Post on 19-Mar-2020
transcript
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.
Chapter1. Introduction 3
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))
Chapter1. Introduction 4
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
Chapter1. Introduction 5
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.
Chapter1. Introduction 6
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
Chapter1. Introduction 7
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.
Chapter 2. Literature Review 10
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.
Chapter 2. Literature Review 11
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.
Chapter 2. Literature Review 12
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
Chapter 2. Literature Review 13
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
Chapter 2. Literature Review 14
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)
Chapter 2. Literature Review 15
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))
Chapter 2. Literature Review 16
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
Chapter 2. Literature Review 17
(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))
Chapter 2. Literature Review 18
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
Chapter 2. Literature Review 19
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.
Chapter 2. Literature Review 20
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)
Chapter 2. Literature Review 21
According to EUROCODE3 (2006):
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)
Chapter 2. Literature Review 22
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))
According to EUROCODE3 (2006):
For :748.0p 2
188.0
p
p
(2.38)
For :748.0p 1 (2.39)
Chapter 2. Literature Review 23
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))
Chapter 2. Literature Review 24
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.
Chapter 2. Literature Review 25
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.
Chapter 2. Literature Review 26
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)
Chapter 2. Literature Review 27
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.
Chapter 2. Literature Review 28
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
Chapter 2. Literature Review 29
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
Chapter 2. Literature Review 30
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)
Chapter 2. Literature Review 31
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.
Chapter 2. Literature Review 32
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))
Chapter 2. Literature Review 33
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
Chapter 2. Literature Review 34
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:
Chapter 2. Literature Review 35
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.
Chapter 2. Literature Review 36
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.
Chapter 2. Literature Review 37
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.
Chapter 2. Literature Review 38
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.
Chapter 2. Literature Review 39
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)).
Chapter 2. Literature Review 40
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
Chapter 2. Literature Review 41
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))
Chapter 2. Literature Review 42
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)
Chapter 2. Literature Review 43
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:
Chapter 2. Literature Review 44
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:
Chapter 2. Literature Review 45
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
Chapter 2. Literature Review 46
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)).
Chapter 2. Literature Review 47
(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.
Chapter 2. Literature Review 48
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.
Chapter 2. Literature Review 49
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
Frame 2nominal 74.0 1.85 1.05
measured 72.8 1.82 1.05experimental 71.5 1.87
Frame 3nominal 50.8 15.40 1.09
measured 50.0 15.10 1.09experimental 45.7 13.80
Load at ultimate (kN)
Chapter 2. Literature Review 50
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).
Chapter 2. Literature Review 51
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).
Chapter 2. Literature Review 52
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.
Chapter 2. Literature Review 53
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.
Chapter 2. Literature Review 54
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).
Chapter 2. Literature Review 55
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.
Chapter 2. Literature Review 56
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
Chapter 2. Literature Review 57
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
3.0 Chapter Synopsis
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
Chapter 3. Test Procedure of Cold-formed Channel Sections under Pure Bending 60
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.
Chapter 3. Test Procedure of Cold-formed Channel Sections under Pure Bending 61
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.
Chapter 3. Test Procedure of Cold-formed Channel Sections under Pure Bending 62
(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
Chapter 3. Test Procedure of Cold-formed Channel Sections under Pure Bending 65
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
Chapter 3. Test Procedure of Cold-formed Channel Sections under Pure Bending 66
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
Chapter 3. Test Procedure of Cold-formed Channel Sections under Pure Bending 67
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
Chapter 3. Test Procedure of Cold-formed Channel Sections under Pure Bending 68
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
Chapter 3. Test Procedure of Cold-formed Channel Sections under Pure Bending 69
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.
Chapter 3. Test Procedure of Cold-formed Channel Sections under Pure Bending 70
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.
Chapter 3. Test Procedure of Cold-formed Channel Sections under Pure Bending 71
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)
Chapter 3. Test Procedure of Cold-formed Channel Sections under Pure Bending 72
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’
Chapter 3. Test Procedure of Cold-formed Channel Sections under Pure Bending 73
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.
Chapter 3. Test Procedure of Cold-formed Channel Sections under Pure Bending 74
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)
Chapter 3. Test Procedure of Cold-formed Channel Sections under Pure Bending 75
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.
Chapter 3. Test Procedure of Cold-formed Channel Sections under Pure Bending 76
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
Chapter 3. Test Procedure of Cold-formed Channel Sections under Pure Bending 77
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.
Chapter 3. Test Procedure of Cold-formed Channel Sections under Pure Bending 78
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.
Chapter 3. Test Procedure of Cold-formed Channel Sections under Pure Bending 79
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
Chapter 3. Test Procedure of Cold-formed Channel Sections under Pure Bending 80
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
4.0 Chapter Synopsis
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.
Chapter 4. Experimental Results and Discussions 83
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.
Chapter 4. Experimental Results and Discussions 84
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
Legend: NC:Non-Compact
C:Compact
Chapter 4. Experimental Results and Discussions 87
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
Chapter 4. Experimental Results and Discussions 88
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-
Chapter 4. Experimental Results and Discussions 89
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.
Chapter 4. Experimental Results and Discussions 90
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
Beginning of stage 2
Beginning of stage 3
Beginning of stage 4
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).
Chapter 4. Experimental Results and Discussions 91
(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)
Chapter 4. Experimental Results and Discussions 92
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
Chapter 4. Experimental Results and Discussions 93
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.
Chapter 4. Experimental Results and Discussions 94
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
Figure 4.9: Normalised moment-curvature diagram with plastic moment and plastic curvature respectively for section 38
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.
Chapter 4. Experimental Results and Discussions 95
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: Normalised moment-curvature diagram with plastic moment and plastic curvature respectively for section 39
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
Chapter 4. Experimental Results and Discussions 96
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
Chapter 4. Experimental Results and Discussions 97
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
Figure 4.12: Normalised moment-curvature diagram with plastic moment and plastic curvature respectively for section 40
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
Chapter 4. Experimental Results and Discussions 98
( 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)
Chapter 4. Experimental Results and Discussions 99
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.
Chapter 4. Experimental Results and Discussions 100
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.
Chapter 4. Experimental Results and Discussions 101
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.
Chapter 4. Experimental Results and Discussions 102
(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
Chapter 4. Experimental Results and Discussions 103
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
Figure 4.17: The failure modes of section 24
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.
Chapter 4. Experimental Results and Discussions 104
(a) Front view
(b) Top view
Figure 4.18: The failure modes of section 4
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.
Chapter 4. Experimental Results and Discussions 105
Figure 4.19: The failure modes of section 37
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.
Chapter 4. Experimental Results and Discussions 106
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
Chapter 4. Experimental Results and Discussions 107
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
Chapter 4. Experimental Results and Discussions 108
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
Chapter 4. Experimental Results and Discussions 109
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
Chapter 4. Experimental Results and Discussions 110
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
Chapter 4. Experimental Results and Discussions 111
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
Chapter 4. Experimental Results and Discussions 112
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
.
Chapter 4. Experimental Results and Discussions 113
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
Chapter 4. Experimental Results and Discussions 114
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
Test ResultMsx/MpMbdistortional/MpM=MyFully effectivek=ky
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
Test ResultMsx/MpMbdistortional/MpM=MyFully effectivek=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 ResultMsx/MpMbdistortional/MpM=MyFully effectivek=ky
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
pTest ResultMsx/MpMbdistortional/MpM=MyFully effectivek=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 ResultMsx/MpMbdistortional/MpM=MyFully effectivek=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 ResultMsx/MpMbdistortional/MpM=MyFully effectivek=ky
Figure4.21: Comparison between test results and EWM and distortional buckling check results graph for selected tested sections (continued)
Chapter 4. Experimental Results and Discussions 117
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
Chapter 4. Experimental Results and Discussions 118
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
Chapter 4. Experimental Results and Discussions 119
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.
Chapter 4. Experimental Results and Discussions 120
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
Chapter 4. Experimental Results and Discussions 121
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:
Assume web is fully effective:
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
Chapter 4. Experimental Results and Discussions 122
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
Chapter 4. Experimental Results and Discussions 123
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
Chapter 4. Experimental Results and Discussions 124
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
3. 126400 mmZZk fullxc
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
Chapter 4. Experimental Results and Discussions 125
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
.
mkNfZM ccnalbdistortio 008.6
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.
Chapter 4. Experimental Results and Discussions 126
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
Chapter 4. Experimental Results and Discussions 127
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.
Chapter 4. Experimental Results and Discussions 128
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
Mean(Pm)= 1.19Cov(VP)= 0.09
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
Chapter 4. Experimental Results and Discussions 129
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
Chapter 4. Experimental Results and Discussions 130
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:
Assume web is fully effective:
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
Chapter 4. Experimental Results and Discussions 131
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
Chapter 4. Experimental Results and Discussions 132
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.
Chapter 4. Experimental Results and Discussions 133
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.
Chapter 4. Experimental Results and Discussions 134
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.
Chapter 4. Experimental Results and Discussions 135
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
Chapter 4. Experimental Results and Discussions 136
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
Chapter 4. Experimental Results and Discussions 137
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
Mean(Pm)= 1.19Cov(VP)= 0.03
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
Chapter 4. Experimental Results and Discussions 138
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
Chapter 4. Experimental Results and Discussions 139
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
Chapter 4. Experimental Results and Discussions 140
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
5.0 Chapter Synopsis
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
Chapter 5. Revising Existing Design Rules and Slenderness Limits 143
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)).
Chapter 5. Revising Existing Design Rules and Slenderness Limits 144
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)
Chapter 5. Revising Existing Design Rules and Slenderness Limits 145
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.
Chapter 5. Revising Existing Design Rules and Slenderness Limits 146
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).
Chapter 5. Revising Existing Design Rules and Slenderness Limits 147
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.
Chapter 5. Revising Existing Design Rules and Slenderness Limits 148
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
Chapter 5. Revising Existing Design Rules and Slenderness Limits 149
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.
Chapter 5. Revising Existing Design Rules and Slenderness Limits 150
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)
Chapter 5. Revising Existing Design Rules and Slenderness Limits 151
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.
Chapter 5. Revising Existing Design Rules and Slenderness Limits 152
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
Chapter 5. Revising Existing Design Rules and Slenderness Limits 153
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:
Assume web is fully effective:
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
Chapter 5. Revising Existing Design Rules and Slenderness Limits 154
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
Chapter 5. Revising Existing Design Rules and Slenderness Limits 155
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.
Chapter 5. Revising Existing Design Rules and Slenderness Limits 156
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
Chapter 5. Revising Existing Design Rules and Slenderness Limits 157
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.
Chapter 5. Revising Existing Design Rules and Slenderness Limits 158
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 ).
Chapter 5. Revising Existing Design Rules and Slenderness Limits 159
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.
Chapter 5. Revising Existing Design Rules and Slenderness Limits 160
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
3. 15720 mmZZk fullxc
Chapter 5. Revising Existing Design Rules and Slenderness Limits 161
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
.
mkNfZM ccnalbdistortio 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
.
mkNfZM ccnalbdistortio 021.1
Chapter 5. Revising Existing Design Rules and Slenderness Limits 162
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)
Chapter 5. Revising Existing Design Rules and Slenderness Limits 163
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
Chapter 5. Revising Existing Design Rules and Slenderness Limits 164
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.
Chapter 5. Revising Existing Design Rules and Slenderness Limits 165
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
Chapter 5. Revising Existing Design Rules and Slenderness Limits 166
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)
Chapter 5. Revising Existing Design Rules and Slenderness Limits 167
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
Chapter 5. Revising Existing Design Rules and Slenderness Limits 168
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
Mean(Pm)= 1.13 1.09 1.19Cov(VP)= 0.08 0.08 0.09Reliability Index(β)= 2.76 2.62 2.91
φ=0.9, γL=1.5, γD=1.2
Chapter 5. Revising Existing Design Rules and Slenderness Limits 169
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.
Chapter 5. Revising Existing Design Rules and Slenderness Limits 170
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
Mean(Pm)= 1.14 1.09 1.19Cov(VP)= 0.07 0.08 0.09Reliability Index(β)= 2.83 2.62 2.91
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
Chapter 5. Revising Existing Design Rules and Slenderness Limits 171
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.
The following computations show how to calculate the ultimate member moment
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 .
Distortional buckling:
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:
The theoretical local buckling stress is determined by using thin wall program:
776.035.0461.0
85.0,002.4,2546 .
lol
bel
ybeolfullxolol
M
M
mkNMMfZMMPaf
Chapter 5. Revising Existing Design Rules and Slenderness Limits 172
:35.0776.0 l mkNMM
MMM be
be
olbebl
98.035.047.02.15.0
Distortional buckling:
53.0461.0
85.0,002.4,2546 .
dod
yd
yodfullxodod
M
M
mkNMmkNfZMMPaf
:53.0d mkNMM ybd 02.12.1
mkNMMMinM bdblDSM 98.0),(
Proposed Method 2 (DSM)
Local buckling:
The theoretical local buckling stress is determined by using thin wall program:
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
Distortional buckling:
53.0461.0
85.0,002.4,2546 .
dod
yd
yodfullxodod
M
M
mkNMmkNfZMMPaf
For :53.0d mkNMM ybd 02.12.1
mkNMMMinM bdblDSM 02.1),(
Chapter 5. Revising Existing Design Rules and Slenderness Limits 173
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
M<MyMy<M<MpM>MpM>Mp, R>3
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
M<MyMy<M<MpM>MpM>Mp, R>3
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
Chapter 5. Revising Existing Design Rules and Slenderness Limits 174
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
Chapter 5. Revising Existing Design Rules and Slenderness Limits 175
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
Legend: NC:Non-Compact
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
Chapter 5. Revising Existing Design Rules and Slenderness Limits 178
The following computations show how to calculate the ultimate member moment
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):
28.0,82,115 webwebwebey
eepey
33.0,30,40 flangeflangeflangeey
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:
29.0,22,110 webwebwebey
eepey
37.0,8,35 flangeflangeflangeey
eepey
Chapter 5. Revising Existing Design Rules and Slenderness Limits 179
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
Chapter 5. Revising Existing Design Rules and Slenderness Limits 180
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
Chapter 5. Revising Existing Design Rules and Slenderness Limits 181
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
6.0 Chapter Synopsis
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.
Chapter 6. Yield Line Mechanism (YLM) Analysis of Cold-formed Channel Sections under Bending 184
(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.
Chapter 6. Yield Line Mechanism (YLM) Analysis of Cold-formed Channel Sections under Bending 185
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
Chapter 6. Yield Line Mechanism (YLM) Analysis of Cold-formed Channel Sections under Bending 186
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.
Chapter 6. Yield Line Mechanism (YLM) Analysis of Cold-formed Channel Sections under Bending 187
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:
Chapter 6. Yield Line Mechanism (YLM) Analysis of Cold-formed Channel Sections under Bending 188
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:
Chapter 6. Yield Line Mechanism (YLM) Analysis of Cold-formed Channel Sections under Bending 189
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)
Chapter 6. Yield Line Mechanism (YLM) Analysis of Cold-formed Channel Sections under Bending 190
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:
Chapter 6. Yield Line Mechanism (YLM) Analysis of Cold-formed Channel Sections under Bending 191
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.
Chapter 6. Yield Line Mechanism (YLM) Analysis of Cold-formed Channel Sections under Bending 192
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.
Chapter 6. Yield Line Mechanism (YLM) Analysis of Cold-formed Channel Sections under Bending 193
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.
Chapter 6. Yield Line Mechanism (YLM) Analysis of Cold-formed Channel Sections under Bending 194
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
Chapter 6. Yield Line Mechanism (YLM) Analysis of Cold-formed channel Sections under Bending 197
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:
Chapter 6. Yield Line Mechanism (YLM) Analysis of Cold-formed channel Sections under Bending 198
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
yyABSTan 27.12.3 22
0
16
0.1
1
1
2
B
BB
BDDB
z
yy
Tan
yyxx
mmzzyyxxl BBBBBBBB 0.522
1
2
1 11
mmSinSinll BBBB 77.427.15211
Rad
BB
ArcTanl
uArcTan 37.0
77.4
83.1
1
22
mmablDI 91.18)( 22
23
mmublGA 34.9111
29.170
07.0 11
ar
mmublIB 1.1623
42.070
07.0 22
ar
32
1034.04
xtF
m yp
0148.0cos
)( 2121
bm
WWW pCDEF
0169.0))(( 212122 uubmWW pAB
0195.02
2 1113
amWWW pCAEA
Chapter 6. Yield Line Mechanism (YLM) Analysis of Cold-formed channel Sections under Bending 199
0127.02
2 2114
amWWW pDBFB
0321.02 15 CGpGEGC lmWWW
0046.02 26 IDpIFID lmWWW
0607.03 1
17
r
lumWW GAp
GA
0079.03 2
28
r
lumWW IBp
IB
0369.02
22
1
11
19 1
r
mau
r
mAWWW pp
ACAAEAC
0088.02
22
2
22
210 1
r
mau
r
mAWWW pp
BDBBFBD
0007.02 211 bmWW pGH
0046.0612 WWWWWW IFIDHLHK
0098.013
IB
HJIBHJ l
lWWW
0001.02 414 bmWW pIM
0045.022 242415 bmbmWWWW ppLPJOKN
2346.015
1
iW
mkNW
Mi
87.52
15
1
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
Chapter 6. Yield Line Mechanism (YLM) Analysis of Cold-formed channel Sections under Bending 200
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.
Chapter 6. Yield Line Mechanism (YLM) Analysis of Cold-formed channel Sections under Bending 201
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
Chapter 6. Yield Line Mechanism (YLM) Analysis of Cold-formed channel Sections under Bending 202
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
Chapter 6. Yield Line Mechanism (YLM) Analysis of Cold-formed channel Sections under Bending 203
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.
Chapter 6. Yield Line Mechanism (YLM) Analysis of Cold-formed channel Sections under Bending 204
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
Chapter 6. Yield Line Mechanism (YLM) Analysis of Cold-formed channel Sections under Bending 205
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
Chapter 6. Yield Line Mechanism (YLM) Analysis of Cold-formed channel Sections under Bending 206
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.
Chapter 6. Yield Line Mechanism (YLM) Analysis of Cold-formed channel Sections under Bending 207
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
Chapter 6. Yield Line Mechanism (YLM) Analysis of Cold-formed channel Sections under Bending 208
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).
Chapter 6. Yield Line Mechanism (YLM) Analysis of Cold-formed channel Sections under Bending 209
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:
Chapter 6. Yield Line Mechanism (YLM) Analysis of Cold-formed channel Sections under Bending 210
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.
Chapter 6. Yield Line Mechanism (YLM) Analysis of Cold-formed channel Sections under Bending 211
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.
Chapter 6. Yield Line Mechanism (YLM) Analysis of Cold-formed channel Sections under Bending 212
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.
Chapter 6. Yield Line Mechanism (YLM) Analysis of Cold-formed channel Sections under Bending 213
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.
Chapter 6. Yield Line Mechanism (YLM) Analysis of Cold-formed channel Sections under Bending 214
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.
Chapter 6. Yield Line Mechanism (YLM) Analysis of Cold-formed channel Sections under Bending 215
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).
Chapter 6. Yield Line Mechanism (YLM) Analysis of Cold-formed channel Sections under Bending 216
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.
Chapter 6. Yield Line Mechanism (YLM) Analysis of Cold-formed channel Sections under Bending 217
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
Chapter 6. Yield Line Mechanism (YLM) Analysis of Cold-formed channel Sections under Bending 218
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.
Chapter 6. Yield Line Mechanism (YLM) Analysis of Cold-formed channel Sections under Bending 219
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
Chapter 6. Yield Line Mechanism (YLM) Analysis of Cold-formed channel Sections under Bending 220
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
Chapter 6. Yield Line Mechanism (YLM) Analysis of Cold-formed channel Sections under Bending 221
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
7.0 Chapter Synopsis
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).
Chapter 7. Finite Element Method (FEM) Analysis of Cold-formed Channel Sections under Bending 224
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.
Chapter 7. Finite Element Method (FEM) Analysis of Cold-formed Channel Sections under Bending 227
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.
Chapter 7. Finite Element Method (FEM) Analysis of Cold-formed Channel Sections under Bending 228
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
Chapter 7. Finite Element Method (FEM) Analysis of Cold-formed Channel Sections under Bending 229
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
Chapter 7. Finite Element Method (FEM) Analysis of Cold-formed Channel Sections under Bending 230
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.
Chapter 7. Finite Element Method (FEM) Analysis of Cold-formed Channel Sections under Bending 231
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.
Chapter 7. Finite Element Method (FEM) Analysis of Cold-formed Channel Sections under Bending 232
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
Chapter 7. Finite Element Method (FEM) Analysis of Cold-formed Channel Sections under Bending 233
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.
Chapter 7. Finite Element Method (FEM) Analysis of Cold-formed Channel Sections under Bending 234
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
Chapter 7. Finite Element Method (FEM) Analysis of Cold-formed Channel Sections under Bending 235
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.
Chapter 7. Finite Element Method (FEM) Analysis of Cold-formed Channel Sections under Bending 236
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.
Chapter 7. Finite Element Method (FEM) Analysis of Cold-formed Channel Sections under Bending 237
Different stages on normalised moment-curvature graph for section 9
Stage1 Stage2 Stage3
Stage4 Stage5 Stage6
Deformation of the web
Figure 7.13: Deformation of section 9
Chapter 7. Finite Element Method (FEM) Analysis of Cold-formed Channel Sections under Bending 238
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.
Chapter 7. Finite Element Method (FEM) Analysis of Cold-formed Channel Sections under Bending 239
Different stages on normalised moment-curvature graph for section 42
Stage1 Stage2 Stage3
Stage4 Deformation of the web
Figure 7.14: Deformation of section 42
Chapter 7. Finite Element Method (FEM) Analysis of Cold-formed Channel Sections under Bending 240
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.
Chapter 7. Finite Element Method (FEM) Analysis of Cold-formed Channel Sections under Bending 241
Different stages on normalised moment-curvature graph for section 40
Stage1 Stage2 Stage3
Stage4 Stage5 Deformation of the web
Figure 7.15: Deformation of section 40
Chapter 7. Finite Element Method (FEM) Analysis of Cold-formed Channel Sections under Bending 242
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.
Chapter 8. Conclusions and Recommendations 245
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:
Chapter 8. Conclusions and Recommendations 246
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
Chapter 8. Conclusions and Recommendations 247
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
Chapter 8. Conclusions and Recommendations 248
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).
Chapter 8. Conclusions and Recommendations 249
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
Chapter 8. Conclusions and Recommendations 250
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-
Walled Structures, Volume 39(9): pp 807-820.
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-
Walled Structures, Volume 15(3): pp 209-233.
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)
Appendix A. Stress-Strain Diagrams of Coupon tests 264
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
Appendix A. Stress-Strain Diagrams of Coupon tests 265
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)
Appendix A. Stress-Strain Diagrams of Coupon tests 266
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)
Appendix A. Stress-Strain Diagrams of Coupon tests 267
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)
Appendix A. Stress-Strain Diagrams of Coupon tests 268
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
Appendix B. Tested Sections Normalised Moment-Curvature Diagrams 271
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
Appendix B. Tested Sections Normalised Moment-Curvature Diagrams 272
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
Appendix B. Tested Sections Normalised Moment-Curvature Diagrams 273
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
Appendix B. Tested Sections Normalised Moment-Curvature Diagrams 274
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
Appendix B. Tested Sections Normalised Moment-Curvature Diagrams 275
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
Appendix B. Tested Sections Normalised Moment-Curvature Diagrams 276
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
Appendix B. Tested Sections Normalised Moment-Curvature Diagrams 277
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
Appendix B. Tested Sections Normalised Moment-Curvature Diagrams 278
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
Appendix B. Tested Sections Normalised Moment-Curvature Diagrams 279
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
Appendix B. Tested Sections Normalised Moment-Curvature Diagrams 280
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
Appendix B. Tested Sections Normalised Moment-Curvature Diagrams 281
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
Appendix B. Tested Sections Normalised Moment-Curvature Diagrams 282
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
Appendix B. Tested Sections Normalised Moment-Curvature Diagrams 283
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
Test ResultMsx/MpMbdistortional/MpM=MyFully effectivek=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 ResultMsx/MpMbdistortional/MpM=MyFully effectivek=ky
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
Test ResultMsx/MpMbdistortional/MpM=MyFully effectivek=ky
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
Test ResultMsx/MpMbdistortional/MpM=MyFully effectivek=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 ResultMsx/MpMbdistortional/MpM=MyFully effectivek=ky
Appendix C. Tested Sections Normalised Moment-Curvature Diagrams Comparison With the North 286 American and Australian Standards Results
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
pTest ResultMsx/MpMbdistortional/MpM=MyFully effectivek=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 ResultMsx/MpMbdistortional/MpM=MyFully effectivek=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 ResultMsx/MpMbdistortional/MpM=MyFully effectivek=ky
Appendix C. Tested Sections Normalised Moment-Curvature Diagrams Comparison With the North 287 American and Australian Standards Results
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 ResultMsx/MpMbdistortional/MpM=MyFully effectivek=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 ResultMsx/MpMbdistortional/MpM=MyFully effectivek=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 ResultMsx/MpMbdistortional/MpM=MyFully effectivek=ky
Appendix C. Tested Sections Normalised Moment-Curvature Diagrams Comparison With the North 288 American and Australian Standards Results
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
pTest ResultMsx/MpMbdistortional/MpM=MyFully effectivek=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 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
Test ResultMsx/MpMbdistortional/MpM=MyFully effectivek=ky
Appendix C. Tested Sections Normalised Moment-Curvature Diagrams Comparison With the North 289 American and Australian Standards Results
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 ResultMsx/MpMbdistortional/MpM=MyFully effectivek=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 ResultMsx/MpMbdistortional/MpM=MyFully effectivek=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 ResultMsx/MpMbdistortional/MpM=MyFully effectivek=ky
Appendix C. Tested Sections Normalised Moment-Curvature Diagrams Comparison With the North 290 American and Australian Standards Results
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
pTest ResultMsx/MpMbdistortional/MpM=MyFully effectivek=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 ResultMsx/MpMbdistortional/MpM=MyFully effectivek=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 ResultMsx/MpMbdistortional/MpM=MyFully effectivek=ky
Appendix C. Tested Sections Normalised Moment-Curvature Diagrams Comparison With the North 291 American and Australian Standards Results
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 ResultMsx/MpMbdistortional/MpM=MyFully effectivek=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 ResultMsx/MpMbdistortional/MpM=MyFully effectivek=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 ResultMsx/MpMbdistortional/MpM=MyFully effectivek=ky
Appendix C. Tested Sections Normalised Moment-Curvature Diagrams Comparison With the North 292 American and Australian Standards Results
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
pTest ResultMsx/MpMbdistortional/MpM=MyFully effectivek=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 ResultMsx/MpMbdistortional/MpM=MyFully effectivek=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 ResultMsx/MpMbdistortional/MpM=MyFully effectivek=ky
Appendix C. Tested Sections Normalised Moment-Curvature Diagrams Comparison With the North 293 American and Australian Standards Results
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 ResultMsx/MpMbdistortional/MpM=MyFully effectivek=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 ResultMsx/MpMbdistortional/MpM=MyFully effectivek=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 ResultMsx/MpMbdistortional/MpM=MyFully effectivek=ky
Appendix C. Tested Sections Normalised Moment-Curvature Diagrams Comparison With the North 294 American and Australian Standards Results
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
pTest ResultMsx/MpMbdistortional/MpM=MyFully effectivek=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 ResultMsx/MpMbdistortional/MpM=MyFully effectivek=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 ResultMsx/MpMbdistortional/MpM=MyFully effectivek=ky
Appendix C. Tested Sections Normalised Moment-Curvature Diagrams Comparison With the North 295 American and Australian Standards Results
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 ResultMsx/MpMbdistortional/MpM=MyFully effectivek=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 ResultMsx/MpMbdistortional/MpM=MyFully effectivek=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 ResultMsx/MpMbdistortional/MpM=MyFully effectivek=ky
Appendix C. Tested Sections Normalised Moment-Curvature Diagrams Comparison With the North 296 American and Australian Standards Results
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
pTest ResultMsx/MpMbdistortional/MpM=MyFully effectivek=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 ResultMsx/MpMbdistortional/MpM=MyFully effectivek=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 ResultMsx/MpMbdistortional/MpM=MyFully effectivek=ky
Appendix C. Tested Sections Normalised Moment-Curvature Diagrams Comparison With the North 297 American and Australian Standards Results
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 ResultMsx/MpMbdistortional/MpM=MyFully effectivek=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 ResultMsx/MpMbdistortional/MpM=MyFully effectivek=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 ResultMsx/MpMbdistortional/MpM=MyFully effectivek=ky
Appendix C. Tested Sections Normalised Moment-Curvature Diagrams Comparison With the North 298 American and Australian Standards Results
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
pTest ResultMsx/MpMbdistortional/MpM=MyFully effectivek=ky
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
Appendix D. Tested Sections Normalised Moment-Strain Diagrams 301
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
Appendix D. Tested Sections Normalised Moment-Strain Diagrams 302
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
Appendix D. Tested Sections Normalised Moment-Strain Diagrams 303
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
Appendix D. Tested Sections Normalised Moment-Strain Diagrams 304
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
Appendix D. Tested Sections Normalised Moment-Strain Diagrams 305
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
Appendix D. Tested Sections Normalised Moment-Strain Diagrams 306
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
Appendix D. Tested Sections Normalised Moment-Strain Diagrams 307
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
Appendix D. Tested Sections Normalised Moment-Strain Diagrams 308
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
Appendix D. Tested Sections Normalised Moment-Strain Diagrams 309
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
Appendix D. Tested Sections Normalised Moment-Strain Diagrams 310
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
Appendix D. Tested Sections Normalised Moment-Strain Diagrams 311
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
Appendix D. Tested Sections Normalised Moment-Strain Diagrams 312
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
Appendix D. Tested Sections Normalised Moment-Strain Diagrams 313
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
Appendix E. Comparison of the Normalised Moment-Curvature Graphs of the Tested 316 Sections, with Edge Stiffener, Based on YLM Analysis, FEM and the Test Results
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
Appendix E. Comparison of the Normalised Moment-Curvature Graphs of the Tested 317 Sections, with Edge Stiffener, Based on YLM Analysis, FEM and the Test Results
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
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 318 Sections, with Edge Stiffener, Based on YLM Analysis, FEM and the Test Results
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 10.0k/kp
M/M
pTest Result
FEM
elastic
YLM
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 10.0k/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
Appendix E. Comparison of the Normalised Moment-Curvature Graphs of the Tested 319 Sections, with Edge Stiffener, Based on YLM Analysis, FEM and the Test Results
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 10.0k/kp
M/M
p
Test Result
FEM
elastic
YLM
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 10.0k/kp
M/M
p
Test Result
FEM
elastic
YLM
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 10.0k/kp
M/M
p
Test Result
FEM
elastic
YLM
Appendix E. Comparison of the Normalised Moment-Curvature Graphs of the Tested 320 Sections, with Edge Stiffener, Based on YLM Analysis, FEM and the Test Results
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 10.0k/kp
M/M
pTest Result
FEM
elastic
YLM
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
FEM
elastic
YLM
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
FEM
elastic
YLM
Appendix E. Comparison of the Normalised Moment-Curvature Graphs of the Tested 321 Sections, with Edge Stiffener, Based on YLM Analysis, FEM and the Test Results
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 10.0k/kp
M/M
p
Test Result
FEM
elastic
YLM
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 10.0k/kp
M/M
p
Test Result
FEM
elastic
YLM
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 10.0k/kp
M/M
p
Test Result
FEM
elastic
YLM
MAS4100/Mp
Appendix E. Comparison of the Normalised Moment-Curvature Graphs of the Tested 322 Sections, with Edge Stiffener, Based on YLM Analysis, FEM and the Test Results
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 10.0k/kp
M/M
pTest Result
FEM
elastic
YLM
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 10.0k/kp
M/M
p
Test Result
FEM
elastic
YLM
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 10.0k/kp
M/M
p
Test Result
FEM
elastic
YLM
MAS4100/Mp
Appendix E. Comparison of the Normalised Moment-Curvature Graphs of the Tested 323 Sections, with Edge Stiffener, Based on YLM Analysis, FEM and the Test Results
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 10.0k/kp
M/M
p
Test Result
FEM
elastic
YLM
MAS4100/Mp
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 10.0k/kp
M/M
p
Test Result
FEM
elastic
YLM
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
FEM
elastic
YLM
MAS4100/Mp
Appendix E. Comparison of the Normalised Moment-Curvature Graphs of the Tested 324 Sections, with Edge Stiffener, Based on YLM Analysis, FEM and the Test Results
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
FEM
elastic
YLM
MAS4100/Mp
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 10.0k/kp
M/M
p
Test Result
FEM
elastic
YLM
MAS4100/Mp
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 10.0k/kp
M/M
p
Test Result
FEM
elastic
YLM
MAS4100/Mp
Appendix E. Comparison of the Normalised Moment-Curvature Graphs of the Tested 325 Sections, with Edge Stiffener, Based on YLM Analysis, FEM and the Test Results
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 10.0k/kp
M/M
p
Test Result
FEM
elastic
YLM
MAS4100/Mp
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 10.0k/kp
M/M
p
Test Result
FEM
elastic
YLM
MAS4100/Mp
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 10.0k/kp
M/M
p
Test Result
FEM
elastic
YLM
MAS4100/Mp
Appendix E. Comparison of the Normalised Moment-Curvature Graphs of the Tested 326 Sections, with Edge Stiffener, Based on YLM Analysis, FEM and the Test Results
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 10.0k/kp
M/M
p
Test Result
FEM
elastic
YLM
MAS4100/Mp
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 10.0k/kp
M/M
p
Test Result
FEM
elastic
YLM
MAS4100/Mp
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 10.0k/kp
M/M
p
Test Result
FEM
elastic
YLM
MAS4100/Mp
Appendix E. Comparison of the Normalised Moment-Curvature Graphs of the Tested 327 Sections, with Edge Stiffener, Based on YLM Analysis, FEM and the Test Results
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 10.0k/kp
M/M
p
Test Result
FEM
elastic
YLM
MAS4100/Mp
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 10.0k/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
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
Section 4
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
θ (Rad)
M (
kN-m
)
Test
YLM
Appendix F. Divided Test and YLM Graphs According to the Simpson Rules 329
Section 5
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
0.00 0.01 0.03 0.04 0.05 0.06 0.08 0.09
θ (Rad)
M (
kN-m
)
Test
YLM
Section 6
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 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11
θ (Rad)
M (
kN-m
)
Test
YLM
Section 7
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
θ (Rad)
M (
kN-m
)
Test
YLM
Appendix F. Divided Test and YLM Graphs According to the Simpson Rules 330
Section 8
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11
θ (Rad)
M (
kN-m
)Test
YLM
Section 9
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
θ (Rad)
M (
kN-m
)
Test
YLM
Section 10
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
θ (Rad)
M (
kN-m
)
Test
YLM
Appendix F. Divided Test and YLM Graphs According to the Simpson Rules 331
Section 11
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
θ (Rad)
M (
kN-m
)
Test
YLM
Section 12
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
θ (Rad)
M (
kN-m
)
Test
YLM
Section 13
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11
θ (Rad)
M (
kN-m
)
Test
YLM
Appendix F. Divided Test and YLM Graphs According to the Simpson Rules 332
Section 14
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 0.04 0.05 0.06 0.07 0.08 0.09
θ (Rad)
M (
kN-m
)Test
YLM
Section 15
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
θ (Rad)
M (
kN-m
)
Test
YLM
Section 16
0.0
2.0
4.0
6.0
8.0
10.0
12.0
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
θ (Rad)
M (
kN-m
)
Test
YLM
Appendix F. Divided Test and YLM Graphs According to the Simpson Rules 333
Section 17
0.0
2.0
4.0
6.0
8.0
10.0
12.0
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07
θ (Rad)
M (
kN-m
)
Test
YLM
Section 18
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
θ (Rad)
M (
kN-m
)
Test
YLM
Section 19
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 0.04 0.05 0.06 0.07 0.08 0.09
θ (Rad)
M (
kN-m
)
Test
YLM
Appendix F. Divided Test and YLM Graphs According to the Simpson Rules 334
Section 20
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 0.04 0.05 0.06 0.07
θ (Rad)
M (
kN-m
)Test
YLM
Section 21
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
θ (Rad)
M (
kN-m
)
Test
YLM
Section 22
0.0
1.0
2.0
3.0
4.0
5.0
6.0
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
θ (Rad)
M (
kN-m
)
Test
YLM
Appendix F. Divided Test and YLM Graphs According to the Simpson Rules 335
Section 23
0.0
1.0
2.0
3.0
4.0
5.0
6.0
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
θ (Rad)
M (
kN-m
)
Test
YLM
Section 24
0.0
1.0
2.0
3.0
4.0
5.0
6.0
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
θ (Rad)
M (
kN-m
)
Test
YLM
Section 25
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
θ (Rad)
M (
kN-m
)
Test
YLM
Appendix F. Divided Test and YLM Graphs According to the Simpson Rules 336
Section 26
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11
θ (Rad)
M (
kN-m
)Test
YLM
Section 27
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
θ (Rad)
M (
kN-m
)
Test
YLM
Section 28
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07
θ (Rad)
M (
kN-m
)
Test
YLM
Appendix F. Divided Test and YLM Graphs According to the Simpson Rules 337
Section 29
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
θ (Rad)
M (
kN-m
)
Test
YLM
Section 30
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
θ (Rad)
M (
kN-m
)
Test
YLM
Section 31
0.0
0.5
1.0
1.5
2.0
2.5
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
θ (Rad)
M (
kN-m
)
Test
YLM
Appendix F. Divided Test and YLM Graphs According to the Simpson Rules 338
Section 32
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11
θ (Rad)
M (
kN-m
)Test
YLM
Section 33
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11
θ (Rad)
M (
kN-m
)
Test
YLM
Section 34
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11
θ (Rad)
M (
kN-m
)
Test
YLM
Appendix F. Divided Test and YLM Graphs According to the Simpson Rules 339
Section 35
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07
θ (Rad)
M (
kN-m
)
Test
YLM
Section 36
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11
θ (Rad)
M (
kN-m
)
Test
YLM
Section 37
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
θ (Rad)
M (
kN-m
)
Test
YLM
Appendix F. Divided Test and YLM Graphs According to the Simpson Rules 340
Section 38
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
θ (Rad)
M (
kN-m
)Test
YLM
Section 39
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
θ (Rad)
M (
kN-m
)
Test
YLM
Section 40
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.00 0.02 0.04 0.06 0.08 0.10
θ (Rad)
M (
kN-m
)
Test
YLM
Appendix F. Divided Test and YLM Graphs According to the Simpson Rules 341
Section 41
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.00 0.02 0.04 0.06 0.08 0.10
θ (Rad)
M (
kN-m
)
Test
YLM
Section 42
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
θ (Rad)
M (
kN-m
)
Test
YLM
342
Appendix G
COMPARISON BETWEEN SIMPLIFIED MODEL AND 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
Linear part
Curve part
M/Mp = 1.03(k/kp)-0.665
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
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
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 = 1.09(k/kp)-0.665
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
k/kp
M/M
p
Test Result
Linear part
Curve part
M/Mp = 0.94(k/kp)-0.665
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
k/kp
M/M
p
Test Result
Linear part
Curve part
M/Mp = 0.89(k/kp)-0.665
Appendix G. Comparison between Simplified Model and Test Results 344
Section8
M/Mp = 0.98(k/kp)-0.665
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
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
Appendix G. Comparison between Simplified Model and Test Results 345
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
Appendix G. Comparison between Simplified Model and Test Results 346
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
Appendix G. Comparison between Simplified Model and Test Results 347
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
Appendix G. Comparison between Simplified Model and Test Results 348
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
Appendix G. Comparison between Simplified Model and Test Results 349
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
Appendix G. Comparison between Simplified Model and Test Results 350
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
Appendix G. Comparison between Simplified Model and Test Results 351
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
Appendix G. Comparison between Simplified Model and Test Results 352
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
Appendix G. Comparison between Simplified Model and Test Results 353
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
Appendix G. Comparison between Simplified Model and Test Results 354
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
Curve part M/Mp = 2.02(k/kp)-0.665
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
Curve part M/Mp = 1.65(k/kp)-0.665
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
Appendix G. Comparison between Simplified Model and Test Results 355
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
Appendix H. Failed Shape of the Tested Sections 358
Appendix H. Failed Shape of the Tested Sections 359
Appendix H. Failed Shape of the Tested Sections 360
Appendix H. Failed Shape of the Tested Sections 361
Appendix H. Failed Shape of the Tested Sections 362