 6.01 6.01 1.13

10 6.76 6.50 6.13 6.13 1.1011 6.09 6.11 5.53 5.53 1.1012 7.48 7.40 6.53 6.53 1.1513 6.60 6.80 5.84 5.84 1.1314 7.97 7.52 6.98 6.98 1.1415 8.76 7.72 6.63 6.63 1.3216 8.57 7.96 7.65 7.65 1.1217 8.73 7.91 7.22 7.22 1.2118 6.38 6.41 5.57 5.57 1.1519 8.37 8.14 6.69 6.69 1.2520 7.82 7.65 6.18 6.18 1.2721 5.78 6.40 5.03 5.03 1.1522 4.98 4.41 4.19 4.19 1.1923 4.97 4.83 4.42 4.42 1.1324 4.91 4.99 4.53 4.53 1.0825 3.95 3.60 3.63 3.60 1.1026 4.26 3.82 3.70 3.70 1.1527 4.46 4.18 3.92 3.92 1.1428 3.11 2.63 2.63 2.63 1.1829 3.30 3.00 3.00 3.00 1.1030 3.40 3.18 3.15 3.15 1.0831 2.24 1.83 1.83 1.83 1.2232 2.50 2.22 2.22 2.22 1.1333 2.72 2.44 2.44 2.44 1.1234 1.58 1.24 1.24 1.24 1.2835 1.70 1.42 1.42 1.42 1.2036 1.88 1.61 1.61 1.61 1.1737 0.91 0.67 0.67 0.67 1.3638 1.07 0.85 0.85 0.85 1.2639 1.22 0.97 0.97 0.97 1.2640 0.52 0.33 0.33 0.33 1.5941 0.64 0.45 0.45 0.45 1.4442 0.73 0.54 0.54 0.54 1.35


The forth method used in this thesis to determine the nominal member moment capacity

of tested sections is EUROCODE3 method. In this method the effective section should

be determined. However, the methods to calculate the effective section are not similar

to the methods in NASPEC and AS/NZS4600. The following computations show how


the nominal member moment capacity based on EUROCODE3 method for section 9 is

determined.

EUROCODE3

33 63.1149.0,414.1637.0,485.357.1

22.22

,3

mmrImmrcmmru

mmt

rrmmtrr

c

iie

mmrbbmmrbdmmrbb

mmtbbmmtbdmmtbdmmtbb

elteltet

ll

78.6,06.122,65.502

22.8,94.142,88.952,53.532

432

4312

Flange:

1,4 k

mmbbkt

b

Fbef

p

pb

py 055.44

823.0)3(55.0

673.0926.04.28

,652.0235 2

Element b3:

717.405.1

2.8

10

688.022 31

kd

tbb

mmddd

mmd

d

mmdd

kt

d

lelele

lele

lble

b

lp

01.8

93.65

2

94.14

1

673.0238.04.28

12

1

Element b4:

1,43.0 k

mmbbkt

b

lble

blp 22.8

1748.0434.0

4.28

2493.702

mmb

dbtA efleles


33

66

63626

32

5555

3444

33

13

1313

32322

32

1311

632.2312

775.112

57.6

175.412

78.02

59.205.0

63.1586.1485.3

79.1312

745.52

49.5

63.1474.16485.3

375.112

28.172

78.6

mmL

ImmL

rbymmdL

mmtL

Immt

ymmbL

mmIImmcrymmuL

mmd

Immd

rymmrdL

mmIImmrcbymmuL

mmtb

Immt

bymmrbL

ele

ef

ce

leleeile

ce

lefilef

422 3314,575.6 mmLYIyLtImmL

yLY iciiis

i

iics

MPaA

EIK

YdY

tEKk

s

scsr

cscs

f 49732,724.4614

,0 1322

3

1

165.0333.0 dcsr

yd X

F

513.0)(

65.0,333.0

0

0

flangee

e

flangeedeflange

Web element:


33

6666

32

5555

3444

33

13

1313

32322

32

1311

6703012

5.492

932

35.812

78.02

175.412

26.32586.197.62

79.1312

745.52

49.5

63.1474.16485.3

375.112

28.172

78.6

mmL

Immd

tymmrdL

mmtL

Immt

ymmrbL

mmIImmcrymmuL

mmd

Immd

rymmrdL

mmIImmrcbymmuL

mmtb

Immt

bymmrbL

i

ief

ce

leleeile

ce

lefilef


38188

32

7177

26.3241.9797.62

272.1012

22.982

65.50

mmIImmrcbymmuL

mmtb

Immt

bymmbL

ce

tt

33

1212

12312212

32

11311111

310311010

33

9199

63.2312

78.112

57.6

375.112

72.812

78.6

63.153.82485.3

17.14612

97.892

06.12

mmL

ImmL

rbymmrdL

mmtb

Immt

bbymmbL

mmIImmrcbbymmuL

mmd

Immd

rbymmdL

eile

ltlt

ce

ltltelt

mmL

yLY

i

iic 396.51

422 31.589 mmELYIyLtI iciiiex

mkNfZMmmY

IZ exs

c

exex 327.6,11420 *3

If the section is fully effective, then it can be design in-elastically as follows:

MPaY

ftYf

cc 24.535

**

1

924.0,52.494 *1

*2

**

2

f

fMPa

Y

fYtdf

cc

97.2178.929.681.701 2 k

1)3(055.0

,1min673.0708.04.28 2

p

pbp

kt

d

Web is fully effective.

815.0)(

869.0)3(055.025.05.0,708.0

0

0

webe

e

webepeweb

815.0)(),(max000

max

webflange

e

e

e

e

e

e


mkNE

ZSZ

FMmimM e

efullxxfullx

ypcRd

811.7811.7,102.8min61

14)(

, 0

max..

3.

3 12640,11420,327.6,396.51 mmZmmZmkNMmmY fullxexsc

mkNMMZZ scRdfullxex 327.6. (Non-fully effective section)

As with the previous methods, the ultimate moment capacities of the tested sections

based on EUROCODE3 design rules were tabulated and are set out in Table 4.7. These

are also compared with the test results. By reviewing Table 4.7 it is evident that the

calculated ultimate moment capacities of the tested sections based on EUROCODE3

design rules are conservative in comparison to the test results.


Table 4.7: Ultimate moment capacities of the tested sections based on EUROCODE3 and AS4100 design rules

EUROCODE3 EUROCODE3 AS4100inelastic

sections Mtest MsEurocode3 Mtest/MsEurocode3 McRd Mtest/McRd MAS4100 Mtest/MAS4100

kN-m kN-m kN-m kN-m1 5.03 5.71 0.88 5.71 0.88 3.61 1.392 4.45 4.40 1.01 4.40 1.01 2.26 1.973 7.90 7.72 1.02 7.72 1.02 8.12 0.974 4.85 4.97 0.98 4.97 0.98 4.26 1.145 7.56 6.66 1.13 7.50 1.01 7.41 1.026 8.17 8.05 1.01 8.07 1.01 8.10 1.017 8.60 8.12 1.06 8.12 1.06 7.76 1.118 7.45 6.97 1.07 6.97 1.07 7.33 1.029 6.80 6.33 1.07 6.33 1.07 5.78 1.18

10 6.76 6.18 1.09 6.18 1.09 6.28 1.0811 6.09 5.92 1.03 5.92 1.03 5.16 1.1812 7.48 7.13 1.05 7.13 1.05 6.99 1.0713 6.60 6.58 1.00 6.58 1.00 5.96 1.1114 7.97 6.43 1.24 6.43 1.24 8.15 0.9815 8.76 6.73 1.30 6.73 1.30 7.84 1.1216 8.57 6.38 1.34 6.38 1.34 7.26 1.1817 8.73 6.66 1.31 6.66 1.31 7.19 1.2118 6.38 5.46 1.17 5.46 1.17 5.54 1.1519 8.37 6.77 1.24 6.77 1.24 8.02 1.0420 7.82 6.31 1.24 6.31 1.24 7.06 1.1121 5.78 5.47 1.06 5.47 1.06 5.41 1.0722 4.98 3.80 1.31 3.80 1.31 4.95 1.0123 4.97 4.04 1.23 4.04 1.23 4.97 1.0024 4.91 4.17 1.18 4.17 1.18 4.80 1.0225 3.95 3.11 1.27 3.11 1.27 4.30 0.9226 4.26 3.34 1.28 3.34 1.28 4.27 1.0027 4.46 3.56 1.25 3.56 1.25 4.33 1.0328 3.11 2.23 1.39 2.23 1.39 3.20 0.9729 3.30 2.64 1.25 2.64 1.25 3.59 0.9230 3.40 2.82 1.21 2.82 1.21 3.56 0.9531 2.24 1.57 1.43 1.57 1.43 2.27 0.9932 2.50 1.93 1.30 1.93 1.30 2.70 0.9333 2.72 2.18 1.25 2.18 1.25 2.92 0.9334 1.58 1.09 1.46 1.09 1.46 1.54 1.0335 1.70 1.27 1.34 1.27 1.34 1.73 0.9936 1.88 1.45 1.29 1.45 1.29 1.92 0.9837 0.91 0.59 1.55 0.59 1.55 0.84 1.0938 1.07 0.76 1.41 0.76 1.41 1.03 1.0439 1.22 0.88 1.38 0.88 1.38 1.15 1.0640 0.52 0.29 1.77 0.29 1.77 0.43 1.2141 0.64 0.41 1.58 0.41 1.58 0.56 1.1442 0.73 0.51 1.45 0.51 1.45 0.67 1.10

Mean(Pm)= 1.24 Mean(Pm)= 1.23 Mean(Pm)= 1.08Cov(VP)= 0.15 Cov(VP)= 0.15 Cov(VP)= 0.16

AS4100 is the final method used to determine the nominal member capacity of tested

sections in this thesis. In AS4100, sections are classified into three classes: slender,

non-compact and compact. Each classification uses different equations to calculate the

nominal member moment capacity of the section. The following computations show

how the nominal section moment capacity, based on the AS4100 method, is determined

for section 9.


AS4100

25.48250

2,56.88

250

2 21

yee

yee

F

t

rbflange

F

t

rbweb

From Table 5.2 AS4100 (1998):

25.48250

2,56.88

250

2 21

yee

yee

F

t

rbflange

F

t

rbweb

77.0,82,115 webwebwebey

eepey

206.1,30,40 flangeflangeflangeey

eepey

30

40

25.48

206.1,max

flange

flange

flange

flangewebflange

epsp

eysy

es

ey

e

ey

e

ey

e

sys Section 9 is slender.

mkNZFM

mmZZ

exys

s

syfullxex

78.510*10480552

1048025.48

4012640

6

3.

Finally, the ultimate moment capacities of the tested sections based on AS4100 design

rules were tabulated and compared with the test results. These are displayed in Table

4.7. As pointed out in the previous chapter, the material properties of the tested sections

are not in a range to satisfy some of the plastic design limitations in the AS4100 (1998).

However, from Table 4.7, it can be concluded that the expected ultimate moment

capacities of the tested sections, based AS4100 design rules, are much closer to the test

results, particularly in comparison to the four other design rules.

The test results compared with AS/NZS4600 (distortional buckling checks), NASPEC

(distortional buckling checks) and DSM design methods which are shown in Figure

4.22. In addition, the ratio of the ultimate moment capacity based on AS/NZS4600 due

to distortional buckling mode, DSM and AS4100 over the yield moment as well as the

values are tabulated in Table 4.8.


0.4

0.6

0.8

1.0

1.2

1.4

0.000 0.337 0.674 1.011 1.348

λd(Hand-method)

M/M

yMy<Mtest<Mp

Mtest>Mp

Mtest<My

AS/NZS4600 Distortional check

(a): Comparison between test and AS/NZS4600 due to distortional buckling check results

0.4

0.6

0.8

1.0

1.2

1.4

0.000 0.337 0.673 1.010 1.346

λd(FSM)

M/M

y

My<Mtest<Mp

Mtest>Mp

Mtest<My

NASPEC Distortional check

(b): Comparison between test and NASPEC due to distortional buckling check results


0.4

0.6

0.8

1.0

1.2

1.4

0.000 0.388 0.776 1.164 1.552

λl(FSM)

M/M

y

My<Mtest<Mp

Mtest>Mp

Mtest<My

DSM Local Buckling

(c): Comparison between test and DSM due to local buckling results

0.4

0.6

0.8

1.0

1.2

1.4

0.000 0.337 0.673 1.010 1.346

λd(FSM)

M/M

y

My<Mtest<Mp

Mtest>Mp

Mtest<My

DSM Distortional Buckling

(d): Comparison between test and DSM due to distortional buckling results

Figure 4.22: Comparison between test and existing design rules results

Figure 4.22 shows that all the tested sections had an ultimate moment capacity greater

than the design capacity. Therefore, the DSM, NASPEC and AS/NZS4600 design


methods are conservative. From Figure 4.22(a) and (b), it is also evident that the

ultimate moment capacities of the sections based on FSM, for calculating the

distortional slenderness ratio, provides closer results to the test results compare to the

hand method.

Table 4.8: The ratio of the ultimate moment capacity over the yield moment with the values based on AS/NZS4600 and NASPEC due to distortional buckling mode and

DSM. AS4600 NASPEC DSM DSM Hand method

FSM FSM AS/NZS4600

sections Mtest/My Mbdistortional/My Mbdistortional/My Mbl/My Mbd/My λl λd λd MAS4100/My Mp/My

1 0.53 0.50 0.49 0.57 0.49 1.77 1.77 1.68 0.38 1.202 0.52 0.42 0.39 0.47 0.39 2.35 2.35 2.13 0.26 1.133 1.01 0.80 0.89 0.99 0.89 0.79 0.82 0.97 1.04 1.194 0.84 0.77 0.82 0.90 0.82 0.92 0.93 1.03 0.74 1.155 1.13 0.77 0.95 1.00 0.95 0.72 0.73 1.01 1.11 1.216 1.01 0.80 0.90 0.96 0.90 0.82 0.80 0.97 1.01 1.207 1.03 0.79 0.90 0.93 0.90 0.87 0.81 0.97 0.93 1.218 1.04 0.80 0.91 1.00 0.91 0.74 0.79 0.97 1.02 1.189 0.97 0.78 0.86 0.95 0.86 0.84 0.87 0.99 0.83 1.16

10 1.04 0.80 0.94 1.00 0.94 0.75 0.75 0.96 0.97 1.1911 0.90 0.76 0.82 0.91 0.82 0.90 0.93 1.03 0.77 1.1512 0.99 0.78 0.86 0.98 0.86 0.80 0.86 1.00 0.93 1.1713 0.91 0.75 0.80 0.93 0.80 0.87 0.96 1.05 0.82 1.1514 1.00 0.79 0.88 0.94 0.88 0.85 0.86 0.99 1.02 1.1815 1.02 0.75 0.77 0.90 0.77 0.92 1.01 1.05 0.91 1.1916 0.94 0.79 0.84 0.88 0.84 0.95 0.90 0.99 0.80 1.2317 0.94 0.75 0.77 0.85 0.77 1.00 1.01 1.06 0.77 1.2118 0.87 0.72 0.76 0.88 0.76 0.96 1.04 1.11 0.76 1.1419 0.93 0.70 0.75 0.91 0.74 0.90 1.07 1.15 0.89 1.1820 0.94 0.69 0.74 0.92 0.74 0.88 1.07 1.17 0.85 1.1521 0.76 0.65 0.66 0.84 0.66 1.01 1.24 1.28 0.71 1.1322 1.13 0.85 0.95 1.00 0.95 0.70 0.74 0.88 1.12 1.1823 1.03 0.83 0.91 1.00 0.91 0.76 0.79 0.92 1.03 1.1724 0.95 0.81 0.87 0.96 0.87 0.82 0.85 0.96 0.93 1.1625 1.10 0.89 1.00 1.00 1.00 0.61 0.66 0.83 1.20 1.2026 1.11 0.86 0.97 1.00 0.97 0.68 0.72 0.86 1.12 1.1827 1.07 0.85 0.94 1.00 0.94 0.73 0.76 0.89 1.04 1.1728 1.18 0.89 1.00 1.00 1.00 0.58 0.61 0.82 1.21 1.2129 1.10 0.90 1.00 1.00 1.00 0.59 0.63 0.80 1.20 1.2030 1.07 0.88 0.99 1.00 0.99 0.66 0.68 0.84 1.12 1.1831 1.22 0.90 1.00 1.00 1.00 0.51 0.55 0.81 1.24 1.2432 1.13 0.92 1.00 1.00 1.00 0.53 0.56 0.78 1.22 1.2233 1.12 0.92 1.00 1.00 1.00 0.57 0.59 0.78 1.20 1.2034 1.28 0.99 1.00 1.00 1.00 0.48 0.48 0.69 1.24 1.2435 1.20 0.99 1.00 1.00 1.00 0.52 0.52 0.69 1.21 1.2136 1.17 0.97 1.00 1.00 1.00 0.57 0.57 0.71 1.19 1.1937 1.36 1.00 1.00 1.00 1.00 0.41 0.41 0.56 1.25 1.2538 1.26 1.00 1.00 1.00 1.00 0.46 0.46 0.57 1.21 1.2139 1.26 1.00 1.00 1.00 1.00 0.53 0.53 0.61 1.19 1.1940 1.59 1.00 1.00 1.00 1.00 0.35 0.35 0.55 1.31 1.3141 1.44 1.00 1.00 1.00 1.00 0.36 0.36 0.52 1.27 1.2742 1.35 1.00 1.00 1.00 1.00 0.41 0.41 0.53 1.22 1.22


As Table 4.8 shows, the plastic over the yield moment ratio for the tested sections

varies between 1.13 and 1.31 with the average value of 1.19 and the covariance value


of 0.03. This means that the plastic section modulus of channel sections is almost 1.2

times greater than its elastic section modulus. The following computations show the

reason behind why the yM

Mvalue for compact sections is assumed to be equal to 1.2.

fullxyfullx

y

y

p ZSFZ

SF

M

M.

.

2.12.1 (4.12)

fullxfullx ZSZSMin .. 2.15.1, (4.13)

Therefore, in Figure 4.23, which is comparing the test result with AS4100 design rules

results, the y

p

M

Mvalue for compact sections is equal to 1.2.

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0 20 40 60 80 100 120 140

λs(AS4100)

M/M

y

AS4100 Web

AS4100 Flange

Web is controlling according to AS4100

Flange is controlling according to AS4100

Figure 4.23: Comparison between test and AS4100 design rules results

The sections in Figure 4.23 are divided into two groups. In one group webs are

controlling the sections failure with flanges controlling the sections failure in the

second group. In the first group, the eye of the sections’ web is greater than the

eye of the sections’ flange. However, in latter the eye of the sections’ flange

is greater than the eye of the sections’ web. A reliable criterion for determining

controlling failure elements from the tests could not be established, thus this


comparison was not made. The test result is defined as web or flange controlled

according to AS4100.

4.8 Conclusions

The bending behaviour of fourteen slender, twenty seven non-compact and one

compact cold-formed channel sections were explored and analysed. This was achieved

by performing pure bending tests. The common observed failure modes from the tests

are as follows:

1. For sections where the width to depth ratio is less than 0.5, the distortional buckling

failure mode is more pronounced when compared to the local buckling failure

mode.

2. For the sections with the width to depth ratio from 0.5 to 0.7, the observed failures

in these sections are most likely due to the combination of the local and the

distortional buckling failure mode.

3. The only section that its local buckling failure mode was more pronounced when

compared to the distortional buckling failure mode was section 4 with the width to

depth ratio of 0.8.

The experimental results were compared with the different Standards design methods

results which led to a number of conclusions:

Firstly it was shown that DSM, NASPEC (2007), AS/NZS4600 (2005) and

EUROCODE3 (2006) standards were conservative for calculating the cold-formed

channel sections ultimate moment capacity.

Secondly, the outcome of the tension test in chapter 3 shows that the material properties

of the tested sections are not in a satisfying range for the plastic design limitations in

the AS4100 (1998). However, the expected ultimate moment capacities of the tested

sections, based on the AS4100 design rules, match well with the test results.

Finally, the sections which are classified as compact sections do not have the

appropriate rotation capacity for plastic design. The section classifications, which have


been defined in the AS4100, therefore were not accurate for the cold-formed channel

sections.

The research gap this thesis seeks to address is identified by reviewing these results. As

a result, the following chapter sets out original works that introduces inelastic reserve

capacity to cold-formed channel sections, modifying in-elastic reserve capacity design

method for channel sections with edge stiffener, the AS/NZS4600 design rules for

distortional buckling check, DSM as well as revising a new slenderness limits for cold-

formed channel sections in AS4100.

141

Chapter 5

REVISING EXISTING DESIGN RULES AND

SLENDERNESS LIMITS


Revisiting the focus of this research is particularly important for this chapter. The

purpose is to study the inelastic behaviour of cold-formed channel sections with edge

stiffener. To provide the context for the research results in this chapter, the reliability

analysis concept is briefly explained. This is then followed with the reliability analysis,

which is based on the previous chapter’s test results. Using the test results, the inelastic

reserve capacity of cold-formed channel sections has been introduced for channel

sections with edge stiffener. The revised design rules for nominal member moment

capacity due to AS/NZS4600’s distortional buckling check and also DSM are then

presented. In addition, new elastic and plastic slenderness limits for cold-formed

channel sections are proposed. In order to test and subsequently evaluate the proposed

design rules, reliability analysis is used.

5.1 Reliability Analysis

According to the limit state design, if the load effect ( S ) exceeds the resistance of the

section ( R ) failure occurs. Therefore, a structure is safe if:

iin SR (5.1)

where is the resistance factor and is usually less than unity. For sections under

bending, according to the AS/NZS4600Supp1 (1998), the value is equal to 0.9 and

is the load factor and varies for different loads. For example, D which is the dead load

factor is equal to 1.2 and L which is the live load factor is equal to 1.5 (AS1170.1

(2002)).

Chapter 5. Revising Existing Design Rules and Slenderness Limits 142

According to the First Order Second Moment (FOSM) method, which was defined by

Ravindra and Galambos (1978), the reliability index ( ), which is the relative measure

for the safety of the design, is equal to:

22

ln

SR

m

m

VV

S

R

(5.2)

where mR and mS are mean values of the resistance and the load effect respectively. RV

and SV are the corresponding Coefficient Of Variation (COV). COV is the ratio of the

standard deviation value over the mean value. Based on their research mR , RV , SV and

mm SR are determined by using the following equations:

)( mmmnm FMPRR (5.3)

222FMPR VVVV (5.4)

mm

LmDmS LD

VLVDV

22

(5.5)

1

n

m

n

m

n

n

n

m

Ln

nD

m

m

R

R

L

L

L

D

D

D

L

D

S

R (5.6)

where nR is the nominal resistance and in this study is considered to be equal to ye fZ .

mP is the mean ratio of the experimental results to the predicted results for the actual

material and cross sectional properties of the tested sections.

Furthermore, mM is the mean ratio of the yield point to the minimum specified value

and mF is the mean ratio of actual section modulus to the nominal value. mD and mL

are the mean values for dead and live loads respectively. DV and LV are the COV values

for dead and live loads respectively. Hsiao et al. (1990) stated in their paper that the


value of mM , mF , MV and FV were developed from two publications (Rang et al.

(1979a; 1979b))and were given as:

1.1mM

1.0MV

0.1mF

05.0FV

The load-statistic studies were analysed in Ellingwood et al. (1980) and the values of

mD , mL , DV and LV were given as:

DDm 05.1

1.0DV

LLm 0.1

25.0LV

Cold-formed steel structures normally have a smaller dead load value compared with

the live load value. According to AS/NZS4600Supp1 (1998), the live load value is five

times greater than the dead load value (5

1nn LD ). This assumption is used to

simplify the reliability index computations.

To determine the value of the reliability index for each proposed design method, the

mean ratio of the experimental results to the proposed design results ( mP ) and the COV

ratio of the experimental results to the proposed design results ( PV ) are identified. mP

and PV values for all of the proposed methods are shown in Tables 5.1 to 5.6.

According to the AS/NZS4600Supp1 (1998) design methods the larger the reliability

index the more reliable the design methods. It is to be noted that for a simply supported

beam, recommended lower limit for reliability index is equal to 2.5 according to the

AISI LRFD Specifications (AS/NZS4600Supp1 (1998)).


5.2 Inelastic Reserve Capacity

As discussed in chapter 4, the curvature for all of the tested sections, and therefore their

strain at the failure point, are greater than the yield curvature and the yield strain. The

ultimate compressive strain for a section with an inelastic behavior is yC (compression

strain factor) times the yield strain ( y ). Figure 5.1 shows the normalised moment-

strain and moment-curvature diagrams for section 2.

Section2

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

k/ky(ε/εy)

M/M

y

k/ky

ε/εy

εult=3338.1, Cy=εult/εy=1.2

Figure 5.1: Normalised moment-strain and moment-curvature diagrams of section 2

Figure 5.1 shows that the ultimate strain at the failure point is 1.2 times greater than the

yield strain. The following computations explain the reason why the normalised

moment-curvature diagram does not match with the normalised moment-strain diagram.

In addition, Equation (3.14) shows the calculation for the ultimate compressive strain

from the ultimate curvature:

c

cult Y

k

(5.7)

The yield curvature can be calculated as follows:

fullx

yy EI

Mk

.

(5.8)


xfull

yxfull

c

c

y

ult

EI

FZY

k

k

(5.9)

yy

E

F (5.10)

Since the tested sections are symmetric along their x axis therefore:

21b

I

Z

xfull

xfull (Half the section’s depth) (5.11)

cy

c

y

ult

Y

b

k

k 21

(5.12)

For fully effective sections:

21b

Yc (5.13)

For not fully effective sections:

21b

Yc (5.14)

y

c

y

ult

k

k

(5.15)

Appendix D shows the entire tested section’s normalised moment-strain diagrams, with

the yield moment and yield strain, for all of the tested sections. In addition, the

compression strain factor for each section is also shown on their normalised moment

strain diagram.

Table 5.1 shows the yC value of up to 2.56 for the tested sections. However, the

yC value can reach up to 3 on hat shape cold-formed beams with the stiffened

compression elements (Reck et al. (1975) and Yener and Pekoz (1985)). In this study all

the sections are open channel sections and the compression flanges are partially

stiffened. Therefore, all the sections have the compression strain of less than three times

the yield strain. The following steps explain how to calculate the ultimate moment

capacity of a section based on yC , being values of greater than one.


1- Using the effective width method that is explained in chapter 2, the area of the

effective elements (webs, flanges and lip) is calculated.

2- Using the following equation, the position of the neutral axis ( cY ) is then found:

cii AYyA (5.16)

where iA is the area of each effective element and A is the area of the effective section

(Figure5.2).

Y

5 54

61 2

3

4

7

8

9 10 9

11

1213

c

y 1 y 2 y 3y 4 y 5

y 6y 7

y 8y 9

y 10 y 11y 12

y 13

Figure 5.2: Position of neutral axis

3- The strain distribution is linear with the maximum compression strain of yyC on the

effective section. The stress distribution is therefore determined by using yC value:

For 1yC , elastic-plastic stress distribution is used (Figure 5.3).


Figure 5.3: Elastic-plastic stress distribution (Cy>1)

For 1yC , elastic stress distribution is used (Figure 5.4).

Figure 5.4: Elastic stress distribution (Cy<1)

In current design standards for cold-formed open sections with partially stiffened

compression elements, yC is equal to one. However, as mentioned earlier, these test

results illustrate that some sections have the yC value up to 2.56.


Table 5.1: Proposed inelastic reserve capacity model data Inelastic model Inelastic model

Cy=1(Existing) Cy(proposed)

sections λFlange λWeb Mtest Mdesign=Ms(EWM) Mtest/Mdesign Cy(test) Cy(proposed) Mdesign=Ms(inelastic) Mtest/Mdesign

kN-m kN-m kN-m

1 2.44 1.34 5.03 5.88 0.86 1.30 1.00 5.88 0.86

2 3.42 1.02 4.45 4.54 0.98 1.20 1.00 4.54 0.98

3 0.72 0.83 7.90 7.69 1.03 1.63 1.00 7.69 1.03

4 1.12 0.56 4.85 4.85 1.00 1.58 1.00 4.85 1.00

5 0.63 0.75 7.56 6.66 1.13 1.87 1.12 7.06 1.07

6 0.63 0.88 8.17 8.05 1.01 1.21 1.13 8.55 0.96

7 0.52 0.95 8.60 8.32 1.03 1.39 1.55 9.52 0.90

8 0.75 0.74 7.45 6.99 1.07 1.27 1.00 6.99 1.07

9 0.91 0.68 6.80 6.39 1.06 1.60 1.00 6.39 1.06

10 0.78 0.68 6.76 6.24 1.08 1.76 1.00 6.24 1.08

11 1.04 0.67 6.09 5.88 1.04 1.59 1.00 5.88 1.04

12 0.84 0.80 7.48 7.14 1.05 1.69 1.00 7.14 1.05

13 1.06 0.72 6.60 6.36 1.04 1.54 1.00 6.36 1.04

14 0.95 0.87 7.97 7.16 1.11 1.76 1.00 7.16 1.11

15 0.89 1.04 8.76 7.42 1.18 1.35 1.00 7.42 1.18

16 0.74 1.13 8.57 8.85 0.97 1.08 1.00 8.85 0.97

17 0.76 1.21 8.73 8.49 1.03 1.10 1.00 8.49 1.03

18 1.20 0.77 6.38 5.63 1.13 1.70 1.00 5.63 1.13

19 1.05 1.09 8.37 7.38 1.13 1.11 1.00 7.38 1.13

20 1.18 0.94 7.82 6.45 1.21 1.35 1.00 6.45 1.21

21 1.44 0.80 5.78 5.28 1.09 1.14 1.00 5.28 1.09

22 0.84 0.63 4.98 4.14 1.20 1.49 1.00 4.14 1.20

23 0.90 0.65 4.97 4.39 1.13 1.42 1.00 4.39 1.13

24 0.98 0.66 4.91 4.52 1.09 1.12 1.00 4.52 1.09

25 0.78 0.54 3.95 3.45 1.14 1.72 1.00 3.45 1.14

26 0.84 0.56 4.26 3.59 1.19 1.96 1.00 3.59 1.19

27 0.89 0.56 4.46 3.80 1.17 1.61 1.00 3.80 1.17

28 1.37 0.50 3.11 2.22 1.40 1.41 1.00 2.22 1.40

29 0.78 0.46 3.30 2.89 1.14 1.92 1.00 2.89 1.14

30 0.84 0.48 3.40 2.98 1.14 1.82 1.00 2.98 1.14

31 1.08 0.40 2.24 1.65 1.36 2.25 1.00 1.65 1.36

32 1.32 0.41 2.50 1.87 1.33 2.50 1.00 1.87 1.33

33 0.78 0.39 2.72 2.35 1.16 1.79 1.00 2.35 1.16

34 0.77 0.31 1.58 1.21 1.31 1.89 1.00 1.21 1.31

35 0.62 0.32 1.70 1.42 1.20 2.56 1.17 1.53 1.11

36 0.61 0.32 1.88 1.61 1.17 2.30 1.19 1.74 1.08

37 0.49 0.22 0.91 0.67 1.36 2.35 1.66 0.80 1.14

38 0.38 0.23 1.07 0.85 1.26 2.10 2.04 1.01 1.06

39 0.41 0.22 1.22 0.97 1.26 1.63 1.90 1.12 1.09

40 0.25 0.15 0.52 0.33 1.59 2.40 2.50 0.42 1.24

41 0.47 0.16 0.64 0.45 1.44 1.70 1.71 0.54 1.20

42 0.38 0.15 0.73 0.54 1.35 1.51 2.03 0.65 1.13

Mean(Pm) 1.16 1.11

Cov(VP) 0.12 0.10

Reliability Index (β) 2.67 2.62

φ=0.9, γL=1.5, γD=1.2

AS/NZS4600


4- Based on the resultant axial force position of neutral axis is checked ( 0 iiA ).

Figure 5.5 shows how to find the neutral axis based on the resultant axial force.

5 54

61 2

3

4

8

9

10

14

11

12

13

Fc

10

FcFc

FcFc

Fc

Fc

Ft

Fc

Ft

FtFt

Ft

Ft

Fc = Ftii

7

9

12

1413

1011

8

7

6

54

1

32

Figure 5.5: Position of neutral axis based on the resultant axial force

5- With the new position of neutral axis the ultimate moment capacity of the sections is

determined using following equation:

iii dA (5.17)

where iA is the area of each element, i is the average value of stress on each section

and id is the distance from centre of each element to the neutral axis. Figure 5.6 shows a

section which is divided into the smaller elements. Dividing the section is based on the

distance to the neutral axes and being either in the elastic or plastic range.


5 54

61 2

3

4

8

9

10

14

11

12

13

10

d7

dd

d

dd

d

d

dd

dd

dd

9

12

14 13

10

11

8

7

6

5

4

1

3

2

Figure 5.6: Dividing a typical section into smaller elements

5.2.1 Proposed inelastic design model for partially stiffened

compression members

As discussed, the test results show that the compression strain factor can be greater than

one for partially stiffened compression elements (Table 5.1). To this end, an inelastic

design model is proposed. This model is aligned with the existing inelastic method in

the Australian and American standards. The compression strain factor varies from one

to 2.5 and the proposal for partially stiffened compression elements is:

For :1 5.2yC (5.18)

For :21 12152.15.2 yC (5.19)

For :2 1yC (5.20)

25.01 (5.21)

673.02 (5.22)

where is slenderness ratio of the partially stiffened compression elements; and can be

calculated with the following equations:

cr

y

f

F (5.23)


crf is the plate elastic theoretical buckling stress.

As shown in Equations 5.18 to 5.22, in order to apply the inelastic behavior on the

partially stiffened compression elements, (not only on their compression flanges) the

slenderness ratio should be less than 0.673. The sections should also be fully effective.

The inelastic design capacity method depends on the flange slenderness ratio. This is

due to the fact that the compression flanges of the channel sections in major axis

bending are under the maximum strain, when compared to the other elements of the

sections. Figure 5.7 compares the proposed inelastic model with the experimental

results.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.00 0.34 0.68 1.02 1.36 1.70

λFlange

Cy

Not fully effective

Fully effective

Proposed model

Figure 5.7: Comparison between the proposed inelastic model and the experimental results

Figure 5.7 shows the Flange verses yC for the tested samples and the proposed inelastic

capacity model. Figure 5.7 shows that there are some inelastic strains in all sections.

However, it is not considered appropriate to apply an inelastic design procedure to the

sections that buckle elastically. Therefore, the proposed model is conservative

compared to the test data. The following computations show the calculation of the

ultimate moment capacity for section 38 with the proposed yC value of 2.04.


EWM:

3.0,1.541,190200

55.1,0,63.9,75.19,4.39,45.1 4321

MPaFMPaE

mmtbmmbmmbmmbmmr

y

i

mmrbdmmrbdmmrbb

mmrImmrcmmru

mmt

rrmmtrr

elee

c

iie

63.6,4.332,75.132

641.1149.0,417.1637.0,493.357.1

225.22

,3

312

33

Flange element:

25.2745115

,167.0328.0399,996.2328.1 42

34

1

tS

btI

tS

btI

F

ES aa

y

49.0333.0,max,167.0,min,49.04

582.0 1211 nnIIItS

bn aaa

0.1,1min,64.37

a

sIs I

IRI

748.143.0582.425.07.0 33

n

IRb

bk

b

b

MPab

tEkfcrb 3818

112

2

2

2

673.0376.0 crb

y

f

F

mmbbFor

For

ef 75.13122.0

1673.0

1673.0


Element b3:

613.034.0

578.0,603.0

22 31

kd

rbb e

673.0307.0,5758112

2

2

2

l

ll

crd

yd

lcrd f

FMPa

d

tEkf

mmdd llef 06.12,1

Web element:


333

444

32

333

3222

33

111

310512

7.192

4.33

753.212

775.02

75.13

283.32583.1987.62

286.2412

315.62

63.6

mmmmd

Immrd

ymmdL

mmtb

Immt

ymmbL

mmIImmcrymmuL

mmd

Immd

rymmdL

e

efef

ce

leflefelef

36166

32

5155

283.32817.37987.62

753.212

625.382

75.13

mmIImmrcbymmuL

mmbt

Immt

bymmbL

ce

33

7177 286.2412

085.332

63.6 mmd

Immd

rbymmdL llel

MPaY

frYfmm

L

yLY

cec

i

iic 7.458,7.19

**

1

1,7.458 *1

*2

**

2

f

fMPa

Y

fYrdf

cce

MPad

tEkfk crd 8884

112,2412124

2

2

23


673.0227.0*

1 crdf

f

If 673.0 , the web is fully effective.

422 396.30,7.19 mmELYIyLtImmY iciiiexc

mkNFZMmmY

IZ yexs

c

exex 85.0,1572 3

673.0376.025.0 21 04.252.15.2 121 yC

Since the section is fully effective and symmetric, the position of the neutral axis is

based on the resultant axial force ( 0 iiA ) which does not need to be checked.

Finally, the ultimate moment capacity of the section is determined using Equation

(5.17).

MPaF

mmdmmtA

MPaFmmrrYdmmrtA

MPaFmmt

YdmmtbA

MPaFmmcrYdmmutA

MPaFmmd

rYdmmtdA

y

yeece

ycef

yec

ylef

eclef

55.2702

47.63

27.9035.15)(7.9

1.5412.13)10(5.085.10)10(

1.541925.182

313.21

1.541117.18)(83.102

1.541385.13)2

(28.10

552

5

442

4

332

3

222

2

112

1


mkNdAM iiidesign 006.12

Table 5.1 shows the ratio of the ultimate moment capacity for the test results over the

ultimate moment capacity based on proposed yC value and also over the ultimate

moment capacity based on yC value equal to one. By reviewing this work, it is

concluded that the estimated ultimate moments based on the proposed method for

majority of the sections are smaller than the test results. It is however less conservative

when compared to the existing method which is based on initiation of yielding. The

reliability index for the proposed method is 2.62 that meets the lower limit for reliability

index according to the AISI LRFD Specifications.

The values of flange and web slenderness ratio (in accordance to AS/NZS4600) are also

tabulated in Table 5.1. Based on the test results, Figure 5.8 shows slenderness ratio

limits of web and flange elements for compact and non-compact sections. It is evident

in Figure 5.8(a) that fully effective sections (sections with the web and flange

slenderness ratio of less than 0.673) have an ultimate moment capacity of greater than

their plastic moment. It is shown in Figure 5.8(b) that by decreasing the flange

slenderness ratio the sections rotation capacity is increased. Section 40 with the flange

slenderness ratio of 0.25 has the rotation capacity of 4.2. This section is therefore

suitable for plastic mechanism analysis which can increase the structural assemblies’

capacities by up to 30% of when the first hinge is formed.


AS/NZS4600

0.000

0.168

0.336

0.504

0.672

0.840

1.008

1.176

1.344

0.000 0.168 0.336 0.504 0.672 0.840 1.008 1.176 1.344

λWeb

λ Fla

nge

M<MyMy<M<MpM>MpM>Mp, R>3

λWeb =0.673λWeb =0.15

λFlange =0.25

λFlange =0.673

(a) Slenderness limits, according to AS/NZS4600, for compact and non-compact sections

0.0

0.6

1.2

1.8

2.4

3.0

3.6

4.2

0.000 0.084 0.168 0.252 0.336 0.420 0.504 0.588 0.672 0.756 0.840 0.924

λFlange (AS/NZS4600)

Rot

atio

n ca

paci

ty

(b) Rotation capacity verses flange’s slenderness ratio for fully-effective sections

Figure 5.8: Slenderness limits for plastic mechanism analysis

5.3 AS/NZS4600 Design Rules

In terms of the AS/NZS4600 design rule, the test results and the proposed inelastic

model demonstrates that the design methods in North American and Australian

standards are conservative. Therefore this section, revises the design method to

calculate the nominal member moment capacity due to distortional buckling. The


member moment capacity is equal to the minimum value of the section moment

capacity ( sM ) and the member moment capacity due to the distortional buckling.

5.3.1 A proposed revision for the AS/NZS4600 design model

As discussed in chapter 4, the ratio of the bending moment based on test results over the

yield moment can be greater than one. Therefore, a revised design model of the member

moment capacity with the distortional buckling check is proposed. It is important to

note that the proposal is aligned with the existing method in the Australian standard. For

the sections which are not fully effective (being slender sections) the existing design

method in the Australian standard is used. However, for fully effective sections the

existing design model is revised. This is due to the inelastic behaviour of these sections.

For sections with a d (slenderness ratio subject to the distortional buckling) value of

greater than 0.674 the existing method, being part 3.3.3.3 of the Australian Standards

(AS/NZS4600), are used. However, for sections with a d value of smaller than 0.674

the critical moment ( cM ) calculation is revised. The proposed method for

determining cM is as follows:

For :674.0d )22.0

1(dd

yc

MM

(5.24)

For :641.0674.0 d yd

yc MMM

641.0674.0

641.021.02.1

(5.25)

For :641.0d yc MM 2.1 (5.26)

od

yd f

F (5.27)

odf is the elastic distortional buckling and has been calculated using Equations 2.3 to

2.15.

Figure 5.9 shows the normalised moment by the yield moment, verses the distortional

slenderness ratio diagrams of the proposed method, for nominal member capacity due to

the distortional buckling, the existing design method and the test results.


0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.000 0.321 0.641 0.962 1.282 1.603 1.923 2.244

λd

M/M

y

proposed methodExisting methodTest result

λd=0.674

Figure 5.9: Comparison between the proposed AS/NZS4600 model for nominal member capacity due to distortional buckling and the experimental results

From Figure 5.9, it is evident that in the proposed methods, the member capacity can

reach 1.2 times the yield moment. This is approximately equal to the plastic moment

(Table 4.8). This figure also shows that for sections with the d value of less than

0.674, the proposed design method’s results are closer to the test results compared to

the existing design method.

Table 5.2 shows the ratio of the moment capacity from the test results over the proposed

section moment capacity. It also shows the ratio of the moment capacity from the test

results over the proposed member moment capacity due to the distortional buckling. By

comparing the mean value, COV and reliability index of the existing design method

(Table 4.5) with the proposed method (Table 5.2), it can be concluded the proposed

method is less conservative compared to the existing method but it is still reliable

( 5.218.3 ).


Table 5.2: Proposed AS/NZS4600 model data Proposed Proposed Proposed

Ms(inelastic) Mbdistortional Min(Ms&Mbdistortional)

sections Mtest MEWM Mtest/MEWM MAS4600 Mtest/MAS4600

kN-m kN-m kN-m kN-m1 5.03 5.88 0.86 4.73 4.73 1.062 4.45 4.54 0.98 3.60 3.60 1.243 7.90 7.69 1.03 6.21 6.21 1.274 4.85 4.85 1.00 4.40 4.40 1.105 7.56 7.06 1.07 5.16 5.16 1.466 8.17 8.55 0.96 6.41 6.41 1.277 8.60 9.52 0.90 6.61 6.61 1.308 7.45 6.99 1.07 5.72 5.72 1.309 6.80 6.39 1.06 5.46 5.46 1.24

10 6.76 6.24 1.08 5.21 5.21 1.3011 6.09 5.88 1.04 5.13 5.13 1.1912 7.48 7.14 1.05 5.90 5.90 1.2713 6.60 6.36 1.04 5.50 5.50 1.2014 7.97 7.16 1.11 6.27 6.27 1.2715 8.76 7.42 1.18 6.47 6.47 1.3516 8.57 8.85 0.97 7.16 7.16 1.2017 8.73 8.49 1.03 6.95 6.95 1.2618 6.38 5.63 1.13 5.30 5.30 1.2019 8.37 7.38 1.13 6.30 6.30 1.3320 7.82 6.45 1.21 5.77 5.77 1.3521 5.78 5.28 1.09 4.92 4.92 1.1822 4.98 4.14 1.20 3.75 3.75 1.3323 4.97 4.39 1.13 4.01 4.01 1.2424 4.91 4.52 1.09 4.17 4.17 1.1825 3.95 3.45 1.14 3.19 3.19 1.2426 4.26 3.59 1.19 3.30 3.30 1.2927 4.46 3.80 1.17 3.54 3.54 1.2628 3.11 2.22 1.40 2.35 2.22 1.4029 3.30 2.89 1.14 2.71 2.71 1.2230 3.40 2.98 1.14 2.80 2.80 1.2231 2.24 1.65 1.36 1.64 1.64 1.3632 2.50 1.87 1.33 2.04 1.87 1.3333 2.72 2.35 1.16 2.25 2.25 1.2134 1.58 1.21 1.31 1.23 1.21 1.3135 1.70 1.53 1.11 1.40 1.40 1.2136 1.88 1.74 1.08 1.57 1.57 1.2037 0.91 0.80 1.14 0.80 0.80 1.1438 1.07 1.01 1.06 1.02 1.01 1.0639 1.22 1.12 1.09 1.16 1.12 1.0940 0.52 0.42 1.24 0.39 0.39 1.3341 0.64 0.54 1.20 0.53 0.53 1.2042 0.73 0.65 1.13 0.65 0.65 1.13

Mean(Pm)= 1.11 Mean(Pm)= 1.25Cov(VP)= 0.10 Cov(VP)= 0.07

Reliability Index(β)= 2.62 Reliability Index(β)= 3.18

φ=0.9, γL=1.5, γD=1.2

The following computations show how to calculate the ultimate member moment

capacity of section 38, in which its section moment capacity has been calculated

previously.


mkNMM Inelasticsdesign 006.1)(

mkNMmkNM py 032.1,85.0

Distortional buckling check:

The theoretical distortional buckling stress can be calculated according to the Appendix

D of AS/NZS4600 (2005).

2

4321

54.45)(

0.0,63.9,75.19,4.39

mmtdbbA

bbmmbdmmbbmmbb

llf

llfw

mmbdb

dbdymm

bdb

bbbbdbx

fll

lll

fll

lflflf 578.12

5.0,112.132

)2()2(2

43

47.363

mmt

dbbJ llf

43

232

2 98.34712122

mmtb

ydtbtdd

ytdytbI fll

lllfx

432

23

2 197412212

5.0 mmtb

xb

btbbxtdtb

xbtbI llflfl

fffy

1.4775.05.05.0 ydbxbtbxbytbydxbtdI llflfflflxy

9.222,4055.5,73.1392

8.4 21

225.0

3

2

A

IIxE

t

bbI yxwfx

065.12

,371.0039.01

222

11

xyfyfx IybIJbI

433.92142

,268.0 32

212122

113

A

EfIbI odxyfy

0332.539.13192.256.12

11.11

06.046.5

22244

24

3

3

E

bb

b

Et

f

b

Etk

ww

wod

w



516.0,619.0039.0 22

113

1

22

11

xyfyfx IbI

E

kJbI

MPaA

Efod 16804

2 32

2121

567.0od

yd f

F

Existing Method:

ycd MMFor ,674.0

dd

ycd

MMFor

22.0

1,674.0

MPaZ

MfmkNM

fullx

ccc 1.541,85.0

.


mkNMinMMMin inelasticsnalbdistortio 85.0)85.0,85.0(),( )(

Proposed Method:

For :641.0d yc MM 2.1

For :641.0674.0 d yd

yc MMM

641.0674.0

641.021.02.1

For :674.0d

dd

yc

MM

22.0

1

MPaZ

MfmkNM

fullx

ccc 58.649,02.1)85.0(2.1

.



mkNMinMMMin inelasticsnalbdistortio 01.1)01.1,021.1(),( )(

5.4 Direct Strength Method Design Rules

The Direct Strength Method (DSM) is only adopted in the American, Australian and

New Zealand standards. However, the EWM method is used world-wide. It is

recommended to use DSM for sections with complex edge stiffener (Schafer et al.

(2006)). Despite the fact that the use of DSM is recommended for designing cold-

formed sections, DSM is quite conservative for non-slender sections. In the following

section, a revised DSM for non-slender sections, based on the test results in the

previous chapter, is proposed.

5.4.1 A proposed revised DSM design model

In DSM the minimum value of the member moment capacity under the lateral buckling,

local buckling and distortional buckling is considered to be the ultimate capacity of the

member. Since all the tested sections are fully restrained in this study, the effect of the

lateral buckling is ignored. Therefore, Mb due to the lateral buckling of beM is equal

to yM . To this end, in this chapter revised DSM methods for local and distortional

buckling are proposed. These proposals align with the existing method in Australian

and American standards.

5.4.2 Revised proposed methods for local buckling failure

Two revised DSM methods for determining the member moment capacity under the

local buckling failure are proposed. The first method is aligned with the existing

method for slender sections but the second method is a new method and simpler to use.

In the first proposed method, for sections with the l value of less than 0.776, a revised

method is proposed. However, the second proposed method is a revision for all the l

values.

The following formulas explain the first method:

For :776.0l bebe

ol

be

olbl M

M

M

M

MM

4.04.0

15.01

(5.28)


For :35.0776.0 l bebe

olbebl M

M

MMM

35.047.02.15.0

(5.29)

For :35.0l bebl MM 2.1 (5.30)

ol

bel M

M (5.31)

olfullxol fZM . (5.32)

The olf has been calculated using the Thin Wall (TW) program which is based on

Papangelis and Hancock (1995) research.

The second suggested method is as follows:

For :35.0l bebebe

ol

be

olbl MM

M

M

M

MM 2.103.0207.01

5.05.0

(5.33)

For :35.0l bebl MM 2.1 (5.34)

Figure 5.10 compares the test results, the existing design method and the two proposed

design methods by plotting their normalised moment with the yield moment verses

slenderness ratio.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.00 0.35 0.70 1.05 1.40 1.75 2.10 2.45

λl

M/M

y

Proposed method 1Proposed method 2Existing methodTest result

λl=0.776λl=0.5

Figure 5.10: Comparison between the proposed DSM models and the experimental results for local buckling


In Figure 5.10 it can be seen that in the proposed methods, the bending capacity for

some sections can reach up to 1.2 times the yield moment. This moment is almost equal

to the plastic moment (Table 4.8). This figure also shows that for sections with the l

value of less than 0.776, compared to the existing design method, the proposed design

methods predict closer results to the test results.

Table 5.3 shows the ratio of the member moment capacity from the test results over the

both proposed and existing methods for local buckling failure.


Table 5.3: Proposed DSM model data for local buckling Proposed Proposed ExistingMethod1 Method1 Method2 Method2

sections Mtest Mbl Mtest/Mbl Mbl Mtest/Mbl Mtest/Mbl

kN-m kN-m kN-m1 5.03 5.46 0.92 5.04 1.00 0.922 4.45 3.97 1.12 3.56 1.25 1.123 7.90 7.68 1.03 7.50 1.05 1.034 4.85 5.16 0.94 5.02 0.97 0.945 7.56 6.83 1.11 6.79 1.11 1.136 8.17 7.76 1.05 7.57 1.08 1.057 8.60 7.74 1.11 7.54 1.14 1.118 7.45 7.26 1.03 7.16 1.04 1.049 6.80 6.62 1.03 6.45 1.05 1.03

10 6.76 6.59 1.03 6.49 1.04 1.0411 6.09 6.11 1.00 5.95 1.02 1.0012 7.48 7.40 1.01 7.21 1.04 1.0113 6.60 6.80 0.97 6.62 1.00 0.9714 7.97 7.52 1.06 7.33 1.09 1.0615 8.76 7.72 1.14 7.50 1.17 1.1416 8.57 7.96 1.08 7.73 1.11 1.0817 8.73 7.91 1.10 7.66 1.14 1.1018 6.38 6.41 1.00 6.22 1.03 1.0019 8.37 8.14 1.03 7.92 1.06 1.0320 7.82 7.65 1.02 7.45 1.05 1.0221 5.78 6.40 0.90 6.19 0.93 0.9022 4.98 4.57 1.09 4.58 1.09 1.1323 4.97 4.87 1.02 4.78 1.04 1.0324 4.91 4.99 0.98 4.86 1.01 0.9825 3.95 3.87 1.02 4.00 0.99 1.1026 4.26 3.99 1.07 4.02 1.06 1.1127 4.46 4.26 1.05 4.22 1.06 1.0728 3.11 2.88 1.08 3.01 1.03 1.1829 3.30 3.26 1.01 3.39 0.97 1.1030 3.40 3.34 1.02 3.39 1.00 1.0731 2.24 2.06 1.09 2.18 1.03 1.2232 2.50 2.47 1.01 2.62 0.95 1.1333 2.72 2.67 1.02 2.80 0.97 1.1234 1.58 1.41 1.12 1.49 1.06 1.2835 1.70 1.59 1.07 1.69 1.01 1.2036 1.88 1.77 1.06 1.85 1.02 1.1737 0.91 0.78 1.16 0.80 1.14 1.3638 1.07 0.98 1.10 1.02 1.05 1.2639 1.22 1.08 1.13 1.15 1.07 1.2640 0.52 0.39 1.33 0.39 1.33 1.5941 0.64 0.53 1.21 0.53 1.20 1.4442 0.73 0.64 1.15 0.64 1.15 1.34

Mean(Pm)= 1.06 1.06 1.12Cov(VP)= 0.07 0.07 0.12Reliability Index(β)= 2.52 2.52 2.53

φ=0.9, γL=1.5, γD=1.2

The second method is easier to use compared to the first method. However, it provides

more conservative results for slender sections compare to both the first and the existing

design methods. The average values for the ratio of the test results over the first, second


and existing methods’ member moment capacity due to local buckling are 1.06, 1.06

and 1.12 with the COV of 0.07, 0.07 and 0.12 respectively. This demonstrates that the

first and second proposed methods provide less conservative answers than the existing

design method by having smaller ratios. Therefore the proposed methods are more

economical in comparison with the existing method. The reliability index for the first,

second and existing methods are 2.52, 2.52 and 2.53 respectively. They all meet the

lower limit for reliability index according to the AISI LRFD Specifications.

5.4.3 Revised proposed methods for distortional buckling failure

Two revised DSM methods for determining the member moment capacity due to the

distortional buckling failure are suggested. The first method is aligned with the existing

method for slender sections but the second method is a new method and simpler to use.

In the first proposed method, only for sections with the d value of less than 0.673, the

existing method is revised. However, the second proposed method is a revision for all

the d values.

The first suggested method is formulated as follows:

For :673.0d yy

od

y

odbd M

M

M

M

MM

5.05.0

22.01

(5.35)

For :53.0673.0 d yy

odybd M

M

MMM

53.04.12.1

5.0

(5.36)

For :53.0d ybd MM 2.1 (5.37)

od

yd M

M (5.38)

odfullxod fZM . (5.39)

The dof has been calculated using the Thin Wall (TW) program.

Furthermore, the formulation for the second suggested method is:

For :53.0d yy

od

y

odbd M

M

M

M

MM

5.05.0

19.01

(5.40)


For :53.0d ybd MM 2.1 (5.41)

Following the earlier layout, Figure 5.11 shows three graphs and a scatter plot. In the

scatter plot, the slenderness ratio verses the test moment results is normalised with the

yield moment. In the first graph, the slenderness ratio verses the existing design method

moment results is normalised with the yield moment. Finally, the second and third

graphs are plots of the slenderness ratio verses the proposed design methods moment

results normalised by the yield moment.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.00 0.27 0.53 0.80 1.06 1.33 1.59 1.86 2.12 2.39

λd(FSM)

M/M

y

Proposed method 1

Proposed method 2

Existing method

Test result

λd=0.673

Figure 5.11: Comparison between the proposed DSM models and the experimental results for distortional buckling

Figure 5.11 shows that the sections with d value of less than 0.53 bending capacity are

equal to 1.2 times the yield moment in the proposed methods. This moment is close to

the plastic moment (Table 4.8). This figure also shows that for sections with the d

value of less than 0.673, compared to the existing design method, the proposed design

methods predict closer results to the test results.

It needs to be noted that, the design rules in the first method, for slender sections, are

similar to the existing design rules. This means that the first method is more familiar to

users than the second. Table 5.4 shows the ratio of the member moment capacity from


test results for both the proposed and existing DSM methods due to distortional

buckling failure.

Table 5.4: Proposed DSM model data for distortional buckling Proposed Proposed ExistingMethod1 Method1 Method2 Method2

sections Mtest Mbd Mtest/Mbd Mbd Mtest/Mbd Mtest/Mbd

kN-m kN-m kN-m1 5.03 4.71 1.07 4.80 1.05 1.072 4.45 3.29 1.35 3.33 1.34 1.353 7.90 6.96 1.13 7.31 1.08 1.134 4.85 4.71 1.03 4.91 0.99 1.035 7.56 6.36 1.19 6.73 1.12 1.196 8.17 7.27 1.12 7.64 1.07 1.127 8.60 7.46 1.15 7.84 1.10 1.158 7.45 6.51 1.14 6.85 1.09 1.149 6.80 6.01 1.13 6.29 1.08 1.13

10 6.76 6.13 1.10 6.48 1.04 1.1011 6.09 5.53 1.10 5.77 1.06 1.1012 7.48 6.53 1.15 6.84 1.09 1.1513 6.60 5.84 1.13 6.07 1.09 1.1314 7.97 6.98 1.14 7.21 1.11 1.1415 8.76 6.63 1.32 6.88 1.27 1.3216 8.57 7.65 1.12 7.99 1.07 1.1217 8.73 7.22 1.21 7.49 1.17 1.2118 6.38 5.57 1.15 5.77 1.11 1.1519 8.37 6.69 1.25 6.92 1.21 1.2520 7.82 6.18 1.27 6.40 1.22 1.2721 5.78 5.03 1.15 5.17 1.12 1.1522 4.98 4.19 1.19 4.43 1.12 1.1923 4.97 4.42 1.13 4.65 1.07 1.1324 4.91 4.53 1.08 4.74 1.04 1.0825 3.95 3.66 1.08 3.88 1.02 1.1026 4.26 3.70 1.15 3.93 1.08 1.1527 4.46 3.92 1.14 4.14 1.08 1.1428 3.11 2.88 1.08 2.98 1.04 1.1829 3.30 3.19 1.03 3.33 0.99 1.1030 3.40 3.15 1.08 3.36 1.01 1.0831 2.24 2.16 1.04 2.19 1.02 1.2232 2.50 2.57 0.97 2.62 0.96 1.1333 2.72 2.71 1.00 2.79 0.97 1.1234 1.58 1.49 1.06 1.49 1.06 1.2835 1.70 1.71 1.00 1.71 1.00 1.2036 1.88 1.84 1.02 1.88 1.00 1.1737 0.91 0.80 1.14 0.80 1.14 1.3638 1.07 1.02 1.05 1.02 1.05 1.2639 1.22 1.16 1.05 1.16 1.05 1.2640 0.52 0.39 1.33 0.39 1.33 1.5941 0.64 0.53 1.20 0.53 1.20 1.4442 0.73 0.65 1.12 0.65 1.12 1.34


φ=0.9, γL=1.5, γD=1.2


The average values for the ratio of the test results over the first, second and existing

methods for the member moment capacity, due to distortional buckling, are 1.13, 1.09

and 1.19. These are with the COV of 0.08, 0.08 and 0.09 respectively. Therefore, the

second method provides a less conservative answer compared to the first as well as the

existing design methods by having the smallest average ratio. The reliability index for

the first, second and existing methods are 2.76, 2.62 and 2.91 respectively. They all

meet the lower limit for reliability index according to the AISI LRFD Specifications.

The minimum value of the member moment capacity due to local buckling and

distortional buckling are defined as the member moment capacity of the section.Table

5.5 shows the ratio of the member moment capacity from the test results over both the

proposed DSM as well as the existing methods.


Table 5.5: Proposed DSM model data Proposed Proposed Proposed Proposed Proposed Proposed Proposed Proposed ExistingMethod1 Method1 Method1 Method2 Method2 Method2 Method1 Method2

sections Mbl Mbd MDSM Mbl Mbd MDSM Mtest/MDSM Mtest/MDSM Mtest/MDSM

kN-m kN-m kN-m kN-m kN-m kN-m1 5.46 4.71 4.71 5.04 4.80 4.80 1.07 1.05 1.072 3.97 3.29 3.29 3.56 3.33 3.33 1.35 1.34 1.353 7.68 6.96 6.96 7.50 7.31 7.31 1.13 1.08 1.134 5.16 4.71 4.71 5.02 4.91 4.91 1.03 0.99 1.035 6.83 6.36 6.36 6.79 6.73 6.73 1.19 1.12 1.196 7.76 7.27 7.27 7.57 7.64 7.57 1.12 1.08 1.127 7.74 7.46 7.46 7.54 7.84 7.54 1.15 1.14 1.158 7.26 6.51 6.51 7.16 6.85 6.85 1.14 1.09 1.149 6.62 6.01 6.01 6.45 6.29 6.29 1.13 1.08 1.1310 6.59 6.13 6.13 6.49 6.48 6.48 1.10 1.04 1.1011 6.11 5.53 5.53 5.95 5.77 5.77 1.10 1.06 1.1012 7.40 6.53 6.53 7.21 6.84 6.84 1.15 1.09 1.1513 6.80 5.84 5.84 6.62 6.07 6.07 1.13 1.09 1.1314 7.52 6.98 6.98 7.33 7.21 7.21 1.14 1.11 1.1415 7.72 6.63 6.63 7.50 6.88 6.88 1.32 1.27 1.3216 7.96 7.65 7.65 7.73 7.99 7.73 1.12 1.11 1.1217 7.91 7.22 7.22 7.66 7.49 7.49 1.21 1.17 1.2118 6.41 5.57 5.57 6.22 5.77 5.77 1.15 1.11 1.1519 8.14 6.69 6.69 7.92 6.92 6.92 1.25 1.21 1.2520 7.65 6.18 6.18 7.45 6.40 6.40 1.27 1.22 1.2721 6.40 5.03 5.03 6.19 5.17 5.17 1.15 1.12 1.1522 4.57 4.19 4.19 4.58 4.43 4.43 1.19 1.12 1.1923 4.87 4.42 4.42 4.78 4.65 4.65 1.13 1.07 1.1324 4.99 4.53 4.53 4.86 4.74 4.74 1.08 1.04 1.0825 3.87 3.66 3.66 4.00 3.88 3.88 1.08 1.02 1.1026 3.99 3.70 3.70 4.02 3.93 3.93 1.15 1.08 1.1527 4.26 3.92 3.92 4.22 4.14 4.14 1.14 1.08 1.1428 2.88 2.88 2.88 3.01 2.98 2.98 1.08 1.04 1.1829 3.26 3.19 3.19 3.39 3.33 3.33 1.03 0.99 1.1030 3.34 3.15 3.15 3.39 3.36 3.36 1.08 1.01 1.0831 2.06 2.16 2.06 2.18 2.19 2.18 1.09 1.03 1.2232 2.47 2.57 2.47 2.62 2.62 2.62 1.01 0.96 1.1333 2.67 2.71 2.67 2.80 2.79 2.79 1.02 0.97 1.1234 1.41 1.49 1.41 1.49 1.49 1.49 1.12 1.06 1.2835 1.59 1.71 1.59 1.69 1.71 1.69 1.07 1.01 1.2036 1.77 1.84 1.77 1.85 1.88 1.85 1.06 1.02 1.1737 0.78 0.80 0.78 0.80 0.80 0.80 1.16 1.14 1.3638 0.98 1.02 0.98 1.02 1.02 1.02 1.10 1.05 1.2639 1.08 1.16 1.08 1.15 1.16 1.15 1.13 1.07 1.2640 0.39 0.39 0.39 0.39 0.39 0.39 1.33 1.33 1.5941 0.53 0.53 0.53 0.53 0.53 0.53 1.21 1.20 1.4442 0.64 0.65 0.64 0.64 0.65 0.64 1.15 1.15 1.35


φ=0.9, γL=1.5, γD=1.2

Furthermore, the average values for the ratio of the test results over the first, second and

existing methods for the member moment capacity are 1.14, 1.09 and 1.19. These are

with the COV of 0.07, 0.08 and 0.09 respectively. Therefore, the second method

provides the least conservative answer compared to the first and also existing design


methods. This is evident from it having the smallest average ratio. In addition, the

second method, when compared to the first method, is simpler and more economical to

use. The reliability index for the first, second and existing methods are 2.83, 2.62 and

2.91 respectively. They all meet the lower limit for reliability index according to the

AISI LRFD Specifications.


capacity of section 38 based on both the proposed methods and also the existing DSM.

Existing Method (DSM)

Local buckling:

The theoretical local buckling stress is determined by using thin wall program:

mkNMMM

M

mkNMMfZMMPaf

beblol

bel

ybeolfullxolol

85.0776.0461.0

85.0,002.4,2546 .


The theoretical distortional buckling stress is determined by using thin wall program:

mkNMMM

M

mkNMmkNfZMMPaf

ybdod

yd

yodfullxodod

85.0673.0461.0

85.0,002.4,2546 .

mkNMMMinM bdblDSM 85.0),(

Proposed Method 1 (DSM)

Local buckling:


776.035.0461.0

85.0,002.4,2546 .

lol

bel

ybeolfullxolol

M

M

mkNMMfZMMPaf


:35.0776.0 l mkNMM

MMM be

be

olbebl

98.035.047.02.15.0


53.0461.0

85.0,002.4,2546 .

dod

yd

yodfullxodod

M

M

mkNMmkNfZMMPaf

:53.0d mkNMM ybd 02.12.1


Proposed Method 2 (DSM)

Local buckling:


35.0461.0

85.0,002.4,2546 .

lol

bel

ybeolfullxolol

M

M

mkNMMfZMMPaf

For :35.0l

mkNMinM

mkNM

mKNMM

M

M

MM

bl

be

bebe

ol

be

olbl

02.1)02.1,043.1(

02.12.1

043.103.0207.015.05.0


53.0461.0

85.0,002.4,2546 .

dod

yd

yodfullxodod

M

M

mkNMmkNfZMMPaf

For :53.0d mkNMM ybd 02.12.1



5.5 Elastic and Plastic Slenderness Limits in AS4100 (1998)

The revised design methods have already been proposed for EWM with distortional

buckling check as well as DSM. However in chapter 4, it was shown that section

classifications, which have been defined in the AS4100, is not accurate for cold-formed

channel sections. Therefore, based on the test results in the previous chapter, new

slenderness ratio limits for web and flange elements are proposed.

Figure 5.12 shows two graphs of the AS4100 and the proposed slenderness limits. It is

evident in Figure 5.12(a) that the test results do not fit with the existing slenderness

limits.

AS4100 slenderness limit

0

20

40

60

80

0 20 40 60 80 100 120 140 160 180 200

λweb

λfla

nge


Plastic limitsElastic limits

Proposed slenderness limit

0

20

40

60

80

0 20 40 60 80 100 120 140 160 180 200

λweb

λfla

nge


Plastic limitsElastic limits

(a) (b)

Figure 5.12: Comparison between the existing and the proposed slenderness limits

Furthermore, Figure 5.12(a) shows that the ultimate moment in some sections in the

plastic range do not reach the plastic moment. Therefore, new slenderness limits which

result in graph 5.12(b) are as follows:

For cold-formed stiffened compression elements: 8

35

ep

ey

(5.42)

For cold-formed stiffened elements under stress gradient: 22

110

ep

ey

(5.43)

The following steps explain how to check the accuracy of the proposed slenderness

limits. The first step is to classify the tested sections into two different groups. The

failure of the sections in the first group is controlled by the web and in the second group

by the flange. In the former, the eye of the section’s web is greater than the eye of


the section’s flange. However, in latter the eye of the section’s flange is greater than

the eye of the section’s web. This is where the slenderness ratio of the elements is e

and where ey is the elastic slenderness ratio. Based on the controlling element in failure

in Figure 5.13, the sections are divided into two individual groups.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0 25 50 75 100 125 150 175

λs

M/M

y

AS4100 λs (Flange)AS4100 λs (Web)Proposed λs FlangeProposed λs Web

Figure 5.13: Sections classification into two individual groups

The calculation of the sections ultimate moment using their slenderness ratio, elastic

and plastic slenderness limits has been discussed in the previous chapters. Figure 5.14

shows the normalised moment by the yield moment, versus the slenderness ratio by the

elastic slenderness limit. The graphs in this figure show the proposed and existing

design method (AS4100) results.

Proposed slenderness limit

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0

λs/λsy

M/M

y

Proposed design result

Test results

Existing slenderness limit

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0

λs/λsy

M/M

y

AS4100 design results

Test results

Figure 5.14: Comparison between the proposed and the existing slenderness limits


The values in Table 5.6 show that the ratio of the moment capacity from test results

over the existing AS4100 method results and also the proposed design results. Table 5.6

compares the sections classifications based on the test results with the sections

classifications based on the proposed slenderness limits. By reviewing this table it is

evident that while sections 3, 6 to 8, 10, 15, 23 and 27 are classified as slender sections

they behaved as non-compact sections. Therefore, it can be concluded that the proposed

elastic slenderness limits are conservative. The reliability analyses of the proposed

models show that the reliability indexes ( ) are 2.26 and 2.67 for the existing and the

proposed slenderness limits respectively. The reliability index for the existing

slenderness limits is less than 2.5. Therefore, according to the AISI LRFD

Specifications, the existing slenderness limits in AS4100 are not reliable.

The average values for the ratio of the test results, over the proposed and existing

methods, are 1.17 and 1.07 with the COV of 0.14 and 0.16 respectively. Therefore, the

existing method provides a less conservative answer compared to the proposed method.

Note that the existing sections classifications are not neither accurate nor reliable. It is

to be mentioned that while sections 4, 5, 9, 11 to 14, 16 to 22, 24 to 26, 28 to 39, 41 and

42 are classified similarly in the proposed and existing method, their moment capacity

are not similar. This is due to their different elastic and plastic slenderness limit in

existing and proposed methods. As shown in equations 2.77 to 2.79 the effective

modulus of elasticity of sections depends on their plastic and elastic slenderness limits.

176

Table 5.6: Proposed AS4100 model data

AS4100 Proposed Proposed Existing

sections λeFlange λeWeb λeyFlange λeyWeb λepFlange λepWeb Max(λe/λey) λs λsy λsp Classification MAS4100 Mtest/MAS4100 Mtest/MAS4100

Proposed (Test) kN-m

1 39.55 148.27 15 110 8 22 2.64 39.55 15 8 S (S) 3.61 1.39 1.39

2 56.64 108.39 15 110 8 22 3.78 56.64 15 8 S (S) 2.26 1.97 1.97

3 36.06 107.61 35 110 8 22 1.03 36.06 35 8 S (NC) 7.57 1.04 0.97

4 54.06 70.34 35 110 8 22 1.54 54.06 35 8 S (S) 3.73 1.30 1.14

5 32.88 97.43 35 110 8 22 0.94 32.88 35 8 NC (NC) 6.77 1.12 1.02

6 32.68 113.99 35 110 8 22 1.04 113.99 110 22 S (NC) 7.77 1.05 1.01

7 26.69 123.26 35 110 8 22 1.12 123.26 110 22 S (NC) 7.42 1.16 1.11

8 38.64 95.18 35 110 8 22 1.10 38.64 35 8 S (NC) 6.48 1.15 1.02

9 48.25 88.58 35 110 8 22 1.38 48.25 35 8 S (S) 5.06 1.34 1.18

10 41.42 89.63 35 110 8 22 1.18 41.42 35 8 S (NC) 5.49 1.23 1.08

11 52.20 85.11 35 110 8 22 1.49 52.20 35 8 S (S) 4.51 1.35 1.18

12 43.20 103.61 35 110 8 22 1.23 43.20 35 8 S (S) 6.12 1.22 1.07

13 48.89 90.79 35 110 8 22 1.40 48.89 35 8 S (S) 5.22 1.26 1.11

14 38.78 107.31 35 110 8 22 1.11 38.78 35 8 S (S) 7.20 1.11 0.98

15 34.02 125.88 35 110 8 22 1.14 125.88 110 22 S (NC) 7.50 1.17 1.12

16 23.83 143.73 35 110 8 22 1.31 143.73 110 22 S (S) 6.95 1.23 1.18

17 29.66 148.93 35 110 8 22 1.35 148.93 110 22 S (S) 6.88 1.27 1.21

18 52.89 91.17 35 110 8 22 1.51 52.89 35 8 S (S) 4.85 1.32 1.15

19 39.44 128.70 35 110 8 22 1.17 128.70 110 22 S (S) 7.67 1.09 1.04

20 47.10 108.78 35 110 8 22 1.35 47.10 35 8 S (S) 6.18 1.27 1.11

21 56.09 91.46 35 110 8 22 1.60 56.09 35 8 S (S) 4.73 1.22 1.07

S: Slender


C:Compact

177

Table 5.6: Proposed AS4100 model data (continued)

AS4100 Proposed Proposed Existing

sections λeFlange λeWeb λeyFlange λeyWeb λepFlange λepWeb Max(λe/λey) λs λsy λsp Classification MAS4100 Mtest/MAS4100 Mtest/MAS4100

Proposed (Test) kN-m

22 33.34 81.41 35 110 8 22 0.95 33.34 35 8 NC (NC) 4.46 1.12 1.01

23 38.25 82.37 35 110 8 22 1.09 38.25 35 8 S (NC) 4.42 1.13 1.00

24 43.15 82.35 35 110 8 22 1.23 43.15 35 8 S (S) 4.20 1.17 1.02

25 27.53 70.05 35 110 8 22 0.79 27.53 35 8 NC (NC) 3.79 1.04 0.92

26 33.54 72.57 35 110 8 22 0.96 33.54 35 8 NC (NC) 3.86 1.10 1.00

27 37.75 71.60 35 110 8 22 1.08 37.75 35 8 S (NC) 3.87 1.15 1.03

28 23.51 62.82 35 110 8 22 0.67 23.51 35 8 NC (NC) 2.87 1.08 0.97

29 27.52 60.84 35 110 8 22 0.79 27.52 35 8 NC (NC) 3.16 1.04 0.92

30 33.32 62.77 35 110 8 22 0.95 33.32 35 8 NC (NC) 3.21 1.06 0.95

31 18.64 51.88 35 110 8 22 0.53 18.64 35 8 NC (NC) 2.10 1.07 0.99

32 22.74 52.01 35 110 8 22 0.65 22.74 35 8 NC (NC) 2.43 1.03 0.93

33 27.53 51.63 35 110 8 22 0.79 27.53 35 8 NC (NC) 2.57 1.06 0.93

34 13.19 41.29 35 110 8 22 0.38 13.19 35 8 NC (NC) 1.48 1.07 1.03

35 18.63 43.25 35 110 8 22 0.53 18.63 35 8 NC (NC) 1.61 1.06 0.99

36 23.51 43.25 35 110 8 22 0.67 23.51 35 8 NC (NC) 1.74 1.08 0.98

37 8.33 30.56 35 110 8 22 0.28 30.56 110 22 NC (NC) 0.82 1.11 1.09

38 13.05 31.70 35 110 8 22 0.37 13.05 35 8 NC (NC) 1.00 1.07 1.04

39 17.73 30.85 35 110 8 22 0.51 17.73 35 8 NC (NC) 1.09 1.12 1.06

40 4.22 20.98 35 110 8 22 0.19 20.98 110 22 C (C ) 0.43 1.21 1.21

41 8.07 22.31 35 110 8 22 0.23 8.07 35 8 NC (NC) 0.56 1.14 1.14

42 12.86 21.83 35 110 8 22 0.37 12.86 35 8 NC (NC) 0.64 1.13 1.10

Mean(Pm)= 1.17 1.08

Cov(VP)= 0.14 0.16

Reliability Index (β)= 2.67 2.26

φ=0.9, γL=1.5, γD=1.2



capacity of section 38 based on both the proposed and existing AS4100 design methods.

Existing Method (AS4100)

051.13250

2,702.31

250

2 21

yee

yee

F

t

rbflange

F

t

rbweb

From Table 5.2 AS4100 (1998):


eepey


eepey

30

40

051.13

33.0,max

flange

flange

flange

flangewebflange

epsp

eysy

es

ey

e

ey

e

ey

e

sys Section 38 is compact. However, according to the test result, this section

behaved as a non-compact section.

3. 1906)2358,1906()15725.1,1906()5.1,( mmMinxMinZSMinZ fullxxex

mkNEFZM yexAS 03.1)1.541)(61906(4100

Proposed Method (AS4100)

051.13250

2,702.31

250

2 21

yee

yee

F

t

rbflange

F

t

rbweb

Revised slenderness limits:


eepey


eepey


8

35

051.13

37.0,max

flange

flange

flange

flangewebflange

epsp

eysy

es

ey

e

ey

e

ey

e

syssp Section 38 is non-compact.

3.. 5.184315721906

835

051.13351572 mmZSZZ fullxx

spsy

ssyfullxex

mkNEFZM yexAS 00.1)1.541)(65.1843(4100

5.6 Conclusions

Building on chapter 4 experimental analyses, this chapter revises the AS/NZS4600,

DSM and AS4100 design rules, for determining the ultimate moment capacity of cold-

formed channel sections in bending. In chapter 4, the experimental test results were

compared with the inelastic reserve capacity, AS/NZS4600 with distortional buckling

check, EUROCODE, DSM and AS4100. Based on this testing, it was concluded that

the inelastic reserve capacity, AS/NZS4600 and NASPEC with distortional buckling

check, EUROCODE and DSM are quite conservative due to predicting much smaller

results compare to the test results, specially for non-slender sections. On the other hand,

section classifications for AS4100 were not found to be accurate. To this end, revisions

of these design methods have been the subject of this chapter. In inelastic reserve

capacity, AS/NZS4600 with distortional buckling check and DSM revised methods, to

determine the inelastic behaviour of cold-formed channel sections with partially

stiffened compression flange were considered.

Non-fully effective sections display some inelastic strains (Figure 5.7), however due to

the fact that it is not considered appropriate to apply an inelastic procedure to a section

that buckles elastically, the design procedures for such sections have not been modified.

For fully-effective sections the design methods have been developed that allows

increases in moment capacity of up to 20% above first yield designs, to account for the


development of inelastic strains in the sections. The modifications decrease the

conservatism for such sections in the effective width method from 25% to 9%, in

AS/NZS4600 with distortional buckling check from 34% to 22%, in DSM (first

method) from 27% to 14% and in DSM (second method) from 27% to 10% (Table 5.7).

To supplement the AS4100 method, new elastic and plastic slenderness limits are

proposed. A slenderness limit has been defined in accordance to both AS4100 and

AS/NZS4600 which channel sections display full plastic capacity with rotational

capacity greater than 3 (compact sections), and which are currently considered

acceptable for plastic design. Considering a portal frame may achieve increases in

failure loads using plastic mechanism analysis of around 30% compared with elastic

first yield analysis (depending on the frame dimensions), this could lead to overall

increases in capacity predictions for cold-formed channel section portal frames of 56%

(1.2 x 1.3).

Table 5.7 summarises the mean values, COV and reliability index of proposed and

existing design methods. Also Table 5.7 shows drop of the conservatism from the

existing to the proposed design methods for fully-effective sections.

Table 5.7: Mean values, COV and reliability index of proposed and existing design

methods Conservatism for

Design method Mean Cov Reliability Index fully effective sectionsInelastic Reseve Capacity (Existing) 1.16 0.12 2.67 25%Inelastic Reseve Capacity (Proposed) 1.11 0.10 2.62 9%

AS/NZS4600 with Distortional buckling check(Existing) 1.27 0.07 3.27 34%AS/NZS4600 with Distortional buckling check(Proposed) 1.25 0.07 3.18 22%

DSM for Local buckling failure (Existing) 1.12 0.12 2.53DSM for Local buckling failure (Proposed method1) 1.06 0.07 2.52DSM for Local buckling failure (Proposed method2) 1.06 0.07 2.52

DSM for Distortional buckling failure (Existing) 1.19 0.09 2.91DSM for Distortional buckling failure (Proposed method1) 1.13 0.08 2.76DSM for Distortional buckling failure (Proposed method2) 1.09 0.08 2.62

DSM(min(Local failure, Distortional failure))(Existing) 1.19 0.09 2.91 27%

DSM(min(Local failure, Distortional failure))(Proposed method1) 1.14 0.07 2.83 14%

DSM(min(Local failure, Distortional failure))(Proposed method2) 1.09 0.08 2.62 10%

AS4100 section classification (Existing) 1.08 0.16 2.26AS4100 section classification (Proposed) 1.17 0.14 2.67


Table 5.7 shows that, except for the AS4100 method, the reliability index for all the

proposed methods decreased slightly. All the propose methods meet the AISI LRFD

Specifications requirement due to the lower limit of reliability index. They therefore

provide less conservative results compared to the existing methods. It can also be

concluded that the proposed section classifications in AS4100 provides a more

conservative result compare to the existing classification. Note that the existing section

classifications are neither accurate nor reliable.

After revising the different design methods for calculating the ultimate moment

capacity of the cold-formed channel sections, it is valuable to investigate the collapse

response of the tested sections after reaching their ultimate capacity (collapse point) as

well. The following chapter therefore, using Yield Line Mechanism (YLM), will

simulate the collapse behaviour of the tested sections. This study is valuable for

determining the energy from the impact which can be dissipated by the cold-formed

channel sections.

182

Chapter 6

YIELD LINE MECHANISM (YLM) ANALYSIS OF

COLD-FORMED CHANNEL SECTIONS UNDER

BENDING


The test results presented in the chapter 3 signify that some cold-formed channel

sections with edge stiffener have a capacity beyond their yield moment. These sections

were then classified as non-compact or compact sections. Therefore, for the design of a

non-compact or a compact section under extreme loads, when the load re-distribution of

cold-formed steel members needs to be considered, the collapse behaviour of the

section needs to be examined. Apart from experimental analysis, which is costly and

not analytical, Yield Line Mechanism (YLM) is another option that can provide the

collapse response of sections. This is when a section fails the YLM of failure forms at

its localised plastic hinge point. YLM analysis is mostly used for thin wall structures

which have local failure mechanisms.

This chapter therefore applies a YLM model for cold-formed channel sections under

bending which is defined using the test observations. After defining an accurate model,

by using the energy method, the failure curve for each tested section is plotted. The

ultimate moment capacities of the slender tested samples are then determined by using

elastic and failure curves. Based on test results, a method to determine the rotation

capacity for cold-formed channel sections under bending is proposed. In addition, the

energy absorption due to the failure based on test results and YLM results are

compared. Finally, a simpler method compared to the YLM analysis is proposed to

determine the failure curve.

Chapter 6. Yield Line Mechanism (YLM) Analysis of Cold-formed Channel Sections under Bending 183

6.1 YLM Model for Cold-Formed Channel Beams

To define a basic YLM model, experimental observations and finite element analysis

are employed. Figure 6.1 shows the common failure mode for the tested simple channel

sections.

Figure 6.1: Common observed failure mode for the tested simple channel sections

It is evident from Figure 6.1 that the V-shape mechanism, similar to the proposed

model by Koteko (2004), can be used for the web of the simple channel sections under

bending (Figure 6.2).

Figure 6.2: YLM model in channel-section columns and beams (Koteko (2004))

Figure 6.3 then shows the common failure mode for the tested channel sections with

edge stiffener.


(a) (b)

Figure 6.3: (a) Common observed YLM model for the edge stiffener and the flange (b) Common observed YLM model for the web and the flange

In cold-formed channel sections with edge stiffener, the web and flange mechanisms

are similar to simple channels. However, additional yield lines are introduced for the

stiffeners and tension flanges. Figure 6.4 shows the YLM model used in this chapter for

cold-formed channel sections with simple and complex edge stiffener.


F

EB

AC

D

G

H

I

JK

L

A1

a

a

uB1

u

22

1

1

(a) Simple edge stiffener

F

EB

AC

D

G

H

I

JK

L

A1

a

a

uB1

u

M

N OP

2

1

1

2

(b) Complex edge stiffener

Figure 6.4: YLM model for cold-formed channel sections with edge stiffener

By establishing the YLM model for the sections and calculating the energy absorption

of each hinge line, the total absorbed energy for causing the failure can be estimated

and failure curve can be plotted. The following section explains the calculation of the


total energy absorption of the defined model for different angle rotations in order to

plot the failure curve.

6.2 Failure Curve

The energy method is used to estimate the failure curve of the sections under bending.

The total energy absorption for the YLM model is the sum of each individual hinge line

works.

n

iWW1

)()( (6.1)

This is where n is number of hinge lines in the model.

Then the bending moment can be obtained by solving following equation.

d

dWM )( (6.2)

where is the rotational angle of the beam. Therefore, for different values of the

rotation angle, the bending moment can be determined and failure curve (moment-

rotation graph) can be plotted. The following paragraphs explain the calculation of each

individual hinge line works that are proposed with the YLM model.

The energy absorption for the compression flange hinges are calculated based on the

work components defined by Kecman (1983). For the V shape mechanism (in webs and

stiffeners) Koteko (2007) work components are used. All of the work components are

shown in Figure 6.4.

These work components for each plastic hinge are defined as follows:

cos

)( 2121

bmWWW p

CDEF (6.3)

2

21arctanb

aa (6.4)

1

111

cos)tan(arccos

a

ba (6.5)

2

322

cos)tan(arccos

a

ba (6.6)

where 1 is shown in Figure 6.5.


E

A

C

G

a

A A1

l

(a -b Tan)1 1

(a -b Tan)Cos1 1

u

X

(a -b Tan)Sin1 1

1b /C

os1

1

1

1

Figure 6.5: Longitudinal cross-section of the web YLM model

To determine the u value following equations are used:

211

211 ))(( CosTanbaaX (6.7)

1111

1 )( XSinTanbaCos

bl

(6.8)

1

21

21

12

12

112

1 2)(

b

lbulubu

(6.9)

11

11 ub

uArcSin ,

23

22 ub

uArcSin (6.10)

))(( 212122 uubmWW pAB (6.11)

22 1113

amWWW pCAEA (6.12)

22 2114

amWWW pDBFB (6.13)

CGpGEGC lmWWW 15 2 (6.14)

1 is GCAGCA ,1 which is determined by following equations. Furthermore, the

coordination of pointsG ,C , 1A , A and A which are shown in Figure 6.6 are as

follows:


E

A

C

G

a1

A A1

l

a Cos1

u

X

b 1C

os

G

a Sin1

b -u 1 1

Y

b Sin1

A''

A'''

F

EB

A

C

D

G

H

I

JK

L

A1

a

a

uB1

uA''

A'''

A''

A'''

11

2

2

1

1

Figure 6.6: Angle η1

0,)(,:0,,:0,0,: 111111111 CosubCosaASinaCosbSinbCCosaG (6.15)

The slope of CG is:

GC

GC

xx

yyABSTan 1 (6.16)

By obtaining the coordination of C and 1A with the slope of CG , the coordination of

A is:

01

1

1

A

AA

ACCA

z

yy

Tan

yyxx

(6.17)

The length of AA 1 and AA 1 is:

22

1

2

1 11 AAAAAAAA zzyyxxl (6.18)

111Sinll AAAA (6.19)

AAl

uArcTan

1

11 (6.20)

)( 21

21 ablCG (6.21)

IDpIFID lmWWW 26 2 (6.22)

2 is IDBIDB ,1 which is determined with the same process as follows:


Furthermore, and as previously outlined, the coordination of point I , D , 1B , B and B

which are shown in Figure 6.4 are as follows:

0,)(,:0,,:0,0,: 223212332 CosubCosaBSinaCosbSinbDCosaI (6.23)

The slope of ID is:

ID

ID

xx

yyABSTan 2 (6.24)

By obtaining the coordination of D and 1B with the slope of DI , the coordination of

B is:

01

1

2

B

BB

BDDB

z

yy

Tan

yyxx

(6.25)

The length of BB 1 and BB 1 is:

22

1

2

1 11 BBBBBBBB zzyyxxl (6.26)

211Sinll BBBB (6.27)

BBl

uArcTan

1

22 (6.28)

)( 22

23 ablDI (6.29)

G

l

GA

rpGA dl

r

lmWW

GA

0

7 (6.30)

11 & rl

lru

l

ll

G

GAGA

GA

Gr (6.31)

GAGAl

G

GA

pG

l

GA

GlpGA

l

lr

umdl

lr

ulmW

0

3

21

1

02

1

12

3

(6.32)


1

1

3r

lumW GAp

GA (6.33)

L

l

IB

rpIB dl

r

lmWW

IB

0

8 (6.34)

22 & rl

lru

l

ll

I

IBIB

IB

Ir (6.35)

IBIBl

I

IB

pI

l

IB

IBpIB

l

lr

umdl

lr

ulmW

0

3

22

2

02

2

22

3

(6.36)

2

2

3r

lumW IBp

IB (6.37)

11 7007.0 ar

(6.38)

22 7007.0 ar

(6.39)

19 1

2r

mAWWW p

ACAAEAC (6.40)

211

1

auAACA (6.41)

210 1

2r

mAWWW p

BDBBFBD (6.42)

222

1

auABDB (6.43)

211 2 bmWW pGH (6.44)

IFIDHLHK WWWWW 12 (6.45)

IB

HJIBHJ l

lWWW13 (6.46)

414 2 bmWW pIM (6.47)

242415 22 bmbmWWWW ppLPJOKN (6.48)

where pm is the plastic moment capacity of the steel elements and is determined with

the following equation:


4

2tFm y

p (6.49)

It is assumed that 1a is the smallest value between 1/3 of the web’s depth and 1/3 of the

flange’s width and 2a is the smallest value of the 1/3 of stiffener’s depth and 1/3 of the

flange’s width. Figure 6.7 shows the measured 1a and 2a values of sections 10 and 27.

Section 10 Section 27

Figure 6.7: Measured a1 and a2 values

The assumed and test measurements values of 1a and 2a are shown in Table 6.1. Also,

the test measurement over the assumed ratio of 1a and 2a values are tabulated and

shown in Table 6.1. The average values of these two ratios are 0.96 and 1.28 with the

COV values of 0.12 and 0.31 respectively. It should be noted that 1a and 2a values are

manually measured and may not be accurate. It is important to note that sections 1 and

2 are simple cold-formed channel sections and are not included in the following tables.


Table 6.1: Comparison between test and assumed values for 1a and 2a Min(b1/3, b2/3) Min(b2/3, b3/3) Test Test

sections b4 b3 b2 b1 a1 a2 a1 a2 a1(test)/a1(assumed) a2(test)/a2(assumed)

mm mm mm mm mm mm mm mm3 12.32 15.94 44.92 122.14 14.97 5.31 14.50 7.00 0.97 1.324 14.20 14.94 62.75 79.85 20.92 4.98 20.00 7.50 0.96 1.515 12.62 21.67 41.49 111.16 13.83 7.22 12.50 7.50 0.90 1.046 12.51 16.29 41.27 129.03 13.76 5.43 12.00 7.00 0.87 1.297 12.39 15.78 34.99 139.88 11.66 5.26 10.00 5.00 0.86 0.958 11.82 17.66 48.23 110.04 16.08 5.89 17.00 7.50 1.06 1.279 9.78 18.06 56.65 99.00 18.88 6.02 20.00 10.00 1.06 1.66

10 17.12 17.98 49.36 99.83 16.45 5.99 15.00 6.00 0.91 1.0011 10.85 16.19 60.10 94.21 20.03 5.40 20.00 7.00 1.00 1.3012 10.85 16.50 50.93 113.76 16.98 5.50 20.00 10.00 1.18 1.8213 9.98 14.27 58.18 102.90 19.39 4.76 20.00 12.00 1.03 2.5214 22.74 47.59 121.10 15.86 7.58 15.00 10.00 0.95 1.3215 13.34 42.49 141.02 14.16 4.45 12.50 5.00 0.88 1.1216 18.67 31.40 159.19 10.47 6.22 9.00 5.00 0.86 0.8017 12.44 37.01 161.69 12.34 4.15 11.00 5.00 0.89 1.2118 17.34 62.09 102.68 20.70 5.78 20.00 7.00 0.97 1.2119 12.45 47.50 141.42 15.83 4.15 15.00 5.00 0.95 1.2020 14.53 55.88 121.20 18.63 4.84 20.00 10.00 1.07 2.0621 12.88 65.86 103.61 21.95 4.29 20.00 10.00 0.91 2.3322 20.00 39.99 89.00 13.33 6.67 14.00 10.00 1.05 1.5023 19.96 45.00 89.98 15.00 6.65 12.00 7.50 0.80 1.1324 19.96 49.99 89.96 16.66 6.65 12.50 10.00 0.75 1.5025 19.97 35.00 79.80 11.67 6.66 10.00 7.00 0.86 1.0526 20.00 40.20 79.99 13.40 6.67 12.00 6.00 0.90 0.9027 19.97 45.00 79.98 15.00 6.66 15.00 7.50 1.00 1.1328 19.96 29.97 70.05 9.99 6.65 12.00 7.00 1.20 1.0529 19.95 34.99 70.10 11.66 6.65 10.00 5.00 0.86 0.7530 19.99 39.97 70.00 13.32 6.66 10.00 7.00 0.75 1.0531 20.00 25.00 58.90 8.33 6.67 10.00 5.00 1.20 0.7532 19.97 29.96 60.80 9.99 6.66 10.00 6.00 1.00 0.9033 19.97 35.00 60.40 11.67 6.66 10.00 8.00 0.86 1.2034 14.80 19.90 49.50 6.63 4.93 7.50 5.00 1.13 1.0135 14.96 24.99 50.10 8.33 4.99 8.00 6.00 0.96 1.2036 14.95 29.97 50.10 9.99 4.98 9.00 6.00 0.90 1.2037 9.75 14.78 38.20 4.93 3.25 5.00 5.00 1.01 1.5438 9.63 19.75 39.40 6.58 3.21 7.50 3.00 1.14 0.9339 9.83 24.68 38.50 8.23 3.28 7.50 5.00 0.91 1.5340 9.20 10.45 28.10 3.48 3.07 0.00 0.0041 9.70 14.50 29.50 4.83 3.23 0.00 0.0042 9.73 19.55 29.00 6.52 3.24 6.00 4.00 0.92 1.23

Mean= 0.96 1.28COV= 0.12 0.31

Figure 6.8 shows ratio of the 1a from the test measurement over the assumed 1a value

verses the width to depth ratio of the tested sections.


0.50

0.75

1.00

1.25

1.50

0.20 0.30 0.40 0.50 0.60 0.70 0.80

Width/Depth

a 1(t

est)/a

1(as

sum

ed)

Figure 6.8: a1 from test measurement over assumed a1 ratio verses width to depth ratio of the tested sections

It can be seen in Figure 6.8 that the 1a values from test measurement over the assumed

1a values ratio vary from 0.75 to 1.20. This means that there is little difference between

the assumed and measured 1a values. Therefore, it has been decided to use the assumed

1a values in this chapter as their values are close to the test results. Figure 6.9 shows

ratio of the 2a from the test measurement over the assumed 2a value verses edge

stiffener to width ratio of the tested sections.


0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

2.50

2.75

0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90

Edge stiffener/Width

a 2(t

est)/a

2(as

sum

ed)

Figure 6.9: a2 from test measurement over assumed a2 ratio verses edge stiffener to width ratio of the tested sections

By reviewing Figure 6.9 it is evident that the 2a values from the test measurement over

the assumed 2a values ratio varies from 0.75 to 1.5 for sections in which their edge

stiffener to width ratio is greater than 0.32. However, for sections with the edge

stiffener over width ratio of less than 0.32 (sections 12, 13, 20 and 21) the assumed 2a

values are significantly smaller than the tested values. Therefore, the collapse curves

for these sections are determined using the 2a values from the test measurement and the

assumed 2a values. This is to determine whether or not the 2a values have a significant

effect on the collapse behaviour of the sections. Figure 6.10 shows the collapse curves

of sections 12, 13, 20 and 21 based on the proposed YLM model with two different

values for 2a .

195

Section 12

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

elastic

assumed a2

measured a2

Section 13

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

elastic

assumed a2

measured a2

Section 20

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

elastic

assumed a2

measured a2

Section 21

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

elastic

assumed a2

measured a2

Figure 6.10: Comparing collapse curves of sections 12, 13, 20 and 21 based on different values of a2

Chapter 6. Yield Line Mechanism (YLM) Analysis of Cold-formed channel Sections under Bending 196

It is to be noted that the hinge lines, which are influenced by length of 2a , are shaped in

edge stiffeners; and the depth of the edge stiffeners are small when compared to the

width and depth of the sections. Therefore, as illustrated in Figure 6.10, the 2a value

does not have a significant effect on the collapse behaviour of the sections with edge

stiffener to width ratio of less than 0.32.

The following computations are to calculate the applying moment of section 9 based on

the YLM method for a rotation angle of 0.02Rad.

Rad

y MPaFMPaE

mmtmmbmmbmmbmmb

02.0

552,198416

56.1,15.10,93.17,26.55,99 4321

mmbb

Mina 5.1842.183

,3

211

mmbb

Mina 0.698.53

,3

232

Rad

b

aa22.0arctan

2

21

RadArcCosa

CosTanbaArcCos 45.047.0

5.18

52.16)(1

1

111

RadArcCosa

CosTanbaArcCos 33.035.0

6

64.5)(2

2

322

mmCosTanbaaX 33.852.165.18))(( 22211

211

mmCosTanbaaX 05.264.56))(( 22232

222

mmSinTanCos

XSinTanbaCos

bl

02.9133.802.0)02.0995.18(02.0

99

)( 1111

1


mmSinTanCos

XSinTanbaCos

bl

0.1605.202.0)02.093.176(02.0

93.17

)( 2323

2

mmb

lbu 66.7

2 1

21

21

1

mmb

lbu 83.1

2 3

22

23

2

Rad

ub

uArcSin 08.0

11

11

Rad

ub

uArcSin 11.0

23

22

1 is GCAGCA ,1 which is determined by following equations. The coordination of

pointsG ,C , 1A , A and A which are shown in Figure 6.6, are as follows:

0,02.91,5.180,)(,:

0,35.99,98.10,,:

0,0,9.180,0,:

11111

111

1

CosubCosaA

SinaCosbSinbC

CosaG

The slope of CG is:

Rad

GC

GC

xx

yyABSTan 41.102.6 11

0

02.91

37.3

1

1

1

A

AA

ACCA

z

yy

Tan

yyxx

mmzzyyxxl AAAAAAAA 13.1537.35.18 222

1

2

1 11

mmSinSinll AAAA 93.1441.113.15111

Rad

AA

ArcTanl

uArcTan 47.0

93.14

66.7

1

11

mmablCG 71.100)( 21

21

2 is IDBIDB ,1 which is determined by the same process as follows:


Furthermore, the coordination of points I , D , 1B , B and B are as follows:

0,16,60,)(,:

0,05.18,36.00,,:

0,0,60,0,:

22321

233

2

CosubCosaB

SinaCosbSinbD

CosaI

The slope of ID is:

Rad

ID

ID

xx

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6.3 Estimating the Ultimate Moment Capacity

The intersection of the failure curve and the elastic curves of the effective sections

represent the ultimate capacity of the slender section. However, for non-compact and

compact sections the failure curves need to be shifted in order to proceed to the plastic

stage. Therefore, the intersection of the elastic and failure curves cannot represent the

ultimate capacity of the section. Different methods can be used to obtain the ultimate

capacity of non-compact and compact sections. In this study the AS4100 design method


with the proposed slenderness limits in the previous chapter is used to calculate the

ultimate moment capacity of the non-compact and compact sections.

A comparison of the test results for the bending moment capacity with the ultimate

moment capacities using YLM analysis are shown in Table 6.2. The ratio of the test

result over the YLM results are shown in Table 6.2, with the average value of 1.08 and

COV of 0.07.

It can also be concluded that the YLM with the reliability index of 2.61 provides less

conservative results compared to the proposed AS4100 results for slender sections. The

reliability index for the proposed AS4100 method is 2.67 (shown in Table 5.7).

The ratio of the test result over the YLM results are plotted verses the width to depth

ratio of the tested sections (displayed in Figure 6.11).

0.8

0.9

1.0

1.1

1.2

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Width/Depth

Mte

st/M

YL

M

Figure 6.11: The ratio of test result over the YLM results verses the width to depth ratio of the tested sections

There are only three sections where the ultimate bending capacity based on YLM is

greater than the test results ─ sections 7, 16 and 17. These sections have a width over

depth ratio of less than 0.25.


Table 6.2: Comparison between test results and YLM results

sections MYLM MAS4100 MYLM/MAS4100 Mtest Mtest/MYLM

kN-m kN-m kN-m3 7.30 7.57 0.96 7.89 1.084 4.76 3.73 1.28 4.85 1.025 6.54 6.77 0.97 7.56 1.166 7.88 7.77 1.01 8.17 1.047 9.58 7.42 1.29 8.60 0.908 6.54 6.48 1.01 7.45 1.149 5.80 5.06 1.15 6.80 1.17

10 5.72 5.49 1.04 6.76 1.1811 5.29 4.51 1.17 6.09 1.1512 6.81 6.12 1.11 7.48 1.1013 6.20 5.22 1.19 6.59 1.0614 7.20 7.20 1.00 7.97 1.1115 8.21 7.50 1.09 8.76 1.0716 10.27 6.95 1.48 8.57 0.8317 9.54 6.88 1.39 8.73 0.9118 5.59 4.85 1.15 6.38 1.1419 7.93 7.67 1.03 8.37 1.0620 6.50 6.18 1.05 7.82 1.2021 5.57 4.73 1.18 5.78 1.0422 4.48 4.46 1.01 4.98 1.1123 4.42 4.42 1.00 4.97 1.1224 4.45 4.20 1.06 4.91 1.1025 3.79 3.79 1.00 3.95 1.0426 4.20 3.86 1.09 4.26 1.0127 3.91 3.87 1.01 4.46 1.1428 2.87 2.87 1.00 3.11 1.0829 3.16 3.16 1.00 3.30 1.0430 3.34 3.21 1.04 3.40 1.0231 2.10 2.10 1.00 2.24 1.0732 2.43 2.43 1.00 2.50 1.0333 2.57 2.57 1.00 2.72 1.0634 1.48 1.48 1.00 1.58 1.0735 1.61 1.61 1.00 1.70 1.0636 1.74 1.74 1.00 1.88 1.0837 0.82 0.82 1.00 0.91 1.1138 1.00 1.00 1.00 1.07 1.0739 1.09 1.09 1.00 1.22 1.1240 0.43 0.43 1.00 0.52 1.2141 0.56 0.56 1.00 0.64 1.1442 0.64 0.64 1.00 0.73 1.13

Mean(Pm)= 1.08COV(Vp)= 0.07Reliability Index (β)= 2.61

φ=0.9, γL=1.5, γD=1.2

After calculating the ultimate capacity, the shifting in the failure curve warrants

discussion. Since the shift in the failure curve depends on the rotation capacity of the


section, the following section proposes a method to calculate this rotation capacity for a

section.

6.4 A Proposed Method for Estimating the Rotation Capacity

Rotation capacity ( R ) is a measure of how much the plastic hinge can rotate before the

failure occurs. This can be calculated by using Equations 2.82 and 2.83 as outlined in

Chapter 2.

It is assumed that the rotation capacity ( R ) varies from one to four for non-compact

sections and exceeds four for compact sections. Bambach et al. (2009a) have drawn on

test results to propose a relationship between the rotation capacity of a hat section and

its slenderness value. The experimental results from the chapter 3 are used to generate

an empirical equation for determining the rotation capacity of a channel section from its

slenderness value.

The following equations are proposed for determining the rotation capacity of cold-

formed channel sections:

,sps 4

s

spR

at pM (6.50)

,syssp 4

pssy

ssyR

at yM (6.51)

,ssy 0R (6.52)

By achieving the R value, the extension of the failure curvature, where the bending

moment is equal to or above the yield moment for the non-compact sections and the

plastic moment for the compact sections, is determined. Consequently, the failure curve

is shifted to the ultimate curvature where the bending moment drops below the yield

moment for non-compact sections and the plastic moment for compact sections. Table

6.3 shows the tabulations of the slenderness ratio, elastic slenderness limit, plastic

slenderness limit and calculated rotation capacity of the tested sections. By comparing

the R value of the test results with the proposed equations of 6.50 to 6.52, it can be


concluded that the proposed equations provide a greater value of R compared to the test

results for sections 25, 28, 29, 31, 33 to 35, 37 to 39 and 41. However, the average

value of the ratio for the test results over proposed results is 1.33. In the following

section the moment-curvature diagram of the tested sections based on test and YLM

results are compared to validate the YLM model.


Table 6.3: Calculated rotation capacity value Calculated Test

sections λs λsy λsp R R Rtest/Rcalculated

3 36.06 35.00 8.00 0.00 0.4 at My

4 54.06 35.00 8.00 0.00 0.00

5 32.88 35.00 8.00 0.31 at My 0.80 at My 2.58

6 113.99 110.00 22.00 0.00 0.20 at My

7 123.26 110.00 22.00 0.00 0.60 at My

8 38.64 35.00 8.00 0.00 0.25 at My

9 48.25 35.00 8.00 0.00 0.00

10 41.42 35.00 8.00 0.00 0.65 at My

11 52.20 35.00 8.00 0.00 0.00

12 43.20 35.00 8.00 0.00 0.00

13 48.89 35.00 8.00 0.00 0.00

14 38.78 35.00 8.00 0.00 0.00

15 125.88 110.00 22.00 0.00 0.10 at My

16 143.73 110.00 22.00 0.00 0.00

17 148.93 110.00 22.00 0.00 0.00

18 52.89 35.00 8.00 0.00 0.00

19 128.70 110.00 22.00 0.00 0.00

20 47.10 35.00 8.00 0.00 0.00

21 56.09 35.00 8.00 0.00 0.00

22 33.34 35.00 8.00 0.25 at My 0.70 at My 2.80

23 38.25 35.00 8.00 0.00 0.20 at My

24 43.15 35.00 8.00 0.00 0.00

25 27.53 35.00 8.00 1.11 at My 0.75 at My 0.68

26 33.54 35.00 8.00 0.22 at My 0.85 at My 3.86

27 37.75 35.00 8.00 0.00 0.45 at My

28 23.51 35.00 8.00 1.70 at My 0.45 at My 0.26

29 27.52 35.00 8.00 1.11 at My 0.90 at My 0.81

30 33.32 35.00 8.00 0.25 at My 0.70 at My 2.80

31 18.64 35.00 8.00 2.42 at My 1.85 at My 0.76

32 22.74 35.00 8.00 1.82 at My 2.00 at My 1.10

33 27.53 35.00 8.00 1.11 at My 0.95 at My 0.86

34 13.19 35.00 8.00 3.23 at My 1.80 at My 0.56

35 18.63 35.00 8.00 2.43 at My 1.70 at My 0.70

36 23.51 35.00 8.00 1.70 at My 2.75 at My 1.62

37 30.56 110.00 22.00 3.61 at My 2.50 at My 0.69

38 13.05 35.00 8.00 3.25 at My 2.75 at My 0.85

39 17.73 35.00 8.00 2.56 at My 1.85 at My 0.72

40 20.98 110.00 22.00 4.20 at Mp 4.30 at Mp 1.02

41 8.07 35.00 8.00 3.99 at My 3.15 at My 0.79

42 12.86 35.00 8.00 3.28 at My 5.70 at My 1.74

Mean= 1.33

COV= 0.74


6.5 Comparison between the Test and the YLM Bending-Curvature

Diagrams

To determine if the proposed YLM model can be used for simulating collapse

behaviour of cold-formed channel sections, the moment-curvature diagram of the tested

sections based on test and YLM results is compared.

Hypothesis test technique is a statistical tool to check whether two sets of measurements

are essentially different. Using this technique, the YLM versus experiment graph for

each section has been compared. Typically this technique is supported with the null

hypothesis in which the mean values of the two sets of measurements are equal. In this

case the matched pair t-test is applicable for normally distributed data (parametric test).

If the normality assumption has been violated for the experimental differences, the

Wilcoxon signed-rank test as a nonparametric test procedure has been used.

All the sections graphs have been checked with 95% confidence interval for the

differences. The p-value, lower and upper limits values of differences between the test

and YLM results are tabulated in Table 6.4.

Table 6.4: t-test and Wilcoxon signed rank test results for YLM versus test results Sections p-value Lower limit Upper limit Sections p-value Lower limit Upper limit

3 0.38 -0.42 0.82 23 0.05 -0.6 04 0.75 -0.6 0.45 24 0.023 -0.9 -0.25 0.15 -1.6 0.25 25 0.02 -0.8 -0.056 0.02 -0.98 -0.05 26 0.23 -0.45 0.167 0.0001 -1.8 -0.92 27 0.41 -0.5 0.28 0.11 -0.85 0.54 28 0.11 -1.1 0.159 0.74 -0.48 0.85 29 0.08 -0.66 0.00510 0.11 -0.14 1.27 30 0.014 -0.44 -0.02811 0.57 -1.1 1.67 31 0.06 -0.7 012 0.57 -0.62 0.92 32 0.06 -0.28 013 0.05 -0.7 -0.13 33 0.44 -0.45 0.2414 0.2 -0.33 1.4 34 0.09 -0.27 0.0215 0.04 -1.3 0 35 0.09 -0.58 0.0216 0.0001 -2.6 -1.6 36 0.25 -0.21 0.0717 0.008 -3.8 -1 37 0.84 -0.1 0.0818 0.04 -1 -0.04 38 0.62 -0.08 0.0719 0.06 -1.65 0.13 39 0.95 -0.1 0.120 0.16 -1.2 0.4 40 0.18 -0.1 0.0221 0.8 -0.15 0.2 41 0.72 -0.05 0.0622 0.4 -0.6 0.12 42 0.14 -0.21 0.02


It can be observed from Table 6.4 that about 90% of the cases (i.e. except sections 7, 16,

17and 24) the p-values are greater than or close to 0.05 and the mean of their difference

include zero within the 95% confidence interval.

Non-dimensionalised moment-curvature diagrams for slender, non-compact and

compact sections are shown in Figures 6.12 to 6.14. The bending moments are

normalised with the section’s theoretical plastic bending moment ( pM ). The curvatures

are then normalised with the section’s theoretical plastic curvature ( pk ). To calculate

the theoretical plastic moment and curvature it is assumed that the stress in the whole

depth of the section have reached the yield stress. Figure 6.12 is a sample comparison

between the test and the YLM diagram for a slender section.

Section22

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 6.0 7.0 8.0 9.0 10.0k/kp

M/M

p

Test Result

elastic

YLM

Figure 6.12: Normalised moment-curvature diagram from the test and the YLM results for a slender section

In Figure 6.12 it is shown that, compared to the test graph, the YLM graph is in a good

agreement with the test result.


Section32

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test ResultYLMMAS4100/MpelasticMy/Mp

R=1.82

Figure 6.13: Normalised moment-curvature diagram from the test and the YLM results for a non-compact section

Figure 6.13 shows the extension of the failure curves, in which the bending moment is

equal to the yield moment for non-compact sections due to their rotation capacity. It

can be seen in this diagram that the YLM graph is very similar to the test graph.

Section40

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

YLM

MAS4100/Mp

elastic

R=4.2

Figure 6.14: Normalised moment-curvature diagram from the test and the YLM results for a compact section

Figure 6.14 shows the extension of the failure curves in which the bending moment is

equal to the plastic moment for a compact section due to its rotation capacity. Figure


6.14 shows that the YLM graph is very similar to the test graph. The comparison

between YLM analysis and the test results of the normalised moment-curvature graphs

for all of the forty tested samples with edge stiffener are shown in Appendix E. Based

on this work, it can be concluded that the YLM collapse curves for slender and shifted

collapse curves for compact and non-compact sections are in a good agreement with the

test graphs.

The energy absorption, based on YLM and the test results using the moment rotation

graphs of the tested sections, are calculated in the following section. To provide the

context and value of these calculations this section also outlines the use of energy

absorbers.

6.6 Energy Absorbers

An energy absorber is a device that is designed to dissipate energy during the event of a

crash. Energy absorbers are widely used in car bodies, aircrafts and highway barriers.

They are primarily made from thin wall sections due to these sections being cheap,

efficient and versatile Nagel (2005).

Figure 6.15 shows a car body frame which is predominately made of thin wall steel.

Figure 6.15: A vehicle body structure (Lu and Yu (2003))

Figure 6.15 shows that the upper rails, lower rails, A-pillars, B-pillars and roof rails are

energy absorbing members in an accident. In addition to the car bodies, energy

absorbers are also used to increase highway safety (Figure 6.16).


Figure 6.16: A W beam barrier

The barrier beams are supported by steel posts which are connected to the ground. In

the event of an accident, the energy from the impact is dissipated by barrier and post

deformations and transferred to the ground.

In steel energy absorbers, by increasing the applying load on the steel, the steel reaches

its yield point. Beyond the yield point, the steel starts collapsing plastically which is not

reversible. Determining the energy absorption capacity of energy absorbers is achieved

by analysing their plastic deformation.

Since the emphasis of this chapter is on the collapse behaviour of cold-formed channel

sections with edge stiffener, the amount of absorbed energy due to the deformation of

the tested sections under bending is determined using the following equation:

0

MdE (6.53)

The area under the moment-rotation curve represents the dissipated energy. The

following section describes a viable method used to calculate absorbed energy for each

section.

6.6.1 Energy absorption computation

The Simpson rule, which is a method to calculate the area under a graph, is used in this

thesis to calculate the absorbed energy. The Simpson equation is described as follows:


evenoddn yyyyh

Area 243 0 (6.54)

)(....)()( 131 nodd xfxfxfy (6.55)

)(....)()( 242 neven xfxfxfy (6.56)

n

xfxf

n

yyh nn )()( 00

(6.57)

where n is the number of strips (should be an even number)and h is the width of the

strip.

Dividing a graph based on Simpson rules is shown below, in Figure 6.17:

Figure 6.17: Dividing a graph based on Simpson rules

According to the Simpson rule, the area under a graph needs to be divided into n

individual strips and the strips should have a similar width of equal to h. Therefore, the

area of the all moment-rotation graphs, based on test results and also YLM results, are

divided into equal strips. Figure 6.18 shows an example of a divided graph. All the

divided graphs based on the test and YLM results are shown in Appendix F.


Section 3

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

0.00 0.01 0.02 0.03 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.09 0.10

θ (Rad)

M (

kN-m

)

Test

YLM

Figure 6.18: Divided moment-rotation graph of section 3 based on Simpson rules

Table 6.5 compares absorbed energy for the tested sections based on test and YLM

results. From this table it is evident that the ratio of energy absorption based on test

results over the YLM results varies between 0.59 and 1.19. The average value of that

ratio is 0.91 with the COV value of 0.15. As highlighted in Table 6.5 only the energy

absorption based on the test results for sections 7, 16 and 17 is significantly smaller

than the YLM results.


Table 6.5: Comparison between absorbed energy for the tested sections based on test results and the YLM results

sections width/depth Absorbed Energy test Absorbed Energy YLM Absorbed Energy test/Absorbed Energy YLM

kJ kJ3 0.37 437.00 404.00 1.084 0.79 116.00 115.00 1.015 0.37 272.00 302.00 0.906 0.32 354.00 423.00 0.847 0.25 311.00 458.00 0.688 0.44 319.00 362.00 0.889 0.57 303.00 290.00 1.04

10 0.49 359.00 309.00 1.1611 0.64 154.00 148.00 1.0412 0.45 376.00 352.00 1.0713 0.57 309.00 357.00 0.8714 0.39 398.00 356.00 1.1215 0.30 314.00 394.00 0.8016 0.20 265.00 401.00 0.6617 0.23 209.00 353.00 0.5918 0.60 204.00 236.00 0.8619 0.34 290.00 386.00 0.7520 0.46 217.00 237.00 0.9221 0.64 273.00 269.00 1.0122 0.45 215.00 235.00 0.9123 0.50 204.00 234.00 0.8724 0.56 188.00 235.00 0.8025 0.44 177.00 242.00 0.7326 0.50 237.00 264.00 0.9027 0.56 189.00 219.00 0.8628 0.43 112.00 149.00 0.7529 0.50 149.00 177.00 0.8430 0.57 145.00 157.00 0.9231 0.42 105.00 116.00 0.9132 0.49 156.00 174.00 0.9033 0.58 158.00 185.00 0.8534 0.40 95.00 106.00 0.9035 0.50 58.00 72.00 0.8136 0.60 106.00 110.00 0.9637 0.39 65.57 69.46 0.9438 0.50 100.18 111.53 0.9039 0.64 109.71 115.95 0.9540 0.37 34.65 31.40 1.1041 0.49 46.04 48.78 0.9442 0.67 77.12 64.72 1.19

Mean(Pm)= 0.91COV(Vp)= 0.15

Figure 6.19 shows the ratio of energy absorption from the test results over the YLM

results, verses the width to depth ratio of the tested sections.


0.50

0.75

1.00

1.25

1.50

0.000 0.125 0.250 0.375 0.500 0.625 0.750 0.875

Width/Depth

Ene

rgy

Tes

t/Ene

rgy

YL

M

Figure 6.19: Energy absorption from test results over the YLM results ratio verses the

width to depth ratio

From Figure 6.19, it is evident that the ratio of the test results over the YLM results for

the majority of the sections are between 0.75 and 1.19. For sections with a width to

depth ratio of less than 0.25 (sections 7, 16 and 17), the energy absorption based on the

test results is significantly smaller than the YLM results.

6.7 A Simplified YLM Equation for the Cold-Formed Channel

Sections

Initiating a geometrical model and determining the energy absorption by different hinge

lines can be a complex exercise. Therefore, a simplified equation to estimate the failure

curve of cold-formed channel sections is proposed. From the YLM analysis, it was

determined that there is a logical relationship between the normalised moment by the

plastic moment and the normalised curvature by the plastic curvature during the linear

part as well as the failure curve.

Figure 6.20 shows the best curve fit with the test result for a slender tested section.


Section22

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

Linear part

Curve part

M/Mp = 1.0(k/kp)-0.665

Figure 6.20: The best curve fit with the test result

As shown in Figure 6.20 by example,

M/Mp=X(k/kp) -0.665 (6.58)

By trial and error, values of the X factor which makes the best graph to fit with the test

graphs are determined and tabulated in Table 6.6.


Table 6.6: The value of X factor Best fit with test graphs

sections λs/λsp M/Mp=X(k/kp)-.665 X=2.8(λs/λsp)-.665

X3 4.51 1.00 1.034 6.76 0.75 0.795 4.11 1.00 1.096 5.18 1.00 0.947 5.60 1.25 0.898 4.83 1.00 0.989 6.03 0.83 0.85

10 5.18 1.00 0.9411 6.53 0.78 0.8012 5.40 0.90 0.9113 6.11 0.84 0.8414 4.85 0.84 0.9815 5.72 0.90 0.8816 6.53 1.16 0.8017 6.77 0.97 0.7818 6.61 0.70 0.8019 5.85 0.82 0.8620 5.89 0.70 0.8621 7.01 0.66 0.7722 4.17 1.00 1.0823 4.78 0.86 0.9924 5.39 0.78 0.9125 3.44 1.22 1.2326 4.19 0.94 1.0827 4.72 0.92 1.0028 2.94 1.38 1.3729 3.44 1.19 1.2330 4.16 1.05 1.0831 2.33 1.85 1.6032 2.84 1.55 1.4033 3.44 1.30 1.2334 1.65 1.80 2.0135 2.33 2.00 1.6036 2.94 1.70 1.3737 1.39 2.20 2.2538 1.63 1.85 2.0239 2.22 1.75 1.6540 0.95 3.00 2.8941 1.01 2.40 2.7842 1.61 2.10 2.04

To find the X factor, the X factors of the all tested sections is plotted verses the ratio of

the sections slenderness over their plastic slenderness limit (Figure 6.21).


y = 2.8004x-0.665

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0

λs/λsp

X

M/Mp=X(k/kp)-0.665

Power (M/Mp=X(k/kp)-0.665)

Figure 6.21: The best curve fit for calculating the X factor from the ratio of the sections slenderness over their plastic slenderness limit

From Figure 6.21 it can be concluded that the failure curve section of the diagram

depends on the ratio of the sections slenderness over their plastic slenderness limit. This

relationship is shown in the following equation:

665.0665.0

8.2

ppp k

k

M

M

(6.59)

The coefficient 2.8 was the best fit for the test curves. Figures 6.22 to 6.24 compare the

curves from the proposed equation with the curves from the experimental result for a

slender, non-compact and compact section. The proposed equation compares well with

the test results.


Section22

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

Linear part

Curve part

M/Mp = 1.08(k/kp)-0.665

Figure 6.22: Normalised moment-curvature diagram from the test and the simplified proposed method for a slender section

Section32

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

MAS4100/Mp

Linear part

Curve part

M/Mp = 1.4(k/kp)-0.665

Figure 6.23: Normalised moment-curvature diagram from the test and the simplified proposed method for a non-compact section


Section40

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 6.0 7.0 8.0 9.0 10.0 11.0

k/kp

M/M

p

Test Result

MAS4100/Mp

Linear part

Curve part

M/Mp = 2.89(k/kp)-0.665

Figure 6.24: Normalised moment-curvature diagram from the test and the simplified proposed method for a compact section

Graphs comparing the normalised moment-curvature of the 40 tested samples, based on

simplified equation for the failure curve, and test results are shown in Appendix G.

6.7.1 Estimating the ultimate moment capacity using simplified

method

The intersection of the failure curve, based on the simplified method and the effective

section’s elastic curves represents the ultimate capacity of the slender section. In

addition, the AS4100 design method with the proposed slenderness limits in the

previous chapter is used to calculate the ultimate moment capacity of the non-compact

and compact sections.

The comparison of the test results for the bending moment capacity with the ultimate

moment capacities from simplified method are shown in Table 6.7. The ratio of the test

result over the simplified method results are shown in Table 6.7 with the average value

of 0.97 and COV of 0.1.


Table 6.7: Comparison between test results and simplified method results. Reduction factor of 0.85 Reduction factor of 0.85

sections Msimplified method Mtest Mtest/Msimplified method only for Slender sections only for Slender sections

0.85(Msimplified method) Mtest/Msimplified method

kN-m kN-m kN-m3 9.34 7.89 0.85 7.94 0.994 5.69 4.85 0.85 4.84 1.005 8.48 7.56 0.89 7.21 1.056 9.24 8.17 0.88 7.86 1.047 9.28 8.60 0.93 7.89 1.098 8.45 7.45 0.88 7.18 1.049 7.37 6.80 0.92 6.27 1.09

10 7.27 6.76 0.93 6.18 1.0911 6.74 6.09 0.90 5.73 1.0612 8.31 7.48 0.90 7.07 1.0613 7.45 6.59 0.88 6.34 1.0414 9.23 7.97 0.86 7.84 1.0215 8.97 8.76 0.98 7.62 1.1516 9.60 8.57 0.89 8.16 1.0517 9.49 8.73 0.92 8.06 1.0818 7.17 6.38 0.89 6.10 1.0519 9.28 8.37 0.90 7.89 1.0620 8.61 7.82 0.91 7.32 1.0721 7.17 5.78 0.81 6.10 0.9522 5.48 4.98 0.91 4.65 1.0723 5.60 4.97 0.89 4.76 1.0424 5.53 4.91 0.89 4.70 1.0425 3.79 3.95 1.04 3.79 1.0426 4.70 4.26 0.91 3.99 1.0727 4.74 4.46 0.94 4.03 1.1128 2.87 3.11 1.08 2.87 1.0829 3.16 3.30 1.04 3.16 1.0430 3.83 3.40 0.89 3.25 1.0531 2.10 2.24 1.07 2.10 1.0732 2.43 2.50 1.03 2.43 1.0333 2.57 2.72 1.06 2.57 1.0634 1.48 1.58 1.07 1.48 1.0735 1.61 1.70 1.06 1.61 1.0636 1.74 1.88 1.08 1.74 1.0837 0.82 0.91 1.11 0.82 1.1138 1.00 1.07 1.07 1.00 1.0739 1.09 1.22 1.12 1.09 1.1240 0.43 0.52 1.21 0.43 1.2141 0.56 0.64 1.14 0.56 1.1442 0.64 0.73 1.13 0.64 1.13

Mean(Pm)= 0.97 1.07COV(Vp)= 0.10 0.04Reliability Index (β)= 2.06 2.63

φ=0.9, γL=1.5, γD=1.2

Table 6.7 shows that the ultimate moment capacity for slender sections, based on

simplified method, are greater than the test values. Therefore, the ultimate moment


capacities which are determined by using the simplified method need to be reduced.

The reduction factor of 0.85 is therefore applied on all slender sections ultimate

capacity, based on the simplified method. The ratios of the test result over the reduced

simplified method are shown in Table 6.7 with the average value of 1.07 and COV of

0.04. It can also be concluded that the simplified method, with the reduction factor for

slender sections, and with the reliability index of 2.63, provides less conservative

results compared to the proposed AS4100 results for slender sections.

6.8 Conclusions

Yield Line Mechanism is an analytical option that provides a less costly method to

simulate the collapse response of thin-wall sections. This chapter proposed a YLM

model for cold-formed channel sections under bending. All the dimensions and

parameters for YLM model can be determined using mathematical and geometrical

calculations except two dimensions being 1a and 2a which are shown in Figure 6.6. As

a result, these two dimensions are determined based on assumptions on which provides

the best fit with the test graphs. They are also checked with measuring the tested

samples. The test measurements and the assumed values are in good agreement except

the 2a value for sections 12, 13, 20 and 21. However, the inaccuracy of the 2a value

does not have a significant effect on the final results.

After proposing the YLM model, using the energy method, collapse curves for each

tested section are plotted. The ultimate moment capacities of the slender tested samples

are then determined using elastic and failure curves. The majority of the ultimate

capacity based on YLM for the slender tested sections are slightly smaller than the test

results. It has been verified that this model can be used for determining slender sections

ultimate moment capacity.

After calculating the ultimate capacity for slender sections, shifting the failure curve is

discussed for compact and non-compact sections. Since the shift in the failure curve

depends on the rotation capacity of the section, a method is proposed to determine the

rotation capacity for cold-formed channel sections under bending. The moment-

curvature of the tested sections based on YLM model with shifting the collapse curves


based on the proposed rotation capacity value were subsequently plotted. This enabled

the conclusion to be drawn that the YLM collapse curves are in a good agreement with

the test graphs.

The energy absorption due to the failure based on test results and YLM results are

compared by using Simpson rules. Majority of the tests over YLM results for energy

absorption are between 0.75 and 1.19. The sections with width to depth ratio less than

0.25 (sections 7, 16 and 17) energy absorption based on test results are significantly

smaller that YLM results.

Finally, a simplified YLM method is proposed to determine the collapse curve of the

tested sections. In this method the normalised moment-curvature collapse curve is a

function of the sections slenderness over its plastic slenderness limit ratio. The graphs

of this method are also in a good agreement with the test results graphs. However, the

ultimate moment capacities of the slender sections are greater than the test results.

Therefore, all the slender sections ultimate moment capacity based on simplified YLM

methods are reduced by a reduction factor of 0.85.

All of this work supports the conclusion that both the YLM and the simplified YLM

models are in a good agreement with the test results. After determining the ultimate

moment capacity of the cold-formed channel sections and examining the collapse

responds of them using YLM method, it is valuable to investigate deformation process

of the tested sections as well. The following chapter therefore, using finite element

program, will simulate the deformation process of the tested sections.

222

Chapter 7

FINITE ELEMENT METHOD (FEM) ANALYSIS OF

COLD-FORMED CHANNEL SECTIONS UNDER

BENDING


This chapter describes the finite element procedure and the outcomes for analysing the

behaviour of cold-formed channel sections under bending. In chapter 4 the test results

were described and it was determined that the ultimate capacity for some of the cold-

formed channel sections with edge stiffener can exceed the predicted capacity, based on

existing design standards. Moreover, during the test procedure, the buckling behaviour

of the tested sections during the test could not be monitored. This was due to their

sudden collapse. Therefore, to complement these test results, the behaviour and strength

of the cold-formed channel sections in bending is simulated by using the ABAQUS

program.

7.1 ABAQUS Models

ABAQUS is developed and supported by Hibbitt, Karlsson & Sorensen (Hibbitt et al.

(2009)) and is a very popular Finite Element Method (FEM) simulator used in

academic and research environments due to its nonlinear physical behaviour modelling

capability. ABAQUS has different element types that provide a set of tools for solving

different problems. Each element has a unique name such as T2D2, S4R, C3D8I, or

C3D8R. Figure 7.1 shows the most commonly used elements in ABAQUS.

Chapter 7. Finite Element Method (FEM) Analysis of Cold-formed Channel Sections under Bending 223

Figure 7.1: Commonly used elements in ABAQUS (Hibbitt et al. (2009))

Nonlinear material behaviours can be examined accurately using shell elements. The

shell element, S4R, is found to be the most suitable element type for analysing the

buckling behaviour and is therefore used in this research.

In ABAQUS, the inelastic flow of steel is expressed with the classical metal plasticity

models that use standard Mises or Hill yield surfaces with associated plastic flow. The

general classical metal plasticity models are simple and accurate for cases such as a

collapse behaviour study. In nonlinear analysis, ABAQUS uses the Newton method or

alternative methods such as Riks method. The Riks method is used in cases with

material nonlinearity, geometric nonlinearity prior to buckling or unstable post-

buckling response. The ABAQUS model in this chapter includes the material and also

geometrical nonlinearity.

7.1.1 Mesh Density

Sections 8, 35 and 41 are modelled with three different mesh sizes in ABAQUS (Figure

7.2).


Figure 7.2: FEM for different mesh sizes

The normalised moment-curvature graphs for the different finite element models are shown

in Figure7.3.

225

Section 8

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 6.0 7.0 8.0 9.0 10.0k/kp

M/M

p

Test Result

4x4 mesh size

2x2 mesh size

Refined mesh at regionsof high stress gradients

Section35

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

4x4 mesh size

2x2 mesh size

Refined mesh at regionsof high stress gradientsM=My

Section41

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

4x4 mesh size

2x2 mesh size

Refined mesh at regions ofhigh stress gradientsM=Mp

Figure 7.3: Normalised moment-curvature of sections 8, 35 and 41 for different mesh density.

226

It can be concluded from Figure7.3 that the mesh size did not have a significant effect

on the ultimate moment and also the rotation capacities of the sections. However, in

Section 35 the FEM results for models with the smaller mesh size (2x2 mesh size) and

also denser mesh at the mid span (at the failure point) predicted collapse curves of

closer to the test result.

7.2 Material and Geometrical Nonlinearity

Material properties of the coupon tests in section 3.1 are entered as input data in the

ABAQUS models. However, the engineering stress and strain values obtained from

coupon tests are modified to the true stress and strain as input data for the model.

Wilkinson and Hancock (2002) simulate the behaviour of cold-formed RHS beams

using ABAQUS. They concluded that geometrical imperfection (nonlinearity) has large

influence on the rotation capacity and therefore the plastic behaviour of cold-formed

RHS beams. Therefore, the geometrical nonlinearity is introduced to the FEM model on

this thesis. To introduce the geometrical nonlinearity to the model, the buckled shape

of the perfect model, based on elastic buckling analysis, is used. The elastic buckling

analysis provides different buckling modes. Some of these are a pure local or pure

distortional buckling mode with others being a combination of the local and distortional

buckling mode. Therefore, for each sample different buckling shapes, based on the

elastic buckling analysis, are chosen and normalised by 10 per cent of the sample’s

thickness. After analysing the models, the results are compared with the test results to

verify the ABAQUS model. The verified model is then used to investigate the buckling

behaviour of the tested sections.

7.3 Results of the Simulation

To verify the ABAQUS model, the moment-curvature graphs, based on the ABAQUS

model, are compared with the test results.

The FEM versus experiment graph for each section has been compared using the

hypothesis test technique to check whether two sets of measurements are essentially

different. All the sections graphs have been checked with 95% confidence interval for

the differences. The p-value, lower and upper limits values are tabulated in Table7.1.


Table 7.1: t-test and Wilcoxon signed rank test results for FEM versus test results Sections p-value Lower limit Upper limit Sections p-value Lower limit Upper limit

3 0.54 -0.07 0.04 23 0.74 -0.25 0.094 0.7 -0.62 0.74 24 0.15 -0.14 0.675 0.36 -1.2 0.49 25 0.26 -0.21 0.676 0.001 0.008 0.072 26 0.8 -0.3 0.357 0.8 -0.5 0.1 27 0.016 -0.7 -0.28 0.77 -0.005 0.025 28 0.46 -0.5 19 0.64 -0.75 0.37 29 0.3 -0.35 0.7510 0.25 -2.4 0.04 30 0.73 -0.1 0.1311 0.38 -1.1 0.8 31 0.56 -0.42 0.5712 0.04 -1.7 -0.14 32 0.44 -0.55 0.2913 0.5 -1.6 0.43 33 0.7 -0.5 0.6214 0.25 -3 0.08 34 0.063 0 0.2515 0.008 -1.3 -0.2 35 0.45 -0.31 0.616 0.5 -0.5 0.2 36 0.31 -0.1 0.4717 0.25 -1.1 0.4 37 0.31 -0.04 0.1218 0.8 -0.6 0.25 38 0.03 0.02 0.1119 0.23 -1.4 0.4 39 0.45 -0.04 0.0720 0.33 -0.8 0.4 40 0.008 -0.08 -0.0321 0.03 -1.4 -0.17 41 0.16 -0.08 0.001522 0.2 -0.5 0.4 42 0.68 -0.02 0.015

It can be observed from Table 7.1 that about 90% of the cases (i.e. except sections 6, 15,

27 and 40) the p-values are greater than or close to 0.05 and the mean of their difference

include zero within the 95% confidence interval.

The normalised moment-curvature graphs compared with the test results and the Finite

Element Method (FEM) results are shown in Appendix E. It is concluded that generally

the FEM graphs are in good agreement with the test result. However, before collapse,

the FEM moment-curvature graphs for sections 9, 11, 13, 14, 18 and 20 show smaller

curvature values compared to the test graphs. Section 9 is representative of this group

and therefore selected for discussion. Figure 7.4 shows the normalised moment-

curvature graph for section 9.


Section9

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

FEM

Figure 7.4: Comparison between the normalised moment-curvature graphs of the section 9 based on inclinometers readings and the FEM results

As shown in Figure 7.4, the before collapse section of the moment-curvature graph for

the test results, has a smaller gradient compared to the FEM results. This is mainly due

to two reasons.

The first reason is that in the FEM model for determining the curvature of the tested

sections due to the applying moment, the rotation angle of point A (which is in the top

left side of the samples) is used (Figure 7.5).

Figure 7.5: Rotation at point A


However, in the test procedure, the position of the inclinometers is not exactly on the

edge of the sample and is closer to the mid-span. Due to the local or distortional

buckling along the sample, the rotation angle varies from point to point. Therefore, for

section 9 the rotation angles are measured at point B which, compare to point A, is

50mm closer to the mid-span (Figure 7.6).

Figure 7.6: Rotation at point B Figure 7.7 shows normalised moment-curvature graphs of section 9 at points A and B.

Section9

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 6.0 7.0 8.0 9.0 10.0k/kp

M/M

p

Test ResultCurvature at ACurvature at B

Figure 7.7: Comparison between the normalized moment-curvature graphs of the section 9 at points A and B


It is evident in Figure 7.7 that the moment-curvature graph before collapse at point B is

closer to the test results compared to point A.

The second reason for the slope difference in the test compared to the FEM graph is

due to the location of the failure point. Figure 7.8 shows the position of the local failure

for the section 9 in which is not at the mid-span of the sample. Therefore, the collected

rotation angles from inclinometers reading are affected by the local deformation of the

sample. It is to be noted that also the failure positions for sections 11, 13, 14, 18 and 20

are not exactly at the mid-span. This is due to the fact that cold-formed sections are not

geometrically perfect and have geometrical nonlinear behaviours.

Figure 7.8: Failure position for section 9

Figure 7.9 shows the comparison between the normalised moment-curvature graph of

the section 9 based on strain gauges readings and the FEM result based on the point A

curvature values. It is concluded that based on FEM, the moment-curvature graph are in

good agreement with the two strain gauges reading results.


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 6.0 7.0

k/kp

M/M

p

Tension strain

Compresion strain

FEM

Inclinometers

Figure 7.9: Comparison between the normalised moment-curvature graphs of section 9 based on strain gauges readings and the FEM result

Table 7.2 tabulates the ultimate moment capacity of the tested sections, based on FEM

results, and compares these with the ultimate moment capacities, based on test results.

The ratios of the test results over the FEM results are also shown in Table 7.2. The

average value of the ratios is equal to 1.00 and the COV is equal to 0.04. To gauge the

accuracy of the FEM models, the tested sections are classified into two groups: sections

with a slenderness ratio greater than their elastic slenderness limit (slender sections)

and sections with the slenderness ratio less than their elastic slenderness limit (non-

slender sections). Histograms of these two groups, based on their ratio of test results

over the FEM results, are shown in Figure 7.10.


Table 7.2: Comparison between ultimate moment capacities of the tested sections based on the FEM and the test results

sections Mtest MFEM Mtest/MFEM sections Mtest MFEM Mtest/MFEM

kN-m kN-m kN-m kN-m

3 7.89 8.34 0.95 23 4.97 5.02 0.99

4 4.85 5.12 0.95 24 4.91 5.20 0.94

5 7.56 7.40 1.02 25 3.95 3.92 1.01

6 8.17 8.41 0.97 26 4.26 4.11 1.04

7 8.60 8.98 0.96 27 4.46 4.44 1.00

8 7.45 7.39 1.01 28 3.11 2.94 1.06

9 6.80 6.77 1.00 29 3.30 3.29 1.00

10 6.76 6.73 1.00 30 3.40 3.39 1.00

11 6.09 6.36 0.96 31 2.24 2.16 1.04

12 7.48 7.55 0.99 32 2.50 2.48 1.01

13 6.59 7.22 0.91 33 2.72 2.68 1.01

14 7.97 7.68 1.04 34 1.58 1.58 1.00

15 8.76 8.23 1.06 35 1.70 1.73 0.98

16 8.57 9.24 0.93 36 1.88 1.84 1.02

17 8.73 8.93 0.98 37 0.91 0.88 1.04

18 6.38 6.37 1.00 38 1.07 1.05 1.02

19 8.37 8.21 1.02 39 1.22 1.19 1.03

20 7.82 7.67 1.02 40 0.52 0.53 0.99

21 5.78 6.28 0.92 41 0.64 0.62 1.04

22 4.98 4.65 1.07 42 0.73 0.71 1.02

Mean(Pm)= 1.00

COV(Vp)= 0.04

0%

5%

10%

15%

20%

25%

30%

0.91 0.93 0.95 0.97 0.99 1.01 1.03 1.05 1.07 1.09

Mtest/MFEM

λs ≤ λsy

λs > λsy

Figure 7.10: Histograms of the ratio of test results over the FEM results


Figure 7.10 shows that, for the majority of the slender sections, their moment capacity,

based on FEM results, is greater than the test results. However, for the majority of the

non-slender sections, the moment capacity based on FEM results is slightly smaller

than the test results. Therefore, it can be concluded that this FEM model is perfect for

simulating bending behaviour and also the strength of sections when the slenderness

ratio is less than their elastic slenderness limits. However, this FEM model is less

conservative for sections where the slenderness ratio is greater than their elastic

slenderness limits compared to the test result.

7.4 Simulation Result for Two Compact Sections

In chapter 4 the only section that has been classified as a compact section was section

40. To further analyse compact sections, two more compact sections (sections A and B)

have been defined. Table 7.3 shows a series of dimensions ( 1b , 2b , 3b , 4b , t ), as well

as the yield stress ( yF ), yield moment ( yM ), plastic moment ( pM ), slenderness ratio

subject to distortional buckling ( d ) and the ultimate moment capacities. These are

based on revised North American and Australian design methods for sections A and B.

By comparing the FEM results for these two sections with the revised design methods,

it can be concluded that the proposed design methods are still conservative.

The normalised moment-curvature graph of sections A, B as well as section 40 based

on FEM results are shown in Figure 7.11. The FEM model results for sections A and B

show that by decreasing the section slenderness ratio subject to distortional buckling,

the ultimate moment capacity and also rotation capacity of the section will increase.


Table 7.3: Dimensions and ultimate capacities of sections A and B based on revised design rules Section A Section B

b1 60 mm 50 mm

b2 33 mm 30 mm

b3 20 mm 15 mm

b4 0.0mm 0.0mm

t 5 mm 5 mm

ri 5 mm 5 mm

Fy 541 Mpa 541 Mpa

My 5.31 kN-m 3.54 kN-m

Mp 6.98 kN-m 4.68 kN-m

λd 0.4 0.35

Ms(inelastic) 6.85 kN-m 4.26 kN-m

Mbdistortional 6.37 kN-m 4.24 kN-m

MAS4600 6.37 kN-m 4.24 kN-m

MAS4100 6.98 kN-m 4.69 kN-m

MDSM 6.37 kN-m 4.24 kN-mMFEM 9.58 kN-m 6.68 kN-m

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0

k/kp

M/M

p

Section 40

SectionA

SectionC

λd=0.55

λd=0.40λd=0.35

Figure 7.11: Comparison between the normalised moment-curvature graphs of the sections 40, A and B

7.5 Deformation of the Tested Sections Prior to Their Collapse Point

In chapter 4, collapsed shapes of the tested sections have been investigated by

measuring their rotation angle due to the deformation of the web-compression flange


juncture and also the out of plane deflection for the compression flange elements.

However, the deformation of the tested sections could not be investigated during the

test. This is due to the fact that all the tested sections with edge stiffener collapsed

suddenly. Therefore, using the FEM model the tested sections deformation are

discussed in more detail.

The deformation shapes of channel sections are classified into three different modes by

Rogers (1995) (shown in Figure 7.12). The first mode is the local buckling which is the

out of plane deflection for the compression flange elements. The second mode is the

lip/flange distortional buckling which is the rotation of compression flange around the

web and compression flange juncture. The third mode is the flange/web distortional

buckling. This buckling mode is the rotation of web around the web and tension flange

intersection. In this chapter the definitions of Rogers (1995) are used for the

deformation of the tested sections.

Figure 7.12: Buckling modes (Rogers (1995))

The final deformation shapes for the majority of the tested sections were discussed in

chapter 3. These shapes are a combination of the Rogers buckling modes. Using FEM

the deformation process is discussed. The tested sections are classified into three

different categories based on their deformation process.

In the first category, sections with the d value of greater than 0.71, the local buckling

appears first and then lip/flange distortional buckling appears subsequently. Finally,

flange/web distortional buckling occurs. The deformed shapes of section 9 at different

stages are shown in Figure 7.13, serving as a sample for this category.


It can be seen in Figure 7.13 that section 9 has not buckled before collapsing. At stage

three, the compression flange element starts to deform. By increasing the curvature at

stage four, the flange starts to rotate around the junction of the web and compression

flange. In stages five and six the web element rotates around the intersection of web

and tension flange subsequently.


Different stages on normalised moment-curvature graph for section 9

Stage1 Stage2 Stage3


Deformation of the web

Figure 7.13: Deformation of section 9


In the second category, sections with the d value smaller than 0.71, have distortional

buckling of the lip/flange appearing initially. This is followed by local buckling of the

compression flange elements and then flange/web distortional buckling. The deformed

shapes of section 42 at different stages are shown in Figure 7.14, which serves as a

sample for this category. Figure 7.14 shows that the flange starts to rotate around the

web and compression flange junction before collapsing (stage two). At stage three,

rotation of the flange around web and compression flange junction is obvious. By

increasing the curvature at stage four, the web element rotates around the web and

tension flange intersection as a result and also that the compression flange element

starts to deform slightly.




Stage4 Deformation of the web



In the last category of compact sections 40, A and B, the flange/web and lip/flange

distortional buckling appears at the same time. Figure 7.15 shows different stages of the

deformations for section 40. By reviewing this figure, it can be concluded that, even in the

last deformation stage, the out of plane deflection for the compression flange elements of

section 40 has not been observed.




Stage4 Stage5 Deformation of the web



7.6 Conclusions

The main aim of this chapter is to provide a FEM model to simulate the failure

behaviour and also the strength of cold-formed channel sections with edge stiffener

under bending. To do so, FEM models were developed and verified using the test

results, showing that they were in a good agreement together. For the majority of

sections with the slenderness ratio of greater than their elastic slenderness limit, which

are classified as slender sections, the moment capacity based on FEM results are greater

than the test results. However, for the majority of the non-slender sections, the moment

capacity, based on FEM results are slightly smaller than the test results. Therefore, the

developed models are perfect for investigating the behaviour of the cold-formed non-

slender (compact and non-compact) channel sections whenever there is lack of

experimental data. Therefore, this FEM model was used to analyse two additional

compact sections. From this analysis it was concluded that by decreasing the section’s

slenderness ratio, the ultimate moment capacity of the section will exceed its plastic

moment. This finding proves that plastic design method could be applicable for cold

formed channel sections in structural assemblies.

Finally, the developed models are used to discuss the failure behaviour of the tested

sections evident from testing. The failure processes are classified into three groups

based on their slenderness ratio subject to distortional buckling. It is concluded that

FEM provide accurate results compared to the test results. Therefore, and importantly

for this research, FEM analysis can be used as a less expensive and time consuming

alternative compare to the test analysis for performing “what-if” scenarios.

243

Chapter 8

CONCLUSIONS AND RECOMMENDATIONS

8.0 General

The main aim of this study was to investigate the behaviour of cold-formed channel

sections under bending. To reach this aim, the relevant literature was reviewed and

experimental studies, semi-empirical analysis as well as numerical simulations were

performed according to the following steps:

The literature review focused on the range of design standards for designing cold-

formed channel sections with edge stiffener. From this, a number of conclusions were

evident. Firstly, the EWM, which is the design methods for determining the nominal

section moment capacity of the cold-formed sections in American and Australian

standards, do not include any inelastic reserve capacity for cold-formed sections with

edge stiffener. Secondly, the assumption in the methods for calculating the nominal

member moment capacity of the cold-formed channel sections, DSM and AS/NZS4600

with distortional buckling check, is that the ultimate moment capacity of cold-formed

sections can not exceed their yield moments due to the lack of experimental data

Schafer (2006a). Thirdly, the plastic design method is not applicable for cold-formed

channel sections. This is due to the fact that most studies regarding the behaviour of

steel in plastic range, (Lyse and Godfrey (1935), Haaijer and Thuerlimann (1958), Lay

(1965), Craskaddan (1968), Lukey and Adams (1969), Korol and Hudoba (1972), Holtz

and Kulak (1973), Hasan and Hancock (1988) and Zhao and Hancock (1991)) are

based on hot-rolled steel or cold-formed close sections experimental data. While some

experimental data Baigent and Hancock (1981) demonstrate the inelastic behaviour of

cold-formed channel sections, no studies were performed on the ultimate strength of

cold-formed channel sections in the inelastic and plastic range. Importantly this

supported the focus of this research on the behaviour of cold-formed channel sections

with edge stiffeners. This is specifically to determine whether or not the inelastic

Chapter 8. Conclusions and Recommendations 244

reserve capacity, and even plastic design rules, can be applied on channel sections with

stiffened flange. The inelastic reserve capacity and plastic design methods are more

economical compared to the traditional elastic design method.

Prior to investigating the behaviour of cold-formed channel sections with edge stiffener

under bending, some preliminary tests were conducted. The conclusion after the first

test was that lateral buckling and local instability were influencing the test results

which is not however the concern of this research. To address this effect, subsequent

modification of the Monash pure bending rig was considered to suit this research. In

addition, the samples were grouted by concrete at their both ends. Investigations of the

test results from the next three samples were then used to guide the preparation of

appropriate samples to address the purpose of future experiments.

Following this work, 39 cold-formed channel sections across three different

geometrical categories (simple channel sections, channel sections with simple edge

stiffener and channel sections with complex edge stiffener) were prepared for testing

according to the technique outlined above.

The prepared sections were classified into three different groups according to their

slenderness ratio based on the AS4100 (1998) classifications. In addition, the ultimate

moment capacity of the prepared sections, based on NASPEC (2007) ( NASPECM ),

AS/NZS4600 (2005) ( 4600ASM ), DSM ( DSMM ), EUROCODE3 (2006) ( 3EurocodeM ) and

AS4100 (1998) design rules ( 4100ASM ) were determined. Finally, the elastic portions of

the moment-curvature graphs for the prepared sections were plotted using the Effective

Width Method as well as the AS/NZS4600 method with the distortional buckling

check. All of this analysis and computations were performed to achieve the aim of

comparing the existing design rules with the test results.

While testing, eleven parameters were monitored in order to calculate the moment and

curvature of each sample. Therefore, the moment-curvature diagrams of the tested

sections were plotted using test results. The peak moments on the graphs were selected

as the ultimate moment capacity of the sections.


After failures, the rotation angles due to the deformation of the web flange juncture and

deformation of the compression flange due to local buckling were measured.

The bending behaviour of cold-formed channel sections has been determined for large

deflections and consequently large strains. Since thin steel channels are used

extensively in automotive and aeronautical structures, the results may be used to

determine the large deformation energy absorbing capabilities of such structures. A

theoretical procedure for such an analysis has been outlined. Additionally, the bending

results may be used to investigate the load resistance of such channels under large

deformations experienced under seismic conditions.

By using the test observations, the YLM model for cold-formed channel sections under

bending was proposed. After defining an accurate model, using the energy method, the

collapse curves for each tested section were plotted. Following which the ultimate

moment capacities of the slender tested samples are determined using elastic and

failure curves, and a method was proposed to determine the rotation capacity for cold-

formed channel sections under bending. Finally, a simpler method compared to the

YLM analysis was proposed in order to determine the failure curve.

In order to comprehensively study the behaviour and strength of cold-formed channel

sections with edge stiffener under bending, finite element models were proposed. These

proposed models were used to investigate the behaviour of the cold-formed channel

sections whenever there was inadequate data. This investigation was also used to

discuss the deformation procedures that could not be investigated during the test.

As a result, the following are conclusions and suggestions for future studies.

8.1 Conclusions

The outcome of the experimental studies, semi-empirical analysis and numerical

simulations are summarised below:


Comparisons were made between the ultimate moments of the three sections,

which were fabricated from a steel sheet of the same width, to determine the

efficiency of various edge stiffener configurations. It was concluded that edge

stiffeners have a positive effect on increasing the ultimate moment capacity of

channel sections.

According to both the AS/NZS4600 (2005) and NASPEC (2007) standards, the

inelastic reserve capacity design method is not applicable. However, by

determining the curvatures at the failure points of the tested sections, the

majority of the tested sections behaved in-elastically. This is due to both their

curvature and their strain at the failure point being greater than the yield

curvature and the yield strain. Therefore, the compression strain factors

( yultyC ) were greater than one.

Determining the rotation angles due to the deformation of the web flange

juncture and deformation of the compression flange due to local buckling,

identified a number of specific conclusions:

1. For sections where the width to depth ratio is less than 0.5, the

distortional buckling failure mode is more pronounced when compared

to the local buckling failure mode;

2. For the sections where the width to depth ratio was from 0.5 to 0.7, the

observed failures in these sections was most likely due to the

combination of the local and the distortional buckling failure mode; and

3. The only section where the local buckling failure mode was more

pronounced when compared to the distortional buckling failure mode

was section 4 which had a width to depth ratio of 0.8.

By comparing the test graphs with the EWM and also AS/NZS4600 with

distortional buckling check graphs, it was concluded that the AS/NZS4600

method with distortional buckling check can predict the buckling point

accurately and the graphs prior to the failure are in a good agreement with the

test results. However, the AS/NZS4600 with distortional buckling check is


conservative for calculating the ultimate moment capacity of cold-formed

channel sections.

The ultimate moment capacities of the tested sections from the test results were

compared with the different Standards design methods results. This showed

that:

1. AS/NZS4600 (2005), NASPEC (2007), EUROCODE3 (2006) and DSM

design rules are conservative for calculating the cold-formed channel

sections ultimate moment capacity;

2. The expected ultimate moment capacities of the tested sections, based

on AS4100 (1998) design rules, are much closer to the test results,

particularly in comparison to the four other design rules. However, the

section classifications, which are defined in the AS4100 (1998), are not

accurate for the cold-formed channel sections.

By using the test results, the inelastic reserve capacity to cold-formed channel

sections has been introduced. Non-fully effective sections display some

inelastic strains (Figure 5.7), however due to the fact that it is not considered

appropriate to apply an inelastic procedure to a section that buckles elastically,

the design procedures for such sections have not been modified. For fully-

effective sections the design methods has been developed that allows increases

in moment capacity of up to 20% above first yield designs, to account for the

development of inelastic strains in the sections. The modifications decrease the

conservatism for such sections in the effective width method from 25% to 9%,

in AS/NZS4600 with distortional buckling check from 34% to 22%, in DSM

(first method) from 27% to 14% and in DSM (second method) from 27% to

10% (Table 5.7). New slenderness limits for cold-formed channel sections were

also proposed. All of the revised and existing design methods have been gauged

using the FOSM-based reliability analysis. This led to the following

conclusions:

1. The proposed inelastic reserve capacity design method (NASPEC

(2007) and AS/NZS4600 (2005)) for partially stiffened compression


elements provides much closer results to the test results when compared to

the existing inelastic reserve capacity design method results;

2. The proposed AS/NZS4600 with distortional buckling check design

method provides less conservative results when compared to the existing

AS/NZS4600 with distortional buckling check design method and is still

reliable;

3. Two different methods are proposed for Direct Strength Method (DSM).

The design rules in the first method, for slender sections, are similar to

the existing design rules. This means that the first method is more

familiar to users than the second. On the other hand, for slender

sections, the second proposed method of DSM due to local buckling is

more conservative compared to the existing method. However, the

second method provides the least conservative result compared to the

first as well as existing design methods for sections subject to

distortional buckling; and

4. The proposed section classification in AS4100 provides a more

conservative ultimate capacity result compared to the existing

classification. However, the proposed classification provides design

results of acceptable reliability (β>2.5), where the existing

classifications do not (β<2.5).

5. Slenderness limits have been defined in accordance with both cold-

formed and hot-rolled international steel specifications, below which

cold-formed channel sections may display full plastic capacity with

rotational capacity greater than 3 (compact sections), and which are

currently considered acceptable for plastic design. For cold-formed steel

specifications, the flange and web slenderness values must be below

0.25 and 0.15 respectively according to the effective width method

(Table 5.1), or the section slenderness values for local and distortional

buckling must both be below 0.35 according to the DSM (Table 4.8).

For hot-rolled steel specifications, the flange and web slenderness

values must be below 8 and 22 respectively (Section 5.5).


After revising the different design methods for calculating the ultimate moment

capacity of the cold-formed channel sections, the Yield Line Mechanism

(YLM) was used to simulate the collapse response of the tested sections after

reaching their ultimate capacity (collapse point). The outcomes of comparing

test results with the proposed model were that:

1. Dimensions and parameters for the YLM model could be determined

using mathematical and geometrical calculations except for two

dimensions which were determined base on assumptions. The test

measurements and the assumed values were in good agreement for the

sections with edge stiffener to width ratio of greater than 0.32;

2. For sections with the width over depth ratio of greater than 0.25 the

ultimate bending capacity based on YLM is in a good agreement with

the test results;

3. For sections with width to depth ratio of greater than 0.25, energy

absorption based on YLM results are close to the test results; and

4. A simplified YLM method was also proposed to determine the collapse

curve of the tested sections. In this method the normalised moment-

curvature collapse curve is a function of the sections slenderness over its

plastic slenderness limit ratio.

The finite element models were verified with the test results and it was shown

that the developed models are perfect for investigating the behaviour of the

cold-formed non-slender (compact and non-compact) channel sections. The

main findings are:

1. For sections with a slenderness ratio subject to distortional buckling

( d ) value of less than 0.6, the ultimate moment capacity of the sections

will exceed their plastic moment; and

2. The Finite Element Method (FEM) models are used to discuss the

failure behaviour of the sections tested. The failure processes are

categorised into three groups based on their slenderness ratio, subject to

distortional buckling. In the first category, sections with the d value

greater than 0.71, the local buckling appears first and then lip/flange


distortional buckling appears subsequently. Finally, the flange/web

distortional buckling occurs. In the second category, sections with the

d value of smaller than 0.71, the lip/flange distortional buckling

appears initially. Then the local buckling for the compression flange

elements and also flange/web distortional buckling appears. In the third

category, being compact sections, the flange/web and lip/flange

distortional buckling appears at the same time. However, local buckling

has not been observed.

8.2 Recommendations for Future Study

In this research, it was found that the ultimate moment capacity of some cold-formed

channel sections with edge stiffener can exceed its plastic moment. However, the

behaviour of these sections as a member of a structural assembly such as portal frames

was not investigated. Further study is required to determine if a plastic collapse

mechanism is formed in cold-formed channel sections structures. Investigation of the

actual rotational capacities required for such sections to allow the development of

plastic mechanisms in portal frames. If the limit of 3 could be reduced, it could be

feasible to allow plastic design of channel section portal frames to a wider range of non-

compact but fully effective sections, which demonstrate rotational capacities between 1

and 3. It is valuable to use Aramis which is a 3D image correlation photogrammetry

technique to monitor the strain distribution along the structure in more detail. Since

Aramis is a relatively new technique, the accuracy of the technique requires

investigation in advance.

Haedir et al. (2009) examined the effect of Fibre Reinforcement Polymers (FRP)

strengthening on Circular Hollow Sections (CHS). Their test results indicate that FRP

strengthening can have a significant improvement on strength, buckling behaviour and

rotation capacity of CHS sections. Also Silvestre et al. (2009) investigated the

behaviour of FRP-strengthened cold-formed channel columns. They concluded that the

design formulas are not always reliable. It is therefore recommended that there be

further experimental and numerical analysis on the behaviour of cold-formed channel

sections with FRP strengthening.

251

REFERENCES

AS1170.1 (2002). "Australian Standard. Part 1: Dead and Live Loads and Load

Combinations." Standard Australia, Sydney.

AS1391 (2005). "Australian Standard Metallic Materials-Tensile testing at ambient

temperature." Standard Australia, Sydney.

AS4100 (1998). "Australian Standard. Steel Structures." Standard Australia, Sydney.

AS/NZS4600 (2005). "Australian/New Zealand Standard. Cold-Formed Steel

Structures." Standard Australia, Sydney.

AS/NZS4600Supp1 (1998). "Australian/New Zealand Standard. Cold-Formed Steel

Structures Commentary. (Supplement to AS/NZS4600:1996)." Standard

Australia, Sydney.

Baigent, A. H. and Hancock, G. J. (1981). "The Stiffness and Strength of Portal Frames

Composed of Cold-Formed Members." Civil Engineering Transaction, The

Institution of Engineers, Australia: pp 278-283.

Baker, J. F., Horne, M. R. and Heyman, J. (1956). "The Steel Skeleton, Volume 2"

Cambridge University Press, Cambridge.

Bakker, M. C. M. (1990). "Yield Line Analysis of Post-Collapse Behavior of Thin-

Walled Steel Members." Heron, Volume 35(3): pp 1-50.

Bambach, M. R. (2003). "Thin-Walled Sections with Unstiffened Elements under Stress

Gradients." Civil Engineering. Sydney, University of Sydney. Doctor of

Philosophy.

Bambach, M. R. (2009). "Design of Uniformly Compressed Edge-Stiffened Flanges and

Sections that Contain them." Thin-Walled Structures, Volume 47(3): pp 277-

294.

References 252

Bambach, M. R., Merrick, J. T. and Hancock, G. J. (1998). "Distortional Buckling

Formulae for Thin Walled Channel and Z-Sections with Return Lips."

International Specialty Conference on Cold-Formed Steel Structures: Recent

Research and Developments in Cold-Formed Steel Design and Construction,

University of Missouri-Rolla, Rolla, MO, United States: pp 21-37.

Bambach, M. R. and Rasmussen, K. J. R. (2004). "Tests of Unstiffened Plate Elements

Under Combined Compression and Bending." Journal of Structural

Engineering, Volume 130(10): pp 1602-1610.

Bambach, M. R. and Rasmussen, K. J. R. (2004a). "Effective Widths of Unstiffened

Elements with Stress Gradient." Journal of Structural Engineering, Volume

130(10): pp 1611-1619.

Bambach, M. R., Tan, G., and Grzebieta, R. H. (2009a). "Steel Spot-Welded Hat

Sections with Perforations Subjected to Large Deformation Pure Bending."

Thin-Walled Structures, Volume 47(11): pp1305-1315.

Beale, R. G., Godley, M. H. R. and Enjily, V. (2001). "A Theoretical and Experimental

Investigation into Cold-Formed Channel Sections in Bending with the

Unstiffened Flanges in Compression." Computers and Structures, Volume

79(26-28): pp 2403-2411.

Bryan, B. G. (1891). "On the Stability of a Plane Plate under Thrusts in its Own Plane

with Application on the Buckling of the Sides of Ship." Proceeding of the

London Mathematical Society: P 54.

Chick, C. G. and Rasmussen, K. J. R. (1999). "Thin-Walled Beam-Columns. II:

Proportional Loading Tests." Journal of Structural Engineering, Volume

125(11): pp 1267-1276.

Cimpoeru, S. J. (1992). "The Modelling of the Collapse during Roll-Over of Bus

Frames Consisting of Square Thin-Walled Tubes." Civil Engineering.

Melbourne, Monash University. Doctor of Philosophy.

References 253

Craskaddan, P. S. (1968). "Shear Buckling of Unstiffened Hybrid Beams." Journal of

Structural Engineering, ASCE, Volume 94(ST8): pp 1965-1990.

Davies, P., Kemp, K. O. and Walker, A. C. (1975). "Analysis of the Failure Mechanism

of an Axially Loaded Simply Supported Steel Plate." Proceedings of the

Institution of Civil Engineers (London). Part 1 - Design & Construction,

Volume 59(pt 2): pp 645-658.

Desmond, T. P., Pekoz, T. and Winter, G. (1981). "Edge Stiffeners for Thin-Walled

Members." Journal of Structural Engineering, ASCE Volume 107(2): pp 329-

353.

Elchalakani, M. (2003). "Cycling Bending Behaviour of Hollow and Concrete-Filled

Cold-formed Circular Steel Members." Civil Engineering. Melbourne, Monash

University. Doctor of Philosophy.

Elchalakani, M., Grzebieta, R. and Zhao, X. L. (2002a). "Plastic Collapse Analysis of

Slender Circular Tubes Subjected to Large Deformation Pure Bending."

Advances in Structural Engineering, Volume 5(4): pp 241-257.

Elchalakani, M., Zhao, X. L. and Grzebieta, R. (2002b). "Bending Tests to Determine

Slenderness Limits for Cold-Formed Circular Hollow Sections." Journal of

Constructional Steel Research, Volume 58(11): pp 1407-1430.

Ellingwood, B., Galambos, T. V., MacGregor, J. G. and Cornell, C. A. (1980).

"Developement of a Probability Based Load Criterion for American National

Standard A58: Building Code Requirements for Minimum Design Loads in

Buildings and Other Structures." U.S. Department of Commerce, National

Bureau of Standards, NBS Special Publication 577, Washington, DC.

Enjily, V., Beale, R. G. and Godley, M. H. R. (1998). "Inelastic Behaviour of Cold-

Formed Channel Sections in Bending." Second International Conference On

Thin-walled Structures. Singapore, pp 197-204.

References 254

EUROCODE3 (2006). "Eurocode 3. Design of Steel Structures - Part 1-5: Plated

Structural Elements." European Committee for Standarsation, ENV 1993-1-5,

Brussels.

EUROCODE3 (2006). " Eurocode 3. Design of Steel Structures - Part 1-3: General

Rules - Supplementary Rules for Cold-Formed Members and Sheeting."

European Committee for Standarsation, ENV 1993-1-3, Brussels.

Galambos, T. V. (1968). "Structural Members and Frames." Prentice-Hall, Englewood

Cliffs, NJ.

Haaijer, G. and Thuerlimann, B. (1958). "On Inelastic Buckling in Steel." ASCE -

Proceedings -Journal of the Engineering Mechanics Division, Volume 84: pp 1-

48.

Haedir, J., Bambach, M. R., Zhao, X. L. and Grzebieta, R. (2009). "Strength of Circular

Hollow Sections (CHS) Tubular Beams Externally Reinforced by Carbon FRP

Sheets in Pure Bending." Thin-Walled Structures, Volume 47(10): pp 1136-

1147.

Hancock, G. J. (1988). "Design of Cold-Formed Steel Structures To Australian

Standard AS 1538-1988)." Australian Institute of Steel Construction, Sydney.

Hancock, G. J. (1997). "Design for Distortional Buckling of Flexural Members." Thin-

Walled Structures, Volume 27(1): pp 3-12.

Hasan, S. W. and Hancock, G. J. (1988). "Plastic Bending Tests of Cold-Formed

Rectangular Hollow Sections." Journal of the Australian Institute of Steel

Construction ,Volume 123(4): pp 477-483.

Hibbitt, Karlsson and Sorensen. (2009). "Abaqus Analysis Users Manual, Volume 5."

SIMULIA.

Holtz, N. M. and Kulak, G. L. (1973). "Web Slenderness Limits for Compact Beams."

University of Alberta, Alberta, Report No. 43.

References 255

Hsiao, L.-E., W., Yu, W. and Galambos, T. V. (1990). "AISI LRFD Method for Cold-

Formed Steel Structural Members." Journal of Structural Engineering, Volume

116(2): pp 500-517.

Kalyanaraman, V., Pekoz, T. and Winter, G. (1977). "Unstiffened Compression

Elements." Journal of Structural Engineering, ASCE, Volume 103(9): pp 1833-

1848.

Kato, B. (1965). "Buckling Strength of Plates in the Plastic Range." International

Association for Bridge and Structural Engineering, Volume 25: pp 12 -141.

Kato, B. (1989). "Rotation Capacity of H-Section Members as Determined by Local

Buckling." Journal of Constructional Steel Research, Volume 13(2-3): pp 95-

109.

Kecman, D. (1983). "Bending Collapse of Rectangular and Square Section Tubes."

International Journal of Mechanical Sciences, Volume 25, UK: pp 623-636.

Kemp, A. R. (1996). "Inelastic Local and Lateral Buckling in Design Codes." Journal of

Structural Engineering, Volume 122(4): pp 374-382.

Korol, R. M. and Hudoba, J. (1972). "Plastic Behavior of Hollow Strauctral Sections."

Journal of Structural Engineering, ASCE, Volume 98(ST5): pp 1007-1023.

Koteko, M. (1996). "Ultimate Load and Post Failure Behaviour of Box-Section Beams

under Pure Bending." Eng Trans, Volume 44(2): pp 229-251.

Koteko, M. (2004). "Load-Capacity Estimation and Collapse Analysis of Thin-Walled

Beams and Columns - Recent Advances." Thin-Walled Structures, Volume

42(2): pp153-175.

Koteko, M. (2007). "Load-Carrying Capacity and Energy Absorption of Thin-Walled

Profiles with Edge Stiffeners." Thin-Walled Structures, Volume 45(10-11): pp

872-876.

References 256

Kuhlmann, U. (1989). "Definition of Flange Slenderness Limits on the Basis of

Rotation Capacity Values." Journal of Constructional Steel Research, Volume

14(1): pp 21-40.

Kwon, Y. B. and Hancock, G. J. (1992). "Tests of Cold-Formed Channels with Local

and Distortional Buckling." Journal of Structural Engineering, Volume 118(7):

pp 1786-1803.

Lau, S. C. W. and Hancock, G. J. (1987). "Distortional Buckling Formulas for Channel

Columns." Journal of Structural Engineering, Volume 113(5): pp 1063-1078.

Lay, M. G. (1965). "Flange Local Buckling in Wide-Flange Shapes." Journal of

Structural Engineering, ASCE, Volume 91(ST6, Part 1): pp 95-116.

Lu, G. and Yu, T. (2003). "Energy Absorption of Structures and Materials." Woodhead

Publishing Limited, Cambridge.

Lukey, A. F. and Adams, P. F. (1969). "Rotation Capacity of Beams under Moment

Gradient." Journal of Structural Engineering, ASCE, Volume 95(ST6): pp 1173-

1188.

Lukey, A. F. and Adams, P. F. (1969). "Rotation capacity of beams under moment

gradient." American Society of Civil Engineers, Journal of the Structural

Division, Volume 95(ST6): pp 1173-1188.

Lyse, I. and Godfrey, H. J. (1935). "Investigation of Web Buckling in Steel Beams."

Transaction, American Society of Civil Engineers, Volume 100: pp 675-695.

Murray, N. W. (1984). "The Effect of Shear and Normal Stress on the Plastic Moment

Capacity on Inclined Hinges in Thin-Walled Steel Structures." Festschrift Roik,

Inst. fur Konstruktiven Ingenieurbau, Ruhr Univ. Bochum, Volume 84: pp 237-

48.

Murray, N. W. and Khoo, P. S. (1981). "Some Basic Plastic Mechanisms in the Local

Buckling of Thin-Walled Steel Structures." International Journal of Mechanical

Sciences, Volume 23(12): pp 703-713.

References 257

Nagel, G. (2005). "Impact and Energy Absorption of Straight and Tapered Rectangular

Tubes." Built Environment & Engineering. Queensland, Queensland University

of Technology. Doctor of Philosophy.

NASPEC (2007). "North American Specification for the Design of Cold-Formed Steel

Structural Members." North American Specification Committee.

Papangelis, J. P. and Hancock, G. J. (1995). "Computer Analysis of Thin-Walled

Structural Members." Computers and Structures, Volume 56(1): pp 157-176.

Rang, T. N., Galambos, T. V. and Yu, W. W. (1979a). "Load and Resistance Factor

Design of Cold-Formed Steel: Study of Design Formats and Safety Index

Combined with Calibration of the AISI Formulas for Cold Work and Effective

Design Width." University of Missouri-Rolla, Rolla, Mo, 1st progress Report.

Rang, T. N., Galambos, T. V. and Yu, W. W. (1979b). "Load and Resistance Factor

Design of Cold-Formed Steel: Statistical Analysis of Mechanical Properties and

Thickness of Materials Combined with Calibration of the AISI Design

Provisions on Unstiffened Elements and Connections." University of Missouri-

Rolla, Rolla, Mo, 2nd Progress Report.

Ravindra, M. K. and Galambos, T. V. (1978). "Load and Resistance Factor Design for

Steel." Journal of Structural Engineering, ASCE, Volume 104(9): pp 1337-1353.

Reck, H. P., Pekoz, T. and Winter, G. (1975). "Inelastic Strength of Cold-Formed Steel

Beams." Journal of Structural Engineering, ASCE, Volume 101(11): pp 2193-

2203.

Rogers, C. A. (1995). "Local and Distortional Buckling of Cold Formed Steel Channel

and Zed Sections in Bending." Civil Engineering. Waterloo, Ontario, Canada,

University of Waterloo. Master of Applied Science.

Rusch, A. and Lindner, J. (2001). "Remarks to the Direct Strength Method." Thin-


References 258

Sanders, P. H. and Householder, J. (1978). "Applicability of Limit Design to Cold-

Formed Steel Box Beams." Forth International Specialty Conference on Cold-

Formed Steel Structure: pp 327-361.

Schafer, B. W. (2003). "Cold-Formed Steel Design by the Direct Strength Method: Bye-

bye Effective Width." Baltimore, MD, United States, Structural Stability

Research Council, Bethlehem, PA, United States: pp 357-377.

Schafer, B. W. (2006a). "Review: The Direct Strength Method of Cold-Formed Steel

Member Design." Journal of Constructional Steel Research, Volume 64(7-8): pp

766-778.

Schafer, B. W. and Pekoz, T. (1998). "Direct Strength Prediction of Cold-Formed Steel

Members Using Numerical Elastic Buckling Solutions." International Specialty

Conference on Cold-Formed Steel Structures: Recent Research and

Developments in Cold-Formed Steel Design and Construction: pp 69-76.

Schafer, B. W. and Pekoz, T. (1999). "Laterally Braced Cold-Formed Steel Flexural

Members with Edge Stiffened Flanges." Journal of Structural Engineering,

Volume 125(2): pp 118-126.

Schafer, B. W., Sarawit, A. and Pekoz, T. (2006). "Complex Edge Stiffeners for Thin-

Walled Members." Journal of Structural Engineering, Volume 132(2): pp 212-

226.

Silvestre, N., Camotim, D. and Young, B. (2009). "On the Use of the EC3 and AISI

Specifications to Estimate the Ultimate Load of CFRP-Strengthened Cold-

Formed Steel Lipped Channel Columns." Thin-Walled Structures, Volume

47(10): pp 1102-1111.

Tan, G. (2009). "Perforated Hat Sections Subjected to Large Rotation Pure Bending."

Civil Engineering. Melbourne, Monash University. Master of Engineering

Science.

Timoshenko, S. P. and Gere, J. M. (1961). "Theory of Elastic Stability." McGrawHill

Book Company, New York.

References 259

Ueda, Y. and Tall, L. (1967). "Inelastic Buckling of Plates with Residual Stresses."

International Association for Bridge and Structural Engineering, Volume 27: pp

211-254.

Von Karman, T., Sechler, E. E. and Donnell, L. H. (1932). "The Strength of Thin Plates

in Compression." Transaction ASME, Volume 54(APM 54-5): p 53.

Wilkinson, T. (1999). "The Plastic Behaviour of Cold-Formed Rectangular Hollow

Sections." Civil Engineering. Sydney, University of Sydney. Doctor of

Philosophy.

Wilkinson, T. and Hancock, G. J. (1998a). "Tests to Examine Compact Web

Slenderness of Cold-Formed RHS." Journal of Structural Engineering, Volume

124(10): pp 1166-1174.

Wilkinson, T. and Hancock, G. J. (1998). "Tests of Portal Frames in Cold-Formed RHS.

Tubular Structures VIII." 8th International Symposium on Tubular Structures.

Singapore, Balkema: pp 521-529.

Wilkinson, T. and Hancock, G. J. (2002). "Predicting the Rotation Capacity of Cold-

Formed RHS Beams Using Finite Element Analysis." Journal of Constructional

Steel Research, Volume 58(11): pp 1455-1471.

Winter, G. (1970). "Commentary on the 1968 Edition of the Specification for the

Design of Cold-Formed Steel Structural Members." American Iron and Steel

Institute.

Yan, J. and Young, B. (2002). "Column Tests of Cold-Formed Steel Channels with

Complex Stiffeners." Journal of Structural Engineering, Volume 128(6): pp

737-745.

Yener, M. and Pekoz, T. (1985). "Partial Stress redistribution in Cold-Formed Steel."

Journal of Structural Engineering ,Volume 111(6): pp 1169-1186.

Yiu, F. and Pekoz, T. (2000). "Design of Cold-Formed Steel Plain Channels."

International Specialty Conference on Cold-Formed Steel Structures: Recent

References 260

Research and Developments in Cold-Formed Steel Design and Construction,

University of Missouri-Rolla, Rolla, MO, United States: pp 13-22.

Young, B. and Yan, J. (2004). "Design of Cold-Formed Steel Channel Columns with

Complex Edge Stiffeners by Direct Strength Method." Journal of Structural

Engineering, Volume 130(11): pp 1756-1763.

Yu, C. and Schafer, B. W. (2003). "Local Buckling Tests on Cold-Formed Steel

Beams." Journal of Structural Engineering, Volume 129(12): pp 1596-1606.

Yu, C. and Schafer, B. W. (2006). "Distortional Buckling Tests on Cold-Formed Steel

Beams." Journal of Structural Engineering, Volume 132(4): pp 515-528.

Yu, C. and Schafer, B. W. (2007). "Simulation of Cold-Formed Steel Beams in Local

and Distortional Buckling with Applications to the Direct Strength Method."

Journal of Constructional Steel Research, Volume 63(5): pp 581-590.

Zhao, X. L. (2003). "Yield Line Mechanism Analysis of Steel Members and

Connections." Progress in Structural Engineering and Materials, Volume 5(4):

pp 252-262.

Zhao, X. L. and Grzebieta, R. (1999). "Void-Filled SHS Beams Subjected to Large

Deformation Cyclic Bending." Journal of Structural Engineering, Volume

125(9): pp 1020-1027.

Zhao, X. L. and Hancock, G. J. (1991). "Tests to Determine Plate Slenderness limits for

Cold-Formed Rectangular Hollow Sections of Grade C450." Journal of

Australian Institute of Steel Construction, Volume 25(4): pp 2-16.

Zhao, X. L. and Hancock, G. J. (1993). "Experimental Verification of the Theory of

Plastic-Moment Capacity of an Inclined Yield Line under Axial Force." Thin-


Zhao, X. L. and Hancock, G. J. (1993a). "A Theoretical Analysis of the Plastic-Moment

Capacity of an Inclined Yield Line under Axial Force." Thin-Walled Structures,

Volume 15(3): pp 185-207.

References 261

Zhou, F. and Young, B. (2005). "Tests of Cold-Formed Stainless Steel Tubular Flexural

Members." Thin-Walled Structures, Volume 43(9): pp 1325-1337.

Zhu, J. H. and Young, B. (2006). "Aluminium Alloy Tubular Columns-Part II:

Parametric Study and Design Using Direct Strength Method." Thin-Walled

Structures, Volume 44(9): pp 969-985.

Zhu, J. H. and Young, B. (2009). "Design of Aluminium Alloy Flexural Members Using

Direct Strength Method." Journal of Structural Engineering, Volume 135(5): pp

558-566.

262

Appendix A

STRESS-STRAIN DIAGRAMS OF COUPON TESTS Coupon G1

0

100

200

300

400

500

600

0.0% 0.1% 0.2% 0.3% 0.4% 0.5% 0.6% 0.7% 0.8% 0.9%

Str

ess

(MP

a)

Strain

Fy=535 MPa

E=194198 MPa

Coupon G1

0

100

200

300

400

500

600

0% 2% 4% 6% 8% 10% 12%

Strain

Str

ess

(MP

a)

Appendix A. Stress-Strain Diagrams of Coupon tests 263

Coupon G2

0

100

200

300

400

500

600

0.0% 0.1% 0.2% 0.3% 0.4% 0.5% 0.6% 0.7% 0.8% 0.9%

Strain

Str

ess

(MP

a)

Fy=522 MPa

E=177338 MPa

Coupon G2

0

100

200

300

400

500

600

0% 2% 4% 6% 8% 10% 12%

Strain

Str

ess

(MP

a)


Coupon H1

0

100

200

300

400

500

600

0.0% 0.2% 0.4% 0.6% 0.8% 1.0% 1.2% 1.4%

Strain

Str

ess

(MP

a)Fy=541 MPa

E=176938 MPa

Coupon H2

0

100

200

300

400

500

600

0.0% 0.2% 0.4% 0.6% 0.8% 1.0% 1.2%

Strain

Str

ess

(MP

a)

E=187905 MPa

Fy=544 MPa


Coupon I1

0

100

200

300

400

500

600

0.0% 0.2% 0.4% 0.6% 0.8% 1.0% 1.2%

Strain

Str

ess

(MP

a)

E=196506 MPa

Fy=557 MPa

Coupon I1

0

100

200

300

400

500

600

700

0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0%

Strain

Str

ess

(MP

a)


Coupon I2

0

100

200

300

400

500

600

0.0% 0.1% 0.2% 0.3% 0.4% 0.5% 0.6% 0.7% 0.8% 0.9% 1.0%

Strain

Str

ess

(MP

a)Fy=525 MPa

E=191620 MPa

Coupon I2

0

100

200

300

400

500

600

0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0%

Strain

Str

ess

(MP

a)


Coupon J1

0

100

200

300

400

500

600

0.0% 0.1% 0.2% 0.3% 0.4% 0.5% 0.6% 0.7%

Strain

Str

ess

(MP

a)

E=198834 MPa

Fy=543 MPa

Coupon J1

0

100

200

300

400

500

600

0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0%

Strain

Str

ess

(MP

a)


Coupon J2

0

100

200

300

400

500

600

0.0% 0.1% 0.2% 0.3% 0.4% 0.5% 0.6% 0.7% 0.8% 0.9% 1.0%

Strain

Str

ess

(MP

a)

E=197997 MPa

Fy=561 MPa

Coupon J2

0

100

200

300

400

500

600

700

0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0%

Strain

Str

ess

(MP

a)

269

Appendix B

TESTED SECTIONS NORMALISED MOMENT-CURVATURE DIAGRAMS

Section 1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

k/kp

M/M

p

Test Result

M=My

Fully effective

k=ky

Section 2

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

k/kp

M/M

p

Test Result

M=My

Fully effective

k=ky

Appendix B. Tested Sections Normalised Moment-Curvature Diagrams 270

Section 3

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

k/kp

M/M

p

Test Result

M=My

Fully effective

k=ky

Section 4

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

k/kp

M/M

p

Test Result

M=My

Fully effective

k=ky

Section 5

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

M=My

Fully effective

k=ky


Section 6

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

M=My

Fully effective

k=ky

Section 7

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

M=My

Fully effective

k=ky

Section 8

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

M=My

Fully effective

k=ky


Section 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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

M=My

Fully effective

k=ky

Section 10

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

M=My

Fully effective

k=ky

Section 11

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

M=My

Fully effective

k=ky


Section 12

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

M=My

Fully effective

k=ky

Section 13

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

My/Mp

Fully effective

ky/kp

Section 14

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

M=My

Fully effective

k=ky


Section 15

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

M=My

Fully effective

k=ky

Section 16

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

M=My

Fully effective

k=ky

Section 17

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

M=My

Fully effective

k=ky


Section 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 5.0 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

M=My

Fully effective

k=ky

Section 19

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

M=My

Fully effective

k=ky

Section 20

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

M=My

Fully effective

k=ky


Section 21

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

M=My

Fully effective

k=ky

Section 22

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

M=My

Fully effective

k=ky

Section 23

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

M=My

Fully effective

k=ky


Section 24

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

M=My

Fully effective

k=ky

Section 25

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

M=My

Fully effective

k=ky

Section 26

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

M=My

Fully effective

k=ky


Section 27

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

M=My

Fully effective

k=ky

Section 28

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

M=My

Fully effective

k=ky

Section 29

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

M=My

Fully effective

k=ky


Section 30

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

M=My

Fully effective

k=ky

Section 31

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

M=My

Fully effective

k=ky

Section 32

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

M=My

Fully effective

k=ky


Section 33

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

M=My

Fully effective

k=ky

Section 34

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

M=My

Fully effective

k=ky

Section 35

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

M=My

Fully effective

k=ky


Section 36

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

M=My

Fully effective

k=ky

Section 37

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

M=My

Fully effective

k=ky

Section 38

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

M=My

Fully effective

k=ky


Section 39

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

M=My

Fully effective

k=ky

Section 40

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

M=My

Fully effective

k=ky

Section 41

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

M=My

Fully effective

k=ky


Section 42

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

M=My

Fully effective

k=ky

284

Appendix C

TESTED SECTIONS NORMALISED MOMENT-CURVATURE DIAGRAMS COMPARISON WITH THE NORTH AMERICAN AND AUSTRALIANS STANDARDS RESULTS

Section 1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

k/kp

M/M

p


Section 2

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

k/kp

M/M

p


Appendix C. Tested Sections Normalised Moment-Curvature Diagrams Comparison With the North 285 American and Australian Standards Results

Section 3

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p


Section 4

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p


Section 5

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p



Section 6

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M


Section 7

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p


Section 8

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p



Section 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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p


Section 10

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p


Section 11

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p



Section 12

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M


Section 13

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test ResultMsx/MpMbdistortional/MpMy/MpFully effectiveky/kp

Section 14

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p



Section 15

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p


Section 16

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p


Section 17

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p



Section 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 5.0 6.0 7.0 8.0 9.0 10.0

k/kp

M/M


Section 19

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p


Section 20

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p



Section 21

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p


Section 22

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p


Section 23

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p



Section 24

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M


Section 25

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p


Section 26

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p



Section 27

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p


Section 28

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p


Section 29

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p



Section 30

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M


Section 31

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p


Section 32

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p



Section 33

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p


Section 34

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p


Section 35

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p



Section 36

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M


Section 37

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p


Section 38

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p



Section 39

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p


Section 40

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p


Section 41

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p



Section 42

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M


299

Appendix D

TESTED SECTIONS NORMALISED MOMENT-STRAIN DIAGRAMS

Section1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

ε/εy

M/M

y

εc=3613.7, Cy=εc/εy=1.3

Section2

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

ε/εy

M/M

y

εc=3338.1, Cy=εc/εy=1.2

Appendix D. Tested Sections Normalised Moment-Strain Diagrams 300

Section3

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 6.0 7.0 8.0 9.0 10.0

ε/εy

M/M

yεc=4652.4, Cy=εc/εy=1.63

Section4

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 6.0 7.0 8.0 9.0 10.0

ε/εy

M/M

y

εc=3494.5, Cy=εc/εy=1.58

Section5

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 6.0 7.0 8.0 9.0 10.0

ε/εy

M/M

y

εc=5050.3, Cy=εc/εy=1.87


Section6

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 6.0 7.0 8.0 9.0 10.0

ε/εy

M/M

y

εc=3443.3, Cy=εc/εy=1.21

Section7

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 6.0 7.0 8.0 9.0 10.0

ε/εy

M/M

y

εc=3944.56, Cy=εc/εy=1.39

Section8

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 6.0 7.0 8.0 9.0 10.0

ε/εy

M/M

y

εc=3612.5, Cy=εc/εy=1.27


Section9

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 6.0 7.0 8.0 9.0 10.0

ε/εy

M/M

yεc=4461.8, Cy=εc/εy=1.60

Section10

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 6.0 7.0 8.0 9.0 10.0

ε/εy

M/M

y

εc=4897.3, Cy=εc/εy=1.76

Section11

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 6.0 7.0 8.0 9.0 10.0

ε/εy

M/M

y

εc=4426.7, Cy=εc/εy=1.59


Section12

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 6.0 7.0 8.0 9.0 10.0

ε/εy

M/M

y

εc=4718.0, Cy=εc/εy=1.69

Section13

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 6.0 7.0 8.0 9.0 10.0

ε/εy

M/M

y

εc=4297.7, Cy=εc/εy=1.54

Section14

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 6.0 7.0 8.0 9.0 10.0

ε/εy

M/M

y

εc=5228.4, Cy=εc/εy=1.76


Section15

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 6.0 7.0 8.0 9.0 10.0

ε/εy

M/M

yεc=3920.4, Cy=εc/εy=1.35

Section16

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 6.0 7.0 8.0 9.0 10.0

ε/εy

M/M

y

εc=3223, Cy=εc/εy=1.08

Section17

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 6.0 7.0 8.0 9.0 10.0

ε/εy

M/M

y

εc=3272, Cy=εc/εy=1.10


Section18

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 6.0 7.0 8.0 9.0 10.0

ε/εy

M/M

y

εc=4752.3, Cy=εc/εy=1.7

Section19

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 6.0 7.0 8.0 9.0 10.0

ε/εy

M/M

y

εc=3297, Cy=εc/εy=1.11

Section20

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 6.0 7.0 8.0 9.0 10.0

ε/εy

M/M

y

εc=4028.3, Cy=εc/εy=1.35


Section21

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 6.0 7.0 8.0 9.0 10.0

ε/εy

M/M

y

εc=3177.2, Cy=εc/εy=1.14

Section22

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 6.0 7.0 8.0 9.0 10.0

ε/εy

M/M

y

εc=4240.0, Cy=εc/εy=1.49

Section23

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 6.0 7.0 8.0 9.0 10.0

ε/εy

M/M

y

εc=4030, Cy=εc/εy=1.42


Section24

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 6.0 7.0 8.0 9.0 10.0

ε/εy

M/M

y

εc=3186.6, Cy=εc/εy=1.12

Section25

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 6.0 7.0 8.0 9.0 10.0

ε/εy

M/M

y

εc=4899.2, Cy=εc/εy=1.72

Section26

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 6.0 7.0 8.0 9.0 10.0

ε/εy

M/M

y

εc=5586.4, Cy=εc/εy=1.96


Section27

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 6.0 7.0 8.0 9.0 10.0

ε/εy

M/M

yεc=4580.5, Cy=εc/εy=1.61

Section28

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 6.0 7.0 8.0 9.0 10.0

ε/εy

M/M

y

εc=4005.2, Cy=εc/εy=1.41

Section29

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 6.0 7.0 8.0 9.0 10.0

ε/εy

M/M

y

εc=5477, Cy=εc/εy=1.92


Section30

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 6.0 7.0 8.0 9.0 10.0

ε/εy

M/M

y

εc=5189.8, Cy=εc/εy=1.82

Section31

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 6.0 7.0 8.0 9.0 10.0

ε/εy

M/M

y

εc=6390.3, Cy=εc/εy=2.25

Section32

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 6.0 7.0 8.0 9.0 10.0

ε/εy

M/M

y

εc=6734.8, Cy=εc/εy=2.5


Section33

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 6.0 7.0 8.0 9.0 10.0

ε/εy

M/M

yεc=5086.4, Cy=εc/εy=1.79

Section34

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 6.0 7.0 8.0 9.0 10.0

ε/εy

M/M

y

εc=5378.5, Cy=εc/εy=1.89

Section35

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 6.0 7.0 8.0 9.0 10.0

ε/εy

M/M

y

εc=7292.2, Cy=εc/εy=2.56


Section36

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 6.0 7.0 8.0 9.0 10.0

ε/εy

M/M

y

εc=6506.8, Cy=εc/εy=2.3

Section37

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

ε/εy

M/M

y

εc=6686.7, Cy=εc/εy=2.35

Section38

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

ε/εy

M/M

y

εc=5916.4, Cy=εc/εy=2.1


Section39

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

ε/εy

M/M

y

εc=4641.7, Cy=εc/εy=1.63

Section40

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

ε/εy

M/M

y

εc=6840.8, Cy=εc/εy=2.4

Section41

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

ε/εy

M/M

y

εc=4873.4, Cy=εc/εy=1.7


Section42

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

ε/εy

M/M

y

εc=4291, Cy=εc/εy=1.51

314

Appendix E

COMPARISON OF THE NORMALISED MOMENT-CURVATURE GRAPHS OF THE TESTED SECTIONS, WITH EDGE STIFFENER, BASED ON YLM ANALYSIS, FEM AND THE TEST RESULTS

Section3

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

FEM

elastic

YLM

Section4

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 6.0 7.0 8.0 9.0 10.0k/kp

M/M

p

Test Result

FEM

elastic

YLM

Appendix E. Comparison of the Normalised Moment-Curvature Graphs of the Tested 315 Sections, with Edge Stiffener, Based on YLM Analysis, FEM and the Test Results

Section5

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

FEM

elastic

YLM

Section6

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 6.0 7.0 8.0 9.0 10.0k/kp

M/M

p

Test Result

FEM

elastic

YLM

Section7

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 6.0 7.0 8.0 9.0 10.0k/kp

M/M

p

Test Result

FEM

elastic

YLM


Section8

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 6.0 7.0 8.0 9.0 10.0k/kp

M/M

pTest Result

FEM

elastic

YLM

Section9

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 6.0 7.0 8.0 9.0 10.0k/kp

M/M

p

Test Result

FEM

elastic

YLM

Section10

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 6.0 7.0 8.0 9.0 10.0k/kp

M/M

p

Test Result

FEM

elastic

YLM


Section11

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 6.0 7.0 8.0 9.0 10.0k/kp

M/M

p

Test Result

FEM

elastic

YLM

Section12

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 6.0 7.0 8.0 9.0 10.0k/kp

M/M

p

Test Result

FEM

elastic

YLM

Section13

0.0

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0.6

0.8

1.0

1.2

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0k/kp

M/M

p

Test Result

FEM

elastic

YLM


Section14

0.0

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0.4

0.6

0.8

1.0

1.2

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0k/kp

M/M

pTest Result

FEM

elastic

YLM

Section15

0.0

0.2

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0.6

0.8

1.0

1.2

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0k/kp

M/M

p

Test Result

FEM

elastic

YLM

Section16

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 6.0 7.0 8.0 9.0 10.0k/kp

M/M

p

Test Result

FEM

elastic

YLM


Section17

0.0

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0.6

0.8

1.0

1.2

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0k/kp

M/M

p

Test Result

FEM

elastic

YLM

Section18

0.0

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0.6

0.8

1.0

1.2

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0k/kp

M/M

p

Test Result

FEM

elastic

YLM

Section19

0.0

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1.0

1.2

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0k/kp

M/M

p

Test Result

FEM

elastic

YLM


Section20

0.0

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0.8

1.0

1.2

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0k/kp

M/M

pTest Result

FEM

elastic

YLM

Section21

0.0

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1.0

1.2

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

FEM

elastic

YLM

Section22

0.0

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1.0

1.2

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0k/kp

M/M

p

Test Result

FEM

elastic

YLM


Section23

0.0

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1.0

1.2

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0k/kp

M/M

p

Test Result

FEM

elastic

YLM

Section24

0.0

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1.0

1.2

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0k/kp

M/M

p

Test Result

FEM

elastic

YLM

Section25

0.0

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1.0

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0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0k/kp

M/M

p

Test Result

FEM

elastic

YLM

MAS4100/Mp


Section26

0.0

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1.0

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0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0k/kp

M/M

pTest Result

FEM

elastic

YLM

Section27

0.0

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0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0k/kp

M/M

p

Test Result

FEM

elastic

YLM

Section28

0.0

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0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0k/kp

M/M

p

Test Result

FEM

elastic

YLM

MAS4100/Mp


Section29

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M/M

p

Test Result

FEM

elastic

YLM

MAS4100/Mp

Section30

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0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0k/kp

M/M

p

Test Result

FEM

elastic

YLM

Section31

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k/kp

M/M

p

Test Result

FEM

elastic

YLM

MAS4100/Mp


Section32

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k/kp

M/M

p

Test Result

FEM

elastic

YLM

MAS4100/Mp

Section33

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M/M

p

Test Result

FEM

elastic

YLM

MAS4100/Mp

Section34

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M/M

p

Test Result

FEM

elastic

YLM

MAS4100/Mp


Section35

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0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0k/kp

M/M

p

Test Result

FEM

elastic

YLM

MAS4100/Mp

Section36

0.0

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0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0k/kp

M/M

p

Test Result

FEM

elastic

YLM

MAS4100/Mp

Section37

0.0

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0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0k/kp

M/M

p

Test Result

FEM

elastic

YLM

MAS4100/Mp


Section38

0.0

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0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0k/kp

M/M

p

Test Result

FEM

elastic

YLM

MAS4100/Mp

Section39

0.0

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1.0

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0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0k/kp

M/M

p

Test Result

FEM

elastic

YLM

MAS4100/Mp

Section40

0.0

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1.4

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0k/kp

M/M

p

Test Result

FEM

elastic

YLM

MAS4100/Mp


Section41

0.0

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1.0

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0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0k/kp

M/M

p

Test Result

FEM

elastic

YLM

MAS4100/Mp

Section42

0.0

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1.0

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0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0k/kp

M/M

p

Test Result

FEM

elastic

YLM

MAS4100/Mp

328

Appendix F

DIVIDED TEST AND YLM GRAPHS ACCORDING TO THE SIMPSON RULES

Section 3

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Test

YLM

Section 4

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Test

YLM

Appendix F. Divided Test and YLM Graphs According to the Simpson Rules 329

Section 5

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Test

YLM

Section 6

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YLM

Section 7

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Test

YLM


Section 8

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YLM

Section 9

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Section 10

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Test

YLM


Section 11

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Section 12

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Section 13

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Section 14

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Section 15

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Section 16

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YLM


Section 17

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Section 18

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Section 19

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YLM


Section 20

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Section 21

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YLM

Section 22

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YLM


Section 23

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YLM

Section 24

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Section 25

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Section 26

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Section 27

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Section 28

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Section 29

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Section 30

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Section 31

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Section 32

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Section 33

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Section 34

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Section 35

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Section 36

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Section 37

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Section 38

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Section 39

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Section 40

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Section 41

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Section 42

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342

Appendix G

COMPARISON BETWEEN SIMPLIFIED MODEL AND TEST RESULTS

Section3

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k/kp

M/M

p

Test Result

Linear part

Curve part

M/Mp = 1.03(k/kp)-0.665

Section4

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k/kp

M/M

p

Test Result

Linear part

Curve part

M/Mp = 0.79(k/kp)-0.665

Appendix G. Comparison between Simplified Model and Test Results 343

Section5

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M/M

p

Test Result

Linear part

Curve part

M/Mp = 1.09(k/kp)-0.665

Section6

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k/kp

M/M

p

Test Result

Linear part

Curve part

M/Mp = 0.94(k/kp)-0.665

Section7

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k/kp

M/M

p

Test Result

Linear part

Curve part

M/Mp = 0.89(k/kp)-0.665


Section8

M/Mp = 0.98(k/kp)-0.665

0.0

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1.0

1.2

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

pTest Result

Linear part

Curve part

Section9

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

Linear part

Curve part

M/Mp = 0.85(k/kp)-0.665

Section10

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

Linear part

Curve part

M/Mp = 0.94(k/kp)-0.665


Section11

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

Linear part

Curve part

M/Mp = 0.80(k/kp)-0.665

Section12

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

Linear part

Curve part

M/Mp = 0.91(k/kp)-0.665

Section13

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

Linear part

Curve part

M/Mp = 0.84(k/kp)-0.665


Section14

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

pTest Result

Linear part

Curve part

M/Mp = 0.98(k/kp)-0.665

Section15

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

Linear part

Curve part

M/Mp = 0.88(k/kp)-0.665

Section16

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

Linear part

Curve part

M/Mp = 0.80(k/kp)-0.665


Section17

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

Linear part

Curve part

M/Mp = 0.78(k/kp)-0.665

Section18

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

Linear part

Curve part

M/Mp = 0.80(k/kp)-0.665

Section19

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

Linear part

Curve part

M/Mp = 0.86(k/kp)-0.665


Section20

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

pTest Result

Linear part

Curve part

M/Mp = 0.86(k/kp)-0.665

Section21

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

Linear part

Curve part

M/Mp = 0.77(k/kp)-0.665

Section22

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

Linear part

Curve part

M/Mp = 1.08(k/kp)-0.665


Section23

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

Linear part

Curve part

M/Mp = 0.99(k/kp)-0.665

Section24

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

Linear part

Curve part

M/Mp = 0.91(k/kp)-0.665

Section25

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

MAS4100/Mp

Linear part

Curve part

M/Mp = 1.23(k/kp)-0.665


Section26

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

pTest Result

Linear part

Curve part

M/Mp = 1.08(k/kp)-0.665

Section27

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

Linear part

Curve part

M/Mp = 1.00(k/kp)-0.665

Section28

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

MAS4100/Mp

Linear part

Curve part

M/Mp = 1.37(k/kp)-0.665


Section29

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

MAS4100/Mp

Linear part

Curve part

M/Mp = 1.23(k/kp)-0.665

Section30

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

Linear part

Curve part

M/Mp = 1.08(k/kp)-0.665

Section31

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

MAS4100/Mp

Linear part

Curve part

M/Mp = 1.6(k/kp)-0.665


Section32

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

pTest Result

MAS4100/Mp

Linear part

Curve part

M/Mp = 1.4(k/kp)-0.665

Section33

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

MAS4100/Mp

Linear part

Curve part

M/Mp = 1.23(k/kp)-0.665

Section34

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

MAS4100/Mp

Linear part

Curve part

M/Mp = 2.01(k/kp)-0.665


Section35

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

MAS4100/Mp

Linear part

Curve part M/Mp = 1.60(k/kp)-0.665

Section36

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

MAS4100/Mp

Linear part

Curve part

M/Mp = 1.32(k/kp)-0.665

Section37

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

MAS4100/Mp

Linear part

Curve part

M/Mp = 2.25(k/kp)-0.665


Section38

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

pTest Result

MAS4100/Mp

Linear part


Section39

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

MAS4100/Mp

Linear part


Section40

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 6.0 7.0 8.0 9.0 10.0 11.0

k/kp

M/M

p

Test Result

MAS4100/Mp

Linear part

Curve part

M/Mp = 2.89(k/kp)-0.665


Section41

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

MAS4100/Mp

Linear part

Curve part

M/Mp = 2.78(k/kp)-0.665

Section42

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 6.0 7.0 8.0 9.0 10.0

k/kp

M/M

p

Test Result

MAS4100/Mp

Linear part

Curve part

M/Mp = 2.04(k/kp)-0.665

356

Appendix H

FAILED SHAPE OF THE TESTED SECTIONS

Appendix H. Failed Shape of the Tested Sections 357

inelastic behaviour of cold-formed channel sections in bending · ultimate capacity of cold-formed...

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