Applied Science, Faculty of
Engineering, School of (Okanagan)
DSpace
UBCO
Zhang, Peng
Doctor of Philosophy - PhD
University of British Columbia
For designing Cold-Formed Steel (CFS) structural members under compression, AISI S100 (2016) incorporates the Direct Strength Method (DSM). Although the DSM has been used for quite some time, its buckling strength curves have issues and have not been thoroughly assessed with experimental data. In this study, buckling strength curves in the DSM were first assessed with an extensive experimental database containing axial strengths, failure modes, and design parameters of 453 solid and 228 perforated CFS columns. This exercise helped determine the merits and shortcomings of the buckling strength curves. Then, the buckling strength curves were assessed further, using full-section tensile yield strengths as design parameters instead of the virgin material tensile yield strengths. For determining the full-section tensile yield strengths of the columns, i) the model developed by Karren (1965) to predict the tensile yield strengths of CFS corners was modified to improve the accuracy and extend its application limit, and ii) the expression proposed by Karren (1965) for estimating the full-section tensile yield strengths of CFS members was modified to simplify its application. Strength estimations of the columns calculated using full-section tensile yield strengths were compared with those calculated using virgin material tensile yield strengths. Then its potential was tested whether the former could be more accurate and used than the latter after the buckling strength curves are recalibrated. Afterwards, compression tests were performed on 55 CFS columns having patterned perforations distributed along the length uniformly to enlarge the experimental database of perforated CFS columns. The columns had three different cross-sections, and each cross-section had seven different lengths varying from 250 mm to 2150 mm to capture different buckling failure modes. Before conducting the tests, extensive eigenbuckling analyses were performed for the columns, and a method was proposed to accurately determine the lower bound of elastic local and distortional buckling loads and the associated buckling half-wavelengths of perforated CFS columns. Finally, new buckling strength curves were proposed, and their LRFD (Load and Resistance Factor Design) resistance factors (φc) were calculated for a target reliability index of 2.5.
BUCKLING STRENGTH CURVES FOR ESTIMATING THE STRENGTHS OF THIN-WALLED, COLD-FORMED STEEL COLUMNS UNDER COMPRESSION USING FULL-SECTION TENSILE YIELD STRENGTHS by Peng Zhang M.Sc., Wuhan University of Technology, 2013 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE COLLEGE OF GRADUATE STUDIES (Civil Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Okanagan) December 2021 © Peng Zhang, 2021 ii The following individuals certify that they have read, and recommend to the College of Graduate Studies for acceptance, the dissertation entitled: BUCKLING STRENGTH CURVES FOR ESTIMATING THE STRENGTHS OF THIN-WALLED, COLD-FORMED STEEL COLUMNS UNDER COMPRESSION USING FULL-SECTION TENSILE YIELD STRENGTHS submitted by Peng Zhang in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Civil Engineering Examining Committee: Dr. Shahria Alam, School of Engineering Supervisor Dr. Kasun Hewage, School of Engineering Supervisory Committee Member Dr. Ahmad Rteil, School of Engineering Supervisory Committee Member Dr. Loïc Markley, School of Engineering University Examiner Dr. Kim Rasmussen, Faculty of Engineering, The University of Sydney External Examiner iii Abstract For designing Cold-Formed Steel (CFS) structural members under compression, AISI S100 (2016) incorporates the Direct Strength Method (DSM). Although the DSM has been used for quite some time, its buckling strength curves have issues and have not been thoroughly assessed with experimental data. In this study, buckling strength curves in the DSM were first assessed with an extensive experimental database containing axial strengths, failure modes, and design parameters of 453 solid and 228 perforated CFS columns. This exercise helped determine the merits and shortcomings of the buckling strength curves. Then, the buckling strength curves were assessed further, using full-section tensile yield strengths as design parameters instead of the virgin material tensile yield strengths. For determining the full-section tensile yield strengths of the columns, i) the model developed by Karren (1965) to predict the tensile yield strengths of CFS corners was modified to improve the accuracy and extend its application limit, and ii) the expression proposed by Karren (1965) for estimating the full-section tensile yield strengths of CFS members was modified to simplify its application. Strength estimations of the columns calculated using full-section tensile yield strengths were compared with those calculated using virgin material tensile yield strengths. Then its potential was tested whether the former could be more accurate and used than the latter after the buckling strength curves are recalibrated. Afterwards, compression tests were performed on 55 CFS columns having patterned perforations distributed along the length uniformly to enlarge the experimental database of perforated CFS columns. The columns had three different cross-sections, and each cross-section had seven different lengths varying from 250 mm to 2150 mm to capture different buckling failure modes. Before conducting the tests, extensive eigenbuckling analyses were performed for the columns, and a method was proposed to accurately determine the lower bound of elastic local and distortional buckling loads iv and the associated buckling half-wavelengths of perforated CFS columns. Finally, new buckling strength curves were proposed, and their LRFD (Load and Resistance Factor Design) resistance factors (φc) were calculated for a target reliability index of 2.5. v Lay Summary Columns in light-gauge steel structures like pallet storage racks are often Cold-Formed Steel (CFS) members. These columns are vulnerable to buckling failure under compression. The North American design specification, AISI S100 (2016), incorporates the Direct Strength Method (DSM) to design CFS members under compression. The literature review showed that the DSM design equations have deficiencies and have not been thoroughly assessed with experimental data. In this study, the DSM design equations were first assessed with extensive compression test results of CFS columns, where the results suggest an improvement of the design expressions. Later, to enlarge the experimental database, 55 perforated CFS columns with three different cross-sections and seven different lengths were physically tested under compression. Finally, the new DSM design equations that have been proposed based on the enlarged experimental database proved to be accurate and led to a more economical design of CFS columns compared to those of the old ones. vi Preface A portion of this research has been submitted to peer-reviewed journals and conference proceedings for publication and presented in symposiums. All literature reviews, research ideas, experiments, data analyses, and numerical simulations presented in the following publications have been solely carried out or come up with by the author. The academic supervisor was responsible for the supervision and review of the work only. Publications and presentations relevant to this thesis include: Journal papers 1. Zhang, P., and Alam, M. S. (2017). “Experimental investigation and numerical simulation of pallet-rack stub columns under compression load.” Journal of Constructional Steel Research, 133, 282-299. 2. Zhang, P., and Alam, M. S. (2020). “Elastic buckling behaviour of Σ-shaped rack columns under uniaxial compression.” Engineering Structures, 212, 110469. 3. Zhang, P., and Alam, M. S. (2021). “Accuracy of Buckling Strength Curves using Direct Strength Method in Estimating the Strengths of Cold-formed Steel Columns under Compression: A Critical Review.” Journal of Structural Engineering (ASCE), submitted. 4. Zhang, P., and Alam, M. S. (2021). “Assessment of Buckling Strength Curves in Direct Strength Method for Estimating Axial Strengths of Cold-formed Steel Columns using Full-Section Tensile Yield Strengths.” Journal of Structural Engineering (ASCE), submitted. 5. Zhang, P., and Alam, M. S. (2021). “Compression tests of thin-walled cold-formed steel columns with Σ-shaped sections and perforations.” Journal of Thin-Walled Structures, submitted. vii 6. Zhang, P., and Alam, M. S. (2021). “Buckling Strength Curves for Estimating Axial Strengths of Cold-formed Steel Members using Full-Section Tensile Yield Strengths.” Journal of Structural Engineering (ASCE), to be submitted. Conference papers 1. Zhang, P., and Alam, M. S. (2016). “Numerical simulation strategy for pallet rack stub columns under uniaxial compression load.” Proc., The 11th Pacific Structural Steel Conference, Shanghai, China. 2. Zhang, P., and Alam, M. S. (2019). "Determination of Critical Elastic Buckling Loads and Half-wavelengths of Rack Columns with Patterned Perforations," The 7th Int Engineering Mechanics and Materials Specialty Conference, Canadian Society for Civil Engineering, Laval, QC, Canada, June. Symposium presentations 1. Zhang, P., and Alam, M. S. (2017). “Performance evaluation and structural design of rack clad buildings.” The 1st China-Canada Symposium on Structural & Earthquake Engineering (CCSSEE), UBC Vancouver, Canada, August 20th to 24th. 2. Zhang, P., and Alam, M. S. (2018). “Rack Clad Building and Column Behaviour.” Symposium, 4th Annual Engineering Graduate Symposium (AEGS2018), UBC Okanagan, Canada. viii Table of Contents Abstract ......................................................................................................................................... iii Lay Summary .................................................................................................................................v Preface ........................................................................................................................................... vi Table of Contents ....................................................................................................................... viii List of Tables ................................................................................................................................xv List of Figures ........................................................................................................................... xviii List of Equations ...................................................................................................................... xxiv List of Symbols ........................................................................................................................ xxvii List of Abbreviations .............................................................................................................. xxxii Acknowledgements ..................................................................................................................xxxv Dedication ............................................................................................................................... xxxvi Chapter 1: Introduction and Thesis Organization .....................................................................1 1.1 Background ..................................................................................................................... 1 1.2 Motivations ..................................................................................................................... 2 1.3 Objectives ....................................................................................................................... 3 1.4 Methodology ................................................................................................................... 4 1.5 Research Scope ............................................................................................................... 6 1.6 Thesis Organization ........................................................................................................ 9 Chapter 2: State-of-the-art Review on Cold-Formed Steel Members under Compression ..12 2.1 Background ................................................................................................................... 12 2.2 Direct Strength Method (AISI_S100 2016) .................................................................. 14 2.2.1 Buckling Strength Curves ......................................................................................... 15 ix 2.2.1.1 Solid Columns ................................................................................................... 15 2.2.1.2 Perforated Columns .......................................................................................... 16 2.2.2 Proposal of the Direct Strength Method (DSM) ....................................................... 17 2.2.3 Assessment of the Buckling Strength Curves in the DSM ....................................... 18 2.2.4 Issues of the Buckling Strength Curves in the DSM ................................................ 18 2.2.4.1 The Pnl-curve for Solid Columns ...................................................................... 19 2.2.4.2 The Pnd-curve for Solid Columns ...................................................................... 19 2.2.4.3 The Pne-curve for Solid Columns ...................................................................... 21 2.3 Calculation of Elastic Buckling Loads ......................................................................... 23 2.3.1 Elastic Critical Local and Distortional Buckling Loads ........................................... 23 2.3.2 Elastic Global Buckling Loads ................................................................................. 24 2.4 Factors Affecting the Axial Strengths of CFS Columns ............................................... 25 2.4.1 Cross-Section Geometry ........................................................................................... 26 2.4.1.1 Local Buckling .................................................................................................. 26 2.4.1.2 Distortional Buckling ........................................................................................ 26 2.4.1.3 Global Buckling ................................................................................................ 27 2.4.2 End Support Condition ............................................................................................. 29 2.4.2.1 Local Buckling .................................................................................................. 29 2.4.2.2 Distortional Buckling ........................................................................................ 29 2.4.2.3 Global Buckling ................................................................................................ 29 2.4.3 Increased Material Strength ...................................................................................... 30 2.4.4 Residual Stresses ....................................................................................................... 31 2.4.4.1 Magnitudes and Distributions of Residual Stresses .......................................... 33 x 2.4.4.2 Factors Affecting the Residual Stresses in CFS Members ............................... 37 2.4.4.3 Considerations of Residual Stresses in Numerical Simulations ....................... 38 2.4.4.4 Effects of Residual Stresses on the Axial Strengths of CFS Columns ............. 43 2.4.5 Initial Geometric Imperfection ................................................................................. 47 2.4.5.1 Local Buckling .................................................................................................. 48 2.4.5.2 Distortional Buckling ........................................................................................ 48 2.4.5.3 Global Buckling ................................................................................................ 49 2.4.6 Initial Loading Eccentricity ...................................................................................... 50 2.5 Compression Tests on CFS Columns ........................................................................... 50 2.6 Summary ....................................................................................................................... 50 Chapter 3: Experimental Database of CFS Columns and the Modification of Karren’s Model and Expression .................................................................................................................53 3.1 General .......................................................................................................................... 53 3.2 Experimental Database of Cold-Formed Steel Columns under Compression .............. 54 3.2.1 Collection of Experimental Data .............................................................................. 54 3.2.2 Elastic Buckling Analysis ......................................................................................... 61 3.2.3 Validity of the Experimental Database ..................................................................... 66 3.3 Experimental Database of Material Property Tests of CFS Corners ............................ 67 3.3.1 Karren’s Model ......................................................................................................... 67 3.3.2 Assessment of Karren’s model ................................................................................. 69 3.3.3 Modification of Karren’s Model ............................................................................... 73 3.3.4 CFS Corners with Angles Different from 90 Degrees .............................................. 74 3.4 Experimental Database of Full-Section Tensile Tests of CFS Members ..................... 77 xi 3.5 Summary ....................................................................................................................... 80 Chapter 4: Assessment of the Buckling Strength Curves in the DSM Using Tensile Yield Strengths of Virgin Materials .....................................................................................................83 4.1 General .......................................................................................................................... 83 4.2 Assessment of the Pnl-curve .......................................................................................... 84 4.3 Assessment of the Pnd-curve ......................................................................................... 86 4.4 Assessment of the Pne-curve ......................................................................................... 91 4.5 LRFD Resistance Factors of the Buckling Strength Curves in the DSM ..................... 95 4.6 Summary ....................................................................................................................... 99 Chapter 5: Assessment of the Buckling Strength Curves in the DSM Using Full-Section Tensile Yield Strengths ..............................................................................................................102 5.1 General ........................................................................................................................ 102 5.2 Assessment of the Pnl-curve ........................................................................................ 103 5.3 Assessment of the Pnd-curve ....................................................................................... 105 5.4 Assessment of the Pne-curve ....................................................................................... 108 5.5 LRFD Resistance Factors of the Buckling Strength Curves in the DSM ................... 112 5.6 Summary ..................................................................................................................... 113 Chapter 6: Elastic Buckling Properties of Perforated CFS Columns with Σ-Shaped Sections........................................................................................................................................115 6.1 General ........................................................................................................................ 115 6.2 Specimens ................................................................................................................... 117 6.2.1 Column Profiles ...................................................................................................... 117 6.2.2 Young’s Modulus.................................................................................................... 118 xii 6.3 Generation and Validation of Column Models ........................................................... 118 6.3.1 Buckling Curve and Signature Curve ..................................................................... 119 6.3.2 End Support Condition ........................................................................................... 120 6.3.3 Finite Element Models ............................................................................................ 120 6.3.4 Validation of Column Models ................................................................................ 121 6.4 Elastic Buckling Properties of Σ-Shaped Sections (Solid) ......................................... 123 6.4.1 Generation of the BC and SC.................................................................................. 123 6.4.2 BCs and SCs of Σ-Shaped Sections (Solid) ............................................................ 126 6.4.3 Multi-half-wavelengths Method ............................................................................. 132 6.5 Elastic Buckling Properties of Σ-Shaped Sections (Perforated) ................................. 134 6.5.1 Perforation Patterns Effects .................................................................................... 135 6.5.2 SCs and BCs of Σ-Shaped Sections (Perforated) .................................................... 139 6.5.3 Columns with Pinned-Pinned End Support Condition ........................................... 143 6.5.4 Determination of the Lower Bounds of Elastic Local and Distortional Buckling Loads 146 6.6 Summary ..................................................................................................................... 148 Chapter 7: Compression Tests of Perforated CFS Columns with Σ-Shaped Sections ........151 7.1 General ........................................................................................................................ 151 7.2 Specimens ................................................................................................................... 152 7.2.1 Nominal Sections .................................................................................................... 152 7.2.2 Lengths of Specimens ............................................................................................. 152 7.2.3 Measurement of Specimens .................................................................................... 153 7.2.4 Material Mechanical Properties .............................................................................. 157 xiii 7.2.5 End Support Conditions .......................................................................................... 159 7.2.6 Elastic Buckling Properties ..................................................................................... 161 7.3 Expanded Ramberg-Osgood Model ............................................................................ 165 7.4 Test Setup.................................................................................................................... 168 7.4.1 Test Setup for the 250 and 500-mm Long Specimens ............................................ 168 7.4.2 Test Setup for the Intermediate-Length Specimens ................................................ 170 7.4.3 Test Setup for the Long Specimens ........................................................................ 173 7.5 Experimental Results .................................................................................................. 175 7.5.1 The 250-mm Long Specimens ................................................................................ 177 7.5.2 The 500-mm Long Specimens ................................................................................ 178 7.5.3 The 760-mm Long Specimens ................................................................................ 179 7.5.4 The 1010-mm Long Specimens .............................................................................. 183 7.5.5 The 1520-mm Long Specimens .............................................................................. 185 7.5.6 The 1820-mm Long Specimens .............................................................................. 191 7.5.7 The 2150-mm Long Specimens .............................................................................. 196 7.6 Causes for the Interactive Buckling of Specimens ..................................................... 199 7.7 Unfactored Design Axial Strengths ............................................................................ 202 7.8 Summary ..................................................................................................................... 203 Chapter 8: Proposal of New Buckling Strength Curves ........................................................205 8.1 General ........................................................................................................................ 205 8.2 New Buckling Strength Curves .................................................................................. 206 8.2.1 Local Buckling of Solid Columns........................................................................... 207 8.2.2 Distortional Buckling of Solid Columns with B-F or F-F Ends ............................. 208 xiv 8.2.3 Flexural Buckling of Solid Columns with P-P or P-P-W Ends .............................. 209 8.2.4 Local Buckling of Perforated Columns .................................................................. 212 8.2.5 TF Buckling of Perforated Columns with P-P Ends ............................................... 215 8.3 Resistance Factors of the New Buckling Strength Curves ......................................... 217 8.4 Summary ..................................................................................................................... 218 Chapter 9: Conclusions .............................................................................................................220 9.1 Summary ..................................................................................................................... 220 9.2 Conclusions ................................................................................................................. 222 9.3 Contributions............................................................................................................... 224 9.4 Limitations of This Study ........................................................................................... 225 9.5 Recommendations for Future Research ...................................................................... 226 Bibliography ...............................................................................................................................228 xv List of Tables Table 2.1 Past studies showing experimental evidence of CFS columns failing in interactive buckling under compression ......................................................................................................... 14 Table 2.2 Relative levels of post-G buckling strength reserves of CFS columns......................... 30 Table 2.3 Performance of different strategies for considering residual stresses in column models....................................................................................................................................................... 42 Table 2.4 Variations of axial strengths of CFS columns after removing the residual stresses ..... 45 Table 2.5 Predicted axial strengths of CFS columns with and without considering the cold-forming effects on the material ..................................................................................................... 46 Table 3.1 Experimental database of compression tests of solid column specimens ..................... 58 Table 3.2 Experimental database of compression tests of perforated column specimens ............ 60 Table 3.3 Experimental database of material property tests of cold-formed, right-angle steel corners ........................................................................................................................................... 71 Table 3.4 Database of material property tests of cold-formed steel corners with angles different from 90 degrees............................................................................................................................. 76 Table 3.5 Database of full-section tensile tests of CFS members ................................................ 79 Table 4.1 Statistics of the Tested-to-Nominal Axial Strength Ratios of the Solid Column Specimens ..................................................................................................................................... 96 Table 4.2 Statistics of the Tested-to-Nominal Axial Strength Ratios of the Perforated Column Specimens ..................................................................................................................................... 97 Table 4.3 A summary of shortcomings of DSM buckling strength curves .................................. 99 Table 5.1 Statistics of the Tested-to-Nominal Axial Strength Ratios of the Solid Column Specimens ................................................................................................................................... 112 xvi Table 5.2 Statistics of the Tested-to-Nominal Axial Strength Ratios of the Perforated Column Specimens ................................................................................................................................... 112 Table 6.1 Problems addressed in this chapter and the corresponding subsections ..................... 116 Table 6.2 Dimensions of nominal cross-sections of columns..................................................... 118 Table 6.3 Cross-section properties .............................................................................................. 118 Table 6.4 Young's modulus ........................................................................................................ 118 Table 6.5 Measured cross-section dimensions of column specimens with section COOOI ...... 121 Table 6.6 Results of experiments and eigenbuckling analyses of the column specimens with section COOOI............................................................................................................................ 121 Table 6.7 Lower bounds of elastic local and distortional buckling loads and the corresponding half-wavelengths ......................................................................................................................... 132 Table 6.8 Results of perforation pattern sensitivity analyses ..................................................... 139 Table 6.9 Estimations of the 𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷 − 𝑷𝑷 of the perforated sections using the P-P end support condition ..................................................................................................................................... 146 Table 6.10 Lower bounds of elastic local and distortional buckling loads and the corresponding half-wavelengths of perforated sections ..................................................................................... 147 Table 7.1 Geometric dimensions of ZA30-section specimens ................................................... 154 Table 7.2 Geometric dimensions of ZA32-section specimens ................................................... 155 Table 7.3 Geometric dimensions of ZA42-section specimens ................................................... 156 Table 7.4 Engineering material mechanical properties ............................................................... 159 Table 7.5 Elastic buckling loads, axial strengths and stiffnesses, and unfactored design axial strengths of ZA30 specimens ...................................................................................................... 162 xvii Table 7.6 Elastic buckling loads, axial strengths and stiffnesses, and unfactored design axial strengths of ZA32 specimens ...................................................................................................... 163 Table 7.7 Elastic buckling loads, axial strengths and stiffnesses, and unfactored design axial strengths of ZA42 specimens ...................................................................................................... 164 Table 7.8 Parameters for describing the ZA32-C2 coupon’s true stress-strain curve using the expanded R-O model .................................................................................................................. 168 Table 7.9 Magnitudes of components of the cross-section deformations at the peak loads and failure modes of specimens ......................................................................................................... 182 Table 7.10 Causes of the interactive buckling of specimens ...................................................... 201 Table 8.1 LRFD resistance factors of the proposed buckling strength curves ........................... 217 xviii List of Figures Figure 1.1 Selective pallet storage racks......................................................................................... 1 Figure 1.2 Buckling strength curves in the DSM ........................................................................... 2 Figure 1.3 Research methodology ................................................................................................ 5 Figure 2.1 Deformed cross-sections of different buckling modes ................................................ 13 Figure 2.2 Residual stresses in cold-formed steel members ......................................................... 34 Figure 3.1 Cross-section shapes of the collected column specimens: (a) lipped-C series; (b) rack series; (c) track series; (d) hat series. ............................................................................................ 56 Figure 3.2 The number of failure modes versus different cross-section shapes: (a) solid column specimens; (b) perforated column specimens. .............................................................................. 57 Figure 3.3 Finite element models of columns under different end support conditions: (a) B-F ends; (b) F-F ends; (c) P-P ends; (d) P-P-W ends; (e) P-P-S ends. .............................................. 63 Figure 3.4 The ratios of 𝐅𝐅𝐅𝐅𝐅𝐅𝐅𝐅𝐅𝐅𝐅𝐅𝐅𝐅 versus the ratios of Fuv/Fyv of the corner specimens: (a) Karren’s model; (b) the modified Karren’s model. ...................................................................... 72 Figure 3.5 The ratios of 𝐅𝐅𝐅𝐅𝐅𝐅𝐅𝐅𝐅𝐅𝐅𝐅𝐅𝐅 of the corner specimens versus (a) r/t ratios; (b) yielding types of virgin steel sheets; (c) Fyv values. ............................................................................................. 74 Figure 3.6 The ratios of 𝐅𝐅𝐅𝐅𝐅𝐅𝐅𝐅𝐅𝐅𝐅𝐅𝐅𝐅𝐅𝐅𝐅𝐅 versus the ratios of θ/90 of corner specimens ................... 75 Figure 4.1 Data points of (λl, Pu/Py) of the column specimens failed in L buckling along with the Pnl-curve: (a) solid column specimens; (b) perforated column specimens. .................................. 84 Figure 4.2 Data points of (λl, Pnl/Pu) of the column specimens failed in L buckling: (a) solid column specimens; (b) perforated column specimens. ................................................................. 85 Figure 4.3 Data points of (λd, Pu/Py) of the column specimens failed in D buckling along with the Pnd-curve and/or Pne-curve: (a) solid column specimens with the B-F or F-F ends; (b) solid xix column specimens with pinned ends; (c) perforated column specimens with the B-F or F-F ends; (d) perforated column specimens with the P-P ends. .................................................................... 87 Figure 4.4 Data points of (λd, Pnd/Pu) of the column specimens failed in D buckling: (a) solid column specimens; (b) perforated column specimens. ................................................................. 88 Figure 4.5 Data points of (λc, Pu/Py) of the column specimens failed in G buckling with the Pne-curve: (a) solid column specimens; (b) perforated column specimens. ........................................ 92 Figure 4.6 Data points of (λc, Pne/Pu) of the column specimens failed in G buckling: (a) solid column specimens; (b) perforated column specimens. ................................................................. 93 Figure 5.1 Comparisons between the nominal and tested axial strengths of the column specimens that failed in L buckling: (a) Pu/Pya ratios of the solid column specimens versus λl; (b) Pu/Pya ratios of the perforated column specimens versus λl; (c) Pnl/Pu ratios of the solid column specimens versus the λl; (d) Pnl/Pu ratios of the perforated column specimens versus the λl. .... 104 Figure 5.2 Comparisons between the nominal and tested axial strengths of the columns specimens that failed in D buckling: (a) Pu/Pya ratios of the solid column specimens tested with the B-F or F-F ends versus λd; (b) Pu/Pya ratios of the solid column specimens tested with pinned ends versus λd; (c) Pu/Pya ratios of the perforated column specimens tested with the B-F or F-F ends versus λd; (d) Pu/Pya ratios of the perforated column specimens tested with the P-P ends versus λd. ..................................................................................................................................... 106 Figure 5.3 Comparisons between the nominal and tested axial strengths of the columns specimens that failed in D buckling: (a) Pnd/Pu ratios of the solid column specimens versus λd; (b) Pnd/Pu ratios of the perforated column specimens versus λd. ...................................................... 107 Figure 5.4 Comparisons between the nominal and tested axial strengths of the column specimens that failed in G buckling: (a) Pu/Pya ratios of the solid column specimens versus λc; (b) Pu/Pya xx ratios of the perforated column specimens versus λc; (c) Pne/Pu ratios of the solid column specimens versus λc; (f) Pne/Pu ratios of the perforated column specimens versus λc. ............... 109 Figure 6.1 Profiles and cross-sections of columns...................................................................... 117 Figure 6.2 Cross-section profile and finite element models ....................................................... 121 Figure 6.3 Buckling curves of specimen D1 ............................................................................... 123 Figure 6.4 Selected nodal lines in the FEM column models ...................................................... 125 Figure 6.5 Signature curves and buckling curves of the solid ZA30 section ............................. 127 Figure 6.6 Signature curves and buckling curves of the solid ZA32 section ............................. 128 Figure 6.7 Signature curves and buckling curves of the solid ZA42 section ............................. 130 Figure 6.8 Sensitivity of buckling loads on the number of buckling half-wavelengths ............. 133 Figure 6.9 Buckling curves generated using the type 1 and type 2 models ................................ 134 Figure 6.10 Flowchart of strategy 1 ............................................................................................ 136 Figure 6.11 Perforation patterns of the column with the ZA32 section...................................... 137 Figure 6.12 Perforation pattern effects on elastic buckling loads ............................................... 138 Figure 6.13 Signature curve and buckling curves of the perforated ZA30 section .................... 140 Figure 6.14 Signature curve and buckling curves of the perforated ZA32 section .................... 141 Figure 6.15 Signature curve and buckling curves of the perforated ZA42 section .................... 141 Figure 6.16 The ratios of the maximum buckling loads to the minimum buckling loads .......... 142 Figure 6.17 A FEM model with the perforated ZA42 section under pinned-pinned end support condition ..................................................................................................................................... 144 Figure 6.18 Buckling curves of the three perforated rack sections ............................................. 147 Figure 7.1 Profiles and net sections of columns ......................................................................... 152 Figure 7.2 Mechanical property test of material: (a) the material coupon; (b) the test setup. .... 157 xxi Figure 7.3 Stress-strain curves: (a) engineering stress-strain curves; (b) yield plateaus of engineering stress-strain curves (enlarged). ................................................................................ 159 Figure 7.4 Determination of secant yield strengths .................................................................... 167 Figure 7.5 True stress-strain curve of the ZA32-C2 coupon: (a) the tested curve and the curve generated using the expanded R-O model; (b) yield plateaus of the stress-strain curves (enlarged). ................................................................................................................................... 168 Figure 7.6 Test setup for the 250 and 500-mm long specimens ................................................. 169 Figure 7.7 Test setup for the 760 and 1010-mm long specimens ............................................... 171 Figure 7.8 Layout of LVDTs and symbols describing the cross-section deformation ............... 172 Figure 7.9 Test setup for the 1520, 1820, and 2150-mm long specimens .................................. 174 Figure 7.10 Axial load-axial displacement curves of the 250, 500, 760, and 1010-mm long specimens .................................................................................................................................... 176 Figure 7.11 Axial load-axial displacement curves of 1520, 1820, and 2150-mm long specimens..................................................................................................................................................... 177 Figure 7.12 The 250-mm long specimens after the test .............................................................. 178 Figure 7.13 The 500-mm long specimens after the test .............................................................. 178 Figure 7.14 Evolution of cross-section deformation of the ZA30-500-S2 specimen with loading..................................................................................................................................................... 179 Figure 7.15 The 760-mm long specimens at the end of the test ................................................. 180 Figure 7.16 Evolution of cross-section deformation of the 760-mm long specimens with loading..................................................................................................................................................... 180 Figure 7.17 Deformed configurations of the ZA42-760-S2 specimen at different loading points..................................................................................................................................................... 181 xxii Figure 7.18 The 1010-mm long specimens at the beginning and end of the test ....................... 183 Figure 7.19 Evolution of cross-section deformation with loading and deflection curves (δy) of the 1010-mm long specimens ..................................................................................................... 184 Figure 7.20 Deformed configurations of ZA32-1010-S2 specimen at different loading points . 185 Figure 7.21 Deformed configurations of the ZA42-1010-S1 specimen at different loading points..................................................................................................................................................... 185 Figure 7.22 Evolution of cross-section deformation with loading and deflection curves (δy) of the 1520-mm long specimens ........................................................................................................... 186 Figure 7.23 Evolution of lateral deflections (uncoupled) at the mid-length of the 1520-mm long specimens .................................................................................................................................... 187 Figure 7.24 Deformed configurations of the 1520-mm long specimens at different loading points..................................................................................................................................................... 189 Figure 7.25 Evolution of cross-section deformation of the 1820-mm long specimens with loading..................................................................................................................................................... 192 Figure 7.26 Deflection curves (δy) of the 1820- and 2150-mm long specimens ........................ 193 Figure 7.27 Evolution of lateral deflections (uncoupled) at the mid-length of the 1820-mm long specimens .................................................................................................................................... 193 Figure 7.28 Deformed configurations of the 1820-mm long specimens at different loading points..................................................................................................................................................... 194 Figure 7.29 Evolution of cross-section deformation of the 2150-mm long specimens with loading..................................................................................................................................................... 197 Figure 7.30 Evolution of lateral deflections (uncoupled) at the mid-length of the 2150-mm long specimens .................................................................................................................................... 197 xxiii Figure 7.31 Deformed configurations of the 2150-mm long specimens at different loading points..................................................................................................................................................... 199 Figure 8.1 Solid column specimens failed in L buckling: (a) the data points of (λl, Pu/Pya) along with the proposed Pnl-curve; (b) the data points of (λl, Pnl/Pu). ................................................... 208 Figure 8.2 Solid column specimens with B-F or F-F ends that failed in D buckling: (a) the data points of (λd, Pu/Pya) with the proposed Pnd-curve; (b) the data points of (λd, Pnd/Pu). ............... 209 Figure 8.3 Solid column specimens with P-P or P-P-W ends that failed in F buckling: (a) the data points of (λc, Pu/Py) with the proposed Pne-curve (Equation 8-3); (b) the data points of (λc, Pne/Pu) calculated as per Equation 8-3; (c) the data points of (λc, Pu/Pya) with the proposed Pne-curve (Equation 8-4); (d) the data points of (λc, Pne/Pu) calculated as per Equation 8-4. ..................... 212 Figure 8.4 Perforated column specimens failed in L buckling: (a) the data points of (λl, Pu/Pyn) with the proposed Pnl-curve (Equation 8-5); (b) the data points of (λl, Pnl/Pu) calculated as per Equation 8-5; (c) the data points of (λl, Pu/Pya, net) with the proposed Pnl-curve (Equation 8-6); (d) the data points of (λl, Pnl/Pu) calculated as per Equation 8-6. ..................................................... 214 Figure 8.5 Perforated column specimens with P-P ends that failed in TF or AD + TF buckling: (a) the data points of (λc, Pu/Pyn) with the proposed Pne-curve (Equation 8-7); (b) the data points of (λc, Pne/Pu) calculated as per Equation 8-7; (c) the data points of (λc, Pu/Pya, net) with the proposed Pne-curve (Equation 8-8); (d) the data points of (λc, Pne/Pu) calculated as per Equation 8-8. .............................................................................................................................................. 216 xxiv List of Equations Equation 2-1 .................................................................................................................................. 15 Equation 2-2 .................................................................................................................................. 15 Equation 2-3 .................................................................................................................................. 16 Equation 2-4 .................................................................................................................................. 16 Equation 3-1 .................................................................................................................................. 67 Equation 3-2 .................................................................................................................................. 74 Equation 3-3 .................................................................................................................................. 77 Equation 3-4 .................................................................................................................................. 78 Equation 4-1 .................................................................................................................................. 95 Equation 6-1 ................................................................................................................................ 125 Equation 6-2 ................................................................................................................................ 125 Equation 6-3 ................................................................................................................................ 125 Equation 6-4 ................................................................................................................................ 125 Equation 6-5 ................................................................................................................................ 133 Equation 7-1 ................................................................................................................................ 165 Equation 7-2 ................................................................................................................................ 166 Equation 7-3 ................................................................................................................................ 166 Equation 7-4 ................................................................................................................................ 166 Equation 7-5 ................................................................................................................................ 166 Equation 7-6 ................................................................................................................................ 166 Equation 7-7 ................................................................................................................................ 167 Equation 7-8 ................................................................................................................................ 167 xxv Equation 7-9 ................................................................................................................................ 169 Equation 7-10 .............................................................................................................................. 169 Equation 7-11 .............................................................................................................................. 170 Equation 7-12 .............................................................................................................................. 170 Equation 7-13 .............................................................................................................................. 170 Equation 7-14 .............................................................................................................................. 170 Equation 7-15 .............................................................................................................................. 172 Equation 7-16 .............................................................................................................................. 172 Equation 7-17 .............................................................................................................................. 172 Equation 7-18 .............................................................................................................................. 172 Equation 7-19 .............................................................................................................................. 172 Equation 7-20 .............................................................................................................................. 172 Equation 7-21 .............................................................................................................................. 172 Equation 7-22 .............................................................................................................................. 172 Equation 7-23 .............................................................................................................................. 172 Equation 7-24 .............................................................................................................................. 172 Equation 7-25 .............................................................................................................................. 172 Equation 7-26 .............................................................................................................................. 172 Equation 7-27 .............................................................................................................................. 172 Equation 7-28 .............................................................................................................................. 173 Equation 8-1 ................................................................................................................................ 207 Equation 8-2 ................................................................................................................................ 208 Equation 8-3 ................................................................................................................................ 209 xxvi Equation 8-4 ................................................................................................................................ 210 Equation 8-5 ................................................................................................................................ 213 Equation 8-6 ................................................................................................................................ 213 Equation 8-7 ................................................................................................................................ 215 Equation 8-8 ................................................................................................................................ 215 xxvii List of Symbols Ag Gross area of cross-section Anet Net area of cross-section Ac Cross-section area of the corner Af Cross-section area of the flat element B Maximum width of edge-stiffened plates in the cross-section disregarding intermediate stiffeners C Ratio between the overall area of corners and the gross or net area of cross-section Cw Warping constant Cφ Calibration coefficient Cp Correction factor D Maximum width of unstiffened plates in the cross-section disregarding intermediate stiffeners E Young’s modulus Fy Material tensile yield strength Fu Material ultimate tensile strength Fyf Material tensile yield strength of flat element in the cross-section Fuf Material ultimate tensile strength of flat element in the cross-section Fm Mean value of the fabrication factor Fyc Tensile yield strength of the corner Fyv Tensile yield strength of virgin material Fuv Ultimate tensile strength of virgin material Fya Full-section tensile yield strength xxviii Fya, net Net-section tensile yield strength Fyca Weighted average tensile yield strength of corners in the cross-section Fyfa Weighted average tensile yield strength of flat elements in the cross-section Fyce Tensile yield strength of corner obtained from experimental test Fyc90° Tensile yield strength of corner having the right angle Fyae Full-section tensile yield strength obtained from experimental test H Maximum width of stiffened plates in the cross-section disregarding intermediate stiffeners Ix Moment of inertial about the axis-of-symmetry Iy Moment of inertial about the axis-of-non-symmetry k Strength coefficient Kae Tested axial stiffness L Length or buckling half-wavelength 𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆 Buckling half-wavelength corresponding to the 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆 𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆 Buckling half-wavelength corresponding to the 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆 𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐𝐶𝐶−𝐶𝐶 Buckling half-wavelength corresponding to the 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝐶𝐶−𝐶𝐶 𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐𝐶𝐶−𝐶𝐶 Buckling half-wavelength corresponding to the 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝐶𝐶−𝐶𝐶 Mm Mean value of the material factor m Degrees of freedom or number of buckling half-wavelengths m1, m2, m3, m4 Constants multiplied with E as the slopes for determining the secant yield strengths n Number of tests or strain-hardening exponent n1, n2, k1, k2 Constants xxix Pu Tested axial strength Pn Nominal axial strength Pnl Nominal axial strength of local buckling Pnd Nominal axial strength of distortional buckling Pne Nominal axial strength of global buckling Py Yield strength of gross cross-section Pynet Yield strength of net cross-section Pcr Elastic critical buckling load Pcrl Elastic critical local buckling load Pcrd Elastic critical distortional buckling load Pcre Elastic critical global buckling load Pm Mean value of professional factor 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆 Lower bound of elastic local buckling load with the simply-simply end support condition 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆 Lower bound of elastic distortional buckling load with the simply-simply end support condition 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝐶𝐶−𝐶𝐶 Lower bound of elastic local buckling load with the clamp-clamp end support condition 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝐶𝐶−𝐶𝐶 Lower bound of elastic distortional buckling load with the clamp-clamp end support condition Rotx Rotational degree of freedom about the x-axis Roty Rotational degree of freedom about the y-axis Rotz Rotational degree of freedom about the z-axis rmin Minimum radius of gyration r inside bend radius of corner xxx s1, s2, s3, s4 Secant yield strengths t Thickness us Out-of-plane displacement of the slave node um Out-of-plane displacement of the master node Ux Translational degree of freedom along the x-axis Uy Translational degree of freedom along the y-axis Ux Translational degree of freedom along the z-axis VM Coefficient of variation of the material factor VF Coefficient of variation of the fabrication factor VQ Coefficient of variation of the load effect VP Coefficient of variation of the tested-to-nominal strength ratios W Greatest overall width of the cross-section Zs Z coordinate of the slave node in the model Zm Z coordinate of the master node in the model ν Poisson’s ratio φc Load and resistance factor design resistance factor λ Slenderness factor λl Slenderness factor of local buckling λd Slenderness factor of distortional buckling λc Slenderness factor of global buckling α Modification factor or a coefficient multiplied with E to represent the slope of yield plateau β0 Target reliability index xxxi β Modification factor θ Corner angle θ1, θ2 Rotations of flanges θ3, θ4 Rotations of web σ True stress σeng Engineering stress σe Proportional limit σp Stress at εp σb Bending stress σm Membrane stress ε True strain εeng Engineering strain εy Strain at Fy εp Strain at the intersection between yield plateau and strain hardening portion 𝜀𝜀𝐹𝐹𝑢𝑢 Strain corresponding to Fu Δ1, Δ2 Local deformation of the flange δx, δy Overall deflections xxxii List of Abbreviations AD Antisymmetric distortional ADF Rotation of flange induced by antisymmetric distortional mode ADW Rotation of web induced by antisymmetric distortional mode B Maximum width of edge-stiffened plates in the cross-section excluding corners B-F Bearing-friction BC Buckling curve CFS Cold-formed steel C-C Clamp-clamp DSM Direct strength method D Distortional DIC Digital image correlation system D + G Interaction between distortional and global DSM Direct strength method F Flexural FE Finite element FEM Finite element method FSM Finite strip Method F-F Fixed-fixed G Global H Maximum width of stiffened plates in the cross-section excluding corners IGI Initial geometrical imperfection ID Inward distortional xxxiii LRFD Load and resistance factor design L Local L + D Interaction between local and distortional L + G Interaction between local and global L + D + G Interaction between local, distortional, and global LD Longitudinal direction LYS Lower yield strength LVDT Linear variable differential transducer OD Outward distortional P-P Pinned-pinned P-P-W Pinned ends about the weak axis of cross-section P-P-S Pinned ends about the strong axis of cross-section SD Symmetric distortional SD(1) Symmetric distortional buckling with a single halfwave SD(2) Symmetric distortional buckling with two halfwaves SD(3) Symmetric distortional buckling with three halfwaves SC Signature curve S-S Simply-simply SDF Rotation of flange induced by symmetric distortional mode SDW Rotation of web induced by symmetric distortional mode TD Transverse direction TF Torsional flexural TF + F Interaction between torsional flexural and flexural xxxiv T Torsion induced by torsional mode UYS Upper yield strength xxxv Acknowledgements I want to express profound gratitude to my parents, uncle, aunt, sister, and Dr Yuting Wang, who have braced me throughout my long-term study, mentally and/or financially. Also, this study would not be fulfilled without the support from many people. I would like to thank my PhD supervisor, Dr M. Shahria Alam, first for his ceaseless guidance throughout my study. His in-depth knowledge and enthusiasm have encouraged me in the whole study. Second, I would like to acknowledge my committee members, including Prof. Kasun Hewage and Prof. Ahmad Rteil, for their guidance and encouragement. Third, I would like to thank Prof. Kim Rasmussen from the University of Sydney for being an external examiner of my thesis. Also, I would like to acknowledge the Applied Laboratory for Advanced Materials & Structures (ALAMS) research group members, who made this journey enjoyable and provided me with valuable feedback on my work. UBC’s Okanagan campus has provided me with an excellent academic atmosphere through the high-quality management system. The work presented in the thesis was funded jointly by the China Scholarship Council (CSC) and Natural Sciences & Engineering Research Council (NSERC) of Canada. The comments and conclusions in this thesis are solely by the author. Lastly, I want to thank Dr Shuai Li, Dr Shahidul Islam, Sherif Mohamed Shaaban, Dr Qi Zhang, Dr Anas Issa, Kyle Charles, and the whole lab technician group for their support in this research. xxxvi Dedication To my parents Yong Zhang and Chunxia Zhuang uncle Xuezhong Zhuang aunt Shaojing Xia and sister Qian Zhang1 Chapter 1: Introduction and Thesis Organization 1.1 Background Cold-Formed Steel (CFS) members with thin-walled and singly symmetric cross-sections like lipped-C, hat, track, and rack sections are popular structural elements in residential and commercial structures due to their superior features like the lightweight and material efficiency. For instance, Figure 1.1 shows selective pallet storage racks and their main structural elements, where these structures are often seen in superstores like Costco Wholesale and the Home Depot. Beams, columns, and bracings in these structures are all CFS members; see Figure 1.1(c). Besides, the columns, see Figure 1.1 (b), have patterned holes uniformly distributed along the length so that beams can be easily mounted on them at the required heights. The structural design of CFS members often involves high complexity due to two reasons. First, CFS members are vulnerable to buckling failure under loads, where the buckling modes can be various. Second, the mechanism behind each type of buckling mode could be rather complex. This study focuses on the structural design of CFS members (including solid and perforated) under compression. Figure 1.1 Selective pallet storage racks 2 The current North American specification, the AISI S100 (2016), incorporates a general and efficient method, namely the Direct Strength Method (DSM), to design CFS members under compression. The DSM adopts three buckling strength curves of Pnl-, Pnd-, and Pne-curve, see Figure 1.2, to estimate the axial strengths of a CFS member under three different limit states of local (L) (or the interaction between local and global) buckling, distortional (D) buckling, and global (G) buckling. The minimum of the estimated axial strengths serves as the unfactored design axial strength of the member. The literature review unveiled that although the DSM is being used for quite some time, the accuracy of the buckling strength curves has not been fully explored. Also, several issues and shortcomings of the buckling strength curves were found. Figure 1.2 Buckling strength curves in the DSM 1.2 Motivations This research has three motivations: ⅰ) To provide the technical/scientific and industrial communities with an overview of the accuracy of the buckling strength curves in the DSM, which is currently incorporated in the North American design standard AISI S100 (2016) for the design of CFS members under compression. The literature review showed that the buckling strength curves in the DSM have 3 limitations/shortcomings, which necessitate a thorough assessment with experimental results. The results of the assessment will point out directions for the refinement of these buckling strength curves. ⅱ) To enlarge the experimental database related to CFS members with perforations under compression. The literature review showed that the DSM for designing perforated members was proposed based on numerical results, and the current experimental database related to CFS members with perforations under compression only has a small volume - the larger the experimental database, the more reliable the proposal and assessment of buckling strength curves. ⅲ) To refine and/or propose new buckling strength curves for CFS members under compression. This work leads to more accurate estimations of the axial strengths of CFS members, resulting in improved performance and a more economical design using the Direct Strength Method. 1.3 Objectives The overall objective of this research is to improve the design of thin-walled CFS members subjected to local, distortional, or global buckling failure under compression. The overall research objective is accomplished through the following individual tasks: ⅰ) To conduct a thorough literature review to identify the merits and shortcomings of the buckling strength curves in the DSM (AISI_S100 2016). ⅱ) To establish an extensive experimental database of compression tests on solid and perforated CFS columns. To perform compression tests on a large number of thin-walled CFS specimens with patterned perforations distributed along the length. This work is to increase the experimental database. 4 ⅲ) To assess the accuracy of the buckling strength curves in the DSM (AISI_S100 2016) with experimental data considering two cases. The first case uses the virgin material tensile yield strength as the design parameter. The second case uses the full-section tensile yield strength as the design parameter. To refine the buckling strength curves in the DSM (AISI_S100 2016) or propose new ones. This work aims to obtain a more efficient, safe, and accurate design of CFS columns under compression. 1.4 Methodology A flowchart is presented in Figure 1.3 to show the methodology of this research. The research comprises three phases. Phase 1 assesses the buckling strength curves in the DSM (AISI_S100 2016) with the experimental data collected from the literature, considering two cases: ⅰ) using the virgin material tensile yield strength (Fyv) as the design parameter and ⅱ) using the full-section tensile yield strength (Fya) as the design parameter. Phase 2 deals with experimental tests of 55 cold-formed steel columns under compression, where the columns have three different Σ-shaped sections, patterned perforations distributed along the length, and seven different lengths ranging from 250 to 2150-mm. A portion of the experimental results is used in Phase 3. Phase 3 refines and/or proposes new buckling strength curves to resolve the shortcomings of the buckling strength curves in the current DSM (AISI_S100 2016).5 Figure 1.3 Research methodology6 1.5 Research Scope In this research, multiple tasks were fulfilled to accomplish the overall research objective. The scope and significance of each task are introduced as follows. Task Ⅰ – Literature review: In this task, studies reported in the past six decades related to the topics of ⅰ) the proposal and assessment of the buckling strength curves in the DSM (AISI_S100 2016); ⅱ) compression tests of CFS columns (including solid and perforated) that failed in L, D, or G buckling; ⅲ) the determination of elastic buckling loads and critical local and distortional buckling half-wavelengths of CFS columns (with and without perforations) under compression; ⅳ) the effects of end support conditions and cross-section geometry (including the shape and the ratios between the widths of the flat elements in the cross-section) on the post-buckling strength reserve of the CFS column under compression, and; ⅴ) the influences of cold-forming effects on the material (including the increase in material strength and residual stresses), initial loading eccentricity, and initial geometric imperfection on the axial strength of the CFS columns were reviewed thoroughly. Based on the literature review, the state-of-the-art of the proposed research was understood, methods and procedures for conducting compression tests on CFS columns were acquired, and the ways to refine the buckling strength curves in the DSM (AISI_S100 2016) were developed. Task Ⅱ – Establishment of three experimental databases: In this task, three different experimental databases were established: ⅰ) First experimental database: Experimental data (e.g., axial strengths and failure modes) associated with compression tests of 453 solid and 219 perforated CFS columns were collected from 40 different research articles. Design parameters (e.g., elastic critical buckling loads) of these columns were determined and assembled with the collected experimental data in a database to be 7 used as a source to evaluate the accuracy of the buckling strength curves in the DSM (AISI_S100 2016). ⅱ) Second experimental database: Experimental data associated with tensile material property tests of 218 CFS corners were collected from 21 different research articles. The experimental database was employed to improve the accuracy and extend the application limit of the model proposed by Karren (1965) for predicting the tensile yield strengths of CFS corners. ⅲ) Third experimental database: Experimental data associated with full-section tensile tests of 24 CFS members were collected from three different research articles. The experimental database was used to simplify the expression proposed by Karren (1965) for estimating the full-section tensile yield strengths of CFS members. Task Ⅲ – Modification of Karren’s model and expression: Karren (1965) developed a model (designated as Karren’s model hereafter) to predict the tensile yield strengths of CFS corners. Also, an expression (designated as Karren’s expression hereafter) was proposed by Karren (1965) for estimating the full-section tensile yield strengths of CFS members. Karren’s model and expression have been incorporated in the AISI S100 (2016). In this task, Karren’s model was modified based on the experimental database established in Task Ⅱ to improve the accuracy and extend the application limit. Moreover, Karren’s expression was modified based on the experimental database established in Task Ⅱ to simplify its application. The modified Karren’s model and expression were employed in the following task to estimate the full-section tensile yield strengths of CFS columns. Task Ⅳ – Assessment of the buckling strength curves in the DSM (AISI_S100 2016): In this task, the accuracy of the buckling strength curves in the DSM (AISI_S100 2016) was assessed with the experimental database established in Task Ⅱ. The assessment was performed 8 twice. In the first assessment, the virgin material tensile yield strength was used as the design parameter to estimate the axial strengths of columns. Then, the LRFD resistance factors (φc) of the buckling strength curves were calculated. The second assessment was like the first one, except the full-section tensile yield strength was used as the design parameter instead of the virgin material tensile yield strength. Merits and shortcomings of using the two different design parameters to estimate the axial strengths of CFS columns were identified. Task Ⅴ – Eigenbuckling analyses of CFS columns with Σ-shaped sections: The literature review in Task Ⅰ unveiled a shortage of experimental data associated with compression tests of perforated CFS columns. For increasing the experimental database, compression tests of perforated CFS columns with three different Σ-shaped sections were performed in the following task. Before conducting the physical tests, elastic buckling properties of the three Σ-shaped sections (with and without holes) under compression were investigated in this task by performing extensive eigenbuckling analyses. The influences of holes, the pattern of perforations, the number of buckling half-wavelengths, and end support conditions on the sections’ elastic buckling properties were investigated. An alternative method was proposed to generate the signature curves of solid columns. Besides, the multi-half-wavelengths method was proposed to determine the lower bounds of elastic local and distortional buckling loads and the associated buckling half-wavelengths of perforated columns. Task Ⅵ – Compression tests of perforated CFS columns: This task performed compression tests on perforated CFS columns with three different Σ-shaped sections. Seven different lengths varying from 250 to 2150-mm were tested for each section. In total, 55 columns were tested. During the tests, structural responses of the columns such as the axial load-axial displacement curve, overall deflections, and the cross-section deformation 9 at the most-deformed section were monitored. Part of the experimental data was combined with the experimental database established in Task Ⅱ as a source to be used in the following task to propose new buckling strength curves. Moreover, an expanded Ramberg-Osgood model was proposed in this task, which can accurately describe the stress-strain curve of the steel featuring a discontinuous yielding (i.e., has a relatively long yield plateau). Task Ⅶ – Proposal of new buckling strength curves: In this task, eight new sets of buckling strength curves were proposed to resolve the shortcomings of the buckling strength curves in the DSM (AISI_S100 2016). The LRFD resistance factors of the new buckling strength curves were calculated for a target reliability index of 2.5 following the procedure given in AISI S100 (2016). 1.6 Thesis Organization The thesis consists of nine chapters. Chapter 1 presents the background, overall objective and sub-objectives, and the scope of the study. Brief introductions of the following chapters are as follows. In Chapter 2, an in-depth literature review is presented, which covers the topics of ⅰ) the proposal and assessment of the buckling strength curves in the DSM (AISI_S100 2016); ⅱ) compression tests of solid and perforated CFS columns that failed in local, distortional, or global buckling; ⅲ) the determination of elastic buckling loads and the critical elastic local and distortional buckling half-wavelengths of CFS columns; ⅳ) the effects of end support conditions and cross-section geometry (the shape and the ratios between the widths of the flat elements in the cross-section) on the post-buckling strength reserve of the CFS column, and; ⅴ) the influences of cold-forming effects on the material (i.e., the increase in material strength and residual stresses), 10 initial loading eccentricity, and initial geometric imperfection on the axial strength of the CFS column. Chapter 3 introduces the establishment of three experimental databases associated with ⅰ) compression tests of solid and perforated CFS columns that failed in local, distortional, or global buckling, ⅱ) tensile material property tests of CFS corners, and ⅲ) full-section tensile tests of CFS members. The collection, process, and analysis of the experimental data are introduced in detail. In addition, the modifications of Karren’s model and expression are presented. Chapter 4 presents the assessment of the buckling strength curves in the DSM (AISI_S100 2016), considering the virgin material tensile yield strength as the design parameter. Also, the LRFD resistance factors (φc) of the buckling strength curves are calculated. Chapter 5 presents the assessment of the buckling strength curves in the DSM (AISI_S100 2016), considering the full-section tensile yield strength as the design parameter. The LRFD resistance factors (φc) of the buckling strength curves are calculated. The estimated axial strengths of the columns and the LRFD resistance factors of the buckling strength curves are compared with those obtained in the previous chapter. Merits and shortcomings of using the two different design parameters with the buckling strength curves in the DSM (AISI_S100 2016) are identified. Chapter 6 presents the investigation on the elastic buckling properties of three Σ-shaped CFS sections under compression. The influences of holes, the pattern of perforations, the number of buckling half-wavelengths, and end support conditions on the sections’ elastic buckling properties are determined. Besides, an alternative method is proposed for generating the signature curves of solid CFS columns. Moreover, the multi-half-wavelengths method is proposed to accurately determine the lower bounds of elastic local and distortional buckling loads and the associated buckling half-wavelengths of perforated CFS columns. 11 Chapter 7 introduces the compression tests on CFS columns (count = 55) with three Σ-shaped sections and patterned perforations. Besides, the proposal of the expanded Ramberg-Osgood model that can accurately describe the stress-strain curve of the steel featuring discontinuous yielding is presented. In Chapter 8, new buckling strength curves are proposed. The accuracy and LRFD resistance factors of the new buckling strength curves are presented. Chapter 9 describes the conclusions of this study and the recommendations for future research. 12 Chapter 2: State-of-the-art Review on Cold-Formed Steel Members under Compression 2.1 Background CFS columns are vulnerable to buckling failure under compression, where the buckling mode is mainly governed by the cross-section geometry (shape and dimensions), unrestrained length, end support conditions, initial geometric imperfection, and the yield strength of the steel (Camotim et al. 2018; Hancock 1984; Schafer 2002; Schafer and Peköz 1998). The common buckling modes include local (L), distortional (D), torsional flexural (TF), and flexural (F). The TF and F buckling are jointly termed as global (G) buckling. For illustration, the deformation configurations of the common buckling modes of a Σ-shaped section are shown in Figure 2.1(a). The SD and AD in the figure refer to symmetric distortional and antisymmetric distortional, where SD buckling modes are often seen in practice, while AD buckling modes are rarely observed. Note that when CFS columns have open and singly symmetric cross-sections, the TF buckling mode is accompanied by an AD mode (see Figure 2.1(b)), where the AD mode fades away as the length increases. Similarly, the F buckling mode is accompanied by an SD mode, where the SD mode fades away as the increasing length. Past studies showing experimental evidence of CFS columns that failed in the common buckling modes under compression are summarized in the following chapter. 13 Figure 2.1 Deformed cross-sections of different buckling modes CFS columns may also fail in interactive buckling under compression, such as the interaction between the L and D buckling (L + D), L + G, D + G, L + D + G, and TF + F (Camotim et al. 2018; Dinis et al. 2020b; Kwon and Hancock 1992; Moen and Schafer 2008; Young and Rasmussen 1999). Interactive buckling may occur in a CFS column when ⅰ) elastic buckling loads of two or more buckling modes are close or ⅱ) the column has prominent post-buckling strength reserve so that the compression load keeps increasing after the first buckling until another one or two buckling modes are involved in causing a failure (Camotim et al. 2013; Dinis et al. 2014). For instance, the Σ-shaped section presented in Figure 2.1 may show L + D buckling when the length lies in the transition range from L to D buckling where the elastic L and D buckling loads are close. Experimental evidence from past studies concerning CFS columns failed in interactive buckling under compression is presented in Table 2.1. In semi-rigid steel structural frames where beams are pin connected with columns, columns are mainly subject to compression loads. So, the buckling of the columns could be prevented from happening by limiting the compression loads below their axial strengths. In moment-resisting frames where beams are rigidly connected to columns, columns are subjected to compression loads and bending moments. So, the interaction between the compression load and bending moment 14 needs to be accounted for in the structural design of the columns. In this case, the axial strengths of the columns serve as design parameters in the interactive design formula (AISI_S100 2016). In summary, it is crucial to estimate the axial strengths of CFS columns accurately. This research focuses on accurately estimating the axial strengths of CFS columns under L, D, or G buckling. Table 2.1 Past studies showing experimental evidence of CFS columns failing in interactive buckling under compression Failure mode Condition Experimental studies Count L + D Solid Kwon and Hancock (1992); Young and Rasmussen (1998); Yan and Young (2002); Yang and Hancock (2004b); Lam et al. (2006); Zhang et al. (2007); Yap and Hancock (2008); Moen and Schafer (2008); Kwon et al. (2009); Yao et al. (2010); Yap and Hancock (2011); Pastor et al. (2013); Wang et al. (2013); Young et al. (2013); Wang et al. (2013); He et al. (2014); Dinis et al. (2014); Dinis et al. (2014b); Chen et al. (2017); Chen et al. (2019); Matsubara et al. (2019); Chen et al. (2020); Huang et al. (2020); Guo and Yao (2021). 24 Perforated Moen and Schafer (2008); Trouncer and Rasmussen (2012); de Almeida Neiva et al. (2018); Guo and Yao (2021). 4 L + G Solid Mulligan (1983); Miller (1990); Young and Rasmussen (1998); Yan and Young (2002); Zhang et al. (2007); Kwon et al. (2009); Shi et al. (2010); Kulatunga and Macdonald (2013); Wang et al. (2016); Ye et al. (2018); Yi et al. (2020); Rajkannu and Jayachandran (2020); Xiang et al. (2020); Li et al. (2021). 14 Perforated Kulatunga and Macdonald (2013) 1 D + G Solid Zhang et al. (2007); Bonada et al. (2012); Crisan et al. (2012); Souza dos Santos et al. (2012); Souza dos Santos et al. (2014); Zhao et al. (2017); Rajkannu and Jayachandran (2020). 7 Perforated Casafont et al. (2011); Crisan et al. (2012); Trouncer and Rasmussen (2012); Zhao et al. (2017). 4 L + D + G Solid Young and Rasmussen (1998); Yan and Young (2002); Zhang et al. (2007); Kwon et al. (2009); Young et al. (2013); Wang et al. (2016); Young et al. (2018). 7 Perforated Trouncer and Rasmussen (2012) 1 TF + F Solid Young and Rasmussen (1998) 1 Perforated nothing 0 2.2 Direct Strength Method (AISI_S100 2016) The current North American specification, the AISI S100 (2016), incorporates an advanced and convenient approach, the Direct Strength Method (DSM), to design CFS members under 15 compression. Even though numerous different buckling modes of CFS columns exist, the DSM checks only three limit states of L (when the column is globally braced) or L + G (when the column is globally unbraced), D, and G buckling failure. To be specific, to design a CFS member under compression, the DSM requires to calculate the nominal axial strengths (i.e., Pnl, Pnd, and Pne) of the column under three different buckling modes of L or L + G, D, and G using the given buckling strength curves. Expressions of the buckling strength curves are shown in Equation 2-1 to Equation 2-4. The minimum of the Pnl, Pnd, and Pne values serves as the unfactored design axial strength of the column. Note that the nominal axial strength of L + G buckling is also designated as Pnl in the DSM (AISI_S100 2016), and it is determined as per the buckling strength curve of L buckling (Equation 2-1), but Py being replaced by Pne in the expressions of the buckling strength curve and slenderness. 2.2.1 Buckling Strength Curves Expressions of the buckling strength curves in the DSM (AISI_S100 2016) are presented as follows. 2.2.1.1 Solid Columns The nominal axial strength for L buckling (Pnl) of a column is calculated as follows: For λl ≤ 0.776; Pnl = Py For λl > 0.776; Pnl= �1 - 0.15 �PcrlPy �0.4� �PcrlPy �0.4 Py Equation 2-1 The nominal axial strength for D buckling (Pnd) of the column is calculated as follows: For λd ≤ 0.561; Pnd = Py For λd > 0.561; Pnd= �1 - 0.25 �PcrdPy�0.6� �PcrdPy�0.6Py Equation 2-2 16 The nominal axial strength for G bucking (Pne) of a column is calculated as follows: For λc ≤ 1.5; Pne= �0.658λc2� Py For λc > 1.5; Pne= �0.877λc2 � Py Equation 2-3 where λl = (Py/Pcrl)0.5, λd = (Py/Pcrd)0.5, λc = (Py/Pcre)0.5. The Pcrl, Pcrd, and Pcre are the elastic critical L, D, and G buckling loads of the column, respectively. 2.2.1.2 Perforated Columns The Pnl of a perforated column shall be calculated as per Equation 2-1, except for the Pcrl shall be determined considering the influence of holes and the Pnl should ≤ Pynet. The Pnd of a perforated column shall be calculated as follows, and the Pcrd shall be determined considering the influence of holes. For λd > λd2; use Equation 2-2. For λd ≤ λd1; Pnd = Pynet For λd1 < λd ≤ λd2; Pnd=Pynet - �Pynet - Pd2λd2 - λd1 � (λd - λd1) Equation 2-4 where λd1 = 0.561(Pynet/Py), λd2 = 0.561[14.0(Py/Pynet)0.4-13.0], and Pd2 = [1-0.25(1/λd2)1.2](1/λd2)1.2Py. The Pne of a perforated column shall be calculated as per Equation 2-3, except for the Pcre shall be determined considering the influence of holes. In Equation 2-1 to Equation 2-4, Py = AgFy, Pynet = AnetFy, Ag = gross area of the cross-section, Anet = net area of the cross-section, and Fy = yield strength of the material. The buckling strength curves of L, D, and G buckling are designated hereafter as Pnl-curve, Pnd-curve, and Pne-curve, respectively. 17 2.2.2 Proposal of the Direct Strength Method (DSM) The DSM for solid columns was proposed by Schafer (2002; 2000), where the original idea came from Hancock et al. (1994). The method started to be incorporated in the AISI specifications in the 2001 edition (AISI 2004). Subject to the context at that time, e.g., a limited amount of experimental data and the failure modes of some of the column specimens were unclear (Schafer 2002; Schafer 2000), the proposal of the method had three limitations: ⅰ) column specimens in the database mostly had plain cross-sections like lipped-C and lipped-zed. Nowadays, cross-sections of CFS columns often have additional end stiffeners and/or intermediate stiffener(s) at the web and/or flanges; ⅱ) the buckling strength curves were obtained from the literature, where they were incorporated in the method without further calibration, and; ⅲ) the design parameter of Pcrd is defined as the lower bound of elastic D buckling load of the cross-section, which leads to a constant nominal axial strength of D buckling of a column regardless of the length and end support condition. The DSM for perforated columns was proposed by Moen and Schafer (2011), which was adapted from the DSM for solid columns proposed by Schafer (2002; 2000). This method started to be incorporated in AISI design specifications in the 2012 edition (AISI_S100 2012). Note that in the work of Moen and Schafer (2011), ⅰ) the columns studied were exclusively lipped C-section columns with simply-simply ends, where they had either circular or slotted holes in the web; ⅱ) the axial strengths of the columns were numerically determined, and; ⅲ) the Pnl-curve was proposed based on the axial strengths of L + G buckling rather than L buckling. 18 2.2.3 Assessment of the Buckling Strength Curves in the DSM Although the DSM has been available for quite some time, the accuracy of its buckling strength curves has not been fully explored. The literature review shows that only limited experimental data were employed to assess the DSM buckling strength curves. The assessment of the buckling strength curves for solid CFS columns can be found ⅰ) in (Chen et al. 2019; Kumar and Kalyanaraman 2012; Xu et al. 2014; Yap and Hancock 2008; Yap and Hancock 2011) on the Pnl-curve, ⅱ) in (Kumar and Kalyanaraman 2014; Ranawaka and Mahendran 2009; Yap and Hancock 2011; Young et al. 2013; Zhao et al. 2017) on the Pnd-curve, and ⅲ) in (Gunalan and Mahendran 2013; Weng and Lin 1992; Young and Rasmussen 1998; Young et al. 2013; Zhao et al. 2017) on the Pne-curve. The assessment of the buckling strength curves for perforated CFS columns can be found in (Xu et al. 2014) on the Pnl-curve and in (Zhao et al. 2017) on the Pne-curve. The assessment of the Pnd-curve has not to be seen from the literature. Among the studies mentioned in the above two paragraphs, only Kumar and Kalyanaraman (2012) and Weng and Lin (1992) established comparatively large experimental databases that contained 32 and 59 tested strengths, respectively. The experimental databases utilized in the rest of the studies all had a small number (≤ 16) of tested strengths. As per the author’s knowledge, an overall picture of the accuracy of any buckling strength curve in the DSM is still missing. 2.2.4 Issues of the Buckling Strength Curves in the DSM In addition to the limitations of the DSM (AISI_S100 2016) reported in Section 2.2.2, more issues were identified from the literature review. The results are summarized as follows. 19 2.2.4.1 The Pnl-curve for Solid Columns The Pnl-curve for solid columns (Equation 2-1) was adapted, without modifying the form of expressions, from the bending strength curve proposed by Schafer and Peköz (1998) for designing laterally-braced CFS flexural members. Later, Kumar and Kalyanaraman (2012) assessed the Pnl-curve in the slenderness range of 0.5 ≤ λl ≤ 3.7 with the axial strengths (32 tested and 71 numerical) of the lipped-C section columns that failed in L buckling. They observed that the ratios of true-to-nominal axial strengths were highly scattered. 2.2.4.2 The Pnd-curve for Solid Columns The Pnd-curve for solid columns (Equation 2-2) was proposed by Kwon and Hancock (1992) based on the tested axial strengths of 40 CFS columns that mostly (34 out of 40) had plain cross-sections. The columns were either fixed ended or bore against loading plates/platens directly during the compression. The experimental boundary conditions were accounted for while proposing the Pnd-curve. Among the 40 columns, 32 were identified as failed in D buckling, and the other eight were identified as failed in L + D buckling. As most columns failed in D buckling, Kwon and Hancock (1992) claimed the validity of the Pnd-curve. This conclusion, however, was challenged by Kumar and Kalyanaraman (2014). Upon further analyzing the columns tested by Kwon and Hancock (1992), Kumar and Kalyanaraman (2014) found that among the 32 columns that were reported failed in D buckling, 17 columns show the Pcrl ≤ Pcrd, and five additional columns show the Pu > Pcrl. Hence, they argued that L buckling might also have been involved in the failure of these 22 columns and hence asserted that the Pnd-curve is conservative since it was proposed based on the axial strengths of mainly L + D buckling. Later, they assessed the Pnd-curve against the axial strengths (14 tested and numerous numerical) of fixed-ended columns with web-flange stiffened lipped-C sections that failed in D buckling, where the results were found to support 20 their assertion. Consequently, Kumar and Kalyanaraman (2014) modified the Pnd-curve for designing fixed-ended columns with web-flange stiffened lipped-C sections. The work of Kumar and Kalyanaraman (2014) was later challenged by Landesmann et al. (2016), who pointed out that the modification of the Pnd-curve made by Kumar and Kalyanaraman (2014) was ill-advised from using a small and biased database. In Landesmann et al. (2016)’s work, both the Pnd-curve and the buckling strength curve proposed by Kumar and Kalyanaraman (2014) were assessed with a database containing extensive numerical axial strengths of fixed-ended columns with lipped-C sections or web-flange stiffened lipped-C sections that failed in D buckling. They observed that the Pnd-curve generally provided conservative axial strength estimations for the columns. However, the buckling strength curve proposed by Kumar and Kalyanaraman (2014) offered unsafe axial strength estimations for many columns. As a result, the Pnd-curve was recommended by Landesmann et al. (2016) to be continuously used in the routine design. Note that in the works of Kumar and Kalyanaraman (2014) and Landesmann et al. (2016), consistent boundary conditions were employed to determine the axial strengths and elastic D buckling loads of the columns. A single buckling strength curve of D buckling (e.g., the Pnd-curve) cannot adequately handle CFS columns with different end support conditions. Landesmann and Camotim (2013) numerically unveiled that the Pnd-curve could estimate the axial strengths of fixed-ended columns adequately but was unsafe, in the slenderness range of λd ≥ 1.5, for estimating the axial strengths of the columns with simply-simply, simply-fixed, or fixed-free ends. This is because, in the slenderness range of λd ≥ 1.5, columns with simply-simply, simply-fixed, or fixed-free ends have post-D buckling strength reserves prominently lower than those of fixed-ended columns. In the work of Landesmann and Camotim (2013), consistent boundary conditions were employed to determine the axial strengths and elastic D buckling loads of the columns. 21 In the DSM (AISI_S100 2016), the Pcrd of a CFS column is defined as the lower bound of the elastic D buckling load of the cross-section. This results in two issues: ⅰ) D buckling generally takes place in CFS columns with moderate lengths that can generally accommodate one to three D buckling half-wavelengths. In practice, the end support condition and length could affect the D buckling strengths of the columns to certain extents due to the limited number of buckling half-wavelengths. ⅱ) As mentioned earlier, the column specimens’ experimental boundary conditions (fixed ends) were considered while proposing the Pnd-curve (Kwon and Hancock 1992). So, a question arises whether the Pcrd determined as per the DSM or the Pcrd determined considering the actual end support condition and length should be fed into the Pnd-curve to estimate the D buckling strength of a column. Note that the Pnd values calculated based on two different Pcrd values can be very different. For instance, Yap and Hancock (2011) presented a design case about a fixed-ended column with a web-stiffened lipped-C section experiencing D buckling failure under compression. The column was 700-mm long and could accommodate a single D buckling halfwave. The Pcrd value of the column determined considering the fixed ends and actual length was 60% greater than the Pcrd value determined as per the DSM. The two Pcrd values were then fed into the Pnd-curve to calculate the Pnd value of the column separately. Yap and Hancock (2011) observed that the Pnd value calculated based on the large Pcrd is 28% greater than the one calculated based on the small Pcrd. 2.2.4.3 The Pne-curve for Solid Columns The Pne-curve for solid columns (Equation 2-3) was initially proposed to design hot-rolled steel members used in buildings (AISC 1986). Based on the work of Peköz and Sümer (1992), the 22 cold-formed steel design standard started to apply this curve since it was adopted in the specification of AISI (1996). Gunalan and Mahendran (2013), Dinis et al. (2019), and Dinis et al. (2020) have unveiled that the post-G buckling strength reserve of the CFS column shows sensitivity to the end support condition and buckling mode. Hence, a single buckling strength curve of G buckling (e.g., the Pne-curve) cannot adequately handle CFS columns in different cases, which is described below: ⅰ) The Pne-curve exhibits high accuracy in estimating the axial strengths of CFS columns in three different cases: ⅰ) fixed ends having F buckling (Dinis et al. 2019; Gunalan and Mahendran 2013), ⅱ) pinned ends having TF buckling (Dinis et al. 2020), and ⅲ) simply-simply ends having F or TF buckling (Gunalan and Mahendran 2013; Rajkannu and Jayachandran 2020)). Despite this, Dinis et al. (2019) observed that the Pne-curve, within the slenderness range of λc ≤ 2.0, slightly overestimates the axial strengths of CFS columns in the case of fixed ends having F buckling. ⅱ) The Pne-curve exhibits low accuracy in estimating the axial strengths of CFS columns in three cases: ⅰ) fixed ends having TF buckling, ⅱ) pinned ends about the axis-of-symmetry having TF buckling, and ⅲ) pinned ends about the axis-of-non-symmetry having TF buckling. To be specific, the Pne-curve, beyond the slenderness range of λc >1.5, underestimates the axial strengths of CFS columns in two cases of fixed ends having TF buckling (Dinis et al. 2019; Gunalan and Mahendran 2013) and pinned ends about the axis-of-symmetry having TF buckling (Dinis et al. 2020). Besides, the Pne-curve, beyond the slenderness range of λc > 1.0, overestimates the axial strengths of CFS columns in the case of pinned ends about the axis-of-non-symmetry having TF buckling (Dinis et al. (2020)). To compensate for these deficiencies, Dinis et al. (2019) and Dinis et al. (2020) proposed three new sets of buckling strength curves that are assembled with the Pne-23 curve as a group to address CFS columns considering different conditions. Note that the above studies were based on nearly all numerical simulations. 2.3 Calculation of Elastic Buckling Loads An elastic buckling load of a CFS column presents the effects of the cross-section geometry, length, and end support conditions on the axial strength of the column. Hence, elastic buckling loads of CFS columns are usually key design parameters for estimating their axial strengths. For instance, the elastic buckling load and material yield strength are the two design parameters of the buckling strength curves in the DSM (AISI_S100 2016)). Thus, it is crucial to determine the elastic buckling loads of CFS columns accurately. In practice, elastic buckling loads of CFS columns can be determined using computer programs like CUFSM v4.05 (Li and Schafer 2010), GBTUL 2.0 (Bebiano et al. 2018), and ANSYS 18.1 (2019). The CUFSM v4.05 is a finite strip method-based program, and the GBTUL 2.0 is a generalized beam theory-based program. The two software are user-friendly and freely available, but they only support limited sets of end support conditions and cannot accommodate holes in the column model. The ANSYS 18.1 is a finite element method-based program, which can accommodate various sets of end support conditions and holes, while its application in the routine design is costly. The merits and shortcomings of the three software for determining the elastic L, D, and G buckling loads of CFS columns are summarized as follows. 2.3.1 Elastic Critical Local and Distortional Buckling Loads The CUFSM v4.05 and GBTUL 2.0 can handle solid CFS columns efficiently. The CUFSM v4.05 can accommodate rounded corners in various shapes of sections, from simple to complex. Also, it can generate a signature curve for a CFS section quickly. The signature curve is proposed to quickly determine the lower bounds of elastic L and D buckling loads of a solid CFS 24 section. For generating the signature curve of a CFS section, shape functions are applied to the column model so that the shapes of buckling modes vary in a sinusoidal way along the length (Li 2009). The GBTUL 2.0 can only accommodate rounded corners in simple sections like the lipped-C section. However, it has the advantage that buckling modes can be designated for a CFS section for the eigenbuckling analysis. This function is very helpful when the elastic buckling load of a CFS section associated with a ‘pure’ buckling mode (i.e., L, D, or G) cannot be determined from a conventional eigenbuckling analysis. Note that the CUFSM v4.05 and GBTUL 2.0 do not support the end support condition of pinned ends, where the column ends can rotate about the two principal axes freely but with warping prevented from occurring. The CUFSM v4.05 cannot accommodate holes in column models. Hence, researchers have proposed simplified solutions for determining the lower bounds of elastic L and D buckling loads of perforated CFS columns using the finite strip method; e.g., Casafont et al. (2012) and Smith and Moen (2014). The simplified solutions are to model holes in a cross-section using plates with reduced thickness to approximately consider the effects of holes on elastic buckling loads. They are empirical solutions and are subject to application limits. To date, CFS columns with holes lack a robust method to determine the lower bounds of elastic L and D buckling loads and the corresponding buckling half-wavelengths. This problem was resolved in this study; see Chapter 6. 2.3.2 Elastic Global Buckling Loads The elastic G buckling load (Pcre) of a CFS column can be ⅰ) read off from the signature curve of the cross-section at the effective length, ⅱ) calculated using the closed-form expressions incorporated in AISI S100 (2016), or ⅲ) determined from an eigenbuckling analysis. Researchers often ignore the differences between the application limits of the three methods. The first method 25 applies to the cases where the effective length factors (K) of flexural and torsional are considered equal, e.g., simply supported columns where Kx = Ky = Kt = 1.0 and fixed-ended columns where Kx = Ky = Kt = 0.5. The second method not only applies to the above cases but also applies to the cases where Kx, Ky, Kt are not all equal, e.g., columns with pinned ends where Kx = Ky = 1.0 and Kt =0.5 and columns with the ends pinned only about the axis-of-non-symmetry where Kx = Kt = 0.5 and Ky = 1.0. The third method is more reliable than the first two methods and can be performed in computer software like CUFSM (Li and Schafer 2010), GBTUL (Bebiano et al. 2018), and ANSYS (ANSYS® 2019). CUFSM and GBTUL are user-friendly and freely available, but they only support limited sets of end support conditions and cannot accommodate holes in column models. Column models created in ANSYS can accommodate various sets of end support conditions and holes, while ANSYS application in the routine design is costly. Note that the first two methods have issues. First, when a CFS column has end support conditions different from the simply-simply ends, the critical G buckling mode determined using the first two methods can be wrong. For instance, the critical G buckling mode of a fixed-ended column may be determined as a TF mode using the first two methods, but it is, in fact, an F mode. Second, unlike the first and third methods, the second method does not consider the interferences of L and D deformation modes to the G buckling mode. Hence, the Pcre of a column determined using the second method tends to be larger than that determined using the first or third method, especially when the length is comparatively short. 2.4 Factors Affecting the Axial Strengths of CFS Columns This section summarizes key findings obtained from the past studies concerning the factors affecting the axial strengths of CFS columns. The factors include the cross-section geometry (shape and the ratios between the widths of the flat elements in the cross-section), end support 26 condition, cold-forming effects on the material (i.e., the increase in material strength and residual stresses), initial geometric imperfection, and initial loading eccentricity. These factors affect the axial strengths of CFS columns in different mechanisms. The first two factors act in the means that under a given condition of the same buckling modes and λ values, columns with different cross-section geometries and/or end support conditions could exhibit different post-buckling strength reserves (Pu/Pcr) under compression. The remaining three factors act in the means that when columns have the same buckling modes and λ values, the magnitude and distribution pattern of any one of the factors could vary from column to column, and different columns may show different sensitivities of axial strengths to the factor. 2.4.1 Cross-Section Geometry 2.4.1.1 Local Buckling Among the factors of the ratio of web-to-flange widths, the overall size of the cross-section, and material yield strength, the first one was identified by Kumar and Kalyanaraman (2012) as a critical factor to affect the post-L buckling strength reserve of the lipped-C section column (solid). Later, they proposed a new set of design expressions for estimating the axial strengths of lipped-C section columns. In the new design expressions, the ratio of web-to-flange widths serves as a design parameter to consider the variations in the post-L buckling strength reserves induced by the different ratios of web-to-flange widths. Similar studies on CFS columns showing other cross-section shapes have not been seen from the literature. 2.4.1.2 Distortional Buckling Kumar and Kalyanaraman (2014) numerically demonstrated that the post-D buckling strength reserves of fixed-ended CFS columns (solid) maintain nearly the same as the section varies from the lipped-C section to web-stiffened lipped-C section, web-flange-stiffened lipped-C 27 section, and hat section. They also demonstrated that when fixed-ended columns with web-flange-stiffened lipped-C sections have the same λd values, the post-D buckling strength reserves of the columns were insensitive to the factors including the overall size of the cross-section, the ratio of lip-to-flange widths, the ratio of web-to-flange widths, material yield strength, and length. However, the column models established by Kumar and Kalyanaraman (2014) were later demonstrated by Landesmann et al. (2016) to be overly restrained. This raises a question about the validity of the conclusions drawn by Kumar and Kalyanaraman (2014). Moreover, Landesmann et al. (2016) numerically demonstrated that the ratio of web-to-lip widths is a key factor to affect the post-D buckling strength reserves of fixed-ended columns with web-flange-stiffened lipped-C sections. 2.4.1.3 Global Buckling CFS columns in different cases show different sensitivities of post-G buckling strength reserves to the cross-section geometry. On one hand, post-G buckling strength reserves of CFS columns (solid) are insensitive to the cross-section geometry in five cases: ⅰ) pinned ends having TF buckling; ⅱ) ends pinned about the axis-of-non-symmetry having TF buckling; ⅲ) fixed ends having F buckling and λc ≥ 2.0; ⅳ) fixed ends having TF buckling and λc ≤ 1.5; and ⅴ) ends pinned about the axis-of-symmetry having TF buckling and λc ≤ 1.5. Evidence supporting the above can be seen from Dinis et al. (2019) and Dinis et al. (2020). To be specific, Dinis et al. (2019) and Dinis et al. (2020) numerically demonstrated that in any of the five cases, the data points of (λc, Pu/Py) of CFS columns (solid) having different cross-section geometries are well concentrated vertically, where a single buckling strength curve could well fit the data points. 28 On the other hand, post-G buckling strength reserves of CFS columns (solid) become sensitive to the cross-section geometry in three cases: ⅰ) Post-G buckling strength reserves of CFS columns are insensitive to the cross-section shape but sensitive to the ratios between the widths of the flat elements in the cross-sections in the case of fixed ends having F buckling and λc ≤ 2.0. Evidence supporting this view is given by Dinis et al. (2019). To be specific, Dinis et al. (2019) numerically demonstrated that in the case of fixed ends having F buckling, the accuracy of the buckling strength curve (namely the fnF-curve) in estimating the axial strengths of CFS columns (solid) maintains nearly unaltered as the cross-section shape varies from plain-C section to lipped-Z section and then to the I section (formed by connecting two plain-C sections back-to-back). Note that the same cross-section shape columns exhibited various ratios between the widths of the flat elements in the cross-sections. Upon looking at the data points of (λc, Pu/Py) of the columns, a vertical dispersion of the data points was observed in the slenderness range of λc < 2.0, which might be caused by the various ratios between the widths of the flat elements in the cross-sections. ⅱ) Post-G buckling strength reserves of CFS columns (solid) become sensitive to the cross-section geometry in two cases of ⅰ) fixed ends having TF buckling and λc ≥ 1.5 and ⅱ) ends pinned about the axis-of-symmetry having TF buckling and λc ≥ 1.5. Evidence supporting this standpoint can be seen from Dinis et al. (2020). To be specific, Dinis et al. (2020) numerically unveiled a feature that post-G buckling strength reserves of CFS columns correlate negatively with the βFT values in the above two cases. The βFT is a cross-section normalized geometric parameter and equals to (Ix + Cw/Ag)/Iy. The Ix is the moment of inertial about the axis-of-symmetry, Iy is the moment of inertial about the axis-of-non-symmetry, and Cw is the warping constant. Hence, Dinis et al. (2020) proposed new expressions for estimating the axial strengths of CFS columns in the 29 two cases, where the βFT serves as a design parameter so that the descending slope of the buckling strength curve can be adjusted. 2.4.2 End Support Condition 2.4.2.1 Local Buckling The post-L buckling strength reserve of the CFS column seems to be insensitive to the end support condition. For instance, to determine the L buckling strengths of three different lipped-C section columns, Pu et al. (1998) conducted stub column compression tests using two different end support conditions, i.e., B-F ends and P-P ends. They observed that the tested strengths obtained using the two end support conditions were nearly identical. 2.4.2.2 Distortional Buckling The post-D buckling strength reserve of the CFS column exhibits sensitivity to the end support condition. Landesmann and Camotim (2013) numerically demonstrated that beyond the slenderness range of λd ≥ 1.2, overall, the post-D buckling strength reserves of CFS columns decreased as the end support condition varies from F-F ends to simply-fixed ends, simply-simply ends, and fixed-free ends regardless of the shape of the cross-section i.e., lipped-C, hat, rack, or lipped-Z. Besides, they found that the Pnd-curve could only estimate the axial strengths of fixed-ended columns adequately. Hence, they proposed two new sets of buckling strength curves to deal with columns with simply-simply (or simply-fixed) and fixed-free ends. 2.4.2.3 Global Buckling It has been mentioned in Section 2.2.4.3 that the post-G buckling strength reserve of the CFS column varies with the end support condition, G buckling mode, and slenderness range. Based on the works of Dinis et al. (2019) and Dinis et al. (2020), the relative overall post-G buckling strength reserve of the CFS column in different cases are summarized in Table 2.2 and are 30 represented by numerical values ranging from 1 to 5. The larger the numerical value, the higher the overall post-G buckling strength reserve. Note that in Table 2.2, comparisons can only be made among the numerical values in the same column of the table. The same numerical values shown in one column of the table indicate that a single buckling strength curve is accurate enough for the relevant cases. On the contrary, different numerical values shown in one column of the table indicate that distinct buckling strength curves are required for the relevant cases if accurate strength estimations are required. Table 2.2 Relative levels of post-G buckling strength reserves of CFS columns Case End support condition Buckling mode Global slenderness range λc ≤ 1.0 1.0 < λc ≤ 1.5 1.5 < λc ≤ 2.0 λc > 2.0 a F-F or P-P-S ends F b 1 b 1 b 2 a 2 b F-F ends TF a 2 a 2 b 5 b 4 c P-P or P-P-W ends F c N/A c N/A c N/A c N/A d P-P ends TF a 2 a 2 a 3 a 2 e P-P-W ends TF a 2 b 1 b 1 b 1 f P-P-S ends TF a 2 a 2 b 4 b 3 Note: a means that the Pne-curve exhibits good accuracy in estimating the axial strengths of the columns in such case; b means that new design expressions have been proposed by Dinis et al. (2019) or Dinis et al. (2020) to deal with columns in such case; c means that studies are awaiting to be carried out. 2.4.3 Increased Material Strength A steel sheet needs to go through several successive cold-working steps, i.e., coiling, un-coiling, flattening, and brake pressing (or roll-forming), to become CFS members. During the cold working, plastic strains are generated, accumulated, and eventually exist in CFS members, especially at cold-bent corners. The plastic strains result in two different effects on the material. On one hand, they cause strain hardening and/or strain ageing of the material, leading to increased material strength (including the yield and ultimate) and decreased ductility. On the other hand, they distribute across the thickness and the cross-section perimeter nonuniformly, resulting in 31 residual stresses being locked in the members. The increase in material strength could improve the resistance of CFS members. On the contrary, residual stresses could be detrimental to the resistance of CFS members. Magnitudes of the plastic strains resulted from cold working feature nonuniformity across the cross-section of CFS members, where the magnitudes at corners are generally significantly higher than those at flats. As a result, the increases in material strength are much more prominent at corners than at flats. The increases in material strength tend to be uniform at the flats of brake-pressed members. For roll-formed members, however, the flats adjacent to corners have gone through high pressure from the rollers. Hence, the increases in material strength at those areas are often more significant than those at the areas away from corners (Karren 1965). The AISI_S100 (2016) allows taking advantage of the increase in material yield strength due to cold working to design CFS columns subjected to L buckling under compression and having λl values ≤ 0.776. Karren (1965) has developed a model to predict the tensile yield strengths of CFS corners. Also, an expression has been proposed by Karren (1965) to estimate the full-section tensile yield strengths of CFS members. They have been incorporated in North American CFS design specifications, e.g., the AISI S100 (2016), for a long period. They were assessed with experimental results and modified in Chapter 3. 2.4.4 Residual Stresses Annealing and numerical simulations are two ways to study the influences of residual stresses on the structural responses of CFS columns under compression. Residual stresses in a CFS column can be eliminated through annealing. Thus, the effect of residual stresses on the structural response of the column under compression can be determined by comparing the structural responses between the annealed and unannealed columns; see examples in Dat (1980). However, 32 annealing a CFS column is costly. Also, the annealing temperature needs to be well controlled to avoid altering the material’s mechanical properties. In contrast, numerical simulations are cost-efficient and convenient, and the components of residual stresses can be selected to be introduced into a column model so that the roles played by the different components of residual stresses in affecting the structural response of the column can be determined. As a result, numerical simulations have been used by many researchers to study this topic. However, the following difficulties are involved while numerically investigating the influences of residual stresses on the structural responses of CFS columns under compression: ⅰ) The measurement of residual stresses in a CFS member is tedious and costly. Besides, techniques for measuring the layering component of residual stresses in thin steel sheets (e.g., t ≤ 5 mm) appear still unavailable now. ⅱ) The magnitudes and distribution patterns of residual stresses in CFS members feature uncertainties. They could vary significantly between different CFS members, and they may also vary between the same members prominently. ⅲ) Researchers have introduced different components of residual stresses into column models to do nonlinear analyses. This issue is further discussed in Section 2.4.4.3. ⅳ) CFS members are geometrically imperfect. When modelling a CFS column, the initial geometric imperfection is often approximately represented by one of the column’s eigenmodes or a mixture of multiple eigenmodes. The magnitude of the initial geometric imperfection could be the suggested value from a literature/specification or determined from the experimental measurement. The modelling of the initial geometric imperfection inevitably brings in errors in the numerical results, resulting in difficulties in accurately quantifying the effect of residual stresses on the structural response of the column. 33 Although researchers have conducted numerous works to study the effects of residual stresses on the axial strengths of CFS columns, the author could not draw a quantified conclusion on this topic. Key findings obtained from past studies related to this topic are summarized as follows. 2.4.4.1 Magnitudes and Distributions of Residual Stresses Past studies have demonstrated that at any location of a CFS member, residual stress in either the longitudinal (LD) or transverse (TD) direction varies through the thickness in a complex manner rather than linearly (Amouzegar et al. 2016; Key and Hancock 1993; Quach et al. 2006; Weng and White 1990). Figure 2.2(a) (see the last panel) presents the through-thickness variation patterns of residual stresses in the two directions that were numerically predicted by Quach et al. (2006). The location of the residual stresses is at the center of a lip-to-flange corner of a brake-pressed lipped-C section (namely the PBC 14 section). Through the thickness, residual stresses can be decomposed into three components: bending, membrane, and layering stresses (Key and Hancock 1993). The bending and membrane stresses vary through the thickness linearly, while the layering stress varies in a complex manner. It is difficult to measure all the components of residual stresses in CFS members experimentally. As a result, past studies only focused on measuring the bending and membrane stresses in the LD (Abdel-Rahman and Sivakumaran 1997; Batista and Rodrigues 1992; Dat 1980; Ellobody and Young 2005; Ingvarsson 1975; Kwon and Hancock 1992; Popovic et al. 1999; TRINH et al. 2018; Weng and Pekoz 1990b; Young and Ellobody 2005; Young and Rasmussen 1995). Moreover, sections of CFS members covered in the past studies were limited to plain-C, lipped-C, web-stiffened lipped-C, lipped equal angle, and plain equal angle sections. From the literature review, the author failed to find any evidence showing that there are components of residual stresses that always show small magnitudes in CFS members and hence 34 their influences on the structural responses of CFS members under compression are of no importance. Figure 2.2 Residual stresses in cold-formed steel members The lipped-C section is one of the classical sections of CFS members. So, it could be a good representative to show residual stresses in CFS members. Weng and Pekoz (1990b) experimentally measured the bending and membrane stresses in the LD for five different CFS members with lipped-C sections using the sectioning method. The members had the following features: ⅰ) two of them were roll-formed, and the others were brake-pressed; ⅱ) the material properties featured the 221.05 ≤ Fyf ≤ 379.83 MPa and 1.20 ≤ Fuf/Fyf ≤ 1.54, and ⅲ) the cross-section dimensions featured the 1.63 ≤ t ≤ 3.07-mm, 31.95 ≤ H/t ≤ 42.27, 16.97 ≤ B/t ≤ 23.41, 7.26 ≤ D/t ≤ 9.73, and 1.03 ≤ r/t ≤ 2.92. The Fyf and Fuf are the yield and ultimate tensile strengths tested from material coupons taken from the center of a flat element in the cross-section. The t, H, B, D, 35 and r are the thickness, web width, flange width, lip width, and inner corner radius, respectively. Note that one of the sections was the PBC14 section, where the measured residual stresses across the section are presented in Figure 2.2(a). Residual stresses in these members exhibited four common features: ⅰ) The bending stress in the LD follows a cross-section distribution pattern of tensile stress at the outer face and compressive stress at the inner face. In Figure 2.2, bending stresses following this distribution pattern are denoted with positive signs; otherwise, they are denoted negatively. ⅱ) The membrane stress in the LD follows a cross-section distribution pattern of tensile stress at corners. As one travels from a corner to the adjacent flat, the stress evolves to compressive stress sharply and then may evolve back to tensile stress at the vicinity of the mid-flat when the flat has a comparatively large width. In Figure 2.2, tensile membrane stresses are denoted with positive signs, and compressive membrane stresses are denoted with negative signs. ⅲ) Magnitudes of the bending stresses (in the LD) at corners are prominently larger than those at flats. Bending stresses (σb) (in the LD) in these members show magnitudes (normalized) that fall within the ranges of 0.43Fyf ≤ σb ≤ 0.64Fyf and 0.24Fyf ≤ σb ≤ 0.50Fyf at corners and mid-flats, respectively. ⅳ) Magnitudes of membrane stresses (in the LD) were small compared to bending stresses (in the LD). Membrane stresses (σm) (in the LD) in these members show magnitudes (normalized and absolute) that fall within the ranges of 0.05Fyf ≤ σm ≤ 0.10Fyf and σm ≤ 0.06Fyf at corners and mid-flats, respectively. Note that residual stresses in the TD of the members were not measured by Weng and Pekoz (1990b). However, the member with the PBC14 section has its residual stresses in the two directions predicted by Quach et al. (2006) numerically. The through-thickness variation patterns 36 of the residual stresses at the mid-web and the center of the lip-to-flange corner of the PBC14 section are shown in the last two panels of Figure 2.2(a). It can be observed from the figures that the residual stress at the center of the lip-to-flange corner is even greater in the TD than in the LD. Although residual stresses in CFS members with sections like plain-C, web-stiffened lipped-C, lipped equal angle, and plain equal angle sections are available in the literature, they are not described here for simplicity. However, key observations of the residual stresses in these members are summarized below: ⅰ) In roll-formed members, bending and membrane stresses (in the LD) at the areas adjacent to a corner can be greater than those at the corner; see experimental evidence from Key and Hancock (1993) and Abdel-Rahman and Sivakumaran (1997). ⅱ) Membrane stresses (in the LD) can be significant in CFS members. For instance, Popovic et al. (1999) experimentally observed that the magnitudes of membrane stresses (in the LD) are 0.10Fyf to 0.24Fyf greater than those of the bending stresses (in the LD) at the corners of three roll-formed, in-line galvanized members with plain equal-angle sections. Figure 2.2(b) presents the magnitudes of membrane and bending stresses (in the LD) in one of the members with the L50×50×5.0 section. ⅲ) Cross-section distribution patterns of residual stresses can vary significantly between the members with different cross-section shapes. For instance, the bending and membrane stresses (in the LD) at the center of the corner of the PBC14 section (see Figure 2.2(a)) show signs opposite to their counterparts at the corner of the L50×50×5.0 section (see Figure 2.2(b)). 37 2.4.4.2 Factors Affecting the Residual Stresses in CFS Members Residual stresses in CFS members feature uncertainties. Here, factors affecting the magnitudes and distribution patterns of residual stresses in CFS members are of interest. Past studies concerning this topic are few, where their key findings are summarized below: ⅰ) Rondal (1987) analytically predicted the residual stresses (in the LD) in roll-formed members considering only the cold-rolling step and the elastic spring back. The members had plain equal-angle sections, and their raw material had a thickness of 6-mm and a yield strength of 355-MPa. By performing parametric analysis, Rondal (1987) found that the residual stress (in the LD) at the corner surface has its magnitude (normalized) correlated to the b/t ratio positively but the r/t ratio negatively. The b is the leg width, including the corner. An inverse trend was exhibited by the residual stress (in the LD) at the surface of the leg. ⅱ) Weng and White (1990) experimentally measured the residual stress (in the LD) at the surface of the corners in brake-pressed members with plain equal-angel sections (t ≥ 25.4 mm). They observed that the magnitude (normalized) of the residual stress was sensitive to the r/t ratio and the corner angle but was insensitive to the yield strength of the raw material. ⅲ) Quach et al. (2006) and Amouzegar et al. (2016) numerically identified the factors affecting the residual stresses in CFS members with lipped-C sections. They found that for the residual stress (in either direction) at the flat region of the cross-section, the magnitude (normalized) and the through-thickness variation pattern are governed by the thickness, initial coiling diameter, and the yield strength of raw material, where the magnitude (normalized) correlates to the thickness positively but to the initial coiling diameter and the yield strength of raw material negatively. The residual stress (in either direction) at the corner was found to have the magnitude (normalized) and through-thickness variation pattern insensitive to the initial coiling 38 diameter and the yield strength of raw material. So, significant variations in the residual stresses at the flat region of the cross-section may occur between the same members when they have very different initial coiling diameters. 2.4.4.3 Considerations of Residual Stresses in Numerical Simulations Residual stresses may soften the axial load-axial shortening response of a CFS column by lowering the initial axial stiffness, proportional limit, and/or axial strength (Abdel-Rahman and Sivakumaran 1997; Dat 1980; Key and Hancock 1993; Quach et al. 2010; Weng and Pekoz 1990). The mechanism is that when a CFS column is under compression, the residual stress (in the LD) and the external loading stress at the same region of the cross-section will superpose. As the compression load increases, the material at those regions where the residual stress (in the LD) and the external loading stress are of the same signs will show material nonlinearity earlier than the case of assuming without residual stresses. Different strategies have been adopted by researchers to introduce residual stresses into column models for nonlinear analyses. Four different strategies have been found from the literature; they are introduced as follows: ⅰ) 1st strategy. Residual stresses, including all components in both directions, are introduced into the column model. Mechanical properties of the material at the flat regions and corners of the cross-section are represented by the stress-strain curves tested from material coupons taken from those areas. This strategy has been demonstrated by Key and Hancock (1993), see Table 2.3, to provide reasonable predictions of the axial load-axial shortening curves for CFS columns experiencing L or F buckling under compression. The columns had square hollow sections, which were roll-formed and sealed using electric resistance welding. Besides, Key and Hancock (1993) demonstrated that the inclusion or exclusion of any component of the residual stresses in a column 39 model, except the longitudinal membrane stress, could significantly alter the axial stiffness. Also, the inclusion of membrane and bending stresses (in the LD) in the column models was a must to ensure accurate predictions of axial strengths. ⅱ) 2nd strategy. This strategy is similar to the 1st strategy, except that this strategy requires only the longitudinal bending stress to be introduced into a column model. Note that this strategy ⅰ) only applies to columns with small membrane and layering stresses in the LD and ⅱ) ignores the influences of transverse residual stresses on the column’s structural response under compression. This strategy has been demonstrated by Sivakumaran and Abdel-Rahman (1998), see Table 2.3, to provide accurate predictions of axial load-axial shortening curves for roll-formed, lipped-C section columns experiencing L buckling under compression. ⅲ) 3rd strategy. This strategy is like the 1st strategy, except that this strategy requires only the longitudinal membrane stress to be introduced into a column model. This strategy ⅰ) ignores the influences of transverse residual stresses on the column’s structural response under compression and ⅱ) assumes that the effects of bending and layering stresses (in the LD) on the material’s mechanical behaviour have already been included in the tested stress-strain curves of the material. This strategy was proposed by Ellobody and Young (2005), who claimed that material coupons could regain their straightness during the tension, resulting in longitudinal bending stress to be recovered in the material coupons. This strategy was employed by Ellobody and Young (2005) and Young and Ellobody (2005) to predict the axial load-axial shortening curves of brake-pressed, angle section columns experiencing L or G buckling under compression, where acceptable to accurate predictions were obtained; see Table 2.3. Note that for the column models created by Ellobody and Young (2005) and Young and Ellobody (2005), the mechanical properties of the material across the thickness were represented by a single stress-strain curve. The stress-strain 40 curve was the average of two stress-strain curves measured at the two opposite surfaces of a material coupon. In fact, materials at different layers across the thickness behave differently under tension due to the longitudinal bending and layering stresses. Hence, different stress-strain curves may be necessary for representing the mechanical properties of the materials at different layers across the thickness. Additionally, Ellobody and Young (2005) and Young and Ellobody (2005) claimed that the inclusion of the longitudinal membrane stress in the column models was unnecessary since the prediction results were nearly not varied after excluding the stress from the column models. ⅳ) 4th strategy. This strategy is different from the first three strategies. In this strategy, residual strains (including elastic and plastic) in a CFS member due to the cold working are predicted first. Then, the residual strains are introduced into the column model for nonlinear analysis. In this case, the mechanical properties of the material at the flat regions and corners of the cross-section are represented by the stress-strain curves tested from material coupons taken from those areas; see examples from Bonada et al. (2015). However, when elastic and equivalent plastic residual strains are introduced into the column model, mechanical properties of the material throughout the cross-section are represented by a single stress-strain curve tested from a material coupon taken from the raw steel sheet; see examples from Quach et al. (2010). As for the latter, the increase in material strength due to strain ageing is ignored. The 4th strategy has been demonstrated by Quach et al. (2010), Pastor et al. (2013), and Bonada et al. (2015) to provide accurate predictions of axial strengths for CFS columns experiencing various buckling modes; see Table 2.3. Note that the column models established in the above three studies were slightly different. First, the column models generated by Quach et al. (2010) included residual strains accumulated from the initial coiling till the final brake pressing, while the column models 41 generated by Pastor et al. (2013) and Bonada et al. (2015) included only the residual strains generated in the roll-forming. Second, the initial geometric imperfections of the column models generated by Quach et al. (2010) consisted of three components: the out-of-flatness of plates, the overall non-straightness of the member, and the elastic spring-back after the introduction of the residual strains. However, the initial geometric imperfections of the column models generated by Pastor et al. (2013) and Bonada et al. (2015) included only the elastic spring-back after introducing the residual strains. 42 Table 2.3 Performance of different strategies for considering residual stresses in column models Strategy a Section (mm) No. of columns BC Failure mode b Pu/Py c IGI Absolute errors of Pu (%) d L-D curve Stub Non-stub min max mean Mode (magnitude) mean max 1st (Key and Hancock 1993) SHS76×76×2.0 SHS152×152×4.9 SHS203×203×6.3 SHS254×254×6.3 4 - B-F L 0.98 1.10 1.03 L f (w/1000) 2.6 6.7 Ⅰ SHS203×203×6.3 - 3 P-P F 0.44 0.96 0.72 F (L/5000) - 6.9 Ⅱ 2nd e (Sivakumaran and Abdel-Rahman 1998) LC203×41.6×13.05×1.88 LC101.5×41.6×13.2×1.27 10 - F-F L 0.60 0.86 0.76 L f (0.0725w�Fy E� ) 2.9 6.7 Ⅰ 3rd (Ellobody and Young 2005) PA70×70×1.5 1 - F-F L 0.36 L (measured) 4.5 Ⅱ - 6 F-F F + TF 0.10 0.28 0.17 L + G (measured) 5.7 12.6 Ⅱ or Ⅲ (Young and Ellobody 2005) LA70×70×15×1.9 1 - F-F L 0.81 L (measured) 6.4 Ⅰ - 7 F-F TF or F + TF 0.25 0.64 0.40 L or L + G (measured) 4.1 7.7 Ⅰ 4th (Quach et al. 2010) g LC76.23×41.45×15.37×1.8 1 - B-F L 1.02 L (0.2t) 3.2 - - 2 P-P-W F 0.55 0.88 0.72 L + F (0.2t + L/1000) 12.0 22.4 - (Pastor et al. 2013) RA4 1 - P-P L 0.95 i SB (0.02 mm) 4.8 - - 1 L + D 0.84 SB (0.05 mm) 0.4 - - 4 D 0.68 0.74 0.70 SB (0.17 to 0.4 mm) 1.6 2.1 - - 1 AD + TF 0.60 SB (0.42 mm) 4.3 - h (Bonada et al. 2015) RA4 1 - L 0.90 SB (0.09 mm) 0.1 Ⅱ - 4 D 0.67 0.81 0.74 SB (0.11 to 0.78 mm) 3.1 5.7 - - 1 D + AD + TF 0.63 SB (0.97 mm) 0.79 - - 1 AD + TF 0.59 SB (1.1 mm) 0.35 - Note: a: SHS = square hollow section, LC = lipped-C, PA = plain angle, LA = lipped angle, RA4 = web-stiffened rack with additional lips; b: Py equals Pyn for perforated columns; c: IGI = initial geometric imperfection; d: Ⅰ = accurate, Ⅱ = close, Ⅲ = acceptable; e: the columns included two solid and eight perforated columns; f: w = web width; g: the section is the PBC14 section, h: the columns had holes, i: SB = spring back.43 2.4.4.4 Effects of Residual Stresses on the Axial Strengths of CFS Columns The literature review showed that only limited research works like the Abdel-Rahman and Sivakumaran (1997), Dat (1980), Key and Hancock (1993), Quach et al. (2010), and Pastor et al. (2013) have been devoted to studying the effects of residual stresses and/or increased material strengths at corners on the axial strengths of CFS columns. These studies covered only three cross-section shapes and a few different slenderness values. The results of these studies are summarized in Table 2.4 and Table 2.5. To be specific, Table 2.4 summarizes the variations in the axial strengths of CFS columns after the residual stresses are released. Table 2.5 summarizes the variations in the axial strengths of CFS columns after the residual stresses and the increases in yield strengths at corners being excluded from column models. Table 2.4 shows that the axial strengths increased by 2.4% to 8.8% for the columns experiencing L buckling under compression and by 5.7% to 15.8% for the columns experiencing F buckling under compression. Note that the columns in the table all exhibit high ratios of Pu/Py; hence, their axial strengths are expected to be sensitive to residual stresses. Table 2.5 shows that the combined effects of residual stresses and the increases in yield strengths at corners on the axial strengths of the CFS columns are mostly detrimental and can be significant regardless of the buckling mode. Besides, the combined effects on the axial strengths of the CFS columns tend to be more prominent in the cases of D or G buckling than in the cases of L buckling. On the other hand, Quach et al. (2010) unveiled that the axial strengths of two identical columns can be remarkably different when the columns were fabricated from raw steel sheets having very different initial coiling diameters. To be specific, Quach et al. (2010) demonstrated that the combined effects on the axial strengths of the PBC14-section columns were negligible when the raw steel sheet had the initial coiling diameter of 611t; see Table 2.5. However, 44 when the initial coiling diameter of the raw steel sheet was reduced to 200t, the combined effects on the axial strengths of the columns became prominent, where the variations in the axial strengths of the columns can be as large as 19.0% after removing the residual stresses and the increases in yield strengths at corners from the column models. The mechanism behind this phenomenon is that the residual strains at the flat regions of the PBC14 section were negligible when the initial coiling diameter of the raw steel sheet was 611t, while they became significant when the initial coiling diameter of the raw steel sheet was reduced to 200t (Quach et al. 2006). 45 Table 2.4 Variations of axial strengths of CFS columns after removing the residual stresses Reference Specimen Section BC t a H/t b B/t c D/t d L/W Pu/Py Strategy IGI Fu (MPa) Diff. (mm) Mode (magnitude) with RS without RS (%) Local buckling (Dat 1980) BP PBC14 LC B-F 1.91 33.2 16.0 6.6 4.2 0.96 - 266.1 282.3 6.1 PBC13 2.30 27.6 13.3 5.5 2.4 1.17 - 304.7 323.4 6.1 PBC13 2.30 27.6 13.3 5.5 4.2 1.15 e - 298.5 308.2 3.2 RF RFC14 1.84 34.5 16.6 6.9 4.2 1.00 - 295.8 315.1 6.5 RFC13 2.27 28.0 13.4 5.6 4.2 1.17 - 304.7 331.6 8.8 (Key and Hancock 1993) RF 76×76×2.0 SHS 2.00 36.1 - - f 0.98 1st L (w/1000) 419.5 443.3 5.7 152×152×4.9 4.90 29.1 - - 1.10 1st L (w/1000) 426.2 438.7 2.9 203×203×6.3 6.30 30.3 - - 1.05 1st L (w/1000) 412.5 422.3 2.4 254×254×6.3 6.30 38.3 - - 0.99 1st L (w/1000) 393.4 419.1 6.5 Flexural buckling (Dat 1980) BP PBC13-C4 LC P-P-W 2.30 27.6 13.3 5.5 17.7 0.89 2nd F (L/2500) 232.4 245.6 5.7 (Key and Hancock 1993) RF 203×203×6.3 SHS P-P 6.30 30.3 - - 25.8 0.77 1st F (L/5000) 278.9 323.0 15.8 Note: a: H = the width of the flat region at the web; b: B = the width of the flat region at the flange; c: D = the width of the flat region at the lip; d: W = the maximum overall width of the cross-section; e: the Fu associated with the case of without residual stresses was obtained through testing annealed columns; f: the specimens were stub columns, where the exact lengths were not given in the reference. 46 Table 2.5 Predicted axial strengths of CFS columns with and without considering the cold-forming effects on the material Reference Specimen Section BC t H/t B/t D/t L/W Pu/Py Strategy IGI Pu (kN) Diff. (mm) mode (magnitude) a with CFE a without CFE (%) L buckling (Abdel-Rahman and Sivakumaran 1997) RF A-D LC F-F 1.88 102.0 16.1 3.9 1.8 0.60 2nd L (0.0725𝑤𝑤�𝐹𝐹𝑦𝑦 𝐸𝐸� ) 125.3 127.4 1.7 (Quach et al. 2010) BP PBC14-254 LC B-F 1.80 35.9 16.6 5.3 3.3 1.02 4th b (Dia. = 611t) L (0.2t) 82.6 77.8 -5.8 4th b (Dia. = 200t) L (0.2t) 86.5 77.8 -10.1 (Pastor et al. 2013) RF S4-250 RA4 P-P 1.83 38.1 13.3 3.6 3.1 0.95 4th c (0.02 mm) 169.2 174.2 2.9 D buckling (Pastor et al. 2013) RF S4-600 RA4 P-P 1.83 38.1 13.3 3.6 7.4 0.74 4th c (0.17 mm) 141.1 167.8 18.9 S4-800 9.9 0.70 c (0.26 mm) 134.4 167.2 24.4 S4-1000 12.4 0.69 c (0.34 mm) 129.9 157.9 21.5 S4-1200 14.9 0.68 c (0.4 mm) 124.9 153.0 22.5 G or L + G buckling (Quach et al. 2010) BP PBC14-2000 LC F-F 1.80 35.9 16.6 5.3 26.2 - 4th b (Dia. = 611t) L + G (0.2t + L/1000) 61.6 61.3 -0.5 PBC14-3000 F-F 39.4 - 40.5 40.5 0.0 PBC14-500 P-P-W 6.6 - 78.4 77.6 -1.0 PBC14-1000 P-P-W 13.1 0.88 68.2 66.7 -2.2 PBC14-1500 P-P-W 19.7 - 51.0 50.8 -0.4 PBC14-2000 P-P-W 26.2 0.55 33.5 33.2 -0.9 PBC14-2000 F-F 26.2 - 4th b (Dia. = 200t) 52.9 61.3 15.9 PBC14-3000 F-F 39.4 - 34.2 40.5 18.4 PBC14-500 P-P-W 6.6 - 80.9 77.6 -4.1 PBC14-1000 P-P-W 13.1 0.88 63.7 66.7 4.7 PBC14-1500 P-P-W 19.7 - 43.1 50.8 17.9 PBC14-2000 P-P-W 26.2 0.55 27.9 33.2 19.0 (Pastor et al. 2013) RF S4-1500 RA4 P-P 1.83 38.1 13.3 3.6 19.3 0.60 4th c (0.42 mm) 116.3 142.8 22.8 Note: a: CFE = cold-forming effects; b: Dia. = initial coiling diameter of raw steel sheet; c: the initial geometric imperfection is the deformed shape of the column model obtained from the initial equilibrium analysis after the residual strains have been introduced. 47 2.4.5 Initial Geometric Imperfection During the fabrication, shipping, and storage, Initial Geometrical Imperfections (IGIs) are formed in CFS members (Zeinoddini and Schafer 2012). The IGI is defined as the deviation of a member’s profile from the designed or nominal geometry (Schafer and Peköz 1998). It can be classified into cross-section imperfection and global imperfection (Schafer and Peköz 1998; Zeinoddini and Schafer 2012). The cross-section imperfection includes the errors in dimensions, plate dents, concave plates, and undulations, which can be further categorized into local and distortional imperfections. The global imperfection refers to cross-section twisting (torsion), overall bending, or the combination of the two. Mode and magnitude are two terms to describe the IGI. The IGI may affect a column’s buckling behaviour under compression, especially when the IGI mode is periodic and has a half-wavelength close to the column’s buckling half-wavelength. For instance, Freitas et al. (2005) reported that the IGI might decrease the axial strength of the CFS column. Besides, influences of the IGI on the failure modes of CFS columns were observed by Roure et al. (2011). Moreover, Pastor et al. (2013) emphasized that both the axial strength and post-buckling behaviour of the CFS column are sensitive to the IGI. To date, numerical simulation seems the only feasible way to study the influences of the IGI on the structural response of the CFS column under compression. The eigenmode mapping technique is commonly used by researchers to model the IGI in column models (Bonada et al. 2012; Chou et al. 2000; Crisan et al. 2012; Pastor et al. 2014; Pastor et al. 2009; Roure et al. 2011; Sivakumaran and Abdel-Rahman 1998; Yang and Hancock 2006). The actual IGI of a CFS column is hard to be measured and restored in the column model. So, the eigenmode mapping technique is to affine the IGI mode using one of the column’s eigenmodes or a mixture of multiple eigenmodes. A single eigenmode is used when the column is expected to fail in L, D, or G 48 buckling, where the eigenmode is the one like the failure mode of the column. A mixture of multiple eigenmodes is used when the column is expected to fail in interactive buckling, and the eigenmodes are like the components (pure modes) of the interactive buckling mode. The IGI magnitude can be a value suggested by a literature or design specification, determined from experimental measurement, or determined by trial-and-error when experimental results are available for calibration (Bonada et al. 2012; Pastor et al. 2014; Schafer and Peköz 1998; Zeinoddini and Schafer 2012). Based on the literature review, an overall picture of the effects of the IGI on the axial strength of the CFS column in the cases of L, D, and G buckling are summarized here separately. 2.4.5.1 Local Buckling Van der Neut (1969) numerically demonstrated that the axial strength of the CFS column shows sensitivity to the IGI magnitude (in L buckling mode) when the column shows a Pcre/Pcrl value close to unity. The unfavorable effect of the IGI on the resistance exists up to the Pcre/Pcrl = 2. While in practice, stub column compression tests are generally conducted to determine the L buckling strengths of CFS columns so that the Pcre values of the columns are usually significantly above the Pcrl values where the Pcre/Pcrl > 2.0 is common, e.g., (Chou et al. 2000; Pastor et al. 2014; Pastor et al. 2009). 2.4.5.2 Distortional Buckling D buckling strengths of CFS columns exhibit sensitivities to both the mode and magnitude of the IGI in the following ways: ⅰ) A CFS column features post-D buckling asymmetry when buckled with an odd number of halfwaves, where the buckling mode presenting inwardly rotated flanges at the mid-length has post-buckling strength reserve either greater or lower than that of the buckling mode showing 49 outwardly rotated flanges at the mid-length provided that the two cases have the same IGIs except for the opposite IGI magnitudes. For instance, when a single D buckling halfwave is presented, the lipped-C section column buckled in an outward distortional (OD) mode has the post-D buckling strength reserve lower than that of the same column but buckled in an inward distortional (ID) mode (Dinis et al. 2007). Post-D buckling asymmetry was also observed in columns with cross-sections like the web-flange-stiffened lipped-C section (Silvestre and Camotim 2006), hat section (Dinis et al.), rack section (Dinis and Camotim 2015; Dinis and Camotim 2004), and web-stiffened lipped-C section (Martins et al. 2016), but they exhibited a trend opposite to the one observed in the lipped-C section columns. Note that the post-D buckling asymmetry is not relevant to a column buckled with an even number of halfwaves, and the significance of the post-D buckling asymmetry decreases with the increase in the number of halfwaves (Silvestre and Camotim 2006). ⅱ) Moen and Schafer (2011) demonstrated that the D buckling strength of the lipped-C section column under compression exhibits sensitivity to the IGI magnitude only in the low-to-moderate slenderness range (e.g., λd ≤ 1.29). 2.4.5.3 Global Buckling On one hand, Moen and Schafer (2011) demonstrated that lipped-C section columns with simply-simply ends exhibit post-buckling asymmetry when failed in F buckling under compression. On the other hand, the G buckling strength exhibits sensitivity to the IGI magnitude in the following cases: lipped-C section columns with simply-simply ends having TF buckling and λc ≤ 1.70 (Moen and Schafer 2011), plain-C section columns with F-F ends having TF buckling and λc ≤ 1.50 (Dinis et al. 2019), plain-C section columns with F-F ends having F buckling and λc ≤ 2.00 (Dinis et al. 2019), and lipped-C section columns with F-F ends having TF buckling and λc 50 ≤ 1.50 (Gunalan and Mahendran 2013). Note that the slenderness ranges shown in the above cases are all low-to-moderate. 2.4.6 Initial Loading Eccentricity When a CFS column is tested with pinned ends under compression and is geometrically aligned, initial loading eccentricity is highly likely to involve and reduce the axial strength of the column. Weng and Pekoz (1990) have pointed out that the geometric alignment of CFS columns with pinned ends in compression tests features uncertainty, and hence the experimental results are likely to show large scatter. Wang et al. (2016) observed that the geometric alignment rarely achieved good alignment for the pin-ended CFS columns in compression tests, where the loading eccentricities showed a maximum of 2.2-mm. Initial loading eccentricities could be minimized by aligning the columns mechanically (e.g., using strain gauges). 2.5 Compression Tests on CFS Columns Studies reported in the past six decades concerning compression tests of CFS columns with open and singly symmetric cross-sections were reviewed thoroughly, and experimental data were collected from them. In total, experimental data (e.g., axial strengths and failure modes) of 453 solid and 219 perforated CFS columns were collected from 40 different research articles. The collected column specimens exhibit 19 different cross-section shapes. The data collection, overall descriptions of the column specimens, and the processing of the experimental data are presented in the following chapter. 2.6 Summary This chapter reported an in-depth literature review that covers numerous different topics: ⅰ) the failure modes of CFS columns under compression and the relevant experimental studies in the past; ⅱ) the issues and past assessment associated with the buckling strength curves in the 51 DSM, where the DSM is incorporated in the North American design specification of AISI S100 (2016) for the design of CFS members under compression; ⅲ) the calculation of elastic buckling loads of CFS columns under compression; ⅳ) the factors and their significance in affecting the axial strengths of CFS columns, where the factors include the cross-section geometry, end support condition, cold-forming effects on the material (i.e., the increase in material strength and residual stresses), initial geometric imperfection, and initial loading eccentricity. Based on the literature review, the following conclusions are made: ⅰ) Although the DSM has existed for quite some time, the accuracy of its buckling strength curves has not been fully explored. An overall picture of the accuracy of any buckling strength curve in the DSM is still missing. ⅱ) The AISI S100 (2016) allows taking advantage of the increase in material yield strength due to cold working to design CFS columns under local buckling and having λl values ≤ 0.776. However, studies concerning using the full-section tensile yield strength as a design parameter of the DSM have not been seen from the literature. ⅲ) Among the factors affecting the axial strengths of CFS columns, the increase in material strength due to cold forming is a feasible one that can now be parameterized and considered in estimating the axial strengths of CFS columns. ⅳ) Researchers have demonstrated that different buckling strength curves or design expressions are necessary to address the D buckling of solid CFS columns with different end support conditions (i.e., fixed-fixed, simply-simply, simply-fixed, or fixed-free ends). Similar studies on perforated CFS columns have not been seen from the literature. ⅴ) Researchers have demonstrated that different buckling strength curves or design expressions are necessary to address the G buckling of solid CFS columns in different cases (e.g., 52 end support condition, buckling mode). Similar studies on perforated CFS columns have not been seen from the literature. 53 Chapter 3: Experimental Database of CFS Columns and the Modification of Karren’s Model and Expression 3.1 General This chapter presents the establishment of three different experimental databases and the modification of Karren’s model and expression: ⅰ) First experimental database: Experimental data associated with compression tests of solid and perforated CFS members were collected from the literature as much as possible. Then, the design parameters (e.g., cross-section areas and elastic buckling loads) of the collected column specimens were determined and assembled with the experimental data in a database. This database was employed in Chapters 4 and 5 as a source to assess the buckling strength curves in the DSM (AISI_S100 2016). ⅱ) Second experimental database: Experimental data related to material property tests of CFS corners with right and non-right angles were collected from the literature as much as possible. The yield strengths, geometries (r and t), and the yield and ultimate tensile strengths of the virgin material of the corners were assembled in a database, which was used in this chapter as a source to evaluate the performance of the model proposed by Karren (1965) for predicting the tensile yield strengths of CFS corners. Afterward, the model was modified to improve the accuracy and extend the application limit. ⅲ) Third experimental database: Experimental data associated with full-section tensile tests of CFS members were collected from the literature as much as possible. The full-section tensile yield strengths, the predicted tensile yield strengths of corners, and the areas of corners and cross-sections of the collected members were assembled in a database, which was employed in 54 this chapter as a source to simplify the expression proposed by Karren (1965) for estimating the full-section tensile yield strengths of CFS members. 3.2 Experimental Database of Cold-Formed Steel Columns under Compression An extensive experimental database containing the compression test results and design parameters of 453 solid and 228 perforated CFS column specimens was established. 3.2.1 Collection of Experimental Data Studies reported in the past six decades related to compression tests of CFS columns with open and singly symmetric cross-sections were reviewed to collect the experimental data. The articles selected in this study, with two exceptions of Zhao et al. (2017) and Bandula Heva and Mahendran (2013), meet two requirements. First, column specimens failed in L, D, and/or G buckling under compression are reported. Second, experimental data of column specimens given in an article are complete or become complete after receiving supplemental data from corresponding researchers. Complete experimental data of a column specimen includes the measured (or nominal) cross-section, radii of corners, length, the geometry of holes, tested material properties (Fy and Fu), boundary conditions, axial strength, and failure mode. Zhao et al. (2017) partially omitted the geometric information of the column specimens due to confidentiality issues, but the elastic buckling loads of the column specimens were given in the article. Also, the information provided in the article enabled the author to assess the validity of the elastic buckling loads of the column specimens. Bandula Heva and Mahendran (2013) did not show the radii of the corners of the column specimens. Since experimental data about the G buckling of CFS columns under compression were rare, the article was included in this study as it contains the results of compression tests of six lipped-C section columns that failed in TF buckling. 55 In total, experimental data associated with the compression tests of 453 solid and 219 perforated CFS column specimens were collected from 40 different articles. Besides, experimental data of nine stub columns (perforated) tested in this study (introduced in Chapter 7) were included as well; hence, the number of perforated CFS column specimens becomes 228. The column specimens exhibit 21 different cross-section shapes that are shown in Figure 3.1. Overall descriptions of the column specimens and their sources (references) are given in Table 3.1 and Table 3.2. In the tables, H, B, and D refer to the maximum widths of stiffened plates, edge-stiffened plates, and unstiffened plates, respectively, disregarding intermediate stiffener(s). For illustration, the H, B, and D of the RA1 section are shown in Figure 3.1. W is the greatest overall width of the cross-section, and t is the wall thickness. All the collected column specimens, except the nine column specimens tested by Yang and Hancock (2004b) that failed in L buckling, fall within the geometry and material strength limits imposed by AISI S100 (2016) for the application of the DSM. The nine column specimens have B/t ratios around 220.0 and hence fall outside the geometry limit of B/t ≤ 160 (Yang and Hancock 2004b). They were included in this study since their axial strengths can be accurately estimated by the Pnl-curve. Besides, the column specimens collected from Yang and Hancock (2004b), Yap and Hancock (2006), Yap and Hancock (2008), and Trouncer and Rasmussen (2012) show tested material yield strengths around 700 MPa, but their nominal material yield strengths fall within the material strength limit of Fy ≤ 655 MPa. 56 Figure 3.1 Cross-section shapes of the collected column specimens: (a) lipped-C series; (b) rack series; (c) track series; (d) hat series. Various failure modes (in terms of the number of counts) experienced by the collected column specimens are shown in Figure 3.2 against different cross-section shapes. The figure unveils a shortage of experimental data about the compression tests of perforated CFS columns. It results from two facts: ⅰ) past studies focusing on compression tests of perforated CFS columns were not many, and ⅱ) although there were a few more compression test results of perforated CFS columns, the experimental data of such column specimens given in the literature are incomplete because of confidentiality issues. 57 Figure 3.2 The number of failure modes versus different cross-section shapes: (a) solid column specimens; (b) perforated column specimens. 58 Table 3.1 Experimental database of compression tests of solid column specimens Length (mm) H/t B/t D/t L/W Fy (MPa) λ Pu/Py Reference BC Section Count min max min max min max min max min max min max min max min max Local buckling (Dat 1980) B-F LC1 5 178 305 27.6 34.6 13.2 16.6 5.5 6.9 2.4 4.2 260.4 296.9 0.5 0.7 1.0 1.2 (Mulligan 1983) B-F LC1 22 304 971 60.6 347.8 27.7 94.0 4.9 19.6 1.7 5.7 198.8 412.3 1.0 4.9 0.3 1.0 (Sivakumaran 1987) B-F LC1 6 200 265 51.6 112.2 19.8 26.0 4.9 6.8 1.7 2.2 262.2 340.6 1.0 1.8 0.6 0.9 (Hancock and Lau 1988) B-F LC1, RA1, RA2, HA1 12 300 300 32.9 51.3 5.3 43.0 1.7 15.4 3.2 3.5 220.2 488.6 0.6 1.2 0.7 1.0 (Miller 1990) B-F LC1 20 257 493 38.6 167.2 12.8 32.8 3.1 5.3 2.8 3.5 302.7 366.1 0.8 3.0 0.4 1.1 (Weng and Pekoz 1990) B-F LC1 13 254 432 21.9 37.6 9.2 16.6 3.6 7.3 3.2 3.8 221.0 379.8 0.4 0.7 1.0 1.1 (Young and Rasmussen 1998) B-F LC1 2 280 300 63.0 63.2 22.0 30.2 6.4 6.9 2.9 3.1 515.0 550.0 1.5 1.5 0.7 0.7 (Yap and Hancock 2006; Yap and Hancock 2008) B-F LC9 1 110 110 159.7 159.7 73.1 73.1 13.8 13.8 1.3 1.3 690.0 690.0 2.3 2.3 0.4 0.4 (Abdel-Rahman 1997) F-F LC1 4 215 365 73.9 102.0 16.1 26.8 3.9 7.4 1.8 2.1 319.0 385.2 1.3 1.9 0.6 0.8 (Yan and Young 2002) F-F TR1 5 498 501 77.8 100.0 40.0 79.6 7.9 9.5 3.2 3.3 482.0 522.0 1.7 2.3 0.5 0.7 (Yang and Hancock 2004b) F-F LC8 9 340 341 290.4 295.1 215.9 218.4 27.5 28.8 2.8 2.8 690.0 690.0 3.2 3.2 0.3 0.4 (Zhang et al. 2007) F-F LC1, LC2, LC3 6 499 501 50.8 51.2 23.4 24.3 4.2 5.7 3.1 3.1 320.7 342.4 0.9 1.0 0.8 1.0 (Yap and Hancock 2006; Yap and Hancock 2008) F-F LC9 2 90 90 159.1 161.1 73.1 73.1 13.5 13.6 1.0 1.0 690.0 690.0 2.2 2.2 0.5 0.5 (Chen et al. 2010) F-F LC7 2 400 401 70.0 70.8 69.1 69.9 4.7 4.7 2.6 2.7 401.0 401.0 1.4 1.4 0.9 0.9 (Yap and Hancock 2008b) F-F LC4 3 330 330 108.1 108.3 56.8 56.8 9.4 9.4 2.8 2.8 634.0 634.0 1.6 1.6 0.4 0.5 (Wang et al. 2013) F-F TR1, TR2 4 599 600 72.4 72.6 32.7 51.9 4.2 4.3 3.2 3.2 262.0 262.0 0.9 1.2 0.8 0.9 (Chen et al. 2019) F-F LC1, RA1, TR1, TR4 5 479 480 78.7 101.4 48.8 62.6 6.7 9.7 2.9 3.0 496.0 521.0 1.1 1.9 0.5 0.8 (Pastor et al. 2013) P-P RA4 3 250 250 38.1 38.1 13.3 13.3 3.6 3.6 3.1 3.1 423.0 423.0 0.7 0.7 0.9 1.0 (Wang et al. 2016) P-P-W TR1, TR2 4 700 700 108.6 110.8 41.0 41.7 5.2 5.6 3.0 3.1 385.5 385.5 1.0 2.2 0.5 0.8 (Xiang et al. 2020) P-P-W TR1 6 600 1000 97.7 150.2 33.9 42.9 5.9 8.0 2.0 3.3 349.0 349.0 1.8 2.7 0.4 0.5 (Shi et al. 2010) P-P-W LC1 1 700 700 86.9 86.9 86.0 86.0 10.3 10.3 5.1 5.1 227.0 227.0 1.6 1.6 0.5 0.5 59 Table 3.1 (Continued) Length (mm) H/t B/t D/t L/W Fy (MPa) λ Pu/Py Reference BC Section Count min max min max min max min max min max min max min max min max Distortional buckling (Hancock and Lau 1988) B-F LC1, RA1, RA2, HA1 29 700 1900 32.9 51.3 5.3 43.0 1.7 15.4 7.6 22.3 220.2 488.6 0.6 1.2 0.6 1.0 (Young and Hancock 2003) B-F LC1, LC2, LC3 42 1478 1504 38.6 64.1 15.8 65.7 1.0 6.6 14.6 15.2 415.0 520.0 0.8 2.8 0.3 1.0 (Kumar and Kalyanaraman 2014) B-F LC8 14 503 1220 69.3 90.1 77.8 91.9 0.8 3.8 3.9 10.2 284.0 284.0 1.2 1.9 0.6 0.7 (Kwon and Hancock 1992; Kwon and Hancock 1992b) F-F LC1, LC4 5 780 1780 103.9 105.2 77.3 77.8 4.2 5.1 6.5 14.9 590.3 590.3 2.6 3.2 0.2 0.3 (Yap and Hancock 2008b) F-F LC4 2 680 680 108.4 108.6 56.7 56.7 9.3 9.3 5.7 5.7 634.0 634.0 1.6 1.6 0.4 0.4 (Yao et al. 2011) F-F LC4 8 698 1398 59.8 73.3 21.2 21.3 4.6 4.8 3.0 7.3 363.0 363.0 0.8 1.0 0.8 1.0 (Wang et al. 2013) F-F TR2, TR3 4 997 1000 74.7 74.8 33.2 54.1 4.2 4.5 5.3 5.3 183.0 183.0 0.6 0.6 1.0 1.0 (Young et al. 2013) F-F LC1 7 616 1757 37.8 52.8 15.2 46.4 5.8 10.3 5.9 15.3 335.6 525.9 0.7 1.0 0.7 0.9 (He et al. 2014) F-F LC4 1 1501 1501 107.1 107.1 78.7 78.7 7.6 7.6 9.1 9.1 435.0 435.0 1.9 1.9 0.3 0.3 (He 2014) F-F LC4, LC8 18 998 2002 106.1 108.2 52.5 79.2 8.9 11.8 6.1 12.3 435.0 435.0 1.3 1.7 0.4 0.5 (Bonada et al. 2012; Pastor et al. 2013) P-P RA4 11 600 1200 38.1 38.1 13.3 13.3 3.6 3.6 7.4 14.9 423.0 423.0 0.8 0.8 0.7 0.7 (Crisan 2011; Crisan et al. 2012) P-P RA4 31 300 2400 31.2 34.3 6.3 7.8 1.5 2.0 3.3 19.7 466.8 466.8 0.5 1.1 0.6 1.1 (Wang et al. 2016) P-P-W TR2, TR4 4 1250 1800 109.3 110.6 40.7 41.4 5.0 5.4 5.5 7.9 385.5 385.5 1.1 1.2 0.6 0.8 (Zhao et al. 2017) P-P RA4, RA5, RA7 6 250 1875 37.6 56.1 7.8 9.7 1.2 3.9 3.3 15.6 297.0 342.2 0.5 1.0 0.7 0.9 (Chen et al. 2017) P-P-W LC4 3 1500 2001 104.2 107.1 77.0 78.4 8.1 8.3 9.2 12.4 434.0 434.0 1.8 2.0 0.2 0.3 (Yi et al. 2020) P-P-W TR1 2 1198 1199 70.8 71.7 70.8 71.5 2.7 3.0 8.0 8.0 349.0 349.0 1.8 1.9 0.5 0.5 Global buckling (Hancock and Lau 1988) B-F LC1, RA1, RA2, HA1 27 1100 1900 32.9 50.3 5.3 43.0 1.7 13.0 12.0 22.3 220.2 488.6 0.6 1.4 0.5 0.9 (Young et al. 2013) F-F LC1 2 2099 2102 39.3 47.5 38.7 47.3 9.1 12.8 20.0 20.3 335.6 342.7 0.8 0.8 0.7 0.7 (Bandula Heva and Mahendran 2013) F-F LC1 6 1686 2766 36.4 55.8 23.6 34.7 6.5 7.4 22.5 50.5 271.0 615.0 0.8 2.7 0.2 0.8 (Dat 1980) P-P-W LC1 36 457 2464 27.6 34.6 13.2 16.6 5.5 6.9 6.3 33.6 260.4 296.9 0.5 1.9 0.3 1.2 (Miller 1990) P-P-W, P-P-S LC1 12 1211 2443 42.3 158.8 14.1 30.7 3.7 5.0 13.2 26.5 302.7 358.5 1.3 2.8 0.1 0.5 (Weng and Pekoz 1990) P-P-W LC1 28 643 2756 21.9 38.6 9.2 17.0 3.6 7.3 8.1 25.5 210.9 379.8 0.6 1.6 0.3 1.0 (Bonada et al. 2012; Pastor et al. 2014) P-P RA4 8 1500 2600 38.1 38.1 13.3 13.3 3.6 3.6 18.6 32.2 423.0 423.0 1.1 1.8 0.3 0.7 (Crisan 2011; Crisan et al. 2012) P-P RA4 9 1090 1650 31.2 31.2 7.8 7.8 2.0 2.0 11.9 17.9 466.8 466.8 0.8 1.0 0.6 0.9 (Zhao et al. 2017) P-P RA6 2 2000 2700 45.9 45.9 7.2 7.2 1.8 1.8 22.2 30.0 339.2 339.2 1.2 1.6 0.4 0.6 (Shi et al. 2010) P-P-W LC1 1 2100 2100 37.0 37.0 24.2 24.2 4.5 4.5 19.7 19.7 377.0 377.0 1.1 1.1 0.6 0.6 60 Table 3.2 Experimental database of compression tests of perforated column specimens Length (mm) H/t B/t D/t L/W Fy (MPa) λ Pu/Py Reference BC Section Count min max min max min max min max min max min max min max min max Local buckling (Loov 1984) B-F LC1 35 195 285 26.6 161.1 15.8 27.8 3.6 6.9 1.3 3.1 268.5 384.0 0.5 2.6 0.5 0.9 (Sivakumaran 1987) B-F LC1 42 200 265 51.6 112.2 19.8 26.0 4.9 6.8 1.7 2.4 262.2 340.6 0.8 1.8 0.5 0.8 (Miller 1990) B-F LC1 20 221 541 43.1 165.7 14.4 32.5 3.9 5.8 1.5 3.6 302.7 366.1 0.8 2.9 0.4 1.0 Chapter 7 B-F LC5, LC6 9 252 255 38.2 53.1 31.0 39.7 8.0 13.8 2.5 3.3 422.5 431.4 0.8 1.0 0.7 0.7 (Abdel-Rahman 1997) F-F LC1 16 250 475 73.9 102.0 16.1 26.8 3.9 7.4 2.1 3.0 319.0 385.2 1.2 2.0 0.5 0.7 (Trouncer and Rasmussen 2012) F-F RA3 2 259 260 74.8 74.8 9.5 9.5 28.1 28.1 2.9 2.9 653.8 653.8 1.9 1.9 0.6 0.6 Distortional buckling (Koen 2008) a B-F RA4 3 373 374 30.0 30.0 10.0 10.0 4.4 4.4 4.2 4.2 550.0 550.0 0.8 0.8 0.7 0.8 (Yao et al. 2011) F-F LC4 8 697 1398 59.7 73.3 21.2 21.2 4.7 4.7 3.0 7.4 363.0 363.0 0.8 1.0 0.6 0.7 (Trouncer and Rasmussen 2012) a F-F RA4 2 997 999 75.5 75.5 24.0 24.0 9.4 9.4 11.3 11.4 667.8 667.8 1.9 1.9 0.4 0.5 (Crisan 2011; Crisan et al. 2012) P-P RA4 48 300 2400 31.2 34.3 6.3 7.8 1.5 2.0 3.3 19.7 466.8 466.8 0.5 1.2 0.5 0.9 (Zhao et al. 2017) P-P RA4, RA5, RA7 10 250 1200 37.6 56.1 7.8 9.7 1.2 3.9 3.3 12.0 299.3 371.3 0.5 1.0 0.5 0.7 Global buckling (Miller 1990) P-P-W, P-P-S LC1 6 1124 2445 43.0 43.8 14.3 14.8 3.7 4.0 12.3 26.5 358.5 358.5 1.3 2.6 0.1 0.5 (Crisan 2011; Crisan et al. 2012) P-P RA4 19 1090 1650 31.2 31.2 7.8 7.8 2.0 2.0 11.9 17.9 466.8 466.8 0.8 1.1 0.5 0.7 (Trouncer and Rasmussen 2012) P-P-S RA4 2 1908 1911 75.5 75.5 24.0 24.0 9.4 9.4 21.7 21.7 667.8 667.8 1.8 1.8 0.3 0.3 (Zhao et al. 2017) P-P RA6 6 1500 2700 41.1 45.9 6.4 7.2 1.5 1.8 16.7 30.0 341.7 373.7 1.1 1.8 0.3 0.5 Note: a = it is deduced here that the warping of column ends was failed to be properly restrained during the compression; hence, the B-F and F-F ends are replaced with simply-simply ends and the F-F ends with warping free, respectively to be used in the eigenbuckling analyses.61 3.2.2 Elastic Buckling Analysis The gross and/or net cross-section areas, areas of corners, and cross-section mid-planes of the collected column specimens were determined using a drafting software, AutoCAD (2020). The cross-section mid-planes of the column specimens were then fed into a finite element analysis program, ANSYS 18.1 (2019), to create finite element models. The finite element models replicated the measured (mostly) or nominal profiles of the column specimens and their experimental end support conditions. Later, eigenbuckling analyses were carried out to calculate the elastic critical buckling loads (Pcrl, Pcrd, and/or Pcre) of the column specimens. The details of the finite element modelling are introduced as follows. Column specimens were modelled using four-noded Shell63 elements. Shell63 element adopts linear shape functions; hence, column models were finely meshed to describe buckling modes accurately. Based on the mesh sensitivity analyses, the cross-sections of column models had meshed with sizes of 2 to 3 mm, and at least two elements had been assigned to each corner. Along the length, the column models were meshed with sizes of 2 to 4 mm. Young’s modulus of 203 GPa and Poisson’s ratio of 0.3 were defined for all the column models. These parameters are recommended by AISI_S100 (2016) to be used for designing CFS structural members. The collected column specimens were axially compressed with different end support conditions. The modelling of the end support conditions is described below. Ⅰ) Bearing-friction (B-F) ends: Column ends are milled flat and made parallel first; then, they bear against hard and flat loading plates/platens. During the test, one of the loading plates/platens maintains its position, whereas the other loading plate/platen moves longitudinally in the form of rigid-body motion. So, the column ends during the compression have ⅰ) in-plane deformations restrained by contact 62 friction, ⅱ) warping, torsion, and global rotations restrained by the rigid loading plates/platens, and ⅲ) uniform axial displacement. The column ends described above are designated as bearing-friction (B-F) ends. Be aware that the B-F ends can resist compression stresses only. Thus, they should not be used in those cases where gaps between the column ends and loading plates/platens are expected to occur at the load ascending phase. Gaps between column ends and loading plates/platens may occur when significant warping occurs at column ends; see the experimental evidence in Moen and Schafer (2008). Generally, the B-F ends are suitable for testing the columns that are expected to fail in L buckling or D buckling showing one or two buckling halfwaves. Since the G buckling of CFS columns is often accompanied by significant warping at column ends, one should be cautious about using B-F ends while testing long columns under compression. The finite element model of a lipped-C section column with B-F ends is shown in Figure 3.3(a). In the model, the nodes at the loading end have the translational degrees of freedom in the cross-section plane (Ux and Uy) restrained, and their translational degrees of freedom in the longitudinal direction (Uz) are coupled. Besides, a uniform compression load is applied to the loading end. The nodes at the other end of the model have all the translational degrees of freedom restrained. Ⅱ) Fixed-fixed ends (F-F ends): The F-F ends are like the B-F ends except that the F-F ends have column ends rigidly connected to the loading plates by welding, glueing, or clamping. The finite element model of a lipped-C section column with F-F ends is shown in Figure 3.3(b). In this model, the loading condition and constraints for creating the B-F ends are presented. Besides, the rotational degrees of freedom are restrained for the nodes at the ends of the model. 63 Figure 3.3 Finite element models of columns under different end support conditions: (a) B-F ends; (b) F-F ends; (c) P-P ends; (d) P-P-W ends; (e) P-P-S ends. Ⅲ) Pinned-pinned ends (P-P ends): Column ends are rigidly connected to end fixtures like steel ball-bearing pads or two-directional hinged joints. The centroids of the end fixtures coincide with the centroid or the effective centroid of the column’s cross-section. The end fixtures can rotate freely about any one 64 of the principal axes of the column. Also, they can prevent warping, torsion about the longitudinal axis, and the rotations of local plates about the bearing edges at the column ends. The finite element model of a column with an RA4-shaped section and the P-P ends is presented in Figure 3.3(c). In this model, the end fixtures are modelled using two solid blocks attached to column ends, where the solid blocks have meshed with Solid185 elements. The solid blocks share the nodes with the column ends at the interfaces. Since the nodes of Solid185 elements only support translational degrees of freedom, a layer of Shell63 elements (t = 10 mm, E = 2.05 GPa, and ν = 0.3) is assigned to the interfaces between the solid block and the column end. By doing so, the rotational degrees of freedom of the nodes at the column ends are coupled with those of the nodes at the end fixture. At the loading end, the translation degree of freedom in the direction perpendicular to the axis-of-symmetry of the cross-section (Ux) is restrained for the two nodes at the ends of the axis-of-symmetry of the cross-section. This is to prevent torsion about the longitudinal axis from occurring at the loading end. Besides, the translational degrees of freedom in the cross-section plane (Ux and Uy) are restrained for the node at the centroid of the cross-section, and a concentric compression force is applied there. The constraints applied to the other end of the model are like those applied to the loading end except that the concentric compression force is omitted and the translational degree of freedom in the longitudinal direction (Uz) is restrained for the node at the centroid of the cross-section. Among the collected column specimens, the column specimens with the P-P ends in the tests were geometrically aligned with respect to either the centroid of the gross cross-section or the cross-section’s effective centroid. Ⅳ) Pinned ends about the weak axis (P-P-W ends): 65 The P-P-W ends are like the P-P ends, except that the end fixtures used in the P-P-W ends can rotate freely only about the weak axis of the column. In practice, the end fixture used for achieving the P-P-W ends is equipped with a single-directional hinge like the knife-edge hinge or rod-pinned hinge. A CFS column’s weak axis is defined herein as the principal axis of the cross-section perpendicular to the axis-of-symmetry. This definition is not always true since the weak axis can be the axis-of-symmetry of the cross-section when a column is showing an abnormally large ratio of flange-to-web widths. A finite element model of a lipped-C section column with the P-P-W ends is shown in Figure 3.3(d). This model is like the one with the P-P ends except with different end support conditions. At the loading end of this model, a concentric compression force is applied to the node at the centroid of the cross-section. Besides, for the nodes lying on the weak axis, ⅰ) the translational degrees of freedom in the cross-section plane (Ux and Uy) are restrained, and ⅱ) the translational degree of freedom in the longitudinal direction (Uz) is coupled. At the other end of the model, the translational degrees of freedom (Ux, Uy, and Uz) of the nodes lying on the weak axis are restrained. Out of the 681 collected column specimens, 98 were tested with the P-P-W ends. Among those, 31 were tested by Miller (1990), Shi et al. (2010), Wang et al. (2016), and Yi et al. (2020) after geometrically aligning them with respect to the weak axis of the gross cross-section, whereas 67 were tested by Dat (1980), Weng and Pekoz (1990), and Chen et al. (2017) after aligning them under a preload where uniform strain across the cross-section was applied. Ⅴ) Pinned ends about the strong axis (P-P-S ends): The P-P-S ends are like the P-P-W ends except that the P-P-S ends can rotate freely only about the strong axis of the column. The strong axis of a CFS column is the symmetry axis of the cross-section. A finite element model of a lipped-C section column with the P-P-S ends is 66 presented in Figure 3.3(e). Experimental data about the compression tests of CFS columns with the P-P-S ends were rare, where only seven column specimens could be found from Miller (1990) and Trouncer and Rasmussen (2012). These column specimens were all aligned geometrically with respect to the symmetry axis of the cross-section. 3.2.3 Validity of the Experimental Database Experimental data (Fy, Fu, axial strengths, and failure modes) of the collected column specimens and their design parameters (Ag, An, Ac, Pcrl, Pcrd, and/or Pcre) were assembled in a database to assess the buckling strength curves in the DSM (AISI_S100 2016) in the following two chapters. The validity of the database is crucial to the success of the work. Thus, the database was inspected rigorously before conducting the assessment. The collection and subsequent processing of the raw experimental data were examined two times at different periods. So, the two examinations were independent. To ensure that the geometries of the column specimens were processed correctly, comparisons of the differences were made between the cross-section areas determined using AutoCAD (2020), ANSYS 18.1 (2019), and from the literature if available. Failure modes of the column specimens reported in the literature were examined carefully, where a few of the failure modes were corrected in this study compared to those reported in the literature. The Pcrl, Pcrd, and/or Pcre values of the column specimens determined using ANSYS were validated against those obtained from GBTUL (Bebiano et al. 2018). Sharp corners were assumed for the column specimens while using GBTUL. The D and G buckling modes determined using ANSYS were visually inspected to ensure that the interferences from L deformation modes were minor or negligible. Besides, the Pcrd and Pcre values determined using ANSYS were compared with those obtained from GBTUL (with L buckling modes excluded from the analyses) to check 67 whether they are close. ANSYS failed to determine the Pcrd values for four column specimens. These are T1.9-MS-4, T2.4-HSS-1, T2.4-HSS-2, and T2.4-HSS-3 specimens tested by Young et al. (2013). The Pcrd values of these column specimens were obtained using GBTUL with L deformation modes excluded from the analyses. GBTUL cannot accommodate pinned ends and holes in column models. Hence, the validation of elastic buckling loads and modes was not performed for perforated column specimens as well as the solid column specimens with pinned ends that failed in D or G buckling. The elastic buckling loads and modes of these columns had been thoroughly examined by performing extensive numerical simulations. 3.3 Experimental Database of Material Property Tests of CFS Corners Experimental data of material property tests of 190 CFS corner specimens with right angles were collected from the literature. The performance of the model proposed by Karren (1965) for predicting the tensile yield strengths of CFS corners was evaluated with the experimental data; then, the model was modified to improve the accuracy and extend the application limit. In addition, experimental data of material property tests of 28 CFS corners with angles different from 90 degrees were collected from the literature. Then, Karren’s model was further modified to extend the application limit to non-right-angle corners. 3.3.1 Karren’s Model The North American specification of AISI S100 (2016) incorporates a model (see Equation 3-1) to predict the tensile yield strengths of right-angle corners in CFS members. The model was developed by Karren (1965). Fyc = kb �tr�m Equation 3-1 68 where b = 1.0 - 1.3n, m = 0.855n + 0.035, k = 2.80Fuv - 1.55Fyv, n = 0.225Fuv/Fyv - 0.120, Fyc = tensile yield strength of corner, t = sheet thickness, r = inside bend radius, k = strength coefficient, n = strain-hardening exponent, Fyv = tensile yield strength of virgin material, Fuv = ultimate tensile strength of virgin material. The virgin steel is defined as the steel sheet in the state before being roll-formed or brake-pressed. For instance, for members that were roll-formed or brake-pressed from flat steel sheets, the flat steel sheets are virgin steel. Coiled steel sheets are virgin steel if roll-formed into members directly (i.e., without uncoiling/flattening). When coiled steel sheets were first uncoiled/flattened and then roll-formed or brake-pressed into members, the uncoiled/flattened steel sheets are considered virgin steel. The virgin material properties can be determined through testing coupons taken from the virgin steel. Alternatively, coupons can be taken from the as-formed member at the middle portion of the flat (e.g., web or flange) furthest from corners. Karren (1965) has experimentally demonstrated that material properties tested from the coupons taken from this portion of the member are very close to those tested from the virgin steel, where the tests involved roll-formed and brake-pressed members with plain-C, hat, and track sections. It should be noted that ⅰ) this procedure should not be applied to members having compact sections (i.e., the greatest width of the flats is comparatively small to the inner radial of adjacent corners), ⅱ) multiple coupons should be tested, and the results should be averaged if they only show small variations, and ⅲ) the results from the coupon showing the lowest yield strength should be used if the coupons show prominent variations in their material properties. When Karren’s model was being developed, it was validated against the tested tensile yield strengths of right-angle corners featuring r/t ratios ≤ 6.0 (Karren 1965). Karren (1965) emphasized that the model should not be applied 69 beyond this limit unless further validated. Upon reviewing Karren (1965)’s work, the author found that Karren’s model may still be subjected to another two limitations: ⅰ) The corners tested by Karren (1965) featured Fyv < 300 MPa. However, most of the column specimens collected in Section 3.2 (508 out of 681) show Fyv values ≥ 300 MPa, and the greatest Fyv value is around 700 MPa. Steel material featuring a high Fyv value tends to show a small ratio of Fuv/Fyv. The Fuv/Fyv ratio is a key input parameter in Karren’s model. Hence, the performance of Karren’s model could be sensitive to the Fyv value. ⅱ) Among the corners tested by Karren (1965), 88% were cold-formed from virgin steel sheets presenting sharp yielding, and 12% were cold-formed from virgin steel sheets showing gradual yielding. The increase in yield strength due to cold forming comes from two sources, i.e., strain hardening and strain ageing (Chajes et al. 1963). Steel material of the gradual yielding type is likely to show a higher degree of strain hardening compared to that of the sharp yielding type when subjected to cold working. Hence, the performance of Karren’s model could be affected by the yielding types of virgin steel sheets. In summary, Karren’s model is subjected to the above-mentioned limitations. Thus, it is necessary to assess Karren’s model before applying it to this study. 3.3.2 Assessment of Karren’s model Overall, experimental data associated with material property tests of 190 CFS corner specimens with right angles were collected from the literature. The validity of the experimental data had been examined. An overall description of the collected corner specimens is shown in Table 3.3, where the supplemental information is presented below: ⅰ) They consist of 140 brake-pressed and 50 roll-formed corner specimens. 70 ⅱ) They were either taken from open-section CFS members or independently fabricated through cold-working processes like those used for manufacturing open-section CFS members. ⅲ) Their thicknesses are within 0.60 mm ≤ t ≤ 6.13 mm, and their tensile yield strengths of the virgin materials are within 225.5 MPa ≤ Fyv ≤ 953.7 MPa. ⅳ) They consist of 50 corner specimens that were cold-formed from virgin steel sheets of the gradual yielding type, 116 corner specimens that were cold-formed from virgin steel sheets of the sharp yielding type, whereas the literature did not provide any information about the yielding type of virgin steel sheets for the remaining 24 corner specimens. ⅴ) 75% of them exhibit Fyc > Fuv. ⅵ) There are 24 corner specimens having r and/or t as nominal dimensions rather than the measured dimensions. In addition, 39 corner specimens have their Fyv and Fuv values obtained from testing material coupons taken from the mid-flats of as-formed members rather than the virgin steel sheets. Overall, 57 corner specimens are accompanied by nonstandard input parameters. Despite this, they were kept in the database since the performance of Karren’s model was not affected even after these corner specimens were included in the database.71 Table 3.3 Experimental database of material property tests of cold-formed, right-angle steel corners Count 1 t (mm) r/t Fyv (MPa) Fuv/Fyv Fyce /Fyv Source 2 Reference Grad. Y Sha. Y Unknown min max min max min max min max min max Fyv & Fuv (Karren 1965) PB 8 73 0 1.50 3.56 0.78 6.33 250.97 295.10 1.25 1.56 1.19 1.92 M F 1 (Uribe 1969) 1 2 3 1.44 3.26 1.10 3.31 225.46 315.78 1.31 1.48 1.34 1.65 M F 1 (Britvec et al. 1970) 0 0 2 2.25 2.25 1.77 1.77 295.10 295.10 1.49 1.49 1.65 1.66 M F 0 (Karren and Gohil 1971) 0 12 0 1.59 1.59 1.16 8.64 323.36 323.36 1.24 1.24 1.05 1.43 M F 1 (Charnvarnichborikarn and Polyzois 1992) 0 0 6 1.48 1.49 2.23 4.36 325.28 356.11 1.44 1.52 1.28 1.62 M F 1 (Ellobody and Young 2005) 0 1 0 1.50 1.50 1.74 1.74 530.00 530.00 1.04 1.04 1.15 1.15 M F 0 (Young and Ellobody 2005) 0 2 0 1.89 1.89 1.37 1.37 495.00 495.00 1.08 1.08 1.17 1.19 M F 0 (Chen and Young 2006) 1 0 0 1.89 1.89 1.72 1.72 524.00 524.00 1.05 1.05 1.09 1.09 M C 0 (Young and Ellobody 2007) 0 0 2 1.90 1.90 1.84 1.84 505.00 505.00 1.06 1.06 1.15 1.17 M F 0 (Kwon et al. 2009) 0 2 0 0.60 0.80 1.00 1.00 627.70 632.80 1.01 1.02 1.06 1.12 N N 0 (Huang and Young 2014) 0 2 0 1.67 1.78 0.83 1.25 495.20 495.20 1.05 1.05 1.09 1.14 M C 1 (Zhang et al. 2019) 18 0 0 4.81 4.98 1.50 1.50 635.00 764.00 1.10 1.16 1.21 1.51 M F 0 (Tran et al. 2019) 3 0 0 4.10 6.01 1.46 1.50 762.33 795.00 1.06 1.06 1.10 1.21 N F 1 (Wang et al. 2020) 0 2 0 6.12 6.13 2.45 2.45 953.71 953.71 1.05 1.05 1.08 1.09 M F 0 (Karren 1965) RF 4 14 0 1.50 3.56 0.89 3.23 250.97 279.24 1.25 1.55 1.28 1.86 M F 1 (Uribe 1969) 0 0 7 1.51 3.71 1.39 5.56 251.66 293.03 1.31 1.60 1.20 1.63 M F 1 (Britvec et al. 1970) 0 0 4 1.49 2.29 2.78 4.81 282.69 380.59 1.35 1.41 1.27 1.47 M F 0 (Popovic et al. 1999) 0 3 0 2.31 4.70 1.00 1.04 352.00 369.00 1.17 1.24 1.54 1.72 N F 1 (Popovic et al. 2001) 0 1 0 2.33 2.33 1.17 1.17 416.00 416.00 1.19 1.19 1.37 1.37 M F 0 (Lam et al. 2006) 4 0 0 1.64 1.65 1.94 1.95 505.50 514.00 1.04 1.06 1.09 1.10 N C 0 (Crisan 2011) 0 2 0 2.60 3.20 0.89 1.10 462.79 478.67 1.12 1.16 1.27 1.27 N N 1 (Kyvelou et al. 2015) 1 0 0 1.51 1.51 2.32 2.32 497.20 497.20 1.12 1.12 1.15 1.15 M F 1 (Bonada et al. 2015) 6 0 0 1.83 1.83 1.97 1.97 418.00 418.00 1.15 1.15 1.14 1.22 N N 1 (Kyvelou et al. 2017) 4 0 0 3.02 3.02 0.99 0.99 483.80 483.80 1.06 1.06 1.16 1.21 N F 1 Note: 1 Grad. Y = gradual yielding, Sha. Y = sharp yielding. 2 M = measured, N = nominal, F = measured at flat, C = measured at the corner, 0 – tensile coupons were taken from the flats of as-formed members away from corners; 1 – tensile coupons were taken from virgin sheet steels.72 Karren’s model was applied to predict the tensile yield strengths of the collected corner specimens. Comparisons were made between the predicted (Fyc) and tested (Fyce ) tensile yield strengths of the corner specimens, and the ratios of Fyc Fyce⁄ are presented against the Fuv/Fyv ratios in Figure 3.4(a). Overall, Karren’s model exhibits good accuracy, where the Fyc Fyce⁄ ratios show the minimum, maximum, mean, and standard deviation of 0.82, 1.18, 1.00, and 0.07, respectively. Karren’s model, however, also presents two deficiencies: ⅰ) the Fyc Fyce⁄ ratios, see Figure 3.4(a), spread within a large range between 0.8 and 1.2, and ⅱ) it overestimated the tensile yield strengths of most of the corner specimens that fall beyond the range of Fuv/Fyv > 1.4. The first deficiency may result from an inherent feature of Karren’s model, where the effect of strain ageing due to cold working on the material yield strength was not considered while proposing the model. This deficiency might be difficult to resolve since the effect of strain ageing on the material yield strength is difficult to quantify. Here, an attempt has been made to resolve the second deficiency in the following section. Figure 3.4 The ratios of 𝐅𝐅𝐅𝐅𝐅𝐅 𝐅𝐅𝐅𝐅𝐅𝐅𝐅𝐅⁄ versus the ratios of Fuv/Fyv of the corner specimens: (a) Karren’s model; (b) the modified Karren’s model. 73 3.3.3 Modification of Karren’s Model To resolve the second deficiency of Karren’s model, the expression for calculating the factor b in Equation 3-1 was replaced with b = 1.11 - 1.912n. Tensile yield strengths of the corner specimens predicted using the modified Karren’s model were compared with those of the tested results, and the ratios of Fyc Fyce⁄ of the corner specimens are plotted against the Fuv/Fyv ratios in Figure 3.4(b). By comparing Figure 3.4(b) to Figure 3.4(a), the improvement in the predictions can be observed. The goodness-of-fit between the tested results and the predictions from the Modified Karren’s model was assessed, where the R2 of 0.969 was obtained, which is slightly larger than the R2 of 0.957 obtained between the tested results and the predictions from Karren’s model. Additionally, the Modified Karren’s model exhibits good accuracy and unbiasedness since: ⅰ) The ratios of Fyc Fyce⁄ of the corner specimens obtained from Modified Karren’s model show the minimum, maximum, mean, and standard deviation of 0.84, 1.17, 1.00, and 0.06, respectively. The comparison between the statistical results obtained before and after the modification of Karren’s model only shows a slight improvement in the predictions. However, graphically speaking, see Figure 3.4, the distribution of the data points (i.e., Fyc Fyce⁄ versus Fuv/Fyv) becomes smoother after the modification. Also, because of the modification, the model has the potential to be used beyond the limit of Fuv/Fyv > 1.6. Therefore, it is worthy of modifying Karren’s model. ⅱ) The ratios of Fyc Fyce⁄ of the corner specimens obtained from the Modified Karren’s model are plotted against the ratios of r/t, yielding types of the virgin steel sheets, and the Fyv values in Figure 3.5(a), Figure 3.5(b), and Figure 3.5(c), respectively. Evenly distributed data points can be observed from these figures. 74 Figure 3.5 The ratios of 𝐅𝐅𝐅𝐅𝐅𝐅 𝐅𝐅𝐅𝐅𝐅𝐅𝐅𝐅⁄ of the corner specimens versus (a) r/t ratios; (b) yielding types of virgin steel sheets; (c) Fyv values. In summary, the Modified Karren’s model had been validated successfully. The application of this model is limited to right-angle corners featuring r/t ≤ 9.0 and Fyv ≤ 1000 MPa. 3.3.4 CFS Corners with Angles Different from 90 Degrees Some of the column specimens collected in Section 3.2 have corners showing angles different from 90 degrees. The corner angle could be a factor to affect the increase in material yield strength due to cold forming since a corner with a smaller angle may have its material undergone a larger amount of cold work than the one with a larger angle. Tran et al. (2019) provided experimental evidence to support this standpoint. They observed that the yield and ultimate tensile strengths of brake-pressed steel corners decreased as the corner angle increased. Thus, it is necessary to propose a model to predict the tensile yield strengths of CFS corners with angles different from 90 degrees. A model that fulfils this objective was proposed here; see Equation 3-2. The model was adapted from the modified Karren’s model by adding a factor α to consider the effect of corner angle on the increase in material yield strength due to cold forming. Fyc = αkb �tr�m Equation 3-2 75 where Fyc ≥ Fyv, b = 1.11-1.912n, m = 0.855n+0.035, α = 1.0 when θ ≤ 90 degrees and α = 1.115-0.115θ/90 when θ > 90 degrees, θ is the corner angle, and k and n are calculated using the expressions shown in Equation 3-1. For proposing the expression for calculating the factor α in Equation 3-2, experimental data associated with material property tests of 28 CFS corners with angles different from 90 degrees were collected from the literature and shown in Table 3.4. First, a comparison was made between the tested tensile yield strengths (Fyce ) of the corner specimens and those (Fyc90°) with right angles. The values of the corner specimens were either obtained from the literature or predicted using the modified Karren’s model. Then, the ratios of Fyce Fyc90°⁄ of the corner specimens were presented against the ratios of θ/90, as shown in Figure 3.6. Finally, a curve fitting of the data points, as shown in Figure 3.6, was performed to obtain the expression for calculating the factor α. Figure 3.6 The ratios of 𝐅𝐅𝐅𝐅𝐅𝐅𝐅𝐅 𝐅𝐅𝐅𝐅𝐅𝐅𝐅𝐅𝐅𝐅⁄ versus the ratios of θ/90 of corner specimens Tensile yield strengths of the corner specimens predicted using the proposed model (Equation 3-2) were compared with the experimental results in Table 3.4. The ratios of the predicted-to-tested tensile yield strengths (Fyc Fyce⁄ ) show the minimum, maximum, mean, and standard deviation of 0.82, 1.14, 1.01, and 0.09, respectively. The results show good accuracy of the proposed model (Equation 3-2). 76 Table 3.4 Database of material property tests of cold-formed steel corners with angles different from 90 degrees Reference Specimen θ t (mm) r/t Fyv (MPa) Fuv (MPa) Fyce (MPa) Fyc90° (MPa) Fyc (MPa) Fyc Fyce� (Karren 1965) 9. HRSK9-30.7 RF 105.6 3.81 1.45 211.67 364.73 410.93 375.89 368.40 0.90 9. HRSK9-30.7 105.6 3.81 1.45 211.67 364.73 451.61 375.89 368.40 0.82 9. HRSK9-30.7 105.6 3.81 1.55 211.67 364.73 433.68 369.34 361.97 0.83 9. HRSK9-30.7 105.6 3.81 1.45 211.67 364.73 450.92 375.89 368.40 0.82 (Uribe 1969) 9-C 60.0 2.84 1.26 251.66 396.45 458.02 451.29 451.29 0.99 9-E 60.0 2.84 1.68 251.66 396.45 460.54 421.73 421.73 0.92 10-C 60.0 2.13 2.24 293.03 408.86 380.94 434.24 434.24 1.14 10-E 60.0 2.13 2.24 293.03 408.86 383.87 434.24 434.24 1.13 11-D 60.0 3.07 1.55 277.86 444.02 450.13 476.65 476.65 1.06 11-F 60.0 3.07 1.93 277.86 444.02 422.35 452.28 452.28 1.07 12-F 60.0 3.71 1.60 257.17 396.45 437.20 431.86 431.86 0.99 12-H 60.0 3.71 1.18 257.17 396.45 473.20 463.40 463.40 0.98 (Chen and Young 2006) Inner corner PB 135.0 1.89 2.87 524.00 551.00 545.00 562.33 529.99 0.97 (Crisan 2011) RS125B RF 115.0 3.20 0.84 478.67 537.40 570.76 669.97 648.57 1.14 RS95B 115.0 2.60 1.04 462.79 538.90 558.71 659.84 638.76 1.14 (Tran et al. 2019) C2-4-S1 PB 100.0 4.10 1.46 762.33 805.00 932.00 907.00 889.97 0.95 C2-4-S2 100.0 4.10 1.46 762.33 805.00 927.00 907.00 889.97 0.96 C2-4-S3 100.0 4.10 1.46 762.33 805.00 914.00 907.00 889.97 0.97 C2-6-S1 100.0 6.01 1.50 795.00 843.67 843.00 874.00 932.04 1.11 C2-6-S2 100.0 6.01 1.50 795.00 843.67 861.00 874.00 932.04 1.08 C2-6-S3 100.0 6.01 1.50 795.00 843.67 853.00 874.00 932.04 1.09 C3-4-S1 120.0 4.10 1.46 762.33 805.00 837.00 907.00 866.93 1.04 C3-4-S2 120.0 4.10 1.46 762.33 805.00 865.00 907.00 866.93 1.00 C3-4-S3 120.0 4.10 1.46 762.33 805.00 859.00 907.00 866.93 1.01 C3-6-S3 120.0 6.01 1.50 795.00 843.67 845.00 874.00 907.91 1.07 C4-4-S2 140.0 4.10 1.46 762.33 805.00 839.00 907.00 843.89 1.01 C4-4-S3 140.0 4.10 1.46 762.33 805.00 831.00 907.00 843.89 1.02 C5-6-S3 160.0 6.01 1.50 795.00 843.67 826.00 874.00 859.66 1.04 Mean 1.01 Deviation 0.09 77 3.4 Experimental Database of Full-Section Tensile Tests of CFS Members The North American design specification, AISI S100 (2016), provides an expression (see Equation 3-3) to estimate the full-section tensile yield strengths (Fya) of CFS members. This expression was also proposed by Karren (1965). The Fya values can be used for designing CFS members to take advantage of the increase in material yield strength due to cold forming. The literature review showed that Karren’s expression exhibited good accuracy with prediction errors of ≤ 5.0% in the experimental assessments conducted by Karren (1965), Macadam (1967), and Sloof and Schuster (2000). Fya = CFyca + (1 - C)Fyfa Equation 3-3 where Fya is the full-section tensile yield strength and Fya ≤ Fuv, C = ΣAc/Ag, Fyca = ∑ AcFyc∑ Ac , Fyfa =∑ AfFyf∑ Af, Ac is the area of a corner, Ag is the gross area of the cross-section, Fyc is the tensile yield strength of a corner, Fyf is the tensile yield strength of a flat, and Af is the area of a flat. There are two issues to apply Karren’s expression (Equation 3-3) to a CFS member: ⅰ) First, the Fyca and Fyfa values of the member need to be determined. The Fyca value of the member can be readily determined using the model proposed in this study (Equation 3-2). While for determining the Fyfa value, numerous material coupons need to be taken from the flats of the member at different areas and then tested. This makes the application of Karren’s expression time demanding. ⅱ) None of the column specimens collected in Section 3.2 has the Fyfa value given in the literature. Instead, their material properties given in the literature were tested from material coupons taken from the flat areas away from corners (mostly) or virgin steel sheets. In CFS members, materials at flat areas away from corners are the least affected by the cold working; 78 hence, the tested material properties are deemed identical to those of the virgin steel sheets. Thus, it will be convenient if the Fyfa value of the member can be predicted using the Fyv value directly. An expression that fulfils the above objective is shown in Equation 3-4, which was proposed based on the experimental data of full-section tensile tests of six brake-pressed and 18 roll-formed steel members collected from Karren (1965), Macadam (1967), and Sloof and Schuster (2000). Table 3.5 shows the collected experimental data. Fya = CFyca + β(1 - C)Fyv Equation 3-4 where Fya shall ≤ Fuv, β = 1.00 for brake-pressed members and β = 1.08 for roll-formed members, C and Fyca shall be calculated using the expressions given in Equation 3-3. For calculating the net-section tensile yield strengths (Fya, net) of CFS members with holes, C = ΣAc/Anet. The proposed Equation 3-4 is described as follows: ⅰ) For each member, the tensile yield strengths of corners (Fyc) were predicted using Equation 3-2; then, the Fyca value was calculated as per Equation 3-3. Table 3.5 (11th column) shows the calculated Fyca values of the members. ⅱ) With the tested value of Fya and the calculated value of Fyca in hand, the Fyfa value of each member was inversely calculated using Equation 3-3. The calculated Fyfa values of the members are presented in the 12th column of Table 3.5. Among the collected members, tested Fyfa values of ten members are available from the literature; see the 9th column of Table 3.5. They are different from the calculated Fyfa values by no more than 6%. 79 Table 3.5 Database of full-section tensile tests of CFS members Specimen Section t (mm) Ag (mm2) ΣAc (mm2) Tested strength (MPa) Predicted strength (MPa) Fyfa/Fyv Fya Fyae⁄ Fyv Fuv Fyfa Fyca Fyfa Fya T1 H PB 1.50 268.58 22.00 264.07 352.32 261.31 273.03 437.03 258.40 278.24 0.98 1.02 T2 H 1.50 268.58 22.00 264.07 352.32 261.31 270.96 437.03 256.15 278.24 0.97 1.03 T4 H 1.57 281.66 23.50 258.55 337.84 273.72 284.75 423.17 272.15 272.29 1.05 0.96 T11 C 1.59 143.46 9.91 258.55 337.84 - 282.00 446.58 269.78 271.55 1.04 0.96 T12 H 1.47 209.43 28.92 264.07 352.32 - 292.34 403.27 274.57 283.29 1.04 0.97 0.09-37 LC 2.25 673.84 71.93 295.10 438.55 294.41 308.89 476.51 288.85 314.46 0.98 1.02 T7 C RF 3.56 374.17 55.33 255.11 396.45 314.40 330.26 492.18 302.16 307.56 1.18 0.93 T8 C 3.56 374.17 55.33 255.11 396.45 314.40 328.88 492.18 300.54 307.56 1.18 0.94 T9 C 3.56 374.17 55.33 255.11 396.45 314.40 329.57 492.18 301.35 307.56 1.18 0.93 T10 T 1.59 328.04 47.60 258.55 337.84 301.99 317.16 393.17 304.26 295.77 1.18 0.93 H-2 H 1.52 217.00 22.62 393.69 495.73 - 418.17 615.24 395.24 445.00 1.00 1.06 H-3 H 1.42 216.00 38.36 299.23 381.28 - 343.91 405.52 330.61 337.80 1.10 0.98 H-4 H 1.47 208.00 35.03 315.78 395.07 - 335.76 435.71 315.51 356.99 1.00 1.06 C-1 LC 1.52 342.00 53.27 473.67 543.31 - 534.57 558.61 530.13 518.89 1.12 0.97 LC-1 T 2.62 349.00 93.63 304.75 402.65 - 418.06 509.79 384.42 377.60 1.26 0.90 LC-2 T 1.88 183.00 46.80 328.19 413.69 - 432.67 526.94 400.28 398.55 1.22 0.92 LC-3 T 2.54 445.00 76.01 281.31 399.21 - 345.86 527.20 308.51 341.97 1.10 0.99 U-1 U 1.88 395.00 136.79 446.78 528.14 - 521.40 561.46 500.18 509.86 1.12 0.98 U-2 U 2.51 529.00 214.70 450.92 557.10 - 516.30 608.21 453.51 536.19 1.01 1.04 U-3 U 2.79 796.00 243.46 410.24 519.18 - 457.12 576.48 404.52 483.87 0.99 1.06 U-4 U 3.20 846.00 318.72 470.91 555.72 - 514.23 616.41 452.46 549.21 0.96 1.07 U-5 U 3.20 913.00 304.41 410.24 561.23 - 466.75 621.90 389.14 502.69 0.95 1.08 0.06-37 LC 1.49 444.85 66.22 282.69 380.77 285.81 293.72 361.04 281.94 313.60 1.00 1.07 0.09-50 LC 2.29 672.38 119.03 380.59 537.17 391.28 398.52 530.26 370.18 432.14 0.97 1.08 Mean 1.00 Deviation 0.06 Note: H = hat, C = channel, LC = lipped channel, T = track, U = U shape. ⅲ) The ratios of Fyfa/ Fyv of the collected members were calculated, where the results are shown in the 14th column of Table 3.5. The ratios of Fyfa/Fyv of the six brake-pressed members show the minimum, maximum, mean, and standard deviation of 0.97, 1.05, 1.01, and 0.04, respectively. The ratios of Fyfa/Fyv of the 18 roll-formed members show the minimum, maximum, 80 mean, and standard deviation of 0.95, 1.26, 1.08, and 0.10, respectively. It can be seen here that the values of 1.00Fyv and 1.08Fyv are reasonable estimations of the Fyfa values for the brake-pressed and roll-formed members, respectively. Based on this observation, Karren’s expression (Equation 3-3) was modified to Equation 3-4. ⅳ) The Fya values of the considered members were calculated using Equation 3-4 and were compared with the tested values of Fyae . The ratios of Fya Fyae⁄ of the members are presented in the last column of Table 3.5, where the minimum, maximum, mean, and standard deviation are 0.90, 1.08, 1.00, and 0.06, respectively. It is crucial to emphasize that the proposed Equation 3-4 is empirical and is subjected to the context of limited experimental data. A refined expression could be proposed in the future when more experimental data are available. In Chapter 5, the model and expression developed in this section were employed to estimate the full-section and/or net-section tensile yield strengths of the column specimens. 3.5 Summary This chapter focused on the establishment of three different experimental databases associated with ⅰ) compression tests of solid and perforated CFS columns, ⅱ) material property tests of CFS corners, and ⅲ) full-section tensile tests of CFS members, where the experimental data were collected from the literature as much as possible. Note that the experimental data of the nine stub columns (perforated) tested in this study (see Chapter 7) were also included in the 1st experimental database. Afterward, the performance of Karren’s model (Karren 1965) for predicting the tensile yield strengths of CFS corners was evaluated with the 2nd experimental database, and the model was then modified to improve the accuracy and extend the application limit. Finally, the expression Karren (1965) proposed for estimating the full-section tensile yield 81 strengths of CFS members was modified based on the 3rd experimental database to simplify its application. The following conclusions are made from this study: 1) An experimental database related to compression tests of CFS columns has been established, which contains the compression test results (e.g., axial strength (Pu) and failure mode) and design parameters (e.g., cross-section areas and elastic buckling loads) of 453 solid and 228 perforated CFS column specimens. 2) An experimental database related to material property tests of CFS corners has been established, which contains the experimental results (e.g., tensile yield strengths (Fyce )) and parameters (e.g., r, t, and material properties of the virgin material (Fyv and Fuv)) of 190 corner specimens with right angles and 28 corner specimens with non-right angles. 3) An experimental database associated with full-section tensile tests of CFS members has been established, which contains the experimental results (e.g., full-section tensile yield strengths (Fyae )) and parameters (e.g., the predicted tensile yield strengths of corners (Fyc), cross-section areas of corners (ΣAc), and cross-section areas of members (Ag)) of 24 CFS members, where six of them were brake-pressed, and the rest 18 were roll-formed. 4) Karren’s model (Equation 3-1) is subjected to a narrow application limit, i.e., right-angle corners with r/t ≤ 6.0. Also, the model has been demonstrated to show a tendency to overestimate the tensile yield strengths of corners featuring Fuv/Fyv > 1.4. The model has been modified to Equation 3-2 to extend the application limit to corners with variable angles, r/t ≤ 9.0, and Fyv ≤ 1000 MPa. The performance of Equation 3-2 has been assessed with the 2nd experimental database, where good accuracy was exhibited in dealing with right-angle and non-right-angle corners. Note that the experimental database employed here contains a small number (28) of non-right-angle corner specimens. 82 5) Karren’s expression (Equation 3-3) has been simplified to Equation 3-4 for estimating the full-section tensile yield strengths (Fya) of CFS members. The input parameter of Fyfa of Karren’s expression was replaced with 1.00Fyv for brake-pressed members and with 1.08Fyv for roll-formed members as shown in Equation 3-4 so that the experimental determination of the Fyfa value is avoided. The performance of Equation 3-4 has been assessed with the 3rd experimental database, where the prediction errors were found ≤ 10%. Note that Equation 3-4 was proposed based on a small experimental database containing the experimental data of 24 CFS members. The 1st experimental database will be employed in Chapters 4 and 5 as a source to assess the buckling strength curves in the DSM (AISI_S100 2016). The modified Karren’s model and expression, i.e., Equation 3-2 and Equation 3-4, were employed in Chapter 5 to estimate the full-section and/or net-section tensile yield strengths of the column specimens in the 1st experimental database. 83 Chapter 4: Assessment of the Buckling Strength Curves in the DSM Using Tensile Yield Strengths of Virgin Materials 4.1 General The North American design specification of AISI S100 (2016) incorporates the Direct Strength Method (DSM) to design CFS members under compression. As mentioned in Section 2.2.3, although the DSM is being used for a long period, its buckling strength curves have not been assessed with experimental results thoroughly. In this chapter, a rigorous assessment of the DSM buckling strength curves was conducted based on the 1st experimental database (i.e., Table 3.1 and Table 3.2) established in Chapter 3, using the tensile yield strengths of virgin materials as design parameters. Material properties of the columns in the database were mostly tested from coupons cut from the as-formed members, i.e., the center of the flat element featured the greatest width in the cross-section. In this study, they were used as virgin material properties since Karren (1965) has experimentally demonstrated that material properties at this region are very close to those tested from the virgin steel. The nominal strengths (the Pnl, Pnd, or Pne) of the column specimens were calculated using the DSM (AISI_S100 2016); see the associated expressions in Section 2.2.1. The results were then compared with the tested strengths (Pu). The assessment is being considered rigorous because ⅰ) the work was based on an extensive experimental database, where the compression test data and design parameters of 453 solid and 228 perforated CFS column specimens were included; ⅱ) the column specimens in the database exhibit 21 different cross-section shapes, and ⅲ) a standard procedure was followed to determine the elastic buckling loads of the column specimens. Details about the establishment of the database have been presented in Chapter 3. This chapter highlights the level of accuracy of the buckling strength curves in the 84 DSM (AISI_S100 2016). Here, only axially loaded CFS columns that failed in L, D, or G buckling were considered. 4.2 Assessment of the Pnl-curve In the database, there are 135 solid and 124 perforated column specimens that failed in L buckling. The lengths of these column specimens were short enough to minimize the interferences of D and G deformation modes to their failure modes. To be specific, the lengths of these column specimens were either ≤ 20 times the least radius of gyration of the gross cross-section or ≤ 4.2 times the greatest overall width of the cross-section. The nominal strengths (Pnl) of these column specimens are compared with the tested strengths (Pu) herein. The data points of (λl, Pu/Py) of the solid and perforated column specimens are presented along with the Pnl-curve in Figure 4.1(a) and Figure 4.1(b), respectively. The data points of (λl, Pnl/Pu) of the solid and perforated column specimens are plotted in Figure 4.2(a) and Figure 4.2(b), respectively. Figure 4.1 Data points of (λl, Pu/Py) of the column specimens failed in L buckling along with the Pnl-curve: (a) solid column specimens; (b) perforated column specimens. It can be observed from Figure 4.2 that the axial strengths of the column specimens are estimated by the Pnl values overall accurately regardless of whether the column specimens had 85 holes. The Pnl/Pu ratios show the minimum, maximum, mean, and standard deviation of 0.76, 1.40, 0.98, and 0.10, respectively, for the solid column specimens and of 0.80, 1.21, 0.99, and 0.09, respectively for the perforated column specimens. Surprisingly, the data points of (λl, Pnl/Pu) of the perforated column specimens exhibit a peculiar distribution pattern; see Figure 4.2(b). Figure 4.2 Data points of (λl, Pnl/Pu) of the column specimens failed in L buckling: (a) solid column specimens; (b) perforated column specimens. In Figure 4.2(a), four data points lie adjacent to or above the Pnl/Pu = 1.2, two data points lie below the Pnl/Pu = 0.8, and the remaining data points lie between the Pnl/Pu = 0.8 and Pnl/Pu =1.2. So, six solid column specimens show Pnl/Pu ratios diverging from the others. The reasons are explored herein. First, the two data points lying below the Pnl/Pu = 0.8 in the figure are from the 3A and 3B specimens tested by Chen et al. (2010). The two column specimens had cross-sections like the LC7 section (see Figure 3.1). The LC7 section features high structural efficiency since it has an intermediate stiffener at each flange, which accounts for a large proportion of the flange width so that the flanges are stiffened significantly. The 3A and 3B specimens might have exhibited significant post-L buckling strength reserves during the compression. Secondly, the three data points (F-F ends) lying adjacent to or above the Pnl/Pu = 1.2 in the figure are from the column 86 specimens in the group of SWC350 tested by Yap and Hancock (2008b). The ends of these column specimens were deemed as the F-F ends since they were bonded with steel loading plates using thick pattern stone. Surprisingly, the tested strengths (Pu) of these column specimens were not close but scattered. This raises a question about the flatness of the ends of the column specimens, and the pattern stone might have failed to compensate for this defect. Lastly, the data point (with pinned ends) lying slightly above the Pnl/Pu = 1.2 in the figure is from the SLC90×90-700 specimen tested by Shi et al. (2010). Among the solid column specimens with pinned ends (the P-P or P-P-W ends) (count = 14), see Figure 4.2(a), the SLC90×90-700 specimen is the only one to show a significant difference between the nominal and tested axial strengths. During compression, the SLC90×90-700 specimen might have been subjected to a comparatively large initial loading eccentricity brought by a poor alignment. Consequently, the Pu value of this specimen is lower than the Pnl value prominently. 4.3 Assessment of the Pnd-curve The database contains 187 solid and 71 perforated column specimens that failed in D buckling. Among the solid column specimens, 85 were tested with the B-F ends, 45 were tested with the F-F ends, and 57 were tested with the P-P or P-P-W ends. Among the perforated column specimens, three were tested with the B-F ends, ten were tested with the F-F ends, and 58 were tested with the P-P ends. All these column specimens have their nominal axial strengths (Pnd) compared with their tested axial strengths (Pu) herein. The data points of (λd, Pu/Py) of the column specimens are presented along with the Pnd-curve and/or the Pne-curve in Figure 4.3. In the figure, the data points of (λd, Pu/Py) of the column specimens tested with pinned ends (the P-P or P-P-W ends) are presented separately from those of the column specimens tested with the B-F or F-F ends 87 because the two groups of column specimens exhibit very different trends of Pu/Py ratios versus the λd values. The data points of (λd, Pnd/Pu) of the column specimens are presented in Figure 4.4. Figure 4.3 Data points of (λd, Pu/Py) of the column specimens failed in D buckling along with the Pnd-curve and/or Pne-curve: (a) solid column specimens with the B-F or F-F ends; (b) solid column specimens with pinned ends; (c) perforated column specimens with the B-F or F-F ends; (d) perforated column specimens with the P-P ends. 88 Figure 4.4 Data points of (λd, Pnd/Pu) of the column specimens failed in D buckling: (a) solid column specimens; (b) perforated column specimens. On one hand, the axial strengths of the column specimens tested with the B-F or F-F ends are overall accurately estimated by the Pnd values regardless of whether the column specimens had holes; see Figure 4.3(a), Figure 4.3(c), and Figure 4.4. The Pnd/Pu ratios of the solid column specimens tested with the B-F or F-F ends show the minimum, maximum, mean, and standard deviation of 0.70, 1.27, 0.99, and 0.13, respectively. The Pnd/Pu ratios of the perforated column specimens tested with the B-F or F-F ends show the minimum, maximum, mean, and standard deviation of 0.89, 1.14, 1.01, and 0.07, respectively. On the other hand, the axial strengths of the column specimens tested with pinned ends (the P-P or P-P-W ends) are mostly overestimated by the Pnd values no matter whether the column specimens had holes; see Figure 4.3(b), Figure 4.3(d), and Figure 4.4. The Pnd/Pu ratios of the solid column specimens tested with pinned ends (the P-P or P-P-W ends) show the minimum, maximum, mean, and standard deviation of 0.91, 1.81, 1.13, and 0.19, respectively. The Pnd/Pu ratios of the perforated column specimens tested with the P-P ends show the minimum, maximum, mean, and standard deviation of 0.92, 1.37, 1.07, and 0.11, 89 respectively. Be noted that the data points presented in Figure 4.3(b) have two outliers lying within the range of 1.8 < λd < 2.0 and above the Pnd-curve. The two outliers are the G150-150-1200A and G150-150-1200B specimens tested by Yi et al. (2020). The Pu values of these two column specimens (solid) were found greater than their Pcrl values by 24% to 30%. Thus, the author believed they failed in L + D buckling rather than D buckling, where the two outliers were excluded from the above statistic results. It is worth mentioning that the issue identified above that the Pnd values are generally unsafe strength estimations of the column specimens with pinned ends (the P-P or P-P-W ends) has been somewhat demonstrated earlier by Zhao et al. (2017). To be specific, Zhao et al. (2017) axially compressed many CFS columns (including solid and perforated) with the P-P ends. Among the solid column specimens, seven failed in D buckling and featured λd values ≤ 1.0. Later, Zhao et al. (2017) observed that their data points of (λd, Pu/Py) all lay below the Pnd-curve slightly. As mentioned earlier in Section 2.2.4.2, Landesmann and Camotim (2013) have numerically demonstrated that the Pnd-curve was unsafe to estimate the axial strengths of solid columns with simply-simply, simply-fixed, or fixed-free ends. So, it can be concluded here that under a given condition of constant λd values, columns with the B-F or F-F ends have post-D buckling strength reserves prominently higher than those of the columns with simply-simply, simply-fixed, fixed-free, P-P, or P-P-W ends. Compared to the columns compressed between the B-F or F-F ends, the columns compressed between the simply-simply, simply-fixed, fixed-free, P-P, or P-P-W ends are subject to an additional P-δ effect, which is responsible for their comparatively lower axial strengths to those of the columns with the B-F or F-F ends having the same slenderness values. The δ of a column comes from the initial crookedness and the shift of the effective centroid of the cross-section during the compression. The shift of the effective centroid is the result of the 90 asymmetric redistribution of longitudinal stress in the column due to the appearance and the development of D buckling deformations; see more details from Rasmussen and Hancock (1993), Young and Rasmussen (1999), and Landesmann and Camotim (2013). Therefore, the D buckling of a pin-ended column with singly symmetric and open cross-section induces overall bending. So, the column specimens with pinned ends (the P-P or P-P-W ends) here were essentially failed in D + F buckling rather than ‘purely’ D buckling. An adequate design solution for these columns should account for the interaction effects between the two buckling modes. The DSM (AISI_S100 2016) adopts the minimum of the Pnl, Pnd, and Pne values as the unfactored design axial strength of a CFS member. Hence, it is necessary to check whether the Pne-curve could provide adequate strength estimations for the column specimens tested with pinned ends (the P-P or P-P-W ends) and failed in D buckling (essentially, the D + F buckling). For comparison, the Pne-curve is presented in Figure 4.3(b) and Figure 4.3(d). The λc = λd is assumed while plotting the Pne-curve and calculating the Pne values of the column specimens. The axial strengths of the solid column specimens, see Figure 4.3(b), are overall conservatively estimated by the Pne-curve, where the Pne/Pu ratios show the minimum, maximum, mean, and standard deviation of 0.79, 1.14, 0.96, and 0.10, respectively. Note that the two outliers were excluded from the statistical results. In contrast, the axial strengths of most perforated column specimens, see Figure 4.3(d), are overestimated by the Pne-curve, where the Pne/Pu ratios show the min, max, mean, and standard deviation of 0.93, 1.38, 1.04, and 0.10, respectively. Note that the scenario will worsen when the real λc values of the column specimens are fed in the Pne-curve. To be specific, the column specimens have their Pcre values greater than the Pcrd values; hence, their actual λc values are smaller than the λd values. Hence, when the actual λc values of the column specimens 91 are fed in the Pne-curve, the data points in Figure 4.3(b) and Figure 4.3(d) will shift left with respect to the Pne-curve so that more data points will lie below the Pne-curve. 4.4 Assessment of the Pne-curve In the database, there are 131 solid and 33 perforated column specimens that failed in G buckling. The nominal axial strengths (Pne) of these column specimens are assessed herein with the tested axial strengths (Pu). The data points of (λc, Pu/Py) of the solid and perforated column specimens are presented along with the Pne-curve in Figure 4.5(a) and Figure 4.5(b), respectively. The data points of (λc, Pne/Pu) of the solid and perforated column specimens are shown in Figure 4.6(a) and Figure 4.6(b), respectively. Be aware that among the solid column specimens, ⅰ) 35 were tested with the B-F (count = 27) or F-F ends (count = 8), and they all failed in TF buckling; ⅱ) two were tested with the P-P-S ends and failed in TF buckling; ⅲ) 19 were tested with the P-P ends, where five failed in F buckling and the other 14 failed in TF or AD + TF buckling, and; ⅳ) 75 were tested with the P-P-W ends and failed in F buckling. Among the perforated column specimens, ⅰ) five were tested with the P-P-S ends and failed in TF or AD + TF buckling, ⅱ) 25 were tested with P-P ends where five failed in F buckling and the other 20 failed in TF or AD + TF buckling, and ⅲ) three were tested with the P-P-W ends and failed in F buckling. 92 Figure 4.5 Data points of (λc, Pu/Py) of the column specimens failed in G buckling with the Pne-curve: (a) solid column specimens; (b) perforated column specimens. Overall, the axial strengths of the solid column specimens are somewhat conservatively estimated by the Pne values, where the Pne/Pu ratios show the minimum, maximum, mean, and standard deviation of 0.58, 1.22, 0.94, and 0.12, respectively. In contrast, the axial strengths of most perforated column specimens are overestimated by the Pne values, where the Pne/Pu ratios show the minimum, maximum, mean, and standard deviation of 0.87, 1.33, 1.12, and 0.11, respectively. By the way, it can be seen from Figure 4.6(a) that the data points of (λc, Pne/Pu) of the column specimens tested with pinned ends spread vertically more prominently than those of the column specimens tested with the B-F or F-F ends. This supports the standpoint of Weng and Pekoz (1990) that while testing pin-ended CFS columns under compression, the geometric alignment features uncertainty so that the results are likely to show large scatter. 93 Figure 4.6 Data points of (λc, Pne/Pu) of the column specimens failed in G buckling: (a) solid column specimens; (b) perforated column specimens. The Pne-curve consists of two segments governing two neighbouring slenderness ranges of λc ≤ 1.5 and λc > 1.5. It is necessary to assess the two segments of the Pne-curve separately. The two segments of the Pne-curve are first assessed with the tested axial strengths of the solid column specimens. Figure 4.5(a) shows that the first segment of the Pne-curve fits the data points in the slenderness range of λc ≤ 1.5 (count = 112) well, where the Pne/Pu ratios show the minimum, maximum, mean, and standard deviation of 0.65, 1.22, 0.96, and 0.10, respectively. In contrast, the second segment of the Pne-curve, see Figure 4.5(a), lies below most data points in the slenderness range of λc > 1.5 (count = 19), where the Pne/Pu ratios show the minimum, maximum, mean, and standard deviation of 0.58, 1.09, 0.84, and 0.16, respectively. The Pne-curve has the second segment much more conservative than the first segment, which attributes to two reasons. On one hand, the second segment of the Pne-curve has a feature to provide conservative strength estimations for solid columns in the case of (the F-F ends, TF buckling) (Dinis et al. 2019; Gunalan and Mahendran 2013). For instance, Figure 4.5(a) shows that in the slenderness range of λc > 1.5, 94 there are three solid column specimens in the case of (F-F ends, TF buckling), and they all lie above the Pne-curve. On the other hand, the second segment of the Pne-curve owns another feature to underestimate the axial strengths of solid columns in the case of (the P-P-W ends, F buckling). It can be seen from Figure 4.5(a) that in the slenderness range of λc >1.5, there are 13 solid column specimens in the case of (the P-P-W ends, F buckling). The Pne/Pu ratios of these column specimens show the minimum, maximum, mean, and standard deviation of 0.59, 1.09, 0.82, and 0.15, respectively. Note that as per the author’s knowledge, this is the first time that the second feature of the Pne-curve has been unveiled. It is possible that the free rotations at the ends of the column specimens in the case of (P-P-W ends, F buckling) had been affected to certain extents during the tests, resulting in increased axial strengths. However, the design curve (Pne-curve) is lower than the elastic critical buckling curve in the slenderness range of λc > 1.5 is likely to be a conservative consideration for design since it is well-known that CFS columns with pinned ends like to show small or negligible post-buckling strength reserves under compression. The two segments of the Pne-curve are then assessed with the tested axial strengths of the perforated column specimens separately. On one hand, Figure 4.5(b) shows that there are 26 perforated column specimens in the slenderness range of λc ≤ 1.5, and their data points mostly lie below the first segment of the Pne-curve. The Pne/Pu ratios of these column specimens show the minimum, maximum, mean, and standard deviation of 0.91, 1.33, 1.15, and 0.09, respectively. Note that in the slenderness range of λc ≤ 1.5, the column specimens in the cases of (the P-P ends, TF or TF + AD buckling), (the P-P ends, F buckling), and (the P-P-W ends, F buckling) have their axial strengths all overestimated by the first segment of the Pne-curve. A similar phenomenon has also been observed by Yao and Rasmussen (2017) that the numerical data points (i.e., Pu-FEM/Py versus λc) associated with lipped-C section columns were mostly below the Pne-curve in the 95 slenderness range of λc ≤ 1.5. These columns had patterned web holes distributed along the length and were compressed between the P-P ends through the centroid of the gross cross-section. On the other hand, Figure 4.5(b) shows that there are seven perforated column specimens in the slenderness range of λc > 1.5, where three of them are in the case of (the P-P ends, TF buckling), two of them are in the case of (the P-P-W ends, F buckling), and the remaining two are in the case of (the P-P-S ends, TF + AD buckling). The Pne-curve (the second segment) accurately estimated the axial strengths of the seven column specimens, where the Pne/Pu ratios show the minimum, maximum, mean, and standard deviation of 0.87, 1.08, 0.98, and 0.09, respectively. 4.5 LRFD Resistance Factors of the Buckling Strength Curves in the DSM The reliability of the buckling strength curves in the DSM was calculated. In particular, it is intended to check whether the φc = 0.85 recommended by the AISI_S100 (2016) is safe for designing CFS columns under compression. First-order second-moment reliability analyses were conducted to calculate the LRFD (Load and Resistance Factor Design) resistance factors (φc) of the buckling strength curves using the following equation from Chapter K – Section K2.1.1 of the AISI main specification (AISI_S100 2016): ϕc = Cϕ(MmFmPm)e−β0�VM2 +VF2+CPVP2 +VQ2 Equation 4-1 where Cφ is a calibration coefficient and is equal to 1.52 for LRFD; Mm = 1.10 and Fm = 1.00 are the mean values of the material and fabrication factors, respectively; Pm is the mean value of the professional factor, which is considered as the mean of the tested-to-nominal strength ratios; β0 is the target reliability index and is equal to 2.5 for structural members for LRFD; VM = 0.10, VF = 0.05, and VQ = 0.21 are the coefficients of variation of the material factor, fabrication factor, and load effect, respectively; VP is the coefficient of variation of the tested-to-nominal strength ratios, 96 but not less than 0.065; is a correction factor where n = number of tests, and m = n-1 is the degrees of freedom. The statistics of the tested-to-nominal axial strength ratios (Pu/Pn) of the solid and perforated column specimens are summarized in Table 4.1 and Table 4.2, respectively. These observations prompt the following remarks: ⅰ) The φc = 0.85 is conservative for the Pnl-curve. The calculated φc values of the Pnl-curve in dealing with solid and perforated column specimens are around 0.90. ⅱ) The φc = 0.85 is slightly conservative for the Pnd-curve to design the column specimens tested with the B-F or F-F ends (including solid and perforated), where the calculated φc values are between 0.87 and 0.89. ⅲ) The Pnd-curve is inadequate to be used for designing column specimens with the P-P or P-P-W ends (including solid and perforated), where the calculated φc values are lower than 0.85. The interaction between D and F buckling ought to be considered in the design of these columns. Table 4.1 Statistics of the Tested-to-Nominal Axial Strength Ratios of the Solid Column Specimens Strength curve Condition λ Pu/Pn φc Min Max Min Max Mean SD COV n Pnl-curve - 0.423 4.865 0.713 1.324 1.030 0.099 0.096 135 0.906 Pnd-curve B-F or F-F ends 0.558 3.173 0.790 1.427 1.028 0.139 0.136 130 0.865 P-P or P-P-W ends 0.464 2.038 0.551 1.097 0.904 0.126 0.139 55 0.755 a Pne-curve D buckling with P-P or P-P-W ends 0.464 2.038 0.877 1.261 1.052 0.108 0.103 55 0.918 Pne-curve overall 0.514 2.797 0.816 1.716 1.082 0.155 0.143 131 0.902 λc ≤ 1.5 0.514 1.465 0.816 1.547 1.056 0.116 0.110 112 0.916 λc > 1.5 1.523 2.797 0.915 1.716 1.234 0.248 0.201 19 0.920 Note: a λc = λd is assumed here for calculating φc. 97 ⅳ) By assuming the λc = λd, the φc = 0.85 is adequate for the Pne-curve to design the column specimens (including solid and perforated) tested with the P-P or P-P-W ends that failed in D buckling. Table 4.2 Statistics of the Tested-to-Nominal Axial Strength Ratios of the Perforated Column Specimens Strength curve Condition λ Pu/Pn φc Min Max Min Max Mean SD COV n Pnl-curve - 0.525 2.901 0.824 1.246 1.020 0.095 0.094 124 0.899 Pnd-curve B-F and F-F ends 0.791 1.881 0.878 1.130 0.993 0.070 0.070 13 0.887 P-P ends 0.534 1.159 0.728 1.087 0.943 0.094 0.100 58 0.825 a Pne-curve D buckling with P-P ends 0.534 1.159 0.722 1.081 0.965 0.074 0.077 58 0.863 Pne-curve overall 0.838 2.596 0.751 1.151 0.906 0.099 0.109 33 0.782 λc ≤ 1.5 0.838 1.415 0.751 1.099 0.873 0.073 0.083 26 0.774 λc > 1.5 1.612 2.596 0.929 1.151 1.027 0.092 0.090 7 0.885 Note: a λc = λd is assumed here for calculating φc. ⅴ) The φc = 0.85 is conservative for the Pne-curve to design the solid column specimens. The calculated φc values of the two segments of the Pne-curve are around 0.92 in dealing with the solid column specimens that failed in G buckling. ⅵ) The φc = 0.85 is unsafe for the first segment but is slightly conservative for the second segment of the Pne-curve to design the perforated column specimens. The calculated φc values of the first and second segments of the Pne-curve are 0.77 and 0.89, respectively. The calculated φc value of the Pne-curve (overall) is 0.782; this value is compared to its counterparts determined by Moen and Schafer (2011) and Yao and Rasmussen (2017B) below, separately. On one hand, the φc value (0.782) calculated here is significantly lower than the one (0.89) determined by Moen and Schafer (2011) while using the Pne-curve to estimate the axial strengths (numerically predicted) of lipped-C section columns with simply-simply ends and holes at the web. Upon looking at the data presented in Moen and Schafer (2011), two observations are made here. 98 First, for the columns featuring An/Ag = 0.80, the Pne-curve fitted the data points (i.e., Pu-FEM/Py versus λc) well in the elastic buckling regime (λc > 1.5) but started to go above the data points as the λc decreased from 1.5. This observation agrees with the one obtained in this study. Note that among the 33 perforated column specimens that failed in G buckling and collected in this study, 26 falls within the slenderness range of λc ≤ 1.5 and features 0.78 ≤ An/Ag ≤ 0.86. Second, for the columns featuring An/Ag = 0.90, the divergence between the Pne-curve and the data points (i.e., Pu-FEM/Py versus λc) appeared only in the vicinity of λc = 0.5. A large ϕc value of 0.89 was obtained by Moen and Schafer (2011) for the Pne-curve, which was due to that the columns featuring An/Ag = 0.80 and had their axial strengths been overestimated by the Pne-curve only accounted for a small portion of the database. So, it is concluded here that the Pne-curve owns a feature to overestimate the axial strengths of the columns with cross-section areas being greatly reduced by perforations (e.g., An/Ag < 0.90) and showing λc values ≤ 1.5. This feature might not have been deemed by Moen and Schafer (2011) as a shortcoming of the Pne-curve since they claimed that the nominal strength of a perforated column is eventually limited by the squash load (Pynet). On the other hand, the calculated ϕc value (0.782) of the Pne-curve is close to but slightly lower than those determined by Yao and Rasmussen (2017B). Yao and Rasmussen (2017B) performed extensive nonlinear FE simulations on perforated CFS columns under compression and found that the Pne-curve exhibit ϕc values of 0.786 and 0.814 in estimating the axial strengths of lipped-C section (count = 1651) and rack-section (count = 383) columns failed in G buckling, respectively. These columns had patterned web holes distributed along the length and were compressed between the P-P ends through the centroid of the gross cross-section. The slight differences between the ϕc value calculated here and the two obtained by Yao and Rasmussen (2017B) are mainly attributed to the very different sample sizes. Only 33 perforated column 99 specimens failed in G buckling were collected in this study, and most of them (26) falls within the slenderness range of λc ≤ 1.5. 4.6 Summary In this chapter, the design parameters of the column specimens in the database were fed in the DSM buckling strength curves to calculate their nominal strengths (Pnl, Pnd, or Pne) using the material tensile yield strengths of virgin materials as design parameters. The nominal axial strengths were then assessed with the tested axial strengths. Finally, the LRFD resistance factors (φc) of the DSM buckling strength curves were determined in accordance with the AISI_S100 (2016). Shortcomings of the DSM buckling strength curves have been identified and are summarized in Table 4.3. Based on the study presented in this chapter, the following conclusions are presented: Table 4.3 A summary of shortcomings of DSM buckling strength curves Buckling strength curve Shortcoming Solid columns Perforated columns Pnl-curve 20 data points lie above the Pu/Py = 1.0. Data points of (λl, Pnl/Pu ) show a peculiar distribution pattern. Pnd-curve Inadequate for designing columns with P-P or P-P-W ends. Pne-curve Underestimates the strengths of columns with P-P-W ends showing λc > 1.5 and experiencing F buckling. The first segment, λc ≤ 1.5, is unsafe to design columns in the cases of (P-P ends, TF or TF + AD buckling), (P-P ends, F buckling), and (P-P-W ends, F buckling). Note: Shortcomings showing in italic type are those that have been resolved in Chapter 8. The remaining ones will be resolved in the future. 1) Axial strengths of the column specimens that failed in L buckling were overall accurately estimated by the Pnl values regardless of whether the column specimens had holes. The φc = 0.85 recommended by AISI_S100 (2016) was conservative for designing these column 100 specimens. Interestingly, the perforated column specimens that failed in L buckling exhibited a peculiar distribution pattern of Pnl/Pu ratios versus the λl. 2) Axial strengths of the solid column specimens that were tested with the B-F or F-F ends and failed in D buckling were overall accurately estimated by the Pnd values. This study collected only 13 perforated column specimens that were tested with the B-F or F-F ends and failed in D buckling, and their axial strengths were accurately estimated by the Pnd values. The φc = 0.85 was adequate for designing these solid and perforated column specimens. 3) The axial strengths of the column specimens that were tested with the P-P or P-P-W ends and failed in D + F buckling were generally overestimated by the Pnd values regardless of whether the column specimens had holes. The φc = 0.85 was unsafe for designing these column specimens. 4) Axial strengths of the solid column specimens that failed in G buckling were overall accurately estimated by the Pne values. The φc = 0.85 was conservative for designing these column specimens. Note that the Pne-curve in the slenderness range of λc > 1.5 exhibited a tendency to underestimate the axial strengths of the solid column specimens that were tested with the P-P-W ends and failed in F buckling. 5) On one hand, the first segment of the Pne-curve (λc ≤ 1.5) provided overestimated strength estimations for the perforated column specimens in the cases of (P-P ends, TF or TF + AD buckling), (P-P ends, F buckling), and (P-P-W ends, F buckling), where the φc = 0.85 was unsafe to design these column specimens. On the other hand, the second segment of the Pne-curve (λc > 1.5) accurately estimated the axial strengths of the perforated column specimens that failed in G buckling, where the φc = 0.85 was adequate to design these column specimens. Note that the 101 assessment of the Pne-curve for dealing with perforated columns was based on limited experimental data. 102 Chapter 5: Assessment of the Buckling Strength Curves in the DSM Using Full-Section Tensile Yield Strengths 5.1 General The L, D, and G buckling strength curves in the DSM (AISI_S100 2016) have been assessed in Chapter 4 using the virgin material tensile yield strength (Fy) as the design parameter. Prominent spread of the nominal-to-tested axial strength (Pn/Pu) ratios was observed regardless of which buckling strength curve (i.e., Pnl-, Pnd-, and Pne-curves) was assessed. The LRFD resistance factor (φc) of 0.85 is recommended by AISI S100 (2016) to be used with the DSM to design CFS members under compression. The φc value may be increased if the spread of the Pn/Pu ratios could be reduced so that the design could be more economical. Factors affecting the spread of the Pn/Pu ratios of CFS columns have been identified from the literature review as presented in Chapter 2. They include the cross-section geometry (shape and the ratios between the widths of the flat elements in the cross-section), end support condition, initial loading eccentricity, cold-forming effects on the material (i.e., the increase in material strength and residual stresses), and initial geometric imperfection. Among these factors, the increase in material strength due to cold forming is a feasible one that can now be parameterized and considered in estimating the axial strengths of CFS columns. Hence, this chapter is pursued in this direction. In this chapter, the Pnl-, Pnd-, and Pne-curves in the DSM (AISI_S100 2016) were assessed using the full-section tensile yield strength (Fya) as the design parameter. Here, the objectives were to check ⅰ) whether the limit imposed by AISI S100 (2016) for using the full-section tensile yield strength as the design parameter of the DSM can be extended; ⅱ) whether the nominal axial strengths calculated using the full-section tensile yield strengths could be more accurate than those 103 calculated using the Fy values, and; ⅲ) whether the spread of the ratios of nominal-to-tested axial strengths (Pn/Pu) exhibited in the case of using the Fy values as design parameters could be reduced by using the full-section tensile yield strengths as design parameters. In this work, first, the modified Karren’s model and expression, i.e., Equation 3-2 and Equation 3-4, were employed to estimate the full-section tensile yield strengths (Fya) of the column specimens in the 1st experimental database (i.e., Table 3.1 and Table 3.2). As mentioned in Section 4.1 the material properties of the column specimens in the database were mostly tested from coupons cut from the as-formed members at the mid-portion of the flat showing the greatest width in the cross-section. They were used as virgin material properties herein since Karren (1965) has experimentally demonstrated that material properties at this region are very close to those tested from the virgin steel. Then, the nominal axial strengths (Pnl, Pnd, or Pne) of the column specimens were computed using the Fya values and then compared with the tested axial strengths (Pu). Finally, the LRFD resistance factors (φc) of the buckling strength curves using the full-section tensile yield strength as the design parameter were determined with a target reliability index of 2.5. 5.2 Assessment of the Pnl-curve Comparisons are made here between the nominal (Pnl) and tested (Pu) axial strengths of the column specimens that failed in L buckling. The data points of (λl, Pu/Pya) of the solid (count = 135) and perforated (count = 124) column specimens are presented along with the Pnl-curve in Figure 5.1(a) and Figure 5.1(b), respectively. The elastic critical buckling curve is also shown to distinguish between inelastic buckling and post-buckling in these figures. In addition, the data points of (λl, Pnl/Pu) of the solid and perforated column specimens are plotted in Figure 5.1(c) and Figure 5.1(d), respectively. 104 Figure 5.1 Comparisons between the nominal and tested axial strengths of the column specimens that failed in L buckling: (a) Pu/Pya ratios of the solid column specimens versus λl; (b) Pu/Pya ratios of the perforated column specimens versus λl; (c) Pnl/Pu ratios of the solid column specimens versus the λl; (d) Pnl/Pu ratios of the perforated column specimens versus the λl. Overall, the axial strengths of the solid column specimens are reasonably estimated by the Pnl values, where the Pnl/Pu ratios show the minimum, maximum, mean, and standard deviation of 0.78, 1.41, 1.03, and 0.10, respectively. In contrast, the axial strengths of most perforated column specimens are slightly overestimated by the Pnl values, where the Pnl/Pu ratios show the minimum, maximum, mean, and standard deviation of 0.86, 1.35, 1.08, 0.11, respectively. 105 The Pnl values of the column specimens calculated using the Fya values are compared with their counterparts calculated using the Fy values. The results show that ⅰ) after the Fy values being replaced with the Fya values and being fed into the Pnl-curve, the strength estimations of the solid column specimens become a bit unconservative, and the COV value of the Pnl/Pu ratios increased slightly from 9.77% to 10.00%; ⅱ) the perforated column specimens show a case similar to that of the solid column specimens; ⅲ) in comparison, the use of the Fya values exhibits an advantage over the use of the Fy values in estimating the axial strengths of the solid column specimens. To be specific, the number of data points (e.g., (λl, Pu/Pya)) that lie above the plateau of the Pnl-curve (e.g., Pu/Pya = 1) is reduced significantly from 20 to two while using the Fya values instead of the Fy values as the design parameters. Note that this merit appears again while proposing a new L buckling strength curve for the perforated columns considering the net-section tensile yield strength (Fya, net) as the design parameter; see Section 8.2.4. 5.3 Assessment of the Pnd-curve Comparisons are made here between the nominal (Pnd) and tested (Pu) axial strengths of the column specimens that failed in D buckling. The data points of (λd, Pu/Pya) of the solid and perforated column specimens are shown against the Pnd-curve in Figure 5.2. The elastic critical buckling curve is also shown to distinguish between inelastic buckling and post-buckling in these figures. In addition, the data points of (λd, Pnd/Pu) of the solid and perforated column specimens are presented in Figure 5.3. On one hand, the axial strengths of the column specimens tested with the B-F or F-F ends are overall reasonably estimated by the Pnd values regardless of whether the column specimens had holes; see Figure 5.2(a) and Figure 5.2(c). To be specific, the Pnd/Pu ratios of the solid column specimens tested with the B-F or F-F ends (count = 130) show the minimum, maximum, mean, 106 and standard deviation of 0.72, 1.27, 1.01, and 0.13, respectively. The Pnd/Pu ratios of the perforated column specimens tested with the B-F or F-F ends (count = 13) show the minimum, maximum, mean, and standard deviation of 0.89, 1.18, 1.08, and 0.09, respectively. Figure 5.2 Comparisons between the nominal and tested axial strengths of the columns specimens that failed in D buckling: (a) Pu/Pya ratios of the solid column specimens tested with the B-F or F-F ends versus λd; (b) Pu/Pya ratios of the solid column specimens tested with pinned ends versus λd; (c) Pu/Pya ratios of the perforated column specimens tested with the B-F or F-F ends versus λd; (d) Pu/Pya ratios of the perforated column specimens tested with the P-P ends versus λd. 107 On the other hand, the axial strengths of most solid and perforated column specimens tested with pinned ends (the P-P or P-P-W ends) are overestimated by the Pnd values; see Figure 5.2(b) and Figure 5.2(d). To be specific, the Pnd/Pu ratios of the solid column specimens tested with pinned ends (the P-P or P-P-W ends) (count = 55) show the minimum, maximum, mean, and standard deviation of 0.94, 1.83, 1.24, and 0.18, respectively. The Pnd/Pu ratios of the perforated column specimens tested with the P-P ends (count = 58) show the minimum, maximum, mean, and standard deviation of 1.06, 1.66, 1.22, and 0.14, respectively. Note that there are two outliers in the data points of (λd, Pu/Pya) that are shown in Figure 5.2(b), which lie in the vicinity of λd = 2.0 and above the Pnd-curve. The two outliers were excluded from the above statistical results, and the reason has been explained in Section 4.3. Figure 5.3 Comparisons between the nominal and tested axial strengths of the columns specimens that failed in D buckling: (a) Pnd/Pu ratios of the solid column specimens versus λd; (b) Pnd/Pu ratios of the perforated column specimens versus λd. The Pnd values of the column specimens tested with the B-F or F-F ends calculated using the Fya values are compared here with their counterparts calculated using the Fy values, where the key observations are described as follows: 108 ⅰ) The Pnd values of the solid column specimens calculated using the Fya values show the accuracy comparable to those exhibited by the Pnd values calculated using the Fy values. Also, the COV values of the Pnd/Pu ratios obtained from the two cases are nearly the same, which are about 12.74%. ⅱ) When the Fy values are being replaced with the Fya values to be used for estimating the axial strengths of the perforated column specimens, the strength estimations become slightly unconservative, and the COV value of the Pnd/Pu ratios increased a bit from 7.04% to 7.95%. ⅲ) In comparison, the use of the Fya values exhibits an advantage over the use of the Fy values in estimating the axial strengths of the solid column specimens. To be specific, when the Fy values were used to estimate the axial strengths of the solid column specimens, there are a few data points of (λd, Pu/Py) that lie above the plateau (Pu/Py =1) of the Pnd-curve; see Figure 4.3(a) and Figure 4.3(b) in Chapter 4. This is avoided when the Fya values are used to estimate the axial strengths of these column specimens; see Figure 5.2(a) and Figure 5.2(b). Axial strengths of the column specimens tested with pinned ends (the P-P or P-P-W ends) are significantly overestimated by the Pnd values no matter whether the Pnd values were calculated using the Fy values or the Fya values. So, comparisons are not made here between the Pnd values of these column specimens obtained from the two cases. 5.4 Assessment of the Pne-curve Comparisons are made here between the nominal (Pne) and tested (Pu) axial strengths of the column specimens that failed in G buckling. Note that those six columns tested by Bandula Heva and Mahendran (2013) are not included here since the radii of the corners of these column specimens were not given. The data points of (λc, Pu/Pya) of the solid and perforated column specimens are shown along with the Pne-curve in Figure 5.4(a) and Figure 5.4(b), respectively. The 109 elastic critical buckling curve is also shown in these figures. In addition, the data points of (λc, Pne/Pu) of the solid and perforated column specimens are presented in Figure 5.4(c) and Figure 5.4(d), respectively. Figure 5.4 Comparisons between the nominal and tested axial strengths of the column specimens that failed in G buckling: (a) Pu/Pya ratios of the solid column specimens versus λc; (b) Pu/Pya ratios of the perforated column specimens versus λc; (c) Pne/Pu ratios of the solid column specimens versus λc; (f) Pne/Pu ratios of the perforated column specimens versus λc. Comparisons are first made between the nominal and tested axial strengths of the solid column specimens (count = 125). On one hand, axial strengths of the solid column specimens are 110 overall reasonably estimated by the Pne values, where the Pne/Pu ratios show the minimum, maximum, mean, and standard deviation of 0.59, 1.25, 0.99, and 0.12, respectively. On the other hand, Figure 5.4(a) shows that the first segment of the Pne-curve correlates well to the data points of (λc, Pu/Pya) in the slenderness range of λc ≤ 1.5 (count = 107), where the Pne/Pu ratios show the minimum, maximum, mean, and standard deviation of 0.73, 1.25, 1.01, and 0.10, respectively. In contrast, the second segment of the Pne-curve lies below most of the data points of (λc, Pu/Pya) in the slenderness range of λc > 1.5 (count = 18), where the Pne/Pu ratios show the minimum, maximum, mean, and standard deviation of 0.59, 1.09, 0.85, and 0.15, respectively. Note that the data points of (λc, Pu/Pya) in the slenderness range of λc > 1.5 are obtained mainly from column specimens with the P-P-W ends that failed in F buckling. Comparisons are then made between the nominal and tested axial strengths of the perforated column specimens (count = 33). On one hand, the axial strengths of most perforated column specimens are overestimated by the Pne values, where the Pne/Pu ratios show the minimum, maximum, mean, and standard deviation of 0.87, 1.43, 1.18, and 0.15, respectively. This can also be observed from Figure 5.4(b) that the Pne-curve lies above most data points of (λc, Pu/Pya) of the perforated column specimens. On the other hand, 24 perforated column specimens fall within the slenderness range of λc ≤ 1.5, and 23 of them have their axial strengths being overestimated by the Pne values by at least 13%; see Figure 5.4(b) and Figure 5.4(d). The Pne/Pu ratios of these 24 perforated column specimens show the minimum, maximum, mean, and standard deviation of 0.92, 1.43, 1.25, and 0.10, respectively. In contrast, Figure 5.4(b) shows that the second segment of the Pne-curve correlates well to the data points of (λc, Pu/Pya) in the slenderness range of λc > 1.5 (count = 9), where the Pne/Pu ratios show the minimum, maximum, mean, and standard deviation of 0.87, 1.18, 1.00, and 0.10, respectively. 111 The Pne values of the column specimens calculated using the Fya values are compared here with their counterparts calculated using the Fy values. The results are addressed below for the solid and perforated column specimens separately. After the Fy values are replaced by the Fya values to be fed in the Pne-curve to estimate the axial strengths of the solid column specimens, it is observed that: ⅰ) The accuracy of the first segment of the Pne-curve is improved, where the Pne/Pu ratios in the slenderness range of λc ≤ 1.5 have their mean increased from 0.96 to 1.01. Besides, the vertical spread of the data points of (λc, Pne/Pu) that fall within this slenderness range is reduced slightly, where the COV value of the Pne/Pu ratios is decreased from 10.18% to 9.82%. ⅱ) The accuracy of the second segment of the Pne-curve is nearly the same. The vertical spread of the data points of (λc, Pne/Pu) that fall beyond the slenderness range of λc > 1.5 is reduced slightly, where the COV value of the Pne/Pu ratios is decreased from 18.67% to 17.42%. After the Fy values are replaced with the Fya values and being fed into the Pne-curve to estimate the axial strengths of the perforated column specimens, it is observed that: ⅰ) The first segment of the Pne-curve becomes even more unsafe, where the Pne/Pu ratios that fall within the slenderness range of λc ≤ 1.5 have their mean increased from 1.15 to 1.25. While the vertical spread of the data points of (λc, Pne/Pu) that fall within this slenderness range maintains nearly the same. ⅱ) The second segment of the Pne-curve becomes slightly unconservative, and the Pne/Pu ratios that fall beyond the slenderness range of λc > 1.5 have the COV value increased from 8.90% to 9.96%. 112 In summary, using the Fya values instead of the Fy values leads to improved strength estimations for the solid column specimens but inferior strength estimations for the perforated column specimens. 5.5 LRFD Resistance Factors of the Buckling Strength Curves in the DSM The LRFD (Load and Resistance Factor Design) resistance factors (φc) of the buckling strength curves in the DSM (AISI_S100 2016) using the full-section tensile yield strength (Fya) as the design parameter were calculated for a target reliability index of 2.5, where the calculation equation (Equation 4-1) is the same as the one used in Chapter 4. The results are summarized in Table 5.1 and Table 5.2. For comparison, the φc values of the DSM buckling strength curves using the Fy as the design parameter are also shown in the last column of the tables. Table 5.1 Statistics of the Tested-to-Nominal Axial Strength Ratios of the Solid Column Specimens Buckling strength curve Condition λ Pu/Pn φc Min Max Min Max Mean SD COV n Cp using Fya using Fy Pnl-curve - 0.458 4.903 0.709 1.283 0.983 0.100 0.102 135 1.023 0.860 0.906 Pnd-curve B-F and F-F ends 0.585 3.186 0.787 1.395 1.003 0.136 0.136 130 1.024 0.844 0.865 P-P or P-P-W ends 0.505 2.063 0.545 1.065 0.824 0.107 0.130 55 1.057 0.696 0.755 Pne-curve overall 0.535 2.935 0.797 1.705 1.028 0.147 0.143 125 1.025 0.857 0.902 λc ≤ 1.5 0.535 1.493 0.797 1.369 0.998 0.103 0.104 107 1.029 0.871 0.916 λc > 1.5 1.524 2.935 0.915 1.705 1.208 0.226 0.187 18 1.196 0.922 0.920 Table 5.2 Statistics of the Tested-to-Nominal Axial Strength Ratios of the Perforated Column Specimens Buckling strength curve Condition λ Pu/Pn φc Min Max Min Max Mean SD COV n Cp using Fya using Fy Pnl-curve - 0.559 3.034 0.740 1.162 0.935 0.096 0.103 124 1.025 0.817 0.899 Pnd-curve B-F and F-F ends 0.809 1.881 0.844 1.130 0.934 0.082 0.088 13 1.292 0.819 0.887 P-P ends 0.572 1.241 0.604 0.947 0.831 0.083 0.100 58 1.054 0.727 0.825 Pne-curve overall 0.898 2.758 0.701 1.151 0.861 0.120 0.139 33 1.099 0.717 0.782 λc ≤ 1.5 0.898 1.498 0.701 1.084 0.807 0.073 0.090 24 1.141 0.710 0.774 λc > 1.5 1.503 2.758 0.849 1.151 1.006 0.099 0.099 9 1.481 0.863 0.885 113 Table 5.1 and Table 5.2 show that the resistance factors (φc) of all the DSM buckling strength curves, except for the second segment of the Pne-curve dealing with the solid column specimens, decreased when the Fy values are being replaced with the Fya values to estimate the axial strengths of the column specimens. Three cases exhibit φc values greater than 0.85 while the Fya values are being used to estimate column specimens’ axial strengths: ⅰ) the φc value is 0.860 in the case of solid column specimens that failed in L buckling; ⅱ) the φc value is 0.857 in the case of solid column specimens that failed in G buckling, and ⅲ) the φc value is 0.863 in the case of perforated column specimens with λc values > 1.5 that failed in G buckling. Other cases show φc values below 0.85 while the Fya values are being used to estimate the column specimens’ axial strengths. 5.6 Summary In this chapter, full-section tensile yield strengths (Fya) of the column specimens in the database established in Chapter 3 were calculated first. Afterward, the Fya values were fed in the DSM buckling strength curves (AISI_S100 2016) to calculate the column specimens’ nominal axial strengths (Pnl, Pnd, or Pne). The quality of the strength estimations (nominal axial strengths) was assessed with the tested axial strengths, which was later compared with that obtained from the design case without considering the increase in material yield strength due to cold forming. Finally, the LRFD resistance factors (φc) of the DSM buckling strength curves using the full-section tensile yield strength as the design parameter were calculated. It was demonstrated that under a given condition of φc = 0.85, the Fya values are safer to be used in the DSM buckling strength curves to design solid CFS columns subjected to L or G buckling failure under compression. In contrast, the Fya values are unsafe to be used in the DSM buckling strength curves to design perforated CFS 114 columns under compression regardless of the type of buckling. In Chapter 8, the DSM buckling strength curves were recalibrated, especially for using the full-section tensile yield strength as the design parameter. 115 Chapter 6: Elastic Buckling Properties of Perforated CFS Columns with Σ-Shaped Sections 6.1 General Elastic buckling loads of CFS columns are design parameters of the DSM (AISI_S100 2016). On one hand, the literature review showed that past studies that can present an overview of the influences of perforations on the elastic buckling properties of CFS columns under compression are scarce (Moen and Schafer 2008; Moen and Schafer 2006). On the other hand, researchers like Casafont et al. (2012) and Smith and Moen (2014) have proposed simplified solutions based on the finite strip method to quickly determine the elastic buckling loads of CFS columns with perforations. However, they neglected that the finite strip method adopts trigonometric functions as shape functions, which may not apply to describe the buckling modes of perforated CFS columns. In addition, a robust method for determining the lower bounds of elastic L and D buckling loads and the corresponding buckling half-wavelengths of CFS columns with perforations is still lacking. This chapter investigated the elastic buckling properties of three different Σ-shaped CFS sections under compression in two cases of with and without patterned perforations distributed along the length. Compression tests of the relevant columns are introduced in the following chapter. More than 4000 finite element simulations were performed on these sections using the finite element program, ANSYS 18.1. The influences of perforations, perforation pattern, the number of buckling half-wavelengths, and end support condition on the elastic buckling properties of each section were studied. An alternative method was proposed to generate signature curves for 116 the sections without holes, and its effectiveness was tested to generate signature curves for sections with holes. Four problems were addressed in this chapter, as shown in Table 6.1: Table 6.1 Problems addressed in this chapter and the corresponding subsections No. Problems Subsection (a) What the differences are between the elastic buckling properties of solid and perforated CFS columns. 6.4.2 and 6.5.2 (b) Whether the shape functions used to describe the buckling modes of solid CFS columns fit the perforated CFS columns. 6.5.2 (c) How the lower bounds of elastic L and D buckling loads and the associated buckling half-wavelengths of CFS sections with perforations distributed along the length can be determined accurately. 6.4.3, 6.5.1, and 6.5.4 (d) Whether the pinned-pinned end support condition can be an alternative of the simply-simply end support condition to be used for experimentally determining the D buckling strengths of CFS columns. 6.5.3 ⅰ) While addressing the problem (a), the differences in the elastic buckling properties between the solid and perforated Σ-shaped CFS sections are numerically investigated thoroughly. ⅱ) For problem (b), it is demonstrated in this chapter that trigonometric shape functions are not applicable to describe the buckling modes of perforated CFS columns. ⅲ) For addressing the problem (c), the multi-half-wavelengths method is proposed in this chapter, which can accurately determine the lower bounds of elastic L and D buckling loads and the associated buckling half-wavelengths for CFS columns with perforations distributed along the length. ⅳ) For problem (d), it is demonstrated in this chapter that the pinned-pinned end support condition cannot be an alternative to the simply-simply end support condition to be used for CFS columns while experimentally determining their D buckling strengths. The simply-simply end support condition of CFS columns is difficult to be achieved in the compression tests. The pinned-pinned end support condition was widely employed by researchers for experimentally determining 117 the D buckling strengths of CFS columns, where extensive experimental evidences have been shown in Chapter 3. 6.2 Specimens 6.2.1 Column Profiles Three different CFS members used as columns in selective pallet storage racks in North America were selected for this study. The profiles and nominal cross-sections of the columns are shown in Figure 6.1. Along the length of the columns, patterned holes are opened at the flanges, web, and the intermediate stiffener. The net cross-section of a perforated column is the cross-section that has the minimum area along the length. Dimensions of the columns’ nominal cross-sections are shown in Table 6.2. The dimensions shown in the table are along the outer face of the cross-sections, but the corner radii are along the middle face of the cross-sections. These columns have the same thickness of 1.80 mm, and all the corners in the cross-sections have right angles. Cross-section properties of the solid and perforated members are presented in Table 6.3. The X-axis refers to the principal axis of non-symmetry of the cross-section, which is the minor (weak) axis of all the columns presented in this thesis. Figure 6.1 Profiles and cross-sections of columns 118 Table 6.2 Dimensions of nominal cross-sections of columns Section Dimension (mm) Radius (mm) Area (mm2) S F W R D E a1 a2 a3 a4 An Ag ZA30 19.05 79.38 36.51 9.53 9.53 44.45 4.08 432.75 499.97 ZA32 19.05 63.50 28.58 19.05 31.75 38.10 3.44 3.44 4.96 3.44 408.53 512.06 ZA42 28.58 76.20 28.58 44.45 31.75 44.45 3.44 3.44 4.97 3.44 534.26 637.78 Table 6.3 Cross-section properties Section Minor axis Ix (mm4) Iy (mm4) Cw (mm6) J (mm4) Solid ZA30 X-axis 423524.87 580784.64 775121657.94 536.39 ZA32 230456.48 418797.14 440404999.91 550.64 ZA42 423342.61 979516.24 1674480867.63 686.40 Perforated ZA30 X-axis 356724.42 552978.14 721189195.17 464.25 ZA32 167616.81 395735.54 369531927.29 439.60 ZA42 318783.71 902506.27 1395510127.60 575.39 6.2.2 Young’s Modulus Material property tests were conducted to determine Young’s modulus (E), where the results are shown in Table 6.4. Table 6.4 Young's modulus Section ZA30 ZA32 ZA42 E (× 105 MPa) 2.06 2.08 2.06 6.3 Generation and Validation of Column Models In this section, column models generated using computer programs of CUFSM v4.05 (Li and Schafer 2010) and ANSYS 18.1 (2019) are validated with experimental results. The generation of Finite Element (FE) models of CFS columns using ANSYS 18.1 is described in detail. It is worth mentioning that more than 4000 finite element models of CFS columns (including the 119 numerical models used for the experimental validation) were generated in this study. Eigenbuckling analyses were performed on these column models for generating buckling curves and signature curves of the CFS sections. It was tedious to generate signature curves of solid CFS sections due to the employment of constraint equations on the column models. For obtaining one of the signature curves, 50 to 70 h were required to generate the column models and perform eigenbuckling analyses. Computers with Intel(R) Core (TM) i7 processor, four cores, and 16.0 GB RAMs were used to run the simulations. The generation of buckling curves and signature curves of perforated CFS sections was much more tedious than that of the solid CFS sections since different perforation patterns need to be considered at each length. It took more than a month to generate one of the buckling curves/signature curves of the perforated CFS sections. 6.3.1 Buckling Curve and Signature Curve Eigenbuckling analyses of CFS columns under compression are often performed by engineers or researchers. Results of the eigenbuckling analyses have a certain degree of accuracy in predicting the buckling modes of CFS columns under compression. While predicting the axial strengths of CFS columns, more factors need to be considered, such as initial geometric imperfections and material nonlinearities. Two important characteristics: Buckling Curve (BC) and Signature Curve (SC) of CFS sections can be obtained using eigenbuckling analyses (Casafont et al. 2012; Hancock 1984; Kwon et al. 2009; Li and Schafer 2010; Moen and Schafer 2011; Yang and Hancock 2004b). The BC shows a CFS section’s lowest elastic buckling load (corresponding to the 1st eigenmode) against length under a given boundary condition. The SC (with the simply-simply end support condition) is proposed to quickly determine the lower bounds of elastic L and D buckling loads of a CFS section and the associated buckling half-wavelengths. For generating 120 the SC of a CFS section, constraints or shape functions are applied to the column models so that the shapes of buckling modes vary in a sinusoidal way along the length (Li 2009). 6.3.2 End Support Condition In structural design, the end support conditions of CFS columns are often idealized as Simply-Simply (S-S) or Clamped-Clamped (C-C) end support conditions. For the S-S end support condition, the transverse displacements of a column’s ends in the cross-section plane are restrained while the column’s ends are free to rotate and warp. For the C-C end support condition, a column’s ends are fixed except for one of them (the loading end) that can move longitudinally in the form of rigid-body movements. 6.3.3 Finite Element Models In ANSYS 18.1 (2019), the accuracy of the Shell63 element on eigenbuckling analysis was compared with that of the Shell181 element. It is observed that the Shell63 element provided closer results when compared to those of the CUFSM v4.05 (Li and Schafer 2010) than the Shell181 element. Hence, the Shell63 element was selected for this study. The Shell63 element was also adopted by Casafont et al. (2012) to model CFS columns under compression in eigenbuckling analyses. An FE model with Section COOOI (shown in Figure 6.2(a)) and under the S-S end support condition is shown in Figure 6.2(b). At the two ends of the model, the Ux and Uy of all the nodes are restrained, and a uniform compressive load is applied to them. Moreover, the Uz of one node at the center of the web of the mid-length is restrained. Figure 6.2(c) presents the FE model of the section COOOI under the C-C end support condition. In this figure, at the loading end (left end), all the degrees of freedom of the nodes are restrained except that the Uz is coupled to allow only rigid-body movements. All the degrees of freedom of the nodes at the right end are restrained. The size of 2 mm was selected to mesh the cross-sections of the models. Along the length of the 121 models, the meshing sizes of 1 mm, 2 mm, and 4 mm were used in the cases presenting L, D, and TF buckling, respectively. Figure 6.2 Cross-section profile and finite element models 6.3.4 Validation of Column Models Two column specimens with the section COOOI and the length of 900 mm were experimentally tested under compression by Hancock (1984). Cross-section dimensions of the column specimens are presented in Table 6.5, where the symbols are presented in Figure 6.2(a). During the tests, the specimens failed nearly elastically and presented D buckling modes with two halfwaves. The tested ultimate loads and ultimate stresses of the column specimens are shown in Table 6.6. Table 6.5 Measured cross-section dimensions of column specimens with section COOOI Specimen t (mm) A (mm) B (mm) C (mm) D (mm) E (mm) F (mm) G (mm) Ag (mm2) D1 1.625 15.575 89.600 9.465 38.500 31.164 29.811 7.875 401.030 D2 1.630 15.170 89.500 9.492 38.400 31.672 30.406 7.970 402.557 Table 6.6 Results of experiments and eigenbuckling analyses of the column specimens with section COOOI Specimen Experimental results Elastic buckling stress (MPa) Stress ratio Ultimate load (kN) Ultimate stress (MPa) FSM FEM FSM/Exp FEM/Exp D1 100.8 252.0 249.2 250.5 0.99 0.99 D2 92.9 232.0 247.4 248.7 1.07 1.07 122 Eigenbuckling analyses were performed on the two column specimens using the two computer programs. The S-S end support condition was applied to the column models. During the tests, the specimens were compressed between rigid platens (Hancock 1984). The friction generated between the loading platens and the specimens’ ends might have restrained the transverse displacements of the specimens’ ends in the cross-section plane effectively, but the warping of the specimens’ ends might have failed to be properly restrained. Elastic buckling stresses of the specimens obtained using the two computer programs are compared with the tested ultimate stresses as presented in Table 6.6. Elastic buckling stresses of the specimens were expected to be comparable to the tested ultimate stresses. To be specific, the D1 and D2 specimens show slenderness (λd) values of about 1.45. The nominal axial strengths of the specimens were calculated as per the Pnd-curve in the DSM (AISI_S100 2016) of about 282 MPa, which is about 13% higher than the elastic buckling stresses. For specimen D1, the elastic buckling stresses yielded from the two computer programs are very close to the tested ultimate stress, where only 1% differences are found between them. For specimen D2, the elastic buckling stresses yielded from the two computer programs are 7% greater than the tested ultimate stress. Be noted that the tested ultimate stress of specimen D2 is 7.9% lower than that of specimen D1, which indicates that specimen D2 probably had larger initial geometric imperfections than specimen D1. It is observed that the buckling modes yielded from the two computer programs agree well with the tested failure modes of the column specimens. The elastic buckling mode of specimen D1 yielded from the ANSYS 18.1 is presented in Figure 6.3(a) (the one at L = 900 mm). 123 Figure 6.3 Buckling curves of specimen D1 Buckling curves of the section COOOI under the S-S and C-C end support conditions were generated using the two computer programs considering the cross-section dimensions of specimen D1. The length range of the buckling curves was defined as 30 mm to 4000 mm so that the performance of the column models for dealing with different types of buckling modes of L, D, and TF can be evaluated. The buckling curves under the S-S and C-C end support conditions generated using the ANSYS 18.1 are plotted along with those generated using the CUFSM v4.05 in Figure 6.3(a) and Figure 6.3(b), respectively. Good agreements between the curves can be observed in the figures. Differences between the results (elastic buckling stresses) yielded from the two computer programs are within ±2%, and the buckling modes agree well mutually. 6.4 Elastic Buckling Properties of Σ-Shaped Sections (Solid) In this section, the BCs and SCs of the three Σ-shaped sections (solid) were generated using ANSYS 18.1 (2019) and CUFSM v4.05 (Li and Schafer 2010). 6.4.1 Generation of the BC and SC The procedure for generating the BCs of the three Σ-shaped sections was the same as the one introduced in Section 6.3. For generating the SCs of the sections, constraint equations were applied to the column models. This method was motivated by Casafont et al. (2012) and is described herein in detail. The invention of the constraint equations was referred to the shape 124 functions adopted in the FSM (Li and Schafer 2010). In the FSM, a column model is discretized only in the cross-section; hence, it looks like the column model consists of many longitudinal strips. Each longitudinal strip is a single element in the FSM model. The trigonometric function of u = sin(πZ/L) (adopted for the column model under the S-S end support condition) and u = sin(πZ/L)2 (adopted for the column model under the C-C end support condition) are adopted in the FSM to describe the membrane deformation and out-of-plane displacement of the strip in the longitudinal direction (Li 2009; Li and Schafer 2010). In ANSYS 18.1, the depth-to-width ratios of elements were controlled to be ≤ 3. Sizes between 1.5 mm to 3 mm were employed to mesh the cross-sections of column models. In the longitudinal direction, meshing sizes between 1.5 mm to 6 mm were used. The longer the column models, the larger the meshing sizes. For generating the SCs of the sections under the S-S end support condition, half-length models and the SYMM boundary condition were utilized for efficiency. Multiple nodal lines were selected in the FEM column models, and a constraint equation (Equation 6-1) was applied to the nodal lines. The selected nodal lines are marked with red arrows and are shown in Figure 6.4. The nodal lines were carefully selected such that the deformation of the whole cross-section can be effectively restrained, and the generation of membrane stresses in the plane of the cross-section due to the application of the constraint equation can be avoided. In each nodal line, the node at the cross-section where the SYMM boundary condition is applied is the master node, and the rest of the nodes are the slave nodes. Since the master node has the maximum out-of-plane displacement in each nodal line, it is convenient to assume that the maximum displacement to be unity (um = 1). As a result, Equation 6-1 is converted to Equation 6-2. For generating the SCs of the sections under the C-C end support condition, full-length models were employed, where Equation 6-3 served as the constraint equation. In each nodal 125 line, the master node is located at the mid-length. Likewise, Equation 6-3 is converted to Equation 6-4 when the um is assumed to be unity. A FEM model with the ZA32 section after the constraint equation is being applied is presented in Figure 6.4(d) for illustration. In this model, the nodes at the two ends are excluded from the slave nodes since boundary conditions are applied there. Figure 6.4 Selected nodal lines in the FEM column models ussin(πZm/L + π/2) - umsin(πZs/L + π/2) = 0 Equation 6-1 us = sin(πZs/L + π/2) Equation 6-2 ussin2(πZm/L) - umsin2(πZs/L) = 0 Equation 6-3 us = sin2(πZs/L) Equation 6-4 where us = out-of-plane displacement of the slave node; um = out-of-plane displacement of the master node; Zs = Z coordinate of the slave node in the model, where Zs > 0; Zm = Z coordinate of the master node in the model, where Zm ≥ 0, and L = full length of the model. For the FE column models under the S-S end support condition, the SYMM boundary condition is applied to the cross-section at Z = 0. The plane of Z = 0 can be at any of the ends of the FE column models under the C-C end support condition. 126 6.4.2 BCs and SCs of Σ-Shaped Sections (Solid) The SCs and BCs of the ZA30, ZA32, and ZA42 sections (solid) are shown in Figure 6.5, Figure 6.6, and Figure 6.7, respectively. For the BCs in the figures: ⅰ) column buckling modes are indicated; ⅱ) boundaries between adjacent two different types of buckling mode are presented; and ⅲ) few column buckling modes are selected and shown on the right side of the figures. The lower bounds of elastic L and D buckling loads and the associated half-wavelengths are symbolized as (𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆, 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆), (𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆, 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆), (𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐𝐶𝐶−𝐶𝐶 , 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝐶𝐶−𝐶𝐶), and (𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐𝐶𝐶−𝐶𝐶 , 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝐶𝐶−𝐶𝐶). The subscript of the symbol indicates the buckling mode, where 𝑙𝑙 means local buckling and 𝑑𝑑 means distortional buckling. The superscript of the symbol indicates the end support condition. The abscissas in the figures show the half-wavelength for SCs and actual length for BCs, and they are presented in the form of a logarithmic scale and are all ranged from 30 mm to 4000 mm. From these figures, a few findings are summarized below: (1) From the BCs, it can be observed that with length increases, the three sections experience the L, SD, AD + TF, and TF buckling. Under the S-S end support condition, the L + D buckling is observed only on the ZA30 section (see Figure 6.5(a)). The L + D buckling is observed on the ZA30 and ZA32 sections under the C-C end support condition (see Figure 6.5(b) and Figure 6.6(b)). L + D buckling regions are located between the L and SD buckling regions and are very narrow (widths ≤ 20 mm), and there is no evidence showing that the interactive buckling modes affect the buckling loads. 127 Figure 6.5 Signature curves and buckling curves of the solid ZA30 section (2) The appearance of TF modes of the three sections is always accompanied by an AD mode. At the lengths where the sections experiencing AD + TF buckling, the buckling loads are lower than the counterparts of experiencing pure TF buckling; see the gap between the BC and the TF-FSM curve in each figure. With the increase in length, the participation of the AD mode decreases. The buckling modes of the sections, therefore, approach pure TF modes and hence the gaps close gradually. 128 Figure 6.6 Signature curves and buckling curves of the solid ZA32 section (3) The lower bounds of elastic L and D buckling loads and the associated half-wavelengths are indicated in the figures and are listed in Table 6.7. The SCs in Figure 6.7 fail to show the (𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆, 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆) and (𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐𝐶𝐶−𝐶𝐶 , 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝐶𝐶−𝐶𝐶) of the ZA42 section; consequently, they are determined using the methodology of FSM@cFSM – 𝑙𝑙𝑐𝑐𝑐𝑐 (Li and Schafer 2010). (4) For the BCs (under the C-C end support condition) of the three sections, the buckling loads gradually decrease after the lengths of the 𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐𝐶𝐶−𝐶𝐶 and approach the 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆; see the BCs in Figure 6.5(b), Figure 6.6(b), and Figure 6.7(b). Hancock (1984) pointed out that when multiple L or D buckling half-wavelengths occur in a column under compression, the restraint effect of the end 129 support condition on the buckling load is diminished. This phenomenon is termed herein as the multi-half-wavelengths effect. (5) While experimentally determining the L buckling strength of the CFS column, the AISI_S902 (2017) requires the lengths of column specimens shall not be less than three times the greatest overall width of the cross-section and not exceed 20 times the minimum radius of gyration (rmin). This length range is presented in Figure 6.5(b), Figure 6.6(b), and Figure 6.7(b) by the blue and dashed lines, and the lines are with the ordinate of the 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆. Since the L buckling of the three sections all occurs at flanges, the width of flange (B) is used here instead of the greatest overall width of the cross-section. From Figure 6.5(b), Figure 6.6(b), and Figure 6.7(b), it can be observed that when L = 3B, the ZA30, ZA32, and ZA42 sections have elastic buckling loads that are 5.0%, 5.5%, and 6.6% above their 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆, respectively. Although the differences are not significant, the longer the lengths, the more minor the differences. (6) In practice, CFS columns are usually braced continuously along the length, which allows for the occurrence of multiple buckling half-wavelengths. In the DSM (AISI_S100 2016), for considering the multi-half-wavelengths effect, a column piece between adjacent two bracing points is isolated and assumed to be under the S-S end support condition. In this circumstance, the 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆 and 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆 are the design parameters of the DSM. This is why the SC has been deemed an indispensable tool for designing CFS columns under compression. From Figure 6.5(a), Figure 6.6(a), and Figure 6.7(a), it can be observed that the (𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆, 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆) and (𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆, 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆) are not only shown in the SCs but also shown in the BCs, and the BCs coincide with the SCs at the global buckling regions. Note that the 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆 of the three sections are smaller than their 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆, which results in those distinct minima of the (𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆, 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆) to be shown in the BCs. 130 Figure 6.7 Signature curves and buckling curves of the solid ZA42 section (7) In all the figures, the SCs from FEM coincide well with the SCs from FSM in the regions of D and G buckling, and the relative differences between the FEM results and the FSM results are within ±5%. In L buckling regions, the SCs from FEM shift downwards slightly relative to the SCs from FSM. For the SCs under the S-S end support condition, the shifts are due to the warping caused by the end deformation is considered in the FEM models rather than the FSM models. The FSM overestimated the elastic L buckling loads of the sections under the C-C end support condition, which caused the shifts of the SCs from the FEM. The lower bounds of elastic L and D buckling loads and the associated half-wavelengths of the three sections are given in Table 131 6.7, where the results obtained from the SCs of FEM are compared with those obtained from the SCs of FSM. From Table 6.7, regarding the lower bound of elastic L buckling load and the associated half-wavelength, it can be observed that the FEM results are lower than the FSM results by the relative differences of -2.1% to -7.1% and -2.4% to -4.3%, respectively. In terms of the lower bound of elastic D buckling load and the associated half-wavelength, the FEM results are very close to the FSM results by the relative differences of -2.5% to 1.0% and -1.7% to 0.0%, respectively. The results captured from the BCs of FEM are almost the same as those determined from the SCs of FEM, which are also presented in Table 6.7. (8) The width and initial length of buckling regions of the sections are apparently different under the two end support conditions. For the BCs under the C-C end support condition, the widths of L buckling regions are more than two times those of the BCs under the S-S end support condition. Moreover, the columns under the S-S end support condition show a single D buckling halfwave, while the columns under the C-C end support condition show multiple D buckling halfwaves. (9) The SCs under the C-C end support condition (FEM and FSM) fail to characterize the elastic buckling properties of the sections. Firstly, the SCs fail to indicate the lower bounds of the elastic L buckling load of the sections. Secondly, the SCs fail to show the D buckling region and the lower bound of elastic D buckling load. Moreover, the SCs fail to present the initial length of G buckling regions and even overestimate the elastic buckling loads of the ZA32 and ZA42 sections at the beginning of the G buckling regions. 132 Table 6.7 Lower bounds of elastic local and distortional buckling loads and the corresponding half-wavelengths Parameters ZA30 ZA32 ZA42 S-S C-C S-S C-C S-S C-C 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐 (kN) FSM 259.8 358.7 415.0 567.6 325.6 437.1 FEM SC 241.3 343.2 388.4 545.8 311.4 427.8 D (%) -7.1 -4.3 -6.4 -3.8 -4.4 -2.1 BC 241.2 - 388.1 - 311.0 - D (%) -7.1 - -6.5 - -4.5 - 𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐 (mm) FSM 69.0 105.0 55.0 85.0 71.0 110.0 FEM SC 66.0 102.0 53.0 83.0 69.0 107.0 D (%) -4.3 -2.9 -3.6 -2.4 -2.8 -2.7 BC 66.0 - 53.0 - 69.0 - D (%) -4.3 - -3.6 - -2.8 - 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐 (kN) FSM 164.9 277.8 292.8 500.8 284.2 491.3 FEM SC 165.3 272.2 295.6 489.8 286.2 479.0 D (%) 0.3 -2.0 1.0 -2.2 0.7 -2.5 BC 164.1 - 292.9 - 283.9 - D (%) -0.5 - 0.0 - -0.1 - 𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐 (mm) FSM 641.0 983.0 563.0 868.0 964.0 1470.0 FEM SC 641.0 968.0 561.0 853.0 - - D (%) 0.0 -1.5 -0.4 -1.7 - - BC 640.0 - 563.0 - - - D (%) -0.2 - 0.0 - - - 6.4.3 Multi-half-wavelengths Method The effect of multi-half-wavelengths can be utilized to determine the (𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆, 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆) and (𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆, 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆) of CFS sections. The way is to perform eigenbuckling analyses on a section under the C-C end support condition, where the model has the length of 𝑚𝑚𝑙𝑙 and is constrained or braced at the ends of each buckling half-wavelength. By doing so, fixed half-wavelengths can be defined, and the interaction of L or D buckling with other buckling modes is prevented. The 𝑚𝑚 is the number of half-wavelengths; the 𝑙𝑙 is the half-wavelength. Eigenbuckling analyses are performed by gradually varying the 𝑙𝑙; hence, a buckling curve can be generated. The minimum point of the buckling curve shows the lower bound of the elastic buckling load and the associated half-133 wavelength. Two types of FE models can be adopted here. Type 1 model is the column model constrained using Equation 6-5. Odd numbers are defined for the m; hence, master nodes can be assigned to the cross-sections at the mid-length. Type 2 model is the column model with the out-of-plane displacement of a few nodes restrained. The nodes are at the cross-sections of (n = 1, 2, 3, … m-1), and their locations are shown in Figure 6.4. The reason for proposing the two types of models is that while determining the (𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆, 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆), the type 2 model is only applicable to the sections featuring the 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆 smaller than the 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆. us = sin(mπZs/L)sin(πZs/L) Equation 6-5 where m = 2n + 1, n = 1, 2, 3,… The m should be large enough to fully diminish the restraint effect of the C-C end support condition. Sensitivity analyses of the elastic buckling loads of the sections on the 𝑚𝑚 were performed. The 𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆 and 𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆 were selected for the 𝑙𝑙, respectively, and both the type 1 and type 2 models were used. The results obtained using the two types of models are close; hence, only the results of the type 2 models are presented in Figure 6.8. It can be observed from the figure that when the m reached 11, the ratios of 𝑃𝑃/𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆 and 𝑃𝑃/𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆 are very close to unity. One result of the ZA42 section is not shown in Figure 6.8(b) since this column presented the L buckling when m = 3. Figure 6.8 Sensitivity of buckling loads on the number of buckling half-wavelengths 134 The accuracy of using the multi-half-wavelengths method to determine the 𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆 and 𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆 of the sections was evaluated. Buckling curves generated using the two types of models are presented in Figure 6.9. From Figure 6.9(a), Figure 6.9(c), and Figure 6.9(e), it can be seen that the (𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆, 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆) shown here are almost the same as the FSM results (see Table 6.7). From Figure 6.9(b), Figure 6.9(d), and Figure 6.9(f), it can be seen that the results from the type 1 models show the maximum differences of 3.9% and -2.3% on the 𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆 and 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆, respectively, while compared with those FSM results in Table 6.7. As for the type 2 models, the maximum differences are 1.2% on the 𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆 and 1.8% on the 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆 . In Figure 6.9(f), the buckling curves fail to show the (𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆, 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆) of the ZA42 section, which agrees well with the SC (FSM) as shown in Figure 6.7(a). Figure 6.9 Buckling curves generated using the type 1 and type 2 models 6.5 Elastic Buckling Properties of Σ-Shaped Sections (Perforated) In this section, the elastic buckling properties of the ZA30, ZA32, and ZA42 sections are studied, considering patterned holes distributed along the length. 135 6.5.1 Perforation Patterns Effects When cutting a piece from a CFS column having patterned perforations distributed along the length, the perforations contained in that piece can be different. For generating the SCs and BCs of a perforated section, at each half-wavelength or length, the lowest value among the elastic buckling loads associated with all the possible perforation patterns should be selected. Sensitivity analyses of the elastic buckling loads of the three perforated sections on the perforation pattern were performed. Two strategies were proposed to do the analyses. Strategy 1 is illustrated in Figure 6.10, and the column shown in this figure has the perforated ZA32 section and a length of 90 mm. For this perforation pattern sensitivity analysis, 33 numerical models were generated. It will be time-consuming to use strategy 1 to generate the SCs and BCs of the sections; hence, a simplified strategy (strategy 2) was proposed. For strategy 2, only five perforation patterns (perforation patterns (a) to (e)) are considered for lengths experiencing L buckling. For lengths experiencing D and G buckling, perforation patterns (b) to (e) are considered. The perforation patterns (a) to (e) are described below, and the associated examples are given in Figure 6.11: ⅰ) Perforation pattern (a) has a pair of flange holes opened at the mid-length of the column; see Figure 6.11(c). ⅱ) Perforation pattern (b) has a pair of web holes opened at the mid-length of the column; see Figure 6.11(e). ⅲ) Perforation pattern (c) has a pair of web holes opened at the top end of the column; see Figure 6.11(d). ⅳ) Perforation pattern (d) has a pair of web holes opened at the bottom end of the column; see Figure 6.11(a). 136 Figure 6.10 Flowchart of strategy 1 ⅴ) Perforation pattern (e) has the column ends opened by web holes with the same depth of openings; see Figure 6.11(b). 137 Figure 6.11 Perforation patterns of the column with the ZA32 section For each perforated section, two lengths were randomly selected for the L, D, and G buckling (one for AD + TF buckling and one for TF buckling). Strategy 1 was used to perform the perforation pattern sensitivity analyses, and the ratios of the elastic buckling load to the minimum elastic buckling load (P/Pmin) are plotted against the shift distance in Figure 6.12. From Figure 6.12(a) and Figure 6.12(d), it can be seen that the maximum ratios of the P/Pmin are 1.11, 1.15, and 1.08 for the ZA30, ZA32, and ZA42 sections with lengths experiencing L buckling, respectively. For the sections with lengths experiencing D buckling, the ratios of the P/Pmin are all under 1.02 (see Figure 6.12(b) and Figure 6.12(e)). In the cases of G buckling, see Figure 6.12(c) and Figure 6.12(f), the ratios of the P/Pmin are all less than 1.01. The L buckling load is much more sensitive to the perforation pattern than the D and G buckling loads. 138 Figure 6.12 Perforation pattern effects on elastic buckling loads The performance of strategy 2 was assessed. The results of the perforation pattern sensitivity analyses from using the two strategies are presented in Table 6.8. It can be seen from the table that the buckling loads obtained from the two strategies are almost the same, and the relative differences are no more than 0.3%. Perforation patterns corresponding to the minimum buckling loads in strategy 2 are given in the sixth column of the table. It is interesting to find that in the L buckling cases, all the minimum buckling loads are obtained at the perforation pattern (a). This indicates that flange holes located at the peak of the L buckling halfwave have the most significant effect on the columns’ elastic L buckling loads. It is proved here that strategy 2 is unbiased and accurate for performing the perforation pattern sensitivity analysis; hence, it is employed in the generations of SCs and BCs of the three perforated sections. 139 Table 6.8 Results of perforation pattern sensitivity analyses Buckling mode Column L (mm) Strategy 1 Strategy 2 Difference (%) P (kN) P (kN) Perforation pattern Local ZA30 102 288.44 - (a) - 73 339.48 (a) ZA32 90 413.48 (a) 83 418.11 (a) ZA42 108 341.39 (a) 120 339.62 (a) Distortional ZA30 966 211.74 211.82 (b) 0.04 1125 207.18 207.18 (b) 0.00 ZA32 853 322.97 - (c) - 759 334.37 335.14 (b) & (e) 0.23 ZA42 1470 371.10 - (d) - 700 671.43 671.43 (e) 0.00 Global ZA30 2300 124.04 124.08 (d) 0.03 3000 78.35 78.36 (d) 0.02 ZA32 1650 211.45 211.65 (c) 0.09 3000 87.19 - (c) - ZA42 2500 253.25 - (b) - 3500 158.86 158.90 (b) & (e) 0.02 Note: ‘-‘ means the results obtained using strategy 2 are the same as those obtained using strategy 1; () & () means two perforation patterns occurred in a column model simultaneously. 6.5.2 SCs and BCs of Σ-Shaped Sections (Perforated) The BCs (under the C-C end support condition) of the three perforated sections were generated using the ANSYS 18.1. It was assumed that the shape function (Equation 6-4) of solid CFS columns applies to perforated columns, and the SCs (under the C-C end support condition) of the perforated sections were generated as well. Strategy 2 was employed here for conducting the perforation pattern sensitivity analysis while one change was made: only the perforation pattern (b) was considered for lengths experiencing G buckling. The reason for making this change is that it has been proved that the elastic G buckling loads of the three sections are insensitive to the perforation pattern. It is found that the perforation pattern (the one that yields the minimum buckling load) of a column can be different from that of the same column but with different 140 constraints applied to them. As a result, the perforation pattern sensitivity analysis was performed in the generations of the SCs and BCs of the sections individually. The SCs and BCs of the ZA30, ZA32, and ZA42 sections (perforated) are shown in Figure 6.13, Figure 6.14, and Figure 6.15, respectively. For the BCs: ⅰ) the column buckling modes are indicated in the figures; ⅱ) the boundaries between adjacent two different types of buckling mode are presented, and; ⅲ) few column buckling modes are selected and shown on the right side of the figures. The SCs and BCs here are compared with those of the solid sections under the C-C end support condition, and the influences of the perforations on the elastic buckling loads and modes are discussed below: Figure 6.13 Signature curve and buckling curves of the perforated ZA30 section (1) The D buckling of the perforated sections takes place at relatively shorter lengths compared to those of the solid sections. The lengths that start to show D buckling are 926 mm, 1020 mm, and 2606 mm for the solid ZA30, ZA32, and ZA42 sections, respectively. The perforated sections start to show D buckling at the length of 820 mm for the ZA30 section, 711 mm for the ZA32 section, and 2198 mm for the ZA42 section. Moreover, the widths of the D buckling regions are increased by the perforations from 992 mm to 1149 mm (15.8% increment) for the ZA30 section, from 276 mm to 624 mm (126.1% increment) for the ZA32 section, and from 82 mm to 211 mm (157.3% increment) for the ZA42 section. 141 Figure 6.14 Signature curve and buckling curves of the perforated ZA32 section (2) The initial lengths of G buckling regions are postponed by the perforations from 1918 mm to 1969 mm for the ZA30 section and from 1296 mm to 1335 mm for the ZA32 section. On the contrary, the perforated ZA42 section shows the G buckling firstly at the length of 2409 mm, which is 279 mm shorter than that of the solid ZA42 section (2688 mm). Figure 6.15 Signature curve and buckling curves of the perforated ZA42 section (3) The length ranges (x) of [3B, 20rmin] are presented in Figure 6.13, Figure 6.14, and Figure 6.15 with the ordinate of the 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆 of the corresponding perforated sections. The determinations of the 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆 of the perforated sections are illustrated in Section 6.5.4. At the lengths of 3B, the elastic buckling loads of the perforated ZA30, ZA32, and ZA42 sections are 5.7%, 3.3%, and 8.1% above their 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆, respectively. (4) The SCs fail to characterize the elastic buckling properties of the three perforated sections under the C-C end support condition (see the gap between the SC and BC in Figure 6.13, 142 Figure 6.14, and Figure 6.15). This is similar to what has been found on the solid sections, while the SCs here do not coincide with the BCs even at the lengths before the first minimum of the SCs. By checking the buckling modes (without constraints), it was observed that the buckling shapes of the perforated columns are not following the shape of the sinusoidal curve (Equation 6-4) in the longitudinal direction. The elastic buckling loads of the perforated sections yielded in the perforation pattern sensitivity analyses while generating the BCs are analyzed here. The ratios of the maximum buckling load to the minimum buckling load (Pmax/Pmin) are plotted against the ratios of 𝑙𝑙/𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆 in Figure 6.16. It can be observed from the figure that the ratios of the Pmax/Pmin become less than 1.01 when the ratios of the 𝑙𝑙/𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆 go beyond the 3.8, 3.6, and 4.7 for the ZA30, ZA32, and ZA42 sections, respectively. The shortest specimen length required in the AISI_S902 (2017) for determining the L buckling axial strength of the CFS column is 3B. However, from the current study, it is observed that if the shortest specimen length is changed to 5𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆, the considerations of the effects of the perforation pattern are not necessary, and the yielded elastic buckling loads are more conservative. The columns start to show D buckling after the 𝑙𝑙/𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆 ratios reach the 10.79, 12.26, and 28.55 for the ZA30, ZA32 and ZA42 sections, respectively. Beyond these lengths, the ratios of Pmax/Pmin are very close to unity. This explains why only the perforation pattern (b) was considered for the G buckling cases while generating the SCs and BCs of the perforated sections. Figure 6.16 The ratios of the maximum buckling loads to the minimum buckling loads 143 6.5.3 Columns with Pinned-Pinned End Support Condition It is difficult to test CFS columns under the S-S end support condition physically; therefore, the end support condition of Pinned-Pinned (P-P) is generally employed by researchers to determine the D buckling strengths of CFS columns. The P-P end support condition is to restrain the torsion and warping of a column’s ends, but the column ends are free to rotate about the two principal axes of the cross-section. To date, however, researchers failed to justify the applications of the P-P end support condition in experiments since the erosion of the buckling load of the CFS columns due to the multi-half-wavelengths effect may not be properly considered. Eigenbuckling analyses were performed to generate the BCs of the three perforated sections under the P-P end support condition. An FE model with the ZA42 section is shown in Figure 6.17. In this figure, the column ends are rigidly attached to two steel loading plates, and the thickness of the loading plates is 30 mm. At the top face of each loading plate, there is a steel block at the centre. The depth of the steel blocks is 40 mm, and their centroids coincide with the effective centroid of the column section determined at the ultimate load of L buckling. The effective centroid of a CFS section is on the line of action of the internal force on the cross-section (Rasmussen and Hancock 1993; Young and Rasmussen 1999). In this study, the effective centroid of a perforated section at the ultimate load of L buckling was determined by looking for a point on the cross-section where the column can reach the maximum capacity when a concentrated compressive force is applied on this point. The column should be short enough to ensure an L buckling failure. Thus, nonlinear numerical simulations of the stub columns under compression were performed for the three perforated sections, where different loading points on the axis-of-symmetry were tested. The Solid185 element is used for the loading plates and steel blocks, and the Shell63 element is used for the column. The column ends share the nodes with the loading plates at the two interfaces, and 144 a layer of the Shell63 elements is assigned to each interface since the nodes of the Solid185 element have none of the rotational degrees of freedom, and hence, can not prevent the rotations of the nodes of the Shell63 element. The end support condition P-P is applied to the nodes at the top face of each steel block. The left end of the model is the loading end. At the loading end, a unit force (Fz = 1 kN) is applied to the node at the centroid of the steel block (see the white arrow), and the Ux and Uy of this node are restrained. Another two nodes with their Ux having restrained are collinear with the loading node. The right end of the model is restrained likewise, except the Uz of the node at the centroid of the steel block is restrained. Figure 6.17 A FEM model with the perforated ZA42 section under pinned-pinned end support condition The BCs of the three perforated sections under the P-P end support condition are shown in Figure 6.13, Figure 6.14, and Figure 6.15. The following findings are summarized: (1) The BCs under the P-P end support condition coincide well with the BCs under the C-C end support condition from the very beginning, and then the two curves start to separate from the lengths of 1653 mm, 626 mm, and 1254 mm for the perforated ZA30, ZA32, and ZA42 sections, respectively. After the separations, the BCs under the P-P end support condition drop fast. The sections under the P-P end support condition start to show different buckling modes, which are responsible for the separations. To be specific, at the length of 1653 mm, the perforated 145 ZA30 section under the C-C end support condition is close to the end of its D buckling region, while the section under the P-P end support condition starts to show the AD + TF buckling. At the length of 626 mm, the perforated ZA32 section under the C-C end support condition is close to the end of its L buckling region, whereas the section under the P-P end support condition starts to show D buckling. The perforated ZA42 section under the C-C end support condition remains in the L buckling region at the length of 1254 mm, while the section under the P-P end support condition starts to show SD + F buckling. For those sections, under the P-P end support condition, the D or G buckling occurs at shorter lengths than those sections under the C-C end support condition. (2) The G buckling modes of the perforated ZA32 and ZA42 sections under the C-C end support condition are AD + TF or TF buckling modes, while the two sections showed the SD + F buckling mode under the P-P end support condition. (3) The probable estimation of the 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆 of a section using the P-P end support condition is checked here. Two methods were proposed, and their performances were evaluated. The first method is to perform an eigenbuckling analysis on a section under compression, and the section has the length of 2𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆 and is under the P-P end support condition. The buckling load of the first buckling mode serves as an estimation of the 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆. The determinations of the (𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆, 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆) of the three perforated sections are presented in Section 6.5.4. The second method is to use the buckling load of a section with the length at its end of the D buckling region as the estimation of the 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆. The results (elastic buckling load P and buckling mode) obtained from the two methods are presented in Table 6.9, where the ratios of 𝑃𝑃/𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆 are shown as well. It can be observed from the table that most of the ratios of the 𝑃𝑃/𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆 are far away from 1.0. The results of the two cases are not shown in Table 6.9 since the D buckling mode (SD) of the perforated ZA32 section under the 146 P-P end support condition gradually transfers to the G buckling mode (SD + F) along the length, which makes the end of the D buckling region difficult to determine. For the perforated ZA42 section under the P-P end support condition, D buckling mode was not observed since the column presents G buckling (SD + F mode) right after the L buckling region. The two methods fail to provide accurate estimations of the 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆 of the sections; consequently, it can be concluded that the P-P end support condition is not suitable for determining the lower bounds of elastic D buckling load of CFS sections. Table 6.9 Estimations of the 𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷−𝑷𝑷 of the perforated sections using the P-P end support condition Column Method 1 Method 2 Length (mm) Buckling mode P (kN) 𝑃𝑃 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆⁄ Length (mm) Buckling mode P (kN) 𝑃𝑃 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆⁄ ZA30 1362 SD 179.56 1.33 1652 SD 164.52 1.22 ZA32 1136 SD 198.83 0.94 - - - - ZA42 1928 SD + F 144.88 0.66 - - - - 6.5.4 Determination of the Lower Bounds of Elastic Local and Distortional Buckling Loads The SCs of the three perforated sections under the S-S end support condition were not generated since the results of trial simulations showed that along the length, the out-of-plane displacement of the columns’ buckling modes failed to follow the shape of a single half-sinewave. The multi-half-wavelengths method using the type 2 models, consequently, was selected here to determine the (𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆, 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆) and (𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆, 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆) of the three perforated sections, and the perforation pattern effects were considered. The results are presented in Table 6.10, and the buckling curves are shown in Figure 6.18. It may be argued that like the solid sections, the lower bounds of elastic L and D buckling loads and the associated half-wavelengths could be determined by generating their buckling curves under the S-S end support condition. This method could be more efficient 147 compared to the multi-half-wavelengths method. The reason to use the type 2 model was that the BCs (under the S-S end support condition) underestimated the (𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆, 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆) of the solid sections due to the influence of the warping of the loading end. In addition, holes may appear at the column ends to affect the smoothness of the buckling curve, leading to a difficult identification of the minimum point. Table 6.10 Lower bounds of elastic local and distortional buckling loads and the corresponding half-wavelengths of perforated sections Section Buckling properties 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆 (kN) 𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆 (mm) 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆 (kN) 𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆 (mm) ZA30 229.4 76.0 135.2 681.0 D (%) -11.7 10.1 -18.0 6.2 ZA32 345.0 58.0 212.3 568.0 D (%) -16.9 5.5 -27.5 0.9 ZA42 273.1 77.0 218.6* 964.0* D (%) -16.1 8.5 -23.1 0.0 Note: * means the value is an estimation. Figure 6.18 Buckling curves of the three perforated rack sections 148 The (𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆, 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆) and (𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆, 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆) of the perforated sections are compared with those of the solid sections, and the relative differences are presented in Table 6.10. The results show that after considering the perforations, the lower bounds of elastic L buckling loads are decreased by -11% to -17%. The lower bounds of elastic D buckling loads are dropped more significantly than that of the elastic L buckling loads, where the decrements are -18.0%, -27.5%, and -23.1% for the ZA30, ZA32, and ZA42 sections, respectively. On the contrary, the associated L buckling half-wavelengths and D buckling half-wavelengths are increased due to the perforations, and the increments are close to or less than 10%. In the case of the perforated ZA42 section, it is difficult to determine the (𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆, 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆). The 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆 of this section can be properly estimated using the elastic buckling load of the perforated section at the 𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆 of the solid ZA42 sections. This is because the perforations do not significantly change the D buckling half-wavelengths of the ZA30 and ZA32 sections, whereas the buckling curve of the perforated ZA42 section is quite flat (see Figure 6.18(f)). 6.6 Summary This chapter introduced the elastic buckling properties of three different Σ-shaped sections under compression. Two scenarios were considered: the sections without perforations and the sections with patterned perforations distributed along the length. The elastic buckling properties of the sections in these two scenarios were compared by generating their buckling curves. The ANSYS 18.1 (2019) was successfully used to generate the signature curves of the three solid sections. The sensitivity analyses of elastic buckling loads of L, D, and G buckling to perforation patterns were performed, and the perforation pattern effects were well considered in the generations of the signature curves and buckling curves of the perforated sections. The lower bounds of elastic L and D buckling loads and the associated half-wavelengths of both the solid and 149 perforated sections were accurately determined using the multi-half-wavelengths method. The possibility of using the pinned-pinned end support condition for estimating the lower bounds of elastic D buckling load of the sections was checked. The following conclusions are made from this study: (1) The end support condition has significant effects on CFS sections’ elastic buckling load, buckling mode, and the region of buckling mode. The pinned-pinned end support condition is not suitable to determine the lower bound of elastic distortional buckling load of the CFS section. (2) The multi-half-wavelengths effect should be properly considered in the structural design of CFS columns. The pinned-pinned end support condition should be used with caution for experimentally determining the distortional buckling strengths of CFS columns under compression since it may not be able to properly consider the effect of multi-half-wavelengths. (3) After considering the perforations, the lower bounds of elastic local and distortional buckling loads of the three Σ-shaped sections decreased by more than 10%. On the contrary, the associated local and distortional buckling half-wavelengths increased, where the increments were close to or less than 10%. (4) The multi-half-wavelengths method has been proved to be unbiased and accurate for determining the lower bounds of elastic local and distortional buckling loads and the associated half-wavelengths of perforated Σ-shaped sections, where the type 2 model and the number of buckling half-wavelengths of 11 are suggested. (5) Signature curves under the C-C end support condition failed to characterize the elastic buckling mode and load of the sections under compression; hence, they are not suggested to be the references for predicting the axial strengths of CFS columns under the C-C end support condition. 150 (6) Perforation pattern effects on the elastic local buckling load of the three Σ-shaped sections were found significant when having lengths less than 5𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆. The effects on the distortional and global buckling loads of the three sections were found minor and can be ignored. (7) The trigonometric shape functions adopted in the finite strip method failed to describe the buckling shapes of perforated columns accurately and hence, should be avoided. (8) To determine the local buckling strength of a CFS section experimentally, it is suggested here that the specimen lengths shall not be less than 5𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆 and not exceed 20 times the minimum radius of gyration (𝑟𝑟). Currently, the development of simplified methods for determining the elastic buckling properties of perforated CFS sections under compression is being pursued by researchers. The multi-half-wavelengths method was proposed in this chapter, which can accurately determine the lower bounds of elastic L and D buckling loads and the associated half-wavelengths of Σ-shaped perforated sections under compression. This method can be a robust tool to be employed for validating the simplified methods. Future research will look into its applicability to other shapes of CFS sections (e.g., channel, hat, and rack). The results of this chapter provide a solid basis for the design (the selection of design parameters), experimental tests (the selection of end support conditions), and numerical simulations (the generation of robust numerical models) of both solid and perforated CFS columns under compression. This chapter, however, only focused on the elastic buckling properties of CFS columns under compression. 151 Chapter 7: Compression Tests of Perforated CFS Columns with Σ-Shaped Sections 7.1 General Efforts have been made in Chapters 4 and 5 to assess the accuracy of the buckling strength curves in the DSM (AISI_S100 2016) with a substantial amount of experimental data collected from the literature. Although numerous experimental studies related to perforated CFS members under compression have been carried out, only limited experimental data was collected successfully, especially the experimental data related to D and G buckling. This is because the experimental data reported in the literature is often incomplete due to confidentiality issues. Consequently, an overall picture of the accuracy of the DSM buckling strength curves for perforated CFS members under compression is still missing. In this chapter, first, thin-walled roll-formed steel columns (count = 55) with three different Σ-shaped sections and seven different lengths were tested under compression. The columns had patterned holes uniformly distributed along the length and mostly were tested with pinned ends. The columns’ axial strengths and stiffnesses, failure modes, and structural responses under compression are reported in this chapter. Then, the design parameters required by the DSM for estimating the axial strengths of the columns were determined, and the results are also reported in this chapter. Finally, the accuracy of the DSM was assessed with the tested axial strengths of the columns. This chapter aims at ⅰ) enlarging the experimental database of perforated CFS members under compression and ⅱ) examining whether the DSM could handle pin-ended CFS members with Σ-shaped sections adequately. 152 7.2 Specimens 7.2.1 Nominal Sections CFS columns with three different roll-formed sections manufactured in North America were tested in this chapter, where the sections are shown in Figure 7.1 and are designated as ZA30, ZA32, and ZA42. Note that the literature review showed that CFS columns with such cross-section shapes had not been studied by other researchers. Along the length, patterned holes were punched at the flanges and web following a constant interval of 76.2-mm. The nominal sections’ geometric dimensions, gross areas (Ag), and net areas (An) have been shown in Table 6.2. Figure 7.1 Profiles and net sections of columns 7.2.2 Lengths of Specimens Lengths of specimens were selected considering two conditions. First, the lengths of specimens should include stub columns and less than the maximum length of 2370-mm that can be accommodated in the testing frame. Second, specimens with multiple lengths featuring short 153 intervals were preferred to investigate the influences of length on the axial strength and failure mode of the columns. As a result, seven lengths of 250, 500, 760, 1010, 1520, 1820, and 2150-mm were selected for preparing specimens; at least two identical specimens were prepared for each length for each section. Finally, the specimens’ ends were polished carefully to improve the flatness. 7.2.3 Measurement of Specimens Lengths and cross-sections of the specimens were measured, where the results are presented in Table 7.1, Table 7.2, and Table 7.3. For any measurement in the tables: ⅰ) the length and thickness are the average results of multiple measurements; ⅱ) the thickness (t) is the average thickness minus an approximate coating thickness of 0.04-mm from (ASTM 2013); ⅲ) dimensions of cross-sections and angles of corners are the average results of two measurements conducted at the column ends, and; ⅳ) the gross (Ag) and net (An) cross-section areas were determined using a drafting computer software, AutoCAD. The angles of the corners, except those of the a3 to a5 corners (see Figure 7.1(b)) at the web stiffener of ZA30-section specimens, offset from 90 degrees by -9.5 to 5.0 degrees due to the elastic spring-back of the specimens that occurred after the cold-rolling and cutting. In addition, elastic spring-back altered the shape of the web stiffener of ZA30-section specimens significantly; hence, the actual shape of the sections is presented in Figure 7.1(b). The measurements of ZA30-section specimens presented in Table 7.1 correspond to Figure 7.1(b). 154 Table 7.1 Geometric dimensions of ZA30-section specimens Specimen Dimension (mm) Angle (°) Area (mm2) Length t S1 S2 F1 F2 W1 W2 R D1 D2 a1 a2 a3 a4 a5 a6 a7 a8 An Ag 250-S1 254 1.763 20.19 18.92 79.67 80.00 32.39 30.93 - 14.94 15.99 90.5 88.0 133.0 85.0 140.0 86.5 89.5 - 415.0 487.5 250-S2 254 1.748 25.42 19.22 79.55 79.80 32.06 30.77 - 15.03 16.00 91.0 88.5 136.0 90.0 140.0 86.5 90.5 - 407.4 484.1 250-S3 254 1.750 20.73 20.09 79.66 79.16 32.60 33.95 - 16.05 15.01 88.5 88.0 130.0 68.0 124.0 87.5 91.0 - 408.2 488.1 500-S1 505 1.775 21.29 20.18 79.53 79.35 33.39 33.28 - 15.25 15.49 88.0 89.0 129.0 69.0 129.5 88.0 89.0 - 416.1 496.1 500-S2 506 1.770 21.05 21.09 79.65 79.21 32.20 33.58 - 15.50 14.87 88.0 87.5 133.0 73.5 128.0 87.0 88.0 - 415.9 493.9 500-S3 506 1.770 21.37 20.31 79.58 79.38 33.28 32.82 - 15.06 15.69 88.0 89.0 129.0 70.5 131.5 88.0 88.5 - 414.8 494.6 500-S4 505 1.770 21.51 20.20 79.37 79.39 33.11 33.02 - 15.37 15.50 87.5 88.5 129.5 70.0 129.0 88.0 88.5 - 414.9 494.1 760-S1 761 1.770 20.35 20.18 79.90 80.57 31.57 32.22 - 15.50 15.01 88.5 87.0 137.0 83.5 136.0 86.5 89.0 - 416.4 492.6 760-S2 762 1.800 19.81 20.47 79.45 80.22 33.71 31.84 - 14.57 16.45 89.0 87.0 124.0 72.5 139.0 87.0 90.0 - 422.6 501.2 760-S3 761 1.805 20.05 20.14 80.21 80.22 32.78 31.85 - 15.02 15.70 89.0 86.5 132.0 79.5 137.5 87.0 88.5 - 423.1 502.6 1010-S1 1014 1.790 19.88 21.36 79.04 79.19 32.98 31.65 - 13.38 15.93 88.5 89.0 132.5 83.5 141.0 89.5 87.5 - 422.8 495.3 1010-S2 1012 1.768 19.95 20.14 80.51 81.13 31.57 30.81 - 14.94 15.35 89.5 86.0 140.5 94.0 143.5 86.5 88.0 - 421.2 491.4 1520-S1 1526 1.760 20.65 20.38 79.60 79.03 31.69 33.54 - 15.95 14.20 88.0 89.0 137.5 74.5 127.5 88.5 86.0 - 411.0 488.0 1520-S2 1526 1.762 20.64 20.06 79.24 79.05 31.74 32.75 - 15.84 15.30 88.0 90.5 134.0 75.0 130.5 89.0 87.5 - 409.6 488.5 1820-S1 1827 1.773 20.88 20.85 80.81 79.23 32.07 33.48 - 15.73 14.93 84.5 87.5 136.0 73.0 127.0 87.0 85.5 - 418.0 495.0 1820-S2 1826 1.793 20.26 20.05 80.73 79.83 31.89 32.85 - 15.76 15.04 88.5 88.0 136.0 79.0 132.0 87.0 88.0 - 427.6 500.0 2150-S1 2157 1.785 19.85 20.01 79.85 80.63 32.52 30.94 - 14.44 14.82 89.0 87.0 136.5 89.0 141.5 88.0 88.5 - 420.9 493.6 2150-S2 2158 1.810 20.41 20.40 79.50 79.90 31.41 31.82 - 15.76 14.06 86.0 88.0 139.0 86.0 135.5 88.5 87.5 - 427.7 499.8 155 Table 7.2 Geometric dimensions of ZA32-section specimens Specimen Dimension (mm) Angle (°) Area (mm2) Length t S1 S2 F1 F2 W1 W2 R D1 D2 a1 a2 a3 a4 a5 a6 a7 a8 An Ag 250-S1 254 1.760 18.55 18.92 64.50 64.66 27.94 28.13 19.82 32.62 31.57 80.5 93.0 90.0 89.0 90.5 90.5 91.5 83.0 397.5 496.9 250-S2 254 1.747 18.00 18.25 63.87 64.24 28.07 27.66 19.54 32.82 31.76 85.5 92.5 90.0 88.0 92.5 92.5 90.0 86.0 392.1 491.0 250-S3 254 1.767 17.96 18.56 63.59 64.05 28.04 28.05 19.89 32.75 31.64 85.5 93.0 90.0 89.5 91.5 93.0 89.0 86.5 397.8 496.8 500-S1 503 1.774 18.75 19.28 64.87 64.63 28.44 28.11 19.74 32.04 32.21 84.5 92.5 91.5 89.5 90.5 91.5 92.0 84.5 405.3 505.1 500-S2 504 1.774 18.91 18.76 64.63 64.52 27.82 27.90 19.76 32.34 32.54 85.0 91.0 92.0 91.0 91.0 90.5 94.5 83.5 404.1 503.5 500-S3 504 1.778 18.98 18.93 64.31 64.99 28.04 27.54 19.74 32.32 32.38 84.0 93.5 90.5 90.5 91.5 91.0 94.0 82.0 403.3 504.3 500-S4 504 1.758 18.76 18.98 64.41 64.32 28.07 28.01 20.08 32.32 32.41 84.5 93.0 91.5 90.5 90.5 89.5 95.0 84.0 399.2 499.6 760-S1 760 1.765 18.05 18.26 64.38 64.03 28.24 28.40 19.51 31.81 31.97 89.5 93.0 92.0 89.0 89.5 91.5 92.5 86.0 394.3 498.3 760-S2 758 1.780 18.19 18.69 64.22 64.52 28.19 28.55 19.65 32.04 31.80 84.5 95.0 90.5 90.0 89.5 91.5 92.0 85.5 400.4 503.3 760-S3 760 1.775 18.46 17.92 64.64 64.36 28.03 28.34 19.59 31.98 31.78 84.5 93.5 91.0 90.0 89.5 92.5 92.5 86.0 396.4 500.8 760-S4 759 1.777 18.60 18.58 64.90 64.59 28.38 28.42 19.69 32.36 32.11 83.0 94.0 90.5 89.5 90.0 91.5 92.5 85.0 400.7 505.0 1010-S1 1014 1.773 18.33 18.58 64.82 64.64 28.44 28.47 20.25 32.54 32.27 83.5 93.5 89.5 89.0 88.0 90.0 93.5 84.5 400.3 504.5 1010-S2 1013 1.763 18.66 18.57 64.60 64.71 28.06 28.62 19.91 32.34 32.20 84.5 93.5 91.5 90.5 89.0 91.0 93.5 84.5 398.2 501.6 1520-S1 1523 1.773 18.26 18.09 64.56 64.65 28.25 28.41 19.57 32.10 31.93 85.5 93.0 91.0 89.0 90.0 91.5 93.0 85.0 395.7 501.3 1520-S2 1522 1.783 17.83 18.46 64.49 64.41 28.21 28.52 19.73 31.92 32.02 83.5 93.0 90.5 90.0 88.5 91.5 92.0 84.5 399.2 502.5 1820-S1 1827 1.807 17.99 18.46 64.50 64.39 28.39 28.38 19.73 31.90 32.22 86.0 93.0 91.5 90.0 90.5 91.0 93.0 86.0 404.0 511.3 1820-S2 1825 1.785 17.64 18.69 64.04 64.83 28.31 28.24 19.66 32.04 32.11 84.5 94.5 90.0 90.0 89.0 92.0 92.5 83.5 398.6 503.6 2150-S1 2158 1.810 18.14 18.13 64.13 64.67 28.25 28.04 19.91 32.10 31.90 86.0 93.0 89.5 89.0 89.5 90.5 92.5 84.5 403.2 509.7 2150-S2 2155 1.798 17.95 18.56 64.44 64.88 28.36 28.47 20.48 31.81 32.02 84.5 93.0 90.0 90.5 88.0 89.5 93.0 83.5 401.8 508.8 156 Table 7.3 Geometric dimensions of ZA42-section specimens Specimen Dimension (mm) Angle (°) Area (mm2) Length t S1 S2 F1 F2 W1 W2 R D1 D2 a1 a2 a3 a4 a5 a6 a7 a8 An Ag 250-S1 255 1.763 29.95 30.07 75.75 75.54 28.67 28.63 45.07 31.75 31.60 92.5 91.5 89.5 90.0 90.0 89.0 91.5 92.0 527.1 634.3 250-S2 252 1.770 29.55 29.72 75.91 75.74 29.07 28.78 45.22 31.30 31.60 93.0 89.5 88.5 90.5 90.0 88.5 91.0 92.0 531.2 635.1 250-S3 255 1.750 29.92 29.94 75.69 75.54 28.52 29.02 45.42 31.59 31.49 92.5 90.0 90.0 90.0 89.5 88.5 91.0 91.5 525.8 628.5 500-S1 503 1.768 29.69 29.96 75.61 75.72 28.79 28.96 45.18 31.49 31.73 92.0 91.0 88.5 90.0 90.0 88.5 90.5 92.0 531.3 631.4 500-S2 500 1.815 29.81 29.94 75.92 75.70 29.01 28.83 45.28 31.31 31.70 91.5 91.0 88.0 89.0 91.0 89.0 91.5 92.5 549.2 648.5 500-S3 505 1.784 29.71 29.86 75.80 75.49 29.12 28.88 45.24 31.32 31.72 92.0 90.0 89.0 89.5 90.5 88.5 90.5 92.0 538.4 636.9 500-S4 497 1.797 29.92 29.94 75.65 75.78 28.60 28.29 45.43 33.37 32.60 91.0 94.5 85.5 90.5 90.5 87.5 93.0 91.5 542.3 645.7 760-S1 761 1.776 29.36 29.74 75.52 75.65 28.70 28.62 45.09 31.66 31.29 92.5 91.5 88.5 90.0 90.5 89.0 89.5 92.5 530.1 631.7 760-S2 760 1.778 29.49 29.42 75.66 75.72 28.66 28.54 45.14 31.70 31.72 92.5 92.0 88.0 90.0 90.0 89.0 89.5 93.5 531.9 633.0 760-S3 762 1.803 29.09 29.54 75.50 75.83 28.80 28.71 45.21 31.71 31.72 91.5 91.5 87.0 89.5 90.0 88.5 90.5 91.0 537.9 641.1 1010-S1 1015 1.769 28.68 29.37 75.43 75.87 28.79 28.60 45.16 31.61 31.53 93.0 92.0 89.5 91.0 90.5 89.0 90.0 92.0 527.5 628.6 1010-S2 1014 1.790 28.96 29.16 75.61 75.76 28.93 28.66 45.30 31.44 31.42 93.0 91.5 88.5 90.0 89.0 88.0 89.5 93.0 533.3 635.7 1520-S1 1523 1.773 29.28 28.77 75.64 75.58 28.50 28.55 45.18 31.40 31.69 91.0 90.0 88.5 91.0 90.5 88.0 92.0 92.5 527.2 628.4 1520-S2 1522 1.760 29.21 29.62 75.75 75.71 29.06 28.63 45.20 31.72 31.51 92.5 91.0 87.5 90.0 91.0 89.0 89.5 93.0 526.6 627.3 1820-S1 1827 1.788 29.19 29.55 75.91 75.96 29.03 29.12 45.01 31.78 31.82 93.0 89.5 88.5 90.0 89.5 88.0 91.5 92.0 534.5 638.5 1820-S2 1828 1.778 29.36 28.81 75.85 75.57 28.60 28.60 45.22 31.94 31.90 91.5 90.0 88.0 90.5 90.5 87.5 92.0 93.0 532.1 632.3 2150-S1 2155 1.788 28.67 29.62 75.48 75.80 28.35 28.75 45.27 31.87 31.53 93.0 92.5 86.5 90.5 90.5 88.0 89.5 91.5 533.3 634.6 2150-S2 2156 1.800 29.49 29.45 75.58 75.71 28.75 28.93 45.20 31.64 31.50 92.5 91.5 90.0 90.5 90.0 88.0 89.5 93.5 538.3 641.4 157 7.2.4 Material Mechanical Properties All specimens were made of the same batch of steel coils, where the steel coils were fed in the cold-rolling machine without flattening or stretch-straightening. Karren (1965) experimentally observed that in a thin-walled steel sheet, the yield (Fy) and ultimate (Fu) tensile strengths exhibited sensitivities to the location at where the material coupon was taken. Thus, the material mechanical properties of the three sections were tested separately to gauge the variations in the yield and ultimate tensile strengths. Material coupons were chosen to be taken from the flanges of the sections at the central area between holes and the flange-web junction. Geometries of the material coupon are illustrated in Figure 7.2(a). It was observed from trial tests that the material in that area was the least affected by the cold rolling and punching in the sections, where the trial tests involved numerous coupons taken from the flat plates across all three sections. The material coupon’s length was designed as 240-mm, which was longer than the standard material coupons recommended by ASTM 370 (2020). Long material coupons were adopted in this study to minimize the influences on the tested results from the gripping ends since the material coupons were bent after being taken out from the columns due to the initial coiling. Figure 7.2 Mechanical property test of material: (a) the material coupon; (b) the test setup. 158 Three material coupons were tested for each section under uniaxial tension using an Instron testing system. The test setup is shown in Figure 7.2(b), where the picture was taken during a trial test (not the routine tests). The test procedure followed ASTM 370 (2020). One of the material coupons was tested first with a constant loading rate of 3 mm/min to determine the Fy value. Afterwards, the remaining material coupons were tested with the loading rate of 3 mm/min up to 1.05Fy; then, the loading rate increased to 15 mm/min till the fracture of the material coupons. The load was monitored during the tests by the Instron testing system, and the strain at the two surfaces of the material coupons was monitored using a three-dimensional Digital Image Correlation (DIC) system, where the cameras had the same resolutions of 5MP. The two systems were synchronized to be triggered simultaneously, and their data frequencies were set the same as 10HZ. Note that before the routine test, the performance of the DIC system has been successfully validated by the author by testing cylindrical bars, prismatic bars, and steel sheets; see an example shown in Figure 7.2(b). In these tests, strain gauges and/or an extensometer were utilized along with the DIC system, and the differences between the results were compared after the test. Only the DIC system was used in the routine test, i.e., strain gauges and extensometer were not used. Engineering mechanical properties of the material obtained from the uniaxial tension tests are shown in Table 7.4, and the engineering stress-strain curves are presented in Figure 7.3(a). All the stress-strain curves present discontinuous yielding, where the yield plateaus are enlarged and shown in Figure 7.3(b). Table 7.4 shows that the engineering mechanical properties obtained from different material coupons are close; hence, the ZA32-C2 coupon is selected as the representative one to show the true stress-strain curve in Section 7.3. 159 Figure 7.3 Stress-strain curves: (a) engineering stress-strain curves; (b) yield plateaus of engineering stress-strain curves (enlarged). Table 7.4 Engineering material mechanical properties Column Coupon Properties E (MPa) LYS (MPa) UYS (MPa) Fu (MPa) (%) ZA30 C1 198652 421.53 435.45 520.23 15.44 C2 204886 419.25 434.32 520.36 14.93 C3 197710 422.84 435.65 527.26 13.90 Average 200416 421.21 435.14 522.62 14.76 ZA32 C1 200300 420.31 432.51 520.72 14.71 C2 200596 424.92 437.69 525.76 14.78 C3 199859 419.67 428.63 519.38 14.69 Average 200252 421.63 432.94 521.95 14.73 ZA42 C1 203693 430.45 441.55 527.30 13.01 C2 202660 428.71 442.08 527.83 13.25 Average 203177 429.58 441.81 527.57 13.13 Note: E = Young’s modulus; LYS = lower yield strength; UYS = upper yield strength; Fu = ultimate tensile strength; = the strain corresponding to Fu. 7.2.5 End Support Conditions The specimens were designed to be tested with pinned-pinned (P-P) ends. However, testing the 250 and 500-mm long specimens with bearing-friction (B-F) ends or P-P ends were unlikely to result in evident differences in the axial strengths. First, elastic D buckling loads (PcrD) of the 500-mm long specimens were found at least 35% greater than the elastic L buckling loads (PcrL); 160 see Section 7.2.6 for the determination of elastic buckling properties of the specimens. So, the failure modes of 250 and 500-mm long specimens were expected to be dominated by L buckling. Second, it has been demonstrated by Pu et al. (1998) that the axial strengths of lipped-C section stub columns tested using P-P and B-F ends were nearly the same, where the columns failed in L buckling. Note that in Pu et al. (1998)’s work, the compression load was applied through the effective centroids of the stub columns while testing with pinned ends. Therefore, B-F ends were chosen for the 250 and 500-mm long specimens for convenience, where the test setup is presented in Section 7.4.1. On the other hand, the 760-mm to 2150-mm long specimens were tested with P-P ends with the compression load applied through the effective centroids of the stub columns; see Sections 7.4.2 and 7.4.3 for the test setups. Effective centroids of the stub columns were determined by performing nonlinear FE simulations. To be specific, a stub column with P-P ends was modelled for each section, and multiple analyses were conducted with the loading point varied along the axis-of-symmetry gradually. The loading point where the highest axial strength was obtained was the effective centroid of the stub column. The effective centroids of the ZA30, ZA32, and ZA42 sections were slightly offset from the centroids of the net sections to the web by 2.40, 2.46, and 0.90-mm, respectively. Compression load should be applied to the stiffness center of a column to achieve an axial compression test. Researchers have conducted numerous studies about axial compression tests of pin-ended, perforated CFS columns specially used in pallet storage racks, e.g., Casafont et al. (2011), Crisan et al. (2012), El Kadi and Kiymaz (2015), Kumar and Jayachandran (2016), Zhao et al. (2017), Elias et al. (2018), and Baldassino et al. (2019). Although these tests were all claimed as ‘axial compression tests’, the columns were loaded at different points. To be specific, the effective centroid of the stub column was employed as the loading point by Casafont et al. (2011), 161 Crisan et al. (2012), and El Kadi and Kiymaz (2015). In contrast, the centroid of the gross cross-section was selected as the loading point by Kumar and Jayachandran (2016), Zhao et al. (2017), Elias et al. (2018), and Baldassino et al. (2019). For rack columns, the holes at different areas of the cross-section are often staggered along the length, resulting in the determination of the stiffness center being difficult. To accurately locate the stiffness center of a column, strain gauges can be attached to the column around the cross-section at the mid-length; then, the loading point is adjusted along the symmetry-axis of the cross-section until the readings from the strain gauges become uniform when the column is preloaded. However, this method is costly, especially when numerous columns are going to be tested. So, from a designer’s viewpoint, the centroid of the gross or net section may be a good choice to be the loading point since it can be determined graphically. Also, the tested axial strength tends to be conservative since an initial loading eccentricity (expected to have a small magnitude) is introduced into the test. On the other hand, some researchers preferred the effective centroid of the stub column to be the loading point, which may because they believed that the effective centroid of the stub column is closer to the stiffness center than the centroid of the gross or net section. 7.2.6 Elastic Buckling Properties Eigenbuckling analyses were performed for the specimens using a Finite Element Analysis (FEA) software, ANSYS 18.1 (2020). Mid-lines of the nominal sections and the measured thicknesses (t) were used to establish column models, then meshed using Shell63 elements. Besides, the perforations and end support conditions (i.e., B-F and P-P ends) were appropriately modelled. Procedures for performing the eigenbuckling analyses have been introduced in Chapters 3 and 6, hence omitted here. Elastic buckling modes and loads of the specimens obtained from the eigenbuckling analyses are presented in Table 7.5, Table 7.6, and Table 7.7.162 Table 7.5 Elastic buckling loads, axial strengths and stiffnesses, and unfactored design axial strengths of ZA30 specimens Section Specimen BC Pcr (kN) Experimental results DSM L SD(1) SD(2) SD(3) AD + TF SD + F Failure mode Kae (kN/mm) Pu (kN) Min [Pn] (kN) Buckling mode Pn/Pu ZA30 250-S1 B-F 209.53 L 197.45 147.17 174.80 L 1.19 250-S2 202.39 L 190.22 147.67 171.60 L 1.16 250-S3 203.23 L 195.86 142.79 171.95 L 1.20 500-S1 216.05 314.02 L + SD(1) 117.71 137.30 157.47 SD(1) 1.15 500-S2 213.93 311.55 L + SD(1) 114.76 136.10 157.48 SD(1) 1.16 500-S3 214.23 311.97 L + SD(1) 119.44 136.87 156.92 SD(1) 1.15 500-S4 214.04 311.70 L + SD(1) 118.12 136.96 156.99 SD(1) 1.15 760-S1 P-P 212.86 230.42 310.50 SD(1) 69.68 116.21 151.11 SD(1) 1.30 760-S2 223.53 236.60 320.62 SD(1) 68.58 116.16 153.43 SD(1) 1.32 760-S3 225.34 237.97 322.53 SD(1) 68.85 115.58 153.65 SD(1) 1.33 1010-S1 219.31 201.29 232.21 333.47 269.71 SD(1) 55.20 124.09 150.39 SD(1) 1.21 1010-S2 211.39 196.44 228.29 326.54 264.69 SD(1) 55.50 119.06 149.21 AD + TF 1.25 1520-S1 208.38 157.46 177.56 175.39 244.33 SD(3) + F 40.05 113.98 125.86 AD + TF 1.10 1520-S2 209.07 157.79 177.91 175.65 244.32 SD(3) + AD + TF 37.40 106.79 126.02 AD + TF 1.18 1820-S1 212.80 153.64 157.99 135.20 192.70 SD(3) + AD + TF 33.71 97.23 109.34 AD + TF 1.12 1820-S2 219.68 157.17 161.11 137.06 195.24 AD + TF + SD + F 29.99 99.75 110.70 AD + TF 1.11 2150-S1 216.95 154.06 147.28 102.92 144.59 SD(3) + AD + TF 28.01 84.77 89.26 AD + TF 1.05 2150-S2 225.91 158.33 151.14 104.43 146.71 SD(3) + AD + TF 27.54 80.25 90.54 AD + TF 1.13 Min = 1.05 Max = 1.33 Mean = 1.18 SD = 0.08 Note: SD(1) = symmetric distortional buckling with a single halfwave; SD(2) = symmetric distortional buckling with two halfwaves; SD(3) = symmetric distortional buckling with three halfwaves. 163 Table 7.6 Elastic buckling loads, axial strengths and stiffnesses, and unfactored design axial strengths of ZA32 specimens Section Specimen BC Pcr (kN) Experimental results DSM L SD(1) SD(2) SD(3) AD + TF SD + F Failure mode Kae (kN/mm) Pu (kN) Min[Pn] (kN) Failure mode Pn/Pu ZA32 250-S1 B-F 307.23 L 162.69 152.12 167.60 L 1.10 250-S2 299.26 L 178.34 152.66 165.32 L 1.08 250-S3 309.53 L 169.07 153.54 167.72 L 1.09 500-S1 317.20 * 429.12 L + SD(1) 102.41 138.58 157.98 SD(1) 1.14 500-S2 316.23 * 427.81 L + SD(1) 107.52 143.84 157.50 SD(1) 1.09 500-S3 316.97 427.41 L + SD(1) 107.67 140.69 157.07 SD(1) 1.12 500-S4 307.02 419.88 L + SD(1) 105.38 142.96 155.34 SD(1) 1.09 760-S1 P-P 317.45 307.41 312.96 SD(1) 65.64 131.67 147.41 SD(1) 1.12 760-S2 325.44 312.47 318.03 SD(1) 66.23 135.13 149.91 SD(1) 1.11 760-S3 322.77 310.43 316.05 SD(1) 65.98 132.34 148.28 SD(1) 1.12 760-S4 323.83 311.30 316.89 SD(1) 65.43 128.30 149.84 SD(1) 1.17 1010-S1 320.96 294.46 268.18 238.48 SD(2) 54.08 129.53 146.44 SD + F 1.13 1010-S2 315.67 291.96 265.94 237.03 SD(2) 53.00 127.85 145.58 SD + F 1.14 1520-S1 320.43 242.73 250.42 170.69 124.43 SD(3) + AD + TF 32.81 110.70 103.82 SD + F 0.94 1520-S2 325.50 245.28 252.90 171.94 125.29 SD(3) + AD + TF 36.63 109.13 104.40 SD + F 0.96 1820-S1 338.64 238.82 241.35 130.54 91.58 SD(3) + TF 32.34 105.95 80.32 SD + F 0.76 1820-S2 326.67 233.45 235.73 128.89 90.52 SD + F 27.38 a 96.63 79.38 SD + F - 2150-S1 340.18 220.30 235.27 98.05 67.44 AD + TF 24.87 86.76 59.15 SD + F 0.68 2150-S2 333.33 218.10 232.08 97.55 67.14 AD + TF 28.37 87.46 58.88 SD + F 0.67 Min = 0.67 Max = 1.17 Mean = 1.03 SD = 0.16 Note: * means local buckling mode is involved, but the elastic buckling mode is overall dominated by distortional mode; a means the Pu value overestimates the axial strength of the specimen. 164 Table 7.7 Elastic buckling loads, axial strengths and stiffnesses, and unfactored design axial strengths of ZA42 specimens Section Specimen BC Pcr (kN) Experimental results DSM L SD(1) SD(2) SD(3) AD + TF SD + F Failure mode Kae (kN/mm) Pu (kN) Min[Pn] (kN) Failure mode Pn/Pu ZA42 250-S1 B-F 255.95 L 249.68 190.51 225.86 L 1.19 250-S2 258.59 L 253.80 194.08 227.01 L 1.17 250-S3 250.37 L 249.18 191.23 223.08 L 1.17 500-S1 255.58 * 963.34 L + SD(1) 171.91 182.68 224.30 SD(1) 1.23 500-S2 278.11 * 1028.71 L + SD(1) 173.55 182.91 232.82 SD(1) 1.27 500-S3 263.61 * 991.20 L + SD(1) 170.40 183.70 227.83 SD(1) 1.24 500-S4 269.33 * 1009.59 L 171.85 188.46 229.35 SD(1) 1.22 760-S1 P-P 257.20 * 498.61 L + SD(1) 79.60 171.55 209.85 SD(1) 1.22 760-S2 257.85 * 499.93 L + SD(1) 81.11 167.48 210.66 SD(1) 1.26 760-S3 268.93 * 511.60 L + SD(1) 82.30 175.14 213.23 SD(1) 1.22 1010-S1 253.56 * 356.88 385.01 L + SD(1) 66.12 175.95 200.03 SD(1) 1.14 1010-S2 262.58 * 367.36 392.68 L + SD(1) 65.43 176.65 202.74 SD(1) 1.15 1520-S1 253.99 * 392.01 * 332.10 305.36 207.92 SD + F 33.85 147.99 156.78 SD + F 1.06 1520-S2 248.73 * 389.08 * 331.67 301.92 206.29 SD + F 42.80 a 162.64 155.99 SD + F - 1820-S1 260.02 284.18 * 336.52 264.96 159.33 SD + F + AD + TF 39.01 a 151.07 133.44 SD + F - 1820-S2 255.74 281.50 * 333.06 262.70 158.16 SD + F 41.99 a 151.69 132.36 SD + F - 2150-S1 259.79 251.39 293.26 215.51 121.12 SD + F 38.32 a 148.45 106.22 SD + F - 2150-S2 265.19 254.30 * 296.38 217.25 121.95 SD + F 31.81 a 120.00 106.95 SD + F - Min = 1.06 Max = 1.27 Mean = 1.19 SD = 0.06 Note: * means local buckling mode is involved, but the elastic buckling mode is overall dominated by distortional mode; a means the Pu value overestimates the axial strength of the specimen. 165 7.3 Expanded Ramberg-Osgood Model For steel material featuring gradual yielding, the Ramberg-Osgood (R-O) model (Ramberg and Osgood 1943) was often adopted by researchers to describe the stress-strain curve (Chakrabarty 2006; Gao 2008). The application of the R-O model is limited to the cases where elastic and plastic strains are of comparable magnitudes. For steel material featuring discontinuous yielding, the R-O model is not applicable. In 2008, Gao (2008) proposed a modified R-O model. The model assumes the material is linear elastic before the yield point and ignores the yield plateau; then, it shifts the curve of the strain hardening portion to the yield point. This model is simple, but the shifting of the curve results in overestimated stress. In this section, an expanded R-O model is proposed; see Equation 7-1. ε=⎩⎪⎨⎪⎧σ E,� 0 ≤ σ ≤ σeσ E⁄ +k1(σ E⁄ )n1, σe < σ ≤ Fyεy+ �σ-Fy� αE⁄ , Fy < σ ≤ σpσ E⁄ +k2(σ E⁄ )n2, σp < σ ≤ Fu Equation 7-1 where σ = true stress σe = proportional limit. The point, where the difference between the R-O stress, from ε = σ/E + k1(σ/E)n1, and the linear stress (σ = εE) is larger than 0% but less than or equal to 0.5% (see Equation 7-2), is defined as σe. ε = true strain E = Young’s modulus Fy = yield strength. Here, the lower yield strength is defined as Fy. Fu = ultimate strength εy = the strain at Fy. 166 n1, n2, k1, k2 = constants and are calculated according to Equation 7-3, Equation 7-4, Equation 7-5, Equation 7-6, respectively. α = a coefficient multiplied with E to represent the slope of the yield plateau. It is determined through a linear fitting of the yield plateau. The yield plateau starts from εy to the point where the tangent modulus starts to vary stably (has no negative sign and without significant variations). It can be determined graphically by plotting the tangent moduli against the strain of that portion. εp = the strain at the intersection between yield plateau and strain hardening portion σp = the stress at εp 0 < |σe−εE|σe≤ 0.5% Equation 7-2 n1 = 1 + log m2(1−m1)m1(1−m2)log�s1s2� Equation 7-3 n2 = 1 + log m4(1−m3)m3(1−m4)log�s3s4� Equation 7-4 k1=(1 m2-1⁄ )(s2 E⁄ )1-n1 Equation 7-5 k2=(1 m4-1⁄ )(s4 E⁄ )1-n2 Equation 7-6 where s1, s2, s3, and s4 are secant yield strengths. They are equal to the ordinates of the intersections with the stress-strain curve of lines through the origin having the slopes equal to m1E, m2E, m3E, and m4E, respectively. An example is presented in Figure 7.4 for an illustration. The m1 and m2 are 0.7 and 0.9, respectively. The m3 and m4 are being chosen where the intersections are close to the stresses of Fy + 4(Fu-Fy)/5 and Fy + (Fu-Fy)/4, respectively. Note that the values of m1 to m4 can be determined by trial-and-error so that the best-fit curve can be obtained. Readers are suggested to go to Ramberg and Osgood (1943) for a deep understanding of these equations. 167 Figure 7.4 Determination of secant yield strengths The expanded R-O model was employed here to describe the ZA32-C2 coupon’s true stress-strain curve. Before doing so, the engineering stress and strain of the ZA32-C2 coupon were converted to true stress and true strain using Equation 7-7 and Equation 7-8, respectively (James 1995). The engineering stress-strain curve after the ultimate stress was discarded because necking of the material coupon probably occurred from here (Chakrabarty 2006). The true stress-strain curve generated using the expanded R-O model is presented along with the tested curve in Figure 7.5(a), and their yield plateaus are enlarged to be shown in Figure 7.5(b). The figures show that the two stress-strain curves agree well. For replicating the ZA32-C2 coupon’s true stress-strain curve, readers are suggested to use the parameters given in Table 7.8 with the expanded R-O model. σ = σeng(1+ εeng) Equation 7-7 ε = ln(1+ εeng) Equation 7-8 where σeng = engineering stress εeng = engineering strain 168 Figure 7.5 True stress-strain curve of the ZA32-C2 coupon: (a) the tested curve and the curve generated using the expanded R-O model; (b) yield plateaus of the stress-strain curves (enlarged). Table 7.8 Parameters for describing the ZA32-C2 coupon’s true stress-strain curve using the expanded R-O model Coupon E (MPa) σe (MPa) Fy (MPa) Fu (MPa) σp (MPa) α (×10-3) k1 (×1014) k2 (×1014) n1 n2 m3 m4 ZA32-C2 201000 209.38 430.54 603.48 443.01 3.5650 7.4928 3.9522 6.6896 6.1423 0.0311 0.0760 7.4 Test Setup 7.4.1 Test Setup for the 250 and 500-mm Long Specimens Test setup for the 250 and 500-mm long specimen is shown in Figure 7.6, which was configured as per AISI S902 (2013) and ANSI MH16.1 (2012). The specimens were compressed using an Instron testing frame with the capacity of 3000kN, where steel loading plates were placed between the column ends and the loading platens of the Instron testing frame. Before testing, the specimens were aligned to make the centroids of the compression platens, steel loading plates, and the net sections of the specimens coincide mutually. Afterwards, a preload of 15kN was applied to the specimens and then released so that ⅰ) the angle of the spherical compression platen can be self-adjusted and ⅱ) good contacts were achieved between the loading platens, steel loading plates, and column ends. Finally, a displacement-controlled load with a magnitude of 4-mm was applied 169 to the specimens at a constant loading rate of 0.1mm/min. Note that although the angle of the spherical platen can be adjusted initially, rotations of the spherical platen were unlikely to occur during the test due to its large diameter (i.e., much larger than the greatest overall width of the column section) and the friction between the spherical platen and the adjacent rigid end. During the tests, cross-section deformation was monitored for the 500-mm long specimens using Linear Variable Differential Transformers (LVDTs), which were assigned to the section where the most prominent deformations were anticipated to occur. Figure 7.6(b) presents the layout of the LVDTs with symbols describing the cross-section deformation illustrated in Figure 7.6(c). Based on the LVDTs’ results, the rotation of flanges (θ1 and θ2), the local deformation of flanges (Δ1 and Δ2), and the overall deflections of the specimens (δx and δy) can be calculated using Equation 7-9 to Equation 7-14. Note that Equation 7-14 applies only to the cross-section deformation without the involvement of AD mode deformation. Figure 7.6 Test setup for the 250 and 500-mm long specimens θ1 = arctan[(δ1 - δ3)/h] Equation 7-9 θ2 = arctan[(δ7 - δ5)/h] Equation 7-10 170 Δ1 = δ2 - (δ1 + δ3)/2 Equation 7-11 Δ2 = δ6 – (δ5 + δ7)/2 Equation 7-12 δy = δ4 Equation 7-13 δx = (δ3 – δ5)/2 Equation 7-14 7.4.2 Test Setup for the Intermediate-Length Specimens Figure 7.7 shows the test setup for the 760 and 1010-mm long specimens. For testing a specimen, a square steel plate was welded to each column end with its centroid coincided with the effective centroid of the column section. A steel pressure pad was then bolted to the steel plate. A rounded end was rigidly attached to the centre of the steel pressure pad (see Figure 7.7(c)), which was designed to be seated in the rounded end seat machined at the centre of the steel loading plate. The steel loading plates were rigidly connected to the grips of an MTS uniaxial testing frame with a capacity of 500kN. Besides, two single-universal joints were installed at the perimeter of the steel loading plate symmetrically. Accordingly, two rod-ends were installed at the perimeter of the steel pressure pad symmetrically. The pin of the single-universal joint was designed to be inserted into the rod end. By doing so, the column ends can rotate about the minor-axis freely but with torsion and the rotation about the major-axis restrained. Before testing, a preload of 10kN was applied to the specimen. Then, a compressive load was applied to the specimen at a 0.1mm/min loading rate. The test continued until the load dropped by 30% from the peak. 171 Figure 7.7 Test setup for the 760 and 1010-mm long specimens The relative axial displacement between the two loading plates was monitored using two LVDTs during the first several tests, where the results agreed well with those obtained from the MTS testing system. Therefore, the two LVDTs were omitted in the following tests. On the other hand, eight LVDTs were used for each specimen to measure the local deformation. They were placed in the section where the most prominent deformations were anticipated to occur. Figure 7.8(a) shows the layout of the LVDTs, where symbols describing the cross-section deformation are illustrated in Figure 7.8(b). The cross-section deformation can be decoupled into components corresponding to pure buckling modes. To be specific, rotations of flanges and webs (θ1 to θ4) and the lateral deflections (δx and δy) can be calculated using the LVDTs’ results as per Equation 7-15 to Equation 7-20. Then, θ1 to θ4 can be decoupled into components corresponding to pure buckling modes as per Equation 7-21 to Equation 7-28. Note that the calculation procedure was proposed on two assumptions: ⅰ) rotations of flanges/webs induced by any one of AD, SD, 172 and T modes are of the same magnitudes and ⅱ) rotations of webs induced by AD mode are negligible. Figure 7.8 Layout of LVDTs and symbols describing the cross-section deformation θ1 = arctan[(δ1 – δ2)/h] – θ3 Equation 7-15 θ2 = arctan[(δ8 – δ7)/h] – θ4 Equation 7-16 θ3 = arctan[(δ3 – δ4)/50] Equation 7-17 θ4 = arctan[(δ6 – δ5)/50] Equation 7-18 δx = (δ2 – δ7)/2 Equation 7-19 δy = δ11 Equation 7-20 θ1 = SDF + ADF Equation 7-21 θ2 = SDF – ADF Equation 7-22 θ3 = SDW + ADW + T Equation 7-23 θ4 = SDW – ADW – T Equation 7-24 SDF = (θ1 + θ2)/2 Equation 7-25 ADF = (θ1 – θ2)/2 Equation 7-26 SDW = (θ3 + θ4)/2 Equation 7-27 173 T = (θ3 – θ4)/2 Equation 7-28 where the SDF is the rotation of flange induced by SD mode; the ADF is the rotation of flange induced by AD mode; the SDW is the rotation of web induced by SD mode; the ADW is the rotation of web induced by AD mode, and; the T is the torsion induced by T mode. 7.4.3 Test Setup for the Long Specimens Figure 7.9 shows the test setup for 1520, 1820, and 2150-mm long specimens. It is like the one presented in Figure 7.7 except for three differences: ⅰ) the compressive load here was applied using a hydraulic jack instead of a computer-controlled actuator; ⅱ) the axial displacement of the hydraulic jack was monitored using two LVDTs, and; ⅲ) single-universal joints of no. 1 and no. 2 (see Figure 7.9) here were aligned along the minor-axis of the specimens, and single-universal joints of no. 2 and no. 4 were kept loose so that they were no longer functioning. The long specimens may fail in AD + TF or SD + F buckling; therefore, the single-universal joints were rearranged to enable column ends to rotate about the two principal axes freely and prevent the torsion of column ends from happening. To be specific, the joint of no. 1 (at the top pressure pad) was in the plane of the minor axis of the specimen so that the top pressure pad could rotate freely about the major axis of the specimen. The joint of no. 3 (at the bottom pressure pad) was in the plane of the major axis of the specimen so that the bottom pressure pads could rotate freely about the minor axis of the specimen. Since the joints of no. 2 and no. 4 were no longer functioning, the top and bottom pressure pads can also rotate freely about an axis that nearly coincided with the minor and major axis of the specimen, respectively. This end support condition can be deemed as pinned ends when only the ultimate loads are of concern since end rotations of the specimens were found insignificant at the ultimate loading points. However, it was observed that the development 174 of the deformations of the specimens was affected by this end support condition at the post-ultimate phases. After the alignment of some specimens, the pins of the single-universal joints (no.1 and/or no. 3) were inclined sideways slightly due to the initial crookedness of the specimens. Under this condition, the pins of the single-universal joints (no.1 and/or no. 3) imposed somewhat restraints to the rotations of column ends so that the specimens might have been stiffened. Therefore, nonlinear FE simulations were performed for the specimens that might have been stiffened by the single-universal joints using ANSYS 18.1 (2020). Here, the nonlinear FE simulations were performed to evaluate the effectiveness of the tested axial strengths (Pu) only. The procedure and results of the nonlinear FE simulations are out of the scope of this study and will be reported in the future. Figure 7.9 Test setup for the 1520, 1820, and 2150-mm long specimens 175 7.5 Experimental Results A total of 55 specimens were tested under compression. The axial load-axial displacement curves of the specimens are presented in Figure 7.10 and Figure 7.11. The specimens’ axial strengths (Pu) and axial stiffnesses (Kae) have been shown in Table 7.5 to Table 7.7. Structural responses of the specimens during the test are described as follows. 176 Figure 7.10 Axial load-axial displacement curves of the 250, 500, 760, and 1010-mm long specimens 177 Figure 7.11 Axial load-axial displacement curves of 1520, 1820, and 2150-mm long specimens 7.5.1 The 250-mm Long Specimens The 250-mm long specimens after the test are shown in Figure 7.12, from where it can be observed that all the specimens failed in L buckling. The axial load-axial displacement curves of the specimens are presented in Figure 7.10(a) to Figure 7.10(c). The initial softening portion of the curves indicates that there might be gaps between the loading plates and column ends. However, the gaps seemed only to affect the axial strengths of the specimens slightly. 178 Figure 7.12 The 250-mm long specimens after the test 7.5.2 The 500-mm Long Specimens Figure 7.13 presents the 500-mm long specimens after the test. As the location of the most-deformed cross-section varied with the specimens, deformation of the most-deformed cross-section was monitored successfully for only three specimens, i.e., the ZA30-500-S2, ZA32-500-S1, and ZA32-500-S2. As the three specimens exhibited similar cross-section deformations, the ZA30-500-S2 specimen is selected as the representative one to show the evolution of cross-section deformation with loading in Figure 7.14. Figure 7.13 The 500-mm long specimens after the test Except for the ZA42-500-S4 specimen, all the 500-mm long specimens failed in L + SD(1) buckling. With the increase in load, rotations and local deformations of the flanges of the ZA30-500 and ZA32-500 specimens were captured by the LVDTs. The ZA30-500-S2 specimen is selected as the representative one to show the evolution of cross-section deformation with loading 179 in Figure 7.14(a) and Figure 7.14(b). Second, except for the ZA42-500-S4 specimen, all the 500-mm long specimens exhibited flange rotation and L buckling halfwave(s) at the flanges, lips, and/or web after the test; see Figure 7.13. The ZA42-500-S4 specimen failed in L buckling (see Figure 7.13(c)), where the Pu value (188.46kN) of the specimen is the highest in the group. The Pu value of the specimen is slightly lower than those of the ZA42-250 specimens (i.e., 190.51kN ≤ Pu ≤ 194.08kN) since the constraint effects from the end support conditions diminished due to the increased length. Figure 7.14 Evolution of cross-section deformation of the ZA30-500-S2 specimen with loading 7.5.3 The 760-mm Long Specimens Figure 7.15 presents the 760-mm long specimens at the end of the test before releasing the pressure. Cross-section deformations of the specimens were decoupled into components corresponding to pure buckling modes (i.e., SDF, ADF, SDW, T, δx, and δy). The magnitudes of the components at the peak loads are shown in Table 7.9. Specimens of the same section presented similar failure modes; therefore, one of them is selected as a representative to show the evolution of cross-section deformation with loading in Figure 7.16. The experimental results are summarized as follows: 180 Figure 7.15 The 760-mm long specimens at the end of the test Figure 7.16 Evolution of cross-section deformation of the 760-mm long specimens with loading (1) The ZA30-760 and ZA32-760 specimens failed in SD(1) buckling. The evidence can be observed from Figure 7.15(a) and Figure 7.15(b). Besides, Figure 7.16(a) and Figure 7.16(b) show that the cross-section deformations of the specimens were dominated by the rotations of flanges and web throughout the tests. (2) The ZA42-760 specimens failed in L + SD(1) buckling; see Figure 7.15(c). The S2 specimen is selected as the representative one to show the interactive buckling process in Figure 7.17. It can be observed from the figure that local buckling halfwaves appeared in the lips at the early loading stage, which became increasingly more evident with the increase in load and later formed kinks as the load descended. The early occurrence of the local buckling halfwaves in the lips was triggered by the non-straightness of the lips, which might be the combined results of residual stresses and the relatively large widths of the lips. Rotations of the flanges and web of the 181 specimen were not prominent during the load-ascending phase but became evident during the load-descending phase; see Figure 7.16(c) and Figure 7.17. Figure 7.17 Deformed configurations of the ZA42-760-S2 specimen at different loading points 182 Table 7.9 Magnitudes of components of the cross-section deformations at the peak loads and failure modes of specimens Section Specimen Rotation (×10-2 rad) Deflection (mm) Failure mode SDF SDW ADF T δx δy ZA30 760-S1 2.78 10.12 -1.40 -0.26 -0.16 - SD(1) 760-S2 - 8.19 - 0.22 0.22 - SD(1) 760-S3 2.64 8.70 0.59 0.34 0.00 - SD(1) 1010-S1 3.92 13.95 1.40 -1.65 -0.09 4.21 SD(1) 1010-S2 4.07 12.57 1.36 -0.49 0.30 5.15 SD(1) 1520-S1 -4.07 -3.24 -0.71 -1.42 -0.61 -0.57 SD(3) + F 1520-S2 -2.53 -1.57 2.20 3.86 3.65 -0.85 SD(3) + AD + TF 1820-S1 -2.42 -1.89 -0.32 -2.72 -2.64 -2.18 SD(3) + AD + TF 1820-S2 -3.72 0.03 1.62 -6.54 -7.56 -6.30 AD + TF + SD + F 2150-S1 -2.26 -1.75 1.03 8.15 8.38 -0.86 SD(3) + AD + TF 2150-S2 -1.87 -1.81 1.00 8.40 9.35 -0.22 SD(3) + AD + TF ZA32 760-S1 6.57 6.76 0.45 -0.14 0.04 - SD(1) 760-S2 7.84 5.09 0.24 -0.57 -0.16 - SD(1) 760-S3 6.58 7.28 -0.52 0.20 -0.44 - SD(1) 760-S4 5.13 4.78 0.70 0.53 0.76 - SD(1) 1010-S1 0.66 -0.97 0.57 0.59 0.46 0.04 SD(2) 1010-S2 -1.82 -2.49 0.30 2.52 0.96 -0.08 SD(2) 1520-S1 -2.24 -1.01 1.01 4.81 4.08 -0.20 SD(3) + AD + TF 1520-S2 -3.36 -0.59 -0.84 -7.21 -3.92 1.36 SD(3) + AD + TF 1820-S1 -1.16 -0.83 0.00 4.22 3.78 -0.51 SD(3) + TF 1820-S2 -1.03 -0.52 0.12 0.45 0.29 -3.78 SD + F 2150-S1 -0.58 -0.30 -0.43 3.18 3.31 -2.54 AD + TF 2150-S2 -0.93 -0.15 -1.17 8.76 7.65 -1.65 AD + TF ZA42 760-S1 -2.47 -0.75 0.04 -0.13 -0.13 - L + SD(1) 760-S2 -2.18 -0.48 0.36 0.02 -0.16 - L + SD(1) 760-S3 -2.04 -0.89 0.35 -0.23 0.15 - L + SD(1) 1010-S1 -2.50 -1.62 0.69 0.13 0.30 -1.75 L + SD(1) 1010-S2 -3.76 -3.72 0.14 -0.05 -0.24 -3.85 L + SD(1) 1520-S1 -2.27 -2.65 -0.06 0.03 -1.05 -5.12 SD + F 1520-S2 -4.00 -3.97 -0.08 -0.46 -2.49 -7.17 SD + F 1820-S1 -1.85 -3.28 -0.55 1.24 2.38 -6.97 SD + F + AD + TF 1820-S2 -2.11 -2.97 -0.23 -0.18 0.72 -7.21 SD + F 2150-S1 -1.32 -1.77 -0.49 -0.15 0.11 -4.24 SD + F 2150-S2 -0.81 -1.29 -0.09 0.37 0.66 -4.31 SD + F 183 7.5.4 The 1010-mm Long Specimens Figure 7.18 presents the 1010-mm long specimens at the beginning and end of the test. Specimens of the same section presented similar failure modes; therefore, one of them is selected as a representative to show the evolution of cross-section deformation with loading in Figure 7.19(a) to Figure 7.19(c). Also, Figure 7.19(d) presents the specimens’ deflection curves (δy) at the peak loads. Moreover, the measured cross-section deformations of the specimens were decoupled into components corresponding to pure buckling modes (i.e., SDF, ADF, SDW, T, δx, and δy). The magnitudes of the components at the peak loads are shown in Table 7.9. The experimental results are summarized as follows: Figure 7.18 The 1010-mm long specimens at the beginning and end of the test (1) The ZA30-1010 specimens failed in SD(1) buckling; see Figure 7.18(a), Figure 7.18(b), and Figure 7.19(a). 184 Figure 7.19 Evolution of cross-section deformation with loading and deflection curves (δy) of the 1010-mm long specimens (2) The ZA32-1010 specimens failed in SD(2) buckling; see Figure 7.18(c), Figure 7.18(d), and Figure 7.19(b). Note that the number of buckling halfwaves of the specimens is also presented in their deflection curves (δy), as shown in Figure 7.19(d). Eigenbuckling analyses showed that SD(2) is the third eigenmode of the specimens, where the first and second eigenmodes are SD + F and AD + TF. The specimens’ failure modes were like the third eigenmode instead of the first or second eigenmode because of two reasons. First, the single-universal joints allowed column ends to rotate freely only about the minor-axis so that AD + TF mode was prevented from occurring in the specimens. Second, the column ends were subjected to spring-back after the cold-rolling and cutting, resulting in the openings at column ends being wider than that at the mid-length; see the specimens at the beginning of the test from Figure 7.18(c) and Figure 7.18(d). The initial geometric shapes of the specimens were more compatible with SD(2) mode than with SD + F mode. For 185 demonstrating the buckling process, deformed configurations of the S2 specimen at different loading points are presented in Figure 7.20. Figure 7.20 Deformed configurations of ZA32-1010-S2 specimen at different loading points (3) The ZA42-1010 specimens failed in L + SD(1) buckling; see Figure 7.18(e), Figure 7.18(f), and Figure 7.19(c). The buckling processes of the specimens were like those of the ZA42-760 specimens. Figure 7.21 shows the deformed configurations of the S1 specimen at different loading points. Figure 7.21 Deformed configurations of the ZA42-1010-S1 specimen at different loading points 7.5.5 The 1520-mm Long Specimens Figure 7.22(a) to Figure 7.22 (e) show the evolution of cross-section deformation with the loading of the 1520-mm long specimens. The two ZA30-1520/ZA42-1520 specimens presented 186 failure modes or Pu values very different; therefore, their results are all presented in Figure 7.22. In contrast, the two ZA32-1520 specimens showed similar failure modes and Pu values; hence, only the results of the S1 specimen are presented in Figure 7.22. Deflection curves (δy) of all the specimens at the peak loads are shown in Figure 7.22(f). Figure 7.23 presents the evolution of lateral deflections (uncoupled) at the mid-length of the specimens with the loading. The measured cross-section deformations of all the specimens were decoupled into components corresponding to pure buckling modes, where the magnitudes of the components at the peak loads are shown in Table 7.9. The experimental results are summarized as follows: Figure 7.22 Evolution of cross-section deformation with loading and deflection curves (δy) of the 1520-mm long specimens 187 Figure 7.23 Evolution of lateral deflections (uncoupled) at the mid-length of the 1520-mm long specimens (1) The ZA30-1520 specimens presented different failure modes: the S1 specimen failed in SD(3) + F buckling, and the S2 specimen failed in SD(3) + AD + TF buckling. In the case of the S1 specimen, Figure 7.22(a) shows that the cross-section deformation at the mid-length was dominated by distortional mode (i.e., SDF and SDW). Although AD + TF mode (i.e., ADF and T) was also involved in the cross-section deformation, the involvement was insignificant and might result from unsymmetric deformation caused by Initial Geometric Imperfections (IGIs). Besides, Figure 7.22(f) shows that the deflection curve (δy) of the S1 specimen at the peak load had three halfwaves, and the overall deflection curve shifted towards the web due to an overall bending of the specimen. Indeed, the deflection curve can be approximately decomposed into two sinusoidal curves: one has three halfwaves with 1.5-mm magnitude, and the other one has a single halfwave with 0.925-mm magnitude. The latter was likely induced by the shift of the effective centroid of the specimen due to the stress redistribution caused by the growth of the distortional deformation, 188 where the axial compression became gradually more eccentric (Camotim and Dinis 2013). In the case of the S2 specimen, Figure 7.22(b) shows that the distortional deformation (i.e., SDF and SDW) and AD + TF deformation (i.e., ADF and T) played comparable roles in the cross-section deformation at the mid-length. Moreover, the deflection curve (δy) of the S2 specimen exhibited three halfwaves at the peak load; see Figure 7.22(f). Eigenbuckling analyses showed that the first eigenmode of the ZA30-1520 specimens is SD(2), and the second and third eigenmodes are AD + TF and SD(3). However, SD(3) mode was involved in the failure modes of both specimens instead of SD(2) mode, which might attribute to the spring back of the column ends. On the other hand, the reasons causing the different failure modes of the specimens are explored here. LVDTs’ results showed that the lateral deflection (δx) of the S1 specimen at the mid-length varied within a small range between -0.32 and 0.27-mm during the load ascending phase; see Figure 7.23(a). In contrast, the lateral deflection (δx) of the S2 specimen at the mid-length increased with the load from the beginning of the test and reached 2.90-mm at the peak load; see Figure 7.23(b). The S2 specimen might be subjected to initial lateral loading eccentricity, which triggered the occurrence of AD + TF mode. The buckling process of the S2 specimen is presented in Figure 7.24(a). The specimen failed suddenly after reaching the peak load (106.8kN), leading to a sudden drop in the load to 61.5kN. 189 Figure 7.24 Deformed configurations of the 1520-mm long specimens at different loading points It is worth pointing out that the deflection curves (δy) of the two specimens at the peak loads (see Figure 7.22(f)) have the bottom halfwave showing a smaller half-wavelength and amplitude compared with those of the top halfwave. This represents that the single-universal joint has somewhat restricted the rotation of the bottom end of the specimens during the test. However, the resulting stiffening effects on the specimens appear to be negligible since the tested Pu values of the specimens are close to those obtained from the nonlinear FE simulations of the specimens. (2) The ZA32-1520 specimens failed in SD(3) + AD + TF buckling; see Figure 7.22(c) and Figure 7.22 (f). The S1 specimen is selected as the representative one to show the buckling process during the test in Figure 7.24(b). The specimen buckled right after the load decreased slightly from 190 the peak (110.7kN) to 108.5kN, resulting in a rapid increase in lateral deflection (δx) and a sharp load drop to 59.5kN. Although the failure modes of the S1 and S2 specimens are similar, the deflection shape (δy) of the S2 specimen at the peak load looks peculiar compared to that of the S1 specimen; see Figure 7.22(f). Therefore, the evolution of the S2 specimen’s deflection curve (δy) with the load is inspected. It is observed that the rotation (about the minor-axis) of the specimen’s bottom end was being restrained by the single-universal joint to a certain extent during the test, resulting in the growth of the distortional deformation to be concentrated at the upper portion of the specimen. Despite the different deflection shapes (δy), the Pu values of the two specimens are nearly equal. Eigenbuckling analyses showed that the first and second eigenmodes of the ZA32-1520 specimens are SD + F and AD + TF, where the Pcr values of the AD + TF eigenmode are about 37% greater than the Pcr values of the SD + F eigenmode. The ZA32-1520 specimens showed SD(3) + AD + TF modes rather than SD + F modes in the tests, where the reasons are explored here. As for the S1 specimen, LVDTs’ results show that the lateral deflection (δx) at the mid-length increased with the load since the test started; see Figure 7.23(c). So, the S1 specimen might be subject to initial lateral loading eccentricity or initial crookedness. In contrast, the lateral deflection (δx) of the S2 specimen at the mid-length started to increase until the load went beyond about 100kN; see Figure 7.23(d). So, the occurrence/development of SD + F mode in the S2 specimen might have been restricted by the single-universal joint during the compression test. (3) The ZA42-1520 specimens failed in SD + F buckling; see Figure 7.22(d) to Figure 7.22(f). The specimens’ failure modes agree to their first eigenmodes. Figure 7.24(c) and Figure 7.24(d) illustrate the specimens’ buckling processes during the compression. However, the Pu value of the S2 specimen (162.64kN) is 9.9% greater than that of the S1 specimen (147.99kN). 191 Also, the axial stiffness of the S2 specimen (42.80kN/mm) is 26.4% higher than that of the S1 specimen (33.85kN/mm). LVDTs’ results showed that the deflection (δy) of the S1 specimen at the mid-length was -5.12-mm at the peak load (Pu = 147.99kN), while the deflection (δy) of the S2 specimen at the mid-length was only -2.98-mm at this load level; see Figure 7.23(e) and (f). So, it is believed that the single-universal joints stiffened the S2 specimen during the compression. The Pu value of the S1 specimen was found close to that obtained from the nonlinear FE simulation of the specimen. 7.5.6 The 1820-mm Long Specimens Figure 7.25 shows the evolution of cross-section deformation of the 1820-mm long specimens at the mid-length during the compression. Figure 7.26(a) presents the specimens’ deflection curves (δy) at the peak loads. Figure 7.27 illustrates the evolution of lateral deflections (uncoupled) at the mid-length of the specimens with the loading. The specimens’ buckling processes during the compression are illustrated in Figure 7.28. The measured cross-section deformations of the specimens were decoupled into components corresponding to pure buckling modes, where the magnitudes of the components at the peak loads are shown in Table 7.9. The experimental results are summarized as follows: (1) The ZA30-1820 specimens failed in different buckling modes, which are described separately. The S1 specimen failed in SD(3) + AD + TF buckling; see Figure 7.25(a), Figure 7.26(a), and Figure 7.28(a). Figure 7.26(a) shows that the deflection curve (δy) of the S1 specimen at the peak load is quite different from a regular sinusoidal curve with three halfwaves. Upon inspecting the evolution of the deflection curve (δy) of the specimen during the compression, it is observed that the rotation (about the minor-axis) of the bottom end of the specimen was somewhat restrained 192 by the single-universal joint, resulting in the growth of distortional deformation to be concentrated at the upper portion of the specimen. However, the Pu value of the specimen is valid since it is close to that obtained from the nonlinear FE simulation of the specimen. Figure 7.25 Evolution of cross-section deformation of the 1820-mm long specimens with loading The S2 specimen failed in G + G buckling (i.e., AD + TF + SD + F); see Figure 7.25(b), Figure 7.26(a), and Figure 7.28(b). Ideally, the SD + F mode was not supposed to occur in the specimen for two reasons. First, the specimen’s first eigenmode is AD + TF, and the ratio (1.42) between the Pcr values of SD + F and AD + TF eigenmodes is significantly above unity. Second, the AD deformation in the specimen, see the ADF curve as shown in Figure 7.25(b), maintained small magnitudes with the increase in load until reaching the vicinity of the peak load. So, the AD deformation was only possible to result in a minor offset of the effective centroid of the specimen, which was unlikely to cause the significant deflection (δy) of the specimen. As a result, it is deduced here that the SD + F mode in the S2 specimen was triggered by initial loading eccentricity 193 and/or initial crookedness. Although the S1 and S2 specimens presented different failure modes, their Pu values are close (97.23kN vs 99.75kN). Figure 7.26 Deflection curves (δy) of the 1820- and 2150-mm long specimens Figure 7.27 Evolution of lateral deflections (uncoupled) at the mid-length of the 1820-mm long specimens (2) The ZA32-1820 specimens also failed in different buckling modes; hence, they are introduced here separately. 194 Figure 7.28 Deformed configurations of the 1820-mm long specimens at different loading points The S1 specimen failed in SD(3) + TF buckling; see Figure 7.25(c), Figure 7.26(a), and Figure 7.28(c). While inspecting the evolution of the S1 specimen’s deflection curve (δy) during the compression test, it is found that SD(4) mode appeared in the specimen first and then transformed to SD(3) mode gradually at the advanced loading stage. Indeed, the deflection curve (δy) of the specimen at the peak load presented in Figure 7.26(a) was in the transition from SD(4) mode to SD(3) mode. On the other hand, eigenbuckling analyses showed that the first eigenmode of the specimen is SD + F, and AD + TF is the second eigenmode. The failure mode of the 195 specimen is like the second rather than the first eigenmode, which might be caused by the initial lateral loading eccentricity and/or initial crookedness since the LVDTs’ results showed that the lateral deflection (δx) of the specimen at the mid-length grew with the load since the test started; see Figure 7.27(c). The S2 specimen failed in SD + F buckling; see Figure 7.25(d), Figure 7.26(a), and Figure 7.28(d). The failure mode of this specimen seems to agree with the first eigenmode. However, it can be observed from the specimen’s deflection curve (δy) at the peak load (see Figure 7.26(a)) that the rotation at the bottom end of the specimen is smaller than that at the top end. The single-universal joint might have somewhat restrained the rotation (about the minor-axis) of the bottom end of the specimen during the compression test. This deduction is further proved by inspecting the evolution of the specimen’s deflection curve (δy) during the compression test. Results from the nonlinear FE simulation of the S2 specimen suggested that the axial strength of the specimen is overestimated by the tested Pu value (96.63kN). (3) The ZA42-1820 specimens also failed in different buckling modes, which are introduced here separately. The S1 specimen failed in G + G buckling (i.e., SD + F + AD + TF); see Figure 7.25(e), Figure 7.26(a), and Figure 7.28(e). Ideally, AD + TF mode was not supposed to occur in the specimen since ⅰ) the first eigenmode of the specimen is SD + F and ⅱ) the Pcr value of the AD + TF eigenmode is 66% higher than that of the SD + F eigenmode. LVDTs’ results showed that the specimen’s lateral deflection (δx) at the mid-length grew with the load since the test started; see Figure 7.27(e). So, the AD + TF mode in the S1 specimen might be triggered by initial lateral loading eccentricity and/or initial crookedness. In contrast, the failure mode of the S2 specimen agrees to the first eigenmode of SD + F; see Figure 7.25(f) and Figure 7.26(a). 196 Although the S1 and S2 specimens exhibited different failure modes, their Pu values are nearly equal that are 151.07kN and 151.69kN, respectively. However, the results of nonlinear FE simulations suggested that the tested Pu values overestimated the axial strengths of the specimens. The rotations (about the minor-axis) of the bottom ends of the specimens were restrained by the single-universal joint to certain extents during the compression test, where the evidence is observed while inspecting the evolution of the deflection curves (δy) of the specimens during compression. 7.5.7 The 2150-mm Long Specimens Figure 7.29 presents the evolution of cross-section deformation of the 2150-mm long specimens at the mid-length during the compression. Deflection curves (δy) of the specimens at the peak loads have been shown in Figure 7.26(b). Figure 7.30 presents the evolution of lateral deflections (uncoupled) at the mid-length of the specimens with the loading. Besides, buckling processes of the specimens during the compression are illustrated in Figure 7.31. The measured cross-section deformations of the specimens were decoupled into components corresponding to pure buckling modes, where the magnitudes of the components at the peak loads are shown in Table 7.9. The experimental results are summarized as follows: (1) The ZA30-2150 specimens failed in SD(3) + AD + TF buckling; see Figure 7.26(b), Figure 7.29(a), and Figure 7.29(b). The S2 specimen is selected as the representative one to show the buckling process during the compression test in Figure 7.31(a), where the specimen buckled smoothly after the load reached the Pu and sudden variations in the load and deformation did not take place. 197 Figure 7.29 Evolution of cross-section deformation of the 2150-mm long specimens with loading Figure 7.30 Evolution of lateral deflections (uncoupled) at the mid-length of the 2150-mm long specimens 198 (2) The ZA32-2150 specimens failed in AD + TF buckling; see Figure 7.26(b), Figure 7.29(c), and Figure 7.29(d). The specimens buckled nearly right after the load reached the Pu; see the buckling processes of the specimens presented in Figure 7.31(b) and Figure 7.31(c). Eigenbuckling analyses showed that AD + TF is the second eigenmode of the specimens, and the Pcr values are about 45% higher than the Pcr values of the first eigenmode of SD + F. The specimens failed in AD + TF buckling instead of SD + F buckling, attributing two reasons. First, the single-universal joint restrained the rotations (about the minor-axis) of the bottom ends of the specimens to certain extents during the compression, resulting in difficulty to attain/develop SD + F mode in the specimens. The evidence can be observed from the deflection curves (δy) of the specimens at the peak loads shown in Figure 7.26(b). Second, LVDTs’ results showed that the lateral deflection (δx) of the specimens at the mid-length grew with the load since the tests started; Figure 7.30(c) and (d). So, the specimens might also be subjected to initial lateral loading eccentricities and/or initial crookedness, which triggered the AD + TF buckling. (3) The ZA42-2150 specimens failed in SD + F buckling; see Figure 7.26(b), Figure 7.29(e), Figure 7.29(f), and Figure 7.31(d). The failure modes of the specimens agree to their first eigenmodes. During the compression tests, it was observed that buckling took place in the specimens right after the load reached Pu. Although the specimens showed similar failure modes, their Pu values are very different. The S1 specimen’s Pu value (148.45kN) is about 24% higher than that of the S2 specimen (120.00kN). The evolution of the deflection curve (δy) during the compression is inspected for the two specimens to explore the reasons. It is found that the single-universal joints somewhat restrained the rotations (about the minor-axis) of both ends of the S1 specimen. This can be observed from the deflection curve (δy) of the specimen at the peak load presented in Figure 199 7.26(b). A similar phenomenon happened in the S2 specimen but only at the bottom end. Results of nonlinear FE simulations suggested that the tested Pu values overestimated the axial strengths of both specimens. Figure 7.31 Deformed configurations of the 2150-mm long specimens at different loading points 7.6 Causes for the Interactive Buckling of Specimens Among all the specimens, those presented interactive buckling modes are summarized in Table 7.10. Reasons causing the interaction between the two components (pure buckling modes) in the specimens’ failure modes are explored in this section. First, ratios between the Pcr values 200 (high order/low order) of the two components in the specimens’ failure modes were calculated, where the results are presented in the 7th column of the table. Second, nominal axial strengths (Pnl, Pnd, or Pne) of the two components in the specimens’ failure modes were determined per the DSM (AISI_S100 2016), and the ratios between them (high order/low order) are presented in the 10th column of the table. In addition, the two nominal axial strengths of each specimen were compared with the tested Pu value, where the ratios of nominal-to-tested axial strengths are presented in the last two columns of the table. Six different reasons were found responsible for the interactive buckling of the specimens: (Ⅰ): Pcr values of the two buckling modes are close. This reason was responsible for the SD(3) + AD + TF buckling of the ZA30-1520-S2 specimen, where the ratio between the Pcr values of SD(3) and AD + TF eigenmodes is close to unity and is about 1.01. (Ⅱ): Pn values of the two buckling modes are close. This reason was responsible for the L + SD(1) buckling of the ZA30-500, ZA32-500, ZA42-500, and ZA42-760 specimens, where the ratios between the nominal axial strengths of SD(1) and L buckling are between 0.90 and 0.99. During the compression test, distortional buckling deformation appeared in the ZA30-500, ZA32-500, and ZA42-760 specimens first, while local buckling deformation appeared in the ZA42-500 specimens first. 201 Table 7.10 Causes of the interactive buckling of specimens Section Specimen Experimental results Eigenmodes Pcr/Pcr Pn (kN) Pn/Pn Causes Pn/Pu Failure mode Pu (kN) Low order High order High order/Low order Low order High order High order/Low order Low order High order ZA30 500-S1 L + SD(1) 137.30 L SD(1) 1.45 175.25 157.47 0.90 Ⅱ 1.28 1.15 500-S2 L + SD(1) 136.10 L SD(1) 1.46 175.17 157.48 0.90 Ⅱ 1.29 1.16 500-S3 L + SD(1) 136.87 L SD(1) 1.46 174.71 156.92 0.90 Ⅱ 1.28 1.15 500-S4 L + SD(1) 136.96 L SD(1) 1.46 174.75 156.99 0.90 Ⅱ 1.28 1.15 1520-S1 SD(3) + F 113.98 SD(3) SD + F 1.38 142.52 144.54 1.01 Ⅲ 1.25 1.27 1520-S2 SD(3) + AD + TF 106.79 AD + TF SD(3) 1.01 126.02 141.83 1.13 Ⅰ 1.18 1.33 1820-S1 SD(3) + AD + TF 97.23 AD + TF SD(3) 1.17 109.34 139.16 1.27 Ⅳ 1.12 1.43 1820-S2 AD + TF + SD + F 99.75 AD + TF SD + F 1.42 110.70 134.08 1.21 Ⅵ 1.11 1.34 2150-S1 SD(3) + AD + TF 84.77 AD + TF SD(3) 1.43 89.26 134.69 1.51 Ⅳ 1.05 1.59 2150-S2 SD(3) + AD + TF 80.25 AD + TF SD(3) 1.45 90.54 137.19 1.52 Ⅳ 1.13 1.71 ZA32 500-S1 L + SD(1) 138.58 L SD(1) 1.35 170.89 157.98 0.92 Ⅱ 1.23 1.14 500-S2 L + SD(1) 143.84 L SD(1) 1.35 170.37 157.50 0.92 Ⅱ 1.18 1.09 500-S3 L + SD(1) 140.69 L SD(1) 1.35 170.02 157.07 0.92 Ⅱ 1.21 1.12 500-S4 L + SD(1) 142.96 L SD(1) 1.37 168.33 155.34 0.92 Ⅱ 1.18 1.09 1520-S1 SD(3) + AD + TF 110.70 AD + TF SD(3) 1.47 125.88 143.34 1.14 Ⅳ 1.14 1.29 1520-S2 SD(3) + AD + TF 109.13 AD + TF SD(3) 1.47 126.51 144.89 1.15 Ⅳ 1.16 1.33 1820-S1 SD(3) + TF 105.95 AD + TF SD(3) 1.85 108.00 145.04 1.34 Ⅳ 1.02 1.37 ZA42 500-S1 L + SD(1) 182.68 L SD(1) 3.77 226.06 224.30 0.99 Ⅱ & Ⅴ 1.24 1.23 500-S2 L + SD(1) 182.91 L SD(1) 3.70 235.93 232.82 0.99 Ⅱ & Ⅴ 1.29 1.27 500-S3 L + SD(1) 183.70 L SD(1) 3.76 229.72 227.83 0.99 Ⅱ & Ⅴ 1.25 1.24 760-S1 L + SD(1) 171.55 L SD(1) 1.94 226.61 209.85 0.93 Ⅱ & Ⅴ 1.32 1.22 760-S2 L + SD(1) 167.48 L SD(1) 1.94 227.12 210.66 0.93 Ⅱ & Ⅴ 1.36 1.26 760-S3 L + SD(1) 175.14 L SD(1) 1.90 231.08 213.23 0.92 Ⅱ & Ⅴ 1.32 1.22 1010-S1 L + SD(1) 175.95 L SD(1) 1.41 224.82 200.03 0.89 Ⅴ 1.28 1.14 1010-S2 L + SD(1) 176.65 L SD(1) 1.40 229.09 202.74 0.88 Ⅴ 1.30 1.15 1820-S1 SD + F + AD + TF 151.07* SD + F AD + TF 1.66 133.44 177.84 1.33 Ⅵ - - Min = 1.02 1.09 Max = 1.36 1.71 Mean = 1.22 1.26 SD = 0.09 0.15 Note: * means the Pu value overestimates the axial strength of the specimen.202 (Ⅲ): The effective centroid of the specimen is offset by distortional deformation. This phenomenon induced the flexural deformation (δy) in the ZA30-1520-S1 specimen that failed in SD(3) + F buckling, where the details have been introduced in Section 7.5. (Ⅳ): Column ends spring back after the cold-rolling and cutting, resulting in the openings there wider than that at the mid-length. This phenomenon is believed to trigger the occurrence of SD(3) mode in the ZA30-1820-S1, ZA30-2150, ZA32-1520, and ZA32-1820-S1 specimens that failed in SD(3) + AD + TF or SD(3) + TF buckling. First, Table 7.10 shows that neither the Pcr nor the Pn values of SD(3) and AD + TF buckling modes of the specimens are close. Second, symmetric rotations of flanges appeared in these specimens earlier than torsion (T) during the compression test; the evidence can be observed from the figures showing the evolution of cross-section deformation of the specimens with loading in Section 7.5. (Ⅴ): Column lips are wide and not straight. This phenomenon triggered local buckling halfwaves to occur in the lips of the ZA42-500, ZA42-760, and ZA42-1010 specimens at early loading stages, where the details have been introduced in Section 7.5. (Ⅵ): The column has initial loading eccentricity and/or initial crookedness. This phenomenon induced SD + F mode in the ZA30-1820-S2 specimen and AD + TF mode in the ZA42-1820-S1 specimen, where both specimens presented AD + TF + SD + F failure modes. The details have been given in Section 7.5. Finally, it can be observed from the last two columns of Table 7.10 that the Pn values overestimated the axial strengths of all the specimens, where differences around 20% are common. 7.7 Unfactored Design Axial Strengths The DSM in AISI_S100 (2016) requires the smallest of the nominal axial strengths, i.e., min (Pnl, Pnd, Pne), as the unfactored design axial strength of a CFS member. The unfactored design 203 axial strengths of the specimens tested in this study were determined per the DSM (AISI_S100 2016), where the results have been shown in Table 7.5, Table 7.6, and Table 7.7 for the ZA30, ZA32, and ZA42-section specimens, respectively. Note here that: ⅰ) L + G buckling was not considered while designing these specimens since this buckling mode was not observed during the experiments, and ⅱ) those specimens whose Pu values overestimated the axial strengths due to the boundary condition issue were not included in the database for assessing the DSM. It can be observed from Table 7.5, Table 7.6, and Table 7.7 that the axial strengths of most specimens are overestimated by the unfactored design axial strengths, where overestimations beyond 15% are common. The low accuracy of the DSM exhibited here attributes to two reasons. First, the DSM does not include the solutions for interactive buckling like L + SD, SD + G, and G + G. Second, the DSM was proposed based on a numerical database containing only lipped-C section columns with simply-simply end support condition (Moen and Schafer 2011), while the columns tested in this study have Σ-shaped sections and mostly P-P ends. 7.8 Summary In this study, thin-walled roll-formed steel columns with three different Σ-shaped sections and seven different lengths were tested under axial compression, where the columns have patterned holes uniformly distributed along the length. In total, 55 columns were tested, where the 250 and 500-mm long specimens were tested with bear-friction ends and the 760-mm to 2150-mm long specimens were tested with pinned ends. Axial strengths (Pu) and stiffnesses (Kae) of the columns were obtained from the experiments. Failure modes of the specimens were determined based on the experimental observations, the evolutions of cross-section deformation with loading, and overall deflections (δx and δy) during the compression. Various buckling modes were presented in the specimens, and the causes contributing to the interactive buckling modes, i.e., L + SD, SD(3) 204 + F, SD(3) + AD + TF, and SD + F + AD + TF, of the specimens were explored. The experimental results were employed to assess the Direct Strength Method in the North American design specification of AISI_S100 (2016). Based on this study, the conclusions are made as follows: (1) Axial strengths of the columns were inaccurately and unsafely (mostly) estimated by the nominal axial strengths calculated using the Direct Strength Method. It is suggested in the future to ⅰ) recalibrate the design expressions in the Direct Strength method with an extensive database containing CFS columns with various shapes of sections, ⅱ) incorporate design solutions addressing various interactive buckling modes like L + SD, SD + G, and/or SD + F + AD + TF of CFS columns in the Direct Strength Method, and ⅲ) consider the variations in the axial strengths of CFS columns caused by different end support conditions. (2) The use of pinned-pinned end support conditions and geometric alignment often brought initial loading eccentricities in the compression tests of the columns, which resulted in uncertainties in the tested results, especially the failure modes. Future studies will present detailed FE analyses, validation, and parametric study considering the above-mentioned uncertainties. 205 Chapter 8: Proposal of New Buckling Strength Curves 8.1 General Shortcomings of the buckling strength curves (i.e., Pnl-, Pnd-, and Pne-curve) in the DSM (AISI_S100 2016) have been identified in Chapter 4. This chapter aims at resolving the shortcomings by proposing new buckling strength curves based on the experimental database established in Chapter 3 and part of the experimental data obtained in Chapter 7. Works fulfilled in this chapter are summarized as follows: ⅰ) In Chapter 4, the Pnl-curve for solid columns exhibited reasonable accuracy in estimating the axial strengths of the solid column specimens that failed in L buckling. However, there were 20 data points (i.e., (λl, Pu/Py) ) lay above the plateau of the curve (i.e., Pu/Py = 1.0), leading to an uneconomic design for the column specimens featuring small λl values; see Figure 4.1(a). This shortcoming is resolved in Section 8.2.1 by proposing a new Pnl-curve considering the full-section tensile yield strength (Fya) as the design parameter. ⅱ) In Chapter 4, the Pnd-curve for solid columns presented good accuracy in estimating the axial strengths of the solid column specimens with B-F or F-F ends that failed in D buckling. Even so, an attempt of proposing a new Pnd-curve considering the full-section tensile yield strength (Fya) as the design parameter is made in Section 8.2.2 to see if a more economical design could be obtained. ⅲ) In Chapter 4, the second segment of the Pne-curve (i.e., λc > 1.5) exhibited a tendency to underestimate the axial strengths of the solid column specimens with P-P-W ends that failed in F buckling. This shortcoming is resolved in Section 8.2.3 by proposing two new sets of Pne-curve. The first Pne-curve considers the virgin material tensile yield strength as the design parameter, while the full-section tensile yield strength (Fya) is employed as the design parameter for the second 206 Pne-curve. Each Pne-curve is described by a single expression. A comparison is made between the performance of the two curves. ⅳ) In Chapter 4, the Pnl-curve for perforated column exhibited adequate accuracy in estimating the axial strengths of the perforated column specimens that failed in L buckling. However, the data points of (λl, Pnl/Pu) exhibited a peculiar distribution pattern; see Figure 4.2(b). This deficiency is resolved in Section 8.2.4 by proposing two new sets of Pnl-curve. The first Pnl-curve considers the net cross-section area (Anet) and virgin material tensile yield strength as design parameters. The second Pnl-curve adopts the net cross-section area (Anet) and net-section tensile yield strength (Fya, net) as design parameters. The performance of the two curves is compared mutually. ⅴ) In Chapter 4, the first segment of the Pne-curve (i.e., λc ≤ 1.5) was unsafe to design the perforated column specimens with P-P ends that failed in TF or AD + TF buckling. This deficiency is resolved in Section 8.2.5 by proposing two new sets of Pne-curve. The first Pne-curve considers the virgin material tensile yield strength and net cross-section area (An) as design parameters. The second Pne-curve adopts the net-section tensile yield strength (Fya, net) and net cross-section area (An) as design parameters. Each Pne-curve is described by a single expression. A comparison is made between the performance of the two curves. 8.2 New Buckling Strength Curves Buckling strength curves were proposed with the assist of a curve fitting software package named TableCurve 2D V5.01. While proposing a buckling strength curve, the form of the equation was defined by the author first. Then, the software determined the coefficients of the equation that best fit the data automatically. Finally, the equation was reviewed by the author and adjusted if necessary. 207 8.2.1 Local Buckling of Solid Columns In the database, 135 solid column specimens failed in L buckling. Based on their experimental data, the Pnl-curve was proposed as follows: For λl ≤ 0.568; Pnl = Pya For λl > 0.568; Pnl = �1 − 0.22 �PcrlPya�0.35� �PcrlPya�0.35 Pya Equation 8-1 where λl = (Pya/Pcrl)0.5, Pya = AgFya The Pnl values of the 135 column specimens were calculated as per Equation 8-1; then, the results were compared with the Pu values. The data points of (λl, Pu/Pya) of the column specimens are plotted along with the proposed Pnl-curve in Figure 8.1(a), form where it can be observed that only two data points lie slightly above the plateau of the curve (i.e., Pu/Pya = 1.0). For comparison, the Pnl-curve in the DSM (AISI_S100 2016) is also shown in the figure. Recall that when plotting the data points of (λl, Pu/Py) of these column specimens with the Pnl-curve in the DSM (AISI_S100 2016), 20 data points lay above the plateau of the curve (i.e., Pu/Py = 1.0); see Figure 4.1(a). In addition, the data points of (λl, Pnl/Pu) of the column specimens are plotted in Figure 8.1(b), where the Pnl/Pu ratios show the minimum, maximum, mean, and standard deviation of 0.75, 1.38, 1.01, and 0.09, respectively. 208 Figure 8.1 Solid column specimens failed in L buckling: (a) the data points of (λl, Pu/Pya) along with the proposed Pnl-curve; (b) the data points of (λl, Pnl/Pu). 8.2.2 Distortional Buckling of Solid Columns with B-F or F-F Ends In the database, 130 solid column specimens were tested with B-F or F-F ends and failed in D buckling. Based on their experimental data, the Pnd-curve was proposed as follows: For λd ≤ 0.533; Pnd = Pya For λd > 0.533; Pnd = �1 − 0.25 �PcrdPya �0.55� �PcrdPya �0.55 Pya Equation 8-2 where λd = (Pya/Pcrd)0.5, Pya = AgFya The Pnd values of the 130 column specimens were calculated as per Equation 8-2, where the results were compared with the Pu values. The data points of (λd, Pu/Pya) of the column specimens are plotted in Figure 8.2(a) along with the proposed Pnd-curve, and the corresponding data points of (λd, Pnd/Pu) are plotted in Figure 8.2(b). The Pnd/Pu ratios of the column specimens show the minimum, maximum, mean, and standard deviation of 0.76, 1.32, 1.04, and 0.13, respectively. The proposed Pnd-curve and the Pnd-curve in the DSM (AISI_S100 2016) exhibit comparable accuracy in estimating the axial strengths of these column specimens. 209 Recall that there were three data points lay above the plateau (i.e., Pu/Py = 1.0) while using the Pnd-curve in the DSM (AISI_S100 2016) in estimating the axial strengths of these column specimens; see Figure 4.3(a). In contrast, this phenomenon is avoided here using the proposed Pnd-curve. Figure 8.2 Solid column specimens with B-F or F-F ends that failed in D buckling: (a) the data points of (λd, Pu/Pya) with the proposed Pnd-curve; (b) the data points of (λd, Pnd/Pu). The Pnd-curve in the DSM (AISI_S100 2016) is also plotted in Figure 8.2(a). It can be observed from the figure that the two design curves nearly coincide when the slenderness value (λd) is less than about 1.20, and the DSM curve is conservative than the newly proposed curve when the slenderness value goes beyond 1.20. 8.2.3 Flexural Buckling of Solid Columns with P-P or P-P-W Ends In the database, 80 solid column specimens were tested with P-P or P-P-W ends and failed in F buckling. Based on their experimental data, a new Pne-curve was proposed here, considering the virgin material tensile yield strength as the design parameter; see Equation 8-3. Pne = Py�1+λc4.05�0.51 Equation 8-3 210 where λc = (Py/Pcre)0.5, Py = AgFy Another new Pne-curve was proposed here, considering the full-section tensile yield strength (Fya) as the design parameter; see Equation 8-4. Pne = Pya�1+λc3.12�0.64 Equation 8-4 where λc = (Pya/Pcre)0.5, Pya = AgFya Note that ⅰ) the form of Equation 8-3 and Equation 8-4 referred to that of the design equation in CSA S16 (2019) for G buckling of conventional steel members under compression and ⅱ) the proposed two Pne-curves maintain no higher than the elastic critical buckling curve; see Figure 8.3(a) and Figure 8.3(c). For comparison, the Pne-curve in the DSM (AISI_S100 2016) is also plotted in the two figures. The Pne values of the 80 column specimens were calculated as per Equation 8-3 first. The results were compared with the Pu values. The data points of (λc, Pu/Py) of the column specimens are plotted along with the proposed Pne-curve (Equation 8-3) in Figure 8.3(a), where the corresponding data points of (λc, Pne/Pu) are presented in Figure 8.3(b). The Pne/Pu ratios of the column specimens show the minimum, maximum, mean, and standard deviation of 0.62, 1.29, 0.98, and 0.14, respectively. In addition, the proposed Pne-curve (Equation 8-3) is compared with its counterpart in the DSM (AISI_S100 2016) in Figure 8.3(a). The figure shows that the DSM curve is conservative than the proposed design curve when λc < 1.20 but nearly coincides with the proposed design curve in the slenderness region of λc ≥ 1.20. Likewise, the Pne values of the 80 column specimens were calculated as per Equation 8-4, and the results were compared with the Pu values. The data points of (λc, Pu/Pya) of the column specimens are plotted in Figure 8.3(c) along with the proposed Pne-curve (Equation 211 8-4), where the corresponding data points of (λc, Pne/Pu) are presented in Figure 8.3(d). The Pne/Pu ratios of the column specimens show the minimum, maximum, mean, and standard deviation of 0.65, 1.25, 0.97, and 0.14, respectively. In addition, the proposed Pne-curve (Equation 8-4) is compared with its counterpart of the DSM (AISI_S100 2016) in Figure 8.3(c). It can be observed from the figure that the two curves are close throughout the slenderness region (i.e., 0 ≤ λc ≤ 4). This indicates that the Fya can be readily used as the design parameter of the DSM Pne-curve (AISI_S100 2016) to design solid column specimens with P-P or P-P-W ends experiencing F buckling. The proposed two Pne-curves show comparable accuracy in estimating the axial strengths of the column specimens. However, the second Pne-curve (Equation 8-4) is preferred since it resulted in a vertical dispersion of the data points of (λc, Pne/Pu) slightly less prominent than the one resulting from the first Pne-curve (Equation 8-3). Alternatively, the DSM Pne-curve (AISI_S100 2016) can be retained but with the design parameter of Fyv being replaced with Fya. 212 Figure 8.3 Solid column specimens with P-P or P-P-W ends that failed in F buckling: (a) the data points of (λc, Pu/Py) with the proposed Pne-curve (Equation 8-3); (b) the data points of (λc, Pne/Pu) calculated as per Equation 8-3; (c) the data points of (λc, Pu/Pya) with the proposed Pne-curve (Equation 8-4); (d) the data points of (λc, Pne/Pu) calculated as per Equation 8-4. 8.2.4 Local Buckling of Perforated Columns In the database, 124 perforated column specimens failed in L buckling, where nine of them are from this study. Based on their experimental data, a new Pnl-curve was proposed here, considering the virgin material tensile yield strength and net cross-section area (An) as design parameters; see Equation 8-5. 213 For λl ≤ 0.765; Pnl = Pyn For λl > 0.765; Pnl = �1 − 0.12 �PcrlPyn�0.28� �PcrlPyn�0.28 Pyn Equation 8-5 where λl = (Pyn/Pcrl)0.5, Pyn = AnFy Another new Pnl-curve was proposed here, considering the net-section tensile yield strength (Fya, net) and net cross-section area (An) as design parameters; see Equation 8-6. For λl ≤ 0.583; Pnl = Pya, net For λl > 0.583; Pnl = �1 − 0.2 � PcrlPya,net�0.3� � PcrlPya,net�0.3 Pya,net Equation 8-6 where λl = (Pya, net/Pcrl)0.5, Pya, net = AnFya, net The Pnl values of the 124 column specimens were calculated as per Equation 8-5 first. The results were compared with the Pu values. The data points of (λl, Pu/Pyn) of the column specimens are plotted in Figure 8.4(a) along with the proposed Pnl-curve (Equation 8-5), where the corresponding data points of (λl, Pnl/Pu) are shown in Figure 8.4(b). The Pnl/Pu ratios of the column specimens show the minimum, maximum, mean, and standard deviation of 0.80, 1.14, 0.98, and 0.06, respectively. Likewise, the Pnl values of the 124 column specimens were calculated as per Equation 8-6, and the results were compared with the Pu values. The data points of (λl, Pu/Pya, net) of the column specimens are plotted along with the proposed Pnl-curve (Equation 8-6) in Figure 8.4(c), where the corresponding data points of (λl, Pnl/Pu) are presented in Figure 8.4(d). The Pnl/Pu ratios of the column specimens show the minimum, maximum, mean, and standard deviation of 0.83, 1.12, 0.98, and 0.06, respectively. 214 Figure 8.4 Perforated column specimens failed in L buckling: (a) the data points of (λl, Pu/Pyn) with the proposed Pnl-curve (Equation 8-5); (b) the data points of (λl, Pnl/Pu) calculated as per Equation 8-5; (c) the data points of (λl, Pu/Pya, net) with the proposed Pnl-curve (Equation 8-6); (d) the data points of (λl, Pnl/Pu) calculated as per Equation 8-6. The second Pnl-curve (Equation 8-6) outperformed the first Pnl-curve (Equation 8-5) since the former avoided data points to lie above the plateau (i.e., Pu/Pya, net = 1.0), providing an economical design for the column specimens having small λl values. 215 8.2.5 TF Buckling of Perforated Columns with P-P Ends In the database, only 20 perforated column specimens were tested with P-P ends and failed in TF or AD + TF buckling. Additionally, two specimens (i.e., the ZA32-2150 specimens) in this study presented AD + TF buckling failure in the compression tests. Based on the 22 tested axial strengths, a new Pne-curve was proposed here, considering the virgin material tensile yield strength and net cross-section area (An) as design parameters; see Equation 8-7. Pne = Pyn�1+λc2.92�0.7 Equation 8-7 where λc = (Pyn/Pcre)0.5, Pyn = AnFy Another new Pne-curve was proposed here, considering the net-section tensile yield strength (Fya, net) and net cross-section area (An) as design parameters; see Equation 8-8. Pne = Pya,net�1+λc2.4�0.82 Equation 8-8 where λc = (Pya, net/Pcre)0.5, Pya, net = AnFya, net The Pne values of the 22 column specimens were calculated as per Equation 8-7 first, where the results were compared with the Pu values. The data points of (λc, Pu/Pyn) of the column specimens are plotted along with the proposed Pne-curve (Equation 8-7) in Figure 8.5(a), where the corresponding data points of (λc, Pne/Pu) are shown in Figure 8.5(b). The Pne/Pu ratios of the column specimens show the minimum, maximum, mean, and standard deviation of 0.86, 1.13, 1.00, and 0.06, respectively. 216 Figure 8.5 Perforated column specimens with P-P ends that failed in TF or AD + TF buckling: (a) the data points of (λc, Pu/Pyn) with the proposed Pne-curve (Equation 8-7); (b) the data points of (λc, Pne/Pu) calculated as per Equation 8-7; (c) the data points of (λc, Pu/Pya, net) with the proposed Pne-curve (Equation 8-8); (d) the data points of (λc, Pne/Pu) calculated as per Equation 8-8. Likewise, the Pne values of the 22 column specimens were calculated as per Equation 8-8. Then, the results were compared with the Pu values. The data points of (λc, Pu/Pya, net) of the column specimens are plotted along with the proposed Pne-curve (Equation 8-8) in Figure 8.5(c), where the corresponding data points of (λc, Pne/Pu) are presented in Figure 8.5(d). The Pne/Pu ratios of the 217 column specimens show the minimum, maximum, mean, and standard deviation of 0.84, 1.12, 1.00, and 0.07, respectively. The two Pne-curves proposed here exhibit comparable accuracy in estimating the axial strengths of the column specimens. 8.3 Resistance Factors of the New Buckling Strength Curves The LRFD (Load and Resistance Factor Design) resistance factors (φc) of the proposed buckling strength curves were calculated for a target reliability index of 2.5, where the calculation equation was the same as the one (Equation 4-1) used in Chapter 4. The results are shown in Table 8.1. Table 8.1 LRFD resistance factors of the proposed buckling strength curves Buckling strength curve λ Pu/Pn φc Min Max Min Max Mean SD COV n Pnl-curve (Equation 8-1) 0.458 4.903 0.722 1.325 1.000 0.093 0.093 135 0.881 Pnd-curve (Equation 8-2) 0.585 3.186 0.758 1.319 0.977 0.127 0.130 130 0.828 Pne-curve (Equation 8-3) 0.514 2.797 0.777 1.608 1.041 0.167 0.160 80 0.847 Pne-curve (Equation 8-4) 0.535 2.935 0.800 1.532 1.054 0.161 0.153 80 0.866 Pnl-curve (Equation 8-5) 0.458 2.637 0.876 1.246 1.027 0.067 0.065 124 0.927 Pnl-curve (Equation 8-6) 0.490 2.762 0.891 1.199 1.023 0.061 0.060 124 0.926 Pne-curve (Equation 8-7) 0.762 1.663 0.888 1.166 1.004 0.068 0.067 22 0.902 Pne-curve (Equation 8-8) 0.821 1.856 0.896 1.189 1.003 0.071 0.071 22 0.898 Table 8.1 shows that all the proposed buckling strength curves, except for the Pnd-curve (Equation 8-2) and Pne-curve (Equation 8-3), show φc values greater than 0.85. The φc values of the Pnl-curve (Equation 8-5), Pnl-curve (Equation 8-6), Pne-curve (Equation 8-7), and Pne-curve (Equation 8-8) are even close to or above 0.90, providing additional safety assuming φc = 0.85 is used in the design. The Pnd-curve (Equation 8-2) shows a φc value (0.828) lower than 0.85, which may attribute to the fact that the nonlinear part of the curve was adjusted after the curve fitting so 218 that it can reach the plateau of Pu/Pya = 1.0. Similarly, the Pne-curve (Equation 8-3) shows a φc value (0.847) slightly less than 0.85 since the curve was adjusted after the curve fitting so that the curve maintains no higher than the elastic critical buckling curve. 8.4 Summary Eight new sets of buckling strength curves were proposed in this chapter to resolve the shortcomings of the buckling strength curves in the DSM (AISI_S100 2016) identified in Chapter 4. The proposal of the buckling strength curves was on the basis of the experimental database established in Chapter 3 and part of the experimental data obtained in Chapter 7. The LRFD resistance factors of the proposed buckling strength curves were calculated for a target reliability index of 2.5 using the procedure given in AISI S100 (2016). Based on this study, conclusions are made as follows: ⅰ) The proposed Pnl-curve (Equation 8-1) provided a safe and more economical design for the solid column specimens that failed in L buckling than the Pnl-curve in the DSM (AISI_S100 2016). The φc value of the proposed Pnl-curve was calculated as 0.881. ⅱ) The Pnd-curve in the DSM (AISI_S100 2016) exhibited a good performance in estimating the axial strengths of the solid column specimens with B-F or F-F ends that failed in D buckling. Hence, it is recommended to be continuously used in the routine design. ⅲ) The proposed Pne-curve (Equation 8-4) provided an adequate and safe design for the solid column specimens with P-P or P-P-W ends that failed in F buckling, where the calculated φc value was 0.866. Hence, it is recommended to be used in the routine design. ⅳ) The proposed Pnl-curve (Equation 8-6) provided an economical and safe design for the perforated column specimens that failed in L buckling, where the calculated φc value was 0.926. This buckling strength curve is suggested to be used in the routine design. 219 ⅴ) The proposed two Pne-curves (Equation 8-7 and Equation 8-8) all provided an adequate and safe design for the perforated column specimens with P-P ends that failed in AD + TF or TF buckling, where the calculated φc values of the two curves were close to 0.90. Either one of them is recommended to be used in the routine design. On the other hand, the remaining shortcomings of the DSM (AISI_S100 2016) are summarized below, where the reasons for not addressing them in this study are also presented. ⅰ) In Chapter 4, the Pnd-curves for solid and perforated columns were inadequate to design the column specimens with P-P or P-P-W ends that failed in D + F buckling. For designing these column specimens adequately, it would be necessary to treat their buckling behaviour as the interaction between D buckling under fixed ends and F buckling under pinned ends. Addressing this problem is out of the scope of this study and will be fulfilled in the future. ⅱ) The Pnd-curve for perforated columns with B-F or F-F ends needs to be further assessed with more data. This task will be fulfilled in the future, where an extensive database could be established by conducting more experimental tests, collecting more experimental data from the literature, and/or performing extensive nonlinear FE simulations. ⅲ) In Chapter 4, the first segment of the Pne-curve (i.e., λc ≤ 1.5) was unsafe to design the perforated column specimens with P-P or P-P-W ends that failed in F buckling. Like the solid columns, different buckling strength curves may be necessary for perforated columns with different end support conditions and experiencing different G buckling modes. Addressing this problem requires an extensive database. Hence, this task will be fulfilled in the future. 220 Chapter 9: Conclusions 9.1 Summary This research focused on the topic of accurately estimating the axial strengths of thin-walled, cold-formed steel (CFS) columns under local, distortional, and global buckling. Three objectives were accomplished in this research: Ⅰ) To assess the accuracy of the buckling strength curves in the Direct Strength Method (DSM), where the DSM is incorporated in the North American design specification of AISI S100 (2016) for the design of CFS members under compression. To accomplish this objective, ⅰ) a thorough literature review was carried out to identify the merits and shortcomings of the buckling strength curves in the DSM; ⅱ) an extensive experimental database including compression tests of 453 solid and 228 perforated CFS columns was established, where the experimental data were mainly collected from the literature, and the author determined the design parameters of the columns; ⅲ) the buckling strength curves in the DSM were assessed with the experimental database to determine their accuracy. Besides, the LRFD (Load and Resistance Factor Design) resistance factors (φc) of the buckling strength curves in the DSM were calculated for a target reliability index of 2.5. Ⅱ) To enlarge the experimental database of compression tests of perforated CFS columns. To accomplish this objective, ⅰ) compression tests were performed on 55 CFS columns with patterned perforations distributed along the length uniformly. The columns had three different cross-sections, and each cross-section had seven different lengths varying from 250 mm to 2150 mm to investigate the influences of length on the axial strengths and failure modes of the columns. The experimental results, including axial strengths, failure modes, axial load-axial displacement curves, and failure mechanisms, were reported in the thesis. Also, design parameters of the 221 columns like cross-section areas, material mechanical properties, elastic buckling loads were also given in the thesis so that readers can easily access the experimental data for further research; ⅱ) extensive eigenbuckling analyses were performed for the columns, where the influences of perforations, perforation pattern, the number of buckling half-wavelengths, and end support condition on the elastic buckling properties of each column section were investigated in detail. Ⅲ) To improve the axial strength estimations for CFS columns under local, distortional, and global buckling. To accomplish this objective, ⅰ) a thorough literature review was conducted to develop ways for improving the axial strength estimations of CFS columns. Factors affecting the axial strengths of CFS columns were identified as the cross-section geometry (shape and the ratios between the widths of the flat elements in the cross-section), end support condition, cold-forming effects on the material (i.e., the increase in material yield strength and residual stresses), initial geometric imperfection, and initial loading eccentricity. Among the factors, the increase in material strength caused by cold forming was found a feasible one to be considered in the design as an improvement; ⅱ) the model and expression proposed by Karren (1965) for predicting the tensile yield strengths of CFS corners and estimating the full-section tensile yield strengths of CFS members had their performance assessed with 218 and 24 experimental results collected from the literature, respectively. Later, the model and expression were modified to improve the accuracy, extend the application limit, and/or simplify the application; ⅲ) the buckling strength curves in the DSM (AISI_S100 2016) were assessed again using full-section tensile yield strengths as design parameters. The results were compared with those obtained from the previous assessment to see the potential whether using full-section tensile yield strengths as design parameters could lead to more accurate strength estimations; ⅳ) eight new sets of buckling strength curves were proposed based on the enlarged experimental database, and their LRFD (Load and Resistance Factor Design) 222 resistance factors (φc) were calculated for a target reliability index of 2.5. Comments on using the new buckling strength curves in the routine design were provided. 9.2 Conclusions The following conclusions are made from this research: 1) Karren’s model (Equation 3-1) for predicting the tensile yield strengths of CFS corners has been modified to Equation 3-2 to improve the accuracy and extend the application limit from right-angle corners with r/t ≤ 6.0 to corners with variable angles and featuring r/t ≤ 9.0 and Fyv ≤ 1000 MPa. 2) Karren’s expression (Equation 3-3) for estimating the full-section tensile yield strengths of CFS members has been modified to Equation 3-4 so that the experimental determination of the parameter of Fyfa is avoided. 3) The expanded Ramberg-Osgood model has been proposed (Equation 7-1) to describe steel’s stress-strain curves with yield plateaus. 4) The multi-half-wavelengths method has been proposed to accurately determine the lower bounds of elastic local and distortional buckling loads and the associated buckling half-wavelengths of perforated CFS columns. 5) The trigonometric shape functions adopted in the finite strip method failed to accurately describe the buckling shapes of perforated columns and, hence, should be avoided. 6) For determining the axial strengths of CFS columns under local buckling experimentally, it is suggested that the specimen lengths shall not be less than 5𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐𝑆𝑆−𝑆𝑆 and not exceed 20 times the minimum radius of gyration (𝑟𝑟). By doing so, the constraint effects on columns from the end support conditions (e.g., bearing-friction ends and fixed-fixed ends) are nearly fully eliminated. 223 7) Buckling strength curves in the DSM (AISI_S100 2016) have been assessed with extensive experimental data, where the shortcomings have been identified and summarized as follows: ⅰ) While using the Pnl-curve to estimate the axial strengths of perforated columns that failed in L buckling, the data points of (λl, Pnl/Pu) exhibited a peculiar distribution pattern. ⅱ) The Pnd-curves were inadequate to design the column specimens with P-P or P-P-W ends that failed in D + F buckling regardless of whether the column specimens had holes. ⅲ) The second segment of the Pne-curve (i.e., λc > 1.5) tended to underestimate the axial strengths of the solid column specimens with P-P-W ends that failed in F buckling. ⅳ) The first segment of the Pne-curve (i.e., λc ≤ 1.5) was unsafe to design the perforated column specimens with P-P ends that failed in TF or AD + TF buckling. ⅴ) The first segment of the Pne-curve (i.e., λc ≤ 1.5) was unsafe to design the perforated column specimens with P-P or P-P-W ends that failed in F buckling. 8) Eight new sets of buckling strength curves have been proposed in this study, where five of them are recommended to be used in the routine design: ⅰ) The Pnl-curve (Equation 8-1). It provided a safe and more economical design for the solid column specimens that failed in L buckling than the Pnl-curve in the DSM (AISI_S100 2016), where the calculated φc value was 0.881. ⅱ) The Pne-curve (Equation 8-4). It provided a reasonable and safe design for the solid column specimens with P-P or P-P-W ends that failed in F buckling, where the calculated φc value was 0.866. 224 ⅲ) The Pnl-curve (Equation 8-6). It provided an economical and safe design for the perforated column specimens that failed in L buckling, where the calculated φc value of the curve was 0.926. ⅳ) The two Pne-curves (Equation 8-7 and Equation 8-8). They both provided a reasonable and safe design for the perforated column specimens with P-P ends that failed in AD + TF or TF buckling, where the calculated φc values were all close to 0.90. Either one is recommended to be used in the routine design. 9) Thin-walled, roll-formed steel columns with three different Σ-shaped sections and seven different lengths have been tested under compression, where the columns had patterned holes uniformly distributed along the length. In total, 55 columns were tested, where the 250 and 500-mm long specimens were tested with bear-friction ends, and the 760-mm to 2150-mm long specimens were tested with pinned ends. Axial strengths of the columns were found inaccurately and unsafely (mostly) estimated by the DSM (AISI_S100 2016). 10) The DSM is suggested to incorporate design solutions to address various interactive buckling of CFS columns. 9.3 Contributions The main contributions of this study are as follows: 1) An overview of the accuracy of the buckling strength curves in the Direct Strength Method incorporated in the current North American design specification of AISI S100 (2016) for the design of cold-formed steel members (including solid and perforated) under compression has been provided. The buckling strength curves were assessed with an extensive experimental database, including 453 solid and 228 perforated cold-formed steel columns, where their merits and shortcomings were identified thoroughly. 225 2) Karren’s model (Karren 1965) for predicting the tensile yield strengths of cold-formed steel corners has been modified to improve the accuracy and extend the application limit from right-angle corners only with r/t ≤ 6.0 to corners with variable angles, r/t ≤ 9.0 and Fyv ≤ 1000 MPa. This work was based on an experimental database including 190 right-angle corners and 28 corners with angles different from 90 degrees. 3) Karren’s expression (Karren 1965) for estimating the full-section tensile yield strengths of CFS members has been simplified to avoid the experimental determination of the input parameter of Fyfa. This work was based on an experimental database including six brake-pressed and 18 roll-formed steel members. 4) A model, namely the expanded Ramberg-Osgood model, has been proposed to describe steel’s stress-strain curves showing yield plateaus. 5) A method, namely the multi-half-wavelengths method, has been proposed to accurately determine the lower bounds of elastic local and distortional buckling loads and the associated buckling half-wavelengths of cold-formed steel columns with perforations. 6) Full-section tensile yield strengths have been used as design parameters to estimate the axial strengths of cold-formed steel members. Eight new buckling strength curves have been proposed, where five of them are recommended to be used in the routine design. 7) The experimental database of cold-formed steel columns with perforations under compression has been enlarged. In this research, 55 cold-formed steel columns were tested under compression, where they had three different Σ-shaped sections, seven different lengths ranging from 250 to 2150-mm, and patterned perforations uniformly distributed along the length. 9.4 Limitations of This Study This study has the following limitations: 226 1) Experimental results of a limited number of CFS corners with angles different from 90 degrees (count = 28) were employed to modify Karren’s model (Equation 3-1) to extend its application limit from right-angle corners to corners with various angles. 2) The simplification of Karren’s expression (Equation 3-3) for estimating the full-section tensile yield strengths of CFS members was based on limited experimental results of 24 CFS members. 3) Experimental results of only 13 perforated column specimens with B-F or F-F ends were used to assess the Pnd-curve in the DSM. 4) Limited experimental results of perforated columns were employed to assess the Pne-curve in the DSM. 5) Among the 55 CFS columns tested in this study, six specimens have the free rotations of the ends somewhat restrained by the single-universal joints. The axial strengths of the six specimens were overestimated by the tested results. 6) The proposal of the new Pne-curves (Equation 8-7 and Equation 8-8) for perforated columns with P-P ends experiencing TF or AD + TF buckling was based on limited experimental results of 22 columns. 9.5 Recommendations for Future Research Recommendations for future research are given below: 1) More compression tests related to CFS columns with perforations are required to enlarge the experimental database. Researchers are suggested to report all the design parameters of the specimens. 227 2) Like the solid CFS columns, different buckling strength curves may be necessary for perforated CFS columns in different cases (i.e., end support condition, buckling mode). 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Thesis/Dissertation
eng
Civil Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Alam, M. Shahria
Buckling strength curves for estimating the strengths of thin-walled, cold-formed steel columns under compression using full-section tensile yield strengths
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