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Seismic performance evaluation and design of a novel piston based self-centering bracing system Haque, A. B. M. Rafiqul 2017

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SEISMIC PERFORMANCE EVALUATION AND DESIGN OF A NOVEL PISTON BASED SELF-CENTERING BRACING SYSTEM  by  A. B. M. Rafiqul Haque   B.Sc. (Civil Engineering), Bangladesh University of Engineering and Technology, 2007 M.A.Sc. (Civil Engineering), The University of British Columbia, 2012   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 Campus)  June 2017  © A. B. M. Rafiqul Haque, 2017ii  The undersigned certify that they have read, and recommend to the College of Graduate Studies for acceptance, a thesis entitled: SEISMIC PERFORMANCE EVALUATION AND DESIGN OF A NOVEL PISTON BASED SELF-CENTERING BRACING SYSTEM Submitted by A. B. M. Rafiqul Haque in partial fulfillment of the requirements of The degree of DOCTOR OF PHILOSOPHY. Dr. M. Shahria Alam, Associate Professor, School of Engineering, UBC Supervisor, Professor Dr. Ahmad Rteil, Assistant Professor, School of Engineering, UBC Supervisory Committee Member, Professor Dr. Abbas Milani, Professor, School of Engineering, UBC Supervisory Committee Member, Professor Dr. Keith Culver, Professor, Faculty of Management, UBC University Examiner, Professor Dr. Michael J. Tait, Professor, Faculty of Engineering, McMaster University External Examiner, Professor 22 June 2017 (Date submitted to Grad Studies)  Additional Committee Members include: Dr. Stephen Karrer O’Leary, Associate Professor, School of Engineering, UBC Neutral Chair  iii  ABSTRACT Buildings designed and constructed using current seismic design codes experience nonlinear deformation during strong earthquakes; resulting in hysteretic damping. This behavior is necessary for dissipating energy from an earthquake. Thus, buildings with traditional structural systems experience permanent deformation after seismic events, resulting in enormous economic losses. To resolve this issue, researchers have developed various smart structural systems in the past decades. One such system is the novel piston based self-centering bracing (PBSC) system. This study investigates the cyclic performance of this bracing system in finite element environment to predict its load-deformation response during seismic events. This newly developed bracing system utilizes Nickel Titanium (Nitinol) based shape memory alloy (SMA) bars inside a sleeve-piston assembly for the self-centering mechanism. During cyclic loading, the bars are pulled from opposite directions to avoid compressive loading on the bars. The energy dissipation is achieved through nonlinear load deformation hysteresis. Furthermore, this bracing system can be designed to be buckling restraint. The system exhibits flag shaped force-deformation hysteresis. A unique hysteresis model is proposed from the simulated hysteresis response of this bracing system. This hysteresis model is implemented in a commercial structural analysis and design software known as “S-FRAME.” In the next phase, seismic performance of steel frames equipped with the PBSC bracing system has been evaluated. Overstrength and force reduction factors were determined using FEMA P695 methodology. Using these factors, PBSC braced frames were designed and their seismic performance was assessed in terms of inter-story drift ratios. Furthermore, the seismic performance of PBSC braced frame was also compared against buckling restrained braced frames in terms of fragility function. Finally, this research presents a design methodology for this bracing system.  iv  PREFACE  A significant part of this study has been submitted to peer-reviewed journals and conference proceedings for publication. All literature review, mathematical calculations and analytical models presented in the following journal and conference papers have been exclusively carried out by the author. The academic supervisor was only responsible for guidance and review of the work produced. List of publications related to this thesis Haque, A.B.M.R. & Alam, M.S. (2014). Piston-based self-centering brace apparatus, Patent US 20160326742 A1, CA2935575A1, WO2015100497A1. Haque, A.B.M.R. & Alam, M.S. (2015). Cyclic Performance of a Piston Based Self-Centering Bracing System. ASCE Structural Congress. April 23–25, 2015, Portland, Oregon.  Haque, A.B.M.R. Sasibut, S. Rong, F. & Alam, M.S. (2016). A Hysteresis Model for the Piston-Based Self-Centering Bracing System Considering Residual Deformation. 5th International Structures Specialty Conference, CSCE, London, ON, June 2016, Paper ID: 978. Haque, A.B.M.R. & Alam, M.S. (2017). Hysteretic behaviour of a piston-based self-centering (PBSC) bracing system made of superelastic SMA bars - a feasibility study. Tentatively accepted, Structures, Elsevier. Haque, A.B.M.R. & Alam, M.S. (2017). SMA flag hysteresis model with sliding response for the piston based self-centering bracing system. To be Submitted, Engineering Structures, Elsevier. Haque, A.B.M.R. & Alam, M.S. (2017). Seismic response modification factor of braced frame buildings equipped with piston based self-centering SMA bracing system. To be Submitted, Journal of Constructional Steel Research, Elsevier.  v  TABLE OF CONTENTS Abstract .......................................................................................................................................... iii Preface ............................................................................................................................................ iv Table of Contents ............................................................................................................................ v List of Tables ................................................................................................................................... x List of Figures ................................................................................................................................ xi List of Notations ........................................................................................................................... xvi Acknowledgements ...................................................................................................................... xxi Dedication .................................................................................................................................. xxiii CHAPTER 1: Introduction and Thesis Organization ...................................................................... 1 1.1 General ...................................................................................................................... 1 1.2 Objective of the study ............................................................................................... 5 1.3 Scope of this research ................................................................................................ 6 1.4 Thesis organization ................................................................................................... 9 CHAPTER 2: Overview of the Current State-of-the-art ............................................................... 12 2.1 General .................................................................................................................... 12 2.1.1 Energy dissipative bracing systems ................................................................. 13 2.1.2 Self-centering bracing systems ........................................................................ 16 2.2 Some notable self-centering systems ...................................................................... 17 2.2.1 PBSC brace ...................................................................................................... 17 2.2.2 MANSIDE brace .............................................................................................. 18 2.2.3 SCED brace ...................................................................................................... 19 2.2.4 RHDB device ................................................................................................... 19  vi  2.2.5 SC-BRB ........................................................................................................... 20 2.2.6 Hybrid steel-SMA device ................................................................................ 21 2.3 Other notable studies ............................................................................................... 22 2.4 Shape memory alloy ................................................................................................ 24 2.4.1 Microstructure of Nitinol ................................................................................. 25 2.4.2 Shape-memory effect ....................................................................................... 26 2.4.3 Superelasticity .................................................................................................. 27 2.4.4 Mechanism ....................................................................................................... 28 2.4.5 SMA bar behavior ............................................................................................ 30 CHAPTER 3: Hysteretic Behavior of a Piston Based Self-centering Bracing System ................ 32 3.1 General .................................................................................................................... 32 3.2 PBSC brace design .................................................................................................. 32 3.3 Materials .................................................................................................................. 35 3.4 Brace mechanism .................................................................................................... 37 3.5 Construction of the brace ........................................................................................ 39 3.6 Finite element model ............................................................................................... 41 3.6.1 Meshing ............................................................................................................ 44 3.6.2 Contact surfaces ............................................................................................... 45 3.6.3 Boundary conditions and loading .................................................................... 47 3.7 Analysis and results ................................................................................................. 48 3.8 Summary ................................................................................................................. 59 CHAPTER 4: A Hysteresis Model for the PBSC Bracing System ............................................... 61 4.1 General .................................................................................................................... 61  vii  4.2 SMA bar behavior ................................................................................................... 62 4.3 SMA flag hysteresis rule ......................................................................................... 63 4.4 PBSC brace hysteresis model development ............................................................ 66 4.5 PBSC mechanism .................................................................................................... 67 4.6 Simplified finite element model .............................................................................. 68 4.7 SMA material model ............................................................................................... 70 4.8 Quasi-static response from MATLAB .................................................................... 81 4.9 Validation ................................................................................................................ 83 4.10 Hysteresis rules ....................................................................................................... 85 4.10.1 In positive force direction ................................................................................ 86 4.10.2 In negative force direction ............................................................................... 87 4.11 Response under quasi-static loading ....................................................................... 89 4.12 Analysis model ........................................................................................................ 94 4.13 Results ..................................................................................................................... 98 4.14 Summary ............................................................................................................... 102 CHAPTER 5: Seismic Response Modification Factors for the PBSC Bracing System ............. 104 5.1 General .................................................................................................................. 104 5.2 Force based design procedure ............................................................................... 106 5.3 Necessity of this study ........................................................................................... 110 5.4 Description of the FEMA P695 methodology ....................................................... 112 5.5 Design and modeling of the PBSC braced frames ................................................ 116 5.1 Incremental dynamic time history analysis ........................................................... 119 5.1.1 Ground motion record selection and scaling ................................................. 120  viii  5.1.2 IDA results ..................................................................................................... 124 5.2 Collapse margin ratio calculation .......................................................................... 127 5.2.1 Adjusted collapse margin ratio ...................................................................... 129 5.2.1 Acceptable values of collapse margin ratio ................................................... 135 5.3 Overstrength Factor ............................................................................................... 139 5.4 Summary ............................................................................................................... 140 CHAPTER 6: Seismic Fragility Comparison Between PBSC and BRB Bracing System .......... 141 6.1 General .................................................................................................................. 141 6.2 Research significance ............................................................................................ 142 6.3 Description of the building models ....................................................................... 142 6.4 Finite element modeling and analysis ................................................................... 143 6.5 Earthquake records ................................................................................................ 143 6.6 Results and discussions ......................................................................................... 145 6.7 Buckling restraint bracing system ......................................................................... 150 6.8 Fragility comparison between PBSC and BRB frames ......................................... 153 6.9 Summary ............................................................................................................... 156 CHAPTER 7: Conclusions .......................................................................................................... 157 7.1 Summary ............................................................................................................... 157 7.2 Limitations of this study ........................................................................................ 159 7.3 Conclusion ............................................................................................................. 160 7.4 Recommendations for future research ................................................................... 161 References ................................................................................................................................... 164 Appendices .................................................................................................................................. 173  ix  Appendix A: Design method for PBSC braced frame ...................................................... 173 Appendix B: MATLAB finite element program ............................................................... 183 Appendix C: MATLAB material models .......................................................................... 191 Appendix D: PBSC brace hysteresis model ...................................................................... 193   x  LIST OF TABLES Table 3-1: Normalized energy dissipation comparison. ................................................................ 59 Table 4-1: Description of the flow chart parameters ..................................................................... 77 Table 4-2: Section properties used for the PBSC MATLAB model ............................................. 81 Table 4-3: Spectral acceleration values for Vancouver Soil Class “C” ........................................ 96 Table 5-1: Twenty-two far-field earthquake record details for collapse evaluation ................... 121 Table 5-2: Normalization factor for far-field record set ............................................................. 123 Table 5-3: Calculation of median collapse intensity and collapse margin ratio ......................... 129 Table 5-4: C0 coefficient calculation for 4-storied PBSC braced frame ..................................... 133 Table 5-5: C0 coefficient calculation for 6-storied PBSC braced frame ..................................... 134 Table 5-6: C0 coefficient calculation for 8-storied PBSC braced frame ..................................... 134 Table 5-7: Period based ductility calculation .............................................................................. 135 Table 5-8: Spectral shape factor (SSF) calculation ..................................................................... 135 Table 5-9: Acceptable values of collapse margin ratio ............................................................... 139 Table 5-10: Overstrength Factor (Ro) calculation ....................................................................... 140 Table 6-1: Damage states based on interstory drift ratio percentage .......................................... 147 Table A-1: Sample material properties for NiTinol SMA bars ................................................... 176 Table A-2: PBSC brace link hysteresis parameters .................................................................... 177  xi  LIST OF FIGURES Figure 1-1: Thesis organization flow-chart .................................................................................... 11 Figure 2-1: (a) A typical design of a buckling restrained brace (b)Hysteresis of a BRB brace (adapted from Asgarian and Shokrgozar (2009)) ...................................................... 14 Figure 2-2: Yielding brace prototype test showing deformed shape of the fingers (Gray et al. 2014a) ......................................................................................................................... 15 Figure 2-3: (a) A 7m long full-scale test specimen (b) Hysteresis of the full-scale yielding brace (Gray et al. 2012a) ..................................................................................................... 16 Figure 2-4: A 3D model of the PBSC bracing system ................................................................... 18 Figure 2-5: MANSIDE brace (a) cross section (b) 3D view (adapted from Dolce et al. (2000)) .................................................................................................................................... 18 Figure 2-6: (a) SCED brace (b) Hysteresis model for SCED brace (adapted from Christopoulos et al. (2008)) ............................................................................................................... 19 Figure 2-7: Longitudinal cross section of the RHDB brace (adapted from Zhang and Zhu (2007)) ........................................................................................................................ 20 Figure 2-8: Self-centering buckling restraint brace components (adapted from Miller et al. (2012)) ........................................................................................................................ 21 Figure 2-9: Cross-section of the hybrid device (adapted from Yang et al. (2010)) ....................... 22 Figure 2-10: Three-dimensional microstructure of the Nickel-Titanium alloy (a) Austenite (b) Martensite ................................................................................................................... 25 Figure 2-11: Two-dimensional microstructural of Nitinol in (a) Austenite (b) Twinned martensite and (c) Detwinned martensite phase (adapted from Speicher (2009)) ..... 26 Figure 2-12: Two-dimensional microstructure of Nitinol during superelastic and shape memory effect (adapted from Speicher (2009)) ......................................................... 28 Figure 2-13: 3D stress–strain temperature diagram depicting deformation and shape memory behavior of Nitinol-based shape memory alloy (adapted from Alam et al. (2007)) .. 29  xii  Figure 2-14: 2D stress–strain behavior of (a)Superelastic SMA (b)Shape memory SMA (adapted from Speicher (2009)) ................................................................................. 30 Figure 2-15: Stress–strain plot for 25.4 mm diameter nitinol SMA bar subjected to quasi-static cyclic loading (adapted from DesRoches et al. (2004)). ............................................ 31 Figure 3-1: Internal details of the PBSC brace with optional high damping rubber filler (a) Configuration 1 and (b) Configuration 2 ................................................................... 34 Figure 3-2: The use of the piston-based self-centering bracing system in a steel frame. .............. 35 Figure 3-3: Stress–strain diagram of SE SMA used in the finite element analysis. ...................... 37 Figure 3-4: (a) Semi-transparent view of the PBSC brace from ABAQUS (2014) (b) Longitudinal profile of the PBSC brace with sleeve hidden from view (c) State of the internal elements during tensile loading (sleeve hidden from view) (d) State of the internal elements during compressive loading (sleeve hidden from view). ......... 39 Figure 3-5: Construction of the PBSC Bracing System (a) Individual components (b) attachment of the SMA bars with the piston plate and the couplers (c) attachment of the back plate and the couplers/nuts (d) Attachment of the sleeve (e) attachment of the front plate and the couplers/nuts. ..................................................................... 40 Figure 3-6: Longitudinal section of the PBSC brace. .................................................................... 41 Figure 3-7: (a) Transverse section of the PBSC brace (b) Couplers for connecting SMA bars with piston plate or end caps. ..................................................................................... 42 Figure 3-8: (a) Backplate(b) Piston plate and (c) Front plate with openings to accommodate SMA bars and/or brace shaft. ..................................................................................... 43 Figure 3-9: (a) Connection details between piston plate and SMA bars (a) nuts/couplers excluded from the face. (b) nuts/couplers included. .................................................. 43 Figure 3-10: Boundary conditions of the PBSC brace model in ABAQUS (2014). ..................... 48 Figure 3-11: Quasi-static deformation loading history. ................................................................. 50 Figure 3-12: Velocity time history of the load. .............................................................................. 50 Figure 3-13: Acceleration time history of the load. ....................................................................... 51 Figure 3-14: von Mises stress distribution on the couplers. .......................................................... 52  xiii  Figure 3-15: von Mises stress distribution on the plates. ............................................................... 53 Figure 3-16: Von Mises stress distribution on (a) Sleeve and (b) Shaft. ....................................... 54 Figure 3-17: Von Mises stress distribution on the NiTinol ties ..................................................... 55 Figure 3-18: Hysteretic response of the PBSC bracing system with (a) 4-10mm (b) 4-12mm (c) 4-16mm and (d) 4-20mm dia NiTinol Bars. ......................................................... 56 Figure 3-19: Hysteretic response of the bracing system with (a) 4-10mm diameter steel bar (b) 4-12mm diameter steel bar (c) 4-16mm diameter steel bar (d) 4-20mm diameter steel bar. ..................................................................................................................... 58 Figure 3-20: Hysteretic flow paths for the brace with (a) SMA tie bars (b) Steel ties bars. ......... 58 Figure 4-1: Stress–strain plot for 25.4 mm diameter nitinol SMA bar subjected to quasi-static cyclic loading (adapted from DesRoches et al. (2004)). ............................................ 63 Figure 4-2: Uniaxial SMA flag-shaped hysteresis (adapted from Auricchio and Sacco (1997)) .. 65 Figure 4-3: Graphical rule of flag-shaped hysteresis (adapted from Carr (2008)) ........................ 66 Figure 4-4: Longitudinal section of the PBSC bracing system...................................................... 68 Figure 4-5: PBSC bracing system shown without the sleeve ........................................................ 68 Figure 4-6: Simplified PBSC brace model .................................................................................... 69 Figure 4-7: Further simplification of the PBSC brace model ........................................................ 70 Figure 4-8: SMA uniaxial stress-strain hysteresis model with residual deformation (Tension Cycles Only) .............................................................................................................. 72 Figure 4-9: Validation of SMA material hysteresis model against experimental result by (DesRoches et al. 2004). ............................................................................................ 73 Figure 4-10: Flowchart of the SMA material hysteresis model ..................................................... 74 Figure 4-11: Flow-chart of the MATLAB PBSC analysis program .............................................. 79 Figure 4-12: Input loading history for the MATLAB FE model ................................................... 82 Figure 4-13: Output deformation history from the MATLAB FE model ...................................... 82  xiv  Figure 4-14: Hysteresis response of the PBSC bracing system from MATLAB FE model .......... 83 Figure 4-15: ABAQUS FE model of PBSC brace configuration 2. .............................................. 84 Figure 4-16: Hysteresis Model validation using ABAQUS FE model. ......................................... 85 Figure 4-17: SMA flag-shaped hysteresis model with sliding response. ....................................... 89 Figure 4-18: PBSC brace hysteresis under different types of triangular loading .......................... 91 Figure 4-19: PBSC brace hysteresis under harmonic loading with increasing magnitude ............ 92 Figure 4-20: PBSC brace hysteresis under transient loading ......................................................... 93 Figure 4-21: Three-dimensional model of the steel building ......................................................... 95 Figure 4-22: (a) Nodal Restraint Conditions (b) Uniformly distributed gravity loading on the beam for the 2D model. .............................................................................................. 97 Figure 4-23: Sample input data for PBSC link hysteresis ............................................................. 98 Figure 4-24: Matched response spectra of the ten earthquake records .......................................... 99 Figure 4-25: Maximum and residual roof drift % ........................................................................ 100 Figure 4-26: (a) Maximum interstory Drift % (b) Residual interstory Drift % ........................... 101 Figure 4-27: Brace hysteresis for Trinidad earthquake (a) First story (b)Second story (c)Third story (d)Fourth story (e) Fifth story and (f) Sixth story ........................................... 102 Figure 5-1: Relationships between force reduction (Rd), overstrength (Ro), response modification factor (R) and displacement ductility (µ) ........................................... 109 Figure 5-2: Determination of seismic performance factors (FEMA 2009) ................................. 114 Figure 5-3: Illustrations of seismic performance factors (adapted from FEMA (2009)) ............ 115 Figure 5-4: Vancouver soil class “C” response spectrum ............................................................ 116 Figure 5-5: Three-dimensional model of the (a) four (b) six and (c) eight-storied PBSC building .................................................................................................................................. 118 Figure 5-6: The designed six-storied PBSC frame showing the required section sizes .............. 119  xv  Figure 5-7: FEMA P695 normalized mean spectra compared to the Vancouver soil class “C” response spectra ....................................................................................................... 124 Figure 5-8: Spectral acceleration vs maximum interstory drift ratio IDA curves for (a) 4-storied (b) 6-storied and (c) 8-storied PBSC braced frame .................................................. 126 Figure 5-9: Cumulative distribution function of collapse level spectral acceleration values for (a) 4-storied (b) 6-storied and (c) 8-storied PBSC braced frame ............................. 128 Figure 5-10: Fundamental mode shapes for the (a) 4-storied (b) 6-storied and (c) 8-storied PBSC building archetypes ....................................................................................... 131 Figure 5-11: Mode shape proportional horizontal loading pattern for the (a) 4-storied (b) 6-storied and (c) 8-storied PBSC braced frame ........................................................... 132 Figure 5-12: Pushover curves for the (a) 4-storied (b) 6-storied and (c) 8-storied frame ........... 133 Figure 6-1: Response spectra of the selected 44 far-field ground motion records. ..................... 145 Figure 6-2: ln(ISDR%) vs. ln(SaT) distribution for the (a) four (b)six and (c)eight-storied PBSC braced frame ............................................................................................................. 146 Figure 6-3: Fragility curve for (a) four (b) six and (c) eight-storied PBSC braced frame ........... 149 Figure 6-4: Different damage states compared between three PBSC frames (a) operational (b) immediate occupancy (c) life safety and (d) collapse prevention ............................ 150 Figure 6-5: ln(ISDR%) vs. ln(SaT) distribution for the (a) four-storied (b) six-storied and (c) eight storied BRB frame .......................................................................................... 151 Figure 6-6: Fragility curve for (a) four (b) six and (c) eight-storied BRB braced frame ............ 152 Figure 6-7: Different damage state compared between three BRB frames (a) operational (b) immediate occupancy (c) life safety and (d) collapse prevention ............................ 153 Figure 6-8: Fragility curves compared between PBSC and BRB frames for (a) four (b) six and (c) eight-storied frames ............................................................................................ 154 Figure 6-9: Collapse level fragility curves compared between PBSC and BRB frames for (a) four (b) six and (c) eight-storied frames for the modified PBSC brace design. ...... 155 Figure A-1: Elongated brace length calculation .......................................................................... 173  xvi  LIST OF NOTATIONS Ab Area of the steel brace As Austenite start temperature Asma Area of the SMA bars Af Austenite finish temperature Co Factor relating roof displacement to the fundamental mode displacement CMR Collapse margin ratio Esteel Modulus of elasticity of steel Esma Modulus of elasticity of SMA Ei Initial modulus Ep Post-yield modulus Et Tangent modulus Ev Variable modulus Fy,steel Yield strength of steel Fy,sma Yield strength of SMA g Acceleration due to gravity hn Height of building to roof level H Height IDA Incremental dynamic analysis IE Importance factor   xvii  𝐾𝑖 Initial stiffness k Effective length factor for buckling load calculation Kb Stiffness of the steel brace Keq Equivalent axial stiffness of the brace Ki Initial stiffness Kp Post-yield stiffness Kshaft Stiffness of the brace shaft Ksma Stiffness of the SMA bars L Length Lb Length of the brace 𝐿𝑖 Increased length 𝐿𝑎 Actual length Lshaft Length of the brace shaft Lsma Length of the SMA bars mx Mass at story x Mf Martensite finish temperature Ms Martensite start temperature Mw Moment magnitude of earthquake Mv Higher mode effect nm Nanometer P Axial load  xviii  Py Yield force Pamf Austenite to martensite finishing force PGA Peak ground acceleration PGV Peak ground velocity ry Residual strain coefficient R Response modification factor 𝑅𝑑 Ductility related force reduction factor 𝑅𝑜 Overstrength factor RTR Record to record variability Sa Spectral acceleration SaT Spectral acceleration at fundamental period of the structure; also denoted as S(Ta) SMT Spectral acceleration for MCE ground motion at the fundamental period (T1) SCT Median spectral acceleration at the collapse level ground motion at T1 SDCT Median spectral displacement at the collapse level ground motion at T1 SDMT Spectral displacement for MCE ground motion at T1 SSF Spectral shape factor T Natural period/Temperature T1 Fundamental period from eigenvalue analysis 𝑉𝑑 Design base shear force 𝑉𝐸 Elastic Structure base shear demand Vmax Maximum base shear of the base shear vs. displacement curve  xix  𝑉𝑦 Yield base shear capacity W Seismic weight of the structure α Unloading force turn factor in PBSC hysteresis model  β Residual deformation factor βRTR Uncertainty parameter for record to record variability βTD Test data uncertainty βMDL Modeling uncertainty βDR Design requirement related uncertainty βTOT Total system collapse uncertainty Ɛ1 Current strain Ɛ_limit Limiting strain on the hysteresis backbone ƐL Plateau strain Ɛr Residual strain Ɛri Residual strain of the ith cycle 𝛥𝑦 Yield displacement 𝛥𝑚𝑎𝑥 Maximum/Ultimate displacement δ Displacement δy,eff Effective yield displacement δu Ultimate displacement 𝜇 Displacement Ductility  xx  μT Period based ductility σ1 Current stress point σ2 Next stress point σ_limit Stress limit on the hysteresis backbone σams Austenite to martensite starting stress σamf Austenite to martensite finishing stress σmas Martensite to austenite starting stress σams Martensite to austenite finishing stress σams Martensite to austenite finishing stress λr   Brace elongation ratio (elongated length/original length) Ø1,x Ordinate of the modal displacement at level x Ø1,r Ordinate of the modal displacement at roof       xxi  ACKNOWLEDGEMENTS This research would not have been possible without the help, support and contribution of many individuals. Firstly, I would especially like to thank my Ph.D. supervisor Dr. M. Shahria Alam for his continuous support and mentoring during my research. His deep knowledge and passion for research have inspired me throughout my graduate studies at UBC. I would also like to thank my course teachers and committee members Dr. Kasun Hewage, Dr. Abbas Milani and Dr. Ahmad Rteil for providing valuable knowledge which helped me in every aspect of my research. I would also like to mention the aid and support of Dr. Marinos Stylianou, Dr. Siriwut Sasibut and Dr. Feng Rong of S-FRAME Software Inc. Their knowledge and support were pivotal in the implementation of this research output in a commercial finite element software. Their continuous feedback and review were immensely helpful in developing the algorithms and finite element models necessary for this study. UBC’s Okanagan campus has provided an outstanding scholastic and work atmosphere through their excellent management system which was crucial for my research work. The research presented in this Ph.D. thesis was jointly funded by Natural Sciences and Engineering Research Council of Canada and S-FRAME Software Inc. through the Collaborative Research and Development Grants. I am also very thankful for being a member of an exceptional research group.  It is very challenging to tackle a doctoral thesis. I would like to thank the research teammates and fellow graduate students who made this journey enjoyable. Specifically, I would like to thank Dr. Muntasir Billah, Shahin Zareie, Dr. Shahidul Islam, Dr. Farshad Hedayati Dezfuli and Zillur Rahman, Lukman Sarker for their continued good spirits and help in this research. The highly  xxii  knowledgeable research group mates provided remarkable support in my research through criticism, dialogue, and technical reviews. Through the ups and downs of life and graduate research, my beautiful wife Fahmida Islam was always there for me. Her constant support, encouragement, and love were indispensable in my success. I am also grateful for my beautiful daughter Samia Haque, who is the inspiration of my life. Finally, I would like to thank my parents (Miah Mahmudul Haque and Mrs. Rokeya Haque) and siblings (Kaniz Mahmuda Sharmeen and M. Mohaimenul Haque) for all their support during this long and arduous study life. I would like to express my deepest gratitude to them.        ION   xxiii  DEDICATION      Dedicated to my parents Mr. Miah Mahmudul Haque and Mrs. Rokeya Haque 1  CHAPTER 1: INTRODUCTION AND THESIS ORGANIZATION 1.1 General Earthquake is one of the most destructive catastrophes of nature. A high-intensity earthquake can cause serious havoc by destroying infrastructure and taking away countless lives. Every year, more than a thousand earthquakes are reported in Western Canada by the seismologists and the Geological Survey of Canada (NRC 2016). During the past 70 years, more than one hundred earthquakes with greater than moment magnitude (Mw) 5.0 occurred in the offshore region of the west of Vancouver Island (NRC 2016). For this reason, the Pacific coast is considered as the most earthquake prone region in Canada. This region is a part of the Pacific ring of fire, which is a concentration of earthquake activities related to the presence of active faults in the earth’s crust. Per Natural Resources Canada, all three types of tectonic plate movements such as sliding past each other, colliding and diverging occur in this region; which causes substantial earthquake activity. The Mw 8.1 Queen Charlotte Island earthquake of 1949 happened along the faults in the BC’s offshore region. A Mw 6.5 earthquake occurred in subduction ocean plate boundary under downtown Seattle in 1965. Finally, in 1946, a Mw 7.3 earthquake took place in central Vancouver Island within the continental crust. The frequency and the size of these earthquakes decrease with the distance from the coast and the active tectonic plate boundaries; this makes Saskatchewan and Manitoba the least earthquake-prone zones in Canada (NRC 2016). If we look at some other significant earthquakes which happened between 1990’s and 2010, we can see the earthquakes like the 1994 Northridge, which caused $40 billion in direct damage to Southern California (Eguchi et al. 1998). This event exposed vulnerabilities of welded moment connections in hundreds of buildings which was previously unknown. In 1995’s Kobe earthquake,  2  central Japan was violently shaken, elevated highways collapsed and numerous buildings were destroyed. Around five thousand people lost their lives and the earthquake caused over $130 billion in direct damage (Scawthorn and Yanev 1995). Finally, the Mw 7.9 earthquake which struck the Sichuan Province in China on March 12, 2008, caused tens-of-thousands of casualties; which is a stark reminder on the consequence of constructing weak structures in active seismic hazard zones. In a more recent event like 2010 Maule (Chile)  earthquake, the infrastructure losses were approximately $20.9 billion (Elnashai et al. 2010). Furthermore, this number is around $20 billion in the 2011 Christchurch (New Zealand) earthquake; which is equivalent to 13% of New Zealand’s GDP. The destruction in the wake of this earthquake was substantial, which included the demolition of about 70% of business district buildings and a loss of more than 50% of heritage buildings. The major business district was closed for over 18 months, and thousands of residents were forced to out-migrate (Elwood 2013). These events demonstrate the severity of seismic events and their effect on the economy. These enormous losses from earthquakes triggered the awareness of building owners on how to reduce such damages. In the last century, our knowledge of seismicity and earthquake resistant structures has significantly improved. Scientists and engineers have come up with various methods for resisting seismic loads on structures. Some noteworthy methods include but are not limited to ductile moment resisting frames, base isolation systems, and seismic bracing systems. The ductile moment resisting system is currently the most widely used method for resisting seismic loading. This approach is prevalent in the construction industry due to its simplicity in design and construction. However, the fundamental concept of this method is to allow nonlinear deformation in specific locations on the primary force resisting elements of the structures during an earthquake  3  event; which is known as plastic hinge formation. A ductile moment resisting framing system dissipates earthquake-induced energy by hysteresis energy dissipation in these plastic hinges. This causes severe plastic deformations which are irreversible in most cases. If this deformation goes beyond a certain limit, the structure might have to be demolished and rebuilt afterward. Hence, regarding re-use or reparability, this method is not very efficient. The base isolation system is considered an effective alternative for resisting seismic loads. This system isolates a structure from earthquake excitation by disrupting the force transfer between the ground and the structure. This method is effective in preventing structural damage and saving lives. However, this approach is not efficient for a broad range of structural height, mass, and stiffness; since the difference in these parameters results in different natural frequencies. Sometimes the use of base isolation system could shift this frequency to an unfavorable domain in the earthquake response spectrum. Furthermore, the base isolation system is often too expensive to see a widespread use around the world. The bracing system is another option for resisting seismic loads. However, the use of traditional structural bracings against seismic loading has its challenges. The two key challenges are compression buckling and permanent deformation. Buckling has a dramatic effect on strength and stiffness of a brace. Once it occurs, strength is reduced, and the brace loses stiffness in every successive load cycle. To resolve this issue, different types of seismic bracing systems have been developed. Most notable ones are the buckling restrained braces (BRBs) (Takeuchi et al. 2004) and cast steel yielding fuses (Christopoulos et al. 2008). BRBs resist seismic forces by hysteretic energy dissipation. They exhibit fat bilinear hysteresis response, which is very efficient in dissipating seismic energy. However, this high-energy dissipation comes with a price since the  4  BRBs experience significant nonlinear deformation during the energy dissipation process. This nonlinear deformation is permanent and cannot be easily reverted. It also causes nonlinear deformation in the connecting elements such as the connections, beams, and columns.  In order to prevent such permanent damages, researchers have come up with smart self-centering bracing systems such as the Self-Centering Energy Dissipation Device (SCED) (Tremblay and Christopoulos 2012), Memory Alloys for New Seismic Isolation Devices (MANSIDE) (Dolce et al. 2000) and RHDB (Zhang and Zhu 2007) braces. These braces offer some special advantages over their counterparts. These braces are buckling restrained; they have moderate energy dissipation capacity, and they experience very low permanent or residual deformation after an earthquake. This low residual deformation is due to their self-centering capability. The self-centering mechanism of these braces is either achieved using Nickel-Titanium based shape memory alloys (Nitinol SMA) (MANSIDE and RHDB braces) or high-strength steel or composite materials such as Kevlar or Aramid cables (SCED brace). SMAs are a type of alloy, which can experience significant nonlinear/plastic deformation and can revert to their original shape upon heating or stress removal. The regaining of original shape upon heating is known as the shape memory effect and the returning to original shape under stress removal is known as superelasticity. The one used in braces are usually Nitinol based superelastic SMAs. However, these smart self-centering bracing systems are either difficult to construct, have low strength or energy dissipation capability (Haque and Alam 2015a).  This thesis focuses on a newly developed self-centering bracing system invented by the author. This device is known as the Piston Based Self-Centering brace or in short, PBSC brace (Haque and Alam 2014). The PBSC bracing system is made using a cylinder-piston mechanism. The  5  energy dissipation and self-centering capability are achieved with Nitinol tie bars. Furthermore, the PBSC brace is designed to be fully buckling restrained. The concept of the PBSC brace is novel in terms of design and constructability. The design of this brace allows the use of both SMA bars and wires; whereas previous designs could only accommodate SMA bars. Furthermore, this brace can be very easily constructed and disassembled for repair or recycling purpose as illustrated in the patent document (Haque and Alam 2014). This research focuses on the cyclic performance, design and seismic load resistance of this bracing system. This proposed structural system will have the potential to not only save peoples’ lives but also retain the structure after a massive earthquake. 1.2 Objective of the study The primary purpose of this research is to develop a novel self-centering bracing system which will be able to bring a structure back to its original position after significant nonlinear deformation induced by a seismic event. The secondary objective of this research is to propose seismic response modification factor values required for the seismic design of this new bracing system. To fulfill the objective mentioned above, a step by step development process has been carried out. This study has focused on the following 1. Development of a novel piston based self-centering bracing system: A gradual development process is shown for the piston based self-centering bracing system. The mechanism of this bracing system has been explained in detail. The bracing system has been modeled using a finite-element analysis software ABAQUS (2014). The load-deformation hysteresis shape have been determined and discussed.  6  2. Development of a hysteresis model for the PBSC bracing system: This objective was achieved by creating a hysteresis material model for Nitinol shape memory alloy with residual deformation. This material model was used in a custom made MATLAB (2012) based finite element software developed specifically for the PBSC bracing system. This finite element program was used to generate PBSC brace’s load deformation hysteresis. The hysteresis results were used to develop an algorithm which was implemented in a commercial finite element software. 3. Determination of seismic response modification factors such as overstrength (Ro) and force reduction (Rd) factor: FEMA P695’s (FEMA 2009) quantification of building seismic performance factor methodology was used to determine the overstrength and force reduction factors for the PBSC bracing system. 4. Seismic performance comparison between PBSC and buckling-restrained braced frame: Two different sets of braced frames were designed, one with the PBSC bracing system and the other with the buckling restrained bracing system. These frames were subjected to forty-four different earthquake ground motion records. The seismic performance of these two systems was compared against each other using fragility functions developed using four different damage states. 1.3 Scope of this research With the purpose of accomplishing the objectives mentioned above, first, a comprehensive literature review was carried out to understand various innovative bracing systems developed by the previous researchers. There are different types of buckling restrained and self-centering systems, some of which depend on the hysteretic energy dissipation or post-tensioning of  7  steel/composite materials; whereas, others use various types of self-centering materials. Understanding these different types of systems is essential for the development of the PBSC bracing system proposed by the author. To fulfill the objectives stated in the previous section, the following studies were carried out: 1. ABAQUS (2014) based finite element models of the PBSC bracing system were built. To accurately predict the cyclic performance of the system, these models were constructed using continuum/solid elements. As PBSC bracing system uses both steel and Nitinol SMA components, the hysteresis models of these materials were first calibrated and then incorporated in these models. Displacement based quasi-static analyses were carried out on these brace models to determine the load deformation hysteresis. The resultant hysteresis was found to be flag-shaped. 2. The flag shaped SMA hysteresis model found in the earlier step (ABAQUS analysis) was coded and implemented in S-FRAME (2016) software which follows the loading and unloading rules defined in Auricchio and Sacco (1997). However, as this hysteresis model can not simulate residual deformation, another hysteresis model was developed specifically to include this behavior. First, Auricchio and Sacco (1997)’s Nitinol hysteresis rule was modified to include residual deformation. This novel material hysteresis model was implemented in a specialized finite element program developed for the PBSC bracing system in the MATLAB (2012) environment. This finite element program was used to establish a force deformation hysteresis model for the PBSC bracing system. The hysteresis model was found to exhibit sliding response near the origin of the force-deformation plot. Finally, this new hysteresis model was implemented in the S-FRAME Software.  8  3. Designing buildings equipped with the PBSC bracing system as per NBCC (2010) requires two critical parameters for seismic design, these are known as overstrength (Ro) and ductility related force reduction (Rd) factors. To calculate these parameters, four, six and eight storied PBSC braced frames were designed using tentative/approximate values of Rd and Ro. In the next step, FEMA P695 (FEMA 2009) suggested seismic performance factor determination analysis was carried out. Collapse margin ratios (CMR) were calculated and adjusted for spectral shape factors. These adjusted CMR values were checked against the acceptable CMR values provided in the guideline (FEMA 2009), and they were found satisfactory; indicating the tentative values of Rd and Ro were adequate for the seismic design.  4. To determine the seismic performance of building frames equipped with PBSC bracing system, four, six and eight-storied PBSC braced frames were designed and built in the S-FRAME’s finite element environment. The flag-shaped SMA hysteresis model with sliding response was used to represent the cyclic load deformation behavior of the PBSC braces. Another set of these frames were equipped with the BRB braces and designed as per NBCC (2010). Nonlinear dynamic time history analysis was carried out on these two groups of frames using 44 ground motion records. Fragility functions were calculated from four different damage states to determine seismic vulnerability under a wide range of seismic intensity. Finally, these fragility functions were used to determine the advantages/disadvantages of the PBSC bracing system against the buckling restrained bracing system.  9  1.4 Thesis organization This thesis is composed of seven chapters. The current chapter provides an introduction, objective and scope of the research. Brief descriptions of the contents of these chapters are given below along a flow-chart (Figure 1-1). In Chapter 2, a literature review on various innovative structural force resisting systems (both energy dissipative and self-centering) have been discussed. The topics cover traditional energy dissipative and self-centering bracing systems. First, the traditional energy dissipative braces are reviewed; their advantages and disadvantages are discussed. Then self-centering braces are covered. Finally, some relevant studies are presented in a tabular format. In Chapter 3, the development process of the PBSC bracing system is discussed in detail. The force transfer mechanism is explained. Finite element models are built in ABAQUS (2014) to examine the cyclic behavior of this bracing system. Finally, the results are presented and discussed in detail. In Chapter 4 Two force-deformation hysteresis models are developed. First one is the traditional SMA flag shaped hysteresis model and the second one is specifically developed for capturing the sliding response of the PBSC bracing system. The development and implementation process of these hysteresis models are discussed in details. The algorithm, flow charts, verification, and codes are also presented. Chapter 5 demonstrates the determination of overstrength and ductility related force reduction factors for the PBSC bracing system. These factors were determined using the guideline stated in FEMA P695. Incremental dynamic time history analysis was carried out on three different building  10  frames. Collapse margin ratio values were calculated, and they were used to validate the preliminary values of these factors utilized in the design. Chapter 6 shows the performance comparison between a BRB and a PBSC braced frame. The BRB frame is designed using established code and guidelines such as NBCC (2010) and CSA (2009). Whereas, the PBSC braced frame is designed using the preliminary values of Rd and Ro determined in chapter 5 and the design method proposed by the author. The performance of these designed frames was evaluated using nonlinear dynamic time history analysis. Fragility function was developed for performance comparison using interstory drift ratio responses of these frames. Finally, Chapter 7 presents the research conclusion. Furthermore, some recommendations for future research have been suggested in this chapter.  11   Figure 1-1: Thesis organization flow-chart  Chapter 2: Overview of the current state-of-the-art (Literature review).Chapter 7: ConclusionsChapter 3: PBSC brace concept development and its cyclic performance determination in finite element environment.Chapter 4: A Brace hysteresis model development considering residual deformation of SMA material. Implementation of this hysteresis model in S-FRAME software for frame analysis.Chapter 5: Determination of ductility and overstrength related force modification factors (Rd & Ro) for seismic design of the PBSC bracign system. This is carried out as per FEMA P695 guideline.Chapter 6: Seismic performance comparison between the PBSC and the BRB bracing system in terms of fragility functions.Chapter 1: Introduction and thesis organization. 12  CHAPTER 2: OVERVIEW OF THE CURRENT STATE-OF-THE-ART 2.1 General Various types of seismic bracing system have been developed, and some of them are available in the market. These systems can be grouped into two major categories: energy dissipative and self-centering systems. The following systems can be classified as energy dissipative systems: Traditional tension only braces, Buckling Restrained Brace (BRB), Cast Steel Yielding Fuse (CSYF). These braces resist seismic load by hysteretic energy dissipation characterized by large nonlinear deformation and fat hysteresis loops. On the other hand, the self-centering braces also go to the nonlinear range; however, these bracing systems can recover from large nonlinear deformation, which sets them apart from the others. This results in negligible (not more than 0.1%) residual interstory drift for the connected structure (Zhang and Zhu 2007). After an earthquake, these braces can pull back the structure to its original position resulting in very low post-earthquake repair costs. Some other self-centering systems are as follows: PBSC (Haque and Alam 2014), SCED (Tremblay and Christopoulos 2012), MANSIDE (Dolce et al. 2000), RHDB (Zhang and Zhu 2007) braces or the post-tensioned beam-column joints. Researchers have also developed base isolation systems with good self-centering capabilities (Dezfuli et al. 2013). Most of these systems rely on one of these following materials: high-strength steel strands, composite cables or superelastic shape memory alloy (SE SMA) wires/bars. Among these materials, SE SMA is a smart thermomechanical alloy composed of roughly equal atomic percentage of Nickel and Titanium. This material is also known as Nitinol, which is an abbreviated form of Nickel-Titanium Naval Ordnance Laboratory; named after the laboratory where it was invented. At room temperature, this material can come back to its original shape after large strains (6%-8%) with moderate hysteretic  13  energy dissipation. This lucrative property has made it one of the most interesting materials in the research of smart structures. More details on the above-mentioned bracing systems are given below. 2.1.1 Energy dissipative bracing systems 2.1.1.1 Buckling restrained bracing system Among the numerous seismic load resisting bracing systems, the buckling restrained brace is the most notable one. Although numerous variations exist (Bystricky and Fanucci 2010; Chung-Che et al. 2014; Hinchman 2013; Iwata and Murai 2006) under the name, the one developed by Takeuchi et al. (2004) and produced and marketed by Nippon Steel Corporation has seen widespread use all around the world. This brace is essentially composed of concrete/grout filled steel tube and an inner steel core. The steel core can freely move inside the concrete sleeve thus reducing the possibility of axial load transfer from the core to the sleeve. When buckling load is reached, the core steel is prevented from buckling by the strong concrete sleeve. Figure 2-1 shows the comparison between traditional tension-compression steel brace and buckling restrained brace frame hysteresis; which shows that the traditional steel brace buckles under a compressive load, which is only a fraction of its tensile strength and loses most of its stiffness and load carrying capacity. The stiffness in the subsequent load cycles also diminishes and deformation increases significantly. On the contrary, the buckling restrained brace shows stable bilinear hysteresis loops under cyclic loading condition and exhibits excellent energy dissipation indicated by the large area inside the hysteresis loops. According to the experimental results of Fahnestock et al. (2007), a well detailed BRB frame can withstand significant drifts and ductility demand in the braces with almost no damage. They were subjected to a maximum ductility demand of 26 without any  14  damage. However, one potential limitation observed for the buckling restrained braces is its high residual deformation during design basis earthquake (DBE) and maximum considered earthquake (MCE) simulations. These large residual drifts may pose substantial challenges while restoring buildings with BRB frames to service condition after a major seismic incident. The hysteretic mechanism (Figure 2-1(b)) shows that after a large nonlinear excursion if the load becomes zero the deformation does not reduce by a significant margin. This large residual deformation induces substantial beam-to-column connection deformation resulting in significant repair or rebuilding cost. Besides residual drift, BRBs are also susceptible core plate local buckling if the sleeve width to thickness ratio is not adequate to prevent it (Takeuchi et al. 2010).  Figure 2-1: (a) A typical design of a buckling restrained brace (b)Hysteresis of a BRB brace (adapted from Asgarian and Shokrgozar (2009)) 2.1.1.2 Cast-steel yielding fuse The cast steel yielding fuse (Christopoulos et al. 2008) or commercially known as the Scorpion yielding brace is a device for improved seismic performance of concentrically braced frames (CBF) which is characterized by its Stable, full, symmetric and repeatable hysteretic behavior and Encasing mortar Steel core section Steel sleevePδTensionCompression(a) (b) 15  its large ductility capacity (Gray et al. 2012b) . The concept was presented as a novel seismic energy-dissipative device for CBFs. The use of a cast steel yielding device removes the inelastic compressive buckling and cyclic tensile yielding of traditional braces. It replaces traditional unstable brace hysteresis with a stable symmetric hysteretic response (Gray et al. 2010) In this system, seismic energy is dissipated by yielding fingers of specially engineered connectors made of cast steel. Figure 2-2 shows a prototype yielding bracing system under experimental investigation. Under severe compression and tension loading, the fingers yield in flexure. This way they provide a full symmetric hysteresis.  Figure 2-2: Yielding brace prototype test showing deformed shape of the fingers (Gray et al. 2014a) A full-scale quasi-static testing of yielding brace is shown in Figure 2-3(a). The hysteresis response of the brace is shown in Figure 2-3(b). This system shows stable hysteresis response up to a certain deformation range; after this range is exceeded, the hysteresis drastically degrades. It can also be observed that this bracing system has almost no self-centering capability. It should also be noted that the maximum displacement achieved by this brace is around 75mm, which is approximately 1% of the 7m long brace.  16   Figure 2-3: (a) A 7m long full-scale test specimen (b) Hysteresis of the full-scale yielding brace (Gray et al. 2012a) Gray et al. (2014b) carried out time-history analysis on a twelve-story sample building designed with BRBs and Cast Steel Yielding braces. They demonstrated that the performance of both systems is very similar when designed with a similar design philosophy. Very Similar peak and residual inter-story drift ratios and accelerations were observed for both frames subjected to the same ground motions scaled to the DBE spectrum. A similar number of collapses were observed for both systems when subjected to the same ground motions scaled to the MCE spectrum. 2.1.2 Self-centering bracing systems 2.1.2.1 Advantages of self-centering systems Although the above-mentioned self-centering braces are different in design, they produce an almost similar force-deformation response. Their force-deformation responses are flag-shaped with slight variations in their loading or unloading responses. If properly designed, they can bring back the structures to their original position after an earthquake. (a) (b) 17  2.1.2.2 Disadvantages of self-centering systems Flag-shaped hysteresis generates less hysteretic energy dissipation or hysteretic damping compared to bilinear kinematic or isotropic hysteresis typically observed in buckling restraint braces. This low hysteretic damping results in a higher acceleration during earthquakes. Another important feature common to these systems is that they are never fully self-centering; although low, they experience a small amount of residual deformation. 2.2 Some notable self-centering systems Different types of self-centering bracing systems have been developed by the researchers around the world. Some of these systems are discussed in this section. These smart self-centering bracing systems can recover from significant nonlinear deformation, which sets them apart from traditional bracing systems. Their use can result in insignificant residual interstory drift ratio in buildings. To understand their behavior, some noteworthy self-centering bracing systems are discussed below. 2.2.1 PBSC brace The piston based self-centering bracing system is a novel invention by Haque and Alam (2014). This system is essentially composed of a piston, shaft and SMA bars/wires. Nickel-titanium-based shape memory alloy bars are used to transfer axial load from the shaft to the connected structure directly or through the sleeve for tension or compression forces respectively. From the detailed solid model based analysis done in ABAQUS (2014), it was found out that the brace has a hysteresis response almost similar to the flag shaped hysteresis (Haque and Alam 2015a), which signifies excellent self-centering capability. Figure 2-4 shows a translucent view of the PBSC bracing system taken from ABAQUS (2014).  18   Figure 2-4: A 3D model of the PBSC bracing system 2.2.2 MANSIDE brace A Nitinol wire based self-centering brace was developed under the Memory Alloys for New Seismic Isolation and Energy Dissipation Devices (MANSIDE) project (Dolce et al. (2001), Dolce and Cardone (2006)). This device employs two sets of wire loops; one set is kept at austenite state, and the other set is prestressed to the martensite state. The austenite wire loop provides re-centering capability while the martensite wire loop provides high-energy dissipation capacity. Figure 2-5(a) shows a cross-section of the device and Figure 2-5(b) shows a 3D view of the system.  Figure 2-5: MANSIDE brace (a) cross section (b) 3D view (adapted from Dolce et al. (2000)) (a) (b)Re-Centering wire loops Re-Centering wire loopsDissipating wire loops Dissipating wire loops 19  2.2.3 SCED brace The self-centering energy dissipative device (SCED), developed by Christopoulos and Tremblay (2005), uses high strength Kevlar/Aramid cables and friction between two concentric steel members to achieve its self-centering and energy dissipation capability respectively. Like MANSIDE brace, the SCED brace also exhibits flag shaped hysteresis response, which translates to good self-centering capability. A cross-section (Figure 2-6(a)) and hysteresis model (Figure 2-6(b)) of the SCED brace is shown below.  Figure 2-6: (a) SCED brace (b) Hysteresis model for SCED brace (adapted from Christopoulos et al. (2008)) 2.2.4 RHDB device Reusable Hysteretic Damping Brace (RHDB) developed by Zhang and Zhu (2007) is a passive energy dissipation device, in which the core energy-dissipating components are made of superelastic SMA (Nitinol) wires. RHDB has several performance advantages compared to the conventional bracing systems. As per the inventors, RHDB will experience minimal residual drifts due to its self-centring capability and can survive multiple strong earthquakes without repair or replacement. A seismic performance study on RHDB frames was carried out using a non-linear time history analysis of 3 and 6-storey RHDB building frames subjected to two suites of 20 earthquake ground motions. A similar analysis was carried out on BRB frames and the results were (a) (b)End ConnectionEnd ConnectionEnd PlateSteel MembersPretensioned tendonsEnd PlateFriction interfaceSlotted holeAxial Force (P)Axial Deformation (δ) 20  compared. From this analysis, the authors found out that RHDBs not only can significantly reduce the residual drifts but also can control their peak inter-storey drifts during strong earthquakes. They observed that the residual story drifts of the RHDB frames are almost zero in all cases.   Figure 2-7: Longitudinal cross section of the RHDB brace (adapted from Zhang and Zhu (2007)) 2.2.5 SC-BRB Self-centering buckling restrained brace or SC-BRB developed by Miller et al. (2012) is composed of a typical BRB component and pre-tensioned superelastic nickel-titanium shape memory alloy rods. The system is held inside two concentric hollow steel tubes and a pair of slotted anchorage end plates; a concept introduced by Christopoulos et al. (2008). The BRB provides energy dissipation capacity whereas the shape memory alloy rod provides self-centering and additional energy dissipation capability. The SC-BRB prototype test showed robust performance, and it was found out that even after BRB core fracture, the SC-BRB had significant load carrying capacity due to the redundancy of the SMA rods. The residual deformation of SC-BRB was only half of the maximum brace deformation. The authors also concluded that the SC-BRB provides excellent energy dissipation and equivalent viscous damping capacity. NiTinol StrandAnchor for NiTinol strandSliding interface External guideBlock BBlock A 21   Figure 2-8: Self-centering buckling restraint brace components (adapted from Miller et al. (2012)) 2.2.6 Hybrid steel-SMA device An innovative hybrid steel-SMA device was developed by Yang et al. (2010) to mitigate structural damage induced by seismic events. The device is made of three major components: SMA re-centering wires, two energy absorption struts and two high strength steel tubes for guiding the movement of this device. A three-storied building model was built in OpenSees finite element environment with two different hybrid device installation as follows: Configuration 1: horizontal installation and Configuration 2: inverted “V” brace. Both pushover and nonlinear dynamic time history analyses were carried out to understand its behavior. Based on the analysis results the authors concluded that when designed by their proposed formula the device could re-center while maximizing its energy dissipation. The three-storied building with hybrid brace in configuration 2 showed comparable performance to the BRB frame in terms of peak interstory drift ratio; while configuration 1 showed good performance in terms of both interstory and residual interstory drift ratio. Middle tube Welds HereSMA RodsSteel BRB CoreAnchorage PlatesOuter Tube Welds HereOuter TubeMiddle TubeInner Tube 22   Figure 2-9: Cross-section of the hybrid device (adapted from Yang et al. (2010)) 2.3 Other notable studies Auricchio et al. (2006) carried out a numerical investigation on building frames equipped with Nitinol chevron braces. They compared the performance of these SMA braced structure with that of buckling restrained brace frame structure. To match the yield strength and time periods of the structure, the Nitinol braces were sized to have same yield force and axial stiffness as those of BRBs. Nitinol braces were modeled with a length ranging from 330mm to 453mm whereas the BRBs were in the range of 6000mm. To compensate for these reduced lengths, rigid elements were added to the Nitinol braces to achieve similar lengths. From this analysis, the authors concluded that, although the maximum interstory drift ratio of the SMA braced frames came higher compared to the counterpart, the residual interstory drift ratio was significantly lower. The low residual interstory drift indicates very low repair and rehabilitation cost after an earthquake. However, there is a limitation to this study, as much shorter length SMA braces were used compared to the steel braces. It was found that the SMA braces were elongated close to 30mm, which is around 8% of their lengths. This high elongation pushed SMA braces to martensite range, which resulted in very high axial force. This type of high axial force can severely damage the connected steel components (e.g. connections, beams or columns). Therefore, using shorter length Nitinol bars paired with rigid members may damage the connection. To eliminate this problem, the Nitinol bar length could be calculated based on the expected deformation demand in the braces, and thus their lengths could SMA WireHigh-Strength Steel TubeForce Force 23  be optimized. Furthermore, using longer Nitinol bars could have reduced the stiffness of the braces and decreased the natural frequency of the structure resulting in lower seismic base shear demand. In another study, the researchers (Moradi et al. 2014) carried out incremental dynamic analysis on SMA (Nitinol) braced frames and compared their results against BRB frames. Similar to the previous investigation (Auricchio et al. 2006), they also used shorter Nitinol braces and added rigid links to compensate for the length reduction. They compared the maximum interstory drift and permanent roof drift ratios between these two types of frames. They found that at design ground motion intensity level both Nitinol and BRB braced frames exhibited maximum interstory drift lower than the design drift limits. They also found that the Nitinol braced frames exhibited more evenly distributed nonlinearity along the height of the buildings compared to BRBs. Furthermore, the authors observed that under severe ground motion excitations, the Nitinol braces are more beneficial. Also, Nitinol braced frames were found to experience significantly lower permanent roof drift compared to BRB braced frames. Another group of researchers (Hu et al. 2013) carried out a similar comparative investigation between BRB frames and buckling restrained Nitinol braced frames. They observed that if Nitinol is used at the end of BRBs, the hysteresis of the BRBs become more flag shaped and enables them to re-center after load removal. However, this combination does not ensure full self-centering. Similar to the previous studies, they also found excellent residual interstory drift reduction while using Nitinol in buckling restrained braces. In order to fully understand the mechanism of self-centering bracing systems made with Nitinol alloy, it is of utmost importance to thoroughly understand this material behavior under cyclic  24  loading. Therefore, the next section peeks into the properties and load-deformation behavior of this material in detail. 2.4 Shape memory alloy Materials that can undergo phase transitions have received tremendous attention in the scientific community. This type of behavior has innovative applications in both medical science and engineering. The most notable example of such materials is shape memory alloy; specifically, the Nickel Titanium based one. This unique material although discovered in 1960’s has seen very little or no use in the construction industry because of the high production cost, lack of understanding of their thermo-mechanical processing and very little knowledge for predicting their behavior. Nitinol has only seen some functional use in the last 30 years in biomedical, commercial and aerospace industries. The reason for this increased use can be attributed to a significant reduction in price and improvement in the quality and reliability of the product (DesRoches et al. 2004).  Also, in recent times many structural systems have been developed using SMA wires and bars (Alam et al. 2007). SMA wires have been used mostly for steel braces, for example in MANSIDE braces (Dolce et al. 2000), in RHDB braces (Zhang and Zhu 2007), in structural vibration control (Ma et al. 2004), in pre-strained wire braces (Lafortune et al. 2007) and in seismic isolation devices (Dezfuli et al. 2013). Nitinol bars have been used for reinforced concrete bridge piers (Billah and Alam 2015), bridge restrainers (Alam et al. 2012), and concrete beam-column joints (Youssef et al. 2008). There has been no use of SMA bars in structural braces so far. Only analytical investigation using SMA structural elements at brace-ends has been carried out by some researchers. A study on special concentrically braced frames made of Nitinol showed that SMA  25  braces are effective in reducing both maximum and residual interstory drifts under seismic loading (McCormick et al. 2007). A similar study was carried out on diagonal, split x and chevron brace and the performance of SMA braced structures were compared against BRB braced frames; and it was found out that SMA braces are effective in reducing peak interstory and residual roof drift ratio under seismic loading (Asgarian and Moradi 2011). 2.4.1 Microstructure of Nitinol There are two basic atomic phases of Nickel-Titanium alloy; they are called the austenite and the martensite phase. Nitinol alloy can transform from one of these phases to the other under the application of pressure or heat, without any form of permanent damage. A three-dimensional diagram of these phases is shown in Figure 2-10. Austenite has a simple cubic structure (B2 body-centered atomic structure). This phase is stable at high temperature and low stress. On the other hand, martensite phase has a body-centered tetragonal crystal structure (B19 rhombic geometry); which is stable at low temperature and high stresses.    Figure 2-10: Three-dimensional microstructure of the Nickel-Titanium alloy (a) Austenite (b) Martensite  26  A two-dimensional representation of the austenite and martensite phase is shown in Figure 2-11. martensite can exist in two different forms; twin variant (Figure 2-11(b)) or a single favored variant (Figure 2-11(c)) (Wayman and Duerig 1990). These two variants are crucial in shape memory effect, which will be explained in the next section. More detail about the crystallography and microstructure of nickel-titanium alloy can be found in the literature (Duerig et al. 2013; Gall et al. 1999; Hane and Shield 1999; Otsuka and Shimizu 1986; Otsuka and Wayman 1998; Perkins 1981; Tadaki et al. 1988).    Figure 2-11: Two-dimensional microstructural of Nitinol in (a) Austenite (b) Twinned martensite and (c) Detwinned martensite phase (adapted from Speicher (2009)) The Nitinol based shape-memory alloy has two important properties. One of them is the shape memory effect, and the other one is known as the superelasticity. These two properties are briefly discussed below. More detailed explanation is given in the mechanism section. 2.4.2 Shape-memory effect Typically, at a low temperature (below Mf as shown in Figure 2-12) if the shape memory alloy in twinned martensite phase is stressed/strained beyond yield, significant nonlinear deformation occurs (Up to 10%-15%), and the shape memory alloy goes to the detwinned martensite phase. However, upon heating beyond a certain temperature (Above Af as shown in Figure 2-12), the alloy transforms to austenite state and recovers all plastic deformation (comes back to its original shape). (a) Austenite (b)Twinned Martensite (c) Detwinned Martensite 27  After cooling, the alloy transforms to twinned martensite phase without any residual deformation; which is known as the shape memory effect and is illustrated using green arrowed cycle in Figure 2-12. 2.4.3 Superelasticity  Above a certain high temperature (Above Af in Figure 2-12), after a mechanical loading resulting in substantial plastic deformation (Up to 10%-15% strain), the alloy recovers all plastic deformations and comes back to the original shape with almost no residual deformation instantly after the removal of the load. This loading-unloading path results in hysteretic energy dissipation. This effect is known as superelasticity. The effect is shown using blue cyclic arrows in Figure 2-12. The phase transformation temperatures (As, Af, Ms, and Mf) are explained in Figure 2-13.  28   Figure 2-12: Two-dimensional microstructure of Nitinol during superelastic and shape memory effect (adapted from Speicher (2009)) 2.4.4 Mechanism According to Khachaturyan (1983) and Wayman (1964), both of these behaviors mentioned above are the result of martensitic transformations; which is a diffusionless, solid-solid transition between an austenite parent phase and martensite product phase. The austenite parent phase is crystallographically more ordered than the martensite phase. For SMA, this transformation is reversible and in some cases rate-dependent as well. The thermo-mechanical behavior of SMA as a function of strain, temperature, and stress is summarized in Figure 2-13. SMA exhibits the shape memory effect below the martensite finish Austenite Twinned Martensite Detwinned MartensiteAustenite Twinned Martensite Detwinned MartensiteAustenite Twinned Martensite Detwin ed MartensiteAustenite Twinned Martensite Detwinned MartensiteSuperelasticityShape MemoryUnloadLoadCoolHeatLoadUnloadDeformationTemperature AfAsMfAuste iTwin artensiteDetwinned artensiteD twinn d Mart 29  temperature, Mf. If the SMA is heated beyond austenite finish temperature (Af), any deformation experienced during the martensite phase is instantly recovered. The SMA is in its austenite phase at a temperature above Af. At this phase stress-induced martensite is formed upon loading. However, upon unloading, the material returns to austenite phase at a lower stress level, thus causing the superelastic behavior. This ensuing nonlinear stress–strain relationship results in a hysteresis. Furthermore, the SMA undergoes ordinary plastic deformation with much higher strength at a higher temperature above Md (DesRoches et al. 2004).    Figure 2-13: 3D stress–strain temperature diagram depicting deformation and shape memory behavior of Nitinol-based shape memory alloy (adapted from Alam et al. (2007)) The change in microstructure under loading and unloading at a constant temperature is shown in a two-dimensional plot in Figure 2-14. Figure 2-14(a) shows the stress-strain diagram for superelastic effect. This figure depicts how austenite phase transforms to detwinned martensite phase under the application of stress and thus accommodate the strain; creating a loading plateau. At the end of the loading plateau, the alloy fully transforms to the stress induced detwinned martensite phase. This figure is true for temperature above austenite finish temperature (Af). Upon unloading the stress-strain curve comes back to the origin, thus recovering all plastic deformations. The resulting hysteresis gives Nitinol its energy dissipation capacity. However, if the Nitinol alloy is continued to be stressed beyond the martensite phase, slip planes will form (like common Plastic Behaviour Super Elastic Effect Shape Memory EffectStrainStrainStres StresCEFGBStresT=AsT=AfT>AfT=MdT>MdT=MsT<MfT=Mf100% MartensiteD 30  metals), and the alloy will experience permanent plastic deformation from which it will not be able to recover to its original shape. Figure 2-14(b) shows the stress-strain diagram of Nitinol SMA below martensite finish temperature (Mf). At this temperature, application of stress changes the twinned martensite to detwinned martensite. Upon unloading the alloy experiences significant plastic strain as shown in the figure. However, this plastic strain can be recovered by heating the alloy above austenite finish temperature (Af) thus exhibiting the shape memory effect.  Figure 2-14: 2D stress–strain behavior of (a)Superelastic SMA (b)Shape memory SMA (adapted from Speicher (2009)) 2.4.5 SMA bar behavior Quasi-static and dynamic cyclic loading tests which are typically observed during earthquakes were carried out on SMA wires and large diameter bars (DesRoches et al. 2004). The diameters of the specimens tested include 1.8mm wire and 7.1mm, 12.7mm and 25.4mm bars. The authors observed that contrary to popular belief, SMA bars also show outstanding re-centering capacity (Figure 2-15). The residual strain observed for the test specimens were on an average 0.65% for Austenite Twinned Martensite Detwinned MartensiteAustenite Twinned Martensite Detwinned MartensiteAustenite Twinned Martensite Detwinned MartensiteAustenite Twinned Martensite Detwinned MartensiteAusteniteDetwinned MartensiteDetwinned MartensiteTwinned MartensiteStressStressStrain StrainT>Af T<Mf(a) (b) 31  all the specimens where the maximum strain was 6%. The only disadvantage of using SMA bars over wires is its lower hysteretic energy dissipation capacity. The equivalent viscous damping represented by the fatness of the hysteresis loops (Figure 2-15) was found to vary from 2% for the 12.7mm bar to a maximum of 7.6% for the 1.8mm wire.     Figure 2-15: Stress–strain plot for 25.4 mm diameter nitinol SMA bar subjected to quasi-static cyclic loading (adapted from DesRoches et al. (2004)).  1" dia SMA rodStress (ksi)Strain (in/in)1201008060402000.00 0.01 0.03 0.04 0.05 0.06 0.07 0.08 32  CHAPTER 3: HYSTERETIC BEHAVIOR OF A PISTON BASED SELF-CENTERING BRACING SYSTEM 3.1 General This chapter presents the conceptual design and development process of the piston based self-centering bracing system. In the first step, the concept of the system with different SMA configuration is proposed and depicted using two-dimensional sketches. The possible usage of the brace in various types of structural systems is proposed. The predicted mechanism of the brace is described under cyclic loading conditions. A brief description of the brace construction method is also presented. Finally, a three-dimensional continuum finite element model is developed. The model is used for carrying out quasi-static time history analysis to understand the cyclic performance of the brace. The hysteretic response of this bracing system is compared against a similar system made with steel energy dissipative elements. The advantages and disadvantages of the SMA based system compared the steel tie based system is discussed. The scope of this chapter is limited to finite element investigation only. No experimental investigation has been carried on the brace prototype. 3.2 PBSC brace design To resolve the shortcomings of the current self-centering bracing systems discussed in the previous chapter, the authors have developed a novel system known as the Piston Based Self-Centering Bracing System (PBSC brace) (Haque and Alam 2014). The PBSC bracing system is designed to have an excellent self-centering capability, moderate damping, and easy constructability. Furthermore, the brace will be buckling resistant under earthquake loading.  33  The PBSC brace is made of two concentric steel members connected in a cylinder-piston arrangement as shown in Figure 3-1. The concentric steel members could be made of circular, rectangular or any other standard structural sections. The piston in this assembly is designed to act as the shaft and the sleeve/cylinder is intended to house the Nitinol SMA bars, steel plates, and the locks/nuts. Two configurations are shown in Figure 3-1; in configuration 1 Nitinol bars are continuous through the piston plate and in configuration two the bars are discontinuous. The configuration one can incorporate twice the number of Nitinol bars compared to configuration 2; making it capable of carrying twice the load relative to the study published earlier (Haque and Alam 2015a). Figure 3-1 shows the internal details of the PBSC brace. In this arrangement, the shaft has a smaller cross section compared to the sleeve for ease of installation inside the sleeve. Two sets of equal length Nitinol SMA bars are connected from the piston plate to the front and back caps using couplers or threaded nuts. The diameter of the orifices (openings) on the front cap, back cap, and the piston plate are made slightly larger than the SMA bar diameter. This is done to allow unrestrained rotational movement of the Nitinol tie bars during loading. This also eliminates the possibility of bending moment generation in ties. Moreover, in the case of plate rotation/bending, the ties mostly remain unaffected and straight. Furthermore, the piston plate is also made slightly smaller than the sleeve to reduce excessive friction during sliding movement. This size difference also reduces the possibility of sleeve inner surface damage from piston plate bending. The brace shaft and the sleeve are made of the same material, and the cross-sectional area of the sleeve is designed to be higher than that of the brace shaft; this is done to reduce the possibility of global or local buckling of the sleeve before the brace shaft. Similarly, the Nitinol SMA bars are designed  34  to have lower force carrying capacity than the brace shaft. This ensures that the SMA bars are the weakest link in the system and they go to nonlinear range before the other components  Figure 3-1: Internal details of the PBSC brace with optional high damping rubber filler (a) Configuration 1 and (b) Configuration 2 The PBSC bracing system is designed to work in any standard configuration such as Single/Chevron/V/X etc. A possible arrangement in a building frame is shown in Figure 3-2. In general, the design tensile and compressive load carrying capacity of the brace should be equal. This can be achieved by using the same cross-sectional area for both front and back ties. Furthermore, in order to have equal deformation capacity under compressive and tensile load, both front and back ties should be of equal lengths. The space inside the sleeve could be filled with high damping rubber. This rubber should be cut in a way to accommodate the ties and the brace shaft. This rubber filler could reduce impact energy during the change of loading direction (e.g. seismic (a)(b) 35  load). The diameter of the rubber filler should be smaller than the sleeve diameter to accommodate for the transverse bulging of the rubber filler during compressive deformation. The required gap can be calculated using the Poisson’s ratio and the maximum design axial deformation of the rubber filler. The end plates should be designed with enough thickness to eliminate plate deformation due to forces coming from the Nitinol bars. The Nitinol bars should be connected to the front and black cap/plates in a way that the bars can move out without any resistance; however, they should be prevented from moving inside the sleeve by couplers/nuts. This ensures that during tensile loading, only front ties get locked with the front cap and the back ties get structurally disconnected from the back cap and vice versa. By this mechanism, the front and back ties are alternately loaded during brace tensile and compressive load cycles and the ties only carry a tensile load.  Figure 3-2: The use of the piston-based self-centering bracing system in a steel frame. 3.3 Materials For this study, it is assumed that the brace shaft, sleeve, piston plate and front/back caps are made of structural steel. The following material properties were used for modeling steel in the finite element environment: modulus of elasticity of steel (200 GPa), Poisson’s ration (0.3), yield  36  stress (345 MPa), ultimate stress (508 MPa) and ultimate strain (15.4%). As the finite element analysis package ABAQUS (2014) requires material properties data to be entered as true stress vs. true strain, all of the material properties data were input as such. The density of steel was set to 7850kg/m3.  Various types of shape memory alloys have been developed over the past decades; such as Cu–Zn–Al–Mn–Zr, Ag–Cd, Au–Cd, Cu–Al–Ni, Cu–Zn, Cu–Zn–Al, Fe–Pd, , Cu–Al–Be, Ti–Ni–Cu, Ti–Ni–Hf, Fe–Mn, Mn–Cu, Ni–Ti–Fe, and Ni–Ti (Otsuka and Wayman 1999). However, for this study, only Ni-Ti based shape memory was chosen as the tie bar material for the following reasons: a rich collection of experimental results, availability of material hysteresis models, and ease of availability of the material. However, other SMA alloys could be used instead of Nitinol-based one if test results show similar or superior performance. The material properties used for NiTinol are as follows: density (6500kg/m3), modulus of elasticity (62.5 GPa), Poisson’s ratio (0.33), austenite to martensite starting stress (400 MPa), austenite to martensite finishing stress (510 MPa), martensite to austenite starting stress (370 MPa) and martensite to austenite finishing stress (130 MPa) (Youssef et al. 2008). The superelastic plateau strain (ƐL) was set to 6% as shown using a dashed arrow in Figure 3-3. This material property data is also presented in Table A-1.  37    Figure 3-3: Stress–strain diagram of SE SMA used in the finite element analysis.  3.4 Brace mechanism During compressive loading, when the stress in the back SMA ties/bars exceeds the austenite to martensite starting stress (Figure 3-3), their stiffness decreases and the bars deform significantly. This low tie bar stiffness reduces the global stiffness of the brace. Since the shaft, the SMA bars, and the sleeve are connected in series, the equivalent stiffness of the bracing system will not exceed the lowest stiffness of these three elements. This low stiffness results in less force sharing from the connected structure and thus limits the maximum load on the brace. This mechanism ensures that the buckling load is never reached for the shaft. This assumption remains valid if the shaft is designed for an axial load larger than the austenite to martensite finishing force of the SMA bars. To understand this mechanism clearly, Figure 3-4(a) and Figure 3-4(b) depict a translucent and a longitudinal view of the PBSC bracing system without the sleeve, respectively. -800-600-400-2000200400600800-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08Stress (Mpa)Strain (m/m)SMA Stress-Strainσams: Austenite to Martensite Starting Stressσamf: Austenite to Martensite Finishing Stressσmas: Martensite to Austenite Starting Stressσmaf: Martensite to Austenite Finishing StressεLσamsσamfσmasσmafEiEpEvEi 38  Similarly, during tensile loading, the shaft pulls the front ties while the front ties get locked with the front plate/cap by the couplers/nuts (as shown in Figure 3-4(c)). During this part of the loading cycle, the load flows through the following path: brace shaft  piston plate  front ties  front cap  external support. During this loading phase, the back ties get disconnected from the sleeve back cap, and they can freely move out of the sleeve by approximately the amount of tensile deformation experienced by the front ties. In this way, the back ties do not participate in the load transfer mechanism during tensile loading. During compressive loading, the piston plate pulls the back ties, and they lock with the back cap and thus transfer the load to the sleeve as compression (as shown in Figure 3-4(d)). So, the load path in compression is brace shaft  piston plate  back ties  back cap  sleeve external support. During compressive loading, the front ties get disconnected from the sleeve front cap, and they can freely move out of the sleeve by approximately the amount of tensile deformation experienced by the back ties. This way the front ties never participate during the compressive loading phase. The connection between the ties and the front/end caps/plates are made in a way that they only lock during inward movement of the ties. However, the tie bars can move freely through the openings when ties move outward. The main objective of this mechanism is that the ties are never under compressive loading at any point in time during static or cyclic loading. This way buckling of ties is always prevented.  To make sure that the above mechanism works, only axial load needs to be resisted by the PBSC bracing system. Therefore, the connection between the brace and the structure should be designed as pinned or hinged.  39   Figure 3-4: (a) Semi-transparent view of the PBSC brace from ABAQUS (2014) (b) Longitudinal profile of the PBSC brace with sleeve hidden from view (c) State of the internal elements during tensile loading (sleeve hidden from view) (d) State of the internal elements during compressive loading (sleeve hidden from view). 3.5 Construction of the brace To prevent SMA bars from fully entering the sleeve, the ends of the SMA bars needs to be made larger using either threaded nuts, couplers or centerless grinding. The use of nuts will require pre-threading which is easy to do at the ends but not practical for the middle portions (as the nuts Brace ShaftSleeveCoupler/NutBack CapFront CapPiston PlateSMA tie barsU2U1U3Brace ShaftCoupler/NutBack CapFront CapPiston PlateSMA tie barsUnloaded Back Ties Front Ties Under Tensile LoadUnloaded Front TiesBackTies Under Tensile Load(a)(b)(c)(d) 40  could not be moved there). Therefore, use of nuts will be restricted to configuration 2 (Figure 3-1(b)) only or tie bar ends for configuration 1. A five-step construction method for the PBSC bracing system (configuration 1) is shown in Figure 3-5.  Figure 3-5: Construction of the PBSC Bracing System (a) Individual components (b) attachment of the SMA bars with the piston plate and the couplers (c) attachment of the back plate and the couplers/nuts (d) Attachment of the sleeve (e) attachment of the front plate and the couplers/nuts. (b)(c) (d)(e)(a) 41  3.6 Finite element model The finite element model is shown in Figure 3-6.  Furthermore, for better understanding, every individual part of the PBSC brace has also been labeled in this figure and Figure 3-4(a). The section is taken at exactly mid-depth to illustrate how the SMA bars pass through front, piston, back plate and the couplers/nuts  Figure 3-6: Longitudinal section of the PBSC brace. A transverse cross section of the PBSC brace is shown in Figure 3-7. This figure illustrates the connection between the following elements: piston plate and brace shaft, SMA bars and the couplers/nuts. Adequate gap has been provided between the couplers/nuts and the sleeve. Also, a similar gap is provided between the couplers/nuts and the brace shaft. This gap prevents unwanted friction between the coupler and other brace components in the event of SMA bars and couplers going out of alignment. Brace ShaftSleeveCoupler/NutBack CapFront CapPiston PlateSMA tie bars 42   Figure 3-7: (a) Transverse section of the PBSC brace (b) Couplers for connecting SMA bars with piston plate or end caps. Figure 3-7(b) shows the model of the coupler/nuts used in the finite element environment. A detail coupler/nut model was not constructed to keep the model simple. Therefore, no threads (for nuts) or screws (for couplers) were modeled. A detailed study using threads for nuts or screws for coupler will be carried out in future. Figure 3-8 shows the back, piston and front plate used for the PBSC brace model utilized in this study. Unlike the other two plates, the back cap/plate has a larger opening at the center to allow shaft movement. The opening is made slightly greater than the brace shaft diameter to accommodate shaft diameter expansion during compressive loading. The required opening diameter is calculated based on maximum allowable compressive strain and Poisson’s ratio of the brace shaft material. As the maximum stress in the brace shaft will be kept below the lower of the buckling and yield stress, the strains generated will be below yield and buckling strain. Even if we consider that the yield strain will be smaller than the buckling strain (which is quite uncommon for a slender member), the maximum diameter expansion will be Poisson’s ratio times the yield strain, which is very low for structural steel members. For example, for a 345 MPa steel, the yield strain is only 0.15%, and the transverse strain will be 0.05%. Brace ShaftSleeveCoupler/NutPiston PlateSMA tie barsCoupler/Nut(a) (b) 43   Figure 3-8: (a) Backplate(b) Piston plate and (c) Front plate with openings to accommodate SMA bars and/or brace shaft. Figure 3-9(a) shows a typical example of the gaps kept between the SMA bars and the plates (Front, back and Piston plate) where the couplers/nuts are excluded to show the plate openings. This gap is kept to prevent SMA bars from bending in case of plate deformation. This also prevents unwanted load transfer (loads other than axial) from the plates to the bars.  Figure 3-9: (a) Connection details between piston plate and SMA bars (a) nuts/couplers excluded from the face. (b) nuts/couplers included. Figure 3-9(b) shows the connection between the couplers/nuts and the SMA bars. Here the couplers are connected to the SMA bars using hard contact formulation available in ABAQUS (2014). Hard contact is a pressure overclosure relationship that enforces zero penetration condition (not strict) between the surfaces and any contact pressure can be transmitted between the surfaces (a) (b) (c)SMA tie barsPiston PlateOpeningSMA tie barsPiston PlateCoupler/Nut(a) (b) 44  when they are in contact. However, separable contact is created between the couplers/nuts and the piston plate. 3.6.1 Meshing After the initial construction of the finite element model of PBSC brace, the edges of the part instances were seeded for mesh generation. Seeding was done before the mesh operation to set the approximate element sizes for the mesh. The seeding operation was done using the “seed part instance” tool available in the graphical user interface. Different seed sizes were selected for various parts of the model using the approximate global size option. The seed sizes for the couplers/nuts and SMA bars were made noticeably smaller than the shaft, piston, front, end plates and the sleeve. This size difference was done to ensure better convergence during slave and master contact surface interactions (ABAQUS 2014). The mesh size difference between the coupler/nut (slave) and the piston plate (master) is shown in Figure 3-9(b). It can be observed that the coupler/nut mesh is made denser compared to the piston plate.  An extremely dense mesh can produce highly accurate results. However, very dense mesh requires a significant amount of computer memory and exponentially high computational power. Otherwise, it could take very long time to solve these finite element problems. Therefore, it is necessary to optimize mesh size to achieve a balance between accuracy and analysis runtime. A trial and error based mesh sensitivity analysis was carried out to find out the optimum size of the mesh required for individual elements of the brace. The goal of this analysis was to maximize the mesh size without sacrificing accuracy. The full PBSC brace finite element model consists a total number of 40381 nodes, 6886 quadratic hexahedron elements (C3D20R). C3D20R is a general purpose 20-node quadratic brick element with reduced integration option. This element has 2x2x2  45  integration point and these are about one-quarter of the typical element size away from the element boundary. A trilinear extrapolation is done for converting values between integration points and nodes (Dhondt 2014). C3D20R elements require significantly more computational time compared to linear elements like C3D8R. For this reason, the meshing of the PBSC brace model was carefully optimized to provide a high degree of accuracy without causing too much increase in computation time. 3.6.2 Contact surfaces Three different contact interaction properties were defined for the brace model. These were defined under the following categories: (1) friction sliding, (2) frictionless sliding, and (3) non-separable (welded) contact. These contact surface formulations are described below in details. 1. Friction sliding: The friction sliding contact surface has a penalty friction formulation with isotropic directionality in the tangential direction. The friction coefficient in this direction was taken 0.51. The friction coefficient value was chosen assuming grit blasted A36 steel (Kulak et al. 1987). The normal behavior was defined as hard contact where separation after the contact was allowed.  2. Frictionless sliding: For the frictionless sliding interaction, tangential behavior was set as frictionless and the normal behavior was kept the same as the “sliding with friction” contact.  3. Non-separable (welded) contact: The non-separable contact interaction was defined as follows, Tangential behavior was set to rough, and the normal behavior was set to hard contact without separation after contact.  46  The following interactions were defined between the PBSC brace components. Sliding with friction contact was defined between the following surface pairs: cylinder inner surface to SMA bar surfaces, shaft outer surface to back tie surfaces, back/front/ piston plate holes to SMA bars, shaft outer surface to coupler/nut surfaces, back cap to back couplers/nuts, front cap to front couplers/nuts, piston plate to middle couplers. Frictionless sliding contact was defined between the following surface pairs: middle couplers to cylinder inner, piston plate to cylinder inner, shaft to back cap inner and finally non-separable (welded) contact was defined between these pairs: back ties’ ends to back couplers/nuts, front ties’ ends to front couplers/nuts, cylinder to back cap, cylinder to front cap, shaft to piston plate, middle couplers to front and back ties. Frictionless contact was assumed between the cylinder inner surface and other components as Teflon coating will be used on the cylinder inner surface to allow free movement. No contact was defined between the shaft outer surface and the cylinder/sleeve inner surface as there is no chance of these two surfaces coming into contact. Although ignored in this analysis, friction can develop during the sliding mechanism if the axis of the shaft rotates with respect to the cylinder/sleeve. In this case, the piston plate might encounter the sleeve inner surface during cyclic movement. Depending upon the friction coefficient between these two metal surfaces there can be some friction force development. However, in the current ABAQUS finite element model this friction force is neglected based on the following two assumptions: (a) The shaft will not rotate enough to generate a significant normal force due to guidance through the thick back plate. Therefore, if the normal force is small, friction will be negligible. (b) Teflon coating on the sleeve/cylinder inner surface and the piston plate will reduce friction significantly. Furthermore, a low friction force will be beneficial in dissipating some earthquake energy; however, if the friction force is too large, it might damage the cylinder, piston plate or the shaft.  47  For the interacting contact surfaces, small and finite sliding formulation was defined based on the anticipated movement between the surfaces. Small sliding was defined for non-separable contacts, and finite sliding was defined for the rest (frictionless sliding and sliding with friction). Surface to surface discretization method was used for all interactions. Furthermore, no surface smoothing or slave adjustment was done for any of these interactions. 3.6.3 Boundary conditions and loading Three boundary conditions were defined for the PBSC brace model in the finite element environment (ABAQUS 2014). The boundary conditions are as follows: Pinned for the front cap, displacement restrained for shaft free end (normal surface close to reference point 1 (*RP-1) point in Figure 3-4(a)) and displacement for the reference point (Figure 3-10). The front cap of the PBSC brace was restrained in three translational directions (U1=U2=U3=0). Here, U3 is the brace axial z direction and U1 and U2 are the two transverse x and y directions at 90 degrees apart (Figure 3-4(a)); this ensured that the front end of the brace remained in a fixed location. As the C3D20R solid elements do not have rotational degrees of freedom, no rotational restraint was applied. Secondly, the free end of the shaft was also restrained in two transverse directions (U1=U2=0), but the axial U3 direction was kept free to allow movement in the axial direction. A reference point (*RP-1) was created to represent the entire free end surface of the shaft which is shown in Figure 3-4(a). A constraint was defined between the reference point and the shaft end surface of the brace model so that any movement of the reference point is translated into an equal displacement for the brace shaft end. Later, a displacement boundary condition was defined for this reference point. The boundary condition for this reference point was set as follows: U1=U2=0 and U3=1.  48   Figure 3-10: Boundary conditions of the PBSC brace model in ABAQUS (2014). 3.7 Analysis and results Four finite element models of the PBSC bracing systems were built in ABAQUS (2014). The length of the PBSC device (Sleeve length) was chosen to be 2m. The full shaft was not modeled for this analysis based on the assumption that the shaft will be in the linear range of response during the load cycles and the elastic deformation of the shaft will be insignificant compared to the nonlinear deformation of the PBSC device. The SMA bars are only allowed to go into the nonlinear range in this system to prevent nonlinear deformation in other components. Hence, studying the cyclic behavior of the PBSC brace device with a short part of the brace shaft is adequate in understanding the brace behavior. Besides, for retrofit works, the brace device with a short brace shaft will be manufactured, and then they will be connected to the existing braces. Furthermore, during frame analysis using the PBSC bracing system, the elastic shaft deformation will be calculated and will be added to the PBSC device deformation in finite element analysis software.  49  To eliminate the brace shaft deformation, a very rigid steel section has been used for the brace shaft. The section is made of solid circular steel with 96mm diameter and having a length of 1.5m. The ties were modeled as solid circular sections with a length of 2m. The Bar diameters chosen for this study are as follows: 10mm, 12mm, 16mm, and 20mm. These four different diameters were used for four cyclic analyses. The sleeve is modeled as a hollow circular section with a diameter of 250mm and a wall thickness of 15mm. 75mm thick steel plates have been used for the front, back cap, and the piston plate. The plate openings size was set to 25mm so that it becomes at least 5 mm larger than the largest diameter SMA bar in this analysis; this is done to allow unrestrained rotational movement of the tie bars. A total number of four SMA bars were used for each of these models. Each of these bars spanned between the three plates as shown in Figure 3-4. The superelastic behavior of SMA bars was simulated using NiTinol superelastic-plastic ABAQUS/CAE plugin. The four transformation stresses used for SMA material model is given in the materials section. The remaining elements (shaft, sleeve and the plates) are made of steel. The material properties of steel are provided in the materials section.  Figure 3-11 represents the cyclic loading time history used for the PBSC brace model. This time history with a maximum value of 60mm was associated with the U3 direction of the *RP-1 reference point. The reference point *RP-1 was created for the centroid of the brace shaft and it was offset 250mm (an arbitrary value with no effect on the analysis). During result extraction, this reference point provided an easier way to retrieve both force and deformation results. Any deformation applied to the reference point was transformed to equal deformation for the shaft end surface.  50                  Figure 3-11: Quasi-static deformation loading history. Figure 3-12 shows the velocity history of the deformation load shown in Figure 3-11. This figure was generated by differentiating the deformation time history with respect to time. It can be seen that the velocity time history has constant velocity plateaus where the deformation history has a constant slope. The velocity history shows sudden jumps where the deformation history has a change in slope positive to negative or vice versa. The change in slope in the max/min points of the deformation history plot indicates load reversal points.  Figure 3-12: Velocity time history of the load. Figure 3-13 shows the acceleration time history of the applied load. This history was calculated by differentiating the velocity time history with respect to time. In the velocity time history (Figure -150-100-50050100150-75 -50 -25 0 25 50 75Force (kN)Displacement (mm)-80-60-40-200204060800 5 10 15 20Axial Deformation (mm)Time (Seconds)-80-60-40-200204060800 5 10 15 20Velocity (mm/sec)Time (Seconds) 51  3-12) where the velocity is constant, the corresponding acceleration values are zero. However, the acceleration has a large spike where the velocity plot has a sudden shift in value (e.g. load reversal points). This sudden acceleration occurs when the shaft reverses from loading to unloading state or vice versa.   Figure 3-13: Acceleration time history of the load. Figure 3-4 (c) and Figure 3-4 (d) depict the deformation states of the internal elements of the PBSC bracing system during tensile and compressive loading, respectively. The sleeve is intentionally kept hidden from this view in these figures. For a proper understanding of the deformed shape, the longitudinal deformation was three times exaggerated. From Figure 3-4 (c), it can be observed that the front end of the ties is under tension and this part is tightly locked between the front cap and piston plate. On the other hand, the back ends of the ties are structurally disconnected, and no load is shared by this part. The back end of these ties freely moved out of the back cap in this figure. Figure 3-4 (d) shows the compressive load cycle for the PBSC bracing system. In this loading phase, the back portions of the tie bars are locked between the piston plate and the back cap whereas the front ends of the tie bars are structurally disconnected from the front cap and freely moved out of the front cap.  -10000-8000-6000-4000-200002000400060008000100000 5 10 15 20Acceleration (mm/sec2)Time (Seconds) 52  Figure 3-14 shows the maximum von Mises stresses observed in the middle eight couplers/nuts during the reverse cyclic loading test. The stresses depicted here are in terms of quilt contours. For quilt contours, the variable values are extrapolated to element faces on the surface of the model. No averaging between elements is carried out. Quilt contours are useful when results are evaluated on an element-by-element basis. Only a single color per element is plotted for axisymmetric elements with asymmetric deformation. It can be observed that the maximum stresses are observed on the inside surfaces of the couplers. Figure 3-14 to Figure 3-17 present the brace analysis results with 16mm dia NiTi rods. All the stresses in these figures are quilt stresses to eliminate element local corner stresses from these figures.  Figure 3-14: von Mises stress distribution on the couplers. Figure 3-15 shows the maximum stresses experienced by the plates during the cyclic load test. In this figure, the piston plate is showing the maximum stresses around its openings. The maximum quilt stress was found to be 80.37 MPa.  53   Figure 3-15: von Mises stress distribution on the plates. Figure 3-16 (a) and (b) show the maximum stresses on the sleeve and shaft of the PBSC bracing system, respectively. It was observed that the sleeve experienced very low maximum stress (48.28 MPa) as intended by design. In contrast, the yield and the theoretical buckling strength for the sleeve are 345MPa and 855 MPa (assuming effective length factor k = 2), respectively. This low stress is critical in lowering any chance of global or local buckling in the sleeve. Figure 3-16(b) shows the von Mises stress distribution on the shaft. As per this figure, the maximum stress observed by the shaft is approximately 42 MPa. This stress is lower than the yield (345Mpa) and buckling strength (126.62Mpa assuming k = 2) for this section.  54   Figure 3-16: Von Mises stress distribution on (a) Sleeve and (b) Shaft. Finally, Figure 3-17 shows the maximum stresses on the SMA bars. It was found that the quilt contour values were less than or equal to 391.01 MPa as shown in Figure 3-17. However, the banded contour values (not shown in this figure) were found to be higher. Those local stresses pushed the bars in the nonlinear range of response. The banded contours represent values as color-filled bands. These contours are computed by extrapolating results to the nodes and conditionally averaged. Averaging depends on the options selected and characteristics of the model (ABAQUS 2014). (a)(b) 55   Figure 3-17: Von Mises stress distribution on the NiTinol ties The whole reverse cyclic loading process was repeated for four different bar sizes mentioned earlier, and the load deformation hysteresis results are plotted in Figure 3-18(a)-(d). From this figure, it can be observed that the hysteresis of the PBSC system is very similar to the flag shape hysteresis.  It was observed in previous studies that flag shaped hysteresis signifies superior self-centering capability (Zhang and Zhu 2007). When installed in a structure, this type of smart systems could reduce permanent residual deformation by a very significant amount after an earthquake.  56   Figure 3-18: Hysteretic response of the PBSC bracing system with (a) 4-10mm (b) 4-12mm (c) 4-16mm and (d) 4-20mm dia NiTinol Bars. Figure 3-18 (a) shows the load deformation hysteresis response of the PBSC bracing system with four numbers of 10mm diameter NiTinol based SMA bars. From this figure, it can be seen that the yield force of this brace is around 95kN. After the yielding, the stiffness is reduced. The post-elastic stiffness is also noteworthy which is helpful in resisting excessive plastic deformation. The maximum force and deformation observed for this brace are 110kN and 60mm, respectively. The observed hysteresis behavior is symmetric and stable as they do not show any strength degradation in subsequent load cycles. The flow direction of the hysteresis shape is provided in Figure 3-20(a). The symmetric behavior also indicates buckling has not occurred in the system. Similar behavior was observed for three other PBSC brace models utilizing 12mm, 16mm, and 20mm diameter SMA bars. The global yield forces of the brace with four numbers of 12mm, 16mm and 20mm bars were 140kN, 240kN and 375kNand the maximum forces were 160kN, 272kN, and 453kN, respectively. The highest deformation for these systems was limited to 60mm. This 60mm -150-100-50050100150-80 -40 0 40 80Axial Force (kN)Axial Deformation (mm)-200-150-100-50050100150200-80 -40 0 40 80Axial Force (kN)Axial Deformation (mm)-400-2000200400-80 -40 0 40 80Axial Force (kN)Axial Deformation (mm)-600-400-2000200400600-80 -40 0 40 80Axial Force (kN)Axial Deformation (mm)(a) (b)(d)(c) 57  maximum deformation was chosen based on approximately 6% elongation of the half-lengths (1075mm) of the SMA bars. To understand the advantage of SMA bars used in this bracing system, a similar model was run using similar diameter steel rebar in place of SMA bars. The steel used in this analysis has the following material properties: Modulus of elasticity 200 GPa, Yield Strength 414 MPa and maximum elongation capability of 10.3%. The analyses results are presented in Figure 3-19, and the hysteresis flow paths are shown in Figure 3-20(b). It was observed that the hysteresis with steel rebar is similar to a sliding device. After a full reverse cyclic loading, the brace starts to slide between the unloaded state to the load reversal state. The stiffness of the brace becomes zero between these points. This large zero stiffness plateau makes the brace unsuitable for use against any lateral and vertical loading. This occurs due to permanent elongation of the steel rebars as steel cannot come back to its original size after load removal. Figure 3-19(a) to Figure 3-19(d) show the hysteretic responses for four braces with four numbers of 10mm, 12mm, 16mm and 20mm diameter 60-grade steel rebar.  58   Figure 3-19: Hysteretic response of the bracing system with (a) 4-10mm diameter steel bar (b) 4-12mm diameter steel bar (c) 4-16mm diameter steel bar (d) 4-20mm diameter steel bar.  Figure 3-20: Hysteretic flow paths for the brace with (a) SMA tie bars (b) Steel ties bars. Finally, the energy dissipation capability of the PBSC brace was compared to the same configuration fitted with steel rebar. As the yield strength of the braces made with SMA and steel rebar are not similar, for a more reasonable comparison, the calculated values were normalized with respect to the respective brace yield forces (Table 3-1). It was found out that, the brace with SMA rebar has a higher energy dissipation capacity compared to the same configuration of this brace made with steel rebar. It was found out that the SMA based system has around 10%-20% -250-200-150-100-50050100150200250-80 -40 0 40 80Axial Force (kN)Axial Deformation (mm)-300-200-1000100200300-80 -40 0 40 80Axial Force (kN)Axial Deformation (mm)-600-400-2000200400600-80 -40 0 40 80Axial Force (kN)Axial Deformation (mm)-800-600-400-2000200400600800-80 -40 0 40 80Axial Force (kN)Axial Deformation (mm)(a) (b)(d)(c)-150-100-50050100150-80 -40 0 40 80Axial Force (kN)Axial Deformation (mm)-250-200-150-100-50050100150200250-80 -40 0 40 80Axial Force (kN)Axial Deformation (mm) 59  higher energy dissipation capacity compared to the steel based one. This is due to the sliding hysteresis of steel based configuration relative to a flag hysteresis observed for the SMA based one. Table 3-1: Normalized energy dissipation comparison.   Dissipated Energy (kN-m/kN) Description 4-10mm 4-12mm 4-16mm 4-120mm With SMA Bar 0.15 0.15 0.15 0.14 With Steel Rebar 0.13 0.13 0.13 0.13 % Higher 14.66 15.42 19.19 10.98  3.8 Summary Four different PBSC brace models were built in the finite element environment and were tested under reverse cyclic quasi-static loading. From this analysis, the following observations were made. • The PBSC brace exhibits flag shaped hysteresis response that is an excellent indicator of self-centering capability. • The brace hysteresis was found to be symmetric with respect to both deformation and force response. This indicates that no buckling occurred during the analysis. • Only SMA bars went to the nonlinear range during the cyclic load test while all other parts of the brace remained elastic. This ensures that the steel parts will require almost no repair or replacement after an earthquake event. The PBSC bracing system has the following limitations compared to other bracing systems. • Deformation capacity of the PBSC brace is limited to the deformation capacity of the SMA bars spanning between the piston plate and the end plates. Therefore, the theoretical maximum SMA bar length that could be utilized for the PBSC brace is equal to the half of the brace length. This implies that the maximum superelastic elongation capacity of the PBSC brace will be limited by this length constraint.  60  The current study has few limitations that need to be addressed in future research. No coupler/nut to SMA bar load slip model has been used in this study, which is necessary for determining proper load deformation behavior of these connections. In real world scenario, slippage could affect the self-centering capability. The SMA material model did not account for residual strain, and hence, the PBSC bracing system showed zero residual deformation. This is less conservative than what is expected from a PBSC brace prototype. An experimental investigation needs to be carried out for the validation of the finite element model test results. The cyclic performance of the PBSC bracing system is highly dependent on the shape memory alloy ties. The hysteresis of SMA is dependent upon ambient temperature. If the ambient temperature drops below austenite finish temperature, SMA starts to lose its self-centering capability. If the temperature goes down to martensite finish temperature, SMA looses its superelastic ability. Therefore, the SMA alloy to be used in cold climate should be carefully calibrated/manufactured to have their austenite finish temperature lower than the expected lowest temperature in those regions. Furthermore, hot weather or fire can be detrimental to the PBSC bracing system. After a major earthquake, fire can break out in a building and aftershocks can coincide. At a high temperature, such as Md (a temperature much higher than Af as shown in Figure 2-13), NiTinol becomes elastoplastic and loses both shape memory effect and superelasticity. Therefore, if NiTinol ties get warmed up beyond Md due to fire, the PBSC bracing system will not be able to bring back the structure to its original position after the earthquake.  61  CHAPTER 4: A HYSTERESIS MODEL FOR THE PBSC BRACING SYSTEM 4.1 General The piston based self-centering (PBSC) bracing system uses superelastic shape memory alloy (SE SMA) bars for its energy dissipation and self-centering capability. These SE SMA bars exhibit flag-shaped hysteresis; in theory, these bars are supposed to bring back the bracing system fully to its original position after significant nonlinear deformations. However, shape memory alloy undergoes some residual deformation after large plastic strains. This residual strain occurs when planes of tightly packed atoms slide past each other; individual bonds are broken and reformed with new atoms in a step-by-step process. Under repeated cyclic loading, these residual deformations add up and deviate from the anticipated zero residual strain flag shaped response. To accurately simulate this behavior during nonlinear dynamic time history analysis, an SMA flag shaped hysteresis model with sliding response has been developed by the author. This chapter shows the gradual development process of this new hysteresis model and provides analysis and verification results to support this claim. To validate the material hysteresis model, a MATLAB based superelastic uniaxial SMA material hysteresis model was developed and incorporated into a custom-made finite element program (MATLAB-based) specifically designed and developed for the PBSC bracing system. This custom made finite element program reduced hours of analysis run-time to seconds. For this reason, the SMA material hysteresis model was implemented in the MATLAB based finite element program instead of ABAQUS (2014). This finite element program was utilized to capture the nonlinear quasi-static response of the PBSC brace. Finally, the PBSC brace response from this analysis was used to develop a novel SMA flag shaped hysteresis model  62  with residual deformation and sliding response for implementation in a commercial structural analysis and design software known as S-FRAME. The hysteresis model was developed in a way to simulate not only PBSC bracing system but also other self-centering structural systems such as MANSIDE and RHDB brace. This hysteresis can also be calibrated to simulated SMA reinforced concrete elements; this will help researchers and engineers around the world simulate different types of self-centering structural systems in the finite element environment; which will enable them to design these systems more accurately and efficiently than before. 4.2 SMA bar behavior For this study, we will critically analyze the NiTinol bar test results discussed in the previous chapter. The diameters of the specimens tested include 1.8mm wire and 7.1mm, 12.7mm and 25.4mm bars. DesRoches et al. (2004) confirmed that both SMA bars and wires show excellent re-centering capacity (Figure 4-1). The residual strain observed for the test specimens were on an average 0.65% for all the specimens where the maximum strain was 6%. The only disadvantage of using SMA bars over wires is its lower hysteretic energy dissipation capacity. The equivalent viscous damping represented by the fatness of the hysteresis loops was found to vary from 2% for the 12.7mm bar to a maximum of 7.6% for the 1.8mm wire.   63   Figure 4-1: Stress–strain plot for 25.4 mm diameter nitinol SMA bar subjected to quasi-static cyclic loading (adapted from DesRoches et al. (2004)). The maximum plastic, residual and yield strain of the 25.4mm dia SMA bar specimen were calculated from Figure 4-1, and the values are shown on the plot. From this figure, the accumulated residual strain is around 9.4% (0.0057/.0607) of the maximum strain; which is 12% of the maximum plastic strain. Here plastic strain is the strain value after deducting the yield strain of the specimen. However, other test results (DesRoches et al. 2004, Nemat-Nasser and Guo 2006) exhibited NiTinol having a residual strain in the range of 5%-12% of the maximum strain. Therefore a  slightly lower (10%) residual to plastic strain ratio is used for the rest of this study. 4.3 SMA flag hysteresis rule Based on the behavior observed in experimental investigations researchers have come up with both material and link hysteresis models for SE SMA material and SE SMA built elements respectively. A widely used material model for SMA is shown in Figure 4-2 (Auricchio and Sacco 1997). In this hysteresis model, unloading happens in two different stages. From a stress beyond martensite to austenite starting stress (σmas), the material unloads at initial elastic modulus (Ei) until 25.4mm dia SMA rodStress (MPa)Strain80070060050040000.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07300200100 0.06070.0140.005 64  it reaches the martensite to austenite starting stress (σmas). After arriving at this stress level, the tangent modulus is drastically reduced to variable modulus, (Ev) and the line move towards martensite to austenite finishing stress point (σmas/Ei, σmas) on the backbone curve.  This is very close to what the experimental results suggest (Figure 4-1).  This material hysteresis model has been implemented in SeismoSoft (2016) finite element analysis software. The only shortcoming of this model is the absence of residual strain formulation, which is very important for accurate representation of this material behavior. Some other notable features of this hysteresis models are the austenite to martensite starting stress (σams), plastic modulus (Ep) and austenite to martensite finishing stress (σamf). For structural applications, σams can be considered as yield stress. σamf  is the stress level beyond which SMA becomes fully martensite and the tangent modulus changes back to Ei. The tangent modulus between σams and σamf  points on the backbone curve can be designated as the plastic modulus Ep. ƐL  is known as the plateau strain, and it represents the strain plateau in the austenite to martensite transition zone.  65    Figure 4-2: Uniaxial SMA flag-shaped hysteresis (adapted from Auricchio and Sacco (1997))  The next hysteresis model for representing SMA based structural elements in finite element software is the SMA flag-shaped hysteresis model available in Ruaumoko finite element software (Carr 2008). A diagram of this hysteresis model is shown in Figure 4-3. Comparison between Figure 4-3 and Figure 4-2 indicates that there is a clear distinction between the unloading rules. In the Ruaumoko model, there is no fixed force level representing the martensite to austenite starting force/stress. The unloading stiffness changes when the unloading line reaches a line parallel to the post-yield stiffness line going through ALPHA*YP point. Therefore, this model is less accurate in representing SMA based elements compared to the one implemented in SeismoStruct. Additionally, this model also lacks residual deformation. Nevertheless, this model has some additional parameters for controlling some important behavior; they are as follows: the parameter ‘DELTA’ represents the point where material in the sections finishes transforming to martensite and starts stiffening up again. PHI*YP and PSI*dy point represent the fracture point. Additionally, -800.0-600.0-400.0-200.00.0200.0400.0600.0800.0-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08Stress (Mpa)Strain (m/m)SMA Stress-Strainσams: Austenite to Martensite Starting Stressσamf: Austenite to Martensite Finishing Stressσmas: Martensite to Austenite Starting Stressσmaf: Martensite to Austenite Finishing StressεLσamsσamfσmasσmafEiEpEvEi 66  the unloading line has a stiffness of BETA*Ko. Therefore, the stiffness of this line can be different compared to the initial stiffness line. Furthermore, in this hysteresis model ALPHA, YP, ko, r, BETA, DELTA, PHI and PSI values can be changed.  Figure 4-3: Graphical rule of flag-shaped hysteresis (adapted from Carr (2008)) 4.4 PBSC brace hysteresis model development From Figure 4-1, it can be observed that SMA bars are not entirely capable of re-centering. The residual plastic deformation is around 10% of the maximum plastic deformation. Both Ruaumoko and Auricchio & Sacco’s hysteresis rules are missing this important feature. Due to these limitations mentioned above, the current hysteresis models are not very accurate in capturing residual deformations of elements made of SMA under cyclic loading conditions. To resolve this issue, a new SE SMA material model has been proposed. This model follows the basic hysteresis rule established earlier (Auricchio and Sacco 1997). In addition to the previous rules, it can simulate the residual strain accumulation. This modified model is only for simulating superelasticity but not the shape memory effect; therefore, it will not be able to simulate the FdYPALPHA*YPALPHA*YPYNPHI*YPdydyDELTA*dy DELTA*dy PSI*dyKor*Kor*Kor*KoBETA*KoBETA*Ko 67  thermomechanical behavior of SMA. Furthermore, this model is specifically designed for frame element based finite element analysis; therefore, it cannot be used in solid or shell element modeling, which warrants more complex continuum mechanics and volume fraction based formulations. Furthermore, no strain rate dependent behavior has been incorporated. Therefore, this modified SE SMA hysteresis model will be only useful in specific applications such as accurately determining the PBSC brace hysteresis or modeling beam-column elements made of SE SMA.  The previous study on the PBSC brace (Haque and Alam 2015a) used NiTinol UMAT subroutine available in ABAQUS (2014) for the NiTinol material model. For calibrating the hysteresis parameters, superelastic-plastic ABAQUS/CAE plugin was used. This NiTinol material model was developed from the works of Auricchio and Sacco (1997); therefore, it is also missing the residual strain calculation algorithm. In the PBSC bracing system the SE SMA bars only get loaded in tension. Therefore an SE SMA material model with only tension loading unloading rule is sufficient in simulating the brace behavior. A brief description of the PBSC brace mechanism is given below which is later used to develop a simplified finite element model necessary for MATLAB implementation. 4.5 PBSC mechanism PBSC bracing system is designed to be used in portal frames typically in a Diagonal/Chevron/V configuration. Figure 3-2 shows an example of such arrangement. The details of two different brace configurations have already been presented in Figure 3-1(a) and (b). Please refer to section 3.4 for reviewing the mechanism of this bracing system for understanding the simplified model discussed in section 4.6.  68  4.6 Simplified finite element model In order to develop a simple MATLAB finite element program for the PBSC bracing system, the model was idealized using one-dimensional bar/strut elements. Below is a systematic description of the process by which the model was simplified.   Figure 4-4: Longitudinal section of the PBSC bracing system It can be seen from Figure 3-1 and Figure 4-4 that the PBSC bracing system has three major components; they are the shaft, the sleeve, and the internal SMA tie bars. The plates are provided to connect the ties to the other parts. From brace mechanism section and Figure 4-4, it can be understood that the sleeve is acting as a support for the back cap/plate, which transfers load from the SMA bars to the support when the brace is under compressive loading. Furthermore, the sleeve cross section is designed to take much larger loads than the ties and the shaft; This ensures minimal longitudinal deformation for the sleeve. Therefore, a finite element model made without the sleeve should behave almost similarly as the one with the sleeve if the plates are restrained from translation.    Figure 4-5: PBSC bracing system shown without the sleeve Coupler/NutFront CapSMA tie barsPiston PlateBrace Shaft SleeveBrace ShaftCoupler/NutBack CapFront CapPiston PlateSMA tie bars 69  Figure 4-5 shows the PBSC brace model without the explicit sleeve model. For MATLAB implementation, we can further assume that both the shaft and the SMA bars are bar elements and the plates are the nodal points. This simplified model is shown in Figure 4-6.  Figure 4-6: Simplified PBSC brace model Finite element modeling the simplified representation of the brace shown in Figure 4-6 using one-dimensional bar/truss element is not possible as the right support node will reside in the same line of the shaft. Therefore, the shaft will be prevented from movement. To resolve this issue, this model was further simplified by moving the right support on the left side and reversing the front tie bars. This further simplified model is shown in Figure 4-7. However, moving the front ties to the back side converted the tensile load to compressive and subsequently, the deformations also became compressive (axial shortening for the front tie bars). However, it can be understood that in this configuration, if buckling is ignored, the compressive deformations experienced by the reversed tie bars are same the as that of the tensile deformations of the model shown in Figure 4-6. Hence, the overall load deformation hysteresis of the brace remains the same as the original model.  If we observe this model carefully, it can also be understood that the ties are occupying the same space if they are modeled in one dimension. To resolve this issue, only one set of tie bar could exist at any point in time depending upon the loading condition. Under tensile loading the Bar element for the shaftSupportNodeBar elements representing the SMA TiesGap SupportCyclic Load 70  compression tie bars were deactivated and vice versa. During the deactivated state, all load deformation history of the tie bars was saved in the memory to be used in the subsequent cycle when that tie gets reactivated.  Figure 4-7: Further simplification of the PBSC brace model 4.7 SMA material model In this section, a tension only SE SMA material hysteresis model with residual strain is developed. In the next step, it is incorporated into the MATLAB finite element program discussed above. This section briefly discusses the rules of SE SMA material model with residual deformation. As this material model is for tensile stresses only, no rules are available for the negative stresses. The following definitions and notations should be used for interpreting the hysteresis rules: loading refers to an increase in the absolute value of the stress without a change in sign. On the other hand, unloading refers to a decrease in the absolute value of the stress without a change in sign. Yielding refers to reaching a line with Ep tangent modulus passing through (σams/Ei, σams) point. Ev is the variable tangent modulus line connecting (σmaf/Ei, σmaf) point on the backbone curve to the unloading line coming from σmas stress level. Please check Figure 4-8 for interpreting the other notations. The uniaxial SE SMA hysteresis rules with residual deformation is given below  71  1. Below yield stress 1.1. Loading: Loading occurs on the right side or on the line with “Ei” tangent modulus and passing through (εri, 0) point.  The loading line follows Ei tangent modulus until yielding. After yielding go to rule 2. 1.2. Unloading:  1.2.1.  If unloading from a stress higher than ‘σmas’ then the unloading line follows Ei tangent modulus. When the line reaches ‘σmas’ go to 1.2.2. 1.2.2.  If unloading from a stress less than or equal to ‘σmas’ then unloading line follows Ev tangent modulus. Ev is the variable tangent modulus of a line originating between σamf and σams stress level and directed towards (σamf/Ei, σamf) point on the backbone curve. This line follows Ev tangent modulus until this line intersects a line with Ei tangent passing through (εri, 0) point. εri is calculated using current hysteresis loops maximum strain and residual strain coefficient (Equation 1). After the intersection go to rule 1.2.3 1.2.3.  If unloading from a point on or below the intersection point defined in 1.2.2: The unloading line follows Ei tangent modulus. 2. After Yielding 2.1. The line follows the post-yield tangent (Ep) until it reaches σamf or Austenite to martensite finishing stress. After reaching σamf go to rule 3 2.2. Unloading: Go to rule 1.2.1 3. At or above σamf stress 3.1. Loading: The loading line follows Ei tangent modulus. 3.2. Unloading: Go to 1.2.1  72   Figure 4-8: SMA uniaxial stress-strain hysteresis model with residual deformation (Tension Cycles Only) A typical tension only cyclic response from the developed SE SMA material model is shown in Figure 4-9. The residual strain is calculated using equation (1). For simplification, a linear relationship has been used between the residual and plastic strain. Residual strain = (Maximum strain of the cycle-Yield Strain) *Residual Strain Coefficient (1) Here, the “Maximum strain of the cycle -Yield Strain” is the plastic strain for the current cycle under consideration. Furthermore, the Yield Strain is calculated from the original backbone curve. In this hysteresis model, residual strain is only allowed to increase. Therefore, if the current cycle has a maximum strain value less than the maximum strain of one of the previous cycles then the new residual strain will not be calculated, and the previous residual strain will be used for the current cycle. As per this rule, if the material experiences large strain cycles at the beginning, then the residual strain will be set to a higher value and will not be reduced if the subsequent cycle’s maximum deformations are smaller.  This SMA hysteresis model with residual strain was validated against the experimental results presented in (DesRoches et al. 2004). The result of this validation is shown in Figure 4-9. In the material hysteresis model, the residual strain coefficient can be set to any value between 0 and 1. εσσamsεyEpEiεr1εr2εr31st Cycle2nd Cycle3rd CycleEiσamfσmasσmaf Evσamf = Austenite to Martensite Starting Stressσams = Austenite to Martensite Finishing Stressσmas = Martensite to Austenite Starting Stressσmaf = Martensite to Austenite Finishing Stressεri = Residual deformation of the ith cycleEi = Initial ModulusEp = Post Yield ModulusEv = Variable Modulus 73  However, a value of 0.128 was set for this validation (Figure 4-9). The material hysteresis model response (shown using blue lines) showed excellent agreement with the experimental result. The energy dissipation from of the experimental result and the MATLAB simulation are 11.03 kN-m/m and 10.73 kN-m/m respectively (calculated using the area inside the hysteresis loops and the cross-sectional area of the bar). The energy dissipation from the MATLAB simulation is only 2.8% lower than the experimental result. Therefore, it can be safely concluded that this material hysteresis model could be utilized to simulate SMA with residual strain/deformation.  Figure 4-9: Validation of SMA material hysteresis model against experimental result by (DesRoches et al. 2004). The SE SMA hysteresis model discussed above was coded as a function in MATLAB (2012), and a flow chart of this function is provided in Figure 4-10.  25.4mm dia SMA rodStress (MPa)Strain80070060050040000.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07300200100 0.06070.0140.005MATLAB ResponseTest Result 74   Figure 4-10: Flowchart of the SMA material hysteresis model StartInput current stress and strain: σ1, ε1Input next stress: σ2NoEt = EiBσ2=>σ1Yesσ_limit= σams * Ei + Ei * Ep * (ε1 - σams/Ei) - σ1* Ep) / (Ei - Ep);ε_limit = (ε1 * Ei + σams - Ep * σams/Ei - σ1) / (Ei - Ep);0<=σ1<σ_limitYesσ2<=σ_limitYesNoNoYesσ_limit<σ2<=σamfNoEt = (σ2-σ1)/(ε_limit+(σ2-σmax)/Ep-ε1)Et = (σ2-σ1)/(εamf+(σ2-σamf)/Ei-ε1)CInput Material Properties: Ei, σams, σamf, σmas, σmaf, εL, ry Ep=(σamf - σams) / (εL + σamf / Ei - εams)XLast load step ? EndYesNoXXXε_max=ε1;εr = (ε_max - εams) * ry;ε1>ε_maxNoYes 75   Figure 4-10: Flowchart of the SMA material hysteresis model (Continued)  Figure 4-10: Flowchart of the SMA material hysteresis model (Continued) YesYesBσ1>σmasσ2>σmas EvEi_σ<=σ2<σmasEt = EiEt = (σ1-σ2)/(ε1-EvEi_ε+(EvEi_σ-σ2)/Ei)No NoYesEt = (σmas - σmaf) / (((σmas -σ1) / Ei + ε1) - σmaf / Ei)NoDσ1>σmasYesNoσmaf<σ1<=σmas YesEv = (σmas - σmaf) / (((σmas -σ1) / Ei + ε1) - σmaf / Ei)Ev = (σ1 - σmaf) / (ε1 - σmaf / Ei)EvEi_ε = (Ei / (Ei - Ev)) * εr + σmaf/Ei;EvEi_σ = ((EvEi_ε - εr) * Ei);X XXσ_limit<=σ1<σamfCYesNoEt = Eiσ2<=σamfYesNoEt = EpEt = (σ2-σ1)/(εamf+(σ2-σamf)/Ei-ε1)XXX 76   Figure 4-10: Flowchart of the SMA material hysteresis model (Continued) The explanation of the parameters used in the flow chart (Figure 4-10) is given in Table 4-1.            EvEi_σ<=σ1<σmasDYesNo Et = Eiσ2>=EvEi_σYesNoEt = (σ1 - σmaf) / (ε1 - σmaf / Ei)Et = (σ1-σ2)/(ε1-EvEi_ε+(EvEi_σ-σ2)/Ei)XXX 77  Table 4-1: Description of the flow chart parameters Parameter Description Typical value Ei Initial elastic modulus of SMA 62500 Mpa σams Austenite to martensite starting stress 400 Mpa σamf Austenite to martensite finishing stress 510 Mpa σmas Martensite to austenite starting stress 370 Mpa σmaf Martensite to austenite finishing stress 130 Mpa εL Plateau Strain 0.06 ry Residual to plastic strain ratio 0.1 Ep Modulus during austenite to martensite transition state 3%-4% of Ei ε1 Current Strain N/A ε_max Previous Maximum Strain N/A εr Residual strain N/A σ1 Current Stress N/A σ2 Stress at next loading step N/A σlimit Stress limit of a line with Ei tangent passing through (εams, σams) point N/A εlimit Strain at σlimit point N/A Et Tangent modulus N/A Ev Variable tangent modulus of a line unloading from σmas stress level and directed towards (εmaf, σmaf) point of the backbone curve N/A EvEi_ε Strain at the intersection point between an unloading line with Ev tangent and a line with Ei tangent going through (εr, 0) point N/A EvEi_σ Stress at the intersection point between an unloading line with Ev tangent and a line with Ei tangent passing through (εr, 0) point N/A The function mentioned above was called from a MATLAB (2012) finite element program specifically developed for the PBSC bracing system. Another material model function was used for the steel shaft of the PBSC brace. That material model is based on bilinear kinematic hysteresis typically used for steel sections. The finite element program takes quasi-static loading as input and at every load step calls the corresponding material models for the relevant brace elements. For example, it calls the SE SMA material model for the SMA bars and bilinear kinematic hysteresis model for the steel shaft. Based on the current stresses in the elements, the material hysteresis models return the appropriate tangent modulus values to the solver. The solver calculates the element stiffnesses from the material tangent modulus. These stiffness values are used to create the global stiffness matrix. Finally, the global deformation matrix is calculated based on the global  78  force and stiffness matrix. The brace geometry is updated, and the program fetches the incremental loading data from the next load step. A flow chart of this overall process is shown in Figure 4-11. It can be seen from the flow chart that there is no particular element for representing the gap element. The gap is automatically taken care of by keeping the elongated tie bar lengths in memory. Due to the use of SMA material model with residual deformation, the tie bars are subjected to plastic deformation which changes their lengths permanently. When the load reverses from either tension to compression or vice versa, the previously loaded ties are deactivated, and their final elongated lengths are kept in the memory. The length data is reloaded to the memory when the load reverses its sign again. When this sudden length change occurs inside the brace, this generates a displacement data without any force; which results in a sliding response in the hysteresis shape.  79    Figure 4-11: Flow-chart of the MATLAB PBSC analysis program Start1. Input Material characteristics (e.g. SMA and Steel hysteresis)2. Input Brace element properties3. Input quasi-static loading historyRead the current point in the loading history (Load increment or decrement values only)FinishSend the stress values to the corresponding SMA and steel material models Get the tangent stiffness values  from the material models and calculate the element stiffnessesForm the global stiffness matrixCalculate the global and local deformations from the stiffness and force matrix. Modify the brace geometry based on these deformations.Last load step?YesNoHas the load direction reversed from the last step ?Yes No1. Save the length data of the previously loaded tie and deactivate it.2. Reactivate the tie currently under loading and retrieve the length data frommemory for stiffness calculation.Calculate the brace element stresses due to the  current and next steps load value2121 80    Figure 4-11: Flow-chart of the MATLAB PBSC analysis program (Continued)  Start1. Input Material characteristics (e.g. SMA and Steel hysteresis)2. Input Brace element properties3. Input quasi-static loading historyRead the current point in the loading history (Load increment or decrement values only)FinishSend the stress values to the corresponding SMA and steel material models Get the tangent stiffness values  from the material models and calculate the element stiffnessesForm the global stiffness matrixCalculate the global and local deformations from the stiffness and force matrix. Modify the brace geometry based on these deformations.Last load step?YesNoHas the load direction reversed from the last step ?Yes No1. Save the length data of the previously loaded tie and deactivate it.2. Reactivate the tie currently under loading and retrieve the length data frommemory for stiffness calculation.Calculate the brace element stresses due to the  current and next steps load value2121 81  4.8 Quasi-static response from MATLAB Based on the procedure described above, a load controlled quasi-static analysis was carried out on the PBSC bracing system. Figure 4-12, Figure 4-13 and Figure 4-14 show the input loading history, deformation response and the global brace hysteresis response, respectively. With residual deformation taken into consideration, the brace shows different behavior compared to the one without residual deformation (Haque and Alam 2015a). The system shows a sliding response at the force-deformation origin. In this analysis, residual strain was 10% of the maximum plastic strain. The following section property data shown in Table 4-2 were used for this analysis. Table 4-2: Section properties used for the PBSC MATLAB model Description Length (m) Number Cross section Material Shaft 4 1 HS219x9.5 Steel Tie 1 2 in each direction 20mm diameter SMA For the SMA model, following transformation stresses were used. Austenite to martensite starting stress (400MPa), austenite to martensite finishing stress (510MPa), martensite to austenite starting stress (370MPa) and martensite to austenite finishing stress (130MPa). The modulus of elasticity of SMA was set to 62.5GPa. The plateau strain was assumed to be 0.06%, and the residual to plastic deformation ratio was set to 0.1 or 10%. Using these section properties and the loading history shown in Figure 4-12, a quasi-static analysis was carried out. Figure 4-12 shows the global brace deformation response due to this applied loading and Figure 4-13 shows the global load deformation hysteresis response of the brace. The previously mentioned sliding behavior can be observed near the origin. This is generated due to the 10% residual to plastic deformation ratio included in the SMA material model.  82   Figure 4-12: Input loading history for the MATLAB FE model  Figure 4-13: Output deformation history from the MATLAB FE model  -400-300-200-10001002003004000 5000 10000 15000 20000Axial Force (kN)Data Points-0.08-0.06-0.04-0.0200.020.040.060.080 5000 10000 15000 20000Deformation (m)Data Points 83   Figure 4-14: Hysteresis response of the PBSC bracing system from MATLAB FE model The sliding or slippage at the force-deformation origin could result in an impact loading between the piston plate and the nuts/couplers during load reversal. However, this slippage and related impact could be significantly reduced by using Belleville springs/washers between the couplers/nuts and the piston plates. 4.9 Validation The hysteresis model developed for the PBSC bracing system was validated using an ABAQUS FE model of the PBSC bracing system. As ABAQUS (2014) does not have an SMA material model capable of simulating residual strain, the model was used to test the MATLAB hysteresis model with residual deformation coefficient set to zero. The ABAQUS FE model utilized for this validation is shown in Figure 4-15; which is designed as per configuration 2 in Figure 3-1(b). -400-300-200-1000100200300400-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08Axial Force (kN)Deformation (m) 84   Figure 4-15: ABAQUS FE model of PBSC brace configuration 2. In the model mentioned above, 2-16mm dia SMA tie bars were used for energy dissipation and self-centering of the brace. The shaft was modeled as custom hollow tubular steel section with an external diameter of 44.45mm and wall thickness of 12mm. The hollow circular sleeve section has an outer diameter of 110mm and wall thickness of 15mm. The length of the shaft, the sleeve and the SMA tie bars in the model, are 1500mm, 2000mm and 1075mm respectively. The modulus of elasticity of steel was set at 200GPa and was modeled as linear elastic. The SMA material properties used for the tie bars are provided in Table A-1 of Appendix A. A deformation loading history shown in Figure 3-11 was applied to the RP-1 reference point of the brace, and a quasi-static analysis was carried out. After the analysis, the hysteresis response shown in Figure 4-16 was observed; which is similar to the hysteresis response of configuration 1 of the PBSC bracing system (Figure 3-18). For validation of the MATLAB FE model, the loading history from the ABAQUS analysis output was extracted and provided as input to the MATLAB PBSC brace model. After some minor parameter adjustment using trial and error, the following hysteresis was observed in the MATLAB model (Figure 4-16). It has been found that the MATLAB-based model  85  of the PBSC bracing system can very accurately simulate the hysteresis of the ABAQUS PBSC brace model. Furthermore, the MATLAB-based model requires few seconds of run time compared to multiple hours in ABAQUS. Therefore, the MATLAB model can be used for very rapid analysis and design of the PBSC brace.  Figure 4-16: Hysteresis Model validation using ABAQUS FE model. 4.10 Hysteresis rules Based on the observed response (Figure 4-14), a novel SMA flag hysteresis rule has been developed for simulating the behavior of PBSC bracing system. This novel flag hysteresis rule with sliding response is presented below. The following definitions should be taken into consideration for interpreting the hysteresis rules: loading refers to an increase in the absolute magnitude of the force (disregarding the sign of the force). This means that in positive force quadrants loading relates to an increase in value and vice versa. “Load reversal” refers to a situation where the loading or unloading line crosses the horizontal deformation axis, and the sign of the -150-100-50050100150200-80 -60 -40 -20 0 20 40 60 80Force (kN)Brace Deformation (mm)ABAQUSMATLAB 86  load gets reversed. Yielding refers to reaching a line with k2 stiffness passing through either the positive (dy, Fy)  or negative (-dy, -Fy) yield deformation-action point. Furthermore, Kv is a variable stiffness of any line coming from the constant ‘Fr’ force level and going towards (αFy/k1, αFy) point on the backbone curve. Refer to Figure 4-17 for understanding other parameters discussed in the following hysteresis rules. 4.10.1 In positive force direction 1. Below positive yield force level 1.1. Loading: Loading occurs if the load deformation point is on the right side or on the line with k1 stiffness and passing through (dri, 0) point. Any deformation left of this line follows zero stiffness, which results in zero force/Action.  The loading line follows k1 stiffness and loading continues at this stiffness until the loading line reaches a line with stiffness k2 passing through backbone curve’s yield deformation-action point (Fy/k1, Fy). Arriving at this line yields the system. Go to rule 2. 1.2. Unloading:  1.2.1.  If unloading from a force level higher than ‘Fr’ then the unloading line follows k1 stiffness. When the line reaches Fr force level go to rule 1.2.2. 1.2.2.  If unloading from a force level lower than or equal to ‘Fr’ but higher than the intersection point between a line with stiffness k1 passing through (dri, 0) and the unloading line with kv stiffness coming from Fr force level; the unloading line is directed towards (αFy/k1, αFy) point on the backbone curve. When the line with kv stiffness intersects the line with k1 stiffness passing through (dri,0) point then go to rule 1.2.3 1.2.3.  If unloading from a point on or below the intersection point defined in 1.2.2: The unloading line follows k1 stiffness. 1.3. Load reversal: Go to the section named “In negative force direction”. 2. After Yielding  87  2.1. The line follows the post-yield stiffness (k2) until it reaches the forward (Austenite to martensite finishing force) transformation finishing force level (Fff). After reaching Fff  go to rule 3 2.2. Unloading: Go to rule 1.2.1 3. At or above Fff force level:  3.1. Loading: The loading line follows k1 stiffness. 3.2. Unloading: Go to 1.2.1 4.10.2 In negative force direction 4. Above negative yield force level 4.1. Loading: Loading occurs if the load deformation point is on the left side or on the line with k1 stiffness and passing through (-dri, 0) point. Any deformation on the right side of this line follows zero stiffness, which results in zero force/Action.  The loading line follows k1 stiffness and loading continues at this stiffness until the loading line reaches a line with stiffness k2 passing through the negative yield deformation-action point (-Fy/k1, -Fy). Reaching this line yields the system. Go to rule 5. 4.2. Unloading:  4.2.1.  If unloading from a force level lower than ‘-Fr’, the unloading line follows k1 stiffness. When the unloading line reaches Fr go to 4.2.2. 4.2.2. If unloading from a force level higher than or equal to ‘-Fr’ but lower than the intersection point between a line with stiffness k1 passing through (-dri, 0) and the unloading line with kv stiffness coming from -Fr force level; the unloading line is directed towards (-αFy/k1, -αFy) point on the backbone curve. When the line with kv stiffness intersects the line with k1 stiffness passing through (-dri,0) point then go to rule 4.2.3  4.2.3. If unloading from a point on or above the intersection point defined in 4.2.2: The unloading line follows k1 stiffness. 4.3. Load reversal: Go to rule “In positive force direction”. 5. After Yielding  88  5.1. The line follows the post-yield stiffness (k2) until it reaches the forward (Austenite to martensite finishing force/action) transformation finishing force/action level (-Fff). After reaching Fff go to rule 6 5.2. Unloading: Go to rule 4.2.1 6. At or below -Fff force level:  6.1. Loading: The loading line follows k1 stiffness. 6.2. Unloading: Go to 4.2.1 Residual deformation values are calculated every time a deformation cycle exceeds the previous maximum deformation in the respective direction. This means that positive and negative deformations are independently calculated and they do not affect each other. The residual deformation “dr” is calculated using the following expression.  (Maximum deformation of the cycle - Yield deformation) *Residual deformation Coefficient Figure 4-17 depicts the proposed PBSC brace hysteresis model with sliding deformation. Three consecutive cycles have been shown here with loading/unloading directions marked with arrows. As this hysteresis has many rules and branch prediction equations, for the brevity of this thesis the flow chart is not given here. However, the visual basic macro code developed for generating this hysteresis model plots is provided in the appendix.  89   Figure 4-17: SMA flag-shaped hysteresis model with sliding response. 4.11 Response under quasi-static loading The hysteresis rule presented above was coded in Visual Basic for Application (VBA) for testing/debugging with various loading histories. The program was coded in deformation controlled manner; therefore, the program takes deformation history as input and produces loading history as output; which is necessary for implementation in traditional finite element analysis and design software. Three types of deformation loading histories were used for this exercise; these are Triangular, harmonic and transient. These deformation histories were carefully scaled to produce a nonlinear response, which resulted in a full cyclic response. Full hysteresis responses are very helpful in finding out anomalies/bugs in the code. At the initial stage of this hysteresis model development, the resultant hysteresis response had many outlier points. For some of the test cases, the resultant hysteresis was asymmetric for a symmetric input deformation history; which is an indication of a bug in the code. The test cases shown in Figure 4-18, Figure 4-19 and Figure 4-20 were used for detecting such bugs. After detection of these bugs, they were carefully fixed dFFydyk2k1k2k1αFydr1dr2dr3Fr1st Cycle2nd Cycle3rd Cycledri Residual deformation of ith cycleReverse transformationstarting forceForward transformationstarting forceFrFff-Fy-Fr-αFyk1k1dr1dr2dr3-FffFffkvkvkv kvkvkv 90  until the desired response was found as per the hysteresis rules laid out earlier in section 4.10. Figure 4-18 shows the link hysteresis code response under three types of triangular loading histories. Figure 4-18(a) is the most common loading protocol which starts from zero and gradually increases in magnitude in every subsequent load cycles. The hysteresis response due to this deformation loading history is shown in Figure 4-18(b). The result shows two flag-shaped hysteresis plots separated by both force and deformation axes. The hysteresis response is symmetric; which validates the input deformation history. Figure 4-18(c) shows a deformation history which starts with larger magnitude cycles and eventually decreases to zero and then again increases to the initial values. This loading history was specifically chosen to see the effect of large residual deformation in the initial cycles on the response of latter smaller magnitude cycles. The hysteresis response (Figure 4-18(d)) shows that the initial larger cycles have a profound effect on the subsequent cycles. As residual deformation in the PBSC bracing system cannot decrease, the subsequent cycles even with smaller magnitudes had same large residual deformations as the initial cycle. It can also be observed that the smaller magnitude cycles did not add any residual deformation to the initial cycles which is in agreement with experimental results by Paradis et al. (2009). They found that once SMA has strained to a large value any subsequent cycle to a strain near that value add up residual strain at a much lower rate. This additional residual strain accumulation can be safely ignored in engineering analysis. The last loading history (Figure 4-18 (e)) is saw tooth shaped with only single directional/signed deformation history. The resulting hysteresis (Figure 4-18(f)) shows no negative force even though the unloading lines touched the deformation axis at non-zero deformation points. After reaching the deformation axis, the stiffness changed to zero which resulted in zero forces.  91   Figure 4-18: PBSC brace hysteresis under different types of triangular loading Figure 4-19 (b) and Figure 4-19 (d) show the responses of the PBSC hysteresis model under harmonic excitation shown in Figure 4-19 (a) and Figure 4-19 (c), respectively. Both deformation histories are similar where the only difference is that their signs are reversed. This sign reversal was done to find bugs in the symmetric part of the code as any asymmetric response is a sign of asymmetry in the code. However, the resultant hysteresis was found satisfactory as per expectation.  92   Figure 4-19: PBSC brace hysteresis under harmonic loading with increasing magnitude Figure 4-20 (b), Figure 4-20 (d) and Figure 4-20 (f) show the PBSC hysteresis model response under irregular transient loading histories shown in Figure 4-20 (a), Figure 4-20 (c) and Figure 4-20 (e) respectively. Figure 4-20 (a) shows a highly asymmetric input deformation history and the resultant response is shown in Figure 4-20 (b). Figure 4-20 (b) shows a loading history comparable to an earthquake deformation history record. The resultant hysteresis response (Figure 4-20 (d)) is shown on the right-hand side. As the input deformation is asymmetric, the output hysteresis was also found similar. Figure 4-20 (e) shows a very unusual loading history; this sine plus step function is created using very few number of data points. The gaps between some of the points are intentionally made very large. These large gaps require multiple hysteresis branches to be predicted in advance. To create a hysteresis response for these loading histories, a branch prediction algorithm was developed inside the hysteresis code. This code can accurately predict  93  which hysteresis branch/rule to follow if there is a large gap in the input data. This prediction not only makes response calculations more accurate but also requires much less time history data points than usual. Therefore, during time history analysis, if larger time steps are used than usual, the accuracy does not get affected.  Figure 4-20: PBSC brace hysteresis under transient loading  94  To show the validity of the response obtained from the input deformation history of Figure 4-20 (e), only the points are plotted in Figure 4-20 (f). All the data points in this plot fall inside the flag shaped hysteresis. 4.12 Analysis model A six-storied tall, 4x4 bay steel braced frame was considered in this study. The bay widths are 5m, and story heights are 3m each. Therefore, the total width of the building in two orthogonal directions is 20m, and the total height is 18m. The braces are only installed on the perimeter frames of the building. To design the braces for full lateral-load arising from the seismic events, all beams were connected to the columns using moment released connections; and the columns were restrained to the foundation using hinges. Nevertheless, the columns were modeled as continuous members along their heights. The modeling was done in a way that the structure becomes unstable under lateral loading if braces are not installed. The braces were modeled using pin ended connections and were installed as inverted “V” in the middle two bays. In this configuration, only the braces will resist the lateral load arising from the earthquake. The slabs were modeled using 150mm deep concrete shell sections. However, for clarity, it is hidden from the view in Figure 4-21. The following loading was applied to the floor slabs (except the roof) in the gravity direction. Superimposed Dead Load: 2kN/m2, live load: 2.4kN/m2. On the roof, the dead load was considered as 0.5 kN/m2, and snow load was taken as 2.2 kN/m2. Furthermore, another 1.6kN/m2 on the roof was considered for miscellaneous storage and mechanical services loads. After the structural modeling, the frame was analyzed under both gravity and seismic loading. The seismic zone considered for this analysis was “Vancouver, ” and the soil class was taken as class “C.”  95   Figure 4-21: Three-dimensional model of the steel building The peak ground acceleration value for Vancouver response spectrum is 0.47g. The peak spectral acceleration values for this seismic zone is given in Table 4-3. For this building, both the importance factor and higher mode effect were taken as 1.0. As Rd and Ro are unknown for the PBSC braced frames, they were estimated based on preliminary analysis using FEMA P695 (FEMA 2009) guideline. This preliminary analysis suggested an Rd value of 6.0 and Ro value of 1.1. These values were later confirmed using detailed analysis; the details of which can be found in CHAPTER 5: Furthermore, designing this braced frame with a low Rd value (4 or below), may not induce nonlinearity in the braces; which will prevent us from utilizing the self-centering capability of this bracing system. Furthermore, if this building can resist seismic load and can also self-center after designing with a large Rd value (6.0), then the performance advantage of this novel bracing system will be confirmed.   96  Table 4-3: Spectral acceleration values for Vancouver Soil Class “C” Sa(T) Sa (0.2) Sa (0.5) Sa (1.0) Sa (2.0) Sa (4.0) Acceleration (g) 0.95 0.65 0.34 0.17 0.085 After carrying out the analysis and design using CSA S16-14, it was found that the minimum required sections for the beams and columns are W250x24 and W310x67, respectively. It was also figured out that the minimum required size for the brace for the upper three floors is HS127x4.8. For the 2nd and 3rd floor the required section is HS127x6.4, and finally, on the ground floor, the required size is HS127x8.0. As nonlinear dynamic time history analysis of three-dimensional building frames is computationally intensive, a two-dimensional model was created using the outer perimeter frame. To run the analysis only in two-dimensions, the following restraint condition was applied to the nodes. All the nodes except the foundation were restrained in Uy, Rz and Rx degrees of freedom. The foundation nodes were restrained in Ux, Uy and Uz directions only. Furthermore, the brace ends were moment released in both local orthogonal planes (My and Mz); additionally, the torsion was also released at one end. This end release condition prevents bending moment generation or transfer to and from the building frame. The previously mentioned gravity loading (Dead, Snow, and Storage) and self-weight of the slab were calculated and applied to the beams as uniformly distributed load as a seismic mass. For dead, and self-weight a multiplier of 1.0 was used; whereas, for the snow and storage loads 0.25 and 0.6 were used. The nodal restraint conditions and the gravity loading on the beams are shown Figure 4-22(a) and the uniformly distributed loading condition is shown in Figure 4-22(b).  97   Figure 4-22: (a) Nodal Restraint Conditions (b) Uniformly distributed gravity loading on the beam for the 2D model. The dashpot shapes shown at the end of the braces represents the zero length link elements. To calculate the hysteresis model’s input parameters, three different PBSC braces were designed for the three different brace sections used in this frame. In order to design the PBSC braces, the ultimate design loads which were used to design the brace sections were retrieved from the software. The envelopes of all the design load cases were taken and the PBSC brace was designed for it. The amount of required SMA needed for each brace was calculated using a spreadsheet specifically developed for this task. The PBSC brace design process used in the spreadsheet is as follows: the ultimate design load was divided using the austenite to martensite starting stress (σams) of SMA to find out the required cross-sectional area of SMA bars. For this study, the value of σams was taken as 400 N/mm2. The result gave the necessary cross-sectional area of the SMA bars. Bar diameters were selected in a way to provide an integer value or as close to that as possible. In the next step, a design length of the SMA bars was chosen. The estimated length of the SMA bars was taken as 1/6th of the brace length or approximately 1m. This ratio has been selected based on the following assumption: buildings are generally designed for a maximum interstory drift of 2%-(a) (b) 98  2.5%. As braces are diagonal members, they typically experience 40%-50% of this drift in their axial direction; which results in a drift of approximately 1%. As NiTinol based SMAs can recover from 6%-9% strain (Dolce and Cardone 2001), we can comfortably make the NiTinol bars of the PBSC brace 1/6th to 1/9th of the total brace length. This will also result in a lot of material and cost savings. The parameters mentioned above were provided as input to the MATLAB quasi-static analyzer developed for the PBSC brace, and a hysteresis was generated. The hysteresis results were used to find out the initial and post-yield stiffness as well as the SMA unloading stiffness. These values were provided as input in the S-FRAME Software link hysteresis input window. This process was repeated three times for the three brace sections, and three links were generated. Finally, these links were assigned to the appropriate brace ends. Figure 10 shows the link input parameters used for the HS127x8 brace section.  Figure 4-23: Sample input data for PBSC link hysteresis 4.13 Results The frame mentioned above was analyzed using ten historically significant earthquake records. These records are as follows: Imperial Valley, Chichi, Corralitos, Emeryville, Trinidad, Kobe, Kocaeli, Loma Prieta, Northridge, Sakaria. These records were matched with Vancouver soil class “C” response spectrum before the analysis. The spectrum matching was done using the SeismoMatch (2015) software which utilizes the wavelets algorithm developed by Abrahamson  99  (1992) and Hancock et al. (2006). The scaled spectral acceleration versus time period (Sa-T) plot of the matched records is shown in Figure 4-24.  Figure 4-24: Matched response spectra of the ten earthquake records After the spectrum matching, these ten earthquake records were used to run the nonlinear dynamic time history analysis on the braced frame. From these analyses results, the following parameters were chosen for investigation: Maximum roof drift %, Residual roof drift %, Maximum interstory drift %, and residual interstory drift %. Figure 4-25 shows the comparison between maximum roof drift % and residual roof drift % for the ten earthquake records. 00.511.522.530 1 2 3 4Spectral Acceleration, Sa (g, m/s2) Time Period (T, Sec)EQ1: NorthridgeEQ2: ChiChiEQ3: FruiliEQ4: HollisterEQ5: Imperial ValleyEQ6: KobeEQ7: KocaeliEQ8: LandersEQ9: Loma PrietaEQ10: TrinidadVancouver Spectrum (0.47g) 100   Figure 4-25: Maximum and residual roof drift % It can be seen that the maximum roof drift % was just above 0.44% for the Trinidad earthquake record. Furthermore, the maximum “residual roof drift %” was found to be 0.014% for the Kocaeli earthquake record. For the 18m tall building frame, this equates to a value of 2.5mm. This low residual roof drift ratio can be attributed to the efficacy of this novel self-centering bracing system. Figure 4-26(a) and (b) compares the maximum and residual interstory drift ratios for the ten earthquake records. The maximum interstory drift ratios for all ten earthquake records are very close to one another except the Trinidad earthquake. The maximum interstory drift ratios were observed mostly in the 3m and 12m level, which are the 1st and 4th floor of the building respectively. 0.0000.1000.2000.3000.4000.5000.600Drift %Earthquake RecordMaximum Roof Drift %Residual Roof Drift % 101   Figure 4-26: (a) Maximum interstory Drift % (b) Residual interstory Drift % The maximum value is 0.59% which is around 1/4th of the code specified limit of 2.5% for the “Other Building” category. Furthermore, the average value was found to be around 0.4% only. On the other hand, the residual interstory drift values were found to vary more significantly between the earthquake records. The range is from almost zero to 0.02%, which is a much wider range (w.r.t max to min ratio) compared to the maximum interstory drift ratios. Like the previous plot, the maximum values are mostly observed at 3m and 12m level (First floor and fourth floor respectively). The average of the maximum was found to be around 0.005%. This value is much lower than traditional BRB frames which experience on an average 0.3% residual interstory drift ratio (Zhu and Zhang 2007). Figure 4-27 shows the brace hysteresis for the Trinidad earthquake record. Six figures from (a) to (f) represents brace hysteresis from first to the sixth floor. It can be observed that the nonlinearity is highest on the first floor and it gradually becomes linear at the 5th and 6th floor. It can also be noted that for this low plastic deformation, the residual deformation is insignificant. Therefore, if the braces do not significantly deform in the nonlinear range, the residual deformation can be safely ignored. 024681012141618200 0.01 0.02 0.03Building Height (m)Residual Interstorey Drift (%)TrinidadSakariaNorthRidgeLandersKocaeliKobeImperial ValleyHollisterFruiliChiChiAverage024681012141618200 0.2 0.4 0.6 0.8 1Building Height (m)Maximum Interstorey Drift (%)TrinidadSakariaNorthRidgeLandersKocaeliKobeImperial ValleyHollisterFruiliChiChiAverage(a) (b)Residual Interstory Drif ( )i  y Drif (%) 102   Figure 4-27: Brace hysteresis for Trinidad earthquake (a) First story (b)Second story (c)Third story (d)Fourth story (e) Fifth story and (f) Sixth story 4.14 Summary A tension only uniaxial mechanical SE SMA material model has been developed with residual deformation simulation capability. Next, it was utilized in a custom-built MATLAB finite element solver developed for the PBSC bracing system. Quasi-static input loading history was used, and the resultant hysteresis was found to exhibit sliding behavior. The resultant hysteresis was analyzed, tested and a novel flag-shaped hysteresis with sliding response was developed. This hysteresis model was integrated into the S-FRAME Structural analysis and design software. A six-storied steel building utilizing the PBSC bracing system was designed using Vancouver soil class ‘C’ response spectrum. A braced frame of the building was modeled in two dimensions, and it was subjected to ten earthquake records scaled to Vancouver spectrum. Nonlinear dynamic time history analyses were carried out on the braced frame, and nodal responses were captured. Next, maximum and residual interstory drift ratios were calculated. The maximum interstory drift ratios were found well below the code specified limit. The residual interstory drift ratio was found to be insignificant -300-200-1000100200300-0.015 -0.005 0.005 0.015Axial Force (kN)Deformation (m)-150-100-50050100150200-0.015 -0.005 0.005 0.015Axial Force (kN)Deformation (m)-150-100-50050100150200-0.015 -0.005 0.005 0.015Axial Force (kN)Deformation (m)-100-50050100150-0.015-0.01-0.005 0 0.005 0.01 0.015Axial Force (kN)Deformation (m)-100-50050100150-0.01 -0.005 0 0.005 0.01Axial Force (kN)Deformation (m)-60-40-20020406080-0.005 -0.0025 0 0.0025 0.005Axial Force (kN)Deformation (m)(a) (b) (c)(f)(e)(d) 103  (below 0.05%). Furthermore, a similar trend was observed for the maximum and residual roof drift ratios. Based on the observed response, it can be concluded that the PBSC bracing system is excellent in controlling residual deformation in buildings. This implies that a structure fitted with this bracing system will be capable of self-centering after an earthquake.  104  CHAPTER 5: SEISMIC RESPONSE MODIFICATION FACTORS FOR THE PBSC BRACING SYSTEM 5.1 General The PBSC bracing system is a novel seismic force resisting system developed by the author. Therefore, recommended values of seismic response modification factors are not available in the code for this system. Hence, it is necessary to determine these key parameters which will enable the designers to analyze and design these structures against seismic loading. There are two essential parameters which are needed for designing building frames against seismic loading as per the National Building Code of Canada (NBCC 2010). These factors are known as the Overstrength (Ro) and the ductility related force reduction factors (Rd). Together, they are also known as the response modification factor (R).  In this chapter the seismic response modification factor of steel frames equipped with piston based self-centering SMA bracing system is evaluated using the method outlined in FEMA P695 (FEMA 2009) guideline. The evaluation is done in terms of collapse margin ratio (CMR), which is defined as the ratio between the median collapse earthquake intensity to the maximum considered earthquake intensity at the fundamental period of the building. For this study, four, six and eight storied 4-bay wide PBSC braced frame models are considered. These frames were designed using relevant code standard (NBCC 2010) and trial values of response modification factors (Rd and Ro). Frame elements (e.g. columns and beams) were designed using CSA S16-09 (CSA 2009) standard. The buildings were assumed to be in a high seismic zone in western Canada (Vancouver). Later, this trial value was examined and evaluated using the collapse margin ratio defined in FEMA P695 (FEMA 2009). In order develop the archetypes needed for the evaluation using collapse margin ratio, nonlinear building frame  105  models were built in a state-of-the-art finite element analysis and design software called S-FRAME. Nonlinear hysteresis behavior of these braces was modeled using the SMA flag hysteresis model with sliding response developed in chapter 4. A set of 44 near-field ground motion records from 22 individual seismic events provided in the FEMA P695 (FEMA 2009) document was used to perform incremental dynamic time history analysis (IDA) on the building frames mentioned above. Besides IDA, nonlinear static pushover analysis was also carried out for the determination of seismic overstrength factor. From this analysis, median collapse earthquake intensities were determined, and these values were used to calculate the collapse margin ratio. These collapse margin ratios were adjusted for spectral shape factor and period based ductilities of these frames. The adjusted collapse margin ratios were compared against acceptable limits provided in FEMA P695 guideline. If the values of these adjusted collapse margin ratio fell outside the acceptable range, new Rd and Ro values were selected, and the whole procedure was repeated until they were found acceptable. After multiple trials, the results from all the archetypes showed that the seismic performance factors calculated in this study satisfy the acceptance criteria. Designing PBSC braced buildings using these factors could provide an adequate margin against collapse under the maximum earthquake hazard defined in the code (NBCC 2010). The scope of this study is limited to only four, six and eight storied PBSC braced frames. Furthermore, this study mostly focuses on the seismicity of Western Canada. Therefore, response modification factors determined in this study should only be used for the short and medium tall building in Western Canada. For much taller or shorter buildings, these factors have to be reinvestigated. Furthermore, for a different geographic region with different seismicity, these factors need to be recalculated by taking these regional factors into consideration.  106  5.2 Force based design procedure The current force-based design procedure outlined in the NBCC (2010) permits the designer to reduce the elastic seismic base shear force demand by Rd*Ro factor (Lee et al. 1999). This reduction in the design base shear force rests on the observation that a well-detailed structure is capable of resisting base shear force higher than the design value (Ro) and can also experience significant deformation without collapse (Rd) (Kim and Choi 2005). These factors significantly reduce the design base shear force. The designer then only needs to perform linear elastic (static or dynamic) analysis using this reduced base shear force. A structure thus designed is expected to deform inelastically to dissipate seismic energy during a design level seismic event. The structural components (e.g., beams and columns) that are designed for this lower force would experience much higher deformation. This method eliminates the need for the designer to perform nonlinear analysis on their building models. Due to this simplicity, Rd and Ro are the most important parameters in force based seismic design. Canadian building design code (NBCC 2010) provides empirical values of Rd and Ro for many different structural systems such as reinforced concrete moment resisting frames, steel moment resisting frames, shear walls and braced frames etc. The code provides the following equation for calculating the design base shear force. 𝑉𝑑 =𝑆(𝑇𝑎)𝑀𝑣 𝐼𝐸𝑊𝑅𝑑𝑅𝑜               (5.1) Here, S(Ta) = Spectral acceleration demand Mv = Factor to account for higher mode effects IE = Structural importance factor W = Seismic weight of the structure  107  Finally, Vd is the design base shear The numerator in equation 1 represents the elastic base shear demand or VE. It consists of spectral acceleration demand or S(Ta). This value comes from the design response spectra of the seismic zone and the fundamental period of the structure to be designed. The next parameter is Mv which takes higher mode effect of the structure into account. Recommended values for this parameter are provided in Table 4.1.8.11 of  NBCC (2010). The value of Mv ranges from 1.0 to 2.5. Another important parameter is called the importance factor (IE). The more important the structure (e.g. Post-disaster, hospitals or schools), the higher the value of this parameter. The value of IE ranges from 0.8 (low importance) to 1.5 (post-disaster). This parameter ensures that important structures are designed with higher base shear force compared to less important ones; which is expected to increase the safety of important structures against a seismic hazard.  The last parameter in the numerator is “W” or the seismic weight of the structure. All weights which contribute to the seismic mass of the structure are considered; this includes dead load, parts of the storage and the snow load and full content of any tank supported by the structure. These parameters in the numerator result in the elastic base shear. The response modification factors, “Rd” and “Ro” are in the denominator; therefore, the higher the value of these factors, the lower the value of design base shear force, Vd. Therefore, higher values of these factors result in a much more economical design. Figure 5-1 depicts the definition of Rd and Ro using an example pushover plot of a building. This example pushover curve shows the lateral strength of a structure in terms of base shear force and roof top displacement. An idealized version of this pushover curve is overlaid on top of the actual pushover curve. The idealization is done to find out the yield strength and yield displacement of the system, which are necessary for the Rd and Ro factor calculation. This idealized curve  108  represents the structures lateral strength in an elastic-perfectly plastic manner. FEMA 356 (FEMA 2000) recommendations are followed for this idealization process. In Figure 5-1, Vy is the yield base shear of the structure. As it is hard to locate the exact yield point on a curved surface (actual backbone curve), the idealized base shear vs. roof displacement curve was used to estimate the yield base shear. The corresponding displacement value is the yield displacement (Δy). The ratio of these two parameters (Vy/Δy) represents the initial stiffness of the structure (Ki). Vd represents the design strength of the structure. This is the base shear force the structure was designed to withstand. Due to structural redundancy, safety factors and overdesign (e.g. using same beam or column sections on every floor), building structures yield at a much higher load than their design strength, which is known as the overstrength. The value of the overstrength factor is calculated by dividing the Vy with Vd. Equation 5.2 shows the formula for calculating the overstrength factor (Ro). 𝑅𝑜 =𝑉𝑦𝑉𝑑                 (5.2)  109   Figure 5-1: Relationships between force reduction (Rd), overstrength (Ro), response modification factor (R) and displacement ductility (µ) Ductility related force reduction factor is another crucial parameter in seismic design. If a structure is designed to remain fully elastic under seismic loading, then the structure will exhibit a different response compared to the actual nonlinear response. The relationship between the base shear force and the roof displacement will remain linear. It was observed that for a certain structural period range an elastic structures displacement response under dynamic excitation is almost same as that of a nonlinear structure; also known as the equal displacement approximation (Newmark and Hall 1982). However, the author observed that this does not hold true for a short period (stiff) structures. Therefore, the linear elastic structure, if flexible enough will experience almost same displacement under the same earthquake event; this will, in turn, result in a much higher base shear force. We can call this base shear (VE) or the elastic base shear demand. If we divide this fictitious elastic base shear force (VE) with the yield base shear force (Vy), we get the Top Displacement (Δ)Elastic StrengthYieldStrengthDesign StrengthVdVEVyBase Shear Froce (V)IdealizedResponseΔy ΔmaxΔmax=μ.ΔyActual Capacity EnvelopeVy=Ro.VdVE=Rd.VyVE=R.VdElastic region Inelastic regionDuctilityOverstrengthKi1 110  ductility related force reduction factor (Rd). On the other hand, the ratio between the elastic base shear force (VE) and the design base shear force (Vd) is known as the response modification factor (R). Equation 5.3 and Equation 5.4 shows the formula for Rd and R. As per Kim and Choi (2004), a greater value of ‘R’ indicates higher energy dissipation capacity. 𝑅𝑑 =𝑉𝐸𝑉𝑌                       (5.3) 𝑅 = 𝑅𝑑𝑅𝑜 =𝑉𝐸𝑉𝑑                      (5.4) Another important parameter is the ductility (µ); defined as the ratio between the maximum displacement (Δmax) and the yield displacement (Δy) of the structure under consideration. Similar to the previous parameters, these are also calculated from the idealized bilinear representation of the structure’s backbone curve. 𝜇 =𝛥𝑚𝑎𝑥𝛥𝑌                 (5.5) 5.3 Necessity of this study Nowadays, the new seismic force resisting systems are getting introduced in every code update cycle. The capability of these new systems in resisting seismic load is not clear; therefore, it is vital to quantify the seismic performance of these new systems (FEMA 2009). As PBSC is a novel bracing system and there is no rational seismic design guideline for it, it is of utmost importance that the seismic performance factors be determined for structures equipped with this bracing system.  Use of SMA based self-centering bracing system in buildings has been studied in the past by many researchers (Auricchio et al. 2006; Moradi et al. 2014; Zhang and Zhu 2007). These studies found that utilizing shape memory alloy in the bracing system could decrease the residual damage of the buildings. However, few of them came up with a proper design method or designing  111  structures with these bracing systems. As discussed earlier, Rd and Ro are two essential parameters in force based seismic design of structures. The code provides empirical values for different types of structural force resisting systems. Most of these factors were proposed based on observations or empirical studies. The proposed factors are mostly conservative and result in overdesign. Recently, studies were carried out to investigate the adequacy or propose values of response modification factors for different types of structures like cold-formed steel special bolted moment frames (Sato and Uang 2013), steel special moment-frame buildings (Miyamoto et al. 2011; Zareian et al. 2010), concrete buildings reinforced with superelastic shape memory alloy rebars (Hossain et al. 2015), cold-formed steel strap braced walls (Comeau et al. 2010), self-centering concentrically braced frames (Tahmasebi et al. 2014) and seismic-resistant friction damped braced frame system with buckling restrained columns (Blebo and Roke 2015). All of these studies were carried out using FEMA P695 (FEMA 2009) guideline which provides a rational method for calculating response modification factor for new structural systems or evaluating the code proposed values for well-established structural systems. NBCC (2010) requires that buildings designed using code specified seismic hazard level should have a very low probability of collapse thus preventing loss of lives. However, it is not well defined on how to achieve this collapse safety as experimental validation would require large-scale test setup which could test up to global collapse limit (Haselton et al. 2010). Researcher and practitioners nowadays tend to quantify the performance criteria with which they design the structures (Ghobarah 2001).  Luckily, due to advancement performance-based seismic design tools and technologies it is now easier than ever to determine the collapse safety of various structural systems with moderate to high accuracy using nonlinear collapse simulation techniques on a probabilistic basis. In this chapter, collapse margin ratio (CMR) of 4,6 and 8 storied PBSC braced  112  frames will be calculated using the FEMA P695 methodology to determine the required seismic response modification factors. 5.4 Description of the FEMA P695 methodology The FEMA P695 guideline offers a rational method for determining building response parameters and system performance. If the method is properly implemented in the design process for various seismic force resisting systems, during a seismic event, it will provide equivalent safety against collapse in contrast to inherent safety against collapse intended by the current seismic design codes (FEMA 2009). FEMA P695 utilizes collapse margin ratio (CMR) for determining building seismic performance factors; which is defined in the guideline (FEMA 2009) as the ratio between median spectral acceleration (5% damped) at the collapse level ground motion, SCT (or corresponding displacement, SDCT) to the spectral acceleration of the maximum considered earthquake (MCE) ground motion (5% damped), SMT (or corresponding displacement, SDMT), at the fundamental period of the seismic force resisting system. 𝐶𝑀𝑅 =𝑆𝐶𝑇𝑆𝑀𝑇=𝑆𝐷𝐶𝑇𝑆𝐷𝑀𝑇                         (5.6)  Another definition of the collapse margin ratio (CMR) could be given as follows: CMR is the amount by which SMT must be increased to achieve building collapse by half (50%) of the earthquake ground motion records in a record set suggested by FEMA P695. Collapse margin ratio depends on the following factors: uncertainty in analysis, design, construction and ground motion variability. These factors are collectively considered in a collapse fragility curve. This curve depicts the collapse probability of the seismic-force-resisting system with respect to the ground motion intensity (FEMA 2009).  113  A flow diagram of the FEMA P695 based seismic performance factor determination method is shown in Figure 5-2. The method starts with designing model buildings using a trial value response modification factor, R or the multiplication of Rd and Ro as per Canadian building design code (NBCC 2010). After the design of the building, nonlinear archetype models are created using hysteresis response of the primary force resisting elements, such as beams, columns, braces, and connections. The hysteresis response and the primary force-deformation backbone curves can be determined using experimental investigation, continuum nonlinear 3D finite element analysis or fiber modeling. Once the building archetype models are created, pushover analysis is carried out for the determination of ductility and overstrength factor. In the next step, 22 far-field (recorded at sites located more than 10km from the fault rupture) ground motions record pairs are normalized with respect to the peak ground velocity of the record set and scaled to the design response spectrum. However, scaling to the design response spectrum can be skipped as during analysis the record group is scaled with several scaling factors to generate the IDA curves. In the next step, the selected ground motion records are collectively scaled and nonlinear dynamic time history analysis is carried out on the building archetypes. This process is repeated until median collapse is achieved; which is the intensity at which half of the records in the set causes collapse to the building model. Collapse is determined by dynamic instability or excessive lateral displacements. This process although similar but differs from the incremental dynamic analysis concept proposed by Vamvatsikos et al. (2002). In the next step, a collapse fragility curve is defined from the collapse data obtained from the IDA results using cumulative distribution function. This plot relates the probability of collapse to the ground motion intensity (Ibarra et al. 2002).   114    Figure 5-2: Determination of seismic performance factors (FEMA 2009) This median collapse ground motion intensity is denoted as SCT; whereas, the intensity of maximum considered earthquake spectral acceleration at the fundamental period of the building is denoted as SMT. Maximum considered earthquake is defined as the intensity of an earthquake event Development of nonlinear archetype modelsCalculation of ductility (μ) and overstrength (Ω) for the system using pushover analysisCalculation of median collapse intensity (SCT) for incremental dynamic time history analysisCalculation of collapse margin ratio (CMR) by dividing SCT with SMTCalculation of adjusted collapse margin ratio (adjusted CMR)Determination of acceptable values of adjusted collapse margin ratioACMR > Acceptable value ?Calculation of deflection amplification factorSelect new trial values for Rd and RoDesign of building models using trial values of Rdand RoResults documentation and peer reviewYesNo 115  with 2% probability of exceedance in 50 years. As per Poisson’s distribution, the return period of such an earthquake is once in 2475 years. In ASCE 7-05 (ASCE 2005), the design response spectrum is taken as 2/3rd of the MCE response spectrum. However, in Canadian building code (NBCC 2010), 2% in 50 years earthquake hazard level is used for design. However, for any structural force resisting system having an Rd value higher than or equal to 1.5, the code limits the maximum spectral acceleration for design to 2/3rd of Sa (0.2) or 2/3rd of the maximum value of the response spectrum. Therefore, these two design standards are very similar to each other in this regard. Hence, FEMA P695(FEMA 2009) method is applicable for evaluating building seismic performance factors in Canadian building design context.  Figure 5-3: Illustrations of seismic performance factors (adapted from FEMA (2009)) After the determination of SCT, these values are divided by SMT to get the value of collapse margin ratio (CMR). In the next step, the calculated value of CMR is adjusted for the following Spectral Displacement (SD)Collapse Level Ground MotionsCsSCTSmaxSpectral Acceleration (g)SDMTΩSMTSDMT/1.5RCMR1.5RMCE Ground MotionsSDCT1.5Cd CMR 116  factors: effects of spectral shape, record to record variability, design requirement uncertainty (test data and modeling uncertainty), etc. This adjusted value is known as the adjusted collapse margin ratio (ACMR). After the adjustment, the ACMR value is checked against acceptable limits defined in the guideline. If the ACMR value is higher than the acceptable CMR value, then the trial value of “R” or “RdRo” is accepted. On the other hand, if the ACMR value is not found satisfactory, then the whole evaluation process is repeated with a new trial value of “R” or “RdRo”. 5.5 Design and modeling of the PBSC braced frames Steel braced frames with three different heights (four, six and eight storied) and same layout plans were selected for this study. These buildings were designed using seismic design procedures of National Building Code of Canada (NBCC 2010). The site for these buildings was chosen as the coastal region of the Vancouver city. The maximum spectral acceleration of the design response spectra is 0.95g at the constant acceleration zone. Figure 5-4 illustrates the response spectra chosen for the design.  Figure 5-4: Vancouver soil class “C” response spectrum  The importance factor for these buildings was taken as 1.0 as these are considered “residential constructions.” As regular plans and elevations were chosen, the higher mode effect was also taken 00.10.20.30.40.50.60.70.80.910 1 2 3 4Spectral Acceleration, (g)Time Period (Sec) 117  as 1.0. For soil class “C,” the Fa and Fv values were taken as 1.0. In the first trial, the Rd and Ro values were selected as 8.0 and 1.3 respectively. However, after some preliminary analysis, these values were found to be outside the acceptable limit provided in the FEMA P695 guideline. Therefore, in the second trial, Rd and Ro values were taken as 6.0 and 1.1 respectively. A well-known structural analysis and design software called S-FRAME V11.2.6 (S-FRAME 2016) was used for the seismic design and section size determination for these buildings. The beams and braces were modeled with pin ended connections which prevented them from resisting or transferring end moments. The column supports were also modeled using hinges. This modeling procedure ensures that the braced frames resist lateral forces only through the axial action of the braces. This was done to eliminate lateral strength contribution from elements other than the braces to understand the seismic performance of the structure solely due to this novel bracing system. As PBSC braces have two different materials in the system (Steel and SMA), it is difficult to calculate the brace stiffness before the final design before knowing the length of each element. Furthermore, commercially available structural analysis and design software such as S-FRAME, SAP2000, ETABS, STAAD Pro do not have SMA material built into them. However, the elastic properties could be provided as input for building the model necessary for design calculations. However, modeling such a way sometimes hamper the automatic steel design process. To prevent this, the braces were modeled using standard hollow circular steel sections without the SMA ties. However, as the steel shaft and the SMA ties are arranged in a series the actual stiffness of the PBSC brace (shaft and SMA ties in a series) would be less than a whole section made of steel shaft only. To match the expected stiffness, the PBSC brace stiffness was separately calculated and it was provided as a section property modifier in the S-FRAME Software. For this design, it was found that an axial stiffness modifier of 0.69 or 31% reduction was appropriate. The detailed calculation  118  can be found in Appendix A. Using this stiffness modifier and modeling technique described above, 4, 6 and 8-storied three-dimensional steel building frame models were built in S-FRAME Software. Isometric views of these building frames are shown in Figure 5-5. The slabs are hidden from this view for clarity.  Figure 5-5: Three-dimensional model of the (a) four (b) six and (c) eight-storied PBSC building After the seismic design was done and the section sizes were determined (Figure 5-6), the following steps were carried out. As nonlinear incremental dynamic analysis procedure is very computationally demanding, two-dimensional models were used instead of the three-dimensional ones. Individual perimeter braced frames from each of these three building models were taken for performance evaluation using nonlinear analysis. The nonlinear behavior of these braces was modeled using zero-length link elements positioned at the end of the braces. A detailed PBSC brace design procedure used for this analysis is given in Appendix A. As nonlinearity of the SMA bars is directly taken into consideration by the link elements, the stiffness of the shaft needs to be adjusted to match the actual PBSC bracing system which the shaft and link combination in the model is simulating. As the SMA bars are represented by zero-length elements, the shaft had to be (a) (b) (c) 119  modeled longer than its actual designed size. This increased length decreases the axial stiffness of the brace by a factor of (Li/La), where Li is the increased length and La is the designed length. Therefore, in the nonlinear model, the axial stiffness was modified by a factor equal to the inverse of the previous factor, or (La/Li). In the final step, the nonlinear hysteretic parameters of the PBSC braces were calculated using the method given in Appendix A.  Figure 5-6: The designed six-storied PBSC frame showing the required section sizes 5.1 Incremental dynamic time history analysis The incremental dynamic analysis of the index archetype is a two-step process. In the first step, ground motion records were selected, normalized and scaled. In the next step, several hundreds of nonlinear dynamic time history analyses are carried out by scaling these records. This two-step procedure is discussed in detail below.  120  5.1.1 Ground motion record selection and scaling The FEMA P695 methodology provides two sets of ground motion records for the collapse assessment of the building archetypes using incremental dynamic time history analysis. One record set contains 22 far-field record pairs (Table 5-1) and another set containing 28 near-field (recorded at sites located less than 10km from the fault rupture) record pairs. However, only far-field record set is necessary for collapse evaluation; the near-field record set is provided as supplemental information. If required, this record set could be used to examine issues arising from near-fault directivity effects; which is done for practical reasons since there are numerous issues in the characterization of near-field hazards as well as ground motion effects. On the other hand, for this analysis, only two horizontal orthogonally separated records are used from each event. The vertical component of the excitation is ignored as this direction is not considered of vital importance for collapse assessment.   121  Table 5-1: Twenty-two far-field earthquake record details for collapse evaluation Event ID Mw Event Name Year Recording Station Fault Type Epicentral Distance PGAmax (g) PGVmax (cm/s) 1 6.7 Northridge 1994 Beverly Hills Thrust 13.3 0.52 63 2 6.7 Northridge 1994 Canyon Country Thrust 26.5 0.48 45 3 7.1 Duzce, Turkey 1999 Bolu Strike-Slip 41.3 0.82 62 4 7.1 Hector Mine 1999 Hector Strike-Slip 26.5 0.34 42 5 6.5 Imperial Valley 1979 Delta Strike-Slip 33.7 0.35 33 6 6.5 Imperial Valley 1979 El Centro Array #11 Strike-Slip 29.4 0.38 42 7 6.9 Kobe, Japan 1995 Nishi-Akashi Strike-Slip 8.7 0.51 37 8 6.9 Kobe, Japan 1995 Shin-Osaka Strike-Slip 46 0.24 38 9 7.5 Kocaeli, Turkey 1999 Duzce Strike-Slip 98.2 0.36 59 10 7.5 Kocaeli, Turkey 1999 Arcelik Strike-Slip 53.7 0.22 40 11 7.3 Landers 1992 Yermo Fire Station Strike-Slip 86 0.24 52 12 7.3 Landers 1992 Coolwater Strike-Slip 82.1 0.42 42 13 6.9 Loma Prieta 1989 Capitola Strike-Slip 9.8 0.53 35 14 6.9 Loma Prieta 1989 Gilroy Array #3 Strike-Slip 31.4 0.56 45 15 7.4 Manjil, Iran 1990 Abbar Strike-Slip 40.4 0.51 54 16 6.5 Superstition Hills 1987 El Centro Imp. Co. Strike-Slip 35.8 0.36 46 17 6.5 Superstition Hills 1987 Poe Road (temp) Strike-Slip 11.2 0.45 36 18 7.0 Cape Mendocino 1992 Rio Dell Overpass Thrust 22.7 0.55 44 19 7.6 Chi-Chi, Taiwan 1999 CHY101 Thrust 32 0.44 115 20 7.6 Chi-Chi, Taiwan 1999 TCU045 Thrust 77.5 0.51 39 21 6.6 San Fernando 1971 LA-Hollywood Stor Thrust 39.5 0.21 19 22 6.5 Fruili, Italy 1976 Tolmezzo Thrust 20.2 0.35 31 The near-field record set contains all high-magnitude events (PGA > 0.2g, PGV > 0.15 m/sec and Mw > 6.5) from the Pacific Earthquake Engineering Research Center (PEER) Next-Generation Attenuation (NGA) database. Large-magnitude earthquakes usually have a long duration and dominate collapse risk, which is important for collapse evaluation of nonlinear material or element hysteresis with gradual strength degradation. The record set includes event records from soft rock and stiff soil sites. The records are from shallow crustal sources, mostly thrust or strike-slip faults (FEMA 2009). The records were selected in a way to encounter some conflicting objectives. From a single earthquake event, no more than two strongest records were chosen to avoid event bias. The record  122  set includes an adequate number of earthquake records to allow the calculation of record to record variability (RTR) and median collapse intensity (SCT).  However, an explicit calculation for RTR is not required; as an alternative, previous development studies and research is used to estimate values of RTR, and this is built in the computation process of the total system collapse uncertainty (FEMA 2009). The main objective of the far-field record set is to deliver a fully defined ground motion record set for collapse evaluation of all seismic design categories located in any seismic hazard zone and founded on any soil condition in a consistent manner. All records in the set are from real seismic events, no synthetic record was used, based on the assumption that regional variation of ground motions is not be considered (FEMA 2009). These records were scaled to represent a specific intensity; usually the collapse intensity of the index archetype under consideration. There are two major steps in the ground motion record scaling process. In the first step, the median peak ground velocity (PGV) of the record set was determined. In the next step, this median value was divided by the PGV of each event. This ratio is known as the normalization factor for the ith record (NMi). Later, these normalization factors were multiplied with the corresponding ground motion record data to calculate the normalized records. The normalization factors for all 22 far-field events are provided in Table 5-2. This normalization process removed the unwarranted inconsistency between records due to intrinsic differences in source type, source distance, site conditions and event magnitude. However, this process does not affect record to record variability.   123  Table 5-2: Normalization factor for far-field record set Event ID Comp. 1 Sa (g) Comp. 2 Sa (g) PGVpeer (cm/s) Normalization factor Normalized PGAmax (g) Normalized PGVmax 1 1.02 0.94 57.2 0.65 0.34 41 2 0.38 0.63 44.8 0.83 0.4 38 3 0.72 1.16 59.2 0.63 0.52 39 4 0.35 0.37 34.1 1.09 0.37 46 5 0.26 0.48 28.4 1.31 0.46 43 6 0.24 0.23 36.7 1.01 0.39 43 7 0.31 0.29 36 1.03 0.53 39 8 0.33 0.23 33.9 1.1 0.26 42 9 0.43 0.61 54.1 0.69 0.25 41 10 0.11 0.11 27.4 1.36 0.3 54 11 0.5 0.33 37.7 0.99 0.24 51 12 0.2 0.36 32.4 1.15 0.48 49 13 0.46 0.28 34.2 1.09 0.58 38 14 0.27 0.38 42.3 0.88 0.49 39 15 0.35 0.54 47.3 0.79 0.4 43 16 0.31 0.25 42.8 0.87 0.31 40 17 0.33 0.34 31.7 1.17 0.53 42 18 0.54 0.39 45.4 0.82 0.45 36 19 0.49 0.95 90.7 0.41 0.18 47 20 0.3 0.43 38.8 0.96 0.49 38 21 0.25 0.15 17.8 2.1 0.44 40 22 0.25 0.3 25.9 1.44 0.5 44 In the final step, as per FEMA P695 guideline, the normalized ground motion records were supposed to be collectively scaled (anchored) to a specific ground motion (response spectra of the seismic zone under consideration) intensity so that the median spectral acceleration of the set matches the MCE spectral acceleration at the fundamental period of the index archetype. However, this last step (anchoring) is not mandatory as the evaluation procedure requires scaling the records up until 50% of the records causes collapse. Therefore any initial scaling factor other than the anchoring factor also works for the evaluation. Figure 5-7 shows the normalized mean spectra of the unscaled far-field ground motion record set compared to the Vancouver 2% in 50-year soil class “C” response spectra.  124    Figure 5-7: FEMA P695 normalized mean spectra compared to the Vancouver soil class “C” response spectra 5.1.2 IDA results After the far-field records had been normalized and scaled, nonlinear dynamic time history analyses were carried out starting from a small-scale factor (around 0.5). This scaling factor was incremented by 0.5 for subsequent analysis until median collapse was observed. The collapse of the PBSC index archetype was set at 6.8% elongation of the NiTinol bars. This elongation corresponds to the austenite to martensite finishing stress for the specific NiTinol bars. After this strain, NiTinol bar fully transforms to martensite. At this phase, the stiffness of the NiTinol bar increases to the initial stiffness thus attracting significantly higher forces. Based on the design assumption that the connections of the PBSC brace to the frame corners will not be designed for stresses greater than this, it is expected that the connections will fail at this stress and will result in a side sway collapse for the frame. As NiTinol bars are only 1m long as per the design (Appendix A), which is approximately one-sixth length of the brace, it resulted in 2.66% interstory drift ratio for the frame.  125  Following settings were chosen in the S-FRAME Software for the nonlinear dynamic time history analysis. For numerical integration “Newmark” method has been selected with α and δ values 0.25 and 0.5 respectively. For damping coefficient calculation, “Rayleigh” damping method was chosen with 5% damping for both stiffness and mass proportional coefficients. For Rayleigh damping matrix formation, both mass and stiffness matrix was formed using initial geometry. The seismic mass was formed from the following formula as per FEMA (2009). Seismic mass = 1.05*Dead Load + 0.25 Live Load                (5.1) The resulting IDA curves (Figure 5-8(a)-(c)) show the incremental dynamic analysis results of the four, six and eight storied PBSC index archetypes. The IDA curves have a distinct yield zones near 1% interstory drift ratio, after which the response becomes much flatter. However, around 2.6%-2.8% the curves again stiffen up due to martensite phase stiffening. After this point, the curves seem to stiffen up and attract a significant amount of spectral acceleration. From the calculation presented in Appendix A, it can be understood that at 2.66% interstory drift ratio the NiTinol bars in the PBSC braces exceed Austenite to martensite finishing stress and starts to stiffen up. The effect of the brace stiffening is clearly visible in Figure 5-8 near the above mentioned interstory drift ratio percentage.  126   Figure 5-8: Spectral acceleration vs maximum interstory drift ratio IDA curves for (a) 4-storied (b) 6-storied and (c) 8-storied PBSC braced frame 01234560 1 2 3 4 5 6 7Spectral Acceleration, Sa(g)Maximum Interstory Drif Ratio (%)0123450 1 2 3 4 5 6 7Spectral Acceleration, Sa(g)Maximum Interstory Drif Ratio (%)012340 1 2 3 4 5 6 7Spectral Acceleration, Sa(g)Maximum Interstory Drif Ratio (%)(a)(b)(c)1-0091-2792-0002-2703-0003-0904-0004-0905-2625-3526-1406-2307-0007-0908-0008-0909-1809-27010-00010-09011-27011-36012-LN12-TR13-00013-09014-00014-09015-L15-T16-00016-09017-27017-36018-27018-36019-E19-N20-00020-09021-09021-18022-00022-270 127  5.2 Collapse margin ratio calculation The collapse margin ratio for the individual index archetypes was calculated using the following method. From the individual spectral acceleration (Sa) vs. interstory drift ratio IDA curves, the failure spectral acceleration (Sa) values were determined at 2.66% interstory drift ratio. These failure Sa values were used to plot cumulative distribution function/curves (Figure 5-9). From these curves, the Sa values at 50% cumulative probability were calculated, which is known as the median collapse intensity (SCT) at 5% damping. From the period of individual PBSC index archetypes and Vancouver soil class “C” response spectrum, the MCE spectral acceleration SMT was found. The ratio of SCT/SMT was calculated and presented in Table 5-3. This ratio represents the collapse margin ratio of the PBSC index archetypes.  128   Figure 5-9: Cumulative distribution function of collapse level spectral acceleration values for (a) 4-storied (b) 6-storied and (c) 8-storied PBSC braced frame 00.10.20.30.40.50.60.70.80.911.10 1 2 3 4 5Cumulative ProbabilitySpectral Acceleration, Sa (g)00.10.20.30.40.50.60.70.80.911.10 0.5 1 1.5 2 2.5 3Cumulative ProbabilitySpectral Acceleration, Sa (g)0.00.10.20.30.40.50.60.70.80.91.01.10 1 2 3 4 5 6Cumulative Probability Spectral Acceleration, Sa (g)(a)(b)(c) 129  From Table 5-3, the collapse margin ratios for the three archetypes ranges from 3.50-3.67; which are high ratios indicating the structures have a very low probability of collapse under MCE spectral acceleration. However, adjustment of these values is required for final judgment. Table 5-3: Calculation of median collapse intensity and collapse margin ratio   4 Storied 6 Storied 8 Storied Median Collapse Intensity, SCT (g) 2.75 2.1 1.55 MCE Spectral Acceleration, SMT (g) 0.75 0.6 0.429 Collapse Margin Ratio, CMR 3.67 3.50 3.61 5.2.1 Adjusted collapse margin ratio The collapse margin ratios calculated above were adjusted to compare with the acceptable values provided in FEMA P695 (2009). Validation of trial value for response modification factor has also been performed based on the acceptability of the adjusted collapse margin ratio (ACMR).  Calculation of collapse margin ratio and collapse capacity can be considerably influenced by frequency content (spectral shape) of the ground motion records. For this reason, spectral shape factors (SSF) were calculated for each index archetype models. This factor is dependent on period based ductility (µT), which was calculated using nonlinear static and Eigenvalue analysis of the archetype models. The period based ductility is a critical parameter for this analysis as it is directly related to ACMR using equation 5.5. ACMR = SSF x CMR             (5.5)  SSF values for different seismic design categories, building time periods and period based ductilities (μT) are provided in Table 7-1a and Table 7-1b of  FEMA (2009). The period based ductility (µT) of an index archetype can be calculated using equation 5.6.  μT =δuδy,eff                                (5.6)  130  In this equation, δu is the ultimate roof displacement and δy,eff is the effective roof yield displacement. The parameter δu in equation 5.6 was calculated from the pushover curves discussed later in this chapter and the parameter δy,eff  is calculated using equation 5.7.   𝛿𝑦,𝑒𝑓𝑓 = 𝐶0𝑉𝑚𝑎𝑥𝑊 𝑔4𝜋2(max (𝑇, 𝑇1))2                          (5.7) In equation 5.7, Vmax is the maximum base shear force (from pushover analysis), W is the weight of the building, g is the acceleration due to gravity, T is the fundamental period of the building from the building code formula, and T1 is the period of the building from modal/Eigenvalue analysis. Co is a factor relating roof displacement to the fundamental single- degree-of-freedom mode displacement. Co was calculated using equation 5.8.   𝐶0 = ∅1,𝑟∑ 𝑚𝑥   Ø1,𝑥𝑁𝑥=1∑ 𝑚𝑥   Ø21,𝑥𝑁𝑥=1                      (5.8) In this equation, mx is the mass of level x. φ1,x, and φ1,r  are the ordinates of the modal displacements of level x and the roof respectively. N represents the number of level in the building archetype under consideration. If the building mode shape ordinates are scaled in a way to make the maximum ordinate equal to unity, then Co represents modal participation factor.  In the first step of calculating adjusted collapse margin ratio, modal analyses were carried out on the PBSC building archetypes. The mode shape pseudo-displacement values with respect to frame heights are plotted in Figure 5-10. As per FEMA (2009) guideline, horizontal loading patterns for pushover analysis were generated from the corresponding mode, which is shown in Figure 5-11.  131   Figure 5-10: Fundamental mode shapes for the (a) 4-storied (b) 6-storied and (c) 8-storied PBSC building archetypes This mode shape proportional horizontal load was applied to the left most perimeter column of each frame, and nonlinear quasi-static analysis with a linearly increasing (ramp) function was carried out. This ramp function was created using a duration of 10 seconds. This large load application duration was selected to eliminate any dynamic effect during the analysis. The function value starts from zero at time zero and reaches unity at ten seconds. The quasi-static analysis was run in one hundred equal time steps for smooth output curves. The maximum values of this modal shape proportional load patterns were carefully adjusted to produce expected pushover response. If the load was too low, the archetypes were not reaching yield or nonlinearity; on the contrary, loads with too high values were making the models unstable. Therefore, trial and error analysis were carried out to achieve the optimum load values for the pushover curve. For brevity, the trial and error steps are not shown here. Figure 5-11 shows the final magnitude of the forces applied to the archetypes as per their fundamental mode shapes. 0510152025300 0.001 0.002 0.003 0.004 0.005 0.006Building Height (m)Mode shape values4 Storied Frame6 Storied Frame8 Storied Frame 132   Figure 5-11: Mode shape proportional horizontal loading pattern for the (a) 4-storied (b) 6-storied and (c) 8-storied PBSC braced frame During the pushover analysis, the displacement of the top node of the middle column was monitored along the base shear force. After the analysis, the roof displacements vs. base shears for the 4,6 and 8-storied frames were plotted and are shown in Figure 5-12(a)-(c). It was found out that all three pushover curves are similar in shape. The yield base shear forces are very close to one another. However, the maximum/ultimate displacements were vastly different between the frames. Due to the short height, the 4-storied frame, it has the smallest maximum deformation and vice versa.  (a) (b) (c) 133   Figure 5-12: Pushover curves for the (a) 4-storied (b) 6-storied and (c) 8-storied frame After the ultimate displacement (δu) values had been calculated using the pushover analyses discussed previously, a detailed calculation was carried out to calculate Co which is shown in Table 5-4 to Table 5-6.  Each row except the last one in these tables represents a single level/story of the building. For each level, the weight, mass, and ordinates of the mode shape (φ1,x) are shown. The last row shows the total for the columns required for the calculation of Co. The final column shows the Co value calculated for each building archetype using equation 5.8. The required φ1,r values can be found in Figure 5-10. Table 5-4: C0 coefficient calculation for 4-storied PBSC braced frame Description Weight (kN) Mass (kg) φ1,x φ1,x2 mxφ1x mx*φ1x2 C0 Roof 126.65 12914.42 0.0053000 0.0000281 68.45 0.36 1.305 Story 3 176.83 18031.73 0.0045000 0.0000203 81.14 0.37   Story 2 177.16 18064.97 0.0032000 0.0000102 57.81 0.18   Story 1 177.48 18098.21 0.0016000 0.0000026 28.96 0.05   Total 658.12 67109.33     236.35 0.96   02004006008001000120014000 0.05 0.1 0.15 0.2 0.25 0.3 0.35Base Shear (kN)Roof Displacement (m)0200400600800100012000 0.1 0.2 0.3 0.4 0.5Base Shear (kN)Roof Displacement (m)02004006008001000120014000 0.1 0.2 0.3 0.4 0.5 0.6 0.7Base Shear (kN)Roof Displacement (m)(a)(b)(c)δu = 0.21mδy = 0.0336mδu = 0.3m δy = 0.0569mδu = 0.44mδy = 0.0824m02004006008001000120014000 0.05 0.1 0.15 0.2 0.25 0.3 0.35Base Shear (kN)Roof Displacement (m)0200400600800100012000 0.1 0.2 0.3 0.4 0.5Base Shear (kN)Roof Displacement (m)02004006008001000120014000 0.1 0.2 0.3 0.4 0.5 0.6 0.7Base Shear (kN)Roof Displacement (m)(a)(b)(c)δu = 0.21mδy = 0.0336mδu = 0.3m δy = 0.0569mδu = 0.44mδy = 0.0824m02004006008001000120014000 0.05 0.1 0.15 0.2 0.25 0.3 0.35Base Shear (kN)Roof Displacement (m)0200400600800100012000 0.1 0.2 0.3 0.4 0.5Base Shear (kN)Roof Displacement (m)02004006008001000120014000 0.1 0.2 0.3 0.4 0.5 0.6 0.7Base Shear (kN)Roof Displacement (m)(a)(b)(c)δu = 0.21mδy = 0.0336mδu = 0.3m δy = 0.0569mδu = 0.44mδy = 0.0824m 134  Table 5-5: C0 coefficient calculation for 6-storied PBSC braced frame Description Weight (kN) Mass (kg) φ1,x φ1,x2 mxφ1x mx*φ1x2 C0 Roof 126.34 12883.39 0.0046000 0.0000212 59.26 0.27 1.35 Story 5 176.20 17967.47 0.0041000 0.0000168 73.67 0.30   Story 4 176.50 17998.49 0.0035000 0.0000123 62.99 0.22   Story 3 176.83 18031.73 0.0026000 0.0000068 46.88 0.12   Story 2 177.16 18064.97 0.0017000 0.0000029 30.71 0.05   Story 1 177.48 18098.21 0.0008000 0.0000006 14.48 0.01   Total 1010.52 103044.27     288.00 0.98   Table 5-6: C0 coefficient calculation for 8-storied PBSC braced frame Description Weight (kN) Mass (kg) φ1,x φ1,x2 mxφ1x mx*φ1x2 C0 Roof 126.34 12883.39 0.0041 0.0000168 52.82 0.217 1.37 Story 7 176.50 17998.49 0.0038 0.0000144 68.39 0.260   Story 6 175.90 17936.45 0.0034 0.0000116 60.98 0.207   Story 5 176.20 17967.47 0.0028 0.0000078 50.31 0.141   Story 4 176.50 17998.49 0.0022 0.0000048 39.60 0.087   Story 3 176.83 18031.73 0.0016 0.0000026 28.85 0.046   Story 2 177.16 18064.97 0.001 0.0000010 18.06 0.018   Story 1 177.48 18098.21 0.0005 0.0000003 9.05 0.005   Total 1362.92 138979.21     328.07 0.981   In the next step, effective yield displacements were calculated from equation 5.7. Some of the required parameters e.g. ultimate displacements (δu) and maximum base shear capacities were taken from Figure 5-12. Time periods for these frames were calculated as per equation (5.9) of NBCC (2010). T = 0.025hn      (5.9) Using the frame heights, building weights, time periods, ultimate displacement and base shear capacity it was found out that the effective yield displacements for 4, 6 and 8-storied frames are 0.06m, 0.092m and 0.138m respectively. By calculating the ratio of δu/δy, eff, the following period based ductility values were found (3.50, 3.27 and 3.20) for the 4, 6 and 8-storied frame respectively. As per expectation, the period based ductility decreased with the increase of frame  135  height. A similar trend for decreasing ductility with story height was observed in Haque and Alam (2015b) for rack clad building structures. Table 5-7: Period based ductility calculation Parameters 4-Storied  6-Storied  8-Storied Frame height, hn 12 18 24 Ultimate displacement, δu 0.21 0.3 0.44 Maximum base shear capacity, Vmax 800 760 780 Building weight, W 658.12 1010.52 1362.92 Time period as per code equation, T 0.3 0.45 0.6 Time period from modal analysis, T1 0.39 0.6032 0.84 max(T,T1) 0.39 0.6032 0.84 C0 1.31 1.35 1.37 δy,eff 0.060 0.092 0.138 μT 3.50 3.27 3.20 In the final step, spectral shape factor and adjusted collapse margin ratio were calculated using time periods and period based ductility values calculated earlier. SSF values for different fundamental time periods and period based ductilities are given in Table 7-1b (SDC Dmax) of FEMA (2009). The SSF values were interpolated from the above mention table and are shown in Table 5-8.  Adjusted collapse margin ratios were calculated using equation 5.5, and it was found that the values ranged from 4.24 to 4.47. Average value of ACMR was found to be 4.37. Table 5-8: Spectral shape factor (SSF) calculation   4-Storied 6-Storied 8-Storied Time Period, T 0.39 0.60 0.84 Period based ductility, μT 3.50 3.27 3.20 SSF 1.2 1.21 1.238 CMR 3.67 3.50 3.61 ACMR 4.40 4.24 4.47 Average ACMR 4.37     5.2.1 Acceptable values of collapse margin ratio The ACMR values calculated above were checked against the acceptable values provided in Table 7-3 of FEMA (2009). The table provides acceptable values of adjusted collapse margin ratios  136  (ACMR10% and ACMR20%) for different levels of total system collapse uncertainty (βTOT) and values of acceptable collapse probabilities (10% and 20%). The table mentioned above was generated based on the assumption that the collapse level uncertainty distribution is lognormal with a median of SCT and lognormal standard deviation equal to βTOT. The FEMA table mentioned above also provides acceptable ACMR values for other levels (5%, 15%, and 25%) of collapse probabilities. However, they are provided only for reference and comparison. Higher values of collapse probability and lower values of total system collapse uncertainty requires lower values of adjusted collapse margin ratio and vice versa. 5.2.1.1 Total system collapse uncertainty System collapse capacity is a highly variable parameter, and many sources of uncertainties influence it. Therefore, it is essential to take all significant sources of uncertainty into account in collapse response to incorporate their effects in the collapse evaluation process. There are various sources of collapse uncertainties e.g.: record to record uncertainty (RTR), design requirement uncertainty (DR), test data uncertainty (TD), modeling uncertainty (MDL). Record-to-record uncertainty occurs due to the variability in the response of building archetype models under different earthquake records. This variability is evident in the IDA plots shown in Figure 5-8. This variability in interstory drift ratio response can be attributed to the difference in frequency content and dynamic characteristics of the ground motion records in the far-field record set. Furthermore, this variability is also dependent on the far-field ground motion record set’s hazard characterization. From previous studies (Haselton 2006; Ibarra and Krawinkler 2005a; b; Zareian and Krawinkler 2006) it was found out that the value of βRTR ranges from 0.35 to 0.45 and it is fairly consistent among various building types. It was found from previous research and studies  137  that archetypes with μT values higher than or equal to 3.0 have a fixed βRTR value of 0.40. For μT values lower than 3.0, the βRTR value can be calculated using equation 5.10. However, the βRTR value must be equal to or higher than 0.20. As all μT values for the PBSC index archetypes were found larger than 3.0 (Table 5-7), the βRTR value was found to be 0.40. βRTR = 0.1+0.1 μT   (5.10) The next uncertainty is known as the design requirement related uncertainty (βDR). This uncertainty is dependent on the robustness and completeness of the design requirements. This uncertainty is quantified with respect to the quality of design requirements. Robustness and completeness of test data related uncertainty are taken care of by the test data uncertainty parameter (βTD). This uncertainty is quantified with respect to the test data quality. The last uncertainty parameter is Modeling uncertainty (βMDL). This uncertainty represents how well a full range of structural response characteristics and associated design parameters are represented by the index archetype models. This parameter also represents how well the archetypes simulate collapse behavior by direct simulation or non-simulated checks of the components. βMDL is measured by the quality of archetype models. The total collapse uncertainty is denoted by the lognormal standard deviation parameter (βTOT). It is calculated by combining the uncertainties mentioned above and expressed using the following equation. The range of values for the parameters in Equation 5.11 are given below.  βTOT = √𝛽𝑅𝑇𝑅2 + 𝛽𝐷𝑅2 + 𝛽𝑇𝐷2 + 𝛽𝑀𝐷𝐿2           (5.11) Where, βTOT = total system collapse uncertainty (0.275 - 0.950) βRTR = record-to-record collapse uncertainty (0.20 - 0.40)   138  βDR = design requirements-related collapse uncertainty (0.10 – 0.50) βTD = test data-related collapse uncertainty (0.10 – 0.50) βMDL = modeling-related collapse uncertainty (0.10 – 0.50) For the PBSC archetypes the following values were selected for the above-mentioned uncertainty parameters: βRTR = 0.40, βDR = 0.2, βTD = 0.35, and βMDL = 0.35. These values were chosen based on the assumptions discussed below. As per equation 5.10 and μT values in Table 5-8, the calculated value of βRTR is 0.40.  For the other three uncertainty parameters related to design requirements, test data, and nonlinear models, the following scale was used for selecting values for the corresponding β parameters. The scale is as follows for the qualities of each uncertainty parameter: (A) Superior, β = 0.10; (B) Good, β = 0.20; (C) Fair, β = 0.35; and (D) Poor, β = 0.50.  As Canadian building design requirement is strict and of a very high standard and the parameter value of 0.2 was chosen for βDR. Although no tests were performed for the PBSC bracing system, good quality test data is available in the literature for shape memory alloy used in the system; therefore, a fair rating with value 0.35 was chosen for the βTD parameter. The archetype model was carefully designed and modeled with nonlinear lumped plasticity based hysteresis model specifically developed for this bracing system. However, nonlinear connection models were not modeled due to lack of data; therefore, a quality rating of fair (β = 0.35) was chosen for the nonlinear model uncertainty parameter βMDL. Putting this values in equation 5.11 gives a value of 0.67 for βTOT. 5.2.1.2 Acceptable values of collapse margin ratio In this section, acceptable values of collapse margin ratios are calculated using the total collapse uncertainty and adjusted collapse margin ratio calculated in the previous sections of this chapter. Table 5-9 shows the acceptable CMR values for 10% and 20% probabilities of collapse.  139  These values are taken from Table 7-3 of FEMA (2009). The minimum ACMR value of the individual archetypes was checked against ACMR20%, and the average ACMR across the performance group was checked against ACMR10%. Both of the ACMR values were found to be higher than the acceptable CMR values, which indicate that collapse probability of the PBSC index archetypes are lower than the FEMA (2009) suggested limit; which means under MCE ground motion an individual archetype has lower than 20% probability of collapse and the whole group on average has less than 10% probability of collapse. Therefore, the Rd and Ro value selected for designing the PBSC index archetypes are adequate for ensuring safety against collapse under MCE ground motion intensity. Table 5-9: Acceptable values of collapse margin ratio Description Value Total System Collapse Uncertainty, βTOT 0.67 ACMR 10% 2.35 ACMR 20% 1.75 ACMR average across performance group 4.37 Minimum of individual ACMR 4.24 5.3 Overstrength Factor The overstrength factor (Ro) was checked using pushover analysis to compare against the initial value assumed at the beginning of this study. To find out the overstrength factor the yield base shear force was divided by the base shear force at first component yielding for all three frames. These base shear forces along with the calculation of overstrength factor are shown in Table 5-10. It was found that the average value of the overstrength factor for the PBSC braced frame is 1.07 which is very close to 1.10 assumed in design. Hence, no re-design/re-analysis is required for the determination of Ro factor.  140  Table 5-10: Overstrength Factor (Ro) calculation   4 Storied 6 Storied 8 Storied Yield Force 420 390 355 Vd (first yield) 371.9 372 343 Overstrength 1.13 1.05 1.03 Average 1.07 5.4 Summary Three PBSC braced frame models were created in S-FRAME Structural analysis and design software. These building frames were designed using (NBCC 2010) building code and (CSA 2009) design standard. These buildings were designed using arbitrary values of Rd = 6.0 and Ro = 1.1. After the design, nonlinear archetypes were built using hysteresis model developed for the PBSC bracing system. Finally, incremental dynamic time history analysis was carried out on the nonlinear archetypes using 44 far-field ground motion records which were normalized with respect to the record set’s median PGV. From this analysis, the median collapse earthquake intensities were calculated, and they were used to calculate collapse margin ratios. These collapse margin ratios were adjusted for spectral shape factor and finally compared against acceptable limits. It was found out that all individual archetypes and the group passed the acceptable limit of collapse margin ratios. Therefore, it could be concluded that the initial seismic performance factor values (Rd = 6.0 and Ro = 1.1) are adequate for seismic design of this system.  141  CHAPTER 6: SEISMIC FRAGILITY COMPARISON BETWEEN PBSC AND BRB BRACING SYSTEM  6.1 General This chapter concentrates on the seismic vulnerability assessment of PBSC braced frames using fragility functions. Fragility functions were developed to determine the probability of exceedance of certain damage states for this bracing system. The damage states were defined with respect to maximum interstory drift ratios. These fragility functions were generated using probabilistic seismic demand models (PSDM). The data for the fragility functions were calculated from nonlinear dynamic time history analysis with a suite of 44 far-field ground motion records. To understand the advantages and disadvantages of the PBSC bracing system, a similar analysis was carried out on the buckling restraint bracing (BRB) system. Both structural force resisting systems were designed using NBCC 2010 building code and CSA S16-14 standard. Bilinear kinematic hysteresis model was used to represent the axial force-deformation hysteresis of the buckling restrained braces; on the other hand, modified SMA flag hysteresis model with residual and sliding response was utilized to represent the nonlinear response of the PBSC bracing system. The fragility functions provided a great insight into the vulnerability of PBSC brace frames against different levels of earthquake intensity and helped in understanding the impact of using SMA in the bracing system in contrast to steel. This study found that the use of SMA in a bracing system (PBSC) could significantly reduce the vulnerability of steel frames in terms of transient interstory drift ratio compared to the counterpart for most of the damage states under consideration. The scope of this study is limited to only short to medium-tall buildings and the seismicity of Western  142  Canada. Further studies should be carried out if the application of the outcome of this study falls outside the scope mentioned above. 6.2 Research significance Canada’s western coast, particularly Vancouver Island, and Vancouver area are considered as the most earthquake hazard prone areas. Vancouver is Canada’s gateway to the Pacific, and a large portion of the Canadian economy is dependent on the well-being of this region; therefore, the economic impact of a major earthquake event in this region could be severe. Any high-intensity earthquake event could cause major damage to the infrastructure which could in turn cripple the economy of this nation. Therefore, seismic vulnerability assessment of the building infrastructure of this region is of high importance. This study focuses on the seismic fragility assessment of the novel PBSC braced frames designed for this region and provides a recommendation on how to improve their performance. 6.3 Description of the building models Similar to the building seismic performance factor study (Chapter 5), three 4x4 bay wide building frames with different story heights (four, six and eight storied) were selected for this study (Figure 5-5). Two sets of these models were created; one set for the PBSC bracing system and the other for the buckling restrained bracing system. The BRB and  PBSC building models were analyzed and designed as per NBCC (2010) seismic design provisions. The steel sections were designed using CSA (2009). However, the PBSC braced frames were designed using Rd = 6.0 and Ro =1.1, which was found from the building seismic performance factor study done earlier. The PBSC braces were designed using the design method shown in Appendix A.  143  6.4 Finite element modeling and analysis All of the building frames models mentioned above were built in the S-FRAME (2016) structural analysis and design software. This software utilizes finite element method for creating analysis models. After the design of the buildings, nonlinear analysis models were created using lumped plasticity models. Link elements were introduced at the brace ends for representing their load deformation hysteresis under cyclic loading. Consistent mass matrix formulation was used rather than the lumped mass matrix to achieve better accuracy. Furthermore, for Rayleigh damping matrix formulation, both stiffness and mass matrices were derived from the initial geometry of the structure. Nonlinear dynamic time history analysis was carried out on the three PBSC braced frame models to find out maximum interstory drift response under different earthquake records. The Newmark (1959) time history integration method was used for this analysis. Similar to the incremental dynamic analysis done for the building seismic performance factor calculation, the following values for alpha and delta parameters were chosen: alpha = 0.25 and delta = 0.50. Geometric nonlinearities were also taken into consideration which included large P-delta effect. To reduce analysis complexity and limit the number of finite element models, structural geometric variabilities (plan and elevation) and material strength uncertainties are not taken into consideration in this study. 6.5 Earthquake records Due to the scarcity of a vast number of strong earthquake records for British Columbia, strong ground motion records were chosen from all over the world instead. The records were selected from the far-field record set provided in the (FEMA 2009) guideline.  Using the ground motion  144  record information available in the guideline, the records were downloaded from PEER NGA strong motion database. The record set contains ground motion records from twenty-two individual earthquake events. Every one of these events has two horizontal and one vertical component. For this analysis, only the horizontal components of the ground motions were used. In total 44 ground motion records were used for analyzing the PBSC and BRB building frames. The records are from rocky soil type with a moment magnitude (Mw) greater than or equal to 6.5. Only far-field record set was selected due to the inherent complexity associated with the characterization of the near-field records. Figure 6-1 represents the response spectra of the 44 far-field ground motion records. In the same figure, the mean and the mean plus one and two standard deviation response spectra are also plotted. These ground motion records were applied in the in-plane direction of the PBSC and BRB braced frames. After each time history analysis, if nonlinearity was not observed, then the earthquake records were scaled up until significant nonlinear behavior or collapse were observed. This ensures that building responses were recorded at higher PGA values not covered in the ground motion record set.  145   Figure 6-1: Response spectra of the selected 44 far-field ground motion records. 6.6 Results and discussions The logarithmic relationship between spectral acceleration (Sa) and interstory drift ratio percentage for the three PBSC and three BRB braced frames are shown in Figure 6-2(a)-(c). A regression analysis was carried out using this data to find out two important parameters “a” and “b”, required for the development of fragility functions. These parameters are calculated from the tangent and intercept values of the ln(ISDR%) vs. ln(SaT) plot. The equation for “a” and “b” are given in equations 6.1 and 6.2. If the equation of the plot is written in the “y = mx+c” format, then 𝑎 = 𝑒𝑐  (6.1) 𝑏 = 𝑚       (6.2) 00.511.522.50 0.5 1 1.5 2 2.5 3 3.5 4Spectral Acceleration, Sa (g)Time Period (Seconds)1-009 1-279 2-0002-270 3-000 3-0904-000 4-090 5-2625-352 6-140 6-2307-000 7-090 8-0008-090 9-180 9-27010-000 10-090 11-27011-360 12-LN 12-TR13-000 13-090 14-00014-090 15-L 15-T16-000 16-090 17-27017-360 18-270 18-36019-E 19-N 20-00020-090 21-090 21-18022-000 22-270 Mean SpecMean Spec + StDev Mean Spec + 2StDev 146   Figure 6-2: ln(ISDR%) vs. ln(SaT) distribution for the (a) four (b)six and (c)eight-storied PBSC braced frame The fragility curves for the four, six and eight storied PBSC braced frames are shown in Figure 6-3(a)-(c) respectively. Each figure compares four different damage states; they are as follows: Operational (O), immediate occupancy (IO), life safety (LS), and collapse prevention (CP). These damage states were defined using FEMA (2000) guideline and the work of Fahnestock et al. (2006). The interstory drift ratio values for these damage states are shown in Table 6-1. For the PBSC bracing system, unlike the other three damage states, the collapse prevention (CP) damage state was not directly adopted from the above-mentioned guidelines and reports; instead this limit was calculated from the brace design procedure shown in Appendix A. The CP limit state of the PBSC bracing system is a variable quantity which can be manipulated by changing the brace design. For example, designing the brace with longer NiTinol bar would increase the collapse prevention ISDR% and vice versa. For the current design with 1m long NiTinol bars, the ISDR% at CP limit state was found to be 2.66%. If 2m long SMA bars were used in the design, then this limit would increase by two-folds to 5.33%. However, before reaching that high interstory drift ratio other structural components (beams, columns, and connections) would fail, making the use of longer NiTinol bar inefficient. Therefore, the length of the NiTinol bars should be selected based on the required/demand ISDR% and other component limit states while minimizing the cost of the brace. y = 1.0601x - 0.4315R² = 0.8266-3-2-10123-2 -1 0 1 2 3ln (ISDR%)ln (SaT)y = 0.8637x + 0.0725R² = 0.7915-3-2-10123-2 -1 0 1 2 3ln (ISDR%)ln (SaT)y = 0.8482x + 0.3948R² = 0.7686-3-2-10123-3 -2 -1 0 1 2 3ln (ISDR%)ln (SaT)(a) (b) (c) 147  Table 6-1: Damage states based on interstory drift ratio percentage Damage State PBSC BRB Operational (O) 0.5 0.5 Immediate Occupancy (IO) 1 1 Life Safety (LS) 2 2 Collapse Prevention (CP) 2.66 4 Figure 6-3(a) shows fragility curves with four different damage states for the four-storied PBSC braced frame. From this figure, it can be comprehended that the four-storied frame is very robust and has almost no probability of collapse up to a spectral acceleration of 2.0g. After this spectral acceleration value, the likelihood of collapse gradually increases to 0.72 for a Sa value of 14.0g; which is a very high spectral acceleration and unlikely to occur during the lifetime of the building. On the other hand, the life safety damage state curve is much steeper compared to the collapse prevention damage state curve. There is no probability of life safety damage until 1.0g spectral acceleration; however, there is around 10% probability at Sa = 2.0g. This curve gradually reaches a value close to unity at Sa = 14.0g. The third curve under discussion is the Immediate occupancy damage state curve. It can be seen that, at Sa = 1.0g, there is approximately 5% probability of damage for the four-storied frame. This curve is much steeper compared to the previous two curves discussed earlier. At Sa = 2.0g, the probability of damage is around 50%, which is very high compared to the previous curves. The final curve is for operational damage state, which is the steepest among the four. From this figure, it can be seen that there is approximately 97% probability of damage even at a low Sa value of 0.5g. Fragility curves for six-storied PBSC frame are shown in Figure 6-3(b). From this figure, it can be seen that the six-storied frame has a slightly higher probability of collapse for all four damage states compared to the four-storied frame.  At Sa =1.0g there is a very slight (around 1%)  148  probability of collapse for the six-storied frame which increases close to 10% at Sa = 2.2g. The life safety damage state reaches the same 10% probability of damage at Sa = 0.7g. For Immediate occupancy damage state, it requires only 0.4g spectral acceleration to arrive at the same probability of damage.  Finally, the operational damage state curve has a 99% probability of exceedance at Sa equal to 0.225g. Figure 6-3(c) depicts the four-damage state fragility curve for the eight-storied PBSC braced frame. The fragility curves in this figure are similar to the six-storied frame. However, the curves are slightly steeper compared to the previous ones. If we compare the Sa values for 10% probabilities of exceedance for each damage states, we can see that the collapse prevention damage state curve achieves this at 1.1g. The life safety and immediate occupancy damage state achieve this same probability at 0.35g and 0.22g respectively; while the operational damage state achieves this at less than 0.05g spectral acceleration.  149   Figure 6-3: Fragility curve for (a) four (b) six and (c) eight-storied PBSC braced frame The above mentioned three frames were compared against each other with respect to the four damage states discussed above. It was found out that for these damage states, the four-storied PBSC frame has the lowest probability of damage. On the other hand, the eight-storied frame has the highest probability of damage while the six-storied frame has damage probabilities in between these two frames. Therefore, for the PBSC frame, an increase in building height equates to an increase in damage probability for the same earthquake intensity level.  00.20.40.60.810 2 4 6 8 10 12 14Probability of damageSpectral Acceleration, Sa (g)O IOLS CP00.20.40.60.810 1 2 3 4 5 6 7 8 9Probability of damageSpectral Acceleration, Sa (g)O IOLS CP00.20.40.60.810 1 2 3 4 5 6Probability of damageSpectral Acceleration, Sa (g)O IOLS CP(a) (b)(c) 150   Figure 6-4: Different damage states compared between three PBSC frames (a) operational (b) immediate occupancy (c) life safety and (d) collapse prevention 6.7 Buckling restraint bracing system The fragility analysis mentioned above was carried out on buckling restrained braced frames for comparison with the PBSC bracing system. The four, six and eight-storied building frames used for PBSC bracing system were designed using standard star seismic braces. ETABS (2015) building analysis and design software was utilized for the analysis and design of the BRB frames. Similar to the PBSC braced frame design, the buildings were assumed to be in a soil class “C” site in Vancouver. The frames were designed using an Rd value of 4.0 and Ro value of 1.2. From the design, it was found out that Star Seismic BRB section 1.0 was adequate in resisting the design earthquake loading. This section has a core area of 645mm2. The initial stiffness is 30230kN/m and the post-yield stiffness is 513.9kN/m, yield force is 197.94kN and tension strength 233.53kN/m. 00.20.40.60.810 1 2 3 4 5 6Probability of damageSpectral accelration, Sa(g)4 Storied Frame6 Storied Frame8 Storied Frame00.20.40.60.810 1 2 3 4 5 6Probability of damageSpectral accelration, Sa(g)4 Storied Frame6 Storied Frame8 Storied Frame00.20.40.60.810 1 2 3 4 5 6Probability of damageSpectral accelration, Sa(g)4 Storied Frame6 Storied Frame8 Storied Frame00.20.40.60.810 1 2Probability of damageSpectral accelration, Sa(g)4 Storied Frame6 Storied Frame8 Storied Frame(a)(c) (d)(b) 151  After the design, same 44 far-field earthquake records (Figure 6-1) from PEER NGA database were used to carry out nonlinear dynamic time history analysis on the three BRB frame models which were used earlier for the PBSC building frames. Similar to the previous analysis, some of the earthquake records were scaled to include higher spectral acceleration values not covered in the original unscaled record set. From the incremental dynamic time history analysis results and equation 6.1, equation 6.2 and Figure 6-5(a), the values of “a” and “b” are 0.7888 and 0.9367 respectively. These parameters were used to calculate mean and standard deviation for the four damage states (Operational, Immediate occupancy, life safety and collapse prevention) discussed earlier. Figure 6-5(a)-(c) shows the natural logarithmic relationship between the earthquake spectral intensity and interstory drift ratio percentage for the four, six and eight storied BRB frames.   Figure 6-5: ln(ISDR%) vs. ln(SaT) distribution for the (a) four-storied (b) six-storied and (c) eight storied BRB frame Figure 6-6(a)-(c) shows the fragility curves for the four, six and eight-storied brb frame. The above mentioned four damage states are shown in this figure. The four storied BRB frame was analyzed up to a spectral acceleration value of 7g. It was found out that even at that high intensity, there was almost no probability of collapse. However, at the same intensity, there is approximately 60% probability of exceeding the life safety damage state. The other two damage states, for example, the operational and immediate occupancy level achieved 100% probability of y = 0.9367x - 0.2372R² = 0.8542-2.5-2-1.5-1-0.500.511.52-2 -1 0 1 2 3ln (ISDR%)ln (SaT)y = 0.9267x + 0.2443R² = 0.8327-2-1.5-1-0.500.511.52-2 -1 0 1 2ln (ISDR%)ln (SaT)y = 0.968x + 0.6384R² = 0.8067-1.5-1-0.500.511.52-2 -1 0 1 2ln (ISDR%)ln (SaT)(a) (b) (c) 152  exceedance at 1.0g and 4.0g respectively. Similar fragility curves were found for the six-storied BRB frame. However, the six-storied frame was subjected to a maximum spectral intensity of 4.0g. At this intensity, it was observed that the frame has less than 10% probability of exceeding the collapse damage state; however, the life safety damage state reached approximately 80% probability at the same intensity level. The remaining two damage states: operational and immediate occupancy level achieved 100% probability at around 0.6g and 2.0g respectively. Finally, if we analyze the fragility curve of the eight-storied BRB frame then we can see that at 2.5g spectral acceleration the frame reached almost 20% probability of exceeding the collapse prevention damage state. At the same spectral acceleration level, the life safety damage state achieved over 90% probability of exceedance; whereas the immediate occupancy and operational level achieved 100% probability at 1.0g and 0.4g respectively.  Figure 6-6: Fragility curve for (a) four (b) six and (c) eight-storied BRB braced frame 00.20.40.60.810 1 2 3 4 5 6 7Probability of damageSpectral Acceleration, Sa (g)O IOLS CP00.20.40.60.810 1 2 3 4Probability of damageSpectral Acceleration, Sa (g)O IOLS CP00.20.40.60.810 0.5 1 1.5 2 2.5Probability of damageSpectral Acceleration, Sa (g)O IOLS CP(b)(c)(a) 153  The above mentioned three frames were compared against each other with respect to the individual damage states. It can be observed that for all four damage states (O, IO, LS, and CP), the probability of damage is lowest for the four-storied frame and highest for the eight-storied frame while six-storied frame remaining in the middle. It was found out that with increasing building height the probability of damage for each damage state increases.  Figure 6-7: Different damage state compared between three BRB frames (a) operational (b) immediate occupancy (c) life safety and (d) collapse prevention 6.8 Fragility comparison between PBSC and BRB frames The fragility functions were compared between the PBSC and the BRB frames in Figure 6-8. Figure 6-8(a) shows the fragility comparison between four-storied PBSC and BRB frames. It was observed that for operational, immediate occupancy and life safety damage states the PBSC braced frame has a higher fragility curve than the BRB frame; which indicates less probability of exceedance for these three damage states at the same earthquake intensity level. However, for the 00.20.40.60.810 1 2Probability of damageSpectral acceleration, Sa(g)4 Storied Frame6 Storied Frame8 Storied Frame00.20.40.60.810 1 2 3 4 5 6Probability of damageSpectral acceleration, Sa(g)4 Storied Frame6 Storied Frame8 Storied Frame00.20.40.60.810 1 2Probability of damageSpectral acceleration, Sa(g)4 Storied Frame6 Storied Frame8 Storied Frame00.20.40.60.810 1 2Probability of damageSpectral acceleration, Sa(g)4 Storied Frame6 Storied Frame8 Storied Frame(a)(c) (d)(b) 154  collapse prevention damage state, the BRB frame outperformed the PBSC frame. This is due to higher interstory drift capacity of BRB frame (ISDR 0.04) compared to the PBSC frame (ISDR 0.0266) at this damage state.   Figure 6-8: Fragility curves compared between PBSC and BRB frames for (a) four (b) six and (c) eight-storied frames However, for the above analysis, the PBSC braces were designed with only 1m long SMA bars for most economic design; which limited its collapse prevention interstory drift ratio capacity to 2.66% (Appendix A). However, this collapse prevention interstory drift limit can be increased to 4% by either of the following three modifications: (a) Increasing the SMA tie bar lengths to (4/2.66) or 1.5 times the previous length or 1500mm. (b) a different composition of the SMA alloy with a higher recoverable strain (e.g. 9% or greater (Dolce and Cardone 2001)) or (c) a combination of (a) and (b) for an economical alternative. Using method (c) stated above; the brace design was modified. The recoverable strain of SMA bars was set at 9%, and their lengths were increased to 00.20.40.60.810 2 4 6 8Probability of damageSpectral acceleration, Sa(g)PBSC OPBSC IOPBSC LSPBSC CPBRB OBRB IOBRB LSBRB CP00.20.40.60.810 1 2 3 4 5Probability of damageSpectral acceleration, Sa(g)PBSC OPBSC IOPBSC LSPBSC CPBRB OBRB IOBRB LSBRB CP00.20.40.60.810 1 2 3Probability of damageSpectral acceleration, Sa(g)PBSC OPBSC IOPBSC LSPBSC CPBRB OBRB IOBRB LSBRB CP(a) (b)(c) 155  1100mm from 1000mm to have the least impact on the overall stiffness of the braced frame. This modification resulted in a maximum elongation capacity of 1100*.09 or 99mm for the NiTinol bars; including the elastic deformation of the brace shaft the total elongation is approximately 101mm or 1.73% for a 5830mm long brace. Converting this brace elongation to interstory drift ratio using the equation provided in Appendix A, we get a maximum ISDR capacity of 4%. Using this design modification, the fragility analysis was repeated to check the performance against BRB bracing system for the collapse prevention damage state. The result of this comparison is shown in Figure 6-9. In this figure, all three PBSC frames have less probability of collapse for the same spectral acceleration compared to the BRB frames. Therefore, the design of the PBSC bracing system can be easily modified to increase or decrease its damage limit states, which gives great flexibility to the engineers to optimize their design.  Figure 6-9: Collapse level fragility curves compared between PBSC and BRB frames for (a) four (b) six and (c) eight-storied frames for the modified PBSC brace design. 00.0250.050.0750.10 2 4 6Probability of damageSpectral acceleration, Sa(g)PBSC CP BRB CP00.0250.050.0750.10 1 2 3 4Probability of damageSpectral acceleration, Sa(g)PBSC CP BRB CP00.0250.050.0750.10 0.5 1 1.5 2Probability of damageSpectral acceleration, Sa(g)PBSC CP BRB CP 156  6.9 Summary Four, six and eight-storied braced frames were analyzed and designed with both PBSC and BRB bracing systems to compare their seismic performance with respect to fragility function. The PBSC frames were designed with the Rd and Ro values calculated in the previous chapter; whereas, the BRB braces were designed using the code provided values. After the design, nonlinear dynamic time history analysis was carried out on this frames to determine their fragility functions against four damage states. The results revealed the following: 1. For the PBSC bracing system, taller frames are more vulnerable to seismic loading compared to the shorter ones for the same spectral acceleration demand. The similar trend was observed for the BRB frames. However, taller frames usually attract less spectral acceleration from an earthquake compared to a shorter frame. This is due to the higher fundamental period of the taller frames compared to the shorter ones. 2. Both PBSC and BRB frame groups have almost no probability of collapse against spectral acceleration values near 1.0g, which is the maximum probable spectral acceleration in Canada. For operational, immediate occupancy and life safety performance levels the PBSC frames outperformed the BRB frames in terms of damage probability. However, for the collapse prevention damage state, the BRB showed better performance compared to the PBSC bracing system. However, the collapse fragility of PBSC bracing system could be increased by simply increasing the length of SMA ties or by changing the plateau strain of SMA; which could be achieved by changing the composition of the SMA alloy.   157  CHAPTER 7: CONCLUSIONS 7.1 Summary Most of the current seismic force resisting systems utilize energy dissipation mechanism for resisting seismic loading. The piston based self-centering bracing system is taking a different approach in resisting this load. Instead of heavily depending on nonlinear hysteretic energy dissipation this system is relying on its superior self-centering capability. As the system is novel and there are no seismic design guidelines for this system, this thesis is a pioneering effort in shedding some light into this area. To understand this system and other competing systems in this space, a comprehensive literature review is also provided for the readers. The author has provided schematic diagrams of the system and explained the mechanism of the system in detail. In the next step, the author has built detailed nonlinear finite element models of this bracing system in an advanced finite element analysis software (ABAQUS 2014). The model was developed using solid elements for high accuracy. SMA material hysteresis model developed by Auricchio and Sacco (1997) was used for modeling the SMA bars. Nonlinear quasi-static analysis with cyclic loading on the system revealed excellent self-centering capability indicated by flag-shaped hysteresis. However, the hysteresis material model discussed above does not simulate the residual deformation of SMA, therefore in the next step, the author developed a new hysteresis model for NiTinol based SMAs with residual deformation. This material model was integrated into a MATLAB based finite element program developed for analyzing the PBSC bracing system. After carrying out the nonlinear quasi-static analysis with the PBSC bracing system, the author discovered a novel hysteresis for the brace. This hysteresis exhibits both residual deformation and sliding response. To understand the seismic behavior of buildings  158  utilizing the PBSC bracing system, this hysteresis model was coded and implemented in a commercial structural analysis and design software know as S-FRAME. National building code of Canada utilizes force based seismic design procedure for designing buildings against seismic loading; which requires the determination of overstrength and ductility related force reduction factors. The author adopted FEMA P695 methodology for determining these factors for this novel structural system. Three building models were designed, and nonlinear models were built in S-FRAME Software environment. During the design process, the initial values of Rd and Ro were chosen to be 6.0 and 1.1 respectively. Then the author carried out incremental dynamic time history analysis on these building frame models using 44 normalized strong motion records retrieved from the PEER NGA strong motion database. From the analysis, it was found out that the adjusted collapse margin ratios were higher than the acceptable limits; therefore, the tentative values of Rd and Ro selected during the design are satisfactory. This study also presented the performance comparison between the PBSC and the BRB frames designed under same seismic and gravity loading conditions. Three frames from each group were designed as per NBCC (2010), and their performances were evaluated and compared with respect to fragility functions. Four limit states (Operational, immediate occupancy, life safety and collapse prevention) were defined as per FEMA (2000) guideline. From the fragility analysis, it was found out that, for both PBSC and BRB frames the seismic vulnerability increases with the increase in building height for a specific spectral acceleration. Furthermore, it was found that both systems have very low probability of achieving the collapse damage states even at the high seismic spectral intensity of 3.0g. Finally, it was found that the BRB frame is more vulnerable under operational, immediate occupancy and life safety damage states but less vulnerable for collapse prevention  159  damage state compared to the PBSC braced frame under the same spectral acceleration level. However, it was also found out that the seismic fragility of collapse prevention damage state for the PBSC bracing system can be improved if longer SMA bars are utilized. This would make PBSC bracing system provide similar or better margin of safety against collapse at the same earthquake intensity. 7.2 Limitations of this study The limitations of this study are as follows: 1. This study mostly focused on one configuration of the brace where SMA bars are continuous through the piston plate. Another configuration where the bars are discontinued at the piston plate was not studied in detail. However, results published earlier showed similar hysteresis behavior. 2. Sensitivity studies were not carried out for finding out the optimal part sizes for the PBSC brace; this was done based on the assumption that these parts/components (e.g. shaft, sleeve, and nuts) will be designed as per the established codes/standards. These components should be capacity protected to ensure that yielding and nonlinear deformation only takes place in the SMA tie bars. 3. Two-dimensional frame analyses were carried out instead of three-dimensional ones. This was done to reduce analysis time. 4. Only braces were considered nonlinear in the analysis. However, the beam-to-column and column-to-foundation connections could exhibit some moment-rotation stiffness and hysteresis even they are not designed for that purpose. This behavior was ignored in this study.  160  5. No experimental investigation was carried out on the PBSC bracing system. However, the residual deformation based SMA material hysteresis model was validated against experimental results. Furthermore, a small scale PBSC brace prototype has recently been built in the UBC Okanagan’s ALAMS lab, and its test results will be published in subsequent studies.  7.3 Conclusion The following conclusions can be drawn from this study 1. The finite element analysis on the PBSC bracing system showed flag-shaped hysteresis response. This hysteresis indicates good self-centering capability with moderate hysteresis damping. 2. The PBSC bracing system is more flexible or can be designed to be more flexible compared to traditional or buckling restrained bracing system due to the one-third elastic modulus of NiTinol compared to steel. Furthermore, the length of the NiTinol SMA bars could also be adjusted and the stiffness of bracing system and the structure could be adjusted. A lower stiffness results in a lower period and eventually lower base shear force from the response spectrum analysis; which could eventually lead to smaller sections required from the design and thus cost savings. 3. Although the PBSC bracing system has exhibited a sliding response, no perceptible negative impact was found on the seismic performance of the buildings. Excellent self-centering was observed after the earthquake loading was over. The residual interstory drift ratios were found insignificant which indicates almost no or meager repair cost after an earthquake event.  161  4. The overstrength and ductility related force related force reduction factors were found to be 6.0 and 1.1. The ductility related force reduction factor was found higher than most other systems given in NBCC (2010). Therefore, more detailed analysis with a greater number of stories and bays with many different brace configurations should be carried out to determine a more conservative value. 5. The fragility analysis of the PBSC bracing system showed good performance against all limit states (except collapse prevention) for earthquake intensities covered in the design response spectra of Canada. The collapse level fragility of the PBSC braced frame was found to be underperforming against BRB frame however that can be improved by tweaking the brace design (e.g. increasing the SMA bar lengths) 7.4 Recommendations for future research The following recommendations are given for future research regarding the PBSC bracing system 1. Prototype models should be built in the lab for testing the cyclic performance of the PBSC bracing system. 2. The self-centering capability of portal frames equipped with the PBSC braces should be investigated experimentally using quasi-static cyclic load tests. 3. Shake table testing should be carried out to find out the seismic performance of single storied steel buildings equipped with the PBSC bracing systems. 4. Sensitivity analysis should be conducted on the coupler design to check their slip deformation relationship with the SMA bars and how it could be prevented by changing shape, size, and number of screws. The seismic performance factor determination study  162  should be extended to cover many different brace configurations (X, EBF, V etc.), the number of bays, the number of stories, etc. 5. The use of PBSC brace in a moment frame should be investigated. This would require nonlinear model creation with nonlinear beams, columns and foundation elements. Soil structure interaction models could be utilized for more accurate results. 6. The self-centering capability of reinforced concrete moment resisting frames utilizing PBSC bracing system should be investigated. Similar studies should be carried out on timber frames to find out the braces applicability in these structural systems. 7. Seismic performance evaluation of multi-column bridge bents strengthened/retrofitted with the PBSC bracing system should also be investigated. 8. Effect of extreme temperatures on the cyclic performance of PBSC bracing system should be investigated. 9. Effect of strain rate on the hysteresis behavior of the PBSC bracing system should be determined using experimental investigation. In the next step, a hysteresis model capable of simulating the strain rate effect should be developed from these experimental results.  10. The performance of the PBSC braced frames against wind load should be investigated. 11. 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(2007). “Seismic behaviour of self‐centring braced frame buildings with reusable hysteretic damping brace.” Earthquake engg & structural dynamics, 1329–1346.  173  APPENDICES Appendix A: Design method for PBSC braced frame A.1 Brace elongation to interstory drift ratio  Figure A-1: Elongated brace length calculation Assuming 𝛳 is a very small angle, the vertical projection of the column does not change after lateral displacement Δ (Figure A-1).  Therefore, interstory drift ratio (ISDR) as per equation A-1, 𝐼𝑆𝐷𝑅 =𝛥𝐻= 𝑆𝑖𝑛(𝛳)  (A-1) We know that, for a very small angle (in radians), Sin(𝛳) = 𝛳 Therefore,  𝛳 =𝛥𝐻           (A-2) HLLBΔ 90-  174  The original length of the brace, LB is given by  𝐿𝐵 = √𝐿2 +𝐻2           (A-3) Horizontal projection of the brace after lateral deformation/brace elongation = L+H*Cos(90-𝛳) Vertical projection of the brace after lateral deformation/Brace elongation = H*Sin(90-𝛳) Elongated length of the brace = [{ L+H*Cos(90-𝛳)}2+{ H*Sin(90-𝛳)}2]0.5 =  [{ L+H*Sin𝛳}2+{ H*Cos𝛳}2]0.5 = [L2+2*L*H*Sin𝛳+H2*Sin2𝛳+H2*Cos2𝛳]0.5 = [L2+2*L*H*Sin𝛳+H2*(Sin2𝛳+Cos2𝛳)]0.5 = [L2+H2+2LHSin𝛳]0.5 Brace elongation ratio, λr =[ 𝐿2+𝐻2+2𝐿𝐻𝑆𝑖𝑛𝛳𝐿2+𝐻2]0⋅5 λr = [1 + 2𝐿𝐻𝑆𝑖𝑛𝛳𝐿2+𝐻2]0⋅5 λr = [1 + 2𝐿𝐻∗𝛥𝐻𝐿2+𝐻2]0⋅5 λr = [1 + 2𝐿𝐻∗𝛥𝐻𝐿𝐵2 ]0⋅5      (A-4) Let’s assume L = 5000mm and H = 3000mm, for an ISDR or Δ/H value of 0.025 or 2.5% ISDR%  175  Brace elongation ratio = [1 + 2𝐿𝐻∗𝛥𝐻𝐿2+𝐻2]0⋅5 = [1 + 2∗5000∗3000∗.02550002+30002]0⋅5  = [1.022059]0⋅5  = 1.010969 ≈ 1.011 Amount of elongation 1.010969-1 = .010969 or 1.0969% Therefore, for a 5000mm wide by 3000mm high frame configuration the brace elongation percentage divided by interstory drift ratio percentage is 1.0969/2.5 = 0.4387. If we use NiTinol bars with the following material properties (Table A-1), we can see that for a plateau strain value of 0.06, the strain at Austenite to martensite finishing stress (εamf) is 0.06816;  which can be calculated using equation A-5. 𝜀𝑎𝑚𝑓 = 𝜀𝐿 + 𝜎𝑎𝑚𝑓/𝐸𝑠𝑚𝑎  (A-5) If 1000mm long NiTinol bars are used, the maximum elongation will be 68.16mm at σamf. If we divide this elongation by the brace length (LB) of 5830.95mm we get an elongation of 1.16%. Therefore, the expected interstory drift ratio at this brace elongation value is 1.16/0.4387 = 2.66%. As the stiffness of NiTinol bars beyond σamf stress level increases drastically, it can be assumed that the PBSC brace to steel frame connections will not be designed for this high degree of stress. Therefore, connections between the brace and the frame might start to fail after σamf stress in the SMA bars. Therefore, for 1000mm long NiTinol bars, the maximum interstory drift ratio at the collapse prevention damage state could be conservatively set to 2.66%. However, if 4% interstory  176  drift ratio at this limit state is desired then the following modifications could be made to the design: (a) brace length should be increased to (4*1000/2.66) or 1503mm. (b) a different composition of the SMA alloy with a higher recoverable strain (e.g. 9%) could be used. A.2 Hysteresis parameters for PBSC braced frame The following equations (A6 to A12) were used to calculate the PBSC link hysteresis parameters in the S-FRAME Structural analysis and design software. The SMA material properties used for the calculation of these parameters are given in Table A-1. Table A-1: Sample material properties for NiTinol SMA bars Properties Values Unit Austenite to martensite starting stress, σams 400 N/mm2 Austenite to martensite finishing stress, σamf 510 N/mm2 Martensite to austenite starting stress, σmas 370 N/mm2 Martensite to austenite finishing stress, σmaf 130 N/mm2 Modulus of elasticity, Esma 62500 N/mm2 Yield Strain, εams 0.0064   Plateau Strain, εL 0.06   Strain at austenite to martensite finishing stress, εamf 0.06816   The required nonlinear parameters are yield force of the brace (Py), initial stiffness (Ki), post-yield stiffness (Kp), austenite to martensite finishing force (Pamf), martensite to austenite starting force (Pmas), unloading force turn factor (α) and residual deformation factor (β). 𝑃𝑦 = 𝜎𝑎𝑚𝑠 ∗ 𝐴𝑠𝑚𝑎  (A-6) 𝑘𝑖 = 𝐸𝑠𝑚𝑎 ∗ 𝐴𝑠𝑚𝑎/𝐿𝑠𝑚𝑎  (A-7) 𝑘𝑝 = (𝜎𝑎𝑚𝑓−𝜎𝑎𝑚𝑠𝜀𝑎𝑚𝑓−𝜀𝑎𝑚𝑠) ∗ 𝐴𝑠𝑚𝑎/𝐿𝑠𝑚𝑎  (A-8) 𝑃𝑎𝑚𝑓 = 𝜎𝑎𝑚𝑓 ∗ 𝐴𝑠𝑚𝑎  (A-9)  177  𝑃𝑎𝑚𝑓 = 𝜎𝑚𝑎𝑠 ∗ 𝐴𝑠𝑚𝑎  (A-10) 𝛼 =𝜎𝑚𝑎𝑓𝜎𝑎𝑚𝑠  (A-11) 0 ≤ 𝛽 ≤ 1  (A-12) Here, Asma = Total cross-sectional area of SMA bars resisting either tension or compression Lsma = Length of SMA bars for tension or compression A sample of calculated values for these parameters are given in Table A-2. These values have been computed using the material property data provided in Table A-1. For this calculation, it was assumed that the brace has 2-10.1mm dia SMA bars each having a length of 1000mm.  Table A-2: PBSC brace link hysteresis parameters Properties Values Unit Yield force, Py 64 kN Initial Stiffness, Ki 10014 kN/m Post-Yield Stiffness, Kp 285.40 kN/m Austenite to martensite finishing force, Pamf 81.72 kN Martensite to austenite starting force, Pmas 59.29 kN Unloading force turn factor, α 0.33   Residual deformation factor, β 0.1  A.3 Axial stiffness modifier for design During building design, it is not feasible to explicitly model the PBSC bracing system with the cylinder/sleeve, internal NiTinol tie bars and end plates in traditional structural design software like S-FRAME, SAP2000 or ETABS etc. To resolve this issue, a simplified method has been proposed to incorporate the actual stiffness of the PBSC brace in the software. The method requires the use of axial stiffness modification factors. These factors can be manually calculated before the analysis of the PBSC braced frames.   178  For the PBSC braced frame design model, the braces could be modeled using elastic truss elements. If the braces are modeled as single elements, then their stiffness do not match the PBSC brace which has stiffness contribution from the brace shaft and the NiTinol tie bars. The stiffness of the cylinder can be ignored as it is much stiffer compared to the other two elements. If we model the brace in finite element software with a prismatic steel member, then the stiffness of the brace can be expressed using equation A-13 𝐾𝑏 =𝐴𝑏∗𝐸𝑠𝑡𝑒𝑒𝑙𝐿𝑏  (A-13) Where, Ab = Cross sectional area of steel brace Esteel = Modulus of elasticity of steel (200 GPa) Lb = Length of the brace However, the brace stiffness comes from two parts connected in series: NiTinol bar part and the shaft. The stiffness of the NiTinol part and the shaft can be expressed using equation A-14 and A-15 𝐾𝑠𝑚𝑎 =𝐴𝑠𝑚𝑎∗𝐸𝑠𝑚𝑎𝐿𝑠𝑚𝑎  (A-14) 𝐾𝑠ℎ𝑎𝑓𝑡 =𝐴𝑠ℎ𝑎𝑓𝑡∗𝐸𝑠𝑡𝑒𝑒𝑙𝐿𝑠ℎ𝑎𝑓𝑡  (A-15) The equivalent stiffness of the brace (Keq) can be calculated as follows 1𝐾𝑒𝑞=1𝐾𝑠𝑚𝑎+1𝐾𝑠ℎ𝑎𝑓𝑡     (A-16) 1𝐾𝑒𝑞=1𝐴𝑠𝑚𝑎∗𝐸𝑠𝑚𝑎𝐿𝑠𝑚𝑎+1𝐴𝑠ℎ𝑎𝑓𝑡∗𝐸𝑠𝑡𝑒𝑒𝑙𝐿𝑠ℎ𝑎𝑓𝑡     (A-17)  179  Assumptions, Asma = (Fy,steel/Fy,sma)Ashaft = fAshaft Modular ratio = Esteel/Esma = r NiTinol tie length to brace length, (Lsma/Lb) = n Steel shaft length to brace length, (Lshaft/Lb) = m Substituting the above-mentioned relationships in equation A-17 we get 1𝐾𝑒𝑞=1𝑓𝐴𝑠ℎ𝑎𝑓𝑡𝐸𝑠𝑡𝑒𝑒𝑙𝑟𝑛𝐿𝑏+1𝐴𝑠ℎ𝑎𝑓𝑡𝐸𝑠𝑡𝑒𝑒𝑙𝑚𝐿𝑏 1𝐾𝑒𝑞=1𝑓𝐾𝑏𝑟𝑛+1𝐾𝑏𝑚 1𝐾𝑒𝑞=𝑟𝑛𝑓𝐾𝑏+𝑚𝐾𝑏 1𝐾𝑒𝑞= (𝑟𝑛𝑓+𝑚)1𝐾𝑏 1𝐾𝑒𝑞= (𝑟𝑛 + 𝑓𝑚𝑓)1𝐾𝑏 𝐾𝑒𝑞 = (𝑓𝑟𝑛+𝑓𝑚)𝐾𝑏             (A-18) Assuming Fy,steel = 350 MPa, Fy,sma = 400 MPa, Esteel = 200 GPa, Esma = 62.5 GPa, Lb = 5.83m (assuming 5m wide by 3m tall bay), Lsma = 1m and Lshaft = 4.83m, we get the following values for f, r, n and m. f = 0.875 r = 3.2  180  n = 0.1715 m = 0.8284 Substituting these values in equation A-18 we get 𝐾𝑒𝑞 = 0.687 ∗ 𝐾𝑏 This multiplier (0.687) was used as an axial stiffness modified for the braces in the finite element model. A.4 Axial stiffness modifier for nonlinear analysis During the nonlinear dynamic time history analysis, link elements are used at the end of the braces to simulate the nonlinear load deformation hysteresis and stiffness of the NiTinol bars. However, the link elements in most of the structural analysis and design software are modeled as zero-length elements. Which means the SMA ties are getting represented without any length in the model. This makes the shaft longer than their actual length and reduces the brace axial stiffness. To correct the stiffness, a different axial stiffness multiplier should be used for nonlinear analysis. This ratio can be simply calculated using the ratio between the modeled length of the brace and the actual length of the brace shaft if links represented the actual length of the NiTinol bars. This equivalent brace stiffness is given by equation A-19. 𝐾𝑒𝑞 = (𝐿𝑏𝐿𝑠ℎ𝑎𝑓𝑡) ∗ 𝐾𝑏             (A-19) Using, Lb = 5.83m (assuming 5m wide by 3m tall bay), and Lshaft = 4.83m (assuming 1m long SMA bars), we get the following expression from equation A-19. 𝐾𝑒𝑞 = 1.207 ∗ 𝐾𝑏    181  This higher than one axial stiffness modification factor increases the brace stiffness and corrects for the extra length modeled. A.5 A step-by-step guideline for PBSC brace frame design The following step-by-step guideline should be used to design a PBSC braced frame 1. Analysis of the frame: Analysis should be carried out on the desired/selected frame with braces as modeled as elastic truss elements. The braces should be pin-connected to the frame; therefore, they should only carry axial forces. The axial stiffness of the braces should be modified to take the stiffness of the SMA bars into account as per section A.3. 2. Seismic analysis: Response spectrum, linear dynamic time history or equivalent lateral force based analysis should be carried out to determine the seismic demand forces on the braces. On the other hand, a sophisticated nonlinear analysis could also be performed for the brace frame design. However, this would require nonlinear modeling of the braces using hysteresis models. The parameters for the link hysteresis would be difficult to guess without prior linear analysis; therefore, this would require numerous trial and error. Therefore, it is recommended to use the linear static or linear dynamic analysis for the design. 3. Brace design: The PBSC braces should be designed using the forces calculated in step 2. For the design of the braces, the SMA bars should be designed using factored demand and unfactored resistance. However, the steel components should be designed using factored demand and factored resistance. Furthermore, an additional load amplification factor of 1.1-1.2 (based on engineering judgment) should be utilized for the design of the shaft and the sleeve. This additional load amplification factor should ensure capacity protection for  182  these elements. The design of these components should follow the axial compression member design procedure of the relevant code/standard of the jurisdiction. However, the design of the SMA ties should be based on simple tensile stress check. The required area of the ties should be calculated using equation A-20. Atie =Tfσams          (A-20) Where, Atie = Required area of SMA tie bars Tf = Factored force demand on the brace (max of tension/compression) σams = Austenite to martensite starting stress 4. SMA Tie Bar length: The length of the SMA tie bars should be determined based on the required interstory drift ratio capacity. Equation A-4 can be used to determine the elongated length of the brace for a design interstory drift ratio. The amount of elongation should be designed to occur mostly in the SMA tie bars. The shaft and sleeve should deform only in the elastic range. Therefore, the SMA tie bar lengths should be chosen in a way that this elongation falls within the recoverable strain limit of the SMA bar. Equation A-21 should be used for calculating the required length of the SMA tie bar. Lsma = (λr -1) * LB/εamf         (A-21) Here, Lsma = Required length of SMA bars for tension or compression λr = Brace elongation ratio (elongated length/original length) LB = Brace length εamf = Austenite to martensite finishing strain  183  Appendix B: MATLAB finite element program clc; clear all; close all;   %Information about external force array input xls file filename = 'loads.xlsx'; sheet = 'LH'; xlRange = 'B:B'; Load_History = xlsread(filename, sheet, xlRange);   %Variable used for turning on SMA hysteresis for the ties sma_ties_used=1;   %Number of data points in the excel file Load_History_Size=size(Load_History); Data_Points=Load_History_Size(1); Tie_Length=1; Shaft_Length=0.01;   %Calculation for the Interior and Exterior Ties. In the stiffness matrix %double the number of ties is used. One set for taking tension and the %other set for taking compression No_Of_Ties=2; Tie_dia =.018; %Diameter of each ties Area_Of_Ties=pi*Tie_dia^2*0.25*No_Of_Ties;   %Calculation for the Shaft Dshft_centerline = 0.11905; %Centerline dimension t_shft=.00795; %Wall thickness Dos=Dshft_centerline+t_shft/2; %Outer diameter Dis=Dshft_centerline-t_shft/2;  %Inner diameter Area_of_Shaft=pi*(Dos^2-Dis^2)*.25; %This area should come from a standard section   %disp('Cross sectional area of bar segments') A=[Area_Of_Ties, Area_of_Shaft]; %First data is for cylinder and second data is for the brace shaft A_backup=A; %disp('Lengths of bar segments'); L=[Tie_Length,Shaft_Length];%First item is for cylinder length and the second item is for brace shaft length L_bckp=L; %Populating L_bckup matrix   %Creating a deformed length matrix for corresponding loading data points Total_Deformed_length=zeros(1,Data_Points); Tie_Deformed_length=zeros(1,Data_Points); Shaft_Deformed_length=zeros(1,Data_Points); Tie_Load_History=zeros(1,Data_Points);   %Number of Bar segments, n n=2; Undeformed_length=L(1)+L(2);   %Modulus of elasticity of the material in Ties if sma_ties_used==0     Em_ties=2e11;     Fy_ties=350e6;     Fu_ties=450e6; else     Em_ties=62.5e9;  184      Fy_ties=400e6;     Fu_ties=510e6; end   %Modulus of elasticity of the material in Shaft Em_shaft=2e11; %Poissons's Ratio, use zero if engineering stresses are used in the material model. Use %0.3 (Steel) 0.33 (SMA) if true stress strain curves are used in the material model.  %v=0.33; v_Ties=0; v_Shaft=0; v = [v_Ties, v_Shaft]; % Member initial material elasticity kept for backup Em=[Em_ties,Em_shaft]; %disp('Variable Modulus of elasticity of bar segments needed for stiffness matrix'); Ev=[Em(1),Em(2)]; %Yield stress of steel Fy=[Fy_ties,350e6]; Fu=[Fu_ties,450e6]; %Positive and negative yield stress matrix Fypos=[1*Fy(1),1.0*Fy(2)]; Fyneg=[-1*Fy(1),-1.0*Fy(2)]; %Positive ultimate stress matrix Fupos=[1*Fu,1.0*Fu]; %disp('Yield strain (m/m):'); ey_pos=[Fypos(1)/Em(1),Fypos(2)/Em(2)]; %disp('Yield displacement (m):'); dy_pos=[ey_pos(1)*L(1),ey_pos(2)*L(2)]; dy_neg=[-ey_pos(1)*L(1),-ey_pos(2)*L(2)]; %Strain at ultimate stress eu=[0.1,0.1]; %Plastic Strain ep=eu-ey_pos; %Post yield elasticity ratio r=[((Fupos(1)-Fypos(1))/ep(1))/Em(1),((Fupos(2)-Fypos(2))/ep(2))/Em(2)]; %Positive yield forces for the elements Pypos=[Fypos(1)*A(1),Fypos(2)*A(2)];   %Buckling load of the shaft k_shaft=1.0; I_shaft=(pi*(Dos^4-Dis^4))/64; r_shaft=(I_shaft/A(2))^.5; Pbuck_shaft=-pi^2*Em(2)*A(2)/(k_shaft*L(2)/r_shaft)^2;   %Buckling monitoring variables shaft_buckled=0;   %Ultimate strain monitoring variables tie_ult_strain_excd=0; shaft_ult_strain_excd=0;   %disp('Stiffness of bar segments'); Kcoeff=[A(1)*Ev(1)/L(1),A(2)*Ev(2)/L(2)]; k=zeros(2,2,n); %Global Displacement Matrix GlobalU=zeros(n+1,1);   Load_History=Load_History*A(1)*Fupos(1)*1.0;   %Local Stiffness matrix formation for i=1:n,  185      k(1,1,i)=1;     k(1,2,i)=-1;     k(2,1,i)=-1;     k(2,2,i)=1; end   Tie_tensile_length_bckp=L(1); Tie_compressive_length_bckp=L(1);   %Global Variable for sharing with the sma hysteresis function global Strain_Max;  global Residual_Strain; global Ei; global Sams; global Samf; global Smas; global Smaf; global Ep; global SMA_Yield_Strain; global residual_ratio; global Smax global Strain_Crspnd_Pos; global Strain_amf; global Brace_Under_Tension; global Strain_Max_Tension; global Strain_Max_Compression; global Residual_Strain_Tension; global Residual_Strain_Compression;   Ei = 62.5e9; Sams = Fy_ties; Samf = Fu_ties; Smas = 370e6; Smaf = 130e6; Smax=Sams; Strain_Plateau=0.06; SMA_Yield_Strain = Sams / Ei; Strain_Max = SMA_Yield_Strain; Strain_Max_Tension = SMA_Yield_Strain; Strain_Max_Compression = SMA_Yield_Strain; Residual_Strain=0; Residual_Strain_Tension = 0; Residual_Strain_Compression = 0; Strain_L = Samf / Ei; Ep = (Samf - Sams) / (Strain_Plateau + Strain_L - SMA_Yield_Strain); residual_ratio = 0.1; Strain_Crspnd_Pos = SMA_Yield_Strain; Strain_amf = Strain_Plateau + Strain_L; Brace_Under_Tension = (Load_History(2)-Load_History(1))/abs((Load_History(2)-Load_History(1)))     for j=1:Data_Points, %Program's main counter for going through every data point          %Implementation of hysteresis rules     if j>1 %Only evaluate starting from second data point         for i=1:n, %This loop goes through the bar elements one by one              if Load_History(j)>=Load_History(j-1) %If load is increasing                                if Load_History(j)>0 %If the arrangement is in tension                     if (i==1)                         if sma_ties_used==1                             if (Load_History(j-1)<=0)  186                                  L(i)=Tie_tensile_length_bckp;                             end                             if j<Data_Points                                 Ev(i)=sma_hysteresis((Load_History(j)/A(i)),(L(i)/Tie_Length-1),(Load_History(j+1)/A(i)),(L_bckp(i)/Tie_Length-1));                             else                                 Ev(i)=sma_hysteresis((Load_History(j)/A(i)),(L(i)/Tie_Length-1),(Load_History(j)/A(i)),(L_bckp(i)/Tie_Length-1));                             end %                             if (Load_History(j-1)<=0) %                                 L(i)=Tie_tensile_length_bckp; %                             end                             Kcoeff(i)=A(i)*Ev(i)/L(i);                               else%If only steel ties are used                             if (Load_History(j-1)<=0)                                 L(i)=Tie_tensile_length_bckp;                             end                             [Ev(i),Fypos(i),Fyneg(i)]=steel_hysteresis(Load_History(j)/A(i),Load_History(j-1)/A(i),Fypos(i),Fyneg(i),Fy(i),r(i)); %                             if (Load_History(j-1)<=0) %                                 L(i)=Tie_tensile_length_bckp; %                             end                             Kcoeff(i)=A(i)*Ev(i)/L(i);                         end                       elseif (i==2)                         [Ev(i),Fypos(i),Fyneg(i)]=steel_hysteresis(Load_History(j)/A(i),Load_History(j-1)/A(i),Fypos(i),Fyneg(i),Fy(i),r(i));                         Kcoeff(i)=A(i)*Ev(i)/L(i);                       end                                                   else %If the arrangement is in compression                     if (i==1) %Calculation for the ties                         if sma_ties_used==1                             if j<Data_Points                                 Ev(i)=sma_hysteresis((Load_History(j)/A(i)),(L(i)/Tie_Length-1),(Load_History(j+1)/A(i)),(L_bckp(i)/Tie_Length-1));                             else                                 Ev(i)=sma_hysteresis((Load_History(j)/A(i)),(L(i)/Tie_Length-1),(Load_History(j)/A(i)),(L_bckp(i)/Tie_Length-1));                             end                             Kcoeff(i)=A(i)*Ev(i)/L(i);                         else %If only steel ties are used                             [Ev(i),Fypos(i),Fyneg(i)]=steel_hysteresis(Load_History(j)/A(i),Load_History(j-1)/A(i),Fypos(i),Fyneg(i),Fy(i),r(i));                             Kcoeff(i)=A(i)*Ev(i)/L(i);                         end                           elseif i==2                         [Ev(i),Fypos(i),Fyneg(i)]=steel_hysteresis(Load_History(j)/A(i),Load_History(j-1)/A(i),Fypos(i),Fyneg(i),Fy(i),r(i));                         Kcoeff(i)=A(i)*Ev(i)/L(i);                     end                 end                   187              elseif Load_History(j)<Load_History(j-1)  %if Load is decreasing                 if Load_History(j)>=0 %If the arrangement is in tension                       if (i==1)                         if sma_ties_used==1                             if j<Data_Points                                 Ev(i)=sma_hysteresis((Load_History(j)/A(i)),(L(i)/Tie_Length-1),(Load_History(j+1)/A(i)),(L_bckp(i)/Tie_Length-1));                             else                                 Ev(i)=sma_hysteresis((Load_History(j)/A(i)),(L(i)/Tie_Length-1),(Load_History(j)/A(i)),(L_bckp(i)/Tie_Length-1));                             end                             Kcoeff(i)=A(i)*Ev(i)/L(i);                         else  %If only steel ties are used. negative sign is applied before the load argument as the compression for the whole brace is tension for the ties.                             [Ev(i),Fypos(i),Fyneg(i)]=steel_hysteresis(Load_History(j)/A(i),Load_History(j-1)/A(i),Fypos(i),Fyneg(i),Fy(i),r(i));                             Kcoeff(i)=A(i)*Ev(i)/L(i);                         end                     elseif (i==2)                           [Ev(i),Fypos(i),Fyneg(i)]=steel_hysteresis(Load_History(j)/A(i),Load_History(j-1)/A(i),Fypos(i),Fyneg(i),Fy(i),r(i));                           Kcoeff(i)=A(i)*Ev(i)/L(i);                     end                 else %If the arrangement is in compression and getting loaded in compression                     if (i==1) %For the ties                         if sma_ties_used==1                             if (Load_History(j-1)>=0)                                 L(i)=Tie_compressive_length_bckp;                             end                             if j<Data_Points                                 Ev(i)=sma_hysteresis((Load_History(j)/A(i)),(L(i)/Tie_Length-1),(Load_History(j+1)/A(i)),(L_bckp(i)/Tie_Length-1));                             else                                 Ev(i)=sma_hysteresis((Load_History(j)/A(i)),(L(i)/Tie_Length-1),(Load_History(j)/A(i)),(L_bckp(i)/Tie_Length-1));                             end %                             if (Load_History(j-1)>=0) %                                 L(i)=Tie_compressive_length_bckp; %                             end                             Kcoeff(i)=A(i)*Ev(i)/L(i);                         else %If only steel ties are used. negative sign is applied before the load argument as the compression for the whole brace is tension for the ties.                             if (Load_History(j-1)>=0)                                 L(i)=Tie_compressive_length_bckp;                             end                             [Ev(i),Fypos(i),Fyneg(i)]=steel_hysteresis(Load_History(j)/A(i),Load_History(j-1)/A(i),Fypos(i),Fyneg(i),Fy(i),r(i)); %                             if (Load_History(j-1)>=0) %                                 L(i)=Tie_compressive_length_bckp; %                             end                             Kcoeff(i)=A(i)*Ev(i)/L(i);  188                          end                                              elseif (i==2)                         [Ev(i),Fypos(i),Fyneg(i)]=steel_hysteresis(Load_History(j)/A(i),Load_History(j-1)/A(i),Fypos(i),Fyneg(i),Fy(i),r(i));                         Kcoeff(i)=A(i)*Ev(i)/L(i);                     end                     end             end         end     end      %Keeping a backup of the previous steps length matrix     L_bckp=L;   %     disp('Calculated Modulus of elasticity of bar segments') %     Ev; %     disp(['Iteration Step :', num2str(j)] )     %Global Stiffness matrix formation     GlobalK=zeros(n+1,n+1);     %k2 is a temporary stiffness matrix holder     k2=zeros(n+1,n+1); %     disp('Bar lengths :') %     L;     for i=1:n,         k2(i:i+1,i:i+1)=k(:,:,i);         GlobalK=GlobalK+Kcoeff(i)*k2(:,:);         k2=zeros(n+1,n+1);     end     GlobalK;       ReducedK=zeros(n,n);     %Elimination of first row-column of the global stiffness matrix     ReducedK=GlobalK(2:n+1,2:n+1);     ReducedK;     F=zeros(1,n+1);       %     disp('Applied force at bar end')     if j==1         F(1,n+1)=Load_History(j);     else         %Load increment is applied only         F(1,n+1)=Load_History(j)-Load_History(j-1);     end          %Transpose of F matrix %     disp('Force Matrix')     F=F';     ReducedF=zeros(n,1);     ReducedF=F(2:n+1,1:1);       %Global Displacement matrix %     disp('Global Displacement matrix (m)')     ReducedGlobalU=zeros(n,1);     ReducedGlobalU=ReducedK\ReducedF;          GlobalU(2:n+1,1:1)=ReducedGlobalU(1:n,1:1);     GlobalU;       GlobalF=zeros(n+1,1);     for i=1:n+1,  189          GlobalF(i,1)=GlobalK(i:i,1:n+1)*GlobalU;     end       for i=1:n,             L(i)=L(i)+GlobalU(i+1)-GlobalU(i);     end %     disp('Bar lengths :')     L;          Tie_Deformed_length(j)=L(1);     Shaft_Deformed_length(j)=L(2);     if j>1         if (Load_History(j)>0)             Tie_tensile_length_bckp=L(1);         elseif (Load_History(j)<0)             Tie_compressive_length_bckp=L(1);         end     end     %Tie Area change due to poisson's ratio. Absolute value of the percent     %elongation is taken as ties are always in tension (Positive strain).     A(1)=pi*(Tie_dia*(1-abs(Tie_Deformed_length(j)/Tie_Length-1)*v(1)))^2*0.25*No_Of_Ties;          %Shaft Area change due to poisson's ratio     if Dis == 0         A(2)=pi*(Dos*(1-(Shaft_Deformed_length(j)/Shaft_Length-1)*v(2)))^2*.25;     else         Dos=Dshft_centerline+t_shft*(1-(Shaft_Deformed_length(j)/Shaft_Length-1)*v(2)); %Outer diameter         Dis=Dshft_centerline-t_shft*(1-(Shaft_Deformed_length(j)/Shaft_Length-1)*v(2));  %Inner diameter         A(2)=pi*(Dos^2-Dis^2)*.25; %This area should come from a standard section     end          %Total deformed brace length calculation     for i=1:n,         if (Load_History(j)>=0)%If the brace is under tension             if (i==1)                 Total_Deformed_length(j)=Total_Deformed_length(j)+L(i);             elseif (i==2)                 Total_Deformed_length(j)=Total_Deformed_length(j)+L(i);             end         else %If the brace is under compression             if (i==1)                 Total_Deformed_length(j)=Total_Deformed_length(j)+L(i);             elseif (i==2)                 Total_Deformed_length(j)=Total_Deformed_length(j)+L(i);             end         end     end           %Monitoring buckling occurance     if (Load_History(j)<=Pbuck_shaft)         shaft_buckled=1;     end          %Monitoring ultimate strain exceedance     if (abs(L(1)/Tie_Length)>=1.1)         tie_ult_strain_excd=1;     elseif (abs(L(2)/Shaft_Length)>=1.1)         shaft_ult_strain_excd=1;     end  190  end   %%%%%%%%%%%%%%%%%%% %Plots and Outputs% %%%%%%%%%%%%%%%%%%%   %Removing the tie compressive loads as they were not used for the %calculation and reversing the sign for the global load to make it tensile %for the ties for k=1:Data_Points,     if Load_History(k)<0         Tie_Load_History(k)=-Load_History(k);     else         Tie_Load_History(k)=0;     end end %Coordinate calculation for the buckling load plot Deformation_Range=zeros(1,2); Deformation_Range(1)=(max(Total_Deformed_length)-Undeformed_length); Deformation_Range(2)=(min(Total_Deformed_length)-Undeformed_length); % Deformation_Range; Shaft_Buckling_Plot=[Pbuck_shaft,Pbuck_shaft];   %Coordinate calculation for ultimate strain points Load_Range=zeros(1,2); Load_Range(1)=max(Load_History); Load_Range(2)=min(Load_History); Tie_Strain_Range=[eu(1),eu(1)]; Shaft_Strain_Range=[eu(2),eu(2)];   subplot(1,4,1) plot(Load_History/1000) ylabel('Axial Force (kN)'); xlabel('Iteration Step'); grid on;   subplot(1,4,2) plot((Total_Deformed_length-Undeformed_length)*1000) ylabel('Axial Deformation (mm)'); xlabel('Iteration Step'); grid on;   subplot(1,4,3) plot( (Total_Deformed_length-Undeformed_length)*1000, Load_History/1000) ylabel('Axial Force in the brace (kN)'); xlabel('Axial Deformation (mm)'); grid on;   subplot(1,4,4) if (shaft_buckled==1)     plot( (Shaft_Deformed_length-Shaft_Length)*1000, Load_History/1000,Deformation_Range*1000,Shaft_Buckling_Plot/1000) else     plot( (Shaft_Deformed_length-Shaft_Length)*1000, Load_History/1000) end   ylabel('Axial Force in Shaft (kN)'); xlabel('Axial Deformation (mm)'); grid on;     disp('Printing some initially calculated matrix for review')  191  A_backup; Undeformed_length; ep; r; Pypos;  if (shaft_buckled==1)     disp('Shaft buckled') end   Appendix C: MATLAB material models Tension only SE SMA uniaxial material model function Esma = sma_hysteresis(Stress1,Strain1,Stress2,Strain0) %Function requires current stress strain point, Next stress point and one step earlier strain  %Global Variables global Strain_Max; global Residual_Strain; global Ei; global Sams; global Samf; global Smas; global Smaf; global Ep; global SMA_Yield_Strain; global Smax; global residual_ratio; global Strain_Crspnd_Pos; global Strain_amf; global Brace_Under_Tension; global Strain_Max_Tension; global Strain_Max_Compression; global Residual_Strain_Tension; global Residual_Strain_Compression; E1=0; Q1Q4_Strain = Smaf/Ei; Q1Q4_Stress = Smaf;  if (Stress2>0)     Brace_Under_Tension = 1;     Strain_Max = Strain_Max_Tension;     Residual_Strain = Residual_Strain_Tension; elseif (Stress2<0)     Brace_Under_Tension = 0;     Strain_Max = Strain_Max_Compression;     Residual_Strain = Residual_Strain_Compression; end  Stress2=abs(Stress2); Stress1=abs(Stress1); Strain1=abs(Strain1); Strain0=abs(Strain0);  if (Stress2>=Stress1)     %Stress increasing          192      if (Strain1 > Strain_Max)           Strain_Max = Strain1;         Residual_Strain = (Strain_Max - SMA_Yield_Strain) * residual_ratio;         if (Brace_Under_Tension == 1)             Strain_Max_Tension = Strain_Max;             Residual_Strain_Tension = Residual_Strain;         elseif (Brace_Under_Tension == 0)             Strain_Max_Compression = Strain_Max;             Residual_Strain_Compression = Residual_Strain;         end     end          Smax= (Sams * Ei + Ei * Ep * (Strain1 - SMA_Yield_Strain) - Stress1 * Ep) / (Ei - Ep);     Strain_Crspnd_Pos = (Strain1 * Ei + Sams - Ep * SMA_Yield_Strain - Stress1) / (Ei - Ep);     if (Stress1 >= 0 && Stress1 < Smax)          if (Stress2 <= Smax)              Esma = Ei;         elseif (Stress2 > Smax && Stress2 <= Samf)             Esma =(Stress2-Stress1)/(Strain_Crspnd_Pos+(Stress2-Smax)/Ep-Strain1);         elseif (Stress2 > Samf)             Esma = (Stress2-Stress1)/(Strain_amf+(Stress2-Samf)/Ei-Strain1);         end     elseif (Stress1 >= Smax && Stress1 < Samf)          if (Stress2 <= Samf)              Esma = Ep;         else             Esma = (Stress2-Stress1)/(Strain_amf+(Stress2-Samf)/Ei-Strain1);             %Esma = Ei;         end     elseif (Stress1 >= Samf)          Esma = Ei;     end elseif (Stress2 < Stress1) %Stress decreasing     if (Brace_Under_Tension == 1)         Q1Q4_Strain = (Ei / (Ei - E1)) * Residual_Strain_Tension + Smaf/Ei;         Q1Q4_Stress = ((Q1Q4_Strain - Residual_Strain_Tension) * Ei);     elseif (Brace_Under_Tension == 0)         Q1Q4_Strain = (Ei / (Ei - E1)) *Residual_Strain_Compression + Smaf/Ei;         Q1Q4_Stress = ((Q1Q4_Strain -Residual_Strain_Compression) * Ei);     end      if (Stress1 > Smas)          if (Stress2>=Smas)             Esma = Ei;         elseif (Stress2<Smas && Stress2>=Q1Q4_Stress)             Smas_Strain=(Smas - Stress1) / Ei + Strain1;             E1 = (Smas - Smaf) / (Smas_Strain - Smaf / Ei);             Esma = E1;         elseif (Stress2<Q1Q4_Stress)             Esma = (Stress1-Stress2)/(Strain1-Q1Q4_Strain+(Q1Q4_Stress-Stress2)/Ei);         end      elseif (Stress1 > Q1Q4_Stress && Stress1 <= Smas)          if (Stress2 >= Q1Q4_Stress)              E1 = (Stress1 - Smaf) / (Strain1 - Smaf / Ei);             Esma = E1;         else             Esma = (Stress1-Stress2)/(Strain1-Q1Q4_Strain+(Q1Q4_Stress-Stress2)/Ei);         end     elseif (Stress1 <= Q1Q4_Stress)   193          Esma = Ei;     end end  Material model for steel function [Esteel,Fy_pos_out,Fy_neg_out] = steel_hysteresis(Stress1,Stress0,Fy_pos,Fy_neg,Fy,r) Ei = 200e9; Ep=Ei*r;  if (Stress1>=Stress0)     %Loading in tension                                                                   if (Stress1<Fy_pos)         Esteel=Ei;     elseif (Stress1>=Fy_pos)         Esteel=Ep;         Fy_pos=Stress1;         Fy_neg=Fy_pos-2*Fy;     end elseif (Stress1 < Stress0) %Loading in compression     if (Stress1>Fy_neg)         Esteel=Ei;     elseif (Stress1<=Fy_neg)         Esteel=Ep;         Fy_neg=Stress1;         Fy_pos=Fy_neg+2*Fy;     end end Fy_pos_out=Fy_pos; Fy_neg_out=Fy_neg;  Appendix D: PBSC brace hysteresis model Private Sub CommandButton1_Click()      Dim i, N As Integer                                       'Loop Variables     Dim k1, k2, k3, k4, k5, k6, Fy, PMax, PMin, NMin, NMax As Double 'Positive Initial stiffness = k1, Positive Post elatic stiffness = k2,   Yield force in positive direction= Fy, Positive force limit = PMax, Negative force limit = NMin     Dim Deformation_Y_Pos, Deformation_Y_Neg As Double     Dim Variable_Deformation_Y_Pos, Variable_Deformation_Y_Neg As Double     Dim Deformation_Max_Pos, Deformation_Min_Pos As Double     Dim Deformation_Min_Neg, Deformation_Max_Neg As Double     Dim Force() As Double     Dim Deformation() As Double     Dim work As Double     Dim Location() As Double     Dim Tangent_Stiffness() As Double     Dim Deformation_Before_Corner As Double     Dim Q2Q3_Junction_Def, Q1Q4_Junction_Def As Double     Dim Q2Q3_Junction_Force, Q1Q4_Junction_Force As Double     Dim r1, ry As Double     Dim Historical_Deformation_Max, Historical_Deformation_Min As Double     Dim SMA_Rule As Integer     Dim Slippage_Rule As Integer     Dim SMA_Cutoff_Force As Double     Dim SMA_Pos_Cutoff_Force, SMA_Neg_Cutoff_Force As Double  194      Dim SMA_Pos_Cutoff_Def, SMA_Neg_Cutoff_Def As Double     Dim SMA_Max_Force, SMA_Max_Pos_Force, SMA_Min_Neg_Force As Double     Dim SMA_Max_Def, SMA_Min_Def As Double     Dim Res_Def_Pos, Res_Def_Neg As Double     Dim PMax_Array(), Deformation_Max_Pos_Array(), NMin_Array(), Deformation_Min_Neg_Array(), PMin_Array(), Deformation_Min_Pos_Array(), k5_Array(), k6_Array(), Q1Q4_Junction_Def_Array(), Q1Q4_Junction_Force_Array() As Double     Dim Q2Q3_Junction_Def_Array(), Q2Q3_Junction_Force_Array(), SMA_Pos_Cutoff_Force_Array(), SMA_Neg_Cutoff_Force_Array(), NMax_Array(), Historical_Deformation_Max_Array(), Historical_Deformation_Min_Array(), Res_Def_Pos_Array(), Res_Def_Neg_Array() As Double     Dim Variable_Deformation_Y_Pos_Array(), Variable_Deformation_Y_Neg_Array() As Double               For k = 1 To 18     load_num = k          Fy = Worksheets("Final").Cells(1, 2).Value     k1 = Worksheets("Final").Cells(2, 2).Value     r1 = Worksheets("Final").Cells(3, 2).Value     r2 = Worksheets("Final").Cells(4, 2).Value     alpha = Worksheets("Final").Cells(5, 2).Value     beta = Worksheets("Final").Cells(6, 2).Value     ry = Worksheets("Final").Cells(7, 2).Value     N = Worksheets("Load Cases").Cells(1, 2 * load_num - 1).Value     SMA_Rule = Worksheets("Final").Cells(11, 2).Value     SMA_Cutoff_Force = Worksheets("Final").Cells(12, 2).Value     SMA_Max_Force = Worksheets("Final").Cells(13, 2).Value     Slippage_Rule = Worksheets("Final").Cells(14, 2).Value                   If (k1 <= 0 Or k2 < 0 Or Fy < 0 Or k2 >= k1 Or beta > 1 Or load_num <= 0 Or load_num > 18 Or SMA_Rule < 0 Or SMA_Rule > 1 Or Slippage_Rule < 0 Or Slippage_Rule > 1 Or SMA_Cutoff_Force > Fy Or SMA_Cutoff_Force < alpha * Fy Or SMA_Max_Force <= Fy Or ry > 1 Or ry < 0) Then         MsgBox "Invalid Entry", vbOKOnly         Exit Sub     End If        ReDim Force(N) As Double     ReDim Deformation(N) As Double     ReDim Tangent_Stiffness(N) As Double     ReDim Location(N) As Double          For i = 0 To N - 1 'Preloading deformation data for faster testing and debugging         Deformation(i) = Worksheets("Load Cases").Cells(i + 3, 2 * load_num - 1).Value     Next i          ReDim PMax_Array(N), Deformation_Max_Pos_Array(N), NMin_Array(N), Deformation_Min_Neg_Array(N), PMin_Array(N), Deformation_Min_Pos_Array(N), k5_Array(N), k6_Array(N), Q1Q4_Junction_Def_Array(N), Q1Q4_Junction_Force_Array(N) As Double     ReDim Q2Q3_Junction_Def_Array(N), Q2Q3_Junction_Force_Array(N), SMA_Pos_Cutoff_Force_Array(N), SMA_Neg_Cutoff_Force_Array(N), NMax_Array(N), Historical_Deformation_Max_Array(N), Historical_Deformation_Min_Array(N), Res_Def_Pos_Array(N), Res_Def_Neg_Array(N), Variable_Deformation_Y_Pos_Array(N), Variable_Deformation_Y_Neg_Array(N) As Double          k2 = r1 * k1     Tangent_Stiffness(0) = k1     Location(0) = 0  195      PMax = Fy     NMin = -Fy     Deformation_Y_Pos = Fy / k1     Deformation_Y_Neg = -Fy / k1     k3 = k1 * beta     k4 = r2 * k1     work = 0     Deformation_Max_Pos = Fy / k1     Deformation_Min_Neg = -Fy / k1     Deformation_Before_Corner = 0          Historical_Deformation_Max = Deformation_Y_Pos     Historical_Deformation_Min = Deformation_Y_Neg          Q1Q4_Junction_Def = alpha * Deformation_Y_Pos     Q2Q3_Junction_Def = alpha * Deformation_Y_Neg          'Needed for SMA hysteresis rule     SMA_Pos_Cutoff_Force = SMA_Cutoff_Force     SMA_Neg_Cutoff_Force = -SMA_Cutoff_Force          SMA_Pos_Cutoff_Def = SMA_Pos_Cutoff_Force / k1 'Test code     SMA_Neg_Cutoff_Def = SMA_Neg_Cutoff_Force / k1 'Test code          SMA_Max_Pos_Force = SMA_Max_Force     SMA_Min_Neg_Force = -SMA_Max_Force          SMA_Max_Def = (SMA_Max_Pos_Force - Fy) / k2 + Deformation_Y_Pos     SMA_Min_Def = (SMA_Min_Neg_Force + Fy) / k2 + Deformation_Y_Neg          Res_Def_Pos = 0     Res_Def_Neg = 0          Variable_Deformation_Y_Pos = Fy / k1     Variable_Deformation_Y_Neg = -Fy / k1            'Start of the program's main loop     For i = 0 To N - 2 Step 1         If (Deformation(i + 1) = Deformation(i)) Then             Force(i + 1) = Force(i)             Tangent_Stiffness(i + 1) = Tangent_Stiffness(i)             Location(i + 1) = 0         ElseIf (Deformation(i + 1) > Deformation(i)) Then             If (i > 0) Then                 If (Deformation(i) < Deformation_Before_Corner) Then                     If (Deformation(i) < Historical_Deformation_Min) Then                         Historical_Deformation_Min = Deformation(i)                      'Set the current rotation as the minimum of all times                         Res_Def_Neg = (Historical_Deformation_Min - Deformation_Y_Neg) * ry                     End If                     If (Force(i) <= SMA_Neg_Cutoff_Force) Then                         SMA_Neg_Cutoff_Def = (SMA_Neg_Cutoff_Force - Force(i)) / k1 + Deformation(i)                         If (alpha * Deformation_Y_Neg = SMA_Neg_Cutoff_Def) Then                             k6 = 0                         Else                             k6 = (-alpha * Fy - SMA_Neg_Cutoff_Force) / (alpha * Deformation_Y_Neg - SMA_Neg_Cutoff_Def) 'Slope of the line connecting (SMA negative  196  cutoff force, corresponding deformation) point and (-alpha*Fy, -alpha*deformation_y) point                         End If                     ElseIf (Force(i) < Q2Q3_Junction_Force) Then                         If (alpha * Deformation_Y_Neg = Deformation(i)) Then                             k6 = 0                         Else                             k6 = (-alpha * Fy - Force(i)) / (alpha * Deformation_Y_Neg - Deformation(i)) 'Slope of the line connecting (SMA negative cutoff force, corresponding deformation) point and (-alpha*Fy, -alpha*deformation_y) point                         End If                     End If                                               If (k6 = k1) Then                         Q2Q3_Junction_Def = Deformation_Y_Neg * alpha                         Q2Q3_Junction_Force = Q2Q3_Junction_Def * k1                     Else                         Q2Q3_Junction_Def = (k1 / (k1 - k6)) * Res_Def_Neg + Deformation_Y_Neg * alpha                         Q2Q3_Junction_Force = (Q2Q3_Junction_Def - Res_Def_Neg) * k1                     End If                 End If             End If                          If (Force(i) >= 0) Then                 PMax = (Fy * k1 + k1 * k2 * (Res_Def_Pos - Deformation_Y_Pos)) / (k1 - k2)                 Deformation_Max_Pos = (Res_Def_Pos * k1 + Fy - k2 * Deformation_Y_Pos) / (k1 - k2)                 If (Deformation(i) < SMA_Max_Def) Then                     If (Force(i) = 0) Then                         If (Deformation(i + 1) <= Res_Def_Pos) Then                             Force(i + 1) = 0                             Tangent_Stiffness(i + 1) = 0                             Location(i + 1) = 1.211                         ElseIf (Deformation(i + 1) > Res_Def_Pos And Deformation(i + 1) <= SMA_Max_Def) Then                             If ((k1 * (Deformation(i + 1) - Res_Def_Pos)) <= PMax) Then                                 Force(i + 1) = k1 * (Deformation(i + 1) - Res_Def_Pos)                                 Tangent_Stiffness(i + 1) = k1                                 Location(i + 1) = 1.212                             Else                                 Force(i + 1) = k2 * (Deformation(i + 1) - Deformation_Max_Pos) + PMax                                 Tangent_Stiffness(i + 1) = k2                                 Location(i + 1) = 1.213                             End If                         ElseIf (Deformation(i + 1) > SMA_Max_Def) Then                             Force(i + 1) = k1 * (Deformation(i + 1) - SMA_Max_Def) + SMA_Max_Pos_Force                             Tangent_Stiffness(i + 1) = k1                             Location(i + 1) = 1.214                         End If                     ElseIf (Force(i) < PMax) Then 'if current force is less than the limit                         If (k1 * (Deformation(i + 1) - Deformation(i)) + Force(i) <= PMax) Then 'if next force is also less than the limit                             Force(i + 1) = k1 * (Deformation(i + 1) - Deformation(i)) + Force(i)                             Tangent_Stiffness(i + 1) = k1  197                              Location(i + 1) = 1.215                         Else    'if next force is more than the limit                             If (k2 * (Deformation(i + 1) - Deformation_Max_Pos) + PMax <= SMA_Max_Pos_Force) Then                                 Force(i + 1) = k2 * (Deformation(i + 1) - Deformation_Max_Pos) + PMax                                 Tangent_Stiffness(i + 1) = k2                                 Location(i + 1) = 1.216                             Else                                 Force(i + 1) = k1 * (Deformation(i + 1) - SMA_Max_Def) + SMA_Max_Pos_Force                                 Tangent_Stiffness(i + 1) = k1                                 Location(i + 1) = 1.217                             End If                         End If                     ElseIf (Force(i) >= PMax) Then                         If (Deformation(i + 1) <= SMA_Max_Def) Then                             Force(i + 1) = k2 * (Deformation(i + 1) - Deformation(i)) + Force(i)                             Tangent_Stiffness(i + 1) = k2                             Location(i + 1) = 1.218                         Else                             Force(i + 1) = k1 * (Deformation(i + 1) - SMA_Max_Def) + SMA_Max_Pos_Force                             Tangent_Stiffness(i + 1) = k1                             Location(i + 1) = 1.219                         End If                     End If                 ElseIf (Deformation(i) >= SMA_Max_Def) Then                     Force(i + 1) = k1 * (Deformation(i + 1) - Deformation(i)) + Force(i)                     Tangent_Stiffness(i + 1) = k1                     Location(i + 1) = 1.22                 End If             ElseIf (Force(i) < 0) Then 'if current force is negative and deformation is increasing                 If (k1 * Deformation(i) - Force(i) - k1 * Res_Def_Neg < 0) Then 'Any point on the left side of the line with stiffness k1 and going through (Res_def_neg, 0) point                     If (Force(i) <= SMA_Neg_Cutoff_Force) Then 'If current force level is less than the Martensite to Austenite starting stress                         If (k1 * (Deformation(i + 1) - Deformation(i)) + Force(i) <= SMA_Neg_Cutoff_Force) Then 'If next force level is also less than the Martensite to Austenite starting stress                             Force(i + 1) = k1 * (Deformation(i + 1) - Deformation(i)) + Force(i)                             Tangent_Stiffness(i + 1) = k1                             Location(i + 1) = 2.1                         ElseIf (k1 * (Deformation(i + 1) - Deformation(i)) + Force(i) > SMA_Neg_Cutoff_Force) Then 'If next force level is more than the Martensite to Austenite starting stress                             If (Deformation(i + 1) <= Q2Q3_Junction_Def) Then                                 Force(i + 1) = k6 * (Deformation(i + 1) - SMA_Neg_Cutoff_Def) + SMA_Neg_Cutoff_Force                                 Tangent_Stiffness(i + 1) = k6                                 Location(i + 1) = 2.11                             ElseIf (Deformation(i + 1) > Q2Q3_Junction_Def) Then                                 PMax = (Fy * k1 + k1 * k2 * (Res_Def_Pos - Deformation_Y_Pos)) / (k1 - k2)                                 Deformation_Max_Pos = (Res_Def_Pos * k1 + Fy - k2 * Deformation_Y_Pos) / (k1 - k2)  198                                  If (k1 * (Deformation(i + 1) - Q2Q3_Junction_Def) + Q2Q3_Junction_Force <= 0) Then                                     Force(i + 1) = k1 * (Deformation(i + 1) - Q2Q3_Junction_Def) + Q2Q3_Junction_Force                                     Tangent_Stiffness(i + 1) = k1                                     Location(i + 1) = 2.14                                 ElseIf (k1 * (Deformation(i + 1) - Q2Q3_Junction_Def) + Q2Q3_Junction_Force > 0) Then                                     If (Deformation(i + 1) <= Res_Def_Pos) Then                                         Force(i + 1) = 0                                         Tangent_Stiffness(i + 1) = 0                                         Location(i + 1) = 2.15                                     ElseIf (Deformation(i + 1) > Res_Def_Pos) Then                                         If ((k1 * (Deformation(i + 1) - Res_Def_Pos)) <= PMax) Then                                             Force(i + 1) = k1 * (Deformation(i + 1) - Res_Def_Pos)                                             Tangent_Stiffness(i + 1) = k1                                             Location(i + 1) = 2.16                                         Else                                             If ((k2 * (Deformation(i + 1) - Deformation_Max_Pos) + PMax) <= SMA_Max_Pos_Force) Then                                                 Force(i + 1) = k2 * (Deformation(i + 1) - Deformation_Max_Pos) + PMax                                                 Tangent_Stiffness(i + 1) = k2                                                 'Variable_Deformation_Y_Pos = Deformation_Max_Pos                                                 Location(i + 1) = 2.17                                             Else                                                 Force(i + 1) = k1 * (Deformation(i + 1) - SMA_Max_Def) + SMA_Max_Pos_Force                                                 Tangent_Stiffness(i + 1) = k1                                                 Location(i + 1) = 2.18                                             End If                                         End If                                     End If                                 End If                             End If                         End If                     ElseIf (Force(i) > SMA_Neg_Cutoff_Force And Force(i) < Q2Q3_Junction_Force) Then                         If ((k6 * (Deformation(i + 1) - Deformation(i)) + Force(i)) <= Q2Q3_Junction_Force) Then                             Force(i + 1) = k6 * (Deformation(i + 1) - Deformation(i)) + Force(i)                             Tangent_Stiffness(i + 1) = k6                             Location(i + 1) = 2.2                         Else                             PMax = (Fy * k1 + k1 * k2 * (Res_Def_Pos - Deformation_Y_Pos)) / (k1 - k2)                             Deformation_Max_Pos = (Res_Def_Pos * k1 + Fy - k2 * Deformation_Y_Pos) / (k1 - k2)                             If (k1 * (Deformation(i + 1) - Q2Q3_Junction_Def) + Q2Q3_Junction_Force <= 0) Then                                     Force(i + 1) = k1 * (Deformation(i + 1) - Q2Q3_Junction_Def) + Q2Q3_Junction_Force                                     Tangent_Stiffness(i + 1) = k1                                     Location(i + 1) = 2.23                             Else                                 If (Deformation(i + 1) <= Res_Def_Pos) Then                                     Force(i + 1) = 0                                     Tangent_Stiffness(i + 1) = 0  199                                      Location(i + 1) = 2.24                                 ElseIf (Deformation(i + 1) > Res_Def_Pos) Then                                     If ((k1 * (Deformation(i + 1) - Res_Def_Pos)) <= PMax) Then                                         Force(i + 1) = k1 * (Deformation(i + 1) - Res_Def_Pos)                                         Tangent_Stiffness(i + 1) = k1                                         Location(i + 1) = 2.25                                     Else                                         If ((k2 * (Deformation(i + 1) - Deformation_Max_Pos) + PMax) <= SMA_Max_Pos_Force) Then                                             Force(i + 1) = k2 * (Deformation(i + 1) - Deformation_Max_Pos) + PMax                                             Tangent_Stiffness(i + 1) = k2                                             'Variable_Deformation_Y_Pos = Deformation_Max_Pos                                             Location(i + 1) = 2.26                                         Else                                             Force(i + 1) = k1 * (Deformation(i + 1) - SMA_Max_Def) + SMA_Max_Pos_Force                                             Tangent_Stiffness(i + 1) = k1                                             Location(i + 1) = 2.27                                         End If                                     End If                                 End If                             End If                          End If                     ElseIf (Force(i) >= Q2Q3_Junction_Force) Then                         PMax = (Fy * k1 + k1 * k2 * (Res_Def_Pos - Deformation_Y_Pos)) / (k1 - k2)                         Deformation_Max_Pos = (Res_Def_Pos * k1 + Fy - k2 * Deformation_Y_Pos) / (k1 - k2)                         If (k1 * (Deformation(i + 1) - Deformation(i)) + Force(i) <= 0) Then                             Force(i + 1) = k1 * (Deformation(i + 1) - Deformation(i)) + Force(i)                             Tangent_Stiffness(i + 1) = k1                             Location(i + 1) = 2.32                         Else                             If (Deformation(i + 1) <= Res_Def_Pos) Then                                 Force(i + 1) = 0                                 Tangent_Stiffness(i + 1) = 0                                 Location(i + 1) = 2.33                             ElseIf (Deformation(i + 1) > Res_Def_Pos) Then                                 If ((k1 * (Deformation(i + 1) - Res_Def_Pos)) <= PMax) Then                                     Force(i + 1) = k1 * (Deformation(i + 1) - Res_Def_Pos)                                     Tangent_Stiffness(i + 1) = k1                                     Location(i + 1) = 2.34                                 Else                                     If ((k2 * (Deformation(i + 1) - Deformation_Max_Pos) + PMax) <= SMA_Max_Pos_Force) Then                                         Force(i + 1) = k2 * (Deformation(i + 1) - Deformation_Max_Pos) + PMax                                         Tangent_Stiffness(i + 1) = k2                                         Location(i + 1) = 2.35                                     Else                                         Force(i + 1) = k1 * (Deformation(i + 1) - SMA_Max_Def) + SMA_Max_Pos_Force                                         Tangent_Stiffness(i + 1) = k1  200                                          Location(i + 1) = 2.36                                     End If                                 End If                             End If                         End If                     End If                 ElseIf (k1 * Deformation(i) - Force(i) - k1 * Res_Def_Neg >= 0) Then                     PMax = (Fy * k1 + k1 * k2 * (Res_Def_Pos - Deformation_Y_Pos)) / (k1 - k2)                     Deformation_Max_Pos = (Res_Def_Pos * k1 + Fy - k2 * Deformation_Y_Pos) / (k1 - k2)                     If (k1 * (Deformation(i + 1) - Deformation(i)) + Force(i) <= 0) Then                         Force(i + 1) = k1 * (Deformation(i + 1) - Deformation(i)) + Force(i)                         Tangent_Stiffness(i + 1) = k1                         Location(i + 1) = 2.42                     Else                         If (Deformation(i + 1) <= Res_Def_Pos) Then                             Force(i + 1) = 0                             Tangent_Stiffness(i + 1) = 0                             Location(i + 1) = 2.43                         ElseIf (Deformation(i + 1) > Res_Def_Pos) Then                             If ((k1 * (Deformation(i + 1) - Res_Def_Pos)) <= PMax) Then                                 Force(i + 1) = k1 * (Deformation(i + 1) - Res_Def_Pos)                                 Tangent_Stiffness(i + 1) = k1                                 Location(i + 1) = 2.44                             Else                                 If ((k2 * (Deformation(i + 1) - Deformation_Max_Pos) + PMax) <= SMA_Max_Pos_Force) Then                                     Force(i + 1) = k2 * (Deformation(i + 1) - Deformation_Max_Pos) + PMax                                     Tangent_Stiffness(i + 1) = k2                                     'Variable_Deformation_Y_Pos = Deformation_Max_Pos                                     Location(i + 1) = 2.45                                 Else                                     Force(i + 1) = k1 * (Deformation(i + 1) - SMA_Max_Def) + SMA_Max_Pos_Force                                     Tangent_Stiffness(i + 1) = k1                                     Location(i + 1) = 2.46                                 End If                             End If                         End If                     End If                 End If             End If             Deformation_Before_Corner = Deformation(i)         ElseIf (Deformation(i + 1) < Deformation(i)) Then                                 'deformation decreasing             If (i > 0) Then                 If (Deformation(i) > Deformation_Before_Corner) Then                         ' Corner point determination                     If (Deformation(i) > Historical_Deformation_Max) Then                         Historical_Deformation_Max = Deformation(i) 'At every corner point the historical deformation max is changing!!! I have to check the logic behind this                         If (Slippage_Rule = 0) Then                             Res_Def_Pos = (Historical_Deformation_Max - Variable_Deformation_Y_Pos) * ry + Res_Def_Neg                         Else  201                              Res_Def_Pos = (Historical_Deformation_Max - Deformation_Y_Pos) * ry                         End If                     End If                     If (Force(i) >= SMA_Pos_Cutoff_Force) Then                         SMA_Pos_Cutoff_Def = (SMA_Pos_Cutoff_Force - Force(i)) / k1 + Deformation(i)                         If (alpha * Deformation_Y_Pos = SMA_Pos_Cutoff_Def) Then                             k5 = 0                         Else                             k5 = (alpha * Fy - SMA_Pos_Cutoff_Force) / (alpha * Deformation_Y_Pos - SMA_Pos_Cutoff_Def)                         End If                     ElseIf (Force(i) > Q1Q4_Junction_Force) Then                         If (alpha * Deformation_Y_Pos = Deformation(i)) Then                             k5 = 0                         Else                             k5 = (alpha * Fy - Force(i)) / (alpha * Deformation_Y_Pos - Deformation(i))                         End If                     End If                     If (k5 = k1) Then                         Q1Q4_Junction_Def = alpha * Deformation_Y_Pos                         Q1Q4_Junction_Force = Q1Q4_Junction_Def * k1                     Else                         Q1Q4_Junction_Def = (k1 / (k1 - k5)) * Res_Def_Pos + Deformation_Y_Pos * alpha                         Q1Q4_Junction_Force = ((Q1Q4_Junction_Def - Res_Def_Pos) * k1)                     End If                 End If             End If              If (Force(i) <= 0) Then                 NMin = (-Fy * k1 + k1 * k2 * (Res_Def_Neg - Deformation_Y_Neg)) / (k1 - k2)                 Deformation_Min_Neg = (Res_Def_Neg * k1 - Fy - k2 * Deformation_Y_Neg) / (k1 - k2)                 If (Deformation(i) > SMA_Min_Def) Then                     If (Force(i) = 0) Then                         If (Deformation(i + 1) >= Res_Def_Neg) Then                             Force(i + 1) = 0                             Tangent_Stiffness(i + 1) = 0                             Location(i + 1) = -1.2                         ElseIf (Deformation(i + 1) < Res_Def_Neg And Deformation(i + 1) >= SMA_Min_Def) Then                             If ((k1 * (Deformation(i + 1) - Res_Def_Neg)) >= NMin) Then                                 Force(i + 1) = k1 * (Deformation(i + 1) - Res_Def_Neg)                                 Tangent_Stiffness(i + 1) = k1                                 Location(i + 1) = -1.21                             Else                                 Force(i + 1) = k2 * (Deformation(i + 1) - Deformation_Min_Neg) + NMin                                 Tangent_Stiffness(i + 1) = k2                                 Location(i + 1) = -1.22                             End If                         ElseIf (Deformation(i + 1) < SMA_Min_Def) Then                             Force(i + 1) = k1 * (Deformation(i + 1) - SMA_Min_Def) + SMA_Min_Neg_Force                             Tangent_Stiffness(i + 1) = k1                             Location(i + 1) = -1.23                         End If  202                      ElseIf (Force(i) > NMin) Then                         If (k1 * (Deformation(i + 1) - Deformation(i)) + Force(i) >= NMin) Then                             Force(i + 1) = k1 * (Deformation(i + 1) - Deformation(i)) + Force(i)                             Tangent_Stiffness(i + 1) = k1                             Location(i + 1) = -1.24                         Else                             If (k2 * (Deformation(i + 1) - Deformation_Min_Neg) + NMin >= SMA_Min_Neg_Force) Then                                 Force(i + 1) = k2 * (Deformation(i + 1) - Deformation_Min_Neg) + NMin                                 Tangent_Stiffness(i + 1) = k2                                 Location(i + 1) = -1.25                             Else                                 Force(i + 1) = k1 * (Deformation(i + 1) - SMA_Min_Def) + SMA_Min_Neg_Force                                 Tangent_Stiffness(i + 1) = k1                                 Location(i + 1) = -1.26                             End If                         End If                     ElseIf (Force(i) <= NMin) Then                         If (Deformation(i + 1) >= SMA_Min_Def) Then                             Force(i + 1) = k2 * (Deformation(i + 1) - Deformation(i)) + Force(i)                             Tangent_Stiffness(i + 1) = k2                             Location(i + 1) = -1.27                         Else                             Force(i + 1) = k1 * (Deformation(i + 1) - SMA_Min_Def) + SMA_Min_Neg_Force                             Tangent_Stiffness(i + 1) = k1                             Location(i + 1) = -1.28                         End If                     End If                 ElseIf (Deformation(i) <= SMA_Min_Def) Then                     Force(i + 1) = k1 * (Deformation(i + 1) - Deformation(i)) + Force(i)                     Tangent_Stiffness(i + 1) = k1                     Location(i + 1) = -1.29                 End If             ElseIf (Force(i) > 0) Then 'if current force is positive and deformation is decreasing                 If (k1 * Deformation(i) - Force(i) - k1 * Res_Def_Pos > 0) Then 'Any point on the right side of the line with stiffness k1 and going through (Res_def_pos, 0) point                     If (Force(i) >= SMA_Pos_Cutoff_Force) Then 'If current force level is more than the Martensite to Austenite starting stress                         If (k1 * (Deformation(i + 1) - Deformation(i)) + Force(i) >= SMA_Pos_Cutoff_Force) Then                             Force(i + 1) = k1 * (Deformation(i + 1) - Deformation(i)) + Force(i)                             Tangent_Stiffness(i + 1) = k1                             Location(i + 1) = -2.1                         Else 'if next force/force is less than the SMA_pos_cutoff_force                             If (Deformation(i + 1) >= Q1Q4_Junction_Def) Then                                 Force(i + 1) = k5 * (Deformation(i + 1) - SMA_Pos_Cutoff_Def) + SMA_Pos_Cutoff_Force                                 Tangent_Stiffness(i + 1) = k5                                 Location(i + 1) = -2.21                             Else  203                                  NMin = (-Fy * k1 + k1 * k2 * (Res_Def_Neg - Deformation_Y_Neg)) / (k1 - k2)                                 Deformation_Min_Neg = (Res_Def_Neg * k1 - Fy - k2 * Deformation_Y_Neg) / (k1 - k2)                                 If (k1 * (Deformation(i + 1) - Q1Q4_Junction_Def) + Q1Q4_Junction_Force >= 0) Then                                     Force(i + 1) = k1 * (Deformation(i + 1) - Q1Q4_Junction_Def) + Q1Q4_Junction_Force                                     Tangent_Stiffness(i + 1) = k1                                     Location(i + 1) = -2.24                                 ElseIf (k1 * (Deformation(i + 1) - Q1Q4_Junction_Def) + Q1Q4_Junction_Force < 0) Then                                     If (Deformation(i + 1) >= Res_Def_Neg) Then                                         Force(i + 1) = 0                                         Tangent_Stiffness(i + 1) = 0                                         Location(i + 1) = -2.25                                     ElseIf (Deformation(i + 1) < Res_Def_Neg) Then                                         If ((k1 * (Deformation(i + 1) - Res_Def_Neg)) >= NMin) Then                                             Force(i + 1) = k1 * (Deformation(i + 1) - Res_Def_Neg)                                             Tangent_Stiffness(i + 1) = k1                                             Location(i + 1) = -2.26                                         Else                                             If ((k2 * (Deformation(i + 1) - Deformation_Min_Neg) + NMin) >= SMA_Min_Neg_Force) Then                                                 Force(i + 1) = k2 * (Deformation(i + 1) - Deformation_Min_Neg) + NMin                                                 Tangent_Stiffness(i + 1) = k2                                                 Location(i + 1) = -2.27                                             Else                                                 Force(i + 1) = k1 * (Deformation(i + 1) - SMA_Min_Def) + SMA_Min_Neg_Force                                                 Tangent_Stiffness(i + 1) = k1                                                 Location(i + 1) = -2.28                                             End If                                         End If                                     End If                                 End If                             End If                         End If                     ElseIf (Force(i) < SMA_Pos_Cutoff_Force And Force(i) > Q1Q4_Junction_Force) Then                         If ((k5 * (Deformation(i + 1) - Deformation(i)) + Force(i)) >= Q1Q4_Junction_Force) Then                             Force(i + 1) = k5 * (Deformation(i + 1) - Deformation(i)) + Force(i)                             Tangent_Stiffness(i + 1) = k5                             Location(i + 1) = -2.3                         Else                             NMin = (-Fy * k1 + k1 * k2 * (Res_Def_Neg - Deformation_Y_Neg)) / (k1 - k2)                             Deformation_Min_Neg = (Res_Def_Neg * k1 - Fy - k2 * Deformation_Y_Neg) / (k1 - k2)                             If ((k1 * (Deformation(i + 1) - Q1Q4_Junction_Def) + Q1Q4_Junction_Force) >= 0) Then                                 Force(i + 1) = k1 * (Deformation(i + 1) - Q1Q4_Junction_Def) + Q1Q4_Junction_Force                                 Tangent_Stiffness(i + 1) = k1                                 Location(i + 1) = -2.33                             Else                                 If (Deformation(i + 1) >= Res_Def_Neg) Then  204                                      Force(i + 1) = 0                                     Tangent_Stiffness(i + 1) = 0                                     Location(i + 1) = -2.34                                 ElseIf (Deformation(i + 1) < Res_Def_Neg) Then                                     If ((k1 * (Deformation(i + 1) - Res_Def_Neg)) >= NMin) Then                                         Force(i + 1) = k1 * (Deformation(i + 1) - Res_Def_Neg)                                         Tangent_Stiffness(i + 1) = k1                                         Location(i + 1) = -2.35                                     Else                                         If ((k2 * (Deformation(i + 1) - Deformation_Min_Neg) + NMin) >= SMA_Min_Neg_Force) Then                                             Force(i + 1) = k2 * (Deformation(i + 1) - Deformation_Min_Neg) + NMin                                             Tangent_Stiffness(i + 1) = k2                                             Location(i + 1) = -2.36                                         Else                                             Force(i + 1) = k1 * (Deformation(i + 1) - SMA_Min_Def) + SMA_Min_Neg_Force                                             Tangent_Stiffness(i + 1) = k1                                             Location(i + 1) = -2.37                                         End If                                     End If                                 End If                             End If                         End If                     ElseIf (Force(i) <= Q1Q4_Junction_Force) Then                         NMin = (-Fy * k1 + k1 * k2 * (Res_Def_Neg - Deformation_Y_Neg)) / (k1 - k2)                         Deformation_Min_Neg = (Res_Def_Neg * k1 - Fy - k2 * Deformation_Y_Neg) / (k1 - k2)                         If (k1 * (Deformation(i + 1) - Deformation(i)) + Force(i) >= 0) Then                             Force(i + 1) = k1 * (Deformation(i + 1) - Deformation(i)) + Force(i)                             Tangent_Stiffness(i + 1) = k1                             Location(i + 1) = -2.43                         Else                             If (Deformation(i + 1) >= Res_Def_Neg) Then                                 Force(i + 1) = 0                                 Tangent_Stiffness(i + 1) = 0                                 Location(i + 1) = -2.44                             ElseIf (Deformation(i + 1) < Res_Def_Neg) Then                                 If ((k1 * (Deformation(i + 1) - Res_Def_Neg)) >= NMin) Then                                     Force(i + 1) = k1 * (Deformation(i + 1) - Res_Def_Neg)                                     Tangent_Stiffness(i + 1) = k1                                     Location(i + 1) = -2.45                                 Else                                     If ((k2 * (Deformation(i + 1) - Deformation_Min_Neg) + NMin) >= SMA_Min_Neg_Force) Then                                         Force(i + 1) = k2 * (Deformation(i + 1) - Deformation_Min_Neg) + NMin                                         Tangent_Stiffness(i + 1) = k2                                         Location(i + 1) = -2.46                                     Else                                         Force(i + 1) = k1 * (Deformation(i + 1) - SMA_Min_Def) + SMA_Min_Neg_Force                                         Tangent_Stiffness(i + 1) = k1                                         Location(i + 1) = -2.47  205                                      End If                                 End If                             End If                         End If                     End If                 Else 'Any point on the left side of the line with stiffness k1 and going through (Res_def_pos, 0) point                     NMin = (-Fy * k1 + k1 * k2 * (Res_Def_Neg - Deformation_Y_Neg)) / (k1 - k2)                     Deformation_Min_Neg = (Res_Def_Neg * k1 - Fy - k2 * Deformation_Y_Neg) / (k1 - k2)                     If (k1 * (Deformation(i + 1) - Deformation(i)) + Force(i) >= 0) Then                         Force(i + 1) = k1 * (Deformation(i + 1) - Deformation(i)) + Force(i)                         Tangent_Stiffness(i + 1) = k1                         Location(i + 1) = -2.53                     Else                         If (Deformation(i + 1) >= Res_Def_Neg) Then                             Force(i + 1) = 0                             Tangent_Stiffness(i + 1) = 0                             Location(i + 1) = -2.54                         ElseIf (Deformation(i + 1) < Res_Def_Neg) Then                             If ((k1 * (Deformation(i + 1) - Res_Def_Neg)) >= NMin) Then                                 Force(i + 1) = k1 * (Deformation(i + 1) - Res_Def_Neg)                                 Tangent_Stiffness(i + 1) = k1                                 Location(i + 1) = -2.55                             Else                                 If ((k2 * (Deformation(i + 1) - Deformation_Min_Neg) + NMin) >= SMA_Min_Neg_Force) Then                                     Force(i + 1) = k2 * (Deformation(i + 1) - Deformation_Min_Neg) + NMin                                     Tangent_Stiffness(i + 1) = k2                                     Location(i + 1) = -2.56                                 Else                                     Force(i + 1) = k1 * (Deformation(i + 1) - SMA_Min_Def) + SMA_Min_Neg_Force                                     Tangent_Stiffness(i + 1) = k1                                     Location(i + 1) = -2.57                                 End If                             End If                         End If                     End If                 End If             End If             Deformation_Before_Corner = Deformation(i)         End If                  PMax_Array(i) = PMax         Deformation_Max_Pos_Array(i) = Deformation_Max_Pos         NMin_Array(i) = NMin         Deformation_Min_Neg_Array(i) = Deformation_Min_Neg         PMin_Array(i) = PMin         Deformation_Min_Pos_Array(i) = Deformation_Min_Pos         k5_Array(i) = k5         k6_Array(i) = k6         Q1Q4_Junction_Def_Array(i) = Q1Q4_Junction_Def         Q1Q4_Junction_Force_Array(i) = Q1Q4_Junction_Force         Q2Q3_Junction_Def_Array(i) = Q2Q3_Junction_Def         Q2Q3_Junction_Force_Array(i) = Q2Q3_Junction_Force         SMA_Pos_Cutoff_Force_Array(i) = SMA_Pos_Cutoff_Force  206          SMA_Neg_Cutoff_Force_Array(i) = SMA_Neg_Cutoff_Force         NMax_Array(i) = NMax         Historical_Deformation_Max_Array(i) = Historical_Deformation_Max         Historical_Deformation_Min_Array(i) = Historical_Deformation_Min         Res_Def_Pos_Array(i) = Res_Def_Pos         Res_Def_Neg_Array(i) = Res_Def_Neg         Variable_Deformation_Y_Pos_Array(i) = Variable_Deformation_Y_Pos         Variable_Deformation_Y_Neg_Array(i) = Variable_Deformation_Y_Neg     Next i                   Dim Test_Case As Integer     Dim Sheet_Name As String     Test_Case = Worksheets("Final").Cells(16, 2).Value     If (load_num > 9) Then         Sheet_Name = "SDOF_crv (" & CStr(Test_Case) & CStr(load_num) & ")"     Else         Sheet_Name = "SDOF_crv (" & CStr(Test_Case) & "0" & CStr(load_num) & ")"     End If     Sheets.Add.Name = Sheet_Name          Worksheets(Sheet_Name).Cells(1, 1).Value = SMA_Rule + 6     Worksheets(Sheet_Name).Cells(1, 2).Value = Fy     Worksheets(Sheet_Name).Cells(1, 3).Value = k1     Worksheets(Sheet_Name).Cells(1, 4).Value = r1     Worksheets(Sheet_Name).Cells(1, 5).Value = r2     Worksheets(Sheet_Name).Cells(1, 6).Value = alpha     Worksheets(Sheet_Name).Cells(1, 7).Value = beta     Worksheets(Sheet_Name).Cells(1, 8).Value = ry     Worksheets(Sheet_Name).Cells(1, 9).Value = SMA_Cutoff_Force     Worksheets(Sheet_Name).Cells(1, 10).Value = SMA_Max_Force     Worksheets(Sheet_Name).Cells(1, 11).Value = Slippage_Rule          For i = 0 To N - 1         Worksheets(Sheet_Name).Cells(i + 2, 1).Value = Deformation(i)         Worksheets(Sheet_Name).Cells(i + 2, 2).Value = Force(i)     Next i     DrawChart          Erase Force, Deformation, Tangent_Stiffness, Location     Erase PMax_Array, Deformation_Max_Pos_Array, NMin_Array, Deformation_Min_Neg_Array, PMin_Array, Deformation_Min_Pos_Array, k5_Array, k6_Array, Q1Q4_Junction_Def_Array, Q1Q4_Junction_Force_Array, Variable_Deformation_Y_Pos_Array, Variable_Deformation_Y_Neg_Array     Erase Q2Q3_Junction_Def_Array, Q2Q3_Junction_Force_Array, SMA_Pos_Cutoff_Force_Array, SMA_Neg_Cutoff_Force_Array, NMax_Array, Historical_Deformation_Max_Array, Historical_Deformation_Min_Array, Res_Def_Pos_Array, Res_Def_Neg_Array          Next k     End Sub  Public Function Larger(ByVal Num1 As Double, ByVal Num2 As Double) As Double     If (Num1 >= Num2) Then         Larger = Num1     Else         Larger = Num2     End If End Function  Public Function Smaller(ByVal Num1 As Double, ByVal Num2 As Double) As Double     If (Num1 <= Num2) Then  207          Smaller = Num1     Else         Smaller = Num2     End If End Function  Sub DrawChart()     Dim LastRow As Long     Dim Rng1 As Range     Dim ShName As String     With ActiveSheet         LastRow = .Range("A" & .Rows.Count).End(xlUp).Row         Set Rng1 = .Range("A2:A" & LastRow & ", B2:B" & LastRow)         ShName = .Name     End With          Charts.Add     With ActiveChart         .ChartType = xlXYScatterLines         .SetSourceData Source:=Rng1         .Location Where:=xlLocationAsObject, Name:=ShName     End With          With ActiveChart.SeriesCollection(1)         .MarkerStyle = xlMarkerStyleNone     End With          ActiveChart.SeriesCollection(1).Select     With Selection.Format.Line         .Visible = msoTrue         '.DashStyle = msoLineSysDash         .ForeColor.RGB = RGB(60, 60, 60) 'now it works     End With  End Sub   

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