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Performance-based seismic design and assessment of concrete bridge piers reinforced with shape memory… Billah, Abu Hena MD Muntasir 2015

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PERFORMANCE-BASED SEISMIC DESIGN AND ASSESSMENT OF CONCRETE BRIDGE PIERS REINFORCED WITH SHAPE MEMORY ALLOY REBAR  by  Abu Hena MD Muntasir Billah   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)  August 2015   © Abu Hena MD Muntasir Billah, 2015 ABSTRACT Recent advancements in numerical analysis and computational power have pushed the current bridge design specifications towards a more descriptive performance-based seismic design (PBSD) approach as compared to the conventional force-based method. One major attributes of this PBSD is to keep bridges operational and reduce the repair cost by limiting the global and local deformations of a bridge to acceptable levels under design loads. Shape memory alloy (SMA), with its distinct superelasticity, shape memory effect and hysteretic damping, is a promising material for the application in bridge piers to attain the objectives of PBSD. The objective of this research is to develop a performance-based seismic design guideline for concrete bridge pier reinforced with different types of SMAs. With the aim of providing a comprehensive design guideline, this study started with the experimental investigation of bond behavior of smooth and sand coated SMA rebar in concrete using pushout specimens. The test results were explored to evaluate the influence of concrete strength, bar diameter, embedment length, and surface condition. In addition, a plastic hinge length expression for SMA-RC bridge pier was developed which can be used for calculating the flexural displacement capacity and design of SMA-RC bridge pier. Using Incremental Dynamic Analysis (IDA), this study developed quantitative damage states corresponding to different performance levels (cracking, yielding, and strength degradation) and specific probabilistic distributions for RC bridge piers reinforced with different types of SMAs. Based on an extensive numerical study, the author proposed residual drift based damage states for SMA-RC pier. Based on the proposed damage states, a sequential procedure for the performance-based design of SMA-RC bridge pier is developed using a combination of residual and maximum drift. Finally, in order to elucidate the potential benefit and applicability of the proposed guideline, fragility curves and seismic hazard curves for different SMA-RC bridge piers are developed considering maximum and residual drift as engineering demand parameters. It is found that the SMA-RC bridge piers designed following the proposed design guideline have very low probability of damage resulting in a lower annual loss which will provide significant financial benefit in the long run.  ii  PREFACE • A version of chapter 2 has been submitted in Engineering Structures, Elsevier. Billah, A.H.M.M. and Alam, M.S. 2015. Application of Shape Memory Alloy in Bridges: Research, Application and Opportunities, Engineering Structures. I wrote the manuscript which was further edited by Dr. Alam. A version of chapter 2 has been published in World Research & Innovation Convention on Engineering & Technology 2014. Alam. M.S. and Billah, A.H.M.M. 2014. Utilizing Shape Memory Alloys (SMAs) for safer and sustainable civil infrastructures. In World Research & Innovation Convention on Engineering & Technology 2014, Putrajaya, Malaysia, 25-26 November 2014. • A version of chapter 3 has been published in Structure and Infrastructure Engineering, Taylor and Francis. Billah, A.H.M.M. and Alam, M.S. 2014. Seismic Fragility Assessment of Highway Bridges: A State-of-The-Art Review. In Press: Structure and Infrastructure Engineering. DOI:10.1080/15732479.2014.912243. I wrote the manuscript which was further edited by Dr. Alam. • A version of chapter 4 has been submitted in Structures, Elsevier. Billah, A.H.M.M. and Alam, M.S. 2015. Bond behavior of plain and modified Shape Memory Alloy rebar in concrete. Submitted in: Structures, Manuscript ID D-15-00078.R1. I conducted experimental investigation and wrote the manuscript which was further edited by Dr. Alam. • A version of chapter 5 has been submitted in Engineering Structures, Elsevier. Billah, A.H.M.M. and Alam, M.S. 2015. Plastic hinge length of Shape Memory Alloy reinforced concrete column. Submitted in: Engineering Structures, Manuscript ID ENGSTRUCT-D-15-00849 S-2015-048. I conducted the numerical analysis and wrote the manuscript which was further edited by Dr. Alam. • A version of chapter 6 has been submitted in Journal of Structural Engineering, ASCE. Billah, A.H.M.M. and Alam, M.S. 2014. Performance based seismic design of concrete bridge pier reinforced with Shape Memory Alloy- Part 1: Development of Performance-Based Damage States. Submitted in: ASCE Journal of Structural Engineering, Manuscript ID: STENG-4011. I conducted the numerical analysis and wrote the manuscript which was further edited by Dr. Alam. iii  • A version of chapter 7 has been submitted in Journal of Structural Engineering, ASCE. Billah, A.H.M.M. and Alam, M.S. 2014. Performance based seismic design of concrete bridge pier reinforced with Shape Memory Alloy- Part 2: Methodology and Application. Submitted in: ASCE Journal of Structural Engineering, Manuscript ID: STENG-4012. I conducted the numerical analysis and wrote the manuscript which was further edited by Dr. Alam. A version of chapter 7 has been accepted in Structures Congress 2015 Conference. Billah, A.H.M.M. and Alam, M.S. 2015. Damping-Ductility relationship for performance based seismic design of shape memory alloy reinforced concrete bridge pier. in ASCE Structures Congress, 2015, Portland, Oregon. I conducted the numerical analysis and wrote the manuscript which was further edited by Dr. Alam. • A version of chapter 8 has been submitted in Journal of Structural Engineering, ASCE. Billah, A.H.M.M. and Alam, M.S. 2015. Probabilistic seismic risk assessment of concrete bridge piers reinforced with different types of shape memory alloys. Submitted in: ASCE Journal of Structural Engineering, Manuscript ID: STENG-4249. I conducted the numerical analysis and wrote the manuscript which was further edited by Dr. Alam. • A version of chapter 8 has been accepted in 11 Canadian Conference on Earthquake Engineering. Billah A.H.M.M. and Alam, M.S. 2015.Seismic performance evaluation of a highway bridge reinforced with different types of shape memory alloy rebar. In 11 CCEE, Victoria, BC, Canada, July 21-24, 2015. I conducted the numerical analysis and wrote the manuscript which was further edited by Dr. Alam.     iv  TABLE OF CONTENTS ABSTRACT……. ............................................................................................................... ii PREFACE……… .............................................................................................................. iii TABLE OF CONTENTS ....................................................................................................v LIST OF TABLES ............................................................................................................ xi LIST OF FIGURES......................................................................................................... xiii LIST OF SYMBOLS AND ABBREVIATIONS .......................................................... xviii ACKNOWLEDGEMENTS .............................................................................................. xx DEDICATION ..................................................................................................................xxi Chapter 1. INTRODUCTION AND THESIS ORGANIZATION ..............................1 1.1 General ...................................................................................................................1 1.2 Objectives of the Study ...........................................................................................3 1.3 Scope and Significance of Research ........................................................................3 1.3.1 Bond behaviour of SMA rebar with concrete ............................................................ 3 1.3.2 Plastic hinge length expression for SMA-RC bridge pier ........................................... 4 1.3.3 Performance-based damage states for SMA-RC bridge pier ...................................... 4 1.3.4 Performance-based design of SMA-RC bridge pier ................................................... 4 1.3.5 Probabilistic seismic risk assessment of SMA-RC bridge pier ................................... 5 1.4 Outline of the Thesis ...............................................................................................5 Chapter 2. APPLICATION OF SHAPE MEMORY ALLOY IN BRIDGES: RESEARCH, APPLICATION AND OPPORTUNITIES .................................................9 2.1 General ...................................................................................................................9 2.2 Shape Memory Alloy ............................................................................................ 11 2.3 Shape Memory Alloy in Bridges ........................................................................... 13 2.3.1 Application in bridge pier ....................................................................................... 16 v  2.3.2 Seismic isolation of bridges .................................................................................... 18 2.3.3 Dampers in bridges ................................................................................................. 20 2.3.4 Prestressing in bridge girders .................................................................................. 21 2.3.5 Retrofitting of bridge girders .................................................................................. 22 2.3.6 Application in bridge expansion joints .................................................................... 22 2.3.7 Restrainer in bridges ............................................................................................... 23 2.4 Comparison of SMA based and Conventional Bridge Component Performance .... 24 2.5 Promising SMAs for Application in Bridges.......................................................... 25 2.6 Future of Smart Bridges ........................................................................................ 28 2.7 Summary ............................................................................................................... 30 Chapter 3. SEISMIC FRAGILITY ASSESSMENT OF HIGHWAY BRIDGES:     A STATE-OF-THE-ART REVIEW ................................................................................. 31 3.1 General ................................................................................................................. 31 3.2 Seismic Fragility Analysis ..................................................................................... 32 3.3 Methods for Fragility Curve Development ............................................................ 35 3.3.1 Expert based/judgmental fragility curves ................................................................ 36 3.3.2 Empirical fragility curves ....................................................................................... 38 3.3.3 Experimental fragility curves .................................................................................. 39 3.3.4 Analytical fragility curves ....................................................................................... 40 3.3.5 Hybrid Fragility curves ........................................................................................... 45 3.4 Intensity Measure and Demand Parameter for Fragility Analysis ........................... 46 3.5 Regional Fragility analysis .................................................................................... 49 3.6 Condition Specific Fragility Assessment ............................................................... 50 3.6.1 Fragility analysis for retrofitted bridge .................................................................... 50 3.6.2 Fragility analysis considering aging effect .............................................................. 52 3.6.3 Fragility analysis considering SSI and liquefaction ................................................. 54 vi  3.6.4 Fragility analysis of isolated bridges ....................................................................... 55 3.6.5 Fragility analysis of irregular, curved and skewed bridges ....................................... 56 3.6.6 Fragility analysis considering effect of scouring ...................................................... 57 3.7 Effect of Ground Motion on Fragility Analysis ...................................................... 58 3.8 Possible Future Development ................................................................................ 58 3.9 Summary ............................................................................................................... 62 Chapter 4. BOND BEHAVIOR OF SMOOTH AND SAND-COATED SHAPE MEMORY ALLOY (SMA) REBAR IN CONCRETE .................................................... 63 4.1 General ................................................................................................................. 63 4.2 Experimental Program ........................................................................................... 64 4.2.1 Variables ................................................................................................................ 64 4.2.2 Materials ................................................................................................................ 65 4.3 Specimen Preparation and Testing ......................................................................... 66 4.4 Experimental Results ............................................................................................. 68 4.4.1 Failure modes ......................................................................................................... 68 4.4.2 Load-slip relationship and bond strength ................................................................. 70 4.4.3 Influencing factor analysis ...................................................................................... 71 4.5 Empirical Relationship for Bond Strength of SMA Rebar ...................................... 78 4.6 Comparison with Bond Behavior of Sand Coated FRP Bars .................................. 78 4.7 Summary ............................................................................................................... 81 Chapter 5. PLASTIC HINGE LENGTH OF SHAPE MEMORY ALLOY (SMA) REINFORCED CONCRETE BRIDGE PIER ................................................................. 82 5.1 General ................................................................................................................. 82 5.2 Design and Geometry of Bridge Pier ..................................................................... 83 5.3 Analytical Modeling .............................................................................................. 85 5.4 Model Validation .................................................................................................. 86 vii  5.5 Analytical Approach for Predicting Plastic Hinge Length ...................................... 87 5.5.1 Effect of axial load ................................................................................................. 88 5.5.2 Effect of aspect ratio ............................................................................................... 89 5.5.3 Effect of SMA properties ........................................................................................ 90 5.5.4 Effect of longitudinal reinforcement ratio................................................................ 91 5.5.5 Effect of transverse reinforcement .......................................................................... 92 5.5.6 Effect of concrete strength ...................................................................................... 93 5.6 Plastic Hinge Length Expression for SMA-RC Bridge Pier ................................... 94 5.7 Validation of the Proposed Equation ..................................................................... 95 5.8 Summary ............................................................................................................... 97 Chapter 6. PERFORMANCE-BASED SEISMIC DESIGN OF SHAPE    MEMORY ALLOY REINFORCED CONCRETE BRIDGE PIER:     DEVELOPMENT OF PERFORMANCE-BASED DAMAGE STATES ........................ 98 6.1 General ................................................................................................................. 98 6.2 Design and Geometry of Bridge Piers .................................................................... 99 6.3 Analytical Modeling of Bridge Piers ................................................................... 102 6.4 IDA- Based Approach for Developing Performance-Based Damage States ......... 104 6.4.1 Selection of ground motions ................................................................................. 104 6.4.2 Performance-based damage states criterion ........................................................... 107 6.4.3 Probabilistic distribution of drift based damage states ........................................... 109 6.4.4 Maximum drift based damage states ..................................................................... 113 6.4.5 Residual drift based damage states for SMA-RC bridge piers ................................ 115 6.5 Prediction of Residual Drift ................................................................................. 118 6.6 Summary ............................................................................................................. 120 Chapter 7. PERFORMANCE-BASED SEISMIC DESIGN OF SHAPE    MEMORY ALLOY (SMA) REINFORCED CONCRETE BRIDGE PIER: METHODOLOGY AND DESIGN EXAMPLE ............................................................. 121 viii  7.1 General ............................................................................................................... 121 7.2 Performance-Based Design of SMA Reinforced Bridge Pier ............................... 122 7.2.1 Step 1: Define seismic hazard ............................................................................... 122 7.2.2 Step-2: Define target residual drift ........................................................................ 123 7.2.3 Step-3: Calculate maximum drift based on target residual drift .............................. 123 7.2.4 Step-4: Select initial parameters ............................................................................ 125 7.2.5 Step-5: Calculate expected ductility demand ......................................................... 125 7.2.6 Step-6: Determine equivalent hysteretic damping .................................................. 126 7.2.7 Step 7: Determine effective time period (Teff) ........................................................ 128 7.2.8 Step 8: Determine effective stiffness (Keff) ............................................................ 130 7.2.9 Step 9: Compute design base shear (Vbase) and design moment (Md) ...................... 130 7.2.10 Step 10: Design the bridge pier ............................................................................. 130 7.3 Illustrative example ............................................................................................. 131 7.4 Bridge Pier Performance Evaluation .................................................................... 135 7.5 Summary ............................................................................................................. 138 Chapter 8. PROBABILISTIC SEISMIC RISK ASSESSMENT OF CONCRETE BRIDGE PIERS REINFORCED WITH DIFFERENT TYPES OF SHAPE    MEMORY ALLOYS....................................................................................................... 139 8.1 General ............................................................................................................... 139 8.2 Probabilistic Seismic Performance Assessment ................................................... 142 8.3 Design of SMA-RC Bridge Piers ......................................................................... 144 8.4 Finite Element Modeling of Bridge Piers ............................................................. 145 8.5 Seismic Hazard and Selection of Ground Motions ............................................... 146 8.6 Fragility Analysis of Different SMA-RC Bridge Piers ......................................... 149 8.6.1 Probabilistic seismic demand model ..................................................................... 150 8.6.2 Characterization of damage states ......................................................................... 152 ix  8.6.3 Fragility Curves .................................................................................................... 154 8.7 Seismic Demand Hazard of Different SMA-RC Bridge Piers .............................. 157 8.8 Summary ............................................................................................................. 159 Chapter 9. SUMMARY, CONCLUSIONS AND FUTURE WORKS..................... 160 9.1 Summary ............................................................................................................. 160 9.2 Core Contributions .............................................................................................. 161 9.3 Conclusions ......................................................................................................... 162 9.3.1 Bond behavior of smooth and sand coated SMA rebar in concrete......................... 162 9.3.2 Plastic hinge length of SMA-RC bridge pier ......................................................... 162 9.3.3 Performance-based seismic design of Shape Memory Alloy reinforced concrete   bridge pier………. ................................................................................................ 163 9.3.4 Probabilistic seismic risk assessment of SMA-RC bridge piers.............................. 165 9.4 Recommendation for Future works ...................................................................... 167 REFERENCES ................................................................................................................ 169 APPENDICES. ................................................................................................................ 198 Appendix A ................................................................................................................... 198 Appendix B ................................................................................................................... 207 Goodness-of-fit test............................................................................................................... 207 Appendix C ................................................................................................................... 212 Curve fitting ......................................................................................................................... 212    x  LIST OF TABLES   Table 2.1. Summary of SMA application in bridge engineering .......................................... 15 Table 2.2. Performance comparison of SMA-based and conventional bridge components ... 25 Table 2.3. Potential SMAs for application in bridge engineering ......................................... 27 Table 2.4. Summary of SMA properties for bridge engineering application and their       effects ............................................................................................................... 29 Table 3.1.Comparison of different methods for development of fragility curves .................. 36 Table 3.2. Comparison of empirical fragility curve parameters ........................................... 39 Table 3.3. Summary of threshold values of different demand parameters ............................ 48 Table 3.4. Key features of modern bridge fragility curve development efforts ..................... 61 Table 4.1. Pushout test specimens ....................................................................................... 65 Table 4.2. Comparison of Bond Strength Sand Coated SMA bars with Sand Coated FRP      Bars .................................................................................................................. 81 Table 5.1. Details of variable parameters ............................................................................ 84 Table 5.2. Details of SMA-RC bridge piers ......................................................................... 85 Table 5.3. Properties of different types of SMA .................................................................. 91 Table 5.4. Comparison of experimental and measured plastic hinge length ......................... 95 Table 5.5. Comparison of measured and calculated ultimate drift ........................................ 97 Table 6.1. Properties of different types of SMA ................................................................ 101 Table 6.2. Material properties for SMA-RC bridge pier .................................................... 102 Table 6.3. Selected earthquake ground motion records ...................................................... 106 Table 6.4. Proposed damage state framework.................................................................... 108 Table 6.5. Damage states of different SMA-RC bridge pier and their associated      distribution ...................................................................................................... 111 Table 6.6. Residual drift damage states of SMA-RC bridge pier........................................ 117 Table 7.1. ATC55/FEMA440 earthquake ground motions* (Miranda, 2003) .................... 127 Table 7.2. Material Properties ........................................................................................... 132 Table 8.1. Selected earthquake ground motion records ...................................................... 149 Table 8.2. PSDMs for different EDPs ............................................................................... 152 xi  Table 8.3. Limit state capacity of SMA-RC bridge pier in terms of maximum and        residual drift .................................................................................................... 153 Table 8.4. Comparison of median PGA (g) ....................................................................... 157 Table 8.5. Annual rate and probability of collapse (DS-4) in terms of maximum drift ....... 159 Table 8.6. Annual rate and probability of DS-2 in terms of residual drift ........................... 159 Table A.0.1. Summary of seismic fragility assessment studies of bridges .......................... 198 Table A.0.2. Summary of regional fragility analysis of highway bridges ........................... 203 Table B.0.1. Results of K-S goodness-of-fit tests for spalling drift limit………………… 208 Table B.0.2. Results of K-S goodness-of-fit tests for yielding drift limit………………….209 Table B.0.3. Results of K-S goodness-of-fit tests for crushing drift limit………………….210 Table C.0.1. List of equations tested……………………………………………………….211      xii  LIST OF FIGURES  Figure 1.1. Outline of the thesis ............................................................................................6 Figure 2.1. Flag shaped hysteresis of Shape memory alloy.................................................. 10 Figure 2.2. Comparison of elastic modulus and recovery strain of different SMAs .............. 12 Figure 2.3. Comparison among different commonly used construction material and    different types of SMAs (adapted from Ma and Karaman 2010) ........................ 13 Figure 2.4. Application of SMA in bridge engineering (a) active confinement of bridge   pier, (b) Post- tensioning in segmental bridge pier, (c) Yielding device in segmental bridge pier, (d) Reinforcement in the plastic hinge region, (e) Restrainer, (f) Isolation bearing, (g) Post-tesioned bridge girder, (h) Expansion joint and (i) Damper in stay cables. ................................................................... 14 Figure 2.5. Statistics of application of SMA in bridge engineering ...................................... 15 Figure 2.6. Comparison of hysteretic response of different SMAs ....................................... 26 Figure 3.1. Statistics of publications on seismic fragility analysis of bridges since 1990 ..... 34 Figure 3.2. Various applications of seismic fragility curves ................................................ 34 Figure 3.3. Methodology for developing seismic fragility curves ........................................ 35 Figure 3.4. Typical survey technique for developing expert based fragility curve ................ 37 Figure 3.5. Comparison of empirical fragility curves developed by Shinozuka et al. (2001) [S] and Yamazaki et al. (2000) [Y] using damage data from Kobe earthquake ... 39 Figure 3.6. Probabilistic Representation of Capacity and Demand Spectra (Mander and ..... 41 Figure 3.7. Schematic Representation of the NLTHA procedure used to develop fragility curves................................................................................................................ 42 Figure 3.8. Comparison of empirical fragility curves for MSC Concrete bridges for   different regions ................................................................................................ 50 Figure 3.9. (a) Fragility curves for as-built and retrofitted bridge (b) Fragility curves for retrofitted bridge bent using different retrofitting techniques (Billah et al. 2013).......................................................................................................................... 51 Figure 3.10. Effect of (a) aging (Ghosh and Padgett, 2010), (b) soil liquefaction (Aygun      et al. 2011), (c) isolation (Zhang and Huo 2009), (d) horizontal curve xiii  (AmiriHormozaki et  al. 2013), (e) skew angle (Sullivan and Nielson 2010) and (f) scour depth (Prasad and    Banarjee 2013) on fragility curves ....................... 53 Figure 3.11. Proposed methodology for developing hybrid fragility curves ......................... 59 Figure 4.1. Bond failure of concrete section having smooth SMA rebar (adapted from    Youssef et al. 2008)........................................................................................... 64 Figure 4.2. Specimens after casting ..................................................................................... 66 Figure 4.3. Sand coating of SMA rebar (a) bonded length, (b) epoxy application, (c) sand coating and (d) sand coated rebars ..................................................................... 67 Figure 4.4. Test setup for bond behavior SMA rebar with concrete ..................................... 68 Figure 4.5. Specimens (smooth) (a) before testing, (b) after testing and (c) inside view ...... 69 Figure 4.6. Failure pattern of sand coated bars (a) radial cracking, (b) crack propagation in concrete and (c) inside view .............................................................................. 70 Figure 4.7. Load-slip curves for pushout test of smooth SMA rebar .................................... 71 Figure 4.8. Effect of concrete compressive strength on average (a) maximum and (b) residual bond strength of smooth SMA bar ........................................................ 72 Figure 4.9. Effect of bar diameter on average (a) maximum and (b) residual bond strength   of smooth SMA bar ........................................................................................... 74 Figure 4.10. Effect of embedment length on average (a) maximum and (b) residual bond strength of smooth SMA bar.............................................................................. 75 Figure 4.11. Effect of concrete cover to bar diameter ratio on average (a) maximum and     (b) residual bond strength of smooth SMA bar .................................................. 76 Figure 4.12. Effect of sand coating on bond strength of SMA rebar (a) bond stress-slip  curve, (b) effect of bar diameter and (c) effect of embedment length ................. 77 Figure 4.13. Comparison between experimental and predicted values of τmax/√fc’ .............. 79 Figure 5.1. Geometry of SMA-RC bridge pier (a) Cross section, (b) Elevation and (c)    Finite element modeling .................................................................................... 83 Figure 5.2. (a) Comparison of predicted and measured strain on SMA rebar (Nakashoji     and Saiidi 2014) and (b) Comparison of predicted and measured curvature (O’Brien et al. 2007) ......................................................................................... 86 Figure 5.3. Effect of axial load on (a) curvature profile and (b) longitudinal rebar strain profile ............................................................................................................... 89 xiv  Figure 5.4. Effect of aspect ratio on (a) curvature profile and (b) longitudinal rebar strain profile ............................................................................................................... 90 Figure 5.5. Effect of  fy-SMA on (a) curvature profile and (b) longitudinal rebar strain      profile ............................................................................................................... 91 Figure 5.6. Effect of longitudinal reinforcement ratio on (a) curvature profile and (b) longitudinal rebar strain profile ......................................................................... 92 Figure 5.7. Effect of transverse reinforcement ratio on (a) curvature profile and (b) longitudinal rebar strain profile ......................................................................... 93 Figure 5.8. Effect of concrete compressive strength on (a) curvature profile and (b) longitudinal rebar strain profile ......................................................................... 94 Figure 5.9. Comparison of measured and predicted plastic hinge lengths ............................ 96 Figure 6.1. Cross section and elevation of SMA reinforced concrete bridge pier ............... 100 Figure 6.2. (a) Moment curvature relationship of RC sections with different types of     SMAs and (b) Static pushover curves for bridge piers reinforced with different types of SMAs ................................................................................................ 102 Figure 6.3. Comparison of experimental and numerical results (a) SMA-RC (SMA-1)   bridge pier (b) SMA-RC (SMA-4) beam ......................................................... 103 Figure 6.4.  Flowchart for the development of performance based damage states for      SMA-RC bridge pier ....................................................................................... 105 Figure 6.5. Design and mean response spectrum of 10 records used for IDA analysis matching the three different CHBDC spectrum (2%, 5%, and 10% in 50 years)........................................................................................................................ 107 Figure 6.6. Dynamic pushover response and different damage states with distribution for SMA-RC-1 for (a) 2% in 50 years (b) 5% in 50 years and (c) 10% in 50 years probability    of exceedance ............................................................................. 111 Figure 6.7. Dynamic pushover response and different damage states with distribution for SMA-RC-2 for (a) 2% in 50 years (b) 5% in 50 years and (c) 10% in 50 years probability    of exceedance ............................................................................. 112 Figure 6.8. Dynamic pushover response and different damage states with distribution for SMA-RC-3 for (a) 2% in 50 years (b) 5% in 50 years and (c) 10% in 50 years probability    of exceedance ............................................................................. 112 xv  Figure 6.9. Dynamic pushover response and different damage states with distribution         for SMA-RC-4 for (a) 2% in 50 years (b) 5% in 50 years and (c) 10% in 50    years probability of exceedance ....................................................................... 112 Figure 6.10. Dynamic pushover response and different damage states with distribution       for SMA-RC-5 for (a) 2% in 50years (b) 5% in 50 years and (c) 10% in 50    years probability of exceedance ....................................................................... 113 Figure 6.11. Fragility curves in terms of residual drift at (a) 10% in 50 years (b) 5% in        50 years and (c) 2% in 50 years probability of exceedance .............................. 117 Figure 6.12. Comparison of residual drift prediction with experimental results                     (a) O’Brien    et al. (2007) and (b) Youssef et al. (2008) .................................. 119 Figure 7.1. Flow diagram of PBSD of SMA-RC bridge pier ............................................. 124 Figure 7.2. Damping-Ductility relation for SMA-RC bridge pier (a) SMA-1, (b) SMA-2,    (c) SMA-3, (d) SMA-4 and (e) SMA-5 ............................................................ 128 Figure 7.3. Comparison of Damping-Ductility curve ........................................................ 129 Figure 7.4. Design Acceleration Response Spectrum ........................................................ 131 Figure 7.5. Determination of effective period from reduced displacement spectrum .......... 133 Figure 7.6. (a) Moment-Shear force interaction diagram and (b) Moment-Axial Load interaction diagram.......................................................................................... 135 Figure 7.7. Displacement spectra of ten earthquake records matched with target response spectrum ......................................................................................................... 136 Figure 7.8. (a) Maximum and (b) residual drift value obtained from time history analysis    of the designed pier (Red line showing the target maximum and                 residual drift)................................................................................................... 137 Figure 8.1. Flowchart of the methodology for seismic risk assessment of SMA-RC bridge  piers ................................................................................................................ 141 Figure 8.2. (a) Cross section, (b) elevation and (c) finite element model of SMA-RC     bridge pier ....................................................................................................... 145 Figure 8.3. Seismic hazard curve for site soil class C in Vancouver (a) Peak ground acceleration and (b) spectral acceleration......................................................... 147 xvi  Figure 8.4. (a) Comparison of UHS, CMS-Crustal, CMS-Interface, and CMS-Inslab at        T1 = 0.7 s, (b-d) comparison of response spectra of the selected records with     the target spectra for individual earthquake types ............................................ 148 Figure 8.5. Comparison of the PSDMs for (a) SMA-RC-1, (b) SMA-RC-2,                          (c) SMA-RC-3, (d) SMA-RC-4 and (e) SMA-RC-5 considering maximum      drift as EDP..................................................................................................... 151 Figure 8.6. Comparison of the PSDMs for (a) SMA-RC-1, (b) SMA-RC-2,                             (c) SMA-RC-3, (d) SMA-RC-4 and (e) SMA-RC-5 considering residual drift     as EDP ............................................................................................................ 151 Figure 8.7. Fragility curves for the five SMA-RC bridge piers for: (a) slight, (b) moderate, (c) extensive and (d) collapse damage state considering maximum drift .......... 154 Figure 8.8. Fragility curves for the five SMA-RC bridge piers for: (a) slight, (b) moderate, (c) extensive and (d) collapse damage state considering residual drift .............. 156 Figure 8.9. Hazard curves for five SMA-RC bridge piers (a) maximum drift and                   (b) residual drift .............................................................................................. 158    xvii  LIST OF SYMBOLS AND ABBREVIATIONS Af Austenite finish temperature of SMA c Concrete cover db Bar diameter E Elastic modulus of SMA  Fy-SMA Yield strength of SMA Fy Yield force fc' Concrete compressive strength fP1 Austenite to martensite finishing stress of SMA fT1 Martensite to austenite starting stress of SMA fT2 Martensite to austenite finishing stress of SMA kr Surface roughness factor H Height of pier Keff Effective stiffness LP Plastic hinge length L/d Aspect ratio L Length/Height of bridge pier d Diameter of pier ld Embedment length Me Effective mass of the pier M f Martensite finish temperature of SMA M s Martensite start temperature of SMA  M d Design moment Pmax Maximum load Pres Residual load Rξ Damping modification factor Sc Median of capacity τmax Maximum average bond strength τres Residual average bond strength Teff Effective time period Vbase Design base shear α Sand size coefficient βc Logarithmic standard deviation of capacity βEDP|IM Logarithmic standard deviation of demand εs Superelastic strain in SMA εr Recovery strain of SMA εSMA Strain in SMA wires εsm Steel strain at maximum tensile stress Δy Yield displacement ΔyT Target yield displacement Δmax Maximum displacement λEDP Mean annual frequency of exceedance μd Displacement ductility demand ξ0 Nominal viscous damping ξeq Equivalent viscous damping ρl Longitudinal reinforcement ratio ρs Transverse reinforcement ratio φy Yield curvature φu Ultimate curvature xviii    AAE Average Absolute Error ACI American Concrete Institute AI Arias Intensity CFRP Carbon Fiber-Reinforced Polymer CMS Conditional Mean Spectrum CSA Canadian Standard Association CSM Capacity Spectrum Method DDBD Direct Displacement Based Design DS Damage State EDP Engineering Demand Parameter ECC Engineered Cementitious Composite IDA Incremental Dynamic Analysis IM Intensity Measure LS Limit State MCE Maximum Considered Earthquake MD Maximum Drift MMI Modified Mercalli Intensity NLTHA Non Linear Time History Analysis PBSD Performance-Based Seismic Design PBEE Performance-Based Earthquake Engineering PDF Probability Density Function PGA Peak Ground Acceleration PGD Peak Ground Displacement PGV Peak Ground Velocity PSDA Probabilistic Seismic Demand Analysis PSDM Probabilistic Seismic Demand Model PSHA Probabilistic Seismic Hazard Analysis RC Reinforced Concrete RD Residual Drift Sa Spectral Acceleration Sd Spectral Displacement SMA Shape Memory Alloy SSI Soil Structure Interaction UHS Uniform Hazard Spectra    xix  ACKNOWLEDGEMENTS I convey my profound gratitude to the almighty Allah for allowing me to bring this effort to fruition. I express my sincere gratitude to my advisor, Dr. M. Shahria Alam for providing me with an opportunity to work with him at The University of British Columbia, Okanagan. I couldn’t have asked for a better mentor and guide for my Doctoral program and I really appreciate all the support, guidance, and motivation that he has provided me through my academic career. He has been instrumental with knowledge, support, and mentoring that made my graduate experience at UBC so impeccably productive and rewarding, and made a great contribution to the success of this research. I would like to thank my doctoral dissertation committee members, Dr. Abbas Milani and Dr. Ahmad Rteil for always supporting my research work and providing me with great feedback from time to time, helping me improve the quality of my work immensely. Graduate school and experimental research facility at UBC’s Okanagan campus has provided an excellent educational experience, and I would like to acknowledge the support I have received for pursuing a graduate degree at this Institution from Natural Sciences and Engineering Research Council of Canada (NSERC) and my industry sponsor Bourcet Engineering Ltd.  I feel privileged to get the opportunity to work with such an excellent group of graduate students in the research group especially Anant, Shahidul, Kader, and Rafiqul who helped me during my experimental works, offered technical knowledge, and friendship. I would also like to acknowledge Dr. Nouroz Islam for his generous help in setting up the data acquisition system. I offer my enduring gratitude to the assistance of Ryan Mandu, UBC Structures Laboratory Technician, for assistance with the test setup.  I am truly grateful for the unconditional support of my family, without which I would likely not be here today. My parents have offered endless support, confidence in me, wise advice, and love. I am especially indebted to my wife, Sumaiya, for being with me and supporting me through the past years. Her support, encouragement, and enduring love have meant the world to me throughout this process and always.   xx  DEDICATION    This disserTaTion is dedicaTed To The memory of my beloved faTher. his words of inspiraTion and encouragemenT in pursuiT of excellence, sTill linger on.    xxi   CHAPTER 1. INTRODUCTION AND THESIS ORGANIZATION 1.1 General In recent years, the seismic design guidelines have been focusing on performance-based design in order to predict and better manage the post-earthquake functionality and condition of structures. Recent developments in performance-based seismic design and assessment approaches have emphasized the importance of properly assessing and limiting the residual (permanent) deformations that are typically sustained by a structure after a seismic event (Pettinga et al. 2006). If reinforced concrete (RC) structures are designed in such a way that they are capable of withstanding large displacement with adequate energy dissipation capacity during a seismic event which will not only eliminate the problem of permanent deformation, but also make the structures safer against earthquakes. Thus, it will substantially scale down the repair and maintenance cost of structures. Superelastic Shape Memory Alloy (SMA) possesses the distinct ability to experience large deformation and retrieve its original shape upon load removal along with high resistance to corrosion (Alam et al. 2009). This is a distinct property that makes SMA a smart material and a strong contender for reinforcement in RC structures particularly at critical locations (plastic hinge region), which is prone to more damage during an earthquake.  Very often the seismic design of structures is carried out considering a simple configuration which allows simplified analysis and design procedure. Within this simplified procedure critical response parameters are identified and checked against design guidelines. In contrast to buildings, the seismic response of highway bridges is controlled by the nonlinear behavior of bridge piers. Bridge piers are one of the most vulnerable elements in a bridge whose failure can have catastrophic consequences. Therefore, ensuring an acceptable performance of bridge piers during a seismic event with adequate energy dissipation, ductility and resistance to residual drift is of paramount importance. However, conventional design approaches are focused on the strength and serviceability requirements and do not consider the performance objectives. On the other hand, performance-based seismic design (PBSD) aims to adopt a wider range of design scope that results in more predictable seismic performance over the full range of earthquake demand (Marsh and Stringer 2013). Destructive earthquakes 1  of Northridge (1994) and Kobe (1995) enhanced interest in PBSD as an alternative to the conventional approaches prescribed by the majority of the codes (AASHTO 2012, CSA-S6-10). The evolution of PBSD has established the option for relating post-earthquake structural performance with engineering demand parameters that allows the owner to determine the potential functionality of a bridge following a major earthquake.  Bridge infrastructure represents a significant portion of the transportation network of any country. Keeping bridges safe and operational is a major challenge. Conventional structural systems are prone to excessive residual deformation under seismic loading and their performance cannot be fully characterized without paying due attention to residual deformation. During a seismic event, bridge piers are subjected to large lateral deformations while supporting gravity loads from superstructure, and can experience severe damage in plastic hinge regions. Identifying the plastic hinge length and a proper design and detailing of the plastic hinge region is critical for ensuring adequate flexural deformation capacity and limiting the residual drift in a bridge pier.  Considering the importance of a bridge, it is necessary to minimize the loss of bridge functions as much as possible during earthquakes by reducing or controlling the residual drift in bridge piers (Billah and Alam 2014c) . Conventional seismic design of bridge piers allows yielding of the longitudinal rebar in the plastic hinge region in combination with cracking and crushing of the concrete during a seismic event, which results in severe damage and large permanent deformations in bridge piers. One promising solution as evident from previous research is the application of high performance innovative materials such as shape memory alloy in the plastic hinge region of bridge pier. While the previous studies proved the potential of using shape memory alloys in bridge piers, before large scale industrial implementation it is required to develop a comprehensive design guideline and perform a complete performance-based evaluation of this novel structural system in light of performance-based earthquake engineering (PBEE). To this end, it is necessary to investigate the ability of such novel structural system in reducing the failure probability as well as the annual rate of exceeding some structural demand parameters given an earthquake scenario. 2  1.2 Objectives of the Study The overall goal of this research is to introduce a performance-based design procedure for design of concrete bridge piers using SMA as longitudinal reinforcement in the plastic hinge region, and assess the accuracy and reliability of the method in lights of performance-based earthquake engineering (PBEE). The specific objectives of the current research include: 1. Experimentally investigate the bond behaviour of SMA rebar with concrete. 2. Develop an expression for the plastic hinge length of SMA reinforced concrete (RC) bridge pier. 3. Develop performance-based damage states for SMA-RC bridge piers considering different SMAs and different earthquake hazard levels. 4. Propose a performance-based seismic design guideline for SMA-RC bridge piers. 5. Probabilistic seismic performance and risk assessment of SMA-RC bridge piers. 1.3 Scope and Significance of Research This research addresses a very important issue that affects the seismic performance of bridge structures. This study will introduce the application of SMA as reinforcement in designing bridge piers following a performance-based approach which has emerged as a promising alternative to the traditional design techniques.  This study provides a first step by investigating the influence of SMA as reinforcement in bridge piers in design issues, as well as its failure probability through the development and comparison of fragility curves. The significance of this research is highlighted below: 1.3.1 Bond behaviour of SMA rebar with concrete Adequate bond strength between concrete and reinforcing bars has been identified as a cardinal parameter to the satisfactory performance of RC structures (ACI 408R-03). Over the past few years researchers have proposed and developed SMA reinforced concrete structures for improved seismic resistance. But no study has been undertaken to evaluate the bond behaviour of SMA rebars with concrete. In order to increase the practical application of SMA rebars in concrete structures, it is required to identify the bond stress-slip behaviour with concrete. Identification of the bond properties of SMA bars in concrete will allow for safe, reliable, and efficient use of SMA. 3  1.3.2 Plastic hinge length expression for SMA-RC bridge pier Compared to conventional bridge pier, behaviour of SMA-RC bridge pier is significantly different and governed by the distinct superelastic and thermo-mechanical properties of SMA. Estimating the plastic hinge length is a major step in predicting the load-drift response of a bridge pier. As very limited test results are available on SMA-RC bridge piers, this study developed an analytical expression for estimating the plastic hinge length of SMA-RC bridge pier using the results of comprehensive nonlinear finite element analyses. In order to limit the use of SMA rebar only in the plastic hinge region (i.e. to confine damages within the region that will eventually recover), the proposed equation will help determine the amount of SMA reinforcement to be used in the SMA-RC bridge pier. 1.3.3 Performance-based damage states for SMA-RC bridge pier  As a prerequisite to the implementation of performance-based design for SMA-RC bridge pier, the performance objectives and their corresponding limit state criteria need to be properly defined. To implement such procedures, it is necessary to define damage in terms of engineering performance criteria. In this study, various performance-based damage states corresponding to different performance levels (cracking, yielding, and strength degradation) were developed for SMA-RC bridge piers reinforced with different types of SMAs under various earthquake hazard levels. The developed damage states and the proposed residual drift prediction equation will help designers choose the right SMA from various types while designing SMA-RC bridge piers under certain seismic hazard condition. 1.3.4 Performance-based design of SMA-RC bridge pier There exists no proper design guideline for designing bridge pier using SMA. Hence, this study aims at developing a performance-based seismic design guideline for SMA-RC bridge pier considering residual drift as the key performance indicator. This study develops step by step procedure, with useful flow charts and graphs, for designing SMA-RC bridge pier along with a design example. 4  1.3.5 Probabilistic seismic risk assessment of SMA-RC bridge pier In addition to the development of performance-based design specifications, a consistent performance-based seismic design approach for bridges requires a detailed probabilistic seismic risk assessment. This study is intended to elucidate the potential benefit and compare the performance of different SMA-RC bridge piers in light of PBEE. This study developed fragility curves and seismic hazard curves for different SMA-RC bridge piers, designed following the proposed design guideline, considering maximum and residual drift as engineering demand parameters. The developed fragility curves express the probability of reaching or exceeding certain damage states corresponding to a certain intensity of ground motion. The hazard curves relate the mean annual rate of exceeding certain damage states. 1.4 Outline of the Thesis This thesis is arranged in nine chapters. The outline of the thesis is depicted in Figure 1.1. In the present chapter a short preface and the objectives and scope are presented. The content of the dissertation is organized into the following chapters: In Chapter 2, a comprehensive literature review is presented on application of SMA in bridge engineering by providing a brief summary of SMA, highlighting different types of SMAs, their comparisons and application in structural engineering. The chapter discusses the existing application of SMAs in different bridge components such as bridge piers, isolation bearing, girders, expansion joints, restrainer, and dampers. This chapter concludes by attempting to highlight the promise and potential of future smart bridges using SMA. Chapter 3 provides a comprehensive review of the existing methodology and identify current trends in the seismic fragility assessment of highway bridges. Based on the existing literature this chapter illustrates, in a systematic manner, a summary of different fragility assessment methodologies for highway bridges, features, and limitations and a critical review of the state-of-the-art currently existing application of fragility assessment methods.   5                       Figure 1.1. Outline of the thesis Performance-Based Seismic Design and Assessment of Concrete Bridge Pier Reinforced with Shape Memory Alloy Rebar Title Introduction Chapter-1 Probabilistic seismic risk assessment of concrete bridge piers reinforced with different types of shape memory alloys Chapter-8 Summary, Conclusions, and Future Works Chapter-9 Application of Shape Memory Alloy in Bridges: Research, Application and Opportunities Chapter-2 Seismic Fragility Assessment of Highway Bridges: A State-of-The-Art Review Chapter-3 Plastic Hinge Length of Shape Memory Alloy Reinforced Concrete Bridge Pier Chapter-5 Performance-based seismic design of Shape Memory Alloy reinforced concrete bridge pier: Development of Performance-Based Damage States Chapter-6 Performance-based seismic design of Shape Memory Alloy (SMA) reinforced concrete bridge pier: Methodology and Design Example Chapter-7 Bond Behavior of Smooth and Sand-Coated Shape Memory Alloy Rebar in Concrete Chapter-4 Review Core Contribution Application 6  Before going to the development of design guidelines, it is critical to have an appropriate understanding of the behaviour of SMA rebar with concrete. Chapter 4, as the first step, intends to experimentally investigate the bond behaviour of SMA rebars embedded in concrete. The objective of this experimental investigation is to study the bond behaviour of SMA rebar where the variables include SMA bar diameter, concrete strength, bonded length, concrete cover, and surface condition. Based on the experimental results, empirical equation for predicting the average maximum bond strength of SMA rebar has been developed.  Chapter 5 develops a plastic hinge length expression for SMA-RC bridge pier using an analytical method. Using a well-calibrated finite element model, this chapter develops a plastic hinge length expression for SMA-RC bridge pier by investigating the distribution of curvature and strain in the longitudinal rebar (both steel and SMA rebar) along the height of the pier. Considering different parameters such as the level of axial load, aspect ratio, concrete strength, SMA properties and ratio of longitudinal and transverse reinforcement, a parametric study is conducted to derive a plastic hinge length expression for SMA-RC bridge pier. Finally, the proposed equation is used to estimate the drift capacity of SMA-RC bridge pier and compared with test results. Using an incremental dynamic analysis (IDA) based analytical approach (Vamvatsikos and Cornell 2002), Chapter 6 develops performance-based damage states (based on drift limits) for SMA-RC bridge piers reinforced with five different SMAs considering different earthquake hazard levels. This chapter also develops residual drift based damages states for the SMA-RC bridge piers and propose an analytical expression that can be used for predicting the residual drift in SMA reinforced concrete elements. This chapter provides a technical basis for the development of performance-based seismic design and evaluation methodologies for the SMA-RC bridge piers. Using the performance-based damage states and associated performance levels developed in previous chapter, this chapter (Chapter 7) develops a performance-based seismic design (PBSD) guideline for SMA-RC bridge pier considering residual drift as the key performance indicator. This chapter also develops the damping-ductility relationship for SMA-RC bridge piers in support of the proposed PBSD methodology. 7  In order to assess the reliability of the proposed design methodology, Chapter 8 evaluates the probabilistic seismic risk of concrete bridge piers reinforced with different types of SMA rebars designed following the proposed guideline and using the developed bond-slip relation and plastic hinge length equation. Considering maximum drift and residual drift as demand parameters, fragility curves are developed for five different SMA-RC bridge piers considering different probable earthquake hazard scenarios. Finally, seismic hazard curves, which compare the mean annual rate of exceedance of different damage states of the different bridge piers, are generated. Finally, Chapter 9 presents the summary and conclusions attained from this research study. Few specific recommendations for future research have also been suggested.   8  CHAPTER 2. APPLICATION OF SHAPE MEMORY ALLOY IN BRIDGES: RESEARCH, APPLICATION AND OPPORTUNITIES 2.1 General The advancement in material science along with the technology has pushed us towards adaptive and intelligent structures and created a growing interest among researchers and structural engineers to introduce smart materials in civil engineering applications. Shape memory alloy (SMA), a smart material with distinct thermomechanical properties and flag shaped hysteresis, has received much attention from researchers as a potential candidate for use in structural engineering applications. The first application of SMA in structural engineering can be traced back to 1991, when Graesser and Cozzarelli (1991) first introduced SMA as a new material for seismic isolation device. Since then, application of SMA has significantly expanded and researchers conducted extensive investigations exploring different structural applications and developing innovative devices making use of the distinctive characteristics of this smart material. According to a recent research report (McWilliams 2015) the global market for smart materials has an annual growth rate of 12.8% for the period from 2011 to 2016. Although a significant portion of this market is occupied by automotive and actuator industry, the market contribution of structural application is predicted to rise from 5.8% in 2010 to 8.5% in 2016. Shape memory effect (SME), superelasticity (SE) and damping capacity, are three of the many distinct properties of SMAs that make them suitable for structural engineering applications. Moreover, the flag shaped hysteresis of SMA (Figure 2.1) allows reinforced concrete and steel members as well as other structural components equipped with SMA to regain its original shape upon load removal while encountering negligible or no residual drift. With the advancement of design methods, most of the design codes around the world are approaching towards a performance-based design. Moreover, there is a consensus among the researchers and earthquake engineering community that structural performance cannot be fully characterized without paying attention to residual deformation. Because of its distinctive characteristics, SMA can undergo excessive deformations and can revert to their parent shape through heat application or by removing the load (Alam et al. 2008a). Evidences from past 9  seismic events demonstrate that bridges undergoing large deformation are susceptible to large residual deformation thereby rendering the bridges to be unusable and requiring major rehabilitation or replacement. In order to maintain structural integrity and functionality of a bridge after an earthquake, it is necessary that the bridge components avoid excessive residual deformation or permanent damage (Kawashima et al. 1998). In an attempt to improve the seismic performance of bridges during extreme events, researchers came up with the idea of mitigating damages by using SMA in different bridge components. In order to take the advantage of the intrinsic properties of SMAs, researchers have investigated their application in bridge piers as reinforcement (using SE) (Saiidi et al. 2009), as supplementary materials in dampers and isolators (using damping and SE) (Dezfuli and Alam 2013), and as reinforcement and prestressing tendons in bridge girders (using SME) (Soroushian et al. 2001). Moreover, researchers are now focusing on practical applications and developing design guidelines for developing structural systems using SMA.  Figure 2.1. Flag shaped hysteresis of Shape memory alloy A good number of studies reported in the literature on the application of SMAs in civil infrastructure are available (Saadat et al. 2002; Dong et al. 2002; DesRoches and Smith 2004; Janke et al. 2005; Wilson and Wesolowsky 2005; Song et al. 2006; Alam et al. 2007, Ozbulut et al. 2011a, Cladera et al. 2014) which mostly highlight the application on building structures and their vibration control. Dong et al. (2011) presented the first overview of existing application of SMA in bridges. However, this study only focused on summarizing the existing applications without providing any insight into the potential of different types of SMAs in -500-400-300-200-1000100200300400500-0.1 -0.05 0 0.05 0.1Stress (MPa)Strain10  bridge engineering applications and future of smart bridges. Moreover, a significant amount of research works were conducted over the last 5 years and many new SMAs have been developed that are suitable and promising for bridge engineering applications. This chapter is aimed at providing a comprehensive review of the existing application of different types of SMAs in bridge engineering, and identifies the current and future trends of smart bridges using SMA.  2.2 Shape Memory Alloy Smart materials like Shape Memory Alloys (SMAs) have demonstrated a wide range of engineering applications namely, biomedical, robotics, aerospace, civil, and mechanical engineering. Two distinct properties such as the shape memory effect (SME, ability to recover plastic strain upon heating) and superelasticity (SE, ability to recover plastic strain upon load removal) make SMA a strong contender against conventional metals and alloys for application in various sectors. Several compositions of SMAs have been developed to date such as Ni-Ti, Cu-Zn, Cu-Zn-Al, Cu-Al-Ni, Fe-Mn, Mn-Cu, Fe-Pd, and Ti-Ni-Cu etc. Numerous applications of SMAs in civil engineering field have been documented (Ocel et al. 2004, Saiidi and Wang 2006, Lindt and Potts 2008, Alam et al. 2009, Araki et al. 2010, Billah and Alam 2012, Dezfuli and Alam 2013). Most of the applications have been focusing on the use of Ni-Ti alloy while very few focused on the application of other alloys such as Cu-based SMAs (Araki et al. 2010, Shrestha et al. 2013), and Fe- based SMAs (Dezfuli and Alam, 2013, Czaderski et al. 2014). Although Ni-Ti SMA shows large recoverable strain, good superelasticity and exceptionally good resistance to corrosion, high cost of Ni-Ti SMA and machinability restrict its large scale applications. In an attempt to reduce the cost of SMA, researchers have come up with various Fe-based and Cu-based low cost SMAs such as Fe-Mn-Si, Fe-Ni-Co-Ti, Fe-Ni-Nb, Cu-AL-Mn, etc. Iron (Fe) based SMAs show good workability, machinability, weldability, and wide transformation hysteresis as compared to Ni-Ti SMA. Although several compositions of Fe-based SMAs have been developed, large scale application is still limited due to the poor shape recovery limit and associated costly ‘training’ treatment. Recently Tanaka et al. (2010) developed a ferrous polycrystalline SMA (Fe-Ni-Co-Al-Ta-B) which has a very high superelastic strain range of over 13% at room temperature. This SMA has approximately 20 times higher SE than Fe-Ni-Co-Ti alloy and almost double that of conventional Ni-Ti alloy. 11  This Fe-based SMA has higher ductility, greater strength, and also energy dissipation capacity several times higher than that of commercially available Ni-Ti SMA. More recently, Omori et al. (2011) developed another Fe-based SMA (Fe-Mn-Al-Ni) which has superelasticity similar to the conventional Ni-Ti SMA but with much lower Austenite finish temperature, which allows this SMA to operate in superelastic range even at very low temperature. In order to improve the machinability and reduce the cost, a Cu- based SMA (Cu-Mn-Al) has been developed (Araki et al. 2010) which has comparable superelasticity to that of NiTi SMAs. Moreover, these Cu–Al–Mn SMAs have comparatively lower strain rate effects than Ni–Ti SMAs (Araki et al 2012) and also been reported to provide recentering and crack recovery capabilities (Shrestha et al. 2013).  Figure 2.2 shows the comparison of elastic modulus and recovery strain of different SMAs. From Figure 2.2, it can be observed that Fe-Ni-Co-Al-Ta-B has very high recovery strain on the other hand the other Fe-based SMA, Fe-Mn-Al-Ni, has very high elastic modulus. However, nitinol alloys have reasonable recovery strain and elastic modulus.   Figure 2.2. Comparison of elastic modulus and recovery strain of different SMAs  Figure 2.3 shows the comparison among different commonly used construction material with different types of SMAs in terms of stress limit and recovery strain. From Figure 2.3, it is evident that the most common construction materials such as steel and aluminium has very low recovery strain although they have high strength. On the other hand, elastomers or rubbers can readily recover the shape but have much less strength. However, SMAs have a good Ni-TiNi-Ti-NbCu-Al-MnCu-Al-BeFe-Ni-Co-Al-Ta-BFe-Mn-Al-Ni02468101214160 20 40 60 80 100 120Recovery Strain (εr), %Elastic Modulus (GPa)12  combination of strength and recoverability and the Fe-based SMA, Fe-Ni-Co-Al-Ta-B possesses relatively very high strength and high recoverability.  Figure 2.3. Comparison among different commonly used construction material and different types of SMAs (adapted from Ma and Karaman 2010) 2.3 Shape Memory Alloy in Bridges Bridge infrastructure represents a significant portion of the transportation network of any country. If a bridge is to maintain its structural integrity and functionality after an earthquake, severe damage to its structural components must be avoided during an earthquake. Development and implementation of innovative structural systems and materials in bridge construction can improve their performance under seismic loads and ensure post-earthquake functionality. In order to mitigate the residual/permanent displacement of bridge piers, researchers have suggested innovative structural systems such as Shape Memory Alloy (SMA) reinforced concrete (RC) bridge columns and bridge decks with prestressed SMA wires. In addition, development of different types of composite materials, isolation devices, and supplemental damping devices incorporating SMA are becoming alternative options for improving the performance of bridges during an extreme natural hazard like earthquake, tsunami, etc. Over the last two decades, SMA has received significant attention from structural engineers and researchers which is reflected through increasing number of research conducted on SMA equipped structural members and elements. Among different applications of SMAs, a significant portion of research and application is focused on bridge engineering. A number of different applications of SMAs in bridge have been investigated to improve the structural 1101001000100000.1 1 10 100 1000Stress Limit (MPa)Recoverable strain (%)ElastomersWoodSteelAluminiumalloysFe-Ni-Co-Al-Ta-BNi-TiCu-Al-MnFe-Mn-Al-Ni13  performance, a synopsis of which is given in the following sections. Figure 2.4 shows the different applications of SMAs in bridge engineering. A major portion of SMA application is focused on bridge piers such as active confinement (Figure 2.4a), prestressing strands (Figure 2.4b), yielding device (Figure 2.4c) and longitudinal reinforcement (Figure 2.4d). Other bridge components which have attracted much attention are the isolation bearing and restrainer. Few applications of SMA in expansion joints, dampers in stay cables, posttensioning tendon in girders have been reported.  Figure 2.4. Application of SMA in bridge engineering (a) active confinement of bridge pier, (b) Post- tensioning in segmental bridge pier, (c) Yielding device in segmental bridge pier, (d) Reinforcement in the plastic hinge region, (e) Restrainer, (f) Isolation bearing, (g) Post-tesioned bridge girder, (h) Expansion joint and (i) Damper in stay cables.  A summary of the statistics of application of SMAs in bridge engineering research found in existing literature is depicted in Figure 2.5. From Figure 2.5 it is evident that, most of the research to date, on the application of SMAs in different bridge components, is focused on developing smart isolation bearings (37%) followed by bridge pier (25%) and dampers (19%). Although seems promising, very little research has been conducted on application of SMAs in bridge girders (4%) and expansion joints (3%). Table 2.1 summarizes the application of SMAs SMA restrainerSMA StrandSMA Yielding deviceSMA RebarSMA TendonSMA damperSMA wire confinementSMA spring in exapnsion joint(a) (b) (c) (d) (e)(f)(g) (h) (i)SMA Wire14  in bridge engineering in different forms (bars, cables, wires) along with the property used in those applications.  Figure 2.5. Statistics of application of SMA in bridge engineering Table 2.1. Summary of SMA application in bridge engineering Alloy Application Type Size (mm) Propoerty Used Study Method* Reference Ni-Ti Bridge Pier Bar 12.7 Superelasticity E+N Saiidi and Wang 2006, Saiidi et al. 2009, Cruz Noguez and Saiidi 2012, 2013 Ni-Ti Bridge Pier Bar 25.4 Superelasticity A Roh and Reinhorne 2010 Ni-Ti Bridge Pier Wire 3 Shape memory E+N Shin and Andrawes 2011 Ni-Ti Bridge Pier Bar 20.6 Superelasticity N Billah and Alam 2014c Cu-Al-Mn Bridge Pier Bar 25 Superelasticity N Gencturk and Hosseini 2014 Ni-Ti Isolation device Bar 150 Superelasticity N+A Wilde et al. 2000 Cu-Al-Be Isolation device Bar 3.5 Superelasticity E Casciati et al. (2007) Ni-Ti Isolation device Wire 10 Superelasticty+ Damping E+N Choi et al. 2005 Ni-Ti Isolation device Wire 2 Superelasticity N Dolce et al. (2007) Ni-Ti Isolation device Wire 1.5 Superelasticty+ Energy Dissipation N+A Ozbulut and Hurlebaus (2010, 2011b) Cu-Al-Be, Ni-Ti Isolation device Wire 2.76 Superelasticty+ Damping N  Bhuiyan and Alam (2013) Fe-Ni-Co-Al-Ta-B Isolation device Wire 2.5 Superelasticity N Dezfuli and Alam (2013) Bridge Pier25%Isolation Bearing37%Restrainer12%Damper19%Expansion Joint3%Bridge Girder4%15  Ni-Ti Isolation device Coil spring 1 Superelasticity N Attanasi and Auricchio (2011) Ni-Ti Damper Plate 5 Damping E+N Adachi and Unjoh (1999)  Ni-Ti Damper in stay cable Spring 0.6 Superelasticity + Damping N+A Liu et al. (2007) Ni-Ti Damper Wire 1 Superelasticity + Damping E+A Suduo and Xiongyan (2007) Cu-Al-Be Restraining damper Wire 1.4 Superelasticity N  Zhang et al. (2009)  Ni-Ti Damper in stay cable Wire 0.2 Superelasticity + Damping N+A Mekki and Auricchio (2010) Ni-Ti Damper in stay cable Wire 2.46 Damping E+N Dieng et al. (2013)  Fe-Mn-Si-Cr Bridge Girder Bar 10.4 Shape memory E Soroushian et al. (2001) Ni-Ti Bridge Girder Bundled Wire 15.3 Shape memory E Li et al. (2007) Ni-Ti-Nb Bridge Girder Wire 3.5 Shape memory E Ozbulut (2013) Ni-Ti Expansion joint Spring 51 Superelasticity E Padgett et al. (2013) Ni-Ti Restrainer Bar 25.4 Superelasticity N DesRoches and Delmont (2002) Ni-Ti Restrainer Bar 12.7 Superelasticity N Andrawes and DesRoches (2005) Ni-Ti Restrainer Cable  Superelasticity N Andrawes and DesRoches (2007a) Ni-Ti Restrainer Cable 0.584 Superelasticity E+N Johnson et al. (2008) Ni-Ti Restrainer Cable 1.584 Superelasticity E+N Padgett et al. (2009) Ni-Ti Restrainer Bar 25.4 Superelasticity + Damping N  Choi et al. (2009)  Ni-Ti Restrainer Bar 40 Superelasticity N Alam et al. (2012)  Ni-Ti Restrainer Cable 2 Superelasticity E+N Cardone and Sofia (2012) Ni-Ti Restrainer Wire 1.2 Superelasticity E Anxin et al. (2012) *Note: E= Experimental; N= Numerical; A= Analytical 2.3.1 Application in bridge pier Bridge piers are one of the most vulnerable elements in a bridge and their failure can have catastrophic consequences. Experience and research on reinforced concrete bridge piers have necessitated that the response of bridges during earthquake should be stable and able to reduce the seismic damage and return to its original position after a seismic event. According to current seismic design guidelines, bridge piers are designed to resist a significant portion of the lateral load during a seismic event while dissipating a significant amount of energy. As a result, once the steel rebar yields, the bridge pier experiences significant permanent damage or residual deformation thereby rendering the bridge susceptible to collapse. In an attempt to 16  reduce the damage of bridge pier and limit the residual deformation, researchers came up with the idea of using SMA in the plastic hinge region of the bridge pier which is subjected to significant nonlinear deformation under ground motion (Saiidi and Wang 2006).  The feasibility of application of SMA in bridge pier was first investigated by Saiidi and Wang (2006). They incorporated Ni-Ti SMA bars in the plastic hinge region of RC piers and conducted shake table tests on quarter scale RC bridge piers. They found that SMA reinforced piers encountered very negligible residual deformation (0.2%) which is important for keeping the bridge pier functional following an earthquake. Later, Saiidi et al. (2009) investigated the performance of bridge pier incorporating SMA and engineered cementitious composite (ECC) in the plastic hinge region. They tested the bridge piers under reverse cyclic loading and concluded that incorporation of SMA and ECC reduced the residual deformation by 83% as compared to conventional bridge pier and increased the drift capacity significantly. Andrawes et al. (2010) and Shin and Andrawes (2011) experimentally investigated the feasibility of using SMA spirals for seismic retrofitting of bridge piers. They concluded that this active confinement technique is more effective and reliable as compared to conventional passive confinement techniques. Shin and Andrawes (2011) concluded that retrofitting using SMA spirals can be cost effective as compared to conventional FRP or steel jackets as it requires small amount of SMA and limited labor as well as the damaged bridge can be restored within a short period of time. Roh and Reinhorne (2010) incorporated SE SMA bar at the base segment of precast segmental bridge pier to improve the energy dissipation capacity and self centering capacity of unbounded post-tensioned segmental columns. They developed new modeling techniques of SMA bar comprising of four springs and analyzed segmental bridge piers with SMA rebar under quasi-static cyclic loading. They found that the inclusion of SMA bar provides good recentering, high ductility and stable energy dissipation. Cruz Noguez and Saiidi (2012, 2013) conducted shake table tests on a four span bridge system with conventional and advanced details. The bridge had three column bents each consisting of two bridge piers and each bent had a different unconventional detailing in the plastic hinge region. One of the bents had a combination of SMA and ECC in the plastic hinge region while the other two had elastomeric bearing pads and posttensiong tendons. Their results showed that, bridge piers with SMA-ECC exhibited higher ductility and experienced minimal damage. They found that the rotational deformations were higher for bridge pier detailed with SMA-ECC as compared to 17  conventional RC pier. However, the residual deformation was significantly reduced which allowed the bridge to remain serviceable after the maximum design earthquake. Billah and Alam (2014c) investigated the seismic vulnerability of bridge pier reinforced with SMA in the plastic hinge region and compared with conventional RC pier. They found that conventional bridge pier is less vulnerable when ductility is considered as the demand parameter. On the contrary, when the residual drift is considered as the demand parameter, the SMA-RC pier possesses significantly less vulnerability as compared to conventional bridge pier. Moreover, SMA-RC pier was reported to improve the performance in terms of different performance criteria (yielding, concrete spalling, and crushing). Gencturk and Hosseini (2014) utilized Cu-based SMA in the plastic hinge region of concrete bridge pier and analyzed under the combined action of shear, flexure and axial loading. They concluded that the application of Cu-Al-Mn SMA eliminates the residual deformation significantly but results in a significant reduction in the energy dissipation capacity.  2.3.2 Seismic isolation of bridges Base isolation systems have been proven as one of the most effective and attractive techniques for seismic response control of bridges. Over the past years, a wide variety of seismic isolation devices have been developed (Ozbulut and Hurlebaus 2010, 2011b, Attanasi and Auricchio 2011, Dezfuli and Alam 2013, Bhuiyan and Alam 2012) and researchers are continuously working on the development of novel isolation devices to overcome the shortcomings of the existing ones. The following section describes different application of SMAs in developing smart isolation bearings. 2.3.2.1 SMA bar based devices The first application of SMA in bridge isolation bearing can be traced back to 1996 when Bondonet and Filiatrault (1996) analytically investigated the feasibility of using SMA in a bridge bearing. They found that incorporation of SMA in the bearing reduced the deck acceleration by 90% as well as significantly reduced the bearing residual deformation. Wilde et al. (2000) proposed a laminated rubber bearing with SMA bar and compared its performance with conventional lead core rubber bearing. They concluded that SMA based isolation device reduced the vulnerability of bridge and dissipated more energy as compared to the conventional system. Using a combination of three inclined SMA bars with two disks, Casciati et al. (2007) 18  and Casciati and Hamdaoui (2008) proposed a new innovative isolation device and conducted shake table experiments. Their result showed that the proposed system could dissipate significant amount of energy while providing sufficient recentering. Billah et al. (2010) investigated the seismic performance of a multi span bridge fitted with SE SMA bar based isolator and compared the performance with conventional lead rubber and high damping rubber bearing. They found that the SMA isolating system increased the deck acceleration, however, reduced the relative displacement between deck and pier. 2.3.2.2 SMA Wire based devices In an attempt to increase the recentering ability of elastomeric isolation bearing, Choi et al. (2005) proposed an elastomeric isolation bearing with SMA wires in the longitudinal direction. Although, the proposed system reduced the relative displacement between deck and pier, at very large shear deformation (200%), the system becomes unusable as it experiences strain higher that its SE strain range. Since then, a number of researchers have proposed and investigated different SMA wire based isolation systems. Based on the superelastic behaviour of pre-tensioned SMA wires, Dolce et al. (2007) proposed an SMA wire based isolation system and compared the performance with steel and rubber based isolation devices. Although the SMA based system provided supplemental recentering thus reducing residual deformation, the system dissipated inadequate energy and was sensitive to temperature variation. Liu et al. (2008) conducted shake table tests on rubber bearings with large diameter diagonal SMA strands. The result showed improvement in damping whereas reduction in residual deformation was negligible as compared to original rubber bearing. Ozbulut and Hurlebaus (2010, 2011b) explored optimum design parameters for an isolation bearing consisting of steel-Teflon sliding bearing and an SMA wires considering temperature effect. They investigated the performance of the proposed isolation system under near fault ground motion which effectively reduced the peak deck displacement but increased the deck acceleration. In another study, Ozbulut and Hurlebaus (2011c) investigated the effectiveness of a SMA-rubber based isolation system under near fault ground motion using sensitivity analysis. They concluded that SMA wire combined with sliding bearing performs better as compared to SMA-rubber based isolation system. Bhuiyan and Alam (2013) assessed the seismic performance of a three span highway bridge equipped with two types of SMA based isolation devices under moderate to strong earthquake motions. The SMA based rubber bearing was composed of two types of SMA wires 19  (Ni-Ti and Cu-Al-Be) in natural rubber bearing and the other one was high damping rubber bearing. They concluded that SMA based bearing was effective in controlling residual deformation and pier displacement under moderate ground motions but under strong ground motion their effectiveness reduced significantly. Dezfuli and Alam (2013) proposed a diagonal configuration of SMA wire based isolation device incorporating FeNiCoAlTaB-SMA, with 13.5% superelastic strain and a very low austenite finish temperature (-620C). They concluded that the proposed system performed effectively under varying temperature condition with sufficient energy dissipation capacity. Recently, Dezfuli and Alam (2014) proposed a performance-based design and assessment methodology for high damping rubber bearing incorporating SMA wires. They presented a design methodology and example for determining the pre-strain and cross section of wires in the SMA wire-based rubber bearings. 2.3.2.3 SMA Spring based devices Masuda et al. (2004) proposed a constitutive equation for an SMA spring based base isolation device using finite element analysis. Attanasi and Auricchio (2011) proposed an SMA spring based isolation device consisting of eight SMA springs in combination with a flat sliding bearing. They presented a design example which satisfied all the design requirements. They concluded that the proposed spring based isolation device performed better than the other SMA based isolation devices. 2.3.3 Dampers in bridges The superelasticity and damping property of SMA along with excellent corrosion resistance have attracted researchers to develop and investigate SMA-based damping devices for vibration control of multi span bridges and cable stayed bridges. Adachi and Unjoh (1999) developed a NiTi SMA based damping device for seismic response control of bridges and conducted shake table tests. Test results showed that the SMA damping device is efficient in shape memory phase and can significantly improve the seismic performance of a bridge through enhanced damping. Liu et al. (2007) conducted an experimental investigation on combined stay cable/SMA damper system under sinusoidal excitations. They found that the SMA damper could effectively supress the vibration in first few dominant modes and the efficiency of the damper is dependent on the damper stiffness, its energy dissipation capability, the yielding deflection and the location. Xu and Zhuo (2007) developed a novel SMA based 20  adjustable fluid damper for vibration control in cable stayed bridges. They proposed a design procedure for selecting the adjustable fluid dampers for vibration mitigation in stay cables.  In an attempt to reduce the cable vibration and increase damping, Casciati et al. (2008) investigated several combinations of steel cable-SMA wire systems. They observed a decrease in vibration amplitude and increase in damping coefficient by up to 124% when a combination of steel cable-SMA wire is used as opposed to steel cable. Sharabash and Andrawes (2009) conducted an analytical investigation on the effectiveness of an SMA damper to control the deck displacement and shear and bending moment demand on towers of a cable stayed bridge. They found that application of SMA damper reduced the maximum bridge displacement, towers base shear, and towers base moment by up to 65%, 65%, and 69%, respectively compared to that of the bridge without SMA damper. Zhang et al. (2009) developed a superelastic Cu-Al-Be SMA wire based passive control device considering wide temperature range from -800C to 1200C. The proposed control device significantly reduced the overall bridge deformation and bearing deformation when subjected to strong ground motions. Mekki and Auricchio (2010) proposed and investigated the performance of a passive control device for stay cable in cable stayed bridges by utilizing the superelasticity and damping property of SMA. They concluded that the proposed device could effectively dampen the high free vibration of stay cables as compared to conventional tuned mass dampers. Dieng et al. (2013) experimentally investigated the efficiency of Ni-Ti SMA damper in reducing the vibration of cables in cable stayed bridges. Experimental result proved the efficacy of SMA dampers in reducing the oscillation periods and their amplitudes. 2.3.4 Prestressing in bridge girders Although a significant amount of research has been conducted on the application of SMA in bridge piers, isolation bearings, active and passive dampers, very few research works have been conducted on SMA’s application in bridge girders. Maji and Negret (1998) pioneered the concept of smart prestressing using SE SMA in bridge girders. They used SMA strand-wires as prestressing tendons which showed good bonding strength with concrete. They concluded that this smart prestressing can actively accommodate additional loading and overcome prestress loss over time. Li et al. (2007) experimentally investigated the application of bundled SMA wire in smart bridge girders. They concluded that using the shape memory property of 21  SMA bundle, the load bearing capacity of bridge girders can be improved by applying current. Ozbulut (2013) experimentally investigated the feasibility of using shape memory alloys for developing self-post-tensioned concrete bridge girders. They aimed at eliminating the jacking force by developing self-stressing capacity using the shape memory effect of SMAs developed from the heat of hydration of grout. 2.3.5 Retrofitting of bridge girders Using the shape memory effect of SMA, Soroushian et al. (2001) investigated the feasibility of using iron based martensite SMA rebar for rehabilitation of shear deficient bridge girders. They developed a design methodology and verified through experimental tests simulating the real bridge scenario. They applied this rehabilitation method on U.S. Route 31 bridge in Michigan using posttensioned SMA rod which reduced the crack width in girders by 40%.   2.3.6 Application in bridge expansion joints Bridge expansion joints are one of the vulnerable components in highway bridges when subjected to moderate to severe ground motion. Although different design guidelines are limiting or eliminating the application of expansion joints in bridges, however, application of smart modular expansion joints can enhance the overall bridge seismic response. In order to eliminate the limitations of current expansion joints and make the use of SMA’s distinct thermomechanical properties, Padgett et al. (2013) developed and tested an SMA based smart expansion joint. They tested different configurations of SMA enhanced modular expansion joints including rings, single stacked bevels, double stacked bevels, triple stacked bevels, round bar S shapes, dollar signs, flat plate s shapes, solid section springs, hollow section springs, and omega shape SMAs. They found that a solid section of SMA spring met all the design and performance objectives. Based on the experimental results, analytical model of the smart expansion joint was developed and validated against test results. They also assessed the comparative vulnerability of conventional and the SMART expansion joints which revealed superior seismic response of the SMART expansion joint. Finally, a comparative life cycle analysis revealed that the developed SMART bridge expansion joint offers a cost effective solution to supplement large capacity joints typically adopted in critical lifeline bridges. 22  2.3.7 Restrainer in bridges Deck unseating resulting from the excessive longitudinal deformation at in-span hinges or supports has been identified as a common bridge damage scenario during recent earthquakes (1994 Northridge, 1999 Chi Chi, 2010 Chile, and 2011 Christchurch earthquake). Since early 1970, steel restrainers have been used as an effective means of reducing pounding or deck unseating. However, the poor performance of steel restrainers during the 1994 Northridge and the 1995 Kobe earthquake triggered the need for more efficient restrainers for improved seismic performance (DesRoches and Delemont 2002). In an attempt to reduce the seismic vulnerability of bridges, DesRoches and Delemont (2002) numerically investigated the performance of SMA restrainer bars. Using 25.4 mm SMA bar as restrainer at the intermediate hinges and abutments, they evaluated the seismic performance of bridge under near fault motions. Comparison of SMA restrainer with conventional steel restrainer cable revealed the superior performance of SMA restrainer in limiting relative hinge displacements at the abutment and deck movement. After that, Andrawes and DesRoches (2005) investigated the performance of 12.7 mm SMA restrainer in preventing the unseating and limiting the relative hinge deformation of multiple frame RC box girder bridge. They concluded that SMA restrainers outperformed conventional steel cable restrainer without increasing the ductility demand in the bridge. Subsequently, Andrawes and DesRoches (2007a) compared the effectiveness of SMA restrainer as a retrofit measure with conventional steel restrainers, metallic dampers, and viscoelastic dampers. They concluded that the effectiveness of retrofit measure is dependent on the bridge geometry and ground motions characteristics. They found that SMA restrainer was effective in limiting residual joint opening as well as restricting unseating. In another study, Andrawes and DesRoches (2007b) investigated the effect of varying temperature on the performance of SMA restrainer in bridges. They found that the effect of temperature is more pronounced near the austenite finish and the effectiveness of SMA restrainer increases at higher ambient temperatures. Johnson et al. (2008) conducted large scale shake table test to investigate the performance of in-span hinges equipped with SMA restrainers and steel cable restrainer. Although, both SMA and steel restrainer experienced similar forces, SMA restrainer experienced limited residual strain while showing little strength and stiffness degradation. Padgett et al. (2009) developed an SMA restrainer cable, connected at the deck-abutment interface of a RC slab bridge and conducted shake table 23  tests. Test results revealed the efficacy of SMA cables which reduced the unseating potential through reduction in the as-built openings by 47% and 32% for low-level and high-level loading, respectively. Choi et al. (2009) experimentally investigated the bending behavior of large diameter SMA bars under various loading speed to determine its feasibility to use as restrainer in bridges to overcome the shortcomings of SMA cable restrainers. They conducted a numerical study on a three span bridge in a moderate seismic zone using SMA bending bar as restrainer. They concluded that the bar restrainer was effective in reducing the hinge opening and the pounding force on abutments. Anxin et al. (2012) conducted shake table tests on SMA wire restrainers connected in the form of deck-deck and deck-pile connections. Their result showed that, SMA restrainers installed in the form of deck-pile connections can significantly decrease the displacement responses of the isolators in the highway bridge. Alam et al. (2012) investigated the seismic fragility of isolated bridge equipped with SMA restrainer under strong ground motions. Two types of isolation bearings, namely, high damping rubber bearings and lead rubber bearings were used in combination with SMA restrainer. They concluded that when the bridge is isolated using lead rubber bearing, inclusion of SMA restrainer increases the failure probability. Cardone and Sofia (2012) conducted shake table tests to evaluate the effectiveness of SMA-based cable restrainers in controlling the displacement response of simply supported deck bridges. Test results revealed that, inclusion of SMA restrainer provides additional protection to the isolation bearings. 2.4 Comparison of SMA based and Conventional Bridge Component Performance Discussions in the previous sections showed that significant amount of research has been conducted to improve the performance of different bridge components using different forms and types of SMAs. This section is intended to provide a brief summary of the comparative performance of different SMA-based and conventional bridge components. Table 2.2 shows the performance comparison of different conventional and SMA-based smart bridge components. In Table 2.2 the performance of SMA-based and conventional bridge components are compared in terms of residual drift and energy dissipation. From Table 2.2 it can be concluded that SMA as reinforcement in bridge pier significantly reduces the residual deformation irrespective of the type (Ni-Ti or Cu-based) and form (bar and wire) used. When SMA wires are used in isolation bearing, except Cu-Al-Be SMA, other SMAs have shown 24  improvement in isolation bearing performance in terms of reducing pier displacement, residual deformation and viscous damping. Table 2.2. Performance comparison of SMA-based and conventional bridge components Bridge Component Alloy Type Performance indicator  SMA-based component Conventional Reference Pier Ni-Ti Rebar Residual Drift (%) 0.36 2.66 Saiidi et al. 2009 Cu-Al-Mn Rebar Residual Drift (%) 0.39 2.78 Shrestha et al. 2015 Ni-Ti Wire Displacement ductility  8 2.8 Shin and Andrawes 2011 Hysteretic energy (kJ) 16.1 75.9 Shin and Andrews 2011 Isolation Bearing Ni-Ti Wire Pier displacement (mm) 55.8 98.3 Bhuiyan and Alam 2013 Cu-Al-Be Wire Pier displacement (mm) 129 98.3 Bhuiyan and Alam 2013 FeNiCoAlTaB Wire Residual deformation (mm) 10.2 5.4 Dezfuli and Alam 2013 Viscous damping (%) 7.5 9.2 Dezfuli and Alam 2013 Restrainer Ni-Ti Cable Maximum displacement (mm) 32 61 Jhonson et al. 2008 Energy dissipation (kN-mm) 263 112 Jhonson et al. 2008 Ni-Ti Cable Residual joint opening (mm) 43 87 Andrawes and DesRoches 2007 Ni-Ti Bar Relative deck and abutment displacement (mm) 63.8 84.6 DesRoches and Delemont 2002 Dampers Ni-Ti Cable Deck displacement (mm) 54 200 Sharabash and Andrawes 2009 Tower base shear (MN) 10.29 30.59 Ni-Ti Cable Damping ratio (%) 1.08 0.41 Liu et al. 2007 Expansion Joint Ni-Ti Spring Column displacement (inch) 0.17 2.17 Padgett et al. 2013  2.5 Promising SMAs for Application in Bridges To date several compositions of SMAs have been developed and their application in different fields of civil engineering have been investigated. There are several compositions of SMA that have strong potentials for application in bridge engineering, especially in seismic prone areas and in areas where the temperature changes form extreme hot to extreme cold. Again, some of those SMAs have been developed as wires or thin sheets, but not as rebars. Figure 2.6 shows the hysteretic response of three different SMAs that have potential for civil 25  engineering application. From Figure 2.6 it can be observed that, the Ni-Ti alloy (Alam et al. 2008a) has lower maximum strain (~6%) but very high strength (> 500MPa) and fatter hysteresis loop. On the other hand, the Cu-Al-Mn alloy (Varela et al. 2014) has maximum strain of 8% but much less strength (< 400 MPa) and thin hysteresis loop. Table 2.3 provides a summary of six SMAs that have very good potential for application in bridge engineering. In Table 2.3 compositions of Ni-based, Cu-based and Fe-based SMAs are provided along with their properties desirable for different bridge engineering applications. All the alloys presented in that table has austenite finish temperature (Af) less than -100C which indicates that all of them can be used in cold regions where temperature varies over a wide range.   Figure 2.6. Comparison of hysteretic response of different SMAs The Ni-Ti alloy (Alam et al. 2008a) possess reasonable elastic modulus, yield strength comparable to conventional steel, good recovery strain of 6% and Af  on the negative side. Ni-Ti SMAs with similar composition and mechanical properties have been used by several researchers as reinforcement in bridge piers (Saiidi and Wang 2006, Billah and Alam 2014c) and as restrainers which (Andrawes and DesRoches 2005, Padgett et al. 2009) performed very well under extreme earthquake events. One drawback of this alloy is that it may not be used in cold regions where the temperature goes beyond -100C unless manufactured with lower Af. The second alloy, Ni-Ti-Nb can be used as spirals for retrofitting of bridge piers, tendons for prestressing of bridge girders as well as for isolation bearing with SMA wires. This alloy 01002003004005006007000 5 10 15 20Stress (MPa)Strain (%)26  can be used as self-heating posttensioning tendons in bridge girders using the heat of hydration of concrete and external heat application. The Cu-based SMAs (Cu-Al-Mn and Cu-Al-Be) have very low Af which is good for cold weather application but have lower elastic modulus and yield strength as compared to Ni-based SMAs. However, these SMAs have much low cost as compared to Ni-based SMAs. The Cu-Al-Mn SMA has very high recovery strain (7%) which holds promise for application in bridge girders as prestreesing tendons as well as in post-tensioned segmental bridge pier construction. Although the Cu-Al-Be has very low recovery strain (3.2%) it is appropriate for damping application in the austenite phase. Table 2.3. Potential SMAs for application in bridge engineering Alloy Composition E (Gpa) Fy-SMA (MPa) εmax (%) εr (%) Af (0C) Reference Ni-Ti 50.02-49.98 62.5 401 6.8 6 -10 Alam et al. 2008a Ni-Ti-Nb 47.45-37.86-14.69 20 250 7 3.2 -22 Park et al. 2010 Cu-Al-Mn 71.9-16.6-9.3 31.2 210 8 7 -39 Araki et al. 2010 Cu-Al-Be 87.68-11.7-0.62 32 230 3 2.4 -65 Zhang et al. 2010 Fe-Ni-Co-Al-Ta-B 59.05-28-17-11.5-2.5-0.05 46.9 750 15 13.5 -62 Tanaka et al. 2010 Fe-Mn-Al-Ni 43.5-34-15-7.5 98.4 320 6.1 5.5 <-50 Omori et al. 2011 E= elastic modulus, Fy-SMA= austenite to martensite starting stress, εmax= maximum strain, εr= recovery strain, Af= austenite finish temperature Recently Tanaka et al. (2010) developed a ferrous polycrystalline SMA (Fe-Ni-Co-Al-Ta-B) which has a very high superelastic strain range of over 13% at room temperature. This SMA has approximately 20 times higher SE than other Fe-based alloys and almost double that of conventional Ni-Ti alloy. This Fe based SMA has extremely high ductility, greater strength, and energy damping capacity several times higher than commercially available Ni-Ti SMA. All these criteria make this an outstanding candidate for all types of bridge engineering applications especially in seismic regions. More recently Omori et al. (2011) developed another Fe based SMA (Fe-Mn-Al-Ni) which has superelasticity similar to the conventional Ni-Ti SMA but with temperature insensitive superelasticity. This temperature insensitive property is very important for cold region engineering especially in North America and Europe where the temperature varies over a wide range. The Fe based SMA have low temperature sensitivity which will allow good bonding and compatibility with cement based matrix. This 27  SMA is also another strong contender for various applications in bridge engineering as discussed in this study. Moreover, this alloy is composed of commonly available low cost metals with good workability and corrosion resistance.  On the other hand, this alloy does not require any thermo-mechanical treatment or ‘training’ to improve its SME like other Fe based SMA which will substantially reduce the production cost (Omori et al. 2011). Moreover, application of this alloy in construction will reduce the difficulties of forming and machining associated with commercial Nitinol alloys. Table 2.4 summarizes the properties of SMA applicable for different bridge engineering applications and their advantages. 2.6 Future of Smart Bridges The rapidly increasing interest in SMAs, both in research and commercial applications, indicate the increasing potential of smart infrastructures. The application of this smart material in bridge engineering is expected to develop in the following three levels: (a) development of low cost SMAs with improved machinability and properties suitable for civil engineering application, (b) development of hybrid structures combining the functional properties of SMAs with the structural properties of other materials, and (c) development of novel ideas and design guidelines for civil engineering applications. The potential of developing smart bridges using different shape memory alloy based components have been investigated by researchers over the last 15 years. Li et al. (2007) proposed a conceptual smart RC bridge using bundled SMA in the bridge girder. Their concept was to use the shape recovery property of SMA when subjected to excessive loading. Alam et al. (2008b) proposed another SMA-based smart simply supported bridge system where different bridge components such as the bridge deck, girders, and piers are reinforced with SMA rebars, the bridge is equipped with SMA-based isolators, dampers, and fiber optic sensors. They concluded that the proposed smart configuration will allow continuous monitoring, permit excessive vehicle loading beyond design level on the bridge, and possess improved energy dissipation and recentering ability during extreme seismic event. However, all the ideas are still in conceptual phase but not have been implemented in real applications due to the high cost of SMA. However, researchers are continuously exploring various combinations of SMA and came up with various low cost Fe-based and Cu-based SMAs with excellent properties which hold great promise and enormous potential for large scale 28  application in bridge engineering ensuring enhanced safety. Improved seismic performance of bridge piers with SMA as reinforcement attracted Washington Department of Transportation and they decided to use SMA in one of the piers of the three span SR-99 bridge in Seattle, WA (Kosowatz 2014). Moreover, researchers are working towards development of design guidelines for SMA based bridges. For example, Dezfuli and Alam (2014) developed performance-based design guidelines for FRP-based high damping rubber bearing incorporating SMA wire.  Table 2.4. Summary of SMA properties for bridge engineering application and their effects SMA Properties Consequences Practical Application in Bridges Shape memory effect Material can be used as post tensioning or prestressing tendons, providing force during shape recovery Active confinement of bridge piers, Prestressing of bridge decks and girders, Post-tensioning of segmental bridge piers, Post-tensioning of bridge pier-cap beam joint, Isolation Bearing. Superelasticity Elastic recovery of strain and material can be stressed to provide large, recoverable deformations at relatively constant stress levels Reinforcement in bridge piers and bridge beam-column joints, Connection between footing and first segment of segmental bridge pier, Short fibers in bridge girders, Bridge restrainer, Isolation Bearing. Hysteresis Allows for significant energy dissipation without permanent deformation under cyclic loading Reinforcement in bridge piers and bridge beam-column joints, Connection between footing and first segment of segmental pier, Active confinement of bridge piers. Recovery Strain Little or no permanent deformation Reinforcement in bridge piers, girdrs, and bridge beam-column joints, Connection between footing and first segment of segmental bridge pier, Active confinement of bridge piers, Bridge restrainer, Isolation Bearing. Damping Allows to work as a passive system which dissipates energy during every cycle of the oscillating system without requiring external control. Structural cables in stay cable bridges, suspension bridges, and prestressed concrete bridges, Isolation bearings. Fatigue Allows the material to undergo several thousands of cycles  Structural cables, Dampers, Isolation Bearings Corrosion Resistance Allows application in harsh environment Reinforcement in bridge piers, Active confinement of bridge piers, dampers and structural cables in aggressive marine environment. 29  2.7 Summary Shape memory alloys (SMAs) are special materials with distinct thermomechanical properties that allow them to ‘memorize’ or retain their original shape when subjected to load or temperature. In recent years, SMAs have drawn significant attention and interests among researchers and structural engineers for diverse civil engineering applications. Superelasticity, shape memory effect and hysteretic damping, are the three major attributes of SMAs that make them ideally suited for bridge engineering applications. The increasing interest on SMAs in bridge engineering research indicates the emerging potential of SMA in construction industry. This chapter provided a review of existing applications of SMAs in bridge engineering, summarized the research results on different SMA based components, categorized the applications in different bridge component domain, and highlighted the sectors of potential development and future application opportunities.       30  CHAPTER 3. SEISMIC FRAGILITY ASSESSMENT OF HIGHWAY BRIDGES: A STATE-OF-THE-ART REVIEW  3.1 General Earthquake induced damages in recent years have exposed bridges as one of the most susceptible components of the transportation system. Failure of bridges during an earthquake (for example severe damages or collapse during the 1994 Northridge, the 1995 Kobe earthquake and the 2011 Christchurch earthquake) can severely disrupt continuous transport facilities, emergency and evacuation routes. To mitigate potential economic losses and loss of lives during a seismic event, performance evaluation of existing bridges, and strengthening of the critical components is crucial for the stakeholders. Development of fragility curves provide a probabilistic assessment of the seismic risk to highway bridges which is critical in pre-earthquake planning and post-earthquake response of transportation systems. A vast majority of the highway bridges around the world were not designed according to any seismic design criteria and thus do not meet the seismic detailing requirement imposed by current guidelines (CHBDC 2010, CALTRANS 2013, Eurocode 2005). These factors lead to the reconsideration of three important issues such as (i) the seismic performance of those non-seismically designed bridges, (ii) potential economic losses, and (iii) selection of risk mitigation and performance improvement techniques, i.e. retrofitting or rehabilitation. The ramification and diversity in bridge design and construction practices all over the world do not allow adopting a single methodology that can be applicable for the seismic vulnerability assessment of highway bridges. Uncertainties arising from myriad of bridge components, material characteristics, and regional seismicity along with the need for better predicting the seismic performance of bridges have resulted in the development of different vulnerability assessment methodologies for highway bridges. Although these different methodologies were targeted for specific purposes and adopted particular mathematical framework, the overall objective was to assess the seismic vulnerability of highway bridges to ensure the safety and security of bridge infrastructure and its management against seismic loading.  The objective of this chapter is to provide a comprehensive review of the existing methodologies and identify current trends in the seismic fragility assessment of highway 31  bridges. Based on the existing literature this study illustrates, in a systematic manner, a summary of different fragility assessment methodologies for highway bridges, features, and limitations and a critical review of the state-of-the-art currently existing application of fragility assessment methods. This study also provides general information about the different aspects of fragility assessment and how different researchers have developed this tool as a means of better-informed decision making. This is a comprehensive but not necessarily an exhaustive study. To the best of the authors’ knowledge no such study has been conducted so far that summarizes the state-of-the-art fragility assessment of highway bridges. 3.2 Seismic Fragility Analysis The inception of lifeline earthquake engineering in the early 1970’s have influenced numerous researchers (ATC, 1985, 1991; King et al., 1997; Shinozuka et al., 1997; Veneziano et al., 2002; Werner et al., 1997) to develop and propose a wide variety of seismic risk assessment methodologies for highway transportation systems. With the advancement of the performance based earthquake engineering, the site specific deterministic design criteria are transitioning towards probabilistic design criteria as a means of describing the performance at different levels of seismic input intensity (Mackie and Stojadinovic, 2005). Fragility curves describe the conditional probability, i.e. the likelihood of a structure being damaged beyond a specific damage level for a given ground motion intensity. Therefore, current seismic performance assessment methodologies are tending toward fragility curves as a means of describing the fragility of structures, such as highway bridges, under uncertain input. The fragility or conditional probability can be expressed as: Fragility= P[LS|IM=y]        (3.1) where, LS is the limit state or damage state of the structure or structural component, IM is the ground motion intensity measure, and y is the realized condition of the ground motion intensity measure. The development of fragility curves for seismic risk assessment can be traced back to 1975 when Whitman et al. (1975) formalized the seismic risk assessment procedure. Subsequently the Applied Technology Council (ATC) and the Federal Emergency Management Agency (FEMA) contributed significantly towards the development of fragility functions and vulnerability assessment procedures. The concept of continuous fragility function was first put forward by ATC 25 report (ATC 1991) by introducing continuous 32  damage functions. Using a regression analysis of the different damage probability matrices, the damage functions or the fragility curves were generated. Later in 1997, FEMA introduced a risk assessment software package, Hazard United States (HAZUS 1997), which is based on the geographical information system (GIS), by involving a panel of experts. Over the years HAZUS has undergone significant development and the most recent version HAZUS-MH 2.1 (HAZUS 2012) is capable of assessing potential risk and losses from earthquakes, floods, and hurricanes. Over the last two decades fragility curve has emerged as an efficient tool for critical decision making for structure and infrastructure safety. Figure 3.1 shows the statistics of research publications related to the seismic fragility assessment of bridges in the last few decades. The number of relevant publications were obtained from different refereed journals. This figure shows an increasing trend of publications which indicates the growing interest of researchers in this field. A total of 350 documents including journal papers, conference proceedings, and dissertations have been published since 1990 where journal publications constitute a significant portion (51%). A large increase in the number of publications took place in 2006-2007 when the number increased by almost 400% from the period of 2004-2005. During the period of 2010-2011 the number of publication was 102 and in 2012-2013 it is 90 where more are expected to come out in the coming months. This increasing growth of publication confirms that there is a widespread interest among the research community and industry to investigate the seismic fragility of existing bridges all over the world. Fragility curves can be used for decision making in both the pre-and post-earthquake disaster management, to make informed decisions on the allocation of resources for retrofit, design, and the improved redundancy of a highway network (Mackie and Stojadinovic 2005).  Figure 3.2 illustrates different applications of fragility curves for bridges. 33   Figure 3.1. Statistics of publications on seismic fragility analysis of bridges since 1990   Figure 3.2. Various applications of seismic fragility curves 010203040506070No. of PublicationsJournalConferenceDissertationSeismic fragility of highway bridgesBridge damage-functionality relation shipRepair and replacement cost estimationAssessment of potential consequences and riskRisk mitigation effortEmergency / disaster response planningEmergency route selectionRetrofit selectionRetrofit prioritizationDirect economic lossLoss of bridge functionalityInformed decision makingandIncreased safety of highway bridges34  3.3 Methods for Fragility Curve Development Over the last two decades, fragility curves have transitioned from empirical to analytical methods. Different methods and approaches have been developed by different researchers for developing fragility curves such as judgemental, field observations, advance analysis using analytical models, as well as hybrid methods. Different researchers have developed and employed different methodologies for assessing the seismic fragility of bridges, a brief outline of which is given in the following sections. Figure 3.3 shows the methodology that is commonly used in generating different types of fragility curves and Table 3.1 shows the comparative assessment of different methodology.   Figure 3.3. Methodology for developing seismic fragility curves Methodology for Fragility Curve DevelopmentDevelopment of system/bridge fragility curveExpert based Experimental Analytical HybridSelection of expert panelPreparation of questionnaireSurvey and Compilation of resultsFormation of damage probability matrixDevelopment of component fragility curveExperimentalSelection of Ground Motion SuiteBridge inventory and classification of bridgeIdentification of material properties and structural configuration (variables)Real SyntheticGenerating synthetic ground motionCollecting real ground motion from different sourcesScaling of ground motionSelection of appropriate IMNonlinear analytical modeling of representative bridgesFunctional and physical definition of different damage statesIdentification of appropriate EDPCapacity Determinationof Bridge ComponentsDetermination of component damage statesNonlinear time history/ Incremental dynamic analysisCalculation of component demandDevelopment of component PSDMShake table experiment of bridge or bridge componentsRelationship between observed damage and IMHybrid Simulation/ combination of statistical data and NLTHA resultsCombination of mean IM from hybrid test and dispersion from literatureAnalyticalHybridEmpiricalSelection of bridge typeObtain actual bridge damage dataClassify according to observed damageFormation of damage matrixSelection of damage distribution functionSelection of appropriate distribution function35  A brief summary of different studies on seismic fragility assessment of bridges are provided in Table A.0.1 in Appendix A. This table shows the features of different studies such as the demand parameters, intensity measures, uncertain parameters, methodology, and different components considered in different studies. Table 3.1.Comparison of different methods for development of fragility curves Method Advantages Disadvantages Expert Based / Judgmental  -Simple method. -All factors can be incorporated. -Extremely subjective. -Depends of panel expertise. -Often biased and lack of reliability. Empirical -Represent a realistic picture. -Shows the actual vulnerability. -Lack of adequate data. -Region and structure specific. -Discrepancy in damage observation.  Experimental -Provides actual damage condition. -Lack of adequate data. -Subjective definition of damage states. -Weak correlation between geometry and structural properties. Analytical -Increased reliability. -Consideration of all types of uncertainty. -Less biased. -Computational cost. -Time consuming. -Selection of analysis technique. -Definition of damage states.  -Selection of probability distribution function. Hybrid -Combination of experimental and analytical observation. -Involves damage data from post-earthquake survey. -Reduced computational effort. -Requirement of multiple data sources. -Extrapolation of damage data. -Large dispersion in the demand model.  3.3.1 Expert based/judgmental fragility curves One of the oldest and simplest methods of deriving fragility functions is expert based or judgemental fragility curves. In this method, an expert panel with expertise in the field of earthquake engineering are questioned concerning the various components of a typical highway bridge and asked to make estimates of the probable damage distribution when subjected to earthquakes of different intensities (Rossetto and Elnashai, 2003). A survey is conducted among the specialists using a set of questionnaires. Based on the expert opinion, probability distribution functions are updated to represent a particular damage level at various levels of ground motion intensity. Since the experts provide their opinion of exceeding each damage state, it is possible to develop fragility curves for each damage state over a wide range 36  of ground motion intensity. One of the practical examples of the judgemental fragility curve is reported in the ATC-13 (ATC 1985) report. This report documented the damage matrices and associated risk of typical California infrastructure based on opinion from a panel of 42 experts. However, only 4 of the 42 experts were experienced with highway bridges and their seismic performance. Based on their responses, a damage probability matrix based on Modified-Mercalli Intensity (MMI) value was developed and included in the ATC-13 report. Figure 3.4 shows a typical survey technique that can be used to obtain an expert opinion. From the figure it can be observed that based on their expertise and observation from previous earthquake, the experts will select the options. Based on the response from the expert panel a damage matrix comprising of intensity measure and damage scenario can be developed. Using the damage matrix and a suitable distribution function, fragility curves can be generated. Since the expert opinion is the only source of developing this type of fragility curves, this method largely depends on the questionnaire used, experience of the panel, as well as the number of experts consulted (Nielson 2005). Very often these judgements are biased and involve number of uncertainties which are not quantified explicitly in the vulnerability functions. Moreover, these are often developed for certain structural types, typical configurations, detailing, and materials. All these factors render the reliability of judgmental fragility curves questionable.  Figure 3.4. Typical survey technique for developing expert based fragility curve   37  3.3.2 Empirical fragility curves Empirical fragility curves are developed using damage distributions from the post-earthquake field observations or reconnaissance reports. Using the large amount of reconnaissance data from the 1994 Northridge and the 1995 Kobe earthquakes, Basöz and Kiremidjian (1997) and Yamazaki et al. (1999), respectively, developed the concept of empirical fragility curves. Based on the post-earthquake damage data and observations, several other researchers (Der Kiureghian, 2002; Shinozuka et al., 2000, 2001; Elnashai et al., 2004) developed empirical fragility curves using different approaches. Using a damage frequency matrix developed from Northridge earthquake damage data, Basoz and Kiremidjian (1997) performed a logistic regression analysis to develop empirical fragility curves. Using the damage data from Kobe earthquake, Shinozuka et al. (2001) applied the Maximum Likelihood Method to estimate the parameters of a lognormal probability distribution describing the fragility curves while Der Kiureghian (2002) adopted a Bayesian approach in order to develop fragility curves. Although empirical fragility curves represent a more realistic picture, they lack generality and are usually associated with a large degree of uncertainty. Inconsistency of different damage state definitions and discrepancy in observation between different inspection teams add up the uncertainty in the developed curves and significantly reduce the usefulness and reliability of the empirical vulnerability curves. Yamazaki et al. (2000) and Shinozuka et al. (2001) developed empirical fragility curves using damage data from 1995 Kobe earthquake. Although they used the damage data from same earthquake for the Hanshin Expressway, their fragility curves were significantly different from each other as illustrated in Figure 3.5.  Table 3.2 shows the comparison of the two parameter, median, λ and log normal standard deviation, ξ used for deriving the fragility curves. These differences in the fragility curves can be attributed to the number of damaged bridges considered, their structural configurations, and definition of damage states. These errors are difficult to avoid using damage statistics and lead to a large data scatter even in cases where a single event and limited survey area are considered (Rossetto and Elnashai, 2003). All these limitations restrict the application of empirical fragility curves. 38  3.3.3 Experimental fragility curves Development of bridge fragility curves using experimental results is not common. Since large-scale experiments involving entire bridge models or full scale components are expensive, bridge fragility analysis utilizing the observed response from shake table tests has been very limited. Although experimental results provide a basis for defining various damage measures for analytical fragility curves, their application is still very limited. Based on experimental results from shake table and cyclic load tests on bridge piers, Vosooghi and Saiidi (2012) developed experimental fragility curves. They developed a probabilistic relationship between experimental damage data and seismic response parameters in the form of fragility curves. Banerjee and Chi (2013) developed fragility curves for bridges using damage data obtained from shake table test of a near-full scale bridge. However, a lack of adequate data points at all damage states and a weak correlation between geometry and structural properties limit the application of the experimental fragility curves.  Figure 3.5. Comparison of empirical fragility curves developed by Shinozuka et al. (2001) [S] and Yamazaki et al. (2000) [Y] using damage data from Kobe earthquake Table 3.2. Comparison of empirical fragility curve parameters Damage Rank Median Log-normal St. Dev Yamazaki et al. (2000) Shinozuka et al. (2001) Yamazaki et al. (2000) Shinozuka et al. (2001) Minor 0.59 0.47 0.53 0.59 Moderate 0.66 0.69 0.52 0.45 Major 0.81 0.80 0.51 0.43 00.20.40.60.810 0.2 0.4 0.6 0.8 1ProbabilityPGA (g)Minor-SModerate-SMajor-SMinor-YModerate-YMajor-Y39  3.3.4 Analytical fragility curves In the absence of adequate damage data, fragility functions can be developed using a variety of analytical methods such as elastic spectral analysis (Hwang et al. 2000), probabilistic seismic demand model using a Bayesian approach (Gardoni et al. 2002, 2003), nonlinear static analysis (Mander and Basoz 1999; Shinozuka et al. 2000, Moschonas et al. 2009), or linear/nonlinear time-history analysis (Tavares et al. 2012; Bhuiyan and Alam 2012, Ramanathan et al. 2012; Pan et al. 2010a; Kwon and Elnashai 2010; Nielson and DesRoches 2007a,b, Choi et al. 2004) and incremental dynamic analysis (Billah et al. 2013, Alam et al. 2012, Zhang and Huo 2009, Mackie and Stojadinovic 2005). The following sections provide a brief overview of the different analytical approaches used for generating fragility curves. 3.3.4.1 Elastic spectral analysis One of the simplest and least time consuming method for generating bridge fragility curve is the elastic spectral analysis (Yu et al., 1991; Hwang et al., 2000). Because of its simplicity, this method is often adopted in checking the performance during design of critical component such as a bridge pier. In this method the capacity/demand ratios of different components are determined to evaluate their seismic damage potential. Hwang et al. (2000) and Jernigan and Hwang (2002) adopted this method for generating fragility curves for Memphis bridges. The capacities of different bridge components are determined using linear elastic models considering effective stiffness properties. The component demands are calculated using elastic spectral analysis. Once the demand and capacity for each component is determined, the capacity/demand ratios of different components are calculated and correlated to particular damage states for various levels of intensity measure. Thus, a bridge damage frequency matrix is generated which is used for developing fragility curves. Although this technique is the simplest one it has several limitations. This method is suitable for bridges which are expected to perform in the linear elastic range. If the bridge is subjected to severe nonlinearity this method fails to accurately predict the demand which in turns make the reliability of derived fragility function questionable. 3.3.4.2 Nonlinear static analysis The limitations of elastic spectral analysis can be overcome using nonlinear static analysis which provides the benefit of considering nonlinearity in the computational model as 40  well as requires less time. Several researchers (Dutta and Mander, 1998; Mander and Basoz, 1999; Mander, 1999; Shinozuka et al., 2000; Banerjee and Shinozuka, 2007) have adopted this method for generating fragility curves for bridges. In this method the capacity is calculated using nonlinear static pushover analysis and demand is estimated from a scaled down response spectrum. Placing the capacity and demand spectra in the same plot, the maximum response of the structure under the specified seismic ground motion is determined by locating the intersection of the two curves (in deterministic analysis). Whenever, uncertainty in capacity and demand is considered, it is represented by plotting the distributions over the capacity and demand curves. Using the intersection of capacity and demand distribution (Figure 3.6), probability of failure can be estimated for a particular intensity level. Using increasing level of intensity measure and various damage states, fragility curves for the bridges can be generated. Apart from its advantages, this method has few limitations. This method was developed based on the recommendations from ATC 40 (ATC 1996) which was developed for buildings. Moreover, this method lacks in defining the bridge structure types and estimation of effective hysteretic damping, which plays a crucial role in seismic performance evaluation.  Figure 3.6. Probabilistic Representation of Capacity and Demand Spectra (Mander and Basoz, 1999).   Demand SpectrumCapacity SpectrumSpectral DisplacementSpectral Acceleration41  3.3.4.3 Nonlinear time history analysis In spite of being one of the most computationally expensive methods, nonlinear time history analysis (NLTHA) is the most reliable method for generating fragility curves (Shinozuka et al. 2000). This method has been used by many researchers (Billah and Alam 2013, Tavares et al. 2012; Ramanathan et al. 2012; Pan et al. 2010a; Kwon and Elnashai 2010; Nielson and DesRoches 2007a, b, Padgett 2007, Choi et al. 2004, Karim and Yamazaki, 2003) for generating fragility curves which have been proven to provide a reliable estimate of the seismic vulnerability of bridges. This method allows the consideration of geometric nonlinearity and material inelasticity to predict the large displacement behaviour and the collapse load of bridges accurately under dynamic loading. Although the actual application of the analyses may vary, all applications follow the basic approach outlined in Figure 3.7.  Figure 3.7. Schematic Representation of the NLTHA procedure used to develop fragility curves The reliability and accuracy of fragility curves derived in this method largely depends on the ground motion suits used for dynamic analyses. As a first step, it is necessary to select a suitable bin of ground motions that closely represents the seismicity of the bridge location and captures the associated uncertainties (e.g. epicentral distance, magnitude). However, still there is debate among researchers on how many ground motions should be selected for generating reliable fragility curves. Once the ground motions are selected, sample bridge Define component limit statesDevelop PSDMDevelop Fragility Curvesln (DI)ln (IM)Ground Motion Suite FEM ModelEstimate Component Responses    42  geometries are created considering variability in geometric, structural, and material properties. Using suitable probability distributions for different random variables statistically significant yet nominally identical 3D/2D analytical bridge models are developed. After that these bridge models are randomly paired with different ground motions and NLTHA is performed for each ground motion-bridge sample. Maximum component demands those are considered critical for bridge vulnerability are recorded from each sample. Using the peak component response and appropriate intensity measure (IM), a probabilistic seismic demand model (PSDM) can be generated using regression analysis or maximum likelihood method. The capacity limit states of different components can be defined based on expert opinion, experimental investigation or analytical approach. Convolving the capacity model with PSDM, fragility curves for the bridges can be developed for different damage states. This method also suffers from several drawbacks such as the priori assumption about the probabilistic distribution of seismic demand and required number of ground motions which makes it computationally expensive. 3.3.4.4 Incremental dynamic analysis In order to reduce the requirement of large number of ground motions for fragility assessment using NLTHA, researchers have come up with the idea of using Incremental dynamic analysis (IDA). IDA is a special type of NLTHA where ground motions are incrementally scaled and series of analyses is performed at different intensity levels. Intensity levels are selected to cover the entire range of structural response, from elastic behaviour through yielding to dynamic instability (or until a limit state ‘‘failure’’ occurs). This technique was developed by Luco and Cornell (1998) and has been described in detail in Vamvatsikos and Cornell (2002) and Yun et al. (2002). Several researchers (Billah et al. 2013, Bhuiyan and Alam 2012, Zhang and Huo 2009, Mackie and Stojadinovic 2005) have preferred this technique over NLTHA for generating fragility curves. However, this incremental scaling of large set of ground motions may lead to instances wherein the computational demand is several times higher than NLTHA. Although this method demands significant computational effort, no prior assumptions are required in terms of probabilistic distribution of seismic demand for the derivation of fragility functions (Zhang and Huo, 2009). This method is similar to NLTHA approach; however, peak component responses need to be calculated at each scaling factor. Using results from IDA, fragility curves can be 43  generated either by deriving the occurrence ratio at each damage state at each ground motion level or by estimating the probability density function of the IM for ground motions in which the damage state thresholds are exceeded (Bhuiyan and Alam 2012). Typically this method is mostly used for collapse fragility assessment of structures. Like other methods, this method has few drawbacks. Selection of ground motions, number of required ground motions, scaling of ground motions, all these can lead to the over or under estimation of the vulnerability of the structures (Baker 2013). 3.3.4.5 Fragility assessment using Bayesian approach Several researchers (Singhal and Kiremidjian 1996, Der Kiureghian 2002, Gardoni et al. 2002, 2003, Koutsourelakis 2010) have adopted Bayesian technique for developing reliable fragility curves by the convolution of demand and capacity models. Using Park and Ang (1985) damage index, Singhal and Kiremidjian (1996) developed fragility curves using Bayesian analysis of observed damage data for subclasses of structural systems. While Der Kiureghian (2002) used the maximum likelihood method in conjunction with the Bayesian approach, Koutsourelakis (2010) used Markov Chain-Monte Carlo techniques along with the Bayesian approach to develop multi-dimensional fragility surfaces as a function of multiple ground motion characteristics. Using a Bayesian approach Gardoni et al. (2002) updated traditional deterministic predictions of capacity and demand models and introduced reliability for generating fragility curves for RC bridges. This study developed fragility curves for typical one and two column bent reinforced concrete highway bridges in California. Later, Zhong et al. (2008) developed PSDM using Bayesian approach for RC bridges with two column bents considering uncertainty and models errors. Using a Bayesian updating approach based on the virtual experiment demand data, Huang et al. (2010) proposed a new PSDM approach for generating fragility curves for single column RC bridge bent. In this study different types of uncertainties, model errors, variation in soil and ground motion characteristics were considered. Bayesian updating technique allows the formulation of confidence bounds which express the epistemic uncertainty around the median fragility curves. This is one of the fundamental advantages of Bayesian technique. 44  3.3.5 Hybrid Fragility curves Different methods of generating fragility curves have their advantages and disadvantages. In order to compensate for the drawbacks of other methods such as the inadequate damage data from real earthquakes, subjectivity of judgemental data, and uncertainties and modelling deficiencies associated with analytical procedures, researchers have come up with the idea of hybrid fragility curves. The hybrid approach attempts to reduce the computational effort of analytical modelling and compensates for the subjective bias of expert judgment method (Kappos et al. 2006). Kappos et al. (1995) first coined the term hybrid method for vulnerability assessment of buildings in Thessaloniki. Later on Kappos and his co-workers (Kappos et al. 1998, 2006, Kappos and Panagopoulos 2010) developed and employed the hybrid fragility curves for vulnerability assessment of reinforced concrete and unreinforced masonry buildings in Greece.  This method incorporates available damage data that resembles the area and structural typology under consideration and combines with analytical damage statistics obtained using nonlinear analysis of typical structures (Kappos et al. 2006).  Hybrid methods also incorporate results from large-scale experimental tests that can reasonably mimic real structural response. More recently, Network for Earthquake Engineering Simulation (NEES) developed a hybrid method for fragility curve generation based on hybrid simulation results along with the calibrated analytical response (Lin et al. 2012). They develop an analytical model of 2D frame in ZEUS-NL and tested a small scale column in hybrid testing facility. Using the mean PGA from the hybrid tests and dispersions from the references, they developed hybrid fragility curves assuming lognormal distribution. Although hybrid fragility curves provide another option for developing reliable fragility curves yet it suffers from few drawbacks such as extrapolation of damage data and relationship between earthquake intensity and level of structural damage (Kappos 1997). Moreover, this method involves large aleatory and epistemic uncertainty which results in significant dispersion in the probabilistic model. Although this method of generating fragility curves have received much attention from the researchers, applications are still limited for buildings. Recently Frankie (2013) developed hybrid fragility curves for a curved four span bridge using hybrid simulation and nonlinear time history analyses. Limit states for the bridge pier were developed using experimental results obtained from the pier response under combined axial, 45  flexural, shear, and torsional loading. Combining these experimental results with analytical structural response, fragility curves for different damage states were developed. 3.4 Intensity Measure and Demand Parameter for Fragility Analysis Fragility curves express the probability of the seismic demand placed on the structure exceeding a predefined performance state under a chosen intensity measure (IM) representative of the seismic loading. Selection of an appropriate Intensity Measure (IM) is an important step in developing fragility relationship. Selection of an appropriate IM for fragility assessment has been a topic of debate among researchers for a long time. In ATC-13 (ATC 1985) Modified Mercalli Scale was used as the IM whereas FEMA P695 (FEMA 2008) preferred spectral acceleration at the first-mode period, Sa(T1) (or simply Sa) as the IM. Luco and Cornell (2007) suggested three criteria for selecting an appropriate IM, i.e. efficiency, sufficiency, and hazard computability. One of the most commonly used IM is the spectral acceleration at the first-mode period, Sa(T1) (or simply Sa). Several alternatives of IM include PGA, Peak Ground Velocity (PGV), Arias Intensity (AI) etc. as proposed and developed by numerous researchers for instance, Giovenale (2003) and Mackie and Stojadinovic (2007). In an attempt to identify an optimal IM, Mackie and Stojadinovic (2005) investigated the use of 65 IMs classified into three classes. An optimal IM was defined as being practical, effective, efficient, sufficient, and robust. Their study suggested that Sa and spectral displacement (Sd) at the fundamental period are the ideal IMs as they were found to reduce uncertainty in the demand models. On the other hand, the peak ground acceleration (PGA) was identified as the optimum IM by Padgett and DesRoches (2008) to describe the severity of the earthquake ground motion. They recommended PGA as the efficient, practical, and most sufficient IM for seismic hazard computation. Since, a large PGA always does not indicate severe structural damage, other intensity measures such as peak ground velocity (PGV) (Avsar et al. 2011), peak ground displacement (PGD), time duration of strong motion (Td), spectrum intensity (SI) and spectral characteristics can also be considered. Several researchers (Bazzurro and Cornell 2002, Shome and Cornell 1999, Baker and Cornell 2005) have proposed different vector valued intensity measures for probabilistic seismic demand model. Shafieezadeh et al. (2012) proposed a fractional order intensity measure for PSDM of highway bridges. The proposed fractional order IM considered a single degree of freedom (SDF) system with fractional damping and fractional response and combined the peak ground response and spectral accelerations at 0.2 and 1.0 s, 46  respectively.  They concluded that proposed fractional order IM showed superior performance over the traditional IMs. However, this intensity measure, at present, is inappropriate for risk analysis due to lack of regional hazard curves for such fractional order intensity measures.  The probability of entering a particular damage state under a ground motion IM is expressed through fragility curves. Damage states (DS) for bridges should reflect a certain functional level and each damage state should indicate a particular level of bridge performance. Different forms of engineering demand parameters (EDPs) are used to measure the DS of the bridge components. Park and Ang (1985) developed a damage index based on energy dissipation capacity and ductility demand of the structure while Hwang et al. (2000) used the capacity/demand ratio of the bridge columns as EDP to develop fragility curves. HAZUS (FEMA 2003) defined four damage states which are widely used in the seismic vulnerability assessment of engineering structures, namely slight, moderate, extensive, and collapse damages. Based on the drift limits of bridge pier, Dutta and Mander (1998) recommended five different damage states. Mackie and Stojadinovic (2005) classified the EDPs as local (material strain), intermediate (maximum moment), and global (drift ratio) demand parameter. Different researchers have used different demand parameters for fragility assessment of highway bridges, for instance, column curvature ductility (Nielson and DesRoches 2007a, Padgett and DesRoches 2008), displacement ductility (Zhang and Huo 2009, Bhuiyan and Alam 2012, Billah et al. 2013), drift ratio (Shinozuka et al. 2002, Tavares et al. 2012), residual drift (Billah and Alam 2014c,  Billah and Alam 2012,  Mackie and Stojadinovic 2004, Lee and Billington 2011), shear strain in isolation bearing (Zhang and Huo 2009, Bhuiyan and Alam 2012), bearing displacement (Zhang and Huo 2009, Ramanathan et al. 2012, Billah and Alam 2013), abutment deformation (Padgett and DesRoches 2008, Ramanathan et al. 2012, Tavares et al. 2012, Billah and Alam 2013), etc. Table 3.3 shows a summary of different demand parameters and the threshold values used by different researchers for fragility assessment of different components of bridges.     47  Table 3.3. Summary of threshold values of different demand parameters   Threshold Value  Component Demand Parameter Slight Moderate Extensive Collapse Reference Column Curvature Ductility 1.29 2.1 3.52 5.24 Nielson 2005 1 1.58 3.22 4.18 Ramanathan et al. 2012 1 5.11 7.5 9 Ramanathan et al. 2012 4.89 9.15 12.46 13.08 Ramanathan et al. 2010 1.44 2.7 6.92 4.18 Ramanathan et al. 2010 1 2 4 7 Choi et al. 2004 1 2.73 4.54 6.5 Jara et al. 2013 Displacement Ductility 1 1.2 1.76 4.76 Alam et al. 2012, Hwang et al. 2000 1 2 4 7 Alipour et al. 2013 2.25 2.9 4.6 5 Banerjee and Prasad, 2013 1 1.22 1.78 4.8 Billah and Alam 2014c Drift 5 7 11 30 Tavares et al.2012 0.7 1.5 2.5 5 Akbari 2012 1.45 2.6 4.3 6.9 Li et al. 2012 0.7 1.5 2.5 5 Kim and Shinozuka 2004 Rotational Ductility 3.14 3.14-5.9 5.9-9.42 >9.42 Banerjee and Chi, 2013  1.58 3.33 6.24 9.16 Banerjee and Shinozuka, 2011 Residual Drift (%) 0.25 0.25-0.75 0.75-1 >1 Billah and Alam 2014c Elastomeric Bearing Shear Strain (%) 100 150 200 250 Alam et al. 2012; Zhang and Huo 2009; Hwang et al. 2001 Drift Ratio 0.007 0.015 0.025 0.05 Yi et al. 2007 Displacement (mm) 0 50 100 150 Choi et al. 2004 28.9 104.2 136.1 186.6 Ramanathan et al. 2010, Nielson 2005 30 100 150 255 Ramanathan et al. 2012 30 60 150 300 Tavares et al. 2012 Fixed Bearing Displacement (mm) 6 20 40 186.6 Ramanathan et al. 2010, Nielson 2005 6 20 40 255 Ramanathan et al. 2012 Abutment   Displacement (mm)   7 15 30 60 Tavares et al. 2012, Billah and Alam 2013 9.8 37.9 77.2 N/A Ramanathan et al. 2010, Nielson 2005 Pile Foundation Displacement (mm) 28 42 86 115 Aygun et al. 2011   48  3.5 Regional Fragility analysis Different researchers in different parts of the world have developed fragility curves of highway bridges for a particular region. Since the seismic hazard, construction practices, bridge type, etc., varies from region to region, researchers have focused on developing regional fragility curves. There are a number of different regional fragility assessments that have been conducted so far in different parts of the world, a synopsis of which is provided in Table A.0.2 in Appendix A. Extensive study on seismic fragility assessment of highway bridges in different parts of USA have been conducted by different researchers. Using the National Bridge Inventory (NBI), Pan et al. (2010a, 2010b) conducted extensive parametric study to evaluate the seismic response parameters for different bridge components of multi-span simply supported steel highway bridges in New York State. Choi et al. (2004), Nielson and DesRoches (2007a, 2007b), Padgett and DesRoches (2008), developed fragility curves for as-built and retrofitted bridges in central and southern United States (CSUS). Ramanathan et al. (2010a, 2012) developed fragility curves for seismically and non-seismically designed bridges in CSUS. While in western US typically for California, Mackie and Stojadinovic (2005) developed fragility curves for highway overpass bridges and Ramanathan (2012) developed fragility curves for typical California bridge classes along with their evolution over three significant design eras. While in Canada, Tavares et al. (2012) and Billah and Alam (2013) developed seismic fragility curves for highway bridges in eastern and western Canada, respectively. Significant amount of research work has also been carried out in several earthquake prone countries such as, Japan (Akiyama et al. 2013, 2011, Karim and Yamazaki 2007, Tanaka et al. 2000), Italy (Felice et al. 2004, Cardone et al. 2007), Turkey (Avsar et al. 2011), Greece (Moschonas et al. 2009) and Taiwan (Liao and Loh 2004, Sung et al. 2013).  Different regions have different design guidelines, bridge types, construction method, seismicity, and soil conditions. Again different researchers considered different structural systems and adopted different modelling and analysis techniques for developing fragility curves. So it is very difficult to compare the fragility curves developed for different regions. However, in this study a comparison of fragility curves developed for different regions particularly for a specific type of bridge (MSC Concrete) was conducted. A comparison of the fragility curves at extensive damage states for MSC concrete bridges are shown in Figure 3.8. 49  It is beyond the scope of this study to compare and comment on the vulnerability of same types of bridges located in different parts of the world.  Figure 3.8. Comparison of empirical fragility curves for MSC Concrete bridges for different regions 3.6 Condition Specific Fragility Assessment  3.6.1 Fragility analysis for retrofitted bridge Most of the studies regarding development of bridge fragility curves are focused on as- built bridges. Fragility curves can also be used as an assessment tool for retrofitted bridges and selecting an optimal retrofit strategy from a group of available retrofit measures. Shinozuka et al. (2002) developed fragility curves for typical southern California bridge piers retrofitted with steel jacket. Using nonlinear dynamic analysis, fragility curves were developed as a function of PGA. They compared the vulnerability of as built and retrofitted bridges. They proposed an “enhancement curve” which can be applied over empirical fragility curve to develop retrofitted bridge fragility curve. Padgett and DesRoches (2008) developed an analytical methodology for developing fragility curves of retrofitted bridges. They evaluated the impact of retrofitting one component on the response of other key components of the bridge. Considering a typical bridge class in CSUS retrofitted with five different alternatives along with different types of uncertainties, fragility curves were generated. Using three-dimensional nonlinear analysis, Padgett and DesRoches (2009) developed fragility curves for four common classes of multi-span bridges in CSUS and five retrofit methods. They concluded that the effectiveness of retrofit measure in reducing system vulnerability is a function of 00.20.40.60.810 0.2 0.4 0.6 0.8 1P [Extensive I PGA]PGA (g)TaiwanGreeceCSUSEastern CanadaWestern Canada50  bridge type and damage state under consideration. Agrawal et al. (2012) developed fragility curves for retrofitted multi-span continuous steel bridges in New York. Effectiveness of various retrofit measures, such as elastomeric bearing, lead rubber bearing, carbon fiber jacketing, and viscous damper, in reducing the vulnerability of bridges were evaluated and compared with the performance of as built bridges. They concluded that a combination of elastomeric bearing and viscous damper provide an optimal retrofit effect for typical multi-span continuous steel bridges in New York. Billah et al. (2013) developed analytical fragility curves for retrofitted multi-column bridge bent under near fault and far field ground motion. They evaluated the effectiveness of different retrofitting techniques (e.g. steel jacket, concrete jacket, CFRP jacket, ECC jacket) and compared their vulnerability under near fault and far field ground motions. They concluded that both ECC and CFRP jacket were effective in reducing the vulnerability under near fault and far field ground motions. Based on the performance of different bridge components using fragility analysis, Stefanidou and Kappos (2013) proposed a methodology for selecting optimal retrofit strategy for bridges. The main aspect of this methodology is the development of correlation between component limit state threshold values and global limit states. Figure 3.9a shows fragility curves for as built and retrofitted bridges and Figure 3.9b shows the comparative effectiveness of different retrofitting techniques in reducing the seismic vulnerability.  Figure 3.9. (a) Fragility curves for as-built and retrofitted bridge (b) Fragility curves for retrofitted bridge bent using different retrofitting techniques (Billah et al. 2013)  00.20.40.60.810 0.5 1 1.5 2P[ModerateIPGA]PGA (g)As BuiltRetrofitted(a)00.20.40.60.810 0.5 1 1.5 2P[ModerateIPGA]PGA (g)ConcreteECCCFRPSteel(b)51  3.6.2 Fragility analysis considering aging effect Aging and deterioration significantly affects the seismic performance of bridges. The detrimental effect of aging and deterioration on the seismic vulnerability of highway bridges has been overlooked by engineering community for a long time. Although there has been a number of studies focusing on the aging and deterioration of bridges, very few studies have incorporated these effects on the fragility curve generation (Choe et al. 2009, Gardoni and Rosowsky 2011, Zhong et al. 2012, Ghosh and Padgett 2012). The impact of aging and deterioration on bridge fragility is heavily influenced by the exposure condition: whether marine exposure, atmospheric exposure or de-icing salt exposure etc (Ghosh and Padgett 2012).  Different researchers have investigated the effect of deterioration on seismic fragility considering different exposure conditions. Choe et al. (2009) investigated the potential reduction in capacity and increase in fragility due to aging and deterioration of a typical single-bent bridge in California considering a marine splash zone. They extended the existing probabilistic seismic demand model for pristine bridges with a probabilistic model for time-dependent chloride-induced corrosion to include the effect of aging on seismic fragility assessment. This study highlighted the significance of considering the effects of aging on seismic fragility and identifying the crucial material and corrosion parameters that most significantly affect the bridge reliability. Simon et al. (2010) developed fragility curves for deteriorated concrete bridges, located in a marine splash zone, designed according to current guidelines to investigate the chloride exposure level and extent of corrosion on the vulnerability of bridges. They showed that spalling of cover concrete and reduction in reinforcement area affect the seismic vulnerability of bridges. Sung and Su (2011) developed time dependent fragility curves for deteriorated RC bridges. Using pushover analysis they investigated the decayed capacity of deteriorated bridges and developed fragility curves with respect to some representative damage levels. Using the time dependent fragility curve, they developed S-surface diagram to illustrate the relationship between cost, intensity measure and service time. Ghosh and Padgett (2010) investigated the effect of multi-component deterioration on the seismic vulnerability of aging bridges. Figure 3.10a shows the effect of aging on the seismic fragility of bridges. Considering the variations in structural properties, ground motion and corrosion parameters they developed time dependent fragility curves for multi span continuous steel girder bridge. The analyses showed that most of the components 52  (columns, fixed bearing, expansion bearing) experience increased vulnerability due to aging while there is a decrease in the vulnerability of few components (abutment longitudinal and transverse response). They concluded that an aging bridge might experience a shift of 32% in the median value of complete damage fragility near the end of its service life.   Figure 3.10. Effect of (a) aging (Ghosh and Padgett, 2010), (b) soil liquefaction (Aygun et al. 2011), (c) isolation (Zhang and Huo 2009), (d) horizontal curve (AmiriHormozaki et al. 2013), (e) skew angle (Sullivan and Nielson 2010) and (f) scour depth (Prasad and Banarjee 2013) on fragility curves Ghosh and Padgett (2012) explored the effect of different exposure conditions, such as de-icing salt exposure and splash zone and atmospheric zone exposure in marine environment, on the vulnerability of typical multi-span concrete bridges in CSUS. They concluded that consideration of different exposure conditions lead to a significant variation in the vulnerability of aging bridges. Recently, Dong et al. (2013) developed time-variant fragility curves for seismically vulnerable bridges considering multiple hazard scenario. They considered the effects of flood induced scour and effects of corrosion on reinforcement bars and concrete cover spalling in generating the fragility curves. Choine et al. (2013) investigated the effect of chloride induced corrosion of the reinforcement, caused by the application of de-icing salts, on the seismic vulnerability of a three span integral concrete bridge. This study found that 00.20.40.60.810 0.2 0.4 0.6 0.8 1P [ModerateI PGA]PGA (g)Pristine25 Years50 Years75 Years100 Years(a)00.20.40.60.810 0.2 0.4 0.6 0.8 1P [CllapseI PGA]PGA (g)Coulmn w/oliquefactionColumn w/liquefaction(b)00.20.40.60.810 0.3 0.6 0.9 1.2 1.5P[DamageIPGA]PGA (g)Extensive (Isolated)Extensive (Un-isolated)Collapse (Un-isolated)Collapse (Isolated)(c)00.20.40.60.810 0.5 1 1.5 2P[CollapseI PGA]PGA (g)0 deg15 deg30 deg45 deg(e)00.20.40.60.810 0.2 0.4 0.6 0.8 1P[CollapseIPGA]PGA (g)0m0.6m1.5m3m6m(f)00.20.40.60.810 0.5 1 1.5 2P[CollapseISa1]Sa1 (g)Straight30 deg Curve60 deg Curve90 deg Curve(d)53  corrosion and aging significantly affect the seismic vulnerability of bridge piers while other components’ vulnerability are less sensitive to aging and deterioration.  3.6.3 Fragility analysis considering SSI and liquefaction  Lack of homogeneity in the underlying soil can result in wide variety of strength parameters which can significantly affect the seismic response of bridges (Brandenbarg et al. 2011). Due to their complex structural configuration compared to buildings, bridges experience more severe soil-structure interaction (SSI) effects during earthquakes (Chaudhury et al. 2001). Several researchers (Boulanger et al. 1999, Zhang et al. 2008; Elgamal et al. 2008) have investigated the effect of SSI modeling techniques and liquefaction on seismic response of bridge components. Kashighandi et al. (2008) investigated the seismic fragility of older-vintage California bridges to liquefaction and lateral spreading. Kwon and Elnashai (2010) developed fragility curves for a highway overcrossing bridge in USA considering soil structure interaction (SSI) using four different modeling techniques to represent the behavior of abutment and foundation. They concluded that the selection of efficient SSI modeling technique significantly affects the reliability of vulnerability assessment. Aygun et al. (2011) developed a computationally efficient coupled bridge-soil-foundation (CBSF) analyses and fragility curves for typical multi-span continuous steel bridges typical of the central and eastern US (CEUS) considering earthquake-induced soil liquefaction. They reported that the vulnerability of columns depends on the type of soil overlying the liquefiable sands, while the fragility of rocker bearings, piles, embankment soil, and the probability of unseating increases with liquefaction. Figure 3.10b shows the effect of considering liquefaction on the vulnerability of bridge columns. The figure illustrates the fact that liquefaction significantly increases the seismic vulnerability. Brandenbarg et al. (2011) developed demand fragility surfaces for bridges in liquefied and laterally spreading ground. Using a beam on a nonlinear Winkler foundation approach, the SSI effects at the bridge abutment components were modeled while the soil-structure elements included p-y springs for lateral interaction, t-z springs for axial interaction, and q-z springs for pile tip bearing. They concluded that consideration of liquefaction and lateral spreading significantly affects the fragility function. Padgett et al. (2013) investigated the sensitivity of seismic fragility of different bridge components for variation in structural and liquefiable soil modeling parameters. They 54  concluded that the undrained shear strength of soil, structural damping ratio, soil shear modulus, gap between deck and abutment, ultimate capacity of soil and fixed and expansion bearing coefficients of friction significantly affects the seismic fragility of bridges. Ni et al. (2013) proposed a direct displacement based assessment approach for fragility assessment of multi-span continuous concrete bridges considering nonlinear dynamic soil–structure interaction effects. The proposed method was found to be fast and reliable which can be used for screening of large sample of bridges.  3.6.4 Fragility analysis of isolated bridges Seismic isolation of highway bridges has been proven to be an efficient technique to reduce the seismic hazards for designing new bridges or improving the performance of existing bridges. Several researchers have investigated the effect of isolation on the seismic vulnerability of existing bridges. Karim and Yamazaki (2007) developed a simplified approach to generate fragility curves of isolated bridges. Using 30 nonlinear models of isolated bridges using different structural parameter, this study illustrated the contribution of isolators on reducing damage probability of bridge columns. They found that the damage probability of isolated systems tends to be higher for a higher level of pier height compared to non-isolated systems. Using a performance based evaluation approach, Zhang and Huo (2009) investigated the effectiveness and optimum design parameters of isolation devices using fragility analysis. Using PSDA and IDA they developed fragility functions for isolated bridges and determined the optimum combinations of mechanical parameters of isolation devices as a function of structural properties and damage states. Figure 3.10c shows the effect of isolation on the seismic fragility of bridges. From this figure it is evident that isolation significantly reduces the bridge vulnerability. Alam et al. (2012) investigated the seismic vulnerability of a three-span continuous highway bridge, restrained by shape memory alloy (SMA) bars and isolated with laminated rubber bearings. They concluded that the failure probability of the bridge system is dominated by the bridge piers over the isolation bearings and inclusion of SMA restrainers in the bridge system exhibits high probability of failure, especially, when the system is isolated with lead rubber bearings. Using capacity/demand approach, Jara et al. (2013) proposed a methodology for generating fragility curves for isolated irregular bridges. They proposed a simplified approach to obtain fragility curves based on non-linear static analyses. 55  3.6.5 Fragility analysis of irregular, curved and skewed bridges Bridges with unequal column height are often found in highway bridges crossing a basin or valley and behave in an undesirable way during a seismic event. In an irregular bridge with different column heights, the deformation demand in individual piers is usually different while the shortest pier being subjected to maximum demand (Tehrani and Mitchell 2012). Considering 18 different bridge configurations based on the column height, Akbari (2010) generated fragility curves for irregular bridges. He concluded that at high intensity earthquake, the short piers of the irregular bridge experience extensive damage while the long piers remain elastic. Horizontally curved steel bridges have become very popular and more than one third of steel bridges constructed in US are curved (Davidson et al. 2002). Since the seismic response of horizontally curved bridges is different from the straight bridges, several researchers have investigated the seismic fragility of horizontally curved steel girder bridges (Mohseni, 2011; Linzell, 2012, AmiriHormozaki et al. 2013). AmiriHormozaki et al. (2013) identified torsion index as a significant parameter for fragility assessment of curved steel girder bridges. This study showed that the vulnerability of curved bridges is under predicted by the HAZUS fragility curves as compared to the analytically derived fragility curves. Figure 3.10d shows the effect of curvature on the fragility of bridges. From this figure it is evident that horizontal curvature significantly affects the vulnerability of bridges.  Skewed bridged are often encountered in the design of highway bridges and mostly found in multi-level interchanges which show a complicated dynamic behavior as compared to straight bridges (Samman et al. 2007). Several researchers (Pottatheere and Renault 2008, Sullivan and Nielson 2010, Moschonas and Kappos 2011, Huo and Zhang 2013, Zakeri et al. 2013) have investigated the impact of skewness on seismic vulnerability of highway bridges. The effect of skewness on the seismic vulnerability of bridges is depicted in Figure 3.10e.  Pottatheere and Renault (2008) reported that for a skewed reinforced concrete bridge, elastomeric bearing and columns are the most vulnerable component and for the same intensity, the damage probability increases with increased skew angle. Huo and Zhang (2013) reported that the influence of pounding can be devastating in skewed bridges while at large skew angle (600) this affect is reduced. They suggested not to incorporate pounding and skewness simultaneously in the design of highway bridges since pounding can increase the 56  deck rotation and the seismic demand on bridge piers of skewed bridges thus influencing the bridge response.  3.6.6 Fragility analysis considering effect of scouring Seismic performance of highway bridges can be significantly affected due to the combined effect of earthquake and scouring (Ghosn et al. 2003). There is a growing concern among researchers and scientific community to evaluate the performance of bridges under the combination of two or more extreme events (Alampalli and Ettouney, 2008). Scouring around bridge foundation and abutment can result in significant reduction in load carrying capacity and increase the flexibility of the bridge (Alipour and Shafei, 2012) thus affecting the seismic vulnerability of bridges. Wang et al. (2012) developed fragility surfaces for two highway bridges considering the combined effect of earthquake and scour. They concluded that although bridges with pile foundation are capacity protected, increasing scour depth can significantly affect the seismic vulnerability of bridges. Alipour and Shafei (2012) developed fragility curves for RC bridges based on the joint probabilities of scouring and earthquake. Using Monte Carlo simulation they estimated various scour depth. Using nonlinear time history analysis, they investigated the structural response, ductility demand, and estimated various bridge fragility parameters for a range of scour depth. The developed fragility curves indicated that the load bearing capacity significantly decreases with increasing scour depth. More recently, Prasad and Banerjee (2013) and Banerjee and Prasad (2013) investigated the impact of flood induced scour on the seismic fragility of RC bridges. Their results demonstrated that scour depth over 3m does not increase the vulnerability of bridges. Figure 3.10f shows the effect of scour depth on the fragility of bridges. From this figure it is evident that increasing scour depth increases the vulnerability of bridges. Alipour et al. (2013) developed a multi-hazard reliability-based framework to evaluate the structural response of RC bridges under the combined effects of pier scour and earthquake events. Considering different sources of uncertainties in scouring and seismic hazard, they developed fragility curves to estimate the failure probability under the combined effect of scouring and earthquake. They suggested that more analytical and experimental works need to be conducted to investigate the combined effect of scouring and earthquake and develop design guidelines to improve bridge response. 57  3.7 Effect of Ground Motion on Fragility Analysis Selection of ground motion plays an important role in generating fragility curves for highway bridges. The effect of ground motion suites, directionality, angle of incidence, and spatial variation on fragility assessment have been investigated by several researchers (Kim and Feng 2003, Ramanathan et al. 2010b, Banerjee and Shinozuka 2011, Nielson and Pang 2011, Torbol and Shinozuka 2012, Elhowary et al. 2013). Kim and Feng (2003) concluded that ground motions with spatial variation induces increased fragility for long span bridges. They suggested incorporating the effect of ground motion spatial variation for the seismic design of long span bridges. The seismic fragility of a nine span continuous box girder bridge under spatially variable ground motion was investigated by Elhowary et al. (2013). They concluded that the bridge response in transverse direction is more sensitive to the spatial variability of ground motion. Their result illustrated that bridges in soft soils are more vulnerable to spatially variable ground motions.  Banerjee and Shinozuka (2011) investigated the effect of ground motion directionality on the fragility characteristics of highway bridges. Their results showed that ground motion directionality play an important role in estimating the fragility characteristics. Considering seismic incidence angle as an important parameter, Torbol and Shinozuka (2012a, 2012b) developed fragility curves for highway bridges. They illustrated that the vulnerability of a highway bridge may be underestimated if the angle of seismic incidence is not considered. They concluded that, this effect gets aggravated in case of skewed and curved bridges. Nielson and Pang (2011) investigated the effect of ground motion suite size on fragility of highway bridges. They suggested using a suite of 80 or more ground motions in order to keep variation in median and dispersion estimates reasonable. They concluded that less number of ground motions can be used if more selective procedure is adopted to assemble the ground motion suite. The effect of fault distance on fragility estimate was investigated by Billah et al. (2013). Using suites of near fault and far field ground motion, they investigated the seismic fragility of retrofitted bridge bents. Their study showed that, near fault ground motion imposes high ductility demand thereby increases the vulnerability of bridge bents.  3.8 Possible Future Development Although there exist a wide variety of methodologies for fragility curve development, still there is scope for significant improvement in fragility curve development methodology. 58  Key features of the different studies described above are summarized in Table 3.4 in order to illustrate the gradual development of fragility curve methodology. The table reveals that, despite advances in analytical models and risk assessment methods, there still remain scopes to improve the existing fragility curve development methodology. An improved hybrid model for fragility curve development is proposed in this study which involves empirical, experimental, and analytical method. A flow chart showing the proposed methodology is illustrated in Figure 3.11.   Figure 3.11. Proposed methodology for developing hybrid fragility curves  Hybrid Fragility CurvesDevelop damage statisticsEstimate damage at different intensitiesEmpirical Method Experimental Method Analytical MethodStatistical quantification of demand and capacity Develop damage-intensity matrixHybrid Simulation/Damage data from experimental investigationBayesian updating of capacity and demand modelDynamic analysis of appropriately calibrated modelEstimation of different component demandModification factor to allow for material and geometric uncertaintyDevelopment of fragility curves59  This method is more suited for regional fragility assessment. Using post-earthquake reconnaissance data empirical fragility curves are developed which lack generality and are usually associated with a large degree of uncertainty. Moreover, the damage observed are structure specific and cannot be extended to other similar bridges having different geometry and material properties. This limitation can be overcome by combining empirical damage states with experimental observation. From the observed damage, a damage matrix can be developed which will relate the different component damage with intensity measure. An interesting technique can be the use of hybrid simulation using appropriately calibrated model of the damaged bridges. This procedure will enable the updating of the damage states of different bridge components and improvements in accuracy in defining the limit states with data available from experiments and simulations. Moreover, if the hybrid simulation facility is not available, experimental results available in the literature that resembles the configuration of different components of bridges can be used to develop the limit states. One of the major elements in developing fragility relationship is the development of demand and capacity models. Using experimental results an accurate demand and capacity models can be developed. Using statistical quantification the uncertainty associated with the demand and capacity models can be estimated. A Bayesian updating technique can be employed to take into account the changes in material and geometric properties. Once the demand and capacity models are established, using calibrated analytical models, the response of the full bridge can be evaluated using dynamic time history analysis over a wide range of ground motion. In addition development of some modification factor will allow to consider for the changes in material and geometric properties. These appropriately calibrated modification factors can be used to generate the fragility functions for a typical class of bridges in the whole inventory. These modification factors can be generated using different statistical learning techniques available in literature. Although this section provides a brief description of possible future development of a novel fragility curve development technique, further study along with detailed examples are required to check the adequacy of the proposed method.    60  Table 3.4. Key features of modern bridge fragility curve development efforts Author Bridge Type Ground Motion Method Component/System Feature Mander 1998 Different Spectrum CSM System Introduction of new generation bridge fragility curve Yamazaki et al. 2000 Expressway in Japan Kobe Empirical System Empirical fragility curve Shinozuka et al. 2000 4-span straight bridge Synthetic NLTHA+CSM Component Comparison of NLTHA and CSM Hwang et al. 2001 4-span straight bridge Synthetic NLTHA Component Damage state definition Karim and Yamazaki 2003 4-span straight bridge Strong Motion NLTHA and SPO Component Simplified  Gardoni et al. 2003 Multi-Span straight bridge N/A Bayesian Updating +SPO System Probabilistic capacity and demand model Mackie and Stojadinovic, 2003 Multi-Span straight bridge Strong Motion IDA System Optimal PSDM Nielson, 2005 SSC/MSSS/MSSC/MSCC/MSCS/SSS Synthetic NLTHA Component +System Component level approach Padgett, 2007 SSC/MSSS/MSSC/MSCC/MSCS/SSS Synthetic NLTHA Component +System Retrofitted and as built bridges Kwon and Elnashai 2010 Multi-Span steel girder bridge Synthetic+ Strong motion NLTHA Component +System SSI modeling technique Aygun et al. 2011 Multi-Span continuous steel bridge Synthetic NLTHA Component +System Soil Liquefaction Ramanathan et al. 2012 MSSC+MSSS+MSCC+ MSCS Synthetic NLTHA Component +System Seismic and non-seismic detailing Vosooghi and Saiidi 2012 Bridge pier Shake Table Experimental Component Probabilistic performance based design Billah et al. 2013 Multi-column bent Strong Motion IDA Component Near fault and far field motion Banarjee and Prasad 2013 5-span straight concrete bridge Synthetic NLTHA Component Flood induced scour Amirhormozaki et al. 2013 Horizontally curved steel girder bridge Strong Motion NLTHA Component +System Curved girder bridge  61  3.9 Summary This chapter presented a detailed review of the state-of-the-art methodologies for the development of fragility curves of highway bridges. This study provides an insight into the current practice and applications relating to the seismic fragility assessment of highway bridges. Because of its versatile application, fragility curve has evolved as an integral part of seismic risk assessment methodology. It allows the decision makers and stake holders in risk mitigation and management by translating the seismic demand into a probabilistic performance matrix. Since its inception, fragility curves have evolved from simplest to complex approaches. This study summarized the evolution of different mechanical approaches developed for fragility curve generation and their applications in different parts of the world along with their features and limitations. This study also presented the fragility curve methodologies for different bridge components and effect of considering different scenarios such as, retrofitting, isolation, soil-structure interaction, on the bridge fragility curves. Seismic fragility assessment of highway bridges involve a large amount of complexity and uncertainty. It is likely that no such methodology is available to fully and accurately consider all these complexity and uncertainties. Each methodology has its own advantages and disadvantages. Individual methodologies were developed based on different assumptions which emphasize on certain aspect of the problem and minimize or even ignore others. Fragility curves generated following any particular method should be interpreted very carefully and should not be considered as definitive. Although fragility analysis has emerged as a promising tool for seismic performance assessment of highway bridges, as of today it has not been included in any design codes or guidelines as a method for determining the seismic performance of bridges at different hazard levels. More research in this area needs to be conducted in developing methodologies for fragility analysis which can be incorporated in the seismic design of highway bridges.   62  CHAPTER 4. BOND BEHAVIOR OF SMOOTH AND SAND-COATED SHAPE MEMORY ALLOY (SMA) REBAR IN CONCRETE 4.1 General Conventional steel reinforcement possess lugs or surface deformation which transfer the bond forces by mechanical interlock and friction. However, SMA rebars are usually produced in round shapes with smooth surface without any lugs. Moreover, most of the commercially available SMA rebars are made of Ni-Ti alloy which is extremely hard and difficult to machine using conventional equipment (Alam et al. 2007). On the other hand, threading of large diameter SMA rebars reduces the strength significantly (Alam et al. 2007). Although the surface of SMA rebar is similar to the plain steel reinforcement found in historical structures, mechanical behaviour of SMA bars, however, significantly differs from that of the plain steel reinforcing bars. Extensive experimental studies have been carried out by several researchers on the bond behaviour of plain steel reinforcement (Wu et al. 2014, Verderame et al. 2009, Feldman and Bartlett 2005, 2007). However, no study has been undertaken so far to evaluate the bond behaviour of SMA rebars with concrete. This justifies the need to conduct an experimental investigation of the bond behaviour of SMA rebars embedded in concrete. Several researchers have investigated and showed the efficacy of SMA as reinforcement in concrete structures. However, for large scale application in construction industry, different structural aspects of SMA rebars should be investigated to ensure their reliable application. The interfacial bond behaviour between SMA rebar and concrete is a governing factor in controlling the deformation of SMA-RC structures. SMA rebar is currently available with smooth surfaces. While using this smooth rebar as internal reinforcement in critical regions (e.g. plastic hinge region of a beam), a large major crack will be formed under loading. This crack will be flexural bond crack and the concrete section might experience shear failure at this location since no aggregate interlocking is available for resisting shear. Figure 4.1 shows such condition, where SMA was used in the plastic hinge region of a beam-column joint and a large major crack was observed due to the use of smooth surfaced SMA rebar. However, for deformed or properly bonded bar, many small cracks will be formed and distributed over the whole plastic hinge length and can help resist more loading. In order to overcome the drawbacks of smooth SMA rebar, the surface of the smooth SMA bar was roughened using 63  sand coating. Two different granulometries were used to evaluate the effect of surface roughness on the bond behavior of SMA rebar by means of providing improved interlocking in addition to mechanical adhesion. The objective of this experimental investigation is to study the bond behavior of SMA rebar where the variables include SMA bar diameter, concrete strength, bonded length, concrete cover, and surface condition. Based on the experimental results, empirical equation for predicting the average maximum bond strength of SMA rebar has been developed. This research has practical significance since the outcome of the study will provide an understanding of the bond behavior of SMA rebar and will provide a basis for the development length prediction of SMA reinforced concrete members.  Figure 4.1. Bond failure of concrete section having smooth SMA rebar (adapted from Youssef et al. 2008) 4.2 Experimental Program The experimental program conducted in this study involved a series of 56 pushout test specimens (concrete cylinders) with different parameters (Table 4.1). In this study, pushout test was selected since it was simple to conduct and overcome the drawbacks associated with pullout test as described in Feldman and Bartlett (2005).  4.2.1 Variables A review of literature on bond behaviour of reinforcement with concrete dictated that five different parameters need to be investigated to evaluate the bond behaviour of SMA rebar with concrete (Verderame et al. 2009, Feldman and Bartlett 2005, 2007, Wambeke and Shield 2006, Hossain and Lachemi 2008, Hossain et al. 2014). The parameters include: concrete compressive strength (35, 40, 50, and 60 MPa); embedment length (3db, 5db, 7db, 9db), bar diameter, db, (20 mm and 32 mm), concrete cover (34 mm, 40mm, 59mm and 65mm) and 64  surface condition (smooth, sand coated). These parameters were selected based on materials availability, available testing facilities, and practical applications. Table 4.1. Pushout test specimens  Bar Size Bar Finish ld, mm Concrete cover, mm Compressive Strength, MPa Sample No., n 20 mm Smooth 60 40 35 2 100 40 35 2 140 65 35 2 180 65 35 2 60 40 50 2 100 40 50 2 140 65 50 2 180 65 50 2 60 40 40 2 60 40 60 2 Sand-300 60 40 50 2 100 40 50 2 140 40 50 2 Sand-600 60 40 50 2 100 40 50 2 140 40 50 2 32 mm Smooth 96 34 35 2 160 34 35 2 224 59 35 2 280 59 35 2 96 34 50 2 160 34 50 2 224 59 50 2 280 59 50 2 Sand-300 96 34 50 2 160 34 50 2 Sand-600 96 34 50 2 160 34 50 2     Total= 56  4.2.2 Materials In this study, Ni-Ti SMA rebar (nitinol) has been used as reinforcement to investigate the bond behaviour. The austenite finish temperature, Af, which defines the transformation from martensite to austenite phase, ranges from -150C to -100C. All the Ni-Ti bars used in this 65  study were 450 mm long. The yield strength of the SMA rebar was 401 MPa at a strain of 0.75% and the elastic modulus was 62.5 GPa. This values were provided by the SMA manufacturer. Four different concrete mixes were considered for evaluating the effect of concrete compressive strength on the bond-behaviour of SMA rebar. Similar type of cement, fine aggregate and coarse aggregate were used for different concrete mixes, while the proportions were varied accordingly to get the desired compressive strength. 4.3 Specimen Preparation and Testing Cylindrical concrete specimens with dimensions of 100 mm×200 mm and 150 mm × 300 mm (D×L) with SMA rebar at the center were used in this study. Figure 4.2 shows the picture of few specimens after casting. The as-received bars were smooth and later the surface condition was modified using sand of two different granulometries. Two different sizes of sand, 300 µm and 600 µm were used to modify the rebar surface and investigate the effect of surface modification on the bond behavior. G/Flex epoxy (west systems) was used as the adhesive to apply sand coating on the rebar. Using sandpaper, the rebars were cleaned to remove any dirt on the surface and the required embedment length was marked before applying the epoxy (Figure 4.3a). A paint brush was used for applying the epoxy coating on the surface of each rebar (Figure 4.3b), and subsequently, the epoxy coated rebars were rolled over the sand for sand coating (Figure 4.3c). The total thickness of the epoxy and sand were between 1.5 mm-2 mm. Then the rebars were cured for 48 hours for proper bonding (Figure 4.3d). The embedment length of sand coated rebars is also shown in Figure 4.3d.  Figure 4.2. Specimens after casting 66   Figure 4.3. Sand coating of SMA rebar (a) bonded length, (b) epoxy application, (c) sand coating and (d) sand coated rebars  For the pushout test, the concrete cylinder with the SMA bar at its center was placed on a metal frame with a circular plate at the top having a 35 mm hole at the center. Figure 4.4 shows the test setup for the pushout test. The rebar was positioned in the cylinder in such a way that 50 mm of the rebar popped out beyond the top surface of the cylinder (loaded end), a certain length of the bar was embedded in concrete (i.e. the embedment length in Figure 4.4), and the remaining portion protruded from the bottom of the cylinder (free end) to allow connection of the displacement sensors (string potentiometer). The embedment length was varied as shown in Table 4.1. In order to avoid stress concentration, a length of 25 mm at both the top and the bottom of the specimens was wrapped with plastics (i.e. the bond breaker in Figure 4.4). A flat metal plate was placed on top of the SMA bar in order to apply the load evenly on the bar. The test was conducted using Instron testing machine and the projecting bar was pushed down by the actuator, and using a string potentiometer attached to the bottom of the protruding rebar, the slip of the rebar was measured at the free end. An electronic load cell equipped with the testing machine measured the load. Both the load and the rebar slip were recorded through the data acquisition system. The load was applied at a rate of 1-1.5 kN/sec. The test was conducted until a slippage of 30 mm was recorded. 67   Figure 4.4. Test setup for bond behavior SMA rebar with concrete 4.4 Experimental Results 4.4.1 Failure modes The pushout test specimens with smooth SMA bars failed at the concrete-rebar interface without developing any splitting crack. In smooth SMA rebar there was no surface deformation. Therefore, the bond force was transferred only by adhesion between the concrete and SMA rebar before any slip occurred. When the adhesion was lost, the bond mechanism developed due to the friction between the rebar and the small particles that broke free from the concrete upon slip, and the plain rebar simply slipped through the concrete. Figure 4.5 shows the condition of the pushout specimens with plain rebar before and after testing. From Figure 4.5a it can be seen that initially the rebar was protruded 50 mm from the top which finally got reduced to 20 mm at the end of the test (Figure 4.5b) without any sign of splitting cracks. A closer look inside the cylinder (Figure 4.5c) shows that there was no significant bond between the smooth SMA rebar and concrete as shown by the smooth surface of the concrete. 68   Figure 4.5. Specimens (smooth) (a) before testing, (b) after testing and (c) inside view  On the other hand, for all the sand coated SMA rebars, failure took place at the interface between the SMA bar and the surrounding concrete (Figure 4.6). Splitting cracks developed on the concrete bearing surface which extended along the perimeter and continued down the length of the specimens for all the cylinders with 20 mm sand coated SMA rebars. In the case of 32 mm bars coated with 600 µm sand, it showed similar crack pattern while the 32 mm bars coated with 300 µm sand only experienced minor radial cracks developed on the concrete bearing surface. However, the radial cracks did not extend to the specimen perimeter for cylinders with 32 mm SMA rebars coated with 300 µm sand. 69   Figure 4.6. Failure pattern of sand coated bars (a) radial cracking, (b) crack propagation in concrete and (c) inside view 4.4.2 Load-slip relationship and bond strength After processing the data obtained from the pushout tests, the load-slip relationship for each test was obtained. Typical load-slip behavior of smooth SMA rebar is shown in Figure 4.7 for a 100 × 200 mm specimen having a 20 mm diameter bar, 60 mm embedment length, and 40 mm concrete cover. The load-slip curve consists of four parts: (I) elastic stage, (II) ascending branch up to peak load, (III) linearly descending branch, and (IV) residual branch. Figure 4.7 also shows the four stages in the load-slip curve. The elastic stage is defined when there is almost no slip with the increase in load and the adhesion bond mechanism plays the major role in transferring the load between SMA and concrete. When the adhesion bond starts to break, the ascending branch starts and continues upto the maximum load, Pmax at little slip.  In the descending stage, the peak load starts to drop suddenly with significant increase in slip value. As slip increases, the wedging action of small particles provide the sole bond mechanism. At the residual stage, the load dropped asymptotically to a limiting residual load Pres and the slip values increased quite quickly.  70    Figure 4.7. Load-slip curves for pushout test of smooth SMA rebar  In this study, the bond strength (τ) of an SMA bar embedded in concrete is assumed to be distributed uniformly over the embedment length (Ld). At any stage of loading, the maximum average bond strength can be calculated using equation 4.1: db LdPπτ maxmax =           (4.1) where, Pmax is the maximum load obtained from the load slip relation, db is the bar diameter, and Ld is the embedment length. In this study, the bond behavior of SMA rebar is investigated in terms of maximum and residual bond strength. The average maximum bond strength (τmax) can be calculated using equation 4.1 and the residual bond strength (τres) is calculated using equation 4.2:  dbresres LdPπτ =           (4.2) where, Pres is the residual load obtained from load slip curve. 4.4.3 Influencing factor analysis The impact of different variables considered in this study was investigated individually to find their effect on the bond strength variability. The following sections discuss the effect of various parameters on bond strength of SMA rebar in concrete. 051015202530350 2 4 6 8 10Load (kN)Slip (mm)Pmax = 32.38 kN slip = 0.48mmPres = 6.25 kN slip = 10 mm(I)(II)(III)(IV)71  4.4.3.1 Effect of concrete strength For investigating the effect of concrete compressive strength, four different concrete strengths were considered. Keeping the embedment length and concrete cover constant at 60 mm and 40 mm, respectively, a total of eight specimens were tested to evaluate the influence of concrete strength on bond behavior of smooth SMA rebar with concrete. Figure 4.8 shows the effect of concrete strength on the maximum and the residual bond strength. Separate regression analyses revealed that both maximum and residual bond strength are functions of the square root of the concrete compressive strength. This is coherent with the findings of other researchers (Wu et al. 2014, Feldman and Bartlett 2005) on plain rebar and as per the current North American standards (CSA A23.3-14, ACI 318-11).  Figure 4.8. Effect of concrete compressive strength on average (a) maximum and (b) residual bond strength of smooth SMA bar  From Figure 4.8 it can be observed that, both maximum and residual bond strength increase with an increase in concrete compressive strength and this increase is proportional to the square root of compressive strength. A regression analysis of the test results for which the maximum average bond strength of smooth SMA rebar were measured, yielded the following equation (4.3).  225.4 /max −= cfτ          (4.3) 024681012140 2 4 6 8 10Max. Bond Stress, τm(MPa)√fc' (MPa1/2)35 MPa40 MPa50 MPa60 MPa(a)00.511.522.530 2 4 6 8 10Res. Bond Stress, τr(MPa)√fc' (MPa1/2)35 MPa40 MPa50 MPa60 MPa(b)72  where,  τmax is maximum average bond strength in MPa. This equation can predict the bond strength very well for concrete with compressive strength of up to 40 MPa, but at a higher strength there is a variation of approximately +/- 1.5 MPa 4.4.3.2 Effect of bar diameter Figure 4.9 compares the average maximum and residual bond strength of 20 mm and 32 mm smooth SMA rebars. From Figure 4.9a it is evident that, as the bar diameter increases the average maximum bond strength decreases. However, no significant influence of bar diameter was observed in the case of average residual bond strength. Since, the results presented in Figure 4.9 had different concrete strengths, the bond strength is normalized by the square root of the concrete compressive strength. In general, the average maximum bond strength of 20 mm bar was 30%-45% higher than that of 32 mm bar. From the test results, it was observed that the effect of bar diameter was more pronounced for concrete with lower strength (35 and 40 MPa) as compared to high strength concrete (50 and 60 MPa). For low strength concrete, the bond strength increased as high as 45% for 20 mm bar as compared to 32 mm bar. In contrast, the bond strength of 32 mm bar decreased by 30% for high strength concrete. This can be attributed to the fact that larger diameter bars require longer embedment length for developing adequate bond strength. Moreover, the Poisson effect with increasing diameter would reduce the adhesion thereby reduces the bond strength. Using the test results, a relationship between bond strength of smooth SMA bar and its bar diameter can be expressed as:  bcdf025.025.1/max −=τ         (4.4) where, db is the bar diameter. Comparison with experimental result showed that equation 4.4 relates very well for smaller diameter as compared to the large diameter. For 20 mm rebar the average absolute error was 3.2% while that for 32 mm rebar was 6.5%. 73   Figure 4.9. Effect of bar diameter on average (a) maximum and (b) residual bond strength of smooth SMA bar 4.4.3.3 Effect of embedment length Four different embedment lengths (3 db, 5 db, 7 db, 9 db) were considered to evaluate their influence on bond strength of smooth SMA rebar. Figure 4.10 shows the effect of embedment length on the average maximum and residual bond strength of SMA rebar. From Figure 4.10 it is evident that the average maximum and residual bond strength increases as the embedment length decreases. Similar behavior has also been reported in literature for steel (Feldman and Bartlett 2005) and FRP rebar (Sayed et al. 2011). The increase in average maximum bond strength is more pronounced in small diameter bars as compared to the large diameter ones. For instance, the average maximum bond strength of the 3db specimens are almost 40% higher as compared to 7db specimens of 20 mm smooth SMA bars. On the other hand, for the 32 mm bars the same increased by only 27%. This can be attributed to the fact that as the embedment length increases, the surface area over which the SMA bar is bonded to the concrete increases.  This increased surface area results in a reduced average bond stress between the bar and the surrounding concrete and also reduces the average stress transferred into the surrounding concrete. Moreover, a reduction in the bar diameter due to Poisson’s effect, which leads to a reduction in friction along the embedment length results in a reduced bond strength. A regression analysis of the test results yielded the following quadratic relationship between the normalized bond strength (τmax/√fc/) of smooth SMA rebar and its embedment length:  00.20.40.60.811.20 10 20 30 40τ max/√fc'(MPa1/2 )Bar diameter (mm)(a)00.050.10.150.20.250.30 10 20 30 40τ res/√fc'(MPa1/2 )Bar diameter (mm)(b)74   05.1005.010 25/max +−= − ddcllfτ        (4.5) This quadratic relationship is in contrast with the behavior of deformed rebar where there is a liner relation between bond strength and embedment length.   Figure 4.10. Effect of embedment length on average (a) maximum and (b) residual bond strength of smooth SMA bar  4.4.3.4 Effect of concrete cover The test results were used to determine the effect of concrete cover on the bond behaviour of smooth SMA bars. The effect of cover concrete was investigated in terms of cover to bar diameter ratio (c/db). Figure 4.11 shows the variation in average maximum and residual bond strength of smooth SMA bar with changing cover to bar diameter ratios. From Figure 4.11 it is observed that c/db has noticeable impact on maximum bond strength, however, residual bond strength was independent of c/db. The influence of c/db is higher for smaller diameter bars as compared to large diameter ones. From Figure 4.11a it can observed that, for 20 mm bars, as the c/db increases from 2 to 3.25 (1.625 times) the average maximum bond strength increases by 14%. On the other hand, for 32 mm bars, as the c/db increases from 1.06 to 1.84 (1.74 times) the average maximum bond strength increases by 6.5%. A regression analysis of the test results yielded the following quadratic relationship between the normalized bond strength (τmax/√fc/) of smooth SMA bar and its cover to bar diameter ratio (c/db): 00.10.20.30.40.50.60.70.80.90 100 200 300τ max/√fc'(MPa1/2 )Embedment length, ld(mm)60 mm 96 mm100 mm 140 mm160 mm 180 mm225 mm(a)00.050.10.150.20.250.30.350 100 200 300τ res/√fc'(MPa1/2 )Embedment length, ld(mm)60 mm 96 mm100 mm 140 mm160 mm 180 mm225 mm(b)75  82.020.009.02/max +−=bbcdcdcfτ       (4.6)  Figure 4.11. Effect of concrete cover to bar diameter ratio on average (a) maximum and (b) residual bond strength of smooth SMA bar  4.4.3.5 Effect of surface modification Previous research on smooth steel and FRP rebars have shown that surface modification of the plain rebars can improve the bond strength significantly (Feldman and Bartlett 2005, Arias et al. 2012). However, several researchers have concluded that rebar surface does not appear to affect the bond strength of FRP rebars in concrete (Mosley et al. 2008, Wambeke and Shield 2006). The smooth SMA rebars used in this study were modified using two different types of sand in order to improve the bond behavior. Due to the importance of rebar surface on the bond behavior, it is worth investigating the variation in bond behavior with different surface finish. Figure 4.12 shows bond strength- slip curves for specimens having different surface finishes with 20 mm bars, ld of 60 mm and 40 mm cover. Observation from Figure 4.12a revealed that, sand coating significantly improves the bond behavior of smooth rebar. The maximum average bond strength of 600µm sand coated rebar was 45% and 37% higher than the smooth and 300µm sand coated rebar, respectively. The average residual bond strength of 600µm sand coated rebar was 29% and 35% higher than the smooth and 300µm sand coated rebar, respectively. Interestingly, average residual bond strength of 300µm sand coated rebar was 6% lower than that of smooth rebar. 0.40.50.60.70.80.911.11.20 20 40 60 80τ max/√fc'(MPa1/2)Concrete cover (mm)34 mm40 mm59 mm65 mm(a)00.050.10.150.20.250.30.350.40.450 20 40 60 80τ res/√fc'(MPa1/2)Concrete cover (mm)34 mm40 mm59 mm65 mm(b)76  Figure 4.12b and c show the influence of rebar diameter and embedment length on the bond strength behavior of SMA rebars with different surface finishes. Similar trend was observed for all the bars irrespective of bar finish; the bond strength decreases as the bar diameter and embedment length increases. From Figure 4.12b it can be observed that, the 32 mm sand coated bars produced higher maximum average bond strength as compared to smooth 32 mm bars. Similar conclusion can be drawn on the effect of embedment length. Figure 4.12c shows that the 600 µm sand coated bars with different embedment lengths produces higher bond strength as compared to those of smooth rebars and 300 µm sand coated bars. It can be concluded that the friction and interlocking produced by the roughened surface creates a more effective mechanism and improves the bond of smooth SMA rebar significantly.  Figure 4.12. Effect of sand coating on bond strength of SMA rebar (a) bond stress-slip curve, (b) effect of bar diameter and (c) effect of embedment length  0246810120 2 4 6 8 10Bond Stress, τ(MPa)Slip (mm)As- receivedSand-300Sand-600(a)024681012As-Received Sand-300 Sand-600Bond Stress, τ(MPa)Rebar Finish20 mm 32 mm(b)024681012As-Received Sand-300 Sand-600Bond Stress, τ(MPa)Rebar Finish3db 5 db 7db(c)77  4.5 Empirical Relationship for Bond Strength of SMA Rebar The analysis results presented and discussed on previous sections revealed the influence of different factors and surface condition on the bond strength of SMA rebar with concrete. Regression analysis of all the specimens, considering all influential parameters, yields the following equation. /max 015.00025.0004.09.0 cbdbr fdcldk +−−=τ    (4.7) Where, τmax is the average maximum bond strength in MPa, db is the bar diameter in mm, ld is the embedment length in mm, c is the concrete cover in mm, fc/ is the concrete compressive strength in MPa, and kr is the surface roughness factor which is 1 for smooth rebar. In the case of sand coated rebar, kr can be calculated using equation 4.8. 5.692.117.0 2 +−= ααrk         (4.8) where, α is the sand size coefficient and calculated as, α= 2/sand size in mm. The proposed equation 4.7 can be used to estimate the bond strength of SMA rebar in concrete considering both smooth and sand coated surface. To verify the accuracy of the proposed equation, comparison was made with experimental results. Figure 4.13 shows the comparison of normalized bond strength obtained from the test results and the proposed equation. Figure 4.13 shows that the proposed equation predicted the bond strength very well where the correlation coefficient is 0.916. 4.6 Comparison with Bond Behavior of Sand Coated FRP Bars For comparative analysis, the bond strength of sand coated FRP bars provided by different design codes are compared with sand coated SMA rebars tested in this study. The average bond strength determined from experimental results and using equation 4.7 are compared with the bond strength calculated as per CSA S806-12 (CSA 2012) and CSA S6-10 (CSA 2010). ACI 440.1R-06 (ACI 2006) was not considered since the ACI equation warrants the development length to be at least 19db and the equation was developed based on concrete strength between 28 MPa and 45 MPa. Since in this study, the sand coated SMA rebars were 78  tested with 50 MPa concrete and embedment length of 3db - 7db, the ACI equation may not be accurate to predict the bond strength of sand coated SMA rebar.  Figure 4.13. Comparison between experimental and predicted values of τmax/√fc’  Canadian Standards Association CSA S806-12 (CSA 2012) provides the following equation (eqn. 4.9) for calculating the development length of FRP Bars. bcFcsd Affdkkkkkl/543215.1=         (4.9)   Using equation 4.9, the following equation was derived to calculate the bond strength of FRP rebars: bccsdkkkkkfdπτ54321/max 5.1=         (4.10) Where, dcs= smallest of the distance from the closest concrete surface to the center of the bar being developed or two-thirds the center to center spacing of the bars being developed (mm), fc/ = compressive strength of concrete (MPa); k1 = bar location factor (1.3 for horizontal reinforcement placed so that more than 300 mm (11.81 inch) of fresh concrete is cast below the bar; 1.0 for all other cases); k2= concrete density factor (1.3 for structural low-density 00.20.40.60.811.21.40 0.2 0.4 0.6 0.8 1 1.2 1.4Predicted (τmax/√fc')Experimental (τmax/√fc')79  concrete; 1.2 for structural semi-low-density concrete; 1.0 for normal density concrete); k3 = bar size factor (0.8 for Ab< 300 mm2); 1.0 for Ab > 300 mm2); Ab is the cross-sectional area of an individual bar in mm2; k4 = bar fibre factor (1.0 for CFRP and GFRP; 1.25 for AFRP); k5 = bar surface profile factor (1.0 for surface roughened or sand coated or braided surfaces; 1.05 for spiral pattern surfaces or ribbed surfaces; 1.8 for indented surfaces). According to the Canadian Highway Bridge Design Code CSA S6-10 (CSA 2010), the expression for the bond strength of FRP rebar is calculated as: crbsfrptrcsfdkkEEkdπτ61max 45.0+= ;         (4.11) snfAk ytrtr 5.10= ; bsfrptrcs dEEkd 5.2≤+       (4.11) Where, Atr = area of transverse reinforcement normal to the plane of splitting through the bars (mm²); fy = yield strength of transverse reinforcement (MPa); s = center to center spacing of the transverse reinforcement (mm); n = number of bars being developed along the plane of splitting; EFRP = modulus of elasticity of FRP bar (MPa); Es = modulus of elasticity of steel (MPa); k6 is bar surface factor, fcr is the flexural strength of concrete in MPa (0.4√f’c for normal density concrete, 0.34√f’c for semi-low density concrete, 0.30√f’c for low-density concrete). Table 4.2 shows the comparison of sand coated SMA rebars obtained from pushout tests and prediction equation with those obtained from the two design codes. Table 4.2 shows that the embedment length and concrete cover have no influence on the bond strength according to CSA S806-12 (CSA 2012) and CSA S6-10 (CSA 2010). Since no transverse reinforcement were provided in pushout specimens, the confinement effect provided by lateral reinforcement index, ktr, in CSA S6-10 (CSA 2010) can be neglected. From Table 4.2 it can be observed that the bond strength obtained using CSA S6-10 (CSA 2010) have a closer match with the experimental and predicted bond strength of sand coated SMA bars. On the contrary, the bond strength calculated using CSA S806-12 (CSA 2012) varies by a large margin. From the results presented in Table 4.2, it can be concluded that with few modifications, the CSA S6-10 (CSA 80  2010)  equation for bond strength prediction of sand coated FRP rebar can be used for the bond strength prediction of sand coated SMA rebar. However, the proposed bond strength equation is not suggested to be used for sand coated FRP rebar since design of FRP reinforced concrete members would require certain other considerations. Table 4.2. Comparison of Bond Strength Sand Coated SMA bars and FRP Bars Rebar Type Experiment Prediction CSA S6-10 CSA S806-12  MPa MPa MPa MPa 20-300-3db 6.12 6.24 6.25 4.89 20-300-5db 5.28 5.35 6.25 4.89 20-300-7db 4.54 4.45 6.25 4.89 20-600-3db 9.82 9.80 6.25 4.89 20-600-5db 8.36 8.40 6.25 4.89 20-600-7db 7.11 7.00 6.25 4.89 32-300-3db 5.04 4.88 3.91 3.06 32-300-5db 3.37 3.46 3.91 3.06 32-600-3db 7.61 7.67 3.91 3.06 32-600-5db 5.48 5.43 3.91 3.06  Sample designation: bar dia-sand size-embedment length  Concrete cover= 40 mm and Concrete strength = 50MPa  4.7 Summary The distinct superelastic properties and flag shape hysteresis of Shape Memory Alloys (SMAs) make them an ideal candidate for the design and development of various structural components in civil infrastructure. Due to the fact that SMA reinforcement has significantly different properties than conventional steel, structures reinforced with SMA will behave differently. The design equations used for steel reinforced concrete structures are not applicable while using SMA as reinforcement in concrete. This chapter investigated the bond behavior of SMA rebars in concrete using 56 pushout specimens. The test results are explored to evaluate the influence of concrete strength, bar diameter, embedment length, and surface condition. Surface modification using sand coating notably improved the bond strength of SMA rebar. Finally, empirical equation based on statistical analyses is presented to predict the maximum average bond strength. The proposed equation appear to be reasonable for calculating the average bond strength of SMA reinforcing bars in concrete. 81  CHAPTER 5. PLASTIC HINGE LENGTH OF SHAPE MEMORY ALLOY (SMA) REINFORCED CONCRETE BRIDGE PIER 5.1 General Shape memory alloys (SMAs) have been emerging as an alternative to conventional steel reinforcement in concrete structures due to its distinct shape recovery and superelastic properties. Considering the importance of bridge pier, it is necessary to predict the displacement capacity of bridge piers during earthquakes. Past researches have shown that SMA could significantly improve the seismic performance of bridge piers through recentering thereby significantly reducing the permanent damage. Previous researchers mostly used the Paulay and Priestely (1992) equation for calculating the plastic hinge length in SMA-RC bridge pier and reported that this equation provides a reasonable estimate of the plastic hinge of SMA-RC pier. However, for seismic design of SMA-RC pier, it is necessary to identify the plastic hinge length of the pier which can be used for calculating the flexural displacement capacity. Most of the previous studies on plastic hinge length focused on beams and columns (Mattock 1964; Corley 1966; Priestley and Park 1987; Paulay and Priestley 1992; Bae and Bayrak 2008) where only a few studies were conducted for bridge piers (Hines et al. 2004, Alemdar 2010). A review of existing plastic hinge equations showed that the plastic hinge length of a bridge pier depends on many factors such as mechanical properties of longitudinal and transverse reinforcement, concrete strength, level of axial load, aspect ratio, reinforcement ratio, and level of confinement. Since the mechanical properties of SMA and its behavior under lateral load are significantly different from conventional RC piers, it warrants a specific plastic hinge expression for SMA-RC bridge pier. Although researchers have investigated the seismic performance of bridge piers considering different types of SMA (Saiidi et al. 2009, Gencturk and Hosseini 2014, Billah and Alam, 2014b), only one study (Nakashoji and Saiidi 2014) has been conducted so far to estimate the plastic hinge length in SMA reinforced concrete (RC) bridge pier. However, their proposed equation does not consider the effect of different parameters and could estimate the plastic hinge length with 11.6% error. Using a well-calibrated finite element model, this chapter developed a plastic hinge length expression for SMA-RC bridge pier by investigating the distribution of curvature and 82  strain in the longitudinal rebar (both steel and SMA rebar) along the height of the pier. This study adopted an analytical method to develop a plastic hinge length expression for SMA-RC bridge pier due to the absence of adequate experimental results and limitations in conducting experiments due to high cost of SMA. Considering different parameters such as the level of axial load, aspect ratio, concrete strength, SMA properties and the ratio of the longitudinal and transverse reinforcement, a parametric study was conducted to derive a plastic hinge length expression for SMA-RC bridge pier. Finally, the proposed equation was used to estimate the drift capacity of SMA-RC bridge pier and compared with test results. 5.2 Design and Geometry of Bridge Pier This section briefly describes the design and configurations of different SMA-RC bridge piers used in this study. Since SMA is a costly material it is only used in the bottom plastic hinge region of the bridge pier. The bridge pier is assumed to be located in Vancouver, BC and was seismically designed following the Canadian Highway Bridge Design Code (CSA-S6-10).  Figure 5.1 shows the cross section of the column. The diameter of all the columns was fixed to be 1.524 m. Several parameters govern the design and the behavior of the bridge piers. These parameters also affect the spread of plasticity along the length of the pier.   Figure 5.1. Geometry of SMA-RC bridge pier (a) Cross section, (b) Elevation and (c) Finite element modeling  (a)(b) (c)83  The primary variables of the parametric study were selected as the aspect ratio (L/d) of the column, axial load ratio (P/f’cAg), longitudinal reinforcement ratio (ρl), transverse reinforcement ratio (ρs), yield strength of SMA rebar (Fy-SMA) and concrete compressive strength (fc’). These parameters were selected based on existing literature on plastic hinge length of reinforced concrete elements (Paulay and Priestley 1992, Hines et al. 2004, Bae and Bayrak 2008, Alemdar 2010, Bohl and Adebar 2011, Kazaz 2013). Table 5.1 shows the list of considered parameters and their associated values. For each parameter three different values were considered. Table 5.2 shows the summary of the SMA-RC pier specimens analyzed in this study. A total of 18 piers were designed. In order to investigate the effect of different parameters on the plastic hinge length of SMA-RC pier, one parameter at a time was varied and others were kept constant. In this study, the interaction effect was not considered as it was found that interaction between parameters do not have any significant impact. Apart from the investigated parameter, the plastic hinge length of the piers was also varied and three different plastic hinge lengths were considered: 0.5 LP/d, 0.75 LP/d and 1 LP/d. These three lengths were selected as previous studies on SMA-RC bridge piers (O’Brien et al. 2007, Nakashoji and Saiidi 2014) showed that the plastic hinge length varies from 0.5 LP/d to 1.1 LP/d. The diameter and number of longitudinal reinforcement of different bridge piers were varied for different reinforcement percentages and 15.875 mm (#5) spirals were used at different spacing as lateral reinforcement.  In this study, in order to ensure flexure dominated behavior and avoid shear failure, three different aspect ratios (3, 5, 7) were considered. Table 5.1. Details of variable parameters Parameters Values Axial Load (%) 5 10 20 ρl (%) 1 2 3 Aspect Ratio (L/d) 3 5 7 fc' (MPa) 35 50 60 ρs (%) 0.8 1 1.2 Fy-SMA (MPa) 210 450 750    84  Table 5.2. Details of SMA-RC bridge piers Variable Pier P/fc’Ag H (m) fc' (MPa) ρl (%) fy-SMA (MPa) Lp (m) ρs (%) Axial Load P1-1 0.05 7.62 35 1 401 0.762 1.2 P1-2 0.1 7.62 35 1 401 1.143 1.2 P1-3 0.2 7.62 35 1 401 1.524 1.2 Aspect Ratio P2-1 0.05 4.572 35 1 401 0.762 1.2 P2-2 0.05 7.62 35 1 401 1.143 1.2 P2-3 0.05 10.668 35 1 401 1.524 1.2 SMA fy P3-1 0.05 7.62 35 1 210 0.762 1.2 P3-2 0.05 7.62 35 1 401 1.143 1.2 P3-3 0.05 7.62 35 1 750 1.524 1.2 ρl (%) P4-1 0.05 7.62 35 1 401 0.762 1.2 P4-2 0.05 7.62 35 2 401 1.143 1.2 P4-3 0.05 7.62 35 3 401 1.524 1.2 fc' P5-1 0.05 7.62 35 1 401 0.762 1.2 P5-2 0.05 7.62 50 1 401 1.143 1.2 P5-3 0.05 7.62 60 1 401 1.524 1.2 ρs (%) P6-1 0.05 7.62 35 1 401 0.762 0.8 P6-2 0.05 7.62 35 1 401 1.143 1 P6-3 0.05 7.62 35 1 401 1.524 1.2  5.3 Analytical Modeling One of the main objectives of this study was to develop a fibre-based numerical model capable of predicting the nonlinear behavior in terms of strain and curvature distribution of SMA-RC bridge piers. The modeling and nonlinear analyses of SMA-RC bridge piers were conducted using fibre element based nonlinear analysis program SeismoStruct (Seismosoft, 2014). Using force based inelastic beam-column element, the circular bridge piers were modeled. The Mander et al. (1988) concrete constitutive model was used to describe the confined and unconfined concrete and the steel reinforcement was represented using the Menegotto–Pinto (1973) steel model. The superelastic SMA was modeled following the constitutive relation developed by Auricchio and Sacco (1997). Mechanical couplers were used to connect SMA with steel rebars (Alam et al. 2010) which is represented by introducing a zero length rotational spring at the bottom of the column section (Figure 5.1c). The stress-slip relationship of the bars inside the coupler and the details of the splicing can be found elsewhere (Billah and Alam 2012a). 85  5.4 Model Validation The accuracy of the adopted finite element modeling program in predicting the seismic response of bridge structures has been demonstrated by several researchers through comparisons with experimental results (Alam et al. 2009; Billah and Alam, 2014a). However, in order to investigate the accuracy of the modeling technique in predicting the strain and curvature distribution, comparisons were made with experimental results of SMA-RC bridge piers. Nakashoji and Saiidi (2014) conducted experimental investigation on SMA-RC bridge piers and extensive measurements of rebar strains were made along the height of the pier. Specimen SR-99 LSE was a square column having a 457 mm square cross section and a height of 1575 mm. The plastic hinge length (457 mm) of the specimen was reinforced with 16-12.7mm diameter Ni-Ti SMA rebar and the remaining portion was reinforced with 16-16mm steel rebar. The vertical strains measured over a 508 mm gauge length from the base of Specimen SR-99 LSE are shown in Figure 5.2a at two drift levels: 1% and 2% for strain gauges 2, 8, 18, 28, and 38. The predicted SMA rebar strains at 1% and 2% drift are also shown in Figure 5.2a. Observation from Figure 5.2a shows that, there is good agreement between the measured and predicted strains. From Figure 5.2a it is evident that the analytical model was also able to predict the nonlinear strain profile observed from the experiment. This comparison shows that the local response of SMA-RC bridge pier can be determined satisfactorily with the adopted nonlinear finite-element modeling technique.  Figure 5.2. (a) Comparison of predicted and measured strain on SMA rebar (Nakashoji and Saiidi 2014) and (b) Comparison of predicted and measured curvature (O’Brien et al. 2007) 05101520250 4000 8000 12000Height (inch)Strain (µ)1% drift (experiment)2% drift (experiment)1% drift (predicted)2% drift (predicted)(a)02468101214160 0.002 0.004 0.006Height (inch)Curvature (rad/inch)1.5 % drift (experiment)3% drift (experiment)1.5% drift (predicted)3% drift (predicted)(b)86  Since this study used both rebar strain and curvature profile to predict the plastic hinge length of SMA-RC bridge pier, the ability of the adopted modeling technique in accurately predicting the curvature distribution was also investigated. O’Brien et al. (2007) investigated the performance of a 1/5-scale circular SMA-RC bridge pier having a diameter of 254 mm and the height of the column was 1143 mm. The column was reinforced with 15.9 mm diameter Ni-Ti SMA in the plastic hinge region. They tested the column under reverse cyclic loading  and measured the curvature distribution over a 355.6 mm gauge length from the base of Specimen RNC. Figure 5.2b shows the comparison of the measured and predicted curvature at two different drift levels: 1.5% and 3% over the height of the specimen. From Figure 5.2b it can be observed that, the profile of the curvature distribution predicted along the length of the pier not only matches closely to the measured response, but also mimics the trend in the curvature profile along the section. 5.5 Analytical Approach for Predicting Plastic Hinge Length Accurate estimation of plastic hinge lengths in RC bridge piers using analytical approach can be complicated. Typically plastic hinge lengths are calculated using experimental results. However, several researchers (Bae and Bayrak 2008, Kazaz 2013) have derived plastic hinge lengths of RC elements using analytical approach based on strain and curvature. This study adopted an analytical approach for deriving an expression for plastic hinge length of SMA-RC pier as there is lack of adequate test results. In this study, two different methods, the longitudinal rebar compressive strain profile and the curvature profile along the height of the pier, were used to calculate the plastic hinge length of SMA-RC bridge pier. During an earthquake, bridge piers are subjected to lateral displacements while supporting gravity loads and plastic hinges usually form at the maximum moment region. This inelastic portion causes a significant increase in inelastic curvature near the base of the bridge pier and forms the plastic hinge zone. As the curvature increases, the compression side of the member experiences increased strain and subsequently reaches a critical value when the concrete cover spalls off. After that the longitudinal bars on the compression side experience yielding and subsequently core concrete starts to crush. Under increasing compressive strain damage starts to accumulate and forms plastic hinges. The compressive strain in the longitudinal rebar is equal to the compressive strain in the outer core concrete fibre. Therefore, a rebar compressive strain profile along the height should give a clear indication on the formation of the plastic hinge. In 87  this study, the SMA-RC bridge piers were analyzed under reverse cyclic loading and the compressive strain profiles in the longitudinal rebar were plotted. By tracking the onset of the yielding of longitudinal rebar in compression, the most damaged area i.e. the plastic hinge was identified.  This study also used the curvature profile along the height of the pier to determine the plastic hinge length. After analyzing the bridge pier under reverse cyclic loading, the curvature profile of the piers were plotted to identify the zone where inelastic curvatures are localized. By tracking the yield curvature in the curvature profile, the plastic hinge was identified. The following section describes the effect of the different parameters on the plastic hinge length of SMA-RC bridge pier.  5.5.1 Effect of axial load Several researchers (Bae and Bayrak 2008, Légeron and Paultre 2000) have considered axial load level an important parameter for plastic hinge estimation of RC columns. However, researchers have reported contradictory conclusions regarding the effect of axial load. Mendis (2001) and Park et al. (1982) reported that the level of axial load does not have any influence on plastic hinge lengths. However, Tanaka and Park (1990) and Légeron and Paultre (2000) found that as the axial load increases the plastic hinge length increase. Except Berry et al. (2008), most of the researchers considered very high levels of axial load which are unusual for bridge piers and most of them were for columns in a frame structure. In this study, three different axial load levels were considered to study the effect of axial load on the plastic hinge length. The range of axial loads (5%, 10% and 20%) was selected based on design codes or common practices. Keeping the other parameters constant, the piers were analyzed under reverse cyclic loading.  Figure 5.3 shows the variation of rebar compressive strain and curvature profile along the height of the pier. From Figure 5.3a it is evident that the curvature profiles are not influenced by the axial load on the plastic hinge length. However, the compressive strain profile, as shown in Figure 5.3b, clearly depicts the effect of increasing axial load on the compressive strain in the longitudinal reinforcement. It is evident from Figure 5.3b that with the increase in axial load, the plastic hinge length increases. The strain profile in the significantly damaged zone drastically changes with the axial load as identified in the plastic hinge region. Yield strain of longitudinal rebar was used to determine the plastic hinge 88  length. For different level of axial load the plastic hinge length varied between 0.78d to 1.18d where d is the diameter of the pier.   Figure 5.3. Effect of axial load on (a) curvature profile and (b) longitudinal rebar strain profile  5.5.2 Effect of aspect ratio Previous researchers (Mattock 1967; Corley 1966; Priestley and Park 1987; Mendis 2001) identified that the plastic hinge length of a RC member is influenced by the aspect ratio (L/d). However, the widely used plastic hinge length equation proposed by Paulay and Priestley (1992) does not account for the effect of the aspect ratio. In order to investigate the influence of the aspect ratio on the plastic hinge length, circular SMA-RC piers with varying aspect ratios (3, 5, and 7) were considered keeping other parameters constant. The results of the analyses are summarized in Figure 5.4. As can be observed in the curvature profile (Figure 5.4a), plastic hinge length is independent of the aspect ratio of the pier. However, the plastic hinge length increases with the increasing aspect ratios as evident from the strain profile (Figure 5.4b). As the aspect ratio increased from 3 to 7, the plastic hinge lengths were found to increase from 0.82d to 1.25d. Bae and Bayrak (2008) and Alemdar (2010) also reported that lp increases with the increasing L/d for a given axial load level. Bae and Bayrak (2008) found that the effect of change in aspect ratio is less pronounced in columns with small aspect ratio (2<L/d<3) as compared to columns having larger aspect ratio. They also concluded that the change in plastic hinge length with increasing aspect ratio are insignificant for columns under low axial load. 01234567890 0.02 0.04 0.06 0.08 0.1Distance from base (m)Curvature (1/m)0.2 Po0.1 Po0.05 Po(a)01234567890 0.005 0.01 0.015 0.02 0.025 0.03Distance from base (m)Longitudinal rebar strain (εs)0.2 Po0.1 Po0.05 Poεy-sma=0.0064(b)89  However, in this study it was found that the aspect ratio contributes to the plastic hinge zone in SMA-RC bridge pier.   Figure 5.4. Effect of aspect ratio on (a) curvature profile and (b) longitudinal rebar strain profile 5.5.3 Effect of SMA properties Since SMA possesses significantly different mechanical properties than conventional steel, it might affect the plastic hinge formation in the SMA reinforced bridge pier. In addition, several compositions of SMAs have been developed which have potential for application in bridge pier such as Ni-Ti, Fe-based and Cu-based. Most of the applications have been focusing on the use of Ni-Ti alloy while very few focused on the application of the alloys such as Cu-based SMAs (Shrestha et al. 2015, Araki et al. 2010), and Fe- based SMAs (Dezfuli and Alam 2013). This study employed three different types of SMA’s having different composition, yield strength, and superelastic strain to investigate the effect of SMA properties on the plastic hinge length. In this study, one nickel–titanium, one Cu-based, and one Fe- based shape memory alloys have been selected for the use in bridge piers. The selected SMAs along with their mechanical properties such as the elastic modulus (E), austenite to martensite starting stress (fy); austenite to martensite finishing stress (fP1); martensite to austenite starting stress (fT1); martensite to austenite finishing stress (fT2); superelastic strain (εs) are listed in Table 5.3. As the three different types of SMAs were used, the bridge piers were designed in such a way that they have comparable moment capacities. Figure 5.5 shows the effect of different types of SMA on the curvature and rebar compressive strain profile. From Figure 5.5a it can be 0246810120 0.005 0.01 0.015 0.02 0.025 0.03Distance from base (m)Longitudinal rebar strain (εs)AR-3AR-5AR-7εy-sma=0.0064(b)0246810120 0.02 0.04 0.06 0.08 0.1Distance from base (m)Curvature (1/m)AR-3AR-5AR-7(a)90  observed that the different types of SMA affects the curvature profile thereby affecting the plastic hinge length. Figure 5.5b depicts that as the yield strength of SMA rebar increases the plastic hinge length increases. As the yield strength of SMA increased from 210 MPa to 750 MPa, the plastic hinge length increases from 0.8d to 1.06d. Previous researchers (Berry et al. 2008, Alemdar 2010) also concluded that the plastic hinge length of concrete bridge pier increases as the yield strength of the reinforcement increases. Table 5.3. Properties of different types of SMA Alloy εs (%) E (GPa) fy (MPa) fp1 (MPa) fT1 (MPa) fT2 (MPa) fy/E Reference NiTi45 6 62.5 401.0 510 370 130 0.0065 Alam et al. (2008a) FeNCATB 13.5 46.9 750 1200 300 200 0.0159 Tanaka et al. (2010) CuAlMn 9 28 210.0 275.0 200 150 0.0075 Shrestha et al. (2013) fy (austenite to martensite starting stress); fP1(austenite to martensite finishing stress); fT1(martensite to austenite starting stress); fT2(martensite to austenite finishing stress), εs (superelastic plateau strain length); and E  (modulus of elasticity).    Figure 5.5. Effect of fy-SMA on (a) curvature profile and (b) longitudinal rebar strain profile  5.5.4 Effect of longitudinal reinforcement ratio The effect of longitudinal reinforcement ratio (ρl) on the plastic hinge length has been ignored by many researchers. However, several researchers investigated the effect of ρl on the plastic hinge length and reported contradictory conclusions. Mattock (1964) concluded that, as the net tension reinforcement increases, the plastic hinge length decreases. On the contrary, 01234567890 0.02 0.04 0.06 0.08 0.1Distance from base (m)Curvature (1/m)SMA-210SMA-450SMA-750(a)01234567890 0.005 0.01 0.015 0.02 0.025 0.03Distance from base (m)Longitudinal rebar strain (εs)SMA-210SMA-450SMA-750(b)91  Mendis (2011) found that the plastic hinge length increases with increasing amount of tension reinforcement. These conclusions were based on beam test results. However, Bae and Bayrak (2008) concluded that the plastic hinge length of a column tend to increase with increasing longitudinal reinforcement ratio (ρl). To study the effect of ρl on the plastic hinge length of SMA-RC pier, three different reinforcement ratios (1%, 2% and 3%) consistent with current seismic design guidelines were selected. Figure 5.6 shows the effect of longitudinal reinforcement ratio (ρl) on the curvature and strain profile. As evident from both curvature and strain profile, the plastic hinge length tends to decrease with increasing longitudinal reinforcement ratio (ρl). The change in plastic hinge length is more pronounced from longitudinal rebar strain profile (Figure 5.6b) as compared to the curvature profile (Figure 5.6a).  Figure 5.6. Effect of longitudinal reinforcement ratio on (a) curvature profile and (b) longitudinal rebar strain profile  5.5.5 Effect of transverse reinforcement Most of the available plastic hinge equations do not consider the effect of transverse reinforcement ratio (ρs). Corley (1966) and Kazaz (2013) did not consider ρs in their proposed plastic hinge expression. Only few researchers (Mendis 2001, Hines et al. 2004) considered the effect of ρs on the plastic hinge length. Mendis (2001) and Hines et al. (2004) have concluded that as ρs increases the plastic hinge length decreases as evident from the plastic hinge equation proposed by Mendis (2001) and Hines et al. (2004). Figure 5.7 shows the variation in curvature and strain profile with changes in the transverse reinforcement ratio (ρs). 01234567890 0.02 0.04 0.06 0.08 0.1Distance from base (m)Curvature (1/m)1%2%3%(a)01234567890 0.005 0.01 0.015 0.02 0.025 0.03Distance from base (m)Longitudinal rebar strain (εs)1%2%3%(b)92  The change in plastic hinge length is more pronounced from strain profile as compared to the curvature profile. From the curvature profile (Figure 5.7a) the plastic hinge length varied from 0.84d to 0.88d. However, from longitudinal rebar strain profile (Figure 5.7b) the plastic hinge length varied from 0.76d to 1.02d. This can be attributed to the fact that as the amount of transverse reinforcement increases, the core concrete experiences less damage thereby reduce the plastic hinge length.  Figure 5.7. Effect of transverse reinforcement ratio on (a) curvature profile and (b) longitudinal rebar strain profile  5.5.6 Effect of concrete strength Several researchers considered the effect of concrete strength on the plastic hinge length of RC members. However, only the plastic hinge expression proposed by Berry et al. (2008) and Alemdar (2010) consider the effect of concrete strength. They found that the plastic hinge length decreases as the concrete compressive strength increases as evident from their plastic hinge equations. This study also considered three different concrete strength (35, 50 and 60 MPa) to investigate the variation in plastic hinge length of SMA-RC pier with varying concrete strength. Figure 5.8 shows the changes in curvature and strain profile as the compressive strength varied from 35 to 60 MPa. The curvature profile depicts that (Figure 5.8a) the change in plastic hinge length is independent of concrete strength as the plastic hinge length varied between 0.75d to 0.78d. On the other hand, the strain profile shows that as the concrete strength increased from 35 to 60 MPa, the plastic hinge length decreased from 1.08d to 0.68d. 01234567890 0.02 0.04 0.06 0.08 0.1Distance from base (m)Curvature (1/m)0.8%1%1.2%01234567890 0.005 0.01 0.015 0.02 0.025 0.03Distance from base (m)Longitudinal rebar strain (εs)0.8%1%1.2%(b)93   Figure 5.8. Effect of concrete compressive strength on (a) curvature profile and (b) longitudinal rebar strain profile  5.6 Plastic Hinge Length Expression for SMA-RC Bridge Pier The results presented in the previous sections showed that the compressive strain profile of the longitudinal rebar facilitates a clearer observation of the plastic hinge length as compared to the curvature profile. As a result, this study utilized the compressive strain profile of the longitudinal rebar to develop the plastic hinge length expression for SMA-RC bridge pier. The discussions presented in preceding sections showed that several factors influence the length of the plastic hinge in SMA-RC pier such as, the level of axial load, the aspect ratio, the yield strength of SMA rebar, the concrete compressive strength, the longitudinal and transverse reinforcement ratio. Considering the effect of different parameters, a new expression for calculating the plastic hinge length of SMA-RC pier was derived by regression analysis. In this study, multivariate linear regression was used as it allows simultaneous testing and modeling of multiple independent variables. Using the multivariate regression analysis technique the following linear expression (equation 5.1) was derived for estimating the plastic hinge length of SMA-RC pier: ( ) ( ) ( ) ( )sclSMAygcP ffdLAfPdLρρ 24.0019.016.00002.008.025.005.1 //−−−+++= −  (5.1) From the proposed equation it can be observed that the plastic hinge length of SMA-RC pier is mostly influenced by the level of axial load, longitudinal and transverse reinforcement ratio and less sensitive to the aspect ratio. Although, the regression coefficients associated with 01234567890 0.005 0.01 0.015 0.02 0.025 0.03Distance from base (m)Longitudinal rebar strain (εs)35 MPa50 MPa60 MPa(b)01234567890 0.02 0.04 0.06 0.08 0.1Disytnce from base (m)Curvature (1/m)35 MPa50 MPa60 MPa(a)94  the yield strength of SMA and concrete compressive strength look insignificant, a small change in fy-SMA or fc/ will result in a significant change in the plastic hinge length. 5.7 Validation of the Proposed Equation To verify the accuracy of the analytically derived expression for plastic hinge length of SMA-RC bridge pier, comparisons were made with plastic hinge length measured from experimental investigations. Since very limited number of test results are available on SMA-RC bridge pier which measured the plastic hinge length, a database composed of four SMA-RC pier test results was compiled. Table 5.4 shows the comparison of the measured and calculated plastic hinge length which illustrates that the use of proposed equation results in good estimates of plastic hinge length for all test specimens. From Table 5.4 it can be observed that the maximum variation was observed in Specimen SR99-LSE (Nakashoji and Saiidi 2014) which was 7.84%. This can be attributed to the fact that all other piers had circular section while SR99-LSE was a square column. Moreover, the proposed equation was derived based on the analyses on circular columns. Best match was observed for Specimen RNE (O’Brien et al. 2007) where the measured and predicted value differed by only 0.87%.  Table 5.4. Comparison of experimental and measured plastic hinge length  Specimens Parameter RNC O'Brien et al.(2007) RNE O'Brien et al.(2007)  SR-99-LSE Nakashoji and Saiidi (2014) SMAC-1 Saiidi and Wang (2006) Axial load ratio (P/fc’Ag) 0.1 0.1 0.0864 0.25 Aspect Ratio (L/d) 4.5 4.5 3.44 4.5 Fy-SMA (MPa) 413.7 413.7 352 379.2 ρl 0.02 0.02 0.01 0.026 fc' (MPa) 31.03 35.8 49.6 43.8 ρs 0.024 0.024 0.015 0.0068 Lp/d (measured) 0.98 0.84 0.44 0.75 Lp /d (calculated) 0.92 0.83 0.47 0.71 Lp (measured) (mm) 249.9 212.3 199 229 Lp (calculated) (mm) 233.48 210.46 214.61 216.66 Error (%) 6.57 0.87 -7.84 5.39  95  Figure 5.9 compares the Lp/d values measured from experimental results with those predicted using equation 5.1. Statistical parameters (mean, standard deviation and COV) displaying the degree of correlation between the measured and predicted values is also shown in the same figure. From Figure 5.9 it is evident that the proposed equation provides a reasonable estimate of the plastic hinge length of SMA-RC bridge pier. From this figure it can be observed that the standard deviation of the predicted plastic hinge length from the measured plastic hinge length is only 0.059. Moreover, the coefficient of variation is only 6% which shows the efficacy of the proposed equation in predicting the experimentally measure plastic hinge length. The proposed plastic hinge equation was also used to calculate the maximum drift of a SMA-RC bridge pier (RNE) tested by O’Brien et al. (2007). Using the plastic hinge length and the yield and ultimate curvature, the ultimate drift of a cantilever bridge pier can be calculated using the following equation: ( ) ( )ppyuyu LLLL 5.031 2 −−+=∆ φφφ       (5.2)  Figure 5.9. Comparison of measured and predicted plastic hinge lengths In order to predict the accuracy of the proposed plastic hinge expression in predicting the ultimate drift capacity of SMA-RC bridge pier, comparisons were made with experimental results and other plastic hinge expression available in literature. Table 5.5 shows a comparison of the measured ultimate drift value and ultimate drift calculated with different plastic hinge equations. From Table 5.5, it is evident that the proposed plastic hinge equation provides a reasonable estimate of the drift capacity of SMA-RC pier. The proposed Lp equation could 00.20.40.60.810 0.2 0.4 0.6 0.8 1Lp/d (Predicted)Lp/d (Experiment)y/x: µ=0.98σ= 0.059COV= 6%96  predict the ultimate drift of the specimen RNE with only 5.09% error, which was the second most accurate among all the compared equations. The plastic hinge equation proposed by Nakashoji and Saiidi (2014) predicted the drift capacity with higher accuracy where the difference was only 3.20%. The plastic hinge equation proposed by Paulay and Priestley (1992) also predicted the ultimate drift with only 10.4% error. The other equations differed by a large margin where the largest difference was 30.5% as predicted by the equation proposed by Alemdar (2010). Table 5.5. Comparison of measured and calculated ultimate drift Reference Lp (mm) Ultimate displacement (mm) % difference RNC, O'Brien et al. (2007)- Test Data - 137.4 - Paulay and Priestley (1992) 207.10 123.05 10.44 Alemdar (2010) 141.45 95.70 30.50 Nakashoji and Saiidi (2014) 232.20 133.02 3.20 Berry et al. (2008) 151.51 100.01 29.25 Mander (1983) 182.60 113.06 17.71 Proposed Equation 233.48 133.52 5.09  5.8 Summary It is often assumed that the maximum seismic damage in a bridge pier will concentrate in the regions subjected to maximum inelastic curvature known as its plastic hinge length. Predicting the plastic hinge length accurately is an important part of seismic design of bridge piers. This chapter focused on deriving an analytical expression for the plastic hinge length of shape memory alloy (SMA) reinforced concrete (RC) bridge pier based on the results from well calibrated nonlinear finite element models.  A parametric study was performed to investigate the effect of different parameters on the plastic hinge length, including axial load ratio, aspect ratio, concrete strength, SMA properties, longitudinal and transverse reinforcement ratio. Multivariate regression analysis was performed to develop an expression to estimate the plastic hinge length in SMA-RC bridge pier and compared with existing plastic hinge length equations. The proposed equation was verified against test results which showed reasonable accuracy.   97  CHAPTER 6. PERFORMANCE-BASED SEISMIC DESIGN OF SHAPE MEMORY ALLOY REINFORCED CONCRETE BRIDGE PIER: DEVELOPMENT OF PERFORMANCE-BASED DAMAGE STATES 6.1 General Numerous experimental and numerical studies proved the efficiency of SMA reinforced structures in seismic regions. However, there exists no proper design guideline for utilizing SMA in highway bridges. Moreover, most of the design guidelines are moving forward to performance-based design. AASHTO has already developed performance-based design guidelines for bridges referred as AASHTO SGS (AASHTO 2011). Moreover, the recent edition of Canadian highway bridge design code (CSA-S6-14) has also adapted performance-based design and defined some performance levels and performance criteria for different types of bridges. To successfully apply the performance-based design concept to SMA reinforced concrete (SMA-RC) bridge pier, the performance objectives and their associated limit state criteria must be clearly defined first. Most of the current researches on SMA-RC bridge piers are focused on the seismic performance assessment and comparison with regular RC bridge pier (Billah and Alam 2014c, Cruz and Saiidi 2012, Saiidi et al. 2009). Although there exists a good number of studies on the performance-based damage states for steel RC bridge piers (Lehman et al. 2004, Hose et al. 2000), no study so far has focused on the performance-based damage states for SMA-RC piers. This is mainly due to limited number of experimental studies performed on SMA-RC piers where high cost of SMAs was the main restraining factor. Since the behavior of SMA-RC piers are significantly different from their steel counterpart, using those damage states for SMA-RC piers might lead to faulty design. Moreover, the mechanical properties of SMAs vary widely where several compositions of SMAs have been developed and used by different researchers in civil engineering applications (Alam et al. 2007). Hence, this chapter aims at developing performance-based damage states for SMA-RC bridge piers considering five different SMAs with three different earthquake hazard levels. The ultimate goal of this study is to provide a technical basis for the development of performance-based seismic design and evaluation methodologies for the SMA-RC bridge piers. 98  Using an incremental dynamic analysis (IDA) based analytical approach (Vamvatsikos and Cornell 2002), performance-based damage states (based on drift limits) have been developed for five different SMA-RC bridge piers and validated against experimental data. Application of such technique may palliate the burden of gathering a large amount of test data and cost of experiments, which were used in the past to develop different damage sates for RC bridge piers. Past studies have demonstrated that it is necessary to consider the residual drifts to fully characterize the performance of a structural system after a seismic excitation and the potential damage that the system can experience (Christopoulos et al. 2003; Erochko et al. 2011). Since SMA has the ability to reduce the residual drift significantly after unloading, the residual drift of different SMA-RC bridge piers under varying intensity of earthquake need to be investigated. This study also developed residual drift based damages states for the SMA-RC bridge pier and proposed an analytical expression that can be used for predicting the residual drift in SMA reinforced concrete elements. 6.2 Design and Geometry of Bridge Piers This section briefly describes the design and configurations of different SMA-RC bridge piers used in this study. Since SMA is a costly material, it is only used in the bottom plastic hinge region of the bridge piers. Five different SMAs are used in this study to develop the performance-based damage states for SMA-RC bridge piers. The bridge pier is assumed to be located in Vancouver, BC and was seismically designed following Canadian Highway Bridge Design Code (CSA-S6-10).  Figure 6.1 shows the cross section and elevation of the bridge pier. The diameter of all the columns was fixed to be 1.83 m; the columns were reinforced with 48 longitudinal reinforcement of different diameter bars for different SMAs and 16 mm-diameter steel spirals at 76 mm pitch. The height of the pier is 9.14m with an aspect ratio of 5 which ensured the flexure dominated behavior. A constant mass of 85 ton was applied at the top which represents the weight of the superstructure. Different diameter bars were used for different SMAs since different SMAs have different elastic modulus and yield strength. Although SMA does not have a yielding process, “yield” is being used to refer to the initiation of phase transformation of SMA and the yield strain was calculated by defining the austenite to martensite starting stress (fy) by the elastic modulus (E). Five different SMA rebars as shown in  99  Table 6.1 are used to design the different bridge piers. The bridge piers are designated as SMA-RC-1 (reinforced with SMA-1), SMA-RC-2 (reinforced with SMA-2), and so on. SMA-RC-1 and SMA-RC-2 is reinforced with 48-28M SMA-1 and SMA-2 bars, SMA-RC-3 is reinforced with 48-20M SMA-3 bars, SMA-RC-4 is reinforced with 48-35M SMA-4 bars, and SMA-RC-5 is reinforced with 48-32M SMA-5 bars, respectively. The sizes of the rebars were selected in such a way that the axial forces developed in the rebar are almost similar. The bridge piers are designed in such a way that they have comparable moment capacities. Figure 6.2a shows the moment-curvature response of different SMA-RC sections. From this figure it is evident that all the sections have similar initial stiffness and comparable moment capacity. Since SMA-5 has higher elastic modulus SMA-RC-5 showed higher initial stiffness which is 1.78, 1.72, 2.21, and 3.87 times higher than that of SMA-RC-1, SMA-RC-2, SMA-RC-3, and SMA-RC-4, respectively. Moment-curvature response of all the sections revealed that this design process led to comparable moment capacities for the five different SMA reinforced bridge piers. The elastic periods of the SMA-RC-1, SMA-RC-2, SMA-RC-3, SMA-RC-4, and SMA-RC-5 were calculated as 0.513 sec, 0.513sec, 0.514, 0.515, and 0.511 sec, respectively which were close and expected to attract similar earthquake forces. Figure 6.2b shows the pushover response curves for the five different SMA-RC bridge piers. From this figure it can be observed that all the bridge piers have similar stiffness and load carrying capacity.   Figure 6.1. Cross section and elevation of SMA reinforced concrete bridge pier 100  Table 6.1. Properties of different types of SMA  Alloy εs (%) E (GPa) fy (MPa) fp1 (MPa) fT1 (MPa) fT2 (MPa) fy/E Ref SMA-1 NiTi45 6 62.5 401.0 510 370 130 0.0065 Alam et al. 2008a SMA-2 NiTi45 8 68 435.0 535.0 335 170 0.0063 Ghassemieh et al. 2012 SMA-3 FeNCATB 13.5 46.9 750 1200 300 200 0.0159 Tanaka et al. 2010 SMA-4 CuAlMn 9 28 210.0 275.0 200 150 0.0075 Shrestha et al. 2013 SMA-5 FeMnAlNi 6.13 98.4 320.00 442.5 210.8 122 0.0033 Omori et al. 2011 fy (austenite to martensite starting stress); fP1(austenite to martensite finishing stress); fT1(martensite to austenite starting stress); fT2(martensite to austenite finishing stress) , εs (superelastic plateau strain length); and E  (modulus of elasticity). The material properties of concrete and steel rebar used in the bridge piers are summarized in Table 6.2. In the SMA-RC bridge piers, SMA was used as longitudinal reinforcement only at the plastic hinge region. In the remaining part, steel rebars were used as reinforcement. The plastic hinge length, Lp was calculated according to the Paulay and Priestley (1992) equation: Lp = 0.08 L+ 0.022dbfy        (6.1) where, L is the length of the member in mm, db represents the bar diameter in mm and fy is the yield strength of the rebar in MPa. Previously, Alam et al. (2008a), O’Brien et al. (2007) showed that the Paulay and Priestley (1992) equation can reasonably estimate the plastic hinge length of SMA reinforced concrete element. Moreover, Saiidi and Wang (2006), Saiidi et al. (2009) and Cruz and Saiidi (2012) also used the Paulay and Priestley (1992) equation to calculate the plastic hinge length for their experimental studies where SMA rebars were placed in the bottom plastic hinge region of bridge piers. Therefore, the Paulay and Priestley (1992) expression for plastic hinge length calculation in SMA-RC elements can be considered reasonably accurate. 101   Figure 6.2. (a) Moment curvature relationship of RC sections with different types of SMAs and (b) Static pushover curves for bridge piers reinforced with different types of SMAs  Table 6.2. Material properties for SMA-RC bridge pier Material Property  Concrete Compressive Strength (MPa) 42.4 Corresponding strain 0.0029 Tensile strength (MPa) 3.5 Elastic modulus (GPa) 23.1 Steel Elastic modulus (GPa) 200 Yield stress (MPa) 475 Ultimate stress (MPa) 692 Ultimate strain  0.14 Plateau strain 0.016  6.3 Analytical Modeling of Bridge Piers In this study, a fiber element based nonlinear analysis program SeismoStruct (Seismosoft, 2014) has been employed to develop performance-based damage states for SMA-RC bridge piers. Incremental dynamic analyses (IDA) have been performed to determine the various damage states of the bridge piers. The program has the ability to determine the large displacement behaviour and the collapse load of framed structures accurately under either static or dynamic loading, while taking into account both geometric nonlinearities and material inelasticity (Pinho et al. 2007). The bridge piers were modelled with 3D inelastic beam–column 04008001200160020000 0.2 0.4 0.6 0.8 1 1.2Base Shear (kN)Displacement (m)SMA-1 SMA-2SMA-3 SMA-4SMA-50200040006000800010000120001400016000180000 0.02 0.04 0.06Moment (kN-m)Curvature (1/m)SMA-1SMA-2SMA-3SMA-4SMA-5102  element (force based element), with circular section for the piers; the constitutive laws of the reinforcing steel and concrete were, respectively, the Menegotto–Pinto (1973) and Mander et al. (1988) models. The superelastic SMA model developed by Auricchio and Sacco (1997) has been employed for modeling SMAs using the parameters provided in Table 6.1.  The accuracy of the program in predicting the strain and curvature response of bridge piers has been demonstrated in previous chapter (Chapter 5). However, this chapter shows the accuracy of the program in predicting the structural response under reverse cyclic loading with two different SMAs. Figure 6.3 shows the comparison of experimental and analytical results from two different studies using two different SMAs. Figure 6.3a shows the comparison of shake table test results and analytical results of a SMA-steel RC bridge pier where SMA was particularly used in the plastic hinge region. The numerical results obtained from SeismoStruct could predict the experimental result of Saiidi and Wang (2006) accurately where the variations were only 5.6%, 6.1%, and 9.4% for base shear, tip displacement, and amount of energy dissipation, respectively. Figure 6.3b shows the load-rotation response of concrete beam reinforced with Cu-Al-Mn SMA (SMA-4) in the mid span under four point reverse cyclic loading (Shrestha et al. 2013). From this figure it is evident that the adopted analytical model was capable of predicting the experimental response very well where the variations were only 3.4% and 5.9% for maximum force and beam rotation, respectively.  Figure 6.3. Comparison of experimental and numerical results (a) SMA-RC (SMA-1) bridge pier (b) SMA-RC (SMA-4) beam -20-15-10-505101520-0.02 -0.01 0 0.01 0.02Force (kN)Rotation (rad)103  6.4 IDA- Based Approach for Developing Performance-Based Damage States For successful implementation of the performance-based design concept in SMA-RC bridge pier, the performance objectives and their corresponding damage state criteria need to be clearly defined. Extensive experimental investigations on bridge piers performed in the past were utilized to develop the damage states for reinforced concrete bridge piers (Berry and Eberhard, 2003; Hose et al. 2000). Due to the fact that very limited experimental results are available for SMA reinforced bridge pier, an IDA-based approach, as illustrated in Figure 6.4, was developed in this study to generate the necessary data used to develop performance-based damage states for bridge pier reinforced with different types of SMAs.  Incremental dynamic analysis (IDA) (Vamvatsikos and Cornell, 2002) was employed to determine the performance limit states of different bridge piers using an ensemble of ten selected ground motions. IDA is a useful method for more detailed seismic performance predictions of structures subjected to different seismic excitation levels. In IDA, the finite element model is subjected to numerous inelastic time history analyses using one or a set of ground motion record(s), each scaled (up and/or down) to study different seismic intensity levels while tracking the response of the structure (e.g., displacements, accelerations, etc.). This procedure of scaling and time history analysis is repeated until dynamic instability in the form of large drifts occurs, indicating structural collapse.  6.4.1 Selection of ground motions The incremental dynamic analyses were carried out using the 10 selected ground motions as shown in Table 6.3. These ground motion records were obtained from the PEER (2011) ground motion database. These accelerograms were chosen such that they represent the seismic characteristics of the site of the structure. The ratio between the peak ground acceleration (PGA) and peak ground velocity (PGV) is an indicator of the frequency content of seismic motion. The characteristic seismic motions for the western part of Canada have a PGA/PGV ratio around 1.0 (Naumoski et al. 1988). The selected ensemble of earthquake records is presented in Table 6.3 where the PGA/PGV ratio varies between 0.8 and 1.3.   104   Figure 6.4.  Flowchart for the development of performance based damage states for SMA-RC bridge pier       105  Table 6.3. Selected earthquake ground motion records No Event Year Record Station M*1 R*2 (km) PGA (g) PGA/PGV 1 Imperial Valley 1979 El Centro Array#11 6.5 21.9 0.36 0.8 2 Imperial Valley 1979 Chihuahua 6.5 28.7 0.254 0.84 3 Kobe 1995 Takatori 6.9 4.3 0.56 0.9 4 Kobe 1995 JMA 6.9 3.4 0.77 1.02 5 Loma Prieta 1989 Holister South & Pine 6.9 28.8 0.371 0.97 6 Loma Prieta 1989 16 LGPC 6.9 16.9 0.605 1.19 7 Nothridge 1994 Rinaldi 6.7 7.5 0.87 0.93 8 Nothridge 1979 Olive View 6.7 6.4 0.721 0.95 9 Superstition Hill 1987 Wildlife liquefaction array 6.7 24.4 0.134 1.0 10 Superstition Hill 1987 Wildlife liquefaction array 6.7 24.4 0.132 1.03 1Moment Magnitudes, 2Closest Distances to Fault Rupture Source: PEER Strong Motion Database, http://peer.berkeley.edu/svbin  These 10 ground motion records were obtained from the PEER strong motion database. The recent edition of Canadian Highway Bridge Design Code (CSA-S6-14) requires that highway bridges should meet target performance levels under seismic ground motions with different return periods. In this study, three different levels of seismic ground motions were considered according to CHBDC 2014 (CSA-S6-14). These records correspond to three different hazard levels with a 2%, 5%, and 10% probability of exceedance in 50 years. The respective return periods are 2475 years, 975 years, and 475 years. For each hazard level 10 ground motions shown in Table 6.3 were used. The selected ground motions were scaled to specific hazard levels using SeismoMatch (Seismosoft 2013). This software is able to adjust any ground motion accelerograms to match a specific design response spectrum using wavelet algorithm proposed by Abrahamson (1992) and Hancock et al. (2006). Matching was done with in the period range of interest which was 0.05 sec to 4 sec as suggested by Baker et al. (2011). The mean spectra and the target spectra corresponding to different hazard levels are shown in Figure 6.5.  106   Figure 6.5. Design and mean response spectrum of 10 records used for IDA analysis matching the three different CHBDC spectrum (2%, 5%, and 10% in 50 years)  6.4.2 Performance-based damage states criterion Performance-based seismic design largely relies on the identification and selection of proper limit/damage states. Often damage states are defined in terms of drift or displacement. Damages are usually defined as discrete observable damage states (e.g., rebar yielding, concrete spalling, longitudinal bar buckling, bar fracture) (Marsh and Stringer 2013).  In this study, four quantitative performance limit states were defined for the SMA-RC bridge piers based on the performance levels and damage states proposed by Hose et al. (2000). Table 6.4 shows the four performance limit states and their associated functional level definition adopted in this study.  The performance limit states considered here are, the drift (%) at the onset of hairline cracks, longitudinal rebar yielding, cover concrete spalling, and crushing of core concrete. In this study, a strain based damage detection approach was used for defining the drift levels at different damage states. The yielding of SMA rebar was monitored by defining the yield strain of SMA bar and tracking the occurrence of first yield in SMA rebar. The spalling strain was assumed to be 0.004 as suggested by Priestley et al. (1996). Paulay and Priestley (1992) found that the crushing strain of confined concrete ranges between 0.015 and 0.05. In this study, the crushing strain of confined concrete for different SMA-RC bridge piers was calculated using the Paulay and Priestley (1992) equation: 00.10.20.30.40.50.60.70.80.910 1 2 3 4Spectral Acceleration (g)Time (sec)2%/50 Year (Target)5%/50 Year (Target)10%/50Year (Target)2%/50 Year (Mean)5%/50 Year (Mean)10%/50 Year (Mean)107  //4.1004.0 csmyhscu ff ερε +=         (6.1) where, εcu is the ultimate compression strain, εsm is the steel strain at maximum tensile stress, fc’ is the concrete compressive strength in MPa, fyh is the yield strength of transverse steel in MPa, and ρs is the volumetric ratio of confining steel.  Table 6.4. Proposed damage state framework Damage Parameter Damage State Functional Level  Description  Cracking DS-1 Immediate Onset of hairline cracks Yielding DS-2 Limited Theoretical first yield of longitudinal rebar Spalling DS-3 Service disruption Onset of concrete spalling Core Crushing DS-4 Life safety Crushing of core concrete  Most of the damage states available in literature are discrete in nature and quantifies the damage deterministically (Marsh and Stringer 2013). Practically, the drift level corresponding to certain damage is not a discrete deterministic quantity and each damage level is associated with a distribution of values. The drift limits defined at different damage states should clearly indicate whether it represents the lower bound, median, or some intermediate value for the onset of damage. In order to develop a comprehensive performance-based damage states, in this study, the probabilistic distribution of each damage state is also identified and the median of the distribution is defined as the drift limit corresponding to each damage state. In order to determine the limit state drift values for different performance levels, the drift limits corresponding to the strain values were determined using IDA for different hazard levels for the five different SMA-RC bridge piers. The drift limits at various performance levels were identified using the dynamic pushover curves obtained from IDA. Dynamic pushover curves represent the relation between maximum drift and corresponding base shear obtained from IDA while being subjected to an earthquake record (Elnashai and Luigi 2008). These curves represent the structural capacity under specific earthquake loading. Dynamic pushover curves, obtained from IDA, take into account progressive structural stiffness degradation, change of modal characteristics, and period elongation of the structure for increasing values of external action which is not achievable through static pushover analysis. Inelastic characteristics such 108  as strength degradation and energy dissipation largely affect the seismic performance of structures which are also required for developing performance-based damage states for performance-based design. The drift levels for different performance levels obtained from IDA were used to find a suitable distribution for each damage state that describes the statistical distribution of the developed damage state. Statistical analyses were carried out to find the most suitable probability density function (PDF) to represent the data related to each damage state. Using statistical tools and analysis, suitable distribution for each damage state were determined using goodness-of-fit tests. The following section discusses the development of performance-based damage states for five different SMA reinforced bridge piers. 6.4.3 Probabilistic distribution of drift based damage states Using the results obtained from IDA, the probabilistic distribution of each damage state corresponding to different hazard levels are determined to represent the statistical variability of damage states at different hazard levels. The probabilistic distribution of each damage state i.e. yielding, spalling, and crushing, is superimposed on the dynamic pushover curves obtained from IDA which are shown in Figures 6.6-6.10. The expected (median) drift level at a particular damage state is represented by the vertical solid line. From Figures 6.6-6.10 it can be observed that the uncertainty of each damage state is unique, as indicated by the dispersion or width of the distribution. The median drift level of each damage state is defined as the limiting drift value for each performance level. The drift levels corresponding to different damage states for different hazard levels are shown in Table 6.5. Table 6.5 also shows the probabilistic distribution of each damage state. From IDA, measurements of drift levels corresponding to each damage state were obtained and statistically processed to find out the most suitable distribution. The suitability of the selected distributions for representing each damage state was evaluated using Kolmogorov-Smirnov (K-S) goodness-of-fit test. Details of the K-S goodness-of-fit test and the results are presented in Appendix-B. The following conclusions are derived from the distribution of different damage states: • Irrespective of the type of SMAs and earthquake hazard level, cracking occurs at a drift of 0.28% and it can be represented better with a uniform distribution. Since the cracking strain of concrete depends only on the tensile strength of concrete, small variation in concrete cracking drift was observed. Uniform distribution is a preferable 109  one when all of the outcomes have an equal probability of occurring. Since the cracking drift of all the SMA-RC bridge piers ranged between 0.28% to 0.30% and have equal probability of occurrence, the cracking drift is assumed to follow a uniform probability distribution. Results of the K-S goodness-of-fit test also confirmed the suitability of uniform distribution for representing the distribution of crushing drift. This drift value of 0.28% can be identified as damage state-1(DS-1).  • From the statistical analysis it was found that the log-normal distribution better represents the uncertainty in drift limits for DS-2 (yielding). Usually the variation in metal strength, such as yield strength of steel is better represented by a log-normal distribution (Ellingwood, 1977, Ghobarah et al. 1998). Similar distribution for yield drift limits for SMA-RC bridge piers was obtained which is largely dependent on the yield strength of SMA. • Normal distribution was found to be the best fit for representing the variability in drift limits corresponding to DS-3 (spalling) based on K-S goodness-of-fit test. Normal distribution is better suited for representing the spalling drift since all the SMA-RC bridge piers showed a strong tendency towards the central value of spalling drift as well as the positive and negative deviations from this central value are equally likely. The selected distribution seems reasonable since concrete strength can be better represented by a normal distribution (Ellingwood, 1977; Mirza et al., 1979). • K-S goodness-of-fit test was performed to identify the most suitable distribution for defining the variation of DS-4 (crushing). The K-S goodness-of-fit test indicated that the gamma distribution, which usually indicates an extreme event, provides best fit to the data and was the most suitable for representing the crushing drift.  110   Table 6.5. Damage states of different SMA-RC bridge pier and their associated distribution   SMA-RC-1 SMA-RC-2 SMA-RC-3 SMA-RC-4 SMA-RC-5 Distribution Damage Parameter Damage State Drift (%) Drift (%) Drift (%) Drift (%) Drift (%) Probability of Exceedance Probability of Exceedance Probability of Exceedance Probability of Exceedance Probability of Exceedance 2% 50 5% 50 10% 50 2% 50 5% 50 10% 50 2% 50 5% 50 10% 50 2% 50 5% 50 10% 50 2% 50 5% 50 10% 50  Cracking DS-1 0.28 0.28 0.28 0.30 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 Uniform Yielding DS-2 1.68 1.76 1.86 1.66 1.72 1.80 2.28 2.42 2.58 1.74 1.83 1.95 1.10 1.16 1.21 Lognormal Spalling DS-3 2.66 2.79 2.88 2.69 2.77 2.87 1.64 1.72 1.80 2.52 2.61 2.68 1.97 2.02 2.10 Normal Crushing DS-4 5.05 5.68 5.94 5.51 5.91 6.05 7.65 7.81 7.94 5.56 5.63 5.72 4.73 4.79 4.84 Gamma    Figure 6.6. Dynamic pushover response and different damage states with distribution for SMA-RC-1 for (a) 2% in 50 years (b) 5% in 50 years and (c) 10% in 50 years probability of exceedance 0500100015002000250030000 2 4 6 8 10Base Shear (kN)Drift (%)SpallingYieldingCrushing(a)0500100015002000250030000 2 4 6 8 10Base Shear (kN)Drift (%)SpallingYieldingCrushing(b)0500100015002000250030000 2 4 6 8 10Base Shear (kN)Drift (%)SpallingYieldingCrushing(c)111    Figure 6.7. Dynamic pushover response and different damage states with distribution for SMA-RC-2 for (a) 2% in 50 years (b) 5% in 50 years and (c) 10% in 50 years probability of exceedance   Figure 6.8. Dynamic pushover response and different damage states with distribution for SMA-RC-3 for (a) 2% in 50 years (b) 5% in 50 years and (c) 10% in 50 years probability of exceedance    Figure 6.9. Dynamic pushover response and different damage states with distribution for SMA-RC-4 for (a) 2% in 50 years (b) 5% in 50 years and (c) 10% in 50 years probability of exceedance 0500100015002000250030000 2 4 6 8 10Base Shear ()kN)Drift (%)SpallingYieldingCrushing(a)0500100015002000250030000 2 4 6 8 10Base Shear (kN)Drift (%)SpallingYieldingCrushing(b)0500100015002000250030000 2 4 6 8 10Base Shear (kN)Drift (%)SpallingYieldingCrushing(c)05001000150020002500300035000 2 4 6 8 10 12Base Shear (kN)Drift (%)SpallingYieldingCrushing(a)05001000150020002500300035000 2 4 6 8 10 12Base Shear (kN)Drift (%)SpallingYieldingCrushing(b)05001000150020002500300035000 2 4 6 8 10 12Base Shear (kN)Drift (%)SpallingYieldingCrushing(c)0500100015002000250030000 2 4 6 8 10Base Shear (kN)Drift (%)SpallingYieldingCrushing(a)0500100015002000250030000 2 4 6 8 10Base Shear (kN)Drift (%)SpallingYieldingCrushing(b)0500100015002000250030000 2 4 6 8 10Base Shear (kN)Drift (%)SpallingYieldingCrushing(c)112    Figure 6.10. Dynamic pushover response and different damage states with distribution for SMA-RC-5 for (a) 2% in 50years (b) 5% in 50 years and (c) 10% in 50 years probability of exceedance 6.4.4 Maximum drift based damage states Figures 6.6-6.10 show the dynamic pushover curves for SMA-RC-1 through SMA-RC-5 under different levels of earthquakes, respectively. The dynamic pushover curves derived from 10 earthquakes (for each bridge pier) were statistically processed to obtain the median, 5 percentile, and 95 percentile capacity curves. Comparisons of Figures 6.6-6.10 reveal that: • SMA-RC-3 (Figure 6.8) has higher deformation and strength capacity as compared to the other SMA-RC bridge piers.  • For seismic hazard level of 2% in 50 years, the median capacity of SMA-RC-3 was 2743kN which was 16%, 15%, 20%, and 17% higher than that of SMA-RC-1, SMA-RC-2, SMA-RC-4, and SMA-RC-5, respectively.  • Maximum base shear demand is also significantly influenced by the earthquake hazard level. For example, the median maximum base shear of SMA-RC-1, for 2% in 50 years is 2305 kN which is 5% and 7% higher than that of 5% and 10% in 50 years records, respectively. Evaluation of the results presented in Table 6.5 provides a valuable insight on the damage states developed for different SMA-RC bridge piers. The damage states are defined for different hazard levels. From Table 6.5 it can be observed that: • Damage state-2 or yielding occurs at a drift level below 2% except for SMA-RC-3. At DS-2, there is significant variation in drift limits for different SMA-RC bridge piers. 0500100015002000250030000 2 4 6 8 10Base Shear (kN)Drift(%)SpallingYieldingCrushing(a)0500100015002000250030000 2 4 6 8 10Base Shear (kN)Drift (%)SpallingYieldingCrushing(c)0500100015002000250030000 2 4 6 8 10Base Shear (kN)Drift (%)SpallingYieldingCrushing(b)113  For SMA-RC-1 and SMA-RC-2, the drift limit is quite similar irrespective of the earthquake hazard levels which ranges between 1.68% to 1.86% and 1.66% to 1.80%, respectively. Since SMA-RC-1 and SMA-RC-2 are reinforced with Ni-Ti SMAs (different compositions) with similar mechanical properties, they tend to have similar drift limits at DS-2.  • Before yielding, SMA-RC-3 sustained higher drift compared to other SMA-RC piers. At DS-2, under 2% in 50 years hazard level, the drift limit for SMA-RC-3 was 2.28% which was significantly higher than the drift limits obtained for other SMA-RC bridge piers. This is expected since SMA-3 has higher yield strength and post yield stiffness as compared to the other SMAs considered in this study.  • Although SMA-4 has low elastic modulus, its yield strength is very high which eventually increased the yield strain and resulted in higher drift values. At DS-2, the drift limits for SMA-RC-4 were 3.4%, 3.8%, and 4.6% higher than that of SMA-RC-1 for 2%, 5%, and 10% in 50 years probability of exceedance, respectively.  • Exceptional performance was observed for SMA-RC-5 which yielded at a very low drift level (1.1%-1.2%) as compared to the other SMA-RC piers. This is due to SMA-5’s very low yield strength to elastic modulus ratio (0.0033), which reduced the drift capacity of SMA-RC-5.  • Although, both SMA-3 and SMA-5 are Fe-based, due to the variation in their yield strength and elastic modulus, the drift limits for SMA-RC-3 at DS-2 were 52% higher than that of SMA-RC-5 irrespective of the hazard level.  • From Table 6.5, it can be observed that except for SMA-RC-3, yielding occurred in all the bridge piers before the initiation of cover spalling. The delayed rebar yielding of SMA-RC-3 can be attributed to its higher yield strength and very high superelastic strain. A similar observation of the SMA-RC column has been reported by Saiidi and Wang (2006) where spalling of cover concrete took place before the initiation of SMA yielding. • DS-3, is considered at the onset of cover concrete spalling. All the piers experienced yielding before spalling where the only exception was SMA-RC-3. For SMA-RC-1, DS-3 occurred at a drift level of 2.66%, 2.79%, and 2.88% for 2%, 5%, and 10% in 50 years hazard level, respectively. This is expected since a hazard level with lower 114  probability indicates more damaging earthquake. Similar trend is also observed for the other SMA-RC bridge piers where the limiting drift value increased with decreased return period. In terms of drift limit, SMA-RC-1 and SMA-RC-2 performed better than the other three SMA-RC piers as they could sustain more drift before entering into DS-3.  • At DS-4 (crushing of concrete), all the SMA-RC bridge piers sustained more than 5% drift under various hazard levels whereas the SMA-RC-3 exceeded 7.5%. For a hazard level with 2% of probability of exceedance in 50 years, SMA-RC-3 sustained a drift of 7.65% before crushing which was 34%, 28%, 27% and 38% higher than that of SMA-RC-1, SMA-RC-2, SMA-RC-4, and SMA-RC-5, respectively.  • For a particular SMA-RC bridge pier, crushing drift also varied significantly at different hazard levels. In the case of SMA-RC-1, the crushing drift corresponding to 2% in 50 years hazard level is 11.5% and 15% lower than the crushing drift at 5% and 10% in 50 years hazard level, respectively.  However, in the case of SMA-RC-5, the crushing drift at 2% in 50 years hazard level was 2.3% and 1.25% lower than the crushing drift at 5% and 10% in 50 years hazard level, respectively.  The drift limits presented in Table 6.5 can be used for performance-based design of SMA-RC bridge pier. Based on the design earthquake scenario, the designer can define the target performance level and associated drift limits. Since, performance-based damage states are proposed for different types of SMA, the designer can select any particular SMA and design the bridge pier according to the owners expected performance level. 6.4.5 Residual drift based damage states for SMA-RC bridge piers Residual drift has been considered as one of the significant performance indicators in judging a structure’s post-earthquake safety and the economic feasibility for repairing (Ramirez and Miranda 2012). Although residual drift dictates the post-earthquake functionality of highway bridges, no other design guidelines except the Japanese code for highway bridge design (JRA 2006) provide any residual drift limit of bridge piers. In a recent study, Saiidi and Ardakani (2012) found that bridge piers meeting current seismic requirements can withstand larger traffic loads even when the residual drift is 1.2% or more. Lee and Billington (2011) considered 1% residual drift large enough for bridge replacement. In order to develop the 115  damage states (DS) for SMA-RC bridge pier a probabilistic approach has been adopted in this study. Based on the existing literature (O’Brien et al. 2007, Billah and Alam 2014c), four different damage states have been identified and a range of limiting residual drifts were considered. It was assumed that a residual drift below 0.25% would meet the serviceability requirement (DS-1) while a residual drift larger than 1% would be characterized as a collapse damage state (DS-4). The intermediate damage states DS-2 and DS-3 are assumed to take place at a residual drift larger than 0.5% and 0.75%, respectively. DS-1 requires that no structural realignment is necessary and the bridge is fully operational. DS-2 consists of minor structural repairing and requires the bridge to be operational without requiring bridge closure. A pier experiencing DS-3 will require major repair and may require bridge closure but should be usable for restricted emergency traffic after inspection. DS-4 corresponds to the case when the residual drift is sufficiently large that the structure is in danger of collapse from earthquake aftershocks. Once the damage sates have been identified, fragility curves for residual drifts were developed using the IDA results for three different seismic hazard levels. In this study, fragility functions were developed using Equation 6.2 which take the form of lognormal cumulative distribution functions having a median value of θ and logarithmic standard deviation or dispersion of β.  =βθφ/ln()(RDRDF         (6.2) where, F(RD) represents the conditional probability that the bridge pier will be damaged to a given DS as a function of the residual drift (RD); F denotes the standard normal cumulative distribution function; and θ and β are the median value of the probability distribution and the logarithmic standard deviation corresponding to the DS, respectively. Figure 6.11 shows the fragility curves for SMA-RC bridge piers for different damage states at three different hazard levels. Here, the fragility curves are plotted irrespective of the SMA types to generalize the associated damage states. Using these fragility curves, the residual drift based damage states for SMA-RC bridge pier have been developed. From the fragility curves corresponding to each damage state, the RD value with a 50% probability of occurrence 116  indicates the limiting value for the corresponding damage state. For example, in Figure 6.11a (10% in 50 years), the 50% probability of occurrence of DS-2 corresponds to a RD of 0.48% while the limiting RD values for DS-2 for 5% in 50 years and 2% in 50 years hazard level correspond to 0.55% and 0.62%, respectively. It can be observed that the limiting RD value for DS-2 was assumed to be 0.5% and the values obtained from the median probability of exceedance are quite close. Similarly, the limiting RD values with a 50% probability of occurrence at different damage states and hazard levels were developed as outlined in Table 6.6.   Figure 6.11. Fragility curves in terms of residual drift at (a) 10% in 50 years (b) 5% in 50 years and (c) 2% in 50 years probability of exceedance Table 6.6. Residual drift damage states of SMA-RC bridge pier  Damage State Functional Level Description Residual Drift, RΔ (%) Probability of Exceedance 10% in 50 5% in 50 2 % in 50 Slight (DS=1) Fully Operational No structural realignment is necessary 0.24 0.28 0.33 Moderate (DS=2) Operational Minor structural repairing is necessary  0.48 0.55 0.62 Extensive (DS=3) Life safety Major structural realignment is required to restore safety margin for lateral stability 0.73 0.82 0.87 Collapse  (DS=4) Collapse Residual drift is sufficiently large that the structure is in danger of collapse from earthquake aftershocks 1.04 1.16 1.22  00.10.20.30.40.50.60.70.80.910 0.5 1 1.5 2P (DS I RD)Residual Drift (%)(a)00.10.20.30.40.50.60.70.80.910 0.5 1 1.5 2P (DS I RD)Residual Drift (%)(b)00.10.20.30.40.50.60.70.80.910 0.5 1 1.5 2P (DS I RD)Residual Drift (%)(c)117  From Table 6.6 it can be observed that as the ground motion return period decreases (probability of occurrence increases) the limiting residual drift corresponding to different DS decreases. For example, at DS-4, the limiting drift value for an earthquake with 2475 years return period is 1.22% which is 6.5% and 13.1% higher than an earthquake with 975 and 475 years return period, respectively. Observation from Table 6.6 indicates that, as the damage level increases (DS-1 to DS-4) the difference in limiting RD values at different hazard levels decreases. For instance, at DS-2, the limiting RD value corresponding to 2475 years return period is 11% and 22.5% higher than that of 975 and 475 years return period, respectively. However, this difference goes down to 6.5% and 13.1% for DS-4. 6.5 Prediction of Residual Drift For performance-based design, prediction of residual drift as a function of the target or maximum drift would be very useful. Previous research has shown that residual drift predictions using non-linear analysis are highly variable and subjected to different modeling features (ATC-58). Recently ATC-58 (2012) recommended some general equations for predicting residual drift using peak transient drift and yield drift. ATC-58 (2012) suggested that, prediction of residual drift requires advanced non-linear simulation with careful attention to cyclic hysteretic response of the models and numerical accuracy of the solution. In this study, the residual drift responses were obtained using IDA which is one of the most advanced non-linear analysis techniques and the models were validated with experimental results. From the residual drift responses of the SMA-RC bridge piers it was found that, the residual drift in SMA-RC bridge pier is a function of maximum drift and superelastic strain of the SMA used. Using the residual drift response obtained from different SMA-RC bridge piers under a wide range of ground motions, a non-linear regression analysis was conducted to investigate the effect of maximum drift and superelastic strain on the residual drift response. Using a non-linear regression analysis the following equation is developed for predicting residual drift of SMA-RC bridge pier: +−=sss MDMDRDεεε 11001005.0 2       (6.3) Where, RD= residual drift (%), εs= superelastic strain, MD= maximum drift (%). 118  In order to investigate the accuracy of the proposed residual drift prediction equation, comparison was carried out with experimental results. Figure 6.12 shows the comparison of residual drifts obtained from experimental investigation and prediction equation. Figure 6.12a shows the comparison between the predicted responses and experimental results of O’Brien et al. (2007) where the SMA-RC bridge pier was tested under reverse cyclic loading. The bridge pier was constructed using Ni-Ti SMA in the plastic hinge region and the superelastic strain of SMA rebar was 6%. Maximum drift values and the corresponding residual drifts were obtained from experimental results. Using the maximum drift value and the superelastic strain of the SMA rebar, the residual drifts were predicted. Figure 6.12a shows that the proposed equation predicted the residual drift very well with an average absolute error (AAE) of 4.65% and average standard deviation of 0.03. Figure 6.12b shows the comparison of residual drift prediction with experimental results of Youssef et al. (2008) where Ni-Ti SMA with superelastic strain of 6% was used as reinforcement in the beam column joint. From Figure 6.12b it is evident that the proposed equation is capable of predicting the residual drift with reasonable accuracy with an AAE of 2.06% and average standard deviation of 0.015.   Figure 6.12. Comparison of residual drift prediction with experimental results (a) O’Brien et al. (2007) and (b) Youssef et al. (2008)  00.511.522.530 0.5 1 1.5 2 2.5 3Predicted RD(%)Experimental RD(%)(a)00.10.20.30.40.50 0.1 0.2 0.3 0.4 0.5Predicted RD(%)Experimental RD(%)(b)119  6.6 Summary Performance-based seismic design aims to dictate the structural performance in a predetermined fashion given the possible seismic hazard scenarios the structure is likely to experience. Identifying and assessing the probable performance is an integral part of performance-based design. Before implementation, accurate and practical definition of different performance levels and the corresponding limit states must be defined properly. This chapter aimed to develop performance-based damage states for shape memory alloy (SMA) reinforced concrete bridge piers considering different types of SMAs and seismic hazard scenarios. Using Incremental Dynamic Analysis (IDA), this chapter developed quantitative damage states corresponding to different performance levels (cracking, yielding, spalling and crushing) and specific probabilistic distributions for RC bridge piers reinforced with different types of SMAs. Based on extensive numerical study, this study also proposed residual drift based damage states for SMA-RC pier. Finally, an analytical expression is proposed to estimate the residual drift of SMA reinforced concrete elements as a function of the expected maximum drift and superelastic strain of SMA. Comparison with experimental results revealed that the proposed equation could very well predict the residual drift obtained from the experimental results.       120  CHAPTER 7. PERFORMANCE-BASED SEISMIC DESIGN OF SHAPE MEMORY ALLOY (SMA) REINFORCED CONCRETE BRIDGE PIER: METHODOLOGY AND DESIGN EXAMPLE 7.1 General The current bridge design specifications in North America (CSA-S6-10, AASHTO LRFD 2012) and Europe (EC8-2) follow the well-established force-based design methodology. However, these prescriptive design methodologies rarely relate the seismic performance of bridges to the important design parameters. With the existing specifications, the designer has little control over the expected seismic performance of the bridge (Marsh and Stringer 2013). In the last decade, seismic design of bridges has transitioned from the conventional force-based method towards more descriptive performance-based seismic design (PBSD) approach with an aim of limiting the global and local deformations of a structure to acceptable levels under design earthquakes (Priestley et al. 2007). The advancement of PBSD allows the designer and owner to interact by selecting a desired performance level and instill the expected performance in the design process. With the development of PBSD, the bridge design guidelines in North America are moving forward to performance-based design. The unique features of PBSD allow the designer to consider different seismic hazard levels along with different functional classifications (Marsh and Stringer 2013). However, most of the PBSD approaches available in literature are based on the direct displacement based design (DDBD) approach developed by Priestley et al. (2007) where a structure is designed for a target maximum displacement under a specified design earthquake. It is well known that the PBSD procedure emphasize the determination of target drift for a selected performance level. However, observation from recent earthquakes and research results have evidenced that residual drift sustained by a structure after an earthquake plays a significant role in defining the seismic performance of a structure and needs to be considered in the seismic design (Christopoulos et al. 2008, Ruiz-Garcia and Miranda 2010, Erochko et al. 2011, Billah and Alam 2014c). In this chapter, a performance-based seismic design methodology has been developed for shape memory alloy (SMA) reinforced concrete (RC) bridge pier considering residual drift as the performance indicator. 121  The response of SMA-RC bridge pier is significantly different from conventional piers due its inherent recentering ability. Moreover, equivalent viscous damping is an essential parameter that affects the behavior of a structural system under seismic excitations (Dawood and Elgawady 2013). Previous researchers (DesRoches et al. 2004, Roh et al. 2012) have shown that the hysteretic damping of SMA rebar is different from conventional steel rebar. This study also developed the damping-ductility relationship for SMA-RC bridge piers in support of the proposed PBSD of SMA-RC pier. Details of the different types of SMAs and the material characteristics can be found in previous chapter (Chapter 6).  This chapter shows a step by step procedure, with useful flow charts and graphs, for designing SMA-RC bridge pier along with a design example. The ability of the designed bridge pier (in the trial application), to meet the performance objectives, has been evaluated by performing nonlinear dynamic time history analyses using ten ground motions. The following section introduces the proposed methodology and step by step description. The subsequent sections provide a detailed design example and seismic performance evaluation of the SMA-RC pier. 7.2 Performance-Based Design of SMA Reinforced Bridge Pier The performance-based design of SMA-RC bridge pier is developed following a displacement-based approach. Unlike other displacement-based approach, the required design base shear is calculated corresponding to a target residual drift and target performance level corresponding to a selected seismic hazard. The procedure adopted in this study follows the procedure developed by Kowalsky et al. (1995) and Priestley et al. (2007), but is specifically tailored to SMA- RC bridge pier using the damping-ductility relationship developed in this study. The design steps adopted in this study are outlined in a simple flowchart in Figure 7.1. 7.2.1 Step 1: Define seismic hazard Performance-based seismic design (PBSD) explicitly evaluates the probable structural performance given the potential hazard it is likely to experience (FEMA 445, 2006). Since the seismic hazard level changes in different parts of a country, a site specific seismic hazard level must be defined as the starting point of PBSD. The seismic hazard level, which is usually expressed as a probability of exceedance in certain number of years or return period, can play a significant role in PBSD. For example, the CALTRANS Seismic Design Criteria (Caltrans 122  2010a) specifies a maximum hazard level of 5% in 50-years seismic event (975-years return period) while the Japan Road Association (JRA 2006) defines two levels of seismic hazard, Type-I and Type-II. In Eurocode 8, Part 2-Seismic Design of Bridges (EC8-2, 2008), usually a single-level seismic hazard level is considered which corresponds to a 475-years return period or a ground motion with 10% probability of exceedance in 50 years. However, both AASHTO LRFD Bridge Design Specifications (AASHTO 2012) and AASHTO Guide Specifications for Seismic Bridge Design (AASHTO 2011) suggest a single seismic hazard level which corresponds to 7% probability of exceedance in 75 years (i.e., 1000 years return period). Previous Canadian Highway Bridge Design Code (CSA-S6-10) specified the hazard level with a 10% probability of exceedance in 50 years while the recent edition of CHBDC (CSA-S6-14) requires that bridges should not collapse when subjected to earthquakes with 2% probability of exceedance in 50 years. The recent CHBDC 2014 (CSA-S6-14) defines acceptable levels of performance corresponding to different hazard levels. In this study, the seismic hazard levels proposed in CHBDC 2014 (CSA-S6-14) are considered. 7.2.2 Step-2: Define target residual drift The second step involves defining the target residual drift based on the selected target performance level and seismic hazard level. In order to ensure an acceptable post-earthquake functionality of the bridge pier, the residual drift for the specified earthquake hazard level must not exceed the target residual drift of the pier, which can be established based on the existing literature or experimental study. As a part of this study, concrete bridge piers reinforced with five different types of SMAs were extensively analyzed under an ensemble of ground motion to establish different performance levels corresponding to different seismic hazard levels. Details of the procedure and residual drifts limits can be found in the previous chapter (Chapter 6). 7.2.3 Step-3: Calculate maximum drift based on target residual drift Step 3 in the flowchart for PBSD of SMA-RC bridge pier focuses on the calculation of maximum drift based on the residual drift. In Chapter 6 Equation 7.1 was proposed from which the maximum drift can be calculated for a given residual drift.  sss MDMDRDεεε 11001005.0 2 + ×− ×=       (7.1) 123  where, RD= target residual drift (%), εs= superelastic strain of the SMA, MD= maximum drift (%).   Figure 7.1. Flow diagram of PBSD of SMA-RC bridge pier From Equation 7.1, it can be seen that, in order to calculate the maximum drift based on the target residual drift, the designer needs to select the superelastic strain of the SMA. This step is critical since decision needs to be made on the selection of SMA since different SMAs                                  SMA εs  (%) Af (°C) NiTi45 6 -10 NiTi45 8 - FeNCATB 13.5 -62 CuAlMn 9 -39 FeMnAlNi 6.13 -50 Performance Level Residual Drift (%) Probability of exceedance in 50 years 2%  5%  10%  Full Operation 0.24 0.28 0.33 Operational 0.48 0.55 0.62 Life safety 0.73 0.82 0.87 Collapse 1.04 1.16 1.22   SMA-1 SMA-2 SMA-3 Damage Parameter Drift (%) Drift (%) Drift (%) Cracking 0.28 0.30 0.28 Yielding 1.68 1.66 2.28 Spalling 2.66 2.69 1.64 Crushing 5.05 5.51 7.65 Define site location and seismic hazard  Select performance level and target residual drift (RD) Select SMA and calculate maximum drift (∆m)    Select initial column parameters  Determine equivalent damping (ξeq)  Determine equivalent time period (Teff) Determine effective stiffness Determine design base shear Determine design moment Verify target RD and MD    Acceptable Complete structural detailing Not Acceptable Design bridge pier  Verify shear and moment capacity  Select yield drift and calculate ductility demand, ym ∆∆= /µ   124  have different range of superelastic strain. Moreover, the performance of the bridge pier is also correlated to maximum drift since this drift value is well correlated to the structural damage of the bridge pier as well as it is a kinematic value directly available from the analysis and/or design process. To ensure satisfactory behavior in a major earthquake, the maximum drift expected to occur in the SMA-RC pier should not exceed the superelsatic strain limit of the SMA. 7.2.4 Step-4: Select initial parameters Choose initial design parameters: height (H) and diameter (D) of the column, mass of the superstructure (M), material properties of the SMA, concrete, and steel reinforcement. 7.2.5 Step-5: Calculate expected ductility demand This step involves selection of the target yield drift based on the selected seismic hazard level for calculating the expected ductility demand. Priestley et al. (2007) proposed equations for calculating the yield curvature and yield displacement of circular RC pier for calculating the expected ductility demand. Priestley et al. (2007) concluded that the yield curvature φy can be calculated using Equation 7.2. Dyyεφ 25.2=           (7.2) Where, εy is the yield strain of the flexural reinforcement and D is the diameter of the section. The yield displacement ∆y can be calculated using Equation 7.3, where α is equal to 1/3 for a cantilever column.  2Hyy αφ=∆           (7.3) Since these equations were developed for regular steel-RC bridge piers, application of these equations for SMA-RC bridge piers is questionable. Moreover, the ductility demand calculated using these equations does not correlate with the selected seismic hazard level. Chapter 6 developed yield drift limits for different SMA-RC bridge pier that correspond to different seismic hazard levels. Based on the seismic hazard level, the target yield drift (∆𝑦𝑦𝑇𝑇) can be selected, which can be used for calculating the expected ductility demand (𝜇𝜇𝑑𝑑) using the following equation: 125  ytmd ∆∆=µ           (7.4) 7.2.6 Step-6: Determine equivalent hysteretic damping Establishing damping-ductility relationship is an important step for the performance-based design of SMA-RC bridge pier.  The unique response of such a bridge pier under seismic loading warrants a completely different damping-ductility relationship which is unlikely to match with traditional steel reinforced bridge pier or post-tensioned bridge pier. The hysteretic response of SMA-RC pier is expected to be similar to flag shaped hysteresis. Several researchers have proposed equations for calculation the equivalent damping of flag shaped hysteresis. For example, Priestely et al. (2007) and Dwairi et al. (2007) proposed Equations 7.5 and 7.6, respectively for flag shaped hysteresis: Priestley Equation for flag shape:  −+=µπµξ 1186.005.0eq    (7.5) Dwairi Equation for flag shape:  −+=µπµξ 1305eq      (7.6) However, no researchers have investigated the damping-ductility relationship for SMA-RC bridge pier. Hence, this study established the damping-ductility relationship for concrete bridge piers reinforced with SMA rebar in its plastic hinge region. The damping-ductility relationship was generated using large number of real ground motions following the method described by Dwairi et al. (2007). In order to develop the damping-ductility relationship comprehensively, five different bridge piers reinforced with five different types of SMAs were selected as described in previous chapter (Chapter 6). A total of 100 ATC55/FEMA440 ground motions (Miranda, 2003) were used for each bridge pier (Table 7.1). Using the results obtained from each nonlinear time history analysis (NLTHA), the ductility demand and corresponding damping value was obtained which provided a single point in the damping-ductility curve. For each SMA-RC pier a series of 100 damping-ductility points were obtained which are shown as dots in Figure 7.2a-e. For each set of points, nonlinear regression analyses were carried out to establish the damping-ductility curves for the SMA-RC piers (shown as solid lines in Figure 7.2a-e). A set of new damping-ductility equations, in 126  accordance with the previous expressions developed by other researchers (Priestely et al. 2007, Dwairi et al. 2007), were developed in order to best approximate the damping-ductility relationship. Equation 7.7 represents the general form of the proposed equivalent viscous damping equation based on ductility for the SMA-RC bridge pier: −+=beqaµπξξ 110          (7.7) In this equation a and b are the two regression coefficients and µ is the ductility demand. The equivalent damping (ξeq) is the sum of two contributions: the nominal viscous damping ratio, ξ0, normally taken as 5% for all types of structures, and the hysteretic damping, which depends on the dissipative capacity of a structure (Priestley et al. 2007). In order to obtain a generic damping-ductility relationship for SMA-RC bridge pier, all the examined bridge piers were considered together and the following expression was developed for the SMA-RC bridge pier: −+=56.011325µπξeq         (7.8) Table 7.1. ATC55/FEMA440 earthquake ground motions* (Miranda, 2003) Date  Earthquake Name Magnitude (Mw) 02/09/1971 San Fernando 6.5 10/15/1979 Imperial Valley 6.8 04/24/1984 Morgan Hill 6.1 07/08/1986 Palm Springs 6.0 10/01/1987 Whittier 6.1 10/17/1989 Loma Prieta 7.1 03/13/1992 Erzican, Turkey 6.9 06/28/1992 Landers 7.5 01/17/1994 Northridge 6.8 01/16/1995 Kobe 6.9 11/12/1999 Duzce, Turkey 7.8 08/17/1999 Kocaeli, Turkey 7.8 *Source: PEER ground motion database  127  The coefficient of determination or R2 value obtained from this expression was higher than 85%. However, the developed relationship is limited to SMA-RC piers having a flexure mode of failure and affected by the adopted ground motions. For the expected ductility demand (calculated in step-5), based on the target drift, the equivalent viscous damping for SMA-RC pier for the selected seismic hazard level can then be determined using the proposed equation.   Figure 7.2. Damping-Ductility relation for SMA-RC bridge pier (a) SMA-1, (b) SMA-2, (c) SMA-3, (d) SMA-4 and (e) SMA-5 Figure 7.3 shows the equivalent viscous damping and ductility curve developed in this study along with the curves proposed by Priestely et al. (2007) and Dwairi et al. (2007) for flag shaped hysteresis. From this figure it can be observed that, the proposed relationship is in well accordance with the existing literature.  Previous researchers (DesRoches et al. 2004, Roh et al. 2012) have shown the hysteretic damping of large diameter of superelastic SMA bars ranges between 2%-7% under dynamic loading. Similar observation was also found in this study. 7.2.7 Step 7: Determine effective time period (Teff) Knowing the maximum displacement (Δm) for the equivalent SDOF of the bridge pier and the equivalent viscous damping (ξeq), the effective time period (Teff) of the pier can be obtained using the displacement response spectrum of the site under consideration at the selected hazard level. The acceleration response spectrum at the selected hazard level can be 02468101214161 2 3 4 5 6Equivalent Damping (%)Ductility(a)02468101214161 2 3 4 5 6Equivalent Damping (%)Ductility(b)02468101214161 2 3 4 5 6Equivalent Damping (%)Ductility(c)02468101214161 2 3 4 5 6Equivalent Damping (%)Ductility(d)02468101214161 2 3 4 5 6Equivalent Damping (%)Ductility(e)128  transformed into the corresponding displacement response spectrum using the following relationship. 224πeffadgTSS =           (7.9) where, Sd is the spectral displacement, Sa is the spectral acceleration, g is the acceleration due to gravity and Teff is the effective time period. Spectral accelerations in the design codes typically represent equivalent viscous damping equal to 5% of critical damping. In order to convert the 5% damped response spectrum to the target damping value obtained in previous step (step-6), a modification factor (Rξ) needs to be determined using the following equation adopted in Eurocode-8 (EC8-2, 2008). 5.005.010.0+=ξξR          (7.10) Using this modification factor the modified displacement spectrum can be calculated using the following equation: ξξ RSS dd ×= %5,,          (7.11)   Figure 7.3. Comparison of Damping-Ductility curve 024681012141 2 3 4 5 6Equivalent Damping (%)DuctilityPriestley-flag shapedDwairi and Kowalsky-flag shapedSMA-RC pier129   7.2.8 Step 8: Determine effective stiffness (Keff) The effective stiffness (Keff) based on the effective period (Teff) is calculated as: 224effeeff TMKπ=         (7.12) Where Me is the effective mass of the pier. 7.2.9 Step 9: Compute design base shear (Vbase) and design moment (Md) Using the relationship between effective stiffness and design displacement, the design base shear can be calculated using the following equation: meffbase KV ∆=          (7.13) Determine the design moment using the following relation:                     (7.14) 7.2.10 Step 10: Design the bridge pier 7.2.10.1 Design longitudinal reinforcement The required longitudinal reinforcement can be calculated based on the design moment and axial load ratio using moment curvature analysis or using design requirement of relevant bridge design codes (i.e. AASHTO, CHBDC). The longitudinal steel ratios should be between 0.7% and 4% to comply with the common design practice. 7.2.10.1.1 Design transverse reinforcement In order to satisfy the confinement and shear strength requirements, the transverse reinforcement needs to be designed properly. Confinement requirements can be obtained from the required displacement ductility as described by Kowalsky et al. (1995) or using design requirement of relevant bridge design codes (i.e. AASHTO, CHBDC). 7.2.10.1.2 Check shear strength requirement The shear strength of the column must be checked to ensure that the shear capacity is greater than the shear demand calculated in step-9. The shear capacity of the pier can be LVM based ×=130  checked using modified compression field theory (Vecchio and Collins, 1986) or modified UCSD shear model (Kowalsky and Priestley, 2000). If the shear strength does not satisfy the requirement, the transverse reinforcement ratio should be revised. 7.3 Illustrative example The following example is presented to demonstrate the performance-based design procedure for SMA-RC bridge pier.  The bridge pier is assumed to be located at Vancouver, BC with site soil class-C (stiff soil). The corresponding design spectrum is selected according to CHBDC-2014 (CSA S6-14) which corresponds to 2% probability of exceedance in 50 years with a return period of 2475 years (Figure 7.4).  Figure 7.4. Design Acceleration Response Spectrum The considered bridge is a lifeline bridge and according to CHBDC-2014 (CSA-S6-14) performance requirement, the bridge should be operational with limited service at the selected seismic hazard level. For the considered damage level, a target residual drift of 0.6% is selected to meet the performance objective. To restrict the residual drift within the target level, a Nitinol shape memory alloy with 6% superelastic strain is selected. The maximum drift based on the target residual drift and selected SMA is calculated using Equation 7.1: 00.10.20.30.40.50.60.70.80.910 1 2 3 4Spectral Acceleration (g)Time (sec)Design ResponseSpectrum131  sss MDMDRDεεε 11001005.0 2 + ×− ×=  or, 61100610065.06.0 2 + ×− ×= MDMD  Solving this quadratic equation we get, maximum drift, MD= 4.92% Maximum displacement, Δm= 0.0492 × 5= 0.246 m  Initial column parameters: Height of the pier = 5m Lumped mass at the top of pier = 500,000 kg Selected material properties of concrete, steel, and SMA are provided in Table 7.2.  Table 7.2. Material Properties Material Property  Concrete Compressive Strength (MPa) 42.4 Elastic modulus (GPa) 23.1 Steel Elastic modulus (GPa) 200 Yield stress (MPa) 400 Ultimate stress (MPa) 672 SMA Modulus of Elasticity (GPa) 58.8 Austenite-to-martensite starting stress (MPa) 401 Austenite-to-martensite finishing stress (MPa) 510 Martensite-to-austenite starting stress (MPa) 370 Martensite-to-austenite finishing stress (MPa) 130 Superelastic strain (%) 6.0  For the considered hazard level, the yield drift is selected as 1.68% as developed in the previous chapter (Chapter 6). Yield displacement, ΔyT = 0.0168 × 5= 0.084 m Ductility demand, μd = 93.2084.0246.0==∆∆Tmy 132  Equivalent viscous damping value corresponding to the design ductility is calculated using the following damping-ductility relation developed in this study: −+=56.011325µπξeq = %6.993.21132556.0= −+π The spectral reduction factor (Rξ) is calculated as: 83.0096.005.010.005.010.05.05.0=+=+=ξξR  Using the spectral reduction factor (Rξ) of 0.83, the displacement spectrum corresponding to 9.6% damping is obtained using Equation 7.5 which is shown in Figure 7.5. Figure 7.5 shows the 5% damped displacement spectrum and the reduced displacement spectrum. With this reduced displacement spectrum and the maximum displacement, Δm, the effective time period of the pier (Teff) is calculated as 3.42 sec.  Figure 7.5. Determination of effective period from reduced displacement spectrum  The effective stiffness (Keff) based on the effective period (Teff) is calculated as:  mMNTMKeffeff /68.142.3500000442222=×==ππ 00.050.10.150.20.250.30.350.40 1 2 3 4 5Displacement (m)Time (sec)5%9.6%133  The design base shear is calculated as: kNKV meffbase 3.413246.01068.16 =××=∆=  The design moment is calculated as: mkNLVM based −=×=×= 5.206653.413  Finally, for the design moment of 2066.5 kN-m, the column section is designed according to CHBDC 2014 (CSA-S6-14) considering a column diameter of 1 m. For the design moment a longitudinal steel ratio of 1.73% is obtained which is provided using 28-25M SMA rebar (24.9 mm diameter) in the plastic hinge region and 28-25M steel (diameter 25.2 mm) rebar in the remaining portion. The shear reinforcement was design following CHBDC 2014 (CSA-S6-14) seismic design requirements which yielded 15M spirals at 50mm pitch providing a spiral reinforcement ratio of 1.49%. The shear capacity of the column is checked using modified compression field theory (Vecchio, and Collins,1986)  which predicts the experimentally determined shear failure within 1% error (Bentz et al. 2006). The shear resistance of the pier is found to be 2264 kN which is much higher than the applied shear force. Figure 7.6a shows the moment-shear force interaction diagram of the designed pier. From Figure 7.6a, it is evident that the maximum moment and shear force are within the safe region. Wang et al. (2008) recommended that the shear capacity of the pier should be greater than 1.6 times the base shear corresponding to the design moment which has also been satisfied.  04008001200160020000 1000 2000 3000Shear (kN)Moment (kN-m)(a)-500005000100001500020000250000 1000 2000 3000 4000 5000Axial Load (kN)Moment (kN-m)(b)134  Figure 7.6. (a) Moment-Shear force interaction diagram and (b) Moment-Axial Load interaction diagram  The axial load versus moment interaction diagram of the designed pier is developed as shown in Figure 7.6b. From the interaction diagram, it is observed that, the applied maximum axial load and moment are within the safe boundary. 7.4 Bridge Pier Performance Evaluation  In order to validate the proposed design approach, the performance of the designed bridge pier is evaluated using NLTHA with ten earthquake records. The bridge pier was modeled in Seismostruct (Seismosoft 2014), a fiber based finite element software. The bridge piers were modelled through a 3D inelastic beam–column element (force based element), with a circular section for the piers; the constitutive laws of the reinforcing steel and of the concrete were, respectively, the Menegotto–Pinto (1973) and Mander et al. (1988) models. The superelastic SMA model developed by Auricchio and Sacco (1997) has been employed for modeling SMA. The objective of this evaluation is to compare the performance objectives (residual drifts and maximum drifts) with the predicted performance under the ensemble of selected ground motions. The selected ground motions were first scaled to match the displacement response spectrum of the location of the bridge pier (Figure 7.7). The results of the analyses in terms of maximum and residual drifts are presented in Figure 7.8a and b, respectively. These figures show the maximum and residual drift response obtained from each nonlinear time history analysis and also the target maximum and residual drift (horizontal line) used in the design. 135   Figure 7.7. Displacement spectra of ten earthquake records matched with target response spectrum  05101520253035400 1 2 3 4Spectral Displacement (cm)Time (sec)Target SpectraChiChiFruiliHollisterImperial ValleyKobeKocaeliLandersLoma PrietaNorthridgeTrinidad136   Figure 7.8. (a) Maximum and (b) residual drift value obtained from time history analysis of the designed pier (Red line showing the target maximum and residual drift) From these figures it is evident that the bridge pier sustained maximum and residual drifts within 15% of the target maximum and residual drift. It was found out from the analysis that among ten earthquake records, two exceeded the target residual drift of 0.6% and maximum drift of 4.92%. The remaining eight are below the design level residual drift and targeted maximum drift. These discrepancies can be attributed to the linearization of the displacement spectrum adopted during the design and the scaling of ground motions. However, the average response in terms of both residual and maximum drifts was very close to the 0123456Maximum Drift (%)(a)00.10.20.30.40.50.60.7Residual Drift (%)(b)137  targeted drift levels. Previous researchers (Kowalsky et al. 1995, Priestley et al. 2007, Haque and Alam 2013) also observed similar differences when NLTHA carried out on structures designed following displacement-based approach. Priestley et al. (2007) suggested that the differences in the target drift and obtained drift from NLTHA is acceptable if the mean of the peak drifts remains close to the design drift. 7.5 Summary In this chapter, a performance-based seismic design method is presented for shape memory alloy (SMA) reinforced concrete (RC) bridge pier. The proposed design method is developed based on the existing displacement-based procedure where the expected performance is quantified by linking material strains and deformations to damage states and to the probable post-earthquake functionality of a bridge. Based on the performance-based damage states developed in chapter 6, this chapter presents the sequential procedure for the performance-based design of SMA-RC bridge pier using a combination of residual and maximum drift. Here, guidelines are provided to determine the target residual drift, which is correlated to the target drift/ductility. From the effective damping-ductility relationship developed in this study for the SMA-RC bridge pier, the time period of the structure is calculated based on target ductility. The proposed design framework is used for designing a trial SMA-RC bridge pier. The SMA-RC pier designed using the presented procedure was subjected to nonlinear time history analyses using a suite of selected earthquake records. The nonlinear analyses showed that the designed pier behave according to design expectations and provided very promising results in terms of the effectiveness and applicability of the proposed design method.      138  CHAPTER 8. PROBABILISTIC SEISMIC RISK ASSESSMENT OF CONCRETE BRIDGE PIERS REINFORCED WITH DIFFERENT TYPES OF SHAPE MEMORY ALLOYS 8.1 General Current seismic design guidelines, followed in North America (CHBDC 2014, AASHTO LRFD 2012) and Europe (EC8-2), allow bridges other than life line bridges to undergo large inelastic deformation while maintaining the load carrying capacity without being completely collapsed during a design level earthquake. However, past experiences (Kobe 1995, Northridge 1994) have shown that bridges undergoing large lateral drift are prone to large residual deformation which renders the bridges to be unusable and require major rehabilitation or replacement. In order to maintain the structural integrity and functionality of a bridge after an earthquake, it is necessary that the bridge components avoid excessive residual deformation or permanent damage (Kawashima et al. 1998). Bridge pier is one of the most critical components of a bridge since the overall seismic response of a bridge is largely dependent on the response of the piers. The extent of residual or permanent deformation sustained by the bridge piers prescribes the likelihood of allowing traffic over the bridge and dictates the amount of repair works and expected loss. Evidences from recent earthquakes and field reports demonstrated the importance of considering residual deformation as an indicator for defining the overall seismic performance of a structure.  Over the last few years, researchers have experimentally and numerically investigated the potential application of shape memory alloys in bridge piers and found promising results (Saiidi et al. 2009, Billah and Alam 2014c, Cruz and Saiidi 2012). However, all the previous studies were focused on the application of Ni-Ti SMA. However, Chapter 6 and 7 of this research developed performance-based damage states and a performance-based seismic design guideline for bridge piers reinforced with different types of superelastic SMAs in the plastic hinge region. While the previous chapters proved the potential of using this smart material in bridge piers and proposed some design guidelines, adoption of these guidelines and successful implementation require a complete performance-based evaluation of this structural system in light of performance-based earthquake engineering (PBEE). To this end, it is necessary to investigate the ability of such novel structural system in reducing the failure probability as well 139  as the annual rate of exceeding some structural demand parameters given an earthquake scenario. The objective of this chapter is to perform fragility based probabilistic seismic risk assessment of concrete bridge piers reinforced with different types of SMA rebar in the plastic hinge region. Figure 8.1 illustrates the methodology adopted in this study. First, the bridge piers are designed following the performance-based design guideline developed in Chapter 7. Later, a detailed description of the finite element model is provided to elucidate the details of bridge pier models. Instead of using code-specified design level earthquakes, this study considered three different earthquake scenarios which resembles the regional seismicity of Vancouver, British Columbia (BC), where the bridge is located. The performance and hazard assessment is conducted by considering shallow crustal, mega-thrust interface, and deep in-slab earthquake events (Atkinson and Goda 2011). Next, incremental dynamic analysis (IDA) (Vamvastikos and Cornell 2002) are conducted on each SMA-RC bridge pier model using 30 selected ground motions scaled to the conditional mean spectra of crustal, in-slab and interface earthquakes. The performance parameters of interest, which are maximum and residual drift in this study, are recorded for each motion. Next, the seismic performances of five different SMA-RC bridge piers are evaluated and compared using fragility curves. The fragility curves are developed using the Probabilistic Seismic Demand Model (PSDM). Finally, a probabilistic risk assessment is conducted to evaluate the mean annual frequency of exceeding different damage levels in terms of the selected demand parameters. 140   Figure 8.1. Flowchart of the methodology for seismic risk assessment of SMA-RC bridge piers  -1-0.500.510 20 40 60Acceleration (g)Time (sec)-0.1-0.0500.050.10 10 20 30 40Displacement (m)Time (sec)Maximum deformationResidual deformationSpectral AccelerationVibration Period (s)UHSCMS-CrustalCMS-InslabCMS-Interface Spectral Acceleration Vibration Period (s)P [DSIPGA]PGA (g)y = 1.03x + 0.36R² = 0.80LN (IM)LN (EDP)Annual rate of exceedanceEDPDS-4DS-3DS-2DS-1141  8.2 Probabilistic Seismic Performance Assessment A commonly used method for probabilistic seismic performance assessment is the Pacific Earthquake Engineering Research (PEER) Centre PBEE methodology (Cornell and Krawlinkler 2000) which attempts to address structural performance in terms of life safety, capital losses and functional losses (Aslani 2005). This PBEE methodology is composed of hazard analysis, structural analysis, damage analysis, and loss analysis. However, most of the applications of PBEE have been focused on buildings and few of them focused on bridge structures (Lee and Billington 2011). Moreover, no study has been conducted to date for probabilistic seismic performance assessment of SMA reinforced bridge piers. This chapter is intended to elucidate the potential benefit and compare the performance of different SMA-RC bridge piers in light of PBEE. This study conducted three steps of PBEE involving hazard, structural and damage analyses. However, the loss analysis was not performed because of limited information regarding the cost of different types of SMAs considered in this study. This research developed fragility curves and seismic hazard curves for different SMA-RC bridge piers considering maximum and residual drift as engineering demand parameters. The developed fragility curves express the probability of reaching or exceeding certain damage states corresponding to a certain intensity of ground motion. The hazard curves relate the mean annual rate of exceeding certain damage states. Instead of proposing a new methodology for the fragility assessment, this chapter offers critical insight on the performance-based evaluation of SMA-RC bridge piers using fragility curves. In this assessment different types of SMAs and uncertainties in the seismic hazard of the site are considered. Details of different methods of fragility curve development can be found in (Billah and Alam 2014b). In this study, the fragility curves are developed using a probabilistic seismic demand model (PSDM) and limit state model. The PSDM which relates the median demand to the intensity measure (IM) is developed using the results obtained from IDA and the power law function (Cornell et al. 2002). The PSDM provides a logarithmic correlation between median demand and the selected IM: EDP = a (IM)b           (8.1) In the log transformed space, Equation 8.1 can be expressed as 142  ln (EDP) = ln (a) + b ln (IM)        (8.2) where, a and b are unknown coefficients which can be estimated from a regression analysis of the response data collected from IDA. Effectiveness of a demand model is determined by the ability of evaluating Equation 8.2 in a closed form. In order to accomplish this task, it is assumed that the EDPs follow log-normal distributions. The dispersion (βEDP|IM) accounting for the uncertainty in the relation which is conditioned upon the IM, is estimated using Equation 8.3 (Baker and Cornell 2006): IMEDP /β  =2))ln()(ln(12−−∑=NaIMEDPNib      (8.3) where, N is the number of simulations. With the probabilistic seismic demand models and the limit states corresponding to various damage states, it is now possible to generate fragilities (i.e. the conditional probability of reaching a certain damage state for a given IM) using Equation 8.4 (Nielson 2005). ]/[ IMLSP =  −ΦcompnIMIMβ)ln()ln(       (8.4)  Φ  [.] is the standard normal cumulative distribution function and  baSIM cn)ln()ln()ln(−=         (8.5) ln(IMn) is defined as the median value of the intensity measure for the chosen damage state, a and b are the regression coefficients of the PSDMs, and the dispersion component is presented in Equation 8.6 (Nielson 2005).   bcIMEDPcomp2/ βββ+=         (8.6) where, Sc is the median and βc is the dispersion value for the damage states of different components of a bridge. 143  By combining the seismic responses obtained from IDA, in terms of maximum and residual drift, with the seismic hazard curve, it is possible to calculate the annual rate of exceeding various levels of demand for each EDP monitored. Using the uniform seismic hazard curve for the site under consideration, and maximum and residual drift responses obtained from IDA, the maximum and residual drift hazard of the SMA-RC piers are calculated based on the convolution integral (Equation 8.7) presented by Deierlein et al. (2003) ( ) ( ) ( )IMdvIMedpEDPPedpEDP ∫ >=λ       (8.7) In this equation, IM denotes the intensity measure of the ground motion, EDP refers to the engineering demand parameter (maximum and residual drift), λEDP(edp) represents the mean annual frequency of exceeding a predefined engineering demand parameter, edp. 8.3 Design of SMA-RC Bridge Piers In this study five concrete bridge piers reinforced with five different SMAs are designed following the performance-based design guidelines proposed in previous chapter (Chapter 7). The bridge piers are assumed to be located at Vancouver, BC with the site soil class-C (stiff soil). The corresponding design spectrum is selected, with a 2% probability of exceedance in 50 years that corresponds to a return period of 2475 years, according to the CHBDC-2014 (CSA S6-14). The bridge is considered to be a lifeline bridge according to the bridge classification described in CHBDC-2014 (CSA S6-14). For the selected seismic hazard level (2% in 50 years), the bridge should be operational (repairable damage) with limited service to meet the performance requirements. Since this design method starts with selecting a target residual drift, to meet the performance objectives and develop a comparable design of five different SMA-RC bridge piers, a target residual drift of 0.6% is selected. The height and diameter of all the bridge piers are assumed to be 5m and 1m, respectively. Five different SMAs having different combinations of alloys and mechanical properties are selected which are shown in Table 6.1. The material properties of concrete and steel reinforcement are listed in Table 6.2. The final design yielded all the bridge piers to be reinforced with 28 longitudinal SMA rebars of different diameter in the plastic hinge region and the remaining portion was reinforced with 28-25M steel (diameter 25.2 mm) rebar. To meet the current seismic design requirements, shear reinforcement was provided using 15 mm spirals at 50 mm pitch. The 144  bridge piers are specified as SMA-RC-1 (reinforced with SMA-1), SMA-RC-2 (reinforced with SMA-2), and so on. SMA-RC-1 and SMA-RC-2 are reinforced with 28-25 mm SMA-1 and SMA-2 bars, respectively. SMA-RC-3 is reinforced with 28-22.5 mm SMA-3 bars whereas SMA-RC-4 is reinforced with 28-35mm SMA-4 bars, and SMA-RC-5 is reinforced with 28-30 mm SMA-5 bars. Figure 8.2 shows the cross section and elevation of the bridge pier. In this study, the plastic hinge length of the SMA-RC bridge piers are calculated using the plastic hinge expression (Equation 8.8) developed in Chapter 5. ( ) ( ) ( ) ( )sclSMAygcP ffdLAfPdLρρ 24.0019.016.00002.008.025.005.1 //−−−+++= −   (8.8) Where, Lp is the plastic hinge length, d is the diameter of the pier, L/d is the aspect ratio, P/fc’Ag is the axial load ration, ρl =longitudinal reinforcement ratio, ρs = transverse reinforcement ratio, fy-SMA = yield strength of SMA rebar and fc’= concrete compressive strength. This equation showed reasonable accuracy in predicting the plastic hinge length measured from experimental investigations.  Figure 8.2. (a) Cross section, (b) elevation and (c) finite element model of SMA-RC bridge pier 8.4 Finite Element Modeling of Bridge Piers In this study, the bridge piers are modeled using a fiber based finite element program Seismostruct (Seismosoft 2014) to explicitly model the concrete, SMA and reinforcing steel materials. This program is able to accurately predict the large displacement and collapse behavior of frame structures under static and dynamic loading considering both geometric  (a) (b) (c) 145  nonlinearity (P-Δ effect) and material inelasticity (Pinho et al. 2007). The fiber sections of confined and unconfined concrete are simulated using the Mander et al. (1988) concrete constitutive model. The longitudinal reinforcement is modeled using the Menegotto–Pinto (1973) steel model with Filippou (1983) hardening rules. The superelastic SMA is modeled based on the constitutive relation developed by Auricchio and Sacco (1997). Mechanical couplers are used to connect SMA with steel rebars (Alam et al. 2010) which is represented by introducing a zero-length rotational spring at the bottom of the column section (Figure 8.2c). The stress-slip relationship of bars inside the coupler and the details of the splicing can be found in (Billah and Alam 2012). This study employed another zero-length inelastic spring to simulate the bond-slip behavior of SMA rebar in concrete. The bond slip spring was modeled based on the experimental bond strength-slip relation developed in Chapter 4. Using the modified Takeda hysteresis curve, described by Otani (1974) which follows the unloading rules proposed by Emori and Schonobrich (1978), the bond-slip spring was modeled. 8.5 Seismic Hazard and Selection of Ground Motions For the considered location of the bridge pier, seismic hazard is calculated using the probabilistic seismic hazard analysis (PSHA). Current Geological Survey of Canada model (NRC 2010), as described in Atkinson and Goda (2011), is used for assessing the seismic hazard of Vancouver. In this study, the hazard curves are obtained considering both peak ground acceleration (PGA) and spectral acceleration at the first mode period (Sa,T1) as intensity measures (IMs). Here, both PGA and Sa,T1 are selected for the seismic hazard curves as these two IMs are commonly available for the site under consideration. Based on the eigenvalue analysis, the fundamental periods of all the bridge piers are found to be around 0.7 sec. Figure 8.3 illustrates the seismic hazard curves for the location of bridge pier. For probabilistic seismic performance assessment, selection of appropriate ground motions which are representatives of the seismic hazard of the site under consideration is very important. In this study, the ground motion records are selected for seismic fragility and hazard assessment of SMA-RC bridge piers located in site soil class-C (VS30 = 550 m/s), in Vancouver, BC, Canada. For the seismicity in Vancouver, consideration of shallow crustal, subcrustal, and mega-thrust Cascadia subduction events are important since they have very different event and ground motion characteristics due to different source and path effects (Goda and Atinson 146  2011). In this study, the ground motions are selected by developing conditional mean spectrums (CMS) for the three different earthquake scenarios (crustal, inslab and interface) that significantly contribute to the seismic hazard of Vancouver.  Figure 8.3. Seismic hazard curve for site soil class C in Vancouver (a) Peak ground acceleration and (b) spectral acceleration The CMS for three different earthquake events are developed following the method described in Baker et al. (2011). In this study, the model proposed by Baker and Cornell (2006) is used for the inter-period correlation of crustal events while for inslab and interface events Goda and Atkinson (2009) model is adopted. Figure 8.4a shows the uniform hazard spectra (UHS) for site soil class-C along with the target CMS for crustal, inslab and interface events at T1=0.7 sec. Here, the vibration period of 0.7 sec is considered since all five SMA-RC bridge piers have their fundamental period of vibration around 0.7 sec. The UHS and the CMS of three events correspond to 2% probability of exceedance in 50 years which represents a return period of 2475 years. From Figure 8.4 it can be observed that the UHS and all the CMS has similar spectral acceleration at the target vibration period of 0.7 sec. In this study 30 ground motion suits (10 from each earthquake scenario) are selected representing crustal, inslab and interface earthquakes in the site under consideration. These records are selected from PEER NGA and K-NET/KiK-NET database. The records are selected in such a way so that they have similar spectral shape as the target CMS and the period range of interest are considered as 0.2T1 to 2T1. Similarity in the spectral shape is determined by selecting the record with the smallest average misfit between the target CMS and the ground motion corresponding to that particular event (i.e. inslab, crustal or interface). The selected records corresponding to 1.E-061.E-051.E-041.E-031.E-021.E-011.E+001.E-03 1.E-02 1.E-01 1.E+00 1.E+01Annual frequency of exceedencePGA (g)2% in 50 years(a)1.E-061.E-051.E-041.E-031.E-021.E-011.E+001.E-03 1.E-02 1.E-01 1.E+00 1.E+01Annual frequency of exceedenceSa (g)2% in 50 years(b)147  different earthquake types along with the target CMS and UHS are shown in Figure 8.4b-d. The records selected for performance assessment of the SMA-RC bridge piers are listed in Table 8.1.  Figure 8.4. (a) Comparison of UHS, CMS-Crustal, CMS-Interface, and CMS-Inslab at T1 = 0.7 s, (b-d) comparison of response spectra of the selected records with the target spectra for individual earthquake types     0.050.550.05 0.5 5Spectral Acceleration (g)Vibration Period (s)UHSCMS-CrustalCMS-InslabCMS-Interface1 20.1 0.2120.1(a)0.050.550.05 0.5 5Spectral Acceleration (g)Vibration Period (s)210.1 0.2120.1(b)0.050.550.05 0.5 5Spectral Acceleration (g)Vibration Period (s)0.10.10.2 1 221(c)0.050.550.05 0.5 5Spectral Acceleration (g)Vibration Period (s)0.10.10.2 1 221(d)148  Table 8.1. Selected earthquake ground motion records No Eq. Name Record ID Event ID Type Mw Epi. Dis (km) PGA (g) PGV (cm/s) Source 1 Northridge 953 127 Crustal 6.69 17.15 0.46 54 PEER 2 Duzce, Turkey 1602 138 Crustal 7.14 12.04 0.72 59 PEER 3 Hector mine 1787 158 Crustal 7.16 11.66 0.31 34 PEER 4 Imperial Valley 169 50 Crustal 6.53 22.03 0.28 28 PEER 5 Kocaeli,Turkey 1158 136 Crustal 7.51 15.37 0.3 54 PEER 6 Landers 900 125 Crustal 7.28 23.62 0.21 38 PEER 7 Loma Prieta 752 118 Crustal 6.93 15.23 0.48 34 PEER 8 Manjil, Iran 1633 144 Crustal 7.37 12.56 0.52 47 PEER 9 Chi Chi, Taiwan 1485 137 Crustal 7.62 26 0.47 39 PEER 10 Kobe, Japan 1106 129 Crustal 6.9 0.96 0.71 78 PEER 11 Tohuku, Japan 27538 368 Inslab 6.8 111.88 0.85 23 K-KIK 12 27451 368 Inslab 6.8 114.01 0.48 16 K-KIK 13 27454 368 Inslab 6.8 112.09 0.48 12 K-KIK 14 9813 184 Inslab 7 117.21 0.75 19 K-KIK 15 9837 184 Inslab 7 52.16 0.72 15 K-KIK 16 9831 184 Inslab 7 79.59 0.58 20 K-KIK 17 20480 294 Inslab 6 52.26 0.15 13 K-KIK 18 19650 285 Inslab 6.2 79.79 0.14 10 K-KIK 19 Tokachi-oki, Japan 6306 148 Inslab 6.8 58.31 0.41 33 K-KIK 20 6267 141 Inslab 6.8 46.89 0.39 25 K-KIK 21 Tokachi-oki, Japan 19085 276 Interface 7 76.98 0.66 24.60 K-KIK 22 19004 276 Interface 7 93.02 0.34 20.18 K-KIK 23 11026 194 Interface 7.9 119.95 0.56 36.6 K-KIK 24 11025 194 Interface 7.9 62.65 0.38 60.15 K-KIK 25 21598 301 Interface 7.1 97.14 0.38 13.28 K-KIK 26 Tohuku, Japan 169 - Interface 9 83.70 1.75 7.090 K-KIK 27 175 - Interface 9 71 0.96 44.43 K-KIK 28 237 - Interface 9 69.14 0.90 56.84 K-KIK 29 323 - Interface 9 62.49 0.67 27.09 K-KIK 30 168 - Interface 9 66.35 0.62 28.47 K-KIK  8.6 Fragility Analysis of Different SMA-RC Bridge Piers This section describes the development of PSDMs, characterization of damage states, and fragility curve generation for different SMA-RC bridge piers considering two different demand parameters. The developed PSDMs and fragility curves are used to examine the impact of different SMA properties on the seismic demand and to estimate the relative vulnerability of different SMA-RC bridge piers. 149  8.6.1 Probabilistic seismic demand model Selection of an appropriate intensity measure (IM) and an effective engineering demand parameter (EDP) is one of the most challenging tasks for probabilistic seismic performance and vulnerability assessment of structures as it dictates the reliability of the vulnerability assessment. An appropriate EDP selection is a function of the structural system and desired performance objectives (Zhang and Zirakian 2015). In this study, maximum drift (MD) of the bridge pier, which represents different performance-based limit states, is considered as one of the EDPs. A review of recent literature (Lee and Billington 2011, Billah and Alam 2014c) revealed that residual drift (RD) should be considered as an EDP to fully characterize the seismic performance of structures in light of the performance-based earthquake engineering. Accordingly, this study considered residual drift as the second EDP for the comparative seismic vulnerability assessment of different SMA-RC bridge piers. Selection and definition of an appropriate IM has been a debatable issue among the researchers. Some researchers suggested acceleration based IMs such as PGA (Padgett and DesRoches 2008) or spectral acceleration at the first mode (Sa-T1) (Mackie and Stojadinovic 2005) while other suggest velocity based IMs (e.g. peak ground velocity, PGV, and spectrum intensity, SI) (Bradley et al. 2009). Because of the efficiency, practicality, sufficiency, and hazard computability of PGA, many researchers (Padgett and DesRoches 2008, Billah et al. 2013) have suggested PGA as the optimal intensity measure for fragility assessment of bridges and bridge piers. Accordingly, for the purpose of this study, PGA is selected as the optimal IM. Incremental Dynamic Analyses (IDAs) are performed using the selected 30 earthquake records for the five different SMA-RC bridge piers. The maximum drift and the residual drift monitored from IDA are incorporated into a PSDM which establishes a linear regression of demand (EDP)–intensity measure (IM) pairs in the log-transformed space. This linear regression model is used to determine the slope, intercept, and dispersion of the EDP-IM relationship. Figure 8.5 shows the PSDMs of different SMA-RC piers in terms of maximum drift. Each figure also depicts the corresponding linear regression equation and R2 value. From Figure 8.5, it is evident that all the PSDMs have a R2 value higher that 0.7 which indicates a strong correlation between the considered EDP and IM (MD-PGA). Similarly, the PSDMs for different SMA-RC bridge piers in terms of residual drift is shown in Figure 8.6. The R2 values shown in these figures also reveal a strong correlation between this EDP-IM pair (RD-PGA).  150   Figure 8.5. Comparison of the PSDMs for (a) SMA-RC-1, (b) SMA-RC-2, (c) SMA-RC-3, (d) SMA-RC-4 and (e) SMA-RC-5 considering maximum drift as EDP  Figure 8.6. Comparison of the PSDMs for (a) SMA-RC-1, (b) SMA-RC-2, (c) SMA-RC-3, (d) SMA-RC-4 and (e) SMA-RC-5 considering residual drift as EDP Using the linear regression model expressed in Equation (8.2), the regression coefficients for various SMA-RC bridge piers in terms of different EDPs are computed and shown in Table 8.2. The parameters listed represent the regression parameters from Equation 8.2 along with the dispersion. From the results, it is evident that in the case of maximum drift, the SMA-RC-1 bridge pier yielded an increase in dispersion in the demand (βD|IM), while the SMA-RC-3 y = 1.0037x + 0.4739R² = 0.7071-5-4-3-2-101234-4 -2 0 2LN (PGA)LN (MD)(a)y = 1.0498x + 0.3913R² = 0.7954-5-4-3-2-10123-4 -2 0 2LN (PGA)LN (MD)(b)y = 1.0492x + 0.3949R² = 0.7955-5-4-3-2-10123-4 -2 0 2LN (PGA)LN (MD)(c)y = 1.0421x + 0.3526R² = 0.7286-5-4-3-2-10123-4 -2 0 2LN (PGA)LN (MD)(d)y = 1.0492x + 0.5347R² = 0.7955-5-4-3-2-101234-4 -2 0 2LN (PGA)LN (MD)(e)y = 1.00x - 0.91R² = 0.71-6-5-4-3-2-1012-4 -2 0 2LN (PGA)LN (RD)(a)y = 1.05x - 1.00R² = 0.80-6-5-4-3-2-1012-4 -2 0 2LN (PGA)LN (RD)(b)y = 1.05x - 1.21R² = 0.80-6-5-4-3-2-1012-4 -2 0 2LN (PGA)LN (RD)(c)y = 1.04x - 1.16R² = 0.73-6-5-4-3-2-1012-4 -2 0 2LN (PGA)LN (RD)(d)y = 1.05x - 0.85R² = 0.80-6-5-4-3-2-1012-4 -2 0 2LN (PGA)LN (RD)(e)151  exhibited a reduction in dispersion in the demand. On the other hand, it is evident from the regression model that the SMA-RC-5 tends to increase the median value and the slope (b) of the demands placed on the piers. This can be attributed to the higher elastic modulus and lower yield strength of SMA-5. It reveals that SMA-RC-3 and SMA-RC-4 are effective in reducing the maximum drift of the bridge pier. Similar observation can be made from the regression coefficients of RD-PGA relationship. From Table 8.2 it is evident that SMA-RC-3 is the most effective pier in reducing the residual drift demand. This can be attributed to the higher recovery strain of SMA-3, which eventually helps reduce the residual deformation of SMA-RC-3. Table 8.2. PSDMs for different EDPs   Maximum Drift Residual Drift Pier Type a b β EDP| IM a b β EDP| IM SMA-RC-1 1.6 1.02 0.71 0.4 1.02 0.71 SMA-RC-2 1.48 1.05 0.58 0.37 1.05 0.58 SMA-RC-3 1.44 1.03 0.56 0.3 1.05 0.55 SMA-RC-4 1.42 1.04 0.59 0.35 1.04 0.58 SMA-RC-5 1.71 1.05 0.68 0.43 1.05 0.71  8.6.2 Characterization of damage states An important aspect of PBEE for fragility curve development is the definition of appropriate damage states in relation to the functionality of the structure. Four damage states as defined by HAZUS-MH (FEMA 2003) are commonly adopted in the seismic vulnerability assessment of engineering structures, namely, slight, moderate, extensive, and collapse damages. Damage states are often developed based on expert judgment, post-earthquake survey, and experimental results. However, in absence of sufficient experimental results or post-earthquake reconnaissance report, analysis based methods are often adopted for developing damage states that corresponds to different functional level. Since very limited experimental results are available on SMA-RC bridge piers and all of them focus on Ni-Ti SMA, performance-based damage states for SMA-RC bridge piers developed in Chapter 6 has been considered in this study. In Chapter 6, performance-based damage states for five different SMA-RC bridge piers in terms of maximum and residual drift as well as considering different seismic hazard levels were developed.  152  As indicated by the closed form of fragility function in Equation 8.4, a reliable capacity limit state model is required for developing dependable fragility curves. For the selected demand parameters, each limit state model is assumed to follow a two-parameter lognormal distribution (median SC and dispersion βC). Table 8.3 lists parameter values used to define the limit state models on the basis of the maximum drift (%) and residual drift (%). The component limit states developed in Chapter 6 has been used in this chapter. Since the previous chapter (Chapter 6) only provides the median values (SC), a prescriptive approach described by Nielson (2005) is followed to define dispersions of limit state models (βC).  The dispersion values are calculated using the following equation provided by Nielson (2005). ( )21ln COVc +=β          (8.9) In this equation the COV values for different limit states are calculated based on the probabilistic distribution of different limit states described in Chapter 6. The COV values were found to be 0.21, 0.26, 0.45 and 0.52 for DS-1, DS-2, DS-3 and DS-4, respectively. These values yielded in similar dispersion values (βC) as described by other researchers (Nielson 2005). Table 8.3. Limit state capacity of SMA-RC bridge pier in terms of maximum and residual drift Damage state Maximum Drift SMA-RC-1 SMA-RC-2 SMA-RC-3 SMA-RC-4 SMA-RC-5 Sc βc Sc βc Sc βc Sc βc Sc βc DS-1 0.28 0.21 0.30 0.21 0.28 0.21 0.28 0.21 0.28 0.21 DS-2 1.68 0.26 1.66 0.26 2.28 0.26 1.74 0.26 1.10 0.26 DS-3 2.66 0.43 2.69 0.43 1.64 0.43 2.52 0.43 1.97 0.43 DS-4 5.05 0.50 5.51 0.50 7.65 0.50 5.56 0.50 4.73 0.50  Residual Drift  Sc βc Sc βc Sc βc Sc βc Sc βc DS-1 0.33 0.21 0.33 0.21 0.33 0.21 0.33 0.21 0.33 0.21 DS-2 0.62 0.26 0.62 0.26 0.62 0.26 0.62 0.26 0.62 0.26 DS-3 0.87 0.43 0.87 0.43 0.87 0.43 0.87 0.43 0.87 0.43 DS-4 1.22 0.50 1.22 0.50 1.22 0.50 1.22 0.50 1.22 0.50  153  8.6.3 Fragility Curves Using the linear PSDMs developed in previous section and limit state models presented in Table 8.3, fragility curves are developed for different SMA-RC bridge piers for each EDP using the closed form of fragility function shown in Equation 8.4. Fragility curves for the two different EDPs are shown in Figure 8.7 and Figure 8.8. The relative vulnerability of different SMA-RC bridge piers are compared in terms of reducing their probability of entering into different damage states. Fragility curves for different SMA-RC piers considering different EDPs are also compared by evaluating the relative change in the median value of the fragility curves.  Figure 8.7. Fragility curves for the five SMA-RC bridge piers for: (a) slight, (b) moderate, (c) extensive and (d) collapse damage state considering maximum drift  00.20.40.60.810 0.5 1 1.5 2P [Yielding I PGA]PGA (g)(b)00.20.40.60.810 0.5 1 1.5 2P [Spalling I PGA]PGA (g)(c)00.20.40.60.810 0.5 1 1.5 2P [CrushingI PGA]PGA (g)(d)00.20.40.60.810 0.5 1 1.5 2P [Cracking I PGA]PGA (g)(a)154  The evaluation of the fragility curves offered a valuable insight on the performance of different SMAs in reducing the probability of damage considering the maximum drift. Figure 8.7 presents the fragility curves of the five bridge piers considering maximum drift as the EDP. From Figure 8.7a it is evident that all the piers have similar probability of cracking damage irrespective of the intensity level. However, the effect of different SMAs is more pronounced in the other damage states. As depicted in Figure 8.7b, SMA-RC-5 is more likely to experience yielding at a lower intensity while SMA-RC-3 showed much better performance as it showed only 47% probability of yielding even at a PGA of  2g. This can be attributed to the very high yield strength of SMA-3 as compared to other SMAs. However, an interesting behavior is observed in spalling damage state where SMA-RC-3 has the highest probability of damage and SMA-RC-2 has the lowest. This can be attributed to the capacity limit states of spalling damage state considered in this study where SMA-RC-3 has the lowest drift limit before entering the spalling damage state. In general, all the SMA-RC piers show better performance at collapse/ crushing damage state as evident from the probability of collapse at maximum considered earthquake (MCE) level, which usually corresponds to 2% probability of exceedance in 50 years (PGA 0.46g), which is only 0.5%, 0.1%, 0.08%, 0.3%, and 0.8% for SMA-RC-1, SMA-RC-2, SMA-RC-3, SMA-RC-4, and SMA-RC-5, respectively. Plots of the fragility curves for the bridge piers for residual drift as the EDP are shown in Figure 8.8, and illustrate the relative vulnerability of the five bridge piers over a range of earthquake intensities and damage states. Unlike maximum drift fragility curves, there are marked differences in fragilities of different bridge piers in terms of residual drift at all damage states. Irrespective of damage states, the SMA-RC-3 showed lower probability of exceeding certain damage level. This can be attributed to the higher recovery strain of SMA-3 which reduced the residual drift in the bridge pier by bringing back the pier close to its original position at the end of ground motion. Moreover, none of the bridge piers showed 50% probability of exceeding DS-2, for which the bridge piers are designed, even at a PGA of 1g. It also indicates that the bridge piers are performing according to the design performance objective. As evident form Figure 8.8, the probability of collapse (DS-4) at maximum considered earthquake (MCE) level is only 1.5%, 0.45%, 0.2%, 0.68%, and 1.4% for SMA-RC-1, SMA-RC-2, SMA-RC-3, SMA-RC-4, and SMA-RC-5, respectively. 155   Figure 8.8. Fragility curves for the five SMA-RC bridge piers for: (a) slight, (b) moderate, (c) extensive and (d) collapse damage state considering residual drift  The different SMA-RC bridge piers are also compared in terms of the relative change in the median value of the fragility curves which indicates the PGA associated with a 50% probability of reaching a certain limit state. Table 8.4 compares the median PGA for different damage states of five different SMA-RC piers in terms of both EDPs. The median PGA in terms of maximum drift for different bridge piers at DS-1 ranges from 0.03g to 0.05g. However, at higher damage states, i.e. at DS-2 and DS-3, the median PGA varies over a wide range from 0.45g-2.16g and 1.21g-3.05g for DS-2 and DS-3, respectively. On the other hand, at DS-4, only SMA-RC-5 has a median PGA lower than 3g, while the other four SMA-RC piers have median PGA around 3.5g and SMA-RC-3 has as high as 3.98g. This can be 00.20.40.60.810 0.5 1 1.5 2P [DS-1 I PGA]PGA (g)(a)00.20.40.60.810 0.5 1 1.5 2P [DS-2 I PGA]PGA (g)(b)00.20.40.60.810 0.5 1 1.5 2P [DS-3 I PGA]PGA (g)(c)00.20.40.60.810 0.5 1 1.5 2P [DS-4I PGA]PGA (g)(d)156  attributed to the fact that except SMA-RC-5 all other SMA-RC piers have collapse drift limit (DS-4) over 5% whereas the same for SMA-RC-5 is 4.73%. However, in terms of residual drift no such big difference is observed at any damage state for different SMA-RC piers. Table 8.4. Comparison of median PGA (g) EDP Maximum Drift Residual Drift  Damage State Damage State Pier Type DS-1 DS-2 DS-3 DS-4 DS-1 DS-2 DS-3 DS-4 SMA-RC-1 0.031 1.113 2.880 3.502 0.357 1.440 3.486 - SMA-RC-2 0.053 1.236 3.045 3.562 0.6 1.595 3.482 - SMA-RC-3 0.056 2.16 1.208 3.980 0.882 2.310 3.781 - SMA-RC-4 0.047 1.470 3.040 3.48 0.662 1.822 3.610 - SMA-RC-5 0.037 0.456 1.325 2.88 0.456 1.234 2.695 2.88  8.7 Seismic Demand Hazard of Different SMA-RC Bridge Piers In order to fully implement the performance-based earthquake engineering (PBEE) methodology for SMA-RC bridge pier, it is necessary to conduct the probabilistic seismic demand analysis (PSDA) in terms of annual rate of exceeding some structural demand parameter such as maximum drift or residual drift. In this study, the annual rate of exceeding various levels of demand for the five considered SMA-RC piers are estimated by aggregating the EDP|IM relationship obtained from seismic response analysis with the seismic hazard curve. Using the convolution integral presented in Equation 8.7, the demand hazard curves for five different SMA-RC bridge piers are developed in terms of maximum and residual drift. Figure 8.9 a and b show the maximum drift and residual drift hazard curves for five SMA-RC bridge piers, respectively. The residual drift hazard curves depict the annual probability of exceeding different damage states for different SMA-RC piers (shown with vertical dashed lines). It should be noted that, the same type of results for different damage states are not presented for the maximum drift since different maximum drift limits were considered for different SMA-RC piers. The probability of collapse (probability of exceeding DS-4) of each bridge pier in terms of maximum drift are summarized in Table 8.5. Here, DS-4 is selected to compare the probability of collapse of different SMA-RC piers. Results show that all the bridge piers have very low probability of collapse while the SMA-RC-3 has the lowest probability of 1.27%. Among the five different SMA-RC piers, SMA-RC-5 has the 157  highest probability of collapse which is 31%, 1%, 33% and 23% higher that that of SMA-RC-1, SMA-RC-2, SMA-RC-3 and SMA-RC-4, respectively. This is due to SMA-5’s very low yield strength to elastic modulus ratio (0.0033), which reduced the drift capacity of SMA-RC-5. The probability of exceeding DS-2 in terms of residual drfit are presented in Table 8.6. Here, the probability of residual drift exceeding DS-2 is presented since all the bridge piers were designed considering a target residual drift of 0.6% which is the limitng value of DS-2. A comparison of the five bridge piers in terms of exceeding DS-2 reveals that SMA-RC-3 has the lowest probablity of exceeding DS-2 in 100 years which is only 2.84%. On the other hand, SMA-RC-5 resulted in highest annual rate of exceeding DS-2 which is 6.53%. A closer look into the annual rate of excceding DS-2 for different SMA-RC bridge pier shows that the annual rate of exceedance is influenced by the superelastic strain of the SMA rebar.  Figure 8.9. Hazard curves for five SMA-RC bridge piers (a) maximum drift and (b) residual drift Estimating the loss-hazard relationship is another integral part of PBEE which can be considered as the ultimate measure of seismic performance for decision making. However, the commercial availability of the Cu-based and Fe-based SMAs are very limited and adequate information on their costing is not available. As a result, the comparative seismic loss estimation of different SMA-RC bridge piers was not conducted in this study.   1.E-041.E-031.E-021.E-011.E+000 2 4 6 8 10Annual rate of exceedanceMaximum Drift (%)SMA-RC-1SMA-RC-2SMA-RC-3SMA-RC-4SMA-RC-5(a)1.E-041.E-031.E-021.E-011.E+000 0.5 1 1.5 2 2.5Annual rate of exceedanceResidual Drift (%)SMA-RC-1SMA-RC-2SMA-RC-3SMA-RC-4SMA-RC-5DS-4DS-3DS-2DS-1(b)158  Table 8.5. Annual rate and probability of collapse (DS-4) in terms of maximum drift  SMA-RC-1 SMA-RC-2 SMA-RC-3 SMA-RC-4 SMA-RC-5 Annual rate of DS-4 1.31E-04 1.88E-04 1.27E-04 1.46E-04 1.90E-04 Prob. Of DS-4 in 100 years 1.31% 1.88% 1.27% 1.46% 1.90%  Table 8.6. Annual rate and probability of DS-2 in terms of residual drift  SMA-RC-1 SMA-RC-2 SMA-RC-3 SMA-RC-4 SMA-RC-5 Annual rate of DS-2 5.35E-04 3.98E-04 2.84E-04 3.39E-04 6.53E-04 Prob. Of DS-2 in 100 years 5.35% 3.98% 2.84% 3.39% 6.53% 8.8 Summary Shape memory alloy (SMA) has emerged as an alternative to conventional steel reinforcement for improving the seismic performance of bridges during an extreme earthquake. This chapter presents the probabilistic seismic risk assessment of concrete bridge piers reinforced with different types of SMA (e.g. Ni-Ti, Cu-Al-Mn, and Fe-based) rebars. To achieve this objective, the bridge piers are designed following a performance-based approach. Ground motions with different probable earthquake hazard scenarios at the site of the bridge piers are considered. Probabilistic seismic demand models are generated using the response parameters obtained from incremental dynamic analysis. Considering maximum drift and residual drift as demand parameters, fragility curves are developed for five different SMA-RC bridge piers. Finally, seismic hazard curves are generated in order to compare the mean annual rate of exceedance of different damage states of different bridge piers. It is observed that all the bridge piers perform according to the design objective, and the performance of SMA-RC piers is significantly affected by the type of SMA used. The results show that all the SMA-RC piers have very low probability of collapse at maximum considered earthquake level. It is found that the bridge pier reinforced with FeNCATB-SMA (SMA-3) performed better as compared to the other SMA-RC piers.     159  CHAPTER 9.  SUMMARY, CONCLUSIONS AND FUTURE WORKS 9.1 Summary This thesis presented a comprehensive summary of the existing applications of shape memory alloys in bridge engineering along with the future of smart bridges using SMA. This study provides an insight into the current applications of SMA in the bridge engineering field. This thesis also presented a review of the different methodologies developed for seismic fragility assessment of highway bridges along with their features, limitations, and applications. This study presented a review of available methodologies and identifies opportunities for future development. This study mainly focused on the key features of different methods and applications rather than penetrating down to a critique of the associated analysis procedure or mathematical framework. This research was aimed at developing a performance-based seismic design guideline for concrete bridge piers reinforced with different types of superelastic shape memory alloy rebar. As a first step, this research experimentally investigated the bond behavior of SMA rebar with concrete. This study also experimented ways to improve the bond behavior of smooth SMA rebar using different types of sand coating. Finally, empirical equation based on statistical analyses is presented to predict the maximum average bond strength. The proposed equation appear to be reasonable for calculating the average bond strength of SMA reinforcing bars in concrete. As a next step, this research developed an analytical expression for plastic hinge length of SMA-RC bridge piers using well calibrated finite element models. A parametric study was performed to investigate the effect of different parameters on the plastic hinge length, including axial load ratio, aspect ratio, concrete strength, SMA properties, longitudinal and transverse reinforcement ratio. Multivariate regression analysis was performed to develop an expression to estimate the plastic hinge length in SMA-RC bridge pier and compared with existing plastic hinge length equations. The proposed equation was verified against test results which showed reasonable accuracy. In the next step, this research developed performance-based damage states for shape memory alloy (SMA) reinforced concrete bridge piers considering different types of SMAs and seismic hazard scenarios. Based on extensive numerical study, this study also proposed 160  maximum and residual drift based damage states for SMA-RC pier. Finally, an analytical expression is proposed to estimate the residual drift of SMA reinforced concrete elements as a function of the expected maximum drift and superelastic strain of SMA. Comparison with experimental results revealed that the proposed equation could very well predict the residual drift obtained from the experimental results. Based on the developed performance-based damage states, a sequential procedure for the performance-based design of SMA-RC bridge pier is presented. This study also developed damping-ductility relationship for different SMA-RC bridge piers. Using the proposed design framework a trial SMA-RC bridge pier was designed and analyzed using a suite of selected earthquake records. The nonlinear analyses showed that the designed pier behave according to design expectations and provided very promising results in terms of the effectiveness and applicability of the proposed design method. Finally, a comprehensive probabilistic seismic risk of assessment of different SMA-RC bridge piers was conducted with the aim of evaluating the performance of the SMA-RC bridge piers in light of performance-based earthquake engineering. Considering maximum drift and residual drift as demand parameters, fragility curves were developed for five different SMA-RC bridge piers. Finally, seismic hazard curves were generated in order to compare the mean annual rate of exceedance of different damage states of different bridge piers. It was found that all the bridge piers performed according to the design objective. The results showed that all the SMA-RC piers have very low probability of collapse at maximum considered earthquake level. 9.2 Core Contributions The outcomes of this research work is expected to initiate practical application of shape memory alloys in bridge engineering especially in bridge piers in seismically active regions. The core contributions of this study are: • Development of plastic hinge length expression for SMA-RC bridge piers. • Prediction of residual drift of SMA-RC elements using maximum drift and superelastic strain of SMA. • Development of a performance-based seismic design guideline for SMA-RC bridge pier. • Bond behavior of smooth and sand coated SMA rebar in concrete. 161  9.3 Conclusions 9.3.1 Bond behavior of smooth and sand coated SMA rebar in concrete This study investigated the bond behaviour of smooth and sand coated shape memory alloy bars in concrete. Experimental investigations were carried out using pushout tests to investigate the influence of concrete strength, bar diameter, concrete cover, embedment length, and surface condition on the bond strength of SMA rebar. The results from 56 pushout tests lead to the following conclusions: • The stress-slip curve of SMA rebar can be divided/idealized into four stages: elastic stage, ascending stage, linearly descending stage and residual stage. • The surface roughness of SMA rebar significantly affects the failure pattern as well as the bond strength. Concrete with smooth SMA rebars resulted in simple pushout failure whereas sand coated rebars resulted in splitting failure.  • The bond strength of both smooth and sand coated SMA rebar is significantly influenced by the concrete strength, bar diameter and embedment length but is independent of concrete cover. • The application of sand coating increased the bond strength between concrete and SMA rebar by developing friction and interlocking forces in addition to the adhesion mechanism. The coarser the sand size, the more is the improvement in bond strength. • A new bond stress prediction equation is proposed for SMA rebar based on the experimental study for different strengths of concrete, bar diameters, surface condition and embedment length. The proposed equation is in good agreement with the experimental results. 9.3.2 Plastic hinge length of SMA-RC bridge pier This study proposed a new expression for estimating the plastic hinge length in SMA-RC bridge pier. The finite element model was validated with different experimental results to ensure the accuracy of the adopted modeling technique. A parametric study was conducted to evaluate the effect of different factors on the plastic hinge length of an SMA-RC bridge pier. A multivariate regression analysis was performed to develop the proposed plastic hinge expression. The proposed equation was verified against test results of SMA-RC piers to check its accuracy. The accuracy of the proposed equation in predicting the drift capacity of SMA-162  RC pier was validated against test result and compared with other plastic hinge expressions. Based on the analysis results, the following conclusions are drawn: • The effect of different parameters are more pronounced when plastic hinge length is estimated in terms of longitudinal rebar strain profile as opposed to the curvature profile. • Compressive strain profile of longitudinal rebar provides a better estimate of plastic hinge length as compared to the curvature profile of SMA-RC bridge pier. • Plastic hinge length of SMA-RC pier increases as the axial load, aspect ratio and the yield strength of SMA rebar increases. On the contrary, plastic hinge length decreases with an increase in concrete compressive strength and the ratio of longitudinal and transverse reinforcement.   • The proposed equation showed reasonable accuracy in predicting the plastic hinge length measured from experimental investigations. The proposed equation predicted the experimental plastic hinge lengths with a COV of only 6%. • The ultimate drift capacity of SMA-RC bridge pier can be predicted with reasonable accuracy using the proposed plastic hinge length equation as compared to other existing expressions. 9.3.3 Performance-based seismic design of Shape Memory Alloy reinforced concrete bridge pier In order to develop a performance-based design guideline for SMA-RC bridge pier, structural performance objectives and their corresponding limit state criteria must be clearly defined. Due to the significant differences in the behavior of SMA reinforced bridge piers as compared to conventional piers, damage states for typical bridge piers may not be applicable for SMA-RC bridge piers. In this study, a set of performance-based damage states for bridge piers reinforced with five different types of SMAs were developed in terms of both maximum and residual drift. Using an IDA-based approach, this study proposed performance-based drift levels for different damage states considering three different hazard levels. To predict the residual deformation of SMA-RC bridge pier, a relationship between maximum drift, residual drift, and superelastic strain of SMA was developed. The prediction equation was validated 163  against experimental observations from SMA reinforced concrete members. Based on the results of this study, the following major conclusions can be drawn: • The progression of damage was similar for all the RC bridge piers reinforced with different SMAs (except for SMA-RC-3): concrete cracking, longitudinal reinforcement yielding, cover spalling, and core crushing. For all SMA-RC bridge piers cracking occurred at the same level of drift (due to same cross-section) while the drift at other performance levels varied based on the mechanical properties of SMA used. • Different performance-based drift limits, i.e. cracking, spalling, yielding, and crushing of SMA-RC bridge pier strongly follow uniform, normal, log-normal, and gamma distribution, respectively. These distributions can be used for selecting the target drift levels for performance-based design of SMA-RC pier. • Except for DS-1(cracking), other three damage states are significantly influenced by the type of SMA used. For DS-2 (yielding), the limiting maximum drift varies from 1.18% (SMA-5) to 2.28% (SMA-3) and for DS-3 (spalling), the limiting maximum drift varies from 1.64% (SMA-3) to 2.69% (SMA-2) for hazard level corresponding to 2475 years return period. • In terms of maximum drift, consideration of different hazard levels does not have any significant impact on DS-1 and DS-2. On the other hand, different hazard levels have substantial impacts on DS-3 and DS-4. • The proposed residual drift limit states tend to decrease with increased probability of occurrence (decreased return period). The damage states developed in terms of residual drift correlate well with damage observed from experimental studies. • Residual drift can be expressed as a function of maximum drift and the superelastic strain of SMA rebar. • Based on the residual drift responses of all the SMA-RC piers under different levels of ground motions, a prediction equation was developed to predict the residual drift response of an SMA-RC bridge pier. The proposed equation can correlate very well with experimental observations. 164  • The proposed equation can be used for predicting the residual drift of SMA-RC bridge pier when designing the pier for a target residual drift. Based on the maximum drift and residual drift the designer would be able to select an SMA with the required superelastic strain, thereby ensuring the safety of bridges under extreme earthquake event. Based on the developed performance-based damage states, this study proposed a new residual drift based design method for shape memory alloy reinforced concrete bridge pier. The approach outlined in this study is a comprehensive approach for performance-based design of SMA-RC bridge pier. This study developed necessary design equations and graphs for PBSD of SMA-RC bridge pier. The proposed method provides the owner to select expected performance of the bridge pier and allows the designer/engineer to select multiple hazard and performance expectation combinations. Following the DDBD guidelines of Priestley et al. (2007) this study developed a new design method and damping-ductility relation of SMA-RC bridge pier which is a key step for performance-based design. In contrast to the conventional DDBD approach, the proposed procedure anticipates a target residual drift based on the expected performance during design earthquake, calculates the maximum drift demand and ensures that those drift demands (maximum and residual) remain below acceptable limits for the design level earthquakes. The performance of the bridge pier was validated using NLTHA, and the maximum and residual drifts at the design level earthquakes were found to satisfy the performance expectations. The design procedure developed in this study is expected to meet engineers’ requirements for a robust and easy to apply performance-based design methodology for SMA reinforced bridge piers in seismic regions. 9.3.4 Probabilistic seismic risk assessment of SMA-RC bridge piers This study conducted a probabilistic performance-based risk assessment of five SMA-RC bridge piers when subjected to three different earthquake scenarios (crustal, inslab and interface) that significantly contribute to the seismic hazard of Vancouver. The piers were designed following a performance-based design guidelines developed in this research. In order to ensure a comprehensive seismic performance and risk assessment, this study developed maximum and residual drift hazard curves and fragility curves for different SMA-RC bridge piers. The influence of application of different SMAs and their properties in the seismic hazard 165  curve was also investigated. Based on the results obtained from the risk assessment, the following conclusions are drawn: • The EDPs considered in this study, i.e., maximum drift and residual drift, are shown to be well correlated with the intensity measure (PGA) considered in this study which provided a basis for a reliable probabilistic seismic risk assessment. • Mechanical properties of different shape memory alloys, specifically the recovery strain, significantly affects the seismic fragility and risk of SMA reinforced concrete bridge piers in terms of both residual and maximum drift. • In general, all the SMA-RC bridge piers met the design objectives in terms of residual drift. Although, the bridge piers were designed following the design spectra corresponding to 2% in 50 years probability of exceedance, the bridge piers performed satisfactorily under the considered three different earthquake scenarios (crustal, inslab and interface). • All the SMA-RC bridge piers, in general, are quite effective in controlling the seismic response and reducing the vulnerability which is exhibited by the low probability (less than 1%) of collapse in terms of maximum drift at the maximum considered earthquake level. • In terms of residual drift, SMA-RC-3 outperformed all other SMA-RC bridge piers at all damage states and significantly reduced the overall vulnerability of the bridge pier. This can be attributed to the higher superelastic strain and low residual strain of SMA-3. • Comparing maximum and residual drift hazard curves for different SMA-RC piers revealed that in both cases SMA-RC-5 has the highest probability of exceeding DS-4 and DS-2 as it was evident from the mean annual rate of exceedance which is 1.90 × 10-4 and 6.53 × 10-4, for maximum and residual drift, respectively. • From the hazard analysis of different SMA-RC bridge piers, it is expected that the SMA-RC bridge piers will incur a lower annual loss and will provide significant financial benefit in the long run since these SMA-RC piers showed very low probability of damage. However, a detailed loss estimation needs to be carried out before highlighting the potential financial benefit of SMA-RC piers.  166  9.4 Recommendation for Future works The present study only considered pushout tests for investigating the bond behavior of SMA rebar in concrete. Further study need to be conducted considering SMA reinforced beams with and without lateral reinforcement. Further study needs to be carried out considering different types of SMA rebar to develop a more comprehensive bond-slip relationship for SMA rebar in concrete. In this research, the performance-based design for SMA-RC bridge piers has been demonstrated which is limited to flexure dominated columns. However, further studies need to be conducted combining different factors that influence the bridge seismic performance, in particular for the shear critical bridge piers. Moreover, the design procedure for the bridge as a system including soil structure interaction needs to be developed. However, since the application of SMA in real life application remains a challenge, development of low cost SMAs along with simplified design procedure will pave the way towards widespread application of SMA in practical applications. Further experimental investigations of SMA-RC bridge piers, designed following the proposed guideline, under unidirectional and bidirectional seismic loading are required to provide a solid, reliable, and valid conclusion regarding the applicability of the proposed guideline. Since the behavior of SMA is also temperature dependent, future studies should also focus on the effect of temperature changes on the seismic response of SMA-RC piers.   The present research assessed the seismic risk of SMA-RC bridge piers without considering material and geometric uncertainties and soil foundation interaction. Incorporating such effects and considering a bridge as a system will shed light on additional issues and are likely to change the dynamics of the response of the entire bridge structure. In future, it will be of great interest to investigate the response of whole bridge by considering different SMAs. Moreover, performing further study considering the construction, repair, and maintenance cost of SMA-reinforced bridge, as well as comparing the smart bridge with a conventional bridge along with the development of a loss-hazard relationship will shed more light on the potential economic benefit of this smart bridge system. To date, researchers have identified many potential applications of SMA in bridge engineering which are mainly focused on using SMA as a supplementary reinforcement or 167  materials in different bridge components but less on the design perspective. Researchers have shown that SMA can be effectively used in not only for developing smart bridges but also for generating a resilient and damage tolerant highway infrastructure system. Although research in SMA related material science has advanced a lot, its application in structural engineering is still limited because of the high cost of SMA and lack of adequate knowledge and interest among the practitioners. In particular, for bridge engineering application, SMA based devices and reinforcement can reduce the overall life cycle cost of the bridge. However, research on smart bridges must concentrate on ensuring that these ‘smart’ devices or reinforcements are compatible with the current industry practice and adequate guidelines are available for practitioners. According to the author, following actions need to be taken to increase the application of SMA in structural as well as in bridge engineering sector: • An integrated effort by the material scientists and structural engineers to ensure a considerable progress in application of SMAs in bridge engineering.  • Development of an efficient and comprehensive database of SMA properties for knowledge sharing that can be used for designing SMA based structural components.  • More concerted effort is required to develop low cost SMAs with excellent superelastcity, high elastic modulus and superior fatigue performance.  • Research should be carried out to find ways for modifying the smooth surface of SMA rebars which affects the bond behavior with concrete. • Development of new compositions of SMAs and hybridization of SMA with other smart materials. • Development of more refined, robust and easy to use computational model of SMA for analysis and design process.         168  REFERENCES Abrahamson, N.A. 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Seismic fragility estimates for corroding reinforced concrete bridges, Structure and Infrastructure Engineering: Maintenance, Management, Life-Cycle Design and Performance, 8(1): 55-69. 197   APPENDICES Appendix A Table A.0.1. Summary of seismic fragility assessment studies of bridges Authors Component Demand Parameter Intensity Measure Uncertain parameters Method Agrawal, A.K., Ghosn, M., Alampalli, S. and Pan, Y. (2012) Column, bearing Curvature ductility, bearing displacement PGA fc', fy, W, ΔT, μB Analytical Akbari, R. (2012)  Column Curvature, Drift, Displacement ductility PGA -* Analytical Alam, M.S., Bhuiyan, A.R., and Billah, A.H.M.M. (2012) Column, Bearing Displacement ductility, Shear strain PGA -* Analytical AmiriHormozaki, E., Pekcan G., and Itani, A. (2013) Column, Bearing, Abutment Curvature ductility, Bearing deformation, Abutment deformation PGA, Sa fc', fy, μB, Ki, ξ, G, θ, Ka Analytical Alipour, A., Shafei, B., and Shinozuka, M. (2013) Column Displacement ductility PGA Ys Analytical Avsar, O., Yakut, A. and Caner, A. (2011).  Column, Cap beam, Deck column and cap beam curvature, shear in both principal axes, and deck displacement PGA, PGV, ASI Ls, H, θ Analytical Aygün, B., Dueñas-Osorio, L., Padgett, J.E. and DesRoches, R. (2011).  Column, Abutment, Bearing, Pile, Deck Column curvature, Bearing deformation, Abutment displacement, Deck unseating, Pile cap displacement PGA fc', fy, μB, Ki, ξ, G, Sg, Su, Φ, p-y spring Analytical Banarjee, S. and Prasad, G.G. (2013) Column Displacement ductility PGA Ys, Flood return period Analytical 198  Banerjee, S. and Chi, C. (2013).  Column Rotational Ductility PGA -* Experimental and Analytical Banerjee S. and Shinozuka M. (2007).  Column Drift ratio, Displacement ductility demand PGA -* Analytical Banerjee S. and Shinozuka, M. (2011). Column Rotational Ductility PGA α Analytical Berry, M. P., Eberhard, M. O. (2003).  Column Cover spalling, Bar buckling Pr, ρ, fc',fy, L/D   Experimental Billah, A.H.M.M., Alam, M.S. and Bhuiyan, A.R. (2013).  Column Displacement ductility PGA -* Analytical Billah, A.H.M.M. and Alam, M.S. (2013) Column, Bearing, Wing wall, Back wall Displacement ductility, Bearing deformation, Wing wall and back wall displacement,  PGA fc', fy, μB, Ki, ξ, G, θ, Kr, Kθ, Ka Analytical Billah, A.H.M.M. and Alam, M.S. (2014) Column Displacement ductility, Residual drift, Maximum Drift PGA -* Analytical Billah, A.H.M.M. and Alam, M.S. (2012).  Column Residual Drift PGA   Analytical Bhuiyan, A.R. and  Alam, M.S. (2012). Column, Bearing Displacement ductility, Shear strain PGA -* Analytical Brandenberg, S.J., Zhang, J., Kashighandi, P., Huo, Y. and Zhao, M. (2011). Column, Bearing, Pile cap, Abutment Curvature ductility, Shear strain, Pile curvature ductility, Abutment displacement and rotation. PGA Crust thickness, crust strength, Axial tip capacity, Liquefied sand thickness, p-y spring Analytical Choe, D., Gardoni, P., Rosowsky, D., and Haukaas, T. (2008, 2009).  Column Deformation and shear force demand Sa Ls, L/H, D/Ds, fc', fy, Ksoil, ρ Analytical Choi, E., DesRoches, R. and Nielson, B.G. (2004).  Column, Fixed bearing, Expansion bearing, Dowel Curvature ductility, bearing displacement, Dowel displacement PGA fc', fy, G Analytical 199  Dong, Y., Frangopol, D.M.  and Saydam, D. (2013) Column Displacement ductility PGA fc', fy, Cover depth, Diffusion coefficient, Chloride concentration Analytical Elnashai, A., Borzi, B., Vlachos, S. (2004) Column Displacement ductility PGA fc', fy,  Analytical Frankie (2013) Column Cracking, Yielding, Peak load, Loss of load capacity PGA   Hybrid Gardoni, P., Der Kiureghian, A., Mosalam, K. M. (2002) Column Drift ratio -* fc', fy, fsu, ρ, Pr Experimental and statistical Gardoni, P., Der Kiureghian, A., Mosalam, K. M. (2003) Column Column deformation Sa fc', fy, ρ, Ksoil, D/Ds, L/H  Analytical and Bayesian method Gardoni, P and Rosowsky, D. (2011).  Column Column deformation Sa fc', fy, ρ, Ksoil, D/Ds, L/H  Bayesian Updating Ghosh, J. and Padgett, J.E. (2010).  Column, Bearing, Abutment Curvature ductility, Bearing displacement, Abutment displacement PGA Cover depth, Diffusion coefficient, Chloride concentration, Rate of corrosion Analytical Huo, Y. and Zhang, J. (2013) Column Section curvature PGA θ, T, G Analytical Huang, Q., Gardoni, P. and Hurlebaus, S. (2010).  Column Column deformation PGV fc', fy, θ, L, H, ρ, W, Ksoil, D/Ds, Ka Analytical Jara, J. M., Galvn, A., Jara, M. and Olmos, B. (2013) Column, Isolation Bearing Curvature ductility, Bearing displacement PGA -* Analytical Karim K.R. and Yamazaki F. (2003) Column Park-Ang damage index PGA, PGV, SI -* Analytical Kwon, O.S. and Elnashai, A.S. (2010) Bearing, Bent, Abutment Bent deformation, Abutment deformation, Bearing deformation PGA fc', fy, Sg, Su, Ksoil Analytical 200  Mackie, K., and Stojadinovic, B. (2005).  Column Peak steel strain, peak concrete strain, Peak column curvature, Curvature ductility, Displacement ductility, Drift ratio, Residual deformation index, Plastic rotation, Hysteretic energy, Normalized hysteretic energy Is, Iv, FR1, FR2, Td, arms, EPD, EPV, EPA, Sd, R, M fc', fy, θ, L, H, ρ, W, Ksoil, D/Ds Analytical Moschonas, I.F.,  Kappos, A.J.,  Panetsos, P., Papadopoulos, V., Makarios, T., and Thanopoulos, P. (2009) Column Column displacement PGA -* Analytical Nielson, B.G. and DesRoches, R. (2007a,b) Column, Bearing, Abutment Curvature ductility, Bearing displacement, Abutment displacement PGA fc', fy, μB, Ki, ξ, G, θ, Kr, Kθ, Ka, Loading direction Analytical Padgett, J.E., Ghosh, J. and Dueñas-Osorio, L. (2013) Column, Expansion bearing, Fixed bearing, Abutment piles Curvature ductility, Bearing deformation, Abutment deformation, Pile deformation PGA  fy, μB, Ki, ξ, G, Sg, Su, Φ, p-y spring, da, db Analytical Padgett, J.E. and DesRoches, R. (2008, 2009) Column, Bearing, Abutment, Shear key, Restrainer Curvature ductility, Bearing deformation, Abutment deformation PGA fc', fy, μB, Ki, ξ, G, θ, Kr, Kθ, Ka, Loading direction, Restrainser cable length and slack. Analytical Pan, Y., Agrawal, A. K., Ghosn, M., and Alampalli, S. (2010a,b) Column, Bearing, Abutment Curvature ductility, Bearing deformation, Abutment deformation PGA fc', fy, W, ΔT, μB Analytical Ramanathan, K., DesRoches, R. and Padgett, J.E. 2012 Column, Expansion bearing, Fixed bearing, Abutment Curvature ductility, Bearing deformation, Abutment deformation PGA fc', fy, μB, Ki, ξ, G, θ, Kr, Kθ, Ka, α, Gb, Loading direction, Dowel bar trength Analytical Shinozuka et al. 2001 Column Displacement ductility PGA fc', fy Analytical, Empirical 201  Shinozuka, M., Feng, M. Q., Kim, H.-K., Kim, S.-H. (2000) Column Displacement ductility PGA -* Analytical Sung, Y.C. and Su, C.K. (2011)  Column Displacement PGA -* Analytical Tavares, D.H., Padgett, J.E. and  Paultre, P. 2012 Column, Bearing, Wing wall, Back wall, Abutment footing Displacement ductility, Bearing deformation, Wing wall and back wall displacement, Abutment deformation PGA fc', fy, μB, Ki, ξ, G, θ, Kr, Kθ, Ka Analytical Torbol, M. and Shinozuka, M. (2012a,b) Column Rotational ductility PGA α Analytical Vosooghi, A. and Saiidi, M.S. (2012) Column Maximum drift, Residual drift, Frequency ratio, Inelasticity index, Maximum steel strain -* L/H, D, ρ, Scale factor Experimental Yamazaki, F., Motomura, H. and Hamada, T. (2000). Bridge Observed damage PGA, PGV -* Empirical Zhang, J. and Huo, Y. (2009) Column, Isolation Bearing Curvature ductility, Bearing shear strain PGA -* Analytical Zhong, J., Gardoni, P and Rosowsky, D. (2012). Column Deformation and Shear deformation Sa Model Uncertainty, Cover depth, Diffusion coefficient, Chloride concentration, Age factor, Environment factor, Test method factor, Curing factor Analytical Note: *no intensity measure or uncertain parameters was considered   202  Table A.0.2. Summary of regional fragility analysis of highway bridges Region Author Bridge Type Features Eastern US: New York Pan et al. (2010a, 2010b) MSSS-SG Identification of vulnerable components and effect of different retrofit measures Central and southern US Choi et al. (2004)  MSSS-SG, MSC-SG,MSSS-PSC, MSC-PSC Identified MSSS and MSC steel girder bridges as the most vulnerable ones Nielson and DesRoches (2007a, 2007b MSC concrete, MSC Slab, MSC Steel, MSSS concrete, MSSS Slab, MSSS conc . Box, MSSS Steel, SS Concrete, SS steel Using a component level approach this study identified the steel girder bridges as the most vulnerable ones followed by concrete girder bridges and single span bridges of all types Padgett and DesRoches (2008) MSC concrete, MSC Slab, MSC Steel, MSSS concrete, MSSS Slab, MSSS conc . Box, MSSS Steel, SS Concrete, SS steel Impact of different retrofit measures on bridge component vulnerability as well the bridge as a system. This study developed framework for the use of the fragility curves in retrofit selection including performance-based retrofit and cost-benefit analyses Ramanathan et al. (2010a, 2012)  MSC concrete, MSC Steel, MSSS concrete, MSSS Steel Investigated the influence of seismic detailing on the seismic vulnerability of four typical bridge classes in CSUS. Compared their fragility curves with HAZUS fragility curves and developed confidence bounds to characterize the uncertainty associated with the median fragility curve Western US: California Mackie and Stojadinovic (2005)  Concrete Highway Overpass Bridges Developed demand, damage, and decision fragility curves. These curves were so developed that they were conditioned on an arbitrary intensity measure that can be varied to best suit the structure and site of interest Zhang and Huo (2009)  MSC concrete box girder Investigated the efficacy and optimal design parameters of isolation devices using a performance based evaluation approach based on PSDA and IDA. 203  Ramanathan (2012)  MSC Conc. Box Girder, MSC Slab, MSC Concrete Girder Developed fragility curves for typical California bridge classes along with their evolution over three significant design eras. This study developed different damage states for different bridge components in alignment with CALTRANS design and operational guidelines Dukes et al. (2013)  MSC Conc. Box Girder Proposed a new methodology to incorporate fragility analysis in the design of new bridges and suggested the use of the fragility curves as  a design check which will enable the design engineer to determine if performance criteria have been met, and also provide information on potential uncertainty of the performance of the design Eastern Canada Tavares et al. (2012) MSC Slab, MSC Steel, MSC Concrete, MSSS Concrete, MSSS Steel Developed component and system fragility curves for five different bridge classes in Eastern Canada and concluded that the concrete girder bridges have relatively high vulnerability as compared to steel girder bridges Lau et al. (2012)  MSC-PSC Proposed a methodology for developing fragility curves for bridges assuming that bridges having same structural configuration and designed and constructed at the same period will have similar vulnerability during a seismic event Western Canada Billah and Alam (2013) MSC Concrete Girder Considering soil-structure interaction along with all types of uncertainties, this study developed fragility curves for MSC concrete girder bridges which represent a significant portion of highway bridges in BC Japan Yamazaki et al. (2000) Expressway bridges Developed fragility curves based on actual damage data. Tanaka et al. (2000) Hanshin Expressway Utilizing the actual damage data from the 1995 Hyogoken-Nanbu earthquake, this study developed the damage database with GIS. With this database, the fragility curves were developed assuming normal distributions and were evaluated in comparing with the probability damage matrix of ATC-13. 204  Karim and Yamazaki (2007) MSC Concrete Developed a simplified approach to generate fragility curves of isolated bridges and illustrated the contribution of isolators on reducing damage probability of bridge columns. They found that the damage probability of isolated systems tends to be higher for a higher level of pier height compared to non-isolated systems. Akiyama et al. (2013a) Tohuku-Shinkansen Viaduct Developed limit states for as-built and retrofitted viaducts, investigated  the effectiveness of the seismic retrofit against the strong ground motions and compared fragility curves for as-built and retrofitted viaducts. Italy De Felice and Giannini (2010) MSSS Concrete, MSC Concrete Assesses the seismic reliability of three Italian Highway bridges using Effective Fragility Analysis (EFA) methodology. Cardone et al. (2007) Existing Highway bridges in Italy Proposed a numerical procedure for the evaluation of the seismic vulnerability and seismic risk of highway bridges that combines elements from the Direct Displacement based design method and the Capacity Spectrum Method. The proposed method provided the possibility to consider possible modifications of strength and ductility due to decay of materials and/or seismic retrofit interventions. Turkey Avsar et al. (2011)  MSMC, MSSC Developed fragility curves for bridges constructed after 1990 and clustered them into four different groups based on their structural attributes. They identified bridges with larger skew angles and single column bent as the most vulnerable ones Greece Moschonas et al. (2009)  Greek motorway bridges Defined different damage states for the bridge components based on energy dissipation mechanism and proposed a new method for generating fragility curves using nonlinear pushover analysis. They reported that the bridges were more vulnerable in the longitudinal direction and the derived fragility curves are heavily influenced by the demand spectra used. 205  Algeria Kibboua et al. (2011) Typical Algerian RC bridge piers They found that cross sectional geometry and longitudinal reinforcement significantly affects the vulnerability of bridge piers. They concluded that bridges supported on wall piers have lower probability of damage as compared to the others Korea Lee et al. (2007)  Expressway bridges in Korea Based on the capacity demand ratio of  different bridge components, they defined three damage states for the Korean bridges. Using logistic curve equations, they developed relationship between peak ground acceleration and vulnerability. Taiwan Liao and Loh (2004)  16 types of highway bridges Defined five different damage states based on the ductility and displacement demand. Although they carried out an extensive study they did not provide any conclusive remarks regarding the most vulnerable types of bridges. Sung et al. (2013)  Existing Highway bridges in Taiwan Proposed a rapid vulnerability assessment method for assessing the seismic vulnerability of existing bridges in Taiwan. The proposed system is capable of estimating and visually demonstrating different damage levels that bridges have encountered due to a specific seismic event and figure out the corresponding economic loss due to the damage of bridges. MSSS=multi-span simply supported, MSC=multi-span continuous, SG= steel girder, PSC= prestressed concrete girder, MSMC= multi span, multi-column, MSSC= multi span single column  206   Appendix B Goodness-of-fit test The goodness-of-fit of a statistical model describes how well it fits a set of observations. Measures of goodness-of-fit typically summarize the discrepancy between the observed values and the values expected under the model in question. Usually goodness-of-fit tests are based on a null hypothesis that the sample data is taken from a larger population that follows a given mathematical distribution. If the null hypothesis is accepted at a given level of significance (α), than it is concluded that the chosen distribution fits the sample data. In this study, two different levels of significance α= 2% and α=5% were considered to evaluate the fit of the considered statistical distributions. The significance levels indicate that the chosen distribution has been selected with a confidence level of 98% and 95%.  Different types of goodness-of-fit tests are available. One of the most commonly used goodness-of-fit tests is the chi-squared test. It is often used to test if a sample of data came from a population with a specific distribution (Snedecor and Cochran, 1989). The limitation associated with this goodness-of-fit test is that it deals with data having only discrete values. For non-discrete or continuous data (i.e. dispalcement and base shear), the chi-squared test requires binning the sample data into arbitrary histogram cells which can directly impact the results of the chi-squared test (D’Agostino and Stepehens, 1986). Another shortcoming of the chi-squared test is that it requires a sufficient sample size in order for the chi-square approximation to be valid.  Goodness-of-fit tests based on the empirical density functions (EDF) provide more powerful goodness-of-fit tests for continuous data. Conover (1971) and D’Agostino and Stepehens (1986) provided a detailed review of the goodness-of-fit tests based on EDF statistics. In the present study, the Kolmogorov-Smirnov (K-S) “D” test was adopted (Mood et al., 1974). This test compares the empirical cumulative distribution function with the cumulative distribution function (CDF) of an assumed theoretical model. An advantage of this test is that the distribution of the K-S test statistic itself does not depend on the underlying cumulative distribution function being tested. Moreover, it is an exact test as compared to the chi-squared test. The limitation of K-S test is that it only applies to continuous distribution and tends to be more sensitive near the center of the distribution. The Kolmogorov-Smirnov (K-S) “D” test 207  statistics is based on a single, maximum vertical offset between the EDF and CDF over the range of sample data. The maximum offset will always occur just to the left or right of an observation point on the EDF. The value of “D” can be computed using equation B.1. D= max (|F(x_i)-(i-1)/n|,|F(x_i )-i/n|)   (B.1) where, n is the sample size, xi is the sample data arranged in ascending order, and F(xi) is the cumulative density function at xi for the statistical distribution under consideration. The first term in equation 1 represents the vertical offset between the EDF and the CDF to the left of xi, while the second term is the offset to the right of xi. The value of “D” represents the maximum of all offsets computed for the entire sample. In particular, the maximum difference (D) between the empirical (F (x)), based on n data, and the assumed theoretical (F (x)) (with known parameters) over the entire range of the random variable X (e.g., the yield displacement)are used as statistics. Then at various significance levels (2% and 5%), which are identified by the scalar α, Dst is compared with the critical value Dcritical, defined as P[Dst ≤ Dcritical ] = 1 - α. If the observed Dst is less than the critical value Dcritical, the assumed theoretical model is acceptable at the specified significance level α.           208   Table B.0.1. Results of K-S goodness-of-fit tests for Spalling Drift Limit Distribution Bridge Pier SMA-RC-1 SMA-RC-2 SMA-RC-3 SMA-RC-4 SMA-RC-5 Normal Dst 0.0711 Dst 0.0853 Dst 0.0962 Dst 0.0619 Dst 0.0764 α 0.05 0.02 α 0.05 0.02 α 0.05 0.02 α 0.05 0.02 α 0.05 0.02 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Fit Yes Yes Fit Yes Yes Fit No Yes Fit Yes Yes Fit Yes Yes Lognormal Dst 0.0727 Dst 0.0871 Dst 0.0986 Dst 0.0696 Dst 0.0791 α 0.05 0.02 α 0.05 0.02 α 0.05 0.02 α 0.05 0.02 α 0.05 0.02 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Fit Yes Yes Fit Yes Yes Fit No Yes Fit Yes Yes Fit Yes Yes Gamma Dst 0.0722 Dst 0.086 Dst 0.0978 Dst 0.0671 Dst 0.0787 α 0.05 0.02 α 0.05 0.02 α 0.05 0.02 α 0.05 0.02 α 0.05 0.02 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Fit Yes Yes Fit Yes Yes Fit No Yes Fit Yes Yes Fit Yes Yes Weibull Dst 0.0834 Dst 0.1098 Dst 0.1173 Dst 0.0798 Dst 0.0871 α 0.05 0.02 α 0.05 0.02 α 0.05 0.02 α 0.05 0.02 α 0.05 0.02 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Fit No Yes Fit No No Fit No Yes Fit Yes Yes Fit Yes Yes Best Fit   Normal   Normal   Normal   Normal   Normal     209  Table B.0.2. Results of K-S goodness-of-fit tests for Yielding Drift Limit Distribution Bridge Pier SMA-RC-1 SMA-RC-2 SMA-RC-3 SMA-RC-4 SMA-RC-5 Normal Dst 0.1029 Dst 0.1064 Dst 0.1056 Dst 0.0979 Dst 0.0965 α 0.05 0.02 α 0.05 0.02 α 0.05 0.02 α 0.05 0.02 α 0.05 0.02 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Fit No Yes Fit No Yes Fit No Yes Fit Yes Yes Fit Yes Yes Lognormal Dst 0.1023 Dst 0.1012 Dst 0.1005 Dst 0.0689 Dst 0.0832 α 0.05 0.02 α 0.05 0.02 α 0.05 0.02 α 0.05 0.02 α 0.05 0.02 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Fit No Yes Fit No No Fit No No Fit Yes Yes Fit Yes Yes Gamma Dst 0.1039 Dst 0.1171 Dst 0.1083 Dst 0.0698 Dst 0.0887 α 0.05 0.02 α 0.05 0.02 α 0.05 0.02 α 0.05 0.02 α 0.05 0.02 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Fit No Yes Fit No No Fit No No Fit Yes Yes Fit Yes Yes Weibull Dst 0.1056 Dst 0.1153 Dst 0.1019 Dst 0.0703 Dst 0.1111 α 0.05 0.02 α 0.05 0.02 α 0.05 0.02 α 0.05 0.02 α 0.05 0.02 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Fit No Yes Fit No Yes Fit No Yes Fit No Yes Fit No No Best Fit   Lognormal   Lognormal   Lognormal   Lognormal   Lognormal    210  Table B.0.3. Results of K-S goodness-of-fit tests for Crushing Drift Limit Distribution Bridge Pier SMA-RC-1 SMA-RC-2 SMA-RC-3 SMA-RC-4 SMA-RC-5 Normal Dst 0.0686 Dst 0.0965 Dst 0.073 Dst 0.0724 Dst 0.07916 α 0.05 0.02 α 0.05 0.02 α 0.05 0.02 α 0.05 0.02 α 0.05 0.02 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Fit Yes Yes Fit Yes Yes Fit Yes Yes Fit Yes Yes Fit Yes Yes Lognormal Dst 0.0681 Dst 0.0887 Dst 0.0725 Dst 0.0733 Dst 0.07952 α 0.05 0.02 α 0.05 0.02 α 0.05 0.02 α 0.05 0.02 α 0.05 0.02 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Fit Yes Yes Fit Yes Yes Fit Yes Yes Fit Yes Yes Fit Yes Yes Gamma Dst 0.0659 Dst 0.0832 Dst 0.0699 Dst 0.0706 Dst 0.0787 α 0.05 0.02 α 0.05 0.02 α 0.05 0.02 α 0.05 0.02 α 0.05 0.02 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Fit Yes Yes Fit Yes Yes Fit Yes Yes Fit Yes Yes Fit Yes Yes Weibull Dst 0.0976 Dst 0.1111 Dst 0.1035 Dst 0.0971 Dst 0.0871 α 0.05 0.02 α 0.05 0.02 α 0.05 0.02 α 0.05 0.02 α 0.05 0.02 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Dcritical 0.0955 0.1068 Fit No Yes Fit No No Fit No Yes Fit No Yes Fit Yes Yes Best Fit   Gamma   Gamma   Gamma   Gamma   Gamma 211   Appendix C Curve fitting In this study several regression analyses were conducted for developing different equations such as, bond stress of SMA rebar in concrete, plastic hinge length equation for SMA-RC bridge pier, residual drift prediction of SMA-RC elements. All these equations were developed based on data from experimental and numerical studies. However, all these equations contain several independent variable. In this study, different forms of regression equations were tested to find the "best fit" line or curve for a series of data points. The criteria for selecting the suitable equation type was the minimum square of the error between the original data and the values predicted by the equation. Although technique may not be the most statistically robust method of fitting a function to a data set, it has the advantage of being relatively simple. Table C.0.1 provides a list of equation tested throughout this study. Table C.0.1. List of equations tested Equation Category Equation Name Sample Equation Standard curves Linear axyy += 0  Quadratic 20 bxaxyy ++=  Logarithm 2 parameter  xayy ln0 +=  3 parameter ( )00 ln xxayy −+=  Polynomial Linear axyy += 0  Quadratic 20 bxaxyy ++=  Power 3 parameter baxyy += 0  212  Modified pareto function ( )baxy +−= 111   213  

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