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Hysteretic behaviour of steel- and fibre-reinforced elastomeric isolators fitted with superelastic shape… Hedayati Dezfuli, Farshad 2015

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HYSTERETIC BEHAVIOUR OF STEEL- AND FIBRE-REINFORCED ELASTOMERIC ISOLATORS FITTED WITH SUPERELASTIC SHAPE MEMORY ALLOY WIRE  by  Farshad Hedayati Dezfuli  B.A.Sc. (Aerospace Engineering), Sharif University of Technology, Iran, 2007 M.A.Sc. (Aerostructures), Sharif University of Technology, Iran, 2009  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)  April 2015   © Farshad Hedayati Dezfuli, 2015 ii  Abstract Among different types of earthquake protective mechanisms, elastomeric base isolators, also called rubber bearings (RBs), are one of the most well-known systems that are widely used in buildings and bridges. They can regulate the seismic response of structures, increase the public safety, and reduce the cost of repair and rehabilitation by providing lateral flexibility and dissipating the earthquake’s energy. RBs consist of elastomeric layers which are reinforced with steel shims or fibre-reinforced polymer composites. Seeking performance improvements, as well as cost and weight reduction led scientists to introduce different types of RBs. However, most RBs possess weaknesses such as limited shear strain capacity, un-recovered residual deformation, and instability due to large deformations. Using superelastic (ability to regain original shape upon unloading) shape memory alloy (SMA) in the form of wire, bar, or spring is a solution to partially overcome the aforementioned limitations. Its unique characteristics such as a flag-shaped hysteresis with zero residual deformation, superelastic effect (up to 13.5% recoverable strain) and a suitable fatigue property make it an ideal candidate for such applications. Objectives of this thesis are to propose a new generation SMA wire-based RBs (SMA-RB) and develop a novel constitutive model for such smart isolators in order to accurately capture their shear hysteretic behaviour. With the purpose of evaluating the performance of SMA-RBs in structural applications, the seismic fragility of a highway bridge isolated by SMA-RBs was assessed. First, a number of scaled carbon fibre-reinforced elastomeric isolators (C-FREIs) were manufactured and tested. Then, based on the experimental observations, numerical simulations were generated using finite element method (FEM). Results showed that incorporating SMA wires into natural and high-damping rubber bearings (NRB, HDRB) slightly improves the re-centring capability and energy dissipation capacity. However, equipping lead rubber bearing (LRB) with double cross SMA wires significantly reduces the residual deformation and noticeably enhances the energy damping property. It was also depicted that the developed hysteresis of SMA model can be characterized by three stiffnesses and two shear strain limits upon activation of SMA wires. Findings revealed that SMA wires can increase the reliability of elastomeric bearings and bridge system.   iii  Preface A version of Chapter 3 (sections 2 and 3) has been published. Hedayati Dezfuli, F., and Alam, M.S. 2012. Material modelling of high damping rubber in finite element method. In proceedings of the 3rd International Structural Specialty Conference, Edmonton, Alberta, Canada. June 6-9, 2012. I wrote the manuscript which was further edited by Dr. Alam. A version of Chapter 3 (sections 3 to 5) has been published. Hedayati Dezfuli, F., and Alam, M.S. 2013. Multi-criteria optimization and seismic performance assessment of carbon FRP-based elastomeric isolator. Engineering Structures, 49: 525-540. I wrote the manuscript which was further edited by Dr. Alam. A version of Chapter 4 (sections 4, 6, 8, and 9) has been published in two articles. Hedayati Dezfuli, F., and Alam, M.S. 2013. Sensitivity analysis of carbon fiber-reinforced elastomeric isolators based on experimental tests and finite element simulation. Bulletin of Earthquake Engineering, 12(2): 1025-1043; Hedayati Dezfuli, F., and Alam, M.S. 2014. Performance of carbon fiber-reinforced elastomeric isolators manufactured in a simplified process: experimental investigations. Structural Control and Health Monitoring. 21(11): 1347-1359. I conducted some parts of experimental tests in the Applied Dynamics laboratory at McMaster University. I wrote the manuscript which was further edited by Dr. Alam. A version of Chapter 4 (section 7) has been published. Hedayati Dezfuli, F., and Alam, M.S. 2014. Finite element simulation of carbon fiber-reinforced elastomeric isolators manufactured through a cold-vulcanization process. In Proceedings of 9th International Conference on Short and Medium Span Bridges, SMSB, Calgary, Alberta, Canada. July 15-18, 2014. A version of Chapter 5 (section 2) has been published. The journal paper has been considered as the featured and the most downloaded article in Smart Materials and Structures journal and has been selected by the editors of the journal in the ‘Highlights of 2013’ collection on the basis of referee endorsement, novelty and scientific impact. Hedayati Dezfuli, F., and Alam, M.S. 2013. Shape memory alloy wire-based smart natural rubber bearing, Smart Materials and Structures, 22(4) 045013. I wrote the manuscript which was further edited by Dr. Alam. iv  A version of Chapter 5 (sections 3 and 5) has been published. Hedayati Dezfuli, F., and Alam, M.S. 2014. Performance-based assessment and design of FRP-based high damping rubber bearing incorporated with shape memory alloy wires. Engineering Structures, 61: 166-183. I wrote the manuscript which was further edited by Dr. Alam. A version of Chapter 5 (sections 2 and 3) has been published. Hedayati Dezfuli, F., and Alam, M.S. 2013. Performance comparison between SMA-based natural rubber bearing and SMA-based high damping rubber bearing. In Proceedings of 7th National Seismic Conference on Bridges & Highways, Oakland, California, USA. May 20-22, 2013. I wrote the manuscript which was further edited by Dr. Alam. A version of Chapter 5 (section 4) has been submitted to Journal of Bridge Engineering (ASCE). Hedayati Dezfuli, F., and Alam, M.S. 2015. Hedayati Dezfuli, F., and Alam, M.S. 2015. Smart lead rubber bearings equipped with ferrous shape memory alloy wires for seismically isolating highway bridges. Submitted to Earthquake Engineering and Structural Dynamics, Manuscript ID: EQE-15-0127. I wrote the manuscript which was further edited by Dr. Alam. A version of Chapter 6 has been accepted for publication in Smart Materials and Structures journal. Hedayati Dezfuli, F., and Alam, M.S. 2015. Hysteresis model of shape memory alloy (SMA) wire-based laminated rubber bearing under compression and unidirectional shear loadings. Smart Materials and Structures, In Press. I wrote the manuscript which was further edited by Dr. Alam. A version of Chapter 7 has been accepted in Structures Congress 2015 Conference. Hedayati Dezfuli, F., and Alam, M.S. 2015. Vulnerability assessment of multi-span continuous steel-girder bridges isolated by SMA wire-based natural rubber bearing (SMA-NRB). Structures Congress 2015 Conference, Portland, Oregon, USA. April 23-25, 2015. I wrote the manuscript which was further edited by Dr. Alam. Publications arising from the work presented in this dissertation are listed as follows: 1- Hedayati Dezfuli, F., and Alam, M.S. 2015. Hysteresis model of shape memory alloy (SMA) wire-based laminated rubber bearing under compression and unidirectional shear loadings. Smart Materials and Structures, In Press. 2- Hedayati Dezfuli, F., and Alam, M.S. 2015. Vulnerability assessment of multi-span continuous steel-girder bridges isolated by SMA wire-based natural rubber bearing v  (SMA-NRB), Structures Congress 2015 Conference, Portland, Oregon, USA (April 23-25, 2015). 3- Hedayati Dezfuli, F., and Alam, M.S. 2015. Smart lead rubber bearings equipped with ferrous shape memory alloy wires for seismically isolating highway bridges. Submitted to Earthquake Engineering and Structural Dynamics, Manuscript ID: EQE-15-0127. 4- Hedayati Dezfuli, F., and Alam, M.S. 2014. Performance-based assessment and design of FRP-based high damping rubber bearing incorporated with shape memory alloy wires. Engineering Structures, 61: 166-183. 5- Hedayati Dezfuli, F., and Alam, M.S. 2014. Performance of carbon fiber-reinforced elastomeric isolators manufactured in a simplified process: experimental investigations. Structural Control and Health Monitoring. 21(11): 1347–1359. 6- Hedayati Dezfuli, F., and Alam, M.S. 2014. Finite element simulation of carbon fiber-reinforced elastomeric isolators manufactured through a cold-vulcanization process. In Proceedings of 9th International Conference on Short and Medium Span Bridges, SMSB, Calgary, Alberta, Canada. July 15-18, 2014. 7- Hedayati Dezfuli, F., and Alam, M.S. 2013. Sensitivity analysis of carbon fiber-reinforced elastomeric isolators based on experimental tests and finite element simulation. Bulletin of Earthquake Engineering, 12(2): 1025-1043. 8- Hedayati Dezfuli, F., and Alam, M.S. 2013. Shape memory alloy wire-based smart natural rubber bearing,” Smart Materials and Structures, 22(4): 045013. 9- Hedayati Dezfuli, F., and Alam, M.S. 2013. Multi-criteria optimization and seismic performance assessment of carbon FRP-based elastomeric isolator. Engineering Structures, 49: 525-540. 10- Hedayati Dezfuli, F., and Alam, M.S. 2013. Performance comparison between SMA-based natural rubber bearing and SMA-based high damping rubber bearing. In Proceedings of 7th National Seismic Conference on Bridges & Highways, Oakland, California, USA. May 20-22, 2013. 11- Hedayati Dezfuli, F., and Alam, M.S. 2012. Material modeling of high damping rubber in finite element method. In Proceedings of the 3rd International Structural Specialty Conference, Edmonton, Alberta, Canada. June 6-9 2012.  vi  Table of Contents Abstract .................................................................................................................................... ii Preface ..................................................................................................................................... iii Table of Contents ................................................................................................................... vi List of Tables ........................................................................................................................... x List of Figures ....................................................................................................................... xiv List of Symbols and Abbreviations .................................................................................. xxiii Acknowledgements ............................................................................................................. xxv Dedication ........................................................................................................................... xxvi Chapter 1 Introduction and Thesis Organization ............................................................ 1 1.1 General ....................................................................................................................... 1 1.2 Objectives ................................................................................................................... 2 1.2.1 Performance Evaluation of Carbon Fibre-Reinforced Elastomeric Isolators                 (C-FREIs) ....................................................................................................................... 2 1.2.2 Development of a Novel Shape Memory Alloy-based Rubber Bearing (SMA-RB)...... 3 1.2.3 Development of a New Hysteresis Model for SMA-RBs .............................................. 3 1.2.4 Seismic Fragility Assessment of a Highway Bridge Isolated by SMA-RBs .................. 3 1.3 Outline of the Thesis .................................................................................................. 3 Chapter 2 Literature Review ............................................................................................. 7 2.1 General ....................................................................................................................... 7 2.2 Steel-Reinforced Elastomeric Isolators (SREI).......................................................... 9 2.2.1 Natural Rubber Bearing (NRB) ...................................................................................... 9 2.2.2 High Damping Rubber Bearing (HDRB) ..................................................................... 11 2.2.3 Lead Rubber Bearing (LRB) ........................................................................................ 12 2.2.4 Ball Rubber Bearing (BRB) ......................................................................................... 13 2.3 Fibre-Reinforced Elastomeric Isolators (FREI) ....................................................... 14 2.4 Smart Rubber Bearings ............................................................................................ 19 2.4.1 Shape Memory Alloy (SMA) ....................................................................................... 19 2.4.2 Shape Memory Alloy-based Rubber Bearings (SMA-RB) .......................................... 21 2.5 Summary .................................................................................................................. 25 Chapter 3 Multi-Criteria Optimization of Rubber Bearings Reinforced with                 CFRP Composites ........................................................................................... 27 vii  3.1 General ..................................................................................................................... 27 3.2 Material Modelling ................................................................................................... 27 3.2.1 Bilinear Model .............................................................................................................. 28 3.2.2 Hyperelastic Model ...................................................................................................... 29 3.2.3 Viscoelastic Model ....................................................................................................... 31 3.2.4 Viscoplastic Model ....................................................................................................... 32 3.2.5 Hyper-Viscoelastic Model ............................................................................................ 32 3.2.6 Comparing Material Models ......................................................................................... 33 3.3 Numerical Validation and Verification .................................................................... 36 3.4 Sensitivity Analysis .................................................................................................. 39 3.4.1 Performance of CFR-HDRB ........................................................................................ 45 3.4.2 Regression Models ....................................................................................................... 47 3.4.3 Effect of Number of Rubber Layers ............................................................................. 50 3.4.4 Effect of Thickness of FRP Reinforcement .................................................................. 51 3.4.5 Effect of shear modulus of rubber layers ...................................................................... 52 3.5 Multi-Criteria Optimization ..................................................................................... 53 3.5.1 Theory .......................................................................................................................... 53 3.5.2 Optimization of CFR-HDRB ........................................................................................ 56 3.6 Summary .................................................................................................................. 58 Chapter 4 Performance of Fibre-Reinforced Elastomeric Isolators:                Experimental and Numerical Investigations ................................................ 60 4.1 General ..................................................................................................................... 60 4.2 Manufacturing Process ............................................................................................. 61 4.3 Test Setup ................................................................................................................. 64 4.4 Experimental Tests ................................................................................................... 65 4.4.1 Failure Tests ................................................................................................................. 65 4.4.2 Vertical Compression Test ........................................................................................... 67 4.4.3 Lateral Cyclic Test ....................................................................................................... 70 4.5 Experimental Parametric Study ................................................................................ 76 4.5.1 Vertical Pressure ........................................................................................................... 77 4.5.2 Lateral Cyclic Rate ....................................................................................................... 83 4.5.3 Number of Rubber Layers ............................................................................................ 87 4.5.4 Thickness of Fibre-reinforced Layers ........................................................................... 88 4.6 Possibility of Using C-FREIs in Bonded Applications ............................................ 91 4.7 Numerical Validation and Verification .................................................................... 91 viii  4.7.1 Finite Element Modelling ............................................................................................. 92 4.7.2 Delamination ................................................................................................................ 94 4.7.3 Comparison................................................................................................................... 96 4.8 Performance of Full Scale C-FREIs ......................................................................... 98 4.9 Numerical Parametric Study .................................................................................. 101 4.9.1 Number of Elastomeric Layers ................................................................................... 101 4.9.2 Thickness of Elastomeric Layers ................................................................................ 103 4.9.3 Thickness of Fibre-Reinforced Sheets ........................................................................ 106 4.10 Summary ............................................................................................................. 108 Chapter 5 Smart Elastomeric Isolators Equipped with Shape Memory Alloy                Wires ............................................................................................................... 110 5.1 General ................................................................................................................... 110 5.2 SMA-based Natural Rubber Bearings (SMA-NRB) .............................................. 111 5.2.1 SMA-NRB equipped with Straight Wires .................................................................. 112 5.2.2 SMA-NRB equipped with Cross Wires ...................................................................... 113 5.2.3 Efficiency of SMAs .................................................................................................... 115 5.2.4 Finite Element Modelling ........................................................................................... 117 5.2.5 Results and Discussions ............................................................................................. 122 5.3 SMA-based High Damping Rubber Bearing (SMA-HDRB) ................................. 137 5.3.1 SMA-HDRB Equipped with Straight Wires .............................................................. 138 5.3.2 SMA-HDRB equipped with Cross Wires ................................................................... 139 5.3.3 Efficiency of Wires ..................................................................................................... 139 5.3.4 Finite Element Modelling ........................................................................................... 141 5.3.5 Results and Discussions ............................................................................................. 145 5.4 SMA-based Lead Rubber Bearings (SMA-LRB) .................................................. 156 5.4.1 Finite Element Validation ........................................................................................... 157 5.4.2 Performance of SMA-LRB ......................................................................................... 160 5.5 Design of SMA-based Rubber Bearings ................................................................ 166 5.6 Summary ................................................................................................................ 169 Chapter 6 Constitutive Model of SMA-based Elastomeric Isolators ......................... 171 6.1 General ................................................................................................................... 171 6.2 SMA-based Rubber Bearings ................................................................................. 173 6.2.1 Superposition Method ................................................................................................ 175 6.3 Hysteresis Model .................................................................................................... 177 ix  6.3.1 Rubber Bearing Model ............................................................................................... 177 6.3.2 SMA Wires Model ..................................................................................................... 179 6.3.3 Verification of SMA Wires Model ............................................................................. 194 6.3.4 Hysteresis of SMA-LRB ............................................................................................ 197 6.4 Summary ................................................................................................................ 200 Chapter 7 Seismic Fragility Assessment of Multi-Span Continuous                 Steel-Girder Bridges ..................................................................................... 201 7.1 General ................................................................................................................... 201 7.2 Seismic Fragility Methodology .............................................................................. 203 7.2.1 Limit/Damage States .................................................................................................. 206 7.2.2 Ground Motion Suite .................................................................................................. 209 7.2.3 System Fragility Curves ............................................................................................. 211 7.3 Fragility Assessment of a Highway Bridge............................................................ 214 7.3.1 Finite Element Modelling ........................................................................................... 214 7.3.2 Probabilistic Seismic Demand Models (PSDM) ........................................................ 219 7.3.3 Component Fragility Curves ...................................................................................... 224 7.3.4 System Fragility Curves ............................................................................................. 233 7.4 Summary ................................................................................................................ 244 Chapter 8 Summary, Conclusions, and Future Works ............................................... 246 8.1 Summary ................................................................................................................ 246 8.2 Conclusions ............................................................................................................ 248 8.2.1 Carbon Fibre-Reinforced Elastomeric Isolator (C-FREI) .......................................... 248 8.2.2 SMA Wire-based Rubber Bearing (SMA-RB) ........................................................... 250 8.3 Future Works .......................................................................................................... 254 8.3.1 Experimental Study .................................................................................................... 254 8.3.2 Numerical Study ......................................................................................................... 257 References ............................................................................................................................ 258 Appendices ........................................................................................................................... 269 Appendix A: Design procedure of determining the radius and pre-strain of                        SMA wires .................................................................................................. 269 Appendix B: Bilinear kinematic hardening model flowchart ........................................... 271    x  List of Tables  Table ‎2.1.      Mechanical characteristics of different shape memory alloys (SMAs) ......................... 20 Table ‎3.1.      Hyper-viscoelastic material model constants ................................................................ 38 Table ‎3.2.      Material properties for CFRP composite material (Howie and Karbhari, 1994) ........... 41 Table ‎3.3.      Parameters and their levels considered in the sensitivity analysis ................................. 43 Table ‎3.4.      A 33 full factorial design with 3 factors and 3 levels ..................................................... 44 Table ‎3.5.      Geometrical properties of CFR-HDRBs ........................................................................ 44 Table ‎3.6.      Responses of CFR-HDRBs in 33 full factorial design ................................................... 46 Table ‎3.7.      Coefficients of each factor in the regression models ..................................................... 47 Table ‎3.8.      Physical properties of CFR-HDRBs .............................................................................. 48 Table ‎3.9.      Performance characteristics of eight CFR-HDRBs obtained from FE analyses  and regression models ................................................................................................... 49 Table ‎3.10.    Assignment of values for a 10-point scale ..................................................................... 54 Table ‎3.11.    Assigned normalized weights to criteria ........................................................................ 56 Table ‎3.12.    Properties of 1st to 5th ranked CFR-HDRBs ................................................................... 57 Table ‎3.13.    Properties of the worst CFR-HDRB .............................................................................. 57 Table ‎4.1.      Physical and geometrical properties of C-FREIs ........................................................... 63 Table ‎4.2.      Performance characteristics of C-FREIs in the vertical direction.................................. 69 Table ‎4.3.      Horizontal operational characteristics of C-FREIs at three shear strain amplitudes ..... 73 Table ‎4.4.      Effective horizontal stiffness and equivalent viscous damping of C-FREIs at   different shear strain amplitudes and vertical pressures ................................................ 80 Table ‎4.5.      Effective horizontal stiffness and equivalent viscous damping of C-FREIs  at  different lateral cyclic rates (20 mm/s, 30 mm/s, and 75 mm/s) ................................... 86 Table ‎4.6.      Material constants of Ogden-Prony model .................................................................... 92 Table ‎4.7.      Hyperelastic material constants of the glue ................................................................... 95 Table ‎4.8.      Results obtained from experimental tests and FE numerical simulations ..................... 97 Table ‎4.9.      Geometrical properties of C-FREIs with different sizes................................................ 99 Table ‎4.10.    Performance specifications of C-FREIs in the horizontal and vertical directions   for different lengths, widths and heights of laminated core ........................................ 100 Table ‎4.11.    Stiffnesses and damping coefficient of C-FREIs with different numbers of  rubber layers ................................................................................................................ 101 xi  Table ‎4.12.    Normalized operational characteristics and their change rates  for  different numbers of rubber layers .............................................................................. 103 Table ‎4.13.    Stiffnesses and damping coefficient of C-FREIs with different  thicknesses of rubber layers ......................................................................................... 104 Table ‎4.14.    Normalized operational characteristics and their change rates  for  different thicknesses of rubber layers .......................................................................... 105 Table ‎4.15.    Stiffnesses and damping coefficient of C-FREIs  with different thicknesses of  fibre-reinforced layers. ................................................................................................ 106 Table ‎4.16.    Normalized operational characteristics and their change rates for different  thicknesses of carbon fibre-reinforced layers .............................................................. 107 Table ‎5.1.      Geometrical properties of NRBs.................................................................................. 111 Table ‎5.2.      Required length for SMA wires in the straight configuration (Figure  5.2a) ................ 113 Table ‎5.3.      Required length for SMA wires in the cross configuration (Figure  5.2b) ................... 114 Table ‎5.4.      Strain in SMA wires for two configurations and two aspect ratios  at  different shear strain amplitudes .................................................................................. 116 Table ‎5.5.      Mechanical characteristics of different shape memory alloys (SMAs) ....................... 116 Table ‎5.6.      Superelastic range of SMAs for different aspect ratios and shear strain  amplitudes in cross configuration ................................................................................ 117 Table ‎5.7.      Material constants of hyper-viscoelastic model ........................................................... 118 Table ‎5.8.      Four cases of SMA-based NRBs considered in the FE simulations ............................ 123 Table ‎5.9.      Operational characteristics of NRB and SMA-NRBs for different  wire configurations and aspect ratios .......................................................................... 125 Table ‎5.10.    Operational characteristics of SMA-NRB-C2 with different wire radii  compared to NRB-2 ..................................................................................................... 131 Table ‎5.11.    Operational characteristics of NRB-2 and SMA-NRB-C2 for different  amounts of pre-strain ................................................................................................... 136 Table ‎5.12.    Geometrical properties of CFR-HDRBs ...................................................................... 138 Table ‎5.13.    Required length of SMA wire for seven CFR-HDRBs in the straight and cross configurations .............................................................................................................. 138 Table ‎5.14.    Strain in SMA wires of SMA-HDRBs with different wire configurations  and aspect ratios........................................................................................................... 140 Table ‎5.15.    Operational range of SMAs for different shear strains and aspect ratios in cross configuration ................................................................................................................ 140 xii  Table ‎5.16.    Hyper-viscoelastic material model constants .............................................................. 141 Table ‎5.17.    Effective horizontal stiffness and residual deformation of SMA-HDRB-C1s  with FeNCATB and NiTi45 wires ................................................................................ 146 Table ‎5.18.    Effective horizontal stiffness of CFR-HDRB and SMA-HDRBs (R = 0.12) .............. 148 Table ‎5.19.    Residual deformation of CFR-HDRB and SMA-HDRBs (R = 0.12) .......................... 148 Table ‎5.20.    Effective horizontal stiffness of CFR-HDRB and SMA-HDRB (R = 0.36) ................ 151 Table ‎5.21.    Residual deformation of CFR-HDRB and SMA-HDRBs (R = 0.36) .......................... 151 Table ‎5.22.    The effective horizontal stiffness and the residual deformation of  SMA-HDRB-C1 compared to those of the CFR-HDRB for different  radii of SMA wire (R = 0.12) ...................................................................................... 153 Table ‎5.23.    Characteristics of CFR-HDRB and SMA-HDRB-C2 for different amounts  of pre-strain  in SMA wires (R = 0.36, γ = 200%) ....................................................... 155 Table ‎5.24.    Material constants of the hyper-viscoelastic model ..................................................... 158 Table ‎5.25.    Residual deformations of SMA-LRB and SMA-LRB (PS) at different  shear strains ................................................................................................................. 166 Table ‎6.1.      Properties of LRB and SMA wire................................................................................ 175 Table ‎6.2.      Properties of bilinear kinematic hardening model used for LRB ................................ 178 Table ‎6.3.      Components of nodal forces in x, y, and z directions .................................................. 183 Table ‎6.4.      Input parameters and characteristics of DC-SMAW model ........................................ 194 Table ‎7.1.      Qualitative limit states (FEMA, 2003) ........................................................................ 207 Table ‎7.2.      Limit/damage states of bridge components ................................................................. 208 Table ‎7.3.      Capacity of RC piers and elastomeric isolation bearings ............................................ 208 Table ‎7.4.      Characteristics of the earthquake records .................................................................... 210 Table ‎7.5.      Material properties of concrete and steel reinforcement .............................................. 216 Table ‎7.6.      Properties of bilinear model with kinematic hardening ............................................... 219 Table ‎7.7.      Regression coefficients of displacement ductility of pier ............................................ 221 Table ‎7.8.      Regression coefficients of shear strain of elastomeric bearings .................................. 224 Table ‎7.9.      Mean and standard deviation of fragility functions for the bridge pier ....................... 225 Table ‎7.10.    Damage probabilities of bridge pier fitted with smart rubber bearings at  1.0g PGA ..................................................................................................................... 228 Table ‎7.11.    Mean and standard deviation of fragility functions for the elastomeric bearing ......... 229 Table ‎7.12.    Damage probabilities of rubber bearings at PGAs of 0.5g, 1.0g, 1.5g, and 2.0g ........ 231 xiii  Table ‎7.13.    Damage probabilities of smart rubber bearings at PGAs of 0.5g, 1.0g, and 2.0g ........ 233 Table ‎7.14.    Damage probabilities of the bridge isolated by conventional rubber bearings   at four PGA values ...................................................................................................... 238 Table ‎7.15.    Damage probabilities of the bridge isolated by SMA-based rubber bearings   at four PGA values ...................................................................................................... 240 Table ‎7.16.    Damage probabilities of the bridge isolated by HDRB and SMA-HDRB   at four PGA values and four damage states ................................................................. 243 Table ‎7.17.    Median values of PGA for the bridge system equipped with different  isolation systems .......................................................................................................... 244   xiv  List of Figures Figure ‎1.1.       Summary of the goals and topics covered in the thesis ................................................. 6 Figure ‎2.1.       Applications of base isolation systems in (a) buildings and (b) bridges ....................... 8 Figure ‎2.2.       Elastomeric isolators in rectangular shapes................................................................... 9 Figure ‎2.3.       Laminated rubber bearing ........................................................................................... 10 Figure ‎2.4.       Lead-plug rubber bearing (LRB)................................................................................. 13 Figure ‎2.5.       Stress-strain curve for SMAs; (a) superelastic effect, (b) shape memory effect ......... 19 Figure ‎2.6.       Earthquake protective systems .................................................................................... 26 Figure ‎3.1.       Nonlinear material models in ANSYS (ANSYS Documentation, Release 14.0) ....... 28 Figure ‎3.2.       Force-displacement relation for bilinear material  (adapted from (Ozkaya et al., 2011)) ......................................................................... 28 Figure ‎3.3.       Cyclic horizontal displacement (f = 0.25 Hz, γ = 100%) ............................................ 34 Figure ‎3.4.       Normalized shear force versus lateral displacement for HDR layer simulated  using different material models; (a) Hyperelastic (Mooney-Rivlin), (b) Bilinear with kinematic hardening, (c) Viscoplastic (Perzyna),  (d) Hyper-viscoelastic (Ogden-Prony), (e) Hyper-viscoelastic  (Mooney-Rivlin and Prony), (f) Hyper-viscoelastic (Bergstrom-Boyce) .................. 34 Figure ‎3.5.       Stable hysteresis loops at different strain amplitudes   (adapted from (Dall’Asta and Ragni, 2006)) .............................................................. 36 Figure ‎3.6.       Top and side views of a HDRB (adapted from (Dall’Asta and Ragni, 2006)) ........... 36 Figure ‎3.7.       HDRB with mapped mesh in ANSYS (ANSYS Mechanical APDL, Release 14) ..... 37 Figure ‎3.8.       Lateral force-deflection hysteresis curve (P = 0 MPa, f = 0.49 Hz, γ = 90%) (Experimental results are adapted from (Dall’Asta and Ragni, 2006)) ...................... 38 Figure ‎3.9.       Lateral force-deflection hysteresis curves of HDRB at γ = 43%, 90%, 155%,  and 200% obtained through FEM and experiment (Dall’Asta and Ragni, 2006) ...... 39 Figure ‎3.10.     Fibre-reinforced elastomeric isolator; (a) plan view, (b) side view ............................ 41 Figure ‎3.11.     Shear modulus of rubber as a function of temperature   (adapted from (GoodCo Z-Tech, 2010)) .................................................................... 43 Figure ‎3.12.     Input and output of the system for performance analysis ........................................... 45 Figure ‎3.13.     Normalized performance characteristics for different CFR-HDRBs .......................... 47 Figure ‎3.14.     Vertical stiffness calculated through FE analysis and regression model   for 8 CFR-HDRBs ...................................................................................................... 49 xv  Figure ‎3.15.     Effective horizontal stiffness calculated through FE analysis and  regression model for 8 CFR-HDRBs ......................................................................... 49 Figure ‎3.16.     Equivalent viscous damping calculated through FE analysis and  regression model for 8 CFR-HDRBs ......................................................................... 50 Figure ‎3.17.     Effect of number of rubber layers on the CFR-HDRB’s behaviour;   (a) tf = 0.31 mm, Gr = 0.6 MPa, (b) tf = 0.62 mm, Gr = 0.7 MPa,  (c) tf = 0.93 mm, Gr = 0.8 MPa ................................................................................... 50 Figure ‎3.18.     Effect of thickness of fibre-reinforced plates on the CFR-HDRB’s behaviour;   (a) ne = 8, Gr = 0.6 MPa, (b) ne = 9, Gr = 0.7 MPa, (c) ne = 10, Gr = 0.8 MPa .......... 51 Figure ‎3.19.     Effect of shear modulus of elastomer on the CFR-HDRB’s behaviour;   (a) ne = 8, tf = 0.31 mm, (b) ne = 9, tf = 0.62 mm, (c) ne = 10, tf = 0.93 mm ............... 52 Figure ‎3.20.     Flow chart of multi-objective optimization ................................................................. 56 Figure ‎3.21.     Lateral force-deflection hysteresis curve for the 1st and 2nd best CFR-HDRBs   (f = 0.2 Hz, γ = 1.0) .................................................................................................... 58 Figure ‎4.1.       Manufacturing process of C-FREIs; (a) bi-directional carbon fibre fabric  between two rubber layers, (b) attaching fibre fabrics to rubber layers  by adding glue (rubber cement), (c) cured laminated pad,  (d) laminated pad cut with water-jet technology ........................................................ 62 Figure ‎4.2.       C-FREIs manufactured with different properties;  (a) A1; (b) B1; (c) C1;  (d) D1; (e) E1; (f) F1; (g) A3; (h) B4; (i) C2.............................................................. 63 Figure ‎4.3.       C-FREI fixed in the test setup ..................................................................................... 64 Figure ‎4.4.       Test setup and related equipment ................................................................................ 65 Figure ‎4.5.       C-FREI-F1 under vertical pressures of (a) 0 MPa, (b) 3 MPa, and (c) 6.1 MPa ......... 66 Figure ‎4.6.       C-FREI-C1 under a combination of 3 MPa vertical pressure and  lateral displacements of (a) 100% tr, (b) 125% tr, (c) 200% tr, and (d) 0% ................ 66 Figure ‎4.7.       Variation of vertical pressure over time for three design pressures ............................ 67 Figure ‎4.8.       Vertical force-deflection curves under 0.75 MPa, 1.5 MPa, and  3.0 MPa pressures;  (a) C-FREI-A1; (b) C-FREI-B1; (c) C-FREI-D1;  (d) C-FREI-E1 ............................................................................................................ 68 Figure ‎4.9.       Input load in the cyclic tests; (a) vertical pressure, (b) cyclic displacements  at  shear strains of 25%, 50%, and 100% ................................................................... 70 Figure ‎4.10.     C-FREIs under cyclic tests at different shear strains; (a) applying constant  pressure,  (b) shear strain of 25%, (c) shear strain of 50%,  (d) shear strain of 100% ............................................................................................. 72 xvi  Figure ‎4.11.     Shear hysteretic response of C-FREIs at 25%tr, 50%tr, and 100%tr ........................... 73 Figure ‎4.12.     Deformation of C-FREI-E1 under the maximum applied shear strain amplitude (100%)  and a vertical pressure of 3 MPa .................................................................. 74 Figure ‎4.13.     Effective shear modulus decay versus shear strain for four C-FREIs ......................... 75 Figure ‎4.14.     Input loads in the cyclic tests; (a) variation of vertical pressure over time,   (b) variation of shear strain over time (25%, 50%, and 100%) .................................. 77 Figure ‎4.15.     Lateral force-displacement hysteresis curves of manufactured C-FREIs under different vertical pressures (P = 1 MPa, 2 MPa, and 3 MPa) and shear strains  (γ = 25%, 50%, and 100%) ......................................................................................... 79 Figure ‎4.16.     Effective horizontal stiffness of C-FREIs under different vertical pressures   (1 MPa, 2 MPa, and 3 MPa) and shear strains of (a) 25%, (b) 50%, and  (c) 100% ..................................................................................................................... 81 Figure ‎4.17.     Equivalent viscous damping of C-FREIs under different vertical pressures   (1 MPa, 2 MPa, and 3 MPa) and shear strains of (a) 25%, (b) 50%, and  (c) 100% ..................................................................................................................... 82 Figure ‎4.18.     Lateral cyclic displacement (50% tr) for three different cyclic rates ........................... 84 Figure ‎4.19.     Lateral force-displacement hysteresis curves at different lateral rates   (20 mm/s, 30 mm/s, and 75 mm/s) and 50% shear strain ........................................... 85 Figure ‎4.20.     Effective horizontal stiffness of C-FREIs under different lateral cyclic rates   (20 mm/s, 30 mm/s, and 75 mm/s) at 50% shear strain ............................................. 86 Figure ‎4.21.     Equivalent viscous damping of C-FREIs under different lateral cyclic rates   (20 mm/s, 30 mm/s, and 75 mm/s) at 50% shear strain ............................................. 87 Figure ‎4.22.     Effect of number of rubber layers on the performance of bearings  at  different shear strain amplitudes (25%, 50%, and 100%);   (a) effective horizontal stiffness, (b) equivalent viscous damping. ............................ 88 Figure ‎4.23.     Effect of fibre-reinforced layers on the performance of bearings  at  different shear strain amplitudes (25%, 50%, and 100%);   (a) effective horizontal stiffness, (b) equivalent viscous damping ............................. 89 Figure ‎4.24.     Carbon fibre-reinforced elastomeric pad (B1) after being tested ................................ 90 Figure ‎4.25.     Carbon fibre-reinforced elastomeric isolator, C-FREI-E1 .......................................... 93 Figure ‎4.26.     C-FREI-E1 with a mapped mesh in ANSYS (ANSYS Mechanical APDL,  Release 14.0); (a) full model, (b) half model ............................................................. 93 Figure ‎4.27.     C-FREI-E1 before and after delamination; (a) γ = 50%, (b) γ = 100% ....................... 96 xvii  Figure ‎4.28.     C-FREI-E1 under 3 MPa vertical pressure; (a) FE half model,  (b) manufactured sample ............................................................................................ 96 Figure ‎4.29.     Shear hysteretic response of C-FREI-E1 under 3 MPa vertical pressure at   (a) γ = 25%, (b) γ = 50%, and (c) γ = 100% obtained from experimental and  FE numerical results ................................................................................................... 97 Figure ‎4.30.     Shear hysteretic response of (a) C-FREI-A1 and (b) C-FREI-D1 at  γ = 50% and P = 3 MPa .............................................................................................. 98 Figure ‎4.31.     The effect of number of rubber layers on; (a) vertical stiffness,  (b) effective horizontal stiffness, (c) equivalent viscous damping ............................................... 102 Figure ‎4.32.     The effect of thickness of rubber layers on; (a) vertical stiffness,  (b) effective horizontal stiffness, (c) equivalent viscous damping ............................................... 105 Figure ‎4.33.     The effect of thickness of carbon fibrereinforced layers on;  (a) vertical stiffness,  (b) effective horizontal stiffness,  (c) equivalent viscous damping ................................................................................ 107 Figure ‎5.1.       Schematic view of the elastomeric isolator; (a) Plan view of NRB-1 and  NRB-2,  (b) Side view of NRB-1, (c) Side view of NTB-2 ..................................... 112 Figure ‎5.2.       Smart rubber bearing; (a) straight SMA wires, (b) cross SMA wires ....................... 113 Figure ‎5.3.       Variation of strain in SMA wire as a function of shear strain amplitude and   aspect ratio for (a) straight configuration and (b) cross configuration ..................... 114 Figure ‎5.4.       Steel-reinforced NRB; (a) side view, (b) plan view  (adapted from (Dehghani Ashkezari et al., 2008)) ................................................... 119 Figure ‎5.5.       NRB with a mapped mesh in ANSYS (ANSYS Mechanical APDL,  Release 14.0) ............................................................................................................ 119 Figure ‎5.6.       Lateral force-deflection curves of steel-reinforced NRB   (experimental results are adapted from (Dehghani Ashkezari et al., 2008)) ............ 120 Figure ‎5.7.       Idealized stress-strain curve of NiTi45 and FeNCATB SMAs at  room temperature ..................................................................................................... 121 Figure ‎5.8.       Decoupled systems; (a) Elastomeric isolator, (b) SMA wires with  internal forces ........................................................................................................... 122 Figure ‎5.9.       Lateral force-deflection curve of (a) NRB-1, (b) SMA-NRB-C1, and  (c) SMA-NRB-S1;  at γ = 100%, 150%, and 200% ................................................. 124 Figure ‎5.10.     Lateral force-deflection curve of (a) NRB-2, (b) SMA-NRB-C2, and  (c) SMA-NRB-S2;  at γ = 100%, 150%, and 200% ................................................. 127 xviii  Figure ‎5.11.     Effective horizontal stiffness of NRBs and SMA-NRBs with straight and cross configurations of wires; γ = 100%, 150%, and 200% .............................................. 128 Figure ‎5.12.     Residual deformation of NRBs and SMA-NRBs with straight and cross configurations of wires; γ = 100%, 150%, and 200% .............................................. 128 Figure ‎5.13.     Dissipated energy of NRBs and SMA-NRBs with straight and cross  configurations of wires; γ = 100%, 150%, and 200% .............................................. 129 Figure ‎5.14.     Equivalent viscous damping of NRBs and SMA-NRBs with straight and cross configurations of wires; γ = 100%, 150%, and 200% .............................................. 129 Figure ‎5.15.     Lateral force-deflection curve of SMA-NRB-C2 for different thicknesses of  SMA wires; (a) rSMA = 1.25 mm, (b) rSMA = 2.5 mm, and  (c) rSMA =  4 mm; γ = 100%, 150%, and 200% ........................................................ 130 Figure ‎5.16.     Operational characteristics of NRB-2 and SMA-NRB-C2 with different  wires’ thickness; (a) Effective horizontal stiffness, (b) Residual deformation,  (c) Dissipated energy per cycle, (d) Equivalent viscous damping ........................... 133 Figure ‎5.17.     Stress-strain curve of ferrous SMA (FeNCATB);  (a) regular  (non-pre-strained) wire, (b) 2% pre-strained wire .................................................... 134 Figure ‎5.18.     Lateral force-deflection curve of SMA-NRB-C2 for different amounts of  pre-strain in SMA wires; (a) ε0 = 0%, (b) ε0 = 2%, and  (c) ε0 = 4%; γ = 100%, 150%, and 200% ................................................................. 135 Figure ‎5.19.     Operational characteristics of NRB-2 and SMA-NRB-C2s with different  amounts of  pre-strain in SMA wires; (a) effective horizontal stiffness,  (b) residual deformation,  (c) dissipated energy per cycle,  (d) equivalent viscous damping ................................................................................ 137 Figure ‎5.20.     Variation of strain in SMA wire as a function of shear strain amplitude  and aspect ratio for (a) straight configuration and (b) cross configuration .............. 139 Figure ‎5.21.     Schematic of steel hook and SMA wires in contact; (a) loose, (b) tight ................... 142 Figure ‎5.22.     SMA-HDRB with a cross arrangement of wires subjected to  displacements in x, y, and z directions; (a) 3D view, (b) orthographic views  (top, front, and side views) ....................................................................................... 143 Figure ‎5.23.     Hysteresis curves of SMA-HDRB-C1; (a) FeNCATB SMA, (b) NiTi45 SMA   (γ = 50%, 100%, 150%, 200%) ................................................................................ 146 Figure ‎5.24.     Hysteresis curves of CFR-HDRB; (a) R = 0.12, (b) R = 0.36   (γ = 50%, 100%, 150%, 200%) ................................................................................ 147 xix  Figure ‎5.25.     Hysteresis curves of (a) SMA-HDRB-C1, (b) SMA-HDRB-S1   (γ = 50%, 100%, 150%, 200%) ................................................................................ 147 Figure ‎5.26.     Hysteresis curves of (a) SMA-HDRB-C2, (b) SMA-HDRB-S2   (γ = 50%, 100%, 150%, 200%) ................................................................................ 149 Figure ‎5.27.     Hysteresis curves of SMA-HDRB-S2 (γ = 150%) .................................................... 150 Figure ‎5.28.     Hysteresis curves of Ferrous SMA-HDRB-C1; (a) rSMA = 2.5 mm,  (b) rSMA = 5 mm  (γ = 50%, 100%, 150%, and 200%) ............................................. 152 Figure ‎5.29.     Adjustable mechanism for fixing the SMA wire and applying pre-strain;   (a) side view of the mechanism, (b) 3D view of slotted hexagonal head bolt  with a hole in the middle .......................................................................................... 154 Figure ‎5.30.     Lateral force-displacement curves of (a) CFR-HDRB, SMA-HDRB-C2, and   SMA-HDRB-C2 with 2% pre-strain, (b) SMA-HDRB-C2 with 2% and  3% pre-strains (γ = 200%) ........................................................................................ 154 Figure ‎5.31.     SMA-LRB; (a) decoupled systems, (b) integrated SMA-LRB ................................. 157 Figure ‎5.32.     LRB used in the experimental tests; (a) side view, (b) top view  (dimensions are in mm) (adapted from (Abe et al., 2004)) ...................................... 157 Figure ‎5.33.     LRB modelled in ANSYS software .......................................................................... 159 Figure ‎5.34.     Hysteretic shear response of LRB at 50% and 150% shear strains obtained  through FEM and experimental tests conducted by Abe et al. (2004) ..................... 160 Figure ‎5.35.     Half model of LRB equipped with double cross SMA wires .................................... 161 Figure ‎5.36.     Procedure of decoupling SMA wires from LRB ....................................................... 162 Figure ‎5.37.     Shear force-strain hysteresis curves for LRB and SMA-LRB subjected to  different  shear strains (50%, 100%, 150%, and 200%); (a) rSMA = 1.5 mm,  (b) rSMA = 2.5 mm ..................................................................................................... 163 Figure ‎5.38.     Stress-strain behaviours of non-pre-strained and 3% pre-strained SMA wires   in SMA-LRB subjected to (a) 100% shear strain and (b) 150% shear strain .......... 165 Figure ‎5.39.     Shear force-strain hysteresis curves of LRB, SMA-LRB, and SMA-LRB (PS)  (ε0 = 3%) under (a) 100% shear strain and (b) 150% shear strain ............................ 165 Figure ‎5.40.     Flow chart of design procedure to determine the diameter and  pre-strain of SMA wires ........................................................................................... 168 Figure ‎6.1.       Half model of SMA-LRB (dimensions are in mm) ................................................... 174 Figure ‎6.2.       Superimposing SMA wires onto LRB ...................................................................... 175 Figure ‎6.3.       Shear hysteresis curves of decoupled systems at shear strains of  (a) 100% and (b) 150% ............................................................................................ 176 xx  Figure ‎6.4.       Shear hysteretic responses of integrated and superimposed systems at  shear strains of  (a) 100% and (b) 150% .................................................................. 176 Figure ‎6.5.       Flow chart of SMA-LRB constitutive model ............................................................ 177 Figure ‎6.6.       Excitations (E) in terms of shear strain over time, and corresponding  shear hysteretic responses (R) in terms of force versus displacement of LRB ........ 179 Figure ‎6.7.       SMA-RB with double cross configuration ................................................................ 180 Figure ‎6.8.       Idealized stress-strain diagram of SMA (Auricchio, 2001) ...................................... 180 Figure ‎6.9.       Response of DC-SMAW to a general excitation; (a) unidirectional lateral displacement and the corresponding strain generated in wires,  (b) idealized stress-strain relation in the wires, (c) variation of axial stress  in the wires and corresponding resultant forces in x and z directions  over the time, (d) resultant forces generated by the wires in x and z  directions versus lateral displacement of bearing ..................................................... 181 Figure ‎6.10.     SMA-LRB with DC arrangement under a unidirectional displacement of ΔX   (in x direction) .......................................................................................................... 184 Figure ‎6.11.     Shear hysteretic response of DC-SMAW for different lateral displacements ........... 188 Figure ‎6.12.     Flowchart of determining model characteristics for DC-SMAW hysteresis............. 190 Figure ‎6.13.     Algorithm of DC-SMAW model hysteresis (part 1) ................................................. 191 Figure ‎6.14.     Typical shear hysteresis of DC-SMAW .................................................................... 193 Figure ‎6.15.     Normalized input displacement ................................................................................. 195 Figure ‎6.16.     Shear hysteretic response of DC-SMAW excited by different  input displacements and evaluated through computer code (MCODE) and  ANSYS (FEM) ......................................................................................................... 196 Figure ‎6.17.     Shear hysteretic responses of LRB, DC-SMA, and SMA-LRB excited by  input displacements E5, E6, E7, and E8 ................................................................... 198 Figure ‎6.18.     Shear hysteretic response of SMA-LRB subjected to a ramp  input displacement (E9) scaled with different factors .............................................. 199 Figure ‎6.19.     Shear hysteretic response of SMA-LRB subjected to a sinusoidal  input displacement (E10) scaled with different factors ............................................ 199 Figure ‎7.1.       Fragility function of a structure at a specified limit state .......................................... 204 Figure ‎7.2.       Spectral acceleration versus time period for 30 fear-field earthquake records ......... 210 Figure ‎7.3.       Methodology of seismic fragility assessment for the MSCS bridge ......................... 213 xxi  Figure ‎7.4.       Multi-span continuous steel-girder (MSCS) bridge; (a) elevation view,   (b) side view of footing, piers, and pier cap, (c) superstructure  consisting of deck and steel girders .......................................................................... 215 Figure ‎7.5.       Reinforcement details of the bent and the column .................................................... 216 Figure ‎7.6.       Actual and idealized shear hysteretic responses of rubber bearings ......................... 218 Figure ‎7.7.       PSDMs for displacement ductility of pier equipped with different  isolation systems;  (a) NRB, (b) HDRB, (c) LRB, (d) SMA-NRB,  (e) SMA-HDRB, (f) SMA-LRB ............................................................................... 221 Figure ‎7.8.       PSDMs for shear strain of elastomeric bearings; (a) NRB, (b) HDRB,  (c) LRB,  (d) SMA-NRB, (e) SMA-HDRB, (f) SMA-LRB ..................................... 223 Figure ‎7.9.       Fragility curves of the bridge pier isolated with NRB (regular) and  SMA-NRB (smart) ................................................................................................... 225 Figure ‎7.10.     Fragility curves of the bridge pier isolated with HDRB and SMA-HDRB ............... 226 Figure ‎7.11.     Fragility curves of the bridge pier isolated with LRB and SMA-LRB ..................... 226 Figure ‎7.12.     Fragility curves of the bridge pier isolated with SMA-NRB, SMA-HDRB,  and  SMA-LRB at (a) slight, (b) moderate, (c) extensive,  and (d) collapse limit states ...................................................................................... 228 Figure ‎7.13.     Fragility curves of NRB with and without SMA wires ............................................. 230 Figure ‎7.14.     Fragility curves of HDRB with and without SMA wires .......................................... 230 Figure ‎7.15.     Fragility curves of LRB with and without SMA wires ............................................. 230 Figure ‎7.16.     Fragility curves of SMA-NRB, SMA-HDRB, and SMA-LRB at  (a) slight,  (b) moderate, (c) extensive, and (d) collapse limit states ......................................... 232 Figure ‎7.17.     Fragility curves of the bridge isolated with six different isolation systems   at slight damage state ............................................................................................... 234 Figure ‎7.18.     Fragility curves of the bridge isolated with six different isolation systems   at moderate damage state ......................................................................................... 235 Figure ‎7.19.     Fragility curves of the bridge isolated with six different isolation systems   at extensive damage state ......................................................................................... 235 Figure ‎7.20.     Fragility curves of the bridge isolated with six different isolation systems   at collapse damage state ........................................................................................... 236 Figure ‎7.21.     Fragility curves of the bridge isolated with NRB, HDRB, and LRB ........................ 237 Figure ‎7.22.     Fragility curves of the bridge isolated with SMA-NRB, SMA-HDRB,  and SMA-LRB ......................................................................................................... 239 Figure ‎7.23.     Fragility curves of the bridge isolated with NRB and SMA-NRB ............................ 240 xxii  Figure ‎7.24.     Fragility curves of the bridge isolated with HDRB and SMA-HDRB ...................... 241 Figure ‎7.25.     Fragility curves of the bridge isolated with LRB and SMA-LRB ............................ 242 Figure ‎7.26.     Differences between damage probability of HDRB and SMA-HDRB ..................... 243 Figure ‎7.27.     Bar chart of median values of PGA for the MSCS bridge equipped with   six different isolation systems .................................................................................. 244  Figure B.1.       Flow chart of bilinear kinematic hardening model ................................................... 271   xxiii  List of Symbols and Abbreviations Af Austenite finish temperature of SMA As Austenite start temperature of SMA  Af Plan area of reinforcement dy Yield dispalcement EA Elastic modulus of SMA at austenite phase Ec Compression modulus Ef Elastic modulus of reinforcement EM Elastic modulus of SMA at martensite phase Fmax Maximum shear force Fmin Minimum shear force Fy Yield force f Cyclic frequency Gr Shear modulus of rubber K0 Initial stiffness KHeff Effective horizontal stiffness KV Vertical stiffness L Length of laminated pad M f Martensite finish temperature of SMA M s Martensite start temperature of SMA  ne Number of rubber layers P Vertical pressure Qd Characteristic strength r Post-yield hardening ratio rSMA Radius of cross section of SMA wire S Shape factor of rubber bearing Sd Median of demand Sc Median of capacity te Thickness of one rubber layer tr Total thickness of rubber layers tf Thickness of reinforcement Ud Energy dissipated per cycle Ue Energy restored in the rubber bearing VH Horizontal velocity of lateral displacement W Width of laminated pad βc Logarithmic standard deviation of capacity βD|IM Logarithmic standard deviation of demand βeq Equivalent viscous damping εs Superelastic strain in SMA εSMA Strain in SMA wires γ Shear strain Δmax Maximum lateral displacement Δmin Minimum lateral displacement λ Median of intensity measure μd Displacement ductility ξ Standard deviation of intensity measure ν Poisson’s ratio ω Angluar frequency   xxiv  C-FREI Carbon fibre-reinforced elastomeric isolator CDF Cumulative distribution function CFR Carbon fibre-reinforced EDC Energy dissipated per cycle EDP Engineering demand parameter FEM Finite element method FEA Finite element analysis FPB Friction pendulum bearing FRP Fibre-reinforced polymer HDR High damping rubber HDRB High damping rubber bearing IM Intensity measure LDT Laser displacement transducer LRB Lead rubber bearing LS Limit state NRB Natural rubber bearing PGA Peak ground acceleration PGD Peak ground displacement PGV Peak ground velocity PSDM Probabilistic Seismic demand model SMA Shape memory alloy SPOT String potentiometer SREI Steel-reinforced elastomeric isolator SMA-RB SMA wire-based rubber bearing SMA-HDRB SMA wire-based high damping rubber bearing SMA-LRB SMA wire-based lead rubber bearing SMA-NRB SMA wire-based natural rubber bearing    xxv  Acknowledgements I am deeply grateful for precious support and helpful feedbacks from my supervisor, Dr. M. Shahria Alam. With an optimistic attitude, he helped me think critically, deal with tough situations, and improve the professional level of my PhD research. I particularly thank my committee members, Dr. Abbas S. Milani and Dr. Ahmad Rteil, whose valuable guidance helped me advance my research quality. I also acknowledge the support of my research group members, especially Mr. Muntasir Billah, Mr. Moein Ahmadipour, Mr. Salamah Meherier, fellow students, and staffs at the UBC. Special thanks to my parents, who have supported me throughout my years of education, morally, emotionally, and financially. The financial contributions of the Natural Sciences and Engineering Research Council (NSERC) of Canada through Engage and Discovery Grants programs were critical to conduct this research and are gratefully acknowledged. I am grateful to GoodCo Z-Tech Company, Laval, QC for providing data and its support in the manufacturing stage. In the experimental part, the support provided by Mr. Niel Van Engelen from the Department of Civil Engineering, McMaster University is gratefully acknowledged.   xxvi  Dedication  I would like to dedicate this doctoral dissertation to my parents, the most respectful, helpful, and compassionate in my whole life.     1  Chapter 1 Introduction and Thesis Organization 1.1 General Earthquakes are one of the most unpredictable and difficult-to-control phenomena, which have catastrophic consequences to human civilization. In order to eliminate or reduce the disastrous effects of earthquakes, one effective way is to use protective systems in structures (e.g. buildings and bridges) such as base isolation mechanisms. Base isolators play an important role in vibration attenuation and seismic response control of civil structures like buildings or bridges against earthquakes. They can significantly reduce seismic damages and prevent structures from collapse. Comprehensive research has been carried out on history and development of isolation systems (Kelly, 1986; Buckle and Mayes, 1990). In this system, a device with high vertical and bending stiffnesses but very low horizontal stiffness is mounted between the substructure and the superstructure. Rubber bearings are one of the most common base isolators with a cubic or cylindrical shape. Their application in ordinary low-rise buildings and highway bridges of developing countries is increasing considerably (Kelly, 2002). In steel-reinforced elastomeric isolators (SREIs), steel shims can be replaced with fibre-reinforced polymer (FRP) composite plates in order to reduce their weight and make them easy to handle during transportation and placement (Kelly, 1999 and 2002). The production cost of fibre-reinforced elastomeric isolators (FREIs) is also reduced, as a potential saving, due to automated manufacturing process (Kelly, 1999; Tsai and Kelly, 2002). SREIs have axial and flexural rigidity while, FREIs are completely flexible under bending due to the presence of fibres (Kelly, 1999). Due to high strength-to-weight ratio of carbon fibre-reinforced polymer (CFRP) composite materials, carbon-FREIs are much lighter than SREIs with superior performance (Tsai and Kelly, 2002). Hence, they can be implemented into a wide range of applications such as bridges, buildings, and other civil infrastructures. Shape memory alloys (SMAs) are one kind of smart and functional materials that can restore to their pre-determined and original shape after deformation via unloading or applying thermal load. They have two solid phases; martensite or unstable phase in which 2  material is at low temperature and austenite, parent, or high-temperature phase. In this regard, four characteristic temperatures are defined to determine the temperature ranges for starting and finishing the phase transformation between martensite and austenite. Superelastic and shape memory effects are two unique properties of SMAs. In superelastic effect, the generated strain due to mechanical loading is fully recovered after unloading while in shape memory effect, the mechanical deformation should be removed by applying thermal load and increasing its temperature. Thanks to the remarkable characteristics of SMAs such as high damping performance and energy dissipation capacity, significant stiffness hardening (variable stiffness), large ductility, long fatigue life and corrosion resistance capability, they are excellent candidates as damper or actuator (Graesser and Cozzarelli, 1991; Soong and Dargush, 1997). More details will be presented in Chapter 2, section 2.4. SMA, as supplementary component, can improve the re-centring capability of elastomeric isolators and as a result, extend their service life (Choi et al., 2005; Andrawes and DesRoches, 2007; Ozbulut and Hurlebaus, 2010).  1.2 Objectives The primary goal and the original contribution of this thesis to knowledge is to analytically develop a constitutive model for new smart steel- or fiber-reinforced elastomeric isolators which are equipped with superelastic SMA wires. The proposed model can be implemented in structural finite element softwares in order to accurately simulate the shear behaviour of such SMA-based rubber bearings (SMA-RBs) and capture their nonlinear hysteretic response. The objectives of this PhD research work are classified as follows:  1.2.1 Performance Evaluation of Carbon Fibre-Reinforced Elastomeric Isolators (C-FREIs) The cold-vulcanization process, which is known as a fast and cost effective manufacturing process, has been used for producing fibre-reinforced elastomeric isolators (FREIs) in unbonded applications where the bearing is not fixed in its place. Here, the cold-vulcanization process is used to fabricate a number of scaled size carbon-FREIs for bonded applications for the first time. The effectiveness and performance of C-FREIs are explored by conducting different types of tests including 3  performance and sensitivity tests. Then, parametric (sensitivity) analyses are carried out at two levels; experimental and numerical. This objective is defined in order to attain an appropriate understanding of the behaviour of FREIs and correctly perceive their advantages and limitations. 1.2.2 Development of a Novel Shape Memory Alloy-based Rubber Bearing (SMA-RB) With the goal of improving the energy dissipation capacity, the re-centring capability and as a result, the service life of existing elastomeric bearings, a novel seismic base isolator is developed using shape memory alloy (SMA) wires. In this passive earthquake protective system, SMA wires are wound around the rubber bearings with different configurations (e.g. straight, cross, and double cross). 1.2.3 Development of a New Hysteresis Model for SMA-RBs By proposing a novel SMA wire-based elastomeric isolator, it is highly beneficial to properly simulate its hysteresis. Existing material models cannot accurately capture the response of such smart bearings. It becomes highly important when the seismic performance of a structure isolated by SMA-RBs is evaluated. Here, as a complementary part of the previous objective, a new hysteresis model is developed for SMA wire-based rubber bearings. 1.2.4 Seismic Fragility Assessment of a Highway Bridge Isolated by SMA-RBs Seismic fragility assessment of a structure is a common tool to evaluate the failure probability of the structure under seismic events. In fact, the probability that a structural demand reaches or exceeds the capacity of a structure is estimated at different levels of damage (limit/damage states). In order to study the effect of SMA-RBs, as a new developed isolation system, on the seismic fragility of isolated structures, the vulnerability of a multi-span continuous steel-girder bridge isolated with SMA-RBs is assessed analytically, as the last objective of this thesis.  1.3 Outline of the Thesis In Chapter 2 of this thesis, a literature review is performed on the base isolation concept and different types of elastomeric isolators including steel-reinforced elastomeric 4  isolators (SREIs), fibre-reinforced elastomeric isolators (FREIs), and smart SMA-based rubber bearings (SMA-RB). Before going to the experimental phase, it is critical to have an appropriate understanding of the behaviour of elastomeric isolators. Chapter 3, as the first step, intends to facilitate this consideration through a design of experiment, sensitivity and regression analyses, and a multi-criteria optimization process. In this chapter, the effect of several factors such as the number and the shear modulus of rubber layers, as well as the thickness of reinforcement are investigated on the performance of carbon fibre-reinforced high damping rubber bearings (CFR-HDRBs) using finite element method (FEM). HDR has a complicated behaviour compared to other types of rubber such as natural low damping rubber. As a result, it is very challenging to simulate the response of HDR. However, by achieving a clear vision of the behaviour of such elastomer, it will be easier to use the attained knowledge and extend it to other types of rubbers. To accurately simulate the highly nonlinear behaviour of HDR, several material models are selected and compared together.      The achievements of the previous chapter help think systematically of possible scenarios to be defined in the experimental part of the thesis as presented in Chapter 4. A number of reduced scale carbon fibre-reinforced elastomeric isolators (C-FREIs) having different numbers and thicknesses of elastomeric and reinforced layers are manufactured and tested under different loading conditions. A parametric study is conducted based on the experimental results. However, because of the small size of samples, and limited number of factors and specimens, it is not appropriate to use the results at the design level. Therefore, it is necessary to perform a comprehensive study on the behaviour of full-size C-FREIs by considering an acceptable number of specimens. This goal is accomplished by modelling and analyzing C-FREIs using FEM. Similar procedure followed in material modelling of HDR (in Chapter 3) is used here to verify and validate the numerical results with experimental ones. Then a parametric study (sensitivity analysis) is conducted based on FEM. After studying the behaviour of elastomeric isolators reinforced with carbon fibre fabrics, in Chapter 5, new shape memory alloy wire-based rubber bearings (SMA-RB) are proposed with different reinforcements; steel shims and carbon fibre-reinforced composites. The goal of introducing SMA-RBs is to overcome weaknesses (e.g. limited shear deformation capacity) of steel- and fibre-reinforced rubber bearings and improve their self-5  centering and energy damping properties. The performance of such SMA-RBs is evaluated using FEM. Three different types of SMA-RBs are considered in this part; SMA-based natural rubber bearing (SMA-NRB), SMA-based high damping rubber bearing (SMA-HDRB), and SMA-based lead rubber bearing (SMA-LRB).  By proposing novel SMA wire-based elastomeric isolators, it is necessary to properly comprehend and analyze their mechanical response through hysteresis. Therefore, as a complementary part of the previous chapter, a new hysteresis model is developed in Chapter 6 for SMA-RBs in order to accurately capture their nonlinear behaviour. It should be noted that this model, as a link element, can be implemented in any structure under static or dynamic loadings in finite element environment. In order to find out that such SMA-RBs are reliable to be used in structural applications, their effect, as new isolation systems, should be investigated on the seismic response of structures (e.g. buildings, bridges). Therefore, to check whether or not these new rubber bearings are efficient enough in seismic isolation, the fragility of a highway bridge isolated by SMA-RBs is assessed in Chapter 7. Finally, in Chapter 8 of the thesis, a summary along with concluding remarks are presented and then, future works are discussed.  Figure  1.1 summarizes the goals of this PhD thesis and relates main chapters to each other by showing the topics which are covered in each section.     6   Figure ‎1.1. Summary of the goals and topics covered in the thesis    Correctly perceiving advantages and limitations of modern fiber-reinforced rubber bearings  Manufacturing and Testing Specimens   Experimental Parametric Study  Numerical Parametric Study  Attaining an appropriate understanding of  the behavior of elastomeric isolators  Design of Experiment (DOE)  Sensitivity and Regression Analyses  Multi-Criteria Optimization  Overcoming the limitations of elastomeric isolators by introducing a new SMA-based rubber bearing reinforced by steel shims or composites plates  Numerical Investigation on SMA-NRB  Numerical Investigation on SMA-HDRB  Numerical Investigation on SMA-LRB Developing a constitutive model for the  SMA-based rubber bearing  Assessing the effect of SMA-based rubber bearings on the seismic fragility of a highway bridge   Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Primary Objectives  7  Chapter 2 Literature Review 2.1 General In order to seismically protect a structure from the devastating effects of earthquake, different protective systems including active, hybrid, and passive vibration control systems have been developed. Passive systems have been extensively implemented in civil engineering applications due to their easier operation where there is no need for external power supplies (Ozbulut et al., 2007). Such systems are categorized into two types; rubber bearings and sliding bearings. Conventional elastomeric isolators or rubber bearings are laminated devices consisting of alternating layers of rubber and reinforcement. In sliding bearings, either flat or curved surfaces are in contact with each other in order to dissipate the energy through a frictional mechanism (Kunde and Jangid, 2003).  The operation of important structures such as hospitals, fire stations and emergency control centres during an earthquake is one of the most important parameters that should be considered in the construction or retrofitting. Consequently, uncoupling a structure (e.g. buildings or bridges) from devastating effects of earthquakes has been one of the major concerns for engineers for a long time. Although, many efforts have been made by introducing numerous devices based on seismic isolation of structures, they are mostly intricate and a limited number of them have been applied into buildings and bridges (Kelly, 1986; Buckle and Mayes, 1990). Kelly (1986) studied a wide range of publications from 1900 to 1984, which were related to seismic base isolators. He presented an extensive literature review about seismic isolation and various types of base isolators used in construction or rehabilitation of buildings. He conducted his study by focusing on characteristics and applications of base isolators.  Seismic isolation systems can prevent or minimize the structural damages of structures (e.g. buildings, bridges, and viaducts) to provide a continuous operation by regulating the seismic response of the structure (Ozkaya et al., 2011). As shown in Figure  2.1, these systems are placed between the substructure (e.g. foundation or pier) and the superstructure (e.g. columns or dock).  8                 Figure ‎2.1. Applications of base isolation systems in (a) buildings and (b) bridges Using base isolation systems in bridges can improve the seismic performance of the structure, increase the public safety, and reduce the cost of repair and rehabilitation (Ghobarah and Ali, 1988; Kikuchi and Aiken, 1997; Wilde et al., 2000; Chaudhary et al., 2000; Hwang et al., 2002; Zhang et al., 2009; Ozbulut and Hurlebaus, 2010 and 2011; Sarrazin et al., 2013; Siqueira et al., 2014). Base isolators can considerably decrease and dissipate the earthquake energy transmitted to the structure by providing a damping mechanism. Due to a low horizontal stiffness (high lateral flexibility), they can shift the fundamental horizontal frequency of an isolated structure away from the dominant frequency range of earthquake by increasing the base period of structure. The main goals of using base isolation techniques are to:  Prevent the structural collapse in severe earthquakes  Avoid or minimize the structural damage in moderate earthquakes  Provide continuous operation in important buildings. The efficiency of an elastomeric isolator is determined by evaluating its horizontal flexibility, vertical stiffness and damping capacity. Indeed, the shear behaviour of a rubber bearing under a combination of vertical pressure and cyclic horizontal loadings is a criterion to indicate its performance. It was shown that by reducing the horizontal stiffness of the rubber bearing, the period of the base-isolated structure increases (Toopchi-Nezhad, 2008a). In fact, when the horizontal stiffness decreases, the lateral flexibility increases and as a result, it takes a longer time for the structure to come back to its initial position in one cycle. Elastomeric isolators, which are usually produced in a rectangular (see Figure  2.2) or circular shape, can be divided to conventional and modern devices depending on the type of the reinforcement (e.g. steel shim or fibre-reinforced composites).   (a)                                                                (b) old.enea.it 9   Figure ‎2.2. Elastomeric isolators in rectangular shapes In elastomeric bearings, reinforcements provide adequate vertical rigidity to carry on the compressive loads due to the weight of the superstructure and also prevent the horizontal bulging of rubber layers. The elastomeric layers provide lateral flexibility as well as damping property. 2.2 Steel-Reinforced Elastomeric Isolators (SREI) In conventional rubber bearings, also called steel-reinforced elastomeric isolators (SREIs), steel shims are bonded to rubber layers in order to provide a high compressive stiffness.  Depending on the material properties and auxiliary elements which are used in rubber bearings to improve their energy dissipation capacity, SREIs were developed and categorized into different types such as low-damping natural rubber bearing (NRB), lead-plug rubber bearings (LRB), high damping rubber bearings (HDRB) and ball rubber bearing (BRB). There are major concerns about these types of base isolators such as size, weight and cost. Consequently, they are applied to large bridges and high-rise buildings equipped with expensive and important appliances. NRB, HDRB, and LRB have been widely used in seismic response mitigation and control of structures subjected to ground motions (Warn and Whittaker, 2004; Andrawes and DesRoches, 2007; Bhuiyan et al., 2009; Alam et al., 2012; Bhuiyan and Alam, 2013). 2.2.1 Natural Rubber Bearing (NRB) In low-damping natural rubber bearings (NRB), also called synthetic rubber bearings, natural rubber or neoprene is reinforced with steel shims through a hot-vulcanization process 10  under heat and pressure in a mold (Naeim and Kelly, 1999). Figure  2.3 shows a laminated rubber bearing. Steel shims are vulcanized to the rubber and surrounded by an elastomeric cover layer. Two fixing plates made of steel are attached to the top and bottom of the laminated pad. In order to mount the rubber bearing on the structure, steel supporting end plates are used.  Figure ‎2.3. Laminated rubber bearing These type of elastomeric isolators are extensively being used in buildings and bridges where supplementary components such as steel bars, viscous dampers, and frictional devices are implemented. NRBs possess critical damping ratios about 2-3% (Ozkaya et al., 2011). Natural low-damping rubber follows the behaviour of a hyperelastic material with low amount of energy damping capacity. This type of elastomer has a low sensitivity to the environmental conditions (e.g. temperature), loading rate, loading history (scragging), and aging. Scragging refers to a behaviour change (i.e. stiffness and damping reduction) during the initial cycles of motion, which is stabilized as the number of cycles increases. Simplicity in manufacturing of such bearings is considered as another advantage. NRB shows almost a linear behaviour up to shear strains above 100% and its hysteretic response encounters negligible changes (Naeim and Kelly, 1999). As a result, NRBs can be easily modelled. The main disadvantage of NRBs is their need to a complementary element such as lead core for providing extra amount of damping. Rubber Steel Shim Supporting End Plates Fixing Plates 11  2.2.2 High Damping Rubber Bearing (HDRB) High damping rubber bearings (HDRBs) consist of steel shims as reinforcement and high damping rubber (HDR) as elastomer for providing horizontal flexibility and damping capacity. HDRBs have 10-20% equivalent viscous damping (Marioni, 1998). Elastomer layers in the HDRB has much higher damping capacity compared to natural low damping rubbers (Ozkaya et al., 2011). The high damping property of the elastomeric isolator is due to adding specific materials like extra-fine carbon black, oils or resins and other proprietary fillers to the natural rubber (Naeim and Kelly, 1999). They also possess a high initial shear modulus compared to the NRBs (Skinner et al., 1993). By designing such base isolators, a rubber compound with sufficient damping property without auxiliary damping devices such as lead core was developed. HDRBs can undergo large shear strain levels (around 400%). Because of the materials added to the natural rubber during vulcanization process, HDRs have specific characteristics such as energy absorption and hardening properties within a wide strain range from 1 to 400% (Yoshida et al., 2004). As a result, it is difficult to capture their mechanical properties like stress-strain relation and fatigue accurately. Many studies have been performed for modelling the behaviour of HDR materials and HDRBs based on the numerical and analytical approaches as well as experimental tests (Yoshida et al., 2004; Amin et al., 2006a; Bhuiyan et al., 2009). Results showed that although the hyperelastic material model can simulate the response of natural low damping rubber, this model cannot accurately capture the mechanical behaviour of HDR materials. HDRBs possess a highly nonlinear and complex behaviour (e.g. scragging referred to a reduction in the stiffness and damping during the initial cycles of motion) at large deformations. Consequently, several considerations and assumptions should be taken into account to analytically model them. Many theoretical and experimental works have been performed to investigate the effects of different parameters on the dynamic performance of HDRBs under various conditions such as shear strain and amplitude (Amin et al., 2006a; Bhuiyan et al., 2009; Dall’Asta and Ragni, 2006; Tsai et al., 2003; Yoshida et al., 2004).  Tsai et al. (2003) proposed a model to capture the rate-dependent effects of HDRBs using analytical Wen’s model (Wen, 1976). In order to validate the proposed mathematical model, different experimental tests were carried out and compared with results obtained from numerical finite element formulations. It was observed that, the lateral force-deflection 12  hysteresis loop calculated by the modified model is so close to that obtained by experimental tests under different shear strains. The model could also predict the nonlinear behaviour of HDRBs at various shear strains and frequencies by simulating the stiffening and velocity dependency. Dall’Asta and Ragni (2006) proposed a viscoelastic material model to simulate the behaviour of rubber bearing under pure cyclic shear loads. Experimental results showed that the dynamic behaviour of HDRB includes a transient response followed by stable hysteresis loops as steady-state response. The lateral force-displacement hysteresis loop with “butterfly” shape is a function of strain-rate, strain amplitude and Mullins’ effect. Mullin’s effect in filled rubbers refers to a variation of the hysteretic (stress-strain) curve depending on the maximum load previously applied. This phenomenon is commonly applied to stress softening. Based on experimental achievements a constitutive analytical model without the limitations of previous models in capturing the nonlinearity behaviour was proposed using a rheological model (Dall’Asta and Ragni, 2006). 2.2.3 Lead Rubber Bearing (LRB) Among different passive earthquake protective systems, lead rubber bearings (LRBs) with high energy dissipation capacity are extensively used in seismic isolation of structures (Turkington et al., 1989; Ozdemir et al., 2011; Bhuiyan and Alam, 2013). They consist of elastomeric layers bonded to steel shims, fixing and supporting steel plates at the top and the bottom, and a lead core located in the central part as shown in Figure  2.4. The main role of the lead core is to dissipate the earthquake’s energy. Supporting steel plates restrain the whole elastomeric isolator and also confine the lead-plug in the middle of LRB.   13   Figure ‎2.4. Lead-plug rubber bearing (LRB) Several analytical studies have been done on the efficiency of lead-plug rubber bearings (Hwang and Chiou, 1996). Doudoumis et al. (2005) verified the accuracy of LRB models analyzed through finite element method (FEM). They considered two models to investigate the effect of lead cores constraint on the behaviour of rubber bearings under vertical and cyclic horizontal loadings. They recommended using such micromodels since lead core changes the internal stress and strain distribution (Doudoumis et al., 2005). Abe et al. (2004) experimentally studied the response of LRB, NRB and HDRB considering different loading types; small amplitude uniaxial load and large amplitude biaxial and triaxial loads. They showed that the vertical pressure has a significant effect on the restoring force of LRB. Experimental results revealed that under a combination of constant vertical pressure and lateral displacement in two directions (triaxial loading), the interaction effect of the loadings noticeably increases the effective lateral stiffness and the equivalent viscous damping in the cases of NRB and HDRB.  However, the interaction of loading had a negligible effect on the stiffness and damping ratio of LRB. Therefore, they pointed out that the interaction effect cannot be ignored at the design level. 2.2.4 Ball Rubber Bearing (BRB) A new type of steel-reinforced laminated bearing, called ball rubber bearing (BRB), has been designed and manufactured by implementing small steel balls in a central hole of NRB (Ozkaya et al., 2011). In this type, lead core is replaced with steel balls. BRBs can works as rubber and sliding bearings simultaneously due to the damping property of elastomeric layers and friction generated between steel balls (Ozkaya et al., 2011). Results Rubber Lead Core Steel Shim 14  obtained from more than 200 different experiments through full-scale cyclic shear tests showed that the effective horizontal and vertical stiffnesses, and damping capacity of BRBs are higher than those of NRBs. The equivalent viscous damping of BRBs varies from 15 to 25%. It was also observed that almost 50% of the vertical pressure is resisted by its central core. It means that BRBs could have lower shape factor (i.e. ratio of loaded area to force-free area of one elastomeric layer) than NRBs. As a result, steel balls can carry larger portion of the vertical compressive load and thus, internal friction and energy dissipation is enhanced (Ozkaya et al., 2011). High weight and relatively high horizontal stiffness compared to other types of RBs (e.g. NRB, HDRB, and LRB) are two main disadvantages of such elastomeric bearings. 2.3 Fibre-Reinforced Elastomeric Isolators (FREI) SREIs are the most common rubber bearings in use. However, they are large, heavy and expensive (Kelly, 2002). They are produced through a high cost process due to vulcanization bonding of steel shims and rubber layers in a mold. The main concern about these types of base isolators is their limited applications which are in large bridges and high-rise buildings having heights of greater than 23 m with expensive and important equipment because of their large size, high weight and cost (Kelly, 2002). Kelly suggested that both price and weight of SREIs can be decreased by replacing steel shims with fibre-reinforced composite plates (Kelly, 1999). Compared to a SREI which should be produced according to a designed size, fibre-reinforced elastomeric isolators (FREIs) can be produced in the form of long rectangular strips and then cut to the required size using a standard band-saw. Kelly also suggested to use micro-wave heating mechanism inside an autoclave instead of applying heat and pressure in a mold. This technique can be done through an automated process and as a result, significantly reduce the overall manufacturing cost in a mass production (Kelly, 1999). Rubber and fibre-reinforced layers can be bonded together using a cold-vulcanized bonding compound without any mold. As a result, labour expenses will decrease remarkably via an automated manufacturing process. Therefore, it is understood that the main goal of designing and producing FREIs is to reduce the cost and weight of elastomeric base isolators in order to extend their applications to ordinary and low-cost residential and public buildings 15  throughout the world especially in developing countries and high-risk seismic regions with severe earthquakes. A big difference between the two types of reinforcement used in conventional and modern rubber bearings is the flexibility of the reinforcement. In contrast to the steel shims with a high flexural rigidity, fibre-reinforced sheets are completely flexible under bending (Kelly, 1999). This characteristic causes the FREI to show a rolling deformation under lateral shear force and as a result, it produces lower forces in the transverse direction compared to the SREI. Therefore, FREI can be laterally deformed with a higher flexibility. FRP composite materials have low density and high strength-to-weight ratio. The density of epoxy matrix composite reinforced with 70% carbon fibres is 1600 kg/m3 while, mild steel has a density of 7850 kg/m3. As a result, FREIs, with superior performance, are significantly lighter than SREIs (Moon et al., 2002). Reducing the weight of rubber bearings can significantly facilitate manufacturing, shipping, handling and installation processes. Consequently, a wide range of applications (e.g. public, residential, and low-rise buildings) have been found for such modern and light isolators (Toopchi-Nezhad et al., 2008b). Kelly (2002) studied the possibility of implementing FRP composite layers in rubber bearings by considering weight and cost. He clarified that FREI and SREI have comparable performances and it is possible to produce such isolators with suitable mechanical properties (Kelly, 1999 and 2002). Tsai and Kelly (2002) studied the effect of fibres on the flexibility of base isolators by presenting formulations for compressive and bending stiffnesses of rectangular FREIs based on analytical method (Tsai and Kelly, 2002). They assumed that the elastomer is incompressible and isolator is in the form of infinite strip pad. Results indicated that the lateral stiffness increases with increasing the shape factor and decreases by using more flexible reinforcement. Carbon fibres exhibit excellent mechanical characteristics such as high elastic modulus (200-800 GPa), high tensile strength (2500-6000 MPa) and suitable fatigue life without creep or relaxation, so they are desirable candidates as reinforcement of FRP composite plates used in rubber bearings (Moon et al., 2002). Moon et al. (2002) fabricated FREIs consisting of different fibres (e.g. carbon, aramid, and glass) and compared them with SREIs. Experimental results revealed that FREIs are superior to SREIs in terms of vertical stiffness, effective horizontal stiffness, and equivalent viscous damping. They observed that, 16  compared to glass and aramid fibres, carbon fibres are more effective in increasing the vertical stiffness and the equivalent viscous damping. Moon et al. also manufactured circular carbon fibre- and steel-reinforced rubber bearings with a same size in order to evaluate and compare their performance. The diameter and the total height of both isolators are around 700 mm and 350 mm, respectively. They showed that the carbon FREI has a vertical stiffness of 3100 kN/mm which is three times higher than SREI. Moreover, the equivalent viscous damping of FREI (15.85%) was found to be 2.5 times greater than that of SREI (6.2%). Another finding was that FREI has an effective horizontal stiffness of 3.24 kN/mm which is lower than SREI with 3.43 kN/mm effective lateral stiffness. These characteristics depict that the carbon FREI is more efficient in terms of both stiffness and energy dissipation capacity. Dehghani Ashkezari et al. (2008) designed and manufactured different specimens of elastomeric bearings using layers of woven carbon fibres to study their mechanical characteristics and dynamic behaviour under compressive and shear loads. They found that carbon fibres can dissipate energy through frictional movements and provide additional damping to the system. It was also determined that the vertical pressure can considerably affect the damping coefficient of FREI, however, it has negligible influence on the shear response of FREIs (Dehghani Ashkezari et al., 2008). Another important finding was that, if cyclic lateral loading is repeated with amplitude less than the maximum load previously applied, horizontal flexibility and energy damping properties will decrease due to stress softening phenomenon. Kang et al. (2003) probed the effect of lead-plug in fibre-reinforced seismic isolators based on experimental tests and analytical approaches. According to their findings, presence of lead-plug does not change the performance of rubber bearing significantly. Mordini and Strauss (2008) conducted experimental work on FREIs made of glass-fibre fabrics and HDR to provide required information in their numerical simulations and analytical model. The robustness and consistency of the proposed model was investigated by seismic response analysis of a liquid storage tank equipped with a FRP-reinforced HDRB. Rubber bearings with different geometries (e.g. number and thickness of rubber and reinforced layers), material models and loading and boundary conditions were tested to investigate their operational characteristics (e.g. effective horizontal and vertical stiffnesses) as well as internal stresses in glass fibres and elastomeric layers. They applied FREI in a full-scale 17  structure using simple nonlinear elements rather than complex FE model to significantly reduce the calculation time of seismic analysis. Results obtained from finite element analyses showed that the acceleration is decreased and the period is altered in the base-isolated structure (Mordini and Strauss, 2008). Zhang et al. (2011) studied the mechanical properties of FREIs after manufacturing and testing a number of samples. Specimens were subjected to vertical pressure for calculating the effective vertical stiffness and compressive modulus. The effective horizontal and damping capacity were determined by applying cyclic horizontal displacements. The hysteretic curves for three FREIs with different thickness and number of elastomeric and reinforced layers subjected to vertical and cyclic shear loads illustrated that the operational characteristics of FREIs are comparable to those of traditional ones. FREIs have adequate efficiency in terms of the energy dissipation capacity (i.e. capacity of the device in damping the earthquake’s energy) and the effective vertical stiffness. Therefore, implementing them in the seismic base isolation is an applicable idea. Rubber bearings are either fixed in between the superstructure and the substructure using steel supporting plates (bonded application) or mounted without any connecting mechanism and supporting plates (unbonded application). With the purpose of studying the behaviour of unbonded C-FREI, Toopchi-Nezhad et al. (2008b) performed experimental tests and observed that a rollover deformation occurs in the laminated pad due to a very low flexural rigidity of the fibre-reinforced layers. As a consequence, the lateral flexibility of such unbonded C-FREI increases under cyclic shear displacements. In order to improve the low damping capacity and inadequate (very low) horizontal stiffness of such rubber bearings, they suggested that HDR or supplementary elements can be used. Focusing on the compressive behaviour of unbonded elastomeric bearings, Van Engelen et al. (2014) explored the effect of geometric modifications on the vertical stiffness and the compressive modulus of rectangular FREIs. They validated a 3-dimensional finite element (FE) model using experimental results and performed a parametric study. They observed that the vertical performance degrades with both interior and exterior modifications, but it is more sensitive to the exterior modification. The modifications were meant to improve the performance of the unbonded FREIs in the horizontal direction by reducing the effective lateral stiffness and increasing the energy dissipation capacity.               18  In addition to experimental work, several analytical studies have been done to describe the behaviour of fibre-reinforced bearing pads under bending, compression, and shear. Russo et al. (2013) proposed a geometric model to predict the deformation of fibre-reinforced pads (unbonded applications) under shear and compression. Using experimental tests conducted on a number of specimens, they considered different types of rubber (e.g. low and high damping neoprene); various reinforcements (e.g. bi-directional and quadri-directional carbon fibre fabrics); as well as aging and shape factor, and presented an expression for the lateral stiffness of the fibre-reinforced isolators.   The performance of FREIs made of carbon fibre fabrics and high-damping rubber was assessed through sensitivity analyses by Hedayati Dezfuli and Alam (2013a). They considered 27 C-FREIs and performed several finite element simulations validated by experimental results in order to propose the most efficient rubber bearing through a multi-objective optimization process. They developed regression models to predict the response of C-FREIs. The seismic response of a three-span continuous steel girder reinforced concrete pier supported bridge isolated by the optimized C-FREI was conducted through dynamic time history analyses. Hedayati Dezfuli and Alam (2013a) observed that the effective horizontal stiffness and the equivalent viscous damping are highly dependent on the shear modulus of the elastomeric layers. In addition, the number of rubber layers and the thickness of carbon fibre-reinforced sheets were found to have large effects on the vertical stiffness. Based on experimental tests and through a parametric study, Hedayati Dezfuli and Alam (2013b) examined the effect of mechanical and physical properties (e.g. shear modulus and thickness of elastomer) on the response of the scaled size C-FREIs in bonded applications. Findings revealed that the equivalent viscous damping and the effective horizontal stiffness are very sensitive to the shear modulus of the elastomer and the vertical stiffness is sensitive to the shape factor. In another experimental study, Hedayati Dezfuli and Alam (2014a) manufactured bonded carbon fibre-reinforced bearing pads in a cold vulcanization process in order to investigate the effectiveness of the process and the performance of the bonded C-FREIs. They observed that at 100% shear strain amplitude, a partial debonding occurs between exterior rubber layers and supporting plates due to the rollover deformation. This phenomenon did not lead to a malfunction in the bearing pads however; more comprehensive 19  work needs to be performed on full size specimens by conducting 3-dimensional excitation tests with extreme loading conditions. 2.4 Smart Rubber Bearings 2.4.1 Shape Memory Alloy (SMA) Shape memory alloys (SMAs) are considered as smart and functional materials that can restore their pre-determined and original shape after deformation via unloading or by applying thermal load. They have two solid phases; martensite or unstable phase in which material is at low temperature, and austenite, parent or high-temperature phase. In this regard, four characteristic temperatures are defined to determine the temperature ranges for starting and finishing the phase transformation. The martensite start temperature, M s, and the martensite finish temperature, M f, respectively represent the starting and finishing phase transformation from austenite to martensite. Similarly, for starting and finishing phase transformation from martensite to austenite, the austenite start temperature, As, and the austenite finish temperature, Af, are defined, respectively. Superelastic and shape memory effects are two unique characteristics of SMAs. In the superelastic effect (Figure  2.5a), the generated strain due to the mechanical loading is fully recovered after unloading while in shape memory effect (Figure  2.5b), the mechanical deformation should be removed by applying thermal load and increasing temperature of the alloy. The SMA materials will show the superelastic behaviour if they are in the austenite phase. In other words, when the temperature of SMA is above the austenite finish temperature, the strain generated in the SMA will be fully recovered if it is lower than the maximum superelastic strain.            Figure ‎2.5. Stress-strain curve for SMAs; (a) superelastic effect, (b) shape memory effect (a)                                                         (b) 20  When the temperature of SMA is below the austenite finish temperature, the generated strain is not fully recovered upon unloading because a fraction or all of the alloy remains in the martensite phase depending on the temperature and the reverse phase transformation (martensite to austenite). In such a situation, increasing the temperature of SMA completes the transformation and the strain is fully recovered (Figure ‎2.5b). SMAs have a larger hysteretic deformation and a higher elastic (superelastic) strain compared to conventional alloys and metallic materials (Lagoudas, 2008). The maximum superelastic strain, εs, in such materials can even reach up to 13.5% (Tanaka et al., 2010). SMAs are excellent candidates as dampers or actuators due to their remarkable characteristics such as high damping performance, large recoverable strain (up to 13%), significant stiffness hardening (variable stiffness), large ductility, long fatigue life, and corrosion resistance capability (Soong and Dargush, 1997; Alam et al., 2007).  There are different types of SMAs such as Nickel-Titanium, Cu-based shape memory alloys and ferrous shape memory alloys which have the potential for smart structural applications. Some mechanical properties like the elastic modulus (EA), the austenite finish temperature (Af) and the superelastic strain (εs) under the maximum applied strain (εmax) for a number of SMAs are listed in Table ‎2.1. Table ‎2.1. Mechanical characteristics of different shape memory alloys (SMAs) Alloy εmax (%) εs (%) EA (GPa) Af (°C) Reference Ni Ti49.1 5.0 3.6 40.4 44.6 Strnadel et al. 1995 Ni Ti49.5 5.7 4.6 45.3 53.0 Strnadel et al. 1995 Ni Ti50 3.1 2.2 117.8 77.8 Strnadel et al. 1995 Ni Ti 8.2 6.8 30.0 42.9 Boyd and Lagoudas 1996 Ni Ti45 6.8 6.0 62.5 -10.0 Alam et. Al. 2008 Ni Ti44.1 6.5 5.5 39.7 0 Alam et. Al. 2008 Ti Ni40 Cu10 4.1 3.4 72.0 66.6 Strnadel et al. 1995 Ti Ni41 Cu10 4.1 3.1 91.5 50.0 Strnadel et al. 1995 Ti Ni41.5 Cu10 3.4 2.8 87.0 60.0 Strnadel et al. 1995 Ti Ni25 Cu25 10.0 2.5 14.3 73.0 Liu 2003 CuAlBe 3.0 2.4 32.0 -65.0 Zhang et al. 2009 FeMnAlNi 6.1 5.5 98.4 < -50°C Omori et al. 2011 FeNiCoAlTaB 15.0 13.5 46.9 -62.0 Tanaka et al. 2010 The elastic modulus of the SMA represents the stiffness of the material in the austenite phase. The maximum strain, εmax, is defined as a strain at which the deformation in the material can be fully recovered after unloading. 21  2.4.2 Shape Memory Alloy-based Rubber Bearings (SMA-RB) Elastomeric bearings are extensively being used in several applications, however, they have some weaknesses such as limited shear strain capacity, unrecovered residual deformation, and instability due to a large deformation (Choi et al., 2005). Using SMA in the form of wire (Dolce et al., 2000; Choi et al., 2005; Ozbulut and Hurlebaus, 2010; Hedayati Dezfuli and Alam, 2013c and 2014b), bar (Wilde et al., 2000; DesRoches and Delemont, 2002), or spring (Attanasi and Auricchio, 2011) is a solution to partially overcome the limitations of conventional rubber bearings. SMAs can undergo an inelastic deformation due to stress-induced phase transformation occurred in microscopic scale. Compared to the other alloys and metallic materials, they have a larger hysteretic deformation without entering to the plastic region and consequently, their energy dissipation capacity is higher. Therefore, shape memory alloy, as a supplementary passive damper, can enhance the re-centring capability as well as the energy damping capacity. They can reduce forces and relative displacements transmitted from the substructure to the superstructure (Attanasi et al., 2008; Ozbulut and Hurlebaus, 2011). Variable properties of SMAs (e.g. stiffness) make them suitable candidates to be used under various exciting forces with different magnitudes and frequencies. In small external loadings such as wind or small earthquakes, SMA-based rubber bearings can supply a stiff link between the substructure and superstructure to prevent the damage in the elements of the structure. In mid-size earthquakes, SMA-based elements enhance the damping capacity of the rubber bearing due to stress induced martensitic (SIM) transformation. In strong ground motions, in addition to providing additional hysteretic damping, they can confine the relative displacement of the superstructure as a controller owing to its stiffness hardening after finishing the phase transformation (Wilde et al., 2000). Wilde et al. (2000) combined a shape memory alloy device with a laminated rubber bearing in order to increase the energy dissipation capacity of the isolator and control the relative displacement of the elevated highway bridges isolated by such bearings. According to the results, they found that although the energy transmitted to the bridge through the proposed SMA-based isolator is large compared to that of a structure equipped with a LRB, the damage energy of the bridge is small. 22  Several researchers have proposed different types of superelastic SMA-based smart isolation devices in the past. Choi et al. (2005) performed numerical study considering NiTi SMA wire wrapped around an elastomeric bearing to improve its re-centring capability over the LRB. However, at very large shear deformation (200% shear strain), this device will malfunction since wires experience axial strain beyond the NiTi’s superelastic strain range. Although Dolce et al. (2000) implemented SMA wires effectively in a base isolation device, the manufacturing of the device was quite complex. Another SMA-based isolation device, developed by Dolce et al. (2000), showed high sensitivity and considerable variation in forces with temperature, and inefficiency in energy dissipation capacity. Liu et al. (2008) used a diagonal arrangement of large diameter SMA strands around the rubber bearing. However, this arrangement did not improve the re-centring capability or the level of damping compared to the original rubber bearing. Attanasi and Aurichhio (2011) proposed an isolation device equipped with eight SMA coil springs, which is expensive due to its complex manufacturing process and the use of expensive large diameter SMA springs. Attanasi et al. (2008) investigated the possibility of using shape memory alloys in base isolation systems. They compared the behaviour of a proposed smart isolator with that of a traditional LRB and an equivalent linear elastic model. According to their results, the behaviour of the smart isolation device with flag-shaped hysteretic loops was similar to a system with elasto-plastic hysteresis. They concluded that it is possible to replace existing LRBs with SMA-based bearing systems considering the amount of energy dissipation capacity. They suggested that SMA-based restrainers can be applied to rubber bearings or friction pendulum systems in order to provide re-centring force and control the relative horizontal displacement and upward force transmitted to the superstructure. DesRoches and Delemont (2002) showed that utilizing elastomeric bearings with SMA bars rather than conventional steel cable restrainers increases the efficiency of the isolation system. SMA-based bearing mechanisms have high re-centring capability as a consequence of the superelastic and shape memory effects in SMAs. Attanasi et al. (2008) investigated the possibility of using SMAs in base isolation systems. They compared the behaviour of a proposed smart isolator with that of LRBs and an equivalent linear elastic model. They showed that it is possible to replace SMA-based bearing systems with existing LRBs regarding the amount of energy dissipation capacity. They found that SMA-based 23  restrainers can be applied to rubber bearings in order to provide re-centring force and control the relative horizontal displacement and upward force transmitted to the superstructure. In addition to some limitations related to durability and aging, LRBs encounter a large amount of residual deformation because of the plastically deformed lead core under severe ground motions (Dolce et al., 2000). Choi et al. (2005) proposed a new type of seismic base isolator using shape memory alloy wires to overcome the disadvantages of LRBs. They applied an SMA-based rubber bearing to a three-span continuous steel bridge in order to evaluate the seismic performance of the proposed smart isolator and compare it with LRB. Results showed that LRB experienced a large unrecoverable deformation while, the proposed SMA-based elastomeric isolator could restrain the deck from large relative displacement without any permanent deformation under strong earthquake records. On the other hand, they found that the amount of the energy dissipated through the proposed smart rubber bearing is less than that of LRB. In this regard, they explained that by increasing the size of SMA wires or changing the heat treatment process, the energy dissipation capacity can be increased.  Suduo and Xiongyan (2007) introduced three types of SMA-based dampers and one base isolator using nickel-titanium SMA wires. They proposed theoretical models to estimate the behaviour of the devices with high amounts of damping capacity. In order to evaluate the performance of these smart mechanisms on the seismic response control of structures, they implemented SMA-MR damper and SMA-based rubber bearing into structures. According to the findings, SMA-MR damper could regulate the seismic responses of a canopy roof structure. Suduo and Xiongyan (2007) compared the performances of lattice shell structures using SMA-based (smart) elastomeric isolator and conventional rubber bearings with those of fixed supported structures. They found that not only the smart base isolators can efficiently mitigate the seismic response in terms of acceleration, displacement, and internal forces but also, they have superior performances relative to the existing rubber bearings. They concluded that these intelligent systems have many advantages such as stability, high energy dissipation capacity, good fatigue and corrosion resistance capabilities and as a result long service life. Ozbulut and Hurlebaus (2011) probed the performance of a novel smart base isolator consisting of rubber bearing and an auxiliary device made of SMA wires on the seismic response of bridges against near-field earthquakes. Elastomeric bearing provides vertical 24  stiffness and horizontal flexibility while, the SMA-based device supplies additional energy dissipation capacity as well as re-centring capability. The SMA-based rubber bearing (SMA-RB) was implemented into a three-span continuous bridge and the whole system was numerically analyzed under several near-field ground motions matched to a design spectrum. For simulating the behaviour of nickel-titanium superelastic wires, a temperature- and rate-dependent model was used. They carried out several time-history analyses as well as a sensitivity analysis on the seismic response of the bridge by considering different factors such as the forward transformation strength and displacement, the pre-strain amount of SMA wires, the ambient temperature and the horizontal stiffness of the elastomeric isolator. It was found that by changing the deck displacement, the acceleration and the base shear change inversely at high values of forward transformation strength. Also, results demonstrated that the system is more sensitive to the negative changes of temperature than the positive ones. Ozbulut and Hurlebaus (2011) concluded that the influence of temperature on the isolation performance is very important and cannot be ignored. Another interesting finding was the improved efficiency of the system by using pre-strained SMA wires (1-1.5%).  Recent numerical studies on the performance of SMA-based elastomeric isolators show that using SMA in forms of wire and bars can effectively advance the efficiency of base isolators in terms of re-centring capability and energy dissipation capacity (Bhuiyan and Alam 2013; Hedayati Dezfuli and Alam, 2013c). Hedayati Dezfuli and Alam (2013c) performed numerical analyses on smart SMA wire-based NRBs considering different arrangements for the wires. They investigated the effect of several parameters (e.g. type of SMA, aspect ratio of the rubber bearing defined as the ratio of the height to the length, thickness of SMA wires, and pre-strain in wires) on the performance of the device. They found that for low-aspect-ratio NRBs (ratio of height to length = 0.38), it is more efficient to use pre-strained SMA wires in the straight configuration, while for high-aspect-ratio NRBs (0.38) using cross SMA wires with 2% pre-strain leads to a better performance. They concluded that the diameter of wires should be designed based on the requirements (i.e. lateral stiffness and equivalent viscous damping) in order to have SMA wires with the best performance as complementary dampers. HDRBs behave quite differently compared to NRBs, especially at high shear strain amplitudes. Hedayati Dezfuli and Alam (2013c) explored the effect of SMAs on the performance of NRBs and concluded that using SMA 25  wires can be beneficial to NRBs in terms of energy dissipation capacity and re-centring capability.  2.5 Summary In order to reduce the seismic demand on civil infrastructures several techniques are commonly employed in seismic prone regions. Seismic isolation, which is one of the most popular techniques, is widely used to safeguard bridges from severe damages due to strong earthquake events. A historical appraisal on seismic isolation systems was provided in the literature. Various types of bridge isolators have been developed, tested and are in use all over the world as effective earthquake resisting systems, including lead-rubber bearing (LRB), high damping rubber bearing (HDRB), friction pendulum bearing (FPB), magneto-rheological damper, and steel plate damper. The most popular isolation devices are LRB and HDRB. They have been used in buildings and bridges for both new constructions and retrofit projects. The premise of using these devices is to have high flexibility, which shifts the natural period of the bridge structure to a value beyond the critical period range of the earthquake event. In addition, they are endowed with damping properties that prevent the bridge piers and decks from undergoing excessive displacement. However, because of high weight and cost of steel-reinforced elastomeric isolators (SREIs), they are mostly implemented in large and expensive structures (e.g. high-rise buildings, important structural centres, and bridges). With the goal of extending the application of such base isolators to residential and low-rise buildings in both developed and developing countries, FREIs were introduced. FREIs with much lower weight can be manufactured in the form of long laminated pads through cost-effective automated processes.        Nevertheless, most of these devices have known limitations related to ageing and durability (e.g., rubber-based dampers), maintenance (e.g., viscous fluid dampers), long-term reliability (e.g., friction dampers), temperature dependent mechanical performance (e.g., rubber-based dampers, viscoelastic dampers), and geometry restoration after a strong earthquake (for most dampers). In this regard, the use of superelastic shape memory alloy (SMA) can provide an effective solution to overcome several of these problems. Superelastic SMA is a unique material with the ability to undergo large deformation and potentially recover its inelastic deformation upon stress removal. Its flag-shaped hysteresis, large 26  recoverable strain and deformation capability, excellent endurance against fatigue, and corrosion resistance make superelastic SMA an ideal material for utilizing it in seismic protection devices. Figure ‎2.6 presents a big picture of earthquake protective systems. By introducing smart SMA-based rubber bearings (SMA-RBs), as illustrated in the figure, it is highly important to establish an appropriate analytical model for such RBs in order to simulate their behaviour. As discussed in Chapter 1, section 1.2, this PhD thesis aims to address this topic which has not been covered in the literature.   Figure ‎2.6. Earthquake protective systems    27  Chapter 3 Multi-Criteria Optimization of Rubber Bearings Reinforced with CFRP Composites 3.1 General A material model was proposed for high damping rubber (HDR) in order to capture its highly nonlinear behaviour under shear and compressive loadings with different magnitudes and frequencies. In this regard, a comparison between six different nonlinear material models available in finite element software, ANSYS (ANSYS Mechanical APDL, Release 14.0) was carried out for modelling the HDR. The best-fitted and the most accurate material model was identified according to the behaviour of HDRB which has been evaluated through experimental study. The proposed model was validated by applying it to a steel-reinforced HDRB and comparing the lateral force-deflection hysteretic curve of the elastomeric isolator obtained from experimental tests and numerical simulations. In the next step, various carbon fibre-reinforced high damping rubber bearings (CFR-HDRBs) were simulated using FEM. The effects of different parameters such as number of rubber layers, thickness of fibre-reinforced polymer (FRP) composite plates and shear modulus of elastomer were investigated on the performance of CFR-HDRBs through a sensitivity analysis by proposing regression models for the response of the device. First, operational characteristics (effective horizontal and vertical stiffnesses, and equivalent viscous damping) of CFR-HDRBs were calculated. Then, depending on the importance level of each characteristic (criterion), a weight was assigned to each of them and the performance of the base isolator was optimized. 3.2 Material Modelling Mechanical properties of HDRB are modelled in ANSYS (ANSYS Mechanical APDL, Release 14.0). Some of the nonlinear material models available in ANSYS are classified in Figure  3.1. According to this chart, there are different models for hyperelastic, viscoelastic, viscoplastic and inelastic rate-independent materials (ANSYS Documentation, Release 14.0). Each material can be applied to a component using several models which are indicated by grey boxes. Each model should be defined with a set of material constants. In 28  addition to the following models, hyper-viscoelastic material models which are a combination of the hyperelastic and viscoelastic models can be defined as well.  Figure ‎3.1. Nonlinear material models in ANSYS (ANSYS Documentation, Release 14.0) 3.2.1 Bilinear Model A bilinear material is considered as a nonlinear, inelastic and strain-rate independent material. According to the Figure  3.2 which shows the stress-strain behaviour of a bilinear material, three different stiffnesses can be defined; the elastic stiffness, K1, in the elastic region, the post-elastic stiffness, K2, in the plastic region, and the effective stiffness which is ratio of the maximum force (Fmax) to the maximum displacement (dmax). The terms dy, Fy, and Qd denote yield displacement, yield force, and characteristic strength, respectively.  Figure ‎3.2. Force-displacement relation for bilinear material (adapted from (Ozkaya et al., 2011)) The Bauschinger effect which represents a decrease in the yield strength of the material by changing the direction of the strain can be included in this model (Bauschinger, Nonlinear Materials Elastic Prony Inelastic Viscoelastic Hyperelastic Rate Independent Rate Dependent Viscoplastic Perzyna Chaboche Neo-Hookean Mooney-Rivlin Ogden Isotropic Hardening Kinematic Hardening Maxwell Curve Fitting Curve Fitting dmax Fmax Fy Qd dy Keff K2 K1 29  1881). Bilinear model can be used for applications with small strain levels but, it cannot accurately predict the response of the materials which undergoes a large amount of strain (ANSYS Documentation, Release 14.0). Since two equations of state can be easily defined with a limited number of material properties, a bilinear model can be easily applied to a material. Also, the finite element analyses (FEA) are done in a shorter time compared to the cases in which other material models are used.  In ANSYS, the bilinear model can be used with isotropic hardening plasticity or kinematic hardening plasticity. In the kinematic hardening plasticity, the Bauschinger effect is considered while in the isotropic one it is assumed that the yield strength does not change by altering the strain direction. For both options, the yield stress and the tangential modulus should be defined in addition to the elastic modulus and Poisson’s ratio. 3.2.2 Hyperelastic Model Hyperelasticity is usually assigned to the elastic, isotropic and nonlinear materials which are almost incompressible in volume and undergo large deformations. Elastomers such as rubber and some polymer materials can be categorized in this class. There are different models for simulating the behaviour of such materials in ANSYS such as Neo-Hookean, Mooney-Rivlin and Ogden models. Neo-Hookean model with two material constants; the initial shear modulus (μ) and the incompressibility parameter (d), has the simplest form of strain energy potential which is defined as follows (Hoss and Marczak, 2010).  𝑊 =𝜇2(𝐼1 − 3) (‎3.1) where I1 is the first strain invariant of the right Cauchy-Green deformation tensor.  Mooney-Rivlin model represents a strain energy function which contains compressible and incompressible parts. Due to taking the higher order parameters into account, this model can predict the behaviour of rubber materials more accurately compared to the Neo-Hookean model. The incompressible part is written as a polynomial of the strain invariants while compressible part is a function of bulk modulus. Since it is assumed that the hyperelastic materials are fully incompressible, the strain energy function, W, is simplified according to Equation (‎3.2) (Wadham-Gagnon et al., 2006; Hoss and Marczak, 2010) 30  𝑊 = ∑ 𝐶𝑖𝑗(𝐼1 − 3)𝑖(𝐼2 − 3)𝑗𝑛𝑖+𝑗=1,   𝑛 = 1,2,3 (‎3.2) where Cij is the material constant describing the deviatoric deformation of the material (ANSYS Documentation, Release 14.0), I1 and I2 are the first and second deviatoric strain invariants, respectively. i and j can vary from 0 to n. By increasing the value of n, the accuracy of the prediction is enhanced. 3-parameter, 5-parameter and 9-paramnter models correspond to values of n equal to 1, 2, and 3, respectively. As an example, the strain energy function for the 9-parameter Mooney-Rivlin model can be written as follows: 𝑊 = 𝐶10(𝐼1 − 3) + 𝐶01(𝐼2 − 3) + 𝐶20(𝐼1 − 3)2 + 𝐶11(𝐼1 − 3)(𝐼2 − 3) + 𝐶02(𝐼2 − 3)2+ 𝐶30(𝐼1 − 3)3 + 𝐶21(𝐼1 − 3)2(𝐼2 − 3) + 𝐶12(𝐼1 − 3)(𝐼2 − 3)2+ 𝐶03(𝐼2 − 3)3 (‎3.3) and  𝐶10 + 𝐶01 =(1 − 2𝜈)𝑑=𝜇2 (‎3.4) where ν, d, and μ are the Poisson’s ratio, the material incompressibility parameter, and the initial shear modulus of the material, respectively. The Ogden material model proposed by Ogden (1972) is capable of approximating large strain levels accurately. The strain can be increased up to 700% (ANSYS Documentation, Release 14.0). Five different combinations from 1 to 5 terms can be defined for the strain energy function. Although a higher parameter value can predict the behaviour of material more precisely, it may increase the numerical complexity in fitting the material constants. The strain energy function is written in terms of the principal stretches λ1, λ2 and λ3 (Equation (‎3.5)). 𝑊 =∑𝜇𝑖𝛼𝑖(𝜆1𝛼𝑖 + 𝜆2𝛼𝑖 + 𝜆3𝛼𝑖  –  3)𝑛𝑖=1,   𝑛 = 1,2,… ,5 (‎3.5) where μi and αi are the material constants. The material constants can be also calculated by the software when the curve fitting option is used rather than pre-defined models. In such a situation, different sets of results obtained from experimental tests should be provided as input. After specifying a material 31  model by which the behaviour of the material is simulated, the material constants are computed via solving the constitutive equations corresponding to the selected material model. In order to calculate the material constants of a hyperelastic model in ANSYS, uniaxial test, biaxial test, shear test, simple shear test and volumetric test should be carried out. 3.2.3 Viscoelastic Model Viscoelastic materials are nonlinear, elastic and viscous. By applying a shear or tensile load, a viscous deformation is generated as a function of time. Prony and Maxwell models are two viscoelastic material models in ANSYS.  Prony model can be applied for estimating the shear and volumetric responses. For the shear response, variation of the shear modulus is calculated by changing the time while, for predicting the volumetric response, the bulk modulus decay is considered as a function of time. In Prony model, by defining a Prony series, the shear or bulk modulus is computed over the time. The Prony series for the shear modulus is given according to Equation (‎3.6) (Christensen, 1982; Mottahedi et al., 2010). 𝐺(𝑡) = 𝐺∞ +∑𝐺𝑖𝑒−𝑡𝜏𝑖𝑛𝑖=1 (‎3.6) where τi and Gi are the relaxation time constant and the corresponding shear modulus at τi, respectively. Number of terms in the Prony series can be increased in order to enhance the accuracy of the prediction. On the other hand, the complexity of the response is increased and as a result, the processing time goes up considerably. A dimensionless material constants, αi is defined as a ratio of the shear modulus at τi to the initial shear modulus, G0. The initial shear modulus G0 is: 𝐺0 = 𝐺∞ +∑𝐺𝑖𝑛𝑖=1 (‎3.7) By rewriting the Equation (‎3.6): 𝛼(𝑡) =𝐺(𝑡)𝐺0= 1 +∑𝛼𝑖 (𝑒−𝑡𝜏𝑖 − 1)𝑛𝑖=1 (‎3.8) where 32  𝛼𝑖 =𝐺𝑖𝐺0 (‎3.9) After performing a relaxation shear test and determining the variation of shear modulus versus time, the material constants for the Prony series are specified.  In addition to the mentioned models, curve fitting option in which material constants are calculated by solving the constitutive equations and fitting curves to the mechanical properties of the material, also can be used in ANSYS. For instance, in order to model the shear and volumetric behaviours of a rubber using curve fitting option, respectively, the changes of the shear and bulk moduli versus time should be obtained through relaxation tests and these experimental results are used as the input data. The maximum number of terms for the Prony series used in the curve fitting option is 10. 3.2.4 Viscoplastic Model A viscoplastic material is nonlinear, inelastic and strain-rate dependent. In such materials, the deformation is a function of the loading rate (Perzyna, 1966). When a permanent deformation happens in a viscoplastic material, it continues to undergo a creep flow through the time before removing the load. Such behaviour is not observed in the rate-independent plastic materials (Perzyna, 1966). Rate-dependency is very important in transient analyses where the applied load is a function of time.  In ANSYS, there are different models for viscoelastic materials such as Perzyna and Chaboche. It is recommended to combine these models with bilinear or multi-linear models with isotropic hardening plasticity. In the Perzyna model, two material constants are used; the strain-rate hardening parameter and the material viscosity parameter. The strain rate hardening effect is activated after plastic yielding by considering a yield surface. For simulating the monotonic hardening and the Bauschinger effect, the Chaboche model can be used (ANSYS Documentation, Release 14.0). Yield stress, two material constants for the first kinematic model and two material constants for the second kinematic model should be defined for this model. 3.2.5 Hyper-Viscoelastic Model Hyper-viscoelastic models can be applied to materials which have both hyperelastic and viscoelastic properties. ANSYS software has a capability of modelling such materials by 33  combining hyperelastic and viscoelastic models. Bergstrom–Boyce, combined Ogden–Prony and combined Mooney-Rivlin–Prony are three models used in ANSYS. The hyper-viscoelastic model could capture the behaviour of the highly nonlinear materials such as HDR for the reason that, both hyperelastic behaviour and inelastic rate-dependent shear and volumetric responses of rubber subjected to a compressive loads and cyclic shear loads are considered (Bergstrom and Boyce, 1998). The Bergstrom-Boyce material model can simulate the hysteretic behaviour of a strain-rate-dependent material (e.g. elastomers) with large elastic and inelastic strain levels (Bergstrom and Boyce 1998). The inelastic shear response of the material can be simulated through this model while just the elastic volumetric deformations can be captured. The Bergstrom-Boyce model describes a system with two parallel springs and one damper which is series with one of the springs (ANSYS Documentation, Release 14.0). The Prony viscoelastic model can be combined with the Ogden or Mooney-Rivlin hyperelastic material models as well. 3.2.6 Comparing Material Models In order to specify an appropriate material model for HDR, a number of FEAs using different material models for HDR are carried out to check which model can be fitted more accurately with the actual response of the elastomer obtained from experimental tests. The complexity of the nonlinear behaviour of HDR and a large amount of elements used in the FEAs lead to a time-consuming simulation and the solution may diverge in some cases. Therefore, a HDR layer with thickness-to-length ratio of 0.05 as a simplified 2-D model is analyzed under a cyclic shear loading via ANSYS software. The bottom surface of the elastomer layer is completely fixed and the top surface is under a cyclic lateral displacement with frequency of 0.25 Hz and amplitude of γ = 100%. Shear strain, γ, is the ratio of the elastomer thickness to the maximum lateral displacement of the top surface (Figure  3.3). The element used for the HDR is PLANE182. 34   Figure ‎3.3. Cyclic horizontal displacement (f = 0.25 Hz, γ = 100%)   Figure ‎3.4. Normalized shear force versus lateral displacement for HDR layer simulated using different material models; (a) Hyperelastic (Mooney-Rivlin), (b) Bilinear with kinematic hardening, (c) Viscoplastic (Perzyna), (d) Hyper-viscoelastic (Ogden-Prony), (e) Hyper-viscoelastic (Mooney-Rivlin and Prony),(f) Hyper-viscoelastic (Bergstrom-Boyce) Horizontal force-deflection hysteretic curve for each material model is plotted in Figure  3.4. By comparing the hysteretic response of the HDR layer modelled and analyzed with different material models, and according to the typical shear behaviour of this nonlinear material, the most applicable model can be specified. In Figure ‎3.4, the vertical axes are the normalized lateral force calculated by dividing each value of the load by the maximum value of the load. As a result, the responses can be compared together in an easier way. The rubber layer modelled by 9-paramater Mooney-Rivlin model which is hyperelastic and rate-independent does not show any hysteresis effect in the force-deflection -5.0-2.50.02.55.00 1 2 3 4 5 6 7 8Lateral Displacement (mm) Time -1-0.500.51-5 -2.5 0 2.5 5-1-0.500.51-5 -2.5 0 2.5 5-1.2-0.8-0.400.40.81.2-5 -2.5 0 2.5 5-1-0.500.51-5 -2.5 0 2.5 5-1-0.500.51-5 -2.5 0 2.5 5-1-0.500.51-5 -2.5 0 2.5 5(a)                                                 (b)                                                 (c)                       (d)                                                 (e)                                                 (f)                       35  response. For the bilinear model with kinematic hardening plasticity, the HDR layer undergoes a plastic deformation when the lateral deflection exceeds the yield displacement. Figure  3.4c, d, and e illustrate the hysteresis responses of hyper-viscoelastic rubber. The Bergstrom-Boyce model and the combined 3-parameter Ogden and 2-parameter Prony models could not capture the shear behaviour in the plastic region accurately while, the combined 9-parameter Mooney-Rivlin and 2-parameter Prony models can simulate the shear hysteretic response precisely. The HDR layer modelled with the viscoelastic Perzyna material model which is inelastic and rate-dependent shows a quasi-static hysteresis effect. Thus, the second loop of the force-deflection curve is different from the first one (Burtscher et al., 1998). By comparing the load-deflection curve for each material model obtained from the FEAs with the typical hysteresis behaviour of a HDRB subjected to the pure cyclic lateral displacement with frequency of 1 Hz and strain ranges from 40% to 200% (Figure  3.5) (Dall’Asta and Ragni, 2006), it can be found that the hyper-viscoelastic models have a better estimation of the behaviour of HDR materials. The main reason is that both hyperelasticity and rate-dependency of the HDR under cyclic shear deformations are considered in the FE modelling. Among three suggested material models; Mooney Rivlin-Prony, Ogden-Prony and Bergstrom-Boyce, the first one can predict the hysteretic behaviour of HDR material under cyclic shear loadings more precisely. The Mooney-Rivlin option models hyperelasticity and the Prony model could describe the rate-dependency of HDR under pure shear loads. Using curve fitting option in ANSYS, material constants of both Mooney-Rivlin and Prony models can be determined from experimental data obtained from ASTM or ISO standard tests. Here, data are gathered from the uniaxial tension-compression tests, the biaxial tension test, and the creep test conducted on HDR (Ibrahim, 2005). The uniaxial and biaxial tests are used for material characterization of hyperelastic model while, the material constants of viscoelastic model are determined through the creep test. 36   Figure ‎3.5. Stable hysteresis loops at different strain amplitudes  (adapted from (Dall’Asta and Ragni, 2006)) 3.3 Numerical Validation and Verification The proposed material model for HDR is used in a steel-reinforced HDRB which is numerically simulated in ANSYS. Lateral load-deflection hysteretic loop of the HDRB is determined through FEA and compared by experimental results obtained from Dall’Asta and Ragni (2006). The HDRB used in the experiments has two rubber layers, each of them with thickness of 5 mm, and one steel shim with thickness of 2 mm. Two supporting steel plates with 15 mm thickness restrain the device in the test setup. Figure  3.6 shows the dimensions of the HDRB.   Figure ‎3.6. Top and side views of a HDRB (adapted from (Dall’Asta and Ragni, 2006))  To model the HDRB in ANSYS, solid element, SOLID185, with 8 nodes and three degrees of freedom at each node is selected for the steel and the rubber. The large-deflection effect is considered in FE analysis in order to simulate the large deformation of rubber layers -100-75-50-250255075100-250 -200 -150 -100 -50 0 50 100 150 200 250Force (kN) Shear Strain (%) 37  at large shear strain amplitudes. Figure  3.7 shows a HDRB with 8 rubber layers and 0.31 mm thick CFRP plates with a mapped mesh in ANSYS (ANSYS Mechanical APDL, Release 14). Blue, purple and red elements respectively illustrate steel, rubber, and CFRP materials. The lower surface of the bottom steel supporting plate is fixed in all degrees of freedom. In order to model the top steel plate to remain perfectly straight during loadings, all nodes on the upper surface of this plate are constrained to move together in the z (vertical) direction while they are free in the two other directions. The HDRB is analyzed under a given cyclic horizontal displacement with frequency of f = 0.49 Hz, and amplitude of γ = 90% with no vertical pressure since no vertical load has been applied in the experiment (Dall’Asta and Ragni, 2006). The hysteretic shear response of the device is evaluated through a full transient analysis during which the cyclic lateral displacement is linearly interpolated for each substep from the value of the previous load step.         Figure ‎3.7. HDRB with mapped mesh in ANSYS (ANSYS Mechanical APDL, Release 14) The material constants of hyper-viscoelastic model which is used for simulating the behaviour of high damping rubber are listed in Table ‎3.1.     38  Table ‎3.1. Hyper-viscoelastic material model constants Mooney-Rivlin Model Prony Model C10 = 2.147 α1 = 0.765 C01 = 0.193 τ1 = 0.041 C11 = -0.01 α 2 = 0.061 C20 = 0.108 τ 2 = 65.82 C02 = 0.047  C30 = 0.003  C21 = -0.013  C12 = 0.0001  C03 = 0.000  After modelling and analyzing the HDRB, the hysteresis curves are plotted and then, compared to the experimental results (see Figure  3.8). Figure  3.8 shows a good consistency between numerical and experimental results.   Figure ‎3.8. Lateral force-deflection hysteresis curve (P = 0 MPa, f = 0.49 Hz, γ = 90%) (Experimental results are adapted from (Dall’Asta and Ragni, 2006)) Verification refers to the correctness of result and validation indicates the accuracy of prediction. Therefore, in order to verify the hyper-viscoelastic material model proposed for HDRB, the hysteretic response of HDRB at different shear strain levels are estimated numerically and then compared to the experimental results. As can be observed in Figure  3.9, the FE model is capable of accurately simulating the hysteresis of HDRB. -60-40-200204060-10 -8 -6 -4 -2 0 2 4 6 8 10Force (kN) Lateral Displacement, Ux (mm) ExperimentNumerical Model-1010 1 2 3 4 5Shear Strain Time (s) 39   Figure ‎3.9. Lateral force-deflection hysteresis curves of HDRB at γ = 43%, 90%, 155%, and 200% obtained through FEM and experiment (Dall’Asta and Ragni, 2006) Based on the numerical results obtained from FE analyses, the effective horizontal stiffness of the HDRB is 7.76 kN/mm, 5.67 kN/mm, 4.56 kN/mm, and 4.23 kN/mm for 43%, 90%, 155%, and 200% shear strain amplitudes, respectively. The maximum difference between the numerical and experimental results in the horizontal stiffness is 5%, which happens at 200% shear strain. According to both experimental and numerical results, the maximum horizontal loads experienced at 43%, 90%, 155%, and 200% shear strains are 34 kN, 51 kN, 71 kN, and 85 kN, respectively. 3.4 Sensitivity Analysis In order to evaluate the performance of CFR-HDRBs, three operational characteristics can be considered. These specifications which identify the vertical stiffness, the horizontal flexibility and the energy dissipation capacity of the seismic base isolators are vertical stiffness, effective horizontal stiffness, and equivalent viscous damping, respectively. The effective horizontal stiffness, which is denoted by KH, is calculated according to Equation (‎3.10) (Kelly, 1997): 𝐾𝐻 =𝐹𝑚𝑎𝑥 –𝐹𝑚𝑖𝑛 Δ𝑚𝑎𝑥 − Δ𝑚𝑖𝑛 (‎3.10) where Fmax and Fmin are the maximum positive and negative shear forces, and Δmax and Δmin are the maximum positive and negative shear displacements, respectively. -100-80-60-40-20020406080100-25 -20 -15 -10 -5 0 5 10 15 20 25Force (kN) Lateral Displacement (mm)           43%           90%           155%           200% FEM       Exp 40  To measure the equivalent viscous damping, βeq, the energy dissipated in each cycle (EDC) which is the area inside the hysteresis loop of shear force-displacement curve should be computed (Naeim and Kelly, 1999). 𝛽𝑒𝑞 =𝐸𝐷𝐶2𝜋 𝐾𝐻 Δ𝑚𝑎𝑥2  (‎3.11) The vertical stiffness of a fibre-reinforced elastomeric isolator is given by Equation (‎3.12) (Tsai and Kelly, 2002): 𝐾𝑉 =𝐸𝑐𝐴𝑓𝑡𝑟 (‎3.12) where Af is the cross-sectional area of the reinforcement, tr is the total thickness of rubber in the bearing and Ec is the instantaneous compression modulus of the FREI under a specific level of vertical load. The effective compressive modulus of a rectangular isolator with the plane size (length and width) of 2a by 2b, is calculated using Equation (‎3.13) (Tsai and Kelly, 2002). 𝐸𝑐 =𝑆2𝐸𝑓𝑡𝑓𝑡𝑒𝑎2(1 − 𝜈2)(1 −tanh𝛼𝑎𝛼𝑎)𝐶 (‎3.13) 𝐶 = {1 +𝑎𝑏[−0.59 + 0.026𝛼𝑎 + 0.074(𝛼𝑎)2 − 0.022(𝛼𝑎)3 + 0.0019(𝛼𝑎)4]} (‎3.14) in which Ef and ν are the elastic modulus and Poisson’s ratio of the reinforcement, respectively. Here, tf and te are the thickness of the equivalent sheet of reinforcement (total thickness of reinforcement layers) and the thickness of one rubber layer, respectively.  𝛼 = √12𝐺𝑟(1 − 𝜈2)𝐸𝑓𝑡𝑓𝑡𝑒 (‎3.15) where Gr is the shear modulus of the elastomer.  S is the shape factor and defined as the ratio of the bonded (loaded) area to the force-free area of one elastomeric layer. For a rectangular rubber bearing, this geometrical parameter is calculated from Equation (‎3.16). 𝑆 =𝐿 ×𝑊2𝑡𝑒(𝐿 +𝑊) (‎3.16) 41  where L and W are length and width of each rubber layer, respectively. The lateral force-deflection curve of each CFR-HDRBs is calculated. Then, the performance characteristics are obtained using Equations (‎3.10) to (‎3.12).  Mechanical properties of fibre-reinforced polymer (FRP) composite material made of carbon and epoxy are listed in Table ‎3.2. Table ‎3.2. Material properties for CFRP composite material (Howie and Karbhari, 1994) FRP Composite Elastic Modulus (MPa) Shear Modulus (MPa) Poisson’s Ratio Tensile Strength (MPa) Thickness (mm) Carbon/Epoxy Ex = 73300 Gxy = 1761.0 νxy = 0.310 755 0.31 Ey = 4613.8 Gyz = 1659.5 νyz = 0.390 Ez = 4613.8 Gzx = 1761.0 νzx = 0.019 Different physical and mechanical properties such as the thickness and the number of rubber layers and FRP composite sheets as well as the shear modulus of elastomer can affect the response of the carbon fibre reinforced elastomeric isolator. In order to probe the effect of each parameter on the behaviour of device, different regression models will be proposed for each operational characteristic as different outputs of the system.  Total height of the laminated elastomeric isolator, H, is a function of different variables and is computed according to the Equation (‎3.17): 𝐻 = 𝑡𝑒𝑛𝑒 + 𝑡𝑓𝑛𝑓 + 2𝑡𝑠 (‎3.17) in which, te, ne, tf and nf are the thickness of one rubber layer, the number of rubber layers, and the thickness and number of FRP composite plates, respectively. Ts is the thickness of supporting steel plates mounted at the top and bottom of the device (Figure  3.10).   Figure ‎3.10. Fibre-reinforced elastomeric isolator; (a) plan view, (b) side view (a)                                                  (b) 42  Since the thickness of supporting plates is assumed to be constant in all cases and number of reinforced sheets is nr – 1, the total height would be: 𝐻 = 𝑓(𝑡𝑒 , 𝑛𝑒 , 𝑡𝑓) = 𝑡𝑒𝑛𝑒 + 𝑡𝑓(𝑛𝑒 − 1) + 2𝑡𝑠 (‎3.18) Here, the performance of different elastomeric base isolators with fixed length, width and height is compared to each other. Under this condition, for a constant height, thickness of rubber layers can be computed as follows: 𝑡𝑒 = 𝑓′(𝑡𝑓 , 𝑛𝑒) =𝐻 − 2𝑡𝑠 − 𝑡𝑓(𝑛𝑒 − 1)𝑛𝑟 (‎3.19) According to Equation (‎3.19), thickness of elastomeric layers is a function of reinforcement thickness and number of rubber layers. Therefore, two physical characteristics (tf and ne) in addition to shear modulus as a mechanical property are considered in the sensitivity analysis. It should be mentioned that other parameters such as the temperature and the speed of loading also affect the performance of the elastomeric base isolator. The shear modulus of the elastomer is a function of the service temperature and the hardness. When the temperature rises, the shear modulus decreases for different values of hardness of rubber according to Figure  3.11. Here a constant temperature of 23°C is considered. The strain rate was considered in the process of validation of the HDR material model, however, it is not considered as a variable (factor) in the optimization process. Variation of temperature is also not considered. Taking these two factors into account in addition to three mentioned factors leads to a total number of 125 runs in a full factorial experiment with five factors and three levels for each. Hence, further study is required to in order to conduct a more comprehensive investigation with this amount of alternatives (rubber bearings).   43   Figure ‎3.11. Shear modulus of rubber as a function of temperature  (adapted from (GoodCo Z-Tech, 2010)) For each factor, three levels as low, medium, and high values are determined. As a result, a full factorial experiment with three factors and three levels can be designed. By considering all combinations of three factors, each of them with three levels, a total number of 27 runs should be observed in this experiment. The factors and their levels are listed in Table ‎3.3. Table ‎3.3. Parameters and their levels considered in the sensitivity analysis Factor Symbol Level 1 2 3 nr A 8 9 10 tf (mm) B 0.31 0.62 0.93 Gr (MPa) C 0.6 0.7 0.8 The 33 full factorial design will be arranged in Table ‎3.4. In this design, the number of rubber layers, thickness of reinforced plates and shear modulus of elastomer are denoted by nr, tf and Gr, respectively.     0.00.51.01.52.02.5-40 -30 -20 -10 0 10 20Shear Modulus, MPa Temperature, °C 50 duro55 duro60 duro44  Table ‎3.4. A 33 full factorial design with 3 factors and 3 levels Run Factor nr tf Gr 1 8 0.31 0.6 2 9 0.31 0.6 3 10 0.31 0.6 4 8 0.62 0.6 5 9 0.62 0.6 6 10 0.62 0.6 7 8 0.93 0.6 8 9 0.93 0.6 9 10 0.93 0.6 10 8 0.31 0.7 11 9 0.31 0.7 12 10 0.31 0.7 13 8 0.62 0.7 14 9 0.62 0.7 15 10 0.62 0.7 16 8 0.93 0.7 17 9 0.93 0.7 18 10 0.93 0.7 19 8 0.31 0.8 20 9 0.31 0.8 21 10 0.31 0.8 22 8 0.62 0.8 23 9 0.62 0.8 24 10 0.62 0.8 25 8 0.93 0.8 26 9 0.93 0.8 27 10 0.93 0.8 In order to obtain an accurate model for the response of the carbon-FREI, the main factor effects (A, B and C), the second order effects (A2, B2 and C2) and the interaction effects (AB, AC, BC, and ABC) are considered in the calculations. The size of rubber bearings (length, width and height) and the thickness of supporting steel plates are listed in Table ‎3.5.  Table ‎3.5. Geometrical properties of CFR-HDRBs Item Symbol Value Unit Length L 200 mm Width W 200 mm Height H 80 mm Thickness of supporting plate ts 20 mm 45  Figure  3.12 demonstrates the process of determining three outputs or responses (performance characteristics of the laminated elastomeric isolator) for the system by considering three factors as different inputs. Responses are calculated through a number of numerical simulations in ANSYS (ANSYS Mechanical APDL, Release 14).  Figure ‎3.12. Input and output of the system for performance analysis 3.4.1 Performance of CFR-HDRB By determining two physical properties and one mechanical property for CFR-HDRB as inputs of the system (Figure  3.12), 27 numerical simulations are performed using FEM. The output will be the system response in terms of vertical and horizontal stiffnesses, as well as equivalent viscous damping. For elastomeric bearings used in the bridges, the allowable pressure under permanent load is 4.5 MPa for the serviceability limit state (SLS) and 7.0 MPa for the ultimate limit state (ULS) (GoodCo Z-Tech, 2010). Here, each rubber bearing is subjected to a constant vertical pressure of 6 MPa and a cyclic lateral displacement with amplitude of γ = 100% and frequency of f = 0.2 Hz. The amplitude is ratio of maximum horizontal deflection to the total thickness of rubber layers. Since number and thickness of rubber layers change for each CFR-HDRB, the magnitude of lateral displacement, which is equal to the relative displacement of the supporting steel plates, varies in each run. In Table ‎3.6, the performance characteristics (outputs) are specified for each rubber bearing as an alternative.      Physical and Mechanical Properties of C-FREI - Number of rubber layers, nr - Thickness of reinforcement, tf  - Shear modulus of rubber layer, Gr Numerical Simulation of C-FREI Finite Element Method Performance of C-FREI - Vertical stiffness, KV - Horizontal stiffness, KH - Eq. viscous damping, βeq Input System Output 46  Table ‎3.6. Responses of CFR-HDRBs in 33 full factorial design CFR-HDRB KV (kN/mm) KH (kN/mm) β (%) 1 114.1 0.410 15.03 2 147.3 0.414 14.95 3 185.5 0.417 14.88 4 132.3 0.434 14.60 5 174.5 0.441 14.56 6 224.6 0.447 14.52 7 153.5 0.458 14.45 8 207.3 0.468 14.42 9 273.5 0.479 14.40 10 132.7 0.471 15.18 11 171.2 0.475 15.10 12 215.6 0.478 15.04 13 154.2 0.497 14.77 14 203.3 0.504 14.73 15 261.6 0.511 14.70 16 178.9 0.523 14.63 17 241.6 0.535 14.60 18 318.8 0.547 14.58 19 151.1 0.530 15.28 20 195.0 0.534 15.20 21 245.6 0.538 15.14 22 175.9 0.558 14.88 23 232.0 0.566 14.85 24 298.5 0.574 14.81 25 204.3 0.587 14.75 26 275.9 0.600 14.72 27 363.9 0.614 14.70 Based on the 33 full factorial design as arranged in Table ‎3.4, the changes of three normalized outputs are compared together (Figure  3.13). For each performance characteristic, the normalized values in each column are obtained by dividing the output of each rubber bearing by the summation of 27 elastomeric bearings’ outputs. These changes show that the vertical stiffness is much more sensitive to the number and shear modulus of rubber layer and the thickness of FRP composite plates compared to two other criteria. Moreover, the equivalent viscous damping fluctuates within a limited range while, the effective horizontal stiffness has an increasing trend.   47   Figure ‎3.13. Normalized performance characteristics for different CFR-HDRBs 3.4.2 Regression Models The importance level of a factor for estimating the response (stiffness or damping of rubber bearings) of the system can be evaluated based on the t-statistic value of that factor which is calculated by dividing the coefficient of the factor by its standard error. The coefficient and t-statistic value for each factor including main effects (A, B and C) and their second order and interaction effects (A2, B2, C2, AB, AC, BC and ABC) are calculated using regression toolbox in Excel (Table ‎3.7). Table ‎3.7. Coefficients of each factor in the regression models Effect KV (kN/mm) KH (kN/mm) β (%) Coeff. t-Stat Coeff. t-Stat Coeff. t-Stat Intercept 396.9 4.2 0.005 0.2 14.77 36.1 nr -87.1 6.3 -0.003 0.1 -0.18 3.0 tf -41.0 0.4 0.018 0.7 -3.52 7.3 Gr -66.1 1.0 0.352 22.7 2.53 8.5 nr² 4.9 7.9 0.0 0.8 0.004 1.4 tf² 35.7 5.5 0.008 5.5 1.27 45.3 Gr² -1.6 0.1 -0.018 5.0 -0.75 11.1 nr ·tf -0.9 0.1 0.005 0.2 0.10 1.8 nr ·Gr 18.2 3.1 -0.002 1.4 0.02 0.7 tf ·Gr -212.6 2.7 -0.038 2.1 0.32 0.9 nr ·tf ·Gr 33.4 3.8 0.009 4.7 -0.02 0.4 For predicting the vertical stiffness, number of rubber layers is the dominant factor while, the shear modulus of elastomeric layers is the most effective factor for the effective horizontal stiffness. The equivalent viscous damping is influenced by all three main factors. 0.020.030.040.050.060.070 6 12 18 24 30Normalized Output Alternative (CFR-HDRB) V. StiffnessH. StiffnessViscous Damping48  Higher order and interaction effects play an important role in estimating the horizontal and vertical stiffnesses of CFR-HDRBs. By comparing the predicted responses calculated from regression models with actual values obtained from numerical simulations and based on the t-statistic value for each coefficient, the simplified regression model for each response by considering the more significant factors are proposed as follows: 𝐾𝑉 = 396.9 − 87.1𝑛𝑟 − 41.0𝑡𝑓 − 66.1𝐺𝑟 + 4.9𝑛𝑟2 + 35.7𝑡𝑓2 + 18.2𝑛𝑟𝐺𝑟 − 212.6𝑡𝑓𝐺𝑟+ 33.4𝑛𝑟𝑡𝑓𝐺𝑟   (𝑘𝑁/𝑚𝑚) (‎3.20) 𝐾𝐻 = 0.005 + 0.352𝐺𝑟 + 0.008𝑡𝑓2 − 0.018𝐺𝑟2 − 0.038𝑡𝑓𝐺𝑟 + 0.009𝑛𝑟𝑡𝑓𝐺𝑟   (𝑘𝑁/𝑚𝑚) (‎3.21) 𝛽𝑒𝑞 = 14.77 − 0.18𝑛𝑟 − 3.52𝑡𝑓 + 2.53𝐺𝑟 + 1.27𝑡𝑓2 − 0.75𝐺𝑟2 + 0.10𝑛𝑟𝑡𝑓 + 0.32𝑡𝑓𝐺𝑟 (‎3.22) where, nr, tf, and Gr are the number of rubber layers, thickness of FRP composite plates, and shear modulus of elastomeric layers, respectively.  In order to validate the proposed regression models and check their accuracy, eight new CFR-HDRBs with different values of nr, tf and Gr from the previously assumed values, are considered and the predicted responses are compared to values computed through FEM.  Length, width, height and the thickness of supporting steel plates are kept constant. Specifications of 8 new elastomeric isolators are listed in Table ‎3.8. Table ‎3.8. Physical properties of CFR-HDRBs Factor CFR-HDRBs #1 #2 #3 #4 #5 #6 #7 #8 L (mm) 200 200 200 200 200 200 200 200 W (mm) 200 200 200 200 200 200 200 200 H (mm) 80 80 80 80 80 80 80 80 te (mm) 5.11 4.09 4.67 3.64 5.11 4.09 4.67 3.64 tf (mm) 0.5 0.5 1 1 0.5 0.5 1 1 ts (mm) 15 15 15 15 15 15 15 15 ne 9 11 9 11 9 11 9 11 nf 8 10 8 10 8 10 8 10 The relative error between the predicted responses (operational characteristics) calculated from the regression model and the results obtained from the FEAs are listed in Table ‎3.9.  49  Table ‎3.9. Performance characteristics of eight CFR-HDRBs obtained from FE analyses and regression models   Vertical stiffness (kN/mm) Effective horizontal stiffness (kN/mm) Equivalent viscous damping (%) KV FEM KV Reg. Δrel (%) KH FEM KH Reg. Δrel (%) βeq FEM  βeq Reg. Δrel (%) 1 203.8 203.1 -3.32 0.523 0.523 -1.30 14.88 14.91 2.72 2 325.2 328.7 -3.65 0.535 0.536 -1.58 14.80 14.82 4.04 3 269.4 271.2 -4.87 0.575 0.575 0.88 14.64 14.67 2.04 4 465.8 446.0 1.44 0.604 0.603 1.07 14.60 14.66 2.94 5 270.3 270.8 -4.01 0.679 0.671 -0.98 15.15 14.99 4.27 6 431.2 431.3 -2.59 0.694 0.687 -1.46 15.08 14.91 5.60 7 358.4 360.9 -4.64 0.745 0.735 0.88 14.94 14.80 3.21 8 619.5 587.2 2.67 0.782 0.770 0.86 14.90 14.78 4.11 Figure  3.14 to Figure  3.16 present the FE results along with regression models proposed for each characteristic.  Figure ‎3.14. Vertical stiffness calculated through FE analysis and regression model  for 8 CFR-HDRBs  Figure ‎3.15. Effective horizontal stiffness calculated through FE analysis and regression model  for 8 CFR-HDRBs 01002003004005006007001 2 3 4 5 6 7 8Vertical Stiffness, KV  (kN/mm) CFR-HDRB FEM SimulationRegression Model0.50.550.60.650.70.750.81 2 3 4 5 6 7 8Horizontal Stiffness, KH (kN/mm) CFR-HDR FEM SimulationRegression Model50   Figure ‎3.16. Equivalent viscous damping calculated through FE analysis and regression model  for 8 CFR-HDRBs Low relative errors for all criteria show that the proposed simplified regression models could accurately predict the response of the system. Since in the simplified regression models unimportant factors are eliminated, the maximum error for the vertical stiffness, the effective horizontal stiffness, and the equivalent viscous damping increase to 4.9%, 1.6% and 5.6%, respectively.  3.4.3 Effect of Number of Rubber Layers The effect of number of elastomeric layers (ne) on the performance of CFR-HDRBs is investigated in this part. Figure  3.17 depicts the changes of each output by increasing the number of rubber layers from 8 to 10 for three different cases in which two other factors (thickness of reinforcement and shear modulus of rubber layers) are fixed at their low, intermediate and high levels, respectively.   Figure ‎3.17. Effect of number of rubber layers on the CFR-HDRB’s behaviour;  (a) tf = 0.31 mm, Gr = 0.6 MPa, (b) tf = 0.62 mm, Gr = 0.7 MPa, (c) tf = 0.93 mm, Gr = 0.8 MPa 12131415161 2 3 4 5 6 7 8Damping Capacity, βeq (%) CFR-HDRB FEM SimulationRegression Model0.00.20.40.60.81.08 9 10Normalized Output No. of Rubber Layers KvKhBeta0.00.20.40.60.81.08 9 10Normalized Output No. of Rubber Layers KvKhBeta0.00.20.40.60.81.08 9 10Normalized Output No. of Rubber Layers KvKhBeta(a)                                                 (b)                                                 (c) 51  For all three cases, by increasing the number of rubber layers, the vertical stiffness increases considerably while the effective horizontal stiffness rises slightly and the equivalent viscous damping almost remains constant. For the first case, when the thickness of reinforcement (tf) and the shear modulus of rubber layers (Gr) are 0.31 mm and 0.6 MPa, respectively, if the number of rubber layers increases from 8 to 10, the vertical stiffness and the effective horizontal stiffness increase about 62% and 2%, respectively. For the second case, these changes are 70% for the vertical stiffness and 3% for the horizontal stiffness. In the last case in which tf and Gr are at their high levels, the changes of KV and KH are 78% and 5%, respectively. 3.4.4 Effect of Thickness of FRP Reinforcement The second physical property which affects the efficiency of the CFR-HDRB with unidirectional carbon fibres is the thickness of FRP composite plates. Figure  3.18 shows the changes of each output by increasing the thickness of reinforced sheets from 0.31 mm to 0.93 mm for three different cases in which ne and Gr are fixed at their low, intermediate and high levels. For all three cases, by increasing the thickness of reinforced plates, the vertical and horizontal stiffnesses increase while the equivalent viscous damping decreases marginally.  Figure ‎3.18. Effect of thickness of fibre-reinforced plates on the CFR-HDRB’s behaviour;  (a) ne = 8, Gr = 0.6 MPa, (b) ne = 9, Gr = 0.7 MPa, (c) ne = 10, Gr = 0.8 MPa In the first case, when 8 rubber layers with shear modulus of 0.6 MPa are used, by increasing tf from 0.31 mm to 0.93 mm, the vertical stiffness, the effective horizontal stiffness and the equivalent viscous damping changes about +35%, +12% and -4%, respectively. For the second case, these variations are +41% for the vertical stiffness, +13% for the horizontal stiffness and -3% for the damping capacity. In the last case in which nr and 0.00.20.40.60.81.00.31 0.62 0.93Normalized Output Thickness of Reinf. (mm) KvKhBeta0.00.20.40.60.81.00.31 0.62 0.93Normalized Output Thickness of Reinf. (mm) KvKhBeta0.00.20.40.60.81.00.31 0.62 0.93Normalized Output Thickness of Reinf. (mm) KvKhBeta(a)                                                 (b)                                                 (c) 52  Gr are at their high levels, the changes of KV, KH and βeq are +48%, +14% and -3%, respectively. 3.4.5 Effect of shear modulus of rubber layers Shear modulus of elastomer changes the behaviour of a base isolator. Figure  3.19 exhibits the changes in the performance characteristics by increasing the shear modulus of rubber layers (Gr) from 0.6 MPa to 0.8 MPa for three different cases in which ne and tf are fixed at their low, intermediate and high levels.   Figure ‎3.19. Effect of shear modulus of elastomer on the CFR-HDRB’s behaviour;  (a) ne = 8, tf = 0.31 mm, (b) ne = 9, tf = 0.62 mm, (c) ne = 10, tf = 0.93 mm All three outputs; the vertical and horizontal stiffnesses, and damping capacity, ascend by increasing the shear modulus of elastomer. The rate of changes in βeq is lower than that of KV and KH. For the first case, when 8 rubber layers and 0.31 mm thick FRP composite sheets are used, by increasing Gr from 0.6 to 0.8 MPa, the vertical stiffness, the effective horizontal stiffness and the equivalent viscous damping increased by 32%, 29% and 2%, respectively. In the case of CFR-HDRB with 9 elastomer layers and 0.62 mm thick reinforced sheets, the changes include; 33%, 28% and 2% increase in vertical stiffness, horizontal stiffness and equivalent viscous damping, respectively. When ne and tf are at their high levels, the variations of KV, KH, and βeq are almost same as the second case. 0.00.20.40.60.81.01.20 1.40 1.60Normalized Output Shear Modulus of Rubber (MPa) KvKhBeta0.00.20.40.60.81.01.20 1.40 1.60Normalized Output Shear Modulus of Rubber (MPa) KvKhBeta0.00.20.40.60.81.01.20 1.40 1.60Normalized Output Shear Modulus of Rubber (MPa) KvKhBeta(a)                                                 (b)                                                 (c) 53  3.5 Multi-Criteria Optimization 3.5.1 Theory In each multi-criteria decision making (MCDM) problem there are a number of alternatives (Ai) and criteria (Cj). According to the main goal which is defined by the decision maker or physics of the problem, each criterion should be maximized, minimized or calibrated (when a target is defined) and the best case or the optimized alternative is specified when all of the criteria are in their best conditions.  There are different weighting and scoring methods by which the ranking of alternatives in a MCDM problem is determined. Weights can be assigned to each attribute by the decision maker (direct assignment method) or can be calculated according to the statistical data in the problem through Entropy method (Hwang and Yoon, 1981). The score of each alternative is computed by taking all criteria into account using different scoring methods such as weighted sum method (WSM) or TOPSIS which are two well-known techniques (Yoon and Hwang, 1995). The alternative with the highest score is selected as the best one and is placed in the first rank when the goal of MCDM is maximizing all criteria. On the other hand, the worst alternative has the lowest score.  The first step in the multi-criteria decision making is to transform data in order to have a problem with “the higher, the better” condition or “the lower, the better” condition for all attributes. For example, if one criterion (Cj) is needed to be minimized while the others should be maximized, by reversing Cj, the goal would be maximizing all criteria. Also, the unit and scale of one attribute may be different from those of other ones. So, in the next step, each criterion should be normalized. There are many methods for normalization. Here, each criterion is divided by the summation of that criterion for all alternatives. In the direct assignment method which is used here, the weight of each attribute is determined according to the qualitative evaluations tabulated in Table 3.10.       54  Table ‎3.10. Assignment of values for a 10-point scale Attribute evaluation Value Extremely Unimportant 0 Very Unimportant 1 Unimportant 3 Average 5 Important 7 Very Important 9 Extremely Important 10 The normalized weight for each criterion is calculated by dividing the evaluation value of each one by the summation of values. If the normalized criterion j for alternative i is denoted by ijCˆ  and the normalized weight for criterion j is indicated by jWˆ , the score of alternative i (Si) using WSM method is calculated by Equation (‎3.23): 𝑆𝑖 =∑?̂?𝑗?̂?𝑗𝑛𝑗=1 (‎3.23) where n is the number of total criteria considered in the MCDM problem. The lower and upper limits of each operational characteristic are determined based on the type of base isolator (NRB, HDRB etc.) and also the application in which the rubber bearing is used (buildings or bridges). Here, HDRBs reinforced with carbon fibre fabrics are implemented in a three-span continuous steel bridge. The equivalent viscous damping of HDRBs is 10-16% (Marioni, 1998). The lower and upper bounds of the effective horizontal and vertical stiffnesses are identified according to the elastomeric bearings datasheet of GoodCo Z-Tech Company, which is the largest manufacturer of the elastomeric and sliding bearings for bridges in Canada (GoodCo Z-Tech, 2010). The minimum and the maximum effective horizontal stiffness of laminated bearing (Series EL) with a minimum available plan size of 300 mm by 200 mm and a height of 80 mm is 0.82 kN/mm and 1.22 kN/mm, respectively. Since dimensions of CFR-HDRBs are 200 mm × 200 mm × 80 mm, the upper and lower limits of the horizontal stiffness should be recalculated. According to the analytical formula of KH = GrA/te (Kelly, 1997), if the width of the rubber bearing decreases from 300 mm to 200 mm, the minimum and the maximum effective lateral stiffness respectively reduces to 0.547 kN/mm and 0.813 kN/mm. Similarly, using Equation (‎3.12), the vertical stiffness decreases to 109.5 kN/mm. 55  The performance characteristics are selected as attributes for each rubber bearing which is considered as an alternative. As a result, a multi-criteria decision making problem has been defined with 27 alternatives and three criteria. Since the elastomeric isolators should have sufficient vertical stiffness under a wide range of compressive loads, the vertical stiffness (KV) should be maximized. Also, they should be more flexible in the horizontal direction so, the horizontal stiffness (KH) should be minimized. The higher damping capacity means the higher amount of energy dissipation. Accordingly, the damping capacity (βeq) should be maximized. Since the second criterion (the effective horizontal stiffness) should be minimized while two others should be maximized, the effective horizontal stiffness for all 27 alternatives is reversed in order to transform the MCDM problem to “the higher, the better” condition for all attributes. Consequently, the ranking of CFR-HDRBs are easily determined based on the highest and lowest scores.  According to the WSM method, the score of each alternative (rubber bearing) is found as follows: 𝑆𝑖 = ?̂?1 ?̂?𝑉𝑖 + ?̂?2 ?̂?𝐻′𝑖 + ?̂?3 ?̂?𝑒𝑞𝑖 (‎3.24) where, iVKˆ, iHK ˆ, and ieqˆ are normalized vertical stiffness, normalized inversed horizontal stiffness, and normalized equivalent viscous damping, respectively. 1Wˆ , 2Wˆ , and 3Wˆ  are the normalized weights corresponded to the first, second and third criteria, respectively. The optimized CFR-HDRB which is the most efficient rubber bearing in terms of vertical stiffness, lateral flexibility and energy dissipation capacity should have the highest score among 27 alternatives with different performance specifications.  Figure  3.20 shows the procedure of optimizing the performance of CFR-HDRBs through a multi-criteria decision making process. 56   Figure ‎3.20. Flow chart of multi-objective optimization 3.5.2 Optimization of CFR-HDRB After inversing the effective horizontal stiffness (K’H) and normalizing all criteria for each CFR-HDRB, using weighted sum method (Equation (‎3.25)), the score of each alternative (rubber bearing), Si, is calculated based on the normalized weights of each criteria in Table ‎3.11. It should be noted that the equivalent viscous damping and the horizontal stiffness are considered as extremely important and very important criteria, respectively, and the evaluation value of vertical stiffness is assumed to be 4 (according to Table 3.10). The weights listed in Table ‎3.11 are normalized using the summation of evaluation values.  Table ‎3.11. Assigned normalized weights to criteria Criterion Weight Vertical Stiffness 0.18 Horizontal Stiffness 0.39 Damping Capacity 0.43  𝑆𝑖 = 0.18 ?̂?𝑉𝑖 + 0.39 ?̂?𝐻′𝑖 + 0.43  ?̂?𝑒𝑞𝑖 (‎3.25) By sorting the alternatives from highest to lowest scores, the best rubber bearing which is placed in the 1st rank is determined. In Table ‎3.12, five best CFR-HDRBs are listed. 33 Full Factorial Design  The Optimized CFR-HDRB with Highest Score SA Optimum = Smax Alternatives, Ai 27 CFR-HDRBs Defining Criteria    C1:  Vertical Stiffness, KV   C2:  Horizontal Stiffness, KH   C3:  Eq. Viscous Damping, βeq Normalizing  Weights and Criteria  →   →   →  Calculating Score of Alternatives using WSM   Assigning Weights  to Criteria  W1 → C1 W2 → C2 W3 → C3 57  Table ‎3.12. Properties of 1st to 5th ranked CFR-HDRBs  Rank Specifications Stiffness Damping ID nr tf (mm) Gr (MPa) KV (kN/mm) KH (kN/mm) β (%) 1st  A3 10 0.31 0.6 185.5 0.417 14.88 2nd  A9 10 0.93 0.6 273.5 0.479 14.40 3rd  A27 10 0.93 0.8 363.9 0.614 14.70 4th  A18 10 0.93 0.7 318.8 0.547 14.58 5th  A6 10 0.62 0.6 224.6 0.447 14.52 According to this ranking, the optimized CFR-HDRB, as the most efficient rubber bearing, has 10 rubber layers with shear modulus of 0.6 MPa and nine FRP composite plates with thickness of 0.31 mm. Since the total height of all base isolators is 80 mm, the thickness of each rubber layer would be 4.72 mm for the optimized CFR-HDRB. For the five best cases, the elastomeric isolator has the maximum number of rubber layers (nr = 10).  On the other hand, the worst alternative with the lowest score is an elastomeric bearing with characteristics listed in Table ‎3.13. Table ‎3.13. Properties of the worst CFR-HDRB Rank Specifications Stiffness Damping ID nr tf (mm) Gr (MPa) KV (kN/mm) KH (kN/mm) β (%) 27th  A22 8 0.62 0.8 175.9 0.558 14.88 Based on the applied load which is a combination of 6 MPa vertical pressure and cyclic lateral displacement with frequency of 0.2 Hz and shear strain of γ = 100%, the lateral force-displacement hysteretic loops for the optimized CFR-HDRB and the 2nd best rubber bearing are plotted in Figure  3.21. 58   Figure ‎3.21. Lateral force-deflection hysteresis curve for the 1st and 2nd best CFR-HDRBs  (f = 0.2 Hz, γ = 1.0) 3.6 Summary Elastomeric base isolators are able to minimize structural damage in moderate seismic events and prevent structural collapse in extreme conditions such as severe earthquakes. They considerably decrease and dissipate the earthquake energy transmitted to the structure by providing a damping mechanism between the substructure and the superstructure due to their very low horizontal to vertical stiffness ratio. HDRB is one type of conventional elastomeric isolators in which thin layers of high damping rubber (HDR) are bonded to steel shims. In this chapter, a material model was proposed for HDR in order to capture its highly nonlinear behaviour under compressive and cyclic shear loadings. Among different kinds of material models including bilinear, hyperelastic, viscoelastic and viscoplastic ones; hyper-viscoelastic model was the best-fitted and the most accurate one to predict the hyperelasticity and strain-rate-dependent behaviour of HDRB subjected to the cyclic lateral displacement.  The efficiency of carbon fibre-reinforced high damping rubber bearing (CFR-HDRB) was numerically optimized through a multi-criteria optimization process. After validating and verifying the hyper-viscoelastic material model with experimental results, the effect of different parameters on the performance of FREIs was investigated through a sensitivity analysis. In this regard, regression models were established for predicting the behaviour of rubber bearings. The performance of CFR-HDRB was optimized by assigning different -25-20-15-10-50510152025-50 -40 -30 -20 -10 0 10 20 30 40 50Force (kN) Lateral Displacement (mm) Optimized C-FREI2nd C-FREI59  weights to the operational specifications, which are the effective horizontal and vertical stiffnesses, and the equivalent viscous damping. Results showed that the effective horizontal stiffness and viscous damping are highly dependent on the shear modulus of the elastomer layers. Also, the number of rubber layers and thickness of FRP composite plates had large effects on the vertical stiffness.     60  Chapter 4 Performance of Fibre-Reinforced Elastomeric Isolators: Experimental and Numerical Investigations 4.1 General In Chapter 3, with the purpose of acquiring an insight into the cause-and-effect relations between the input and output of the system (C-FREI), numerical investigations (regression and sensitivity analyses) were performed using design of experiments (DOE) method through which the effect of three factors was assessed on the output (performance) of the C-FREI.  In this chapter, experimental investigations were conducted on C-FREIs as a complementary part of the previous chapter.   This chapter aimed to introduce a simple and fast manufacturing process developed for fabricating carbon fibre-reinforced elastomeric isolators (C-FREIs) in bonded applications. Moreover, the performance of such fibre-reinforced isolators was evaluated through experimental and numerical investigations. In order to show the efficiency of C-FREIs under different loading conditions, operational specifications of C-FREIs in the vertical and horizontal directions were determined through experimental tests. In this regard, nine 1/4 scale C-FREIs were produced and then the performance characteristics including vertical and horizontal stiffnesses, as well as energy dissipation capacity and equivalent viscous damping were assessed through experimental investigations. Bonded C-FREIs were fixed to the substructure and superstructure using steel supporting plates. Long strip laminated pads consisting of rubber layers and carbon fibre fabrics were fabricated without using a mold and cut to small sizes. This technique reduces the time of the manufacturing process and makes it simple. The performance characteristics were determined by conducting vertical pressure and horizontal cyclic displacement tests. Furthermore, future investigations were suggested regarding the performance variation of C-FREIs through a sensitivity analysis. The performance sensitivity of C-FREIs was experimentally assessed. The main motivation was the lack of adequate information on the response sensitivity of bonded FREIs which were manufactured in the proposed process. The effect of different parameters, including the number of rubber layers, the thickness of carbon fibre fabric, the vertical pressure, and the rate of lateral cyclic displacements, was investigated on the operational 61  characteristics of the C-FREIs. The sensitivity analyses were performed by conducting pressure and rate sensitivity tests. In the analyses, the shear hysteretic responses of C-FREIs were evaluated in order to calculate the effective horizontal stiffness and the equivalent viscous damping.  In the next step, a detailed parametric study on the effect of various parameters affecting the performance of full size C-FREIs was conducted numerically. First, full-scale C-FREIs were modelled using finite element method (FEM) in ANSYS (ANSYS Mechanical APDL, Release 14.0). Then, C-FREIs were validated and verified through the performed experimental tests. In a comprehensive study, a wide range of C-FREIs were considered by changing the plan size of elastomeric layers (length and width) as well as the height of the laminated pad. The vertical and effective horizontal stiffnesses as well as the equivalent viscous damping were determined for various combinations. In order to assess the sensitivity of the response of C-FREIs in the vertical and horizontal directions, the effect of three factors; the number and the thickness of rubber layers, and the thickness of carbon fibre reinforced layers were investigated on the behaviour of rubber bearings. Since, three levels as low, medium, and high, were defined for each factor, nine cases were evaluated for C-FREIs with the same plan size, and a total number of 36 specimens were considered for four different sets of plan size. 4.2 Manufacturing Process Nine 1/4 scale C-FREIs are manufactured using the commercial high quality neoprene with a hardness of 55 Shore A and a minimum tensile strength of 17 MPa specified by the CHBDC CAN/CSA S6-06 (CHBDC, 2006), bi-directional carbon fibre fabrics (orientations 0/90°) with a tensile strength of 4413 MPa (ACP Composites, 2012), and steel supporting end plates. All specimens have an identical width and length of 70 mm by 70 mm but with different numbers and thicknesses of elastomeric and reinforcement layers. The shape factor (i.e. ratio of loaded (plan) area to load free (side) area of a rubber layer) is 11.7, 5.8, and 3.9 for base isolators with elastomeric layers’ thickness of 1.5 mm, 3 mm, and 4.5 mm, respectively.  62            Figure ‎4.1. Manufacturing process of C-FREIs; (a) bi-directional carbon fibre fabric between two rubber layers, (b) attaching fibre fabrics to rubber layers by adding glue (rubber cement), (c) cured laminated pad, (d) laminated pad cut with water-jet technology Elastomeric layers are bonded to bi-directional (orientations 0/90°) carbon fibre fabrics using rubber cement. Rubber cement is a cold bonding compound made of elastic polymers (typically latex). It is used to attach elastomeric layers to bi-directional carbon fibre fabrics (see Figure  4.1). After fabricating a laminated pad consisting of alternating layers of elastomer and reinforcement, in the curing process, laminated pads are subjected to a uniform pressure of 4 MPa for 24 hours at the room temperature without using a mold. According to the rubber cement manufacturer’s instruction, 70% to 80% of the bond strength is developed during 24 hours at room temperature and the rest is developed over the next 14 days. Then, the laminated pads in the form of long strips are cut to the required size (70 mm by 70 mm) using the water-jet (see Figure  4.1d) in order to create very smooth side surfaces and prevent delamination between layers during the cutting process. Finally, in order to improve the bonding between layers and prevent premature delamination that might occur during shipping, installation, or testing, side faces are coated with two layers of adhesive (rubber cement). The same bonding compound is used to attach steel supporting plates to the C-Bi-directional carbon fibre  fabric Rubber layer (a)                                                                      (b) (c)                                                              (d) 63  FREI. Nine C-FREIs with different numbers and thicknesses of rubber and carbon fibre-reinforced layers are depicted in Figure  4.2. Table ‎4.1 demonstrates the geometrical properties of the manufactured rubber bearings.                       Figure ‎4.2. C-FREIs manufactured with different properties;  (a) A1; (b) B1; (c) C1; (d) D1; (e) E1; (f) F1; (g) A3; (h) B4; (i) C2 Table ‎4.1. Physical and geometrical properties of C-FREIs C-FREI Plan size of steel plates (mm × mm) Plan size of reinforcement (mm × mm) H  (mm) ts  (mm) te  (mm) tf   (mm) ne nf S A1 150 × 150 70 × 70 31.8 6.35 1.5 0.25 8 7 11.7 B1 150 × 150 70 × 70 41.9 6.35 1.5 0.25 12 11 11.7 C1 150 × 150 70 × 70 43.4 6.35 3.0 0.25 8 7 5.8 D1 150 × 150 70 × 70 40.5 6.35 3.0 0.50 7 6 5.8 E1 150 × 150 70 × 70 41.8 6.35 3.0 0.75 7 6 5.8 F1 150 × 150 70 × 70 45.5 6.35 4.5 0.25 6 5 3.9 A3 150 × 150 70 × 70 50.6 6.35 1.5 0.25 16 15 11.7 B4 150 × 150 70 × 70 54.5 6.35 1.5 0.25 17 16 11.7 C2 150 × 150 70 × 70 45.9 6.35 3.0 0.25 9 8 5.8 H: total height of C-FREI; ts: thickness of steel supporting plates; te: thickness of elastomeric layers; tf: thickness of carbon fibre fabrics; ne: number of elastomeric layers; nf: number of fibre-reinforced layers; S: shape factor. (a)                                                  (b)                                                   (c) (d)                                                  (e)                                                   (f) (g)                                                  (h)                                                   (i) 64  4.3 Test Setup The test setup is equipped with vertical and horizontal hydraulic jacks. Force and displacement in the horizontal and vertical directions are measured in each test. Three load cells, each of them with a capacity of 10,000 lb (44.5 kN), measure the load applied through the vertical hydraulic jack. The computer receives the electrical signal generated by the load cells and estimates the corresponding force. The same mechanism is implemented for evaluating the lateral force applied through the horizontal hydraulic jack using one load cell. Four laser displacement transducers (LDT) mounted at four sides of the C-FREI are used to measure an average value for vertical displacement (see Figure  4.3). The horizontal displacement is determined by a string potentiometer (SPOT), a transducer for measuring the linear position of rubber bearings.  Figure ‎4.3. C-FREI fixed in the test setup Different parts of test setup are identified in Figure  4.3 and Figure  4.4. Eight bolts are used to connect the steel supporting plates to the test setup and fix the C-FREI in its place. In all tests, the vertical pressure is applied through the upper supporting plate which does not move in the horizontal direction. The cyclic lateral displacements are applied to rubber bearings through the lower supporting plate which has no movement in the vertical direction. LDT LDT Bearing (for uniformly distributing the load) C-FREI 65    Figure ‎4.4. Test setup and related equipment  4.4 Experimental Tests In order to explore the functionality of C-FREIs, performance tests are conducted in the vertical and horizontal directions. 4.4.1 Failure Tests Before defining the loading scenarios in cases of vertical pressure and lateral cyclic displacements, failure tests are conducted in order to determine the capacity of the manufactured C-FREIs in both vertical and horizontal directions. In this regard, two specimens are chosen to be tested under extreme loading conditions.  In order to evaluate the vertical stiffness capacity of C-FREIs, specimen F1 as the worst case is selected since rubber layers with maximum considered thickness (4.5 mm) are used. C-FREI-F1 is subjected to a maximum of 6.1 MPa vertical pressure. Figure  4.5 shows three stages of the tests in which the base isolator is subjected to 0 MPa, 3 MPa, and 6.1 MPa. At 3 MPa vertical load, rubber layers bulges but no failure was observed. However, when the pressure increased to 6.1 MPa, elastomeric layers encountered an extra bulging because fibre-reinforced layers could not prevent them from extra deformation (see Figure  4.5c). After removing the load, local delamination between CFR and rubber layers and also complete detachments between rubber layers and steel supporting plates were observed.   Horizontal Hydraulic Jack Load Cell Load Cells C-FREI 66      Figure ‎4.5. C-FREI-F1 under vertical pressures of (a) 0 MPa, (b) 3 MPa, and (c) 6.1 MPa To determine the capacity of C-FREIs in the horizontal direction, C-FREI-C1 is subjected to a lateral displacement equals to 200% of the total thickness of rubber layers (tr = 24 mm) while a constant 1.5 MPa vertical pressure is applied to the base isolator. Figure  4.6 demonstrates different stages at which the shear strain, defined as the ratio of lateral displacement to total thickness of elastomeric layers, increases from 100% to 200% and then is removed. It is observed that at all shear strain levels (from 100% to 200%), first and last rubber layers are locally detached from the steel plates as a result of rollover deformation happened in the laminated pad. In addition, at 200% shear strain (see Figure  4.6c), delamination occurred between rubber layers and fibre-reinforced sheets. Consequently, it can be observed that when the lateral displacement reaches zero (Figure  4.6d), a permanent deformation happened in the rubber bearing due to a complete debonding between CFR and rubber layers.       Figure ‎4.6. C-FREI-C1 under a combination of 3 MPa vertical pressure and lateral displacements of (a) 100% tr, (b) 125% tr, (c) 200% tr, and (d) 0% Based on the results obtained from the failure tests, it was decided to limit the maximum vertical pressure and the shear strain amplitude to 3 MPa and 100%, respectively, in order to complete all tests scenarios without any global failure. It should be noted that (a)                                                  (b)                                                   (c) (a)                                                                            (b) (c)                                                                           (d) 67  since specimens C-FREIs C1 and F1 encountered unrecoverable damage during failure tests, they cannot be used in experimental tests. Moreover, specimen C-FREI-C2 was damaged during the shipping process and as a result, it was not used. 4.4.2 Vertical Compression Test 4.4.2.1 Test Procedure The objectives of the vertical compression test are to evaluate the vertical stiffness and the vertical deflection of rubber bearing. This test is performed under load control since the vertical force is controlled during the tests. For each C-FREI, three tests with different values of design vertical pressure, PD, including 0.75, 1.50, and 3.00 MPa (3.7, 7.4, and 14.7 kN) were conducted. C-FREI is loaded monotonically up to the design pressure. Then, three fully reversed cycles with a variation of 20% of the design pressure is applied with a frequency of fV = 0.2 Hz. Finally, C-FREI is monotonically unloaded. Figure  4.7 shows the behaviour of pressure changes versus time for three considered design pressures.  Figure ‎4.7. Variation of vertical pressure over time for three design pressures After conducting the vertical tests, the operational characteristics including the vertical stiffness, KV, the compressive modulus, Ec, and the maximum vertical deflection at the design pressure, ΔV are determined. According to Equation (‎4.1), the compressive modulus can be calculated from the vertical stiffness obtained from the tests (Naeim and Kelly, 1999). 𝐸𝑐 =𝐾𝑉𝑡𝑟𝐴𝑓 (‎4.1) 012340 10 20 30 40 50Vertical Pressure (MPa) Time (s) P = 3.0 MPaP = 1.5 MPaP = 0.75 MPa68  where tr is the total thickness of rubber layers and Af is the cross-sectional area of the fibre-reinforced layer which is bonded to the elastomer. 4.4.2.2 Vertical Characteristics In order to determine the vertical stiffness as well as the maximum vertical deflection, changes of vertical force are plotted versus vertical deflection for each C-FREI. Figure  4.8 depicts the corresponding results for three different pressure levels (0.75 MPa, 1.5 MPa, and 3.0 MPa). The slope of the dashed lines is the tangent vertical stiffness of C-FREI at the corresponding design pressure. Table ‎4.2 represents the vertical operational characteristics.         Figure ‎4.8. Vertical force-deflection curves under 0.75 MPa, 1.5 MPa, and 3.0 MPa pressures;  (a) C-FREI-A1; (b) C-FREI-B1; (c) C-FREI-D1; (d) C-FREI-E1 Results show that for all C-FREIs, the vertical stiffness increases with increasing the design pressure. This fact is due to, first, the stiffening of rubber layers and second, the 03691215180 0.4 0.8 1.2Vertical Force (kN) Vertical Displacement (mm) 03691215180 0.4 0.8 1.2Vertical Force (kN) Vertical Displacement (mm) 03691215180 0.4 0.8 1.2Vertical Force (kN) Vertical Displacement (mm) 03691215180 0.4 0.8 1.2Vertical Force (kN) Vertical Displacement (mm) 69  presence of bi-directional carbon fibres. Elastomeric layers bulge under vertical pressure and as a result lateral in-plane forces are applied to the fibre-reinforced layers in both directions. When the vertical pressure increases, the transverse shear forces between rubber and reinforcement layers cause fibres to be stretched more tightly. Therefore, the vertical stiffness of fibre-reinforced layers enhances due to the stress stiffening. The nonlinear response of C-FREIs in monotonic loading and unloading parts is mainly because of the nonlinear behaviour of elastomer under compression. When the pressure reaches the target value, PD, before starting the cyclic variations, a creep happens in the C-FREI. In fact, the vertical deflection continues to increase under a constant vertical load (i.e. the design pressure). Rubber bearings undergo a plastic deformation in the vertical direction as the pressure is released in the unloading phase. This behaviour can be seen in Figure  4.8. Table ‎4.2. Performance characteristics of C-FREIs in the vertical direction C-FREI tr (mm) FV (kN) PD (MPa) KV (kN/mm) EC (MPa) ΔV (mm) A1 12 3.7 0.75 52.9 129.4 0.28 7.4 1.5 61.7 151.0 0.37 14.7 3.0 147.0 360.0 0.50 B1 18 3.7 0.75 24.7 90.6 0.63 7.4 1.5 38.9 143.1 0.89 14.7 3.0 56.5 207.7 1.17 D1 21 3.7 0.75 28.5 122.0 0.47 7.4 1.5 41.1 176.2 0.67 14.7 3.0 47.4 203.2 1.01 E1 21 3.7 0.75 28.5 122.0 0.55 7.4 1.5 38.9 166.9 0.81 14.7 3.0 54.4 233.3 1.15 The maximum amount of the vertical stiffness, at 3.0 MPa vertical pressure, is 147 kN/mm for A1 which has the minimum total thickness of rubber layers (12 mm) among four rubber bearings (see Table ‎4.2). When the total thickness of rubber layer increases, the flexibility of C-FREI in the vertical direction increases and accordingly, the vertical stiffness decreases. Although C-FREIs D1 and E1 have the same total thickness of rubber layers, the effective vertical stiffness of E1 is higher than that of D1 since E1 is made of thicker carbon fibre-reinforced sheets (tf = 0.75 mm). Therefore, increasing the thickness of reinforcement layers causes an increase in the vertical stiffness of C-FREIs against the vertical compressive loads. Variation of compressive modulus can be described similarly. E1 has higher total 70  thickness of elastomeric layers and compressive modulus compared to B1. This fact can be interpreted by considering Equation (‎4.1). Both C-FREIs have identical cross-sectional areas while, E1 has a greater tr, which causes EV to increase. The vertical deflection goes up with increasing both the vertical load and the thickness of rubber layers. The maximum displacement reaches around 1.2 mm for B1 and E1 under a vertical pressure of 3.0 MPa. 4.4.3 Lateral Cyclic Test 4.4.3.1 Test Procedure Cyclic test is performed under vertical load control and horizontal displacement control by applying a vertical pressure and lateral cyclic displacements simultaneously. The horizontal stiffness and the equivalent viscous damping are two performance specifications of rubber bearing evaluated through this test. The horizontal stiffness determines the flexibility of base isolator in the lateral direction. The equivalent viscous damping represents the capability of the device in dissipating the earthquake’s energy transmitted to the elastomeric isolator.  While the C-FREI is under a constant vertical pressure of 3.0 MPa, the cyclic horizontal displacements are applied. At each amplitude of horizontal deflection including 25% tr, 50% tr, and 100% tr, three fully reversed sinusoidal cycles are applied at constant horizontal rate of VH = 20 mm/s. Variation of vertical pressure and cyclic horizontal displacement versus time are demonstrated in Figure  4.9a and Figure ‎4.9b, respectively.      Figure ‎4.9. Input load in the cyclic tests; (a) vertical pressure, (b) cyclic displacements at  shear strains of 25%, 50%, and 100% 012340 10 20 30Vertical Pressure (MPa) Time (s) -100-500501000 10 20 30Horizotnal Displacement (% tr) Time (s) (a)                                                                               (b) 71  Knowing that the effective horizontal stiffness of C-FREIs, KH eff, is a function of shear strain and is calculated from Equation (‎3.10), the effective shear modulus, Geff, at each shear strain amplitude (γ), is computed according to Equations (‎4.2) (Kelly, 1997). 𝐺𝑒𝑓𝑓(𝛾) =𝐾𝐻𝑒𝑓𝑓(𝛾)𝑡𝑟 𝐴 (‎4.2) A is the cross-sectional area of the elastomeric layer which is in contact with the fibre-reinforced layer. The equivalent viscous damping of rubber bearing, β, is defined as a ratio of the dissipated energy to the elastic energy restored in the C-FREI (Naeim and Kelly, 1999). 𝛽 =14𝜋𝑈𝑑𝑈𝑒 (‎4.3) in which Ud is the energy dissipated per cycle and equals to the area inside the lateral force-deflection hysteresis curve in each cycle and Ue is the energy restored in the rubber bearing measured according to Equation (‎4.4). 𝑈𝑒 =12𝐾𝐻𝑒𝑓𝑓Δ𝑎𝑣𝑔2  (‎4.4) where Δavg = (Δmax + |Δmin|)/2. In order to show the whole cyclic test for each specimen, four stages are selected as shown in Figure  4.10. At the first stage, the rubber bearing is subjected to a constant vertical pressure. Then, the maximum lateral displacement during the cyclic sinusoidal deflections is depicted at stages two, three, and four by increasing the shear strain amplitude from 25% tr to 100% tr while the vertical pressure remains constant.       72           Figure ‎4.10. C-FREIs under cyclic tests at different shear strains; (a) applying constant pressure,  (b) shear strain of 25%, (c) shear strain of 50%, (d) shear strain of 100% 4.4.3.2 Horizontal Characteristics The operational characteristics of C-FREIs are obtained from the lateral force-deflection hysteresis curves plotted at different amplitudes. Figure  4.11 shows the hysteretic shear behaviours of four C-FREIs. Table ‎4.3 represents the effective horizontal stiffness, the effective shear modulus, the dissipated energy, and the equivalent viscous damping of C-FREIs at three shear strains. The energy dissipated by C-FREIs is tabulated for one cycle at each shear strain amplitude, Ud.  (a)                                     (b)                                       (c)                                     (d) 73              Figure ‎4.11. Shear hysteretic response of C-FREIs at 25%tr, 50%tr, and 100%tr Table ‎4.3. Horizontal operational characteristics of C-FREIs at three shear strain amplitudes C-FREI γ  (%) KH (kN/mm) Geff  (MPa) Ud  (J) Ue  (J) β (%) A1 25 0.375 0.918 3.1 2.1 11.8 50 0.287 0.702 8.2 6.3 10.4 100 0.239 0.586 18.3 16.3 9.0 B1 25 0.226 0.829 4.8 2.9 13.1 50 0.177 0.649 11.3 7.1 12.6 100 0.128 0.471 31.1 19.5 12.7 D1 25 0.203 0.869 4.9 3.5 11.1 50 0.165 0.706 11.5 8.8 10.3 100 0.132 0.564 30.9 26.7 9.2 E1 25 0.212 0.911 5.4 3.7 11.6 50 0.170 0.731 12.3 9.3 10.6 100 0.135 0.578 33.1 27.6 9.6 4.4.3.2.1 Effective Horizontal Stiffness By increasing the amplitude of lateral displacement, a nonlinear behaviour is observed in the hysteresis curves (Figure  4.11). The effective horizontal stiffness has a decreasing rate when the shear strain increases. This is mainly due to a nonlinear softening -3-2-10123-21 -14 -7 0 7 14 21Shear Force (kN) Lateral Displacement (mm) -3-2-10123-21 -14 -7 0 7 14 21Shear Force (kN) Lateral Displacement (mm) -3-2-10123-21 -14 -7 0 7 14 21Shear Force (kN) Lateral Displacement (mm) -3-2-10123-21 -14 -7 0 7 14 21Shear Force (kN) Lateral Displacement (mm) 74  behaviour in elastomer as a result of the rollover deformation which is demonstrated in Figure  4.12 for C-FREI-E1, as an example. At high shear strains (γ = 100%), the top and bottom rubber layers are detached from the steel plates. Unlike the steel shims used in the conventional elastomeric isolators, fibre-reinforced layers almost have no flexural rigidity and as a result, they can be deformed under large lateral displacements. This deflection which can be seen in Figure  4.12, for all C-FREIs, applies a peel-off force due to the tensile stress at the top and bottom rubber layers near the edges generated from coupling moment at large shear strains. Under such a condition, the stress between the elastomeric layer and the supporting steel plate exceeds the bonding strength of the glue used for attaching the rubber and steel and consequently, detachment starts to increase from the edges. Results show that the local delamination due to the rollover deformation mainly affects the effective horizontal stiffness of the base isolators.     Figure ‎4.12. Deformation of C-FREI-E1 under the maximum applied shear strain amplitude (100%)  and a vertical pressure of 3 MPa The reduction in the effective horizontal stiffness with increasing the shear amplitude is mostly due to the decrease in the shear modulus of elastomer and the rollover deformation. Figure  4.13 shows the reduction in the effective shear modulus of natural rubber as a function of shear strain. Rollover deformation 75   Figure ‎4.13. Effective shear modulus decay versus shear strain for four C-FREIs Rubber bearings D1 and E1 have the lowest effective horizontal stiffness since they have the maximum amount of total thickness of rubber layers (tr) among four C-FREIs. It shows that increasing the thickness of elastomers increases the lateral flexibility of the device. Increasing the lateral flexibility of the elastomeric isolator leads to an increase in the fundamental periods of the isolation system and accordingly improves its performance in regulating the behaviour of the structure by shifting its natural period. On the other hand, the lateral flexibility of the base isolator should be greater than a minimum value determined through a design process; otherwise, the device undergoes a permanent residual deformation and cannot work efficiently under cyclic loads. In such a situation, by determining the lower and upper limits of the horizontal stiffness in the design procedure, the lateral flexibility (inverse of the horizontal stiffness) can be maximized by increasing the total thickness of rubber layers.    4.4.3.2.2 Energy Dissipation Capacity and Damping Ratio  Although the dissipated energy in all C-FREIs increases with increasing the amplitude of the lateral displacement, the damping ratio, β, decreases. This fact can be clarified according to the definition of the equivalent viscous damping which is proportional to the ratio of the dissipated energy to the restored energy (Equation (‎4.3)). Since the increase rate of the energy restored in the C-FREIs is greater than that of the dissipated energy, the damping ratio decreases as listed in Table ‎4.3. Pinched hysteresis loops are another reason of decreasing the damping ratio when the lateral displacement increases. Pinching refers to a behaviour in which hysteresis loops get thinner in the middle due to a sudden reduction in the stiffness caused by the delamination. This behaviour can be clearly observed in the hysteresis 00.20.40.60.8125 50 75 100Effective Shear Modulus, Geff (MPa) Shear Strain, γ (%) D1E1B1A176  curves in Figure  4.11. When the lateral displacement increases to 100% of tr (γ = 100%), the central part of the loops get thinner compared to the loops with a shear strain of 25%.       Increasing the thickness of rubber layer increases the energies which are dissipated and restored by the base isolators. If the amount of energies for D1 and E1, having same tr, are compared to each other, it will be understood that E1 has a higher capability in dissipating and restoring energy at all shear strain levels. The reason is that, E1 has a higher thickness of fibre-reinforced layer (0.75 mm) compared to D1 (0.5 mm). It can be interpreted that, carbon fibre-reinforced layers with almost no flexural rigidity are the second source of energy absorption and dissipation due to their frictional damping as a result of interfacial slip between carbon fibres (Kelly, 2002).  Here, an important finding is that the damping ratios of C-FREIs are larger than the damping of the natural rubber itself. This is due to the type of the reinforcement and the interaction between the reinforced plates and rubber layers. In fact, the additional amount of equivalent viscous damping is attributed to the carbon fibre-reinforced polymer composite layers. The bi-directional carbon fibre fabrics are bonded to the elastomeric layers using the rubber cement (glue) which is considered as the matrix for the CFRP layers. After curing, the fibre-reinforced layer still has a specific amount of flexibility under shear forces. Therefore, in addition to the frictional damping due to the slip between fibres, the matrix (the cured glue) absorbs and dissipates a certain amount of energy and as a result, the reinforced layers can increase the damping ratio of the elastomeric isolator compared to a case in which rigid steel shims are used. 4.5 Experimental Parametric Study In this section, two operational characteristics of C-FREIs in the lateral direction, (the effective horizontal stiffness and the equivalent viscous damping), are evaluated through sensitivity analyses. The vertical pressure, the lateral cyclic rate, the number of rubber layers, and the thickness of fibre-reinforced layers are factors considered in the parametric study.  77  4.5.1 Vertical Pressure 4.5.1.1 Pressure Sensitivity Test Procedure The objective of the pressure sensitivity tests is to evaluate the effect of vertical pressure on the effective horizontal stiffness and the equivalent viscous damping of the base isolator. The horizontal stiffness is a measure of lateral flexibility and the equivalent viscous damping represents the capability of the device in dissipating the earthquake’s energy. In order to explore the effect of pressure on the efficiency of the device, different cyclic tests are performed by changing the vertical compressive load (1 MPa, 2 MPa, and 3 MPa) while other parameters are kept constant. Cyclic tests are performed under vertical load control and horizontal displacement control by applying vertical pressure and lateral cyclic displacements, simultaneously. In each test, while the C-FREI is subjected to a constant vertical pressure, P, the cyclic horizontal displacements are applied. At each amplitude of horizontal deflection including 25% tr, 50% tr, and 100% tr (tr is the total thickness of rubber layers), three fully reversed sinusoidal cycles are applied at constant horizontal rate of VH = 20 mm/s. Variations of vertical pressure and cyclic horizontal displacements versus time are demonstrated in Figure  4.14a and b, respectively. It should be mentioned that at shear strain levels higher than 100%, unrecoverable deformations occured.Consequently, the shear strain amplitude limited to 100% in all tests.      Figure ‎4.14. Input loads in the cyclic tests; (a) variation of vertical pressure over time,  (b) variation of shear strain over time (25%, 50%, and 100%) 012340 5 10 15 20 25 30Vertical Pressure (MPa) Time (s) P=1 MPaP=2 MPaP=3 MPa-100-500501000 10 20 30Shear Strain (%) Time (s) (a)                                                                                      (b) 78  For each C-FREI, three lateral force-deflection hysteresis curves are presented for three vertical pressures. Figure  4.15 shows the shear behaviour of the rubber bearings at different shear strains and pressure levels.    -3-2-10123-27 -18 -9 0 9 18 27Shear Force (kN) Lateral Displacement (mm) -3-2-10123-27 -18 -9 0 9 18 27Shear Force (kN) Lateral Displacement (mm) -3-2-10123-27 -18 -9 0 9 18 27Shear Force (kN) Lateral Displacement (mm) -3-2-10123-27 -18 -9 0 9 18 27Shear Force (kN) Lateral Displacement (mm) -3-2-10123-27 -18 -9 0 9 18 27Shear Force (kN) Lateral Displacement (mm) -3-2-10123-27 -18 -9 0 9 18 27Shear Force (kN) Lateral Displacement (mm) -3-2-10123-27 -18 -9 0 9 18 27Shear Force (kN) Lateral Displacement (mm) -3-2-10123-27 -18 -9 0 9 18 27Shear Force (kN) Lateral Displacement (mm) -3-2-10123-27 -18 -9 0 9 18 27Shear Force (kN) Lateral Displacement (mm) P = 1 MPa                                           P = 2 MPa                                          P = 3 MPa P = 1 MPa                                          P = 2 MPa                                           P = 3 MPa P = 1 MPa                                           P = 2 MPa                                          P = 3 MPa 79      Figure ‎4.15. Lateral force-displacement hysteresis curves of manufactured C-FREIs under different vertical pressures (P = 1 MPa, 2 MPa, and 3 MPa) and shear strains (γ = 25%, 50%, and 100%) By comparing the hysteretic responses, it can be observed that C-FREI-A1, with the lowest thickness of rubber layers, has the maximum amount of effective horizontal stiffness. This behaviour shows that the elastomeric layers mainly contribute to the lateral flexibility of the device. On the other hand, the area inside the hysteresis loops, which represents the energy dissipated by the device, increases for C-FREIs with a higher total thickness of rubber layers (tr) such as A3 and B4.  -3-2-10123-27 -18 -9 0 9 18 27Shear Force (kN) Lateral Displacement (mm) -3-2-10123-27 -18 -9 0 9 18 27Shear Force (kN) Lateral Displacement (mm) -3-2-10123-27 -18 -9 0 9 18 27Shear Force (kN) Lateral Displacement (mm) -3-2-10123-27 -18 -9 0 9 18 27Shear Force (kN) Lateral Displacement (mm) -3-2-10123-27 -18 -9 0 9 18 27Shear Force (kN) Lateral Displacement (mm) -3-2-10123-27 -18 -9 0 9 18 27Shear Force (kN) Lateral Displacement (mm) -3-2-10123-27 -18 -9 0 9 18 27Shear Force (kN) Lateral Displacement (mm) -3-2-10123-27 -18 -9 0 9 18 27Shear Force (kN) Lateral Displacement (mm) -3-2-10123-27 -18 -9 0 9 18 27Shear Force (kN) Lateral Displacement (mm) P = 1 MPa                                          P = 2 MPa                                           P = 3 MPa P = 1 MPa                                           P = 2 MPa                                         P = 3 MPa P = 1 MPa                                           P = 2 MPa                                          P = 3 MPa 80  When C-FREIs A3 and B4 are subjected to a vertical pressure of 1 MPa, the first cycle of the lateral loading at 50% and 100% shear strain levels is different from the second and third cycles. This fact is due to scragging and also the delamination which occurs in the exterior layers of the laminated pad after the first cycle. It should be noted that no scragging was observed for other C-FREIs. As a result, the slope of the lateral force-deflection curve decreases. Since the total number of elastomeric layers in A3 and B4 is higher than that of four other C-FREIs (see Table ‎4.1), the possibility of debonding of either the rubber layer from the steel supporting plate or the rubber layer from the reinforcement noticeably increases. Such a behaviour is not observed when the pressure increases to 2 MPa and 3 MPa because the vertical load postpones the delamination (debonding). The effective horizontal stiffness and the equivalent viscous damping are measured using Equations (‎3.10) and (‎4.3), respectively. The average of the three hysteresis loops, at each displacement, was calculated and the results are listed in Table ‎4.4. Table ‎4.4. Effective horizontal stiffness and equivalent viscous damping of C-FREIs at  different shear strain amplitudes and vertical pressures C-FREI  KH (kN/mm) β (%) P (MPa) γ (%) 1 2 3 1 2 3 A1 25 0.375 0.380 0.375 11.7 11.9 11.7 50 0.287 0.289 0.287 10.3 10.4 10.3 100 0.239 0.240 0.239 9.1 9.2 9.1 B1 25 0.226 0.244 0.233 13.0 13.0 13.2 50 0.177 0.188 0.181 12.6 12.3 12.5 100 0.128 0.141 0.137 12.7 11.9 11.6 D1 25 0.203 0.211 0.203 11.1 11.0 11.1 50 0.165 0.169 0.165 10.3 10.2 10.3 100 0.132 0.132 0.132 9.3 9.3 8.9 E1 25 0.212 0.223 0.212 11.6 11.4 11.6 50 0.170 0.176 0.170 10.5 10.3 10.5 100 0.135 0.136 0.135 9.7 9.7 9.8 A3 25 0.209 0.171 0.165 12.5 12.6 12.6 50 0.153 0.132 0.128 11.2 11.6 11.7 100 0.095 0.096 0.094 13.2 11.2 11.2 B4 25 0.213 0.173 0.166 12.8 12.9 13.1 50 0.158 0.133 0.129 11.4 11.8 12.0 100 0.099 0.099 0.097 12.3 10.7 10.9 81  4.5.1.2 Effective Horizontal Stiffness In this part, the effect of vertical pressure on the lateral flexibility of C-FREIs is investigated while rubber bearings are subjected to cyclic lateral displacements. At each shear strain (25%, 50%, and 100%), the effective horizontal stiffness of each C-FREI is calculated under three vertical pressures (1 MPa, 2 MPa, and 3 MPa).  Figure  4.16 shows that the vertical pressure has a negligible influence on the lateral flexibility of the C-FREIs regardless of the shear strain amplitude. At low lateral displacements (γ = 25%), minor changes are observed in the horizontal stiffness with increasing the pressure. However, at 100% shear strain, the fluctuation of the effective horizontal stiffness almost vanishes. This characteristic demonstrates that the manufactured C-FREIs are almost insensitive to the vertical pressure. This insensitivity is due to material properties of elastomer (neoprene) used in the C-FREIs. Compared to the HDR, which has a highly nonlinear behaviour depending on the loading and environmental conditions, the neoprene used in this study has a low sensitivity to the vertical pressure. Hence, almost the same responses are observed for C-FREIs when the vertical pressure changes.    Figure ‎4.16. Effective horizontal stiffness of C-FREIs under different vertical pressures  (1 MPa, 2 MPa, and 3 MPa) and shear strains of (a) 25%, (b) 50%, and (c) 100% 00.10.20.30.4A1 B1 D1 E1 A3 B4Effective Horizontal Stiffness (kN/mm) C-FREI 1 MPa2 MPa3 MPa00.10.20.30.4A1 B1 D1 E1 A3 B4Effective Horizontal Stiffness (kN/mm) C-FREI 1 MPa2 MPa3 MPa00.10.20.30.4A1 B1 D1 E1 A3 B4Effective Horizontal Stiffness (kN/mm) C-FREI 1 MPa2 MPa3 MPa(a)                                                                    (b)  (c) 82  At 25% and 50% shear strains, the effective horizontal stiffnesses of A3 and B4 are reduced when the vertical pressure increases from 1 MPa to 2 MPa. The reason is that these two C-FREIs have higher lateral flexibility under low vertical pressure (1 MPa) due to their higher thickness of rubber layers compared to other bearings. 4.5.1.3 Equivalent Viscous Damping The second considered specification is the equivalent viscous damping of rubber bearing. Here, results are presented in three bar charts; each plot corresponds to one shear strain amplitude where the vertical pressure increases from 1 MPa to 3 MPa (Figure  4.17).    Figure ‎4.17. Equivalent viscous damping of C-FREIs under different vertical pressures  (1 MPa, 2 MPa, and 3 MPa) and shear strains of (a) 25%, (b) 50%, and (c) 100% By increasing the pressure, the equivalent viscous damping changes between 9.1% and 13.2% (see Table ‎4.4). With respect to the magnitude of the damping coefficients, the variation of this parameter is negligible. When the vertical pressure changes, at low shear strain levels (25% and 50%), the equivalent viscous damping remains almost constant (see Figure  4.17a and b). However, at 100% shear strain, the damping coefficient of B1, A3, and 05101520A1 B1 D1 E1 A3 B4Equivalent Viscous Damping (%) C-FREI 1 MPa2 MPa3 MPa05101520A1 B1 D1 E1 A3 B4Equivalent Viscous Damping (%) C-FREI 1 MPa2 MPa3 MPa05101520A1 B1 D1 E1 A3 B4Equivalent Viscous Damping (%) C-FREI 1 MPa2 MPa3 MPa(a)                                                                              (b)  (c) 83  B4 encounter a slight reduction when the vertical pressure increases from 1 MPa to 2 MPa. The reason is that these three C-FREIs have a higher capability in dissipating the energy under low vertical loads (1 MPa) due to their higher number of rubber layers compared to other bearings. As a result of using the commercial high quality neoprene, known as the primary source of energy dissipation, C-FREIs have a higher damping capacity compared to low-damping (unfilled) rubber bearings. However, compared to HDRBs, the C-FREIs have a lower capacity. 4.5.2 Lateral Cyclic Rate Different earthquakes with different magnitudes and frequency contents hit the base-isolated structures. In such a situation, investigating the effect of lateral cyclic rate or frequency on the performance of the elastomeric bearings will be of great interest. 4.5.2.1 Rate Sensitivity Test Procedure In the rate sensitivity test, the influence of the lateral cyclic rate on the performance of the C-FREIs is investigated. Cyclic tests are conducted by changing the rate of the cyclic horizontal displacements (20 mm/s, 30 mm/s, and 75 mm/s) while the pressure and the lateral displacement amplitude are kept constant. The procedure of performing the tests is the same as the horizontal cyclic tests. While the C-FREI is under a constant vertical pressure of 1.5 MPa, the cyclic horizontal displacements are applied. At a lateral amplitude of 50% tr, three fully reversed sinusoidal cycles are applied. Variation of cyclic horizontal displacements versus time is depicted in Figure  4.18 for three different rates.       84     Figure ‎4.18. Lateral cyclic displacement (50% tr) for three different cyclic rates In a sinosoidal harmonic motion, the amplitude of the linear velocity, V, is related to the amplitude of the displacement using the angular frequency, ω, according to Equation (‎4.5). Since ω can be expressed in terms of cyclic frequency, f, (ω = 2πf), and the amplitude of cyclic lateral displacement, A, is equal to the shear strain, γ, multiplied by the total thickness of rubber layers, tr, (A = γtr), the lateral cyclic frequency, fH, at 100% shear strain, can be calculated from the horizontal rate, VH, as follows. 𝑉 =  𝜔𝐴 (‎4.5) 𝑓𝐻 =𝑉𝐻2𝜋𝑡𝑟 (‎4.6) Based on the test procedure defined for the rate sensitivity experiment, the hysteretic shear response at 50% shear strain are illustrated for six elastomeric isolators (Figure  4.19). For each C-FREI, three hysteresis curves are plotted in one figure in order to compare the behaviours of the laminated rubber bearing by changing the cyclic rate, VH. -50-25025500 5 10 15 20 25 30Horizotnal Disp.  (% tr) Time (s) 20 mm/s -50-25025500 5 10 15 20 25 30Horizotnal Disp.  (% tr) Time (s) 30 mm/s -50-25025500 5 10 15 20 25 30Horizotnal Disp.  (% tr) Time (s) 75 mm/s 85    Figure ‎4.19. Lateral force-displacement hysteresis curves at different lateral rates  (20 mm/s, 30 mm/s, and 75 mm/s) and 50% shear strain The effective horizontal stiffness and the equivalent viscous damping are calculated at different lateral cyclic rates, VH, as listed in Table ‎4.5. It should be noted that the horizontal frequency, fH, is computed according to Equation (‎4.6). By considering a constant shear strain (e.g. 100%) and a constant lateral cyclic rate, the horizontal frequency changes for different elastomeric bearings because C-FREIs have different total thicknesses of rubber layer (tr).        -2-1012-14 -7 0 7 14Sheaar Force (kN) Lateral Displacement (mm) 20 mm/s30 mm/s75 mm/s -2-1012-14 -7 0 7 14Shear Force (kN) Lateral Displacement (mm) 20 mm/s30 mm/s75 mm/s -2-1012-14 -7 0 7 14Shear Force (kN) Lateral Displacement (mm) 20 mm/s30 mm/s75 mm/s-2-1012-14 -7 0 7 14Shear Force (kN) Lateral Displacement (mm) 20 mm/s30 mm/s75 mm/s -2-1012-14 -7 0 7 14Shear Force (kN) Lateral Displacement (mm) 20 mm/s30 mm/s75 mm/s -2-1012-14 -7 0 7 14Shear Force (kN) Lateral Displacement (mm) 20 mm/s30 mm/s75 mm/sC-FREI-A1                                         C-FREI-B1                                        C-FREI-D1 FREI-E1                                              FREI-A3                                            FREI-B4 86  Table ‎4.5. Effective horizontal stiffness and equivalent viscous damping of C-FREIs  at different lateral cyclic rates (20 mm/s, 30 mm/s, and 75 mm/s) C-FREI VH (mm/s) fH  (Hz) KH (kN/mm) β  (%) A1 20 0.27 0.284 10.5 30 0.40 0.293 10.5 75 1.00 0.317 9.9 B1 20 0.18 0.171 14.1 30 0.27 0.176 13.5 75 0.66 0.187 12.2 D1 20 0.15 0.165 10.3 30 0.23 0.167 10.1 75 0.57 0.174 9.6 E1 20 0.15 0.171 10.6 30 0.23 0.173 10.5 75 0.57 0.181 10.1 A3 20 0.13 0.128 12.2 30 0.20 0.130 11.9 75 0.50 0.134 11.5 B4 20 0.12 0.129 12.1 30 0.19 0.131 11.8 75 0.47 0.135 11.4 4.5.2.2 Effective Horizontal Stiffness In order to determine whether or not the loading rate (VH) affects the horizontal stiffness of C-FREIs, lateral cyclic displacements with three different rates including 20 mm/s, 30 mm/s, and 75 mm/s are applied to the rubber bearings. The following bar chart compares this parameter at different rates (Figure  4.20).  Figure ‎4.20. Effective horizontal stiffness of C-FREIs under different lateral cyclic rates  (20 mm/s, 30 mm/s, and 75 mm/s) at 50% shear strain 00.10.20.30.4A1 B1 D1 E1 A3 B4Effective Horizontal Stiffness (kN/mm) C-FREI 20 mm/s30 mm/s75 mm/s87  For all C-FREIs, increasing the rate leads to an increase in the effective horizontal stiffness. However, the amount of change is negligible when the lateral rate is low (20 mm/s and 30 mm/s). By increasing the cyclic rate to 75 mm/s, a greater increase is observed in the lateral stiffness. In order to clarify this behaviour, it can be mentioned that, when the rate of the lateral cyclic loading increases, the elastomeric layers are stiffened and as a result, the rubber bearings show a lower flexibility in the horizontal direction. 4.5.2.3 Equivalent Viscous Damping In contrast to the effective horizontal stiffness, the equivalent viscous damping decreases by increasing the lateral cyclic rate. This fact can be observed in Figure  4.21 that depicts the variation of the damping coefficient for each C-FREI by changing the loading rate. The flexibility of rubber layers is reduced at high lateral loading rates and as a result, the energy restored in the C-FREIs increases according to Equation (‎4.4). On the other hand, the capability of the device in dissipating the earthquake’s energy degrades because the rubber layers are stiffened by increasing the lateral loading rate. Therefore, the equivalent viscous damping, which is proportional to the ratio of the dissipated energy to the restored energy (see Equation (‎4.3)), decreases by increasing the horizontal loading rate.  Figure ‎4.21. Equivalent viscous damping of C-FREIs under different lateral cyclic rates  (20 mm/s, 30 mm/s, and 75 mm/s) at 50% shear strain 4.5.3 Number of Rubber Layers In order to study the effect of the number of rubber layers, ne, on the response of C-FREIs, this parameter changes while the thicknesses of elastomeric and fibre-reinforced layers remain constant. Consequently, C-FREIs A1, B1, and A3, with ne equals to 8, 12, and 05101520A1 B1 D1 E1 A3 B4Equivalent Viscous Damping (%) C-FREI 20 mm/s30 mm/s75 mm/s88  16, respectively, are selected among nine specimens (Table ‎4.1). Figure  4.22 illustrates the effect of the number of rubber layers on the effective horizontal stiffness and the equivalent viscous damping of C-FREIs. As expected, when the number of elastomeric layers increases, the lateral flexibility of the rubber bearings increases and the horizontal stiffness decreases. Figure 13a shows similar behaviours for three shear strain levels.   Figure ‎4.22. Effect of number of rubber layers on the performance of bearings  at different shear strain amplitudes (25%, 50%, and 100%);  (a) effective horizontal stiffness, (b) equivalent viscous damping. By increasing the number of rubber layers, the performance of the C-FREIs improves in terms of the damping capacity (Figure  4.22b). This behaviour can be clearly observed when ne changes from 8 to 12. However, by increasing ne from 12 to 16, the equivalent viscous damping is slightly reduced. The reason is that when ne increases from 12 to 16, both the restored elastic energy and the dissipated energy are increased, but the restored energy changes more than the dissipated energy. Therefore, the ratio of the restored energy to the dissipated energy, which denotes the equivalent viscous damping, decreases. 4.5.4 Thickness of Fibre-reinforced Layers Among geometrical properties considered in this study, only the thicknesses of reinforcement (tf) of C-FREIs C1, D1, and E1 are different (see Table ‎4.1). However, specimen C-FREI-C1 experience unrecoverable damages during the failure tests. For that reason, C-FREIs D1 and E1 are selected in order to perform a sensitivity analysis on the thickness of the carbon fibre fabrics. In D1 and E1, two and three layers of bi-directional 00.10.20.30.40.56 8 10 12 14 16 18Effective Horizontal Stiffness (kN/mm) Number of rubber layers, ne 25%50%100%051015206 8 10 12 14 16 18Equivalent Viscous Damping (%) Number of rubber layers, ne 25%50%100%(a)                                                                            (b) 89  carbon fibre fabrics are used, respectively. Since a single fibre fabric has a thickness of 0.25 mm, the thickness of reinforcements in D1 and E1 is 0.50 mm and 0.75 mm, respectively. The effective horizontal stiffness of C-FREI-E1 is higher than that of the D1. This difference disappears by increasing the lateral deflection (see Figure  4.23a). Increasing the thickness of the reinforced layers decreases the lateral flexibility. The reason is that fibre-reinforced layers become stiffer when their thickness increases.   Figure ‎4.23. Effect of fibre-reinforced layers on the performance of bearings  at different shear strain amplitudes (25%, 50%, and 100%);  (a) effective horizontal stiffness, (b) equivalent viscous damping Slight changes are observed in the equivalent viscous damping when the thickness of reinforcement increases. The minor increase in the damping coefficient is due to using a greater amount of adhesive for three layers of fibre fabrics (tf = 0.75 mm) compared to two layers (tf = 0.50 mm) (Figure  4.23b). In the case of tf = 0.75 mm, three coats of adhesive are applied to three fibre fabric layers (one coat for each layer). The combination of bi-directional carbon fibre fabric and the adhesive, as the matrix, provides a flexible reinforcement. As a result, fibre-reinforced layers can slightly contribute to the energy dissipation and be considered as a minor source of energy dissipation. In order to accurately investigate the condition of C-FREIs after conducting the shear and the compression tests, the elastomeric pad of C-FREI-B1 is detached from the supporting plates and then cut as shown in Figure  4.24. The laminated pad is checked for any internal delamination between rubber and fibre-reinforced layers. The specimen is cut using the water-jet in order to minimize or even eliminate any damage (e.g. debonding) that might 00.10.20.30.40.50.4 0.5 0.6 0.7 0.8Effective Horizontal Stiffness (kN/mm) Thickness of carbon fibre-fabric (mm) 25%50%100%051015200.4 0.5 0.6 0.7 0.8Equivalent Viscous Damping (%) Thickness of carbon fibre-fabric (mm) 25%50%100%(a)                                                                            (b) 90  occur during the cutting phase. As indicated in Figure  4.24, an internal delamination is observed between the exterior rubber layer and the first reinforcement at the top. The reason is that the interfacial stress in the layers, close to the top and bottom of the pad, exceeds the bonding strength of the adhesive and detachment occurs. Another point is that no delamination between internal layers shows the acceptable performance of the fibre-reinforced pads under shear and compression.     Figure ‎4.24. Carbon fibre-reinforced elastomeric pad (B1) after being tested In order to check whether the manufactured C-FREIs possess advantages over steel-reinforced elastomeric isolators (SREIs), their performances (e.g. stiffness and damping ratio) are compared. Since the SREI was not fabricated through this study, the comparison is made between the manufactured C-FREIs and a SREI manufactured by Dehghani Ashkezari et al. (2008). It should be noted that the loading conditions (e.g. vertical pressure and shear strain magnitude) are the same for both cases. Based on the experimental results in this study, at 3 MPa vertical pressure and 100% shear strain magnitude, the equivalent viscous damping of the manufactured C-FREIs ranges between 8.9% and 13.2%, and the effective lateral stiffness varies from 0.094 kN/mm to 0.240 kN/mm. Under a 3 MPa vertical pressure, the maximum vertical stiffness of C-FREIs is 147 kN/mm (Hedayati Dezfuli and Alam, 2014a). For the SREI, the equivalent viscous damping, and the effective horizontal and vertical stiffnesses are 8.0%, 0.381 kN/mm, and 153 kN/mm, respectively. Results show that the C-FREIs have higher damping ratio and lateral flexibility, and a vertical stiffness which is comparable to that of the SREI. In terms of weight, the laminated pad of a SREI is 2.6 times heavier than that of a C-FREI. Delamination 91  4.6 Possibility of Using C-FREIs in Bonded Applications The most well-known local failures in FREIs are delamination between reinforced and elastomeric layers and debonding between rubber layer and steel supporting plate. These modes of failure usually occur due to a very low flexural rigidity of reinforcement. If the bonding strength of adhesive used for attaching rubber layers to reinforcement and steel plates is lower than the interfacial stress generated between layers, it can be another reason of failure. For the C-FREIs manufactured through the cold-vulcanization process no delamination or debonding was observed during and after vertical tests when the compressive load increased from 0.75 MPa to 3 MPa. In order to increase the capacity of base isolators in carrying the vertical pressure, their vertical stiffness should be enhanced by either increasing the number or the thickness of fibre-reinforced layers through a design process.       In the cyclic displacement tests, a partial debonding between rubber layers and supporting plates were observed at shear strains greater than 50%. However, this local failure did not lead to a malfunction up to 100% shear strain and after performing tests, the rubber bearings did not undergo a global failure. It should be mentioned that since neither delamination nor debonding is acceptable, the manufacturing process should be modified in a way that no failure (partial or global) occurs. If the bonded C-FREIs are properly designed and manufactured through the proposed process, they can be used rather than steel-reinforced NRBs or HDRBs in buildings and bridges. In addition, this method of manufacturing can be extended to LRBs which are implemented in bridges, viaducts and buildings.  4.7 Numerical Validation and Verification A limited number of specimens were manufactured in scaled size. Therefore, in order to perform a comprehensive study on the behaviour of full size C-FREIs, a numerical method, FEM, is used. In this regard, one of the manufactured C-FREIs (CFREI-E1) is chosen to be modelled and analyzed using FEM in ANSYS and then, the FE modelling is validated with experimental results in order to assess the correctness of numerical simulations. In the next step, the accuracy of FE model in predicting the behaviour of C-FREIs is evaluated. After verifying the FEM (Hedayati Dezfuli and Alam, 2014c), real-size C-FREIs are modelled and analyzed in ANSYS. 92  4.7.1 Finite Element Modelling The first step in numerical simulations using FEM is to determine the type of elements by considering the number of nodes and degrees of freedom at each node. Defining the material properties is the most challenging part of the process since the behaviour of materials used in the device should be correctly captured in order to have an accurate simulation. In fact, the accuracy of FE results is highly dependent on the material properties of each component.  In the next step, the model is created and then discretized into finite number of elements by meshing. Finally, boundary conditions (BCs) and loading conditions are applied to the model and the whole system is solved and analyzed. Modelling, solving and analyzing the C-FREIs are performed in ANSYS (ANSYS Mechanical APDL, Release 14.0). Element SOLID185 with eight nodes and three degrees of freedom at each node is selected for both steel shims and rubber layers. Carbon fibre-reinforced (CFR) layers are modelled as shells, 4-noded rectangular element, SHELL181, with six degrees of freedom at each node is chosen. This element has a capability to be used in layered composite shells and capture the in-plane bending. SHELL181 is suitable for thin to moderately-thick shell structures used in nonlinear applications with large strain and/or rotation (ANSYS Documentation, Release 14.0). Among different types of nonlinear material models available in ANSYS, Ogden-Prony as a hyper-viscoelastic model is used to simulate the nonlinear behaviour of natural rubber under combined vertical pressure and cyclic lateral displacements. The material constants of the model used for rubber are listed in Table ‎4.6. Amin et al. (2006a and b) also conducted a research on the modelling of the response of natural and high damping rubbers based on experimental and numerical results. Steel shims are modelled as an isotropic material with the Young’s modulus of 210 GPa and a Poisson’s ratio of 0.3. The large-deflection effect is considered in full transient analyses in order to simulate the large deformation of rubber layers at large shear strain amplitudes. A perfect bonding is assumed between rubber and CFR layers.  Table ‎4.6. Material constants of Ogden-Prony model Ogden Prony μ1 0.056 α1 0.375 α1 2.564 τ1 0.062 μ2 4041.7 α2 0.061 α2 0.00095 τ2 65.82 93  Figure  4.25 demonstrates the top, side and 3D views of C-FREI-E1. The FE model with a mapped mesh in ANSYS is plotted in Figure  4.26. Mapped mesh is an organized type of meshing in which the size and the number of elements can be controlled. Purple and blue elements, respectively, illustrate the elastomeric layers and supporting steel plates. Since CFR layers are modelled as shell (area), their elements cannot be seen in the figure.                                                               Figure ‎4.25. Carbon fibre-reinforced elastomeric isolator, C-FREI-E1     Figure ‎4.26. C-FREI-E1 with a mapped mesh in ANSYS (ANSYS Mechanical APDL, Release 14.0); (a) full model, (b) half model (a)                                                                            (b) 94  Since C-FREIs have a plane of symmetry according to the geometry, BCs and loading conditions, half of the rubber bearing is modelled by applying a symmetry BC, on the xz plane at y = 0, in order to significantly reduce the processing time of FE analyses (Figure  4.26b). 4.7.2 Delamination Based on the experimental tests, at 100% shear strain amplitude, a detachment occurs between the rubber layers (the first and the last) and the steel plates when the lateral displacement exceeds 50% of the maximum deflection. This delamination is due to (1) very small flexural rigidity in carbon fibre-reinforced layers, and (2) insufficient bonding strength in the glue used for attaching the laminated core to the steel plates. Unlike steel shims used in the SREIs, the CFR layers in the manufactured C-FREIs almost have no flexural rigidity and as a result, they can be deformed (rollover deformation) under large lateral displacements. Consequently, the first and the last rubber layers are subjected to a peel-off force near the edges. When the bearing is subjected to lateral displacement, tensile stresses are developed near the edges of the laminated pad due to the moment. Under such a condition, the stress between the elastomeric layer and the steel plate exceeds the bonding strength of the glue and detachment starts from the edges. This local delamination due to the rollover deformation mainly affects the lateral flexibility of the base isolators (Toopchi-Nezhad et al. 2008b). In order to model the debonding behaviour in FE numerical simulation, two different approaches could be applied. In the first one, a layer of glue is modelled between rubber layer and steel plate at the top and bottom of the laminated core. The material properties of the glue, which behaves as a hyperelastic material, degrade over time (time steps) and as a result, the bonding strength of glue decreases. When the base isolator reaches a certain amount of lateral deflection, the glue cannot tolerate the peeling force introduced by the tension and the shear force generated between rubber layer and steel plate. Consequently, the rubber layer is detached from the steel supporting plate. The glue is modelled using a two-parameter Mooney-Rivlin material model with an initial shear modulus of 0.87 MPa and a constant Poisson’s ratio of 0.499 (close to 0.5) representing the behaviour of a nearly incompressible material. The material constants of the glue are listed in Table ‎4.7 at different time steps. In the second method, instead of modelling the glue as a continuous material 95  (solid element) between the rubber layer and the steel plate, glue is modelled as beam elements. In this case, when the lateral displacement reaches a certain value, a number of beam elements are destroyed in the delamination region due to a high amount of shear force generated between the rubber layer and the steel plate. This technique is applied by using birth and death concepts in ANSYS. In fact, element birth and death options reactivate and deactivate selected elements, respectively. The number of killed elements increases when the horizontal deflection reaches its maximum value. As a result, there will be no element (material) to resist against the shear force and accordingly, debonding is started. Since modelling the glue with degradable material properties is closer to the reality and provides more accurate results, the first approach was selected in FE simulations.  Table ‎4.7. Hyperelastic material constants of the glue Time Step C10 C01 1 0.180 0.253 2 0.130 0.165 3 0.090 0.125 4 0.070 0.105 5 0.060 0.095 6 0.054 0.089 7 0.050 0.085 8 0.048 0.083 9 0.046 0.083 10 0.045 0.080 11 0.044 0.079 Figure  4.27 shows the C-FREI-E1 before and after detachment. When the lateral deflection reaches 100% of the total thickness of rubber layers (γ = 100%), the glue between rubber layer and steel plate undergoes a very large deformation because of the increased amount of peel-off force.  96   Figure ‎4.27. C-FREI-E1 before and after delamination; (a) γ = 50%, (b) γ = 100% 4.7.3 Comparison While the C-FREI-E1 is under a constant vertical pressure of 3.0 MPa, the cyclic horizontal displacements are applied. At each amplitude of horizontal deflection including 25% tr, 50% tr, and 100% tr, three fully reversed sinusoidal cycles are applied at constant horizontal rate of VH = 20 mm/s. Variation of vertical pressure and cyclic horizontal displacement versus time are demonstrated in Figure  4.9, respectively. Figure  4.28 depicts the FE half model of C-FREI-E1 and the manufactured specimen under 3 MPa vertical pressure. In the FE simulation, the lower supporting plate is fixed in all directions and the vertical load is applied to the upper steel plate. The lateral bulging of elastomeric layers due to the vertical compressive load is clearly observed in both cases.   Figure ‎4.28. C-FREI-E1 under 3 MPa vertical pressure; (a) FE half model, (b) manufactured sample In order to compare the results obtained from the FEM with those of the experimental tests, shear force-deflection hysteresis curves of C-FREI-E1 evaluated from both Debonding (Glue Rupture) (a)                                                                            (b) (a)                                                     (b) 97  experimental tests and numerical simulations are plotted at different shear strain levels (25%, 50%, and 100%) in Figure  4.29.   Figure ‎4.29. Shear hysteretic response of C-FREI-E1 under 3 MPa vertical pressure at  (a) γ = 25%, (b) γ = 50%, and (c) γ = 100% obtained from experimental and FE numerical results Based on the experimental results, the maximum force at 25%, 50%, and 100% shear strains are 1.17 kN, 1.74 kN, and 2.69 kN, respectively. The peak shear forces obtained from the FE simulations at same shear strain amplitudes (25%, 50%, and 100%) are 1.10 kN, 1.74 kN, and 2.82 kN, respectively. The maximum relative difference, Δmax, defined as a ratio of difference between experimental and numerical results to the experimental result (in percentage), is found to be 6% which happens at γ = 25%. When the energy dissipation capacity per cycle defined as the area inside the force-displacement hysteresis curve is measured though experiment and FEM, the maximum relative difference is 11.5%. Table ‎4.8 shows the peak shear force, Fmax, the energy dissipation capacity per cycle, EDC, and their corresponding relative differences at each shear strain amplitude obtained from the experimental tests and FEM. It should be mentioned that at 25% shear strain, the maximum lateral displacement exceeds 5.25 mm which is 25% of the total thickness of rubber layers (tr = 21 mm). This happened because of the limitations and initial calibrations in the test setup. Table ‎4.8. Results obtained from experimental tests and FE numerical simulations γ (%) Fmax (kN) EDC (N.m) FEM Exp. Δ* (%) FEM Exp. Δ* (%) 25 1.09 1.24 11.6 4.8 5.4 11.5 50 1.74 1.74 0.4 11.4 12.3 7.5 100 2.82 2.69 4.8 35.8 33.1 8.0 Δ: ratio of difference between experimental and numerical results to the experimental result in percentage -3-1.501.53-24 -12 0 12 24Shear Force (kN) Lateral Displacement (mm)         Exp.         FEM -3-1.501.53-24 -12 0 12 24Shear Force (kN) Lateral Displacement (mm)         Exp.         FEM -3-1.501.53-24 -12 0 12 24Shear Force (kN) Lateral Displacement (mm)         Exp.         FEM (a)                                                  (b)                                                (c) 98  A good agreement between experimental and numerical results shows the accuracy of the FE model. In order to validate the FE results further, the shear hysteretic behaviour of C-FREI-D1 obtained from the numerical simulations is compared to that of C-FREI-D1 determined from the experimental tests at 50% shear strain. Figure  4.30 depicts the FEM results along with the experimental results, which provides the validity of the model for extending it to other models.     Figure ‎4.30. Shear hysteretic response of (a) C-FREI-A1 and (b) C-FREI-D1  at γ = 50% and P = 3 MPa It should be noted that the maximum displacement values in the lateral force-deflection curve of C-FREI-A1 obtained from the experimental test exceed the 50% shear strain (6 mm) because of the limitations in the test setup. Therefore, the difference between experimental and numerical results increases for this rubber bearing. It is worthy to mention that the shear strain amplitude in FEM can be matched to that in the experimental tests in order to make a direct comparison. 4.8 Performance of Full Scale C-FREIs Due to limited capacity of the actuator and reaction frame, it was not possible to perform full scale testing of the C-FREIs. This section, therefore, will determine the performance of full scale C-FREIs produced through the proposed manufacturing process in finite element environment. In this regard, full-size elastomeric isolators are modelled and analyzed using FEM in ANSYS based on the half model of C-FREI-E1 used for experimental results verification.  -2-1012-12 -8 -4 0 4 8 12Shear Force (kN) Lateral Displacement (mm) ExperimentFE Simulation -2-1012-12 -8 -4 0 4 8 12Shear Force (kN) Lateral Displacement (mm) ExperimentFE Simulation(a)                                                                           (b)  99  C-FREIs with different sizes (length, width, and height), number and thickness of rubber layers, and thickness of carbon fibre-reinforced sheets are considered and their specifications are calculated from shear force-deflection hysteretic loops and vertical force-displacement curves obtained from numerical simulations. Table ‎4.9 presents the geometrical properties of C-FREIs. Table ‎4.9. Geometrical properties of C-FREIs with different sizes Rubber Bearing L × W (mm × mm) H (mm) ne te (mm) tf (mm) S C-FREI-23NE6 200 × 300 31 6 4.5 0.75 13.3 C-FREI-23NE9 47 9 4.5 0.75 13.3 C-FREI-23NE12 62 12 4.5 0.75 13.3 C-FREI-23NE15 78 15 4.5 0.75 13.3 C-FREI-34NE6 300 × 400 31 6 4.5 0.75 19.0 C-FREI-34NE9 47 9 4.5 0.75 19.0 C-FREI-34NE12 62 12 4.5 0.75 19.0 C-FREI-34NE15 78 15 4.5 0.75 19.0 C-FREI-45NE6 400 × 500 31 6 4.5 0.75 24.7 C-FREI-45NE9 47 9 4.5 0.75 24.7 C-FREI-45NE12 62 12 4.5 0.75 24.7 C-FREI-45NE15 78 15 4.5 0.75 24.7 C-FREI-56NE6 500 × 600 31 6 4.5 0.75 30.3 C-FREI-56NE9 47 9 4.5 0.75 30.3 C-FREI-56NE12 62 12 4.5 0.75 30.3 C-FREI-56NE15 78 15 4.5 0.75 30.3 The full-scale dimensions of C-FREIs and the performance characteristics of elastomeric isolators in the vertical and horizontal directions are presented in Table ‎4.10. The name of each rubber bearing contains three parts. The first letter (C) refers to the material of fibres used in the reinforced layers and the second part is the abbreviation of fibre-reinforced elastomeric isolator. The first two digits in the last part are for length and width of the laminated pad, respectively; NE represents the number of rubber layers since at each plan size, the height changes as the number of rubber layers alters; and the last number refers to the number of rubber layers.      100  Table ‎4.10. Performance specifications of C-FREIs in the horizontal and vertical directions  for different lengths, widths and heights of laminated core Rubber Bearing Dimensions Horizontal  Specifications Vertical Specifications L × W (mm × mm) H (mm) Δmax (mm) KH (kN/mm) β (%) KV (kN/mm) Ec (MPa) C-FREI-23NE6 200 × 300 31 13.5 1.74 18.2 329.7 0.148 C-FREI-23NE9 47 20.3 1.05 14.5 220.7 0.149 C-FREI-23NE12 62 27.0 0.73 12.3 166.0 0.149 C-FREI-23NE15 78 33.8 0.55 11.0 133.0 0.150 C-FREI-34NE6 300 × 400 31 13.5 3.60 16.9 1592.9 0.358 C-FREI-34NE9 47 20.3 2.23 12.9 1069.8 0.361 C-FREI-34NE12 62 27.0 1.61 10.4 805.4 0.362 C-FREI-34NE15 78 33.8 1.25 8.7 645.7 0.363 C-FREI-45NE6 400 × 500 31 13.5 6.06 16.6 4858.3 0.656 C-FREI-45NE9 47 20.3 3.77 12.5 3278.7 0.664 C-FREI-45NE12 62 27.0 2.75 9.9 2474.2 0.668 C-FREI-45NE15 78 33.8 2.16 8.2 1986.8 0.671 C-FREI-56NE6 500 × 600 31 13.5 9.12 16.5 11250.0 1.013 C-FREI-56NE9 47 20.3 5.69 12.4 7627.1 1.030 C-FREI-56NE12 62 27.0 4.15 9.7 5769.2 1.038 C-FREI-56NE15 78 33.8 3.27 8.0 4639.2 1.044 In Table ‎4.10, the vertical stiffness, KV, and the compressive modulus, Ec, are measured from vertical tests simulated at 6 MPa vertical pressure and a vertical frequency of 0.2 Hz. The effective horizontal stiffness, KH, as well as the equivalent viscous damping are calculated from cyclic tests at a shear strain amplitude of 50%, a vertical pressure of 6 MPa, and a lateral rate of 50 mm/s. The maximum horizontal displacement (Δmax) at which the cyclic tests are performed is also provided. By increasing the number of elastomeric layers from 6 to 15, the equivalent viscous damping reduces from 18.2% to 11.0% for C-FREIs with the smallest plan size (200 mm by 300 mm) and from 16.5% to 8.0% for C-FREIs with the largest plan size (500 mm by 600 mm). The reason is that the increase rate of the elastic energy restored by the elastomeric isolator is more than that of the energy dissipated per cycle and as a result the equivalent viscous damping decreases (see Equation (‎4.3)). When the number of rubber layers increases from 6 to 15, the effective horizontal stiffness reduces from 1.74 kN/mm to 0.55 kN/mm and the vertical stiffness decreases from 329.7 kN/mm to 133.0 kN/mm for C-FREI with the smallest plan size. For C-FREIs with 500 mm by 600 mm plan size, increasing the number of elastomeric layers from 6 to 15 causes the effective lateral stiffness and the vertical stiffness 101  to be decreased from 9.12 kN/mm to 3.27 kN/mm and from 11250.0 kN/mm to 4639.2 kN/mm, respectively. 4.9 Numerical Parametric Study In this section, the effects of different factors are investigated on the performance of C-FREIs through sensitivity analyses. Here, three factors including the number of elastomeric layers (ne), the thickness of rubber layers (te), and the thickness of CFR sheets (tf) are chosen. The sensitivity of the effective horizontal stiffness, the equivalent viscous damping, and the vertical stiffness of manufactured C-FREIs is assessed. By considering three levels for each factor, nine C-FREIs are defined for each plan size. Effect of each factor is studied separately according to the following sections. 4.9.1 Number of Elastomeric Layers Low, medium, and high levels are considered as six, nine, and twelve, respectively for the number of rubber layers. Table ‎4.11 depicts the operational specifications of C-FREIs considered by changing ne. Thicknesses of rubber layer and carbon fibre-reinforced sheet are 4.5 mm and 0.75 mm, respectively, for all base isolators. If the number of elastomeric layers increases from 6 to 12, the effective horizontal stiffness, the equivalent viscous damping, and the vertical stiffness will reduce 138%, 99%, and 69%, respectively. Table ‎4.11. Stiffnesses and damping coefficient of C-FREIs with different numbers of rubber layers Rubber Bearing L × W  (mm × mm) H  (mm) ne KV (kN/mm) KH  (kN/mm) β (%) C-FREI-23NE6 200 × 300 31 6.0 329.7 1.74 18.2 C-FREI-23NE9 47 9.0 220.7 1.05 14.5 C-FREI-23NE12 62 12.0 166.0 0.73 12.3 C-FREI-34NE6 300 × 400 31 6.0 1592.9 3.60 16.9 C-FREI-34NE9 47 9.0 1069.8 2.23 12.9 C-FREI-34NE12 62 12.0 805.4 1.61 10.4 C-FREI-45NE6 400 × 500 31 6.0 4858.3 6.06 16.6 C-FREI-45NE9 47 9.0 3278.7 3.77 12.5 C-FREI-45NE12 62 12.0 2474.2 2.75 9.9 C-FREI-56NE6 500 × 600 31 6.0 11250.0 9.12 16.5 C-FREI-56NE9 47 9.0 7627.1 5.69 12.4 C-FREI-56NE12 62 12.0 5769.2 4.15 9.7 102  Figure  4.31 shows the effect of number of rubber layers on the operational characteristics of C-FREIs for different lengths and widths of elastomers. By increasing this parameter, the vertical stiffness decreases since the strength of the device degrades against the vertical loads. When the total thickness of rubber layers (which are mainly responsible for providing the lateral isolation) goes up, the lateral flexibility increases and accordingly, the effective horizontal stiffness reduces. However, increasing ne causes the lateral displacement of rubber bearing to be increased and as a result, the elastic energy restored in the C-FREI remarkably enhances since it is proportional to the square of lateral displacement, Δavg, (see Equation (‎4.4)). The energy dissipated by the device also increases when a higher amount of elastomer is used. However, the restored energy increases with a higher rate compared to the energy dissipated per cycle. Hence, the equivalent viscous damping decreases according to Equation (‎4.3).    Figure ‎4.31. The effect of number of rubber layers on; (a) vertical stiffness,  (b) effective horizontal stiffness, (c) equivalent viscous damping In order to determine how much each performance specification (response) is sensitive to the variation of number of rubber layers, Table ‎4.12 is constructed. Since the unit and the magnitude of order of responses are not the same, first, performance characteristics are normalized by dividing each specification by the maximum value. Then, the rate of change, R, of each specification is measured at each plan size using Equation (‎4.7). 𝑅𝑉 = |Δ?̂?𝑉Δ𝑛𝑒|,   𝑅𝐻 = |Δ?̂?𝐻Δ𝑛𝑒|,   𝑅𝛽 = |Δ?̂?Δ𝑛𝑒| (‎4.7) 0246810126 9 12KV (MN/mm) Number of Rubber Layers C-FREI-23C-FREI-34C-FREI-45C-FREI-5602468106 9 12KH (kN/mm) Number of Rubber Layers C-FREI-23C-FREI-34C-FREI-45C-FREI-56051015206 9 12β (%) Number of Rubber Layers C-FREI-23C-FREI-34C-FREI-45C-FREI-56(a)                                                (b)                                                   (c) 103  where VKˆ , HKˆ , and ˆ  are normalized vertical stiffness, normalized effective horizontal stiffness, and normalized equivalent viscous damping, respectively. Δne refers to the changes in the number of rubber layers between each two base isolators. Table ‎4.12. Normalized operational characteristics and their change rates  for different numbers of rubber layers No. Rubber Bearing VKˆ (kN/mm) HKˆ  (kN/mm) ˆ  (%) Two way comparison RV RH Rβ 1 C-FREI-23NE6 1 1 1 1 – 2 0.11 0.13 0.07 2 C-FREI-23NE9 0.67 0.60 0.80 2 – 3 0.06 0.06 0.04 3 C-FREI-23NE12 0.50 0.42 0.68 1 – 3 0.08 0.10 0.05 4 C-FREI-34NE6 1 1 1 4 – 5 0.11 0.13 0.08 5 C-FREI-34NE9 0.67 0.62 0.76 5 – 6 0.06 0.06 0.05 6 C-FREI-34NE12 0.51 0.45 0.61 4 – 6 0.08 0.09 0.06 7 C-FREI-45NE6 1 1 1 7 – 8 0.11 0.13 0.08 8 C-FREI-45NE9 0.67 0.62 0.75 8 – 9 0.06 0.06 0.05 9 C-FREI-45NE12 0.51 0.45 0.60 7 – 9 0.08 0.09 0.07 10 C-FREI-56NE6 1 1 1 10 – 11 0.11 0.13 0.08 11 C-FREI-56NE9 0.68 0.62 0.75 11 – 12 0.06 0.06 0.05 12 C-FREI-56NE12 0.51 0.46 0.59 10 – 12 0.08 0.09 0.07 It should be mentioned that R is calculated between each two C-FREIs as shown in the three last columns of Table ‎4.12. For example, RV in the second row is obtained by comparing the vertical stiffnesses of C-FREI-23NE6 and C-FREI-23NE9. At each plan size, every two base isolators (from three considered C-FREIs) should be compared together in order to check whether increasing ne from 6 to 9 and 9 to 12 has a higher effect or increasing ne from 6 to 12 is more significant. In the “Two way comparison” column, every three cases (rows) having the same plan sizes are compared to each other. For example, in the two way comparison of 2 – 3, C-FREI-23NE9 is compared with C-FREI-23NE12. The operational characteristic which exhibits the maximum rate of variation has the most sensitivity to the factor ne. Based on the results presented in Table ‎4.12, the effective horizontal stiffness is the most sensitive response and the equivalent viscous damping is the least sensitive response regardless of the length and width of rubber sheets. 4.9.2 Thickness of Elastomeric Layers While the number of rubber layers (ne = 9) and the thickness of CFR sheets (tf = 0.75 mm) are kept constant, the thickness of elastomer was increased from 3.0 mm to 6.0 mm 104  with an increment of 1.5 mm. Stiffnesses and equivalent viscous damping of C-FREIs for different plan sizes and heights of C-FREIs are listed in Table ‎4.13. Table ‎4.13. Stiffnesses and damping coefficient of C-FREIs  with different thicknesses of rubber layers Rubber Bearing L × W  (mm × mm) H  (mm) te KV (kN/mm) KH  (kN/mm) β (%) C-FREI-23TE3 200 × 300 33 3.0 730.2 1.79 17.2 C-FREI-23TE4 47 4.5 220.7 1.05 14.5 C-FREI-23TE6 60 6.0 98.7 0.67 14.3 C-FREI-34TE3 300 × 400 33 3.0 3529.4 3.63 16.6 C-FREI-34TE4 47 4.5 1069.8 2.23 12.9 C-FREI-34TE6 60 6.0 452.3 1.57 10.8 C-FREI-45TE3 400 × 500 33 3.0 10256.4 6.09 16.4 C-FREI-45TE4 47 4.5 3278.7 3.77 12.5 C-FREI-45TE6 60 6.0 1400.2 2.72 10.1 C-FREI-56TE3 500 × 600 33 3.0 23076.9 9.15 16.4 C-FREI-56TE4 47 4.5 7627.1 5.69 12.4 C-FREI-56TE6 60 6.0 3327.2 4.13 9.9 The variations of operational specifications by changing the thickness of elastomeric layers are observed in Figure  4.32. Similar to the first factor (number of rubber layers), the vertical and the effective horizontal stiffnesses, as well as the equivalent viscous damping decrease when rubber sheets with higher thicknesses are used. The reason is that a higher total thickness of elastomer leads to a device with a higher lateral flexibility and a lower vertical rigidity. The results also show that using thicker rubber layers does not improve the energy dissipation capability of a base isolator. When the thickness of rubber sheets double to 6, in the extreme conditions, the effective lateral stiffness, the equivalent viscous damping and the vertical stiffness decrease 167%, 66%, and 640%, respectively.  105     Figure ‎4.32. The effect of thickness of rubber layers on; (a) vertical stiffness,  (b) effective horizontal stiffness, (c) equivalent viscous damping Rate of change of performance characteristics calculated by changing the thickness of rubber layers are listed in Table ‎4.14. By comparing the change rates at each plan size, it is observed that the vertical stiffness is more sensitive to the thickness of rubber layers. However, for C-FREIs with the largest plan size (500 mm by 600 mm), both the vertical and horizontal stiffnesses experience similar amount of changes when the thickness of elastomer increases. Table ‎4.14. Normalized operational characteristics and their change rates  for different thicknesses of rubber layers No. Rubber Bearing VKˆ  (kN/mm) HKˆ  (kN/mm) ˆ  (%) Two way comparison RV RH Rβ 1 C-FREI-23TE3 1 1 1 1 – 2 0.47 0.28 0.11 2 C-FREI-23TE4 0.30 0.59 0.84 2 – 3 0.11 0.14 0.01 3 C-FREI-23TE6 0.14 0.37 0.83 1 – 3 0.29 0.21 0.06 4 C-FREI-34TE3 1 1 1 4 – 5 0.46 0.26 0.15 5 C-FREI-34TE4 0.30 0.61 0.78 5 – 6 0.12 0.12 0.09 6 C-FREI-34TE6 0.13 0.43 0.65 4 – 6 0.29 0.19 0.12 7 C-FREI-45TE3 1 1 1 7 – 8 0.22 0.25 0.16 8 C-FREI-45TE4 0.67 0.62 0.76 8 – 9 0.26 0.12 0.10 9 C-FREI-45TE6 0.29 0.45 0.61 7 – 9 0.24 0.18 0.13 10 C-FREI-56TE3 1 1 1 10 – 11 0.21 0.25 0.17 11 C-FREI-56TE4 0.68 0.62 0.75 11 – 12 0.25 0.11 0.10 12 C-FREI-56TE6 0.30 0.45 0.60 10 – 12 0.23 0.18 0.13 Equation (‎4.8) represents the variation rates of three specifications. 05101520253.0 4.5 6.0KV (MN/mm) Thickness of Rubber Layers C-FREI-23C-FREI-34C-FREI-45C-FREI-5602468103.0 4.5 6.0KH (kN/mm) Thickness of Rubber Layers C-FREI-23C-FREI-34C-FREI-45C-FREI-56051015203.0 4.5 6.0β (%) Thickness of Rubber Layers C-FREI-23C-FREI-34C-FREI-45C-FREI-56(a)                                                (b)                                                 (c) 106  𝑅𝑉 = |Δ?̂?𝑉Δ𝑡𝑒|,   𝑅𝐻 = |Δ?̂?𝐻Δ𝑡𝑒|,   𝑅𝛽 = |Δ?̂?Δ𝑡𝑒| (‎4.8) 4.9.3 Thickness of Fibre-Reinforced Sheets In order to investigate the effect of thickness of fibre-reinforced layers on the performance of C-FREIs, CFR sheets with thicknesses of 0.50 mm, 0.75 mm, and 1.25 mm are selected. Two other factors (ne and te) are kept constant at 6 and 4.5 mm, respectively. Table ‎4.15 shows the vertical and horizontal stiffnesses, as well as the equivalent viscous damping for twelve considered C-FREIs. Table ‎4.15. Stiffnesses and damping coefficient of C-FREIs  with different thicknesses of fibre-reinforced layers. Rubber Bearing L × W  (mm × mm) H  (mm) tf KV (kN/mm) KH  (kN/mm) β (%) C-FREI-23TF5 200 × 300 30 0.50 327.3 1.73 18.3 C-FREI-23TF7 31 0.75 329.7 1.74 18.2 C-FREI-23TF12 33 1.25 331.2 1.75 18.1 C-FREI-34TF5 300 × 400 30 0.50 1568.6 3.59 17.0 C-FREI-34TF7 31 0.75 1592.9 3.60 16.9 C-FREI-34TF12 33 1.25 1610.7 3.61 16.9 C-FREI-45TF5 400 × 500 30 0.50 4724.4 6.05 16.6 C-FREI-45TF7 31 0.75 4858.3 6.06 16.6 C-FREI-45TF12 33 1.25 4979.3 6.07 16.6 C-FREI-56TF5 500 × 600 30 0.50 10843.4 9.11 16.5 C-FREI-56TF7 31 0.75 11250.0 9.12 16.5 C-FREI-56TF12 33 1.25 11688.3 9.13 16.5 Figure  4.33 demonstrates the effect of thickness of CFR layers on the stiffnesses and damping coefficient of C-FREIs with different plan sizes. An increasing trend is observed for the vertical stiffness and the effective horizontal stiffness when the thickness of CFR sheets goes up from 0.5 mm to 1.25 mm. however, the changes are very small compared to the cases in which two other factors (ne and te) vary. The equivalent viscous damping encounters negligible alteration while increasing the thickness of reinforcement. It can be understood that CFR sheets have almost no contribution to the energy dissipating. By changing the carbon fibre-reinforced layers’ thickness from 0.5 mm to 1.25 mm, the vertical stiffness and the effective horizontal stiffness increase 7.2% and 0.9%, respectively and the equivalent viscous damping diminishes 0.9%. 107   Figure ‎4.33. The effect of thickness of carbon fibrereinforced layers on; (a) vertical stiffness,  (b) effective horizontal stiffness, (c) equivalent viscous damping When the change rates of the operational specifications are compared for different plan sizes (see Table ‎4.16), it can be observed that the vertical stiffness has the maximum variation when CFR sheet with higher thicknesses are used in the rubber bearing. This fact shows that the fibre-reinforced layers are mainly responsible for providing vertical stiffness. Although the effect of CFR sheets on the performance of C-FREIs is small compared to the rubber layers, increasing the thickness of fibre-reinforced layers can increase the vertical stiffness up to 7.2% when the thickness increases from 0.5 mm to 1.25 mm.  Table ‎4.16. Normalized operational characteristics and their change rates  for different thicknesses of carbon fibre-reinforced layers No. Rubber Bearing VKˆ  (kN/mm) HKˆ  (kN/mm) ˆ  (%) Two way Comparison RV RH Rβ 1 C-FREI-23TF5 0.988 0.991 1 1 – 2 0.029 0.017 0.015 2 C-FREI-23TF7 0.995 0.995 0.996 2 – 3 0.009 0.010 0.011 3 C-FREI-23TF12 1 1 0.991 1 – 3 0.016 0.012 0.012 4 C-FREI-34TF5 0.974 0.994 1 4 – 5 0.060 0.010 0.007 5 C-FREI-34TF7 0.989 0.997 0.998 5 – 6 0.022 0.006 0.005 6 C-FREI-34TF12 1 1 0.996 4 – 6 0.035 0.007 0.005 7 C-FREI-45TF5 0.949 0.996 1 7 – 8 0.108 0.003 0.002 8 C-FREI-45TF7 0.976 0.997 0.999 8 – 9 0.049 0.006 0.003 9 C-FREI-45TF12 1 1 0.998 7 – 9 0.068 0.005 0.003 10 C-FREI-56TF5 0.928 0.998 1 10 – 11 0.139 0.007 0.003 11 C-FREI-56TF7 0.963 0.999 0.999 11 – 12 0.075 0.001 0.001 12 C-FREI-56TF12 1 1 0.999 10 – 12 0.096 0.003 0.002 024681012140.50 0.75 1.00 1.25KV (MN/mm) Thickness of CFR Layers C-FREI-23C-FREI-34C-FREI-45C-FREI-5602468100.50 0.75 1.00 1.25KH (kN/mm) Thickness of CFR Layers C-FREI-23C-FREI-34C-FREI-45C-FREI-56051015200.50 0.75 1.00 1.25β (%) Thickness of CFR Layers C-FREI-23C-FREI-34C-FREI-45C-FREI-56(a)                                                (b)                                               (c) 108  Change rates of the operational characteristics can be calculated using Equation (‎4.9) when the thickness of CFR layers varies. 𝑅𝑉 = |Δ?̂?𝑉Δ𝑡𝑓|,   𝑅𝐻 = |Δ?̂?𝐻Δ𝑡𝑓|,   𝑅𝛽 = |Δ?̂?Δ𝑡𝑓| (‎4.9) 4.10 Summary FREIs are relatively new elastomeric bearings in which fibre-reinforced polymer (FRP) composite plates are used as reinforcements rather than steel shims. Producing FREIs in the form of long laminated pads without using a mold and cutting them to the required size significantly reduces the complexity, time and overall cost of the manufacturing process. In addition, FREIs are much lighter than SREIs. Due to the lack of adequate information on the performance of bonded FREI, which are manufactured using the proposed cold-vulcanization process, the goal of this chapter was to assess the efficiency and performance sensitivity of bonded carbon-FREIs (C-FREIs) based on experimental and numerical investigations. Nine 1/4 scale C-FREIs were fabricated using a simple and fast manufacturing process, which has a potential to be applied in developing countries. Experimental results showed that under cyclic displacements, although a partial delamination occurs between rubber layer and steel supporting plate due to the rollover deformation at shear strains greater than 50%, the rubber bearings perform properly up to 100% shear strain. The vertical stiffness increases with increasing the fibre-reinforced layers’ thickness and with decreasing the elastomers’ thickness. The flexibility in the horizontal direction increases by increasing the total thickness of rubber layers while, the energy dissipation capacity enhances with increasing the thickness of both fibre-reinforced and elastomeric layers. The effect of several factors including the vertical pressure, the lateral cyclic rate, the number of rubber layers and the thickness of carbon fibre-reinforced layers were explored on the behaviour of rubber bearings. Results revealed that the effect of vertical pressure on the response of base isolators is negligible. However, decreasing the cyclic loading rate increases the lateral flexibility and the damping capacity. Another finding was that, carbon fibre-reinforced layers can be considered as a minor source of energy dissipation. A numerical parametric study was performed by exploring the effect of number and thickness of rubber layers, as well as 109  thickness of carbon fibre reinforced sheets on the performance of C-FREIs. The results showed that by increasing the number and thickness of rubber layers, the efficiency of C-FREIs degrades in terms of vertical stiffness and damping capacity, however, the performance improved in terms of lateral flexibility. Another important observation was that the increasing thickness of fibre-reinforced layers can increase the vertical rigidity of the base isolator. The vertical stiffness had the most sensitivity to the thickness of elastomeric layers and the thickness of CFR sheets. On the other hand, the effective lateral stiffness was mostly affected by increasing the number of rubber layers.    110  Chapter 5 Smart Elastomeric Isolators Equipped with Shape Memory Alloy Wires 5.1 General Based on the results obtained in Chapter 4, it was found that fiber-reinforced rubber bearings have limited shear deformation capacity and can undergo local failures (e.g. partial delamination) under shear strain levels equal to and greater than 100%. In order to provide a solution for the aforementioned limitations, in this chapter, shape memory alloy (SMA), as a supplementary component, was implemented in elastomeric bearings.  A diagonal (cross) configuration of SMA wires was proposed for NRBs and its performance was compared to the straight arrangement suggested by Choi et al. (2005). First, the most efficient SMA was determined for smart elastomeric isolators based on the superelastic strain range and compatibility with environmental conditions (temperature). In order to simulate the nonlinear behaviour of the elastomer, a hyper-viscoelastic material model was used and validated by experimental results. After evaluating the hysteretic shear response of SMA-based natural rubber bearings (SMA-NRB) through finite element method (FEM), the effective horizontal stiffness, the residual deformation, the energy dissipation capacity, and the equivalent viscous damping of SMA-NRB were calculated according to analytical equations. The effect of aspect ratio of rubber bearing (i.e. ratio of the height to the length), arrangement and thickness of wires, and pre-strain in SMA wires were investigated on the performance of base isolators. Similar cross configuration of SMA wire was used for carbon fibre-reinforced high damping rubber bearings (CFR-HDRB) by considering two types of SMA (NiTi- and ferrous-based). Then, the most efficient SMA in terms of the superelastic strain range and compatibility with environmental thermal conditions was identified through a performance assessment. The effective horizontal stiffness and the residual deformation of the proposed smart base isolators were calculated from the lateral force-deflection hysteresis curves through numerical simulations. After validating the FE results obtained from the numerical simulations, the hysteretic shear response of SMA-based CFR-HDRB was evaluated. The effect of different factors, including the aspect ratio of rubber bearing, the SMA types, and the arrangement and thickness of wires was investigated on the performance of the device.  111  In order to study the effect of SMA on the LRB, SMA wires were wrapped around the LRB with a symmetric double cross configuration. The finite element (FE) model of LRB was validated with experimental tests and then extended to SMA-LRB. Hysteretic shear response of SMA wire-based LRB was determined through FEM in ANSYS (ANSYS Mechanical APDL, Release 14.0) at different shear strain levels and with different radii of SMA wire.  Finally, in order to appropriately determine the pre-strain and the radius of cross section of SMA wires, a performance-based design approach was developed along with a design example for SMA wire-based rubber bearings. 5.2 SMA-based Natural Rubber Bearings (SMA-NRB) SMA-based smart base isolators have many advantages such as stability, re-centring capability, high energy dissipation capacity and long service life. They not only will mitigate the seismic response of structures in terms of acceleration, displacement and internal forces but also, they will have superior performance in terms of fatigue property and energy dissipation capacity compared to existing rubber bearings (Suduo and Xiongyan, 2007).  In this section, SMA wires are used as a supplementary element to improve the performance of steel-reinforced NRBs in terms of energy dissipation capacity and residual deformation, which occurs at large shear strain amplitudes. Regarding the maximum superelastic strain in SMA wires, two different configurations are considered for wires. The effect of several factors such as the aspect ratio of rubber bearing, the thickness of wires, and the pre-strain in SMA wires in addition to the arrangement of wires is investigated on the performance of the smart base isolator. Geometrical properties of two NRBs with different numbers of elastomeric layers are listed in Table ‎5.1.  Table ‎5.1. Geometrical properties of NRBs Specimen Horizontal dimensions of isolator (mm × mm) Horizontal dimensions of steel shims (mm × mm) tE (mm) tr (mm) ts (mm) nr ns R S NRB-1 240 × 240 200 × 200 15 4.5 1.0 8 7 0.22 0.075 NRB-2 240 × 240 200 × 200 15 4.5 1.0 14 13 0.38 0.075 tE: thickness of supporting steel plates; tr: thickness of rubber layers; ts: thickness of steel shims; nr: number of rubber layers; ns: number of steel shims; R: aspect ratio of the rubber bearing, i.e. the ratio of the height to the length; S: shape factor, i.e. the ratio of loaded area to force-free area of one rubber layer 112  The thickness of elastomer layers and steel shims are 4.5 mm and 1 mm, respectively, for both cases. While, the number of rubber layers, nr, is increased from 8 to 14. The aspect ratio of rubber bearing is defined as a ratio of the effective height to the length.  𝑅 =𝐻𝑒𝑓𝑓𝐿 (‎5.1) where the effective height is the total thickness of rubber layers and steel shims. 𝐻𝑒𝑓𝑓 = 𝑡𝑟𝑛𝑟 + 𝑡𝑠𝑛𝑠 (‎5.2) The schematic view of NRB-1 and NRB-2 consisting of 8 and 14 rubber layers, respectively, is plotted in Figure ‎5.1.                 Figure ‎5.1. Schematic view of the elastomeric isolator; (a) Plan view of NRB-1 and NRB-2,  (b) Side view of NRB-1, (c) Side view of NTB-2 5.2.1 SMA-NRB equipped with Straight Wires In the SMA-NRB with straight wire (SMA-NRB-S), two continuous SMA wires with a radius of 2.5 mm are wounded in two opposite sides of the rubber bearing as shown in Figure  5.2a. This type of arrangement of SMA wires was previously proposed by Choi et al. (2005). Each wire passes through four steel hooks which are mounted at each corner. In such configuration, since one continuous wire is used instead of four wires fixed at each corner of the supporting plates, the induced strain along the SMA wire due to the cyclic lateral displacement of rubber bearing noticeably decreases.  (a)                                                              (b)                   (c) 113   Figure ‎5.2. Smart rubber bearing; (a) straight SMA wires, (b) cross SMA wires The total length of SMA wires, LSMA, which is required for the straight configuration, is presented in Table ‎5.2 for each aspect ratio.  Table ‎5.2. Required length for SMA wires in the straight configuration (Figure  5.2a) Specimen R LSMA (mm) SMA-NRB-S1 0.22 1092 SMA-NRB-S2 0.38 1224 The strain in SMA wires (εSMA) is a function of the shear strain amplitude, γ, and the aspect ratio, R. When the shear strain is increased from 25% to 200%, the strain noticeably goes up in the SMA wires. ΕSMA also increases with the increase in aspect ratio of the base isolator. 5.2.2 SMA-NRB equipped with Cross Wires In the SMA-NRB with cross wires (SMA-NRB-C), two SMA wires with a radius of 2.5 mm are wounded around the rubber bearing diagonally as shown in Figure ‎5.2b. A steel hook is mounted at each corner on the lower and upper surfaces of the top and bottom supporting plates, respectively. The SMA wires pass through these hooks (see Figure ‎5.2b). Compared to the straight configuration, the main reason of using wires in such an arrangement is to effectively reduce the maximum strain in the wires due to large shear strain amplitudes of rubber bearing. The total length of wires needed for this arrangement is presented in Table ‎5.3. Although a larger length of SMA wire is required for this configuration compared to the SMA Wire 1 SMA Wire 2 Hook x y z Supporting Steel Plate (a)                                                                                  (b) 114  straight arrangement, the generated strain in cross wires due to the lateral deflection of rubber bearing is much lower than that of straight wires. The strain in cross and straight SMA wires will be calculated in the section of efficiency of SMAs (next section) for different shear strain amplitudes and two aspect ratios. Table ‎5.3. Required length for SMA wires in the cross configuration (Figure  5.2b) Specimen R LSMA (mm) SMA-NRB-C1 0.22 1871.9 SMA-NRB-C2 0.38 1937.9 Figure  5.3 depicts the variation of strain in straight and cross SMA wires by increasing the shear strain amplitude. Strain in the SMA wires is geometrically calculated by increasing the shear strain and by considering the dimensions of the base isolator and the configuration of wires. Figure  5.3 illustrates that at each shear strain amplitude the strain in the SMA wires is higher when the smart elastomeric isolator has a larger aspect ratio. It can be also observed that when the shear strain amplitude is lower than 150%, the strain induced in SMA wires will not exceed 15% for both SMA-NRBs. However, at 200% shear strain amplitude, the SMA wires in SMA-NRB-S1 (R = 0.22) and SMA-NRB-S2 (R = 0.38) experience about 15% and 23% strain, respectively (Figure  5.3a).    Figure ‎5.3. Variation of strain in SMA wire as a function of shear strain amplitude and  aspect ratio for (a) straight configuration and (b) cross configuration In the cross configuration (Figure  5.3b), there is a significant reduction in the SMA wire strain compared to that of the straight configuration. In this case, the SMA strain is lower than 8% for both aspect ratios. The maximum strain generated in cross SMA wires is 051015202525 50 75 100 125 150 175 200SMA Strain (%) Shear Strain, γ (%) R = 0.22R = 0.38051015202525 50 75 100 125 150 175 200SMA Strain (%) Shear Strain, γ (%) R = 0.22R = 0.38(a)                                                                            (b) 115  7.2% which occurs in the SMA-NRB-C2 with an aspect ratio of 0.38 at 200% shear strain. It shows that when the aspect ratio of the elastomeric isolator is increased, unlike the straight configuration, the cross SMA wires in smart NRBs can operate in a superelastic range under large shear strain amplitudes. 5.2.3 Efficiency of SMAs For each aspect ratio, the SMA strain (εSMA) in wires with 2.5 mm radius is calculated at eight different levels of shear strain amplitudes, form 25% to 200%, in the case of cross and straight configurations (Table ‎5.4). Since most of SMAs have a superelastic strain below 6% (see Table ‎5.5), SMA wires with the straight configuration cannot operate in a superelastic range at large shear strain amplitudes, especially when the height of the rubber bearing is increased. Whereas, the cross SMA wires can operate within its superelastic range at large lateral cyclic displacements. This fact demonstrates the effectiveness of the cross configuration over the straight one. In the straight configuration, when the shear strain is higher than 150% (see Figure  5.3a), using any type of SMA listed in Table ‎5.5 is inefficient for R = 0.38 since the strain in wires exceeds the superelastic strain range. However, a comparison is performed between the straight and the cross configurations, with R = 0.38 at 200% shear strain, in order to investigate the behaviour of SMA wires and consequently the hysteretic shear response of rubber baring when the wires are subjected to strains above the superelastic strain range. It should be mentioned that a different alloy, which is not considered in this thesis, with different properties (e.g. yield stress and superelastic strain range) may lead to different results when it is implemented in the straight configuration. Therefore, further investigations can be conducted in the future works.        116  Table ‎5.4. Strain in SMA wires for two configurations and two aspect ratios  at different shear strain amplitudes  εSMA (%) R = 0.22 R = 0.38    Wire       γ (%) Straight Cross Straight Cross 25 0.3 0.04 0.5 0.1 50 1.3 0.2 2.1 0.5 75 2.9 0.3 4.4 1.0 100 4.8 0.6 7.4 1.8 125 7.1 1.0 10.9 2.9 150 9.5 1.4 14.8 4.1 175 12.2 1.9 18.9 5.6 200 15.0 2.4 23.3 7.2 Table ‎5.5. Mechanical characteristics of different shape memory alloys (SMAs) Alloy εmax (%) εs (%) EA (GPa) Af (°C) Reference Ni Ti49.1 5.0 3.6 40.4 44.6 Strnadel et al. 1995 Ni Ti49.5 5.7 4.6 45.3 53.0 Strnadel et al. 1995 Ni Ti50 3.1 2.2 117.8 77.8 Strnadel et al. 1995 Ni Ti 8.2 6.8 30.0 42.9 Boyd and Lagoudas 1996 Ni Ti45 6.8 6.0 62.5 -10.0 Alam et. Al. 2008 Ni Ti44.1 6.5 5.5 39.7 0 Alam et. Al. 2008 Ti Ni40 Cu10 4.1 3.4 72.0 66.6 Strnadel et al. 1995 Ti Ni41 Cu10 4.1 3.1 91.5 50.0 Strnadel et al. 1995 Ti Ni41.5 Cu10 3.4 2.8 87.0 60.0 Strnadel et al. 1995 Ti Ni25 Cu25 10.0 2.5 14.3 73.0 Liu 2003 CuAlBe 3.0 2.4 32.0 -65.0 Zhang et al. 2009 FeMnAlNi 6.1 5.5 98.4 < -50°C Omori et al. 2011 FeNiCoAlTaB 15.0 13.5 46.9 -62.0 Tanaka et al. 2010 Table ‎5.6 presents the effectiveness of various types of SMA wires in cross configuration applied to the rubber bearing for a range of aspect ratios. It shows that all six types of SMA can operate in the elastic range when the aspect ratio is smaller than 0.24. The TiNi40Cu10 will remain within its superelastic range for all shear strain amplitudes when the aspect ratio is equal to or smaller than 0.27. For SMA-NRBs with 0.38 aspect ratio, strain in TiNi40Cu10 and CuAlBe exceed the superelastic limit at shear strain amplitudes equal to and higher than 150%. If FeMnAlNi wires are used, the lateral displacement of the rubber bearing can go up to 175% of the height without causing plastic deformation in the SMA wires. However, when the elastomeric base isolator with an aspect ratio of 0.38 is subjected 117  to 200% shear strain, only NiTi, NiTi45, and FeNiCoAlTaB (FeNCATB) can operate in their superelastic range. It means that these three types of SMA can be considered as good candidates to be used in cross configuration. Table ‎5.6. Superelastic range of SMAs for different aspect ratios and shear strain amplitudes  in cross configuration  TiNi40Cu10 CuAlBe FeMnAlNi NiTi NiTi45  FeNCATB         R γ (%) ≤0.27 0.38 ≤0.24 0.38 ≤0.33 0.38 ≤0.38 ≤0.38 ≤0.38 ≤ 125          150          175          200          Different environmental conditions such as temperature and humidity can affect the performance of elastomeric base isolators. The operational temperature range varies according to the location in which a rubber bearing operates. Since, the superelastic effect of SMA wires occurs at temperatures above the austenite finish temperature, in order to have a smart elastomeric bearing with superelastic SMA wires, the austenite finish temperature of the SMA wires should be lower than the ambient temperature. In such circumstances, since the minimum ambient temperature in countries with cold climatic conditions often gets below 0°C and in some places reaches -40°C, the austenite finish temperature of the SMA wire should be lower than this minimum temperature. Therefore, NiTi45, CuAlBe, FeMnAlNi, and FeNCATB with Af lower than zero (see Table ‎5.5) can be implemented in elastomeric base isolators.  When both the superelastic strain and the austenite finish temperature are considered as two important criteria for choosing the most efficient SMA, FeNCATB with 13.5% superelastic strain and -62°C austenite finish temperature will be the best candidate to be used in SMA-NRBs. Hence, FeNCATB SMA wires are implemented in smart NRBs. 5.2.4 Finite Element Modelling 5.2.4.1 Material Model Modelling, meshing and analyzing the NRB is performed in ANSYS (ANSYS Mechanical APDL, Release 14.0). In this regard, among different types of nonlinear material 118  models available in ANSYS, Mooney Rivlin – Prony (hyper-viscoelastic) model is used to simulate the nonlinear behaviour of natural rubber under combined vertical pressure and cyclic lateral displacements. In FE analyses, the shear modulus of rubber is 0.50 MPa and the Poisson’s ratio is assumed to be 0.4998. The material constants of the hyper-viscoelastic model are listed in Table ‎5.7. Steel shims are modelled as an isotropic material with the Young’s modulus of 210 GPa and a Poisson’s ratio of 0.3. Element SOLID185 with 8 nodes and three degrees of freedom at each node is selected for both steel shims and rubber layers. The large-deflection effect is considered in full transient analyses in order to simulate the large deformation of rubber layers at large shear strain amplitudes. In order to validate the material models used for rubber layers and steel shims, the lateral force-deflection hysteresis curves of a NRB assessed using FEM are compared with the experimental results.  Table ‎5.7. Material constants of hyper-viscoelastic model Mooney-Rivlin Model Prony Model C10 0.502 α1 0.565 C01 0.307 τ1 0.130 C11 -0.018 α2 0.061   τ2 65.82 Figure  5.4 demonstrates a schematic view of an NRB consisting of 16 elastomer layers with a thickness of 2.73 mm and 15 steel shims with a thickness of 1 mm. Two supporting steel plates with 9.8 mm thickness are mounted at the top and the bottom. The NRB is covered by a layer of rubber with a thickness of 5 mm. Figure  5.5 shows the NRB with a mapped mesh in ANSYS. Elements with purple and blue colours respectively illustrate the elastomeric cover layer and supporting steel plates. The hysteretic shear behaviour of NRB is evaluated under 6 MPa vertical pressure, three different shear strain amplitudes (100%, 150%, and 184%), and a horizontal frequency of 0.2 Hz through full transient analyses.  119               Figure ‎5.4. Steel-reinforced NRB; (a) side view, (b) plan view  (adapted from (Dehghani Ashkezari et al., 2008))  Figure ‎5.5. NRB with a mapped mesh in ANSYS (ANSYS Mechanical APDL, Release 14.0) According to the hysteresis curves plotted in Figure  5.6, a good consistency is observed between the results obtained from FE simulations and experimental tests conducted by Dehghani Ashkezari et al. (2008). Based on the numerical results, the effective horizontal stiffness of the NRB is 0.38, 0.34, and 0.33 kN/mm for 100%, 150%, and 184% shear strain amplitudes, respectively. The maximum difference between the numerical and experimental results in the horizontal stiffness is 4% which occurs at 150% shear strain. Also, the maximum horizontal loads corresponding to 100%, 150%, and 184% shear strains are 16.7, 22.3, and 26.4kN, respectively which closely match with the experimental results.  (a)                                (b) 120   Figure ‎5.6. Lateral force-deflection curves of steel-reinforced NRB  (experimental results are adapted from (Dehghani Ashkezari et al., 2008)) 5.2.4.2 Modelling of SMA wires In generating the FE model of the SMA-NRB, a method of superposition is implemented in order to simplify the system by decoupling the rubber bearing and SMA wires. Here, a smooth contact is assumed between the steel hook and the SMA wire. Instead of modelling steel hooks, and the contact between the hooks and continuous SMA wires, exerted forces to the elastomeric isolator due to SMA wires are considered. These assumptions considerably reduce the complexity of the FE simulation. Otherwise, running nonlinear full transient analysis for determining the hysteretic shear behaviour of base isolators with different aspect ratios and wire configurations might not converge, especially at high shear strain amplitudes. In reality, a frictional force is generated between the wire and the hook in the contact area. In such a situation, the relative displacement between the wire and the hook will be limited or even fixed in the worst case. In a future work, more realistic comparisons can be performed by considering two types of contact including smooth (current case) and friction (real case). This further study can indicate how much the smooth contact assumption is close to the real case. In order to simplify the FE model, before analyzing the system, first, the strain generated in SMA wires at each pre-defined time step is calculated according to the geometry of the device and the arrangement of wires. In the next step, the axial stress in SMA wires can be determined form the stress-strain relationship of shape memory alloy based on the Auricchio’s superelasticity model (Auricchio, 2001). Since Auricchio’s model is utilized for -30-20-100102030-90 -60 -30 0 30 60 90Lateral Force (kN) Lateral Displacement (mm)         Exp.         FEM 121  SMAs in ANSYS, the superelastic behaviour of SMA wire is simulated using this model and by considering the properties of SMA obtained from experimental results (Tanaka et al., 2010). Here, we assume that the stress-strain hysteresis of FeNCATB SMA wire does not change by increasing the number of loading cycles. However, further experimental study is required in order to accurately simulate the dynamic behaviour of SMA wires and take into account the strain time history. Using the axial stress in SMA wires and the direction of wires at each time step, the force vectors exerted from the SMA wire to the hook are computed. The idealized stress-strain curve of FeNCATB (Tanaka et al., 2010) at room temperature is plotted in Figure  5.7. In such a situation, instead of modelling the SMA wires, the steel hooks, and the contact between them, the equivalent forces are applied to the rubber bearing at each time step while running the nonlinear transient analysis. In fact, the rubber bearing and the SMA wires are decoupled as two separate systems in FE simulations. Then, by measuring the force generated in SMA wires as a function of time, the effect of one system (SMA wires) is estimated on the other one (elastomeric isolator). Decoupled systems for the smart rubber bearing with cross SMA wires are depicted in Figure  5.8.  Figure ‎5.7. Idealized stress-strain curve of NiTi45 and FeNCATB SMAs at room temperature 020040060080010000 2 4 6 8 10 12 14 16Stress (MPa) Strain (%)          NiTi45          FeNCATB 122   Figure ‎5.8. Decoupled systems; (a) Elastomeric isolator, (b) SMA wires with internal forces Since SMAs have thermo-mechanical behaviour, both thermal and mechanical loadings affect the response of SMAs. Here, it is assumed that the environmental temperature does not change during cyclic loading. As a result, the coupling between the thermal and the mechanical loads can be neglected during an earthquake. However, the yield stress and consequently the hysteretic behaviour of an SMA wire operating at 30°C will be different from the response of the SMA wire which works at temperatures below 0°C. Therefore, the temperature at which the base isolator is operating plays an important role in the behaviour and performance of the device. The operational temperature (temperatures of the environment and SMA wires) is assumed to remain constant at 20°C. It should be mentioned that in order to investigate the effect of this parameter on the overall performance of the smart rubber bearing, further research needs to be conducted. 5.2.5 Results and Discussions By computing the hysteretic behaviour of SMA-NRBs subjected to a frequency of 0.2 Hz, and 6 MPa vertical pressure, the effect of the shear strain amplitude, the aspect ratio of rubber bearing, the configuration and thickness of wires, and the pre-strain in SMA wires have been assessed on the performance of the base isolator. In each case, four operational characteristics of the base isolator including the horizontal stiffness (KH), the residual deformation (RD) (i.e. a lateral displacement at which the shear force becomes zero when the rubber bearing is coming back to its initial position), the energy dissipation (energy dissipated per cycle, EDC), and the equivalent viscous damping (β) are calculated in order to compare the performances of NRBs with those of SMA-NRBs. The effective horizontal SMA Wire 2 SMA Wire 1 (a)                                                                   (b) 123  stiffness and the equivalent viscous damping are obtained according to Equations (‎3.10) and (‎4.3), respectively. Since in each configuration of wires (cross and straight), two aspect ratios (0.22 and 0.38) are considered for smart elastomeric isolators, in total, four cases are investigated (Table ‎5.8). The hysteretic shear response of each SMA-NRB is compared with that of a NRB with the same geometrical and mechanical properties under three different shear strain amplitudes: 100%, 150%, and 200%. As discussed in sections 5.2.2 and 5.2.3, although implementing SMA wires in the straight configuration is ineffective for NRB with R = 0.38 at shear strains higher than 150%, a comparison is performed between the straight and the cross configurations, with R = 0.38 and γ = 200%, in order to investigate the behaviour of SMA wires and the hysteretic shear response of SMA-NRB-S2.  Table ‎5.8. Four cases of SMA-based NRBs considered in the FE simulations Case Wire R Specimen 1 Cross 0.22 SMA-NRB-C1 2 Cross 0.38 SMA-NRB-C2 3 Straight 0.22 SMA-NRB-S1 4 Straight 0.38 SMA-NRB-S2 5.2.5.1 Low Aspect Ratio SMA-NRB Figure ‎5.9 shows the lateral hysteretic behaviour of NRB-1, SMA-NRB-C1 and SMA-NRB-S1. When SMA wires are used in the straight configuration, the response of the base isolator noticeably changes, whereas in the cross configuration these changes are negligible at 100% shear strain amplitude. Under a specific horizontal deflection, the strain generated in SMA wires with straight arrangement is much higher than that with the cross configuration. As a result, the axial stress in the straight wires is considerably increased and a larger force will be applied to the elastomeric isolator from the SMA wires with straight arrangement. This behaviour leads to significant changes in the lateral stiffness and the maximum horizontal force of SMA-NRB-S1 with straight wires. 124   Figure ‎5.9. Lateral force-deflection curve of (a) NRB-1, (b) SMA-NRB-C1, and (c) SMA-NRB-S1;  at γ = 100%, 150%, and 200% Operational characteristics of the natural and the smart rubber bearings are listed in Table ‎5.9. Changes in the performance of SMA-NRBs are also investigated by measuring the difference between each characteristic of the SMA-NRBs and that of the NRB. Each characteristic is calculated at three different shear strain amplitudes (100%, 150%, and 200%). At 100% shear strain, the effective horizontal stiffness of the NRB-1 with an aspect ratio of 0.22 is 0.86 kN/mm. When straight SMA wires are used, the horizontal stiffness increases to 1.95 kN/mm which is 125% higher than that of the NRB-1. On the other hand, SMA wires in the cross configuration (SMA-NRB-C1) increase the effective lateral stiffness to 9% at shear strain of 100%. A similar trend is observed for other shear strain amplitudes. However, the increase in the effective horizontal stiffness of SMA-NRB-S1 follows a regular pattern. At 100%, 150%, and 200% shear strains, the horizontal stiffness is increased by 125%, 131%, and 132%, respectively, compared to that of NRB-1. When the shear strain amplitude increases from 100% to 200%, the stiffness of SMA wires goes up due to the forward phase transformation. As a result, the stress generated in the wires noticeably increases and consequently, the rate of increase of the lateral stiffness of SMA-NRB-S1 rises from 125% to 131%. In terms of lateral flexibility, the SMA-NRB-C1 exhibited superior performance compared to SMA-NRB-S1.    -140-105-70-3503570105140-80 -40 0 40 80Lateral Force (kN) Lateral Displacement (mm) -140-105-70-3503570105140-80 -40 0 40 80Lateral Force (kN) Lateral Displacement (mm) -140-105-70-3503570105140-80 -40 0 40 80Lateral Force (kN) Lateral Displacement (mm) (a)                                               (b)                                                 (c) 125  Table ‎5.9. Operational characteristics of NRB and SMA-NRBs  for different wire configurations and aspect ratios  γ (%) NRB-1 SMA-NRB-S1  (ΔNRB)* SMA-NRB-C1  (ΔNRB) NRB-2 SMA-NRB-S2  (ΔNRB) SMA-NRB-C2  (ΔNRB) Horizontal Stiffness (kN/mm) 100 0.86 1.95 (125%) 0.94 (9%) 0.49 1.13 (129%) 0.71 (44%) 150 0.77 1.77 (131%) 0.91 (18%) 0.44 1.08 (147%) 0.63 (44%) 200 0.71 1.65 (132%) 0.87 (22%) 0.40 1.13 (183%) 0.58 (45%) Residual Deform. (mm) 100 4.8 3.2 (-33%) 4.4 (-9%) 8.5 5.7 (-33%) 7.3 (-14%) 150 6.3 3.9 (-38%) 5.2 (-18%) 11.1 6.7 (-40%) 9.2 (-17%) 200 10.2 5.4 (-47%) 8.7 (-15%) 18.1 16.1 (-11%) 13.7 (-24%) Dissipated Energy (kJ) 100 0.5 1.3 (167%) 0.5 (0%) 0.9 2.3 (174%) 1.2 (43%) 150 0.9 2.8 (203%) 0.9 (0%) 1.6 5.1 (220%) 2.7 (67%) 200 1.7 5.0 (186%) 2.1 (24%) 3.0 11.1 (268%) 4.8 (59%) Viscous Damping  (%) 100 7.0 8.3 (18%) 6.4 (-8%) 7.0 8.4 (19%) 6.9 (-1%) 150 6.5 8.5 (31%) 5.5 (-16%) 6.5 8.4 (30%) 7.5 (16%) 200 7.5 9.2 (23%) 7.6 (1%) 7.6 9.8 (30%) 8.3 (10%) *Difference between operational characteristics of SMA-NRBs and those of the NRB SMA wires can reduce the residual deformation of the NRBs due to their re-centring capability. The SMA-NRB-C1 reduces the residual deformation of the NRB by 9%, 18%, and 15% at 100%, 150%, and 200% shear strain amplitudes, respectively. While, the SMA-NRB-S1 decreases the residual deformation by 33%, 38%, and 47% at 100%, 150%, and 200% shear strain amplitudes, respectively. It shows that the straight configuration of wires is more efficient than the cross wires in reducing the plastic deformation of the rubber bearing after releasing the horizontal shear force. Due to the lower effective length of SMA wires in the straight configuration compared to that of cross wires, straight wires are subjected to a higher strain and as a result, the axial stress induced in these wires is higher. In terms of reducing the residual deformation, the SMA-NRB-S1 is more efficient compared to the SMA-NRB-C1. Another important characteristic of the NRB which is affected by SMA wires is the energy dissipation capacity. This specification is increased when SMA wires are used either in the form of straight or cross. In fact, the flag-shape hysteresis curve of a SMA wire under loading and unloading enlarges the shear hysteresis of the NRB and as a result, the area inside the lateral force-deflection curve which represents the dissipated energy by the elastomeric isolator is enhanced. The SMA-NRB-S1 increases the energy dissipated per 126  cycle more than 160% at all shear strain amplitudes. In contrast, the SMA-NRB-C1 does not change this parameter at shear strains of 100% and 150% while at 200% shear strain amplitude, the dissipated energy encounters a 24% increase. Changes in the equivalent viscous damping are different from the energy dissipation capacity. For the straight configuration, the maximum increase in the damping capacity occurs at 150% shear strain which is 31% more than that of the NRB. At γ = 200%, due to the large amount of the restored energy, the increase in the equivalent viscous damping is lower. In the SMA-NRB-C1, since at 100% and 150% shear strains, the energy dissipated per cycle does not change and the effective horizontal stiffness increases, the equivalent viscous damping decreases. 5.2.5.2 High Aspect Ratio SMA-NRB When the aspect ratio of NRB is increased by increasing the number of elastomeric layers, the elongation in SMA wires is enhanced for both straight and cross configurations. According to Table ‎5.4, for an aspect ratio of 0.38, the strain induced in straight and cross SMA wires reaches 23.3% and 7.2%, respectively, at 200% shear strain amplitude. It means that by increasing the amplitude of cyclic lateral displacement, the straight SMA wires will not remain in the superelastic range and as a result, they encounter a plastic deformation. In other words, when SMA-NRB goes back to its initial position (zero lateral displacement) after a half cycle, a residual strain is generated in the SMA wires. Based on the idealized stress-strain curve of FeNCATB subjected to strains more than 15% (Tanaka et al., 2010), changes of the stress in SMA wires can be estimated as a function of stress. When a maximum strain of 23% is applied to FeNCATB wires due to a cyclic loading, nearly 6.5% strain remains in the wires after a half cycle. This fact causes SMA-NRB-S2 to work ineffectively under large shear strains (e.g. more than 200%). Figure  5.10 represents the hysteretic behaviour of NRB-2, SMA-NRB-C2, and SMA-NRB-S2, respectively. At 200% shear strain, when the SMA-NRB-S2 (Figure  5.10c) goes back to its initial position from the maximum and minimum lateral displacements, it undergoes sudden changes in the lateral force in about 60 mm and -75 mm displacements, respectively. The reason is that the axial stress in the SMA wires reaches zero before finishing each half cycle due to a plastic deformation which is occurred in the wires. As a result, the force exerted from SMA wires to the rubber bearing is removed and consequently, the SMA-NRB-S2 follows the behaviour of 127  NRB-2 (see Figure  5.10a). The maximum shear force in SMA-NRB-S2 significantly increases by increasing the shear strain. It is because of a huge force transferred from the elongated SMA wires having much higher strain compared to wires in SMA-NRB-C2.   Figure ‎5.10. Lateral force-deflection curve of (a) NRB-2, (b) SMA-NRB-C2, and (c) SMA-NRB-S2;  at γ = 100%, 150%, and 200% Figure ‎5.11 to Figure ‎5.14 show the performance characteristics of NRBs and SMA-NRBs in the form of bar charts. By comparing the overall trend of changes in the effective horizontal stiffness, it can be observed that the NRBs and SMA-NRBs with higher aspect ratios are more flexible in the lateral direction. Similar to elastomeric isolators with 0.22 aspect ratios, when straight SMA wires are used in a NRB with an aspect ratio of 0.38 (SMA-NRB-S2), the increase in the horizontal stiffness is higher than a case in which cross SMA wires are implemented (SMA-NRB-C2). It indicates that the straight configuration is not an effective option for NRBs with high aspect ratios since they significantly stiffen the base isolator.   -150-100-50050100150-140 -70 0 70 140Lateral Force (kN) Lateral Displacement (mm) -150-100-50050100150-140 -70 0 70 140Lateral Force (kN) Lateral Displacement (mm) -150-100-50050100150-140 -70 0 70 140Lateral Force (kN) Lateral Displacement (mm) (a)                                                 (b)                                                 (c) 128   Figure ‎5.11. Effective horizontal stiffness of NRBs and SMA-NRBs with straight and cross configurations of wires; γ = 100%, 150%, and 200% The residual deformation of NRB-2 decreases about 33% and 40% for 100% and 150% shear strain amplitudes, respectively, when straight SMA wires are used (see Table ‎5.9). The reduction in the residual deformation of SMA-NRB-C2 at 100%, 150%, and 200% shear strains are 14%, 17% and 24%. As it is observed in Figure  5.12, SMA-NRB-S2 has superior performance in terms of residual deformation reduction. However at large shear strain amplitudes (200% and more), SMA wires encounter a plastic deformation in the straight configuration. Since the flag-shape hysteresis curve of FeNCATB wires noticeably shrinks by increasing the number of cyclic displacements, generating residual deformation in wires causes the SMA-NRB-S2 to have an inferior performance compared to SMA-NRB-C2.  Figure ‎5.12. Residual deformation of NRBs and SMA-NRBs with straight and cross configurations of wires; γ = 100%, 150%, and 200% 0.00.51.01.52.02.5100 150 200 100 150 200H. Stiffness (kN/mm) Shear Strain, γ (%) 05101520100 150 200 100 150 200Res. Deformation (mm) Shear Strain, γ (%)      NRB-1      SMA-NRB-S1      SMA-NRB-C1      NRB-2      SMA-NRB-S2      SMA-NRB-C2       NRB-1      SMA-NRB-S1      SMA-NRB-C1      NRB-2      SMA-NRB-S2      SMA-NRB-C2  129  According to Figure  5.13 and results listed in Table ‎5.9, straight SMA wires in SMA-NRB-S2 can noticeably increase the energy dissipation capacity of NRB-2 up to 268% at shear strain amplitude of 200%. The energy dissipated per cycle in the SMA-NRB-C2 is 1.2kJ, 2.7kJ, and 4.8kJ for 100%, 150% and 200% shear strains, respectively. The maximum enhancement in the energy dissipation capacity of SMA-NRB-C2 is 67%. When the equivalent viscous damping of SMA-NRB-S2 and SMA-NRB-C2 are compared together (Figure  5.14), it is observed that the straight wires are more efficient since SMA-NRB-S1 and SMA-NRB-S2 have higher damping capacity due to a greater strain and consequently a larger hysteresis curve in FeNCATB SMA wires. However, at 200% shear strain, SMA wires in the straight configuration are deformed plastically and exert a large amount force to the rubber bearing. As a result, SMA-NRB-C1 and SMA-NRB-C2 can be considered as better alternatives at high shear strains.  Figure ‎5.13. Dissipated energy of NRBs and SMA-NRBs with straight and cross configurations of wires; γ = 100%, 150%, and 200%  Figure ‎5.14. Equivalent viscous damping of NRBs and SMA-NRBs with straight and cross configurations of wires; γ = 100%, 150%, and 200% 024681012100 150 200 100 150 200Dissipated Energy (kJ) Shear Strain, γ (%) 024681012100 150 200 100 150 200Viscous Damping, β (%)  Shear Strain, γ (%)      NRB-1      SMA-NRB-S1      SMA-NRB-C1      NRB-2      SMA-NRB-S2      SMA-NRB-C2       NRB-1      SMA-NRB-S1      SMA-NRB-C1      NRB-2      SMA-NRB-S2      SMA-NRB-C2  130  5.2.5.3 Thickness of SMA Wires The thickness of SMA wire can affect the hysteretic behaviour of the SMA-NRB. When the radius of the wire’s cross section increases, the force exerted from wires to the base isolator is significantly enhanced. Among four considered cases (Table ‎5.8), the SMA-NRB-C2 in which cross SMA wires are implemented in a NRB with an aspect ratio of 0.38 is selected as a case study. Wires with three different radii; 1.25 mm, 2.5 mm and 4 mm are mounted on this rubber bearing. In order to investigate the effect of SMA wires’ thickness on the performance of the NRB, hysteretic behaviours of smart base isolators are evaluated and the effective horizontal stiffness and the energy dissipation capacity are compared to those of the NRB.  The hysteretic shear behaviours of SMA-NRB-C2 with different wire’s radii (1.25 mm, 2.5 mm, and 4 mm) are plotted in Figure  5.15. As can be seen, when a thicker SMA wire is used, the maximum shear force in smart elastomeric isolators is increased since the force generated in SMA wires increases with their thickness. It shows that the hysteretic shear response of SMA-NRB-C2 is highly dependent on the thickness of SMA wires which are installed with the cross configuration. Four characteristics including the effective horizontal stiffness, the residual deformation, the energy dissipation capacity, and the equivalent viscous damping are listed in Table ‎5.10. The lateral stiffness increases almost equally when the shear strain amplitude changes from 100% to 200%. However, by enhancing the radius of wire’s cross section from 1.25 mm to 4 mm, the effective horizontal stiffness, at 200% shear strain, increases from 12% to 112%.    Figure ‎5.15. Lateral force-deflection curve of SMA-NRB-C2 for different thicknesses of SMA wires; (a) rSMA = 1.25 mm, (b) rSMA = 2.5 mm, and (c) rSMA =  4 mm; γ = 100%, 150%, and 200% -120-80-4004080120-140 -70 0 70 140Lateral Force (kN) Lateral Displacement (mm) -120-80-4004080120-140 -70 0 70 140Lateral Force (kN) Lateral Displacement (mm) -120-80-4004080120-140 -70 0 70 140Lateral Force (kN) Lateral Displacement (mm) (a)                                                   (b)                                                 (c) 131  Table ‎5.10. Operational characteristics of SMA-NRB-C2 with different wire radii  compared to NRB-2  γ (%) NRB-2 SMA-NRB-C2 rSMA = 1.25mm  (ΔNRB)* rSMA = 2.5mm  (ΔNRB)* rSMA = 4.0mm  (ΔNRB)* Horizontal Stiffness (kN/mm) 100 0.49 0.55 (12%) 0.71 (44%) 1.04 (111%) 150 0.44 0.49 (12%) 0.63 (44%) 0.92 (111%) 200 0.40 0.45 (12%) 0.58 (45%) 0.85 (112%) Residual Deform. (mm) 100 8.5 8.2 (-3%) 7.3 (-14%) 6.0 (-29%) 150 11.1 10.6 (-4%) 9.2 (-17%) 7.1 (-36%) 200 18.1 16.5 (-9%) 13.7 (-24%) 10.1 (-44%) Dissipated Energy (kJ) 100 0.9 1.0 (13%) 1.2 (43%) 1.8 (105%) 150 1.6 1.9 (22%) 2.7 (67%) 4.1 (160%) 200 3.0 3.6 (18%) 4.8 (59%) 7.4 (144%) Viscous Damping  (%) 100 7.0 7.0 (0%) 6.9 (-1%) 6.8 (-3%) 150 6.5 7.1 (9%) 7.5 (16%) 8.0 (23%) 200 7.6 7.9 (5%) 8.3 (10%) 8.7 (15%) *Difference between operational characteristics of SMA-NRB-C2s and those of the NRB-2 As it is expected, SMA wires with 4 mm radius of cross section, decreases the residual deformation of NRB-2 more than that of wires with lower thickness. At 200% shear strain, the residual deformation reduces by 9%, 24%, and 44% for SMA-NRBs with 1.25 mm, 2 mm, and 4 mm wire’s radii, respectively. Although the maximum energy dissipation capacity for SMA-NRB-C2 with different thickness of wires is achieved at 200% shear strain amplitudes, the maximum changes in the dissipated energy occur at 150% shear strain amplitude when it is compared to NRB-2. For example in the case of SMA-NRB-C2 with 1.25 mm wires (Figure  5.15a), the dissipated energy at 150% shear strain is 1.9 kJ which is 22% of the dissipated energy in NRB-2 while, SMA-NRB-C2 can dissipates the energy by 3.6 kJ at 200% shear strain which is 18% of the NRB-2. It shows that the rate of increasing the energy dissipation capacity is not constant when the shear strain amplitude increases. The trend of changes in the equivalent viscous damping for shear strains of 150% and 200% is similar to the dissipated energy. For SMA-NRB-C2 with 2.5 mm and 4 mm SMA wires subjected to 100% shear strain amplitude (Figure  5.15b and c), although both the dissipated and the restored energies in SMA-NRB-C2 increase, changes in the equivalent viscous damping is negative compared to that of the 132  NRB-2. The reason is that the rate of changes of the restored energy is more than that of the dissipated energy due to a significant increase in the horizontal stiffness and subsequently, the ratio of the restored energy to the dissipated energy for SMA-NRB-C2 is lower than that of the NRB-2. When the lateral cyclic displacement increases to 150% of the total thickness of rubber layers in SMA-NRB-C2 with 2.5 mm and 4 mm wires, the energy dissipation capacity enhances by 67% and 160%, respectively, and the equivalent viscous damping increases by 16% and 23%, respectively. According to Figure  5.16 which demonstrates the changes trend in four operational characteristics of SMA-NRBs, increasing the thickness of SMA wires causes a reduction in the lateral flexibility and the residual deformation, and an increase in the energy dissipation capacity. The equivalent viscous damping is enhanced by increasing the thickness at shear strain amplitudes greater than 150%. Therefore, using SMA wires with a higher thickness improves the performance of the NRB-2 in terms of residual deformation, energy dissipation capacity, and equivalent viscous damping. Since the lateral flexibility significantly decreases by increasing the thickness of wires, the radius of SMA wires’ cross section should be designed for a required lateral stiffness.          133      Figure ‎5.16. Operational characteristics of NRB-2 and SMA-NRB-C2 with different wires’ thickness; (a) Effective horizontal stiffness, (b) Residual deformation, (c) Dissipated energy per cycle, (d) Equivalent viscous damping  5.2.5.4 Pre-strain in SMA wires The yield strength of the ferrous SMA, FeNCATB, which represents the starting stress in the forward phase transformation from austenite to martensite, is aroud 750 MPa (Figure  5.7). The rate of increasing the stress from 0 to 750 MPa which denotes the initial elastic stiffness in the austenite phase is about 47 GPa. The large amount of yield stress, corresponds to 1.6% strain, induces a large normal force in SMA wires and consequently, a large lateral force is exerted to the NRB in the opposite direction of the cyclic horizontal displacement. In such a situation, the lateral flexibility is noticeably reduced and as a result, the effective horizontal stiffness is significantly increased compared to a NRB. In order to efficiently use the SMA wire as a damper in the rubber bearing, the initial elastic part of ferrous shape memory alloy should be removed by pre-straining the SMA wire (Choi et al., 2005). In the pre-strained SMA wire, the forward phase transformation occurs at a lower strain and as a result, the yield stress considerably decreases. Subsequently, when the SMA 0.00.20.40.60.81.01.2100 150 200H. Stiffness (kN/mm) Shear Strain, γ (%) NRBr = 1.25mmr = 2.5mmr = 4mm05101520100 150 200Res. Deformation (mm) Shear Strain, γ (%) NRBr = 1.25mmr = 2.5mmr = 4mm012345678100 150 200Dissipated Energy (kJ) Shear Strain, γ (%) NRBr = 1.25mmr = 2.5mmr = 4mm0246810100 150 200Viscouse Damping, β (%) Shear Strain, γ (%) NRBr = 1.25mmr = 2.5mmr = 4mm(c)                                                                               (d) (a)                                                                            (b) 134  wire is elongated and subjected to a tension due to the cyclic displacement of the rubber bearing, a smaller force will be transferred to the superstructure. By applying a pre-strain (e.g. 2%) to the SMA wire, the stress-strain curve shifts according to Figure  5.17a.  Figure ‎5.17. Stress-strain curve of ferrous SMA (FeNCATB);  (a) regular (non-pre-strained) wire, (b) 2% pre-strained wire Figure  5.17b describes the stress-strain curve of the SMA wire with a 2% pre-strain. In pre-strained wires, the strain at the completion of forward phase transformation (8%) and the maximum strain in the fully martensite phase (13%) correspond to stresses of 620 MPa and 780 MPa, respectively, which are much lower than those of the regular (non-pre-strained) SMA wire (820 MPa and 980 MPa, respectively). Thus, pre-strained SMA wires have a lower effect on the lateral stiffening of the NRB which can be considered as a desirable effect. In order to investigate the effect of pre-straining on the performance of NRBs, SMA wires with different amount of pre-strain are used in SMA-NRB-C2. Figure  5.18 depicts the hysteresis shear behaviour of SMA-NRB-C2s with 0%, 2%, and 4% pre-strained wires. All elastomeric isolators are subjected to three shear strain amplitudes (100%, 150%, and 200%) with horizontal frequency of 0.2 Hz and 6 MPa vertical pressure. As can be seen in Figure  5.18, the maximum shear force in SMA-NRB-C2 is reduced when pre-strained wires are used. The reason is that the stress generated in the SMA wires decreases when they are installed with a pre-strain due to a shift in the stress-strain curve. Consequently, a lower amount of force is exerted to the base isolator from pre-strained SMA wires. 020040060080010000 2 4 6 8 10 12 14 16Stress (MPa)  Strain (%) 020040060080010000 2 4 6 8 10 12 14Stress (MPa) Strain (%) (a)                                                                             (b)  135   Figure ‎5.18. Lateral force-deflection curve of SMA-NRB-C2 for different amounts of pre-strain in SMA wires; (a) ε0 = 0%, (b) ε0 = 2%, and (c) ε0 = 4%; γ = 100%, 150%, and 200% When 2% pre-strained SMA wires are used, the effective horizontal stiffness of the rubber bearing decreases compared to the SMA-NRB-C2 with 0% pre-strain. According to Table ‎5.11, by comparing the horizontal stiffnesses at 200% shear strain amplitude, it is observed that 45% increase for SMA-NRB-C2 with ε0 = 0 reduces to 32% increase for SMA-NRB-C2 with ε0 = 2%. However, increasing the amount of pre-strain from 2% to 4% has negligible effect on the lateral flexibility of the SMA-NRB-C2. 2% pre-strained SMA wires can considerably reduce the large yield stress though, when the amount of pre-strain increases to 4%, the hysteresis curve of SMA slightly changes compared to 2% pre-strain. In fact, when the pre-strain is greater than the strain corresponding to the forward phase transformation (1.6%), if it increases, the yield and the maximum stresses will not noticeably change. Therefore, the alteration of operational characteristics in SMA-NRB-C2 with 4% pre-strained wires will be negligible. Since the maximum axial stress generated in 4% pre-strained SMA wires is more than that in 2% pre-strained wires at 200% shear strain, the increase in the lateral stiffness of the SMA-NRB-C2 with ε0 = 4% at this shear strain is more (see Table ‎5.11).      -80-60-40-20020406080-140 -70 0 70 140Lateral Force (kN) Lateral Displacement (mm) -80-60-40-20020406080-140 -70 0 70 140Lateral Force (kN) Lateral Displacement (mm) -80-60-40-20020406080-140 -70 0 70 140Lateral Force (kN) Lateral Displacement (mm) (a)                                                 (b)                                                 (c) 136  Table ‎5.11. Operational characteristics of NRB-2 and SMA-NRB-C2  for different amounts of pre-strain  γ (%) NRB-2 SMA-NRB-C2 ε0 = 0%  (ΔNRB)* ε0 = 2%  (ΔNRB)* ε0 = 4%  (ΔNRB)* Horizontal Stiffness (kN/mm) 100 0.49 0.71 (44%) 0.63 (28%) 0.63 (28%) 150 0.44 0.63 (44%) 0.57 (30%) 0.57 (30%) 200 0.40 0.58 (45%) 0.53 (32%) 0.55 (37%) Residual Deform. (mm) 100 8.5 7.3 (-14%) 8.2 (-4%) 8.3 (-2%) 150 11.1 9.2 (-17%) 10.1 (-9%) 10.3 (-7%) 200 18.1 13.7 (-24%) 15.4 (-15%) 15.3 (-16%) Dissipated Energy (kJ) 100 0.9 1.23 (43%) 1.26 (47%) 1.27 (48%) 150 1.6 2.65 (67%) 2.60 (64%) 2.63 (65%) 200 3.0 4.81 (59%) 5.26 (74%) 4.85 (60%) Viscous Damping  (%) 100 7.0 6.9 (-1%) 8.0 (15%) 8.1 (16%) 150 6.5 7.5 (16%) 8.2 (26%) 8.3 (27%) 200 7.6 8.3 (10%) 10.0 (32%) 8.8 (17%) *Difference between operational characteristics of SMA-NRB-C2s and those of the NRB-2 Although pre-straining SMA wires can increases the lateral flexibility as an advantage, it has negative effect on the reduction of the residual deformation of the SMA-NRB-C2 due to a decrease in the maximum stress induced in wires. For 100%, 150%, and 200% shear strain amplitudes, SMA-NRB-C2 with 0% pre-strain decreases the residual deformation of NRB-2 by 14%, 17%, and 24%, respectively, while, the SMA-NRB-C2 with 2% pre-strain decreases this specification by 4%, 9%, and 15%, respectively.  Figure  5.19a, b, c, and d show changes in the horizontal stiffness, the residual deformation, the energy dissipation capacity and the equivalent viscous damping of the NRB-2 and SMA-NRB-C2s by changing the pre-strain level, respectively. It can be observed that at shear strain amplitudes equal and smaller than 150%, the performance of smart NRBs with different pre-strains are almost the same in terms of the dissipated energy. However, SMA-NRB-C2s with pre-strained wires are more efficient in terms of the equivalent viscous damping since their restored elastic energy is lower than that of the SMA-NRB-C2 with 0% pre-strain. Moreover, at 200% shear strain, SMA-NRB-C2 with ε0 = 2% has a higher energy dissipation capacity and equivalent viscous damping. Since the maximum shear force in the 4% pre-strained SMA-NRB-C2 is larger than that in the 2% pre-strained SMA-NRB-C2 at 137  200% shear strain, the lateral stiffness is increased. As a result, the equivalent viscous damping of the 4% pre-strained SMA-NRB-C2 decreases according to Equation (‎4.3). When the performance of SMA-NRB-C2s with different pre-strains are compared together, it is observed that the 2% pre-strained SMA-NRB-C2 is more efficient than the other smart NRBs in terms of the lateral flexibility, the energy dissipation capacity and the equivalent viscous damping.     Figure ‎5.19. Operational characteristics of NRB-2 and SMA-NRB-C2s with different amounts of  pre-strain in SMA wires; (a) effective horizontal stiffness, (b) residual deformation,  (c) dissipated energy per cycle, (d) equivalent viscous damping  5.3 SMA-based High Damping Rubber Bearing (SMA-HDRB) Similar to wire configurations proposed for SMA-NRB, two arrangements of wires (straight and cross) are applied to HDRB reinforced with CFRP composite. The effect of different factors, such as the type of SMA and the aspect ratio of bearing are studied on the performance of SMA-HDRBs. In this regard, seven CFR-HDRB are considered with different aspect ratios I. Their geometrical properties are listed in Table ‎5.12.  0.00.20.40.60.8100 150 200H. Stiffness (kN/mm) Shear Strain, γ (%) NRB0% Pre-Strain2% Pre-Strain4% Pre-Strain05101520100 150 200Res. Deformation (mm) Shear Strain, γ (%) NRB0% Pre-Strain2% Pre-Strain4% Pre-Strain0123456100 150 200Dissipated Energy (kJ) Shear Strain, γ (%) NRB0% Pre-Strain2% Pre-Strain4% Pre-Strain024681012100 150 200Viscous Damping, β (%) Shear Strain, γ (%) NRB0% Pre-Strain2% Pre-Strain4% Pre-Strain(a)                                                                              (b) (c)                                                                              (d) 138  Table ‎5.12. Geometrical properties of CFR-HDRBs CFR-HDRB #1 #2 #3 #4 #5 #6 #7 L (mm) 250 250 250 250 250 250 250 W (mm) 250 250 250 250 250 250 250 H (mm) 60 70 80 90 100 110 120 ts (mm) 15 15 15 15 15 15 15 te (mm) 4.74 4.73 4.72 4.72 4.71 4.71 4.71 tf (mm) 0.31 0.31 0.31 0.31 0.31 0.31 0.31 ne 6 8 10 12 14 16 18 nf 5 7 9 11 13 15 17 R 0.12 0.16 0.20 0.24 0.28 0.32 0.36 The thickness of the elastomer layer, te, is kept constant (4.7 mm in all cases), while the number of HDR layers, ne, is increased from 6 to 18.  5.3.1 SMA-HDRB Equipped with Straight Wires As shown in Figure  5.2a, in the straight configuration, two continuous SMA wires are wound in two opposite sides of the rubber bearing. In the straight configuration, the total length of the SMA wires (LSMA) required at each aspect ratio is presented in Table ‎5.13.  Table ‎5.13. Required length of SMA wire for seven CFR-HDRBs  in the straight and cross configurations CFR-HDRB #1 #2 #3 #4 #5 #6 #7 R 0.12 0.16 0.20 0.24 0.28 0.32 0.36 LSMA (mm) (straight) 1240 1280 1320 1360 1400 1440 1480 LSMA (mm) (cross) 2252.8 2262.7 2275.4 2290.9 2308.9 2329.6 2352.9 The strain in the SMA wires (εSMA) is a function of the shear strain amplitude, γ, and the aspect ratio, R. When the shear strain increases from 50% to 200%, the strain generated in SMA wires increases. The SMA strain also goes up by enhancing the aspect ratio of the base isolator. Figure  5.20a shows the variation of SMA wire strain by increasing the shear strain amplitude and the aspect ratio of the rubber bearing. At 200% shear strain amplitude, the SMA wires in all of the base isolators experience a strain greater than 10%. It can be also observed that the strain induced in SMA wires will not exceed 10% when the base isolator is subjected to shear strains lower than 100%. However, at larger shear strains, the SMA wire in the base isolator with high aspect ratio can experience as high as 27% strain. 139     Figure ‎5.20. Variation of strain in SMA wire as a function of shear strain amplitude and aspect ratio for (a) straight configuration and (b) cross configuration 5.3.2 SMA-HDRB equipped with Cross Wires In the cross configuration proposed in section 5.2.2, two continuous SMA wires are wrapped around the rubber bearing diagonally (see Figure  5.2b).  The total length of wires needed for this arrangement is presented in Table ‎5.13. Although a larger length of SMA wire is required for this configuration relative to the straight one, the generated strain in wires due to the lateral deflection of rubber bearing is much lower.  Figure  5.20 shows that in the case of cross configuration there is a significant reduction in the SMA wire strain compared to the straight configuration. At 200% shear strain amplitude with an aspect ratio of 0.36, the maximum strain induced in the SMA wires does not exceed 9%. It shows that unlike straight configuration in which SMAs can operate in a limited range of γ and R, different types of SMA can be used in the cross arrangement. Figure  5.20b illustrates that at each shear strain level, the strain in the SMA wires reaches its maximum value when the smart elastomeric isolator has the maximum considered aspect ratio. 5.3.3 Efficiency of Wires In order to investigate the efficiencies of different types of SMAs (see Table ‎5.5) in the cross and straight configurations, two aspect ratios, 0.12 and 0.36, and four shear strain amplitudes, 50%, 100%, 150%, and 200%, are considered. According to Table ‎5.14, when the aspect ratio is 0.12, the maximum strain in the SMA wire, εSMA, at 200% shear strain is 0510152025300.12 0.18 0.24 0.3 0.36SMA Strain (%) Aspect Ratio, R          50%          100%          150%          200% 02468100.12 0.18 0.24 0.3 0.36SMA Strain (%) Aspect Ratio, R 50% 100% 150% 200% (a)                                                               (b) 140  1% for the cross configuration and 11.1% for the straight configuration. At same shear strain level, when the aspect ratio increases to 0.36, the SMA strain in the cross and straight configurations reaches 8.8% and 27.5%, respectively. Since most of the SMAs have superelastic strain lower than 6% (Table ‎5.5), SMA wires with the straight configuration cannot operate in a superelastic range at large shear strain amplitudes, especially in high-aspect-ratio elastomeric isolators. Whereas, the cross SMA wires can operate within an elastic range at large lateral cyclic displacements (150% and 200% shear strains). This fact demonstrates the effectiveness of the cross configuration over the straight one.  Table ‎5.14. Strain in SMA wires of SMA-HDRBs with different wire configurations and aspect ratios  Symbol Wire Configuration  γ (%) R 50 100 150 200 εSMA (%) SMA-HDRB-C1 Cross 0.12 0.1 0.3 0.6 1.0 SMA-HDRB-S1 Straight 1.0 3.7 7.2 11.1 SMA-HDRB-C2 Cross 0.36 0.6 2.2 5.0 8.8 SMA-HDRB-S2 Straight 2.6 9.1 17.8 27.5 Table ‎5.15 presents the effectiveness of various types of SMA wires in cross configuration applied to the rubber bearing with two different aspect ratios. For a rubber bearing with a 0.12 aspect ratio, the maximum induced strain in wires with cross configuration is about 1% (see Table ‎5.14). However, in the case of straight arrangement, the strain in SMA wires exceeds the superelastic strain for most of the SMAs at shear strains higher than 150%. Another point is that, if a smart elastomeric base isolator with an aspect ratio of 0.36 is subjected to 200% shear strain, only FeNCATB (with about 13.5% superelastic strain range) can be utilized. Table ‎5.15. Operational range of SMAs for different shear strains and aspect ratios in cross configuration  Ni Ti Ni Ti45  Ti Ni40 Cu10 Cu Al Be Fe Mn Al Ni FeNCATB         R γ (%) ≤0.24 0.36 ≤0.24 0.36 ≤0.24 0.36 ≤0.24 0.36 ≤0.24 0.36 ≤0.24 0.36 ≤ 100             125             150             175             200             141  Based on the fact that the superelastic effect of SMA wires occurs at temperatures above the austenite finish temperature, the austenite finish temperature of the SMA wires should be lower than the ambient temperature. Since the minimum ambient temperature in countries with cold climatic conditions may go below -20°C, the austenite finish temperature of the SMA wire should be lower than this temperature. Therefore, NiTi45 and FeNCATB, with respective Af values of -10°C and -62°C, are chosen for SMA-HDRBs. 5.3.4 Finite Element Modelling FE modelling is performed by considering appropriate element types, material properties, geometry, mesh, and boundary conditions (load and displacement). Element SOLID185 with eight nodes and three degrees of freedom (translation in x, y, and z directions) at each node is selected for both reinforcement and elastomeric layers. Hyperelasticity, stress stiffening, large deflection, and large strain can be modelled using this element. In order to simulate the large deformation of rubber layers at large shear strain amplitudes, the large-deflection effect is considered in full transient analyses. Steel shim used in HDRB is modelled as an isotropic material with a Young’s modulus of 210 GPa and a Poisson’s ratio of 0.3. Material properties of carbon fibre-reinforced sheets (carbon/epoxy), which behave as an orthotropic material, are the same as CFRP composite used in Chapter 3 (see Table ‎3.2). Mooney-Rivlin hyperelastic model is combined with Prony viscoelastic model in order to simulate the behaviour of HDR. As described in Chapter 3, section 3.2.6, material constants of both Mooney-Rivlin and Prony models are determined from experimental data gathered from the uniaxial tension-compression tests, the biaxial tension test, and the creep test conducted on HDR (Ibrahim, 2005). The material constants of hyper-viscoelastic model are listed in Table ‎5.16 for a 9-parameter Mooney-Rivlin model and a 2-parameter Prony model. Table ‎5.16. Hyper-viscoelastic material model constants Mooney-Rivlin Model Prony Model C10 = 1.192 C12 = 0.000 α1 = 0.765 C01 = 0.547 C21 = -0.013 τ1 = 0.124 C11 = -0.038 C03 = 0.000 α 2 = 0.061 C02 = 0.047 C30 = 0.002 τ 2 = 65.82 C20 = 0.108   142  It should be noted that the FE model of HDR, which has been validated and verified in Chapter 3, section 3.3, is used here.  In generating the FE model of SMA-HDRB, same as SMA-NRB, a method of superposition is implemented in order to simplify the system by decoupling the rubber bearing and SMA wires. Here, a smooth contact with no friction is assumed between the steel hook and the SMA wire. The frictional force generated between the wire and the hook in the contact area can be minimized by using steel hooks with a very smooth surface and minimizing the contact area between wire and hook. Figure  5.21 shows a schematic of such a contact surface. Lubricating the contact surface can also reduce the friction generated due to back and forth movements of SMA wire inside the hook.  Figure ‎5.21. Schematic of steel hook and SMA wires in contact; (a) loose, (b) tight Instead of modelling steel hooks and the contact between the hooks and continuous SMA wires, exerted forces to the elastomeric isolator due to SMA wires are considered. This assumption considerably reduces the complexity of the FE simulation. Otherwise, nonlinear full transient analyses for determining the hysteretic shear behaviour of base isolators with different aspect ratios and wire configurations might not converge, especially at high shear strain levels. In order to simplify the FE model, before analyzing the system, the strain generated in SMA wires at each pre-defined time step is calculated according to the geometry of the device and the arrangement of wires. Rubber bearings have deformations along three axes: x, y, and z. Here, it is assumed that the effects of torsion, rotation about all axes, delamination due to the shear deformation in laminated pad, and vibration in the vertical direction are ignored. Based on these simplifications, the formula of SMA strain in two cases is provided; (1) strain induced by displacements in x, y, and z directions and (2) strain induced by a displacement in the x direction. In the latter case, the effect of vertical Steel Hook SMA Wire (a)                                                         (b) 143  displacement, ΔZ, due a constant compressive load is assumed to be negligible on the SMA strain since the vertical deflection is very small compared to the lateral displacements in the xy plane.        Figure ‎5.22. SMA-HDRB with a cross arrangement of wires subjected to displacements in x, y, and z directions; (a) 3D view, (b) orthographic views (top, front, and side views)  Figure  5.22 shows the rubber bearing equipped with cross SMA wires and subjected to a combination of vertical displacement in the z direction and lateral displacement in the xy plane. According to this figure, the strain in the wire is calculated using Equation (‎5.3). In this equation, the initial length, L0, remains constant (Equation (‎5.4)) while, length L changes over the time since it is a function of displacements, ΔX, and ΔY (Equation (‎5.5)). It should (a)      (b) 144  be noted that ΔZ has a constant value due to a constant vertical pressure applied on the isolator. 𝜀𝑆𝑀𝐴 =𝐿𝑆𝑀𝐴 − 𝐿0 𝑆𝑀𝐴 𝐿0 𝑆𝑀𝐴 (‎5.3) 𝐿0 𝑆𝑀𝐴 = 2(√(𝐿 + 2𝑙𝑒)2 +𝐻𝑒𝑓𝑓2 +√(𝑊 + 2𝑤𝑒)2 +𝐻𝑒𝑓𝑓2 ) (‎5.4) 𝐿𝑆𝑀𝐴 = √(𝐿 + 2𝑙𝑒 + Δ𝑋)2 + Δ𝑌2 + (𝐻𝑒𝑓𝑓 − Δ𝑍)2+√(𝐿 + 2𝑙𝑒 − Δ𝑋)2 + Δ𝑌2 + (𝐻𝑒𝑓𝑓 − Δ𝑍)2+√Δ𝑋2 + (𝑊 + 2𝑤𝑒 + Δ𝑌)2 + (𝐻𝑒𝑓𝑓 − Δ𝑍)2+√Δ𝑋2 + (𝑊 + 2𝑤𝑒 − Δ𝑌)2 + (𝐻𝑒𝑓𝑓 − Δ𝑍)2 (‎5.5) If the displacement is in one direction (x), considering aforementioned assumptions, Equation (‎5.5) will be simplified to Equation (‎5.6). 𝐿𝑆𝑀𝐴 = √(𝐿 + 2𝑙𝑒 + Δ𝑋)2 +𝐻𝑒𝑓𝑓2 +√(𝐿 + 2𝑙𝑒 − Δ𝑋)2 +𝐻𝑒𝑓𝑓2+ 2√Δ𝑋2 + (𝑊 + 2𝑤𝑒)2 +𝐻𝑒𝑓𝑓2 (‎5.6) It should be noted that, for decoupling the effect of SMA wires from the rubber bearing in a general case, the effect of displacement in 3D on SMA strains can be calculated in order to find out the force in wires and add to the response of rubber bearing. It is assumed that isolators are subjected to a cyclic lateral displacement only in the x direction. The axial stress in SMA wires can be determined from the strain based on the constitutive model of SMA (Auricchio, 2001). The idealized stress-strain curve for FeNCATB and NiTi45 are plotted in Figure  5.7. Then, using the value of the axial stress and the direction of wires at each time step, the force vectors (with x, y, and z components) exerted from the SMA wire to the hooks are computed. Since the SMA wires with cross arrangement (wrapped around the elastomeric bearing) has a 3-D form, at each corner where the SMA wire is in contact with the steel hook, the generated force along the wires has components in all three directions of x, y, and z. In the decoupled system, all of these components are taken into account for FE simulations.  145  5.3.5 Results and Discussions Based on the nonlinear material model of HDR, which has been verified with the experimental tests (Dall’Asta and Ragni, 2006), finite element analyses (FEAs) have been performed to simulate the behaviour of the SMA-HDRB using ANSYS (ANSYS Mechanical APDL, Release 14.0). By calculating the hysteretic response of SMA-HDRBs subjected to a frequency of 0.2 Hz and 6 MPa vertical pressure, the influence of the shear strain amplitude, the aspect ratio of rubber bearing, the types of SMA, the wire arrangement, the thickness of SMA wires, and the pre-strain in SMA wires on the performance of base isolators have been assessed. 5.3.5.1 Fe- and NiTi-based SMA wires FeNCATB and NiTi45 wires having cross configurations are used in SMA-HDRBs with aspect ratios of 0.12 (SMA-HDRB-C1). Both types of SMA wires have the same cross sectional areas (19.6 mm2). Figure  5.23 shows lateral force-deflection curves of SMA-HDRB-C1s consisting of different types of SMA at different shear strain amplitudes. According to Table ‎5.14, at 200% shear strain, the strain in SMA wires reaches 1.0%. As shown in Figure  5.7, the strain at which the forward phase transformation is started in FeNCATB and NiTi45 wires is 1.6% and 0.6%, respectively. Consequently, when the SMA-HDRB-C1 equipped with NiTi45 is subjected to 200% shear strain, wires enter the phase transformation region (see Figure  5.7). The maximum shear force in smart rubber bearings at γ = 200% is 141 kN and 147 kN when FeNCATB and NiTI45 wires are used, respectively. The reason is that NiTi45 has a higher elastic stiffness (in austenite phase) compared to FeNCATB and as a result, a higher axial stress is generated in NiTi45 wire for same amount of strain in wires. However, the performances of SMA-HDRB-C1s are very similar when ferrous and NiTi-based SMA wires are implemented. This fact can be understood by comparing the effective horizontal stiffnesses and residual deformations of SMA-HDRB-C1s according to Table ‎5.17. This consistency in results is due to a low amount of strain in wires (1%), which is much smaller than the superelastic strain for both types of SMA wires. In other words, if the wire strain increases by increasing the aspect ratio of rubber bearing or changing the wire configuration, the behaviours of SMA-HDRBs will vary. 146   Figure ‎5.23. Hysteresis curves of SMA-HDRB-C1; (a) FeNCATB SMA, (b) NiTi45 SMA  (γ = 50%, 100%, 150%, 200%) Table ‎5.17. Effective horizontal stiffness and residual deformation of SMA-HDRB-C1s  with FeNCATB and NiTi45 wires  FeNCATB NiTi45 γ (%) KH (kN/mm) ΔKH  (%) R.D. (mm) ΔRD (%) KH (kN/mm) ΔKH (%) R.D. (mm) ΔRD  (%) 50 3.89 0 4.3 -11 3.89 0 4 -10.9 100 2.93 0 10.3 -12 2.94 0 10 -11.5 150 2.60 0 18.1 -14 2.61 0 18 -13.4 200 2.49 0 29.5 -16 2.58 0 34 -14.6 Note: Δ is the relative difference between performance characteristics of SMA-HDRBs and CFR-HDRB 5.3.5.2 Low Aspect Ratio SMA-HDRB Figure  5.24 shows the lateral force-deflection curves of the CFR-HDRB with two aspect ratios of 0.12 and 0.36 at four different shear strain amplitudes. Figure  5.25 depicts the hysteresis loops of low-aspect-ratio SMA-HDRBs (R = 0.12), which are equipped with ferrous SMA wires in the cross configuration (SMA-HDRB-C1) and the straight configuration (SMA-HDRB-S1). Wires in both arrangements have the same cross sectional areas (19.6 mm2). -250-150-5050150250-80 -40 0 40 80Lateral Force (kN) Lateral Displacement (mm) -250-150-5050150250-80 -40 0 40 80Lateral Force (kN) Lateral Displacement (mm) (a)                                                                             (b) 147   Figure ‎5.24. Hysteresis curves of CFR-HDRB; (a) R = 0.12, (b) R = 0.36  (γ = 50%, 100%, 150%, 200%)  Figure ‎5.25. Hysteresis curves of (a) SMA-HDRB-C1, (b) SMA-HDRB-S1  (γ = 50%, 100%, 150%, 200%) By comparing the hysteretic behaviours of CFR-HDRB (Figure ‎5.24a) and SMA-HDRB-C1 and SMA-HDRB-S1 (Figure  5.25), it is observed that the effective horizontal stiffness and the residual deformation change due to the use of SMA wires. When the stress along the SMA wires increases, a forward phase transformation from austenite to martensite occurs in the wires. As a result, the horizontal component of the force exerted to the rubber bearing noticeably goes up in the opposite direction of the lateral cyclic displacements. This increase in the lateral force causes horizontal stiffening and re-centring in the whole system. Therefore, the effective horizontal stiffness increases and the residual deformation decreases.  The strain induced in the SMA wires in the straight configuration increases with a very high rate from 1.0% at 50% shear strain to 11.1% at 200% shear strain (Table ‎5.14). As a result, the stress along the SMA wires increases rapidly. In such a situation, using -250-150-5050150250-80 -40 0 40 80Lateral Force (kN) Lateral Displacement (mm) -250-150-5050150250-200 -100 0 100 200Lateral Force (kN) Lateral Displacement (mm) -250-150-5050150250-80 -40 0 40 80Lateral Force (kN) Lateral Displacement (mm) -250-1250125250-80 -40 0 40 80Lateral Force (kN) Lateral Displacement (mm) (a)                                                                             (b) (a)                                                                             (b) 148  FeNCATB wires instead of NiTi45 wires is advantageous since FeNCATB has a higher superelastic strain range (13.5%). Table ‎5.18 and Table ‎5.19 demonstrate the changes in the effective horizontal stiffness and the residual deformation, respectively, by changing the arrangement of wires for the aspect ratio of 0.12. In the cross configuration (SMA-HDRB-C1), slight changes in the stiffness and the residual deformation are observed compared to performance characteristics of the CFR-HDRB. On the other hand, when SMA wires are installed in the straight configuration (SMA-HDRB-S1), the horizontal stiffness increases and the residual deformation considerably decreases. This is because of a large amount of stress generated in SMA wires due to the higher elongation compared to the wires with cross arrangement.  Table ‎5.18. Effective horizontal stiffness of CFR-HDRB and SMA-HDRBs (R = 0.12)  CFR-HDRB SMA-HDRB-C1 SMA-HDRB-S1 γ (%) KH  (kN/mm) KH (kN/mm) ΔKH (%) KH (kN/mm) ΔKH (%) 50 3.93 3.89 0 5.03 28 100 2.95 2.93 0 4.41 50 150 2.61 2.60 0 3.80 46 200 2.48 2.49 0 3.56 43 Note: Δ is the relative difference between effective horizontal  stiffnesses of SMA-HDRBs and CFR-HDRB Table ‎5.19. Residual deformation of CFR-HDRB and SMA-HDRBs (R = 0.12)  CFR-HDRB SMA-HDRB-C1 SMA-HDRB-S1 γ (%) R.D. (mm) R.D. (mm) ΔRD (%) R.D. (mm) ΔRD (%) 50 4.8 4.3 -11 3.1 -36 100 11.6 10.3 -11 8.5 -27 150 21.1 18.1 -14 13.0 -38 200 35.2 29.5 -16 18.3 -48 Note: Δ is the relative difference between residual deformations of SMA-HDRBs and CFR-HDRB As can be seen in Figure  5.25, Table ‎5.18, and Table ‎5.19, at 200% shear strain, although the phase transformation in the FeNCATB wires is not completed, a high amount of yield stress, defined as the starting stress for the forward phase transformation, causes a significant growth of 43% in the effective horizontal stiffness and a high reduction of 48% in 149  the residual deformation. It is also observed that, for rubber bearings with low aspect ratio (0.12), the performance of SMA-HDRB-S1 is superior to that of the SMA-HDRB-C1. 5.3.5.3 High Aspect Ratio SMA-HDRB Figure  5.26b depicts the lateral force-deflection curves of SMA-HDRBs with an aspect ratio of 0.36 and two wires arrangements. Similar to the previous cases, SMA wires used in the cross and the straight configurations have equal cross sectional area (19.6 mm2). In the cross configuration (SMA-HDRB-C2), for shear strain amplitudes up to 150%, the maximum strain induced in the SMA wires is 5% which is lower than the superelastic strain range of FeNCATB. In the SMA-HDRB-S2, when the aspect ratio is 0.36, the strain in the SMA wire increases from 2.6% to 27.5% with increasing the shear strain from 50% to 200%. At 150% and 200% shear strain levels, the strain generated in the straight FeNCATB wires is higher than the superelastic strain (13.5%). As a result, FeNCATB SMA wires undergo a plastic deformation. Since the plastic deformation substantially degrade the performance of SMA wires subjected to cyclic loadings, the lateral force-deflection curves of SMA-HDRB-S2 are plotted for the first cycle at 150% and 200% shear strains.  Figure ‎5.26. Hysteresis curves of (a) SMA-HDRB-C2, (b) SMA-HDRB-S2  (γ = 50%, 100%, 150%, 200%) By comparing the hysteresis curves of two high-aspect-ratio SMA-HDRBs (Figure  5.26), it is observed that the effective horizontal stiffness significantly increases when the SMA wires are mounted in the straight arrangement. In order to investigate the effect of SMA wires subjected to strains higher than superelastic strain range, the hysteretic shear behaviour of SMA-HDRB-S2 is re-plotted at shear strain of 150% in Figure  5.27. As -400-2000200400-200 -100 0 100 200Lateral Force (kN) Lateral Displacement (mm) -400-2000200400-200 -100 0 100 200Lateral Force (kN) Lateral Displacement (mm)  (a)                                                                              (b) 150  can be seen in Figure  5.27, the lateral force does not follow the path indicated by dashed lines. SMA-HDRB-S2 experiences a rapid change in the lateral force at around -35 mm lateral displacement. Considering a complete cycle of lateral displacement, if SMA wires undergo a plastic deformation due to exceeding the superelastic strain, the deformation will remain in the wires when the bearing returns to its initial position in the first half cycle and as a result, the wires get loose at zero lateral displacement. Thus, in the second half cycle, by increasing the lateral displacement, SMA wires will not be effective up to a level of lateral displacement (-35 mm) at which the wires are fitted in their place. In this situation, the strain in SMA wires starts to increase from the plastic strain previously generated and not from zero. When the rubber bearing returns to its initial position in the second half cycle, the stress in SMA wires reaches zero before the lateral displacement reaches zero (at about -50 mm). Therefore, compared to a case in which no plastic deformation happens in SMA wires, the shear force in the base isolator decreases. The shear behaviour of SMA-HDRB-S2 at 200% shear strain can be interpreted similarly.   Figure ‎5.27. Hysteresis curves of SMA-HDRB-S2 (γ = 150%) Knowing the frequency of horizontal cyclic loading, and using Equation (‎5.3), which provides a relation between the lateral displacement and the strain in SMA wires, the exact time and the lateral displacement level at which the strain in SMA wires exceeds the superelastic strain limit can be specified. Subsequently, the time and the displacement at which the axial stress in wires vanishes while the rubber bearing is returning to its initial position can be determined from the stress-strain curve of SMA (Figure  5.7). -300-200-1000100200300-150 -100 -50 0 50 100 150Lateral Force (kN) Lateral Displacement (mm) 151  The maximum lateral forces in SMA-HDRB-C2 and SMA-HDRB-S2 at 200% shear strain are 153 kN and 320 kN, respectively. At this shear strain, the straight wires undergo 27.5% strain, which is much higher than the superelastic strain range of FeNCATB. After finishing the forward phase transformation in SMA wires, at strains above 15%, the stress increases rapidly with a rate equal to the martensitic modulus of elasticity. As a result, a significant amount of stress is generated in SMA wires when the strain reaches 27.5%, and consequently, the lateral stiffness of the base isolator considerably increases. Table ‎5.20 and Table ‎5.21 compare the performance characteristics of CFR-HDRB, SMA-HDRB-C2, and SMA-HDRB-S2 at different shear strain levels. At 50% shear strain, the effective horizontal stiffness of CFR-HDRB increases by 4% and 46%, respectively, when cross and straight wires are used. At the maximum considered shear strain (200%), the effective horizontal stiffnesses of SMA-HDRB-C2 and SMA-HDRB-S2 are 16% and 142% greater than that of CFR-HDRB, respectively. Table ‎5.20. Effective horizontal stiffness of CFR-HDRB and SMA-HDRB (R = 0.36)  CFR-HDRB SMA-HDRB-C2 SMA-HDRB-S2 γ (%) KH  (kN/mm) KH (kN/mm) ΔKH (%) KH (kN/mm) ΔKH (%) 50 1.30 1.36 4 1.89 46 100 0.96 1.14 18 1.52 58 150 0.85 1.00 18 1.70 99 200 0.78 0.90 16 1.89 142 Note: Δ is the relative difference between effective horizontal  stiffnesses of SMA-HDRBs and CFR-HDRB Table ‎5.21. Residual deformation of CFR-HDRB and SMA-HDRBs (R = 0.36)  CFR-HDRB SMA-HDRB-C2 SMA-HDRB-S2 γ (%) R.D. (mm) R.D. (mm) ΔRD  (%) R.D. (mm) ΔRD  (%) 50 14.9 12.4 -17 12.7 -15 100 36.4 33.0 -9 24.7 -32 150 69.2 54.9 -21 60.8 -12 200 109.4 86.6 -22 116.3 6 Note: Δ is the relative difference between residual deformations of SMA-HDRBs and CFR-HDRB  The maximum reduction in the residual deformation of SMA-HDRB-C2 is 32%, which happens in the straight configuration at 100% shear strain (Table ‎5.21). Under this 152  condition, the effective horizontal stiffness increases about 58% (Table ‎5.20). Since SMA wires undergo plastic deformations at 150% and 200% shear strains, they cannot be fully recovered after unloading. As a result, their performance in improving the re-centring capability of rubber bearing is degraded. In the cross configuration, when the residual deformation decreases by 22% at a shear strain of 200%, the effective horizontal stiffness increases by 16%. This fact shows that although the straight configuration of SMA wires can reduce the residual deformation of rubber bearing more than the cross configuration, it has an inferior performance in terms of the lateral flexibility. 5.3.5.4 Thickness of SMA Wires The dimension of SMA wire can affect the behaviour of smart rubber bearing subjected to a compressive and a cyclic shear loading. When the radius of the wire’s cross section increases, the effective force exerted to the base isolator is significantly enhanced. As a case study, the cross FeNCATB wires with two different radii; 2.5 mm and 5.0 mm are chosen to be installed in a CFR-HDRB with an aspect ratio of 0.12. The corresponding cross sectional areas of SMA wires are 19.6 mm2 and 78.5 mm2. In reality, for the second case, two 2.5 mm wires can be used rather than a thick wire with a 5 mm radius. In order to investigate the effect of SMA wires’ thickness on the performance of the rubber bearing, hysteretic behaviours of smart base isolators are compared (Figure  5.28).   Figure ‎5.28. Hysteresis curves of Ferrous SMA-HDRB-C1; (a) rSMA = 2.5 mm, (b) rSMA = 5 mm  (γ = 50%, 100%, 150%, and 200%) -200-1000100200-80 -40 0 40 80Lateral Force (kN) Lateral Displacement (mm) -200-1000100200-80 -40 0 40 80Lateral Force (kN) Lateral Displacement (mm)  (a)                                                                            (b) 153  Compared to CFR-HDRB, changes in the effective horizontal stiffness and the residual deformation of SMA-HDRBs are calculated with increasing the radius of SMA wires and shear strain amplitude (Table ‎5.22). When 2.5 mm SMA wires are used in a HDRB subjected to a 200% shear strain, the effective horizontal stiffness does not change while the residual deformation decreases about 16%. When 5.0 mm SMA wires are incorporated, the effective horizontal stiffness increases by 8% and the residual deformation decreases by 34%. This shows that using an SMA wire with a higher thickness improves the performance of the HDRB in terms of residual deformation reduction. The reason of this behaviour is the increase of the force generated in the SMA wires due to an increase in the cross sectional area of wires. Table ‎5.22. The effective horizontal stiffness and the residual deformation of SMA-HDRB-C1 compared to those of the CFR-HDRB for different radii of SMA wire (R = 0.12)  Effective Horizontal Stiffness (%) Residual Deformation (%) r (mm) γ (%) 2.5 5 2.5 5 50 -0.9 -0.3 -11 -12 100 -0.7 2.0 -12 -16 150 -0.2 5.1 -14 -24 200 0.0 8.0 -16 -34 5.3.5.5 Pre-Strain in SMA Wires In order to investigate the effect of pre-straining on the performance of smart elastomeric bearings, ferrous SMA wires with 2% pre-strain and a cross sectional area of 19.6 mm2 are mounted on a HDRB with an aspect ratio of 0.36 in cross configuration. When SMA wires are wrapped around the rubber bearing by passing through the steel hooks bolted to the steel end plates, both ends of the wire are attached to a slotted hexagonal head bolt which is mounted in a small box (housing). According to Figure  5.29a, ends of the SMA wire are passed through a hole located in the middle of the bolt (see Figure  5.29b). In order to fix the wire from sliding inside the bolt, before applying a pre-stress, wire is wound around the bolt. To apply a pre-strain to SMA wires, the bolt is rotated and as a result, wires are subjected to a tensile stress. Accordingly, a specific amount of pre-strain is generated in the stretched wires (e.g. 2% pre-strain). The advantage of the proposed mechanism is its high accuracy in adjusting the level of pre-strain by accurately tightening the bolt. In fact, the 154  level of pre-strain can be controlled by considering the perimeter of the screw and the number of rotation applied to it.    Figure ‎5.29. Adjustable mechanism for fixing the SMA wire and applying pre-strain;  (a) side view of the mechanism, (b) 3D view of slotted hexagonal head bolt with a hole in the middle  Figure  5.30a depicts the hysteresis shear behaviour of a CFR-HDRB, an SMA-HDRB without pre-strain (SMA-HDRB), and an SMA-HDRB with 2% pre-strain. The areas of cross section of SMA wires are the same (19.6 mm2). All rubber bearings are subjected to 200% shear strain and 6 MPa vertical pressure.   Figure ‎5.30. Lateral force-displacement curves of (a) CFR-HDRB, SMA-HDRB-C2, and  SMA-HDRB-C2 with 2% pre-strain, (b) SMA-HDRB-C2 with 2% and 3% pre-strains (γ = 200%) When 2% pre-strained SMA wires are used, the decrease in the residual deformation is greater and the increase in the effective horizontal stiffness is lower compared to those of -200-1000100200-200 -100 0 100 200Lateral Force (kN) Lateral Displacement (mm) HDRBSMA-RBSMA-RB (2% Pre-Strain) -200-1000100200-200 -100 0 100 200Lateral Force (kN) Lateral Displacement (mm) 2% Pre-Strain3% Pre-StrainHousing SMA Wire SMA  Wire Slotted Hexagonal Head Bolt Hole (a)                                      (b) (a)                                                                              (b) 155  the non-pre-strained SMA-HDRB. These changes show that pre-strained SMA wires can improve the efficiency of the smart rubber bearing more than the regular SMA wires can do. Changes in the effective horizontal stiffness and the residual deformation are listed in Table ‎5.23. The effective horizontal stiffness of the SMA-HDRB-C2 is 16% more than that of the HDRB reinforced with CFRP composite plates. When 2% pre-strained SMA wires are used, the effective horizontal stiffness increases to 0.87 kN/mm which is 14% higher than that of CFR-HDRB. This shows that the reduction in the lateral flexibility of SMA-HDRB can be controlled by pre-straining SMA wires. The reason for this behaviour is the lower amount of stress induced in the pre-strained SMA wires after the completion of the forward phase transformation. On the other hand, compared to the CFR-HDRB, the residual deformations of the SMA-HDRB and 2% pre-strained SMA-HDRB are reduced by 22% and 8%, respectively. When the 2% pre-strained SMA wires are elongated due to the horizontal cyclic displacement, the forward phase transformation in the wires starts and finishes at lower stress and strain levels compared to a case in which regular SMA wires are used. Therefore, a smart HDRB with pre-strained SMA wires cannot reduce the residual deformation of CFR-HDRB as much as SMA-HDRB with regular SMA wires. Table ‎5.23. Characteristics of CFR-HDRB and SMA-HDRB-C2 for different amounts of pre-strain  in SMA wires (R = 0.36, γ = 200%)  Rubber Bearing Horizontal Stiffness Residual Deformation KH (kN/mm) ΔKH (%) R.D. (mm) ΔRD  (%) CFR-HDRB 0.78  109.4  SMA-HDRB 0.90 16% 85.8 -22% SMA-HDRB (2% Pre-Strain) 0.86 14% 100.3 -8% SMA-HDRB (3% Pre-Strain) 0.89 15% 101.1 -8% Note: Δ is the relative difference in operational characteristics of SMA-HDRB-C2 and CFR-HDRB  In order to evaluate how the amount of pre-strain in SMA wires affect the performance of the smart rubber bearing, considering same size of cross section, SMA wires with 3% pre-strain are used in the SMA-HDRB. The hysteretic behaviour and performance characteristics of the SMA-HDRB with 3% pre-strained wires are compared to those of the SMA-HDRB with 2% pre-strained SMA wires. 156  Figure  5.30b demonstrates that changes in the hysteretic response of SMA-HDRBs are negligible when the magnitude of pre-strain increases in the SMA wire. According to Table ‎5.23, the increase in the effective horizontal stiffness of the SMA-HDRB with 3% pre-strained wires is slightly higher than that of the 2% pre-strained SMA-HDRB. Under the maximum lateral displacement, a higher strain is generated in the 3% pre-strained SMA wires compared to the 2% pre-strained wires. As a result, the maximum stress at which SMA wires are fully martensitic will be larger in the SMA wires with 3% pre-strain. Therefore, a higher lateral force is applied to the rubber bearing from the 3% pre-strained wires and consequently, the horizontal stiffness of the 3% pre-strained SMA-HDRB increases by 15%. Since the stresses of starting and finishing phase transformation in the 2% pre-strained SMA wire are close to those of the 3% pre-strained wire, the performances of SMA-HDRBs with 2% and 3% pre-strains in reducing the residual deformation of the CFR-HDRB are the same (see Table ‎5.23). 5.4 SMA-based Lead Rubber Bearings (SMA-LRB) Based on the model proposed for smart NRBs using SMA wires with the cross configuration (Hedayati Dezfuli and Alam, 2013c), similar idea can be applied to LRBs with rectangular cross section. As shown in Figure  5.31, double cross SMA wires are wrapped around the bearing by passing through steel hooks connected to the steel supporting end plates. The difference between this configuration (double cross configuration) and the one proposed in sections 5.2 and 5.3 (cross configuration) is that a symmetric arrangement with a larger amount of wire is used here. As a result, the effective stain in SMA wires generated due to the shear strain in the LRB decreases since wires have a higher initial length.  157              Figure ‎5.31. SMA-LRB; (a) decoupled systems, (b) integrated SMA-LRB 5.4.1 Finite Element Validation In order to establish a valid, reliable, and accurate simulation for the response of SMA-LRB using FEM, the LRB should be validated with experimental tests. In this regard, the LRB experimentally tested by Abe et al. (2004) is modelled and then analyzed in ANSYS (ANSYS Mechanical APDL, Release 14.0). Then, the numerical results are compared to the experiment at different shear strain levels, γ, (50% and 150%). Figure  5.32 depicts the side and the top views of LRB adapted from (Abe et al., 2004).  Figure ‎5.32. LRB used in the experimental tests; (a) side view, (b) top view (dimensions are in mm) (adapted from (Abe et al., 2004)) Defining element types, determining material models and mechanical properties, creating the geometry, meshing the model, defining the boundary and loading conditions as well as contact areas are steps that should be followed before solving and extracting the results in the FEM. The most challenging part is determining the material model since the Double Cross SMA Wires  (DC-SMAW) Lead Rubber Bearing  (LRB) 40 210 200 360 48.8 92.8 22 22 70.8 (a)                                                              (b) (a)                                                                                            (b) 158  responses of LRB and SMA-LRB are nonlinear and highly dependent on the materials behaviour. Homogeneous structural element, SOLID185, with eight nodes and three degrees of freedom (translation in x, y, and z directions) at each node and with capability of modelling hyperelasticity, stress stiffening, large deflection, and large strain is chosen for steel shims, supporting plates, and rubber layers. Since elastomeric layers may undergo a large deformation at large shear strain levels, the large-deflection effect is considered in transient analyses. Reinforcement and supporting end plates are made of mild steel which is modelled as an isotropic material with an elastic modulus of 210 GPa, a yield strength of 247 MPa, and a Poisson’s ratio of 0.3. Among different types of nonlinear material models available in ANSYS (ANSYS Mechanical APDL, Release 14.0), hyper-viscoelastic model could correctly capture the highly nonlinear behaviour of rubber materials under a combination of normal and shear deformations (Hedayati Dezfuli and Alam, 2012 and 2013a). In this regard, the Mooney-Rivlin hyperelastic model is combined with the Prony viscoelastic model and attributed to the elastomer. The material constants of the hyper-viscoelastic model are given in Table ‎5.24.  Table ‎5.24. Material constants of the hyper-viscoelastic model Mooney-Rivlin Model Prony Model C10 = 0.232 α1 = 1.035 C01 = 0.107 τ1 = 0.101 C11 = -0.0004 α2 = 0.061   τ2 = 65.82 After creating the geometry of LRB according to the sizes indicated in Figure  5.32, mapped meshing is applied in order to discretize the whole model into a finite numbers of SOLID185 elements with a regular pattern. This method of meshing significantly decreases the processing time and accordingly shorten the converging time. Nodes at the bottom of the lower supporting end plate are completely fixed. All the nodes at the top of upper supporting plate move together in the vertical direction (z direction), since it is assumed that supporting steel plates and steel shims are rigid compared to the rubber layers which are flexible in the vertical and horizontal directions. Considering the geometry, loading conditions (combination of a uniform vertical compressive load and cyclic lateral displacements in the x 159  direction), as well as boundary conditions, the model has one plane of symmetry (plane xz) as shown in Figure  5.33.   Figure ‎5.33. LRB modelled in ANSYS software In order to noticeably decrease the processing time in transient analyses, the model is divided into two halves with respect to the plane of symmetry and just a half of the model is analyzed. In this regard, the symmetry boundary condition is applied to all nodes located on the plane of symmetry according to Figure ‎5.33.  Here, it is assumed that rubber layers are perfectly bonded to steel shims and steel fixing plates and also there is a perfect bonding between the supporting end plates and fixing plates. Another assumption is that the lead core is glued neither to the elastomer nor to the reinforcement. In this regard, contact areas are defined between the lead and other materials that surround the core. The contact pair consists of two surfaces; target and contact. The contact surface is the one that moves toward and potentially in contact with the target surface. The contact element type is considered to be surface-to-surface. Element CONTA173 with 4 nodes at each corner is used to defined a deformable contact surface which is located on the surface of 3D solid elements, SOLID185. Geometric characteristics of this element is the same as those of SOLID185. Contact occurs when the contact element surface penetrates into one of the target segment elements (TARGE170) on a specified target surface (ANSYS Documentation, Release 14.0). The target surface is discretized by a set of segment elements, TARGE170, and is paired with the associated contact surface.  The hysteresis curves (i.e. shear force versus shear strain) are obtained at 50% and 150% shear strain amplitudes (cyclic loadings in the x direction), a vertical pressure of 7.84 MPa, and a lateral frequency of 0.01 Hz (Abe et al. 2004). Figure ‎5.34 shows a good match between FE simulations and experimental tests. However, a small difference is observed z x y Plane of Symmetry 160  between numerical and experimental results because of the nonlinear material properties defined for elastomeric layers and the boundary condition (contact area) defined for the lead core. The shear strain is defined as the ratio of the lateral displacement to the total thickness of rubber layers. The maximum difference between the peak shear forces obtained through experiment and FEM is 9% at 50% shear strain and 14% at 150% shear strain. In terms of the energy dissipated per cycle, the maximum variation of two approaches is 3% and 1% for shear strains of 50% and 150%, respectively.  Figure ‎5.34. Hysteretic shear response of LRB at 50% and 150% shear strains obtained through FEM and experimental tests conducted by Abe et al. (2004) 5.4.2 Performance of SMA-LRB After validating the FE model through experimental tests, the numerical simulation can be extended to other cases in which shear strain varies while the material properties and the boundary conditions are kept unchanged. In order to model the SMA-LRB in ANSYS, SMA wires are decoupled from the LRB (superposition method) and the FE model of LRB is used. Here, it should be noted that the simplification of symmetry boundary condition considered for LRB does not raise any problem in the FE analysis of SMA-based LRB since the double cross configuration of SMA wires proposed in this study also has the same plane of symmetry (see Figure  5.33). In all simulations, it is assumed that the rubber bearings are subjected to a combination of vertical pressure and uniaxial horizontal cyclic displacements (in the x direction). Figure  5.35 demonstrates the half model of SMA-LRB.  -50-2502550-200 -100 0 100 200Laterla Force (kN) Shear Strain (%) ExperimentFEM161   Figure ‎5.35. Half model of LRB equipped with double cross SMA wires Same as what was discussed and assumed in section 5.2.4.2 for analyzing the SMA-NRB, the procedure of decoupling SMA wires from the LRB, which is depicted in Figure  5.36, is implemented here. In this approach, after defining the load pattern of cyclic lateral displacements, x(t), the loading frequency, fH, and the time increment, Δt, the length of SMA wire is calculated as a function of lateral displacement, X = x(t), and size of LRB (Length, L, Width, W, and Height, H) at each time step. Knowing the initial length of wire, L0, the strain in SMA wires, εSMA is computed and accordingly, the stress in wires is found based on the SMA constitutive model (stress-strain relation). In the next step, the axial force along the SMA wire having a circular cross section with a radius of rSMA is calculated as a function of time. Then, considering the angles that the SMA wires make with respect to principal directions (x, y, and z) at each node (A, B, C …), components of the force (nodal force) are computed at each time step. Finally, the SMA wire-based LRB can be decoupled to two systems: LRB and nodal forces generated due to SMA wires as shown in the last step of the flowchart in Figure  5.36. z x y Lead Core SMA Wires Elastomer Supporting End Plate Steel Fixing Plate 162   Figure ‎5.36. Procedure of decoupling SMA wires from LRB After analyzing the decoupled system through a full transient analysis, the variation of lateral force can be calculated over the time when the shear strain increases from 50% to 200%. Figure  5.37 shows the hysteretic shear responses of LRB and SMA-LRBs with two different SMA wire’s radii; 1.5 and 2.5 mm at different shear strain levels. In SMA-LRBs, ferrous SMA (FeNiCoAlTaB) with 13.5% superelastic strain (Tanaka et al., 2010) is used. This ferrous SMA has outstanding characteristics which make it superior to NiTi. It shows a superelastic strain (over 13%) which is almost double that of NiTi. Having the capability of exceptional cold and hot workability, it has a capacity of reaching 800 MPa in the Input  Load Pattern, X = x (t) Loading Frequency, fH Time Increment, Δt Calculate LSMA = f (X, L, W, H)   Calculate Stress, σSMA, from SMA Constitutive Model Stress, σ  Strain, ε  FSMA = σSMA πr2 = F(t) Determine Components of Force, FSMA, at each Node FA x, FA y, FA z … Finite Element Modelling of Decoupled System LRB + Nodal Forces due to SMA Wires +ΔX -ΔX z x y  X (mm) Time (s) Example + 163  superelastic range. Ferrous SMA can be used along with magnetically activated materials and sensors (e.g. magnetic sensors) due to its ferromagnetic property. As a result, it can be implemented into new applications where such features are necessary such as construction, general manufacturing, and precision machinery (Tanaka et al., 2010; Omori et al., 2011).  At low shear strains (e.g. 50%), the behaviour of SMA-LRB is almost the same as that of the LRB for both wire’s thicknesses (rSMA = 1.5 and 2.5 mm). The reason is that at γ = 50%, SMA wires are not activated yet since no transformation happens from austenite to martensite. When the shear strain goes above 100%, the effect of the flag-shaped hysteresis of SMA can be observed on the overall behaviour of SMA-LRBs. The superelastic effect of this material (i.e. capability of returning back to its initial shape after a large deformation (e.g. up to 15%)) causes an improvement in the re-centring capability of smart rubber bearings. As observed in Figure  5.37, the effective lateral stiffness and energy dissipation capacity (i.e. the area inside the hysteresis loop) of the SMA-LRB increases and the residual deformation decreases at shear strains equal to and greater than 100% compared to those of LRB. When SMA wires with a higher radius (rSMA = 2.5 mm) are used, the shear force noticeably increases as a result of an increase in the axial force in SMA wires, FSMA. Therefore, compared to wires with lower radius, the superelastic effect of SMA is augmented on the response of LRB (see Figure  5.37b).   Figure ‎5.37. Shear force-strain hysteresis curves for LRB and SMA-LRB subjected to different  shear strains (50%, 100%, 150%, and 200%); (a) rSMA = 1.5 mm, (b) rSMA = 2.5 mm SMA wires with 1.5 mm radius increase the effective horizontal stiffness of LRB by 38% and 48% at shear strains of 100% and 200%, repectively. While, using wires with 2.5 -150-100-50050100150-250 -150 -50 50 150 250Shear Force (kN) Shear Strain (%)        LRB        SMA-LRB -150-100-50050100150-250 -150 -50 50 150 250Shear Force (kN) Shear Strain (%)       LRB       SMA-LRB (a)                                                                              (b) 164  mm radius leads to 108% and 133% increase in the effective horizontal stiffness at 100% and 200% shear strains, repectively. Changes of the peak shear force due to implementing SMA wires is same as that of the lateral stiffness. In terms of energy dissipation, SMA wires have significant effect on LRB since at 100% and 200% shear strains they can increase the energy dissipated per cycle of LRB by 12% and 24%, respectively, when the radius of wires is 1.5 mm and by 34% and 74%, respectively, when the radius of wires is 2.5 mm. 5.4.2.1 Pre-Strain in SMA Wires SMA has a high initial elastic modulus in the austenite phase which increases the effective horizontal stiffness of the LRB before starting forward phase transformation (austenite to martensite). In other words, the lateral flexibility of the bearing significantly decreases before activating SMA wires. Since the unique characteristics of SMAs improve the behaviour of LRB when SMA is activated and generates a large hysteresis upon high-amplitude loadings, it will be highly beneficial if the flag-shaped hysteresis area of SMA enlarges and the effect of high initial elastic modulus is diminished. In order to achieve this goal, SMA wire goes through a process by which a specific amount of pre-strain is generated in the wire. In this process the SMA wire is first loaded up to a strain level greater than the defined pre-strain, ε0, and then unloaded to ε0. By generating the pre-strain, a positive initial stress called pre-stress produces in the wire as well.  In Figure  5.38 showing the pre-straining process indicated by dotted lines, the hysteretic behaviours of a regular and a 3% pre-strained SMA wire are compared to each other. In the first case (Figure  5.38a), the wire is elongated up to a strain of 1.84% due to 100% shear strain in the SMA-LRB while in the second case (Figure  5.38b), wire experiences 4.10% strain when the SMA-LRB is subjected to 150% shear strain amplitude. As can be observed, in both cases, the 3% pre-strained wire has a larger hysteresis (indicated by solid lines) compared to non-pre-strained SMA wire (indicated by dashed lines). The reason is that by pre-straining the SMA wire, it partially transforms to the martensite phase and as a result, when the stress increases and the strain goes above the pre-strain level, the forward phase transformation starts and completes sooner compared to a non-pre-strained wire.    165    Figure ‎5.38. Stress-strain behaviours of non-pre-strained and 3% pre-strained SMA wires  in SMA-LRB subjected to (a) 100% shear strain and (b) 150% shear strain  In order to explore the effect of pre-strained SMA wires on the behaviour of SMA-LRB and compare the performance of regular and pre-strained wires, hysteretic shear responses of LRB and smart LRB equipped with non-pre-strained wires (SMA-LRB) and pre-strained wires (SMA-LRB (PS)) are obtained. Same as previous part, the radius of SMA wire’s cross section in the smart elastomeric isolators is considered to be 2.5 mm. The shear force-strain hysteresis curves of LRB and SMA-based LRBs are plotted in Figure  5.39 at two different strain levels; 100% and 150%.  Figure ‎5.39. Shear force-strain hysteresis curves of LRB, SMA-LRB, and SMA-LRB (PS) (ε0 = 3%) under (a) 100% shear strain and (b) 150% shear strain  Shear hysteretic response of SMA-LRB (PS) demonstrates that using pre-strained SMA wires improves the behaviour of SMA-LRB in terms of residual deformation reduction. 02004006008000 1 2 3 4 5 6 7 8σ (MPa) ε (%) Non-Pre-Strained SMAPre-Strained SMAPre-Straining Path02004006008000 1 2 3 4 5 6 7 8σ (MPa) ε (%) Non-Pre-Strained SMAPre-Strained SMAPre-Straining Path-120-80-4004080120-160 -80 0 80 160Shear Force (kN) Shear Strain (%) LRBSMA-LRBSMA-LRB (PS)-120-80-4004080120-160 -80 0 80 160Shear Force (kN) Shear Strain (%) LRBSMA-LRBSMA-LRB (PS)(a)                                                                              (b) (a)                                                                             (b) 166  in order to quantitatively compare SMA-LRB with and without pre-strain effect, the residual deformations (R.D.) of both bearings are compared at 100% and 150% shear strains in Table ‎5.25. According to the results, when pre-strained wires are used, the residual deformation of SMA-LRB is reduced by around 19% and 15% at shear strain amplitudes of 100% and 150%, respectively. Table ‎5.25. Residual deformations of SMA-LRB and SMA-LRB (PS) at different shear strains   Rubber Bearing Shear Strain, γ 100% 150% R.D. (mm) Δ R.D. (mm) Δ SMA-LRB 11.9  12.4  SMA-LRB (PS) 9.7 -18.6% 10.6 -15.1% By comparing the energy dissipation capacity of smart rubber bearings, negligible difference is observed between SMA-LRB and SMA-LRB (PS). The reason is that although the energy dissipation (hysteresis area) of the pre-strained SMA is larger than that of the non-pre-strained SMA (see Figure  5.38), the double cross configuration causes the variation of the resultant force of pre-strained wires in the x direction (i.e. algebraic summation of x-component of nodal forces at A, B, C, … according to Figure  5.36) to be almost same as that of regular wires.  Therefore, it is understood that on one hand, applying pre-strain to SMA wires improves the re-centring capability of SMA-LRB and on the other hand, increasing the energy dissipation capacity of the SMA-LRB by generating pre-strain in wires is highly dependent on the arrangement of wires mounted on the LRB. As discussed in section 5.2.5.4 (SMA-NRB) and section 5.3.5.5 (SMA-HDRB), applying pre-strain to the cross SMA wires could reduce the effective lateral stiffness, however, could not improve the re-centring property of SMA-RBs.  5.5 Design of SMA-based Rubber Bearings NiTi45 wire, in the straight arrangement, undergoes a large strain which is beyond the superelastic strain range (6.8%) at large shear strain amplitudes (Table ‎5.14). Therefore, to study the effect of SMA type, ferrous (FeNCATB) and NiTi-based (NiTi45) wires are 167  considered to be installed in low-aspect-ratio CFR-HDRBs (R = 0.12) with cross arrangement. Ferrous FeNCATB wire, as a more efficient option compared to NiTi45, is chosen to be implemented in all four cases (see Table ‎5.14). The hysteretic shear behaviour of each SMA-HDRB is compared to that of a CFR-HDRB with the same geometrical and mechanical properties under four different shear strain levels: 50%, 100%, 150%, and 200%.  Increasing the diameter of SMA wires increases the effective horizontal stiffness and the total energy dissipation capacity of the base isolation system. On the other hand, increasing the amount of pre-strain in SMA wires causes the horizontal stiffness to be decreased. Since increasing the lateral flexibility and maximizing the damping capacity of a rubber bearing are both beneficial, it is more advantageous to reach the target horizontal stiffness by changing both the diameter and pre-strain of SMA wires rather than just altering the diameter of wires. In a design procedure, in order to determine the diameter and pre-strain of wires, first, a target value for the effective horizontal stiffness (KH d) of elastomeric isolator is determined. Then, for a specific size of base isolator, the diameter of SMA wires increases from an initial value. Based on the geometry of rubber bearing and the configuration of SMA wires (cross), a minimum initial diameter (rw0) is obtained from a force ratio (RF), defined as a ratio of the maximum lateral force generated by SMA wires (FSMA max) to the maximum shear force of the isolator with no SMA wires (Fs max) at 100% shear strain amplitude. The force ratio is set to a value at which the effect of SMA wires can be observed on the hysteretic shear behaviour of the base isolator (here considered 10%). In fact, knowing the maximum strain in SMA wire at γ = 100% from Equation (‎5.3), and accordingly the maximum axial stress in wire, σmax, the initial radius of SMA wire, rw0, can be obtained from FSMA max calculated using the force ratio. The stiffness margin, MK, is defined in order to determine the relative difference between the calculated effective lateral stiffness of SMA-based rubber bearing, KH, and the target value of stiffness, KH d (initially considered 15%). Under this condition, the radius of wires should increase up to a level at which KH is 15% higher than the target value. In this step, by fixing the radius, pre-strain in SMA wires goes up and the effective lateral stiffness is calculated. Considering pre-strain values lower than 5%, if the reduction in shear force due to pre-straining the SMA wires causes the lateral stiffness to be reduced and reaches the target point, with a 10% error, the 168  values of radius and pre-strain are selected and the design procedure is finished. Otherwise, the stiffness ratio decreases by 5%, the radius of wire and the pre-strain are set to their initial values (rw0 and 0, respectively), and then, the effective horizontal stiffness is recalculated again. The flow chart of the design process is shown in Figure  5.40.  Figure ‎5.40. Flow chart of design procedure to determine the diameter and pre-strain of SMA wires Yes No No Yes Yes Set a target for effective horizontal stiffness, 𝐾𝐻𝑑  𝑅𝐹 =𝐹𝑆𝑀𝐴𝑚𝑎𝑥 𝐹𝑠𝑚𝑎𝑥 𝑀௄= 0.15  𝑟𝑤 𝑖+1 = 1.2 𝑟𝑤 𝑖 𝑟 𝑖+1 = 𝑟 𝑖 + Δ𝑟   Calculate 𝐾𝐻  𝐾  𝐾𝐻 ≥ (1 +𝑀௄)𝐾𝐻𝑑  𝐾 ≥ 1.15𝐾𝜀0𝑖+1 = 𝜀0𝑖 + 1%   𝑟𝑤 = 𝑟𝑤 𝑖+1 𝜀0 = 𝜀0 𝑖+1 𝑟𝑤 = 𝑟𝑤 𝑖+1 𝜀 = 𝜀   Calculate the initial value of radius of SMA wire, 𝑟𝑤  0  No  𝐾𝐻 −𝐾𝐻𝑑 𝐾𝐻𝑑≤ 10%  𝐾 − 𝐾  ≤ 10%𝜀0𝑖+1 < 5% 𝜀𝑖+1 ≤ 0.04 𝑟𝑤 𝑖 = 𝑟𝑤  0 𝜀0𝑖 = 0 Reduce 𝑀௄ by 5% Calculate 𝐾𝐻  𝐾  169  Based on the described design procedure, the radius of SMA wires increases from 1 mm, considered as an initial value, to the 2.5 mm. Since the stiffnesses of FeNCATB and NiTi45 at the austenite phase are close to each other (Table ‎5.5), same thicknesses are selected for both types of SMA wires. In order to make the design procedure more clear, an example is given for SMA-HDRB-C2 in Appendix A. 5.6 Summary This chapter dealt with a new generation smart rubber bearings incorporated with shape memory alloy (SMA) wires. Due to the unique characteristics of SMAs such as the superelastic effect and the re-centring capability, the residual deformation in SMA-based rubber bearing (SMA-RB) decreases and the energy dissipation capacity increases. Therefore, SMA wires can make rubber bearings more reliable by extending service life. SMAs in the form of wire were wrapped around three different rubber bearings (NRB, HDRB, and LRB). Two different configurations of wires (e.g. straight and cross) were considered for SMA-NRB and SMA-HDRB. For SMA-LRB, wires were wrapped around the LRB with a double cross configuration. It was because such an arrangement (double cross) was found to be more efficient for the LRB. The effect of several parameters including the shear strain amplitude, the type of SMA, the aspect ratio of base isolator, the thickness of SMA wire, and the amount of pre-strain in wires was investigated on the performance of the SMA-RBs. Isolators were subjected to a vertical pressure and unidirectional cyclic lateral displacements. Hysteretic shear response of SMA-RBs was determined through FEM. Results showed that, ferrous SMA wire, FeNiCoAlTaB, with 13.5% superelastic strain and a very low austenite finish temperature (-62°C), is the best candidate to be used in SMA-RBs subjected to high shear strain amplitudes. In terms of the lateral flexibility and wires’ strain level, the smart rubber bearing with cross configuration of SMA wires is more efficient. Moreover, the cross configuration can be implemented in high-aspect-ratio elastomeric bearings since the strain induced in wires does not exceed the superelastic range. When cross SMA wires with 2% pre-strain is used in a smart NRB, the dissipated energy is increased by 74% and the residual deformation is decreased by 15%. Using cross SMA wires in HDRBs leads to the highest energy dissipation capacity. However, SMA wires have negligible effect on the residual deformation reduction in HDRBs. Results revealed that 170  wrapping SMA wires in the double cross arrangement could significantly improve the re-centring capability of LRB by decreasing the maximum shear strain of LRB up to 59%. Findings showed that the pre-straining process advance the re-centring property of SMA-LRB. It was also observed that the maximum shear strain of SMA-LRB could be reduced when 3% pre-strained SMA wires are used. Another point is that enlarging the flag-shaped hysteresis of SMA as a result of pre-straining process does not lead to an increase in the energy dissipation capacity of SMA-LRB because of the configuration of double cross wires. Finally, a performance-based design flowchart was also provided along with a design example for determining the pre-strain and the radius of cross section of wires in the SMA wire-based rubber bearings.   171  Chapter 6 Constitutive Model of SMA-based Elastomeric Isolators 6.1 General In the process of performance evaluation of rubber bearings, experimental-based data is usually obtained by fabricating real size specimens and performing expensive full-scale tests under different loading and environmental conditions. As a reliable alternative, numerical approaches such as finite element method (FEM) can be used in order to significantly reduce the difficulties and costs associated with the experimental procedures. Considering different steps that are followed in the FEM, material behaviour modelling is one of the most challenging parts. In this regard, the constitutive model used in numerical simulations should properly describe the actual behaviour of the system. Therefore, it is of great importance to implement a well-fitted and accurate hysteresis model by either using an existing model or developing a new one. The hysteretic behaviour of elastomeric isolators is determined based on their components including elastomer, reinforcement (e.g. steel, carbon fibre-reinforced polymers), and supplementary elements (e.g. lead core). In other words, depending on the material model assumed for each component of the isolator, the hysteretic behaviour of the rubber bearing can be identified. By considering different types of elastomer such as low-damping rubber (Takayama and Morita, 2000; Amin et al., 2006a; Gjorgjiev and Garevski, 2013), commercial high quality neoprene (Hedayati Dezfuli and Alam, 2013b and 2014a), and high damping rubber (HDR) (Amin et al., 2006a and b; Hedayati Dezfuli and Alam, 2012), different material models are used. Since HDR is commonly implemented in the base isolation systems and has favourable characteristics (e.g. high damping capacity and resistance to corrosion) but with complex response, it has attracted attentions with experimental (Yoshida et al., 2004; Bhuiyan et al., 2009), analytical (Tsai et al., 2003; Hwang et al., 2002; Bhuiyan and Ahmed, 2007), and numerical (Amin et al., 2006a) perspectives. In FEM, low-damping rubber can be usually simulated using hyperelastic models because it follows the behaviour of a hyperelastic material (Hossa and Marczak, 2010; Ali et al., 2010). Such a model cannot accurately capture the highly nonlinear behaviour of HDR (Hedayati Dezfuli and Alam, 2012 and 2013a). Knowing the fact that HDR shows viscoelastic strain-rate-dependent behaviour under shear deformations, the 172  hyper-viscoelastic material model (i.e a combination of hyperelastic and viscoelastic models) is an appropriate choice (Bergstrom and Boyce, 1998; Hedayati Dezfuli and Alam, 2012). LRBs can provide a considerable amount of equivalent viscous damping ranging from 15% to 35% (Kelly, 2001). The main advantages of LRB, which makes it the most common type of isolator, is satisfactory amounts of rigidity, flexibility, and damping ratio at different load levels (e.g. service and earthquake) (Kelly, 2001; Chen et al., 2011). In the numerical FE simulations, lead is usually modelled either as an elastic-perfectly plastic material (SAP2000 software) or a bilinear elasto-plastic material with a hardening law (Kelly, 2001; Doudoumis et al., 2005). In both cases, a major amount of energy is dissipated when the lead, with a high initial stiffness, yields and enters the plastic region. However, a significant residual deformation, which is defined as a shear displacement at which the shear force becomes zero, occurs in the material after unloading. LRBs undergo a large residual deformation under strong excitations. Therefore, it is highly beneficial to implement SMAs, as auxiliary components, in such isolators in order to extend their service life. This improvement is achieved by controlling the displacement and limiting the force transmitted to the superstructure. Attanasi et al. (2008) proposed an innovative SMA-based isolation device and showed that although there is a significant difference in the hysteretic responses of SMA device (flag-shaped hysteresis model) and LRB (elasto-plastic model), both systems have similar displacement and force demands. However, the main advantage of SMA-based device over LRB was zero residual deformation. They concluded that using SMA as a lateral restrainer can improve the re-centring property and energy dissipation capacity of bearing systems. Based on studies performed by Kelly (1997), Choi et al. (2004), and Bhuiyan and Alam (2013), hystereses of LRBs and SMA-RBs were simulated with bilinear models by considering three characteristics: initial stiffness, post-yield stiffness, and yield force for each of them. In a numerical study conducted by Hedayati Dezfuli and Alam (2014b), SMA wires were implemented into a HDRB with a cross configuration (i.e. diagonal). In order to identify the efficiency of the SMA-HDRB, they evaluated the seismic response of a three-span continuous steel-girder RC-pier supported bridge, which was isolated by the proposed SMA-HDRB. They modelled the hysteretic behaviour of both rubber bearings with bilinear kinematic hardening (BKH) models. Although they considered different characteristics (e.g. 173  initial stiffness and post-yield hardening ratio) for bearings, insignificant differences were observed in the performances. The reason was due to the BKH model which was not able to correctly simulate the actual behaviour of SMA-HDRB. The objective of this chapter was to develop a hysteresis model for shape memory alloy wire-based rubber bearings (SMA-RBs). In Chapter 5, three types of SMA-RBs were introduced and a thorough discussion was carried out on the advantages of these new smart isolators. However, their real performance on the seismic behaviour of structures, specifically bridges, was yet to be investigated. More importantly, hysteresis models available in structural FE softwares such as SAP2000 (SAP2000 software), Seismostruct (Seismostruct, v6.5), or Opensees (McKenna et al., 2000) cannot accurately capture the actual behaviour of SMA-RBs under seismic loadings. Therefore, the necessity of implementing SMA-RBs into multi-span steel-girder bridges and the lack of an appropriate hysteresis model for such isolators led this study to propose a constitutive model for SMA-RB. For developing the model, it was assumed that the bearing is subjected to a compressive loading and unidirectional lateral displacements. Since the vertical deflection due to the compression was much smaller than the lateral displacements, it was neglected. Due to the complexity of the shear behaviour of SMA-RB, the idea of superposition was used to simplify the model by decoupling the smart isolator into two separate systems; SMA wires and RB. As a result, by considering the bilinear kinematic hardening model for the RB, first, a hysteresis model was developed for SMA wires and then, it was superimposed onto the RB hysteresis model. Before presenting an algorithm for the SMA wires model, the superposition method was verified through the FE model validated by experimental results. In the validation procedure the shear hysteretic response of the SMA-RB, as an integrated system, was compared to that of the superimposed system in which the behaviours of decoupled systems were added together. 6.2 SMA-based Rubber Bearings In Chapter 5, when performance of NRB, HDRB, and LRB was compared with that of SMA-NRB, SMA-HDRB, and SMA-LRB, it was observed that implementing SMA wires into LRB leads to more significant improvements in terms of re-centring capability and energy dissipation capacity. Therefore, the focus of this chapter is on the behaviour of SMA-174  LRB and the hysteresis model is developed for this smart elastomeric isolator (Hedayati Dezfuli and Alam, 2015a). However, it should be mentioned that the same procedure, which will be presented for developing the hysteresis model, can be used for SMA-NRB and SMA-HDRB.  Although bilinear kinematic hardening model can be used for SMA-LRB, it cannot precisely capture the nonlinear behaviour of SMA-LRB, especially at high shear strain levels. Hence, the result will not be accurate enough to be attributed to the real case. Therefore, it is of great interest to develop a constitutive model for the SMA-LRB. The mechanical and geometric properties of the SMA-LRB are the same as what Abe et al. (2004) used in their study for LRB (see Figure  6.1). It should be also mentioned that the FE model of LRB, which is used for analyzing and determining hysteretic shear responses, is the one used in Chapter 5 and validated through the experimental tests conducted by Abe et al. (2004).  Figure ‎6.1. Half model of SMA-LRB (dimensions are in mm) In addition to the dimensions specified in Figure  6.1, other physical and material properties of the LRB and SMA wires are provided in Table ‎6.1. Ferrous SMA, FeNiCoAlTaB, is chosen for wires with a cross sectional radius of 2.5 mm. In all the FE models, rubber is simulated with hyper-viscoelastic model, and steel and lead are assumed to follow bilinear behaviours with kinematic hardening law.       92.8 22 70.8 200 210 360 48.8 180 40 175  Table ‎6.1. Properties of LRB and SMA wire LRB SMA Wire nr 7 rSMA (mm) 2.5 ns 6 EA (GPa) 46.9 tr (mm) 5.0 Af (°C) -62.0 ts (mm) 2.3 εs (%) 13.5 nr: number of rubber layers; ns number of steel shims; tr: thickness of rubber layers; ts: thickness of steel shims; rSMA: radius of wires; EA: elastic modulus of SMA in austenite phase; Af: austenite finish temperature; εs: superelastic strain limit  6.2.1 Superposition Method The SMA-LRB has a complex hysteretic behaviour. Therefore, in order to determine its shear hysteresis, the whole system is simplified by applying a superposition method. In fact, double cross SMA wires (DC-SMAW) are decoupled from the rubber bearing and then, the effect of SMA wires is superimposed on the LRB (Figure  6.2). To provide a solid proof for this assumption, the hysteresis curves of the integrated system (i.e. LRB equipped with DC-SMAW) are compared to those of the decoupled systems (i.e. LRB and DC-SMAW). In this regard, first, lateral force-deflection curves of the decoupled systems are calculated through FE analysis by using ANSYS (ANSYS Mechanical APDL, Release 14.0) (see Figure  6.3). Then, shear hysteretic responses of superimposed and integrated systems are evaluated (see Figure  6.4). It should be noted that in the superimposed system, the hysteresis of LRB and SMA wires are calculated separately and then their responses are added together. In the integrated system, the SMA-LRB is modelled and analyzed as one system.  Figure ‎6.2. Superimposing SMA wires onto LRB Here it is assumed that wires are in contact with the hooks through a frictionless mechanism similar to what was explained in section 5.3. Hence, instead of modelling the DC-SMAW with details of the connection system in ANSYS, axial forces in SMA wires are LRB DC-SMAW SMA-LRB 176  calculated as functions of time and applied to the model. In the FE modelling of DC-SMAW, 16 nodes (8 points on the bottom supporting plate and 8 points on the top plate) are created at locations where steel hooks are attached. Then, nodal forces are applied to these nodes. In order to properly capture the variation of axial force in the S