Behaviour and Design of Stainless Steel Columns

BEHAVIOUR AND DESIGN OF STAINLESS STEEL COLUMNS

by Shameem Ahmed

The University of New South Wales August 2017

BEHAVIOUR AND DESIGN OF STAINLESS STEEL COLUMNS

Shameem Ahmed

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

School of Engineering and Information Technology

The University of New South Wales

Australian Defence Force Academy

Canberra ACT 2610, Australia

August 2017

Abstract

Current design standards do not appropriately recognise the characteristic nonlinear stress-strain behaviour with significant strain hardening offered by stainless steel. Loss of effectiveness due to local buckling is dealt with effective width method, which is unjustified for stainless steel as there is no obvious yield plateau. With recent developments, the continuous strength method (CSM) was shown to accurately predict cross-section resistances of stocky sections exploiting the benefit of strain hardening. The current research extended the scope of CSM for slender sections retaining the same base curve, and also develop CSM based design guidelines for predicting the buckling resistance of stainless steel columns.

A new concept of using an equivalent elastic deformation capacity εe,ev, defined as the elastic strain at ultimate load, was proposed for slender sections. A comprehensive FE study was carried out to establish relationships between εe,ev and buckling strain εcsm. Proposed relationships allowed using the same CSM base curve for both stocky and slender sections, and hence, cross-section resistances could be directly determined using

CSM buckling stress fcsm and gross cross-sectional properties.

Once CSM was successfully implemented for all cross-section types, the current study extended its scope to the member level. A series of new buckling curves was proposed for cold-formed RHS and SHS columns using available test results and generated FE results. The proposed technique combined fcsm with Perry type buckling curves. Once the proposed buckling formulas performed well for cold-formed hollow sections, the flexural buckling of welded I columns were thoroughly investigated through experimental and numerical methods. Valuable experimental evidences on column buckling of welded sections were added to currently available limited data. An extensive FE analysis supplemented the test results, and all available results were used to develop CSM based buckling curves for welded I-sections considering the effects of residual stresses and other important parameters. All developed formulas were shown to produce accurate, consistent and reliable predictions.

Acknowledgements

First and foremost, I express my heartiest gratitude to my supervisors Dr Mahmud Ashraf and Dr Safat Al-Deen, for giving me the opportunity to work on this project and their untiring support during my study. They trained me to be an independent researcher and taught me the very important traits a researcher should possess. Their useful suggestions, kind supports, precious discussions and constant encouragements at the various stages throughout the research work are highly appreciated. Without their support and guidance, this work would not have been possible.

I also extend my appreciation to all the laboratory and workshop staffs at the School of Engineering and Information Technology at UNSW Canberra, especially Jim Baxter, David Sharp, Douglas Collier, Mark Dumbrell and Nick Baxter for their help in tackling all experimental challenges.

I would like to give special thanks to my colleagues Mohammad Anwar-Us-Saadat, Shayani Mendis and Xin Li for giving hands during experiments.

I would also like to thank my colleagues and officemates Md. Ashraful Ismal, Md. Abdul Kader, Biruk Hailu Tekle, Zhengliang Liu and Zongjun Li their support, encouragement and friendship.

I am also grateful to Md. Shahidul Islam, Shahana Ferdous, Shahariar Jahan, Shawkat Mozumder, Shafin Rahman and Jamil Ashraf for their support and encouragement throughout my study and stay in Canberra.

Special thanks are due to my parents and all my families for their endless patience, encouragement, continuous support and help during this study.

Finally, I take this opportunity to express my gratitude to the University of New South Wales, Canberra for the financial support in the form of TFS scholarships

Table of Contents Table of Contents

Abstract ...... i Acknowledgements ...... ii Table of Contents ...... iii List of Figures ...... vi List of Tables ...... xiv Notation ...... xvi Chapter 1 Introduction ...... 1 Background ...... 1

Material content and classification ...... 3 Applications in the construction industry ...... 5 Research objectives ...... 8 Thesis outline ...... 10 Chapter 2 Literature Review ...... 12 Introduction ...... 12 Material model ...... 12 Corner strength enhancement due to cold-work ...... 17 Residual stress ...... 20 Geometric imperfections ...... 23 Existing codes ...... 27 Design guidelines for cross-section resistance ...... 28 Design guidelines for flexural buckling resistance ...... 29 Continuous Strength Method ...... 32 Direct Strength Method ...... 38 Summary ...... 41 Chapter 3 The Continuous Strength Method for Stainless Steel Slender Sections. 42 Introduction ...... 42 Experimental results ...... 43 Development of the finite element model ...... 43 Verification of the developed FE model ...... 49 Parametric study ...... 57 iii

Table of Contents

CSM for slender sections ...... 59 Resistance of slender cross-sections against compression and bending ...... 67 Performance of the proposed design technique ...... 68 Reliability analysis ...... 78 Worked out Examples ...... 80 3.10.1 Example I: Axial Compression resistance ...... 80 3.10.2 Example II: In-plane bending resistance ...... 81 Conclusions ...... 82 Chapter 4 Buckling Resistance of Stainless Steel Hollow Columns ...... 84 Introduction ...... 84 CSM based design approach for buckling resistance ...... 85 Development of the finite element model ...... 87 Verification of the FE model ...... 90 Parametric study ...... 94

Effect of λp, e , n and H/B on column curves ...... 96 Correction factors for e and n ...... 102 Reduction factor χ ...... 104 Performance of the proposed CSM method ...... 106 Reliability analysis ...... 112 Worked out example: Buckling capacity of a RHS column ...... 113 Conclusions ...... 115 Chapter 5 Behaviour and Design of Stainless Steel Welded I-columns ...... 117 Introduction ...... 117 Test program ...... 119 Tensile coupon test ...... 119 5.3.1 Procedure adopted for tensile test ...... 119 5.3.2 Results obtained from tensile tests ...... 122 Residual stresses observed in stainless steel welded I-sections ...... 124 5.4.1 Measurement technique adopted for residual stresses ...... 124 5.4.2 Discussion on results obtained from residual stress measurement ...... 128 Initial geometric imperfection measurements ...... 129 Testing of stub columns ...... 134 5.6.1 Test procedure ...... 134

Table of Contents

5.6.2 Discussion of results for stub columns ...... 137 Long column test ...... 138 5.7.1 Test procedure ...... 138 5.7.2 Discussion of results obtained for long columns ...... 143 Summary of the testing scheme ...... 151 Finite element model for long columns ...... 152 Verification of FE models for long columns ...... 154 Parametric study for welded I-columns ...... 158 Analysis of FE results ...... 159 Imperfection factor η for welded I-columns...... 164 Performance of the proposed method ...... 166 Reliability analysis ...... 171 Conclusions ...... 172 Chapter 6 Conclusions and Recommendations ...... 174 Conclusions ...... 174 6.1.1 General ...... 174 6.1.2 The continuous strength method for slender cross-sections ...... 174 6.1.3 Buckling resistance of stainless steel hollow section columns ...... 176 6.1.4 Buckling resistance of stainless steel welded I-columns ...... 178 6.1.5 Advancement of CSM ...... 180 Future recommendations ...... 180 References ...... 182 Appendix A ...... 193 Appendix B ...... 195 Appendix C ...... 196

List of Figures List of Figures

Figure 1.1 Early uses of stainless steel as structural material ...... 7 Figure 1.2 Recent examples of use of stainless steel as structural material ...... 8 Figure 2.1 Comparison of the stress-strain behaviour of carbon steel and stainless steel ...... 13 Figure 2.2 Membrane residual stress distribution of a welded I-section [48]...... 22 Figure 2.3 Typical Load vs End shortening curve of a stub column ...... 33 Figure 2.4 CSM base curve proposed by Ashraf et al.[12] ...... 34 Figure 2.5 Material model proposed by Afshan and Gardner [14] ...... 36 Figure 2.6 CSM base curve proposed by Afshan and Gardner [14] for stocky sections 38 Figure 3.1 Typical mesh distribution used in a hollow section 60×60×3-SC1 [62] ...... 45 Figure 3.2 Typical mesh distribution used in an I-Section I 2205-372 [63] ...... 46 Figure 3.3 Boundary condition applied to the FE model of SHS stub column 60×60×3- SC1 [62] ...... 47 Figure 3.4 Boundary condition applied to the FE model of I-Section stub column I 2205-372 [63] ...... 47 Figure 3.5 Boundary condition applied to the FE model of SHS beam 50×50×1.5L900 [102] ...... 48 Figure 3.6 Boundary condition applied to the FE model of I-beam I-200×140×8×6-2 [100] ...... 49 Figure 3.7 Load vs. end shortening curve comparing test observation and numerical simulation for SHS 80×80×3-SC1 stub column [101] ...... 54 Figure 3.8 Load vs. end shortening curve comparing test observation and numerical simulation for I304-462 stub column [63] ...... 54 Figure 3.9 Moment vs. curvature curve comparing test observation and numerical simulation for SHS 120×120×5-2B beam [103] ...... 55 Figure 3.10 Moment vs. curvature curve comparing test observation and numerical simulation for I-200×140×10×8-2 beam [100] ...... 55 Figure 3.11 Comparison of deflected shapes observed in experiment and FE simulation for 60×60×3-SC1 stub column [62] ...... 56 Figure 3.12 Comparison of deflected shapes observed in experiment and FE simulation for I 2205-372 stub column [63] ...... 56 vi

List of Figures Figure 3.13 Comparison of deflected shapes observed in experiment and FE simulation for 50×50×1.5L900 beam [102] ...... 57 Figure 3.14 Comparison of deflected shapes observed in experiment and FE simulation for I-200×140×10×8-2 beam [100] ...... 57 Figure 3.15 Typical post buckling behaviour observed in slender sections ...... 60

Figure 3.16 Variation of C with cross-section slenderness λp for RHS stub columns .... 61

Figure 3.17 Variation of C with cross-section slenderness λp for SHS stub columns .... 61

Figure 3.18 Variation of C with cross-section slenderness λp for I-section stub columns ...... 62

Figure 3.19 Variation of C with cross-section slenderness λp for RHS beams subjected to major axis bending ...... 62

Figure 3.20 Variation of C with cross-section slenderness λp for RHS beams subjected to minor axis bending ...... 63

Figure 3.21 Variation of C with cross-section slenderness λp for SHS beams ...... 63

Figure 3.22 Variation of C with cross-section slenderness λp for I-beams ...... 64

Figure 3.23 Variation of C with cross-section slenderness λp for compression ...... 65

Figure 3.24 Variation of C with cross-section slenderness λp for in-plane bending ...... 65

Figure 3.25 Variation of C with cross-section slenderness λp for stub column data of all cross-section types ...... 66

Figure 3.26 Variation of C with cross-section slenderness λp for in-plane bending data of all cross-section types ...... 66

Figure 3.27 Variation of C with cross-section slenderness λp for axial compression and in-plane bending ...... 67 Figure 3.28 Performance of the suggested CSM method and other considered design standards in predicting cross-section compression resistances of RHS ...... 71 Figure 3.29 Performance of the suggested CSM method and other considered design standards in predicting cross-section compression resistances of SHS ...... 71 Figure 3.30 Performance of the suggested CSM method and other considered design standards in predicting cross-section compression resistances of I-sections ...... 72 Figure 3.31 Performance of the suggested CSM method in predicting cross-section compression resistances of all types of cross-sections ...... 72 Figure 3.32 Performance of EN 1996-1-4+A1 [68] in predicting cross-section compression resistances of all types of cross-sections ...... 73

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List of Figures Figure 3.33 Performance of AS/NZS 4673 [5] or SEI/ASCE 8-02 [4] in predicting cross-section compression resistances of all types of cross-sections ...... 73 Figure 3.34 Performance of DSM [86] in predicting cross-section compression resistances of all types of cross-sections ...... 74 Figure 3.35 Performance of the suggested CSM method and other considered design standards in predicting in-plane bending resistances of RHS bending about major axis 74 Figure 3.36 Performance of the suggested CSM method and other considered design standards in predicting in-plane bending resistances of RHS bending about minor axis75 Figure 3.37 Performance of the suggested CSM method and other considered design standards in predicting in-plane bending resistances of SHS beams ...... 75 Figure 3.38 Performance of the suggested CSM method and other considered design standards in predicting in-plane bending resistances of I-beams ...... 76 Figure 3.39 Performance of the suggested CSM method in predicting in-plain bending resistances of all types of cross-sections ...... 76 Figure 3.40 Performance of EN 1996-1-4+A1 [68] in predicting in-plane bending resistances of all types of cross-sections ...... 77 Figure 3.41Performance of AS/NZS 4673 [5] or SEI/ASCE 8-02 [4] in predicting in- plane bending resistances of all types of cross-sections ...... 77 Figure 3.42 Performance of DSM [86] in predicting in-plane bending resistances of all types of cross-sections...... 78 Figure 4.1 Typical mesh distribution of the FE model developed for 60×60×3-800 SHS column [62] ...... 88 Figure 4.2 Support conditions with applied deflection of the FE model developed for 60×60×3-800 column [62] ...... 90 Figure 4.3 Comparison of load-deformation curves of hollow section columns obtained from FE models with test results ...... 93 Figure 4.4 Experimental and FE failure mode of C4L1200 column [84] ...... 93

Figure 4.5 Column curves of SHS for different λp values (e = 0.001, n = 7) ...... 96

Figure 4.6 Column curves of SHS for different λp values (e = 0.003, n = 7) ...... 97

Figure 4.7 Column curves of SHS for different λp values (e = 0.002, n = 5) ...... 97

Figure 4.8 Column curves of SHS for different λp values (e = 0.002, n = 10) ...... 98

Figure 4.9 Column curves of SHS for different e values (λp = 0.48, n = 7) ...... 99

Figure 4.10 Column curves of SHS for different e values (λp = 0.88, n = 7) ...... 99

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List of Figures

Figure 4.11Column curves of SHS for different n values (λp = 0.48, e = 0.002) ...... 100

Figure 4.12 Column curves of SHS for different n values (λp = 0.88, e = 0.002) ...... 100

Figure 4.13 Column curves of RHS for different H/B ratios (λp = 0.48, e = 0.002 and n = 7) ...... 101

Figure 4.14 Column curves of RHS for different H/B ratios (λp = 0.88, e = 0.002 and n = 7) ...... 101

Figure 4.15 Column curves of SHS with respect to λm for different e values (λp = 0.48, n = 7) ...... 103

Figure 4.16 Column curves of SHS with respect to λm for different e values (λp = 0.88, n = 7) ...... 103

Figure 4.17 Column curves of SHS with respect to λm for different n values (λp = 0.48, e = 0.002) ...... 104

Figure 4.18 Column curves of SHS with respect to λm for different n values (λp = 0.88, e = 0.002) ...... 104

Figure 4.19 η calculated from FE results and proposed curves of η for different λp .... 105 Figure 4.20 Comparison of CSM and EN 1993- 1-4+A1 [68] predicted buckling resistances with FE and test results for SHS columns ...... 107 Figure 4.21 Comparison of CSM and SEI/ACSE 8-02 [4] predicted buckling resistances with FE and test results for SHS columns ...... 108 Figure 4.22 Comparison of CSM and AS/NZS 4673 [5] predicted buckling resistances with FE and test results for SHS columns ...... 108 Figure 4.23 Comparison of CSM and EN 1993- 1-4+A1 [68] predicted buckling resistances with FE and test results for RHS columns ...... 109 Figure 4.24 Comparison of CSM and SEI/ACSE 8-02 [4] predicted buckling resistances with FE and test results for RHS columns ...... 109 Figure 4.25 Comparison of CSM and AS/NZS 4673 [5] predicted buckling resistances with FE and test results for RHS columns ...... 110 Figure 4.26 Comparison of CSM and EN 1993- 1-4+A1 [68] predicted buckling resistances with FE and test results for SHS columns for different cross-section slenderness ...... 110 Figure 4.27 Comparison of CSM and EN 1993- 1-4+A1 [68] predicted buckling resistances with FE and test results for SHS columns for different e values ...... 111

List of Figures Figure 4.28 Comparison of CSM and EN 1993- 1-4+A1 [68] predicted buckling resistances with FE and test results for SHS columns for different n values ...... 111 Figure 4.29 Comparison of CSM and EN 1993- 1-4+A1 [68] predicted buckling resistances with FE and test results for RHS columns for different aspect ratios ...... 112 Figure 5.1Tension test coupons according to EN ISO 6892-1 [127] ...... 120 Figure 5.2 Test set-up for tensile coupon test ...... 121 Figure 5.3 Installation of the linear electrical resistance strain gauge on one of the test coupons ...... 121 Figure 5.4 Complete stress-strain curves of different stainless steel plates used to fabricate I-sections ...... 123 Figure 5.5 Stress-strain curves (enlarged up to 1% strain) for all plates used to fabricate I-sections ...... 123 Figure 5.6 Punch marks on the flange of 80×80×5×4-500 section ...... 124 Figure 5.7 Taking measurement of initial gauge length of the strips marked for residual stress measurement...... 125 Figure 5.8 All strips cut from I 80×80×5×4 section to measure residual stresses ...... 125 Figure 5.9 Taking measurements of the final gauge length on cut out strips ...... 126 Figure 5.10 Definition of the offset value δ for longitudinally curved strips cut to measure residual stress [48] ...... 127 Figure 5.11 Measurement of the offset values observed in the curved strips cut to measure residual stress ...... 127 Figure 5.12 Residual stresses distribution observed in welded I 80×80×5×4-500 section ...... 128 Figure 5.13 Residual stresses distribution observed in welded I 100×60×6×4-450 section ...... 129 Figure 5.14 Measurement of the initial local geometric imperfections of I 120×60×4×2- 500 column ...... 130 Figure 5.15 A typical distribution of the local imperfections of a cross-section ...... 131 Figure 5.16 Distribution of the initial local geometric imperfection of I 80×60×4×2-320 column ...... 131 Figure 5.17 Measurement of the initial global geometric imperfections of I 120×60×4×2-1000 column ...... 132 Figure 5.18 Typical distribution of the global geometric imperfect ...... 133

List of Figures Figure 5.19 Distribution of the initial global geometric imperfections of I 80×60×4×2- 1500 column about minor axis...... 133 Figure 5.20 Distribution of global imperfection of I 80×60×4×2-1500 column about major axis...... 133 Figure 5.21 Typical cross-section of an I-section defining different dimensions ...... 134 Figure 5.22 Schematic diagram of the instrumentation of stub column test ...... 135 Figure 5.23 Positions of electric strain gauges at the mid-length of stub column specimens of stainless steel welded I-section ...... 136 Figure 5.24 Test set-up for stub column test of I 80×60×6×4-320 column ...... 136 Figure 5.25 Failure modes of the stub columns of stainless steel welded I-sections .... 137 Figure 5.26 Complete axial load vs end shortening curves of the stub columns ...... 138 Figure 5.27 Strain distributions of the stub columns at mid height at different stages of loading ...... 138 Figure 5.28 Schematic diagram of the test set-up for flexural buckling test of welded I- columns ...... 140 Figure 5.29 Typical instrumentation at mid-height cross-section of flexural buckling specimens ...... 140 Figure 5.30 Test set-up of flexural buckling test ...... 142 Figure 5.31 Flexural buckling of I 80×60×4×2-750 column showing a typical failure mode ...... 143 Figure 5.32 Load vs end shortening of columns with I 80×60×4×2 cross-section ...... 145 Figure 5.33 Load vs end shortening of columns with I 80×60×6×4 cross-section ...... 145 Figure 5.34 Load vs end shortening of columns with I 80×80×5×4 cross-section ...... 146 Figure 5.35 Load vs end shortening of columns with I 100×60×6×4 cross-section ..... 146 Figure 5.36 Load vs end shortening of columns with I 120×60×5×3 cross-section ..... 147 Figure 5.37 Load vs end shortening of columns with I 120×60×4×2 cross-section ..... 147 Figure 5.38 Load vs lateral deflection of columns with I 80×60×4×2 cross-section .... 148 Figure 5.39 Load vs lateral deflection of columns with I 80×60×6×4 cross-section .... 148 Figure 5.40 Load vs lateral deflection of the columns with I 80×80×5×4 cross-section ...... 149 Figure 5.41 Load vs lateral deflection of the columns with I 100×60×6×4 cross-section ...... 149 Figure 5.42 Load vs lateral deflection of columns with I 1200×60×5×3 cross-section 150

List of Figures Figure 5.43 Load vs lateral deflection of the columns with I 120×60×4×2 cross-section ...... 150 Figure 5.44 Strain distributions at mid-height of columns with I 80×60×6×4 section at different loading stages...... 151 Figure 5.45 Strain distributions at mid-height of columns with I 80×80×5×4 section at different loading stages...... 151 Figure 5.46 Support conditions of a I-column subjected to major axis buckling ...... 153 Figure 5.47 Support conditions of a I-column subjected to minor axis buckling ...... 154 Figure 5.48 Deflected shape of I 80×60×4×2-750 specimen ...... 155 Figure 5.49 Comparison of experimental load vs. end shortening curves with FE results for welded I-columns ...... 157 Figure 5.50 Comparison of experimental load vs. lateral deflection curves with FE results for welded I-columns ...... 157 Figure 5.51 Column curves for major axis buckling of stainless steel welded I-section for different λp values ...... 160 Figure 5.52 Column curves for minor axis buckling of stainless steel welded I-section for different λp values ...... 160 Figure 5.53 Column curves of stocky stainless steel welded I-sections for different H/B ratios (λp = 0.48) ...... 161 Figure 5.54 Column curves of slender stainless steel welded I-sections for different H/B ratios (λp = 0.88) ...... 161

Figure 5.55 Column curves of stocky stainless steel welded I-sections for different tf/tw ratios (λp = 0.48) ...... 162

Figure 5.56 Column curves of slender stainless steel welded I-sections for different tf/tw ratios (λp = 0.88) ...... 162 Figure 5.57 Comparison of column curves of stocky welded I-sections buckling around major axis with residual stress and without residual stress ...... 163 Figure 5.58 Comparison of column curves of stocky welded I-sections buckling around minor axis with residual stress and without residual stress...... 163 Figure 5.59 Comparison of column curves of slender welded I-sections buckling around major axis with residual stress and without residual stress ...... 164 Figure 5.60 Comparison of column curves of slender welded I-sections buckling around minor axis with residual stress and without residual stress...... 164

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List of Figures

Figure 5.61 η calculated from FE results and proposed curves of η for different λp values for major axis buckling of stainless steel welded I-columns ...... 165

Figure 5.62 η calculated from FE results and proposed curves of η for different λp values for minor axis buckling of stainless steel welded I-columns ...... 166 Figure 5.63 Comparison of CSM and EN1993-1-4+A1 [68] predictions for major axis buckling of stainless steel welded I-columns ...... 168 Figure 5.64 Comparison of CSM and SEI/ASCE 8-02 [4] predictions for major axis buckling of stainless steel welded I-columns ...... 168 Figure 5.65 Comparison of CSM and AS/NZS 4673 [5] predictions for major axis buckling of stainless steel welded I-columns ...... 169 Figure 5.66 Comparison of CSM and EN1993-1-4+A1 [68] predictions for minor axis buckling of stainless steel welded I-columns ...... 169 Figure 5.67 Comparison of CSM and SEI/ASCE 8-02 [4] predictions for minor axis buckling of stainless steel welded I-columns ...... 170 Figure 5.68 Comparison of CSM and AS/NZS 4673 [5] predictions for minor axis buckling of stainless steel welded I-columns ...... 170

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List of Tables List of Tables

Table 2.1 Parameters of the predictive model of the membrane residual stresses for stainless steel I-sections [48] ...... 23 Table 3.1 List of experimental results considered in the current study on stub columns and in-plane bending ...... 44 Table 3.2 Comparison of FE results with experiments for stub column test of RHS and SHS ...... 50 Table 3.3 Comparison of FE results with experiments for four-point bending of RHS and SHS beams ...... 51 Table 3.4 Comparison of FE results with experiments for stub column test of I-sections ...... 52 Table 3.5 Comparison of FE results with experiments for four point bending of I-beams ...... 53 Table 3.6 Cross-sectional properties of Rectangular Hollow Sections considered in FE analysis ...... 58 Table 3.7 Cross-sectional properties of Square Hollow Sections considered in FE analysis ...... 59 Table 3.8 Cross-sectional properties of I-sections considered in FE analysis ...... 59 Table 3.9 Material properties used in the parametric study ...... 59 Table 3.10 Proposed values for coefficients a and b for different cross-section types under axial compression ...... 64 Table 3.11Proposed values for coefficients a and b for different cross-section types subjected to in-plane bending ...... 65 Table 3.12 Proposed values for coefficients a and b for all considered cross-section types ...... 67 Table 3.13 Comparison of the predictions of cross-section compression resistance obtained by CSM and other standards with stub column test and FE results ...... 70 Table 3.14 Comparison of the predictions of in-plane bending resistance obtained by CSM and other standards with in-plane bending test and FE results ...... 70 Table 3.15 Summary of the reliability analysis of the proposed CSM method for predicting cross-section compression resistance of stainless steel slender sections ...... 79

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List of Tables Table 3.16 Summary of the reliability analysis of the proposed CSM method for predicting in-plane bending resistance of stainless steel slender sections ...... 80 Table 4.1 Comparison of FE results of RHS and SHS columns with test results ...... 91 Table 4.2 Thicknesses of 125×125mm SHS used in the parametric study showing corresponding values of p and e...... 95 Table 4.3 Dimensions of RHS cross-sections used in the parametric study ...... 95 Table 4.4 Comparison of the performance of the proposed CSM method for buckling resistance of hollow columns with other standards ...... 107 Table 4.5 Reliability analysis of CSM method proposed for buckling resistance of RHS and SHS columns ...... 113 Table 5.1 Material properties of austenitic stainless steel plates used to fabricate welded I-column specimens ...... 123 Table 5.2 Geometric dimensions and the maximum local imperfections of the stub column specimens considered for experimental study...... 134 Table 5.3 Stub column test results ...... 137 Table 5.4 Geometric dimensions and initial geometric imperfections of the long column specimens of stainless steel welded I-sections ...... 141 Table 5.5 Key results obtained from the flexural buckling test of pin-ended welded I- columns ...... 144 Table 5.6 Comparison of FE results with tests performed on pin-ended stainless steel welded I-columns ...... 156 Table 5.7 Cross-sectional properties of I-sections used in the parametric study ...... 159 Table 5.8 Comparison of the performance of the proposed CSM method and other standards ...... 167 Table 5.9 Results of the reliability analysis of the proposed buckling formulas for stainless steel welded I-columns ...... 172

Notation Notation

A Coefficient of sigmoidal function for η a Coefficient relating C and λp

Ac Coupon cross-sectional area

Aeff Effective cross-sectional area

Ag Gross cross-sectional area B Coefficient of sigmoidal function for η b Coefficient relating C and λp

C Coefficient relating εcsm and εe,ev

Ce Correction factor for e CHS Circular hollow section

Cn Correction factor for n COV Coefficient of variation CSM Continuous strength method DSM Direct strength method E Young's modulus e Non-dimensional proof stress (σ0.2/E)

E0.2 Tangent modulus at 0.2 % proof stress EHS Elliptical hollow section

Esh Slope of the hardening curve

Et Tangent modulus fcsm Buckling stress according to CSM fy Yield stress LVDT Linear variable displacement transducer m Second strain hardening exponent

Mcsm CSM predicted moment capacity

Mel Elastic moment capacity

MFF Moment capacity according to FE analysis

Mpl Plastic moment capacity

Mu Ultimate moment capacity n Ramberg-Osgood strain hardening exponent

Ncr Elastic critical buckling load of the column based on gross area

Ncsm CSM predicted compression capacity

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Notation

NFE Compression capacity according to FE analysis

Nne Nominal member capacity of a member in compression for flexural, torsional or flexural-torsional buckling

Nnl Nominal axial capacity of a member for local buckling

Nu Ultimate axial compression capacity

Ny Yield capacity of a member RHS Rectangular hollow section ri Internal redius of a corner SHS Square hollow section t Thickness tf Flange thickness tw Wed thickness W Coefficient of sigmoidal fuction of η w0 Initial geometric imperfection amplitude

Wel Elastic section modulus

γM0 Partial safety factor for cross-section resistance

γM1 Partial safety factor for member resistance

εcsm Buckling strain according to CSM

εe,ev Eqivalent elastic deformation capacity

εpl,f Plastic strain at fracture

εt,0.2 Total strain at 0.2 % proof stress

εt,1.0 Total strain at 1 % proof stress

εu Strain at ultimate strength

εy Strain at yield stress η Imperfection parameter of Perry curves

κel Curvature at the elastic moment capacity

κu Curvature at the ultimate moment λ Non-dimensional member slenderness based on yield stress

λcsm Non-dimensional member slenderness based on buckling stress

λm Modified non-dimensional member slenderness

λp Cross-section slenderness

σ0.01 0.01% proof stress

σ0.05 0.01% proof stress

σ0.2 0.2 % proof stress

σ0.2,c 0.2 % proof stress of corner material

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Notation

σ0.2,f 0.2 % proof stress of flat material of holoow section

σ0.2,mill 0.2 % proof stress according to mill certificate

σ0.2,v 0.2 % proof stress of virgin material

σ1.0 1 % proof stress

σcr Critical buckling stress of a plate

σcr,cs Elastic buckling capacity of the full cross-section

σp Proportional limit

σu Ultimate strength

σu,c Ultimate strength of corner material

σu,f Ultimate strength of flat material of holoow section

σu,mill Ultimate strength of according to mill certificate

σu,v Ultimate strength of of virgin material

σy Yield stress χ Buckling reduction factor

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Chapter 1 Introduction

Background

Stainless steel, a family of corrosion resistant alloys of iron containing a minimum of 10.5% chromium, is increasingly used in the construction industry for its many desirable characteristics [1, 2]. In addition to its obvious corrosion resistance, durability, higher strength, better fire resistance, low maintenance cost and attractive appearance make stainless steel a unique construction material, which could be deemed as an alternative to ordinary carbon steel if appropriately utilized. Being durable, recyclable, and corrosion resistant, the use of any toxic coating can be avoided making it an environmental friendly construction material.

The invention of stainless steel was a combined effort of scientists and metallurgists from Britain, Germany, France, Poland, U.S.A, and Sweden. The cogs started in 1820 by Englishmen Stoddard and Farraday, and in 1821 by a Frenchman Pierre Berthier. Relevant research studies were continued by Woods and Clark in England (1872), Brustlein in France (1875), Hans Goldschmidt in Germany (1895), Leon Guillet in France (1904) and Giesen in England (1909). In 1911, German metallurgists P. Monnartz and W. Borchers discovered the importance of a minimum chromium content and pointed out that corrosion resistance of steel increased significantly when at least 10.5 percent chromium content was present. In 1912, English metallurgist Harry Brearley, the lead researcher at Brown Firth Laboratories at that time, was given a task by a small arms manufacturer to protect its gun barrels from corrosion. In August 1913, Brearley became successful in his assignment and created a steel with 12.8% chromium and 0.24% carbon, which was the first commercial production of stainless steel, which gave him the recognition as the inventor of stainless steel. During the First World War, stainless steel was used for manufacturing valves of aircraft engines. After that, it was used for regular production of cutlery and surgical instruments from 1919 to 1923. The first structural application of stainless steel was a roof construction in America in 1924. Other significant early structural uses of stainless steel included construction of a reinforcing chain to stabilize the dome of St Paul’s Cathedral in London in 1925, and construction of top seven arches of The Chrysler Building in New York in 1930. At present, stainless steel is widely

Chapter 1: Introduction used in the construction industry, from small but sophisticated glazing castings to heavy load-bearing girders in bridges.

However, the optimal use of stainless steel in structural applications to date is still not achieved due to the lack of appropriate design guidelines. Although stainless steel and carbon steel exhibits fundamentally different material responses, the codified design procedures for structural stainless steel were primarily based on the available design guidelines of carbon steel. Unlike carbon steel, stainless steel alloys typically show nonlinear stress-strain behaviour at relatively low stress levels, do not exhibit any well- defined yield point and demonstrate significant strain hardening strength. But, current design codes [3-5] treat stainless steel as an elastic, perfectly-plastic material like carbon steel, which do not make an efficient use of this costly material by neglecting its strain hardening benefits. On the other hand, the effective width approach [6] is the only codified approach to deal with the local buckling of slender cross-sections. This approach is quite calculation intensive and is difficult to use for cross-sections with complex geometry and/or stress gradients. Moreover, this approach is suitable for carbon steel due to the presence of yield plateau where material loses its stiffness after yielding. But, stainless steel shows continuous stress-strain response and a continuous treatment of its local buckling phenomenon would be more appropriate.

Tangent Stiffness Method and Perry Formulations are typically used for obtaining the buckling resistance of stainless steel columns. The tangent stiffness method [4, 5] involves simple equations and takes account of material nonlinearity, but the process is iterative and does not explicitly consider the inevitable imperfections of a member. Perry type buckling curves that are currently adopted in EN 1993-1-4 [3], use a direct method involving a series of curves for different cross-section types, and explicitly address member imperfections but do not consider material nonlinearity. Rasmussen and Rondal [7], showed that material nonlinearity and non-dimensional proof stress significantly influence column strength, and a single column curve cannot be used to accurately predict the column strength of different grades of stainless steel. Hradil et al. [8] suggested techniques to include material nonlinearity in Perry curves by defining transformed slenderness but the procedure was iterative as they used tangent modulus. Shu et al. [9] recently proposed two base curves and a number of transfer equations which could be used to develop multiple curves to cover all grades of stainless steel but the suggested 2

Chapter 1: Introduction formulations are too complicated to be used in practice. Importantly, all of the aforementioned techniques require calculation of effective sectional properties for slender cross-sections. They also limit the column resistance to its yield strength, which neglects the strain hardening benefits of stainless steel.

A strain based design method, the Continuous Strength Method (CSM) [10-14], has recently been developed to appropriately incorporate the beneficial effects of material nonlinearity. Primary components of CSM are a base curve, which relates the deformation capacity of a given thin-walled section to the cross-section slenderness, and a material model that explicitly recognises strain hardening. With the recent development of CSM [14], cross-section resistances of relatively stocky sections, which experiences considerable strain hardening before failure, can be predicted using a simple technique. CSM demonstrated accurate and consistent predictions for stocky cross-sections. The very first objective of this current research was to extend CSM for slender sections using the same design base curve proposed for stocky sections. It is however worth mentioning that Zhao et al. [15] have very recently proposed an extension to their approach for slender sections by proposing a different base curve to tackle unique challenges offered by slender sections. These aspects will be discussed in detail later in the thesis.

Material content and classification

There is a wide range of stainless steels offering different levels of corrosion resistance and strength. This varying levels of stainless steel properties are obtained by controlling the contents of alloying element, each affects specific attributes of strength and their ability to be corrosion-resistant in varying environmental conditions. The basic corrosion resistance property of stainless steel comes from the chomium content. When the chromium content is more than 10.5 percent, a very thin (5 × 10-6 mm), stable and non- porous flim is produced instantly in an oxidizing environment on the surface of stainless steel. The flim can be reformed spontanuously in the presence of oxygen, if cutting or scratching damages the film.

This corossion resistance characteristic increases with the increase in chromium content, generally up to 17 percent, and could be further increased by adding molybdenum and nitrogen. Besides corrosion resistance, molybdenum also increases the hardness and

Chapter 1: Introduction enhances mechanical strength and promotes a ferritic structure. Other alloying elements enhance specific properties for specific uses, such as, nickel ensures the correct microstructure and increases ductility and toughness of the alloy. It also reduces the rate of corrosion which makes steel advantageous to be used in acid environments.

Since its invention, the number of grades of stainless steel has increased rapidly. Depending on their microscopic structure at room temperature, stainless steel can be classified into five major categories: austenitic, ferritic, duplex, martensitic and precipitation hardening. Brief descriptions of different grades of stainless steel are given below.

The most widely used type of stainless steels is austenitic, which contains 17 to 18% chromium and 8 to 11% nickel. More than 50% of the global production of stainless steel is austenitic type. The atomic structure of ordinary carbon steels is a body-centered cubic (crystal) structure, whereas, austenitic stainless steels have a face-centered cubic atomic structure. Austenitic grades are readily weldable and easily cold-formable, and have high ductility, in addition to their corrosion resistance. Moreover, austenitic stainless steel also have significantly better toughness compared to structural carbon steels over a wide range of temperatures [16]. Corossion resistance can be increased further by increasing chromium content as well as by adding molybdenum and nitrogen. This type of stainless steels can strenghtened by cold-forming but not by heat treatment.

Ferritic stainless steels have a chromium content of 10.5% to 18% with very small or no nickel. Their body-centered atomic structure is similar to that of structural carbon steels. In comparison to austenitic stainless steels, ferritic stainless steels are less ductile, less weldable and less formable. Like the austenitic, they can be strengthened to a limited extent by cold-working, but cannot be strengthened by heat treatment. Ferritic stainless steel can perform well to resist corrosion, and their corrosion resistance performance can be further enhanced by increasing molybdenum content [17].

Stainless steels possessing a microstructure showing a combination of austenite and ferrite are typically classified as duplex stainless steel. Duplex stainless steels typically have 20 to 26% chromium, 0.05 to 5% molybdenum, 1 to 8% nickel, and 0.05 to 0.3% nitrogen. Recently, a variety of duplex grades lean duplex is gaining widespread 4

Chapter 1: Introduction popularity in structural applications due its very low nickel content, which contains around 1.5% of nickel; this offers a more competitive and less fluctuating price tag for lean duplex. Duplex grades offer significantly higher strength than austenitic grades, and offers good corrosion resistance with sufficient ductility. Like austenitic and ferritic, duplex stainless steels have good weldability and they can only be strengthened by cold- working.

Martensitic stainless steels have higher carbon content, and hence can be strengthened by heat treatment. The atomic structure of martensitic stainless steels is similar to that of ferritic and structural carbon steels showing a body-centred cubic atomic structure. Generally, this type of stainless steels are used in a hardened and tempered condition, where they can provide high strength and moderate corrosion resistance. They are used for making surgical instruments, cutlery, industrial knives, turbine blades, wear plates etc., which can take advantage of wear and abrasion resistance and hardness properties of martensitic stainless steel.

Precipitation hardening steels have corrosion resistance similar to austenitic grades with 18% chromium and 8% nickel content. Very high strength properties of precipitation hardening steels can be achieved by heat treatment. Depending on their type, the microstructure of precipitation hardening steels falls into three families i.e. martensitic, semi-austenitic and austenitic. These steels are mostly used in the aerospace industry but are also used for applications that take advantage of their high strength and moderate corrosion resistance, like tension bars, bolts and shafts.

Applications in the construction industry

Due to high initial cost, low availability and absence of efficient design codes, stainless steel has not traditionally been widely used as a structural material in the construction industry. However, the structural use of stainless steel has been increasing in recent years as its beneficial characteristics suppressed its higher cost. Recent development of cost- effective grades like lean duplex makes stainless steel more competitive in the structural industry. The life cycle cost saving is also an important criterion for selection of stainless steel. Due to its corrosion resistance property, stainless steel could be a viable choice for structures situated in aggressive environments such as, structures in very heavily polluted

Chapter 1: Introduction areas, in proximity to saltwater or exposed to de-icing salts. The use of stainless steel as a primary structural component for pedestrian and vehicular bridges exposed to aggressive environments is rapidly increasing, especially around Europe. The most common applications for vehicular bridges are concrete reinforcing bar, expansion joints, seismic components, bumper structural supports, cable sheathing, pins, and railings and stair components. Stainless steel is also increasingly used for piers, seawalls, parking garages and other structures exposed to costal environment or de-icing salts. In the industrial sector, stainless steel is commonly used in the water treatment, pharmaceutical, chemical, nuclear, pulp and paper, and food and beverage industries for constructing barriers/gates, platforms and equipment supports. Due to high strength, ductility, and strain hardening properties, stainless steel could absorb considerable impact without fracturing, which could be beneficial to protect structures against explosion and impact loading. Corrosion resistance, heat resistance and strength characteristics could make stainless steel alloys suitable for industrial and offshore structures. Due to its superior aesthetics and surface finish, stainless steel structural components are a popular choice for structures supporting glass walls and canopies. Overall, stainless steel possesses some unique special features which could be exploited in structural applications through appropriate design rules.

Notable early uses of stainless steel include construction of the top 88 m of the Chrysler Building in New York with stainless steel cladding in 1930 (Figure 1.1 a). The very first large structural application of stainless steel was the 192 m high Gateway Arch in St. Louis, Missouri (Figure 1.1.b), constructed with 804 tons of 6 mm thick stainless steel Type S30400 in 1965; this construction attracted numerous research into the structural performance of stainless steel in the United States at that time.

Some recent examples of structural stainless steel in different parts of the world are shown in Figure 1.2. The Gatineau Preservation Centre of Canadian National Archive was constructed in 2004 for 500 years of service life with minimal material replacement (Figure 1.2 a). The structural support for the outer building, made of total 1,320 tons of S30403 and S31603 stainless steel, consists 24 m tall 34 stainless steel towers connected by curved beams [17]. The use of stainless steel was justified to add additional protection from the environment, fire, terrorism, vermin and water. In 1995, a greenhouse at Bergianska trädgården, Stockholm, Sweden was constructed by using austenitic stainless 6

Chapter 1: Introduction steel Grade 1.4401 (Figure 1.2 b). Low maintenance requirement of stainless steel was one of the main factors in choosing stainless steel for construction; this also allows the plants to remain undisturbed, which would have otherwise been unavoidable for regular structural maintenance. In 2010, the Helix pedestrian bridge in Marina Bay, Singapore (Figure 1.2 c) was opened for the public. The 280 m structure of the bridge was constructed by two helices made of duplex stainless steel hollow sections spiralled around each other. These are only a few examples of the diverse uses of stainless steel in construction industry in recent times

(a) Chrysler building, New York (1930) (b) Gateway Arch, St. Louis (1965)

Figure 1.1 Early uses of stainless steel as structural material

Chapter 1: Introduction

(a) Gatineau Preservation Centre, Canadian (b) Green house at Bergianska, National Archive trädgården, Sweden

Figure 1.2 Recent examples of use of stainless steel as structural material

Research objectives

The primary objective of this current research was to devise a rational design approach for stainless steel members subjected to compression. In codified approaches, design of stainless steel is considered analogous to carbon steel where effective width is the commonly followed approach to deal with the local buckling of slender sections. Effective width method is a calculation intensive method and is suitable for materials showing obvious yielding. But, stainless steel shows continuous nonlinear stress-strain

Chapter 1: Introduction response, which deserves a continuous treatment for local buckling. Recently developed CSM can accurately predict the cross-section resistance of stocky sections by incorporating the benefits of strain hardening. The first objective of this research was to develop a simple design guideline for stainless steel slender sections using the same CSM approach i.e. without modifying the design base curve. Design formulas were developed to predict the compression resistance and the in-plane bending resistance of stainless steel slender sections using full cross-sectional properties; this made CSM applicable for full range of cross-sections.

Once CSM showed promising performance at the cross-section level, the next step was to look into member behaviour with primary focus on flexural buckling of stainless steel columns. Current design rules for stainless steel columns do not appropriately incorporate the nonlinear material behaviour of stainless steel, and limit the buckling resistance to the traditional squash load for a given cross-section. Previous studies showed that material nonlinearity can have significant effect on the buckling resistance of stainless steel columns. A number of proposals emerged from some recent studies to incorporate material nonlinearity in flexural buckling formulations but the suggested techniques are too complicated to be readily used in practice. This led to devising flexural buckling formulas for widely available cold-formed stainless steel hollow sections, SHS and RHS, by using the same CSM design philosophies. A comprehensive numerical study was carried out to investigate effects of all parameters affecting flexural buckling of stainless steel hollow sections and eventually material nonlinearity was successfully incorporated into the proposed buckling curves.

In large structural applications, welded sections are more suitable to meet the requirements of high load bearing capacity members and could be a more viable option in attaining more efficient and cost effective structural design. However, the behaviour of welded sections could be considerably different from readily available cold-formed sections. The presence of residual stresses is reported to significantly reduce their load carrying capacity, and extremely high temperature effects involved in welding could induce higher geometric imperfections. The next and final objective of this current research was to check the suitability of the buckling formulas proposed for SHS and RHS through experimental and numerical approach. Experimental evidences on stainless steel welded sections are quite limited, and hence the conducted experimental investigation on 9

Chapter 1: Introduction welded I-section columns produced some useful data to the field. Finite element models were developed based on experimental data, and were eventually used to devise flexural buckling curves for welded I-section columns produced from stainless steel.

Thesis outline

This chapter presents a brief introduction on stainless steel as a construction material covering its history, advantages and current design philosophy. A brief description of different grades of stainless steel is presented and their prospective uses in construction industries are shown with some examples of existing structures. A brief insight into the significance and objectives of the current research are also presented.

Chapter 2 contains a review of literatures that are relevant to this research project providing an insight into material behaviour and the current design guidelines. Initially, stress-strain response of stainless steel was discussed with special emphasis on different material models. Strength enhancement due to cold-work is a significant phenomenon for stainless steel due to cold-work was discussed in detail. Discussion on the magnitude and the distribution of inevitable initial geometric imperfections and residual stresses observed in stainless steel members were presented briefly. A summary of major stainless steel design codes covering the American, the Australian/New Zealand and the European design standards was presented, which was followed by a brief discussion on the continuous strength method and the direct strength method.

In Chapter 3, local buckling behaviour of stainless steel slender sections was discussed and a new design method was developed for slender sections retaining the core design principles of CSM. The behaviour of slender sections under axial compression and in- plane bending for commonly used stainless steel sections, Rectangular Hollow Sections (RHS), Square Hollow Sections (SHS) and I- sections, produced from a variety of common grades such as austenitic, ferritic, duplex and lean duplex were studied here. To tackle the observed significant post-buckling of slender cross-sections, a new parameter called Equivalent Elastic Deformation Capacity εe,ev was proposed, and all available test and numerical results were used to developed a relationship between the buckling resistance and εe,ev. The suggested technique was further extended to predict the cross-

Chapter 1: Introduction section resistance, and reliability of the proposed method was evaluated following code guidelines.

Chapter 4 presents a comprehensive study on the buckling behaviour of stainless steel Square Hollow Section (SHS) and Rectangular Hollow Section (RHS) columns. With sufficient experimental evidences available in literature, numerical technique was adopted in the current research to devise a CSM based design technique for flexural buckling of SHS and RHS. A series of Perry type column curves was proposed incorporating the key features observed due to material nonlinearity. Finally, the performance of the proposed method was evaluated and the reliability of this method was verified.

In Chapter 5, the structural behaviour of stainless steel welded I-sections was investigated by giving special emphasis on residual stresses and geometric imperfections originated due to the welding process. Recognising the lack of experimental evidences on stainless steel welded sections, an experimental program was carried out including material test, initial imperfection and residual stress measurements, stub column test and flexural buckling test. Buckling resistance of stainless steel welded I-section columns were investigated using both experimental and numerical results, which eventually led to proposing column curves for welded I-sections.

Finally, Chapter 6 provides a summary of the current research, and identifies the limitations and the possible scopes for future research in this area.

Chapter 2 Literature Review

Introduction

This chapter presents a comprehensive review of previous studies on stainless steel structures that are relevant to the research work presented in this thesis. Firstly, the material behaviour of stainless steel and its response to cold-work are discussed. Studies on the magnitude and the distribution of initial geometric imperfections and residual stresses observed in stainless steel members are discussed later. Gradual development of major stainless steel design standards are also briefly presented covering the American, the Australian/New Zealand and the European design standards. This is followed by an overview of the guidelines for predicting the cross-section resistance and the buckling resistance of stainless steel members based on the design standards as well as using proposals from revised design rules. Finally, different aspects of the Continuous Strength Method (CSM) are discussed in detail.

Material model

Being a nonlinear material, the stress-strain behaviour of stainless steel is different from that of ordinary carbon steel. The stress-strain behaviour of austenitic, ferritic and duplex grades of stainless steel are compared with that of ordinary carbon steel in a schematic diagram in Figure 2.1, which clearly shows the material nonlinearity observed in stainless steel in contrast to the linear elastic behaviour observed for ordinary carbon steel which is followed by a yield plateau. Since there is no well-defined yield point in stainless steel, the 0.2% proof stress is defined as the yield stress. It also exhibits significant amount of strain hardening. Austenitic and duplex grades of stainless steel show higher level of ductility as the plastic strain at fracture is approximately 40-60%, but ferritic grades show lower ductility.

Chapter 2: Literature Review

Figure 2.1 Comparison of the stress-strain behaviour of carbon steel and stainless steel

Many researchers proposed various types of mathematical models to replicate the stress- strain relationship of nonlinear materials like aluminium and stainless steel. Holmquist and Nadai [18] were the first to propose a polynomial equation for this type of nonlinear metallic materials to describe the stress-strain behaviour beyond the proportional limit.

The polynomial expression is given in Eq. 2.1, where σy and εy are the yield stress and strain, respectively, E stands for the Young’s modulus of elasticity and σP is the proportional limit. The parameter n determines the nonlinearity of the curve; for an ideal elastic–plastic material n is infinite.

n σ σ−σP ε= +σy ( ) when σ > σP (2.1) E σ−σP

Ramberg and Osgood [19] proposed a similar model for aluminium alloys as given in Eq. 2.2, where E is the Young's modulus, K is a constant and n is the nonlinearity parameter.

Later Hill [20] modified this model for stainless steel. Using 0.2% proof stress σ0.2 as the yield stress and its corresponding offset plastic strain value (0.002) as the constant K; modified Ramberg-Osgood model is shown in Eq. 2.3, which is widely used in different design codes. Value of n is usually calculated by using Eq. 2.4, where σ0.01 is the 0.01% proof stress.

Chapter 2: Literature Review

σ σ n ε= +k ( ) (2.2) E E

σ σ n ε= +0.002 ( ) (2.3) E σ0.2

ln⁡(20) n = (2.4) ln⁡(σ0.2⁄σ0.01)

The Ramberg-Osgood model was observed to produce excellent predictions for stress- strain behaviour of stainless steel material up to the 0.2% proof stress, but tends to over- predict the stresses beyond this point. In order to overcome this, Mirambell and Real [21] split the model into two steps – they adopted Ramberg-Osgood expression up to 0.2% proof stress and proposed a new but similar type of stress-strain relationship beyond that point. The second step of their model is given in Eq. 2.5 where εu is the strain at the ultimate tensile stress, E0.2 is the tangent modulus at the 0.2% proof stress point (εt,0.2, σ0.2) and can be calculated using Eq. 2.6, and n'0.2,u is an additional strain hardening exponent.

In their model, the parameters σu, εu and n'0.2,u are not related to the basic Ramberg- Osgood parameters and, hence, have to be determined from experimental stress-strain curves. On the other hand, use of ultimate stress and strain makes this model applicable in tension only.

′ n 0.2,u σ−σ0.2 σu−σ0.2 σ−σ0.2 ε= + (εu − εt,0.2 − ) ( ) + ε0.2 (2.5) E0.2 E0.2 σu−σ0.2

E E0.2 = (2.6) 1+0.002n(E⁄σ0.2)

Rasmussen [22] simplified this two-stage Ramberg-Osgood model by reducing the number of parameters, which is given in Eq. 2.7. He developed predictive expressions for the ultimate stress, the ultimate strain and the second strain hardening exponent m as given by Eq.s 2.8-2.11 respectively, and reduced the number of required input parameters to three basic Ramberg-Osgood parameters (E, σ0.2 and n). This model showed good agreement with complete stress-strain curve for stainless steel alloys, and was included in EN 1993-1-4, Annex C [3] for the modelling of the stainless steel material behaviour. But this modified model is appropriate for replicating stress-strain behaviour in tension only as it relies on the ultimate tensile stress σu and the strain at the ultimate tensile stress

εu, which do not exist in compression.

Chapter 2: Literature Review m σ−σ0.2 σ−σ0.2 ε= +εu ( ) + εt,0.2 (2.7) E0.2 σu−σ0.2

σ σ = 0.20 + 185 0.2 for austenitic and duplex stainless steel (2.8) u E

σ 0.20+185 0.2 σ = E for all stainless steel (2.9) u 1−0.0375(n−5)

σ0.2 εu = 1 − (2.10) σu

σ m = 1 + 3.5 0.2 (2.11) σu

Recently Arrayago et al. [23] evaluated the performance of the proposed equations for determining n, m , σu and εu using all available test data. They observed that use 0.05% proof stress (σ0.05) instead of 0.01% proof stress, as given in Eq. 2.12, may provide a better prediction of strain hardening exponential n. They also noticed that Eq. 2.11 provides higher values for the second strain-hardening exponent than those obtained from curve fitting. They proposed a revised expression, given by Eq. 2.13, for all stainless steel grades. Rasmussen [22] proposed two expressions (Eq.s 2.8 and 2.9) to predict σu, where Eq. 2.8 is applicable for austenitic and duplex grades and provides very accurate prediction. But the Eq. 2.9, which was proposed for all stainless steel grades, provides inaccurate predictions. One the other hand, the revised proposal of Real et al. [24] given by Eq. 2.14, produces consistent and accurate predictions of σu for ferritic grades. To assess the accuracy of the prediction of εu, they reported that Eq. 2.10 provides good results for austenitic and duplex grades. However, for ferritic grades this equation yields conservative predictions as ductility of ferritic grades are significantly less than austenitic and duplex grades. Bock et al. [25] modified Eq. 2.10 for ferritic grades, as given by Eq. 2.15, which was shown to produce good agreement with experimental data.

ln⁡(4) n = (2.12) ln⁡(σ0.2⁄σ0.05)

σ m = 1 + 2.8 0.2 (2.13) σu

σ σ = 0.46 + 145 0.2 for ferritic stainless steel (2.14) u E

σ0.2 εu = 0.6 (1 − ) (2.15) σu

Chapter 2: Literature Review In order to improve the accuracy of the model at low strains, and to make the model suitable for both tension and compression behaviour, Gardner and Ashraf [26] modified Mirambell and Real’s originally proposed model [21]. They proposed to use 1.0% proof stress and its corresponding total strain εt,1.0 instead of the ultimate stress σu and the ′ ultimate strain εu in the second stage of the model, which leads to Eq. 2.16 where n 0.2,1.0 is the second strain hardening exponent. The model provides excellent agreement with experimental stress–strain data for both tension and compression; approximately up to 10% tensile strain and 2% compression strain.

′ n 0.2,1.0 σ−σ0.2 σ1.0−σ0.2 σ−σ0.2 ε= + (εt1.0 − εt,0.2 − ) ( ) (2.16) E0.2 E0.2 σ1.0−σ0.2

Additional two-stage models were proposed by Gardner et al. [16], and Chen and Young [27] for stress-strain behaviour of stainless steel at elevated temperatures. Gardner et al. [16] modified the material model proposed by Gardner and Ashraf [26], and Chen and Young [27] extended Rasmussen’s model [22].

Two stage material models show good agreement with experimental behaviour when used in FE modelling of common structural members like columns and beams. However, many structural phenomena such as modelling of cold-forming process or behaviour of structural moment resisting connections require an accurate prediction of stress-strain behaviour up to very high strains. Quach et al. [28] developed a three stage material model to address this issue. For the first stage i.e. for stress up to the 0.2% proof stress, the basic Ramberg-Osgood model [19] was used. For the second stage, the Gardner and Ashraf [26] model was proposed covering stresses up to the 2% proof stress. For the final stage, a linear relationship was proposed from the 2% proof stress to the ultimate strength. More recently, Hradil et al. [29] proposed a generalized multistage model based on Mirambell and Real’s model [21].

The comparative study showed that three-stage models give the most accurate stress- strain relationship for full stress-strain curve of stainless steel but they require too many input parameters. Two-stage model proposed by Rasmussen [22] also shows good agreement with experimental results for stainless steel up to σu using easily obtainable three basic input parameters. So this model provides an optimum balance between

Chapter 2: Literature Review accuracy and practicality for modelling stainless steel columns where the maximum strain is relatively low.

Corner strength enhancement due to cold-work

During cold-work processes like cold-rolling and press-braking, thin-walled steel sections go under significant plastic deformation at corners, which causes strength enhancement of the material at those regions. Karren [30] observed this phenomenon in carbon steel while testing specimens formed by cold-rolling and press-braking. He also observed that the ratio of inner corner radius to plate thickness ri/t and σu/σy ratio have significant influence on corner strength, and proposed a power model for predicting the enhance strength in ordinary carbon steel sections. Coetzee et al. [31] tested some press-braked lipped channels produced from stainless steel and obtained similar results. Van den Berg and Van der Merwe [32] further investigated the phenomena on press-braked sections and proposed a relationship between the 0.2% proof stress of the corner material σ0.2,c and the 0.2% proof stress of the virgin material σ0.2,v based on Karren’s methodology.

The proposed relationship is shown in Eq. 2.17 where Bc and m are constants that depend

σu,v on the ratio of ⁄σ0.2,v where σu,v is the ultimate strength of the virgin material

Bcσ0.2,v σ0.2,c = m (2.17) ri ( ⁄t)

Gardner [33] performed tension tests on coupons cut from corner regions and flat regions of SHS and RHS, and observed a linear relationship between the 0.2% proof stress of the corner material and the ultimate strength of the material collected from the flat face. The proposed Eq. 2.18 relates the ultimate strength σu,f of the flat material with the 0.2% proof stress, σ0.2,c of the corner regions.

σ0.2,c ⁡ = ⁡0.85⁡σu,f (2.18)

Analysing all available stainless steel test data, Ashraf et al. [34] proposed two power models to predict σ0.2,c of stainless steel sections for both press-braked and cold-rolled sections. The first one is a simple power model to predict σ0.2,c using the 0.2% proof stress of virgin material σ0.2,v, as is given in Eq. 2.19, where ri is the inner radius of the corner and t is the thickness of the plate. The second model can also predict σ0.2,c but uses σ0.2,v 17

Chapter 2: Literature Review and σu,v. They also proposed a linear equation, as shown in Eq. 2.20, to predict the ultimate strength of the corner material σu,c based on σ0.2,v and σu,v.

1.881σ0.2,V σ0.2,c ⁡ = ⁡ r 0.194 (2.19) ( i) t

σu,v σu,c = 0.75σ0.2,c ( ) (2.20) σ0.2,v

Cruise and Gardner [35] performed tension coupon tests on cold-rolled box sections and press-braked open sections. They observed that 0.2% proof stress and the ultimate strength of flat faces of box sections were significantly greater than those reported in mill documents. They proposed two expressions (Eq.s 2.21 and 2.22) to predict the 0.2% proof stress and the ultimate strength of the flat faces of cold-rolled box sections, where, b and d are width and depth of the section. For 0.2% proof stress of corner material in cold- rolled box sections, they modified Eq. 2.18, and proposed 0.83 instead of 0.85 as given in Eq. 2.23. Considering their test results and all published data, they recalibrated Eq. 2.19 and proposed Eq. 2.24 for predicting enhanced corner strength for pressed-braked sections. They observed that the extent of corner strength enhancement for press-braked sections is limited within the corner radius. In contrast, for cold-rolled sections, it extends twice the thickness of the plate beyond corner radius. Ashraf et al. [36] also reported similar extension for box sections based on numerical studies.

0.85σ0.2,mill σ0.2,f = 1 (2.21) −0.19+ πt 12.42( )+0.83 2(b+d)

σ0.2,f σu,f = σu,mill (0.19 ( ) + 0.85) (2.22) σ0.2,mill

σ0.2,c ⁡ = ⁡0.83⁡σu,f (2.23)

1.673σ0.2,V σ0.2,c ⁡ = ⁡ r 0.1264 (2.24) ( i) t

Recently, Rossi [37] proposed a generic model for nonlinear metallic materials to evaluate the strength enhancement in the flat faces of cold-rolled box sections, and the corner regions of both cold-rolled and press-braked sections. She determined plastic strains associated with different stages of the cold-work and used the inverted Ramberg-Osgood

Chapter 2: Literature Review model proposed by Abdella [38] to predict the enhanced strength. This model requires complicated and lengthy computational efforts, and involves large number of parameters making it difficult to implement in design calculations. Later, Rossi et al. [39] simplified Rossi’s [37] proposal using simple power law model instead of compound Ramberg- Osgood model, and a simple assessment of strain induced during cold-work. They proposed Eq. 2.25 to predict the 0.2% proof stress for flat faces σ0.2,f of cold-rolled sections, and Eq. 2.26 to predict the σ0.2,c of cold-rolled and press-braked sections. In Eq.s

2.25 and 2.26, εf,av is the average strain induced at the flat face, εc,av is the average strain induced at the corner regions, εt,0.2 is the total strain at σ0.2, and p and q are the material model constants. εf,av, εc,av, p and q can be calculated by Eq.s 2.27-2.32 respectively, where b is the section width and h is the section depth.

q σ0.2f = 0.85[p(εf,av + εt,0.2) ] (2.25)

q σ0.2c = 0.85[p(εc,av + εt,0.2) ] (2.26)

t t ( ⁄2) ( ⁄2) εf,av = [ ⁄ ] + [ ⁄ ] (2.27) Rcoilling Rf

t ( ⁄2) εc,av = 0.5 [ ⁄ ] (2.28) Rc

b+h−2t R = (2.29) f π

t R = r + (2.30) c i 2

σ0.,mill p = q (2.31) εt,0.2

ln(σ ⁄σ ) q = 0.2,mill u,mill (2.32) ln(εt,.02⁄εu)

Amongst all available models, Cruise and Gardner [35] model produces more accurate predictions with low scatter, and the corner strength can be calculated from easily obtainable mill data sheet or flat coupon test.

Chapter 2: Literature Review Residual stress

Differential heating and cooling and/or material deformations at inelastic level during the manufacturing process are the main source of residual stresses. Two types of residual stresses are of main concern in structural members - bending residual stress and membrane residual stress. Rasmussen and Hancock [40] and Gardner [33] observed that the test coupons cut from RHS and SHS became curved due to the release of through thickness bending residual stresses. During the test, those coupons were straightened again, which reintroduced the bending residual stresses into the coupons. So, the bending residual stress are automatically incorporated within the coupons test results collected from within the cross-section, and do not need to be defined explicitly in numerical models. However, the membrane residual stresses induced through welding requires to be defined explicitly in numerical modelling.

Having higher coefficient of thermal expansion and lower value of thermal conductivity, it is probable that behaviour of thermal residual stresses in stainless steel section would be different from those for carbon steel. Bredenkamp et al. [41] observed that the magnitude of residual stresses in built-up stainless steel I-sections were of the same order as in the equivalent carbon steel section, although Lagerqvist and Olsson [42] carried out a similar study and found considerably higher residual stresses in stainless steel sections.

Gardner [33] incorporated idealised rectangular residual stress distribution model around the weld like carbon steel into the FE models of cold-formed stainless steel SHS and RHS. In his study, the effect of residual stresses on the load deformation behaviour of stub columns with hollow sections appeared insignificant. Ashraf et al. [36] also found similar results for hollow sections. However, they recognised that residual stress model like carbon steel may not be appropriate for welded I-sections.

Cruise and Gardner [43] measured residual stresses of hot-rolled angles, press-braked angles and cold-rolled box sections of austenitic stainless steel; both membrane and bending residuals stresses were measured. Analysing the results of hot-rolled sections and press-braked sections, they observed that both bending and membrane residual stresses were relatively low compared to the 0.2% proof stress σ0.2 of the material. In both types of sections, both membrane and bending residual stresses for flat faces were typically less

Chapter 2: Literature Review than 10% of σ0.2. However, the magnitude of bending residual stress at corner regions varied from 20% to 30% of σ0.2. Membrane residual stresses for cold-rolled box sections were found slightly higher than those observed in the hot-rolled and press-braked sections. Bending residual stresses were considerably higher varying between 30% to

70% of σ0.2. Huang and Young [44] also measured the membrane and the bending residual stresses of lean duplex RHS, and found maximum membrane residual stress as 32.5% of

σ0.2 and the bending residual stress as 60.8% of σ0.2. Assuming through thickness rectangular stress block and 95th percentile values as characteristic values, Gardner and Cruse [45] proposed bending residual stress models for hot-rolled angles, press-braked angles and cold-rolled box sections. Based on the test results performed by Bredenkamp et al. [41] and Lagerqvist and Olsson [42], they also proposed residual stress model for welded I-sections. For sections produced from austenitic and duplex stainless steel, they modified the model described in the Swedish design code BSK 99 [46] by increasing the magnitude of the tensile stress to 1.3σ0.2 and by widening the tension region in the web to 3 times the web thickness. For ferritic sections, the tensile peaks in web regions were observed to be significantly lower than the material yield stress. So existing carbon steel model of BSK 99 [46] was proposed for ferritic grades.

Jandera et al. [47] measured residual stresses of cold-rolled RHS of austenitic stainless steel by X-ray diffraction method; both surface and half through thickness measurements were carried out. From surface measurements, a thin layer of compressive residual stress on the outer faces of the specimen was revealed. Through-thickness measurements showed that the distribution of residual stresses through the thickness is nearly uniform with the exception of the thin compressive surface layer.

Yuan et al. [48] measured membrane residual stresses of 18 welded stainless steel I- sections and box sections of austenitic grade (EN 1.4301) and duplex grade (EN 1.4462). They compared their obtained results and all other available test results with the ECCS [49] and the BSK 99 [46] models for carbon steel built-up sections. In all the considered cases, the peak values of tensile residual stresses were lower than the material yield strength; however, the magnitudes of the compressive residual stresses were higher than expected. The zone of peak tensile residual stress is narrower than in carbon steel sections but the total tension region appears to be much wider. The reason for the lower peak tensile residual stresses may be justified by the fact that higher strains are required to 21

Chapter 2: Literature Review reach the equivalent yield stress ( i.e. σ0.2) due to the rounded stress-strain behaviour of stainless steel. Furthermore, the electric resistance of stainless steel is higher than carbon steel, which may lead to less heat input requirement during the welding process of stainless steel sections than those for carbon steel. They proposed the peak tensile residual stress for austenitic grades as 0.8σ0.2 and that for duplex and ferritic grades as 0.6σ0.2. Test results also suggested that flange width has more influence on the size of the peak tensile residual stress region rather than flange thickness. They proposed longitudinal residual stress distribution models for stainless steel welded I-section and box section. The residual stress model of Yuan et al. [48] for welded I-sections is shown in Figure 2.2 with all key parameters given in Table 2.1 for I-sections.

Figure 2.2 Membrane residual stress distribution of a welded I-section [48].

Chapter 2: Literature Review Table 2.1 Parameters of the predictive model of the membrane residual stresses for stainless steel I-sections [48]

Stainless

steel σft= σwt σfc σwc a b c d alloys

Austenitic 0.8σ0.2 a + b 2c + d Duplex σft σwt 0.225bf 0.05bf 0.025hw 0.225hw bf − (a + b) hw − (2c + d) and 0.6σ0.2 Ferritic

Geometric imperfections

During the manufacturing process of thin-walled sections, some imperfections are introduced in cross-sections, whose presence could have significant influence on the capacity of cross-sections. Having superior material strength, stainless steel cross- sections tend to be thinner than ordinary carbon steel, which could make them more vulnerable to initial imperfections [50]. Appropriate understanding of the characteristics of geometric imperfections is to be acknowledged to develop rational design rules for stainless steel. This section reviews relevant investigations on geometric imperfections of thin-walled metallic cross-sections.

Dawson and Walker [51] proposed a simple linear relation between local geometric imperfection and the thickness of the plate, i.e. w0 = Kt, where K is a constant and t is the plate thickness. For cold-formed carbon steel sections, they showed that the value of K is 0.2. In their proposal, some important parameters like material strength, fabrication process and boundary conditions were not considered to predict the imperfection amplitude. To overcome this criticism, they also proposed Eq.s 2.33 and 2.34 for the stiffened compression elements of thin-walled sections, where t is the plate thickness, σy is the material yield stress, σcr is the plate critical buckling stress and α and γ are constants..

0.5 w0/t = α (σy/σcr) (2.33) w0/t = γ (σy/σcr) (2.34)

Chapter 2: Literature Review Schafer and Peköz [52] examined the distribution and magnitude of geometric imperfections of cold-formed steel members. Using their own test results and previous research data, they proposed some rules of thumb to predict the maximum geometric imperfections of internal elements and outstand elements in cold-formed steel sections. Eq.s 2.35 and 2.36 were proposed to predict the local imperfection amplitude of internal elements based on plate width and plate thickness respectively, where w is the width of the plate and t is the thickness of the plate. These rules are applicable for sections with plate thickness less than 3 mm and width-to-thickness ratio less than 200. For outstand elements, the imperfection amplitude can be assumed equal to the plate thickness.

w0 = 0.006w (2.35)

−2t w0 = 6te (2.36)

Gardner and Nethercot [53] investigated the applicability of Dawson and Walker’s proposals ( Eq.s 2.33 and 2.34) in predicting the imperfection amplitudes of cold formed stainless steel members. Using linear regression analysis, they found that the Eq. 2.34 was more suitable for predicting the initial local geometric imperfection of RHS and SHS. They proposed the value of γ to be 0.023, and proposed Eq. 2.37 to predict the local geometric imperfection amplitude in numerical modelling of stainless steel RHS and SHS members. As local geometric imperfections data for CHS were too limited, Gardner and Nethercot [53] performed a parametric study assuming local imperfection amplitudes of 0.01t, 0.1t, 0.2t, and 0.5t (where t is the material thickness) and found that using local imperfection amplitude of 0.2t could replicate the test results most accurately. To predict the global geometric imperfections, they also performed parametric study using three imperfection amplitudes – L/1000, L/2000 and L/5000, where L is the length of column, and found that an amplitude value of L/2000 produced the most accurate agreement between FE ultimate load and test ultimate load. The shape of local initial geometric imperfection was assumed to be the same as that of the lowest local buckling mode or eigenmode, whilst a global imperfection shape was assumed to be that of the lowest global buckling mode. The complete initial imperfection field, therefore, comprised a superposition of the lowest local and global modes.

w0/t = 0.023(σy/σcr) (2.37)

Chapter 2: Literature Review Cruise and Gardner [54] thoroughly studied imperfection measurements on austenitic stainless steel angles and hollow sections produced following three different production routes: hot-rolled and two types of cold-formed such as cold-rolled and press-braked. They reported that global imperfections for cold-rolled sections and hot-rolled sections fall within the acceptable tolerance limits proposed in EN 1993-1-5 [55] and they also tabulate the upper limits and lower limits of the values of γ for different types of sections to be used in Eq. 2.34.

Gardner et al. [56] measured initial geometric imperfections of 20 stub columns produced from structural steel hollow sections, with 10 sections hot-rolled and the rest being cold- formed. Using the measured imperfections to date, they determined the values of α and γ used in Eq. 2.33 and 2.34. Their determined values were α = 0.028 and γ = 0.064 for hot- rolled sections, and α = 0.028 and γ = 0.068 for cold-formed sections, which implies that the imperfection amplitudes for cold-formed sections are slightly higher than those for hot-rolled sections.

Ashraf et al. [36] reviewed all previous research on initial geometric imperfections, and numerically investigated the effects of imperfection on stainless steel stub columns with different open sections (Angle, Chanel, Lipped Chanel and I-Section). Each column was analysed using combinations of first three Eigenmodes with maximum amplitude taken using Schafer and Peköz’s [52] and Gardner and Nethercot’s [53] models. Overall, Gardner and Nethercot’s [53] proposal for the maximum amplitude in conjunction with Eigenmode 1 produced the best agreement with the considered test results..

Lecce and Rasmussen [57] measured initial imperfections of eight lipped Channel columns produced from austenitic 304 alloy, chromium weldable 3Cr12 steel alloy and ferritic 403 alloy. Lecce and Rasmussen [58] also performed finite element modelling of their considered lipped channels where initial imperfections were assigned as amplitude of the critical elastic distortional mode with a value based on Eq. 2.33 with constant α = 0.3.

Becque and Rasmussen [59] performed tests on 24 I-columns (back-to-back Channel) produced from austenitic and ferritic stainless steel alloys. Geometric imperfections of the columns were measured as part of the experimentation. In numerical analysis, Becque 25

Chapter 2: Literature Review and Rasmussen [60] modelled distribution of overall imperfection as the shape of flexural buckling mode with an amplitude of L /1500, whilst local imperfections were modelled using a local buckling mode with an amplitude value proposed by Dawson and Walker [51].

Rossi et al. [61] measured geometric imperfections as part of their 48 full-scale tests on 3Cr12 stainless steel lipped channel section columns. The measured overall imperfections showed that major and minor axis flexural imperfections were less than 1/2000 of the length at the centre of the column.

Theofanous and Gardner [62] measured initial geometric imperfections of lean duplex stainless steel SHS and RHS columns. In their numerical model, the first local and the first global mode shapes were introduced as geometric imperfections in the flexural buckling models. Four different magnitudes such as the maximum measured imperfection, 1/10 and 1/100 of the cross-sectional thickness and the imperfection amplitude derived from the predictive model proposed by Gardner and Nethercot [53] for local imperfection amplitude, were considered in their nonlinear analyses. For global imperfection amplitudes, four fractions of the respective buckling length were considered, namely L/500, L/1000, L/1500 and L/2000, noting that L/1500 represented experimental imperfection. The ultimate load and the end shortening at ultimate load for stub columns were best predicted when an imperfection amplitude from Gardner and Nethercot [53] model or t/100 was used. For flexural buckling model, the most accurate and consistent prediction of test response was obtained when the global imperfection amplitude of L/1500 was used with conjunction of the local imperfection amplitude calculated by Gardner and Nethercot [53] model.

Yuan et al. [63] recently measured initial geometric imperfections of 18 welded sections (10 I-Sections, 4 RHS and 4 SHS) produced from austenitic and duplex grades. In their study, it has been observed that the maximum local imperfection amplitudes for I-section specimens were higher than those observed in hollow sections.

Aforementioned discussions clearly suggest that in numerical analysis of stainless steel structures, the effect of geometric imperfection must be considered. Comparing all other methods, the maximum amplitude of local geometric imperfection can be best predicted 26

Chapter 2: Literature Review by the Dawson and Walker model [51] modified by Gardner and Nethercot [53]. For global geometric imperfection, numerical results show, the best agreement with test results is obtained when the maximum amplitude of global imperfection is taken equal to L/1500, where L is the member length. The lowest elastic buckling mode shape was successfully used in a number of previous research to predict the distribution of initial geometric imperfections.

Existing codes

‘Specification for the Design of Light Gauge Cold-Formed Stainless Steel Structural Members’ was the first design guidelines on structural stainless steel published by the American Iron and Steel Institute (AISI) in 1968. With the availability of increased numbers of test data and better understanding of the structural behaviour of stainless steel, this guidance was revised in 1974 [64]. American Society of Civil Engineers (ASCE) published their first standard for cold formed stainless steel structures as ANSI/ASCE-8- 90 in 1990, which was later revised in 2002 as SEI/ASCE-8-2002 [4]. The American Institute of Steel Construction (AISC) published their stainless steel design specification AISC Design Guide 27: Structural Stainless Steel [17] in 2013 with provision for hot- rolled and welded stainless steel structural sections.

The first European stainless steel design standard ‘Design Manual for Structural Stainless Steel’ was published in 1994 on behalf of Euro-Inox/SCI. In 1996, the European Standards organisation CEN issued the pre-standard Eurocode for stainless steel, referred to as ENV 1993-1-4: Design of Steel Structures, Supplementary Rules for Stainless Steels, which was later converted to a full European Standard, EN 1993-1-4:2006 [3]; this version of EN provided supplementary provisions for the design of stainless steel buildings and civil engineering works. The standard only supplements, modifies or supersedes the equivalent carbon steel provisions, and should be used alongside the relevant carbon steel parts, EN 1993-1-1:2005 [65], EN 1993-1-2:2005 [66], EN 1993-1- 3:2005 [67]. EN 1993-1-2 outlines the fire design provisions for stainless steel structures, and EN 1993-1-4 has recently been amended as EN 1993-1-4:2006+A1:2015 [68].

For cold-formed stainless steel structures the Australian/New Zealand stainless steel design standard AS/NZS 4673 [5]was published back in 2001 [5], which is similar to the

Chapter 2: Literature Review ASCE/SEI-8 [4] specifications. Rasmussen reported the background work of this design standard in [69].

Design guidelines for cross-section resistance

The concept of cross-section classification is used in Eurocode EN 1993-1-4 [3] to treat local buckling phenomenon in stainless steel cross-sections. In which, stainless steel is treated like carbon steel where the presence of clearly defined yield point, beyond which instability is triggered by sudden drop of material stiffness, allows cross-sections to be set into discrete behavioural classes; this essentially is a criteria that helps to identify whether the cross-section can attain yield stress before local buckling or not. EN1993-1- 4 [3] adopts the carbon steel cross-section classification approach set out in EN1993-1-1

[65], with the yield stress σy taken as the 0.2% proof stress σ0.2. For a cross-section subjected to pure compression, failure may occur either by material yielding and inelastic local buckling in the case of stocky cross-sections (class 1–3) or by local buckling at an average stress below the yield stress for slender cross-sections (class 4). For class 1-3 cross-sections, the compression capacity is equal to the yield load. However, the bending moment capacity of class 1 and 2 is equal to the plastic moment capacity, where class 1 cross-sections are able to sustain comparatively higher deformation capacity, and for class 3 it is equal to elastic moment capacity. The compression capacity and the bending moment capacity of class-4 cross-sections are less than yield load and elastic moment capacity respectively. A series of limits for the width-to-thickness ratios (b/t), in terms of 0.5 the material properties ε=[(235/σy)(E/210000)] , edge support conditions (i.e., internal or outstand) and the form of the applied stress field, are provided. The overall cross- section classification is assumed to relate to that of its most slender constituent element, thus neglecting the benefits of element interaction. Analysing comprehensive experimental database, Grader and Theofanous [70] has shown that the classifications limits are over conservative. They harmonised these limits with those for carbon steel. Recently their proposal was incorporated in EN 1993-1-4+A1 [68].

Due to the elastic local buckling, capacity of slender cross-sections cannot reach to their yield capacity. The capacity of a slender cross-section is calculated by the effective width approach developed by Johanson and Winter [6], where some parts of such cross-sections are considered ineffective for elastic local buckling. According to SEI/ACSE-8 [4] and 28

Chapter 2: Literature Review Australia/ New Zealand code [5] effective width approach is applicable when the cross- section slenderness exceeds the limiting value of 0.673, whilst in EN1993-1-4 [3] effective width approach is applicable for class 4 cross-sections. Although the effective width approach is widely used in international design codes, it is a calculation intensive method and complexities arise in applying the method for cross-sections with complicated geometry. Due to the resulting shift in neutral axis of cross-sections, calculation of effective sections and the corresponding stress distribution may become an iterative process. Moreover, use of elastic, perfectly-plastic material model for stainless steel in all aforementioned codes, essentially neglects the rounded nature of the stress- strain response as well as significant strain hardening of stainless steel. Recently proposed design method like the Continuous Strength Method (CSM) and the Direct Strength Method (DSM) incorporated appropriate element interaction and material strain hardening benefits. Brief discussions on CSM and DSM are presented in sections 2.9 and 2.10.

Design guidelines for flexural buckling resistance

The tangential stiffness method and Perry-Robertson formulas are two widely used methods to determine the buckling resistance of steel columns. Like carbon steel, the Perry type equations were adopted in EN1993-1-4 [3] for stainless steel columns. Currently used buckling formulas in EN1993-1-4 are presented in Eq.s 2.38 to 2.42, where Ag is the gross cross-sectional area, Aeff is the effective cross-sectional area, fy is the 0.2% proof stress (σ0.2), χ is the buckling reduction factor, λ is the non-dimensional member slenderness of the column and Ncr is the elastic critical buckling load of the column based on gross area. Effective cross-section properties are used to deal with the local buckling of slender cross-sections of class 4. Four column curves were proposed for different cross-section and loading types, and they differ from each other by varying a linear function of imperfection parameter η. The suggested imperfection parameter η is expressed using the relationship, η = α(λ-λ0) where α and λ0 factors vary depending on cross-section types. The effect of residual stress on welded sections is also included in η. Stainless steel is a highly nonlinear material and its nonlinearity significantly varies from grade to grade. Rasmussen and Rondal [7] showed that these nonlinearity has a significant effect on column resistance. However, in the current European guidelines, the effect of material nonlinearity is not recognised. 29

Chapter 2: Literature Review

Nu = χAgfy for Class 1, 2 and 3 cross-sections (2.38)

Nu = χAefffy for Class 4 cross-sections (2.39)

A f λ = √ g y for Class 1, 2 and 3 cross-sections (2.40) Ncr

A f λ = √ eff y for Class 4 cross-sections (2.41) Ncr

1 χ = ⁡ ≤ 1.0 where, ϕ = 0.5[1 + η + λ2⁡] and η = α(λ − λ ) (2.42) ϕ+√ϕ2−λ2 0

Tangential stiffness method is based on Euler formulas [71]. The American [4] and the Australian codes [5] follow the tangential stiffness approach to determine column resistances. This method involves a simple equation as given in Eq. 2.43, where Fn is buckling stress, which can be calculated by using Eq. 2.44. Calculation of Fn requires iterations as Fn and tangent modulus Et are interdependent. By using Et, the effect of material nonlinearity of stainless steel is incorporated into the tangential stiffness method. Although member imperfections have significant effects on column resistance, they are not considered in this approach. All design codes limit the maximum compression capacity of a section to its squash load ignoring the strain hardening strength of stainless steel. To overcome the shortcomings of the codes, Rasmussen and Rondal [7], Hradil et al. [8] and Shu et al. [9] proposed techniques to incorporate material nonlinearity in column curves; their suggested techniques are discussed in the following sub-sections.

Nu = AeffFn (2.43)

π2E F = t ⁡⁡ ≤ f (2.44) n (KL⁄r)2 y

Rasmussen and Rondal [7] numerically investigated the buckling behaviour of nonlinear metallic columns, and showed that proof stress and material nonlinearity have significant effects on column curves. Considering these effects, Rasmussen and Rondal [10] modified the imperfection parameter η into a nonlinear function as shown in Eq. 2.45, and parameters α, β, λ1 and λ0 were expressed as functions of non-dimensional proof

Chapter 2: Literature Review stress e (= σ0.2/E) and strain hardening exponent n. This modification allowed incorporating the material parameters as well as geometric imperfections in column curves. The performance of their proposed formula was very good but too complex to be used in design practice. However, this technique was included in the AS/NZS 4673 [5] as an alternative approach by proposing values of α, β, λ1 and λ0 for some selective grades of stainless steel.

β η = α((λ − λ1) − λ0) (2.45)

Hradil et al. [8] observed that traditional tangent stiffness method and Perry formulas do not appropriately consider geometric imperfections and material nonlinearity. They suggested to incorporate material nonlinearity in Perry formulas by modifying the non- dimensional member slenderness λ; a new parameter called transformed slenderness (λ*) was proposed as shown in Eq. 2.46 where n is the material strain hardening exponent, E is the initial Young’s Modulus and χ is the buckling reduction factor as given in Eq. 2.42.

They recalibrated the imperfection factors α and λ0 given in EN 1993-1-4 [3] for different sets of material properties. But this method requires iteration as the transformed slenderness and the reduction factor depend on each other.

E λ∗ = λ⁡√1 + .002n χn−1 (2.46) σ0.2

Shu et al. [9] observed the effect of non-dimensional proof stress e and strain hardening exponent n on flexural buckling capacity of columns, and concluded that a single column curve cannot represent the flexural buckling response of all different grades of stainless steel. Two base column curves were proposed for two groups of materials depending on their n values with Group A representing column curve for n = 3, 4 and 5, whilst Group B belonging to columns with strain hardening exponent n = 6, 7, 8 and 9. An analytical technique was proposed to convert the non-dimensional member slenderness of any column to their equivalent slenderness, which is subsequently used in the corresponding base curve. In addition, quadratic polynomial functions of λ were proposed to determine η. However, their proposal is calculation intensive, and is only applicable for columns with high slenderness that are susceptible to flexural buckling only. Columns failing due to local buckling and combined effect of local and flexural buckling are not within the scope of the proposed method. 31

Chapter 2: Literature Review It is worth noting that all aforementioned methods limit the column strength to their squash load ignoring the effects of material strain hardening. They also use effective width approach to deal with the loss of effectiveness due to local buckling as commonly observed in slender sections. Huang and Young [72] recently suggested to use stub column properties with full cross-sectional area for both slender and stocky cross-sections to avoid the lengthy technique of calculating effective cross-sectional properties. However, stub column properties are not commonly available for all cross-sections making this proposal valid only for specific research projects.

All current codes and other proposed methods set the upper limit for column resistance to its yield load ignoring the significant strain hardening benefits of stainless steel. These codes also use complicated effective width approach to deal with the loss of effectiveness due to local buckling as commonly observed in slender sections. Some proposals [7-9] were also made to incorporate obvious material nonlinearity in flexural buckling formulas, but their proposals are too complex to be used in practical design. Stainless steel columns require new design formulas, which can incorporate strain hardening benefits and acknowledge material nonlinearity in a simple manner. Recently developed strain based design technique, the continuous strength method (CSM), already allows for strain hardening benefits in predicting the cross-section resistances. Extending the concept of CSM for predict the buckling capacity of long columns is one of the major objectives of the current research. All relevant technical details of CSM are discussed in the following section.

Continuous Strength Method

The Continuous Strength Method (CSM) is a strain based design approach where local buckling of nonlinear metallic cross-sections is treated as a continuous function replacing the traditional codified cross-section classifications. Primary components of CSM are a base curve, which relates the deformation capacity of a section with its cross-section slenderness and a material model that explicitly recognises strain hardening. Gardner and Nethercot [73] first developed the concept for stainless steel hollow sections. Gardner and Ashraf [26] extended the technique for other nonlinear metallic materials such as aluminium and high strength steel showing its general applicability, and Ashraf et al. [50] proposed generalized design rule “Resistance based on Deformation Capacity” for both 32

Chapter 2: Literature Review hollow and open stainless steel cross-sections. Gardner [74] first introduced the term “Continuous Strength Method” and made the technique more generalized for easy application.

In the CSM, a relationship between cross-section deformation capacity and cross-section slenderness has been established based on the result of stub column tests. Initially, the cross-section slenderness (β) was defined by slenderness of the most slender element of the section as given by Eq. 2.47, where b and t are the element centreline width and thickness, respectively, σ0.2 and E are the material 0.2% proof stress and Young's modulus, respectively and kσ is the plate buckling coefficient –taken as 4.0 for internal plate elements in compression, 0.425 for outstand plate elements in compression and 23.9 for internal elements in bending. The cross-section deformation capacity (εlb) was defined as the compression local buckling strain at ultimate load given by Eq. 2.48, where 훿u is the deflection of a stub column at the ultimate load as shown in Figure 2.3 and L is the length of the stub column. The values of εlb of all available stub column test results were normalized by the corresponding yield strain (εy) and plot against β. They followed a common trend as shown in Figure 2.4. The relationship between the cross-section normalised deformation capacity εlb/εy and β was considered as base curve in CSM.

Ashraf et al. [12] proposed Eq. 2.49 as base curve, where εy is the elastic strain corresponding to the 0.2% proof stress (i.e. the yield strain).

Figure 2.3 Typical Load vs End shortening curve of a stub column

Chapter 2: Literature Review

b σ β= √ 0.2⁄ √4⁄ (2.47) t E kσ

δ ε = u (2.48) lb L

εlb 6.44 = 2.85−0.27β (2.49) εy β

Figure 2.4 CSM base curve proposed by Ashraf et al.[12]

Gardner and Theofanous [13] used the standard plate slenderness definition, as given in

Eq. 2.50, to calculate the cross-section slenderness which is denoted by λp, where σcr is the elastic critical buckling stress of the plate element, υ is the Poisson's ratio and other parameters are as previously defined. They establish a relationship between the cross- section normalised deformation capacity εlb/εy and λp. Their relationship is given is Eq. 2.51.

2 σ0.2 b σ0.2 12(1−υ ) 휆푝=√ = √ ⁄ √ (2.50) σcr t E kσ

εlb 1.43 = 2.71−0.69휆 (2.51) εy 푝 휆푝

The buckling stress, the maximum stress attained by a cross-section prior to local buckling, can be derived from the value of εlb using the compound Ramberg-Osgood material model modified by Gardner and Nethercot [10] to determine the cross-section resistance. However, the material model was expressed in terms of strain, and hence a

Chapter 2: Literature Review number of iterations are required to calculate the local buckling stress i.e. cross-section resistance. For cross-sections subjected to bending, Ramberg-Osgood material model deduces a nonlinear bending stress distribution along the section depth, which involves numerical integration; this complicates the determination of cross-section bending resistance. The concept of a generalised shape factor was adopted to avoid this complicated computational technique by Gardner [33], details of which are provided by Gardner [33] and Ashraf et al. [12]. Overall, the continuous strength method clearly produced better predictions of cross-section resistances, but was not straightforward at early stages of development.

Afshan and Gardner [14] used the concept of full cross-section slenderness in CSM instead of using the slenderness of the most slender element of a cross-section; this concept was adopted from the direct strength method (DSM) [75]. The expression of cross-section slenderness is given in Eq. 2.52 where σcr,cs is the elastic buckling capacity of the full cross-section and σ0.2 is the 0.2% proof stress of the material. σcr,cs may be determined using existing numerical tools (CUFSM) [76] or utilising approximate analytical methods reported by Seif and Schafer [77]. Use of σcr,cs in defining λp makes appropriate recognition of element interactions within a cross-section. They analysed a large amount of test data on stainless steel, carbon steel and aluminium alloys, and observed that strain hardening was only beneficiary for sections whose λp are less than or equal to 0.68. This λp value was proposed as the common slenderness limit for stainless steel, carbon steel and aluminium alloys to define the transition between slender and stocky sections. A cross-section with λp value less than or equal 0.68 is considered as a stocky cross-section, and those exceeding this limit are considered as slender cross- sections.

σ0.2 휆푝=√ (2.52) σcr,cs

Instead of using compound Ramberg-Osgood material model [11-13, 26, 73, 74, 78], Afshan and Gardner [14] proposed an elastic, linear hardening model as shown in Figure 2.5. The origin of the new material model is at (0.002, 0) and the curve passes through the yield point (fy, εy) and the point of (fu, 0.16εu), where εu is the ultimate tensile strain and fu is the ultimate tensile strength. The deformation capacity εcsm is obtained by

Chapter 2: Literature Review subtracting the plastic strain at 0.2% proof stress from the actual local buckling strain εlb to make it compatible with the adopted material model. The deformation capacities of cross-sections normalized by the elastic strain εy were determined for stub columns using Eq. 2.53, whilst those for beams subjected to 4-point bending were calculated using Eq.

2.54, where δu is the deflection at the ultimate load of a stub column, L is the total length of the stub column, κu is the curvature at the ultimate moment, κel is the elastic curvature corresponding to the elastic moment capacity and ymax is the distance from the neural axis to the extreme compressive fibre.

Figure 2.5 Material model proposed by Afshan and Gardner [14]

δ u−0.002 εcsm εlb−0.002 L = = for λp ≤ 0.68 (2.53) εy εy εy

εcsm εlb−0.002 κuymax−0.002 = = for λp ≤ 0.68 (2.54) εy εy κelymax

Afshan and Gardner [14] used available stub column and in-plane bending test results of stainless steel sections and stub column test results of carbon steel sections for developing the base curve. As a base curve, the normalized deformation capacity was expressed as a function of λp up to the limiting slenderness 0.68 with two upper limits as shown in Eq. 2.55. Base curve proposed by Afshan and Gardner [14] is shown in Figure 2.6. Bock et 36

Chapter 2: Literature Review al [25] modified the upper limit for less ductile ferritic grades of stainless steel as given in Eq. 2.56. Having determined the normalised deformation capacity of a cross-section

εcsm/εy using the design base curve in conjunction with the proposed material model, the limiting buckling stress fcsm for the cross-section can be calculated using Eq. 2.57 where

Esh is the slope of the hardening curve and can be calculated by Eq.s 2.58 and 2.59. Finally, the cross-section resistance in compression can be predicted using Eq. 2.60, where Ag is the gross cross-sectional area. Assuming plane sections remain plain and normal to the neutral axis in bending, in-plane moment capacity of a cross-section may be derived from Eq. 2.61, where f is stress in the section with a maximum outer fibre value of fcsm, y is the distance from the neutral axis and dA is the differential cross- sectional area. North American design code AISC Design Guide 27 [17] recently incorporated this method as an alternative design approach.

εcsm 0.25 εcsm 0.1εu = 3.6 but ≤ 15, for austenitic and duplex grades for 휆푝 ≤ 0.68 (2.55) εy 휆푝 εy εy

εcsm 0.25 εcsm 0.4εu = 3.6 but ≤ 15, for ferritic grades for 휆푝 ≤ 0.68 (2.56) εy 휆푝 εy εy

εcsm fcsm = fy + Eshεy ( − 1) for 휆푝 ≤ 0.68 (2.57) εy

σu−σy where, Esh = for austenitic and duplex grades (2.58) 0.16εu−εy

σu−σy Esh = for ferritic grades (2.59) 0.45εu−εy

Ncsm = fcsmAg (2.60)

M = fydA (2.61) csm ∫A

Chapter 2: Literature Review

Figure 2.6 CSM base curve proposed by Afshan and Gardner [14] for stocky sections

The proposed CSM technique is a simple, straightforward and an accurate method for predicting cross-section resistances for stocky sections. However, in this current form CSM approach is not applicable for slender sections. As part of the current research, the scope of CSM is extended for slender sections, which is discussed in detail in Chapter 3.

Direct Strength Method

The direct strength method (DSM) was developed for designing cold-formed steel members. One of the major features of this design approach was to overcome the limitation of the effective width method, which is difficult to apply for cross-sections with complex geometry. Based on the earlier work of Hancock et al. [79] covering local, distortional and global buckling on carbon steel members, the direct strength method (DSM) was proposed by Schafer and Peköz [75]. The main idea of DSM is that the capacity of a cold-formed carbon steel member can be predicted directly form the elastic critical stresses of the complete cross-section for different failure modes, i.e. local, distortional and global buckling, using different strength curves. The strength curve relates the nominal resistance of a member to its yield strength and slenderness. The Australian [80] and the North American design specifications [81] for cold-formed carbon steel adopted DSM as an alternative design approach. This procedure eliminates the cumbersome calculation of effective width method commonly used in different codes for slender sections, and made the design process simple for cross-sections with complex 38

Chapter 2: Literature Review geometry as well as for cross-sections with elements under stress gradients. The strength curve for the nominal member capacity of a member in compression for flexural, torsional or flexural-torsional buckling Nne may be calculated using Eq. 2.62, where λc is the member slenderness as expressed by Eq. 2.63. The nominal axial capacity of a member for local buckling Nnl may be calculated using Eq. 2.64, where λl is the cross-section slenderness, which can be calculated by Eq. 2.65. In Eq.s 2.63 and 2.65, Ny is the yield capacity and Ncre and Ncrl are the critical elastic buckling load in corresponding failure mode. Use of critical local buckling load for a full cross-section, makes appropriate recognition of the beneficial effect of element interactions in determining cross-section resistance.

2 λc (0.658 )Ny⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡forλc ≤ 1.5 Nne = { 0.877 (2.62) ( 2 ) Ny⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡forλc > 1.5 λc

Where

Ny λc = √ (2.63) Ncre

Nne⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡forλl ≤ 0.776⁡ Nnl = { 1 0.15 (2.64) ( 0.8 − 1.6 ) Nne⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡forλl > 0.776 λl λl

Where

Nne λl = √ (2.65) Ncrl

In the recent past, a number of researches proposed extensions for the application of the DSM to other metallic material like stainless steel. Becque et al. [82] investigated the compressive behaviour of lipped channel, I-sections and RHS produced from different grades of stainless steel. They did not observe any significant difference between different grades, and proposed a single strength curve following DSM for all grades to determine

Nnl as given in Eq. 2.66, but they lowered the limiting slenderness value to 0.474. Niu et al. [83] showed that Eq. 2.66 can be used conservatively to determine the bending resistance of cross-sections. To predict the compression capacity for overall buckling

Nne, they proposed to use Perry formulas suggested by Rasmussen and Rondal [7] and 39

Chapter 2: Literature Review adopted in AS/NZ 4673 [5] or EN 1993-1-4 [3]. Huang and Young [84] also investigated the behaviour of lean duplex columns and modified Eq.s 2.62 and 2.64.

Nne⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡forλl ≤ 0.474⁡ Nnl = { 0.95 0.22 (2.66) ( 0.8 − 1.6 ) Nne⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡forλl > 0.474 λl λl

In DSM, cross-section resistance of stocky sections is limited to its yield load, which restricts this method to get benefits of strain hardening of stainless steel. Rossi and Rasmussen [85] included the strain hardening effect into the DSM design approach. They considered that the resistance of a cross-section is equal to the ultimate load of the cross- section at zero member slenderness and is equal to the yield load of the cross-section at limiting member slenderness. They suggested that, this cross-section resistance varies linearly from ultimate load to yield load with the variation of member slenderness from zero to limiting slenderness. They proposed Eq.s 2.67 and 2.69, where λLIM is the limiting slenderness and can be calculated by Eq. 2.68, where β, λ0, λ1 are imperfection factors proposed by Rasmussen and Rondal [7]. Although they proposed this expression for compression, it can be used for other loading conditions like bending.

λc σu [(1 − ) + ( − 1)]⁡Ny⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡forλc ≤ λLIM⁡ λLIM σ0.2 Nne = { Ny (2.67) ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡forλc > λLIM 2 2 φ+√(φ −λc)

1 ⁄β Where λLIM = (λ0 − λ1), (2.68)

σ [1 + (1 − 2.11λ ) ( u − 1)]⁡N ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡forλ ≤ 0.474⁡ l σ ne l N = { 0.2 (2.69) nl 0.95 0.22 ( 0.8 − 1.6 ) Nne⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡forλl > 0.474 λl λl

Recently Arrayago et al. [86] showed that the DSM approach for calculating the local buckling resistance considering the carbon steel strength curve as given as in Eq. 2.64 produced better results than those proposed by Becque et al. [82]. For sections with low slenderness, they applied the approach of Rossi and Rasmussen [85] to incorporate strain hardening effects in Eq. 2.64 and proposed Eq. 2.70.

Chapter 2: Literature Review

σu [1 + (1 − 1.29λp) ( − 1)]⁡Nce⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡forλl ≤ 0.776⁡ σ0.2 Nnl = { 1 0.15 (2.70) ( 0.8 − 1.6 ) Nce⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡forλl > 0.776 λp λp

Summary

An overview of the literature relevant to this thesis has been presented in this chapter. A comparative study on different mathematical models used to simulate the stress-strain response of stainless steel alloys showed that the two-stage Ramberg-Osgood model proposed by Rasmussen [22] with modifications recently suggested by Arrayago et al. [23] produce the most accurate predictions and is most widely used in relevant research. Corner strength enhancement is an unavoidable issue in FE modelling of stainless steel structures. A number of proposals were made during the last decade, and Cruise and Gardner’s [35] proposals are easy to use and are also reported to produce reliable results. Studies on the residual stress of stainless steel welded sections are very limited with only one complete predictive model is available in the literature [48]. It was reported that stainless steel sections might possess considerably different residual stresses than carbon steel; further extensive research is required to explore the full extent of residual stresses in welded stainless steel sections. In regards to modelling initial geometric imperfections, it was observed that Dawson and Walker [51] model modified by Gardner and Nethercot [53] produced reliable predictions for member resistances. For global geometric imperfection, L/1500 is the most recommended value. In most of the studies, eigen modes were successfully used for modelling the distribution of geometric imperfections. Analysis of existing design guidelines for cross-section resistance and buckling resistance of stainless steel members showed that the current design standards fail to incorporate the beneficial special characteristics of stainless steel to ensure the optimal use of this costly material. Finally, a discussion on the newly developed CSM and DSM was presented.

Chapter 3 The Continuous Strength Method for Stainless Steel Slender Sections

Introduction

Slender cross-sections are quite common in practice when dealing with high strength material like stainless steel. The effective width method, originally proposed for ordinary carbon steel in recognition to its characteristic yielding phenomenon, is the only codified approach to deal with the loss-of-effectiveness observed in slender sections due to local buckling. This chapter focuses on the local buckling behaviour of stainless steel slender cross-sections and describes a new approach to calculate the cross-section resistance of such sections based on the Continuous Strength Method (CSM). Details of CSM have already been described in Section 2.9. The current simplified CSM [14] does not apply for slender sections as the focus was on exploitation of strain hardening as observed in stocky sections. Hence, the scope of CSM was extended for slender cross-sections so that a similar technique could be used without any need for lengthy computational efforts like the effective width approach. Axial compression and in-plane bending resistance of stainless steel Rectangular Hollow Sections (RHS), Square Hollow Sections (SHS) and I- sections produced from a variety of common grades such as austenitic, ferritic, duplex and lean duplex were studied in the current research. Numerical models were developed and verified against available experimental results to produce additional structural performance data on slender cross-sections. All available experimental and numerical results were used to establish relationships between the cross-section deformation capacity of CSM (csm) and the Equivalent Elastic Deformation Capacity (e,ev), which is a new parameter proposed herein, to accurately capture the deformation capacities of slender sections that experience loss of effectiveness due to local buckling. Gross cross- sectional properties were used in the proposed technique to be in line with CSM; this eliminates the need for calculating the effective cross-sectional properties. Overall, the proposed technique produces accurate, consistent and reliable predictions for compression and bending resistance of slender sections. The outcome of the presented research was published in 2016 [87]. Recently, however, Zhao et al. [15] also proposed a different technique based on CSM for predicting cross-section resistance of slender sections, the details of which will be discussed later.

Chapter 3: The Continuous Strength Method for Stainless Steel Slender sections Experimental results

A total of 177 experimental data on stub columns including 97 slender sections representing RHS, SHS and I-sections were analysed in the current study. Out of 108 in- plane bending tests considered in the current study, including both 3-point and 4-point bending, only 18 specimens fall within the slender category. Table 3.1 lists the number of experimental data used in this study with references, where number of slender sections is shown within parentheses. Considered experimental data were analysed using the corresponding material properties obtained from tension coupon tests as compression coupon tests are not very common due to the obvious difficulty of applying compression to a thin coupon. Corner coupon test results were also considered for cold-formed hollow sections; weighted average of flat and corner coupon properties were considered as the representative material property for the full cross-section. Where properties of corner coupons were not reported, enhanced 0.2% proof stress for corner region was calculated using the model proposed by Cruise ad Gardner [35]. For I-sections, the material properties were calculated as the weighted average of those reported for flange and web. The equations for calculating the weighted average material properties for SHS, RHS and I-sections are given in Appendix C.

Development of the finite element model

A numerical simulation program was carried out using commercial finite element analysis package ABAQUS [88]. Initially, the developed FE models were validated against experimental results, and the sensitivity of FE models to various input parameters were thoroughly investigated. Once verified, the FE models were used to perform parametric analysis generating additional structural performance data to supplement limited experimental evidences on slender sections.

Guidelines proposed by Gardner and Nethercot [53] and Ashraf et al. [36] were followed to develop the FE models including material and geometrical nonlinearity. Four-node doubly curved shell element with reduced integration, known in ABAQUS as S4R, was used in FE modelling with a uniform mesh for I-sections as well as the flat portions of RHS and SHS, with element size not exceeding 5 mm, to achieve reliable results at a

Chapter 3: The Continuous Strength Method for Stainless Steel Slender sections Table 3.1 List of experimental results considered in the current study on stub columns and in-plane bending

Reference Stainless steel Number of stub column Number of in-plane types test bending test RHS SHS I-section RHS SHS I-section Rasmusen and Austenitic (1.4307) - 2 - - 1 - Hancock [40] Talja and Salmi [89] Austenitic (1.4301) 2 (1) 1 (1) 6 3 - Stangenberg [90] Austenitic (1.4301) - - 4 (1) - - - Stangenberg [91] Ferritic (1.4003) - - 7 - - - Austenitic (1.4301, Kuwamura [92] - 12 (7) 16 (12) - - 4 1.4318) Liu and Young [93] Austenitic (1.4301) - 4 (2) - - - - Young and Liu [94] Austenitic (1.4301) 8 (4) - - - - - Gardner and 16 Austenitic (1.4301) 15 (6) - - - - Nethercot [10] (6) Gardner and Austenitic (1.4301) - - - 4 5 (2) - Nethercot [95] Real and Mirambell Austenitic (1.4301, - - - 1 1 1 [21] 1.4318) Young and Lui [96] Duplex and HSA 3 (3) 6 (3) - - - - Zhou and Young [97] Austenitic (1.4301) - - - 7 8 (3) - Gardner et al. [98] Austenitic (1.4318) 4 (4) 4 (3) - 4 2 (2) - Theofanous and Lean Duplex 2 4 - - Gardner [62] (1.4162) Theofanous and Lean Duplex - - - 4 9 (1) - Gardner [99] (1.4162) Huang and Young Lean Duplex 4 (3) 2 (1) - - - - [44] (1.4162) Saliba and Gardner Lean Duplex - - 4 (2) - - 8 (2) [100] (1.4162) Afshan and Gardner Ferritic (1.4003, 4 (2) 6 4 4 - [101] 1.4509) Huang and Young Lean Duplex - - 8 (2) 2 (1) - [102] (1.4162) Austenitic (1.4301) Yuan et al. [63] - - 15 (10) - Duplex (1.4462) Austenitic (1.4301, 1.4307, 1.4404, Zhao et al. [103] 2 3 - 2 3 - 1.4571) Duplex (1.4762) 10 Shu et al. [104] Austenitic (1.4301) 9 (9) - - - - (10) Bock et al. [105] Ferritic (1.4003) 6 (6) 2 (2) - 7 (4) 2 - Arrayoga et al. [106] Ferritic (1.4003) - - - 6 (1) 2 - 60 71 53 42 Total 46 (25) 13 (2) (38) (34) (7) (9) The number in the parenthesises is the number of slender sections.

Chapter 3: The Continuous Strength Method for Stainless Steel Slender sections reasonable computational time. Finer mesh was used at corner regions of RHS and SHS columns and beams. Typical mesh distributions for an SHS and an I-section are shown in Figures 3.1 and 3.2. RHS and SHS examined in this study were cold-rolled. During the forming process, cold-rolled sections experience enhancement in material strength as a result of inevitable plastic deformation. This phenomenon is more prominent in the corner regions and, hence, enhanced corner strengths were used up to 2t beyond the curved portion into the flat region [34, 35]. Two-stage Ramberg–Osgood (R–O) [19] material model proposed by Rasmussen [22], with recent modifications proposed by Arrayago et al. [23] was used in developing the FE models. The material model is presented in Appendix B. Strength enhancement in the flat region is automatically incorporated through material coupons taken from within the cross-sections [34, 35]. The details of material model and the predictive formulas for corner strengths have been discussed in Section 2.2 and 2.3 respectively.

Structural response of thin-walled members is affected by the presence of initial geometric imperfections. In the current FE analysis, only the local geometric imperfections were considered, as global buckling was not expected in any of the stub column specimens used to investigate the cross-sectional behaviour. Eigenvalue analysis was performed and the lowest elastic buckling mode was used to simulate the distribution of imperfections with appropriately chosen amplitude for subsequent nonlinear analysis. To assess the sensitivity of the model, three imperfection amplitudes were considered: 1) Dawson and Walker [51] model modified by Gardner and Nethercot [53], 2) t/100 and 3) t/10 where t was the plate thickness for the considered SHS and RHS, and web or flange thickness (whichever is critical) for I-sections.

Figure 3.1 Typical mesh distribution used in a hollow section 60×60×3-SC1 [62] 45

Chapter 3: The Continuous Strength Method for Stainless Steel Slender sections

Figure 3.2 Typical mesh distribution used in an I-Section I 2205-372 [63]

Membrane residual stresses were incorporated into the FE models of I-sections to account for the effects of welding, which induced tensile stresses in the vicinity of the welds with compressive stresses away from those regions. As discussed in Section 2.4, residual stress model proposed by Yuan et al. [48] for stainless steel welded I-sections has been used in this study with the maximum tensile residual stress for austenitic grade taken as 0.80.2 and for other grades as 0.60.2. Membrane residual stress is relatively small for RHS and SHS, and its influence is reported not to affect their load carrying capacities [43, 47]. On the other hand, bending residual stress is relatively higher than membrane residual stress for RHS and SHS, but it is inherently present in the material stress-strain properties obtained from coupon tests [62, 99, 102, 107].

Stub columns were considered as fixed at both ends allowing longitudinal displacements to take place only at the top end. All the nodes at the bottom and the top were coupled with two reference points located at the centroid of corresponding sections. Support conditions were applied on to those reference points with all degrees of freedom restrained except for the longitudinal translation at the top. Displacements were applied at the top reference point to simulate stub column tests. Typical boundary conditions applied to SHS stub columns and I-section stub columns are shown in Figures 3.3 and 3.4 respectively.

Chapter 3: The Continuous Strength Method for Stainless Steel Slender sections

Figure 3.3 Boundary condition applied to the FE model of SHS stub column 60×60×3- SC1 [62]

Figure 3.4 Boundary condition applied to the FE model of I-Section stub column I 2205-372 [63]

For in-plane bending resistance, four-point bending tests were simulated to obtain cross- section behaviour in pure bending. Instead of modelling the full length of the beam, only

Chapter 3: The Continuous Strength Method for Stainless Steel Slender sections the beam between two supports was modelled. Nodes of end sections were coupled to the reference points located at the mid-points of the bottom flange at their respective ends. All translations and rotations, except for the longitudinal displacement and rotation about bending axis, were restrained. Longitudinal displacement was restrained at the mid- section of beams. For RHS and SHS beams, loads were applied on reference points that were coupled with the loading strip of the top flange. For I-beams, all nodes of loading sections were coupled to the corresponding reference points located at mid-point of the top flange at loading sections (instead of modelling additional stiffeners) to avoid web crippling. Lateral translation was also prevented at the loading points for all beams. Typical boundary conditions applied to the FE models are shown in Figures 3.5 and 3.6 respectively.

Figure 3.5 Boundary condition applied to the FE model of SHS beam 50×50×1.5L900 [102]

Chapter 3: The Continuous Strength Method for Stainless Steel Slender sections

Figure 3.6 Boundary condition applied to the FE model of I-beam I-200×140×8×6-2 [100]

Verification of the developed FE model

The accuracy of the developed FE models was assessed by comparing the FE results with those obtained from experiments for both stub columns and beams reported in available literatures. A wide range of material grades and cross-sections including 26 stub columns and 25 beams with SHS, RHS and I-sections, collected from seven different sources, were simulated for verification. The section dimensions and material properties of the sections used to verify FE models are given in Appendix A. Key parameters such as the ultimate load (Nu) and the corresponding axial deformation (δu) for stub columns, and the ultimate moment (Mu) and the ultimate curvature (κu) (curvature at which moment curvature curve falls below plastic moment capacity (Mpl) on the descending branch) as well as full load- deformation responses reported in experiments were compared with those obtained from numerical analysis. Tables 3.2 and 3.3 compare the key results for stub columns and four point bending tests considering various imperfection amplitudes for RHS and SHS. Results for I-sections are compared in Tables 3.4 and 3.5 for considered imperfection amplitudes with or without residual stress. It was observed that the effect of imperfection amplitudes on the ultimate resistance Nu and Mu obtained from the FE analysis were relatively insignificant, but the corresponding deformation parameters δu and κu seem more sensitive. Residual stress was considered for I-sections only, and a modest effect was observed on the ultimate resistances and the corresponding displacements. In all cases, FE results showed good accuracy in predicting the experimental results particularly 49

Chapter 3: The Continuous Strength Method for Stainless Steel Slender sections in terms of ultimate resistances (Nu and Mu). Overall, imperfection amplitude taken from the modified Dawson and Walker model [53] produced better predictions, and for I- sections residual stresses should be included in FE models. Figures 3.7 and 3.8 compare experimental and numerically obtained load-deflection curves for two stub columns, whilst Figures 3.9 and 3.10 compare moment-rotation behaviour for a SHS and I-section respectively. Deflected shapes of different stub columns and beams are shown in Figures 3.11-3.14. Overall comparisons showed that the adopted FE modelling techniques were capable of simulating compression and bending behaviour for stainless steel slender sections, and hence were used to generate additional results for comprehensive understanding of their failure mechanism.

Table 3.2 Comparison of FE results with experiments for stub column test of RHS and SHS

Reference Specimen Imperfection amplitude designation D-W Model t/100 t/10

Nu,FE/ δu,FE/ Nu,FE/ δu,FEM/ Nu,FE/ δu,FE/

Nu, Test δu, Test Nu, Test δu, Test Nu, Test δu, Test

100×100×4-SC1 0.99 0.77 0.99 0.73 0.97 0.68 Theofanous 100×100×4-SC2 1.00 0.70 0.99 0.69 1.05 1.15 and Gardner [62] 80×80×4-SC1 1.04 1.07 1.03 0.96 1.00 0.82 80×80×4-SC2 1.03 1.13 1.01 0.99 0.98 0.86 RHS 120×80×3-SC1 0.88 0.88 0.94 0.88 0.93 0.87 RHS 120×80×3-SC2 0.97 0.84 0.97 0.87 0.97 0.83 Afshan and RHS 60×40×3-SC1 0.99 0.73 1.00 0.99 0.97 0.55 Gardner [101] RHS 60×40×3-SC2 1.01 0.75 1.03 1.02 0.99 0.56 SHS 80×80×3-SC1 0.99 0.92 0.99 0.84 0.98 0.70 SHS 80×80×3-SC2 0.99 0.84 0.99 0.80 0.98 0.67 100×100×5-1A 1.07 0.92 0.87 0.73 0.88 0.75 120×120×5-2A 1.01 1.18 0.99 1.04 1.00 1.06 Zhao et al. 150×100×6-3A 1.01 0.95 1.01 0.95 1.01 0.94 [103] 150×100×8-4A 1.03 0.99 1.03 0.97 1.03 0.99 150×150×8-5A 1.07 1.00 1.04 0.63 1.07 0.91 Average 1.01 0.91 0.99 0.87 0.99 0.82 COV 0.04 0.15 0.04 0.14 0.05 0.21 Maximum 0.88 0.70 0.87 0.69 0.88 0.55 Minimum 1.07 1.18 1.04 1.04 1.07 1.15

Chapter 3: The Continuous Strength Method for Stainless Steel Slender sections

Table 3.3 Comparison of FE results with experiments for four-point bending of RHS and SHS beams

Imperfection amplitude D-W Model t/100 t/10 Specimen Reference designation Mu,FE/ κu,FE/ Mu,FE/ κu,FE/ Mu,FE/ κu,FE/ Mu,test κu,test Mu,test κu,test Mu,test κu,test

Afshan RHS 120×80×3-4PB 1.04 1.54 1.04 1.55 0.99 1.21 and RHS 60×40×3-4PB 1.18 - 1.16 - 1.08 - Gardner [101] SHS 80×80×3-4PB 1.01 1.37 1.01 1.36 0.96 1.05 SHS 60×60×3-4PB 1.03 1.35 1.03 1.35 1.03 1.37 SHS 100×100×5-1B 0.97 - 0.97 - 0.94 -

Zhao et al. SHS 120×120×5-2B 0.98 1.18 0.99 1.18 0.95 1.02 [103] RHS 150×100×6-3B 1.06 - 1.06 - 1.03 - RHS 150×100×8-4B 1.12 - 1.12 - 1.09 - RHS 150×150×8-5B 1.07 1.05 1.07 1.05 1.06 - 50×30×2.5L900 1.07 - 1.06 - 0.97 - 30×50×2.5L900 0.99 - 0.99 - 0.96 - 50×50×1.5L900 1.05 0.94 1.05 0.94 1.05 0.94 Huang 50×50×2.5L900 0.96 - 0.96 - 0.93 - and 70×50×2.5L1100 1.01 - 1.01 - 0.98 - Young [102] 50×70×2.5L1100 1.05 - 1.05 - 1.03 - 100×50×2.5L1500 1.02 1.29 1.04 1.39 0.98 1.24 50×100×2.5L1500 1.03 - 1.07 - 1.04 - 150×50×2.5L1500 1.01 0.87 1.03 0.89 0.98 0.82 50×150×2.5L1500 0.99 - 1.01 - 0.99 - Average 1.03 1.20 1.04 1.21 1.00 1.09 COV 0.05 0.18 0.05 0.18 0.05 0.16 Maximum 1.18 1.54 1.16 1.55 1.08 1.21 Minimum 0.96 0.87 0.96 0.89 0.93 0.82

Chapter 3: The Continuous Strength Method for Stainless Steel Slender sections

Table 3.4 Comparison of FE results with experiments for stub column test of I-sections

Imperfection amplitude with Residual stress Imperfection amplitude without Residual stress D-W Model t/100 t/10 D-W Model t/100 t/10 Reference Specimen designation Nu,FE/ δu,FE/ Nu,FE/ δu,FE/ Nu,FE/ δu,FE/ Nu,FE/ δu,FE/ Nu,FE/ δu,FE/ Nu,FE/ δu,FE/ Nu,Test δu,Test Nu,Test δu,Test Nu,Test δu,Test Nu,Test δu, Test Nu,Test δu,Test Nu, Test δu,Test I304-260 0.99 0.62 1.00 0.60 0.95 0.69 1.00 0.60 1.01 0.60 0.96 0.69 I304-282 1.05 0.72 1.08 0.72 1.00 0.90 1.12 0.72 1.15 0.76 1.04 0.76 I304-312 1.02 1.17 1.02 1.14 1.02 1.14 1.04 0.97 1.04 0.94 1.04 0.97 I304-320 0.99 0.84 1.00 0.86 0.97 0.81 1.01 0.82 1.02 0.84 0.99 0.82 I304-372 1.04 0.95 1.03 0.94 1.04 0.91 1.07 0.71 1.07 0.69 1.06 0.75 Yuan et.al. [63] I304-462 1.01 1.05 1.01 1.07 1.02 0.99 1.06 0.85 1.06 0.85 1.06 0.87 I2205-150 1.01 0.79 1.02 0.84 0.97 0.53 1.02 0.81 1.02 0.85 0.97 0.53 I2205-192 1.02 0.80 1.04 0.83 0.94 0.75 1.06 0.81 1.08 0.88 0.97 0.75 I2205-200 0.97 0.51 0.98 0.51 0.94 0.56 0.98 0.48 0.99 0.49 0.95 0.52 I2205-252 1.03 1.02 1.03 1.04 1.03 0.99 1.06 0.84 1.06 0.84 1.06 0.84 I2205-372 1.06 1.07 1.05 1.11 1.06 1.06 1.08 0.88 1.08 0.86 1.08 0.88 Average 1.02 0.87 1.02 0.88 1.00 0.85 1.04 0.77 1.05 0.78 1.02 0.76 COV 0.03 0.23 0.03 0.23 0.04 0.22 0.04 0.17 0.04 0.16 0.05 0.18 Maximum 1.05 1.17 1.08 1.14 1.06 1.14 1.12 0.97 1.15 0.94 1.06 0.97 Minimum 0.97 0.62 0.98 0.6 0.94 0.53 0.98 0.48 0.99 0.49 0.95 0.52

Chapter 3: The Continuous Strength Method for Stainless Steel Slender sections

Table 3.5 Comparison of FE results with experiments for four point bending of I-beams

Imperfection amplitude with Residual stress Imperfection amplitude without Residual stress D-W Model t/100 t/10 D-W Model t/100 t/10 Reference Specimen designation Mu,FE/ κu,FE/ Mu,FE/ κu,FE/ Mu,FE/ κu,FE/ Mu,FE/ κu,FE/ Mu,FE/ κu,FE/ Mu,FE/ κu,FE/ Mu,test κu,test Mu,test κu,test Mu,test κu,test Mu,test κu,test Mu,test κu,test Mu,test κu,test I-200×140×6×6-2 0.92 - 0.93 - 0.87 - 0.92 - 0.93 - 0.87 - Saliba and I-200×140×8×6-2 0.93 0.95 0.94 1.00 0.90 0.55 0.93 0.96 0.94 1.01 0.90 0.56 Gardner [100] I-200×140×10×8-2 0.95 1.01 0.95 1.01 0.91 0.91 0.95 1.01 0.95 1.01 0.92 0.91 I-200×140×12×8-2 0.90 1.83 0.90 1.81 0.88 1.73 0.90 1.83 0.90 1.81 0.87 1.38 I-160×160-B0 1.09 - 1.09 - 1.04 - 1.09 - 1.09 - 1.04 - Stangenberg [90] I-160×160 Duplex- B0 1.09 - 1.09 - 1.06 - 1.09 - 1.09 - 1.06 -

I-320×160 - B0 1.06 - 1.06 - 1.00 - 1.06 - 1.06 - 1.00 - Average 0.99 1.26 0.99 1.27 0.95 1.06 0.99 1.27 0.99 1.28 0.95 0.95 COV 0.09 0.39 0.08 0.36 0.08 0.57 0.09 0.38 0.08 0.36 0.08 0.43 Maximum 1.09 1.83 1.09 1.81 1.06 1.73 1.09 1.83 1.09 1.81 1.06 1.38 Minimum 0.92 0.95 0.93 1.00 0.88 0.55 0.92 0.96 0.93 1.01 0.87 0.56

Chapter 3: The Continuous Strength Method for Stainless Steel Slender sections

450 400 350 300 250 200 Load (kN) Load 150 100 Test 50 FE 0 0 0.5 1 1.5 2 2.5 3 End shortening (mm)

Figure 3.7 Load vs. end shortening curve comparing test observation and numerical simulation for SHS 80×80×3-SC1 stub column [101]

1000 900 800 700 600 500

Load (kN) Load 400 300 200 Test 100 FE 0 0 5 10 15 20 25 End shortening (mm)

Figure 3.8 Load vs. end shortening curve comparing test observation and numerical simulation for I304-462 stub column [63]

Chapter 3: The Continuous Strength Method for Stainless Steel Slender sections

1.4

1.2

pl 0.8

M/M 0.6

0.4 Test 0.2 FE 0 0 5 10 15 20 25 30 κ/κ pl

Figure 3.9 Moment vs. curvature curve comparing test observation and numerical simulation for SHS 120×120×5-2B beam [103]

1.4

1.2

pl 0.8

M/M 0.6

0.4 Test 0.2 FE 0 0 5 10 15 20

κ/κpl

Figure 3.10 Moment vs. curvature curve comparing test observation and numerical simulation for I-200×140×10×8-2 beam [100]

Chapter 3: The Continuous Strength Method for Stainless Steel Slender sections

Figure 3.11 Comparison of deflected shapes observed in experiment and FE simulation for 60×60×3-SC1 stub column [62]

Figure 3.12 Comparison of deflected shapes observed in experiment and FE simulation for I 2205-372 stub column [63]

Chapter 3: The Continuous Strength Method for Stainless Steel Slender sections

Figure 3.13 Comparison of deflected shapes observed in experiment and FE simulation for 50×50×1.5L900 beam [102]

Figure 3.14 Comparison of deflected shapes observed in experiment and FE simulation for I-200×140×10×8-2 beam [100]

Parametric study

Verified FE models were used to perform a series of parametric studies to generate additional structural performance data over a wide range of cross-section slenderness to complement currently available limited experimental evidences. A total of 60 different cross-sections were studied including 21 RHS, 16 SHS and 23 I-sections. RHS with four different height-to-width ratios varying between 1.25 to 2.0 were modelled with a 57

Chapter 3: The Continuous Strength Method for Stainless Steel Slender sections thickness variation of 2 to 6 mm. The same variation in thickness was considered for SHS with four different outer dimensions. In case of I-sections, five height-to-width ratios varying between 1.0 to 2.0 were considered, whilst flange-to-web thickness ratios were varied between 1.0 to 2.0. The internal corner radii of hollow sections were taken equal to the cross-section thickness. Details of cross-sectional dimensions are given in Tables 3.6-3.8. Each section was analysed for four stainless steel grades i.e. austenitic (1.4301), ferritic (1.4003), duplex (1.4462) and lean duplex (1.4162). Material properties were selected from four different sources [10, 62, 96, 101] and are tabulated in Table 3.9. Both major axis bending and minor axis bending resistances were determined for RHS. For stub column analysis, the length of each model was set to be equal to three times of the maximum cross-section dimension [108]. For four-point bending analysis, beam length was considered as 2800mm with two concentrated loads applied at a distance of 900mm from end supports. It is worth noting that the beams were restrained against any possible lateral movement. Modified Dawson and Walker model [53] was used to predict the initial local imperfection amplitudes, and residual stresses were incorporated for I- sections. A total 224 stub column models and 288 four-point bending models were analysed. All results obtained from FE analysis, together with available experimental results were used to develop the proposed design expressions for slender sections in the current study.

Table 3.6 Cross-sectional properties of Rectangular Hollow Sections considered in FE analysis

Height Width Thickness mm mm mm 125 100 2, 2.5, 3, 3.5 and 4* 150 100 2, 2.5, 3, 3.5 and 4* 200 120 2, 3, 4, 5 and 6* 240 120 2**, 3, 4, 5, and 6 *Thickness used to determine minor axis bending only ** Thickness used to determine major axis bending only.

Chapter 3: The Continuous Strength Method for Stainless Steel Slender sections

Table 3.7 Cross-sectional properties of Square Hollow Sections considered in FE analysis

Height Width Thickness mm mm mm 125 125 2, 3, 3.5 and 4 150 150 2, 3, 4 and 5 200 200 2, 3, 4 and 5 240 240 3, 4, 5 and 6

Table 3.8 Cross-sectional properties of I-sections considered in FE analysis

Height Width Flange thickness/Web thickness mm mm mm/mm 150 150 4/3, 4/4, 5/4, 6/4 and 6/3 200 160 4/3, 4/4, 5/4, 6/4 and 6/3 260 150 4/3, 4/4, 5/4 and 6/4 300 200 4/3, 4/4, 5/4 and 6/4 320 160 4/3, 4/4, 5/4, 6/4 and 6/3

Table 3.9 Material properties used in the parametric study

Grade Flat material properties Corner material properties Reference

σ0.2 σu n E σ0.2 σu n GPa MPa MPa GPa MPa MPa Austenitic 201.3 382 675 6.6 197.4 587 820 3.5 [10] (1.4301) Ferretic 216.0 423 472 10.2 226.0 535 554 6.0 [101] (1.4003) Duplex 206.3 543 759 6.3 213.5 710 967 3.5 [96] (1.4462) Lean Duplex 198.8 586 761 9.0 206.0 811 917 6.3 [62] (1.4162)

CSM for slender sections

Cross-sections with slenderness λp greater than 0.68 were considered as slender sections in the current study; these cross-sections will experience loss of effectiveness due to local buckling. Although failure of slender sections is typically dominated by elastic local buckling, significant post buckling behaviour is observed resulting in excessive deformation prior to failure. This effect produces erroneous results if the current definition of deformation capacity csm is directly applied for slender sections. Gardner [33] and Ashraf [50] observed this behaviour and modified their proposed design curves to include this effect. The recent CSM proposed Afshan and Gardner [14], however, is

Chapter 3: The Continuous Strength Method for Stainless Steel Slender sections solely focused on stocky sections, and hence this effect was not relevant to their proposed formulation. The current study introduces a new parameter called Equivalent Elastic

Deformation Capacity e,ev to tackle this unique behaviour demonstrated by slender sections. e,ev is defined by the elastic strain at ultimate load as shown in Figure 3.15, and can be calculated by Eq.s 3.1 and 3.2 for stub columns and beams, respectively. Here, Nu is the ultimate load of a stub column, Mu is the ultimate moment of a beam, E is the

Young’s modulus, Ag is the gross cross-sectional area and Wel is the elastic section modulus of gross cross-section.

Nu εe,ev = (3.1) EAg

Mu εe,ev = (3.2) EWel

E fpredicted Material property

factual

Stres Slender section

εe,ev εcsm Strain

Figure 3.15 Typical post buckling behaviour observed in slender sections

Afshan and Gardner [14] proposed a new base curve for stocky stainless steel cross- sections, which forms the very basis of current CSM. This concept is further extended herein for slender cross-sections without changing the formula for the base curve; this generalization is believed as a significant step forward in turning CSM into a complete design tool. The ratio of εe,ev and εcsm, considered as C (shown in Eq. 3.3) was determined for all considered slender sections and were plotted against the corresponding cross- section slenderness λp. Figures 3.16-3.18 show the distribution of C with respect to λp for stub columns, whilst in-plane bending behaviour are shown in Figures 3.19-3.22. It is obvious from Figures 3.16-3.22 that in all considered cases there is a clear trend in the variation of C with λp. Simple power models can be used to establish relationships

Chapter 3: The Continuous Strength Method for Stainless Steel Slender sections between C and λp, as shown in Eq. 3.4 where a and b are two constants that depend on cross-section types.

εe,ev = Cεcsm for λp > 0.68 (3.3)

b C = aλp (3.4)

30 C 20

10 FE data Test data 0 0.68 1.18 1.68 2.18 2.68

λ p

Figure 3.16 Variation of C with cross-section slenderness λp for RHS stub columns

C 30

20 FE data 10 Test data 0 0.68 1.18 1.68 2.18 2.68 3.18

λp

Figure 3.17 Variation of C with cross-section slenderness λp for SHS stub columns

Chapter 3: The Continuous Strength Method for Stainless Steel Slender sections

30 C 20

10 FE data Test data 0 0.68 1.18 1.68 2.18 2.68

λp

Figure 3.18 Variation of C with cross-section slenderness λp for I-section stub columns

C 6

2 FE data

0 0.68 0.88 1.08 1.28 1.48 1.68 λ p

Figure 3.19 Variation of C with cross-section slenderness λp for RHS beams subjected to major axis bending

Chapter 3: The Continuous Strength Method for Stainless Steel Slender sections

C 15

10 FE data 5 Test data 0 0.68 1.18 1.68 2.18

λp

Figure 3.20 Variation of C with cross-section slenderness λp for RHS beams subjected to minor axis bending

30 C 20 FE data 10 Test data

0 0.68 1.18 1.68 2.18 2.68 λ p

Figure 3.21 Variation of C with cross-section slenderness λp for SHS beams

Chapter 3: The Continuous Strength Method for Stainless Steel Slender sections

15 C 10 FE data 5 Test data

0 0.68 0.88 1.08 1.28 1.48 1.68 1.88

λp

Figure 3.22 Variation of C with cross-section slenderness λp for I-beams

Tables 3.10 and 3.11 list the values suggested for a and b for the considered cross-section types as obtained through regression analysis of all test and FE results. Figure 3.23 shows the curves for RHS, SHS and I-sections plotted using the suggested values under compression. It is, however, observed that there is negligible difference between I and RHS sections, and a single curve could be used for both sections. Figure 3.24 shows the variation of C with λp for different types of cross-sections subjected to bending. For in- plane bending, major axis and minor axis bending have to be treated separately for RHS as the difference is significant.

Table 3.10 Proposed values for coefficients a and b for different cross-section types under axial compression

Section Types a b Rectangular Hollow Section (RHS) 3.20 2.90 Square Hollow Section (SHS) 2.90 2.75 I-Section 3.05 3.00

Chapter 3: The Continuous Strength Method for Stainless Steel Slender sections

Table 3.11Proposed values for coefficients a and b for different cross-section types subjected to in-plane bending

Section Types a b Rectangular Hollow Section (Major axis bending) 3.55 2.75 Rectangular Hollow Section (Minor axis bending) 3.70 3.15 Square Hollow Section 3.75 2.95 I-Section 3.40 3.00

60 RHS 50 SHS I-section 40

C 30

0 0.68 1.18 1.68 2.18 2.68

λp

Figure 3.23 Variation of C with cross-section slenderness λp for compression

80 RHS (Major Axis bending) 70 RHS (Minor Axis bending) SHS 60 I Section

C 40

0 0.68 1.18 1.68 2.18 2.68 λ p

Figure 3.24 Variation of C with cross-section slenderness λp for in-plane bending

Chapter 3: The Continuous Strength Method for Stainless Steel Slender sections

Figures 3.25 and 3.26 show the variation of C with λp for all considered cross-section types subjected to compression and bending showing a general trend, overall. In-plane bending responses for the considered slender cross-sections, however, show a very good trend as observed in Figure 3.26. Regression analysis was performed and the obtained values for a and b are listed in Table 3.12, whilst Figure 3.27 shows the variation of C with λp for compression and bending with all cross-sections considered together.

60 I-section FE data 50 I-section Test 40 data RHS FE data

C 30 RHS Test data 20 SHS FE data 10 SHS Test data 0 0.68 1.18 1.68 2.18 2.68 3.18 λ p

Figure 3.25 Variation of C with cross-section slenderness λp for stub column data of all cross-section types

50 I-section FE data 45 40 I-section test data 35 RHS major axis 30 bending FE data RHS minor axis

25C bending FE data 20 RHS minor axis 15 bending test data SHS FE data 10 5 SHS test data 0 0.68 1.18 1.68 2.18 2.68 λ p

Figure 3.26 Variation of C with cross-section slenderness λp for in-plane bending data of all cross-section types

Chapter 3: The Continuous Strength Method for Stainless Steel Slender sections

Table 3.12 Proposed values for coefficients a and b for all considered cross-section types

Section Types a b Axial compression 3.05 2.90 In-plane bending 3.45 3.00

70 Compression 60 Bending 50

40 C 30

0 0.68 1.18 1.68 2.18 2.68 λ p

Figure 3.27 Variation of C with cross-section slenderness λp for axial compression and in-plane bending

Resistance of slender cross-sections against compression and bending

The material model proposed by Afshan and Gardner [14] was adopted in the current study. It is worth noting that previously adopted material models i.e. modified Ramberg- Osgood formulations represent material nonlinearity even at stress levels lower than 0.2% proof stress. On the other hand, the adopted model is a bilinear model giving a purely elastic stress-strain response up to 0.2% proof stress. Once εe,ev is determined, the buckling stress fcsm for slender sections with λp> 0.68 can be calculated by multiplying the obtained εe,ev to Young’s Modulus E as given by Eq. 3.5, where fcsm is the maximum average stress that a section can achieve prior to failure. Hence the cross-section compression resistance Nu may be estimated by Eq. 3.6, where Ag is the gross cross- sectional area. Considering the assumption that plane sections remain plain and are normal to the neutral axis in bending, the in-plane bending resistance for a slender cross-

Chapter 3: The Continuous Strength Method for Stainless Steel Slender sections section Mu can be determined by Eqs. 3.7 and 3.8 for major axis and minor axis bending respectively, where Ag is the gross cross-sectional area , Wel,y and Wel,z are the elastic section modulus of the gross cross-sectional area about the relevant axis. This approach clearly eliminates the need for going through the lengthy process of calculating effective cross-sectional properties for slender cross-sections.

fcsm = εe,evE = CεcsmE for λp > 0.68 (3.5)

Nu = Agfcsm (3.6)

My,u = fcsmWel,z for λp > 0.68 (3.7)

Mz,u = fcsmWel,z for λp > 0.68 (3.8)

This method of calculating cross-section resistance for slender cross-sections using CSM approach was published in Thin Walled Structure in 2016 [87]. Very recently, in 2017, Zhao et al. [15] proposed an alternative way to design slender cross-sections using CSM principles. They used the equivalent elastic deformation capacity εe,ev proposed in the previous section to define the buckling strain for slender sections. However, they proposed a separate base curve for slender sections which is similar to the strength curve of slender sections used in DSM [75]. In the current research, on the other hand, the same base curve was adopted for both stocky and slender sections with appropriate modifications to the deformation capacities were proposed for slender sections.

Performance of the proposed design technique

Cross-section capacities predicted using the proposed coefficients were compared with those obtained from experiments and FE analysis are shown in Tables 3.13 and 3.14 for axial compression and for in-plane bending, respectively. Predictions obtained for the considered slender sections using EN 1993-1-4+A1 [68], AS/NZS 4673 [5], SEI/ASCE 8-02 [4] and DSM [86] are also presented herein. The provision of calculating axial compression and in-plane bending resistance for slender cross-sections are same for AS/NZS and ASCE standards. As discussed in Section 2.10, there are a number of equations proposed in DSM to calculate the cross-section resistance of slender cross- sections. Recently, Arrayago et al. [86] showed that DSM formulas (Eq. 2.70), originally 68

Chapter 3: The Continuous Strength Method for Stainless Steel Slender sections proposed for carbon steel, performed better than other proposed equations for predicting cross-section resistance. Hence, Eq. 2.70 was used as DSM guideline in the current study for comparisons. Figures 3.28-3.38 show all comparisons for predictions determined using the suggested method and all other standards with those obtained experimentally and numerically. It is observed that in every case, the proposed CSM technique for slender sections offers improved mean resistance than other considered standards with less scattered predictions. In most cases, AS/NZS [5], ASCE [4] and DSM [86] over-predict the axial compression resistance making the results unconservative. The performance of the proposed method is significantly better in predicting the in-plane bending resistance; for all cross-section types, proposed CSM can predict with a mean accuracy of 0.93, whereas those for EN 1993-1-4+A1 [68] is 0.80, for AS/NZS [5] or ASCE [4] is 0.84 and for DSM [86] is 0.90. For axial compression, this mean value improves from 0.92 to 0.98 for CSM when compared against EN 1993-1-4+A1 [68]. Comparing the predictions of in-plane bending resistance for SHS with experimental results, it seems that CSM may slightly over-estimate the resistance as the mean is 1.06, but the number of available experimental evidences (only 7) is too small to make a meaningful argument. FE results, however, show good agreement with CSM predictions (mean is 0.97 and COV is 0.04). For the considered data set, the proposed method shows acceptable agreement with both experimental and FE results, and the accuracy of the prediction is better than other standards. All the FE data used in this comparison were also used to develop CSM formula for slender sections and the number of test results of slender sections were also limited. Substantially more FE and test data should be used in future research to validate the proposed approach for general applicability.

Chapter 3: The Continuous Strength Method for Stainless Steel Slender sections Table 3.13 Comparison of the predictions of cross-section compression resistance obtained by CSM and other standards with stub column test and FE results

CSM EN 1993-1-4+A1 AS/NZS (ASCE) DSM Section Type Ncsm/NFE/Test NEC3/NFE/test NAs/NZS/NFE/Test NDSM/NFE/Test Average COV Average COV Average COV Average COV RHS FE data 1.00 0.04 0.88 0.03 1.01 0.05 1.02 0.05 RHS test data 0.99 0.05 0.89 0.05 1.00 0.05 1.00 0.06 SHS FE data 0.95 0.05 0.87 0.05 1.03 0.06 1.09 0.07 SHS test data 0.96 0.11 0.88 0.11 1.00 0.14 1.03 0.14 I-Section FE data 0.97 0.06 1.02 0.09 1.04 0.06 1.03 0.07 I-Section test data 1.02 0.08 1.01 0.08 1.07 0.10 1.11 0.09 All sections FE data 0.97 0.07 0.94 0.10 1.03 0.06 1.05 0.07 All Sections test data 0.98 0.10 0.92 0.10 1.02 0.10 1.04 0.11

Table 3.14 Comparison of the predictions of in-plane bending resistance obtained by CSM and other standards with in-plane bending test and FE results

CSM EN 1993-1-4+A1 AS/NZS (ASCE) DSM Section Type Mcsm/MFE/Test MEC3/MFE/Test MAS/NZS/MFE/Test MDSM/MFE/Test Average COV Average COV Average COV Average COV RHS FE data (major axis) 0.95 0.06 0.79 0.19 0.82 0.08 0.87 0.09 RHS FE data (minor axis) 0.97 0.04 0.75 0.05 0.82 0.05 0.84 0.04 RHS test data (minor axis) 1.06 0.04 0.82 0.06 0.82 0.06 0.98 0.09 SHS FE data 0.98 0.03 0.78 0.09 0.84 0.05 0.85 0.03 SHS test data 1.00 0.04 0.79 0.17 0.82 0.08 0.85 0.05 I-Section FE data 0.97 0.04 0.92 0.07 0.89 0.05 0.95 0.05 I-Section test data 0.80 0.01 0.72 0.01 0.72 0.01 0.78 0.01 All sections FE data 0.94 0.07 0.83 0.13 0.85 0.06 0.89 0.08 All Sections test data 0.93 0.08 0.80 0.13 0.84 0.090 0.90 0.10

Chapter 3: The Continuous Strength Method for Stainless Steel Slender sections

1.2

0.8 FE/test

0.6

pred N 0.4 CSM FE EN 1993-1-4+A1 FE AS/NZS (ASCE) FE DSM FE 0.2 CSM test EN 1993-1-4+A1 test AS/NZS (ASCE) test DSM test 0 0.68 1.18 1.68 2.18 2.68 λp Figure 3.28 Performance of the suggested CSM method and other considered design standards in predicting cross-section compression resistances of RHS

1.6

1.4

1.2

1 FE/test

0.8

/N pred

N 0.6 CSM FE EN 1993-1-4+A1 FE 0.4 AS/NZS (ASCE) FE DSM FE 0.2 CSM test EN 1993-1-4+A1 test AS/NZS (ASCE) test DSM test 0 0.68 1.18 1.68 2.18 2.68 3.18 λp

Figure 3.29 Performance of the suggested CSM method and other considered design standards in predicting cross-section compression resistances of SHS

Chapter 3: The Continuous Strength Method for Stainless Steel Slender sections 1.4

1.2

0.8

FE/test /N

pred 0.6 N

0.4 CSM FE EN 1993-1-4+A1 FE AS/NZS (ASCE) FE DSM FE 0.2 CSM test EN 1993-1-4+A1 test AS/NZS (ASCE) test DSM test 0 0.68 1.18 1.68 2.18 2.68 λp

Figure 3.30 Performance of the suggested CSM method and other considered design standards in predicting cross-section compression resistances of I-sections

1.4

1.2

0.8

FE/test /N

csm 0.6 N I-section FE SHS FE 0.4 RHS FE I-section test 0.2 SHS test RHS test 0 0.68 1.18 1.68 2.18 2.68 λp

Figure 3.31 Performance of the suggested CSM method in predicting cross-section compression resistances of all types of cross-sections

Chapter 3: The Continuous Strength Method for Stainless Steel Slender sections

1.4

1.2

0.8

FE/test /N

EC3 0.6 N

0.4 I-section FE SHS FE RHS FE I-section test 0.2 SHS test RHS test

0 0.68 1.18 1.68 2.18 2.68 λp

Figure 3.32 Performance of EN 1996-1-4+A1 [68] in predicting cross-section compression resistances of all types of cross-sections

1.4

1.2

1 FE/test

/N 0.8

0.6 AS/NZS orASCE AS/NZS

N 0.4 I-section FE SHS FE

RHS FE I-section test 0.2 SHS test RHS test 0 0.68 1.18 1.68 2.18 2.68 λp

Figure 3.33 Performance of AS/NZS 4673 [5] or SEI/ASCE 8-02 [4] in predicting cross-section compression resistances of all types of cross-sections

Chapter 3: The Continuous Strength Method for Stainless Steel Slender sections

1.6

1.4

1.2

1 FE/test

/N 0.8 DSM

N 0.6 I-section FE SHS FE 0.4 RHS FE I-section test 0.2 SHS test RHS test 0 0.68 1.18 1.68 2.18 2.68 λ p

Figure 3.34 Performance of DSM [86] in predicting cross-section compression resistances of all types of cross-sections

1.2

0.8 FE/test

0.6

pred M 0.4 CSM FE EN 1993-1-4+A1 FE 0.2 AS/NZS (ASCE) FE DSM FE 0 0.68 0.88 1.08 1.28 1.48 1.68 1.88 λp

Figure 3.35 Performance of the suggested CSM method and other considered design standards in predicting in-plane bending resistances of RHS bending about major axis

Chapter 3: The Continuous Strength Method for Stainless Steel Slender sections

1.2

0.8 FE/test

0.6

pred M 0.4 CSM FE EC3 FE AS/NZS (ASCE) FE DSM FE 0.2 CSM test EC3 test AS/NZS (ASCE) test DSM test 0 0.68 1.18 1.68 λp

Figure 3.36 Performance of the suggested CSM method and other considered design standards in predicting in-plane bending resistances of RHS bending about minor axis

1.2

0.8 FE/test

0.6

pred M 0.4 CSM FE EN 1993-1-4+A1 FE As/NZS (ASCE) FE DSM FE 0.2 CSM test EN 1993-1-4+A1 test AS/NZS (ASCE) test DSM test 0 0.68 1.18 1.68 2.18 λp Figure 3.37 Performance of the suggested CSM method and other considered design standards in predicting in-plane bending resistances of SHS beams

Chapter 3: The Continuous Strength Method for Stainless Steel Slender sections

1.2

0.8 FE/test

0.6

pred M 0.4 CSM FE EN 1993-1-4+A1 FE AS/NZS (ASCE) FE DSM FE 0.2 CSM test EN 1993-1-4+A1 test AS/NZS (ASCE) test DSM test 0 0.68 0.88 1.08 1.28 1.48 1.68 1.88 λp Figure 3.38 Performance of the suggested CSM method and other considered design standards in predicting in-plane bending resistances of I-beams

1.2

0.8 FE/test

0.6

csm M 0.4 I-section FE SHS FE RHS MJ FE RHS MI FE 0.2 I-section test SHS test RHS MI test 0 0.68 1.18 1.68 2.18 λp Figure 3.39 Performance of the suggested CSM method in predicting in-plain bending resistances of all types of cross-sections

Chapter 3: The Continuous Strength Method for Stainless Steel Slender sections

1.2

0.8 FE/test

0.6

EC3 M 0.4 I-section FE SHS FE RHS MJ FE RHS MI FE 0.2 I-section test SHS test RHS MI test 0 0.68 1.18 1.68 2.18 λp Figure 3.40 Performance of EN 1996-1-4+A1 [68] in predicting in-plane bending resistances of all types of cross-sections

1.2

0.8

FE/test /M

0.6

AS/NZS ASCE or AS/NZS 0.4

M I-section FE SHS FE RHS MJ FE RHS MI FE 0.2 I-section test SHS test RHS MI test 0 0.68 1.18 1.68 2.18 λp

Figure 3.41Performance of AS/NZS 4673 [5] or SEI/ASCE 8-02 [4] in predicting in- plane bending resistances of all types of cross-sections

Chapter 3: The Continuous Strength Method for Stainless Steel Slender sections

1.2

0.8 FE/test

/M 0.6 DSM

M 0.4 I-section FE SHS FE RHS MJ FE RHS MI FE 0.2 I-section test SHS test RHS MI test 0 0.68 1.18 1.68 2.18 λp

Figure 3.42 Performance of DSM [86] in predicting in-plane bending resistances of all types of cross-sections

Reliability analysis

A standard statistical analysis was carried out to assess the reliability of the proposed design formulations for slender cross-sections using the guidance of EN1990-Annex D [109]. Key parameters used in this analysis are summarized in Tables 3.15 and 3.16 for axial compression and in-plane bending, respectively, where n is the total number of tests and FE results, kd,n is the design (ultimate limit state) fractile factor for n number of data, b is the average ratio of experimental or FE to model resistance based on a least squares fit for each set of data. Vδ is the coefficient of variation of tests and FE simulations relative to the resistance model, Vr is the combined coefficient of variation incorporating both model and basic variable uncertainties and γM0 is the partial safety factor for cross-section resistance. According to Afshan et al. [110], the material over-strength factor was taken as 1.3 for austenitic grades, 1.2 for ferritic grades and 1.10 for duplex and lean duplex grades, and the coefficient of variation of material strength was taken as 0.06, 0.045 and 0.03 respectively. The coefficient of variation in geometric properties was taken as the recommended value of 0.05. It is observed that the partial safety factor γM0 with the CSM design resistance function for slender cross-sections are less than 1.10 as recommended in EN 1993-1-4 [3] for austenitic and ferritic grades. Suggested CSM technique for slender cross-sections may be used for austenitic and ferritic grades safely with the code

Chapter 3: The Continuous Strength Method for Stainless Steel Slender sections recommended value of γM0. But most of the cases for duplex and lean duplex grades require a higher value of γM0. The new design proposal may be used safely with a partial safety factor 1.1 for austenitic and ferritic grades, whilst a higher partial safety factor of 1.15 may be adopted for duplex and lean duplex grades, though in some cases calculated

γM0 were slightly over than 1.15. Lower over strength factors proposed for duplex and lean duplex grades may have contributed to this requirement. However, significant experimental evidence on slender cross-sections produced from duplex and lean duplex grades will allow for an appropriate in-depth analysis to obtain a suitable material safety factor.

Table 3.15 Summary of the reliability analysis of the proposed CSM method for predicting cross-section compression resistance of stainless steel slender sections

Section Material n Kd,n b Vδ Vr γM0 Austenitic 47 3.19 1.02 0.05 0.09 1.01 Ferritic 18 3.19 1.01 0.03 0.07 1.04 RHS Duplex 18 3.19 1.03 0.04 0.07 1.10 Lean Duplex 19 3.19 1.05 0.04 0.07 1.09 Austenitic 43 3.20 1.04 0.10 0.12 1.10 Ferritic 16 3.20 1.06 0.05 0.08 1.02 SHS Duplex 18 3.20 1.06 0.04 0.07 1.07 Lean Duplex 17 3.20 1.08 0.04 0.07 1.06 Austenitic 43 3.18 1.02 0.06 0.10 1.03 Ferritic 23 3.18 1.03 0.06 0.09 1.08 I-section Duplex 26 3.18 1.04 0.05 0.08 1.11 Lean Duplex 25 3.18 1.07 0.05 0.08 1.09 Austenitic 133 3.12 1.03 0.09 0.12 1.08 All types Ferritic 57 3.12 1.04 0.07 0.10 1.08 of section Duplex 62 3.12 1.05 0.07 0.09 1.15 Lean Duplex 61 3.12 1.06 0.07 0.09 1.15

Chapter 3: The Continuous Strength Method for Stainless Steel Slender sections Table 3.16 Summary of the reliability analysis of the proposed CSM method for predicting in-plane bending resistance of stainless steel slender sections

Section Material n Kd,n b Vδ Vr γM0 RHS Austenitic 8 3.36 1.03 0.03 0.08 0.97 (major Ferritic 8 3.36 1.02 0.02 0.07 1.02 axis Duplex 11 3.36 1.04 0.08 0.10 1.19 bending) Lean Duplex 12 3.36 1.07 0.07 0.09 1.15 RHS Austenitic 12 3.27 1.04 0.03 0.08 0.96 (minor Ferritic 15 3.27 1.00 0.04 0.08 1.07 Axis Duplex 13 3.27 1.03 0.06 0.09 1.16 bending) Lean Duplex 17 3.27 1.03 0.03 0.07 1.09 Austenitic 19 3.25 1.02 0.04 0.09 0.99 Ferritic 12 3.25 0.99 0.02 0.07 1.04 SHS Duplex 15 3.25 1.02 0.03 0.07 1.09 Lean Duplex 17 3.25 1.04 0.02 0.06 1.06 Austenitic 25 3.20 1.04 0.07 0.10 1.03 Ferritic 23 3.20 1.02 0.04 0.08 1.04 I-section Duplex 23 3.20 1.01 0.05 0.07 1.14 Lean Duplex 22 3.20 1.01 0.05 0.08 1.14 Austenitic 64 3.13 1.06 0.06 0.10 1.00 All type of Ferritic 58 3.13 1.03 0.05 0.08 1.05 section Duplex 62 3.13 1.05 0.08 0.10 1.18 Lean Duplex 68 3.13 1.06 0.07 0.09 1.15

Worked out Examples

3.10.1 Example I: Axial Compression resistance

CSM predicted compression resistance of I-section I2205-252 stub column, tested by Yuan et.al. [63] was determined as follows:

Cross-section geometric and material properties:

H = 252.9 mm Weld size = 3 mm σ0.2 = 605.6 MPa 2 B = 245.3 mm A = 4388.1 mm σu= 797.9 MPa tf = 6.00 mm E=199200 MPa εy =605.6/193200=0.00313 tw = 6.00 mm

Determination of cross-section slenderness:

σ0.2 605.6 λp = √ = √ = 1.48 σcr,cs 274.9 2 where σcr,cs = 274.9 N/mm was obtained directly from the CUFSM software [76]

Chapter 3: The Continuous Strength Method for Stainless Steel Slender sections

Multiplying by (Cflat/Ccl)max, where Cflat is the flat element width and Ccl is the centreline element width. Corrected λp = 1.48×0.976 = 1.45 > 0.68 (so, it is a slender section)

Determination of the cross-section deformation capacity:

εcsm 0.25 εu = 3.6 = 0.066 < min⁡(15, 0.1 )⁡ εy 1.45 εy

Determination of the Equivalent elastic strain

b 3.0 C = aλp = 3.05(1.45) = 9.3

εe,ev = Cεcsm = 9.3 × 0.066 × 0.00313 = 0.00192

Determination of the buckling stress

2 fcsm = εe,evE = 0.00192 × 193200 = 370.9 N/mm

Determination of the cross-section compression resistance

Nu = Agfcsm = 4388.1 × 370.9 = 1627.7⁡kN

[Test ultimate load = 1653 kN; EN 1993-1-4+A1[68] predicted resistance = 1461.7 kN and AS/NZS 4673 [5] or ASCE/SEI 8-02 [4] predicted resistance = 1676.4 kN.]

3.10.2 Example II: In-plane bending resistance

CSM predicted in-plane bending resistance for H 150×150×3 beam, tested by Zhou and Young [97] was determined as follows:

Cross-section geometric and material properties: 2 H = 150.7 mm A = 1614.15 mm σ0.2 = 466.9 MPa 4 B = 150.6 mm Wel,y = 77355 mm σu = 699MPa t = 6.00 mm E = 189000 MPa εy = 466.9/195800=0.00247 ri = 4.8 mm

Determination of cross-section slenderness:

σ0.2 466.9 λp = √ = √ = 1.18 σcr,cs 335.07 2 where σcr,cs = 335.07 N/mm was obtained directly from the CUFSM software [76]

Chapter 3: The Continuous Strength Method for Stainless Steel Slender sections

Multiplying by (Cflat/Ccl)max, where Cflat is the flat element width and Ccl is the centreline element width. Corrected λp = 1.18×0.916 = 1.08 > 0:68 (so, it is a slender section)

Determination of the cross-section deformation capacity:

εcsm 0.25 εu = 3.6 = 0.1895 < min⁡(15, 0.1 )⁡ εy 1.08 εy

Determination of the Equivalent elastic strain

b 2.95 C = aλp = 3.75(1.08) = 4.7

εe,ev = Cεcsm = 4.7 × 0.1895 × 0.00247 = 0.0022

Determination of the buckling stress

2 fcsm = εe,evE = 0.0022 × 189000 = 415.8 N/mm

Determination of the cross-section compression resistance

My,u = fcsm × Wel,y = 415.8 × 77355 = 32164209⁡Nmm = 32.16⁡kNm

[Test ultimate moment = 31.68 kN-m; EN 1993-1-4+A1[68] predicted resistance =

26.35 kN-m and AS/NZS 4673 [5] or ASCE/SEI 8-02 [4] predicted resistance = 28.74 kN-m.]

Conclusions

Effective width approach is traditionally adopted to deal with the loss of effectiveness demonstrated by thin-walled slender cross-sections due to local buckling. The application of the effective width method typically requires lengthy calculation process, and is somewhat ambiguous for cross-sections with complex geometry, and sometimes become iterative for cross-sections with stress gradient. The Continuous Strength Method (CSM) is a new strain based concept originally developed for stocky sections to exploit the effects of strain hardening. Typically available stainless steel sections, however, fall into the slender category. Previous attempts to harmonize this deformation based concept for

Chapter 3: The Continuous Strength Method for Stainless Steel Slender sections both stocky and slender stainless steel sections produced erratic results (especially for slender sections) due to the significant post-buckling behaviour demonstrated by slender sections prior to failure [12]. To develop the CSM technique for slender cross-sections i.e. with cross-section slenderness λp > 0.68, a new parameter named as the Equivalent

Elastic Deformation Capacity e,ev, was introduced to deal with the observed post- buckling effect. A comprehensive numerical study was used to generate reliable results on the load-deformation response of stainless steel slender sections with RHS, SHS and I-sections. All available experimental and FE results on stub columns and beams subjected to in-plane bending were used to develop relationships to predict the maximum attainable elastic strain for slender sections prior to failure; this allowed obtaining the corresponding buckling stress using the newly proposed bilinear material model for stainless steel. Cross-section resistance can be calculated using the failure stress and the gross cross-sectional properties eliminating the need for obtaining an effective area. The performance of this new method was assessed by comparing the predicted results with those obtained experimentally and numerically. Comparing the performance with other standards, it was observed that the proposed method produced more accurate and consistence predictions. Reliability analysis showed that the proposed method can be used rationally with the code recommended partial factor for austenitic and ferritic grades but a higher partial factor was recommended for duplex and lean duplex grades. The proposed design method makes CSM a complete design tool for stainless steel and eliminates the necessity of lengthy traditional process of calculating effective cross-sectional properties for slender sections.

Chapter 4 Buckling Resistance of Stainless Steel Hollow Columns

Introduction

The tangential stiffness method and the Perry formulations are two different approaches generally used to predict the buckling resistance of stainless steel columns. SEI/ASCE 8- 02 [4] uses the tangential stiffness method and EN 1993-1-4 [3] adopts the Perry formulations. However, in AS/NZS 4673 [5] guidelines, there are provisions to use both the tangential stiffness method and the Perry formulations. In tangential stiffness method, the buckling stress is calculated through Euler’s formula [71] using instantaneous tangent modulus (Et); this process considers material nonlinearity of stainless steel through an iterative process and does not take account of the inevitable initial imperfections in structural members. The Perry formulations adopted in EN 1993-1-4 [3] use four different column curves for different cross-section types, in which member imperfections are explicitly addressed but the proposed functions are not related to material properties. It is now well known that stainless steel shows nonlinear stress-strain response and its nonlinearity significantly varies among different grades. Rasmussen and Rondal [7], Hradil et al. [8] and Shu et al. [9] showed that the material nonlinearity has considerable effect on the buckling resistance of column, and they tried to incorporate the material nonlinearity in column curves. Rasmussen and Rondal [7] proposed Perry type formula by modifying overall imperfection parameters where column curves directly depend on material properties. Although this method can accurately predict column resistance against flexural buckling, their formulas are very complicated to be used in practice. Hradil et al. [8] combined the formulas of tangential stiffness method and Perry curves to incorporate material nonlinearity and member imperfections in column curves, but their proposal is iterative. Shu et al. [9] developed another design method by proposing two base column curves based on Perry formula. A number of transfer equations were also proposed, which can convert the column curves of different grades of stainless steel to the base column curves. However, iteration is required in their proposed method making it complex for practical use. All of the methods described above limit the column resistance to its yield capacity ignoring the strain hardening benefits of stainless steel. The Continuous Strength Method, a strain based design approach, was developed to

Chapter 4: Buckling Resistance of Stainless Steel Hollow Columns exploit the benefits of strain hardening observed in stainless steel. With the recent developments of CSM ([14] and Chapter 3), cross-section resistances for both stocky and slender sections can be predicted using simple techniques. With its demonstrated accuracy at the cross-section level, this design concept is further extended herein to predict the buckling resistance of stainless steel columns.

The primary aim of this study is to develop a simple design method to calculate the buckling resistance of stainless steel hollow sections i.e. Square Hollow Section (SHS) and Rectangular Hollow Section (RHS), that can appropriately incorporate the beneficial characteristics of stainless steel. An FE modelling technique has been adopted in the current study; a significant number of nonlinear FE models were developed and were, later, verified using test results available in literature. A comprehensive parametric study was carried out to identify the parameters that affect the buckling resistance. Generated numerical results were used to develop Perry type column curves based on CSM design principles so that material nonlinearity and strain hardening properties could be incorporated without changing the basic forms of the currently used Perry equations in the European code [3]. A series of curves were proposed that integrate all influential parameters to cover a wide range of stainless steel grades, and finally, the accuracy of the proposed method was verified. The research work of this chapter was submitted for publication in 2016, and has recently been published in the Journal of Constructional Steel Research [111].

CSM based design approach for buckling resistance

Between two different methods: tangential stiffness method and Perry curves, Perry curves were chosen to develop the buckling formulas for stainless steel columns in this study. The reason for choosing Perry curves was that it is very versatile for producing a wide range of column curves by varying the imperfection parameter η. It can even replicate the Structural Stability Research Council [108] strength curves with suitable imperfection parameters. Perry formulations have been adopted in EN 1993-1-4 [3]. The proposed equations of EN 1993-1-4 for determining the buckling resistance of columns are presented in Eq.s 4.1-4.5; where, Nu is the buckling resistance of the column, Ag is the gross cross-sectional area, Aeff is the effective cross-sectional area, fy is the material yield stress, λ is the non-dimensional member slenderness, Ncr is the elastic critical 85

Chapter 4: Buckling Resistance of Stainless Steel Hollow Columns buckling capacity of the member based on Ag and η is the imperfection parameter. The details of this method was described in Section 2.8.

Nu = χAgfy for Class 1, 2 and 3 cross-sections (4.1)

Nu = χAefffy for Class 4 cross-sections (4.2)

A f λ = √ g y for Class 1, 2 and 3 cross-sections (4.3) Ncr

A f λ = √ eff y for Class 4 cross-sections (4.4) Ncr

1 χ = ⁡ ≤ 1.0 where, ϕ = 0.5[1 + η + λ2⁡] and η = α(λ − λ ) (4.5) ϕ+√ϕ2−λ2 0

In its current form, CSM can accurately predict the local buckling stress fcsm of a given cross-section, which can be used to predict cross-section resistances under axial compression, in-plane bending as well as combined action of compression and bending for both stocky and slender sections [14, 87, 112, 113]. This study aims to achieve accurate predictions for buckling resistance of stainless steel columns by using fcsm to extend the scope of this design approach in a rational manner. In CSM, fcsm is calculated using a simple material model with linear strain hardening, which incorporates the benefit of strain hardening commonly observed in stainless steel stocky sections. The loss of effectiveness of slender sections due to local buckling is also taken care of in fcsm. As a result, gross cross-sectional properties can used for slender sections instead of those calculated based on effective cross-section by eliminating the need for lengthy computations. Efforts are made herein to obtain column curves for RHS and SHS, which are the most commonly used cross-section types in stainless steel, by exploiting CSM design techniques whilst retaining the Perry formulas that are currently adopted in EN-

1993-1-4 [3]. A simple technique is proposed herein to use fcsm instead of the material yield stress fy in EN-1993-1-4 formulas presented in Eq.s 4.1-4.5. Once fcsm is introduced in flexural buckling equations, the formula to calculate the buckling resistance of a column becomes Eq. 4.6 and the non-dimensional column slenderness λcsm may be obtained using the modified definition as shown in Eq. 4.7, where Nu is the buckling

Chapter 4: Buckling Resistance of Stainless Steel Hollow Columns resistance of the member, χ is the reduction factor for buckling (given in Eq. 4.8), Ag is the gross cross-sectional area, Ncr is the elastic critical buckling capacity of the member based on Ag and η is the imperfection parameter. The behaviour of stainless steel SHS and RHS columns was investigated through a parametric study based on FE modelling technique, and suitable expressions were developed to determine the imperfection parameters for stainless steel SHS and RHS columns.

Nu = χAgfcsm for both stocky and slender sections (4.6)

Agfcsm λcsm = √ (4.7) Ncr

1 2 χ = ⁡ ≤ 1.0 where, ϕ = 0.5[1 + η + λcsm⁡] (4.8) 2 2 ϕ+√ϕ −λcsm

Equations 4.6 to 4.8 simply outline the design philosophy of the proposed CSM technique for flexural buckling. The following sections explain how numerical modelling technique was used to generate reliable results to turn this concept into a complete design guideline to predict the flexural buckling resistance of stainless steel columns.

Development of the finite element model

Numerical simulation was performed using commercial FE package ABAQUS [88] following the guidelines proposed by Gardner and Nethercot [53] and Ashraf et al. [36] Initially, FE models were developed and validated against experimental results. Once verified, FE models were used to carry out a comprehensive parametric study to identify the influence of different parameters such as proof stress σ0.2, strain hardening exponent n and cross-sectional slenderness p on column curves. Finally, the data generated through the parametric study were used to develop a new complete proposal for predicting the buckling resistance of stainless steel columns.

Four-node doubly curved shell element with reduced integration S4R was used in developing the FE models. Uniform mesh sizes were used in all FE models. The length of elements in flat portions of a cross-section was no more than 5 mm in the transverse direction and 10 mm in the longitudinal direction. A finer mesh was used for corner regions in transverse directions, where each corner and the associated strength-enhanced

Chapter 4: Buckling Resistance of Stainless Steel Hollow Columns zone (up to 2t) was divided into ten elements. Figure 4.1 shows a typical mesh distribution of the FE model for 60×60×3-800 column [62]. All FE models were considered as cold- rolled sections as this is the most common type for SHS and RHS. Strength enhancement occurs throughout the whole section due to the cold-rolling process, which enhances material strength but this effect is more prominent in the corner regions. Hence, enhanced corner strengths were used at corner regions up to 2t distance beyond the curved portion into the flat region, where t is the plate thickness [34, 35]. Strength enhancement in the flat region was automatically incorporated through material coupon tests taken from within the cross-sections [34, 35]. Two-stage Ramberg–Osgood (R–O) [19] material model proposed by Rasmussen [22] was used to model the complete stress-strain relationship of stainless steel considering the recent modifications suggested by Arrayago et al. [23], who suggested that austenitic and duplex grades of stainless steel can be modelled using the same model but ferritic grades require a different material model. The detail material model is given in appendix B. Only the material models for austenitic and duplex grades were used in FE models developed as part of the current study.

Figure 4.1 Typical mesh distribution of the FE model developed for 60×60×3-800 SHS column [62]

Chapter 4: Buckling Resistance of Stainless Steel Hollow Columns The presence of initial geometric imperfections is reported to have significant influence on the structural response of thin-walled members. In current FE analysis, both global and local geometric imperfections were considered. Eigenvalue analysis was performed and the lowest global buckling mode shape representing the distribution of global imperfections and the lowest local buckling mode shape were used to simulate the distribution of local imperfections with appropriately chosen amplitudes for subsequent nonlinear analysis. To assess the sensitivity of the model, three global imperfection amplitudes were considered: 1) L/1000, 2) L/1500 and L/2000 where L is the geometric length of the column. For modelling local imperfections, the Dawson and Walker [51] model modified by Gardner and Nethercot [53] was used to calculate the maximum amplitude. Residual stress was not considered as the effect of membrane residual stress is reported to have insignificant effect on the resistance of RHS and SHS columns, and the effect of bending residual stress is automatically included in the material properties obtained from flat coupon tests [62, 99, 102, 114].

Columns were considered as pin supported at both ends. All nodes at the bottom and the top ends were coupled with two reference points located at the centroid of the corresponding section, and relevant support conditions were applied to those reference points. At the bottom, all lateral translations and rotations were restrained except for the rotation around the buckling axis. At the top, all lateral translations and rotations were restrained except for the longitudinal translation and rotation around the buckling axis.

Displacements were applied at the top reference point to simulate actual column tests.

Details of the adopted support conditions are shown in Figure 4.2.

Chapter 4: Buckling Resistance of Stainless Steel Hollow Columns

Figure 4.2 Support conditions with applied deflection of the FE model developed for 60×60×3-800 column [62]

Verification of the FE model

Finite element results were compared with the corresponding test results for assessing the performance of the developed FE models. Tests reported on 51 long columns with RHS and SHS, collected from three different sources [62, 72, 95], were simulated for a comprehensive verification covering a wide range of material grades and cross-sections.

Ultimate load Nu as well as the complete load-deformation behaviour obtained from the numerical analysis was compared with those reported from experiments. Table 4.1 compares the ultimate loads obtained from FE analysis for different imperfection amplitudes to those obtained from test results. It is observed that the effect of imperfection amplitudes on the ultimate resistances Nu is relatively small. In all cases, FE results showed good accuracy in predicting the experimental results. Overall, the global imperfection amplitude L/1500 produced marginally better predictions. Hence, the

Chapter 4: Buckling Resistance of Stainless Steel Hollow Columns maximum global imperfection amplitude was used as L/1500 along with the local imperfection in the models developed for parametric study described in the next section. The maximum amplitude of the local imperfections were calculated by the Dawson and Walker [51] model modified by Gardner and Nethercot [53]. Figure 4.3 compares the experimental and numerical load-deformation curves for one RHS column subjected to buckling about major axis (80×40×4-MJ-1200 [62]), one RHS column subjected to bucking about minor axis (RHS 100×50×6-LC-1m [95]) and for one SHS column (60×60×3-1600 [62]). The failure mode of C4L 1200 column [72] is shown in Figure 4.4. Overall comparisons show that the adopted FE modelling techniques were capable of simulating the buckling behaviour of stainless steel hollow columns, and were subsequently used to generate additional results for comprehensive understanding of their behaviour.

Table 4.1 Comparison of FE results of RHS and SHS columns with test results

Types of Failure Imperfect Amplitude Reference Member ID stainless mode L/1000 L/1500 L/2000 steel NFE/Ntest NFE/Ntest NFE/Ntest 80×80×4-2000 F 1.08 1.11 1.13 80×80×4-1200 F 0.89 0.90 0.91 60×60×3-2000 F 1.03 1.06 1.07 60×60×3-1600 F 1.03 1.06 1.08 Theofanous 60×60×3-1200 F 1.04 1.07 1.08 and 60×60×3-800 Lean F 1.03 1.01 1.06 Gardner Duplex F [62] 80×40×4-MI-1600 (1.4162) 0.89 0.93 0.94 80×40×4-MI-1200 F 1.05 1.06 1.08 80×40×4-MI-800 F 1.12 1.15 1.16 80×40×4-MJ-1600 F 0.83 0.86 0.87 80x40x4-MJ-1200 F 0.93 0.95 0.96 80×40×4-MJ-800 F 1.05 1.07 1.08 SHS 80×80×4-LC-1.9m F 1.09 1.12 1.14 SHS 80×80×4-LC-2m F 0.98 1.00 1.02 Gardner SHS 100×100×2-LC-2m F 0.93 0.94 0.95 and Austenitic Nethercot SHS 150×150×4-LC-2m F 0.98 0.99 1.00 (1.4301) [95] RHS 100×50×6-LC-2m F 1.01 1.04 1.05

RHS 150×100×4-LC-2m F 0.90 0.92 0.93 RHS 100×50×6-LC-1m F 0.97 0.99 1.01

Chapter 4: Buckling Resistance of Stainless Steel Hollow Columns Types of Failure Imperfect Amplitude Reference Member ID stainless mode L/1000 L/1500 L/2000 steel NFE/Ntest NFE/Ntest NFE/Ntest C2L200 L 1.06 1.06 1.06 C2L550 L 1.06 1.08 1.09 C2L900 F 1.00 1.03 1.04 C2L1200 F 0.99 1.02 1.03 C2L1550 F 1.02 1.05 1.07 C3L200 Y 0.92 0.90 0.95 C3L550 L 0.91 0.91 0.93 C3L900 F 0.83 0.85 0.86 C3L1200 F 1.04 1.08 1.09 C3L1550 F 1.03 1.06 1.08 C1L200 Y 0.97 0.98 0.99 C1L550 F 0.99 1.01 1.02 C1L900 F 1.01 1.04 1.05 C1L1200 F 0.98 1.00 1.01 C1L1550 F 0.95 0.96 0.97 Lean Y Huang and C4L200 1.06 1.07 1.07 Duplex Young [72] L C4L550 (1.4162) 0.94 0.96 0.97 C4L900 F 0.96 0.98 0.99 C4L1200 F 0.94 0.97 0.98 C4L1550 F 0.97 1.00 1.01 C5L200 L 1.08 1.08 1.08 C5L550 L 0.98 0.98 0.98 C5L900 L+F 0.94 0.96 0.98 C5L900R L+F 0.93 0.96 0.97 C5L1200 F 1.01 1.03 1.05 C5L1200R F 1.01 1.05 1.07 C5L1550 F 0.96 0.99 1.01 C6L200 L 0.89 1.09 1.08 C6L550 l 1.04 1.05 1.06 C6L900 L+F 0.87 0.89 0.90 C6L1200 L+F 0.87 0.89 0.90 C6L1550 L+F 0.93 0.96 0.99 Mean 0.98 1.00 1.02 COV 0.07 0.07 0.07 Y= material yielding, L = local buckling, F= Flexural buckling

Chapter 4: Buckling Resistance of Stainless Steel Hollow Columns

700 RHS 100×50×6-LC-1m Test 600 FE 500

400 80×40×4-MJ-1200

300 Load, kN Load, 200 60×60×3-1600 100

0 0 10 20 30 40 Lateral deflection at mid height, mm

Figure 4.3 Comparison of load-deformation curves of hollow section columns obtained from FE models with test results

Figure 4.4 Experimental and FE failure mode of C4L1200 column [84]

Chapter 4: Buckling Resistance of Stainless Steel Hollow Columns Parametric study

Verified FE modelling technique was used to identify the parameters that significantly affect the buckling resistance of stainless steel columns through a comprehensive parametric study. Non-dimensional proof stress e (= σ0.2/E) and strain hardening exponent n have significant effects on buckling resistance [7-9, 115]. In this research, effects of e, n as well as the possible effect of cross-section slenderness λp and aspect ratio of RHS on buckling resistance was investigated with a view to proposing reliable buckling curves for stainless steel columns.

In this parametric study, the values of e varies from 0.001 to 0.0035 (0.001, 0.0015, 0.002, 0.0025, 0.003 and 0.0035) and six values of n varying between 5 to 10 were considered. These covered a wide range of material properties for various grades of stainless steel. When the effect of e was investigated, the value of n remain constant as 7. On the other hand, when the influence of n was observed, the value of e remain constant as 0.002. For each material combination, seven different cross-sections were considered where, values of λp ranging from 0.38 to 0.98 (with 0.68 being the transition between stocky and slender sections). λp was calculated according to Eq. 2.52 [14] where the elastic buckling capacity of the full cross-section (σcr,cs) was determined using CUFSM [116].

Cross-section slenderness λp depends on the value of e, hence, a total of 42 different SHS columns were analysed for different sets of λp and e values. Considering a constant nominal outer dimension of 125 mm for SHS, thicknesses chosen for 42 cross-sections are shown in Table 4.2. For each combination of material properties (e and n), corresponding seven cross-sections (with different λp values) were analysed for 15 different column lengths. The column lengths were chosen such as way that non- dimensional member slenderness λcsm varied from 0.2 to 2.0 (0.2, 0.3, 0.35, 0.4, 0.45, 0.5, 0.55, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8 and 2.0). In order to achieve an appropriate transition between long columns and short columns, smaller intervals for λcsm were chosen between 0.2 to 0.6.

RHS columns with four different section depth to width (H/B) ratios such as 1.25, 1.50, 1.75 and 2.00 were considered in the current parametric study. The effect of H/B ratio on buckling resistance of RHS columns was examined for both stocky (λp = 0.48) and slender

Chapter 4: Buckling Resistance of Stainless Steel Hollow Columns

(λp = 0.88) sections. The effect of λp on column curves of RHS was studied on a 150×100 mm section with seven different thicknesses to obtain different λp values varied from 0.38 to 0.98; geometric dimensions of RHS are shown in Table 4.3. Other material parameters such as e and n were taken as 0.002 and 7 respectively. All RHS columns were analysed for a total of 15 different values for λcsm. Both major axis and minor axis buckling resistances were determined for the considered RHS columns. A total of 1545 models were analysed to obtain a thorough understanding of the buckling behaviour of stainless steel columns with hollow sections.

Table 4.2 Thicknesses of 125×125mm SHS used in the parametric study showing corresponding values of p and e.

e λp 0.001 0.0015 0.002 0.0025 0.003 0.0035

0.38 4.6 5.46 6.15 6.73 7.23 7.67 0.48 3.75 4.47 5.06 5.56 5.99 6.38 0.58 3.16 3.79 4.3 4.74 5.12 5.46 0.68 2.74 3.29 3.74 4.13 4.47 4.77 0.78 2.41 2.91 3.31 3.66 3.96 4.24 0.88 2.15 2.6 2.97 3.32 3.56 3.81 0.98 1.95 2.35 2.69 2.98 3.23 3.46

Table 4.3 Dimensions of RHS cross-sections used in the parametric study

Height Width Thickness

(mm) (mm) (mm)

200 100 7.18, 4.20

175 100 6.15, 3.60

150 100 6.72, 5.51, 4.67, 4.05, 3.58, 3.20, 2.90

125 100 4.79, 2.80

Chapter 4: Buckling Resistance of Stainless Steel Hollow Columns

Effect of λp, e , n and H/B on column curves

Typical variations of column curves due to changes in cross-section slenderness as well as material parameters e and n are shown in Figures 4.5-4.12. Non-dimensional column

Nu,FE strength or reduction factor χ was calculated as , where Nu,FE is the ultimate buckling Agfcsm load of the column, Ag is the gross cross-section area and fcsm is the buckling stress of that cross-section calculated by the CSM formulas described in Chapter 3. In Figures 4.5-4.8, the influence of λp on column curves is shown for different e and n values respectively. It was observed that cross-section slenderness λp has a significant effect on column curves, and column curves move upward with increasing values of λp i.e. for a given set of material properties the reduction factor  is higher for relatively slender cross-sections with higher λp. For example, non-dimensional strength of a column with λp = 0.98 is 35% greater than that of a column with λp = 0.38 (at λcsm = 0.6 for material properties e = 0.003 and n = 7). For slender cross-sections with higher value of λp, the column resistance approaches to its cross-section capacity i.e. χ approaches 1.00 at a relatively higher value of λcsm when compared against stocky cross-sections with smaller λp; this is evident from

Figures 4.5-4.8. The effect of λp is more prominent at the intermediate portion of column curves and diminishes with the increase of λcsm. Figures 4.7 and 4.8 also show that influence of λp is more prominent for higher values of n.

1.2 λp = 0.38 1 λp = 0.48 λp = 0.58 λp = 0.68 0.8 λp = 0.78 λp = 0.88

χ 0.6 λp = 0.98

0.4

0.2

0 0 0.5 1 1.5 2 2.5 λcsm

Figure 4.5 Column curves of SHS for different λp values (e = 0.001, n = 7)

Chapter 4: Buckling Resistance of Stainless Steel Hollow Columns

1.2 λp = 0.38 λp = 0.48 1 λp = 0.58 λp = 0.68 0.8 λp = 0.78 λp = 0.88 λp = 0.98 χ 0.6

0.4

0.2

0 0 0.5 1 1.5 2 2.5 λcsm

Figure 4.6 Column curves of SHS for different λp values (e = 0.003, n = 7)

1.2 λp = 0.38 λp = 0.48 1 λp = 0.58 λp = 0.68 0.8 λp = 0.78 λp = 0.88 λp = 0.98

χ 0.6

0.4

0.2

0 0 0.5 1 1.5 2 2.5 λcsm

Figure 4.7 Column curves of SHS for different λp values (e = 0.002, n = 5)

Chapter 4: Buckling Resistance of Stainless Steel Hollow Columns

1.2 λp = 0.38 λp = 0.48 1 λp = 0.58 λp = 0.68 0.8 λp = 0.78 λp = 0.88 λp = 0.98 χ 0.6

0.4

0.2

0 0 0.5 1 1.5 2 2.5

λcsm

Figure 4.8 Column curves of SHS for different λp values (e = 0.002, n = 10)

Figures 4.9 and 4.10 show the effect of proof stress on column curves for stocky and slender cross-sections respectively; a moderate effect of e was observed. With the increase of e value, column curves move upward and this effect is more prominent for slender cross-sections. For example, reduction factor increases from 0.64 to 0.72 (12% increase) at λcsm=0.80 when e increases from 0.001 to 0.0035 for a stocky section (λp=

0.48), whereas χ increases from 0.72 to 0.86 (19% increase) for a slender section (λp= 0.88). Moreover, for stocky cross-sections, effect of e is more prominent for intermediate columns and its effect diminishes with higher or lower λcsm values. For slender cross- sections, on the other hand, the effect of e diminishes with increase of column slenderness but for columns with smaller λcsm, column curves become almost parallel to each other. From Figures 4.11 and 4.12, it is clear that n showed a similar effect on column curves but its effect is less significant than e. It was observed from Figures 4.9-4.12 that shapes of all column curves with the same λp are very similar irrespective of their material properties. Effects of H/B ratio on column curves of RHS are shown in Figures 4.13 and 4.14 for stocky and slender sections respectively. It was observed that H/B ratio has no significant effect on column curves of stocky sections (λp=0.48). In case of slender sections (λp = 0.88), however, aspect ratio showed some minor effect on column curves at low λcsm values.

Chapter 4: Buckling Resistance of Stainless Steel Hollow Columns

1.2 e=0.001 1 e=0.0015 e=0.002 0.8 e=0.0025 e=0.003

χ 0.6 e=0.0035

0.4

0.2

0 0 0.5 1 1.5 2 2.5 λcsm

Figure 4.9 Column curves of SHS for different e values (λp = 0.48, n = 7)

1.2

e=0.001 1 e=0.0015 e=0.002 0.8 e=0.0025 e=0.003 χ 0.6 e=0.0035 0.4

0.2

0 0 0.5 1 1.5 2 2.5 λcsm

Figure 4.10 Column curves of SHS for different e values (λp = 0.88, n = 7)

Chapter 4: Buckling Resistance of Stainless Steel Hollow Columns

1.2 n= 5 1 n= 6 n=7 0.8 n=8 n=9

χ 0.6 n=10

0.4

0.2

0 0 0.5 1 1.5 2 2.5

λcsm

Figure 4.11Column curves of SHS for different n values (λp = 0.48, e = 0.002)

1.2 n= 5 1 n= 6 n= 7 0.8 n= 8 n= 9

χ 0.6 n= 10

0.4

0.2

0 0 0.5 1 1.5 2 2.5 λcsm

Figure 4.12 Column curves of SHS for different n values (λp = 0.88, e = 0.002)

100

Chapter 4: Buckling Resistance of Stainless Steel Hollow Columns

1.2 H:B=1:2 1 H:B=1:1.75 H:B=1:1.50 H:B=1:1.25 0.8 H:B=2:1 H:B=1.75:1

χ 0.6 H:B=1.50:1 H:B=1.25:1 0.4

0.2

0 0 0.5 1 1.5 2 2.5 λcsm

Figure 4.13 Column curves of RHS for different H/B ratios (λp = 0.48, e = 0.002 and n = 7)

1.2 H:B=1:2 H:B=1:1.75 1 H:B=1:1.50 H:B=1:1.25 0.8 H:B=2:1 H:B=1.75:1

χ 0.6 H:B=1.50:1 H:B=1.25:1 0.4

0.2

0 0 0.5 1 1.5 2 2.5 λcsm

Figure 4.14 Column curves of RHS for different H/B ratios (λp = 0.88, e = 0.002 and n = 7)

Observed variations clearly show that the parameters e, n and λp have considerable effects on buckling resistance, and should be incorporated in column resistance formulas for different stainless steel grades to appropriately predict their structural response. It is worth noting that traditional use of σ0.2 produces χ > 1.0 for stocky stainless steel sections but the use of CSM buckling stress fcsm automatically captures the effects of strain hardening maintaining χ ≤ 1.0.

101

Chapter 4: Buckling Resistance of Stainless Steel Hollow Columns Correction factors for e and n

As part of the current study, the non-dimensional column slenderness λcsm was modified to incorporate the observed effects of e and n on the buckling resistance. Two modification factors Ce and Cn were proposed as shown in Eq. 4.9, where λm is the modified slenderness and, Ce and Cn are the modification factors for e and n respectively.

To diminish the effect of e on column curves, the expression of Ce was developed to move all other column curves towards the column curve for e = 0.002. On the other hand, Cn moves all other column curves towards the column curves for n = 5. Expression of Ce is given in Eq.s 4.10-4.12, whilst that for Cn is given in Eq. 4.13. Ce and Cn are suggested to be used for 0.2 ≤ λcsm ≤ 2.0. Beyond this limit, effects of e and n are negligible and the value of these factors is zero. Figures 4.15 and 4.16 show the effect of non-dimensional proof stress e on column curves with modified slenderness m for stocky and slender cross-sections, and it was observed that all column curves for different e values with the same λp and n values almost overlap each other. Similarly, from Figures 4.17 and 4.18, it was observed that n had no obvious effect on the modified column curves. These proposed factors effectively included the effects of e and n, and a single column curve can represent all columns with a cross-section slenderness λp.

5−n λ = λ + C + C ≥ 0 (4.9) m csm e 5 n

(λ −0.2)(0.185 ln(e)+1.15) C = − csm ⁡⁡⁡for⁡0.2 < λ < 0.6⁡and⁡λ < 0.68 (4.10) e 0.4 csm p

(2−λ )(0.185 ln(e)+1.15) C = − csm ⁡⁡⁡for⁡0.6 ≤ λ < 2.0⁡and⁡λ < 0.68 (4.11) e 1.4 csm p

(2−λ )(0.17 ln(e)+1.06) C = − csm ⁡⁡⁡for⁡⁡0.2 < λ < 2.0⁡and⁡λ ≥ 0.68 (4.12) e 1.4 csm p

2 Cn = −0.25λcsm + 0.6λcsm − 0.22⁡ ≥ 0⁡⁡for⁡⁡0.2 < λcsm < 2.0⁡ (4.13)

102

Chapter 4: Buckling Resistance of Stainless Steel Hollow Columns

1.2 e=0.001 1 e=0.0015 e=0.002 0.8 e=0.0025 e=0.003

χ 0.6 e=0.0035

0.4

0.2

0 0 0.5 1 1.5 2 2.5 λm

Figure 4.15 Column curves of SHS with respect to λm for different e values (λp = 0.48, n = 7)

1.2 e=0.001 1 e=0.0015 e=0.002 0.8 e=0.0025 e=0.003 e=0.0035 χ 0.6

0.4

0.2

0 0 0.5 1 1.5 2 2.5 λm

Figure 4.16 Column curves of SHS with respect to λm for different e values (λp = 0.88, n = 7)

103

Chapter 4: Buckling Resistance of Stainless Steel Hollow Columns

1.2 n= 5 1 n= 6 n=7 0.8 n=8 n=9

χ 0.6 n=10

0.4

0.2

0 0 0.5 1 1.5 2 2.5 λm

Figure 4.17 Column curves of SHS with respect to λm for different n values (λp = 0.48, e = 0.002)

1.2 n= 5 1 n= 6 n= 7 0.8 n= 8 n= 9

χ 0.6 n= 10

0.4

0.2

0 0 0.5 1 1.5 2 2.5 λm

Figure 4.18 Column curves of SHS with respect to λm for different n values (λp = 0.88, e = 0.002)

Reduction factor χ

The reduction factor used in the Perry type curves was retained in the current proposal to be in line with current EN 1993-1-4 [3] technique but the non-dimensional member slenderness λ has been replaced by the suggested modified non-dimensional member

104

Chapter 4: Buckling Resistance of Stainless Steel Hollow Columns slenderness λm as given in Eq. 4.14. To develop an appropriate expression for the imperfection parameter η, the reduction factors determined from the FE models were used to calculate the exact values of η required for a perfect fit to the proposed Perry curves. Values of η for different cross-section slenderness are plotted in Figure 4.19; this clearly shows that the variation of η with λm is nonlinear and the effect of λp should be considered in the expression of η. A sigmoidal function of λm would be a better fit than a linear expression. The suggested expression of η is given in Eq. 4.15 where A, B and W are the coefficients of the chosen sigmoidal function. Values of A and B depend on λp as given in Eq. 4.16 and 4.17 and the constant value of W = 0.25 is proposed for both RHS and

SHS. The proposed method produced separate column curves for different λp.

1 2 χ = ⁡ ≤ 1.0 where, ϕ = 0.5[1 + η + λm⁡] (4.14) 2 2 ϕ+√ϕ −λm

0.6

λp=0.58 0.5 λp=0.38

0.4

η 0.3

0.2

λp=0.98 0.1

0 0 0.5 1 1.5 2 2.5 λm

Figure 4.19 η calculated from FE results and proposed curves of η for different λp

A η = − 0.25⁡ ≥ 0⁡ (4.15) 1+exp−(λm−B)⁄W

A = −0.33λp + 0.85 ≥ 0.52 (4.16)

B = 0.38λp + 0.15 ≤ 0.42 (4.17)

105

Chapter 4: Buckling Resistance of Stainless Steel Hollow Columns Performance of the proposed CSM method

The performance of the proposed method was verified with available numerical and test results. A total of 132 test results collected from different sources [40, 62, 72, 93-95, 98, 117] were used. The predictions of the proposed method were also compared with the predictions of EN 1993- 1-4+A1 [68], SEI/ACSE 8-02 [4] standards and Rasmussen and

Rondal’s [7] proposal adopted in AS/NZS 4673 [5]. The values of the parameters α, β, λ1 and λ0, were calculated by equations proposed by Rasmussen and Rondal [7]. Figures 4.20-4.22 show the comparison of the proposed method with EN 1993- 1-4+A1 [68], SEI/ACSE 8-02 and AS/NZS 4673 for SHS columns respectively. Figures 4.23-4.25 also show this comparison for RHS columns. All the figures clearly show that the new CSM formulations produced more accurate and more consistent results. The key features of the comparison are shown in Table 4.4. For SHS columns, the ratio of CSM prediction and FE results lies in the range of 0.92 to 1.06 with an average of 0.98 and the coefficient of variation (COV) is only 0.03, whereas this mean for EN 1993-1-4+A1 [68] is 0.92 , for ASCE is 1.05 and for AS/NZS is 0.97. Table 4.4 also shows that the performance of the proposed CSM technique is better than EN 1993-1-4+A1 [68] and ASCE [4] standards when compared against available test results of SHS and RHS. The average of the ratio of AS/NZS 4673 [5] prediction to test results for SHS columns is marginally better than the prediction of the proposed method but the predictions of the proposed method are less scattered. More importantly, the computational effort required in the current CSM technique is considerably less than any other design rules, especially in the case of slender cross-sections. From Figures 4.21 and 4.24, it was also observed that ASCE [4] over predicts the buckling resistance especially for columns with higher member slenderness. Figures 4.26-4.29 show the ratio of the predicted resistance and FE results for the variation of λp, e, n and H/B respectively for newly proposed CSM method and for EN 1993- 1-4+A1 [68]. All those figures clearly show that the new CSM design technique produced similar level of accuracy throughout the studied range of different parameters. From Figures 4.22 and 4.23, it is observed that the accuracy of EN 1993- 1-4+A1 [68] decreases with increasing values of λp and e but the proposed CSM method showed similar level of accuracy for all λp and e values.

106

Chapter 4: Buckling Resistance of Stainless Steel Hollow Columns Table 4.4 Comparison of the performance of the proposed CSM method for buckling resistance of hollow columns with other standards

Section CSM EN 1993-1-4+A1 ASCE AS/NZS

types Ncsm/NFE/Test NEC3/NFE/Test NASCE/NFE/Test NAS/NZS/NFE/Test

Average COV Average COV Average COV Average COV

SHS FE 0.98 0.03 0.92 0.09 1.06 0.06 0.97 0.05 data

SHS Test 0.95 0.10 0.92 0.12 1.05 0.14 0.97 0.12 data

RHS FE 1.00 0.04 0.94 0.08 1.05 0.06 0.97 0.05 data

RHS Test 0.97 0.14 0.89 0.15 1.00 0.14 0.94 0.14 data

1.4

1.2

FE/Test 0.8 /N

0.6 csm/EC3

N EN 1993-1-4+A1 FE data 0.4 EN 1993-1-4+A1 test data CSM FE data 0.2 CSM test data

0 0 0.5 1 1.5 2 2.5 λm or λ

Figure 4.20 Comparison of CSM and EN 1993- 1-4+A1 [68] predicted buckling resistances with FE and test results for SHS columns

107

Chapter 4: Buckling Resistance of Stainless Steel Hollow Columns 1.6

1.4

1.2

1 FE/Test

/N 0.8

csm/ASCE 0.6

N ASCE FE data 0.4 ASCE test data CSM FE data 0.2 CSM test data

0 0 0.5 1 1.5 2 2.5 λm or λ Figure 4.21 Comparison of CSM and SEI/ACSE 8-02 [4] predicted buckling resistances with FE and test results for SHS columns

1.4

1.2

FE/Test 0.8 /N

0.6

csm/AS/NZS AS/NZS FE data N 0.4 AS/NZS test data CSM FE data 0.2 CSM test data

0 0 0.5 1 1.5 2 2.5 λm or λ Figure 4.22 Comparison of CSM and AS/NZS 4673 [5] predicted buckling resistances with FE and test results for SHS columns

108

Chapter 4: Buckling Resistance of Stainless Steel Hollow Columns

1.4

1.2

1 FE/Test

/N 0.8

csm/EC3 0.6 N EN 1993-1-4+A1 FE data 0.4 EN 1993-1-4+A1 test data CSM FE data 0.2 CSM test data

0 0 0.5 1 1.5 2 2.5 λm or λ Figure 4.23 Comparison of CSM and EN 1993- 1-4+A1 [68] predicted buckling resistances with FE and test results for RHS columns

1.6

1.4

1.2

FE/Test 1 /N

0.8 csm/ASCE N 0.6 ASCE FE data 0.4 ASCE test data CSM FE data 0.2 CSM test data

0 0 0.5 1 1.5 2 2.5 λm or λ Figure 4.24 Comparison of CSM and SEI/ACSE 8-02 [4] predicted buckling resistances with FE and test results for RHS columns

109

Chapter 4: Buckling Resistance of Stainless Steel Hollow Columns

1.4

1.2

1 FE/Test

/N 0.8

0.6 csm/AS/NZS

N AS/NZS FE data 0.4 AS/NZS test data CSM FE data 0.2 CSM test data

0 0 0.5 1 1.5 2 2.5 λm or λ Figure 4.25 Comparison of CSM and AS/NZS 4673 [5] predicted buckling resistances with FE and test results for RHS columns

1.4

1.2

FE /N

0.8 csm/EC3

N 0.6

0.4 EN 1993-1-4+A1 0.2 CSM 0 0.2 0.4 0.6 0.8 1 1.2 λp Figure 4.26 Comparison of CSM and EN 1993- 1-4+A1 [68] predicted buckling resistances with FE and test results for SHS columns for different cross-section slenderness

110

Chapter 4: Buckling Resistance of Stainless Steel Hollow Columns 1.4

1.2

1 FE

/N 0.8 csm/EC3

N 0.6

0.4 EN 1993-1-4+A1 0.2 CSM

0 0 0.001 0.002 0.003 0.004 e Figure 4.27 Comparison of CSM and EN 1993- 1-4+A1 [68] predicted buckling resistances with FE and test results for SHS columns for different e values

1.4

1.2

FE /N

0.8 csm/EC3 N 0.6

0.4 EN 1993-1-4+A1 0.2 CSM 0 4 5 6 7 8 9 10 11 n Figure 4.28 Comparison of CSM and EN 1993- 1-4+A1 [68] predicted buckling resistances with FE and test results for SHS columns for different n values

111

Chapter 4: Buckling Resistance of Stainless Steel Hollow Columns 1.4

1.2

FE /N

0.8 csm/EC3 N 0.6

0.4 EN 1993-1-4+A1 0.2 CSM 0 0 0.5 1 1.5 2 2.5 H/B Figure 4.29 Comparison of CSM and EN 1993- 1-4+A1 [68] predicted buckling resistances with FE and test results for RHS columns for different aspect ratios

Reliability analysis

Reliability of the formulas proposed for predicting the buckling resistance of stainless steel hollow section columns was conducted according to the guidance of EN1990-Annex D [109]. Since, the material model for austenitic and duplex grades are exactly the same, there is no fundamental difference between the FE models of austenitic grades column and those for duplex grades column. In FE models, the value of n was varied between 5 to 10, which covers both austenitic and duplex grades of stainless steel. Hence, FE results were considered in austenitic and duplex grades in reliability analysis. Results of the reliability analysis are summarised in Table 4.5, where n is the total number of tests and

FE results, kd,n is the design (ultimate limit state) fractile factor for n number of data, b is the average ratio of experimental or FE results to model resistance based on a least squares fit for each set of data, Vδ is the coefficient of variation of tests and FE simulations relative to the resistance model, Vr is the combined coefficient of variation incorporating both model and basic variable uncertainties and γM1 is the partial safety factor for member resistance. As recommended by Afshan et al. [110], the material over-strength factor and the coefficient of variation of material strength for austenitic grades of stainless steel are 1.3 and 0.06 respectively. For duplex grades, these values are 1.1 and 0.03. The coefficient of variation of geometric properties was considered as 0.05 [110]. The analysis 112

Chapter 4: Buckling Resistance of Stainless Steel Hollow Columns showed that the partial safety factor γM1 is 1.00 for austenitic grades columns and 1.10 for duplex grades columns. EN 1993-1-4 [3] recommended partial safety faction is also 1.10. Therefore, it could be concluded that the proposed buckling formulas can be safely used with partial safety factor γM1 equal to 1.10 for austenitic and duplex grades of stainless steel hollow section columns.

Table 4.5 Reliability analysis of CSM method proposed for buckling resistance of RHS and SHS columns

Material n Kd,n b Vδ Vr γM1

Austenitic 1663 3.096 1.020 0.045 0.090 1.00

Duplex 1619 3.096 1.018 0.035 0.068 1.10

Worked out example: Buckling capacity of a RHS column

The CSM predicted buckling resistance for a RHS column ‘RHS 120x80x3-LC-1m’ buckling about minor axis, tested by Gardner and Nethercot [95] was determined as follows:

Cross-section geometric and material properties:

2 H =120 mm ro = 7.46 mm E = 209300 N/mm 2 2 B =80.2 mm A =1082.76 mm σ0.2 = 461.9 N/mm 4 2 t =2.86 mm Iy-y =1176943 mm σu = 739 N/mm L=1000.6 mm Ncr=2428.3 kN n = 4.1 εy = 461.9/209300=0.0022

Determination of cross-section slenderness λp:

σ0.2 461.9 λp = √ = √ = 0.912 σcr,cs 555.9 2 where σcr,cs = 555.9 N/mm was obtained directly from the CUFSM software [76]

Multiplying by (Cflat/Ccl)max, where Cflat is the flat element width and Ccl is the centreline element width. Corrected λp = 0.912×0.897 = 0.818 > 0.68 (so, it is a slender section)

113

Chapter 4: Buckling Resistance of Stainless Steel Hollow Columns

Determination of cross-section deformation capacity εcsm:

εcsm 0.25 εu = 3.6 = 0.516 < min⁡(15, 0.1 ⁡) εy 0.818 εy

Determination of equivalent elastic deformation capacity εe,ev

b 2.9 C = aλp = 3.2(0.818) = 1.787

εe,ev = Cεcsm = 1.787 × 0.516 × 0.0022 = 0.002

Determination of buckling stress fcsm:

2 fcsm = εe,evE = 0.002 × 209300 = 418.6 N/mm

Determination of non-dimensional member slenderness λcsm:

Agfcsm 1082.76×418.6 λcsm = √ = √ = 0.436 Ncr 2428.3

As ⁡0.2 < λcsm < 2.0⁡and⁡λp ≥ 0.68 461.9 (2 − λ )(0.17 ln(e) + 1.06) (2 − 0.43) (0.17 ln ( ) + 1.06) C = − csm = ⁡⁡ − 209300 e 1.4 1.4

Ce = −0.0226

2 Cn = −0.25λcsm + 0.6λcsm − 0.22 = ⁡ −0.25 × 0.43 + 0.6 × 0.43 − 0.22

Cn = −0.008⁡ ≥ 0

Cn = 0

Determination of modified non-dimensional member slenderness λm:

5−n λ = λ + C + C = 0.436 − 0.0226 = 0.413 ≥ 0 m csm e 5 n

Determination of imperfection parameter η:

A = −0.33λp + 0.85 = ⁡ −0.33 × 0.818 + 0.85 = 0.58 ≥ 0.52

B = 0.38λp + 0.15 = 0.38 × 0.818 + 0.15 = 0.46 ≤ 0.42 B = 0.42

A 0.58 η = − 0.25 = ⁡ − 0.25 = 0.036 ≥ 0⁡ 1+exp−(λm−B)⁄W 1+exp−(0.413−0.42)⁄0.25

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Chapter 4: Buckling Resistance of Stainless Steel Hollow Columns

2 2 ϕ = 0.5[1 + η + λm⁡] = 0.5[1 + 0.036 + 0.413 ] = 0.60

Determination of buckling reduction factor χ:

1 1 χ = = ⁡ = 0.966 ≤ 1.0 ϕ+√ϕ2−λ2 0.60+√0.602−0.4132

Determination of buckling resistance of the column Nu:

Nu = χAgfcsm = 0.966 × 1082.76 × 418.6 = 437833⁡N = 437.8⁡kN

[Test ultimate load = 448 kN; EN 1993-1-4+A1 [68] predicted buckling resistance is

406.3 kN, SEI/ASCE 8-02 [4] predicted buckling resistance is 454.2 kN and AS/NZS

4673 [5] predicted buckling resistance is 454.2.]

Conclusions

With the development of CSM formulas for determining the cross-section resistance of stainless steel slender sections, CSM is now applicable for full range of cross-section slenderness. The work presented in this chapter was aimed at extending CSM at member level by developing buckling formulas for stainless steel hollow columns. Currently available stainless steel design standards do not appropriately address the effects of material nonlinearity, and do not include any provision for exploiting the benefits of strain hardening. However, stainless steel shows significant material nonlinearity with considerably varying strain hardening properties between different grades. In this study, a new design technique has been proposed to predict column resistance following the basic design concepts of CSM with widely used Perry formulations. In the proposed method, CSM local buckling stress fcsm was used instead of fy in the Perry formulas adopted in EN 1993-1-4 [3].

Cold-formed RHS and SHS are the most commonly used stainless steel sections in the construction industry. In this chapter, the flexural buckling behaviour of RHS and SHS columns were investigated using FE models. Initially, FE models were verified using available test results and later, verified FE models were used to generate a large number

115

Chapter 4: Buckling Resistance of Stainless Steel Hollow Columns of column responses for parametric study. Analysis of the FE results showed that cross- section slenderness λp, non-dimensional proof stress e and material strain hardening exponent n have significant influences on column curves. Effects of e and n on column curves were addressed using two coefficient Ce and Cn to modified non-dimensional member slenderness λcsm to modified non-dimensional member slenderness λm. With this modification, column curves only vary with λp. The imperfection factor η was expressed as a sigmoidal function of λp and λm where separate column curve was generated for each value of λp.

The performance of the proposed CSM formulas was evaluated using available test results and FE results. The predictions of the proposed method were also compare with the predictions of EN 1993-1-4+A1[68], SEI/ASCE 8-02 [4] and AS/NZS 4673 [5] standards and the proposed method appeared more accurate and more consistence than others. Rasmussen and Rondal’s [7] proposal adopted in AS/NZS also produces good predictions of the buckling resistance of RHS and SHS columns. However, EN 1993-1- 4+A1 [68] under predicted the buckling resistance and SEI/ASCE 8-02 [4] over predicted the buckling resistance. Reliability analysis showed that developed CSM formulas can be safely used to predict the buckling resistance of RHS and SHS columns of austenitic and duplex grades of stainless steel with partial safety factor 1.10.

With the development of the buckling formulas in this study, the applicability of CSM was extended for stainless steel members. This method effectively includes the beneficial effects of material nonlinearity and strain hardening in column buckling curves through a simple technique. Importantly, for slender sections, gross cross-sectional properties were used instead of effect cross-sectional properties, which significantly reduced the computational effort. This proposal can be considered as a significant step towards advancement of CSM into predicting resistance of stainless steel structural members.

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Chapter 5 Behaviour and Design of Stainless Steel Welded I-columns

Introduction

Welded sections are often used to meet the high load bearing capacity required for buildings and bridges. Welded sections can be fabricated to meet the exact design requirements, and may yield more economic design for a structure. During the last decade, research on structural stainless steel was mainly focused on cold-formed sections due largely to their easy availability. Very few studies were was conducted on welded sections, and very few design codes [3, 17] have design guidelines for welded sections. In recent years, a number of research projects were reported on stainless steel welded sections. Kuwamura [92], Saliba and Gardner [100] and Yuan et al. [63] studied the local buckling behaviour of stainless steel welded I-sections. Real et al. [118], Saliba and Gardner [119] and Hassanein [120] studied the shear response of stainless steel plate girders. Wang et al. [121] and Yang et al. [122] investigated the lateral torsional buckling of stainless steel welded I-section beams. Yuan et al. [48] measured the residual stresses of welded box sections and I-sections, and observed that the magnitudes and the distribution of longitudinal residual stresses of stainless steel welded sections were different from those in carbon steel welded sections. They also proposed a model for residual stress distribution in stainless steel welded sections. Investigation on the compression resistance of stainless steel welded sections is scarce. Recently, Yuan et al. [123] studied the local-overall interactive buckling of welded box sections by testing eight specimens. Yuan et al. [124] also tested welded I-columns of austenitic and duplex grades of stainless steel to study a similar behaviour. They also performed numerical analysis and observed that residual stresses significantly affect the buckling resistance of welded I-columns. Yang et al. [125] tested stainless steel welded I-columns for flexural buckling and showed that EN 1993-1-4 [3] and AS/NZS 4673 [5] predictions were conservative for predicting the buckling resistance of stainless steel welded I-columns, and ASCE 8- 02 [4] predictions were very scattered. It is noted that, AS/NZS 4673 and ASCE 8-02 design rules are proposed for cold-formed stainless steel structures. Recently, Gardner et al. [126] investigated the behaviour of laser welded stainless steel columns for local buckling and flexural buckling, and observed that the carrying capacity of laser welded

Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns sections were higher than the conventionally welded sections due to lower residual stress magnitudes. It is evident that, there is significant lack of test data for appropriate understanding of the behaviour of stainless steel welded sections, and current design standards produce conservative or erroneous predictions for the buckling resistance of welded sections. This chapter aims to fill up the knowledge gaps through an experimental program, and proposes design formulations for stainless steel welded I-columns based on comprehensive numerical analysis.

In Chapter 4, it was discussed that despite having significant influence on the buckling resistance, the current design standards fail to appropriately recognise the nonlinear material behaviour of stainless steel. The current codes also do not allow for the benefits of strain hardening in determining the buckling resistance of stainless steel columns. This limitation led to devising a new proposal for predicting the buckling resistance of cold- formed stainless steel RHS and SHS columns following CSM approach. This proposal successfully incorporated all the characteristic features of stainless steel through simple equations. However, the behaviour of welded sections is different from cold-formed sections due to the presence of residual stresses [123-125]. Further investigations are required to investigate the suitability of the proposed CSM based technique for stainless steel welded sections.

In this chapter, the structural behaviour of stainless steel welded I-columns are investigated through a comprehensive test program as well as FE analysis. The test program included material test, initial geometric imperfection and residual stresses measurements, stub column tests and flexural buckling tests on austenitic grade stainless steel welded I-sections. Based on the results obtained from this test program, nonlinear FE models were developed and verified, and a comprehensive parametric study was carried out to identify the influential key parameters on the flexural buckling of welded I-columns. The design formulas proposed for hollow sections in Chapter 4 were recalibrated using the test and FE results for welded I-columns. Finally, the performance of the proposed CSM formulas was verified and compared with other standards.

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Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns Test program

A test program was conducted to investigate the structural behaviour of stainless steel welded I-sections, which were made from 316L austenitic grade stainless steel. Flanges and webs were connected by Tungsten Inert Gas (TIG) welding. Most of the recent studies used shielded metal arc welding (SMAW) for the fabrication of welded sections [48, 123- 125]. But compared to SMAW, TIG welding offers better quality and precision. TIG welding is aesthetically good with smaller seam size and the thermal distortion is also significantly low for TIG welded members. Material properties of the stainless steel were determined through tensile coupon tests. Three stub column tests were performed to examine the local buckling behaviour of stainless steel welded I-sections. To examine the buckling behaviour of the welded members, 16 long columns were tested. Prior to the test, the local and the global geometric imperfections were measured. In addition, residual stresses of two representative members were also determined by the sectioning method. The details of the test program are described in the following sections. Designation system adopted for the considered specimens are as follows: “I H×B×tf×tw-L”, where I stands for I-section, H is the nominal depth of the section, B is the nominal width of the section, tf is the nominal flange thickness, tw is the nominal wed thickness and L is the nominal length of column.

Tensile coupon test

5.3.1 Procedure adopted for tensile test

Tensile coupon tests were performed to evaluate accurate material properties for the plate materials used to fabricate the considered I-sections. Plates of five different thicknesses were used for different cross-sections, and plate thicknesses varied from 2 mm to 6 mm. Five coupons, each representing a specific thickness, were cut from 200×200 mm plates that were supplied to represent the same batch as the welded I-columns. The dimensions of all the tensile coupons were set according to EN ISO 6892-1 [127]. All the tensile coupons were necked at the middle as shown in Figure 5.1. To minimise the heat effect during the cutting process, submersible wire cutting technology was used to prepare the test coupons. A digital Vernier calliper was used to measure the width and the thickness of the necked region of the coupon. Three widths and three thickness measurements were

119

Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns taken along the coupon necked length, and the average width and thickness values were used to calculate the cross-sectional area.

Figure 5.1Tension test coupons according to EN ISO 6892-1 [127]

All tensile coupon tests were performed using a Shimadzu Z100 kN electromechanical universal testing machine (UTM), as shown in in Figure 5.2, in accordance with EN ISO 6892-1 [127]. Video extensometer was used to measure the longitudinal strain over a specified gauge length. A linear electrical resistance strain gauge was also attached to the face of each tensile coupon to record more accurate measurements for the initial elastic part of the stress-strain curves as shown in Figure 5.3.

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Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns

Figure 5.2 Test set-up for tensile coupon test

Figure 5.3 Installation of the linear electrical resistance strain gauge on one of the test coupons

121

Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns A pair of flat grips was used to hold the coupons firmly at each end. Load, strain and elongation were all recorded at 0.10 second intervals using the Horizon and Labview data acquisition system. According to the EN ISO 6892-1 [127] requirements, for the determination of the 0.2% proof strength, the strain rate should not exceed 0.0025/s. For the determination of the tensile strength, the strain rate may be increased up to the maximum limit of 0.008/s. During the test, strain rate was maintained at 0.001/s up to the 0.02% proof strength and then raised to 0.005/s until fracture.

According to EN ISO 6892-1 [127], the plastic strain at fracture should be measured over a gauge length of⁡5.65√Ac, where Ac is the cross-sectional area of the coupon. The necked lengths of all the specimens were marked at the specified length using a scribe. In order to measure the plastic deformation at fracture, the two halves of each coupon were fitted back together, and the elongation after fracture was measured between the scribe marks once the test was completed. The measured values were used to calculate the percentage of plastic strain at fracture as εpl,f(%) = [(Lu − L0)⁄L0] × 100, where L0 is the original marked length and Lu is the extended length after fracture.

5.3.2 Results obtained from tensile tests

Key material parameters such as Young’s modulus E, 0.2% proof stress σ0.2, ultimate tensile strength σu and Ramberg-Osgood (R-O) nonlinearity parameters n and m were extracted from the recorded stress-strain curves. The best fit Young's modulus E was calculated based on the strain gauge measurements. Compound Ramberg-Osgood (R-O) nonlinearity parameters n and m were also calculated from the strain gauge data. Plastic strain at fracture εpl,f was measured according to the procedure described in Section 5.3.1. Results obtained from all tested coupons are summarised in Table 5.1. Stress-strain curves are shown in Figure 5.4 and 5.5.

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Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns Table 5.1 Material properties of austenitic stainless steel plates used to fabricate welded I-column specimens

Plate ε E σ σ n m pl,f thickness 0.2 u MPa MPa MPa % 6 mm 198340 310 620.5 9.0 2.1 54 5 mm 197480 301 611.01 8.0 2.2 58 4 mm 199445 290 597.8 6.5 2.2 60 3 mm 206090 311 628.5 16.0 2.1 65 2 mm 201200 309 648 25.0 2.1 68 700

600

500

400 2 mm 300

Stress, Stress, MPa 3 mm 200 4 mm 5 mm 100 6 mm 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Strain

Figure 5.4 Complete stress-strain curves of different stainless steel plates used to fabricate I-sections

400

350

300

250

200 2 mm

Stress, Stress, MPa 150 3 mm 100 4 mm 5 mm 50 6 mm 0 0 0.002 0.004 0.006 0.008 0.01 Strain Figure 5.5 Stress-strain curves (enlarged up to 1% strain) for all plates used to fabricate I-sections 123

Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns Residual stresses observed in stainless steel welded I-sections

5.4.1 Measurement technique adopted for residual stresses

Residual stresses are unavoidable in welded sections. In this study, residual stresses were measured by sectioning method, which has been widely used for many years and is reported to produce accurate and reliable results. In 1888, Kalakoutsky [128] first proposed this method for measuring longitudinal stresses in steel bars. He slit the bars into longitudinal strips; measured the change in length of each strip before and after slitting, and residual stresses were calculated by applying Hooke’s law.

In this study, residual stresses were measured for the following two welded I-sections – I 80×80×5×4-500 and I 100×60×6×4-450. Consistent and uniform welding throughout the member was ensured to obtain reliable results. The strips were cut at least at a distance ‘d’ from both ends to minimise the end effects, where d is the maximum dimension of the cross-section [129]. Each element of I-sections (flange and wed) was divided in to nine strips. Strips were marked and a pair of punch marks was indented on each strip over a gauge length of 200mm using setting out bar as shown in Figure 5.6. Use of the setting out bar for punching ensured better contact between punch holes and demac gauge.

Punch mark

Figure 5.6 Punch marks on the flange of 80×80×5×4-500 section

124

Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns The initial gauge length of each strip was measured using a standard 200mm demac gauge as shown in Figure 5.7. Each length was measured three times and the mean value was recorded as the initial gauge length Li. All specimens were kept in an environmentally controlled room for 24 hours before taking any measurement; the room temperature was fixed at 23±0.5°C for minimising the influence of temperature change. After recording the initial gauge length, flanges and webs were separated from each other, and were subsequently cut into pre-designed strips. All cutting operations were conducted using a submergible wire cutting machine to avoid additional heat input during the sectioning process, which could otherwise affect residual stresses. All the strips cut from within I 80×80×5×4 section are presented in Figure 5.8.

Figure 5.7 Taking measurement of initial gauge length of the strips marked for residual stress measurement.

Figure 5.8 All strips cut from I 80×80×5×4 section to measure residual stresses 125

Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns After sectioning, all the strips were kept in the environment control room for 24 hours and the final gauge readings Lf were recorded as the average of three sets of repeated reading (shown in Figure 5.9). Using the initial and final gauge lengths, the relieved longitudinal residual strains were determined using Eq 5.1 where Li and Lf are the initial and the final gauge length respectively. A negative relived strain indicated a tensile residual stress, whereas a positive relived strain indicated a compressive residual stress.

L −L ε = f i (5.1) Li

Figure 5.9 Taking measurements of the final gauge length on cut out strips

Due to the presence of through thickness residual stress gradient, strips exhibited longitudinal curvature and this was more prominent in strips nearer to the weld. For this curvature, the gauge lengths were chord lengths rather than arc lengths. Chord lengths were corrected to arc length using the offset value δ and the initial gauge length L as defined in Figure 5.10. The offset values were measured by using three dial gauges as shown in Figure 5.11. The corrected relieved strain εc can be approximated as Eq. 5.2 where δ/L is the ratio of the offset δ to the initial gauge length L. The curvature correction may be neglected until this ratio exceeds 0.001 [129].

(δ⁄L)2 ε = ε + (5.2) c 6(δ⁄L)4+1 126

Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns

Figure 5.10 Definition of the offset value δ for longitudinally curved strips cut to measure residual stress [48]

Figure 5.11 Measurement of the offset values observed in the curved strips cut to measure residual stress

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Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns 5.4.2 Discussion on results obtained from residual stress measurement

Residual stresses were calculated by multiplying the Young’s modulus with the measured relieved residual strains. The computed residual stresses and their distribution patterns observed in two stainless steel welded I-sections are shown in Figure 5.12 and 5.13. The observed pattern was in-line with the typical distribution observed in welded I-sections; tensile residual stresses developed near the weld and compressive residual stresses developed away from the weld. In I 80×80×5×4-500, the maximum tensile residual stress was 257MPa, which is 85.4% of the material yield stress (0.2% proof stress σ0.2) and developed in the web. The maximum compressive residual stress was 194 MPa (64.5% of σ0.2) and developed in top flange. In the case of I 100×60×6×4-450 section, the maximum tensile residual stress was 94% of σ0.2 and the maximum compressive residual stress was 53% of σ0.2. In both sections, maximum tensile residual stresses in flanges were significantly lower than those observed in the web.

Figure 5.12 Residual stresses distribution observed in welded I 80×80×5×4-500 section

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Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns

Figure 5.13 Residual stresses distribution observed in welded I 100×60×6×4-450 section

Initial geometric imperfection measurements

The initial local geometric imperfections of all the stainless steel column specimens were measured by a laser scanner (scanCONTROL 2710-100(500)). To measure the imperfections, column specimens were placed on the table of a milling marching as shown in Figure 5.14. The table of the milling machine provided a flat reference surface for the accurate measurement. The laser scanner was set on top of the milling machine, and 3D data were collected automatically using a built-in software. Imperfects of the top surfaces of two flanges and web surface of each specimen were measured. The laser scanner has the ability to measure the vertical profile along a line. The table of the milling machine was moved at a constant rate of 5 mm/sec and the vertical profiles were measured at 0.1 sec interval. As a result, data of the vertical profiles were collected at a distance approximately 0.5 mm from each other along the length of the column specimen. It is worth noting that the milling table was moved manually, which inevitably produced some variability on the speed of the table. However this was not significant, and hence the 129

Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns distance between each of the obtained vertical profiles was also assumed to be constant. Distance between two horizontal measurement points in a profile typically varied from 0.2 to 0.25 mm as it depends on the distance between the surface and the scanner. The raw data were post-processed by Matlab using ‘meshgrid’ option; collected raw data were converted to 1 mm by 1mm grid data, where value of each grid point was determined as a weighted average of the values of the surrounding points.

Figure 5.14 Measurement of the initial local geometric imperfections of I 120×60×4×2- 500 column

The deviation of a point from the flat surface was considered as local imperfection of that point. A typical distribution of the local imperfections of a cross-section is shown in

Figure 5.15, where Wtf is the maximum local imperfection of the top flange at that section,

Wbf is the maximum local imperfection of the bottom flange at that section and Ww is the maximum local imperfection of the web at that section. The maximum value of Wtf, Wbf and Ww was considered as the local imperfection amplitude at that section. To avoid the end effect, the measurements from the end of the specimen to a section, d distance from that end, were not considered for determining the initial local geometric imperfection. This d distance was defined as the larger of the section dimensions. The maximum values of the local imperfections of the considered sections was recorded as the maximum amplitude of the initial local geometric imperfection W0 for that column. The results are presented in Table 5.2 for stub column specimens and in Table 5.4 for long column specimens. The maximum local imperfection value varied from 0.31 mm to 1.05 mm, 130

Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns which are much higher than the EN 1993-1-5 [55] recommended value of b/200, where b is the unsupported length of the element. The cross-section profile at mid length of I 80×60×4×2-320 column is shown in Figure 5.16.

Figure 5.15 A typical distribution of the local imperfections of a cross-section

Figure 5.16 Distribution of the initial local geometric imperfection of I 80×60×4×2-320 column

A digital dial gauge was used to measure the initial global geometric imperfections of long column specimens. The specimens were set on a milling table and a dial gauge was mounted in the place of the drill bit as shown in Figure 5.17. Zhang and Alam [130] used similar set-up to measure the imperfections of pallet-rack stub columns. Measurements

131

Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns were taken along the centre line of two flanges and web. Due to the presence of end plates, the first measurement was recorded at 50 mm away from the end and the other measurements were taken at 50 mm intervals including the middle points (for I 80×60×4×2-1500 column interval was 100 mm). Two end points of each surface was considered to be on a flat surface and the deviation of any point from that flat surface was considered as global imperfection of that point. The maximum value of the global imperfections of a flange or web was considered as the global imperfection of that element as shown in Figure 5.18. The maximum value of the global imperfection of two flanges was reported as the global imperfection amplitude about major axis. The maximum global imperfection of the web was reported as the global imperfection amplitude about minor axis. The maximum global imperfections amplitude of long column specimens are presented in Table 5.4. The global imperfection values about major axis varied from L/490 to L/6250 and those about minor axis varied from L/794 to L/4286. For most of the columns, the maximum global imperfection amplitudes were less than L/1000, which is the recommended global imperfection value of a member according to EN 1993-1-5 [55]. The distribution of the global imperfection of I 80×60×4×2-1500 column about major and minor axis are presented in Figure 5.19 and 5.20 respectively; these are the typical distributions observed in the considered welded I-sections.

Figure 5.17 Measurement of the initial global geometric imperfections of I 120×60×4×2-1000 column 132

Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns

Figure 5.18 Typical distribution of the global geometric imperfect

1.5

0.5

0 Deflection Deflection centerline, from mm 0 200 400 600 800 1000 1200 1400

-0.5 Length, mm

Figure 5.19 Distribution of the initial global geometric imperfections of I 80×60×4×2- 1500 column about minor axis.

2.5

1.5

0.5 Deflection Deflection centerline, from mm 0 0 200 400 600 800 1000 1200 1400 Length, mm

Figure 5.20 Distribution of global imperfection of I 80×60×4×2-1500 column about major axis.

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Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns Testing of stub columns

5.6.1 Test procedure

A total of three stub columns were tested in axial compression. The lengths of the columns were chosen equal to three times the larger nominal dimension of the cross-section, which is long enough to ensure representative distribution of geometric imperfection and residual stress but short enough to avoid global buckling [108]. Two 200×200 mm plates of 16 mm thickness were welded at the top and the bottom end of the stub columns. The geometric dimensions of stub column specimens were measured by digital slide calliper. The mean values of three sets of measured dimensions were recorded in Table 5.2, whereas Figure 5.21 defines different dimensions. The initial local geometric imperfections were measured prior to testing as outlined in Section 5.5.

Figure 5.21 Typical cross-section of an I-section defining different dimensions

Table 5.2 Geometric dimensions and the maximum local imperfections of the stub column specimens considered for experimental study

Section ID H B tf tw L W0 mm mm mm mm mm mm I 80×60×4×2-320 79.3 59.4 4.01 2.00 316 1.05 I 80×60×6×4-320 79.6 59.8 6.08 3.95 315 0.56 I 120×60×5×3-360 118.9 59.6 4.93 2.99 355 0.98 134

Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns Tests were conducted in a Shimadzu 1000kN hydraulic UTM. The displacement of the bottom plate was measured by four LVDTs, whilst additional four LVDTs were used to measure the displacement of the top base plate. Ten linear electric resistance strain gauges were attached at the mid-length. The schematic diagram of the instrumentation is shown in Figure 5.22, and positions of the electric strain gauges are shown in Figure 5.23. A hemispherical swivel head was used at the top of test specimens to ensure appropriate contact and concentric axial loading. After applying 5kN load, the rotations of swivel head were locked using metal wedges. The test set-up is shown in Figure 5.24. The tests were performed under displacement controlled condition where a constant displacement of 0.5 mm/min was applied throughout the test. All relevant data such as time, load, displacements and strains were recorded at 0.1second interval using the data acquisition system of National Instrument and Labview computer package.

Figure 5.22 Schematic diagram of the instrumentation of stub column test

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Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns

Figure 5.23 Positions of electric strain gauges at the mid-length of stub column specimens of stainless steel welded I-section

Figure 5.24 Test set-up for stub column test of I 80×60×6×4-320 column 136

Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns 5.6.2 Discussion of results for stub columns

Key results such as the ultimate load Nu and the corresponding end shortening at Nu obtained from stub column tests are provided in Table 5.3. The end shortening of each specimen was calculated as the difference between the average displacement of the bottom plate and the top plate. All tested stub column specimens failed by local buckling of the comprising elements, as expected. Failure modes of the stub columns are shown in Figure 5.25, and complete load vs end shortening curves are shown in Figure 5.26. From electric strain gauge reading, the strain distribution at mid-height of stub columns at the ultimate load Nu and at 50% of Nu are shown in Figure 5.27. In the case of I 80×60×6×4- 320, electric strain gauges lost contact before reaching ultimate load, and hence strain distribution for this specimen is shown at 90% of Nu. From Figure 5.27, it is clear that strains at the ultimate load are well above the yield strain for all the stub columns.

Table 5.3 Stub column test results

End Section ID Load shortening kN mm I 80×60×4×2-320 220.4 3.22 I 80×60×6×4-320 375.4 2.79 I 120×60×5×3-360 274.1 9.08

Figure 5.25 Failure modes of the stub columns of stainless steel welded I-sections

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Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns

400

350

300

250

200

Load, kN 150 I 80×60×4×2-320 100 I 80×60×6×4-320 50 I 120×60×5×3-360

0 0 5 10 15 20 End Shortening, mm Figure 5.26 Complete axial load vs end shortening curves of the stub columns

Figure 5.27 Strain distributions of the stub columns at mid height at different stages of loading

Long column test

5.7.1 Test procedure

To investigate the flexural buckling behaviour of stainless steel welded I-sections, 16 long columns were tested for minor axis buckling in pin-ended support condition. Considered columns represented six different cross-sections covering a wide range of tf/tw ratio, height-to-width ratio, cross-section slenderness λp and member slenderness λ. Cross- section slenderness λp varied from 0.26 to 0.83, whilst geometric lengths of the columns varied from 500 mm to 1500 mm providing a wide spectrum of λ as calculated accordance with EN 1993-1-4 [3], ranging from 0.45 to 1.39.

138

Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns The geometric dimensions of the column specimens were measured prior to testing. The averages of three sets of measured dimensions are provided in Table 5.4; symbols were previously defined in Figure 5.21. Initial local and global geometric imperfections for all column specimens were measured prior to testing and are reported in Table 5.4.

The specimens were tested using the Shimadzu 1000 kN UTM, which was used for stub columns. Hardened steel knife-edge support systems were used at both ends to provide pin-end support conditions about the axis of buckling and fixed end support conditions about the orthogonal axis. The schematic diagram of the test set-up is shown in Figure 5.28. For appropriate transfer of the applied loading, two plates were welded at the ends of each column. Two fastener plates were used to attach the column specimens to a knife- edge support at each end. Fastener plates can be adjusted to position the column at the centre of the knife-edges. A laser guided level machine was used to align the centre line of the column with the centre of the knife-edges as well as that of the UTM.

The instrumentation consisted of ten LVDTs as shown in Figure 5.28. Four LVDTs were installed to measure the displacements of the bottom plate and additional four LVDTs were used to measure the displacements of the top plate. The other two LVDTs were used to measure the lateral deflection at mid-height. Similar to the stub columns, ten linear electric resistance strain gauges were installed at mid-height cross-section. Instrumentation of the columns at the mid-height cross-section is shown in Figure 5.29, and the test set-up is shown in Figure 5.30. Tests were performed in displacement control condition with a constant rate of 0.5 mm/min, and all relevant data were recorded at 0.1 second intervals. A typical buckling shape for column I 80×60×4×2-750 is shown in Figure 5.31.

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Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns

Figure 5.28 Schematic diagram of the test set-up for flexural buckling test of welded I- columns

Figure 5.29 Typical instrumentation at mid-height cross-section of flexural buckling specimens

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Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns Table 5.4 Geometric dimensions and initial geometric imperfections of the long column specimens of stainless steel welded I-sections

Global Imperfection Section ID D B tf tw L Le w0 Major Minor axis axis mm mm mm mm mm mm mm mm mm I 80×60×4×2-750 79.39 59.44 3.93 2.00 748 965 0.67 0.26 0.47 I 80×60×4×2-1000 79.5 59.4 4.0 2.0 996.0 1213.0 0.38 0.31 1.26 I 80×60×4×2-1500 79.3 59.5 3.9 2.0 1496.0 1713.0 - 2.67 1.68 I 80×60×6×4-750 79.7 59.7 6.0 4.0 743.0 960.0 0.60 0.12 0.4 I 80×60×6×4-1000 79.6 59.8 6.0 4.0 997.0 1214.0 0.31 0.44 0.35 I 80×60×6×4-1200 79.9 59.8 6.0 4.0 1194.0 1411.0 0.75 2.45 0.28 I 80×80×5×4-500 79.3 79.6 4.9 4.0 496.0 713.0 0.50 0.2 0.34 I 80×80×5×4-900 79.3 79.8 4.9 4.0 895.0 1112.0 0.48 0.34 0.42 I 80×80×5×4-1200 79.6 80.2 4.9 4.0 1194.0 1411.0 0.52 2.28 0.46 I 100×60×6×4-450 100.2 60.0 6.1 4.0 446.0 663.0 0.65 0.22 0.5 I 100×60×6×4-900 100.0 59.8 6.0 4.0 897.0 1114.0 0.65 0.32 0.42 I 100×60×6×4-1200 100.4 60.0 6.0 4.0 1197.0 1414.0 0.78 0.92 0.44 I 1200×60×5×3-720 119.0 59.7 4.8 3.0 718.0 935.0 0.48 0.39 0.34 I 1200×60×5×3-1200 119.5 59.6 4.9 3.0 1196.0 1413.0 0.44 0.5 0.38 I 1200×60×4×2-500 119.9 59.6 4.0 2.0 496.0 713.0 0.37 0.18 0.36 I 1200×60×4×2-1000 119.2 59.6 3.9 2.0 996.0 1213.0 0.88 0.37 0.81

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Figure 5.30 Test set-up of flexural buckling test

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Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns Figure 5.31 Flexural buckling of I 80×60×4×2-750 column showing a typical failure mode

5.7.2 Discussion of results obtained for long columns

Key results obtained from the conducted buckling tests such as the ultimate load Nu, the end shortening at ultimate load, the lateral defection at ultimate load and the failure modes of the column specimens, are presented in Table 5.5, where ‘FB’ and ‘LB’ refers to flexural buckling and local buckling respectively. The load vs end shortening curves for the columns of different cross-sections are shown in Figures 5.32-5.37, and the load vs lateral deflection curves are shown in Figures 5.38-5.43. Strain distributions at the mid- height level at different loading stages for columns with I 80×60×6×4 and I 80×80×5×4 sections are presented in Figures 5.44 and 5.45 respectively. To obtain the strain distribution of a cross-section, readings recorded from strain gauges SG2, SG3, SG5 and SG6 were averaged, and were considered as strain at top fibre. Similarly, average of the strain gauge reading of SG4 and SG9 was considered as strain at the middle, and average of the strain gauge readings of SG1, SG7, SG8 and SG10 was considered as the strain at the bottom fibre. In Figures 5.44 and 5.45, it was observed that for all column specimens, strain distributions at 50% of Nu were uniform confirming the concentric loading condition for the tested columns. As the load was increased, cross-sections rotated due to flexural buckling. In case of I 80×80×5×4-500 in Figure 5.45, due to the initiation of local buckling, cross-section is not plane after deformation, hence strain distribution of this specimen was different from others. It was observed that the average strain of a cross- section decreased with increase in column length. As shown in Figure 5.44, for I 80×60×6×4-750 column, strains of the bottom half of the cross-section were greater than the material yield strain at Nu. Whereas, in the case of I 80×60×6×4-1000 column strain at the bottom was slightly over the material yield strain, and in the case of I 80×60×6×4- 1200 column, strains of the full cross-section were below the material yield strain. Strain distributions observed in columns as shown in Figure 5.45 also showed similar trend.

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Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns Table 5.5 Key results obtained from the flexural buckling test of pin-ended welded I- columns

Ultimate End Lateral Failure Section ID Load Shortening Deflection mode kN mm mm I 80×60×4×2-750 112.9 1.21 2.73 FB

I 80×60×4×2-1000 92.9 1.24 5.05 FB

I 80×60×4×2-1500 56.3 1.15 1.58 FB

I 80×60×6×4-750 189.8 1.67 2.39 FB

I 80×60×6×4-1000 149.4 1.45 2.40 FB

I 80×60×6×4-1200 123.8 1.22 1.87 FB

I 80×80×5×4-500 288.0 2.11 1.06 LB+FB

I 80×80×5×4-900 216.0 2.12 2.18 FB

I 80×80×5×4-1200 178.3 1.39 4.92 FB

I 100×60×6×4-450 260.1 1.54 4.28 LB+FB

I 100×60×6×4-900 171.9 1.61 5.88 FB

I 100×60×6×4-1200 127.0 1.25 7.12 FB

I 1200×60×5×3-720 167.0 1.46 3.06 FB

I 1200×60×5×3-1200 104.6 1.10 4.00 FB

I 1200×60×4×2-500 150.9 1.89 0.98 LB+FB

I 1200×60×4×2-1000 93.7 1.43 2.55 FB

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Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns

120

100

I 80×60×4×2-750 Load, kN I 80×60×4×2-1000 40 I 80×60×4×2-1500

0 0 0.5 1 1.5 2 2.5 3 End Shortening, mm

Figure 5.32 Load vs end shortening of columns with I 80×60×4×2 cross-section

200 180 160 140 120 100

Load, kN 80 60 I 80×60×6×4-750 40 I 80×60×6×4-1000 20 I 80×60×6×4-1200 0 0 0.5 1 1.5 2 2.5 3 3.5 End Shortening, mm Figure 5.33 Load vs end shortening of columns with I 80×60×6×4 cross-section

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Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns

350

300

250

200

150 Load, kN

100 I 80×80×5×4-500 50 I 80×80×5×4-900 I 80×80×5×4-1200 0 0 1 2 3 4 5 End Shortening, mm Figure 5.34 Load vs end shortening of columns with I 80×80×5×4 cross-section

300

250

200

150 Load, kN 100 I 100×60×6×4-450 50 I 100×60×6×4-900 I 100×60×6×4-1200 0 0 1 2 3 4 End Shortening, mm Figure 5.35 Load vs end shortening of columns with I 100×60×6×4 cross-section

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Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns

200 180 160 140 120 100

Load, kN 80 60 40 I 120×60×5×3-720 20 I 120×60×5×3-1200 0 0 0.5 1 1.5 2 2.5 3 3.5 4 End Shortening, mm Figure 5.36 Load vs end shortening of columns with I 120×60×5×3 cross-section

160

140

120

100

Load, kN 60

40 I 120×60×4×2-500 20 I 120×60×4×2-1000 0 0 0.5 1 1.5 2 2.5 3 3.5 End Shortening, mm Figure 5.37 Load vs end shortening of columns with I 120×60×4×2 cross-section

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Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns

120

100

60 I 80×60×4×2-750

Load, kN I 80×60×4×2-1000 40 I 80×60×4×2-1500 20

0 0 5 10 15 20 25 Lateral Deflection, mm

Figure 5.38 Load vs lateral deflection of columns with I 80×60×4×2 cross-section

200 180 160 140 120 100

Load, kN 80 60 I 80×60×6×4-750 40 I 80×60×6×4-1000 20 I 80×60×6×4-1200 0 0 5 10 15 20 25 Lateral Deflection, mm Figure 5.39 Load vs lateral deflection of columns with I 80×60×6×4 cross-section

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350

300

250

200

150 Load, kN

100 I 80×80×5×4-500 50 I 80×80×5×4-900 I 80×80×5×4-1200 0 0 5 10 15 20 25 Lateral Deflection, mm Figure 5.40 Load vs lateral deflection of the columns with I 80×80×5×4 cross-section

300

250

200

150 Load, kN 100 I 100×60×6×4-450 50 I 100×60×6×4-900 I 100×60×6×4-1200 0 0 5 10 15 20 25 Lateral Deflection, mm Figure 5.41 Load vs lateral deflection of the columns with I 100×60×6×4 cross-section

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Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns

200 180 160 140 120 100

Load, kN 80 60 40 I 120×60×5×3-720 20 I 120×60×5×3-1200 0 0 5 10 15 20 25 Lateral Deflection, mm Figure 5.42 Load vs lateral deflection of columns with I 1200×60×5×3 cross-section

160

140

120

100

Load, kN 60

40 I 120×60×4×2-500 20 I 120×60×4×2-1000 0 0 5 10 15 20 25 Lateral Deflection, mm Figure 5.43 Load vs lateral deflection of the columns with I 120×60×4×2 cross-section

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Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns

Figure 5.44 Strain distributions at mid-height of columns with I 80×60×6×4 section at different loading stages.

Figure 5.45 Strain distributions at mid-height of columns with I 80×80×5×4 section at different loading stages.

Summary of the testing scheme

In the conducted test program, the behaviour of stainless steel welded I-sections was studied through tensile coupon test, residual stresses and initial geometric imperfection measurement, stub column test and flexural buckling test. It was observed that the measured maximum tensile residual stresses were higher than the values recommended by Yuan et al. [48]. The amplitudes of local geometric imperfect were also higher than the code [55] recommended values; however, for most of the columns the amplitudes of global imperfection was lower than the codified values. All three stub columns failed in local buckling and their stress distributions showed that the average strains at failure were higher than the yield strain. The buckling behaviour was observed through flexural buckling test about minor axis. The buckling test results along with material test data and

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Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns measured imperfection amplitudes were used to verify the FE models developed in the next section.

Finite element model for long columns

Commercial finite element analysis package ABAQUS was used for numerical simulation of the behaviour observed in long column testing. Initially, the developed FE models were validated against test results carried out as part of the current study, and, once verified, the developed FE models were used to perform a comprehensive parametric study to identify the influence of different parameters on column resistance of stainless steel welded I-sections. Finally, results obtained from FE analysis were used to propose new column curves for welded I-columns produced from stainless steel.

Following a similar approach for developing FE models as adopted in Chapters 3 and 4, four-node doubly curved shell element with reduced integration was used with mesh sizes not greater than 5 mm along the transverse direction and 10 mm along the longitudinal direction of I-section. The two-stage Ramberg–Osgood (R–O) [19] material model proposed by Rasmussen [22], with recent modifications proposed by Arrayago et al. [23], was used in FE models. In this study, both local and global geometric imperfections were considered. Eigenvalue analysis was performed and the corresponding elastic buckling modes were used to simulate the distribution of local and global imperfections. In their reported parametric study, Yuan et al. [124] showed that the amplitude of global geometric imperfection has moderate effect on column resistance but the amplitude of local imperfection has no significant effect on column resistance. In this study, four different amplitudes for global imperfection such as measured imperfection, L/1000, L/1500 and L/2000 where L is the length of the column, were considered. In the case of local geometric imperfection, an amplitude of b/200 was used, where b is the unsupported width of the flange. L/1000 and b/200 are the code recommended values for the global geometric imperfection and the local geometric imperfection respectively [55]. Membrane residual stresses are developed in I-sections due to welding, which induce tensile stress in the vicinity of the welds with compressive stress away from those regions. Residual stresses were incorporated in the developed FE models according to the proposal of Yuan et al [48], where the maximum tensile stress was taken as 0.80.2.

152

Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns Columns were simulated as pin supported at both ends. All nodes at the bottom end and at the top end were coupled with two reference points located at the centroid of the corresponding sections. The pin support conditions were applied to those reference points allowing for longitudinal translation at the top end. Columns subjected to buckle around major axis were laterally supported at the quarter lengths and at the mid length to prevent minor axis buckling. Typical support conditions adopted in FE models for major axis buckling and minor axis buckling are shown in Figures 5.46 and 5.47 respectively. Displacement was applied at the top reference point to simulate the column test.

Figure 5.46 Support conditions of a I-column subjected to major axis buckling

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Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns

Figure 5.47 Support conditions of a I-column subjected to minor axis buckling

Verification of FE models for long columns

The accuracy of the FE modelling technique was verified using the test results obtained from the welded I-columns considered in this study. The comparison of ultimate load (Nu) of FE models with those obtained from tests is shown in Table 5.6. FE models that included measured local and global imperfections with residual stress, and the FE models that included global imperfection of L/2000 and local imperfection of b/200 with residual stress showed good accuracy in predicting the test results. In both cases, the average ratio of the ultimate loads obtained from FE models and test results was 1.00 with a coefficient of variation (COV) of 0.04 and 0.02. Finite element models using a global imperfection amplitude of L/1500 also showed good agreement with test results with an average of 0.99. However, FE models with L/1000 global imperfection amplitude slightly under predicted the column capacity as the average of the ratio of FE results and test results is 0.96. On the other hand, if the residual stresses were not considered in the FE model, FE

154

Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns model always over predicted the column capacity; the average over prediction was 16% for the considered columns. Figure 5.48 shows the deflected shape for the FE model of I 80×60×4×2-750 column. In Figure 5.49 and 5.50, complete load-deflection responses of FE models for four columns are compared with test responses showing good agreement. Overall comparisons showed that the developed FE models were capable of predicting the behaviour of welded I-section stainless steel columns, and could be used to generate additional results to develop reliable column curves. Although the amplitude of global imperfection value L/2000 with residual stress showed marginally better agreement with test results, the amplitude value of L/1500 with residual stresses was conservatively used in the parametric study where the amplitude of local imperfection was b/200.

Figure 5.48 Deflected shape of I 80×60×4×2-750 specimen

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Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns Table 5.6 Comparison of FE results with tests performed on pin-ended stainless steel welded I-columns

Section ID Test FE result with residual stress FE results without result residual stress Measured Measured L/1000 L/1500 L/2000 Imperfection Imperfection

Nu,FE/ Nu,FE/ Nu,FE/ Nu,FE/ Nu,FE/ Nu,test Nu,FE Nu,FE Nu,FE Nu,FE Nu,FE Nu,test Nu,test Nu,test Nu,test Nu,test kN kN kN kN kN kN I 80x60x4x2-750 112.9 112.9 1.00 109.4 0.97 112.2 0.99 113.9 1.01 124.4 1.10 I 80x60x4x2-1000 92.9 85.6 0.92 85.9 0.92 88.4 0.95 89.9 0.97 98.8 1.06 I 80x60x4x2-1500 56.3 54.1 0.96 54.8 0.97 56.9 1.01 58.2 1.03 66.5 1.18 I 80x60x6x4-750 189.8 186.4 0.98 178.1 0.94 182.4 0.96 184.9 0.97 213.7 1.13 I 80x60x6x4-1000 149.4 150.5 1.01 139.5 0.93 143.8 0.96 146.4 0.98 185.5 1.24 I 80x60x6x4-1200 123.8 132.9 1.07 119.0 0.96 123.5 1.00 126.3 1.02 165.8 1.34 I 80x80x5x4-500 288.0 281.6 0.98 275.9 0.96 279.4 0.97 281.4 0.98 287.5 1.00 I 80x80x5x4-900 216.0 220.2 1.02 210.7 0.98 215.2 1.00 217.9 1.01 238.7 1.10 I 80x80x5x4-1200 178.3 183.0 1.03 172.6 0.97 177.0 0.99 179.7 1.01 212.2 1.19 I 100x60x6x4-450 260.1 254.2 0.98 251.6 0.97 255.9 0.98 258.4 0.99 266.9 1.03 I 100x60x6x4-900 171.9 170.1 0.99 160.4 0.93 164.9 0.96 167.8 0.98 204.7 1.19 I 100x60x6x4-1200 127.0 134.6 1.06 123.1 0.97 127.9 1.01 130.8 1.03 167.9 1.32 I 1200x60x5x3-720 167.0 169.2 1.01 160.3 0.96 164.5 0.99 166.9 1.00 192.7 1.15 I 1200x60x5x3-1200 104.6 109.4 1.05 98.3 0.94 102.5 0.98 105.0 1.00 134.3 1.28 I 1200x60x4x2-500 150.9 153.3 1.02 150.2 1.00 152.5 1.01 153.8 1.02 162.0 1.07 I 1200x60x4x2-1000 93.7 92.1 0.98 90.5 0.97 93.2 0.99 94.8 1.01 107.5 1.15 Average 1.00 0.96 0.99 1.00 1.16 COV 0.04 0.02 0.02 0.02 0.10

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350 I 80×80×5×4-500 Test FE 300

250 I 100×60×6×4-450 200

Load, kN 150

100 I 80×80×5×4-1200

50 I 120×60×5×4-1200

0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 End Shortening, mm Figure 5.49 Comparison of experimental load vs. end shortening curves with FE results for welded I-columns

350

I 80×80×5×4-500 Test FE 300

250

200 I 100×60×6×4-450

Load, kN 150 I 80×80×5×4-1200 100

50 I 120×60×5×4-1200

0 0 5 10 15 20 25 Lateral Deflection, mm Figure 5.50 Comparison of experimental load vs. lateral deflection curves with FE results for welded I-columns

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Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns Parametric study for welded I-columns

The verified FE modelling technique was used to identify the parameters that could significantly affect the buckling resistance of stainless steel columns through a parametric study. In Chapter 4, it was shown that cross-section slenderness λp has a significant effect on column curves for RHS and SHS columns. Therefore, the effect of λp on column curves of welded I-section buckling about the major and the minor axis was thoroughly investigated herein. Additionally, effects of other cross-sectional properties such as the ratio of height-to-width (H/B) of the section and the ratio of flange thickness-to-web thickness (tf/tw) were also examined. Three values of H/B varying between 1.0 to 2.0 (1.0,

1.5 and 2.0) and three values of tf/tw (1.0, 1.5 and 2.0) were considered to cover a practical range of I-sections. Effects of H/B and tf/tw were observed on both stocky and slender cross-sections where λp of the considered stocky section was 0.48 and that for slender section was 0.88. Seven values of λp ranging from 0.38 to 0.98 were considered to evaluate the effect of λp on column curves of stainless steel welded I-sections. The effect of cross-section slenderness was studied on cross-sections having H/B = 1.5 and tf/tw =

1.5. λp was calculated according to the proposal of Afshan and Gardner [14] where the elastic buckling capacity of the full cross-section (σcr,cs ) was determined using CUFSM [76]. A total of 23 different I-sections were analysed for the major and the minor axis buckling, and their cross-sectional properties are shown in Table 5.7. Basic material properties such as Young’s modulus (E), proof stress (σ0.2) and strain hardening exponent (n) were taken as 200 GPa, 400MPa and 7 respectively. Effect of residual stress on column resistance was also observed on the stocky and the slender cross-section for major and minor axis buckling. For each I-section, columns with 15 different slenderness λcsm (0.2, 0.3, 0.35, 0.4, 0.45, 0.5, 0.55, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8 and 2.0) were analyzed to get a full range of column curves. A total of 720 models were analysed to obtain a thorough understanding of the buckling behaviour of stainless steel welded I- columns.

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Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns Table 5.7 Cross-sectional properties of I-sections used in the parametric study

Sl. No. H B tf/tw (mm) (mm) (mm/mm)

1-6 150 150 8.05/8.05, 4.54/4.54, 9.0/6.0, 5.1/3.4, 9.45/4.73, 5.47/2.74 9.6/9.6, 5.38/5.38, 14.15/9.43, 11.5/7.67, 9.65/6.43, 8.35/5.57, 7-17 225 150 7.35/4.9, 6.56/4.37, 5.93/3.95, 13.83/6.92, 8.03/4.02

18-23 300 150 12.17/12.17, 6.83/6.83, 15.48/10.32, 8.82/5.88, 19.4/9.7, 11.22/5.61

Analysis of FE results

To observe the effect of different parameters, column curves are plotted in Figures 5.51 to 5.60. In these column curves, non-dimensional column strength or reduction factor χ

Nu,FE was calculated as . In Figures 5.51 and 5.52, the influence of λp on column curves Agfcsm is shown for major axis buckling and minor axis buckling respectively. It was observed that cross-section slenderness λp has a significant effect on the column curves for welded I-section, and it is similar to the effect observed on the column curves of RHS and SHS

(Section 4.6). In both cases, column curves move upward with increasing values of λp i.e. the reduction factor χ is higher for relatively slender cross-sections with higher values of

λp. Shapes of column curves also depend on λp. For slender cross-sections (with higher value of λp), the column resistance approaches to its cross-section capacity (i.e. χ approaches 1.00 at a relatively higher value of λm) when compared against stocky cross- sections (with smaller λp). The effect of λp is more prominent at the intermediate portion of column curves and diminishes with the increase of λm.

Figures 5.53-5.56 show the effect of H/B and tf/tw on column curves for both stocky and slender cross-sections. It was observed that H/B and tf/tw have no significant effect on column curves for stocky sections (λp=0.48). In the case of slender sections (λp = 0.88), however, H/B showed some minor effects on column curves at low λm values. From Figure 5.53-5.56, it is also clear that column resistance for major axis buckling is higher than that of minor axis buckling. This effect is more significant at the intermediate portion of the column curves and diminishes with the increase or decrease of λm. From this analysis, it is clear that separate column curves are required for major axis buckling and minor axis buckling, and the effect of λp should be included in the formulas.

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Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns

1.2 λp = 0.38 1 λp = 0.48 λp = 0.58 λp = 0.68 0.8 λp = 0.78 λp = 0.88

χ 0.6 λp = 0.98

0.4

0.2

0 0.0 0.5 1.0 1.5 2.0 2.5 λm

Figure 5.51 Column curves for major axis buckling of stainless steel welded I-section for different λp values

1.2 λp = 0.38 1 λp = 0.48 λp = 0.58 λp = 0.68 0.8 λp = 0.78 λp = 0.88

χ 0.6 λp = 0.98

0.4

0.2

0 0.0 0.5 1.0 1.5 2.0 2.5 λm Figure 5.52 Column curves for minor axis buckling of stainless steel welded I-section for different λp values

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Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns

1.2 MJ-H/B=1.0 1 MJ-H/B=1.5 MJ-H/B=2.0 0.8 MI-H/B=1.0 MI-H/B=1.5

χ 0.6 MI-H/B=2.0 0.4

0.2

0 0.0 0.5 1.0 1.5 2.0 2.5 λm

Figure 5.53 Column curves of stocky stainless steel welded I-sections for different H/B ratios (λp = 0.48)

1.2 MJ-H/B=1.0 1 MJ-H/B=1.5 MJ-H/B=2.0 0.8 MI-H/B=1.0 MI-H/B=1.5

χ 0.6 MI-H/B=2.0

0.4

0.2

0 0.0 0.5 1.0 1.5 2.0 2.5 λm

Figure 5.54 Column curves of slender stainless steel welded I-sections for different H/B ratios (λp = 0.88)

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Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns

1.2 MJ-tf/tw=1.0 1 MJ-tf/tw=1.5 MJ-tf/tw=2.0 0.8 MI-tf/tw=1.0 MI-tf/tw=1.5

χ 0.6 MI-tf/tw=2.0

0.4

0.2

0 0.0 0.5 1.0 1.5 2.0 2.5 λm

Figure 5.55 Column curves of stocky stainless steel welded I-sections for different tf/tw ratios (λp = 0.48)

1.2 MJ-tf/tw=1.0 1 MJ-tf/tw=1.5 MJ-tf/tw=2.0 0.8 MI-tf/tw=1.0 MI-tf/tw=1.5

χ 0.6 MI-tf/tw=2.0

0.4

0.2

0 0.0 0.5 1.0 1.5 2.0 2.5 λm

Figure 5.56 Column curves of slender stainless steel welded I-sections for different tf/tw ratios (λp = 0.88)

The effect of residual stress on column resistance is shown in Figures 5.57 to 5.60. All the figures show that column capacity is considerably reduced due to the presence of residual stresses in welded I-sections. This reduction is more significant for columns buckling about minor axis. In the case of minor axis buckling, maximum compressive stress developed at the tip of flanges, which are already in compression due to the presence of the residual stress, which may lead to early failure. On the other hand, in case 162

Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns of major axis buckling, full flange goes to compression, which is in self-equilibrium under the effect of residual stress. Hence, the effect of residual stress is less severe for major axis buckling. Unlike for the case of cold-formed stainless steel members, the residual stress should be incorporated in the FE models of stainless steel welded members for predicting the buckling resistance of columns.

1.2 With residual stress 1 Without residual stress

0.8

χ 0.6

0.4

0.2

0 0.0 0.5 1.0 1.5 2.0 2.5 λm Figure 5.57 Comparison of column curves of stocky welded I-sections buckling around major axis with residual stress and without residual stress

1.2 With residual stress 1 Without residual stress

0.8

χ 0.6

0.4

0.2

0 0.0 0.5 1.0 1.5 2.0 2.5 λm Figure 5.58 Comparison of column curves of stocky welded I-sections buckling around minor axis with residual stress and without residual stress

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Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns 1.2 With residual stress 1.0 Without residual stress

0.8

χ 0.6

0.4

0.2

0.0 0.0 0.5 1.0 1.5 2.0 2.5 λm Figure 5.59 Comparison of column curves of slender welded I-sections buckling around major axis with residual stress and without residual stress

1.2 With residual stress 1 Without residual stress

0.8

χ 0.6

0.4

0.2

0 0.0 0.5 1.0 1.5 2.0 2.5 λm Figure 5.60 Comparison of column curves of slender welded I-sections buckling around minor axis with residual stress and without residual stress

Imperfection factor η for welded I-columns

In Chapter 4, Perry formulas with the buckling stress of CSM were successfully used to predict the buckling resistance of stainless steel hollow members. In that proposal, CSM buckling stress fcsm was used instead of the material yield stress in basic Perry curves, and the imperfection parameter η was expressed as a sigmoidal function of λm, where the coefficients of that function depend on λp. Similar approach was adopted herein to predict

164

Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns the buckling resistance of stainless steel welded I-columns. Equations 4.6-4.8 described in Section 4.2 are adopted here to predict the resistance of stainless steel I-columns, and the imperfection parameter η was recalibrated for I-sections. To obtain an appropriate imperfection parameter η for stainless steel I-columns, values of η were calculated from the FE results and were plotted against λm for different values of λp as shown in Figures 5.61 and 5.62 for major axis and minor axis buckling respectively. From these figures, it is obvious that like SHS and RHS, λp has a significant influence on η and the variation of

η with λm can be well represented by the same sigmoidal function described in Eq. 4.15. The coefficients A, B and W of the sigmoidal function were determined for stainless steel welded I-columns for both major and minor axis buckling. Values of A and B depend on

λp. For major axis buckling, the values of A and B can be calculated by using Eq.s 5.3 and 5.4. For minor axis buckling, the values of A and B can be calculated following Eq.s 5.5 and 5.6. A constant value of W = 0.4 is proposed for both major axis buckling and minor axis buckling.

0.7

0.6 λp=0.38 0.5

0.4 η 0.3 λp=0.68

0.2 λp=0.98 0.1

0 0 0.5 1 1.5 2 λm

Figure 5.61 η calculated from FE results and proposed curves of η for different λp values for major axis buckling of stainless steel welded I-columns

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Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns

1.4

1.2

1 λp=0.38

0.8 η 0.6 λp=0.68 0.4 λp=0.98 0.2

0 0 0.5 1 1.5 2 λm

Figure 5.62 η calculated from FE results and proposed curves of η for different λp values for minor axis buckling of stainless steel welded I-columns

A = −0.32λp + 1.06⁡ ≥ 0.74 (5.3)

B = 0.3λp + 0.4⁡ ≤ 0.65 (5.4)

A = −0.5λp + 1.55⁡ ≥ 1.05 (5.5)

B = 0.1λp + 0.72⁡ ≤ 0.80 (5.6)

Performance of the proposed method

The accuracy of the proposed method was verified with available test results of stainless steel welded I-columns and the FE results generated in the parametric study. A total of 46 test results, collected from different studies in the literature [124, 125, 131], and 16 test results performed in this study were considered. Among the test results, 19 columns were tested for major axis buckling and 43 columns were tested for minor axis buckling. The performance of the proposed method was also compared with those obtained using EN 1993-1-4+A1[68], SEI/ASCE 8-02 [4] and AS/NZS 4673 [5]. For AS/NZS 4673, Rasmussen and Rondal’s [7] proposal was considered where the values of the parameters α, β, λ1 and λ0 were calculated using their suggested equations. The key features of the comparison are shown in Table 5.8. It was observed that the predictions of the proposed method are more accurate and less scattered than the predictions of other standards. For major axis buckling, the average of the ratio of CSM predictions and FE results is 0.99 and the average of the ratio of CSM predictions and test results is 1.0. For minor axis 166

Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns buckling, the average is 0.98 for both FE results and test results. The coefficient of variation (COV) for CSM predictions is always less than those of other standards predictions except for one case. Comparing test results of minor axis buckling, the COV of the CSM prediction is lightly higher than the COV of other standards predictions. The prediction of the EN 1993-1-4+A1[68] standard shows good agreement with test and FE results for major axis buckling but for minor axis buckling this code is more conservative. ASCE [4] guidelines were proposed for cold form stainless steel and did not consider the effect of residual stress of the welded section. As a results in all cases, ASCE [4] over- predicted the buckling resistance of stainless steel welded I-columns. The effect of residual stress is more prominent for columns subjected to minor axis buckling, and hence the average ratio of ASCE [4] prediction to test or FE results is much higher for minor axis buckling than major axis buckling. For similar reason, AS/NZS [5] also over- predicted the buckling resistance of the welded I-columns. Figures 5.63-5.68 illustrates the comparison of CSM predictions with the predictions of other codes for test and FE results.

Table 5.8 Comparison of the performance of the proposed CSM method and other standards

Loading CSM EN 1993-1-4+A1 ASCE AS/NZS type

Ncsm/NFE/test NEC3/NFE/test NASCE/NFE/test NAS/NZS/NFE/test

Average COV Average COV Average COV Average COV

FE 0.99 0.03 0.95 0.05 1.09 0.09 1.01 0.05 Major data axis Test buckling 1.01 0.06 0.98 0.07 1.10 0.08 1.03 0.07 data

FE 0.98 0.04 0.92 0.06 1.13 0.13 1.05 0.08 Minor data axis Test buckling 0.98 0.10 0.86 0.08 1.14 0.06 1.03 0.06 data

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Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns 1.4

1.2

FE 0.8 /N

csm/EC3 0.6

EC3+A1 FE data N

0.4 EC3+A1 test data CSM FE data 0.2 CSM test data

0 0 0.5 1 1.5 2 2.5 λ or λ m Figure 5.63 Comparison of CSM and EN1993-1-4+A1 [68] predictions for major axis buckling of stainless steel welded I-columns

1.6

1.4

1.2

FE /N 0.8

csm/ASCE ASCE FE data

0.6 N ASCE test data 0.4 CSM FE data

0.2 CSM test data

0 0 0.5 1 1.5 2 2.5 λm or λ Figure 5.64 Comparison of CSM and SEI/ASCE 8-02 [4] predictions for major axis buckling of stainless steel welded I-columns

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Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns 1.4

1.2

FE 0.8 /N

0.6

AS/NZS FE data csm/AS/NZS N 0.4 AS/NZS test data CSM FE data 0.2 CSM test data

0 0 0.5 1 1.5 2 2.5 λm or λ Figure 5.65 Comparison of CSM and AS/NZS 4673 [5] predictions for major axis buckling of stainless steel welded I-columns

1.4

1.2

FE 0.8 /N

csm/EC3 0.6

EC3+A1 FE data N

0.4 EC3+A1 test data CSM FE data 0.2 CSM test data

0 0 0.5 1 1.5 2 2.5 3 λm or λ

Figure 5.66 Comparison of CSM and EN1993-1-4+A1 [68] predictions for minor axis buckling of stainless steel welded I-columns

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Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns 1.8

1.6

1.4

1.2 FE

/N 1

0.8 csm/ASCE

N 0.6 ASCE FE data ASCE test data 0.4 CSM FE data 0.2 CSM test data 0 0 0.5 1 1.5 2 2.5 3 λm or λ Figure 5.67 Comparison of CSM and SEI/ASCE 8-02 [4] predictions for minor axis buckling of stainless steel welded I-columns

1.4

1.2

FE 0.8 /N

0.6

AS/NZS FE data csm/AS/NZS N 0.4 AS/NZS test data CSM FE data 0.2 CSM test data

0 0 0.5 1 1.5 2 2.5 3 λm or λ Figure 5.68 Comparison of CSM and AS/NZS 4673 [5] predictions for minor axis buckling of stainless steel welded I-columns

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Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns Reliability analysis

Reliability analysis of the buckling formulas proposed for stainless steel welded I- columns was performed following the guidelines of EN1990-Annex D [109]. Although the same material model can be adopted for both austenitic and duplex stainless steel, the magnitudes of residual stresses are different for welded sections produced from different grades of stainless steel. In the FE models , the maximum tensile residual stress was taken

0.8σ0.2 considering austenitic grades, however, the maximum tensile residual stress for duplex grades of stainless steel is recommended as 0.6σ0.2 [48]. In the analysis presented in Section 5.12, it was found that the residual stress has a negative effect on the buckling resistance of the stainless steel welded I-columns. Therefore, FE results generated for austenitic grades can also conservatively represent the behaviour of weld I-columns of duplex grades stainless steel. Hence, FE results were considered in both austenitic and duplex grades stainless steel for the reliability analysis. The material over strength factors for austenite and duplex grades were considered as 1.30 and 1.10 respectively according to the proposal of Afshan et al. [110]. They also proposed the coefficient of variation of material strength for austenitic and duplex grades as 0.06 and 0.03, respectively, and the coefficient of variation in geometric properties was considered as 0.05. The results of the reliability analysis are presented in Table 5.9, where n is the total number of tests and FE results, kd,n is the design (ultimate limit state) fractile factor for n number of data, b is the average ratio of experimental (or FE) results to model resistance based on a least squares fit for each set of data, Vδ is the coefficient of variation of tests and FE simulations relative to the resistance model, Vr is the combined coefficient of variation incorporating both model and basic variable uncertainties, and γM1 is the partial safety factor for member resistance. It is observed from Table 5.9 that the partial safety factor for all cases are below 1.10, which is agreement with EN 1993-1-4[3] recommended value. So, the proposed formulas can be used with confidence to predict the buckling resistance of stainless steel welded I-columns produced from austenitic and duplex grades.

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Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns Table 5.9 Results of the reliability analysis of the proposed buckling formulas for stainless steel welded I-columns

Loading Material n K b V V γ types d,n δ r M1

Major axis Austenitic 386 3.115 1.025 0.031 0.084 0.97 buckling Duplex 383 3.115 1.024 0.031 0.066 1.09

Minor axis Austenitic 407 3.114 1.049 0.043 0.089 0.97 buckling Duplex 386 3.114 1.059 0.043 0.072 1.07

Conclusions

Welded sections play a significant role in structural engineering when readily available sections fail to meet the required high load carrying capacities. In many cases, welded sections may provide a better option than cold-formed sections as these sections can be fabricated according to the design requirements and may lead to an economic design. However, the structural response of welded members vary from cold-formed members due to the presence of longitudinal residual stresses. In case of stainless steel, very limited number of test evidences are available on welded sections as most of the studies are focused on commonly used cold-formed hollow sections. In this chapter, the behaviour of stainless steel welded I-columns were studied through experimental investigation and FE analysis. Additionally, the buckling formulas proposed in Chapter 4 for cold-formed hollow stainless steel columns were examined and calibrated for welded I-columns.

The experimental program involved measurements of material properties of 316L stainless steel, residual stress and initial geometric imperfection measurement. A number of stub columns and long columns were tested as part of the current study. Key material properties such as proof stress, ultimate strength, Young’s modulus, ductility and Ramberg-Osgood parameters were determined for the plates used to fabricate the I- columns by tensile coupon test. Longitudinal residual stresses were measure by sectioning method, and it was observed that the maximum tension residual stress may be higher than the currently suggested value of 0.8σ0.2 [48]. Local and global buckling characteristics were examined through testing of stub columns and long columns.

172

Chapter 5: Behaviour and Design of Stainless Steel Welded I-columns The behaviour of stainless steel welded I-columns was further investigated using developed nonlinear FE models. A parametric study was carried out to observe the effects of λp, H/B, tf/tw and residual stress on the buckling resistance of welded columns. It was observed that H/B and tf/tw had no significant effect on column curves. Presence of longitudinal residual stresses had a negative impact on the buckling resistance of welded columns, and this effect was more significant for columns buckling around minor axis.

The effect of λp on the buckling resistance of welded I-columns was similar to the effect observed on RHS and SHS columns. Finally, two sets of equations were proposed for major and minor axis buckling of welded I-columns.

The performance of the proposed method was assessed using available test results and FE results generated in this study. Compared to EN 1993-1-4+A1 [68], SEI/ASCE 8-02 [4] and AS/NZS 4673 [5] standards, this method produced more accurate and more consistent predictions for buckling resistance of stainless steel welded I-columns. Overall, the experimental work presented in this chapter added useful experimental evidences for welded I-columns, and the scope of CSM based design technique was further extended in predicting buckling resistance for stainless steel welded columns.

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Chapter 6 Conclusions and Recommendations

Conclusions

6.1.1 General

Stainless steel alloys have been increasingly used in structural applications for their superior material properties such as corrosion resistance, higher strength, significantly lower maintenance cost and attractive appearance. Despite its obvious advantages over the traditional counterpart, ordinary carbon steel, its usage in structures has been limited primarily due to the lack of appropriate design guidance that can make optimum utilisation of its beneficial properties. Current design codes consider stainless steel as an elastic, perfectly-plastic material like ordinary carbon steel. Stainless steel, in actual practice, shows highly nonlinear stress-strain behaviour offering significant strain hardening. Without appropriate recognition of its special characteristics, stainless steel design guidelines cannot predict member resistances accurately. The primary objective of this thesis has been to develop a number of efficient and rational design rules for stainless steel structures to ensure appropriate use of this promising environmentally friendly construction material.

6.1.2 The continuous strength method for slender cross-sections

The stress-strain behaviour of stainless steel is significantly different from ordinary carbon steel; absence of a well-defined yield point makes the use of traditional design approach adopted in codes unjustified for stainless steel. Significant strain hardening allows stainless steel stocky sections to resist loads much higher than the typical squash load limited by material yielding. The Continuous Strength Method (CSM) was devised to exploit the special features of stainless steel making sure that this expensive yet promising material is used efficiently in structural applications. Following the recent proposals made by Afshan and Gardner [14], CSM can produce accurate predictions of cross-section capacities for stocky sections. Resistance of slender cross-sections is normally below the section yield load as it loses effectiveness due to premature local buckling, but significant post-buckling behaviour was observed prior to failure. Use of the same cross-section deformation capacity, obtained from the design base curve for

Chapter 6: Conclusions and Recommendations stocky sections as proposed in CSM, for slender cross-sections is unrealistic and was reported to produce unreliable predictions for cross-section resistances [10, 11]. Effective width approach is the only codified approach for slender cross-sections but this technique is lengthy and becomes iterative for cross-sections with stress gradient. Referring to codes for slender cross-sections [14] was considered as a shortcoming to promote wider use of CSM in practice, and hence, devising a simple design method for slender cross-sections following the CSM approach was the first objective of this research.

To understand the cross-section behaviour of slender sections, FE models of stainless steel stub columns subjected to axial compression, and beams subjected to in-plane bending were developed and thoroughly verified. A comprehensive parametric study was conducted on slender cross-sections with RHS, SHS and I-sections representing wide variations in cross-section slenderness as well as in material grades such as austenitic, ferritic, duplex and lean duplex stainless steel. It was observed that the maximum elastic strain prior to failure could be used to predict the local buckling capacity of a slender cross-section instead of using the cross-section deformation capacity εcsm obtained directly from the proposed design base curve [14]. As the first stepping stone of the current research, for slender cross-sections with λp > 0.68, this maximum elastic strain was named as the Equivalent Elastic Deformation Capacity e,ev, which is defined as the elastic stain at the ultimate load of a slender cross-section. Using available test results and generated FE results, a rational relationship between csm and e,ev was established through regression analysis; this allowed to obtain the maximum elastic strain for a slender cross-section using the same base curve proposed in CSM for stocky sections. Hence, the maximum stress that a slender section can attain prior to failure as defined by the CSM buckling stress fcsm, can be calculated by simply multiplying e,ev by material Young’s modulus. Cross-section resistance under axial compression and in-plane bending can be determined by using the buckling stress and the gross cross-sectional properties. Use of gross cross-sectional properties to determine the cross-section resistance of slender sections eliminates the necessity of calculating effective cross-sectional properties.

The outcome of this research was published as a technical paper in Thin-Walled Structures in 2016 [87]. It is, however, worth noting that Zhao et al. [15] have very recently proposed a different approach following CSM design principles for slender

175

Chapter 6: Conclusions and Recommendations sections in 2017 based on the concept of e,ev proposed herein. The primary difference between their approach and the technique proposed in the current research lies in the design base curve; Zhao et al [15] proposed a separate base curve for slender sections whilst the current research adopted the same base curve for all sections and suggested modifications to the deformation capacity for slender sections.

The performance of the proposed method was verified using test and FE results, and was compared against those obtained using EN1993-1-4+A1[68], ASCE 8-02 [4], AS/NZS 4673 [5] and DSM [86]. Detailed comparisons showed that the predictions of the proposed method were more accurate and less scattered. Suggested CSM technique for slender sections could predict up to 98% of test compression resistance, on average, with significantly less scatter. EN 1993-1-4+A1[68] predictions were more conservative whilst other considered codes over-predicted the compression resistance of cross- sections. Importantly, the proposed technique is a lot simpler than the other codified techniques. In case of predicting the in-plane bending resistance, all considered codes were proved to be over conservative. The average ratios of predictions to test results were 0.80, 0.84 and 0.90 for EN1993-1-4+A1[68], ASCE [4] or AS/NZS [5] and DSM [86] respectively. However, this average was 0.93 for the proposed CSM method offering a significant 16% improvement when compared with EN1993-1-4+A1[68] predictions. Reliability of the proposed method was also analysed, and it showed that the code recommended partially safety factor 1.10 could be safely used for austenitic and ferritic grades of stainless steel. Higher partial safety factor is required for duplex and lean duplex grades stainless steel to ensure similar level of safety.

Overall, the proposed technique made CSM applicable for all types of stainless steel cross-sections. At the same time, the method requires less computational effort, offers a rational approach in recognising the unique features of stainless steel without compromising on cross-section resistances.

6.1.3 Buckling resistance of stainless steel hollow section columns

Once the scope of CSM was extended for all types of cross-sections, the current research focussed on to investigating member behaviour i.e. columns that primarily fail due to flexural buckling. The obvious material nonlinearity and stain hardening offered by

176

Chapter 6: Conclusions and Recommendations stainless steel alloys vary considerably between different grades. All currently available design guidelines do not take account of material nonlinearity and strain hardening in predicting the buckling resistance of stainless steel columns. In Chapter 4, a new design technique was proposed to accurately predict the buckling resistance of stainless steel hollow section columns by combining the basic design principles of CSM and the widely used Perry type buckling formulas.

Hollow sections are the most widely used type of stainless steel cross-sections, and considerable amount of test data are available in literature. In Chapter 4, the buckling behaviour of stainless steel RHS and SHS columns were investigated using FE modelling technique. Developed FE models were extensively verified using available experimental results. Once verified, a large number of FE models were used to investigate the effect of cross-section slenderness λp, non-dimensional proof stress e, material strain hardening exponent n and depth to width ratio H/B on the buckling resistance of columns. Obtained results were carefully analysed; H/B ratio was found not have any considerable effect on column resistance, but λp, e and n showed significant effects on the flexural buckling resistance of columns. Effects of e and n were incorporated by modifying the non- dimensional member slenderness λcsm using two proposed coefficients Ce and Cn; this modified member slenderness was denoted as λm. Introduction of λm allowed for the column curves to be presented as a function of λp. The imperfection parameter η was expressed as a sigmoidal function of λm and λp using the results obtained from FE models.

Ultimately, a series of column curves were proposed for different values of λp.

The performance of the proposed buckling curves, based on CSM design philosophy, was evaluated using all available test results as well as those generated through FE technique. A relative comparison between the prediction obtained using proposed CSM method and the other codified approaches was also carried out. The average ratio of the predictions obtained using proposed CSM with test and FE results were 0.98 and 0.95, respectively, for SHS columns, whilst those for RHS columns were 0.97 and 1.00. Rasmussen and Rondal’s [7] proposal adopted in AS/NZS [5] also produced good predictions for the buckling resistance of stainless steel hollow columns. EN1993-1-4+A1 [68] under- estimated but ASCE [4] standard over-predicted the resistance of columns subjected to flexural buckling. Overall, the proposed CSM produced significantly less scattered predictions with very good agreement with test and FE results for column buckling. 177

Chapter 6: Conclusions and Recommendations Reliability analysis showed that the proposed CSM could safely be used to predict the buckling resistance of stainless steel RHS and SHS columns of austenitic and duplex grades with a partial safety factor of 1.10.

The proposed technique extended the scope for CSM to be applicable in predicting member resistance. This method successfully incorporated the material nonlinearity and effects of strain hardening in buckling curves with appropriate recognition to cross- section slenderness. Computational effort for slender columns will be much simpler and straightforward as there is no need to calculate effective cross-sectional properties.

6.1.4 Buckling resistance of stainless steel welded I-columns

The behaviour of welded sections is different from that of the cold-formed sections due to the presence of residual stresses, the change in material properties in the heat affected zones and the presence of relatively higher geometric imperfections induced by the welding process. To date, limited number of studies is available on stainless steel welded sections in comparison to widely available cold-formed sections. Chapter 5 was designed to investigate behaviour of welded I-sections produced from austenitic stainless steel through experimentally and numerically, to investigate whether the CSM buckling formulas proposed for cold-formed hollow section columns could be used for stainless steel welded I-columns.

The experimental program included tensile coupon tests, measurements of residual stresses and initial geometric imperfections, testing of stub columns and long columns. Local buckling behaviour was observed through stub column tests, whilst flexural buckling response was studied by performing flexural buckling tests on pin-ended welded long columns produced from stainless steel. From tensile coupon tests, material properties such as Young’s modulus, proof stress, ultimate strength, ductility and Ramberg-Osgood parameters were determined for 316L austenitic grade stainless steel. Measured longitudinal residual stresses showed that the maximum tensile residual stress may be higher than the value proposed by Yuan et al. [48]. For most of the columns, the initial global imperfection amplitudes were less than the code recommended value of L/1000 [55]. However, the initial local imperfection amplitudes were significantly higher than the code recommended value of b/200 [55]. All stub columns failed in local buckling as

178

Chapter 6: Conclusions and Recommendations expected. Out of 16 long column specimens, 13 columns failed in flexural buckling but the rest failed in combined local and flexural buckling.

FE models were developed to generate additional results on welded I-columns produced from different grades of stainless steel. Major parameters considered in the parametric study were residual stress, depth-to-width ratio H/B, flange-to-web thickness ratio tf/tw and cross-section slender λp. No significant effect of H/B and tf/tw ratio was observed on the column resistance. Residual stresses had a negative impact on the column resistance; this effect was more prominent for columns buckling around minor axis. λp showed a similar effect on the buckling resistance of welded I-columns as was observed for SHS and RHS columns in Chapter 4. Series of curves were proposed for I columns based on

λp. Two sets of equations were proposed for major and minor axis buckling recognising the obvious differences in their flexural buckling responses.

The performance of the proposed CSM formulations to predict the buckling resistance of stainless steel welded I-columns were evaluated using test evidences as well as FE results. When compared against the performances of other codified design methods, the proposed method produced more accurate and less scattered predictions for the buckling resistance of welded I-columns produced from stainless steel. EN1993-1-4+A1 [68] predictions were marginally less than the proposed CSM predictions for major axis buckling, but the European code predictions were more conservative for columns buckling about minor axis. Since, ASCE 8-02 [4] and AS/NZS 4673 [5] standards were developed for cold-formed stainless steel members, the residual stresses developed in welded sections were not considered in those guidelines. Hence, both standards over predicted the buckling resistance of stainless steel welded I-columns.

Experiments carried out on welded I columns have added valuable test evidences to currently available limited data on welded sections produced from stainless steels. This investigation also facilitated in understanding the behaviour of stainless steel welded I- columns failing in flexural buckling, and hence the scope of CSM was further extended to accurately predict the buckling resistance of welded I-columns.

179

Chapter 6: Conclusions and Recommendations 6.1.5 Advancement of CSM

CSM is a strain based design approach originally developed for nonlinear metallic materials such as stainless steel, aluminium and high strength steel [26]. The concept went through a number of modifications during the last decade with the latest simplified approach proposed by Afshan and Gardner [14], which primarily focussed on to exploit the strain hardening characteristics of stainless steel for stocky sections only. In the current study, CSM has been successfully extended for slender cross-sections by retaining the same base curve proposed for stocky sections; this made CSM applicable for all cross- section types. Additionally, CSM design philosophy has been extended in predicting member resistances. Buckling curves for cold-formed hollow section columns as well as for welded I-columns were proposed considering the effects of cross-section slenderness, material nonlinearity and extensive strain hardening. This should be considered as a major advancement for CSM towards gradual development of this technique into an effective and efficient design tool for nonlinear metallic materials.

Future recommendations

The proposed design methods in this thesis were focussed on typical cross-sections such as SHS, RHS and welded I-section. The applicability of this method on other types of cross-sections i.e. angle, channel, lipped channel etc. should be investigated to turn this technique into a general design tool. With the increasing use of built-up sections produced from thin-walled cross-sections, the scope of CSM should also be expanded for built-up cross-sections. The behaviour of circular hollow section (CHS) and elliptical hollow section (EHS) is completely different from box sections or open sections. Some initial research was reported on CHS [10], but the recent advancement of CSM has primarily been restricted to plated cross-sections. Extensive research should be conducted to devise CSM formulations for CHS and EHS.

The buckling formulas of stainless steel columns proposed in this thesis cover austenitic and duplex grades of stainless steel. Ferritic grades are becoming popular due to a more affordable price tag but offers a considerably different stress-strain response showing higher nonlinearity but significantly lower ductility. Further research is required to investigate stainless steel sections produced from ferritic grades to devise a unified CSM for all stainless steel alloys. 180

Chapter 6: Conclusions and Recommendations The current research investigated the flexural buckling behaviour of stainless steel columns, but two other common buckling modes i.e. distortional buckling and flexural- torsional buckling were not considered. For unsymmetrical or open cross-sections, these buckling modes are unavoidable to develop a complete design guideline. Moreover, torsional behaviour of stainless steel members as well as lateral torsional buckling behaviour of stainless steel beams needs to be investigated. CSM will only be regarded as a practical design tool once all of the aforementioned cases are appropriately incorporated.

CSM was originally developed for nonlinear metallic materials; however, current study was limited to stainless steel members. Similar study may be conducted for other nonlinear metallic materials like aluminium and high strength steel to extend the scope of CSM for those materials. Gardner and Ashraf [26] and Ashraf and Young [132] used CSM for aluminium and found improvement in predicting the cross-section resistance of hollow sections. This clearly shows the potential of CSM to be developed through further research as a generalized design approach for all nonlinear metallic materials. The scope of CSM can also be extended for structural steel. Liew and Gardner [133] suggested to use CSM formulas to predict the cross-section resistance of stocky sections produced from structural steel with very low strain hardening slope. The applicability of the proposed method for slender cross-sections as well as the buckling formulas can be investigated for structural steel.

In this thesis, only the static behaviour of stainless steel structural elements was investigated. Austenitic and duplex grades stainless steel show high level of ductility, which could make it a favourable material for the structures subjected to dynamic loading. In 1995, stainless steel beam of a research laboratory in Osaka were found undamaged after the nearby earthquake in Kobe [1]. Stainless steel could be a feasible option for building frames under seismic loading. However, very limited research is currently available to investigate the full potential of stainless steel in dynamic loading. Comprehensive study is essential to obtain complete cyclic responses of different stainless steel elements before using in seismic loading.

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192

Appendix A

Table A.1 Section and material properties of the specimens of table 3.2

Sl. Specimen Reference H B t r L E σ σ n No. Designation i 0.2 u 1 Theofanous 100×100×4-SC1 101.5 101.5 3.9 3.8 400.0 200676 644.6 801.6 8.3 2 and 100×100×4-SC2 102.5 102.5 4.0 3.9 400.0 200684 644.9 801.8 8.3 3 Gardner 80×80×4-SC1 80.3 80.3 3.9 3.8 319.7 203253 696.3 834.7 6.2 4 [62] 80×80×4-SC2 80.0 80.0 3.8 3.6 332.2 203168 695.8 833.2 6.2 5 RHS 120×80×3-SC1 119.9 80.0 2.8 3.7 362.0 218044 445.9 488.8 10.8 6 Afshan and RHS 120×80×3-SC2 120.0 80.0 2.8 3.9 362.2 218072 446.2 489.0 10.8 7 Gardner RHS 60×40×3-SC1 59.9 40.0 2.8 3.2 122.1 211414 491.2 524.8 6.5 8 [101] RHS 60×40×3-SC2 59.9 40.0 2.8 3.2 122.1 211414 491.2 524.8 6.5 9 SHS 80×80×3-SC1 80.1 80.1 2.8 3.7 242.0 212571 451.8 465.8 8.5 10 SHS 80×80×3-SC2 80.1 80.1 2.8 3.4 242.0 212511 451.3 465.3 8.5 11 100×100×5-1A 100.0 100.0 4.7 2.1 349.9 193072 479.1 717.7 4.4 12 120×120×5-2A 120.2 120.2 4.7 5.8 399.9 192648 394.2 627.9 7.8 Zhao et al. 13 150×100×6-3A 150.6 100.0 5.9 7.1 450.1 192057 431.7 698.6 7.3 [103] 14 150×100×8-4A 150.1 100.2 7.8 9.7 450.0 198262 440.0 662.9 5.4 15 150×150×8-5A 150.2 150.2 8.0 11.2 449.8 202027 648.0 807.4 6.8

Table A2 Section and material properties of the specimens of table 3.3

Sl. Reference specimen H B t ri L E σ σ n No. 0.2 u 1 RHS 120×80×3-4PB 120.0 79.9 2.8 3.8 1500.0 216772 431.6 478.3 10.4 2 Afshan and RHS 60×40×3-4PB 60.2 39.9 2.9 3.2 1500.0 215745 470.8 497.5 7.2 Gardner 3 [101] SHS 80×80×3-4PB 80.4 80.0 2.8 4.0 1500.0 210900 438.3 453.6 8.6 4 SHS 60×60×3-4PB 60.7 60.7 2.9 2.9 1500.0 219370 528.7 554.9 7.2 5 SHS 100×100×5-1B 100.0 100.0 4.6 2.1 1200.0 193073 478.9 717.6 4.4 6 SHS 120×120×5-2B 120.0 120.2 4.6 5.8 1400.0 192647 393.8 627.8 7.8 Zhao et al. 7 RHS 150×100×6-3B 150.0 100.0 5.8 7.0 1800.0 192067 430.9 698.1 7.3 [103] 8 RHS 150×100×8-4B 100.0 150.1 7.7 9.0 1800.0 198202 437.1 661.3 5.4 9 RHS 150×150×8-5B 150.1 150.1 8.0 11.1 1800.0 202007 647.2 806.9 6.8 10 50×30×2.5L900 50.0 30.0 2.6 2.0 900.0 195249 731.1 860.7 5.6 11 30×50×2.5L900 30.3 50.5 2.6 2.0 900.0 195277 730.6 859.7 5.6 12 50×50×1.5L900 50.2 50.2 1.5 1.0 900.0 195269 655.3 792.8 5.0 13 50×50×2.5L900 50.0 50.1 2.5 1.0 900.0 200410 688.0 769.3 5.7 14 Huang and 70×50×2.5L1100 70.5 50.8 2.6 1.0 1100.0 195553 672.8 804.5 7.2 Young 15 [102] 50×70×2.5L1100 50.7 70.4 2.5 1.0 1100.0 195545 672.5 804.2 7.2 16 100×50×2.5L1500 100.4 50.4 2.5 1.0 1500.0 200564 673.3 784.5 5.8 17 50×100×2.5L1500 50.7 100.1 2.5 1.0 1500.0 200562 673.1 784.3 5.8 18 150×50×2.5L1500 150.4 50.1 2.5 2.0 1500.0 201532 690.1 815.9 4.3 19 50×150×2.5L1500 50.1 149.9 2.5 2.0 1500.0 201531 690.1 816.0 4.3

Appendix A Table A3 Section and material properties of the specimens of table 3.4

Sl. Specimen Reference H B t t L E σ σ n No. Designation f w 0.2 u 1 I304-260 259.0 165.7 10.0 6.0 779.9 188740 323.7 670.6 6.6 2 I304-282 282.9 185.6 6.0 6.0 850.4 188600 312.6 695.7 5.8 3 I304-312 313.7 305.5 6.0 6.0 951.6 188600 312.6 695.7 5.8 4 I304-320 319.6 205.6 10.0 6.0 961.6 188739 323.7 670.7 6.6 5 I304-372 373.3 246.1 6.0 6.0 1117.8 188600 312.6 695.7 5.8 Yuan et al. 6 I304-462 462.6 186.0 6.0 6.0 1400.7 188600 312.6 695.7 5.8 [63] 7 I2205-150 150.7 150.0 10.2 6.0 499.4 191607 581.1 779.7 6.8 8 I2205-192 193.1 125.8 6.0 6.0 600.0 193200 605.6 797.9 7.4 9 I2205-200 200.6 124.9 10.2 6.0 601.4 191796 584.0 781.8 6.9 10 I2205-252 252.9 245.3 6.0 6.0 780.0 193200 605.6 797.9 7.4 11 I2205-372 372.9 245.0 6.0 6.0 1117.0 193200 605.6 797.9 7.4

Table A4 Section and material properties of the specimens of table 3.5

Sl. Specimen Reference H B t ri E σ σ n No. Designation 0.2 u 1 I-200×140×6×6-2 202.1 138.6 6.1 6.0 193500 516.0 727.5 10.7 2 Saliba and I-200×140×8×6-2 200.6 138.3 8.1 6.1 199841 508.0 727.5 11.7 3 Gardner [100] I-200×140×10×8-2 199.3 139.0 10.3 8.0 211996 502.0 754.8 11.9 4 I-200×140×12×8-2 198.9 139.6 12.3 8.1 204775 470.3 724.0 11.0 5 I-160×160-B0 158.8 159.1 9.9 6.0 201162 300.0 612.9 5.5 Stangenberg I-160×160 6 159.2 161.7 10.1 6.8 202000 522.5 761.0 5.2 [90] Duplex- B0 7 I-320×160 - B0 319.6 160.7 9.8 6.0 199912 302.5 618.3 6.0

194

Appendix B

Material model used in this study.

σ σ n ε= +0.002 ( ) for 휎 ≤ 휎0.2 (B.1) E σ0.2

m σ−σ0.2 σ−σ0.2 ε= +εu ( ) + εt,0.2 for 휎 > 휎0.2 (B.2) E0.2 σu−σ0.2

E E0.2 = (B.3) 1+0.002n(E⁄σ0.2)

σ m = 1 + 2.8 0.2 (B.4) σu

σ σ = 0.20 + 185 0.2 for austenitic and duplex stainless steel (B.5) u E

σ σ = 0.46 + 145 0.2 for ferritic stainless steel (B.6) u E

σ0.2 εu = 1 − for austenitic and duplex stainless steel (B.7) σu

σ0.2 εu = 0.6 (1 − ) for ferritic stainless steel (B.8) σu

800

700

600

500 ), MPa),

σ 400

300 Stress Stress (

200

100

0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Strain (ε)

Figure B.1 Stress-strain diagram derived from the material model for an austenitic grade stainless steel with E = 200,000 MPa, σ0.2 = 400 MPa, σu = 701 MPa and n= 7.

Appendix C

Equation for calculation the weighted material properties for RHSH and SHS section

푀푃 (퐴−퐴 )+푀푃 ×퐴 푀푃 = 푓푙푎푡 푐 푐표푟푛푒푟 푐 (C.1) 푠푒푐푡푖표푛 퐴

Where, A is the gross cross sectional area, Ac is the area of the corner region, MPsection is the weighted average material properties for the section, MPflat is the material properties of the flat portion of the section and MPcorner is the material properties of the corner region of the section. Equation for calculation the weighted material properties for I-section

푀푃 ×퐴 +푀푃 ×퐴 푀푃 = 푓푙푎푛푔푒 푓푙푎푛푔푒 푤푒푏 푤푒푏 (C.2) 푠푒푐푡푖표푛 퐴

Where, Aflange is the total area of both flange, Aweb is the area of the web, MPsection is the weighted average material properties for the section, MPflange is the material properties of the flange and MPweb is the material properties of the web.