EVALUATING THE PERFORMANCE-BASED SEISMIC DESIGN OF RC BRIDGES ACCORDING TO THE 2014 CANADIAN HIGHWAY BRIDGE DESIGN CODE by Sepideh Ashtari B.Sc., Sharif University of Technology, 2009 M.A.Sc., the University of British Columbia, 2012 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Civil Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) December 2018 © Sepideh Ashtari, 2018 ii The following individuals certify that they have read, and recommend to the Faculty of Graduate and Postdoctoral Studies for acceptance, the dissertation entitled: Evaluating the performance-based seismic design of RC bridges according to the 2014 Canadian Highway bridge design code submitted by Sepideh Ashtari in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Civil Engineering Examining Committee: Carlos E. Ventura, Department of Civil Engineering Supervisor W.D. Liam Finn, Department of Civil Engineering Supervisory Committee Member Ricardo Foschi, Department of Civil Engineering Supervisory Committee Member Donald Anderson, Department of Civil Engineering University Examiner Annalisa Meyboom, School of Architecture and Landscape Architecture University Examiner Additional Supervisory Committee Members: Don Kennedy, Associated Engineering Supervisory Committee Member Sharlie Huffman, BC Ministry of Transportation Supervisory Committee Member iii Abstract The 2014 edition of the Canadian Highway Bridge Design Code, CSA S6-14, has adopted a performance-based design approach for the seismic design of lifeline and major-route bridges in highly seismic zones. This addition offers many opportunities as well as some challenges with regards to implementing the CSA S6-14 performance-based design provisions in practice. This thesis aims to identify these challenges through a critical review of the CSA S6-14 performance-based design provisions and to address a number of them within the scope of the thesis. The motivation behind conducting the present study is to prepare a reference document for engineers to better comprehend and implement the new provisions in practice. The focus of the thesis is on the performance-based design of new reinforced concrete bridges with ductile substructures. The addressed challenges are related to CSA S6-14 performance verification framework, calibration of performance criteria, and appropriate numerical models to evaluate the established performance criteria. A deterministic and a probabilistic framework are recommended to be used with the CSA S6-14 performance-based design approach. The applications of each of the frameworks are demonstrated through two detailed case studies and the advantages and disadvantages of each framework are discussed. The performance criteria of the code are compared against the recommended criteria in the literature and other design guidelines. Moreover, the strain limits of the code are examined to predict the damage to a number of tested reinforced concrete bridge columns. A thorough comparison of the CSA S6-14 and the updated strain limits of the BC MoTI Supplement to CSA S6-14 is presented. Finally, common modelling techniques for reinforced concrete structures including distributed and concentrated plasticity models are employed to predict the response of a number of tested bridge columns. Mesh-sensitivity issues due to the localization of plastic strains at critical sections or elements of distributed plasticity models are discussed and the methods to rectify the issue are presented and compared. A simple solution is proposed to eliminate the post-processing effort that is required to verify the strain limits of the code in distributed plasticity models, for which material model regularization is used to deal with the mesh-sensitivity issue. iv Lay Summary The 2014 edition of the Canadian Highway Bridge Design Code (CSA S6-14) introduced a major shift in the seismic design of bridges by adopting a performance-based design approach. The performance-based design approach is based on meeting specified performance criteria at multiple seismic hazard levels. While the new design approach provides many opportunities for achieving better seismic designs with more reliable structural performances during earthquake events, it faces a number of challenges too. The present study provides a critical review of the CSA S6-14 performance-based design approach and identifies the challenges of implementing the code in practice. A number of the identified challenges with regards to analysis, design framework, performance criteria, and performance verification are addressed subsequently. The thesis helps practicing engineers better understand and implement the performance-based design provisions in their projects and in that sense, it serves as a reference for practical purposes. v Preface This dissertation is an original intellectual product of Sepideh Ashtari under the primary supervision of Prof. Carlos Ventura. Identification and design of the research program, analysis, result presentations, and writings were done by Sepideh Ashtari. Some parts of this thesis has been already published, as specified below. The rest of the research material in the thesis is unpublished to this date. - Portions of Chapter 5 have been published in Ashtari and Ventura (2018) “Verifying the seismic performance of concrete bridge columns according to the 2014 Canadian Highway Bridge Design Code,” proceedings of the 11th National Conference on Earthquake Engineering, Los Angeles, California, US. I was the principal author of the paper and conducted all the numerical analyses and writing of the paper. Professor Ventura provided assistance in revising the manuscript. - A portion of Chapter 7 has been published in the EGBC Professional Practice Guidelines on the Performance-Based Seismic Design of Bridges in BC as Ashtari et al. (2018) “Appendix D1: Reinforced Concrete Bridge.” This paper contains only the results for the response spectrum analysis of the case study. Mr. Khan provided the proportioning of the bridge model. I conducted all the numerical analyses and wrote the manuscript. Professor Ventura and Mr. Khan reviewed the manuscript. Dr. Atukorala provided technical assistance with the geotechnical inputs for the model. A version of the above material with the addition of nonlinear time-history analysis has been published in Ashtari and Ventura (2017b) “A case study on implementing the performance-based seismic provisions of the 2014 Canadian Highway Bridge Design Code,” proceedings of International Workshop on Performance-Based Seismic Design of Structures, Shanghai, China. - Portions of Chapter 8 has been published in Ashtari et al. (2017) “A case study on evaluating the performance criteria of the 2014 Canadian Highway Bridge Design Code,” Proceedings of 39th IABSE Symposium, Vancouver, Canada. I formulated the concept, performed all the numerical analyses, and wrote the first draft of the paper. Mr. vi Kennedy provided the drawings of the Trent River Bridge and helped with the technical and practical aspects of the paper. Professors Finn and Ventura provided technical and academic inputs and revised the paper. - Section D.1 of Appendix D have been published in Ashtari and Ventura (2017a) “Correlation of damage with seismic intensity measures for ductile concrete bridge columns in British Columbia,” Proceedings of 16th World Conference on Earthquake Engineering, Santiago, Chile. I conducted all the analysis and wrote the paper. Professor Ventura provided technical input and revised the paper. vii Table of Contents Abstract ......................................................................................................................................... iii Lay Summary ............................................................................................................................... iv Preface .............................................................................................................................................v Table of Contents ........................................................................................................................ vii List of Tables ............................................................................................................................... xii List of Figures ............................................................................................................................ xvii Acknowledgements ................................................................................................................... xxii Chapter 1: Introduction ................................................................................................................1 1.1 Motivation and Objectives .............................................................................................. 1 1.2 Scope ............................................................................................................................... 2 1.3 Overview of the Thesis ................................................................................................... 3 Chapter 2: Background on Performance-Based Seismic Design ..............................................5 2.1 A Shift in Seismic Design Philosophy ............................................................................ 5 2.2 Progress of PBD Procedures in Building Industry ......................................................... 7 2.3 Progress of PBD Procedures in Bridge Industry........................................................... 11 2.4 Performance-Based Design vs Assessment .................................................................. 14 2.5 Review of a Number of Existing Design/Assessment Frameworks ............................. 16 2.5.1 Load and Resistance Factor Design (LRFD) ............................................................ 16 2.5.2 PEER Framework ..................................................................................................... 17 2.5.3 ASCE/SEI 7-16 Performance Assessment Framework ............................................ 19 2.6 Overview of the CSA S6-14 PBD Provisions............................................................... 21 Chapter 3: Merits and Challenges of the CSA S6-14 PBD Approach ....................................24 viii 3.1 General Challenges of Performance-Based Procedures ............................................... 24 3.2 Particular Challenges of the CSA S6-14 PBD Approach ............................................. 27 3.3 Merits of CSA S6-14 PBD approach ............................................................................ 28 3.4 What Is Required to Implement the CSA S6-14 PBD Approach in Practice ............... 29 Chapter 4: Assessment Frameworks for CSA S6-14 PBD .......................................................31 4.1 Assessment Framework Requirements for CSA S6-14 PBD ....................................... 31 4.2 Deterministic Assessment Framework for CSA S6-14 PBD ........................................ 31 4.2.1 Implementation Steps................................................................................................ 32 4.3 Probabilistic Assessment Framework for CSA S6-14 PBD ......................................... 36 4.3.1 Demand and Capacity Factored Design (DCFD) ..................................................... 36 4.3.2 Implementation Steps................................................................................................ 41 Chapter 5: Examining Modelling Alternatives of RC Bridges for PBD .................................46 5.1 Modelling Alternatives for RC Bridges ........................................................................ 46 5.1.1 Linear Elastic Models with Cracked Section Properties ........................................... 47 5.1.2 Concentrated Plasticity Models ................................................................................ 49 5.1.3 Distributed Plasticity Models .................................................................................... 50 5.2 Localization in Distributed Plasticity Models of RC Components ............................... 53 5.2.1 Background in Fracture Mechanics .......................................................................... 53 5.2.2 Localization in Distributed Plasticity Beam-Column Elements ............................... 54 5.2.3 The Importance of Addressing Localization Issues for PBD of RC Bridges ........... 55 5.2.4 Options for Dealing with Localization Issues in DBE and FBE .............................. 57 5.3 Validating Distributed and Concentrated Plasticity Models of RC Bridge Columns ... 63 5.3.1 Selected Bridge Column Tests .................................................................................. 64 5.3.2 Bridge Column Models ............................................................................................. 68 ix 5.3.3 Validation of the Distributed Plasticity Models for Test 328 ................................... 72 5.3.4 Sensitivity of Monotonic Response to GFcc Value ................................................... 81 5.3.5 Adjusting Local Strain Response .............................................................................. 84 5.3.6 Comparison of the Options for Dealing with Localization Issues ............................ 90 5.3.7 Validation of the Concentrated Plasticity Model for Test 328 ................................. 91 5.3.8 Comparison of the Distributed and Concentrated Plasticity Models of Column 32891 5.3.9 Comparison of the Models for the Other Tests ......................................................... 94 Chapter 6: Evaluating the Strain Limits of CSA S6-14 for RC Bridges ................................97 6.1 Strain Limits in CSA S6-14 and BC MoTI Supplement............................................... 97 6.2 Performance Criteria for RC Bridges in Literature and Other Design Guidelines ....... 99 6.2.1 Strain Limits............................................................................................................ 100 6.2.2 Curvature Limits ..................................................................................................... 102 6.2.3 Deformation Limits ................................................................................................. 103 6.3 Examining the Strain Limits of CSA S6-14 and BC MoTI Supplement .................... 104 6.3.1 Damage Observed in the Tests ............................................................................... 104 6.3.2 Damage Predicted by the Models ........................................................................... 105 6.3.3 Comparison and Discussion .................................................................................... 106 Chapter 7: Implementation of the Frameworks for CSA S6-14 PBD...................................110 7.1 Description of the Bridge ............................................................................................ 110 7.2 Bridge Site Properties ................................................................................................. 111 7.3 Probabilistic Seismic Hazard Analysis ....................................................................... 112 7.4 Performance Criteria and Analysis Requirements ...................................................... 113 7.5 Structural Analysis Models ......................................................................................... 115 7.6 Fundamental Period of the Bridge .............................................................................. 117 x 7.7 Ground Motion Selection and Scaling ........................................................................ 117 7.8 Response Spectrum Analysis ...................................................................................... 123 7.9 Nonlinear Time-History Analysis ............................................................................... 123 7.10 Pushover Analysis ....................................................................................................... 123 7.11 PBD Assessment Using the Deterministic Framework .............................................. 124 7.12 PBD Assessment Using the Probabilistic Framework ................................................ 128 7.13 Discussion ................................................................................................................... 138 7.13.1 Comparison of the Deterministic vs Probabilistic Frameworks ......................... 138 7.13.2 Question of Uniform Reliability (The Value of Loss Analysis) ......................... 140 Chapter 8: Case Study-Seismic Evaluation of the Trent River Bridge ................................142 8.1 Description of the Bridge ............................................................................................ 142 8.2 Soil Properties ............................................................................................................. 145 8.3 Bridge Model .............................................................................................................. 146 8.3.1 Choice of Analysis Tool ......................................................................................... 146 8.3.2 Material Models ...................................................................................................... 146 8.3.3 Structure .................................................................................................................. 147 8.3.4 Foundations ............................................................................................................. 147 8.3.5 Abutments ............................................................................................................... 148 8.4 Performance Criteria ................................................................................................... 149 8.5 Structural Analysis ...................................................................................................... 149 8.6 PBD Assessment ......................................................................................................... 151 8.6.1 Assuming Major-route Importance Category ......................................................... 153 8.6.2 Assuming Lifeline Importance Category ................................................................ 153 8.6.3 Assuming Other Importance Category ................................................................... 156 xi 8.7 FBD Assessment ......................................................................................................... 157 8.8 Conclusions ................................................................................................................. 157 Chapter 9: Summary and Future Research ............................................................................159 9.1 Summary of Thesis ..................................................................................................... 159 9.2 Main Contributions ..................................................................................................... 162 9.3 Future Research .......................................................................................................... 163 References ...................................................................................................................................165 Appendices ..................................................................................................................................174 Appendix A CSA S6-14 Tables for PBD ............................................................................... 174 Appendix B Deriving Expressions for Post-Processing Local Strain Response .................... 178 Appendix C Acceleration and Displacement Response Spectra of the Selected Records for NTHA in Chapter 7 and Chapter 8 ......................................................................................... 180 C.1 Crustal Suite (Chapter 7) ........................................................................................ 180 C.2 Subcrustal Suite (Chapter 7) ................................................................................... 183 C.3 Subduction Suite (Chapter 7) .................................................................................. 187 C.4 Suite of Motions for Chapter 8 ............................................................................... 191 Appendix D PBD Aid Using the Correlation of Damage with Seismic Intensity Measures .. 195 D.1 Study1: Developing the PBD Aid ........................................................................... 195 D.2 Study2: Examining the PBD Aid for the Implementation Example of Chapter 5 .. 207 xii List of Tables Table 2.1 Rehabilitation objectives in ASCE 41-06 (2006). ........................................................ 10 Table 2.2 Seismic performance criteria in ATC-32 (1996a) and ATC-18 (1997) ........................ 13 Table 2.3 Bridge damage assessment stage of the five-level performance evaluation procedure developed by the University of California, San Diego (Hose and Seible 1999) .......................... 13 Table 2.4 Minimum performance levels for retrofitted bridges in FHWA Seismic Retrofitting Manual (2006) ............................................................................................................................... 13 Table 2.5 Performance goals in ASCE/SEI 7-16 .......................................................................... 19 Table 4.1 ASCE/SEI 7-16 assumed uncertainty values for component deformation demand ..... 42 Table 4.2 ASCE/SEI 7-16 assumed uncertainty values for component deformation capacity .... 42 Table 5.1 Specimen properties of the test columns (Lehman et al. 2004) .................................... 66 Table 5.2 Material properties of the test columns (Lehman et al. 2004) ...................................... 66 Table 5.3 Imposed displacement histories in mm for columns with different aspect ratios (Lehman et al. 2004) ..................................................................................................................... 66 Table 5.4 Empirical plastic hinge length for the selected test columns ........................................ 71 Table 5.5 Calculated regularized strain of the unconfined and confined concrete at 80% strength loss, and the ultimate rupture strain and post-peak slope of the reinforcement steel for Column 328 models .................................................................................................................................... 75 Table 5.6 Failure displacements of the DBE and FBE models of Column 328 with and without material regularization .................................................................................................................. 78 Table 5.7 Calculated values of ε20c-PH for the Column 328 model with LIP=LP=376 mm ............ 84 Table 5.8 Column 328 DBE models: (a) displacements corresponding to the specified strain limits, (b) mapped strain limits based on the regularized material models, (c) displacements corresponding to the modified strain limits .................................................................................. 88 xiii Table 5.9 Column 328 FBE models: (a) displacements corresponding to the specified strain limits, (b) mapped strain limits based on the regularized material models, (c) displacements corresponding to the modified strain limits .................................................................................. 89 Table 5.10 Column 328 fibre hinge model, displacements corresponding to the specified strain limits ............................................................................................................................................. 93 Table 5.11 Average displacements of Table 5.8 and Table 5.9 for the FBE and DBE models of Column 328 ................................................................................................................................... 93 Table 6.1 CSA S6-14 concrete and reinforcing steel strain limits ................................................ 99 Table 6.2 BC MoTI Supplement to CSA S6-14 concrete and reinforcing steel strain limits ....... 99 Table 6.3 Strain limits associated to the flexural damage states of reinforced concrete columns 99 Table 6.4 ATC-32 (1996) strain limits (AC1 2016) ................................................................... 101 Table 6.5 Strain limits by Kowalsky (2000) ............................................................................... 101 Table 6.6 Bridge performance/design parameters SRPH-1 by Hose and Seible (Hose and Seible 1999; Transportation Research Board 2013) .............................................................................. 101 Table 6.7 Mean tensile strains in longitudinal reinforcement corresponding to visual damage states as reported by Vosooghi and Saiidi (2010; ACI 2016) (f’c is in ksi) ................................ 101 Table 6.8 Displacements cycle in meter reported for the first occurrence of the damage states in the tested bridge columns (Lehman et al. 2004) ......................................................................... 105 Table 6.9 Recorded average strain values corresponding to the first occurrence of the damage states in the tested bridge columns (Lehman et al. 2004) ........................................................... 105 Table 6.10 Predictions of the regularized FBE and DBE models for the cyclic displacements in meter corresponding to the first occurrence of the specified strain limits .................................. 106 Table 6.11 Comparison between the measured and predicted displacements by the regularized FBE and DBE models, corresponding to the first occurrence of damage states (all values are in meter) .......................................................................................................................................... 109 xiv Table 7.1 CSA S6-14 and BC MoTI strain limits associated to the performance levels of a major-route bridge ................................................................................................................................. 114 Table 7.2 CSA S6-14 and BC MoTI strain limits associated to the performance levels of a major-route bridge ................................................................................................................................. 114 Table 7.3 Composite section properties of the deck at different sections .................................. 116 Table 7.4 Deaggregation of 2%/50 UHS of Victoria at period of 1.61 s. Results of the deaggregation at the 0.56 were very similar to the 1.61 s and are not reported separately. ....... 119 Table 7.5 Source contribution to the UHS of Victoria at different hazard levels and at different periods (the sum of the contributions in each row is equal to 1). ............................................... 121 Table 7.6 Selected ground motion records for time history analysis .......................................... 121 Table 7.7 Column drift ratio capacities associated to the first occurrence of the damage states 124 Table 7.8 Mean column drift demands from NTHA in the longitudinal (x) and transverse (y) directions for crustal, subcrustal, subduction suites, and all 33 records ..................................... 126 Table 7.9 Column drift demands from RSA and NTHA in the longitudinal (x) and transverse (y) directions, along with the predicted damage (M: minimal, Y: yielding of longitudinal reinforcements, SL1 & SL2: serviceability limit 1 & 2, SP1 & SP2: cover spalling 1& 2, B: reduced buckling)........................................................................................................................ 126 Table 7.10 Ratio of the drift demand to drift capacity of the columns in the longitudinal (x) and lateral (y) directions, and the reserve drift capacity for each hazard level ................................. 127 Table 7.11 Maximum longitudinal and lateral displacement of the deck ................................... 127 Table 7.12 Median drift ratio demands of the bridge columns from NTHA in the longitudinal (x) and lateral (y) directions, and the calculated record-to-record variabilities ............................... 130 Table 7.13 Demand and capacity aleatory and epistemic uncertainty values ............................. 130 Table 7.14 Obtained values for the demand and hazard curve parameters ................................ 130 Table 7.15 Demand and capacity factors calculated for the longitudinal and lateral directions 130 Table 7.16 Confidence factor values calculated for various confidence intervals ..................... 130 xv Table 7.17 Implied factor of safety in evaluating the lateral and longitudinal response of the bridge columns in terms of drift ratio with/without considering the confidence interval .......... 130 Table 7.18 CSA S6-14 performance criteria checks using factored demand and capacity drift ratios of the bridge columns in the longitudinal and lateral direction (P: pass; F: fail; FD: factored drift ratio demand in %, FC: factored drift ratio capacity in %; λ90%: confidence factor for 90% confidence interval; D: median demand; C: median capacity) ................................................... 132 Table 7.19 BC MoTI performance criteria checks using factored demand and capacity drift ratios of the bridge columns in the longitudinal and lateral direction For CSA S6-14 performance criteria, the use of the DCFD framework indicates that the columns do not meet any of the specified criteria in the longitudinal direction, while the deterministic framework suggest that they meet the ones at 2%/50. ...................................................................................................... 132 Table 7.20 Recalculated confidence factors of Table 7.16calculated for lower confidence intervals ....................................................................................................................................... 140 Table 8.1 Calculated SSI spring constants for the Trent River Bridge site Class D model........ 148 Table 8.2 CSA S6-14 and BC MoTI strain limits associated to the performance levels of lifeline and other bridges. ........................................................................................................................ 150 Table 8.3 Selected motions for NTHA of the Trent River Bridge on site class D ..................... 151 Table 8.4 Drift ratio capacities of the columns associated to the first occurrence of the damage states ............................................................................................................................................ 152 Table 8.5 Column maximum drift demands from RSA in the longitudinal (x) and lateral (y) directions, along with the predicted damage (N: none, M: minimal damage, Y: yielding of longitudinal reinforcements) ....................................................................................................... 154 Table 8.6 Column mean drift demands from NTHA in the longitudinal (x) and lateral (y) directions, along with the predicted damage............................................................................... 154 Table 8.7 Ratios of the drift demand to drift capacity of the columns the reserve drift capacities assuming Major-route importance category for the bridge ......................................................... 155 xvi Table 8.8 Ratios of the drift demand to drift capacity of the columns the reserve drift capacities assuming Lifeline importance category for the bridge ............................................................... 155 Table 8.9 Ratios of the drift demand to drift capacity of the columns the reserve drift capacities assuming Other importance category for the bridge ................................................................... 156 Table A.1 CSA S6-14 site classification for seismic site response ..…………………………. 174 Table A.2 CSA S6-14 Seismic performance category based on 2475-year return period spectral values .......................................................................................................................................... 174 Table A.3 CSA S6-14 Requirements for PBD and FBD ............................................................ 174 Table A.4 CSA S6-14 minimum analysis requirements for 2%/50 and 5%/50 hazard levels.... 175 Table A.5 CSA S6-14 minimum analysis requirements for 10%/50 hazard level ..................... 175 Table A.6 CSA S6-14 Minimum performance levels for PBD approach................................... 175 Table A.7 CSA S6-14 Performance criteria (continued) ............................................................ 176 Table D.1 Strain-based damage Criteria ..................................................................................... 197 Table D.2 Selected ground motion records for time history analysis ......................................... 199 Table D.3 Prediction of the damage to the 1.0s period bridge column at different hazard levels using smaller suites of records. PGD was utilized as the intensity measure; μn is the average of the maximum drift ratios for n records with the highest PGD values ........................................ 205 Table D.4 Correlation coefficients of longitudinal and lateral drift ratios at 2%/50 hazard level with PGD, PGV, and PGA of H1 and H2 components ................................................................ 209 Table D.5 Mean longitudinal drift ratios (%) of the smaller suites sampled based on H1-PGV correlation ................................................................................................................................... 209 Table D.6 Mean lateral drift ratios (%) of the smaller suites sampled based on H2-PGA correlation ................................................................................................................................... 209 xvii List of Figures Figure 2.1 SEAOC Vision 2000 recommended seismic performance objectives for buildings (SEAOC 1995) ................................................................................................................................ 9 Figure 2.2 Visualization of the first generation performance-based earthquake engineering (Moehle and Deierlein 2004) .......................................................................................................... 9 Figure 4.1 The deterministic PBD assessment framework flowchart.......................................... 35 Figure 4.2 The probabilistic PBD assessment framework flowchart .......................................... 45 Figure 5.1 Types of nonlinear component models: (a) and (b) are concentrated plasticity models; (c), (d), and (e) are distributed plasticity models (NEHRP 2013) ................................................ 47 Figure 5.2 Priestley et al. chart for finding the effective stiffness of cracked RC circular sections (1996) ............................................................................................................................................ 49 Figure 5.3 Element level and sectional level response of a cantilever beam modelled using a single FBE with (a) elastic strain-hardening (b) elastic-perfectly plastic, and (c) strain-softening section responses from Coleman and Spacone (Coleman and Spacone 2001) ............................. 56 Figure 5.4 Schematic post-peak crushing energy of (a) unconfined concrete and (b) confined concrete (reproduced from Pugh, 2012) ....................................................................................... 60 Figure 5.5 Schematic post-peak energy of reinforcing steel (reproduced from Pugh, 2012) ....... 63 Figure 5.6 (left) geometry and reinforcement of the tested specimens, (right) test configuration and instrumentation (Calderone et al. 2001; Lehman et al. 2004) ................................................ 65 Figure 5.7 Imposed lateral displacement time history (Lehman et al. 2004) ............................... 66 Figure 5.8 Cyclic force-displacement responses of the selected test columns (PEER 2003) ....... 67 Figure 5.9 Schematic distributed and concentrated plasticity models of the columns in OpenSees (left) and SAP2000 (right) ............................................................................................................ 68 Figure 5.10 Concret02 constitutive model in OpenSees (PEER 2017) ........................................ 70 xviii Figure 5.11 Comparison of the simulated cyclic force-displacement response of Column 328 using DBE with (a) 0.076 m, (b) 0.152 m, (c) 0.305 m, (d) 0.610 m element size and 2 integration points, and FBE with (a`) 0.610 m element size and 2 integration points, (b`) 0.914 m element size and 3 integration points with the experimental results ............................................ 73 Figure 5.12 Regularized cyclic force-displacement response of Column 328 using DBE with (a) 0.076 m, (b) 0.152 m, (c) 0.305 m, (d) 0.610 m element size and 2 integration points, and FBE with (a`) 0.610 m element size and 2 integration points, (b`) 0.914 m element size and 3 integration points with the experimental results ........................................................................... 76 Figure 5.13 Monotonic response of the DBE and FBE models of Column 328 before material regularization ................................................................................................................................ 78 Figure 5.14 Monotonic response of the DBE and FBE models of Column 328 after material regularization ................................................................................................................................ 78 Figure 5.15 Comparison of the cyclic and monotonic responses of the (a) DBE model and (b) FBE model with LIP=LP against the test results of Column 328 ................................................... 80 Figure 5.16 Comparison of the cyclic and monotonic responses of the (a) DBE model and (b) FBE model with LIP=0.5 LP against the test results of Column 328 ............................................. 81 Figure 5.17 Sensitivity of the monotonic force-displacement response of the Column 328 (a) DBE and (b) FBE models to GFcc value ....................................................................................... 82 Figure 5.18 Mapping strain limits in the post-peak region of (a) confined concrete and (b) reinforcing steel materials ............................................................................................................. 86 Figure 5.19 Comparison of the monotonic response of the regularized (a) DBE and (b) FBE models of Column 328 with the model satisfying LP=LIP ............................................................ 91 Figure 5.20 Monotonic force-displacement response of the SAP2000 Fibre Hinge model of Column 328 ................................................................................................................................... 92 Figure 5.21 Comparison of the regularized (a) DBE and (b) FBE models with the concentrated plasticity models of Column 328 .................................................................................................. 93 xix Figure 5.22 Comparison of the predictions of the regularized DBE and FBE models with the fibre hinge models for Column (a) 415, (b) 430, (c) 828, and (d) 1028 ...................................... 96 Figure 7.1 Schematic elevation view of the bridge .................................................................... 111 Figure 7.2 Schematic cross-section of the bridge at the pier ..................................................... 111 Figure 7.3 UHS (left) and design spectra (right) for site class D at 2%/50, 5%/50, and 10%/50 hazard levels................................................................................................................................ 112 Figure 7.4 Fibre cross section of the columns in SAP2000 (left) and the moment-curvature response of the section in the plastic hinge region (right) (the response is calculated under the dead load). ................................................................................................................................... 118 Figure 7.5 View of the bridge model in SAP2000..................................................................... 118 Figure 7.6 Mean spectra for the crustal, subcrustal, subduction suites, and all 33 records vs the target spectrum (the range over which the mean spectra are matched to the target spectrum, are shown with the vertical lines). .................................................................................................... 122 Figure 7.7 Longitudinal and lateral drift ratio demands versus the spectral acceleration at periods of 1.61 s and 0.56 s, respectively, using all 33 records. .............................................................. 129 Figure 7.8 Mean annual frequency versus the spectral acceleration at periods of 1.61 s and 0.56 s for Victoria .................................................................................................................................. 129 Figure 7.9 MAF of exceeding limit states for longitudinal deck displacement ......................... 133 Figure 7.10 Column fragility curves for the CSA S6-14 performance criteria in the (a) longitudinal and (b) lateral directions. ........................................................................................ 135 Figure 7.11 Column fragility curves for the BC MoTI Supplement performance criteria in the (a) longitudinal and (b) lateral directions. ........................................................................................ 135 Figure 7.12 Factored demand to factored capacity ratio in the lateral and longitudinal directions versus the mean annual frequency of exceeding the performance limits states of CSA S6-14. . 136 Figure 7.13 Factored demand to factored capacity ratio in the lateral and longitudinal directions versus the mean annual frequency of exceeding the performance limits states of BC MoTI Supplement. ................................................................................................................................ 136 xx Figure 7.14 Factored demand to factored capacity ratio with confidence interval of 90% in the lateral and longitudinal directions versus the mean annual frequency of exceeding the performance limits states of CSA S6-14. .................................................................................... 137 Figure 7.15 Factored demand to factored capacity ratio with confidence interval of 90% in the lateral and longitudinal directions versus the mean annual frequency of exceeding the performance limits states of BC MoTI Supplement. .................................................................. 137 Figure 8.1 Relative location of the Trent River Bridge with respect to Victoria and Vancouver..................................................................................................................................................... 143 Figure 8.2 Elevation view of the Trent River Bridge ................................................................ 144 Figure 8.3 Plan view of the Trent River Bridge ......................................................................... 144 Figure 8.4 Deck, pier diaphragm, and cap-beam cross-sections (dimensions in mm) .............. 144 Figure 8.5 Connection of the deck to the end diaphragm .......................................................... 145 Figure 8.6 Pier 1 and Pier 2 elevation views (looking North) ................................................... 145 Figure 8.7 Spine 3D model of the Trent River Bridge in SeismoStruct .................................... 149 Figure 8.8 Bridge design spectra for site class C and D at 2%/50 hazard level ........................ 150 Figure 8.9 Mean acceleration and displacement spectra for the 11 motions vs the target spectrum (the period range over which the mean spectra are matched to the target spectrum, are shown with the vertical lines). ................................................................................................................ 151 Figure 8.10 Pushover curves for the Trent River Bridge on site class C in the lateral direction152 Figure 8.11 Drift ratios of the bridge columns from NTHA at 2%/50 ...................................... 154 Figure 8.11 Comparison of the axial force-bi-axial bending moment demand at 2%/50 hazard level with the column interaction diagram.................................................................................. 158 Figure B.1 Mapping strain limits in the post-peak region of (a) confined concrete and (b) reinforcing steel ………………………………………………………………………………. 179 Figure D.1 Maximum drift ratios versus PGV and Vmax/Amax for the three bridge columns ... 201 xxi Figure D.2 (left column) Maximum drift ratios versus PGD for the three bridge columns and (right column) maximum drift ratios versus PGD of the individual earthquake sources for the bridge column with fundamental periods of 1.0 s....................................................................... 202 Figure D.3 Maximum longitudinal and lateral drift ratios versus PGD, PGV, and PGA of H1 and H2 components of the crustal suite.............................................................................................. 210 Figure D.4 Maximum longitudinal and lateral drift ratios versus PGD, PGV, and PGA of H1 and H2 components of the subcrustal suite ........................................................................................ 211 Figure D.5 Maximum longitudinal and lateral drift ratios versus PGD, PGV, and PGA of H1 and H2 components of the subduction suite ....................................................................................... 212 Figure D.6 Maximum longitudinal and lateral drift ratios versus PGD, PGV, and PGA of H1 and H2 components of all records ...................................................................................................... 213 xxii Acknowledgements The creation of this research work would not have been possible without the constant support and encouragement of my supervisor Dr. Carlos Ventura. He motivates his students not only to advance in their research but also to develop skills that would further help them succeed in their future paths. It has been a pleasure and privilege to work under his supervision. I would also like to express my deepest gratitude to Dr. Liam Finn, who was a true mentor to me. He patiently devoted hours guiding me to move forward in my research, while keeping my ideas and efforts focused on the main theme of the thesis. His continuous support and kind feedbacks inspired me to excel my work every step of the way. I owe particular thanks to Mr. Don Kennedy from the Associated Engineering, who never hesitated to respond to my questions. His many years of practical seismic design experience helped me to better understand the industry and connect my work to the needs of current practice. He was very kind to provide the Trent River Bridge drawings that were used in one of the case studies of this thesis. I offer my enduring gratitude to other members of my supervisory committee, Dr. Ricardo Foschi, Dr. Anoosh Shamsabadi form Caltrans, and Ms. Sharlie Huffman. Their feedbacks were essential to the success of this work. I also wish to gratefully acknowledge the funding for this research project, which was partly supported through the Four-Year Fellowship award from the University of British Columbia and partly provided by the Natural Sciences and Engineering Research Council of Canada (NSERC). Many thanks go to the wonderful faculty and staff members of UBC Civil Engineering Department, who were always ready to assist me throughout my graduate studies. I also would like to thank all my friends and colleagues at UBC Earthquake Engineering Research Facility. Special thanks are owed to my dear friend, Dr. Armin Bebamzadeh, who inspires every single person around him by his passion for research. I would like to dedicate this work to my parents, whose love and kindness gave me strength and will all the way. I could not make it this far without their endless love and support. xxiii To my parents, who inspired me every step of my life, and to whom I shall always remain grateful. 1 Chapter 1: Introduction 1.1 Motivation and Objectives The release of the new seismic performance-based design (PBD) provisions in the 2014 Canadian Highway Bridge Design Code (2014) marked one of the boldest and most progressive changes ever made to Canadian seismic design provisions. The new provisions stemmed from a radically different seismic design philosophy, compared to the force-based design (FBD) philosophy incorporated in the previous versions of the code. The use of the PBD design approach became mandatory for major-route and lifeline bridges, as well as irregular bridges, in highly seismic zones. The new provisions also aimed to facilitate the seismic design of innovative new systems, for which no elastic seismic force reduction factors were available to be used with the FBD approach. As promising as all the advantages of the new PBD approach were anticipated to be, concerns were expressed about the followings: - Whether the recommended performance criteria were suitable or adequate to guarantee meeting the expected performance levels; - And, whether the practicing engineers were able to fully comprehend and adopt the new provisions, as envisioned by the code committee. The motivation behind conducting the present study was to address the above-mentioned concerns, within the scope established for the thesis. Based on this motivation, the main objectives of this thesis are: - First, to provide a critical review of the recommended provisions and to identify and highlight the challenges of implementing the PBD provisions in practice. - Second, to address a number of the identified challenges, within the established scope of the thesis as specified in Section 1.2. - Third, to provide detailed case studies on implementing the PBD provisions and the recommended solutions to address the identified challenges. - Finally, to serve as a reference for practicing engineers to better understand and implement the PBD provisions of CSA S6-14. 2 The present study is the first of its kind to focus on the PBD provisions of CSA S6-14, with an aim to achieve the above objectives. As such, this thesis serves a great purpose to the progress and improving of PBD practice in the Canadian bridge industry. 1.2 Scope The thesis is focused on the seismic performance-based design of new bridges with ductile reinforced concrete substructures. This class of bridges is one of the most prevailing classes of bridge structures in British Columbia, Canada, and therefore, it was selected for this study. Only the performance criteria relevant to ductile reinforced concrete bridges were studied. Evaluating the seismic performance of existing or retrofitted bridges is beyond the scope and interest of the thesis. In preparing the case studies, multi-span reinforced concrete bridges with steel girders or pre-stressed concrete girders were utilized. The case studies included two types of abutment conditions, namely seat-type abutment with expansions joints and semi-integral abutment. Other types of abutment conditions such as integral abutments were not covered in the examples. It is recognized that there are numerous configurations possible for a bridge within this class and further studies may include archetypes of common ductile reinforced concrete bridge configurations in British Columbia, instead of individual examples. In addition, the limited scope of the case studies does not compromise the applicability of the outcomes to other bridges within this class of bridges. One-of-a-kind bridges or bridges with special seismic resistant systems were not addressed herein. Although, the identified challenges of implementing the PBD provisions and some of the provided solutions are still valid and applicable to these bridges as well. The modelling of soil-structure interaction effects was limited to foundation flexibility effects, using compliance springs. Kinematic interaction and foundation damping were not considered in the case studies, as including their effects were not indispensable to the focus of the thesis. The study of the remaining capacity of bridges in aftershock events is out of the scope of the thesis, and it is not covered. In selecting ground motions for nonlinear-time history analysis, only far-field ground motions were considered and near-fault motions were excluded to 3 narrow the scope of the study. The effects of directionality on the response of the studied bridges were ignored, as the case study bridges were relatively short with regular configurations. 1.3 Overview of the Thesis The thesis opens with a background literature review on the performance-based seismic design of structures with a focus on bridge structures in Chapter 2. The chapter reviews the progress of PBD procedures in both building and bridge industries and summarizes the essence of a number of existing design and assessment frameworks. An overview of the CSA S6-14 PBD provisions is then included to serve as a reference throughout the thesis for those readers who are not familiar with the provisions. Chapter 2 is followed by a critical review of the CSA S6-14 PBD provisions in Chapter 3. The chapter identifies the merits and challenges of implementing the new PBD provisions in practice. Once these challenges are recognized, the chapter lists the actions required to address them and to facilitate the implementation of the PBD provisions. This chapter sets the flow of the thesis, as the following chapters each try to address a part of the identified challenges, within the established scope of the thesis in Section 1.2. The first of these challenges is addressed in Chapter 4, which is dedicated to searching for suitable frameworks for CSA S6-14 PBD approach. The chapter recommends both a deterministic and a probabilistic framework to be utilized with the CSA S6-14 PBD approach to verify the performance objectives. The details and steps of each framework are presented. Chapter 5 encompasses suitable modelling alternatives for verifying the performance of reinforced concrete bridges. The chapter covers the strain localization issues associated with the use of strain-softening materials in distributed plasticity models of RC components. A series of previously tested reinforced concrete bridge columns are then employed to validate the models. The predictions of different modelling alternatives are compared against one another to give a perspective on the capability of each model to accurately predict the test results. Chapter 6 is a follow-up chapter to Chapter 5 and aims to evaluate the strain limits of CSA S6-14 and BC MoTI Supplement to CSA S6-14 (2016). A brief background review on strain and deformation limits recommended in various references for predicting the damage states of ductile reinforced concrete components is presented. The last part of the chapter uses the validated and 4 verified models of Chapter 5 and the strain limits of CSA S6-14 and BC MoTI Supplement to predict the damage to the test columns and compares the predictions with the actual test results to evaluate the strain limits. Chapter 7 provides a detailed application of both of the deterministic and probabilistic assessment frameworks introduced in Chapter 4 to a reinforced concrete bridge with steel girder. This chapter takes all the elements introduced and examined in the previous chapters into account, and combines them together within a case study. A full comparison of the two frameworks and their advantages and disadvantages are discussed at the end of the chapter. Chapter 8 constitutes the second case study of the thesis. The Trent River Bridge, a major-route reinforced concrete bridge with pre-stressed concrete girders, is selected for this study. However, the bridge is treated as a newly designed bridge so that it can be employed for the purpose of this thesis. Moreover, the performance of the bridge is evaluated assuming three different scenarios, where the bridge is assumed to be a lifeline, major-route, and other bridge. The focus of the chapter is to identify the governing performance criteria in each scenario. This is especially important for practicing engineers, since it helps them to reduce the design trial and error process by initiating the design verification at the governing performance level. Chapter 9 concludes the thesis by summarizing the research outcomes and main contributions and discussing the visions of the author for future research in this area. The thesis also includes four appendices. Appendix A simply provides copies of the CSA S6-14 tables used with the PBD approach. Appendix B presents the step-by-step derivation of the expressions in Chapter 4 for adjusting the local monotonic response of RC components modelled with distributed plasticity models. Appendix C contains the acceleration and displacement response spectra of the motions selected for nonlinear time-history analysis in Chapter 7. Finally, Appendix D demonstrates a design aid as a solution to reduce the trial and error effort when nonlinear time-history analysis is employed to verify PBD. The application of the design aid is demonstrated on a number of reinforced concrete bridge columns designed to CSA S6-14, as well as the case study bridge of Chapter 7. This was presented as an appendix, to keep the main body of the thesis coherent. 5 Chapter 2: Background on Performance-Based Seismic Design This chapter presents a brief review of the evolution of performance-based design procedures and their implementation in bridge and building codes and design guidelines. The review summarizes some of the well-recognized challenges that performance-based design procedures have faced and overcome through the last two decades. This is followed by a selective review of the three widely accepted frameworks for design/assessment of structures, including load and resistance factor design, PEER performance-based earthquake engineering framework, and the more recent ASCE/SEI 7-16 (2016) framework. The purpose of the review is to provide a background and a point of comparison for Chapter 4, where a deterministic and a probabilistic assessment framework are selected for the CSA S6-14 PBD approach. Finally, an overview of the CSA S6-14 PBD provisions is presented to conclude the chapter. 2.1 A Shift in Seismic Design Philosophy The eleventh edition of the Canadian Highway Bridge Design Code (CSA S6-14) has introduced performance-based design approach for the seismic design of lifeline and major-route bridges in highly seismic zones, and considers it as an option for the design of other bridges. The force-based design (FBD) approach in the previous edition of the code, CSA S6-06 (2006), was quite similar to AASHTO LRFD Bridge Design Specifications (2004). In both cases, seismic design forces acting on ductile substructure elements were obtained by dividing the resulting forces from elastic analysis by a reduction factor, R, which accounted for the amount of ductility in the elements. The values of R for different ductile substructure elements ranged from 2 to 5 and were in fact, lower than the expected displacement ductility capacities of the elements. This conservatism was due to the fact that the procedure was intended to be applied to designing of a wide range of bridge geometries (Tehrani 2012). The seismic design forces were also affected by the importance of bridge through the importance factor, and by the type of soil profile through the site coefficients. While FBD has long been the standard seismic design approach recommended by various guidelines and codes, and has been widely adopted in practice, several problems have been associated with the use of this design approach. Priestley et al. (Priestley, M. J. N. and Calvi 6 2007; Calvi et al. 2013) has summarized some of the main issues with the FBD approach, including the following: 1. The equal displacement rule, which relates inelastic displacement demands to elastic demands, does not hold in many cases. 2. The implicit assumption of equal displacement ductility demand for different ductile substructure elements when using the same force reduction factor may not hold. For instance, columns with similar cross-section but different heights have different displacement ductility capacities. Nevertheless, the same force reduction factor applies to all of them. 3. In FBD, it is assumed that stiffness is independent of strength, and thus yield curvature is proportional to flexural strength. However, it has been shown that this assumption is invalid, and for a given section the yield curvature is independent of strength, and stiffness is proportional to strength. One major consequence of the aforementioned issues is that the structures designed with similar R values may experience different levels of damage at a given hazard level; i.e. the FBD reduction factors are not fully representatives of the expected level of damage to the structures. The goal of seismic design changed after the occurrence of major earthquakes in the 1990s, such as the 1994 California Northridge earthquake and the 1995 Japan Kobe earthquake1 (Ghobarah 2001; Jalayer 2003). The extent of the damage to buildings and infrastructure due to these earthquakes was unexpectedly significant and enormous direct and indirect financial losses were incurred. It became clear that there was a need for new seismic design approaches that would enable designing structures with more predictable performances. Providing collapse prevention was neither the primary, nor the sole goal of seismic design any more. Instead, controlling and limiting the level of damage to structures and the subsequent losses became the primary design objective. PBD emerged and evolved from this shift in seismic design 1 Although earlier examples of this shift in seismic design practice can be found-such as the two-level seismic design in British Columbia in mid-‘80s, or the displacement-based seismic retrofit of bridges in British Columbia commencing in 1990-the occurrence of the aforementioned events are typically considered as the motivation of sudden movement towards adopting performance-based seismic design philosophy. 7 perspective, and ever since its birth, it has been implemented gradually in various forms in many seismic design and retrofit guidelines and standards around the world. 2.2 Progress of PBD Procedures in Building Industry The progress of seismic PBD procedures can be roughly broken down into three stages: 1. Displacement-based seismic design 2. First generation performance-based earthquake engineering 3. Next generation performance-based earthquake engineering The above classification is based on the evolution of performance-based procedures both in terms of concept and framework. The performance-based design label can be applied to any design approach that ensures that structures meet certain performance objectives. In view of this definition, all limit state design procedures can be considered as performance-based procedures. Conventionally, in seismic design the performance was synonymous to strength, and only strength-based criteria were considered. However, as mentioned earlier, the seismic design philosophy has gradually evolved towards designing for improved seismic performance in terms of enhanced safety and reduced damage. Attention was paid to quantifying the inelastic deformation and ductility capacities of structural components, which were more indicative of the structural damage. In the early stages, the pure seismic FBD was improved by the addition of displacement checks and related modification of the design strength. This improved design approach was referred to as displacement-based design or performance-based design (Priestley, MJN 2000). The early efforts towards improving the seismic design approach evolved into the first generation of performance-based design and assessment procedures. Several conceptual frameworks were proposed such as SEAOC Vision 2000 (1995), ATC-40 (1996b), FEMA 273 (1997) and its later version FEMA 356 (2000a), and FEMA 350 (2000b). The first generation performance-based procedures were different from earlier procedures in the definition of performance objectives and their relation to the expected intensity of seismic ground motions. The performance objective in Vision 2000 was defined as “an expression of the desired performance level for each earthquake design level” (Krawinkler 1999). The coupling of seismic ground motion levels with performance levels as described in Vision 2000 is shown in Figure 8 2.1. As illustrated, the performance levels were labeled with names that were representative of the expected level of overall damage (FEMA 2006). To verify the level of damage from analysis, specific values of structural response parameters were associated to each performance level and the overall level of damage. Examples of such response parameters included ductility, story drift indices, damage indices, floor acceleration, and velocity (Bertero and Bertero 2002). A schematic visualization of the first generation performance based procedures is illustrated in Figure 2.2. While the first generation performance-based earthquake engineering took some major steps towards designing for enhanced seismic performance, they suffered from three main shortcomings as listed by Moehle and Deierlein (2004): 1. Inaccuracy of simplified analysis techniques in predicting engineering demands, including static and linear methods. In case of using more refined analysis methods such as non-linear analysis, a lack of calibration between calculated demands and component performance is a major issue. 2. The relations between engineering demand and component performance are defined based on laboratory test results, analytical models, or engineering judgment, which results in inconsistency in the established relations. Consistent approaches for defining these relations are necessary. 3. The assumption of equality of the overall system performance to the worst performance calculated for any component in the structure. This assumption does not hold for many structural systems, and does not give a realistic prediction of the system capacity. One of the key documents showcasing the implementation of performance-based procedures was ASCE 41-06 (2006), Seismic Rehabilitation of Existing Buildings. The document was the continuation of FEMA 356 and it can be considered as a link between the first and next generation performance-based procedures. The performance assessment approach adopted in this document was quite comprehensive and considered many aspects of the performance of structural and non-structural components. In this approach, the rehabilitation objective selected from Table 2.1 determined how many hazard levels were considered in performance assessment and what performance level was expected at each hazard level. For 9 Figure 2.1 SEAOC Vision 2000 recommended seismic performance objectives for buildings (SEAOC 1995) Figure 2.2 Visualization of the first generation performance-based earthquake engineering (Moehle and Deierlein 2004) instance, if the building was rehabilitated for the basic safety objectives (BSO) of “k” and “p”, then it should have been at life safety performance level under rare earthquakes and at collapse prevention performance level under MCE. The document linked target building performance levels to structural and non-structural performance levels, and indicated which combinations of structural and non-structural performances were acceptable for a target building performance level. Each of the structural and non-structural performance levels were defined qualitatively and quantitatively based on the type of structural element or non-structural system/component. ASCE 10 41 also provided separate acceptance criteria for deformation-controlled and force-controlled actions on an element-by- element basis employing either linear or nonlinear methods. Several shortcomings were identified with the ASCE 41-06 performance assessment approach. The approach did not directly assess the economic losses, which is of great concern when design decisions are made. The performance of the system was assessed in terms of individual structural and non-structural components rather than a system. Also, the reliability of the approach in delivering the expected performances was not characterized. Target Building Performance Levels Operational (1-A) Immediate Occupancy (1-B) Life Safety (3-C) Collapse Prevention (5-E) Design Hazard Level Frequent (50%/50-72 years) a b c d Occasional (20%/50 -225 years) e f g h Rare (10%/50-474 years) i j k l MCE (2%/50-2475 years) m n o p Notes: The rehabilitation objectives in the above table may be used to represent the following three specific rehabilitation objectives: - Basic Safety Objectives (BSO): k and p - Enhanced Objectives: k and m, n, or o; p and i or j, k and p and a, b, e, or f; m, n, or o alone - Limited Objectives: k alone; p alone; c, d, g, h, or l alone Table 2.1 Rehabilitation objectives in ASCE 41-06 (2006). With an aim of overcoming the shortcomings of the first generation performance-based procedures, and to better communicate the performance objectives to stakeholders, the next generation performance-based procedures were developed. The hallmark of the next generation procedures was expressing the performance objectives in terms of the expected value of some decision variable such as amount of loss within a probabilistic framework, which would take into account the sources of uncertainty effectively. The loss in this context, primarily referred to direct economic loss (repair cost), indirect economic loss (downtime and business interruption), and casualties (injuries and death) (Whittaker et al. 2007). Two major projects, namely the ATC-11 58 and later ATC 58-1, were devoted specifically to develop the next generation of performance-based seismic design and assessment guidelines for new and existing buildings. The outcomes of the two projects were documented under FEMA 445 (2006) and FEMA P-58-1 (2012). These two documents contained a generic probabilistic framework and methodology for the next generation performance-based procedures, which were developed based on breaking down the process into the following four distinctive elements: 1. Hazard Analysis 2. Structural Analysis 3. Damage Analysis 4. Loss Analysis The above framework for performance-based earthquake engineering was developed originally at the Pacific Earthquake Engineering Research Center (PEER), and is also referred to as the PEER PBEE framework. There are numerous ongoing studies to develop and enhance fundamental elements of the PEER PBEE framework. The main goal of these studies is to lay out the necessary foundation for implementing the next generation performance-based procedures in design guidelines and ultimately in practice. 2.3 Progress of PBD Procedures in Bridge Industry The PBD procedures in bridge industry progressed in parallel with the building industry, although it had a different manifestation in design guidelines. The basic seismic design philosophy for ordinary bridges used to be a no-collapse-based design, i.e. to prevent collapse during major earthquakes and to withstand more frequent earthquakes with minimal damage. Since 1989, there has been a shift in the California Department of Transportation (Caltrans) seismic design guidelines from FBD towards displacement-based design with an emphasis on capacity design (Duan and Li 2003). For instance, Caltrans Seismic Design Criteria 1.7 (Caltrans 2013) requires that three displacement-based criteria must be met, including global displacement, demand ductility, and capacity ductility. Another example is AASHTO guide specifications for seismic bridge design (AASHTO 2011), which also uses a displacement-based design approach. However, the first distinctive appearance of PBD concept for the seismic design of bridges can be found in ATC-32 (1996a) and ATC-18 (1997) documents. In these documents, performance 12 levels in terms of overall structural damage and serviceability were described for two levels of functional evaluation and safety evaluation ground motions, for ordinary and important bridges (Table 2.2). The structural damage for each performance level was described as minimal, repairable, or significant. The repairable damage referred to a level of damage that could be repaired with minimum risk of functionality. The significant damage on the other hand referred to a level of damage that would require closure for repairs, although exposing minimum risk of collapse. With regards to the descriptions of performance levels, these documents could be considered as the first generation seismic performance-based procedures for bridges. Another appearance of PBD concept was the use of a five-level performance evaluation approach by the University of California, San Diego (UCSD) for the development of a bridge performance database. The approach was developed based on the results of the tests conducted at UCSD on bridge components and systems. The classification of performance in five levels followed the previous work in buildings, while the trend for bridge industry was moving towards a two or three-level approach (Hose and Seible 1999). The performance evaluation consisted of three stages. In the first stage called bridge damage assessment, the classification of structural damage was related to socio-economic descriptions as shown in Table 2.3 . The second stage called bridge performance assessment employed the same five levels in the first stage to describe performance qualitatively and quantitatively. This stage involved field investigations of structural damage after an earthquake, detailed assessment of laboratory, and/or detailed analysis. The final stage of the assessment procedure, called bridge performance/design parameters, involved investigating the correlations of a series of quantitative parameters, including strain limits, curvature ductility, drift ratio, etc., with the qualitative performance levels. The aim of this stage was to identify the parameters that would correlate best with the specified performance levels, and to use these parameters later for design. One other document that specifically employed performance-based concepts was the FHWA Seismic Retrofitting Manual for Highway Structures (2006). The guideline categorized bridges as either standard or essential with three anticipated service life periods of 0-15 (ASL1), 16-50 (ASL2), and >50 years (ASL3). Two seismic hazard levels with 100-year and 1000-year return periods were considered. The minimum performance levels were then defined for a certain type of bridge with an anticipated service life period and a specified hazard level (Table 2.4). 13 Level of Damage and Post-Earthquake Service Levels Important Bridge Ordinary Bridge Ground Motion at Site Service Damage Service Damage Functional Evaluation Ground Motion Immediate Minimal Immediate Repairable Safety Evaluation Ground Motion Immediate Repairable Limited Significant Table 2.2 Seismic performance criteria in ATC-32 (1996a) and ATC-18 (1997) Level Damage Classification Damage Description Repair Description Socio-economic Description I No Barely visible cracking No repair Fully Operational II Minor Cracking Possible repair Operational III Moderate Open cracks, Onset of spalling Minimum repair Life Safety IV Major Very wide cracks, Extended concrete spalling repair Near Collapse V Local Failure/Collapse Visible permanent deformation, buckling/rupture of reinforcement replacement Collapse Table 2.3 Bridge damage assessment stage of the five-level performance evaluation procedure developed by the University of California, San Diego (Hose and Seible 1999) Bridge Importance and Service Life Category Standard Essential Earthquake Ground Motion ASL1 ASL2 ASL3 ASL1 ASL2 ASL3 50%/75-100 years PL0 PL3 PL3 PL0 PL3 PL3 7%/75-1000 years PL0 PL1 PL1 PL0 PL1 PL2 Notes: The tolerable damages in each performance level is as follows: - PL0 (No Minimum): no minimum level of performance is recommended. - PL1 (Life Safety): significant damage is sustained and service is significantly disrupted, but life safety is assured. The bridge may be replaced after a large earthquake. - PL2 (Operational): sustained damage is minimal and after inspection and clearance of debris, full service for emergency vehicles should be available. Bridge should be repairable disregarding the traffic flow restriction. - PL3 (Fully Operational): sustained damage is negligible, and after inspection and clearance of debris, full service for all vehicles is available. Damage to the bridge is repairable without interrupting traffic. Table 2.4 Minimum performance levels for retrofitted bridges in FHWA Seismic Retrofitting Manual (2006) 14 Four performance levels were specified in terms of overall structural damage and serviceability, including no minimum, life safety, operational, and fully operational levels. A description of the structural damage levels termed as negligible, minimal, and significant was provided. Although the retrofitting manual defined performance objectives for different importance, service life, and hazard levels, it did not clarify how the performance levels should be linked to structural damage limit states. It would have been the task of engineers to establish this link and set criteria that would deliver the anticipated performance (Transportation Research Board 2013). In line with the above changes in design and retrofitting guidelines, PBD was adopted in the latest edition of the Canadian Highway Bridge Design Code as a standard design approach. The basic premise of the CSA S6-14 PBD approach is to meet multiple performance criteria defined in terms of tolerable structural damage, and serviceability objectives at multiple hazard levels. CSA S6-14 has indeed taken a significant step towards enhancing seismic design practice of bridges in Canada. It is also fair to say that such explicit adoption of PBD is quite unique among the current bridge design guidelines. If held to its premises, the performance-based design would deliver bridges with more predictable and reliable seismic performances. 2.4 Performance-Based Design vs Assessment The structural design problem under any loading condition such as seismic loading, entails finding a valid structural system that satisfies the design limit state(s) (objective function) subject to a number of limitations (constraints). Two major considerations in structural design process are uncertainties and optimization (Royset et al. 2001). Uncertainties must be properly taken into account to ensure that the safety of design is not compromised due to randomness in materials or input loading, etc., or due to the lack of knowledge or bias. Optimization of design is also desirable since it maximises benefits (minimizes cost) of satisfactorily meeting the design limit states. Therefore, the structural design problem is essentially an optimization problem under uncertainties. As mentioned in Section 2.2, conventionally the seismic design limit state was formulated based on strength criteria. With the progress of PBD procedures, the seismic design problem included more limit states to meet multiple performance criteria at multiple hazard levels. Moreover, the design goal in PBD is often to meet the specified limit states with an acceptable 15 level of confidence over the service life of the structure (Zhang and Foschi 2004; Moller et al. 2015). To evaluate the level of confidence for each limit state, reliability analysis techniques are required to estimate the probability of failure. Consequently, finding the best solution to a PBD problem involves conducting optimization of structural design under reliability-based constraints. Achieving such optimal solution is not straight forward. A lot of research effort has been dedicated to facilitate the solution of these complex optimization problems and a number of techniques have been developed (Madsen et al. 1986; Wen 2001; Zhang and Foschi 2004; Ellingwood, Bruce R. and Wen 2005; Haukaas 2008). Even so, the solution process is very computationally intensive and most often requires using reliability analysis software. This is a downside, since most often the reliability analysis tools are not commercially available or compatible with the common structural analysis tools. All of the above-mentioned hurdles are the reasons for engineers to take a practical approach to achieve an acceptable but not necessarily optimal design solution. Such solution is obtained through trial and error process. For the initial trial, the structure is proportioned based on the best practice and designer’s experience, which are the key factors to reduce the number of trial and error efforts before an acceptable solution is reached. The better the initial design, the less the number of trials. For each design trial, performance limit states are assessed and the probabilities of failures are evaluated. If the trial meets all of the specified performance limit states with the specified minimum levels of confidence, then the design is deemed acceptable. Otherwise, modifications are made to the design trial and the process is repeated. While such approach does not necessarily find an optimal solution, it provides a good enough feasible solution. The initial proportioning of the structure based on experience, guarantees the feasibility of solution in many cases. Throughout the rest of this thesis, the second approach is assumed to solve PBD problems. While the first approach has an evident upside in scientific applications, it is too complex for design code applications with the current state of practice. Therefore, the second approach is more suitable for the purpose of this thesis. Within the context of the second approach, design and assessment are inter-connected, as the design process is comprised a series of assessments on design trials. As a result, in many instances throughout the thesis, the two terms of performance-16 based design framework and performance-based assessment framework are used interchangeably. 2.5 Review of a Number of Existing Design/Assessment Frameworks 2.5.1 Load and Resistance Factor Design (LRFD) Load and resistance factor design or LRFD is a design approach that incorporates both load and resistance factors to achieve an implicit desired level of reliability for the applicable range of structures. LRFD is the term used in the United States to refer to multiple-factor design formats. Other names have been used to refer to such formats in their earlier implementations in design guidelines, such as partial safety factor format in 1940 reinforced concrete and geotechnical standards of Denmark, ultimate strength design in 1950 reinforced concrete standard of the American Concrete Institute, and limit state design in the National Building Code of Canada since 1977 (Madsen et al. 1986). Using LRFD approach, the design is deemed acceptable for the specified limit states if the following inequality is met: ϕRn≥ ∑ γiQnii1 (2.1) where Rn is the normal strength calculated from the given equations in design guidelines and Qni is the load effect, ϕ is the resistance factor, and γi is the load factor for load effect i, respectively. The LRFD was an improvement to the earlier allowable stress design approach, which utilized a single factor of safety in the design equation as follows: Rn/F.S.≥ ∑ Qnii1 (2.2) By using more than one factor, LRFD provided more consistency in the provided level of safety for complex loading conditions and various structural members. The calibration of the load and resistance factors in LRFD is done considering that resistance, R, and load effect, Q, are random variables and by finding optimum ϕ and γi factors, which would satisfy P[R<Q]<Pf (the probability of resistance being exceeded by the load effect is less than Pf ). Pf is the allowable probability of exceeding a limit state. For code calibration purposes, Pf is preferably expressed in terms of a reliability index, β, as follows: 17 β= -Φ-1(Pf) (2.3) In which, Φ is the Gaussian (or Normal) cumulative distribution function2. Typical values of β range from 2 to 6, for which increase of one unit corresponds roughly to the decrease of one order of magnitude in Pf (Galambos 1981). It is possible to calibrate ϕ and γi factors using various reliability analysis techniques. A common practice in code calibration in earlier days was to utilize first-order second-moment reliability analysis, which considered only a central value such as mean and a measure of dispersion such as coefficient of variation for random parameters. This method however, did not consider the actual distribution of random variables. More refined methods such as first-order and second order reliability analysis methods may be utilized instead. A thorough application of such methods to develop the load factors for the American National Standard A58 can be found in Ellingwood et al. (1980). 2.5.2 PEER Framework The Pacific Earthquake Engineering Research Center (PEER) formulated a probabilistic framework for the seismic assessment of structures, which was based on acceptable probability (or the associated frequency) of exceeding specific performance levels (Cornell and Krawinkler 2000). To obtain such probability, PEER broke down the process of performance-based assessment into four elements of hazard analysis, structural analysis, damage analysis, and loss analysis. Subsequently, the annual frequency of exceeding a performance level is obtained through a triple integral, using the total probability theorem as follows: λ(DV)= ∭ G〈DV | DM〉 |dG〈DM | EDP〉| |dG〈EDP | IM〉| |dλ(IM)| (2.4) In the above formula, the IM, EDP, DM, and DV denote intensity measure, engineering demand parameter, damage measure, and decision variable; G(A|B) denotes conditional probability of exceeding B given A; and 𝜆(DV) is the annual frequency of exceeding. The four elements of the PEER framework can be briefly described as below: 2 It should be noted however that in general, R-Q does not necessarily have a normal distribution, but for most practical cases an approximate value of the reliability index β can be obtained this way (see Cornell 1969; Madsen et al. 1986). 18 - Hazard Analysis: In this step, one or more ground motion intensity measures are evaluated and appropriate input ground motions for response history analysis are selected. Standard IMs are peak ground acceleration and spectral acceleration, which can be evaluated through conventional probabilistic seismic hazard analysis (PSHA). IMs are expressed as the mean annual probability of exceedance for the particular location and design characteristics of a facility. - Structural Analysis: Having IM and input ground motions determined, structural analysis is performed next to obtain EDP values. The most common EDPs for structural components in buildings are inter-storey drift ratios, inelastic component deformations and associated forces; and for non-structural components are inter-storey drift ratios, floor accelerations, and floor velocities. To determine the relation between the EDPs and IM, inelastic simulations are typically carried out, in which aspects of structural and geotechnical engineering and soil-structure-foundation-interaction are considered. One of the procedures developed by PEER to systematically calculate the conditional probability of p(EDP|IM) is the incremental dynamic analysis (IDA) (Vamvatsikos and Cornell 2002). - Damage Analysis: In this step the EDPs are related to the physical damage to a facility through damage measures. DMs provide the description of damage to structural/non-structural components and contents, along with the necessary repairs and functionality condition of the facility. The conditional probability of p(DM|EDP) is typically referred to as fragility relation or fragility curve. - Loss Analysis: The last step in the framework is to aggregate the loss due to foreseen damage and calculate decision variables most suitable for decision making. This is achieved by integrating p(DV|DM) with the mean annual DM probability of exceedance p(DM), obtained in the previous step. 19 2.5.3 ASCE/SEI 7-16 Performance Assessment Framework ASCE7/SEI 7-16 (2016) defines the performance goals in terms of the tolerable probability of collapse under the maximum considered earthquake (MCER) ground motions. This probability depends on the risk category of the building, as shown in Table 2.5. The stated collapse performance goals are evaluated implicitly through prescribed set of analysis rules and acceptance criteria, as proposed in Chapter 16 of the standard. Risk Category Tolerable Probability of Collapse Ground Motion Level I or II 10% MCER III 6% MCER IV 3% MCER Table 2.5 Performance goals in ASCE/SEI 7-16 The acceptance criteria ensure that the building meets the performance-goals of Table 2.5, and are categorized as follows: - Global acceptance criteria defined for the average story drifts, maximum story drifts, and residual story drifts - Element-level acceptance criteria including the ones defined for force-controlled (brittle) components and the ones defined for deformation controlled (ductile) components For force-controlled components, the acceptance criteria follow the framework employed by the PEER TBI guidelines (2008) with the following expression (Haselton et al. 2014): λ FmeanDemand ≤ ϕ Fn,e (2.5) In which, 𝜆 is a calibration parameter, ϕ is the strength reduction factor, and Fn,e is the nominal strength computed considering expected material properties. Assuming that both of component demand and capacity follow a lognormal distribution, the ratio of 𝜆/ϕ is calibrated to meet the tolerable probabilities of failure depending on how critical the component is to the collapse of the building. For instance, for critical components the calibration of 𝜆/ϕ is done for 10% probability of failure, and for ordinary components for 25% probability of failure. An overview of the calibration process is presented in the commentary of Chapter 16 of ASCE7/SEI 7-16. The calibrated acceptance criteria for force-controlled components (with the exception of capacity-controlled components) are as follows (Haselton et al. 2015): 20 - For critical components, where the failure of the component would likely lead to a progressive global collapse of the building, it is required that: 2.0 IeFmeanDemand ≤ FmeanStrength (2.6) - For ordinary components, where the failure of the component would lead to a local collapse, then: 1.5 IeFmeanDemand ≤ FmeanStrength (2.7) - For non-critical components, where the failure of the component would not lead to any structural instability, then: 1.0 IeFmeanDemand ≤ FmeanStrength (2.8) In the above expressions ϕ=1 is used to calculate the mean strength, and Ie is the importance factor, which is equal to 1.00 for seismic risk category I and II, 1.25 for III, and 1.5 for IV. The reason for using the Ie factor is to account for the lower tolerable probabilities of collapse for seismic risk categories of III and IV. For critical deformation-controlled components, the acceptance criterion is proposed as (Haselton et al. 2015): DriftmeanDemand ≤ 0.3 to 0.5/Ie Driftcapacity (2.9) For ordinary deformation-controlled components, the criterion is modified as: DriftmeanDemand ≤ 0.5 to 0.7/Ie Driftcapacity (2.10) The lower bound capacities in the above two expressions, correspond to the case where no redistribution path for the gravity loads exist in the structural system, and upper bound values correspond to the otherwise. The deformations of non-critical deformation controlled components are limited by the global acceptance criteria and no element-level capacities are defined for these components. Similar to force-controlled components, drift capacity factors are obtained to ensure tolerable probabilities of collapse. For instance, in Equation (2.9), the factor of 0.5 implies a 40% probability of building collapse if the drift capacity is exceeded in a single component, and the factor of 0.3 implies a 100% probability of collapse. 21 2.6 Overview of the CSA S6-14 PBD Provisions Section 4.4 of CSA S6-14 named as “earthquake effects”, contains the seismic design provisions of both of the PBD and FBD approaches. The following steps encapsulate the essence of the CSA S6-14 PBD approach: 1. Determining importance category: Bridges are categorized in terms of importance as lifeline, major-route, and other bridges. Major-routes are used for emergency response and movement of people after an earthquake, and thus the bridges on these routes have higher priorities. Lifeline and major-route bridges are both located on major-routes, but compared to major-route bridges, life-line bridges are typically more complex. They require more resources, money, and time to restore following an earthquake and their downtime have serious and extensive impacts on the economy of the region of lives of the people. Examples of such bridges are Lions’ Gate Bridge in Vancouver, British Columbia, and Golden Gate Bridge in San Francisco, California. Other bridges refer to those bridges located on minor and local routes. 2. Performing seismic hazard analysis and constructing bridge design spectrum: CSA S6-14 performs PBD at three hazard levels with 475-year, 975-year, and 2475-year return periods. These will correspond to 10%, 5%, and 2% probabilities of exceedance in 50 years, respectively. For brevity, we will refer to these hazard levels with 10%/50, 5%/50, and 2%/50 notations from here on through the rest of this thesis. The 5% damped spectral values of the uniform hazard spectrum (UHS) at the specified hazard levels can be obtained by conducting probabilistic seismic hazard analysis (PSHA) of the bridge site, using programs such as EZ_FRISK (The Fugro Consultants 2015). Alternatively, the values of the uniform hazard spectrum for select locations can be obtained from the 2015 National Building Code of Canada seismic hazard calculator available online at the Natural Resource Canada Website (ref). The 2015 hazard maps of Canada (ref) gives values for the peak ground acceleration (PGA), peak ground velocity (PGV), and peak ground displacement (PGD), and the 5% damped spectral acceleration at periods of 0.2 s, 0.5 s, 1.0 s, 2.0 s, and 10.0 s. The UHS values are calculated for the reference ground condition, i.e. site class C of the code (see Table A.1 for site classifications in CSA S6-14). For constructing bridge 22 design spectrum, the UHS values should be multiplied by the recommended period-dependent site coefficients, to reflect the bridge site ground condition. The 5% damped spectral values of the UHS should also be modified for damping ratios. The damping modification factor is calculated from the following expression: RD= (0.05ξ)0.4 (2.11) In which ξ is the damping ratio of the bridge, but should not be taken greater than 0.1. If ξ < 0.05, then the 5% damped spectral values are increased. On the other hand, if the bridge abutment is designed to mobilize the backfill soil and satisfies the conditions of Clause 4.4.3.5 of CSA S6-14, then the 5% damped spectral values can be reduced. For cases where the abutment is not specifically designed to mobilize the backfill soil, then no modification is necessary. 3. Determining seismic performance category: The seismic performance category indicates the level of seismic design requirements for the bridge. The higher the category number, the higher the requirements of seismic design, either using PBD or FBD. The seismic performance category is determined based on the principal period of the bridge in the considered direction, site specific spectral acceleration values at 2%/50 hazard level (the site specific spectral accelerations are the UHS spectral values for reference site C modified by the site coefficients for the site class at the bridge site), and the importance of the bridge. Table A.2 copies the CSA S6-14 Table 4.10 for determining the seismic performance category of a bridge. In case the seismic performance categories of a bridge for the two principal directions are different, the higher performance category governs. 4. Determining minimum analysis requirements and design approach: CSA S6-14 does not mandate PBD for design of all bridges. Table A.3 copies the design approach requirements of the code (for regular and irregular bridge specification see Clause 4.4.5.3.2 of CSA S6-14). For lifeline bridges regardless of their seismic performance category and regularity, PBD is the only acceptable design approach, while major-route and other bridges can in some cases be designed using FBD approach. It is evident that all bridges may be designed according to PBD requirements. 23 The code also recommends minimum analysis requirements for each of the specified hazard levels, as copied in Table A.4 and Table A.5. In these tables, ESA is elastic static analysis, EDA is elastic dynamic analysis (response spectrum analysis, elastic time-history analysis), ISPA is inelastic static pushover analysis, and NTHA is nonlinear time-history analysis. 5. Determining minimum performance levels and performance criteria: CSA S6-14 defines minimum performance levels in terms of serviceability and tolerable structural damage criteria at each of the three hazard levels, as shown in Table A.6. The minimum performance levels are determined based on the importance of the bridge. For instance, the minimum performance level of a major-route bridge at 10%/50 hazard level is minimal damage and immediate service. Once the minimum performance levels are determined, the PBD criteria can be specified from Table 4.16 of the code. For instance, to ensure minimal damage in RC bridges, the compressive strains of concrete should not exceed 0.004 and the reinforcing steel strains should not exceed yield. The criteria in Table 4.16 are not defined specifically for a particular type of bridge or lateral force resisting system. Additional performance criteria may be established for a bridge depending on the type of the bridge structure and its importance. 6. Verifying that the design meets the performance objectives: In order for the bridge to meet the PBD requirements, it must satisfy all the specified performance criteria at each hazard level. However, the code does not provide specific recommendations regarding how engineers should verify that the design meets the performance objectives. 24 Chapter 3: Merits and Challenges of the CSA S6-14 PBD Approach This chapter highlights the merits of the provisions of seismic performance-based design (PBD) in CSA S6-14 (2014) and identifies the challenges that need to be addressed to successfully implement these provisions in practice. The challenges are categorized as either “general” or “particular”. General challenges are the main challenges that still hinder the implementation of PBD procedures in design guidelines, including CSA S6-14. Particular challenges however, concern only the CSA S6-14 PBD approach and require improvement to the current approach in future editions of the code. Finally, based on the discussion presented in this chapter, a number of the identified challenges are selected to be addressed later in the following chapters of the thesis. 3.1 General Challenges of Performance-Based Procedures Some of the challenges of the first generation PBD procedures were outlined in Section 2.2. The next generation performance-based earthquake engineering also faces a number of challenges ahead of successful implementation in practice. It should also be noted that the necessary basis for implementation of PBD procedures is significantly less developed for bridge structures compared to buildings. With that in mind, some of the major challenges of performance-based procedures with respect to design and assessment of bridges in general are as follows: 1. There is not a unified agreed-upon framework for performance-based design/assessment. The PEER PBEE framework has been the most referenced framework for research applications. However, as explained in Section 2.4, seismic design guidelines and codes tend to simplify the design problem and incorporate simpler deterministic frameworks. 2. The next-generation PBD procedures heavily rely on the use of fragility curves within probabilistic frameworks. There is still a resistance within the engineering community to use probabilistic frameworks that utilize fragility curves. Probabilistic frameworks have gained acceptance for only some specific design problems such as the design of bridge piers against vessel impacts (see Foschi 2007), or in corrosion studies. This lack of interest is partly due to the fact that practicing engineers may not be fully familiar with the probability concepts and reliability analysis and how they can be applied to engineering problems. But primarily, the lack of interest is due to disbelief in the usefulness of fragility curves for design problems. 25 3. Another obstacle in utilizing a probabilistic framework for the next-generation PBD procedures in many instances is the lack of reliable data for defining the intervening random variables. This would result in incorporating inaccurate or subjective statistical inferences to obtain the properties of the corresponding probability distributions of the variables. 4. Frequently, the tools required for implementation of a probabilistic framework for the next-generation PBD procedures are not readily accessible outside of academia or available in commercial software packages. One example is reliability analysis. Despite the many software packages that have been developed to perform reliability analysis, none have been integrated seamlessly into commercial structural analysis programs. In addition, in many cases the developed tools are intended for design and assessment of buildings only. 5. There are very few guidelines and documents dedicated exclusively to the implementation of next generation PBD procedures. The most comprehensive guidelines on the subject are FEMA P-58-1 (2012). The second volume provides a step-by-step guide on the implementation of the methodology introduced in the first volume, and the third volume is dedicated to the supporting electronic material and documentations. No equivalent guidelines have been published so far that specifically addresses performance-based seismic design of bridges. 6. The complexity of the next generation performance-based assessment procedures, demands enormous computational effort even for the design of simple facilities. Designing by trial and error utilizing the next generation PBD procedures could become unfeasible considering the current limitations. 7. To accurately predict the extent of structural damage, damage models are needed that quantify the structural damage in terms of measurable and meaningful engineering parameters. A damage model is essentially a capacity model that relates the structural damage to engineering parameters. Examples of damage models are fragility curves, damage indices, and other predictive expressions. The values of engineering parameters should be obtainable via structural analysis, so that calculated demands can be checked against the capacity obtained from damage models. While there has been a great deal of effort invested in developing more accurate and reliable damage models of structural components, it seems 26 that there is still a huge gap in knowledge for quantifying physical damage. Some components such as concrete columns have been the subject of many studies. For these components, more accurate damage models exist that quantify the level of damage at different damage states of the components. Example of such models can be found in the work of Berry and Eberhard (2003) on the flexural damage of concrete columns. They have developed models that predict the strains corresponding to concrete cover spalling and bar buckling damage states for concrete columns. Employing these models, one is able to calculate the associated drift of a column corresponding to the first occurrence of the two aforementioned damage states using the Priestley et al. (1996) procedure. However, for many other components, predictive damage models have not yet been developed and the quantification of damage relies on subjective interpretations of physical damage in terms of engineering demand parameters. In addition, the developed damage models may not necessarily provide accurate predictions of damage. For instance, all fragility curves, and many of the other damage models take only one engineering parameter as an input. However, it is argued that the state of damage in a component cannot be described accurately utilizing only a single input parameter. Example of such arguments can be found in Gardoni et al. (2002). The aforementioned general challenges of implementing PBD can be summarized as the follows: 1. Need for a unified agreed-upon framework (probabilistic vs deterministic, academic vs codified) 2. Disbelief in usefulness or lack of familiarity with probabilistic performance-based frameworks in practice 3. Need for more reliable data to use with probabilistic performance-based frameworks 4. Need for available commercial tools for enhancing the use of probabilistic procedures 5. Need for guidelines for implementing the PBD procedures for other structures than buildings 6. Need for a systematic performance-based design procedure to avoid high computational costs 7. Need for more accurate predictive damage models for all structural components 27 3.2 Particular Challenges of the CSA S6-14 PBD Approach Aside from the general challenges with the next-generation performance-based design, there are particular challenges in the way of implementing the CSA S6-14 PBD provisions: 1. The code does not provide a clear framework for verifying that a design meets the specified performance criteria of the PBD approach. To be more specific, it is not clear how the performance criteria of the code should be formulated in terms of performance limit states and be evaluated for a structure. From the review of literature presented in Chapter 2:, the lack of such framework becomes evident. 2. There is inconsistency in how the deformation-controlled and force-controlled performance criteria of the code should be verified. To verify force-controlled limit states such as the shear of ductile components, the code recommends using FBD approach, which employs load and resistance factors. In contrast, for verifying the deformation-controlled performance criteria such as the strain limits of RC bridges, no load and resistance factors are specified. This will result in inconsistent reliability achieved for the deformation-controlled versus the force-controlled limit states. 3. The performance criteria of the code are primarily defined in terms of the performance of components. It is not clear if the criteria are met, how the performance of components would relate to the performance of system. It was mentioned earlier in Section 2.2, that ASCE 41 links target building performance levels to structural and non-structural performance levels. Similar concept could also be used in relating the performance of components or sub-assemblies to the performance of bridge structural system. 4. It is not specified whether a bridge designed to meet the code performance levels, will achieve a uniform reliability at all hazard levels or this is of any concern at all. 5. The recommended minimum analysis requirements of the code include elastic dynamic analysis and linear time history analysis (Table A.4 and Table A.5). These methods may provide reasonable estimation of inelastic global displacement demand if the conditions of equal displacement rule hold. Unlike ASCE-41, the code does not recommend modifications to elastic displacements to incorporate the effects of hysteric behaviour. Moreover, elastic 28 analysis procedures cannot be directly used to check local strain response, the amount of dissipated energy, or other engineering demand parameters that are indicators of structural damage. There is no clear recommendation in the code and its commentary on how these methods should be utilized for such purpose. 6. Other than the above challenges, there are some ambiguities in some of the phrases in the code. An important one is in the use of expected versus nominal material properties for performance evaluations. It is stated that for the prediction of the extensive damage and probable replacement performance levels, expected material properties may be used. Following that, two different sets of material properties will be required for evaluating the different performance levels of a bridge. This may lead to inconsistent interpretations of bridge performance at different hazard levels. The aforementioned particular challenges of implementing the CSA S6-14 PBD approach can be summarized as follows: 1. Need for a clear design/assessment framework for the PBD approach 2. Inconsistency in verifying deformation-controlled versus force-controlled performance criteria 3. Vaguely defined relation of component performance to system performance 4. Vaguely definition of design goals and minimum target reliabilities 5. Vague guidelines on using elastic analysis methods to evaluate the structural damage parameters 6. Ambiguity in the use of expected and nominal material properties for performance evaluation 3.3 Merits of CSA S6-14 PBD approach All the limitations of the FBD approach mentioned in Section 2.1, which are addressed by the PBD approach, are only part of the merits of the PBD approach. The most important motivation behind introducing the PBD approach in CSA S6-14 was to encourage and require the discussion of expected post-earthquake performance of bridges between owners and engineers. PBD facilitates a wide range of opportunities in conceptual and detailed design of innovative and low-29 damage structural systems. Most often the seismic performance of such systems cannot be adequately demonstrated using the FBD approach. Kennedy et al. (2017) provide examples of such systems, including system with: Base isolated bearings (lead-rubber bearings, laminated elastomeric bearings, friction-pendulum bearings, sliding systems, or combination of these items) Dampers (shock absorbers) and lock-up devices (shock transmitters) Ductile fuses yielding in flexure or shear Ductile intermediate or end diaphragms Bucking restrained braces or ductile or semi-ductile braces Rocking foundations FRP (for improved ductility or strength) 3.4 What Is Required to Implement the CSA S6-14 PBD Approach in Practice The general and particular challenges of the CSA S6-14 PBD approach should not undermine its merits over the traditional FBD approach and the value that it brings to the seismic design practice. Rather these challenges should be viewed as opportunities to take the next step towards implementing the PBD approach in practice. Some of the general challenges mentioned in Section 3.1 need more resources and time to be addressed, and in some cases, are subjects of many ongoing research studies. Producing more test data and developing more accurate models and predictive equations for damage are under this category. However, the successful implementation of CSA S6-14 PBD approach is not conditioned on addressing all of the challenges at once. In fact, the particular challenges listed in the previous section are the ones, which require immediate attention. In line with these particular challenges, the following actions items are required: 1) Providing an assessment framework for the CSA S6-14 PBD approach; such framework should be simple enough to implement and should incorporate the elements of the code PBD approach. 2) Defining acceptance criteria for verifying the code deformation-controlled limit states. This would entail defining and calibrating load and resistance factors for deformation performance criteria. 30 3) Investigating whether the CSA S6-14 performance levels result in a uniform reliability across hazard levels or certain performance levels will control the design. 4) Investigating appropriate structural models and analysis techniques, which would allow predicting the performance criteria of the code with acceptable accuracy. 5) Providing case studies on implementing the CSA S6-14 approach including the suggested changes for engineers in practice. The subsequent chapters of this manuscript aim to tackle the above action items within the scope of the thesis. 31 Chapter 4: Assessment Frameworks for CSA S6-14 PBD This chapter recommends a deterministic and a probabilistic assessment framework for verifying the PBD of bridges according to CSA S6-14. The chapter starts by introducing assessment framework options and listing the requirements of such a framework for the CSA S6-14 PBD approach. Subsequently, the details of the two frameworks are described. The deterministic assessment framework is formulated based on the reviewed literature. For the probabilistic framework, demand and capacity factor design (DCFD) developed by Jalayer and Cornell (Jalayer 2003; Jalayer and Cornell 2003) is considered suitable for further examination. 4.1 Assessment Framework Requirements for CSA S6-14 PBD Ideally, the assessment framework for CSA S6-14 PBD should have the following features: 1. Should enable engineers to verify that their design meets the specified performance limit states with acceptable probability of exceedance and satisfies minimum target reliabilities as design objectives. 2. Should account for the uncertainties involved in the design problem. 3. Should be simple enough so that it can be easily adopted and implemented in practice. 4. The necessary tools for implementing the framework should be available to practicing engineers. 5. Should be robust enough to be applicable to design of all types of bridges, and accommodate future improvements. 4.2 Deterministic Assessment Framework for CSA S6-14 PBD Deterministic frameworks are referred to as deterministic, since they do not incorporate random variables as inputs, and the outputs of the assessment have deterministic values. The deterministic framework here follows the CSA S6-14 PBD approach step-by-step (see Section 2.6 and Appendix A). However, it introduces additional elements for performance verification. The additional elements that were considered in formulating the framework are as follows: i. Reserve Capacity of a Structural Component or System: Reserve capacity is defined as the ratio of the difference between capacity and demand to capacity, as follows: 32 𝑅𝐶𝑖𝑗=1-𝐷𝑖𝑗/𝐶𝑖 (4.1) where, RCij is the reserve capacity of a structural component or system for action i at hazard level j, Dij is the demand for action i at hazard level j, and Ci is the capacity of the component or system for action i. ii. Target Reserve Capacity of a Structural Component or System: The target reserve capacity represents the design objective, which should be met for design to be deemed acceptable. The target reserve capacity is denoted here as RCij*, such that: 0≤ 𝑅𝐶𝑖𝑗*≤1 (4.2) iii. Acceptance Criteria for Performance Limit States: The performance of a structural component or system with respect to a performance criterion is deemed acceptable if the reserve capacity of that component or system for the action associated to the performance criterion is greater than or equal to the target reserve capacity of the component for the same action. 𝑅𝐶𝑖𝑗≥𝑅𝐶𝑖𝑗* (4.3) In the above definitions, component refers to the individual members of a system and in this case a bridge structural system and actions are deformations and forces associated to independent degrees of freedom of a component. 4.2.1 Implementation Steps The steps of the deterministic framework for CSA S6-14 PBD are summarized in the flowchart of Figure 4.1, which are as follows: 1. Determine importance, regularity, and seismic performance category of the bridge. 2. Perform preliminary design and service load design. Initial proportioning of the members may be carried out based on experience or force-based design equations. 3. Identify possible ductile and non-ductile local and global failure mechanisms of the bridge. For instance, flexural failure of concrete columns is a local ductile failure mechanism, shear failure of concrete columns is a local non-ductile failure mechanism, and unseating of deck is a global failure mechanism. 33 4. Determine appropriate performance criteria at the three hazard levels of 2%/50, 5%/50, 10%/50, based on the minimum performance requirements of the code and the identified failure mechanism. The performance criteria should be expressed in terms of the limit states of measurable displacement quantities such as strains, curvatures, etc., or forces such as shear strength. 5. Obtain the UHS for the bridge site with the 2%, 5%, and 10% probabilities of exceedance in 50 years and calculate the bridge design spectra at those hazard levels. 6. Depending on minimum analysis requirements, obtain seismic demands using either response spectrum analysis (RSA) or non-linear time history analysis (NTHA). If performing nonlinear time history analysis, select and scale a appropriate suite of ground motions records following the recommendations of CSA S6-14 commentary, or more sophisticated approaches as outlined in NIST GCR 11-917-15 (2013) or similar references. 7. Perform structural analysis and determine the seismic demands from response spectrum analysis or the mean demands from nonlinear time history analysis at each of the specified hazard levels. 8. Determine capacities of the structural components for each of the related performance limit states. The drift capacities of ductile components and the structural system can be determined using nonlinear static pushover analysis. To do so, conduct pushover analysis of the entire bridge in each of the two principal axes of the structure. Determine drift ratios in each of the two principal directions that correspond to the first occurrence of the limit states defined in Step 4 that are related to the ductile failure mechanisms. These would be the drift capacities for those limit states. For force-controlled actions such as shear, CSA S6-14 FBD or BC MoTI Supplement equations may be utilized to calculate capacities. 9. Calculate reserve capacities at each hazard level for each of the limit states defined in Step 4. 10. Determine desired target reserve capacities for the considered limit states. It should be noted that the capacities calculated for force-controlled actions using FBD equations, are factored capacities, unlike the drift capacities obtained as explained in Step 8. 34 11. Check the reserve capacities from Step 9 against the target reserve capacities from Step 10. If the reserve capacities are greater than or equal to the target reserve capacities, the design is meeting the specified PBD requirements successfully; if not redesign the components that do not meet the considered criteria and repeat steps 7 to 11. If the period of the modified design changes from the initial design, or if the bridge system changes, return to Step3. If NTHA is conducted to obtain seismic demands, the performance limit states should be also checked for each of the selected ground motion records. Per recommendation of the code commentary, the number of ground motions, for which the bridge does not meet the specified performance criteria, should be limited to one. 35 Figure 4.1 The deterministic PBD assessment framework flowchart 36 4.3 Probabilistic Assessment Framework for CSA S6-14 PBD Upon reviewing the literature on performance-based design procedures and with regards to the requirements of Section 4.1 for a suitable assessment framework, the demand capacity factored design developed by Jalayer and Cornell (Jalayer 2003; Jalayer and Cornell 2003) appears to be a suitable probabilistic framework to be adopted for the CSA S6-14 PBD approach. The selected framework offers several advantages. Both Equation (4.19) and (4.20) provide simplified closed-form solutions for verifying a performance limit state for a desired MAF of exceedance with measurable confidence. The simplicity of these equations makes them advantageous for using in engineering practice. The terms in both equations can be readily obtained with the current available tools in practice and do not demand extra computational beyond what is already required by the code. A summary of the framework formulation and main expressions is given in the following section. 4.3.1 Demand and Capacity Factored Design (DCFD) Demand and capacity factor design is a probability-based framework for seismic design and assessment. Developed by Jalayer and Cornell, the framework is in fact a special case of the PEER framework, where a scalar binary decision variable is employed that takes value of 1 when the capacity for a limit states is exceeded and takes 0, otherwise. The main purpose of the methodology is to ensure that the structural seismic design meets the specified performance objectives with a desired guaranteed degree of confidence. The degree of confidence is measured by setting an upper confidence bound on the probability of exceeding a performance limit state. The work of Jalayer and Cornell has later been improved upon by other researchers, an example of which can be found in Mackie et al. (2008). Here, only the main expressions employed in the framework are reproduced for the convenience of the readers. A full derivation of the framework can be found in Jalayer (2003) and Jalayer and Cornell (2003). 4.3.1.1 Mean Annual Frequency of Exceeding a Limit State Considering Aleatory Uncertainty The mean probability of exceeding a performance limit state is formulated with the following two approaches (Jalayer 2003): 37 1. Displacement-based approach: PLS=P[D>C]= ∑ ∑ P[all xD>C|D=d] P[D=d|Sa=x] P[Sa=x]all d (4.4) 2. IM-based approach: PLS=P[D>C]= ∑ P[all xD>C|Sa=x] P[Sa=x] (4.5) Both of the above formulations, use total probability theorem to decompose the probability of demand exceeding capacity into a number of probabilities, which are easier to calculate. P[Sa=x] is the likelihood that the intensity measure (in this case spectral acceleration) will be equal to a specific value. This term can be obtained from probabilistic seismic hazard analysis (PSHA) of the site. The other intermediate terms are conditional probabilities, which can be calculated from structural analysis and damage analysis. To facilitate the calculation of the intervening terms in Equations (4.4) and (4.5), some simplifying assumptions are typically made. The first is to assume that the hazard curve in the vicinity of Sa=x to follow a power-law relation as below (Jalayer 2003): 𝜈𝑆𝑎(Sa)=P[Sa>x]≅kox-k (4.6) in which 𝜈𝑆𝑎 is the mean annual frequency (MAF) of exceeding the intensity level Sa. Parameters of the power-law relation, ko and k, can be found by fitting a line to the hazard curve in the log-log scale at Sa=x. It is also assumed that the median of the demand at each hazard level has a power-law relation with the intensity measure, and its aleatory uncertainty can be quantified by a lognormal random variable (Jalayer 2003): 𝐷 = ?̂?𝐷|𝑆𝑎(𝑥)𝜀 = 𝑎𝑥𝑏𝜀 (4.7) For the lognormal random variable ε, the median is ?̂?ε=exp(mean(ln(ε))=1, and the lognormal standard deviation is σln(ε)=βD|Sa also denoted by βRD. The parameters for both the median power-law relation and the lognormal random variable ε are extracted from the results of structural analysis performed at the specified hazard levels. Another simplifying assumption is made for limit state threshold or capacity. It is assumed that the random variable C can be also 38 characterized as a lognormal random variable with median ηc, and lognormal standard deviation βRC. With the aforementioned assumptions and using the displacement approach, the MAF of demand exceeding capacity is formulated once only considering the aleatory uncertainty, as follows (Jalayer 2003): ?̅?LS=νP[D>C]=?̂?Sa(Sa?̂?C) exp( k2 2b2βRD2⁄ ) exp( k2 2b2βRC2⁄ ) =ko(?̂?ca)-k/bexp( k22b2βRD2⁄ ) exp( k2 2b2βRC2⁄ ) (4.8) In the above formula, the hazard level corresponding to the median capacity is multiplied by two terms that account for the aleatory uncertainties in demand conditioned on intensity measure, and capacity. 4.3.1.2 Mean Annual Frequency of Exceeding a Limit State Considering Aleatory and Epistemic Uncertainty While Equation (4.8) considers sources of aleatory uncertainty in calculating the MAF of demand exceeding capacity, it does not address epistemic uncertainty3. To improve the equation with this regard, it is assumed that the intervening random variables can be defined in the following form (Jalayer 2003): 𝑋 = ?̂?𝑥 𝜀𝜂 𝜀𝑥 (4.9) where ?̂?𝑥 is the current point estimate of median X, εη and εx are both lognormal random variables with unit median to represent epistemic and aleatory uncertainties respectively. Using the above form for the intervening random variables, the MAF of demand exceeding capacity is obtained as (Jalayer 2003): ν̅LS=?̂?Sa(Saη̂C) exp(1/2 βUH2) exp( k22b2(βRD2+βUD2)⁄ ) exp( k2 2b2(βRC2+βUC2)⁄ ) (4.10) In the above expression, βUH, βUD, and βUC, refer to lognormal standard deviation for epistemic uncertainty in hazard, demand, and capacity, respectively. A comparison between Equations 3 Aleatory uncertainty is due to inherent randomness, while epistemic uncertainty is due to uncertainty in knowledge. 39 (4.8) and (4.10) reveals that the latter equation is essentially the former equation times the terms that account for the epistemic uncertainty in hazard, demand, and capacity. It is then assumed that the epistemic uncertainty in hazard can be accounted for by using mean hazard instead of median, as follows (Jalayer 2003): ν̅LS=ν̅𝑆𝑎(Saη̂C) exp( k2 2b2(βRD2+βUD2)⁄ ) exp( k2 2b2(βRC2+βUC2)⁄ ) (4.11) Where: ν̅𝑆𝑎(Saη̂C)= ?̂?Sa(Saη̂C) exp(1/2 βUH2) =ko(?̂?ca)-k/bexp(1/2 βUH2) (4.12) 4.3.1.3 Factored Demand and Capacity Format Assuming a maximum allowable MAF of exceedance for a limit state denoted by Po, such that: ν̅LS≤Po (4.13) It is possible to rearrange Equation (4.11) as follows (Jalayer 2003): a(Po/ko)-b/kexp( k 2b (βRD2+βUD2)⁄ )≤ η̂c exp( -k 2b (βRC2+βUC2)⁄ ) (4.14) The LHS of the above inequality is the factored demand corresponding to MAF of exceedance of Po, and the RHS is the factored capacity. To distinguish between the involved factors, they can be denoted as the following (Jalayer 2003): - Demand factor representing the aleatory uncertainty (in case of NTHA, record-to-record variability) in demand 𝛾𝑅 = exp( k 2b βRD2)⁄ (4.15) - Demand factor representing epistemic uncertainty in demand 𝛾𝑈 = exp ( k 2b βUD2)⁄ (4.16) - Capacity factor representing aleatory uncertainty in capacity ϕ𝑅= exp( - k 2b βRC2)⁄ (4.17) - Capacity factor representing epistemic uncertainty in capacity 40 ϕ𝑈= exp( - k 2b βUC2)⁄ (4.18) Using the above notations, Equation (4.14) can be written as: γRγUD (Po) ≤ ϕRϕUC γD (Po) ≤ϕC (4.19) The above inequality is the demand and capacity factored design (DCFD) format. This representation is very similar to LRFD format and therefore is very convenient for use in practice. 4.3.1.4 Factored Demand and Capacity Format with Confidence Level The final addition to the DCFD format is to build a confidence interval around the limit state frequency of exceedance. As such, it could be then stated that the structure would meet a performance criterion for an allowable MAF of exceedance Po with a confidence level of x%. The confidence interval is added to the DCFD format of Equation (4.19) through an additional factor as follows (Jalayer 2003): γD (Po) ≤𝜆𝑥ϕC (4.20) where λx is the confidence factor corresponding to confidence level x, defined as (Jalayer 2003): λx=exp(-βUT(Kx - k 2b βUT))⁄ (4.21) In the above equation Kx is the standard Gaussian (Normal) variate associated with probability of x not being exceeded, and βUT is the total epistemic uncertainty calculated as follows: βUT=√βUD2+βUC2 (4.22) Using the DCFD format in Equation (4.20), the performance of a structure for a performance criterion is deemed acceptable if the ratio of factored demand to capacity is less than λx. The difference between Equation (4.20) and (4.19) is the addition of λx. It can be considered that Equation (4.19) is a special case of Equation (4.20), where λx=1. The DCFD format has been implemented in FEMA-351 (2000b), where Equation (4.20) represents the acceptance criteria for evaluating the performance of existing welded steel 41 moment-frame building in terms of interstory drift, column axial load, and column splice tension. In commentary of Section 3.6.1 of FEMA-351, it is explained that the confidence in the above format is “calculated as a function of the number of standard deviations that factored-demand-to-capacity-ratio λ lies above or below a mean value” and provides “a measure of the extent that predicted behaviour is likely to represent reality.” 4.3.2 Implementation Steps To implement the DCFD framework, the following parameters need to be determined: 1) Hazard parameters k and ko - see Equation (4.6) 2) Demand parameters a and b - see Equation (4.7) 3) Estimate of aleatory and epistemic uncertainties βRD, βUD, βRC, and βUC - Calculating hazard parameters k and ko The hazard curve parameters in Equation (4.6) can be obtained readily by fitting a power-law curve in Microsoft Excel or similar tools to the segment of the hazard curve spanning between 2%/50, 5%/50, and 10%/50 hazard levels. To plot this segment, the values of the UHS spectral acceleration at the fundamental period of the bridge should be plotted against the MAF of that hazard level (hazard levels 2%/50, 5%/50, and 10%/50 have MAF of 0.000404, 0.00106, and 0.002105, respectively). - Calculating demand parameters a and b The demand curve parameters in Equation (4.7) can also be obtained by fitting a power-law curve in Microsoft Excel or similar tools to the median of the demand parameter from NTHA at 2%/50, 5%/50, and 10%/50 hazard levels. - Estimate of aleatory and epistemic uncertainties βRD, βUD, βRC, and βUC The aleatory uncertainty in demand, i.e. βRD, is the only parameter that needs to be calculated. For the other three uncertainty parameters, the values recommended in the literature can be utilized. βRD is calculated by taking the standard deviation of the natural logarithm of the demand parameter values in Microsoft Excel or similar tools. Since the NTHA is run at three hazard levels of 2%, 5%, and 10% probability of exceedance in 50 years, three values of βRD will be obtained. The assumption in developing the DCFD framework is that βRD does not change with 42 the change in the intensity measure (Jalayer 2003). As such, the arithmetic mean of the βRD values at the three hazard levels can be used for the DCFD framework. There are a few references in the literature, which provide estimate values for the other three uncertainty parameters, βUD, βRC, and βUC, in case there is no option for more accurate measure of these parameters4. Previously, FEMA-350 (2000b) and FEMA P-695 (FEMA 2009) contained reference values for these parameters, but the most recent recommended values can be found in Chapter 16 of ASCE/SEI 7-16 (2016). The document provides two different sets of values for the uncertainty parameters to be used with component force and deformation. The uncertainty values recommended for component deformation capacity are about twice as large as those of the component force capacity, due to the lack of enough available test data on the former. For later reference in the thesis, the uncertainty values recommended for deformation-controlled actions are reproduced in Table 4.1 and Table 4.2. Source of Uncertainty in the Deformation Value Record-to-record variability for MCER ground motions (βRD) 0.4 Uncertainty from estimating deformation demands using structural model (βUD) 0.2 Variability from estimating deformation demands from mean of only 11 ground motions (βRD) 0.13 Total 0.46 Table 4.1 ASCE/SEI 7-16 assumed uncertainty values for component deformation demand Source of Uncertainty in the Deformation Value Typical variability in prediction equation for deformation capacity from available data (βRC) 0.6 Typical uncertainty in prediction equation for deformation capacity due to extrapolation beyond data (βUC) 0.2 Uncertainty in as-built deformation capacity because of construction quality and errors (βUC) 0.2 Total 0.66 Table 4.2 ASCE/SEI 7-16 assumed uncertainty values for component deformation capacity The steps of implementing the DCFD framework for CSA S6-14 PBD are summarized in Figure 4.2, which are as follows: 4 The value of βRC or the aleatory uncertainty in capacity can be more accurately measured from test data. The epistemic uncertainty in capacity, βUC, can also be estimated from the bias in test data. The source of βUD or the epistemic uncertainty in demand according to the Appendix A of FEMA-350 is the inaccuracies in defining the modelling parameters such as yield strength, viscous damping, foundation flexibility, etc. To calculate this parameter, a series of structural models should be developed, in which the aforementioned parameters are varied and time-history analysis is performed to calculate the demands. 43 1. Determine the bridge importance, regularity, and seismic performance category. 2. Perform preliminary design and service load design. Initial proportioning of the members may be carried out based on experience or force-based design equations. 3. Identify possible ductile and non-ductile, local and global failure mechanisms of the bridge. For instance, flexural failure of concrete columns is a local ductile failure mechanism, shear failure of concrete columns is a local non-ductile failure mechanism, and unseating of deck is a global failure mechanism. 4. Determine appropriate performance criteria at the three hazard levels of 2%/50, 5%/50, 10%/50, based on the minimum performance requirements of the code and the identified failure mechanism. The performance criteria should be expressed in terms of the limit states of measurable displacement quantities such as strains, curvatures, etc., or forces such as shear strength. 5. Obtain the UHS for the bridge site with the 2%, 5%, and 10% probabilities of exceedance in 50 years by performing probabilistic seismic hazard analysis, or alternatively, use the tabulated values by the National Building Code of Canada (2015). 6. Calculate hazard parameters k and ko. 7. Select and scale an appropriate suite of ground motions records, following the recommendations of CSA S6-14 commentary, or more sophisticated approaches as outlined in NIST GCR 11-917-15 (2013) or similar references. 8. Perform NTHA and determine seismic demands at each of the specified hazard levels. 9. Calculate demand parameters a and b. 10. Determine capacities of the structural components for each of the related performance limit states. The drift capacities of ductile components and the structural system can be determined using nonlinear static pushover analysis. To do so, conduct pushover analysis of the entire bridge in each of the two principal axes of the structure. Determine drift ratios in each of the two principal directions that correspond to the first occurrence of the limit states defined in Step 4 that are related to the ductile failure mechanisms. These 44 would be the drift capacities for those limit states. For force-controlled actions such as shear, CSA S6-14 FBD equations may be utilized to calculate capacities. 11. Compute the demand and capacity factors using Equation (4.15) to (4.18). 12. Compute the confidence factor corresponding to the desired level of confidence (for example 90%) using Equation (4.21). 13. Check factored drift demand against factored drift capacity for the desired confidence interval in Step 12, using Equation (4.20). The factored demand at each hazard level is checked against the factored capacity of those limit states, which are specified by the code to be checked at that hazard level. Check force-controlled actions using the CSA S6-14 FBD approach. If the factored capacities are greater than or equal to factored demand, the design is meeting the specified PBD requirements successfully; if not redesign the components that do not meet the considered criteria and repeat steps 8 to 13. If the period of the modified design changes from the initial design then return to Step 6, or if the bridge system changes return to Step3. 45 Figure 4.2 The probabilistic PBD assessment framework flowchart 46 Chapter 5: Examining Modelling Alternatives of RC Bridges for PBD At the heart of performance-based design lies the ability to predict engineering demand parameters that relate to structural performance from analysis. The key to achieve this is generating appropriate models for structural analysis. An “appropriate model” in this manuscript refers to any model, which has the following two main characteristics: 1. It enables evaluating the specified performance criteria of CSA S6-14 through predicting the associated demand and/or capacity parameters. 2. It is computationally affordable and sufficiently accurate to use in practice. The main objective of this chapter is to examine and compare the common structural modelling options used in practice and academia for PBD of RC bridges. Special attention is given to distributed plasticity models with inelastic beam-column elements and fibre sections, as these models are very advantageous for evaluating local strain response of RC bridges. However, the use of these models is challenged by mesh-dependency and localization issues, as will be explained in this chapter. Some techniques such as material model regularization have been proposed in the literature to deal with the mesh-dependency issue will be reviewed later. The distributed plasticity models are subsequently utilized to predict the response of a tested RC bridge column. The recommended methods for dealing with the mesh-dependency issue are examined by comparing the test results with the model predictions and the effectiveness of each method is assessed. Concentrated plasticity models are also utilized to predict the response of the test column and comparison are made between the advantages and disadvantages of using each of the distributed and concentrated plasticity models with respect to evaluating the strain limits of CSA S6-14 PBD. The applicability of the final takeaways is examined on four more bridge column tests. 5.1 Modelling Alternatives for RC Bridges Structural models of bridges can be classified in several different ways with respect to how they represent the geometry and behaviour of the actual structure. Each way of classification, allows comparing models with respect to the feature that the classification is based on. For example, bridge models are sometimes categorized in terms of the hierarchy of bridge structural system into global, frame, and bent models. It is also possible to classify bridge models based on the 47 level of complexity in describing the geometry of structure, into three categories of lumped parameter models, spine or grid models, and continuum models (Priestley, M. J. N. et al. 1996; Caltrans 2015). The focus of this chapter however, is on another way of classification, which is concerned with modelling the inelastic response of bridge components due to material nonlinearity. This classification is especially meaningful for seismic analysis, since it identifies the capabilities of different models in estimating the various inelastic demand and capacity parameters that are used in seismic design evaluation. Nonlinear models are often divided into two categories based on the degree of idealization in the model (Figure 5.1). At one end, there are distributed plasticity models that are detailed in the physical presentation of structures and their components. At the other end, there are concentrated plasticity models, which are primarily phenomenological models of the nonlinear behaviour of structural components (PEER and ATC 2010). It is also possible to estimate inelastic seismic demands using modified elastic models. Each of the above three modelling alternatives, namely concentrated plasticity, distributed plasticity, and linear elastic models with cracked sections properties, are briefly reviewed and discussed in the following sections. Figure 5.1 Types of nonlinear component models: (a) and (b) are concentrated plasticity models; (c), (d), and (e) are distributed plasticity models (NEHRP 2013) 5.1.1 Linear Elastic Models with Cracked Section Properties Linear elastic models are used when an elastic method of analysis such as EDA or ESA is employed to estimate seismic demand parameters, or to obtain structural periods. To account for the nonlinearity due to the initial cracking and yielding of concrete members, effective section properties are assigned to these members. It is important to use reasonable values for effective section properties, as the outcomes of elastic analysis are very sensitive to these values. There 48 are several recommendations regarding how the effective section properties should be determined for various concrete members. Section 4.4.5.3.3 of CSA S6-14 defines the effective flexural stiffness as the slope of the moment-curvature curve between the origin and the point corresponding to the first yielding in the reinforcement steel, as follows: EcIeff = Myϕy (5.1) In the above expression, Ec is the elastic modulus of concrete, Ieff is the effective moment of inertia of the section, My is the moment at first yield of the section, and ϕy is the curvature of the section at first yield. The effective shear stiffness of ductile RC elements is determined based on the effective flexural stiffness: (GA)eff = Gc AcvIeffIg (5.2) Where (GA)eff is the effective shear stiffness of section, Gc is the shear modulus of concrete, Acv is the effective shear area that may be taken equal to cross-sectional area, and Ig is the gross moment of inertia of section. Caltrans Seismic Design Criteria 1.7 (2013) uses a similar definition for the effective flexural stiffness of concrete sections as CSA S6-14. Alternatively, it allows finding the flexural stiffness of reinforced concrete columns from the charts developed by Priestley et al. (1996) for circular and rectangular cross sections. In these charts, the ratio of the effective flexural stiffness to the gross flexural stiffness is a function of axial load ratio and longitudinal reinforcement steel ratio (Figure 5.2). Caltrans also recommends using 0.5Ig-0.75Ig to estimate the effective flexural stiffness of box girder superstructures, noting that the lower bound value represents lightly reinforced sections and the upper bound, heavily reinforced sections. The shear and torsional stiffness of the section should also be modified according to the recommendations of Caltrans SDC. The effective shear area of concrete members can be taken as 80% of the gross area of the sections. Caltrans considers a significant reduction in the torsional moment of area of columns after the onset of cracking and assumes that the effective torsional moment of inertia of concrete columns can be reduced to 20% of the original value. 49 Figure 5.2 Priestley et al. chart for finding the effective stiffness of cracked RC circular sections (1996) 5.1.2 Concentrated Plasticity Models In concentrated plasticity models, the nonlinearity and hysteretic behaviour of structural components are lumped at discrete locations of the structure, while the rest of the structure is elastic. The lumped plasticity is either in the form of a plastic hinge at a critical section of a beam-column element or in the form of nonlinear spring elements (spring hinges). Some commercial analysis tools such as CSI SAP2000 use concentrated plasticity models. In SAP2000, plastic hinges are deformation-controlled hinges assigned to pre-determined locations of beam or column elements, where nonlinear behaviour is expected to be concentrated. There are several options in SAP2000 for defining a plastic hinge, depending on which actions of a component the hinge is defined for. For instance, for the flexural response of frame elements, uncoupled bending moment hinges in either of the principal axes of a section, may be utilized or an axial force-biaxial moment interaction hinge may be used instead. Hinges can be defined in terms of sectional response, i.e. the moment-curvature response of a section or element response, i.e. the moment-rotation response of an element. If the first option is used, an approximate length needs to be defined for the assigned plastic hinges. The program will then automatically integrate the sectional response assigned to the hinges using the given length to produce element response. The hinge properties and the back-bone curve can be defined by the user. Alternatively, one can use predefined auto hinges in the program, which are based on FEMA 356 (2000a) and Caltrans SDC 1.7 (2013) recommendations. Another way of deriving hinge properties is by defining fibres for a hinge section. The sectional response is then calculated by the program from the 50 uniaxial response of individual fibres, which in turn depend on the material constitutive model assigned to each fibre. In that case, the hinge is called a fibre hinge. Fibre hinges can capture the axial force-biaxial moment interaction at the hinge section as well as the post-yield degradation and softening, but are unable to model pinching and bond slip effects. Further details and a thorough comparison of the SAP2000 plastic hinge options for columns can be found in the PEER report by Aviram et al. (2008). Another type of concentrated plasticity models are nonlinear spring hinge models. Spring hinges are individual or a set of zero-length spring elements assigned to the ends of an elastic beam-column element, which together captures the nonlinear response of that element. Just as plastic hinges, spring hinges may be defined for different actions of a component. For instance, three individual springs in series sometimes is employed to capture the interaction of axial, shear, and flexural responses of a column (Elwood and Moehle 2005; Elwood and Moehle 2004). Unlike plastic hinge option, which is not available in all structural analysis tools, spring hinges are available in most of the commercial and academic analysis tools. The strength of concentrated plasticity models is in their ability to capture the strength degradation behaviour due to bar buckling, bond slip, and shear failure. They are also very compatible with force and deformation limit state checks in codes and standards. However, hinge behaviour in these models is primarily based on empirical models rather than theory, and therefore it is dependent on the tests that the model was derived from. Moreover, the empirical models may not be available for all types of components or the existing models may not be applicable to configurations other than the tested ones. In that case, additional tests need to be conducted to develop new empirical models. Alternatively, a continuum model may be used to predict the component behaviour. 5.1.3 Distributed Plasticity Models In distributed plasticity models the nonlinear behaviour is not limited to certain points in the structure, and instead it is captured over the length, area, or volume of structural components. Examples of distributed plasticity models include continuum finite element models, fibre section inelastic beam-column elements, and finite-length hinge-zone inelastic beam-column elements. Fibre section inelastic beam column elements include multiple fibre sections along the length of 51 the element. This allows capturing the inelastic action at several points along the elements as opposed to concentrated plasticity models, where the plasticity is lumped at hinges. There are two common formulations for inelastic beam-column elements, namely displacement-based elements and force-based elements. - Displacement-based elements (DBE): Displacement-based elements are formulated based on the stiffness method. This means that for these elements, compatibility is achieved in the strong form and equilibrium is achieved in the weak form. The element deformations are interpolated from nodal displacements by using shape functions that have the basic assumptions of linear curvature and constant average axial strain along the length of the element (Pugh 2012; Pugh et al. 2015). In DBE formulation, the iterations are performed at the structure level, and there are no internal iterations at the fibre, section, or element levels. - Force-based elements (FBE): Force-based elements are formulated based on the flexibility method. Opposite to DBE, for these elements the equilibrium is satisfied in the strong form and the compatibility is satisfied in the weak form. The basic assumption of FBE formulation is constant axial load and linearly varying moment along the length of the element (Pugh 2012; Pugh et al. 2015). To make the flexibility-based formulation of FBE compatible with the displacement-based finite element programs, the formulation of FBE has additional internal iterations at the element and section levels (Coleman and Spacone 2001) on top of the iterations at the structure level. As a result, the formulation of FBE is noticeably more complex than DBE. Models using FBE require greater computational effort compared to models using DBE for the same number of elements, and may experience convergence issues. However, when the inelastic flexural response of frame elements at plastic hinges need to be calculated, FBE has an advantage over DBE. This is due to the fact that at plastic hinges the curvature is highly nonlinear and FBE assumes linear varying curvature as opposed to the constant curvature of DBE. Consequently, greater number of DBE is needed to accurately estimate the curvature at the plastic hinge compared to FBE. This will compensate considerably for the additional computational effort that individual FBE requires because of the higher number of required iterations compared to DBE. 52 Continuum finite element models are another class of distributed plasticity models that use continuum elements such as shells and solid elements. Continuum models are the most detailed class of models and do not require the simplifying assumptions of the inelastic beam-column elements about stress or strain field at the section level. They are inherently capable of capturing the interaction of various actions such as shear, flexure, and torsion within a component by detailed modelling of the geometry and the interacting mechanism within the component. The material models assigned to continuum elements can range from a simple linear elastic isotropic material model with just a few parameters to complex constitutive models with many parameters. Due to their higher level of complexity, sometimes continuum models are not categorized under the distributed plasticity models and are recognized as a separate class. The challenge with using continuum models for seismic design evaluation purposes stems from three main reasons. Firstly, continuum models are computationally way more expensive than other distributed plasticity models. This is especially problematic when the entire structure is subjected to multiple ground-motion records for NTHA and completing the analysis for a set of 11 or more records as required by CSA S6-14 may take several days. The use of continuum models are further hampered by convergence issues, when an implicit solution is used with NTHA. In that case an explicit solution is often employed, which requires even more computational effort to achieve accurate results, and besides explicit solution algorithms may not be available on all analysis platforms. Secondly, the accuracy of continuum models depends greatly on the constitutive material models, which accuracy in turn depends on the values assigned to the input parameters of the models. As a result, to guarantee accuracy of continuum models, it is often required to calibrate the constitutive model parameters to some test results. However, tests results are not always available for all input parameters and performing additional tests can become expensive. If test results are not available, then the input parameters need to be assigned subjectively. Finally, to perform code-based checks on structural components, post-processing most often is needed to obtain the relevant demand and capacity parameters from continuum models outputs. All the aforementioned reasons, limits the use of continuum models for seismic design evaluation to mostly academic purposes. 53 5.2 Localization in Distributed Plasticity Models of RC Components 5.2.1 Background in Fracture Mechanics Localization in finite element analysis refers to the localization of plastic strain in critical elements (or at sections) where strain softening is expected. The localization term has its roots in the research of fracture mechanics and the use of smeared cracking model with strain softening materials. The smeared cracking model, which was originally introduced by Rashid (1968), replaces discrete crack lines by infinitely many parallel cracks with infinitely small opening that are distributed continuously over the finite element (Bazant, Zdenek P. and Planas 1997). Strain softening is the gradual decline of stress at increasing strain and it is incorporated in the smeared cracking to reflect the growth of cracks until full fracture. As the use of smeared cracking gained popularity in the finite element analysis of concrete structures, localization and mesh sensitivity issues in these models were identified. It was recognized that a fracture cannot be consistently and objectively described by a single softening stress-strain curve, and additional conditions, called localization limiters, are necessary to prevent strain localization in finite element analysis. Bazant and Planas (1997) provide a comprehensive theoretical background on why localization occurs in finite element analysis of softening materials such as concrete. They first examine the localization under static loading in a series of N equal strain softening elements and then on a softening bar, which is the continuum model of the N discrete elements as N goes to infinity. They conclude that the assumption of a simple stress-strain curve with strain softening will result in the softening zone to have zero width and volume, the inelastic strain and fracture energy to be zero, and the finite element computations to be mesh-dependent. They report that the same conclusions apply to dynamic situations as well. Therefore, it is necessary to complement continuum formulation of strain softening materials with some localization limiter. Crack-band model proposed by Bazant and Colleagues (Bazant, Z. P. 1976; Bazant, Z. P. and Cedolin 1979; 1980; 1983) uses the simplest localization limiter. In this model, the strain softening stress-strain curve is associated with a certain width of the crack band as a reference width that can be considered as a material property. Based on this reference width, the stress-strain curve of the softening material for a finite element of any mesh size should be adjusted so that the global response of the finite element model becomes objective (mesh-independent). This 54 adjustment is done based on preserving constant fracture energy in the post-peak region of the softening stress-strain curve. 5.2.2 Localization in Distributed Plasticity Beam-Column Elements Distributed plasticity beam-column elements exhibit similar localization and mesh-dependency issues with strain softening material models. Both DBE and FBE exhibit these issues. However, in FBE damage is localized at a critical section, while in DBE it is localized at a critical member (Pugh 2012). Coleman and Spacone (2001) demonstrated the localization and mesh-dependency in FBE using a simple cantilever beam under imposed transverse tip displacements. They considered three situations: 1. Cantilever beam with elastic-strain hardening section response 2. Cantilever beam with elastic-perfectly plastic section response 3. Cantilever beam with strain softening section response They were interested to compare the force-displacement response at the element level, and the moment curvature response at the section level of each of the above three models, when the number of integration points was increased. Figure 5.3 copies the summary of their observations for the three models. For a strain hardening section response, both of the element level (global) and section level (local) response remained objective, with increasing the integration points from 3 to 8 points. The minor difference in the response of the element with 3 integration points was due to lower accuracy in integrating the element integrals. In the case of an elastic-perfectly plastic section response, the element level response remained objective, while the section response was mesh-dependent. The post yield curvature demand kept increasing with increasing the number of integration points. The loss of objectivity in curvature prediction was argued to be due to the localization of the inelastic curvature at the base integration point. Once the beam reaches its plastic moment capacity, with the addition of the tip displacement, the inelastic curvature at the base integration point increases to correspond to that level of global response, while the curvature in the other integration points remains elastic. When the number of integration points increases, the length associated to each integration point (we may call that the characteristic length) decreases. Therefore, the inelastic 55 curvature at the base point increases for elements with shorter characteristic lengths to result in the same tip displacement. For a softening section, both the element and section responses were mesh-dependent and depended on the number of the integrations points and the associated characteristic length. It was observed that the post-peak global response of the model became more brittle as the number of integration points increased. RC bridge piers and columns with high axial load and subjected to seismic loads exhibit this type of softening section response. In this case, as the characteristic length decreases and the inelastic curvature increases, the compressive strain of concrete fibres also increases. This will lead to more degradation in the material stiffness and consequently the post-peak stiffness. Coleman and Spacone did not demonstrate the localization in displacement-based elements. Similar observations are reported for DBE as well (Pugh 2012), with the difference that the localization occurs in a critical element and not at a section. When the number of DBE is increased, the characteristic length associated to each element including the critical element at the plastic hinge region reduces. This will again lead to mesh-dependent response for a strain softening or elastic perfectly plastic section response. 5.2.3 The Importance of Addressing Localization Issues for PBD of RC Bridges In Chapter 2, it was explained that the performance-based design of RC bridges according to CSA S6-14 requires meeting several local strain limits and global displacement limits at different hazard levels. This performance evaluation is only valid, if objective local and global responses are obtained for the demand and capacity parameters. However, the distributed plasticity models of RC components responding in the post-peak region will face the localization and mesh-dependency issues. Therefore, the response of these models, whether local or global, is not objective and cannot be used in performance evaluations5. Should distributed plasticity models be employed for the analysis of RC bridges, it is inevitable to address the localization issues prior to calculation of any response parameter. In the next section, some of the recommended methods for dealing with the localization issues in DBE and FBE are briefly reviewed. 5 This is because the values of demand or capacity parameters predicted by these models vary depending on the selected mesh size, while they are all checked against the same limits. 56 (a) (b) (c) Figure 5.3 Element level and sectional level response of a cantilever beam modelled using a single FBE with (a) elastic strain-hardening (b) elastic-perfectly plastic, and (c) strain-softening section responses from Coleman and Spacone (Coleman and Spacone 2001) 57 5.2.4 Options for Dealing with Localization Issues in DBE and FBE In the research related to structural analysis, only a few studies have addressed the localization and mesh-dependency issues in concrete and other strain softening materials. There are few analysis guidelines that have mentioned the problem of mesh sensitivity in distributed plasticity models of RC components (NEHRP 2013), and even fewer that have recommended a solution for that. Two techniques have been recommended more frequently so far to deal with the regularization issues: (1) adjusting the mesh size based on an empirical plastic hinge length, (2) material model regularization. Technique 1: Setting the mesh size based on an empirical plastic hinge length This is a more common technique of dealing with mesh sensitivity in FBE and DBE. In this technique, the length associated to the critical section (if FBE is employed) or the critical element (if DBE is employed) is related to an empirical plastic hinge length. This way, the physical aspect of localization is mirrored in the numerical model. One commonly used expression to obtain empirical plastic hinge length can be found in Priestley et al. (1996), which is also adopted in Caltrans SDC 1.7 (2013): Lp= 0.08L+0.022fyedb > 0.044fyedb (mm, MPa) (5.3) In which, L is the member length from the point of maximum moment to the point of contra-flexure, fye is the expected yield strength of the longitudinal rebars, and db is the nominal diameter of the longitudinal rebars. PEER/ATC-72-1 (2010) is one of the few analysis guidelines, which recognizes the localization issue in distributed plasticity models of RC components. In summary recommendations for the modelling of planar and flanges concrete walls, it recommends using an element size equal to the empirical plastic hinge length. Another version of this idea is mentioned by Hachem et al (2003) for modelling circular RC bridge columns. Hachem and colleagues suggest that for FBE to predict curvatures accurately, the following condition must be satisfied: 58 w1= LpLend𝐿 (L-Lp) ≈ LpLend (5.4) Where w1 is the Gauss-Lobatto weight of the first integration point of the end element, L is the total length of element, and Lend is the length of the end element at plastic hinge. If the element has two integration points, the weight of each point is equal to 0.5. This will imply that the length of end element should be 2Lp. Based on this reasoning, he suggests using a 2-point FBE in the plastic hinge zones of the bridge column models. Similar technique is used by Lara (2011) to model several tested RC bridge columns. Calabrese and colleagues (2010) have also looked at the numerical issues in distributed plasticity models of RC frame elements for seismic analysis. They report that if a regularization technique is not available, then for FB elements, it is common to use the above technique. The downside of using this technique is that for short elements, it may require using a small number of integrations points, or small number of elements. In both cases accuracy is compromised and error will be introduced in response calculations. In these cases, material regularization technique can be employed instead. Technique 2: Material model regularization Material regularization technique is based on preserving constant fracture energy of concrete in compression, referred to as crushing energy. The concept of constant fracture energy of concrete in tension is commonly used to address the localization and mesh-dependency issues in the continuum finite element analysis of smeared crack models (Section 5.2.1). However, the results of the experimental research on the compression failure of concrete cylinders suggested that this is also a localized phenomenon and the amount of post-peak fracture energy does not depend on the length of specimens (Jansen and Shah 1997; Lee and William 1997). Using this concept, Coleman and Spacone (2001) suggested a simple regularization technique for fibre section FBE. The crushing energy is defined as the area under the post-peak stress-displacement of concrete, obtained from the following integral: Gfc= ∫ σ dui (5.5) 59 In which, σ is the concrete stress and ui is the inelastic displacement. This energy can be related to the material stress-strain curve through the characteristic length h. For FBE h is the length associated to the critical integration point and is denoted by LIP, as follows: Gfc=h ∫ σ dεi = LIP ∫ σ dεi (5.6) Where, ε is the concrete strain. The above expression suggests that, if Gfc is preserved constant, then the production of the characteristic length and the area under the stress-strain curve of concrete in the post-peak region should remain constant. Consequently, changing the characteristic length by varying either the mesh size or the number of integration points should require adjusting the area under the stress-strain curve correspondingly. The shaded area in Figure 5.4 Part (a) represents this area for a typical compressive stress-strain curve of unconfined concrete and is equal to the ratio of Gfc to LIP. It should be noted that the post-peak softening branch of the model is not necessarily linear. εo is the strain of unconfined concrete at the peak stress of f’c and ε20u is the strain in the post-peak region of the curve corresponding to 80% strength loss (i.e. stress is equal to 20% of f’c). ε20u is the parameter that is adjusted in the concrete uniaxial stress-strain curve for regularization. Coleman and Spacone present an expression for calculating ε20u, assuming the modified Kent and Park model (Pugh 2012) is used for concrete: ε20u=Gfc0.6f'cLIP- 0.8f'c𝐸𝑐+εo (5.7) In this expression, Ec is the elastic modulus of unconfined concrete in compression. The modified Kent and Scott model assumes a linear softening branch. Given Gfc, the ε20u and therefore the material model assigned to each critical section or element, should be changed based on their characteristic length. Similar expressions can be utilized for confined concrete as well. However, the confined concrete parameters should be used in place of the unconfined concrete parameters. Figure 5.4 Part (b) illustrates the equivalent of Part (a) for confined concrete. In this picture, the Gfcc and f’cc are the crushing energy and compressive strength of confined concrete, respectively, εoc is the strain of confined concrete at f’cc, ε20c is the strain in the post-peak region corresponding to 80% strength loss, and Ecc is the elastic modulus of confined concrete in compression. 60 εo ε20uf’c0.2 f’cEcσ GfcLIP εoc ε20cf’cc0.2 f’ccEccσ GfccLIP (a) (b) Figure 5.4 Schematic post-peak crushing energy of (a) unconfined concrete and (b) confined concrete (reproduced from Pugh, 2012) Preserving a constant crushing energy will ensure that the inelastic portion of the global response becomes objective. However, the inelastic portion of the local response remains mesh-dependent, since this response should vary with the size of the mesh so that it produces the same global displacement. Coleman and Spacone suggested a very simple procedure to obtain an objective prediction of curvature in FBE. The idea is simply to scale the inelastic part of the curvature obtained from analysis to the inelastic curvature that is predicted using the empirical plastic hinge length. The inelastic curvature from analysis can be approximated as: ϕimodel=δiLIP (L2-LIP2) (5.8) And the curvature is the sum of the elastic curvature and the scaled inelastic curvature: ϕ=ϕe+SF ϕimodel (5.9) The scale factors are obtained by taking the ratio of ϕipredicted/ϕimodel, where the predicted curvature is obtained by replacing LIP in Equation (5.8) with Lp obtained from Equation (5.3): SF=wIPL2(1-wIP)Lp(L-Lp) (single curvature) (5.10) SF=wIPL2(2-wIP)Lp(2L-Lp) (double curvature) (5.11) 61 Referring to the definition of the scale factors, Coleman and Spacone argued that if LIP = Lp then no further post-processing is necessary to obtain objective curvatures. It becomes clear with this explanation, that the first technique of dealing with location issues described in the previous section can be considered as a special case of the material regularization technique. When it is not possible or reasonable to set LIP equal to Lp, then material regularization technique must be used. The most recent extensive research on this topic was conducted by Pugh (2012; Pugh et al. 2015). He studied the localization issues in distributed plasticity models for numerical simulation of reinforced concrete shearwalls and extended the work done by Coleman and Spacone to include DBE as well as FBE. His main contribution was to recommend a more scientific formulation for the crushing energy of both unconfined and confined concrete. Coleman and Spacone recommended using a value of 20 N/mm for the crushing energy of unconfined concrete based on the recommendation of Jansen and Shah (1997), up to a value of 30 N/mm. The confined concrete crushing energy is approximated then as six times of the crushing energy for the unconfined concrete (150-180 N/mm). Pugh conducted experimental tests on a number of planar wall specimens and observed that the crushing energy can be formulated in terms of the specified strength of unconfined and confined concrete. He recommended using the following crushing energy values with FBE and DBE: Gfc=2f'c ( N mm)⁄ (FBE) (5.12) Gfc=0.56f'c ( N mm)⁄ (DBE) (5.13) And the confined concrete crushing energy for both FBE and DBE can be estimated as: Gfcc=1.7Gfc (5.14) Then Equation (5.7) can be written for the confined concrete properties as: ε20c=Gfcc0.6f'ccLIP- 0.8f'cc𝐸𝑐𝑐+εoc (5.15) The crushing energy recommended to be used with DBE is considerably lower than FBE. Pugh argued that in DBE, the predicted axial load at the critical section is lower than the actual applied 62 load and as a result, the predicted curvature ductility values are greater than those predicted by FBE. The reduction in the crushing energy values accounts for this difference. Pugh made another improvement to the Coleman and Spacone’s work by suggesting material regularization to be applied to reinforcing steel as well as concrete. Steel shows a strain-hardening behaviour in the post-yield portion of the stress-strain curve. This will ensure distribution of plasticity along the beam-column element instead of localization in a single critical element or section. However, in a reinforced concrete element, for which at the critical section concrete exhibits strain-softening, steel deformations also localize at the critical section to conform to compatibility conditions. Consequently, steel strain demands at the critical section become mesh-dependent and require material regularization. The post-yield energy of steel material is referred to as hardening energy. Much like concrete, this energy can be related to the stress-strain curve through a length measure. For steel this length is the length, along which the inelastic deformation localizes and is taken equal to the gage length used in the laboratory test. Figure 5.5 shows a simplified bi-linear steel stress-strain. The shaded area under the post-yield portion of the curve represents the ratio of the hardening energy, Gs, to the gage length, Lgage. Therefore, the hardening energy can be calculated as: Gs=0.5 (εsu,exp-εy) (fu+fy) Lgage (5.16) Where εsu,exp is the expected rupture strain, εy is the yield strain, and fu and fy are the ultimate tensile strength and yield strength, respectively. For regularizing the steel material, the ultimate rupture strain assigned to the steel material model in the analysis should be modified based on the length associated to the critical section or element. The underlying assumption again is that the hardening energy is preserved constant. Using this assumption, Pugh derived the following expression for the ultimate rupture strain in the regularized material model: εsu=εy+(εsu,exp-εy)LgageLIP (5.17) 63 εy εsufuEsε fybEsGsLgage Figure 5.5 Schematic post-peak energy of reinforcing steel (reproduced from Pugh, 2012) Pugh suggested using an 8 in (0.203 m) gage length required by ASTM A370 for numerical modelling of wall specimen, if gage length was not reported in test results. Also, he argued that Equation (5.17) can be used with Menegotto-Pinto steel model (1973) as well, and the simplifications have insignificant impact on numerical results. Finally, it should be noted that the post-yield hardening modulus will be changed based on the value calculated for the ultimate rupture strain of the regularized material. 5.3 Validating Distributed and Concentrated Plasticity Models of RC Bridge Columns The main purpose of this section is to apply the techniques of dealing with localization and mesh-dependency issues discussed in the previous section, to distributed plasticity models of a number of tested RC bridge columns. It is desirable to observe how these models would predict the response of the columns before and after each of the two techniques have been applied to them. The selected columns were tested under lateral quasi-static cyclic loading condition. In this exercise the cyclic force-displacement response of the models is compared against the test results and the monotonic response of the models is predicted using static pushover analysis. In addition to the global force-displacement response, the local strains of concrete and reinforcing steel fibres are checked for consistency. The aim is not to calibrate the models so that they would reproduce the test results with the utmost accuracy; rather, it is an exercise to illustrate how well with the current modelling techniques we are able to predict the performance of RC bridge columns, when testing is not an option. This is in fact the likely case when a new structure is designed for construction and testing of the main components is too expensive to be feasible or simply is not possible. Once consistent local and global responses are achieved, the performance 64 criteria of CSA S6-14 and BC MoTI Supplement will be employed to predict the extent of damage to the columns in the next chapter. 5.3.1 Selected Bridge Column Tests Five reinforced concrete bridge column tests are selected from the PEER structural performance database (2003). All of the selected columns have circular cross-section and are laterally reinforced with spirals. This configuration of bridge columns is very common in the Canadian bridge industry and therefore is relevant to the theme of this thesis. The selected tests are part of the experimental program developed by Lehman and colleagues at PEER (Lehman and Moehle 2000a; Lehman and Moehle 2000b; Calderone et al. 2001; Lehman et al. 2004) to study the cyclic performance of concrete bridge columns detailed for ductile flexural response in high seismicity zones. The main focus of the program was on capturing the performance of the bridge columns in terms of relatable engineering demand parameters at a range of damage states other than failure of the component. The measured engineering demand parameters included global lateral load and displacement and local concrete and steel strains values at different damage states. The availability of these test results makes them appropriate for the purpose of this section. A schematic picture of the test configuration and instrumentation as well as the overall geometry and reinforcement details of the tested specimens are shown in Figure 5.6. Table 5.1 lists the properties of test specimens including the test number, length, reinforcement details, and axial load ratio, and Table 5.2 shows the material properties for the test specimens. The tests specimens are one-third of the full-scale columns. All columns have similar diameter of 2 ft (0.61 m). The spiral pitch is constant for Column 415 and 430 throughout the length of the column, while it varies for Column 328, 828, and 1028. In these columns the pitch is smaller in the plastic hinge zone (Lc in Figure 5.6) and is twice larger at the rest of the columns. Column 415 and 430 are different only in the amount of longitudinal reinforcement, whereas Column 328, 828, and 1028 vary in their length and thus their aspect ratio. The first digit in the test number indicates the aspect ratio value, for instance 328 and 1028 has aspect ratios of 3 and 10, respectively. The rest of the number indicates the longitudinal reinforcement ratio. 65 Figure 5.6 (left) geometry and reinforcement of the tested specimens, (right) test configuration and instrumentation (Calderone et al. 2001; Lehman et al. 2004) The axial load was applied to the columns through the high-strength rods at either side of the test specimens and a spreader beam as shown in Figure 5.6. The lateral displacement history was applied using an actuator attached to the top of the column. All specimens had similar instrumentation and test procedure. First, the axial load was applied and was maintained constant, while the lateral load was applied. Figure 5.7 shows the lateral load time-history in terms of the target displacement ductility values, and Table 5.3 contains the amplitude values at each displacement level that are imposed to the columns with varying aspect ratios. The loading protocol for the post-yield cycles included three cycles at each amplitude and a following cycle with one-third of the amplitude. The recorded cyclic force-displacement responses of the columns are shown in Figure 5.8. Comparing the hysteretic response of the columns, it is notable that Column 328, 415, and 430 demonstrate higher strength degradation at the last three loading cycles, compared to Column 828 and 1028. 66 Reinforcement Column Length (mm) Longitudinal ρl (%) Spiral Spacing (mm) ρs (%) Confined Length (mm) Axial Load Ratio 328 1829 28 No. 6 2.8 25/50 0.87 610 0.1 828 4877 28 No. 6 2.8 25/50 0.87 915 0.1 1028 6096 38 No. 6 2.8 25/50 0.87 1220 0.1 415 2438 22 No. 5 1.5 32 0.7 2438 0.1 430 2438 44 No. 5 3 32 0.7 2438 0.1 Table 5.1 Specimen properties of the test columns (Lehman et al. 2004) Column f'c (MPa) fym (MPa) fum (Mpa) εsh εu fyhm (MPa) 328, 828, 1028 34 448 634 0.02 0.14 607 415, 430 31 497 662 0.02 0.13 607 Table 5.2 Material properties of the test columns (Lehman et al. 2004) Figure 5.7 Imposed lateral displacement time history (Lehman et al. 2004) Aspect Ratio Displacement Level 3 4 8 10 Pre-cracking 1 2 4 5 Pre-yield1 3 3 15 20 Pre-yield2 5 8 45 64 Pre-yield3 10 19 89 127 μΔ≈1 15 25 133 191 μΔ≈1.5 20 38 178 254 μΔ≈2 30 51 267 381 μΔ≈3 51 76 445 635 μΔ≈5 71 127 μΔ≈7 102 178 μΔ≈10 132 Table 5.3 Imposed displacement histories in mm for columns with different aspect ratios (Lehman et al. 2004) 67 Figure 5.8 Cyclic force-displacement responses of the selected test columns (PEER 2003) 68 5.3.2 Bridge Column Models Two separate analysis platforms are employed to model the tested bridge columns including OpenSees and CSI SAP2000. OpenSees is an open source analysis platform, which is widely used for academic purposes and is geared towards fulfilling current research demands, while SAP2000 is a well-established commercial tool used by many engineers in practice. The reason for using both platforms is that they have different strength and limitations, and therefore employing both offers the chance to contrast the capabilities of the academic versus commercial tools with respect to the modelling alternatives described earlier in this chapter. OpenSees is used to generate distributed plasticity models of the test columns, whereas SAP2000 is used to build concentrated plasticity models. A schematic illustration of the column models in both platforms is shown in Figure 5.9. Figure 5.9 Schematic distributed and concentrated plasticity models of the columns in OpenSees (left) and SAP2000 (right) - OpenSees Models: The test columns are modelled using inelastic beam-column elements with fibre sections. Both DBE and FBE are employed for the models. The base of the columns is restrained in all degrees-of-freedom to mimic the fixed-base condition in the tests. A concentrated vertical load, equal to the applied axial load in the tests and a unit lateral load is applied at the top of the columns. The lateral load is imposed at the center of the loading zone at the top of the column, from where the height of the column is measured (Figure 5.6, left). A displacement-controlled integrator is employed in the analysis, for which the control node is the node where the lateral load is applied. 69 To create fibre sections, separate material models needs to be defined for the unconfined and confined concrete, and reinforcement steel. Two sections are defined for the test columns where the plastic hinge zone and the rest of the column have different confinement arrangement. Concrete02 is used for both of the unconfined and confined concrete material models. This model is developed based on the Kent-Scott-Park constitutive relationship and it has a bi-linear tension response with linear tension softening. The compression response has an initial parabolic segment up to the maximum compressive strength, followed by a linear softening segment, and a final plateau, as demonstrated in Figure 5.10. The model takes seven calibration parameters including the compressive strength of concrete (fpc), concrete strain at maximum strength (epsc0), concrete crushing strength (fpcU), concrete strain at crushing strength (epsU), ratio between unloading slope and initial slope (lambda), tensile strength of concrete (ft), and tension softening stiffness (Ets). OpenSees does not automatically apply the confinement effects. Therefore, the confined concrete material properties are manually obtained following the Mander et al. confined concrete model (1988). The strain at the peak stress of the unconfined concrete, the confined concrete in the plastic hinge zone, and outside of it, is assigned as 0.002, 0.0045, and 0.0038, respectively. The ultimate strain capacity of the confined concrete can be calculated using Priestley et al. expression (1996), as follows: εcu=0.004+1.4ρs fyh εfsfcc' (5.18) In the above expression, ρs is the spiral reinforcement ratio, fyh is the spiral yield strength, εfs is the spiral fracture strain, which can be taken equal to εsu,exp, and f’cc is the confined concrete strength. It should be noted this expression has been formulated for confined concrete section under compression. When used for member under bending or combination of bending and axial compression, it tends to underestimate εcu by at least 50% (Priestley, M. J. N. et al. 1996; Kowalsky 2000). Using the above formula, the ultimate strain capacity of Column 328 is obtained as -0.0180 in the plastic hinge zone, and -0.011 outside of it. For the reinforcement steel, Steel02 material model is employed. This is a Giuffre-Menegotto-Pinto material model (1973) with isotropic strain hardening proposed by Filippou et al. (1983). The model takes eleven calibration parameters. The strain hardening parameter, 70 Figure 5.10 Concret02 constitutive model in OpenSees (PEER 2017) defined as the ratio of the post-yield stiffness to the initial elastic stiffness, is calculated as 0.01. The ultimate strain capacity of the reinforcement steel is set to 0.09, based on the recommendation of Caltrans SDC 1.7 (2013) for the reduced ultimate tensile strain of bar #10 or smaller. To impose this limit on the response of the steel fibres, a MinMax uniaxial material model is additionally assigned to the column sections with the limiting strain of 0.09 in tension and infinity in compression. This material model will ensure that the steel fibres strength reaches zero when the strain in those fibres is equal to 0.09, which will mimic the effect of fracture in the reinforcement steel. The beam-column elements are not capable of modelling the shear response of the columns. To account for the shear stiffness loss of the columns, an elastic shear section was integrated to the fibre section using section aggregator in models with FBE. This option does not work with DBE in OpenSees, and instead a zero-length shear spring was added to the base of the models with DBE, as suggested by Pugh (2012). The effective shear stiffness modulus can be taken as follows (Oyen 2006): Geff=0.1G≈0.04 E (5.19) Where G is the elastic shear modulus and E is the Young’s modulus of elasticity. CSA S6-14 recommends obtaining effective shear stiffness using Equation (5.2). It should be noted however, that accounting for the loss of shear stiffness is very important for the modelling of shear 71 dominated elements such as shearwalls. If the columns are dominated primarily by flexure response, the effect of the shear stiffness loss is minimal and can be ignored. - SAP2000 Models: Unlike OpenSees, SAP2000 does not have the option of distributed plasticity models. Instead, non-linear behaviour is modelled with concentrated plasticity models, assigning plastic hinges with a specified length to elastic frame elements, as explained earlier in Section 5.1.2. Fibre hinges are employed here to model the nonlinear response of the columns. The shear and torsion behaviour of the cross section are elastic. So, the loss of shear stiffness should be captured by applying shear area modification factors to the elastic frame elements. The plastic hinge length assigned to fibre hinges can be calculated using Equation (5.3), as listed in Table 5.4. As shown in Figure 5.9, plastic hinges should be assigned to the mid-point of the plastic hinge zone of the columns. This will ensure that the length of the column that undergoes plastic rotation is correct. Column L (mm) Lp (mm) Lp/L 415 2438 369 0.15 430 2438 369 0.15 328 1829 376 0.21 828 4877 578 0.12 1028 6096 676 0.11 Table 5.4 Empirical plastic hinge length for the selected test columns SAP2000 has a built-in model to calculate confined concrete material properties from the inputs for the unconfined concrete and confinement properties of a section, based on Mander et al. confined concrete model (1988). The column sections were defined using the section designer module and radial fibre arrangement was assigned to the sections. A nonlinear static load case was defined for the vertical gravity load applied at the top of the columns. A nonlinear static pushover analysis case was also defined using a displacement-controlled load application for the top node of the columns. While using fibre hinges, it should be noted that there was an elastic softening issue in SAP2000 version 18.0.1 and prior, which was rectified in version 18.1.0 and later. The elastic softening was due to double counting of the elastic flexibility in frame elements with fibre 72 hinges. This was the reason for the recommendations on using increased stiffness factors with elements (see Aviram et al. 2008). To remove this issue, the elastic stiffness of frame elements is now automatically set to zero throughout the tributary length of the fibre hinges. 5.3.3 Validation of the Distributed Plasticity Models for Test 328 The generated distributed plasticity models in OpenSees are used to predict the response of the bridge column in Test 328. The localization and mesh sensitivity issues in distributed plasticity models of the column are demonstrated for both DBE and FBE models and the two methods of dealing with localization issues are applied to correct the predictions of the models. All the investigations are performed on Column 328 model, as the exercise is similar for the rest of the test columns. The final takeaways from the study on Column 328 are examined for Column 828, 1028, 415, and 430 in Section 5.3.8. 5.3.3.1 Cyclic Response without Regularization The effect of changing the mesh size in DBE and FBE models on the cyclic response of the models is demonstrated in Figure 5.11. The graphs show the displacement of the top node of the column versus the base shear. Parts (a) to (d) compare the predicted cyclic response of the DBE models with 0.076, 0.152, 0.305, and 0.610 m element size against the test results (the mesh size is equal to 1/3, 1/6, 1/12, and 1/24 of the height of the column, respectively). All the DBE models use two integration points per element. Therefore, the length associated to each integration point is half of the element size (0.038, 0.076, 0.152, and 0.305 m, respectively). Parts (a’) and (b’) show the FBE model responses with 0.610 and 0.914 m element sizes, having two and three integration points per element, respectively (the mesh size is equal to 1/3 and 1/2 of the height of the column). The length associated to the integration point at the critical section of the element at the base of the column is 0.305 and 0.152 m. The FBE and DBE models are different only in the selected mesh size and otherwise are similar. By examining the graphs, it is readily evident that changing the mesh size significantly affects the force-displacement response of both the DBE and FBE models. Models with smaller mesh size have smaller hysteresis loops, and they quickly degrade and loose strength. We will refer to the displacement, at which there is a sudden drop in base shear at two consecutive cycles, as the failure displacement. 73 (a) (b) (c) (d) (a’) (b’) Figure 5.11 Comparison of the simulated cyclic force-displacement response of Column 328 using DBE with (a) 0.076 m, (b) 0.152 m, (c) 0.305 m, (d) 0.610 m element size and 2 integration points, and FBE with (a`) 0.610 m element size and 2 integration points, (b`) 0.914 m element size and 3 integration points with the experimental results 74 The DBE models in Parts (a), (b), and (c) predicts the failure displacement of the column to be at 0.05, 0.07, and 0.10 m, respectively, while the test results suggest that the failure displacement is about 0.13 m. Therefore, all these models underestimate the displacement capacity of the column at failure. Only the DBE model with 0.610 m mesh size provides a good estimation of the cyclic response of the column. The reason for this, is that the length associated to the integration point at the critical element at the base of the column (LIP = 0.305 m) is close to the empirical plastic hinge length of Column 328 in Table 5.4 (Lp = 0.376 m). While for the other mesh sizes, LIP is considerably smaller than Lp. This means that the length, over which the plastic strain and the plastic curvature localize, is smaller than the physical plastic hinge length. Therefore, when the fibres in the critical section reaches the strain value where the material model starts degrading and eventually fails, the predicted failure displacement at that strain value is smaller than the actual displacement of the column. Comparing the two FBE models, the model with 0.914 m mesh size and three integration points underestimate the displacement capacity of the column at 0.10 m, while the model with 0.610 m mesh size and two integration points provides an acceptable prediction of the response. This again relates to how the length associated to the integration point at the critical section of the column compares to the plastic hinge length of the column. For the FBE model in Part (b’), LIP = 0.152 m, while for model in Part (a’), LIP = 0.305 m, which is closer to Lp = 0.376 m. 5.3.3.2 Cyclic Response with Regularization Material regularization technique should be applied to each of the following four material models individually: (1) confined concrete in the plastic hinge zone, (2) confined concrete outside the plastic hinge zone, (3) unconfined concrete, and (4) reinforcement steel. The steps of the regularization technique were explained in Section 5.2.4. Once the regularization is applied, each of the above material models needs to be updated with the values for the post-peak parameters, including the strain at the 80% loss of strength for unconfined and confined concrete, and the ultimate rupture strain and the post-peak slope of reinforcement steel. The calculated values of these parameters are listed in Table 5.5. In this table Lele is the mesh size, IPs is the number of integration points used per element, LIP is the length associated to the integration point at the critical section of a FBE or the critical DBE at the base of the column, ε20u, ε20c-PH, and ε20c-75 R are the strains at the 80% loss of strength for the unconfined concrete, and the confined concrete inside the plastic zone, and outside of it, respectively, and εsu and b are the ultimate rupture strain and the post-peak slope of the reinforcement steel. It should be noted that the integration scheme for the DBE models is Gauss-Legendre and for the FBE models Gauss-Lobatto, and LIP is calculated based on Lele,, the number of integration points per element, and the integration weights for each type of integration scheme. Type Lele (m) IPs LIP (m) ε20u ε20c-PH ε20c-R εsu b DBE 0.610 2 0.305 -0.0040 -0.0077 -0.0074 0.0607 0.0159 DBE 0.305 2 0.152 -0.0071 -0.0120 -0.0122 0.1193 0.0079 DBE 0.152 2 0.076 -0.0132 -0.0207 -0.0216 0.2363 0.0040 DBE 0.076 2 0.038 -0.0255 -0.0381 -0.0406 0.4703 0.0020 FBE 0.610 2 0.305 -0.0119 -0.0189 -0.0196 0.0607 0.0159 FBE 0.914 3 0.152 -0.0228 -0.0344 -0.0365 0.1193 0.0079 FBE 1.829 4 0.152 -0.0228 -0.0344 -0.0365 0.1193 0.0079 FBE 1.829 6 0.061 -0.0556 -0.0808 -0.0872 0.2948 0.0032 Table 5.5 Calculated regularized strain of the unconfined and confined concrete at 80% strength loss, and the ultimate rupture strain and post-peak slope of the reinforcement steel for Column 328 models The DBE and FBE models of Column 328 used in the previous section are updated with the regularized material properties in Table 5.5 and their cyclic force-displacement response are compared against the test results in Figure 5.12. Comparing Part (a) to (d) of Figure 5.12 with Figure 5.11, reveals how regularizing the material models changes the cyclic response of the DBE models. The following can be observed from this comparison: Regularization of the DBE models allows the hysteretic behaviour to fully develop and prevents the premature failure of the models with smaller mesh sizes in Parts (a) to (c). The predicted failure displacements of the four DBE models are quite similar, unlike Figure 5.11. The hysteretic loops of the four regularized material models have different shapes. The models with smaller mesh sizes tend to develop larger hysteresis loops at similar displacement levels. The regularized FBE models have very similar hysteric response, both in terms of the shape and the failure displacement. However, they both underestimate the failure displacement at 0.1 m (about 23%). 76 (a) (b) (c) (d) (a’) (b’) Figure 5.12 Regularized cyclic force-displacement response of Column 328 using DBE with (a) 0.076 m, (b) 0.152 m, (c) 0.305 m, (d) 0.610 m element size and 2 integration points, and FBE with (a`) 0.610 m element size and 2 integration points, (b`) 0.914 m element size and 3 integration points with the experimental results 77 5.3.3.3 Monotonic Response without and with Regularization Once the cyclic response predictions of Column 328 are compared against the cyclic test results, It is possible to use the same model to predict the monotonic response of the columns, although there are no monotonic test results to compare to. It is of interest to see how material regularization will affect the monotonic response of DBE and FBE models of Column 328 with various mesh sizes. Figure 5.13 shows the monotonic response of DBE models with 0.076, 0.152, 0.305, and 0.610 m mesh size, and FBE models with 0.610, 0.914, 1.828 m mesh size, and 2, 3, and 4 and 6 integration points per element, respectively. The cyclic test result is also shown in the figure so that the monotonic response can be compared against the backbone of the cyclic response. We will define the failure displacement for monotonic response as the displacement, at which the base shear suddenly drops by 20% or more. Using this definition, the monotonic results also suggest that the failure displacement of both of the DBE and FBE models is dependent on the mesh size. These values are listed in Table 5.6 under the “without regularization” column. The difference of the failure displacements between the models with the largest and the smallest mesh sizes is about 300-350%, which is very significant. One cannot conclude, which of these displacements are the failure displacement of the actual column, unless the mesh-sensitivity issue is dealt with. Figure 5.14 shows the monotonic response of the same models but after applying material regularization. For both DBE and FBE models, the regularized models produce almost identical force-displacement responses and have similar failure displacements. The failure displacements of these models are also listed in Table 5.6 under the “with regularization” column. The difference of the failure displacements between the models with the largest and the smallest mesh sizes in this case is about 7-8%. The only outlier is the DBE model with 0.610 m mesh size, which has a difference of about 10% in failure displacement with the 0.075 m model. Also as evident in Part (a) of the figure, it predicts higher strength values as compared to the other 78 (a) (b) Figure 5.13 Monotonic response of the DBE and FBE models of Column 328 before material regularization (a) (b) Figure 5.14 Monotonic response of the DBE and FBE models of Column 328 after material regularization Failure Displacement (m) Type Lele (m) IPs LIP (m) WO Reg. W Reg. DBE 0.610 2 0.305 0.173 0.119 DBE 0.305 2 0.152 0.098 0.121 DBE 0.152 2 0.076 0.057 0.129 DBE 0.076 2 0.038 0.037 0.132 FBE 0.610 2 0.305 0.155 0.096 FBE 0.914 3 0.152 0.081 0.100 FBE 1.829 4 0.152 0.081 0.102 FBE 1.829 6 0.061 0.039 0.104 Table 5.6 Failure displacements of the DBE and FBE models of Column 328 with and without material regularization -700-500-300-100100300500700-0.15 -0.05 0.05 0.15Lateral Force (kN)Displacement (m)Test328DBE 0.076 mDBE 0.152 mDBE 0.305 mDBE 0.610 m-700-500-300-100100300500700-0.15 -0.05 0.05 0.15Lateral Force (kN)Displacement (m)Test328FBE 1.828 m 6IPFBE 1.828 m 4IPFBE 0.914 m 3IPFBE 0.610 m 2IP-700-500-300-100100300500700-0.15 -0.05 0.05 0.15Lateral Force (kN)Displacement (m)Test328DBE 0.076 mDBE 0.152 mDBE 0.305 mDBE 0.610 m-700-500-300-100100300500700-0.15 -0.05 0.05 0.15Lateral Force (kN)Displacement (m)Test328FBE 1.828 m 6IPFBE 1.828 m 4IPFBE 0.914 m 3IPFBE 0.610 m 2IP79 DBE models. This is because DBE models need more elements compared to FBE models to achieve the same level of accuracy. Therefore, the minor discrepancy in the DBE model with 0.610 m mesh size with the other DBE models is due to using small number of elements (3 elements). It is also possible to compare the monotonic failure displacements of the regularized models with the cyclic failure displacement from the test result. Although it should be noted that the two displacements may not necessarily be the same as in the cyclic test, the column undergoes many cycles before reaching the failure displacement and meanwhile dissipates a lot of energy through material nonlinearity. On the contrary, in the monotonic response, the column is consistently pushed until reaching the failure displacement. As a result, the column dissipates less energy through material nonlinearity and is more likely to endure less damage at the same level of displacement when compared to the column under cyclic loading. The cyclic failure displacement of Column 328 is about 0.13 m. Comparing the failure displacements of the regularized models with this value, it can be observed that the regularized FBE models underestimate the failure displacement of the column by 23%, but the regularized DBE models almost predicts the same value of failure displacement with only 4% error on average. However, the FBE models predict the backbone of the cyclic response with accuracy, while the DBE models overestimate the strength of the column in initial cycles by about 9% and underestimate it in the rest of the cycles up to the failure displacement by about 9-10%. 5.3.3.4 Cyclic and Monotonic Response of the Models with Mesh Size Set Based on LP The other method of dealing with localization and mesh sensitivity issues, which was introduced in Section 5.2.4 is to set the mesh size and therefore the length associated to the critical integration point (LIP) based on the empirical plastic hinge of the column (LP). Considering the discussion in Section 5.2.4, first LIP is set equal to LP. The cyclic and monotonic responses of the DBE and FBE models satisfying this condition are shown in Figure 5.15, along with the test result of Column 328. In terms of the cyclic response, it is observed that both DBE and FBE models more or less provide acceptable predictions of the actual response of the column. The DBE model better captures the shape of the hysteresis loops, but overestimates the strength in the 80 (a) (b) Figure 5.15 Comparison of the cyclic and monotonic responses of the (a) DBE model and (b) FBE model with LIP=LP against the test results of Column 328 initial cycles. The situation is not the same for the monotonic response predictions. The DBE model significantly overestimates both the failure displacement and the strength of the column, by about 60% and 25%, respectively. As explained in the previous section, this is due to the fact that DBE models need more elements to accurately predict the response as opposed to FBE models. The FBE model also overestimates the failure displacement by about 46%, but underestimates the strength in the post-peak cycles by about 13% on average. Since the models with LIP= LP overestimates the monotonic response, it is intuitive to choose a smaller mesh size that is not too small to significantly underestimate the response. We will try LIP equal to 0.5 LP. This corresponds using an element size equal to the plastic hinge length having two integration points, and follows the recommendations of PEER/ATC-72-1 (2010). The cyclic and monotonic response of the DBE and FBE models with this condition is shown in Figure 5.16. Considering both the monotonic and cyclic responses, the DBE model provides a reasonable prediction of the failure displacement, while the FBE model underestimates that. However, the underestimation of the response by the FBE model is not too significant to rule it out for design purposes. The DBE model captures the shape of the hysteresis more closely, and the overestimation of strength in the monotonic response is less significant in this case. -700-500-300-100100300500700-0.15 -0.05 0.05 0.15Lateral Force (kN)Displacement (m)Test328PushoverCyclic-700-500-300-100100300500700-0.15 -0.05 0.05 0.15Lateral Force (kN)Displacement (m)Test328PushoverCyclic81 (a) (b) Figure 5.16 Comparison of the cyclic and monotonic responses of the (a) DBE model and (b) FBE model with LIP=0.5 LP against the test results of Column 328 5.3.4 Sensitivity of Monotonic Response to GFcc Value The response of a regularized model depends on the value assumed for the crushing energy of confined concrete, Gfcc, as evident from Equations (5.15). It is desirable however, to understand how this value would affect the predictions of the monotonic force-displacement response of a concrete column. To achieve so, a simple sensitivity analysis on the DBE and FBE models of Column 328 is performed, where the confined concrete crushing energy is varied in a range, while all the other parameters, including the unconfined concrete crushing energy are kept constant. The mesh size for the DBE models is 0.305 m and for the FBE models is 0.610 m with two integration points (although as demonstrated earlier, the regularization process cause models with different mesh sizes to yield similar force-displacement responses). Figure 5.17 shows the monotonic force-displacement response of the DBE and FBE models for four different values of Gfcc. The minimum values for Gfcc are taken equal to the unconfined concrete crushing energy calculated from Equation (5.12) and (5.13), which are 19.0 N/mm for the DBE models and 68.0 N/mm for the FBE models. The maximum considered crushing energy for both of the DBE and FBE models is 180.0 N/mm, which is the value used by Coleman and Spacone (2001) for regularizing FBE models in their work. The middle values are calculated from Equations (5.15) using the unconfined crushing energy of Equation (5.12) and (5.13), and 1.5 times of those values. The results in Part (a) reveal that changing Gfcc value changes the shape of the post-peak monotonic force-displacement response of the DBE models. -700-500-300-100100300500700-0.15 -0.05 0.05 0.15Lateral Force (kN)Displacement (m)Test328PushoverCyclic-700-500-300-100100300500700-0.15 -0.05 0.05 0.15Lateral Force (kN)Displacement (m)Test328PushoverCyclic82 (a) (b) Figure 5.17 Sensitivity of the monotonic force-displacement response of the Column 328 (a) DBE and (b) FBE models to GFcc value -700-500-300-100100300500700-0.15 -0.05 0.05 0.15Lateral Force (kN)Displacement (m)Test328Gfcc=19.0 N/mmGfcc=32.4 N/mmGfcc=48.6 N/mmGfcc=180 N/mm-700-500-300-100100300500700-0.15 -0.05 0.05 0.15Lateral Force (kN)Displacement (m)Test328Gfcc=68.0 N/mmGfcc=115.6 N/mmGfcc=173.4 N/mmGfcc=180 N/mm83 The hump in the initial part of the curves becomes smoother and eventually vanishes as the value of Gfcc increases. The difference in response of the FBE models in Part (b) with varying Gfcc value is subtler, although it can be seen that for Gfcc value of 68.0 N/mm the hump shape of the post-peak response becomes visible. It should be noted that the change in the Gfcc values of the FBE models is much smaller than the DBE models, so the resulting effect is expected to be less significant. Forming of the hump shapes relates to the confined concrete material model losing strength in the post-peak region. Equation (5.15) suggests that by increasing the Gfcc value, the ε20c value also increases, resulting in the confined concrete material model to lose strength at higher strain values. That explains why the hump shape fades with increasing the Gfcc value, while all other parameters are constant. The strength loss in reinforcement steel material model does not occur until rupture, while in confined concrete it occurs right after the peak up to crushing. The sudden significant drop of base shear in the monotonic response of the models, which was used to define the failure displacement, occurs when both reinforcement steel and confined concrete have reached ε20c and εsu, respectively. If confined concrete reaches to its limiting strain earlier than reinforcement steel, the failure displacement is controlled by the rupture of steel and vice versa. Understanding how the value of Gfcc can considerably affect the shape of the monotonic response of the models, the questions posed is what value of Gfcc should be used in analysis. It seems that the Gfcc value suggested by Pugh (2012) for FBE provides reasonable predictions of the monotonic response of concrete columns, but this is not the case for DBE. As mentioned earlier, he recommends a much smaller value to be used with DBE to compensate for the effect of high axial load in the critical elements. The latter needs further investigation, as the results in this chapter do not support using smaller Gfcc values for DBE. In fact, based on the sensitivity analysis presented in this chapter, it seems that a higher value of Gfcc, close to the value used for FBE elements could be used with DBE, as well. One intuitive way to answer this question is to set the Gfcc value such that a model (FBE or DBE) with LIP = Lp, would have ε20c-PH value equal to the empirical crushing strain of confined 84 concrete εcu, obtained from Equation (5.18). To do so, Equation (5.15) may be rewritten for such condition as: ε20c-PH=Gfcc0.6f'ccLP- 0.8f'cc𝐸𝑐𝑐+εoc (5.20) Using the above equation, the values of ε20c-PH can be calculated for varying values of Gfcc as listed in Table 5.7 (for column 328, Lp is 376 mm and εcu is -0.018). The tabulated values shows that the Gfcc of 135 N/mm will results in ε20c-PH= εcu= -0.018. This value is fairly close to the Gfcc value recommended by Pugh for FBE models, which is 115.6 N/mm for Column 328, but it is much larger than his suggested value for DBE models, which is 32.4 N/mm. Setting Gfcc value at 32.4 N/mm implies that the confined concrete at the plastic hinge of the column crushes at the strain value of -0.0069. This strain value is unduly conservative, considering that it is much smaller than the εcu value, and εcu itself can be conservative at least by 50% (see Section 5.3.2). The inconsistency in the regularization process stems from the fact that there are three separate expressions available to calculate the Gfcc, εcu, and Lp values, while using Equation (5.20) only two of these parameters can be assigned independently. In the author’s opinion, while the above presented method somewhat alleviates this inconsistency, this problem needs further in depth studies, which is beyond the scope of this thesis. Gfcc (N/mm) 19.0 32.4 48.6 68.0 115.6 135.0 173.4 180.0 ε20c-PH -0.0054 -0.0069 -0.0086 -0.0108 -0.0159 -0.0180 -0.0222 -0.0229 Table 5.7 Calculated values of ε20c-PH for the Column 328 model with LIP=LP=376 mm 5.3.5 Adjusting Local Strain Response While regularized models produce mesh-independent global force-displacement response, their local response, such as curvature, strain values are still mesh-dependent. To obtain the correct values of these parameters, post-processing of outputs is required. In Section 5.2.4, the post-processing of curvature recommended by Coleman & Spacone (2001) was outlined. However, there have been no recommendations so far on how to perform post-processing on concrete and reinforcing steel strain values of regularized models. This is of paramount importance for verifying the performance of concrete structures according to CSA S6-14, since the performance 85 criteria are defined in terms of strain limits. Without strain post-processing, checking of the strain limits of the code could be misleading. It is possible to post-process the strain response of regularized models using the post-processed curvatures suggested by Coleman and Spacone. However, this would require tremendous amount of post-processing effort and can become inconvenient. In this section, a much simpler and more direct method is formulated for the post-processing of strain values in regularized models. The proposed method is then applied to the regularized DBE and FBE models of Column 328 to confirm validity of the derived expressions. The Gfcc value for regularizing the DBE and FBE models is 135 N/mm, following the discussion in the previous section. The regularization of material models can be viewed as a mapping between the post-peak region of a reference material model and the post-peak region of the regularized model. This is illustrated in Figure 5.18 for the confined concrete and reinforcing steel material models. The shaded areas belong to the reference material models before regularization and the transformed post-peak regions belong to the regularized models. The question posed here is how to check a strain limit defined for the reference concrete model, εc1, or reinforcing material model, εs1, in the corresponding regularized model. One answer is to map the strain limits to the regularized material models similar to how the ultimate strain values are mapped in the regularization process. This is possible considering that the ratio of the post-peak energy up to the specified strain limit to the total post-peak energy is similar for the reference and the regularized material models. In Figure 5.18, the mapped strain limits of concrete and reinforcing steel are indicated by εc2 and εs2, respectively. The simplifying assumption, which also has been used in deriving the regularization expressions in the previous sections, is linear post-peak response for both concrete and steel material models. Using this assumption, the mapped the strain limits can be obtained using the following expressions: - Confined Concrete: εc2= (ε20c-PH-εocεcu-εoc) (εc1-εoc)+εoc (5.21) 86 - Unconfined Concrete: εc2= (ε20u-εoεu-εo) (εc1-εo)+εo (5.22) - Reinforcing Steel: εs2=εy+(εs1-εy)LgageLIP (5.23) In which, εu is the crushing strain of unconfined concrete, typically between -0.004 to -0.006. The detailed derivation of the above expressions is given in Appendix B. εoc εcuf’cc0.2 f’ccEccσ ε εc1 εc2 ε20c-PHf cc (a) εy εsu,expEsfuσ fyfsεs1 εs2 εsu ε (b) Figure 5.18 Mapping strain limits in the post-peak region of (a) confined concrete and (b) reinforcing steel materials 87 To test the derived expressions, they are applied to predict the displacements of the regularized DBE and FBE models of Column 328, which correspond to reinforcing steel strain of 0.0022 (yielding) and 0.025, unconfined concrete strain of -0.004 (cover spalling), and confined concrete strain of -0.018 (crushing). These strain limits are defined with respect to the original material models, with εu of -0.004, εcu of -0.018, and εsu,exp of 0.09. The first step is to find the displacement at the top of the column from monotonic response, corresponding to the first occurrence of these strain limits in the regularized models. In this step the strain limits are not mapped to the regularized material models yet. These displacements are listed in Part (a) of Table 5.8 and Table 5.9, for the DBE and FBE models of Column 328 with various mesh sizes. By inspecting the tabulated values, it is clear that except for the yielding limit, which is not in the post-peak region, both of the regularized DBE and FBE models predict mesh-dependent displacements for the specified strain limits. For instance, the regularized DBE model with 0.610 m mesh size, predicts the crushing of confined concrete (εcc =-0.018) at the displacement of 0.071m, while the regularized DBE model with 0.076 m mesh size, predicts the same displacement at 0.030 m, less than half of the first value. It should be noted that all these models produce mesh-independent global force-displacement response, since they are regularized. So, the mesh-dependency in the obtained displacements relates to strain limits not being post-processed. The second step is to map the strain limits using Equation (5.21) to (5.23). These are listed in Part (b) of Table 5.8 and Table 5.9 for the DBE and FBE models, respectively. For instance, the confined concrete strain limit of -0.018 for the original material model corresponds to strain limit of -0.0215 for the regularized DBE model with 0.610 m mesh size, and -0.1481 for the regularized DBE model with 0.076 m mesh size. This significant difference in the modified strain limits explains why the displacements of the models in Part (a) are considerably different. Once the strain limits are mapped, now the displacements corresponding to the first occurrence of the mapped strain limits are found for the regularized models. These are listed in Part (c) of Table 5.8 and Table 5.9 for the DBE and FBE models, respectively. Comparing the results in Part (c) with Part (a) of each table shows that the discrepancy in predicting the displacements corresponding to the post-peak strain limits is considerably reduced for the FBE models. The DBE models results still show some level of discrepancy in the displacements corresponding to 88 Displacement (m) εs=εy=0.0022 εs=0.025 εc=-0.004 εcc =-0.018 DBE 0.076 m 0.007 0.020 0.013 0.030 DBE 0.152 m 0.007 0.025 0.014 0.041 DBE 0.305 m 0.007 0.035 0.016 0.059 DBE 0.610 m 0.007 0.052 0.018 0.071 (a) Mapped Strain Limits εs=εy=0.0022 εs=0.025 εc=-0.004 εcc =-0.018 DBE 0.076 m 0.0022 0.124 -0.031 -0.148 DBE 0.152 m 0.0022 0.063 -0.016 -0.076 DBE 0.305 m 0.0022 0.033 -0.008 -0.040 DBE 0.610 m 0.0022 0.017 -0.005 -0.022 (b) Corresponding Displacement (m) εs=εy=0.0022 εs=0.025 εc=-0.004 εcc =-0.018 DBE 0.076 m 0.007 0.049 0.037 0.126 DBE 0.152 m 0.007 0.045 0.034 0.113 DBE 0.305 m 0.007 0.041 0.028 0.093 DBE 0.610 m 0.007 0.039 0.021 0.078 (c) Table 5.8 Column 328 DBE models: (a) displacements corresponding to the specified strain limits, (b) mapped strain limits based on the regularized material models, (c) displacements corresponding to the modified strain limits 89 Displacement (m) εs=εy=0.0022 εs=0.025 εc=-0.004 εcc =-0.018 FBE 0.610 2IP 0.007 0.041 0.018 0.071 FBE 0.914 3IP 0.007 0.025 0.014 0.045 FBE 1.828 4IP 0.007 0.025 0.014 0.044 FBE 1.828 6IP 0.007 0.017 0.013 0.027 (a) Mapped Strain Limits εs=εy=0.0022 εs=0.025 εc=-0.004 εcc =-0.018 FBE 0.610 2IP 0.0022 0.017 -0.005 -0.022 FBE 0.914 3IP 0.0022 0.033 -0.007 -0.040 FBE 1.828 4IP 0.0022 0.033 -0.007 -0.040 FBE 1.828 6IP 0.0022 0.078 -0.016 -0.094 (b) Corresponding Displacement (m) εs=εy=0.0022 εs=0.025 εc=-0.004 εcc =-0.018 FBE 0.610 2IP 0.007 0.030 0.021 0.081 FBE 0.914 3IP 0.007 0.030 0.020 0.080 FBE 1.828 4IP 0.007 0.030 0.019 0.078 FBE 1.828 6IP 0.007 0.033 0.023 0.079 (c) Table 5.9 Column 328 FBE models: (a) displacements corresponding to the specified strain limits, (b) mapped strain limits based on the regularized material models, (c) displacements corresponding to the modified strain limits 90 the confined and unconfined concrete strain limits. Compared to the FBE models, the DBE models predict higher displacement values for each of the strain limits except yielding. Considering the failure displacement of each model, the predicted displacements for the mapped strain limits seem more reasonable. For instance, for the crushing of confined concrete at εcc =-0.018, the mapped strain limit predicts displacement values between 0.078 to 0.126 m, and the reference strain limit predicts values between 0.027 to 0.071 m, while the predicted failure displacements are between 0.10 to 0.13 m. The unresolved discrepancy in the predictions of the DBE models is expected to be related to how DBE is formulated as opposed to FBE, considering that DBE ensures constant axial strain and curvature across the element, while FBE ensures constant axial load and linear curvature (Section 5.1.3). Overall, it seems that the suggested method for post-processing the local strain response of regularized models is working very well with the FBE models and is improving the predictions of the DBE models. Without post-processing the strain limits, the performance evaluation based on strain values would be improper. 5.3.6 Comparison of the Options for Dealing with Localization Issues It is worthwhile, to compare the two methods introduced in Section 5.2.4 for dealing with the localization issues in distributed plasticity models. To do so, the FBE and DBE models of Column 328 with LIP=LP and LIP=0.5 LP in the plastic hinge region are employed again. The material properties of the models are not regularized (εcu =-0.018 and εsu,exp =0.09). The monotonic force-displacement responses of these models are compared against those of the regularized DBE and FBE models in Figure 5.19 (the models are regularized using the Gfcc value recommended by Pugh). The results suggest that using either DBE or FBE models with LIP=LP does not provide reasonable and accurate estimation of the response. The results also show that surprisingly, the force-displacement response of the models with LIP=0.5 LP is fairly close to the response of the regularized models, using both DBE and FBE. Nevertheless, the local responses of the former differ from the latter, as they employ different material properties. The material model regularization is a more scientific approach to deal with the localization issue in distributed plasticity models and it allows the freedom to choose a suitable mesh size, which ensures accuracy of the results. However, in many cases the analysis tool does 91 not provide access to the material model properties to perform material regularization. In these cases, setting the mesh size so that LIP=0.5 LP seems to be an acceptable alternative. (a) (b) Figure 5.19 Comparison of the monotonic response of the regularized (a) DBE and (b) FBE models of Column 328 with the model satisfying LP=LIP 5.3.7 Validation of the Concentrated Plasticity Model for Test 328 The monotonic response of the fibre hinge model of Column 328 in SAP2000 is shown in Figure 5.20, along with the cyclic test results. The monotonic response closely follows the backbone of the cyclic response in the linear range and in the post-peak range up to a displacement of roughly 0.06 m, and then gradually degrades until the failure displacement. The predicted failure displacement is very close to the value obtained from the cyclic test result. 5.3.8 Comparison of the Distributed and Concentrated Plasticity Models of Column 328 A comparison is made between the monotonic responses of the concentrated plasticity and regularized distributed plasticity models of Column 328, which were developed in the previous sections (Figure 5.21). The distributed plasticity models are regularized using the crushing energy values suggested by Pugh (2012) and the adjusted confined concrete crushing energy values, as explained in Section 5.3.4. For column 328, the failure displacements predicted by the regularized DBE models is comparable to the fibre hinge model and they are both close to the failure displacement of the column obtained from the cyclic test result. However, the DBE model overestimates the strength in the first few cycles. The DBE models also overestimate the linear -900-700-500-300-100100300500700-0.15 -0.05 0.05 0.15Lateral Force (kN)Displacement (m)Test328DBE Lip=LpDBE Lip=0.5 LpDBE Reg-900-700-500-300-100100300500700-0.15 -0.05 0.05 0.15Lateral Force (kN)Displacement (m)Test328FBE Lip=LpFBE Lip=0.5 LpFBE Reg92 Figure 5.20 Monotonic force-displacement response of the SAP2000 Fibre Hinge model of Column 328 stiffness. Nevertheless, this is related to the assumed stiffness value for the shear spring at the base of the DBE models, and can be adjusted readily. The fibre hinge model on the other hand uses effective section properties values for the linear elastic beam column element to correct the stiffness value, and in this case it has provided a better estimation of the initial stiffness. The regularized FBE models underestimate the failure displacement of the column as discussed in the previous sections, while they closely predict the back bone of the cyclic curve. The monotonic response of the FBE models It is desirable to obtain the displacement values corresponding to the first occurrence of the concrete and reinforcement steel strain limits. Similar strain limits as those used in Section 5.3.5 are considered, and the corresponding displacements of the fibre hinge model are listed in Table 5.10. These values can be compared against average values of displacements predicted by the FBE and DBE models in Table 5.8 and Table 5.9, which are listed in Table 5.11. It is observed that the displacements predicted by the fibre hinge model are fairly close to the predictions of the FBE models. However, the fibre hinge model predicts smaller displacements in the post-peak region compared to the average displacements of the DBE models. The difference in the predicted displacements of the three types of models becomes larger at higher strain limits for concrete, such as the crushing of core concrete. -700-500-300-100100300500700-0.15 -0.05 0.05 0.15Lateral Force (kN)Displacement (m)Test328Fiber Hinge93 (a) (b) Figure 5.21 Comparison of the regularized (a) DBE and (b) FBE models with the concentrated plasticity models of Column 328 Corresponding Displacement (m) εs=εy=0.00224 εs=0.025 εc=-0.004 εcc =-0.018 SAP2000 Fibre Hinge 0.010 0.038 0.020 0.069 Table 5.10 Column 328 fibre hinge model, displacements corresponding to the specified strain limits Average Displacement (m) εs=εy=0.00224 εs=0.025 εc=-0.004 εcc =-0.018 FBE models 0.007 0.031 0.021 0.079 DBE models 0.007 0.044 0.030 0.102 Table 5.11 Average displacements of Table 5.8 and Table 5.9 for the FBE and DBE models of Column 328 -1100-900-700-500-300-100100300500700-0.15 -0.05 0.05 0.15Lateral Force (kN)Displacement (m)Test328DBE 0.310 m Reg Pugh 32.4 N/mmDBE 0.310 m Reg 135 N/mmFiber Hinge-1100-900-700-500-300-100100300500700-0.15 -0.05 0.05 0.15Lateral Force (kN)Displacement (m)Test328FBE 0.610 m 2IP Reg Pugh 115.6 N/mmFBE 0.610 m 2IP Reg 135 N/mmFiber Hinge94 5.3.9 Comparison of the Models for the Other Tests To extend the investigation of the previous section, the monotonic responses of the regularized DBE and FBE models of the other selected bridge columns are compared with those of the fibre hinge models. The DBE and FBE models are regularized using the adjusted confined concrete crushing energy in Section 5.3.4. The results are shown in Figure 5.22 along with the cyclic test results for Column 415, 430, 828, and 1028. These columns, as suggested by their numbers, have higher aspect ratios (4, 8 and 10), which in turn indicates that the columns are more slender and the flexural response is more dominant. Considering the results of Column 415 and 430, all three models perform well in terms of capturing the initial stiffness and the backbone of the cyclic curve. However, the regularized FBE models underestimate the failure displacements of the columns, while the fibre hinge models overestimate that. The regularized DBE models provide acceptable estimation of the failure displacements. The results for Column 828 and 1028 shows that the regularized FBE models underestimate the failure displacements by a greater degree compared to the shorter columns 328, 415, and 430. The Fibre hinge models slightly overestimate the failure displacements of the two. The regularized DBE model of Column 828 underestimates the failure displacement as well, while the regularized DBE model of Column 1028 provides an acceptable prediction of the failure displacement. All three models capture the initial stiffness and the backbone of the cyclic curve satisfactorily. Overall, from the investigations of the regularized distributed plasticity and concentrated plasticity models of the selected test column, the following was observed regarding the monotonic response of the models: 1) The regularized FBE models tend to moderately underestimate the failure displacements of the concrete bridge columns, while they provide accurate estimations of the initial and post-peak backbone of the cyclic response. 2) The regularized DBE models tend to provide accurate estimation of the failure displacements of concrete bridge columns, while they may overestimate the strength. 95 3) The fibre hinge models accurately capture the initial stiffness and backbone of the cyclic response. They may provide an acceptable estimation of the failure displacement or overestimate that. The above observations are made based on a few tested bridge columns. Several assumptions in the models affect the quality of their predictions. The assumed material models for concrete and reinforcing steel as well as the equation for estimating the plastic hinge length of the columns directly affect the response of the columns. Moreover, different analysis tools may have slight differences in terms of implementing the material models and the formulation of inelastic beam-column elements. Therefore, care should be taken in extending the above observations when other analysis tools are employed. 96 (a) (b) (c) (d) Figure 5.22 Comparison of the predictions of the regularized DBE and FBE models with the fibre hinge models for Column (a) 415, (b) 430, (c) 828, and (d) 1028 -600-500-400-300-200-1000100200300400-0.3 -0.1 0.1 0.3Lateral Force (kN)Displacement (m)Test415DBE 0.310 m 132 N/mmFBE 0.610 m 2IP 132 N/mmFiber Hinge-800-600-400-2000200400600-0.3 -0.1 0.1 0.3Lateral Force (kN)Displacement (m)Test430DBE 0.310 m 132 N/mmFBE 0.610 m 2IP 132 N/mmFiber Hinge-400-300-200-1000100200300-1 -0.5 0 0.5 1Lateral Force (kN)Displacement (m)Test828DBE 0.310 m 207 N/mmFBE 0.610 m 2IP 207 N/mmFiber Hinge-400-300-200-1000100200300-1 -0.5 0 0.5 1Lateral Force (kN)Displacement (m)Test1028DBE 0.310 m 242 N/mmFBE 0.610 m 2IP 242 N/mmFiber Hinge97 Chapter 6: Evaluating the Strain Limits of CSA S6-14 for RC Bridges In the previous chapter, appropriate modelling alternatives for PBD of RC bridges were examined and a number of methods for dealing with the localization of plastic strains in distributed plasticity models were examined. This chapter is a subsequent to the previous chapter, but pursues a different objective. The main objective of this chapter is to employ the strain limits of CSA S6-14 and BC Ministry of Transportation and Infrastructure (MoTI) Supplement (2016) to predict the damage to the tested bridge columns introduced in Chapter 5. To achieve so, the regularized FBE and DBE models generated and examined in Chapter 5, are utilized. The chapter starts by introducing the strain limits of the code and BC MoTI Supplement for RC structures. Next, a brief review of a number of performance criteria for RC bridges that are recommended in the literature and other design guidelines is presented. The review serves as a reference and a point of comparison for the recommended strain limits of the code and the supplement. Next the regularized FBE and DBE models of the tests are utilized to predict the damage to the columns based on the strain limits of the code and BC MoTI Supplement. The extent of the predicted damage is compared against the actual observed damage to the columns during the tests, and the results are discussed. This comparison reveals how accurately the combination of the established strain limits and the generated models predicts the actual damage to the columns. 6.1 Strain Limits in CSA S6-14 and BC MoTI Supplement An overview of the CSA S6-14 PBD approach was given in Section 2.6 with the corresponding tables copied in Appendix A . CSA S6-14 performance criteria for RC bridges include quantitative strain limits of concrete and reinforcing steel for specified levels of damage. The values of these limits are summarized in Table 6.1, where εc and εcc are unconfined and confined concrete strains, εcu is the ultimate strain capacity of confined concrete, and εs is reinforcing steel strain. After the release of the code, the strain limits at some damage levels were found to be unduly conservative. Meeting these criteria was particularly challenging for bridges on soft soil sites. Modifications to the CSA S6-14 performance criteria were adopted in the BC MoTI Supplement to CSA S6-14, which was published in late 2016. The document relaxed some of the 98 strain limits of the code and also provided additional strain limits and damage descriptions for various performance levels, as summarized in Table 6.2. Each strain limit in CSA S6-14 and BC MoTI Supplement represents the initiation of a damage state in ductile concrete members, as follows (Table 6.3): (1) Yielding of the longitudinal rebars: This damage state is reached when the reinforcing steel tensile strain exceed yielding. (2, 3) Cover Spalling: This damage state is controlled by the compressive strain of unconfined concrete, and the limiting strain values represent the onset of cover spalling. (4) Serviceability Limit 1: This damage state is reached when the longitudinal reinforcing steel tensile strain exceeds 0.01. This limit replaced the yielding criteria for minimal damage. The name of the damage state is assigned based on the similarity of this damage state to the Serviceability limit stated in Kowalsky (2000). (5) Serviceability Limit 2: This damage state is reached when the longitudinal reinforcing steel tensile strain exceeds 0.015, which corresponds to residual crack width exceeding 1 mm (Kowalsky 2000). (6) Reduced Buckling: This damage state is reached when the longitudinal reinforcing steel tensile strain exceeds 0.025. This limit replaced Serviceability Limit 2 for repairable damage. Since, this value is half of the strain limit established to check the buckling of longitudinal rebars, the damage state is referred to as reduced buckling. (7, 8) Core Crushing: This damage state is controlled by the compressive strain of confined concrete. (9) Reduced Fracture 1: 99 Established by Kowalsky (Kowalsky 2000; Goodnight et al. 2013), the onset of buckling in the longitudinal reinforcing steel is checked by their tensile strain not exceeding 0.05. Since buckling of the longitudinal rebars is a prelude to their fracture, this damage state is called reduced fracture. (10) Reduced Fracture 2: This limit represents the initiation of fracture in the longitudinal reinforcing steel for probable replacement damage, and is controlled by their tensile strain not exceeding 0.060 or 0.075, depending on size. Damage Level Concrete Strain Reinforcing Steel Strain None NS NS Minimal εc > -0.004 εs < εy Repairable NS εs < 0.015 Extensive εcc > εcu εs < 0.050 Probable Replacement NS NS Table 6.1 CSA S6-14 concrete and reinforcing steel strain limits (NS: not specified) Damage Level Concrete Strain Reinforcing Steel Strain Minimal εc > -0.006 εs < 0.010 Repairable NS εs < 0.025 Extensive εcc > 0.8 εcu εs < 0.050 Probable Replacement εcc > εcu εs < 0.075 (30 M or smaller) εs < 0.060 (35 M or larger) Table 6.2 BC MoTI Supplement to CSA S6-14 concrete and reinforcing steel strain limits (NS: not specified) Damage State Strain Limit (m/m) Yielding(1) εs < εy Cover Spalling 1(2) εc < -0.004 Cover Spalling 2(3) εc < -0.006 Serviceability Limit 1(4) εs > 0.010 Serviceability Limit 2(5) εs > 0.015 Reduced Buckling(6) εs > 0.025 80% Core Crushing(7) εcc > 0.8 εcu Core Crushing(8) εcc > εcu Reduced Fracture 1(9) εs > 0.050 Reduced Fracture 2(10) εs > 0.060 or 0.075 Table 6.3 Strain limits associated to the flexural damage states of reinforced concrete columns 6.2 Performance Criteria for RC Bridges in Literature and Other Design Guidelines The flexural performance of reinforced concrete members are commonly measured by using limits on reinforcing steel and concrete strains, section curvatures, or deformations of the 100 members. Energy-based criteria may be used as well. In the following sections, an overview of some of the strain and deformation limits recommended in literature and design guidelines for RC bridges is presented. 6.2.1 Strain Limits One of the early examples of adopting strain limits for the performance-based design of bridges can be found in ATC-32 (1996a). The limits are listed in Table 6.4 for three performance levels of minimal, repairable, and significant damage. In comparison to both CSA S6-14 and BC MoTI Supplement, the steel strain limits are larger, while the concrete limits are comparable to the limits of the both documents. Table 6.5 reproduces the strain limits recommended by Kowalsky (2000) for the two performance levels of serviceability and damage control. These values are one of the reference values used to establish the strain limits of CSA S6-14. The serviceability concrete strain corresponds to initiation of spalling and the steel strain corresponds to residual crack width not exceeding 1 mm. The damage control concrete limit corresponds to repairable damage in concrete and is estimated using the energy balance approach developed by Mander et al. (1988) to obtain the ultimate strain of concrete in compression. Test results have shown that this value is consistently conservative by 50% and so it was recommended for damage control performance level. Hose and Seible (1999) also report reinforcing steel and concrete strain limits for evaluating the performance of RC bridges under the five-level performance assessment procedure of Table 2.3 (Table 6.6). The recommended values are comparable to those of CSA S6-14 and BC MoTI Supplement. Vossoghi and Saiidi (2010) recommended limits for mean tensile strain of longitudinal reinforcing steel based on the shake-table test of scaled models. These values are summarized in Table 6.7. What is unique about the strain limits of this table is the consideration of the effects of shear demand and the type of ground motion on the flexural strain limits of reinforcing steel. 101 Minimal Damage Repairable Damage Significant Damage Concrete Strain Limit (1/3) εcu or 0.004 (1/3) εcu or 0.008 εcu or 0.012 Steel Strain Limit Grade 420 (10-25 mm) 0.03 0.08 0.12 Steel Strain Limit Grade 420 (29-57 mm) 0.03 0.06 0.09 Table 6.4 ATC-32 (1996) strain limits (AC1 2016) Limit State Concrete Strain Limit Steel Strain Limit Serviceability 0.004 0.015 Damage Control 0.018, εcu 0.06 Table 6.5 Strain limits by Kowalsky (2000) Level Description Steel Strain Concrete Strain % Drift Displacement Ductility I Fully Operational <0.005 <0.0032 <1.0 <1.0 II Operational 0.005 0.0032 1.0 1.0 III Life Safety 0.019 0.01 3.0 2.0 IV Near Collapse 0.048 0.027 5.0 6.0 V Collapse 0.063 0.036 8.7 8.0 Table 6.6 Bridge performance/design parameters SRPH-1 by Hose and Seible (Hose and Seible 1999; Transportation Research Board 2013) Visual Damage State DS-1 DS-4 DS-5 Approximate Performance State Ductile Flexural Column Condition Fully Functional Operational Delayed Operation Far-field ground motions Shear stress < 4√𝑓𝑐′ 0.012±0.006 0.035±0.007 0.045±0.014 Far-field ground motions Shear stress > 6√𝑓𝑐′ 0.005±0.003 0.035±0.008 0.051±0.014 Near-field ground motions Shear stress < 4√𝑓𝑐′ 0.013±0.003 0.033±0.016 0.038±0.015 Table 6.7 Mean tensile strains in longitudinal reinforcement corresponding to visual damage states as reported by Vosooghi and Saiidi (2010; ACI 2016) (f’c is in ksi) 102 6.2.2 Curvature Limits The FHWA Seismic Retrofitting Manual (2006) defines performance limits in terms of curvature limits. These curvature limits are based on some reinforcing steel and concrete strain limits. The limits are reproduced here, as follow: - Compression failure of unconfined concrete: ϕp=εuc-ϕy (6.1) - Compression failure of confined concrete: ϕp=εcu(c-d")-ϕy (6.2) - Compression failure due to buckling of the longitudinal reinforcement: ϕp=εb(c-d')-ϕy (6.3) εb=2fyEs (6.4) - Longitudinal tensile fracture of reinforcing bar: ϕp=εsu,exp(d-c)-ϕy (6.5) - Low-cycle fatigue of longitudinal reinforcement: ϕp=2εap(d-d')=2εapD' (6.6) εap=0.08 (2Nf)-0.5 (6.7) Nf=3.5 (Tn)-1/3 (6.8) 103 - Failure in the lap-splice zone: ϕp=(μlapϕ+7)ϕy (6.9) 6.2.3 Deformation Limits In their fundamental study on performance models for flexural damage in RC columns, Berry and Eberhard (2003) generated expressions for predicting the drift ratios corresponding to the onset of cover spalling and bar buckling in RC columns. They used a subset of UW-PEER reinforced concrete column performance database (now PEER performance database) to identify the key parameters that affect the drift ratio, displacement ductility, plastic rotation, and longitudinal strains corresponding to these two damage states. A total of 114 rectangular-reinforced and 52 spiral-reinforced columns were selected, which all met the following three criteria: 1) They were flexure-critical as defined by Camarillo (2003). 2) Their aspect ratio was 1.95 or greater. 3) The longitudinal reinforcing steel was continuous with no splice. The database of the columns used by Berry and Eberhard was employed to derive the AASHTO Guide Specifications for Seismic Bridge Design (2011) implicit formulae to obtain the displacement capacity of RC columns for Seismic Design Categories of B and C (Transportation Research Board 2013). More recently, the ACI 341.4R-16 report (2016) on the seismic design of bridge columns based on drift, utilized Berry and Eberhard`s expressions for cover spalling and bar buckling to obtain the mean drift capacities corresponding to operational and delayed operational performance levels, respectively. For the delayed operational performance level, the document assumed Caltrans SDC 1.6 (2010) minimum lateral reinforcement ratio for columns with diameters greater than 36 in (900 mm) to calculate the effective reinforcement ratio, a 0.8 factor to account for biaxial bending, and a normalized bar diameter of 0.05, to calculate the mean drift limit, as follows: 104 DrO=1.6 (1-PAgf'c) (1+Lcol10hcol) (6.10) DrDO=0.8*3.25 (1.45+1.125PAgf'c) (1-PAgf'c) (1+Lcol10hcol) (6.11) In the above equations, DrO and DrDO are drift ratios corresponding to operational and delayed operational performance levels in %, P/Agf’c is axial load ratio, and Lcol/hcol is the shear span-depth ratio of column6. The equations indicate the effect of axial load ratio and shear span-depth ratio on the flexural drift capacities. As axial load increases in a column, the drift capacity of the column reduces. The shear span-depth ratio affects the drift ratio at the onset of the damage state. With an increase in the shear span-depth ratio, the yield displacement and also the plastic hinge length increase. 6.3 Examining the Strain Limits of CSA S6-14 and BC MoTI Supplement 6.3.1 Damage Observed in the Tests According to Lehman et al (2004), the sequence of the observed damage states were similar for all of the tested bridge columns, and included concrete cracking, yielding of longitudinal reinforcements, initial concrete cover spalling, extensive concrete cover spalling, spiral fracture, longitudinal reinforcement buckling, and fracture. Table 6.8 lists the displacement cycles, at which the damage states were observed for the tested columns. Table 6.9 shows the average compressive strain of concrete and average compressive and tensile strains of reinforcing steel corresponding to the initiation of damage states. The strain values were measured by averaging the values along the gauge length between the instrumentation rods, while local peak strain values were equal or greater than these values. Lehman and colleagues presented experimental cumulative distribution functions (CDF) for the compressive strain of concrete corresponding to initial spalling and initial core crushing, and for the tensile strain of reinforcing steel corresponding to different residual crack width. The test data showed large dispersions in the measured strain values. For cover spalling, the 6 The second equation is developed for heavily reinforced circular concrete columns with spirals or hoops. The document provides also an equation for lightly reinforced columns. 105 compressive strain of concrete ranged between -0.0039 to -0.011 with a mean value of -0.00664. Columns with larger aspect ratios had smaller spalling strains. It is argued that the strain value of -0.004 recommended by ATC-32 for initial spalling (Table 6.4) is the mean minus one standard deviation of the test data. For core crushing, the measured strain values ranged between -0.010 to -0.0297. The mean tensile strain of reinforcing steel was about 0.023 and 0.024 for the residual crack width being greater than 0.13 and 0.25 mm, respectively. For bar buckling, the authors did not present a CDF of the strain values as they are heavily affected by the cyclic history, and so the results would not be generally applicable. It is argued by the authors that bar buckling damage state cannot be adequately captured by a single limiting strain value, and a more refined model is necessary. Test# Yielding Initial Spalling Initial Core Crushing Bar Buckling Failure (20% Strength loss) 328 0.013 0.030 0.071 0.132 0.132 828 0.059 0.178 0.445 NR NR 1028 0.098 0.254 0.889 0.889 0.089 415 0.016 0.038 0.127 0.178 0.178 430 0.017 0.051 0.178 0.178 0.178 Table 6.8 Displacements cycle in meter reported for the first occurrence of the damage states in the tested bridge columns (Lehman et al. 2004) Initial Spalling Initial Core Crushing Bar Buckling Failure (20% Strength loss) Test# εc εcc εs-comp εs-ten εs-comp εs-ten 328 -0.0057 -0.0098 -0.0570 0.0860 -0.0620 0.0440 1028 -0.0043 -0.0175 -0.0310 0.0980 -0.0440 0.0470 415 -0.0068 -0.0220 -0.0470 0.0730 -0.0470 0.0500 430 -0.0110 -0.0170 -0.0510 0.0890 -0.0520 0.0440 Table 6.9 Recorded average strain values corresponding to the first occurrence of the damage states in the tested bridge columns (Lehman et al. 2004) 6.3.2 Damage Predicted by the Models The regularized FBE and DBE models of Chapter 3 were used to predict the damage to the test columns. The models were subjected to the displacement time history of Figure 5.7 and Table 5.3. The strain limits of the code and BC MoTI Supplement in Table 6.3 were checked for each 7 The difference between the minimum and maximum strain values is partly due to the difference in the confinement of the columns. 106 model by adjusting the local strain response, as explained in Section 5.3.5. The displacements corresponding to the first occurrence of the strain limits are listed in Table 6.10. The regularized DBE models for Column 828 and 1028, did not predict the occurrence of some of the damage states, as designated by NA in the table. FBE model Test# 328 828 1028 415 430 εs > 0.00224 0.007 0.048 0.076 0.013 0.014 εs > 0.01 0.015 0.086 0.123 0.023 0.025 εs > 0.015 0.019 0.093 0.143 0.028 0.031 εs > 0.025 0.028 0.125 0.188 0.039 0.042 εs > 0.05 0.051 0.181 0.256 0.068 0.073 εc < -0.004 0.020 0.117 0.190 0.033 0.031 εc < -0.006 0.030 0.164 0.253 0.048 0.044 εcc < -0.0144 0.063 0.327 0.566 0.110 0.098 εcc < -0.018 0.075 0.269 0.403 0.115 0.114 DBE model Test# 328 828 1028 415 430 εs > 0.00224 0.007 0.049 0.076 0.013 0.014 εs > 0.01 0.020 0.102 0.153 0.031 0.033 εs > 0.015 0.028 0.131 0.183 0.398 0.042 εs > 0.025 0.043 0.170 0.239 0.061 0.064 εs > 0.05 0.082 0.294 0.420 0.114 0.117 εc < -0.004 0.029 0.242 0.518 0.051 0.046 εc < -0.006 0.047 0.391 NA 0.082 0.073 εcc < -0.0144 0.093 0.583 NA 0.152 0.126 εcc < -0.018 0.103 NA NA 0.171 0.148 Table 6.10 Predictions of the regularized FBE and DBE models for the cyclic displacements in meter corresponding to the first occurrence of the specified strain limits 6.3.3 Comparison and Discussion It is now possible to compare the predictions of the regularized FBE and DBE models for damage to the columns with the actual observed damage in the tests. This was done by comparing the predicted drift versus observed drift ratios for the reported damage states in the tests, including yielding of the longitudinal reinforcing steel, initial spalling, initial core crushing, and longitudinal bar buckling, as tabulated in Table 6.11. For column 828, tests results were not available for bar buckling drift ratio, and therefore this damage state was not evaluated for the column. The comparison of the values in the table, suggest the following: 107 1) For yielding damage state, both DBE and FBE models provide reasonably conservative predictions of the drift capacities using the yielding criteria of the code. The predictions of both models are fairly close in all of the tests. 2) For initial spalling, the FBE model provides reasonable estimates of the drift capacities using the BC MoTI Supplement criteria with the exception of Column 415, and conservative estimates, using the code criteria. The DBE model tends to overestimate the drift capacities using both the code and BC MoTI Supplement criteria. The difference between the predicted and observed values increases with the aspect ratio of the columns and is significant for Column 828 and 1028. 3) For initial core crushing, the FBE model provides conservative estimates of the drift capacities using the code criteria, with the exception of Column 328. The underestimation of capacities is more significant in columns with larger aspect ratios. The DBE model tends to overestimate the drift capacities. 4) For bar buckling, both models provide considerably conservative estimates of the drift capacities using the code criteria. As presented in Section 6.2.1, discrete deterministic strain limits have been utilized for evaluating RC structures in the first generation performance-based procedures. Due to uncertainty in strain limits at the onset of damage states, typically conservative lower-bound values are recommended. As a result, a level of conservatism is implied in the performance evaluation utilizing these strain limits (Transportation Research Board 2013). However, the level of added conservatism is not accurately quantified nor is it consistent across damage states. To address this issue, it is recommended to use a probabilistic description for the strain limits at the onset of damage states. While a probabilistic solution refines the solution, it may not completely address the issue. A full description of damage in many instances cannot be achieved by a single strain limit, and better models with more than one parameter are necessary. Bucking of the longitudinal reinforcing steel is an example, as explained in Section 6.3.1. Another consideration with using strain limits to predict damage is the numerical model used to predict the strain values. The accuracy of damage predictions cannot be evaluated solely based on how the strain limits are linked to the onset of damage states; rather the accuracy in predicting damage is a combination of the accuracy in the strain limits and the numerical model 108 used to evaluate those limits. In probabilistic terms, the uncertainty in evaluating damage is a product of the uncertainty in the strain limits and modelling uncertainty. The effect of the latter is clearly visible by comparing the predictions of the regularized FBE and DBE models in this section. While both models are supposed to be adequate for predicting damage, their predicted displacements vary a lot and show significant discrepancy with the observed values in the tests. The FBE models tend to underestimate the displacement capacities, while the DBE models tend to overestimate the capacities of those damage states related to concrete compressive strains, in columns with larger aspect ratios. The question is whether the predictions of the models for these damage states can be safely used for performance evaluation. To fully understand this problem, it should be noted that the observed displacements in the tests correspond to the first occurrence of each damage state and not the full development of that damage state. Besides, the observed displacements have inherent uncertainty. One way to check the predicted displacements is by using the Berry and Eberhard`s expressions introduced in Section 6.2.2. The expressions can be employed as the point of comparison for the element level displacement response. 109 Yielding (εs > 0.00224) Test# Measured FBE model DBE model 328 0.013 0.007 0.007 828 0.059 0.048 0.049 1028 0.098 0.076 0.076 415 0.016 0.013 0.013 430 0.017 0.014 0.014 Initial Spalling (εc < -0.004) Initial Spalling (εc < -0.006) Test# Measured FBE model DBE model FBE model DBE model 328 0.030 0.020 0.029 0.030 0.047 828 0.178 0.117 0.242 0.164 0.391 1028 0.254 0.190 0.518 0.253 NA 415 0.038 0.033 0.051 0.048 0.082 430 0.051 0.031 0.046 0.044 0.073 Initial Core Crushing (εcc < -0.018) Test# Measured FBE model DBE model 328 0.071 0.075 0.103 828 0.445 0.269 NA 1028 0.889 0.403 NA 415 0.127 0.115 0.171 430 0.178 0.114 0.148 Bar Buckling (εs > 0.05) Test# Measured FBE model DBE model 328 0.132 0.051 0.082 1028 0.889 0.256 0.420 415 0.178 0.068 0.114 430 0.178 0.073 0.117 Table 6.11 Comparison between the measured and predicted displacements by the regularized FBE and DBE models, corresponding to the first occurrence of damage states (all values are in meter) 110 Chapter 7: Implementation of the Frameworks for CSA S6-14 PBD The two frameworks introduced in Chapter 4: are applied to assess the seismic design of a two-span steel girder concrete bridge. Through this exercise, the step-by-step implementation of the frameworks for practical design purposes is demonstrated. The chapter is concluded by comparing the outcomes of the two assessments and a discussion on advantages and disadvantages of the deterministic versus the probabilistic framework options. 7.1 Description of the Bridge The bridge considered for this study is a hypothetical major-route bridge located in Victoria, British Columbia. The assumed coordinates of the bridge site are 48.4284, -123.3656. It is a two-span reinforced concrete bridge with steel girders. The initial member sizing of the bridge was achieved from force-based design principals and based on experience. A schematic elevation view of the bridge as well as the cross-section of the bridge at the pier are shown in Figure 7.1 and Figure 7.2. The total length of the bridge is 125 m, with the west and east span each being 60 m and 65 m, respectively. The superstructure is comprised of three steel girders topped with a 0.225 m concrete slab and a 0.09 m asphalt overlay. The section of the steel girders changes along each span as shown in the Figure 7.1, and the maximum depth of the girders is 2.9 m. The bridge bent includes two 8 m high circular reinforced concrete columns, connected at the top with a 2.1x1.8 m reinforced concrete cap-beam. The columns are both 1.5 m in diameter and has 35-35M longitudinal rebars, making up a 2% longitudinal reinforcement ratio. They are laterally reinforced with 20M spirals at 0.07 m pitch in the plastic hinge region (1.5m from the top and bottom of the columns) and 0.15 m pitch, elsewhere. The thickness of the cover concrete for both columns is 0.075 m, and their axial force ratio (Pa/f’c Ag) is 0.10. The minimum specified compressive strength of concrete for all members is 35 MPa, with the unit weight of 24 kN/m3. The reinforcing steel grade is 400R with minimum specified yield strength of 400 MPa and ultimate yield strength of 540 MPa. The unit weight of the reinforcing steel is 77 kN/m3. Each column has a 1.5 m deep 6x6.5 m concrete spread footing. At the abutments, the bridge has expansion bearings and it is free to move in the longitudinal direction, until closing of the gap. At the pier, the bridge has pinned bearings and transfers only shear. 111 7.2 Bridge Site Properties The soil profile at the bridge site includes a soft sand layer corresponding to site class D in CSA S6-14 (see Appendix A ). These conditions roughly correspond to a uniform sand layer with assumed shearwave velocity of 180-200 m/s, friction angle of 32 degrees, zero cohesion, Poisson’s ratio of 0.3, and unit weight of 18 kN/m3. For this site condition, the effects of soil-structure interaction must be considered. 15000 mmSection 120000 mmSection 225000 mmSection 325000 mmSection 325000 mmSection 215000 mmSection 18000 mm2-ϕ 1525 mm ColumnsWest Abutment East AbutmentPier Figure 7.1 Schematic elevation view of the bridge 10920 mm225 mm2900 mm90 mm Thick Asphalt OverlayHaunchRailingParapet3700 mm 3700 mm10000 mm 8000 mm1800 mm500 mmCapbeam, 2100 mm wide1525 mm1500 mm6500 mmFootings, 6000 mm wide Figure 7.2 Schematic cross-section of the bridge at the pier 112 7.3 Probabilistic Seismic Hazard Analysis A probabilistic seismic hazard analysis was conducted for the city of Victoria using EZ_FRISK (The Fugro Consultants 2015). Three distinctive sources of earthquakes are active in this region, namely shallow crustal, and deep subcrustal sources, and Cascadia subduction zone. All three sources contribute to the seismic hazard, depending on the fundamental period of structure and site-to-source distance. The proposed probabilistic hazard model for Cascadia subduction zone (Halchuk et al. 2014) enables a full PSHA for the region, and combines the contributions of all the three sources probabilistically at once. Subsequently, uniform hazard spectrum of Victoria was obtained for 10%/50, 5%/50, and 2%/50 hazard levels, corresponding to 475, 975, and 2475-year return periods, respectively (Figure 7.3-a). The calculated UHS values are close to the values obtained from the hazard calculator available online at the Natural Resources Canada Website (2016). The design spectrum was then obtained using the UHS at each hazard level and appropriate site coefficients recommended by CSA S6-14 (Clause 4.4.3.3). Since the abutments were not specifically designed for sustained soil mobilization, 5% damped spectral response acceleration values should be used (Clause 4.4.3.5). The 5% damped design spectra of the bridge at the specified hazard levels are presented in Figure 7.3-b. These were utilized in the response spectrum analysis of the bridge, while the selecting and scaling of ground motion records for time history analysis were performed using uniform hazard spectra. (a) (b) Figure 7.3 (a) UHS and (b) design spectra for site class D at 2%/50, 5%/50, and 10%/50 hazard levels 0.00.20.40.60.81.01.21.40 2 4 6 8 10Spectral Acceleration (g)Period (s)2%/505%/5010%/500.00.20.40.60.81.01.21.40 2 4 6 8 10Spectral Acceleration (g)Period (s)2%/505%/5010%/50113 7.4 Performance Criteria and Analysis Requirements - Seismic Performance Category The fundamental period of the bridge in both of the longitudinal and lateral directions is greater than 0.5 s (see Section 7.6). The seismic performance category of a major-route bridge with T ≥ 0.5 s and S(1.0) ≥ 0.3 is SPC 3 (Clause 4.4.4, Table A.2). - Regularity and Minimum Analysis Requirements According to the definition of Clause 4.4.5.3.2 the case study bridge is a regular bridge. The minimum analysis requirements of a regular major-route bridge in seismic performance category 3 is elastic dynamic analysis at 2%/50 and 5%/50 hazard levels and is elastic static analysis at 10%/50 hazard level (Clause 4.4.5.3.1, Table A.4 and Table A.5). For this case study, response spectrum analysis (RSA) and nonlinear time-history analysis were utilized to obtain the seismic demands on the bridge at the specified hazard levels. In addition, inelastic static pushover analysis was utilized to get the sequence of plastic hinge formation in the ductile members (i.e. columns), and the drift capacities corresponding to the first occurrence of the considered performance criteria (drift capacities). - Minimum Performance Levels The minimum performance levels for major-route bridges in terms of tolerable structural damage is “minimal” at 10%/50, “repairable” at 5%/50, and “extensive” at 2%/50 hazard level (Clause 4.4.6.2, Table A.6). The minimum serviceability objectives for the above performance levels, is “immediate”, “service limited”, and “service disruption”, respectively. - Possible Failure Mechanisms Prior to setting the performance criteria, the possible local and global failure mechanisms should be determined. Here we consider four possible failure mechanisms as follows: - Ductile failure of the columns in flexure (local failure) - Brittle failure of the columns in shear (local failure) - Unseating of the deck at the abutments in the longitudinal direction (global failure) - Pounding between the deck and the abutments (global failure) 114 Other failure mechanisms such as foundation soil failure, abutment backfill soil failure, etc. should also be considered, which are out of the objectives of this case study. - Performance Criteria I. Flexural Failure of the Columns The strain limits of the code and BC MoTI Supplement and the corresponding damage states were presented in Section 6.1. The ultimate strain capacity of confined concrete can be calculated using Equation (5.18). For εfs, a value of 0.09 could be used in the equation, following the recommendation of Caltrans SDC 1.7 (2013) for the reduced ultimate tensile strain of Grade 400 #10 (Metric #32) rebars or smaller. The value of f’cc, can be obtained using Mander et al. (1988) constitutive model. The confinement factor for the column cross section in the plastic hinge region is calculated as 1.288 using Mander model, which multiplied by the expected compressive strength of f`ce=43.75 MPa, gives f’cc=56.35 MPa. Substituting all values in the above expression gives an ultimate compressive strain capacity of -0.0163 for the plastic hinge region. The established strain limits and their corresponding damage states are listed in Table 7.2. Hazard Performance Level CSA S6-14 BC MoTI 10%/50 Minimal Damage εc >-0.004, εs < εy εc >-0.006, εs <0.010 5%/50 Repairable Damage εs <0.015 εs <0.025 2%/50 Extensive Damage εc >-0.0163, εs <0.050 εc >-0.0130, εs <0.050 Table 7.1 CSA S6-14 and BC MoTI strain limits associated to the performance levels of a major-route bridge Damage State Strain Limit (m/m) Yielding(1) εs > 0.0024 Cover Spalling 1(2) εc < -0.004 Cover Spalling 2(3) εc < -0.006 Serviceability Limit 1(4) εs > 0.01 Serviceability Limit 2(5) εs > 0.015 Reduced Buckling(6) εs > 0.025 80% Core Crushing(7) εc < -0.0130 Core Crushing(8) εc < -0.0163 Reduced Fracture(9) εs > 0.05 Table 7.2 CSA S6-14 and BC MoTI strain limits associated to the performance levels of a major-route bridge 115 II. Shear Failure of the Columns The brittle shear failure of the columns is checked by comparing the shear demand versus capacity of the columns. Clause 4.4.10.4.3 of CSA S6-14 defines the shear demand as either the unreduced elastic design shear or the shear corresponding to inelastic hinging of the columns calculated by using probable flexural resistance of the member and its effective height. However, this has been modified in the BC MoTI Supplement to exclude the former method. The shear capacity of concrete can be calculated using either the simplified method with β=0.1 and θ=45̊ (Clause 4.7.5.2.4), or by using the general method, which modifies the shear capacity based on the member axial strain (Clause 8.9.3.7). BC MoTI Supplement allows using more refined methods to calculate seismic shear capacity, which modify the shear capacity based on ductility demands. III. Unseating and Pounding of the Deck with the Abutments To check the last two failure mechanisms, the longitudinal displacement at the deck level should meet the following two criteria: Δdeck≤ Lexpansion (7.1) Δdeck≤ N (7.2) In the above expressions, Lexpansion is the length of the longitudinal gap and N is the provided support length at the abutments. 7.5 Structural Analysis Models A 3D spine model of the bridge was generated in CSI SAP2000 version 18.2.08. Expected material properties were used in the definition of steel and concrete materials. The behaviour of the unconfined and confined concrete was modelled with the Mander et al. (1988) constitutive model. The program automatically calculates and applies the confinement factor to the confined concrete material from the input information of a section. Two models were utilized for the bridge; a nonlinear fibre hinge model for performing nonlinear time-history analysis and 8 CSI SAP2000 was selected as the analysis tool as it is one of the most widely used commercial tools in bridge engineering along with CSI Bridge. Therefore, the implementation example would demonstrate the applicability of the deterministic and probabilistic assessment frameworks in practice. 116 pushover analysis, and an elastic model with effective material properties for response spectrum analysis and modal analysis. The two models differ in how they represent the nonlinear behaviour of the substructure ductile elements (i.e. columns), but both use similar superstructure models and boundary conditions. - Elastic Cracked Model The cracked section properties of the columns were calculated following Section 5.1.1. The effective flexural stiffness was calculated from the moment-curvature response of the column section (Figure 7.4) as the slope of the line connecting the origin to the point of first yield in the longitudinal rebars. This gave EcIeff=0.456 EcIg. Similar stiffness modifier was applied for the effective shear stiffness of the columns. A property modifier of 0.2 was also applied to the torsional constant of the column, following Caltrans SDC 1.7 (2013) recommendations. The flexural stiffness of the cap-beam was modified by a factor of 0.5. Since the super structure steel girders were capacity protected, it was assumed that they remain essentially elastic under seismic loading. Therefore, the steel girders and the concrete deck slab were modelled using elastic frame elements with composite section properties as calculated in Table 7.3. A nominal linear spring was assigned to the ends of the deck in the lateral direction to mimic the restraining effect of shearkeys and remove the unrealistic modes of vibration in that direction. In the longitudinal direction, the deck is free to move and simplified roller boundary conditions were employed to model the seat-type abutments. Section1 Section2 Section3 Equivalent Steel Area (m2) 1.61 1.65 1.82 Dead Load (kN/m) 124 127 140 Ivertical (m4) 0.82 0.91 0.96 Itransverse (m4) 8.20 8.50 9.00 Table 7.3 Composite section properties of the deck at different sections (see Figure 7.1) - Fibre Hinge Model For NTHA and pushover analysis, a fibre hinge model was created in SAP2000, according to Section 5.1.2 and 5.3.2. The fibre discretization of the column section is shown in Figure 7.4. The fibre hinges were assigned to the mid-height of the plastic hinge zone of the columns. Using Equation (5.3), the plastic hinge length for the longitudinal direction with single curvature was 117 obtained as 1089 mm, and for the lateral direction with double curvature as 754 mm. Similar elastic section stiffness modifiers as used with the elastic cracked model, were assumed for the column and cap-beam sections. The deck was assumed to remain elastic. - Modelling Foundation Flexibility to Account for SSI For the site class D model, the soil-structure interaction (SSI) effects should be accounted for properly. The foundation flexibility effects were captured using a set of six uncoupled equivalent springs, recommended by FEMA-356 (2000a) for shallow rigid foundations. FEMA-356 provides expressions for calculating spring constants as well as the embedment correction factors for shallow rigid foundations. The spring constants of the foundations were calculated using degraded shear moduli of the soil. According to FEMA-356, the resulting static stiffness values are sufficient to represent repeated loading conditions in seismic events. The degraded shear modulus of the soil was taken as 0.5 to 0.2 of the maximum shear modulus, following the commentary of CSA S6-14, and was calculated as 17 MPa for the considered site class D soil profile. Kinematic interaction and foundation damping were not considered in the case studies, as including their effects were not indispensable to the focus of the thesis. 7.6 Fundamental Period of the Bridge The results of the modal analysis on the bridge model revealed that in the longitudinal direction, the first mode with the period of T1-long=1.61 s was dominating the response with 98% contribution. In the lateral direction two modes contributed to 90% of the response; the first mode with the period of T1-lat=0.56 s and a contribution of 84%, and the second mode with the period of T2-lat=0.14 s and a contribution of 11%. 7.7 Ground Motion Selection and Scaling The commentary of CSA S6-14 recommends a minimum of 11 ground motions each containing two horizontal components to be selected for NTHA. The selected motions should be representative of the tectonic regime, magnitude and distances that control the seismic hazard, and the site condition. The target spectrum for selection and scaling of ground motion records was site class D UHS at 2%/50 hazard level. The records were linearly scaled to match to the target spectrum by minimizing the MSE within a specified period range. MSE is the mean squared error of the difference between the spectral acceleration of the record and the target 118 Figure 7.4 Fibre cross section of the columns in SAP2000 (left) and the moment-curvature response of the section in the plastic hinge region (right) (the response is calculated under the dead load). Figure 7.5 View of the bridge model in SAP2000 04,0008,00012,00016,0000.00 0.05 0.10 0.15Moment (kN-m)Curvature (1/m) 119 spectrum (PEER 2010). The spectral acceleration of each record was calculated by taking the geometric mean of the two horizontal components of that record. For the period range of interest, the code commentary recommends a period range of 0.2T1 to larger of 2 T1 and 1.5 s, where T1 is the fundamental period of the bridge. This would yield a period range of 0.14-3.22 s for the bridge model. The selected records should include ground motions from crustal, subcrustal, and subduction earthquakes, with proper magnitudes and source to site distances. This information was obtained from deaggregation of the 2%/50 UHS of Victoria at the fundamental periods of the bridge in the longitudinal and lateral directions, as shown in Table 7.4. Because the period range of interest spans from relatively short periods up to much longer periods, it was decided to match the records of each type of earthquake to a portion of the period range of interest, instead of the entire range. This is justified by considering that the contribution of each earthquake source to the hazard varies considerably across the period range of interest. As a result, the shape of the spectrum of the records for that type of earthquake becomes less conforming to the shape of the UHS. This will in turn cause poor matches to the target spectrum, if the records are to be linearly scaled in the entire period range of interest. Earthquake Source Magnitude Distance (km) Crustal 6 - 7.5 10 - 40 Subcrustal 6.5 - 7.5 50-100 Subduction 8.5 - 9.0 50-100 Table 7.4 Deaggregation of 2%/50 UHS of Victoria at period of 1.61 s. Results of the deaggregation at the 0.56 were very similar to the 1.61 s and are not reported separately. Table 7.5 lists the contribution of the crustal, subcrustal, and subduction earthquakes to the UHS of Victoria at 2%/50, 5%/50, and 10%/50 hazard levels at different periods. The tabulated values suggest the following trends: 1) Crustal earthquakes have higher contribution to the Victoria UHS at periods less than 1 s. At these periods the contribution is higher at higher hazard levels (less frequent events). 2) Subcrustal earthquakes have higher contribution to the Victoria UHS at periods less than 1.5 s. At these periods the contribution is higher at lower hazard levels (more frequent events). 3) Subduction earthquakes have higher contribution to the Victoria UHS at periods larger than 0.5 s. At these periods the contribution is higher at higher hazard levels (less frequent events). 120 Based on above-mentioned trends, the following period ranges were chosen for the three types of earthquakes: - Crustal 0.14-0.80 s - Subcrustal 0.20-1.50 s - Subduction 1.00-3.22 s For each type of earthquake, a suite of 11 ground motions (total of 33 records) were selected to match the target spectrum at 2%/50 hazard level within the aforementioned period ranges (the commentary of the National Building Code of Canada (2015) suggest a minimum of 5 records per suite). The selected records excluded velocity pulses and near-fault effects. The crustal records were selected using PEER NGA-West2 database (ref). For the selection of subcrustal and subduction records S2GM online tool (Bebamzadeh 2015; Bebamzadeh and Ventura 2015) developed at the University of British Columbia was utilized. For the NTHA at 5%/50 and 10%/50 hazard levels, the same suites of ground motion records, which had been already selected and scaled for 2%/50 hazard level, were rescaled. The scale factor for each hazard level was calculated as Sa(T1-long) hazard level i / Sa(T1-long) 2% /50, in which Sa(T1-long) is the spectral acceleration for hazard level i at the fundamental period of the bridge in the longitudinal direction. The selected records are listed in Table 7.6, along with the year and location of their corresponding historical event. The acceleration and displacement spectra of individual motions are presented in Appendix C . The commentary of CSA S6-14 recommends not selecting more than two records from the same historical event for a suite of motions. However, subcrustal and subduction earthquakes have far less recorded historical events as compared to crustal earthquakes that would give reasonable match to the target spectrum within the presumed period ranges. As a result, this condition was forgone when selecting motions for these types of earthquakes. Another condition stipulated by the commentary of CSA S6-14, is that the mean spectrum of each suite of motions should not fall below the target spectrum by more than 10% within the presumed period range. Figure 7.6 shows the mean spectra of the crustal, subcrustal, and subduction suites versus the target spectrum. The mean spectrum of each suite is calculated by taking the arithmetic mean of the geometric mean spectra of the horizontal components of the 121 Source Contribution Crustal Subcrustal Subduction T=0.2 s 2%/50 0.38 0.41 0.21 5%/50 0.31 0.47 0.22 10%/50 0.27 0.52 0.21 T=0.5 s 2%/50 0.21 0.42 0.37 5%/50 0.18 0.49 0.34 10%/50 0.16 0.55 0.29 T=1.0 s 2%/50 0.18 0.22 0.60 5%/50 0.17 0.31 0.53 10%/50 0.17 0.40 0.44 T=1.5 s 2%/50 0.13 0.19 0.69 5%/50 0.13 0.29 0.58 10%/50 0.13 0.41 0.46 Table 7.5 Source contribution to the UHS of Victoria at different hazard levels and at different periods (the sum of the contributions in each row is equal to 1). Type Event Year Location Record Number Crustal Cape Mendocino 1992 California, US 1 Crustal Christchurch 2011 New Zealand 2 Crustal El Mayor 2010 Mexico 3,4 Crustal Imperial Valley 1979 California, US 5 Crustal Landers 1992 California, US 6 Crustal Loma Prieta 1989 California, US 7 Crustal Northridge 1994 California, US 8 Crustal Superstition Hills 1987 California, US 9,10 Crustal Victoria 1980 Mexico 11 Subcrustal Miyagi Oki 2005 Japan 12 Subcrustal Nisqually 2001 Washington, US 13 Subcrustal Olympia 1949 Washington, US 14 Subcrustal Geiyo 2001 Japan 15-17 Subcrustal El Salvador 2001 Guatemala 18-22 Subduction Hokkaido 1952 Japan 23-28 Subduction Michoacán Mexico 29,30 Subduction Tohoku 2011 Japan 31-33 Table 7.6 Selected ground motion records for time history analysis 122 records in that suite. It can be observed that a considerably well match to the target spectrum is achieved for the crustal and subduction suites and the match is satisfactory for the subcrustal suite. Figure 7.6 also shows the mean spectrum of all 33 records across the entire period range. It can be seen that the mean spectra for all records falls more than 10% below the target spectrum for periods larger than 2 s. This drop in the mean spectra can be attributed to the spectral shape of the subcrustal and crustal records, and therefore confirms the justification for using potion-wise matching to the target spectrum. Figure 7.6 Mean spectra for the crustal, subcrustal, subduction suites, and all 33 records vs the target spectrum (the range over which the mean spectra are matched to the target spectrum, are shown with the vertical lines). 123 7.8 Response Spectrum Analysis To obtain the seismic demands on the bridge structure, response spectrum analysis (RSA) was initially performed. 5% damped design spectra of Section 7.3 were utilized in the analysis. At each hazard level, two load cases were considered, following Clause 4.4.9.2 of the code: “The horizontal elastic seismic effects on each of the principal axes of a component resulting from analyses in the two perpendicular horizontal directions shall be combined within each direction from the absolute values to form two load cases as follows: (a) 100% of the absolute value of the effects resulting from an analysis in one of the perpendicular directions combined with 30% of the absolute value of the force effects from the analysis in the second perpendicular direction. (b) 100% of the absolute value of the effects from the analysis in the second perpendicular direction combined with 30% of the absolute value of the force effects resulting from the analysis in the first perpendicular direction.” Therefore, the seismic load combination included 125%-80% dead load, 100% seismic load in one direction, and 30% seismic load in the orthogonal direction (see Clause 3.5.1 for load combinations). For modal combination of the seismic effects, SRSS rule was applied, since the contributing modes were well separated. 7.9 Nonlinear Time-History Analysis To further verify the seismic performance of the bridge model, non-linear time history analysis (NTHA) was conducted at the three specified hazard levels. A total of 99 analyses were performed on the bridge model (33 records x 3 hazard levels). The two horizontal components of each ground motion were applied simultaneously to the longitudinal and lateral directions of the bridge model. No preference was given to the individual components of a motion when assigning to the two principal directions. A Rayleigh damping of 5% was considered for the model. P-delta effects were included in the analyses. 7.10 Pushover Analysis The extent of the flexural damage in the columns was predicted by checking the maximum relative drift ratios of the columns from RSA and NTHA against the relative drift ratios 124 corresponding to the first occurrence of each of the damage states9. Separate pushover analyses were conducted on the bridge structure in the longitudinal and lateral directions. P-delta effects were included in the analyses. The structure was pushed to the point of failure, indicated by significant reduction in the strength capacity of the columns. The drift ratios corresponding to the first occurrence of each damage state in the columns were considered as the drift ratios capacities for those damage states. This can be obtained by checking the fibre hinge strains against the strain limits of Table 7.2. Table 7.7 lists the obtained drift ratio capacity of the columns for each of the considered damage states. Performance Criteria Longitudinal Drift (%) Lateral Drift (%) Yielding 0.82 0.51 Cover Spalling 1 1.82 1.15 Serviceability Limit 1 1.80 1.31 Cover Spalling 2 2.40 1.68 Serviceability Limit 2 2.23 1.83 Reduced Buckling 3.26 2.88 80% Core Crushing 5.17 4.01 Core Crushing 6.24 4.93 Reduced Fracture 5.82 5.59 Table 7.7 Column drift ratio capacities associated to the first occurrence of the damage states 7.11 PBD Assessment Using the Deterministic Framework The deterministic framework described in Section 4.2 is applied to evaluate the performance of the bridge in terms of the specified performance criteria of Section 7.4. A target reserve capacity (RC*) of 10%, 0%, and 10% was assumed for checking flexural, unseating/pounding, and shear failure mechanisms at all hazard levels, respectively. The demand parameters were calculated using both RSA and NTHA. For NTHA, the demand parameters values were calculated by taking the maximum of the mean demand values of the three suites of motions.10 The mean drift 9 In NTHA, it is possible to directly check the maximum strain demands against the strain limits to predict damage. In that case, there would be no need to check the maximum displacement demands against the displacement capacities from pushover analysis. 10 The commentary of the NBCC 2015 recommends that “Each structural response parameter should be taken as the mean value as computed from the three ground motions inducing the largest value of that response parameter, for each suite of motions.” Considering that 11 motions were selected for each ground motion suites in this example, the mean value of each suite seemed logical to be used for design assessment instead of the mean of the three largest values in each suite. Using 11 motions guarantees acceptable estimation of the mean demand parameter value. 125 ratios of the columns in the longitudinal and lateral directions of the bridge are reported in Table 7.8 for individual crustal, subcrustal, and subduction suites, and for all 33 records altogether. The tabulated values suggest that in the longitudinal direction, the subduction suite induces the maximum drift demands, while in the lateral direction the subcrustal suite induces the largest response. This outcome could be anticipated, considering that at the fundamental period of the bridge in the lateral direction (0.56 s), subcrustal earthquakes have the highest contribution to the seismic hazard in Victoria. However, at the fundamental period of the bridge in the longitudinal direction (1.61s), subduction earthquakes contribute the most. The maximum drift ratio demands of the columns calculated from RSA and NTHA along with the predicted level of damage under the flexural failure mechanism are summarized in Table 7.9. Overall, the drift values predicted by the NTHA are about 20% and 40% larger than those predicted from RSA in the longitudinal and lateral directions, respectively. Inspecting the predicted level of flexural damage, reveals that the bridge endures less damage in the lateral direction compared to the longitudinal direction. In the lateral direction the bridge undergoes yielding and minor spalling, while in the longitudinal direction it endures major spalling and wide cracks, with high plastic strains in the longitudinal rebars. This difference in performance is due to the fact that in the lateral direction, the bridge benefits from the framing action and the restraining effect of the shear keys. The lower period of the bridge in this direction impose lower displacements demands on the structure as well. However, in the longitudinal direction, the bridge essentially acts as a cantilever, and therefore the imposed displacement demands are considerably larger. To verify the performance of the columns under the flexural failure mechanism, the reserve drift ratio capacity of the columns (RC) were calculated for the performance criteria listed in Table 7.1. The drift demand to capacity ratios were obtained in the longitudinal and lateral directions, considering both of the CSA S6-14 and BC MoTI Supplement performance criteria. The reserve capacities were calculated using the largest drift demand to capacity ratios for the two directions. The performance of the bridge was deemed acceptable if the calculated reserve capacities were equal to or larger than the target reserve capacity of 10% at all hazard levels. Considering the RSA results in Table 7.10, the following was observed: 126 Crustal Subcrustal Subduction All Records Hazard Level Δx (%) Δy (%) Δx (%) Δy (%) Δx (%) Δy (%) Δx (%) Δy (%) 2%/50 3.84 1.48 2.66 1.60 4.65 1.41 3.72 1.50 5%/50 2.53 0.99 1.97 1.08 3.07 0.98 2.53 1.02 10%/50 1.90 0.65 1.46 0.72 2.13 0.67 1.83 0.68 Table 7.8 Mean column drift demands from NTHA in the longitudinal (x) and transverse (y) directions for crustal, subcrustal, subduction suites, and all 33 records CSA S6-14 BC MoTI Hazard Level Δx (%) Δy (%) Damagex Damagey Damagex Damagey RSA 2%/50 3.94 1.05 SL2 Y B M 5%/50 2.58 0.74 SL2 Y SP2 M 10%/50 1.76 0.54 Y Y M M NTHA 2%/50 4.65 1.60 SL2 SP1 B SL1 5%/50 3.07 1.08 SL2 Y SP2 M 10%/50 2.13 0.72 SP1 Y SL1 M Table 7.9 Column drift demands from RSA and NTHA in the longitudinal (x) and transverse (y) directions, along with the predicted damage (M: minimal, Y: yielding of longitudinal reinforcements, SL1 & SL2: serviceability limit 1 & 2, SP1 & SP2: cover spalling 1& 2, B: reduced buckling) - Employing the CSA S6-14 criteria, the bridge meets the specified performance criteria at 2%/50 hazard level with acceptable reserve capacity, while it fails to meet the 5%/50 and 10%/50 performance criteria. - Employing the BC MoTI Supplement criteria, the bridge meets the specified performance criteria at 2%/50 and 5%/50 with reasonable reserve capacity, but fails to meet the 10%/50 performance criteria with acceptable reserve capacity. - The controlling performance criteria using both CSA S6-14 and BC MoTI Supplement is at 10%/50 hazard level. - The calculated reserve capacities at different hazard levels are more uniform using the BC MoTI Supplement criteria compared to the CSA S6-14 criteria. The above observations are valid for the NTHA results as well, except that at 5%/50 the reserve capacity is less than the target reserve capacity using the BC MoTI criteria. The maximum longitudinal displacement of the deck at the three hazard levels is listed in Table 7.11 to check the unseating and pounding failure mechanisms. The provided support length and the longitudinal gap should be checked against these values to meet the 0% target reserve capacity goal. The large longitudinal displacements at all hazard levels indicate the 127 possibility of pounding between the deck and the abutments. This can be rectified by either of the following options: 1) Incorporating elastomeric bearings at the abutments to control the longitudinal displacements of the girders. 2) Redesigning the abutment to semi-integral. 3) Reducing the longitudinal drifts of the columns, by increasing the column longitudinal stiffness. Finally, the shear capacity of the columns should also be checked against the shear demand to control the shear failure mechanism. The comparison needs to be made only at 2%/50 hazard level, which induces the largest shear demand in the columns. The details of such calculations are not presented here, as they were carried out using standard force-based design expressions provided by CSA S6-14 or BC MoTI Supplement. CSA S6-14 BC MoTI Supplement Hazard Level Δd/Δc (%)-x Δd/Δc (%)-y RC (%) Δd/Δc (%)-x Δd/Δc (%)-y RC (%) RSA 2%/50 68 21 32 76 26 24 5%/50 115 40 -15 79 26 21 10%/50 217 104 -117 98 41 2 NTHA 2%/50 80 32 20 90 40 10 5%/50 137 59 -37 94 38 6 10%/50 263 140 -163 119 55 -19 Table 7.10 Ratio of the drift demand to drift capacity of the columns in the longitudinal (x) and lateral (y) directions, and the reserve drift capacity for each hazard level Hazard Level Δdeck-x (m) RSA 2%/50 0.379 5%/50 0.248 10%/50 0.169 NTHA 2%/50 0.423 5%/50 0.282 10%/50 0.197 Table 7.11 Maximum longitudinal and lateral displacement of the deck 128 7.12 PBD Assessment Using the Probabilistic Framework It is desired to evaluate the performance criteria of Table 7.1 using the DCFD framework Equation (4.19) and (4.20). Here, all 33 records are used together to obtain the median demand values, although it is possible to repeat the same process for the individual suites of crustal, subcrustal, and subduction records. The results of the NTHA for the maximum longitudinal and lateral drift of the bridge columns at the three hazard levels of 2%/50, 5%/50, and 10%/50 are shown in Figure 7.7, along with the median drift values. Using Microsoft Excel a power-law curve was fitted to the median values of the demand in each direction, equations of which are shown on the plots. The coefficient and the power numbers give the demand curve parameters a and b, respectively. The median demand values as well as the record-to-record variability of the NTHA results in the lateral and longitudinal directions are listed in Table 7.12. The latter was obtained simply by taking the standard deviation of the natural logarithm of the drift ratio demands. It was observed that the values of βRDx and βRDy varied with the hazard level. Therefore, the mean values of βRDx and βRDy were employed in calculating demand factors, as listed in Table 7.13. This table also lists the assumed values for the epistemic uncertainty in demand and the aleatory and epistemic uncertainties in capacity based on the recommended values by ASCE/SEI 7-16 (see Table 4.1 and Table 4.2). The next step was to obtain the hazard curve parameters. The mean annual frequency is plotted against the spectral acceleration at the fundamental period of the bridge in the lateral and longitudinal directions individually in Figure 7.8. The segment of the hazard curve, which spans between the three hazard levels of 2%/50, 5%/50, and 10%/50 is indicated with the red color. A power-law curve was fitted to the red segments of the hazard curve using Microsoft Excel. The coefficient and the power numbers give the hazard curve parameters ko and k. The values of the obtained demand and hazard curve parameters are tabulated in Table 7.14. Once the values of the demand and hazard curve parameters and the uncertainties in demand and capacity were established, the demand and capacity factors were calculated using Equation (4.15) to (4.18), as shown in Table 7.15. The confidence interval was also calculated using Equation (4.21), the value of which is listed for confidence intervals of 95%, 90%, and 85% in Table 7.16. Implied factor of safety in evaluating the lateral and longitudinal response of the bridge columns were obtained as in Table 7.17 129 Figure 7.7 Longitudinal and lateral drift ratio demands versus the spectral acceleration at periods of 1.61 s and 0.56 s, respectively, using all 33 records. Figure 7.8 Mean annual frequency versus the spectral acceleration at periods of 1.61 s and 0.56 s for Victoria y = 4.819x0.8190.001.002.003.004.005.006.007.008.000.00 0.20 0.40 0.60 0.80Longitudinal Drift (%)Sa (1.61 s) (g)2%/505%/5010%/50MedianPower (Median )y = 1.105x1.2000.000.501.001.502.002.503.000.00 0.50 1.00 1.50Lateral Drift (%)Sa (0.56 s) (g)2%/505%/5010%/50MedianPower (Median )y = 0.0002x-2.0470.00010.0010.010.01 0.10 1.00 10.00Mean Annual FrequencySa (1.61s) (g)y = 0.0007x-2.4550.00010.0010.010.10 1.00 10.00Mean Annual FrequencySa (0.56 s) (g)130 Median Values Δx (%) Δy (%) βRDx βRDy Sa(1.61 s) Sa(0.56 s) 2%/50 3.25 1.38 0.38 0.28 0.628 1.221 5%/50 2.40 0.95 0.33 0.32 0.411 0.857 10%/50 1.67 0.62 0.29 0.32 0.280 0.623 Table 7.12 Median drift ratio demands of the bridge columns from NTHA in the longitudinal (x) and lateral (y) directions, and the calculated record-to-record variabilities βRDx 0.33 βUC 0.28 βRDy 0.31 βUD 0.24 βRC 0.60 βUT 0.37 Table 7.13 Demand and capacity aleatory and epistemic uncertainty values Longitudinal Lateral Hazard Parameters ko 0.0002 0.0007 k 2.047 2.455 Demand Parameters a 4.819 1.105 b 0.819 1.200 Table 7.14 Obtained values for the demand and hazard curve parameters Longitudinal Lateral Capacity Factors ϕR 0.64 0.69 ϕU 0.90 0.92 ϕ 0.58 0.64 Demand Factors γR 1.15 1.10 γU 1.07 1.06 γ 1.23 1.17 Table 7.15 Demand and capacity factors calculated for the longitudinal and lateral directions λx Confidence Kx Longitudinal Lateral 95% 1.65 0.64 0.62 90% 1.28 0.74 0.72 85% 1.04 0.81 0.78 Table 7.16 Confidence factor values calculated for various confidence intervals Factor of Safety Longitudinal Lateral γ/ϕ 2.14 1.83 γ/ϕλx 2.89 2.56 Table 7.17 Implied factor of safety in evaluating the lateral and longitudinal response of the bridge columns in terms of drift ratio with/without considering the confidence interval 131 The final step was to perform the design checks using the factored demand and capacity drift ratios. This is demonstrated in Table 7.18 and Table 7.19 for CSA S6-14 and BC MoTI Supplement performance criteria, respectively. The design checks were carried out using the following three objectives: 1) factored demand < factored capacity 2) factored demand / factored capacity < λ90% 3) median demand < median capacity The first and second objectives correspond to the DCFD with and without considering a 90% confidence interval. The last objective represents the situation where the design is checked using a deterministic approach. The following is observed from the tabulated results in the tables: - For CSA S6-14 performance criteria, the use of both frameworks suggest that the design meets all the specified performance criteria at all hazard levels in the lateral direction, except for the yielding criteria, where the 90% confidence is not met. - For BC MoTI performance criteria, the use of the DCFD framework indicates that the columns do not meet any of the specified criteria in the longitudinal direction, while the deterministic framework suggest that they meet all the criteria, except for the serviceability limit 1 at 10%/50. - For BC MoTI performance criteria, the use of both frameworks suggests that the design meets all the specified performance criteria at all hazard levels in the lateral direction. To check the unseating and pounding failure mechanisms, the longitudinal deck displacement is the demand variable, and the demand parameters a and b are found in the same way as was done for the drift ratio demands of the columns. The capacity variable is the support length for unseating and the longitudinal gap for pounding failure. The capacity in this case may vary only minimally due to construction tolerances and unlike drift ratio capacity, it has a uniform distribution (instead of lognormal distribution) with a small coefficient of variation. Therefore, one may assume βRC= βUC=0 and ϕ=1, when applying the DCFD framework for checking the unseating and pounding failure mechanisms limit states. It would be also useful to plot the MAF of demand exceeding capacity using Equation (4.11) and setting βRC= βUC=0. This is presented 132 Longitudinal Response Hazard Level Performance Criteria FD (%) FC (%) FD<FC FD/FC< λ90% D/C<1 10%/50 Yielding 2.32 0.47 F F F Cover Spalling 1 2.32 1.05 F F F 5%/50 Serviceability Limit 2 3.09 1.29 F F F 2%/50 Core Crushing 4.49 3.60 F F P Reduced Fracture 4.49 3.36 F F P Lateral Response Hazard Level Performance Criteria FD (%) FC (%) FD<FC FD/FC< λ90% D/C<1 10%/50 Yielding 0.32 0.33 P F P Cover Spalling 1 0.32 0.73 P P P 5%/50 Serviceability Limit 2 0.50 1.17 P P P 2%/50 Core Crushing 0.85 3.14 P P P Reduced Fracture 0.85 3.56 P P P Table 7.18 CSA S6-14 performance criteria checks using factored demand and capacity drift ratios of the bridge columns in the longitudinal and lateral direction (P: pass; F: fail; FD: factored drift ratio demand in %, FC: factored drift ratio capacity in %; λ90%: confidence factor for 90% confidence interval; D: median demand; C: median capacity) Longitudinal Response Hazard Level Performance Criteria FD (%) FC (%) FD<FC FD/FC< λ90% D/C<1 10%/50 Serviceability Limit 1 2.32 1.04 F F F Cover Spalling 2 2.32 1.39 F F P 5%/50 Reduced Buckling 3.09 1.88 F F P 2%/50 80% Core Crushing 4.49 2.98 F F P Reduced Fracture 4.49 3.36 F F P Lateral Response Hazard Level Performance Criteria FD (%) FC (%) FD<FC FD/FC< λ90% D/C<1 10%/50 Serviceability Limit 1 0.32 0.84 P P P Cover Spalling 2 0.32 1.07 P P P 5%/50 Reduced Buckling 0.50 1.84 P P P 2%/50 80% Core Crushing 0.85 2.56 P P P Reduced Fracture 0.85 3.56 P P P Table 7.19 BC MoTI performance criteria checks using factored demand and capacity drift ratios of the bridge columns in the longitudinal and lateral direction For CSA S6-14 performance criteria, the use of the DCFD framework indicates that the columns do not meet any of the specified criteria in the longitudinal direction, while the deterministic framework suggest that they meet the ones at 2%/50. 133 in Figure 7.9. To use this figure, one should find the displacements corresponding to the MAF of 0.000404, 0.00106, and 0.002105, which gives 0.41, 0.28, and 0.21 m, respectively. Based on these numbers, the longitudinal deck displacement is excessive, and should be controlled using any of the three options mentioned under the deterministic framework assessment section. Finally, the shear failure mechanism is checked using CSA S6-14 FBD expressions similar to Section 7.11. Figure 7.9 MAF of exceeding limit states for longitudinal deck displacement To this end, the assessment of the bridge design using the DCFD framework is complete. However, it is possible to gain further insights into the design problem using the probabilistic framework. Figure 7.10 and Figure 7.11 show the bridge column fragility curves for CSA S6-14 and BC MoTI supplement, individually. The fragility curves were calculated using the following expression (Nielson and DesRoches 2007): P[D>C|IM]=Φ(ln(η̂D|Sa/η̂C)/√βRD2+βRC2) (7.3) where Φ is the Gaussian (or Normal) cumulative distribution function. Plotting the fragility curves allows inspecting the two sets of performance criteria in CSA S6-14 and BC MoTI Supplement by comparison. A spectral acceleration of 1 g may be considered as a point of reference. At this spectral acceleration, the probability of demand exceeding capacity at each 0.00010.0010.010.110 0.1 0.2 0.3 0.4 0.5 0.6MAF of Exceeding Limit StateLongidutinal Deck Displacement (m)2%/50 10%/50 5%/50 134 performance level is considerably higher if the CSA S6-14 performance criteria is used. For instance, in the longitudinal direction and at Sa=1 g, the probability of demand exceeding Serviceability Limit 2 is about 87%, while the probability of demand exceeding Reduced Buckling is 72%. It is also evident that the fragility curves of the BC MoTI Supplement performance criteria are more evenly spaced compared to the CSA S6-14. This is in line with the previous observation on the uniformity of reserve capacities for the two sets of criteria in Section 7.11. Similar conclusions can be made considering the response in the lateral direction. Another valuable insight can be gained by comparing the fragility curves in the two directions for each of the two sets of performance criteria. This comparison reveals that the conditional probability of demand exceeding each of the specified performance criteria is much higher in the longitudinal direction compared to the lateral direction. For instance, at Sa(1.61 s)=0.28 g (see Table 7.12) the probability of demand exceeding Cover Spalling 1 is about 45% in the longitudinal direction, while at Sa(0.56 s)=0.62 g it is about 15% in the lateral direction. Such information indicates that the bridge columns are performing much better in the lateral direction compared to the longitudinal direction, and therefore suggest a revise in the design of the bridge in the longitudinal direction to control the excessive drift demands on the columns. One interesting representation of the DCFD outputs is to plot the ratio of factored demand to factored capacity versus MAF exceedance for each of the performance limit states. This is presented in Figure 7.12 to Figure 7.15. A factored demand to factored capacity ratio of one specifies the limit of satisfactory performance, which is indicated by a vertical black dashed line in the plots. The points on the LHS of this line define the space of satisfactory performance. Using such plots enables to determine the MAF of exceeding a performance limit state, at which the design of the bridge is deemed acceptable. Conversely, one may find the ratio of factored demand to factored capacity for a given MAF of exceeding a limit state. For instance, for a MAF of 0.002105 (corresponding to 10%/50 hazard level), the ratio of factored demand to factored capacity for yielding criteria is 4.9 and 1.0 in the longitudinal and lateral directions, respectively. This ratio for the relaxed Serviceability Limit 1 criteria in the BC MoTI Supplement is 2.23, and 0.39, respectively. In using these plots, it should be noted that the extrapolation of points outside the range of 10%/50 to 2%/50 hazard levels introduces some error, since both the hazard curve and demand curve parameters were estimated locally for this range. Nevertheless, the 135 extrapolated portions of the plots reveal the estimate trend of the plots at higher and lower hazard levels than those specified by the code. (a) (b) Figure 7.10 Column fragility curves for the CSA S6-14 performance criteria in the (a) longitudinal and (b) lateral directions. (a) (b) Figure 7.11 Column fragility curves for the BC MoTI Supplement performance criteria in the (a) longitudinal and (b) lateral directions. 0.00.10.20.30.40.50.60.70.80.91.00 0.5 1 1.5 2P(D>C|Sa)Sa (1.61s) (g)0.00.10.20.30.40.50.60.70.80.91.00 0.5 1 1.5 2P(D>C|Sa)Sa (0.56 s) (g)0.00.10.20.30.40.50.60.70.80.91.00 0.5 1 1.5 2P(D>C|Sa)Sa (1.61 s) (g)0.00.10.20.30.40.50.60.70.80.91.00 0.5 1 1.5 2P(D>C|Sa)Sa (0.56 s) (g)136 Figure 7.12 Factored demand to factored capacity ratio in the lateral and longitudinal directions versus the mean annual frequency of exceeding the performance limits states of CSA S6-14. Figure 7.13 Factored demand to factored capacity ratio in the lateral and longitudinal directions versus the mean annual frequency of exceeding the performance limits states of BC MoTI Supplement. 0.00010.0010.010.0 5.0 10.0 15.0MAF of Exceeding Limit StateLongitudinal FD/FC0.00010.0010.010.0 2.0 4.0 6.0MAF of Exceeding Limit StateLateral FD/FC0.00010.0010.010.0 5.0 10.0 15.0MAF of Exceeding Limit StateLongitudinal FD/FC0.00010.0010.010.0 2.0 4.0 6.0MAF of Exceeding Limit StateLateral FD/FC137 Figure 7.14 Factored demand to factored capacity ratio with confidence interval of 90% in the lateral and longitudinal directions versus the mean annual frequency of exceeding the performance limits states of CSA S6-14. Figure 7.15 Factored demand to factored capacity ratio with confidence interval of 90% in the lateral and longitudinal directions versus the mean annual frequency of exceeding the performance limits states of BC MoTI Supplement. 0.00010.0010.010.0 10.0 20.0MAF of Exceeding Limit StateLongitudinal FD/(λxFC)0.00010.0010.010.0 5.0MAF of Exceeding Limit StateLateral FD/(λxFC)0.00010.0010.010.0 10.0 20.0MAF of Exceeding Limit StateLongitudinal FD/(λxFC)0.00010.0010.010.0 5.0MAF of Exceeding Limit StateLateral FD/(λxFC)138 7.13 Discussion 7.13.1 Comparison of the Deterministic vs Probabilistic Frameworks The case study demonstrated the step-by-step implementation of the deterministic framework of Section 5.3 and the probabilistic DCFD framework. The question is which one of these frameworks should be preferred. To answer this, it would be helpful to recognize that the DCFD framework can be viewed as an upgrade to the deterministic framework, where the demand to capacity ratio is less subjectively evaluated. The deterministic framework does not provide a systematic approach to decide on the target reserve capacities for different limit states. In fact, the code implies using a zero target reserve capacity for all limit states. This is equivalent to say that following the code, the performance criteria are met if mean deformation demand is less than mean deformation capacity, and that factored force demand is less than factored force capacity. When using the DCFD framework, the discrepancy between how the deformation-controlled and force-controlled limit states are evaluated is addressed by introducing demand and capacity factors for the former. This enhancement though, comes with only minimal effort in addition to what is already required to carry out the deterministic assessment. That is what makes the DCFD a suitable framework for code-based applications and the use of the DCFD framework should be included in the CSA S6-14 PBD approach. Nevertheless, a number of considerations exist when using the DCFD framework for the CSA S6-14 PBD: 1) Assumed values of aleatory and epistemic uncertainties for demand and capacity: The assumed values for aleatory and epistemic uncertainties (Table 4.1 and Table 4.2) determine the values of calculated demand and capacity factors and therefore, directly affect the outcome of performance evaluation as demonstrated in the implementation example. One important number in this table is the value of aleatory uncertainty in deformation demand, which is considerably larger than the other tabulated values. The commentary of the ASCE/SEI 7-16 (2016) explains that: “these βC values are larger than the comparable values for force-controlled components because the uncertainty is quite large when trying to quantify the deformation at which loss of vertical load-carrying capability occurs.” What ASCE/SEI 7-16 is concerned with is an accurate estimate of collapse capacity for the MCER ground motions, and collapse is defined as loss of vertical load-carrying capability. This is a 139 substantially different philosophy than that of the CSA S6-14 PBD approach. The uncertainty in estimation of the deformations of concrete components is larger in the strength-degraded states. However, in CSA S6-14, the strains of concrete and steel are checked at various levels of damage, from minimal damage to extensive damage. It is reasonable to say that the assumed aleatory uncertainty for yielding strain capacity should be less than concrete core crushing, or extensive spalling. Therefore, the use of large βRC value for the performance criteria of minimal damage state may be conservative. 2) Mean vs median response: For lognormal variables the mean and median are related through the following expression: H̅=Ĥeβ22 (7.4) where, H̅ is mean, Ĥ is median, and β is the lognormal standard deviation of the variable. Typically mean response values are used in performance evaluation, as was utilized with the deterministic framework in the implantation example. The DCFD framework uses median response values, and these two should be distinguished. For normally distributed variables, the mean and median values coincide, while for lognormal variables, the use of Equation (7.4) suggest that the mean is about 1.08 times the median if β=0.4, and about 1.2 if β=0.6. 3) Mean vs median strain limits: It is not stated in the code if the values suggested for the strain limits are mean or median values, and whether the original data followed a lognormal or normal distribution. One reference for such information is the PEER report by Berry and Eberhard (2003). They demonstrate that a normal distribution may provide a better fit to the experimental fragility curves for longitudinal bar buckling and cover spalling drift ratios. They also provide mean and coefficient of variation for the compressive strains corresponding to longitudinal bar buckling, and cover spalling. In the implementation example, the strain limits of the code were assumed to be mean values when using the deterministic framework and to be median values when using the DCFD framework. While this assumption may introduce inaccuracies for distributions other than normal, it does not compromise the validity of the outcomes. This issue would have been solved, if there was more information on the stipulated strain limits. 140 4) The choice of confidence interval: In FEMA-351 (2000), the 90% confidence interval is recommended for evaluating (some) of the criteria of collapse prevention performance level, while a 50% minimum confidence interval was recommended for evaluating the immediate occupancy performance criteria11. Table 7.20 shows the values of confidence factor for the implementation example, recalculated using lower confidence intervals of 75%, and 50%. It is notable that if a 50% confidence interval is employed, the confidence factor becomes greater than 1.0. This implies that the factored demand can be greater than the factored capacity by 19% and 15% in the lateral and longitudinal directions. In other words, a lower safety factor is achieved. The confidence factor would be equal to 1.0, for about 80% confidence interval. This indicates that even when the factored demand is directly compared to factored capacity, an 80% confidence interval is implied. Based on the above considerations, it may seem reasonable to employ a lower confidence interval for evaluating the criteria of the minimal and repairable damage performance levels compared to extensive damage and probable replacement levels. λx Confidence Kx Longitudinal Lateral 90% 1.28 0.74 0.72 75% 0.67 0.93 0.90 50% 0.00 1.19 1.15 Table 7.20 Recalculated confidence factors of Table 7.16calculated for lower confidence intervals 7.13.2 Question of Uniform Reliability (The Value of Loss Analysis) The implementation example provided a comparison between the performance criteria in CSA S6-14 and BC MoTI for major-route bridges. It was shown that the use of the BC MoTI criteria will result in more uniform reserve capacities in the deterministic framework and probabilities of exceeding limit states in the DCFD framework across the three hazard levels. The updated criteria in the BC MoTI Supplement relaxed the strains limits at 10%/50 and 5%/50 hazard levels, which were found to be unduly conservative, and tightened the strain limits at 2%/50 hazard level. Consequently, it is easier to satisfy the criteria of BC MoTI than CSA S6-14, and moreover, the design is less dominated by the criteria at lower hazard levels. To this end, it 11 The choice of higher confidence interval for the collapse prevention performance level reflect more stringent evaluation at that performance level. 141 would seem reasonable to conclude that achieving uniform reliability across hazard levels as the performance goal is desired. Since, reliability is chosen as the performance goal, the design approach is often referred to as reliability-based design. However, the choice of uniform reliability has been challenged in the next generation performance-based procedure by considering loss analysis in decision-making. Wen (2001) explains that a cost-based design approach is more logical than a purely reliability-based design and he suggests choosing target reliabilities by minimizing the life-cycle cost of structures. Considering this view, one may argue that although it is easier to satisfy the relaxed criteria of BC MoTI Supplement and they are exceeded less frequently, the consequences of exceeding them in terms of loss is higher than the more conservative CSA S6-14 criteria. For instance, in the implementation example the design may exceed the yielding criteria more frequently than the serviceability limit 1, but it would be less expensive to repair the structure at the yielding damage state than the serviceability limit 1. Consequently, to answer which set of criteria is a preferred choice for PBD, one must first decide on the performance goals. If a uniform reliability across hazard levels is the goal then the BC MoTI Supplement criteria is the preferred choice. However, if minimizing cost is the goal, then further studies considering loss analysis are needed to compare the two sets of criteria. 142 Chapter 8: Case Study-Seismic Evaluation of the Trent River Bridge This chapter presents a case study on the PBD evaluation of the Trent River Bridge following the provisions of CSA S6-14. The chapter aims to provide a comparison between the performance criteria of CSA S6-14 and BC MoTI supplement when the importance category of the bridge varies between life-line, major-route, and other. It also serves the purpose of showcasing the importance of including SSI effects in PBD evaluation. To achieve these objectives, a number of assumptions were made. First, the Trent River Bridge was treated as a new bridge instead of an existing bridge. This assumption helps to keep the focus of thesis on the design of new structures. Second, while the existing Trent River Bridge is a major-route bridge, the performance evaluation of the bridge was carried out assuming two performance categories of life-line and other in addition to major-route category. Third, two soil site classes were considered for the bridge including site class C and D. For the site class C condition, SSI effects were ignored and a fixed-based model was employed, while for the site class D condition, SSI effects were included. From the above assumptions, it is clear that the conditions assumed for the case study do not aim to necessarily mimic the conditions of the actual bridge. The case study borrows the structure of the Trent River Bridge, only to fulfill the above-mentioned objectives. 8.1 Description of the Bridge The Trent River Bridge is a major-route bridge, located at the crossing of the Trent River with Highway 19 in Vancouver Island, British Columbia. An area map in Figure 8.1 shows the location of the bridge relative to Victoria and Vancouver, two major cities of British Columbia province in Canada. The bridge was originally designed using FBD according to CSA S6-88 (1988) and AASHTO (1992), with the reference seismic acceleration of A=0.4 g. The three-span pre-stressed concrete bridge has semi-integral abutments (Figure 8.2 to Figure 8.5). The main span of the bridge is 40-meter-long, and the two side spans are each 33 meter long. The bridge deck is composed of three 2 m pre-stressed concrete girders, topped with a 0.25 m thick concrete slab, and an approximately 0.05 m thick asphalt concrete wearing surface. The width of the superstructure is 12 m and it is fixed at the pier diaphragm-pier cap connections. At the abutments, the girders are resting on 600x425x116 mm rubber bearings pads, which are 143 reinforced with five 3 mm thick, grade 300W steel plates. The lateral movement of the girders at the abutments is controlled with the shear keys. The substructure of the bridge includes two reinforced concrete piers, each consisting of two circular columns of 1.5 meter in diameter, and a pier cap-beam (Figure 8.6). The clear height of the Pier 1 and Pier 2 columns are 14.9 m and 9.83 m, respectively. The thickness of the cover concrete for the columns is 75 mm. The longitudinal reinforcement of the columns includes 28-30M rebars, making up 1.1% reinforcement ratio. For the transverse reinforcement, 15M spirals are utilized with 0.065 m pitch in the plastic hinge region and 0.15 m pitch, elsewhere. Both piers have 1.8-meter-deep 6x12.5 m concrete spread footings. The minimum specified compressive strength of concrete for all members is 35 MPa and the reinforcement steel grade is 400R with minimum specified yield strength of 400 MPa. The actual bridge consists of two separate east and west bridges, carrying the northbound and south bound lane traffics, respectively (Figure 8.3). However, only the structure of the east bridge was considered as an individual bridge for this study. Figure 8.1 Relative location of the Trent River Bridge with respect to Victoria and Vancouver 144 Figure 8.2 Elevation view of the Trent River Bridge Figure 8.3 Plan view of the Trent River Bridge Figure 8.4 Deck, pier diaphragm, and cap-beam cross-sections (dimensions in mm) 145 Figure 8.5 Connection of the deck to the end diaphragm Figure 8.6 Pier 1 and Pier 2 elevation views (looking North) 8.2 Soil Properties The Trent River Bridge rests on soft rock to very dense soil, which matches site class C in CSA S6-14. These conditions roughly correspond to a uniform sand layer with shearwave velocity of 650 m/s. The sand layer has a friction angle of 32 degrees, zero cohesion, Poisson’s ratio of 0.3, and unit weight of 18 kN/m3. To investigate how SSI affects the performance assessment of the bridge, a softer sand layer corresponding to site class D was also considered. The shearwave velocity in this case was 146 assumed to be around 180 m/s to 200 m/s, which represents the softest soil in this class. Similar values were assumed for other soil properties as for the site class C sand layer. 8.3 Bridge Model 8.3.1 Choice of Analysis Tool A 3D spine model of the bridge structure was generated in SeismoStruct. SeismoStruct is an analysis tool developed by SEiSMOSOFT, used frequently both for commercial and academic purposes. It offers the option of distributed plasticity beam-column elements as well as concentrated link elements to capture the nonlinear behaviour of structures. SeismoStruct is especially advantageous for PBD of structures, since it allows direct checking of strain limits at the fibre level and notifies the user when those limits are reached. While being more user-friendly compared to its counterparts, OpenSees and CSI PERFORM 3D, it does not provide the option of manipulating material models for material model regularization discussed in Chapter 5:. Therefore, when using distributed plasticity models, the localization issues need to be addressed by setting the mesh size based on an empirical plastic hinge length (Technique 1 in Section 5.2.4). 8.3.2 Material Models Expected material properties were utilized in the bridge model. The behaviour of confined and unconfined concrete were modelled with con_ma material model that uses Mander et al. (1988) constitutive relationship. The implemented material model takes five calibrating parameters including, the compressive and tensile strength of unconfined concrete, the modulus of elasticity, the strain at peak stress, and the specific weight of concrete. The strain at the peak stress of the unconfined concrete was assumed to be 0.002 and the tensile strength of concrete was assumed to be zero. The cyclic behaviour of the reinforcement steel was modelled using stl_mp material model that uses Menegotto-Pinto constitutive relationship (1973) and isotropic hardening rules proposed by Filippou et al. (1983). When defining a fibre section, the program takes the unconfined concrete and reinforcement steel materials and the lateral and longitudinal reinforcement arrangement, and then it automatically calculates and applies the confinement factor to the confined portion of the section. 147 8.3.3 Structure The bridge columns were modelled with inelastic DBE with fibre sections. Each fibre section included 152 fibres of confined concrete at the core of the columns, unconfined concrete at the cover of the columns, and reinforcement steel. The plastic hinge length, LP, for the Pier 1 and Pier 2 columns was calculated as 2.5 m and 1.7 m, respectively, using Equation (5.3). The columns were meshed into elements such that the length associated to the critical integration point, LIP was 0.5 LP, following Section 5.3.6. The cap-beams were modelled using elastic frame elements with cracked section properties. The connection of the columns to the cap-beams was assumed rigid with no releases. For response spectrum analysis, elastic frame elements with cracked section properties were utilized for the columns as well. The pre-stressed concrete girders and the concrete slab were assumed to remain elastic, following the recommendations of Caltrans SDC 1.7 (2013) for pre-stressed decks. Subsequently, elastic frame elements with equivalent elastic section properties were employed to model the deck. 8.3.4 Foundations For the site class C model, the SSI effects at the foundations were negligible and a fixed-base model was utilized. For the site class D model, similar to Section 7.5 the foundation flexibility effects were captured utilizing a set of six uncoupled equivalent springs per recommendations of FEMA-356 (2000). In calculating the spring constants, degraded shear moduli of the soil were utilized. The maximum shear modulus of the soil for site class D was calculated as 73.4 MPa. The degraded shear modulus of the soil was taken as 0.5 to 0.2 of the maximum shear modulus, following the commentary of CSA S6-14, and was calculated as 14.7 MPa (Table 8.1). To model the spring foundations, the two columns of each pier were connected at the bottom with relatively rigid elastic frame elements. Then a link element comprised of the six uncoupled foundation springs were assigned to the middle of the rigid elements. According to FEMA-356, when the height of effective sidewall contact is taken larger than zero, the resulting stiffness of the springs will include sidewall friction and passive pressure contributions. 148 Embedment Correction Stiffness (MN/m) Footing 1 Footing 2 Translation along x-axis 5153 1.72 2.13 Translation along y-axis 5573 1.72 2.13 Translation along z-axis 6593 1.28 1.37 Rocking about x-axis 4671 2.33 2.40 Rocking about y-axis 181403 1.70 1.73 Torsion about z-axis 163872 2.45 2.45 Table 8.1 Calculated SSI spring constants for the Trent River Bridge site Class D model 8.3.5 Abutments The abutment behaviour, embankment flexibility, and the interaction of the bridge with the embankment soil considerably affect the seismic response of a bridge under moderate to strong levels of shaking. The soil-structure interaction in the abutments is even more pronounced for short span bridges with relatively stiff superstructures, where the inelastic response of the embankment soil primarily controls the response of the bridge and intermediate column bents (Aviram et al. 2008). Therefore, in case of the Trent River Bridge, it is necessary to employ a appropriate model for the abutments that can adequately capture the soil-structure interaction at the abutments. A simplified abutment model proposed by Aviram et al. (2008) was employed to model the semi-integral abutment and the back-fill soil behaviour. In this model, the abutment is replaced with a rigid elastic frame element and a set of translational springs in the longitudinal, transverse, and vertical directions. The rigid elastic frame element has a length equal to the width of the superstructure and is rigidly connected in the middle to the superstructure. The link elements containing the three translational springs are connected to the ends of the rigid element. In the transverse and vertical directions, elastic springs proposed by Wilson and Tan (1990) were used for the abutments. The static stiffness values of these springs were calculated based on the reduced shear moduli of the soil. In the longitudinal direction, a bi-linear spring proposed by Caltrans SDC 1.7 (2013) was utilized to mimic the response of the backfill soil. Caltrans gives single stiffness and strength values for competent soils. These values were utilized for the abutment springs in the site class C model. For the site class D backfill soil in the longitudinal direction, the stiffness and strength of the longitudinal springs for site class C were reduced by the ratio of the maximum shear moduli of the site class D soil to site class C (73.4 MPa to 775.3 149 MPa). The participating mass of the abutments and the shear resistance of the bearing pads were not considered in the model. While these assumptions may be challenged, the simplicity of the model makes it very convenient for practical design purposes. Figure 8.7 Spine 3D model of the Trent River Bridge in SeismoStruct 8.4 Performance Criteria The performance criteria of CSA S6-14 and BC MoTI supplement for a regular major-route bridge was previously determined and tabulated in Table 7.1 and Table 6.3. Assuming importance categories of lifeline and other for the Trent River Bridge requires that the bridge satisfies different set of performance levels at the three hazard levels. The expected performance levels for lifeline and other bridges and the strain limits associated to the performance criteria of each performance level are listed in Table 8.2. The description of “no damage” performance level in the code is unclear and does not specify a limit or a qualitative measure for allowable damage. It is possible to assume that no damage is met if no cracking occurs in cover concrete. But this criterion would be too conservative for a 475-year return period ground motion. A more relaxed criterion equal to half the yielding strain of longitudinal rebars was employed instead for verifying no damage performance level. This limit represents the “minimal” damage state. 8.5 Structural Analysis The seismic demands on the structure were obtained by conducting NTHA and RSA at the three specified hazard levels. NTHA was performed for only site Class D model. The design spectra for both site class C and D models were calculated according to CSA S6-14, using the 5% damped spectral accelerations at the bridge site (Figure 8.8). The periods of the first 150 Life-line Bridges Hazard Performance Level CSA S6-14 BC MoTI 10%/50 None εs < 0.5 εy εc >-0.006, εs <0.010 5%/50 Minimal Damage εc >-0.004, εs < εy εc >-0.006, εs <0.010 2%/50 Repairable Damage εs <0.015 εs <0.025 Other Bridges Hazard Performance Level CSA S6-14 BC MoTI 10%/50 Repairable Damage εs <0.015 εs <0.025 5%/50 Extensive Damage εc >-0.0163, εs <0.050 εc >-0.0130, εs <0.050 2%/50 Probable Replacement εs <0.050 εc >-0.0163, εs <0.075 Table 8.2 CSA S6-14 and BC MoTI strain limits associated to the performance levels of lifeline and other bridges. Figure 8.8 Bridge design spectra for site class C and D at 2%/50 hazard level longitudinal mode, and the first and second lateral modes of the bridge were obtained as 0.37, 0.27, 0.16 s on site class C, and 0.56, 0.6, 0.33 s on site class D, respectively. For NTHA 11 pairs of horizontal ground motion records were selected following the CSA S6-14 commentary and as previously discussed in Section 7.7. The records were linearly scaled and matched to the site class D UHS of the bridge site in the period range of 0.1-1.5s. Table 8.3 lists the type and event of the selected records. Acceleration and displacement spectra of the selected records along with the target spectrum are presented in C.4. Figure 8.9 shows the mean acceleration and displacement spectra of the 11 records and how well they match the target spectrum within the period range of interest. 00.20.40.60.811.21.40 2 4 6 8 10Spectral Acceleration (g)Period (s)class Cclass D151 Type Event Year Location Record Number Crustal Chi-Chi 1999 Taiwan C1 Crustal El Mayor 2010 Mexico C2 Crustal Landers 1992 California, US C3 Crustal Northridge 1994 California, US C4 Subcrustal Miyagi Oki 2005 Japan SC1, SC2 Subcrustal Nisqually 2001 Washington, US SC3 Subduction Maule 2010 Chile SD1, SD2 Subduction Tohoku 2011 Japan SD3, SD4 Table 8.3 Selected motions for NTHA of the Trent River Bridge on site class D Figure 8.9 Mean acceleration and displacement spectra for the 11 motions vs the target spectrum (the period range over which the mean spectra are matched to the target spectrum, are shown with the vertical lines). The drift capacities of the columns were obtained from pushover analysis of the entire bridge system in the lateral and longitudinal directions, similar to Section 7.10. The calculated drift capacities for the site class C and D models are listed in Table 8.4. Figure 8.10 shows the results of the pushover analysis of the Trent River Bridge in the lateral direction on site class C. 8.6 PBD Assessment Table 8.5 summarizes the maximum drift ratio demands of the columns from RSA in the longitudinal and lateral directions, along with the predicted level of damage. The drift ratios are calculated based on the relative deformation of the columns and do not include drift due to the foundations displacements and rotations. Referring to the table, in the longitudinal direction the 152 Figure 8.10 Pushover curves for the Trent River Bridge on site class C in the lateral direction Site class C Long. Drift (%) Lat. Drift (%) Performance Criteria Pier1 Pier2 Pier1 Pier2 Minimal 0.37 0.28 0.32 0.23 Yielding 0.74 0.56 0.63 0.45 Serviceability Limit 1 1.52 1.19 1.51 1.12 Cover Spalling 1 2.18 1.83 1.71 1.22 Cover Spalling 2 2.75 2.19 2.35 1.73 Serviceability Limit 2 2.15 1.88 1.89 1.42 Reduced Buckling 2.71 2.27 2.68 2.08 80% Core Crushing 3.48 2.83 2.99 2.18 Core Crushing 4.73 3.66 3.56 2.56 Reduced Fracture 4.87 4.12 4.63 3.71 Site class D Long. Drift (%) Lat. Drift (%) Performance Criteria Pier1 Pier2 Pier1 Pier2 Minimal 0.35 0.25 0.50 0.43 Yielding 0.70 0.51 1.01 0.86 Serviceability Limit 1 1.60 1.26 1.99 1.63 Cover Spalling 1 2.03 1.61 2.25 1.83 Cover Spalling 2 2.80 2.24 2.86 2.29 Serviceability Limit 2 2.00 1.65 2.42 2.03 Reduced Buckling 2.73 2.29 3.16 2.65 80% Core Crushing 3.57 2.83 3.50 2.70 Core Crushing 4.35 3.43 4.13 3.20 Reduced Fracture 4.62 3.89 5.13 4.32 Table 8.4 Drift ratio capacities of the columns associated to the first occurrence of the damage states -4000-3500-3000-2500-2000-1500-1000-50000 1 2 3 4 5 6Pier Base Shear (kN)Pier Lateral Drift Ratio (%)Site class CPier 1 Pier 2Yielding Spalling 1Serviceability Limit 2 BucklingCrushing Fracture153 bridge sustains minimal or no damage at all hazard levels on site class C. On site class D, Pier 2 undergoes yielding in the longitudinal direction at all hazard levels, and Pier 1 at 2%/50 hazard level. In the lateral, the drift demands are higher. Nevertheless, Pier 1 sustains minimal or no damage at all hazard levels, while Pier 2 undergoes yielding at 2%/50 and 5%/50 hazard levels, on both site classes. The results of the NTHA are shown in Table 8.6 and Figure 8.11. In general the mean drift demands from NTHA are a bit lower than the RSA demands but otherwise they are in reasonable agreement with the RSA results. 8.6.1 Assuming Major-route Importance Category The results of Table 8.5 indicate that the bridge satisfactorily meets all the minimum performance requirements of major-route bridges in CSA S6-14 and BC MoTI Supplement (see Table 7.1). One exception is the yielding criteria of the CSA S6-14 at 10%/50 hazard level, which is not met by Pier 2 in the longitudinal direction on site class D. The ratios of the drift demand to drift capacity and reserve capacities are listed in Table 8.7. Using the CSA S6-14 performance criteria, the reserve capacity of the design varies considerably at different hazard levels, and is the least at 10%/50 hazard level for both of the site class C and D models. Therefore, the performance criteria of 10%/50 hazard level control the design. However, using the BC MoTI Supplement performance criteria, the reserve capacity of the design is relatively even at all hazard levels. The large reserve capacities of the columns at 2%/50 hazard level are primarily due to the contribution of abutments in carrying the seismic load, and partly due to lower displacement demands at the period of the bridge. These findings are in line with those made in Chapter 7: for the implementation example and further confirm the comparison of the CSA S6-14 and BC MoTI Supplement criteria for major-route bridges. 8.6.2 Assuming Lifeline Importance Category If the importance category of the bridge is assumed to be lifeline, the bridge still meets all the performance requirements of BC MoTI Supplement on both site class C and D. However, Pier 2 does not satisfy the CSA S6-14 criteria at 10%/50 and 5%/50 hazard levels. The ratios of the drift demand to drift capacity and reserve capacities for the lifeline bridge are listed in Table 8.8. The reserve capacities for CSA S6-14 criteria suggest that while the bridge meets the requirements of 2%/50 hazard level with sufficient amount of reserve 154 Pier 1 Pier 2 Hazard Level Dx (%) Damage Dy (%) Damage Dx (%) Damage Dy (%) Damage Site class C 2%/50 0.19 N 0.48 M 0.29 M 0.68 Y 5%/50 0.13 N 0.34 M 0.20 N 0.48 Y 10%/50 0.10 N 0.25 N 0.15 N 0.35 M Site class D 2%/50 0.86 Y 0.91 M 1.25 Y 1.25 Y 5%/50 0.60 M 0.64 M 0.87 Y 0.87 Y 10%/50 0.43 M 0.46 N 0.62 Y 0.63 M Table 8.5 Column maximum drift demands from RSA in the longitudinal (x) and lateral (y) directions, along with the predicted damage (N: none, M: minimal damage, Y: yielding of longitudinal reinforcements) Pier 1 Pier 2 Hazard Level Dx (%) Damage Dy (%) Damage Dx (%) Damage Dy (%) Damage Site class D 2%/50 0.64 M 0.87 M 0.95 Y 1.23 Y 5%/50 0.42 M 0.56 M 0.62 Y 0.77 M 10%/50 0.31 M 0.43 M 0.46 M 0.60 M Table 8.6 Column mean drift demands from NTHA in the longitudinal (x) and lateral (y) directions, along with the predicted damage Figure 8.11 Drift ratios of the bridge columns from NTHA at 2%/50 0.000.200.400.600.801.001.201.401 2 3 4 5 6 7 8 9 10 11Longitudinal Drift (%)Record #P1 P2Mean P1 Mean P20.000.200.400.600.801.001.201.401.601.802.001 2 3 4 5 6 7 8 9 10 11Lateral Drift (%)Record #P1 P2Mean P1 Mean P2155 CSA S6-14 BC MoTI Supplement Hazard Level Δd/Δc (%)-x Δd/Δc (%)-y Reserve (%) Δd/Δc (%)-x Δd/Δc (%)-y Reserve (%) Site class C 2%/50 Pier 1 4.2 13.6 86.4 5.5 16.1 83.9 Pier 2 7.8 26.4 73.6 10.3 30.9 69.1 5%/50 Pier 1 6.2 18.0 82.0 4.9 12.7 87.3 Pier 2 10.6 33.7 66.3 8.8 22.9 77.1 10%/50 Pier 1 13.3 39.3 60.7 6.4 16.4 83.6 Pier 2 26.4 77.8 22.2 12.4 31.1 68.9 Site class D 2%/50 Pier 1 19.8 22.1 77.9 24.2 25.9 74.1 Pier 2 36.4 39.1 60.9 44.3 45.6 54.4 5%/50 Pier 1 29.9 26.3 70.1 21.9 20.0 78.1 Pier 2 52.6 42.8 47.4 38.7 32.3 61.3 10%/50 Pier 1 61.6 45.6 38.4 26.9 22.8 73.1 Pier 2 122.5 72.7 -22.5 49.4 37.5 50.6 Table 8.7 Ratios of the drift demand to drift capacity of the columns the reserve drift capacities assuming Major-route importance category for the bridge CSA S6-14 BC MoTI Supplement Hazard Level Δd/Δc (%)-x Δd/Δc (%)-y Reserve (%) Δd/Δc (%)-x Δd/Δc (%)-y Reserve (%) Site class C 2%/50 Pier 1 9.0 25.5 74.5 7.1 18.0 82.0 Pier 2 15.4 47.9 52.1 12.8 32.5 67.5 5%/50 Pier 1 18.0 53.9 46.1 8.7 22.5 77.5 Pier 2 35.8 106.6 -6.6 16.8 42.6 57.4 10%/50 Pier 1 23.0 54.0 46.0 6.4 16.4 83.6 Pier 2 45.9 102.4 -2.4 12.4 31.1 68.9 Site class D 2%/50 Pier 1 43.1 37.8 56.9 31.5 28.7 68.5 Pier 2 75.8 61.5 24.2 55.8 46.4 44.2 5%/50 Pier 1 85.7 63.2 14.3 37.5 31.6 62.5 Pier 2 170.5 100.8 -70.5 68.7 51.9 31.3 10%/50 Pier 1 89.4 79.6 10.6 26.9 22.8 73.1 Pier 2 171.1 122.0 -71.1 49.4 37.5 50.6 Table 8.8 Ratios of the drift demand to drift capacity of the columns the reserve drift capacities assuming Lifeline importance category for the bridge 156 CSA S6-14 BC MoTI Supplement Hazard Level Δd/Δc (%)-x Δd/Δc (%)-y Reserve (%) Δd/Δc (%)-x Δd/Δc (%)-y Reserve (%) Site class C 2%/50 Pier 1 4.1 10.4 89.6 3.4 8.9 91.1 Pier 2 6.9 18.2 81.8 5.8 14.0 86.0 5%/50 Pier 1 2.9 9.6 90.4 3.8 11.4 88.6 Pier 2 5.4 18.6 81.4 7.1 21.8 78.2 10%/50 Pier 1 4.6 13.1 86.9 3.6 9.2 90.8 Pier 2 7.8 24.6 75.4 6.5 16.7 83.3 Site class D 2%/50 Pier 1 18.7 17.8 81.3 15.8 15.7 84.2 Pier 2 32.2 28.9 67.8 25.1 23.9 74.9 5%/50 Pier 1 13.8 15.4 84.6 16.8 18.1 81.9 Pier 2 25.3 27.2 72.8 30.8 31.7 68.3 10%/50 Pier 1 21.5 19.0 78.5 15.7 14.5 84.3 Pier 2 37.8 30.9 62.2 27.8 23.8 72.2 Table 8.9 Ratios of the drift demand to drift capacity of the columns the reserve drift capacities assuming Other importance category for the bridge capacity, it does not meet the requirements of the other two hazard levels. The reserve capacities for the 5%/50 and 10%/50 requirement are in the same order. Therefore, the design of the bridge is controlled by the performance criteria of either 10%/50 or 5%/50 hazard level. Using BC MoTI Supplement criteria, the reserve capacities are more evenly distributed across hazard levels, with the reserve capacity at 5%/50 hazard level being slightly smaller. Therefore, using the criteria at 5%/50 hazard level controls the design of the lifeline bridge. 8.6.3 Assuming Other Importance Category The last scenario is to assume the importance category of the bridge as other. For this scenario, the bridge meets all the performance requirements of both CSA S6-14 and BC MoTI Supplement on both site class C and D. The ratios of the drift demand to drift capacity and reserve capacities for this case are listed in Table 8.9. Unlike the major-route and lifeline bridge scenarios, the reserve capacities for the performance criteria of CSA S6-14 are distributed evenly across the hazard levels, with the performance criteria at 5%/50 being slightly larger. Therefore, the design is controlled by the criteria at either 2%/50 or 10%/50 hazard levels. Using the BC MoTI Supplement criteria, similar to the major-route and lifeline bridge scenarios, the reserve capacities are distributed evenly across the hazard levels. In this case, the criteria at 5%/50 hazard levels with a slight difference control the design. 157 8.7 FBD Assessment CSA S6-14 allows FBD as an alternative approach to PBD for regular major-route and other bridges, but requires PBD for seismic design of life-line bridges (see Table A.3). The importance factor, I, for major-route bridges is 1.5 and for other bridges is 1.0. The response modification factor, R, for multiple-column ductile reinforced concrete bents is 5 and is applicable to both principal directions of the bridge. The elastic load effects on ductile members are reduced by the ratio of R/I, which gives 3.33 for major-route bridges and 5 for other bridges. For FBD assessment only the major-route bridge scenario was considered. Figure 8.12 shows the axial force and bi-axial bending moment demands for the Pier 1 and Pier 2 columns at 2%/50 hazard level along with the column interaction diagram. The demands were obtained for the site class C model. It is observed that the reduced demands are inside the capacity curve and therefore the column is safe in flexure at 2%/50 hazard level. It is also observed that the Pier 2 columns have higher bending moment demands and consequently lower reserve strength capacity. However, unlike the PBD, the FBD approach is not capable of predicting the level of damage to the columns. Moreover, the interaction diagram, which serves as the capacity curve, is calculated with the fundamental assumption that concrete crushes at strain value of 0.0035, which is much smaller than the strain limits recommended for the PBD approach. Using the FBD approach, if the reduced demand crosses the capacity curve, it only signifies that the force demands have reached the strength capacity of the columns and it does not necessarily indicate failure of the columns. Comparing the outcomes of the PBD and FBD for the major-route bridge scenario, one may conclude that the force-based designed bridge satisfies the CSA S6-14 PBD criteria for 2%/50 and 5%/50 hazard levels, but does not meet the yielding criteria at 10%/50 hazard level. 8.8 Conclusions The seismic performance of a pre-stressed reinforced concrete bridge was evaluated using the performance criteria of both CSA S6-14 and the BC MoTI Supplement, assuming three importance category scenarios for the bridge. It was observed that the CSA S6-14 performance criteria for the major-route and lifeline bridges are very conservative at 10%/50 and 5%/50 hazard levels, compared to 2%/50 hazard level. Consequently, the reserve capacity of the bridge 158 is uneven across the hazard levels. This is not the same for other bridge scenario as in this scenario the CSA S6-14 criteria produce relatively even reserve capacities across the hazard levels. In contrast to CSA S6-14 performance criteria, the criteria in BC MoTI Supplement result in relatively similar level of reserve capacity at all hazard levels for all three importance category scenarios. As discussed in Section 7.13.2, to be able to determine, which set of performance criteria has preference over the other for PBD, one may need to include loss analysis and compare the two sets of criteria in terms of incurring total loss. The case study also highlighted the importance of including SSI effects for PBD. Soil-structure interaction noticeably changed the displacement demands on the bridge and reduced the reserve capacity of the columns on site class D. Figure 8.12 Comparison of the axial force-bi-axial bending moment demand at 2%/50 hazard level with the column interaction diagram -100000-80000-60000-40000-200000200000 5000 10000 15000Axial Force (kN)Bending Moment (kN-m)Plastic Hinge RegionP-M Intercation Curve Pier 1 Reduced DemandPier 2 Reduced Demand Pier 1 Elastic DemandPier 2 Elastic Demand159 Chapter 9: Summary and Future Research 9.1 Summary of Thesis The recently adopted performance-based design provisions of CSA S6-14 provides engineers with means to accomplish more sophisticated seismic designs of conventional bridges as well as innovative new systems, for which force-based design provisions would fall short. In addition, the CSA S6-14 performance-based design promotes the discussion of post-earthquake performance of bridges between owners and engineers, and facilitates asset management and post-earthquake resiliency. All the merits of the PBD approach as envisioned by the code committee are conditioned on whether the PBD provisions can be properly implemented in practice. There are several challenges identified in this thesis with regards to implementing the CSA S6-14 provisions. Aside from the general challenges that concern PBD procedures in general, the particular challenges of implementing the CSA S6-14 approach are primarily related to: (1) performance-based design verification framework; (2) calibration of performance criteria; and (3) appropriate analysis models to evaluate the established performance criteria. Moreover, there was a need expressed from the industry for case studies demonstrating the step-by-step implementation of the CSA S6-14 PBD approach. This thesis have managed to address the aforementioned challenges and needs within the scope of the thesis. Two performance-based design verification frameworks, including a deterministic and a probabilistic framework, were introduced to complement the CSA S6-14 PBD approach. The deterministic framework reformulates the demand versus capacity checks in terms of reserve capacity versus target reserve capacity checks. This reformulation gives a perspective on how the reserve capacity of bridge components for each performance limit state would compare at different hazard levels. The target reserve capacities reflect the desirable safety margin in deterministic terms. When verifying the force-controlled limit states such as shear demand versus capacity, the safety margin is implicitly enforced through the demand and capacity factors. For verifying the deformation-controlled limit states such as reinforcing steel or concrete strain limit states, however, the CSA S6-14 does not provide demand and capacity factors. This is incompatible with the force-controlled limit states. The addition of the target reserve capacities 160 helps with resolving this incompatibility and ensures a level of safety for the deformation-controlled limit states of the code. For the probabilistic framework, Demand and Capacity Factored Design developed by Jalayer and Cornell (2003) was adopted. The DCFD framework was developed with an intention for code application and unlike many other probabilistic PBD frameworks does not rely on fragility curves, which makes it a suitable framework option for the CSA S6-14 PBD approach. Using the DCFD framework, engineers are able to calibrate the demand and capacity factors of each performance limit state to achieve desirable minimum reliabilities. This could be an important addition to the CSA S6-14 PBD approach with regards to verifying deformation-controlled limit states. Compared to the deterministic approach, the DCFD provides a systematic approach to include desirable levels of safety in verification of deformation-controlled limit states. This is achieved through determining the demand and capacity factors such that each limit state is met for an allowable mean annual frequency of exceeding. The framework also includes a confidence interval factor to meet the limit state for the allowable mean annual frequency of exceeding. The main formulation of the framework was borrowed in the thesis and was tailored for adoption with the CSA S6-14 PBD approach. All the necessary inputs to the framework were provided and discussed for further clarification. The step-by-step implementation of each of the two frameworks was presented in a detailed case study of a two-span steel girder reinforced concrete bridge. The performance of the case study bridge was evaluated using both response spectrum analysis and nonlinear time-history analysis. The selection and scaling of ground motion records for each suite of crustal, subcrustal, and subduction motions was presented in detail. The case study demonstrated how readily both frameworks can be implemented in practice with the current analysis tools, and no additional cost. Moreover, a comparison of the two frameworks was presented, highlighting the additional advantages of the DCFD framework in verifying the performance-based design compared to the deterministic framework. It was also demonstrated that the factors calculated from the DCFD were comparable to those recommended for deformation-controlled limit states in ASCE/SEI 7-16 (2016). 161 With regards to calibration of performance criteria, a thorough comparison of the strain limits of CSA S6-14 versus the BC MoTI Supplement was presented in a case study on a three-span reinforced concrete bridge with pre-stressed concrete girders. The case study showed that that the CSA S6-14 strain limits are unduly conservative for the minimal damage performance level, such that this performance level controls the design. For major-route and lifeline bridges, this means that the performance criteria at 10%/50 hazard level will control the design, while the performance criteria at 2%/50 hazard level are the least critical. The BC MoTI Supplement on the other hand, employs more relaxed strain limits for the minimal damage performance level and smaller strain limits for the repairable damage performance level. It was demonstrated that this adjustment in strain limits of the BC MoTI Supplement result in a more uniform performance across hazard levels compared to the CSA S6-14. Another important aspect of the performance verification addressed in the thesis was the use of appropriate analysis models to evaluate the established performance criteria. Concentrated plasticity models and distributed plasticity models are commonly used in practice. Distributed plasticity models are commonly considered to be more advantageous in obtaining strain demands. The use of distributed plasticity models with softening materials such as concrete is challenged by the localization of plastic strains and mesh-dependency issue. Verification of both global and local strain and curvature responses are only meaningful if the mesh-dependency issue is addressed first. Common techniques of addressing the mesh-dependency issue in inelastic beam-column elements include choosing a mesh size based on an empirical plastic hinge length or alternatively applying material model regularization. The thesis provided an insightful comparison of concentrated and distributed plasticity models and the two techniques of dealing with mesh-dependency issue by employing them to model a number of tested reinforced concrete bridge columns. It was shown that concentrated plasticity models with fiber-hinges provide acceptable estimation of local strain and global displacement response of reinforced concrete bridge columns, without the concern of mesh-dependency issue. Inelastic force-based or displacement-based beam-column elements provide access to local strain response across the entire element, but require dealing with the localization issue if used for reinforced concrete components. When material model regularization is employed to address the localization issue, verifying the strain limits of CSA S6-14 require additional post-processing effort. A technique 162 was developed in the thesis to verify the strain limits without the additional post-processing effort and was examined on the model of a bridge column test. A summary of the recommended strain and deformation limits in the literature and other design guidelines was included in the thesis to serve as a point of comparison for the strain limits of the CSA S6-14 and the BC MoTI supplement. The strain limits of the CSA S6-14 and the BC MoTI supplement were utilized with appropriate analysis models developed in the previous chapters of the thesis to predict the damage to the selected bridge columns tests. The predications were compared with the test results. This comparison demonstrated how the combination of appropriate modelling and the recommended performance criteria would predict the actual damage to modern well-detailed bridge columns. It was brought to attention that Berry and Eberhard’s drift limits for the spalling of concrete and buckling of longitudinal rebars maybe used as a cross-check for the predictions of distributed plasticity models of reinforced concrete bridge columns. This will exclude the possibility of numerical errors due to mistreatment of the localization issue. 9.2 Main Contributions Each chapter of the thesis elaborated on a relatively different topic and a summary of the main takeaways was presented in the previous section. As such, the main contribution of the thesis to the state of knowledge and practice can be highlighted as follows: 1) The challenges of implementing the CSA S6-14 PBD approach in practice were identified and addressed within the established scope of the thesis. These challenges were related to performance-based design verification framework, calibration of performance criteria and appropriate analysis models to evaluate the established performance criteria. 2) A deterministic and a probabilistic verification frameworks were examined to be adopted for the CSA S6-14 PBD approach. 3) A thorough comparison of the CSA S6-14 and the BC MoTI Supplement strain limits was presented with regards to calibration of the established performance criteria. 4) Appropriate numerical models for verifying the local strain response of reinforced concrete bridge components were investigated, including concentrated and distributed plasticity models. 163 5) Detailed case-studies of reinforced concrete bridges were provided to present the step-by-step implementation of the CSA S6-14 PBD approach and the application of the deterministic and probabilistic verification frameworks. In addition, this thesis serves as a reference document for practicing engineers to better understand the challenges of implementing the CSA S6-14 PBD approach and provides calibrated and demonstrated methods and tools through the contributions listed above. 9.3 Future Research The addition of the PBD approach to CSA S6-14 is only the first step towards the future of seismic design guidelines and as such, the research in this area should be a continuing effort. One major area that is not covered in this research is the performance-based retrofit design and seismic assessment of existing bridges. This area is of great interest as many of the bridges built prior to 1990’s do not have adequate seismic detailing and need retrofit interventions. Both CSA S6-14 and the BC MoTI Supplement aim to achieve similar performance-levels as new bridges with retrofit design. Achieving those performance levels with only retrofit intervention may increase the cost of retrofit design so high that a replacement solution becomes more viable, considering the remaining life the bridge. In that case, the owner should decide on the expected performance levels of the retrofit design. The challenge of this task is to associate appropriate performance criteria to achieve the expected performance levels in the existing bridges. Research is necessary in this area to complement the existing experience with retrofit design in practice. Currently the public review draft of CSA S6-19 has employed more relaxed performance levels for existing bridges with single hazard level. It should be investigated whether the associated performance criteria would actually result in the expected performance levels in existing and retrofitted bridges. For instance, the spalling of concrete may result in extensive damage in an existing bridge, while it is associated to minimal damage for modern code-compliant designs. In addition, the focus of the present research was on PBD of bridges with ductile reinforced concrete substructures. Research is still needed to cover the PBD evaluation of other types of bridges. A more comprehensive study may include studying archetypes of various types of bridges. Another important topic for future studies could be the performance assessment of bridges in aftershock events and their remaining capacity. Finally, the possibility of moving 164 towards loss-based design rather than damage-based or cost-based designs for future generations of PBD provisions should be investigated. As mentioned in the general challenges of PBD procedures, research is still needed in developing better damage and loss predictions models. 165 References AASHTO. (2011). Guide Specifications for LRFD Seismic Bridge Design. 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Probabilistic Engineering Mechanics, 19(3), 259-267. 174 Appendices Appendix A CSA S6-14 Tables for PBD Average properties in top 30 m Site class Ground profile name Shear wave average velocity, Vs (m/s) Standard penetration resistance, N60 Soil undrained shear strength, su A Hard rock Vs > 1500 Not applicable Not applicable B Rock 760 < Vs ≤ 1500 Not applicable Not applicable C Very dense soil and soft rock 360 < Vs ≤ 760 N60 > 50 su > 100 kPa D Stiff soil 180 < Vs ≤ 360 15 ≤ N60 ≤ 50 50 < su ≤ 100 kPa E Soft soil Vs ≤ 180 N60 < 15 su < 50 kPa Any profile with more than 3 m of soil with the following characteristics: Plastic index PI > 20; Moisture content w ≥ 40%; and Undrained shear strength su < 25 kPa F Other soil Site specific evaluation required Table A.1 CSA S6-14 site classification for seismic site response For T < 0.5 s For T ≥ 0.5 s Seismic performance category Lifeline bridges Major-route and other bridges S(0.2) < 0.2 S(1.0) < 0.1 2 1 0.2 < S(0.2) < 0.35 0.1 < S(1.0) < 0.3 3 2 S(0.2) ≥ 0.35 S(1.0) ≥ 0.3 3 3 Table A.2 CSA S6-14 Seismic performance category based on 2475-year return period spectral values Seismic performance category Lifeline bridges Major-route bridges Other bridges Irregular Regular Irregular Regular Irregular Regular 1 No seismic analysis is required 2 PBD PBD PBD FBD FBD FBD 3 PBD PBD PBD FBD* PBD FBD *PBD might be required by the Regularity Authority Table A.3 CSA S6-14 Requirements for PBD and FBD 175 Seismic performance category Lifeline bridges Major-route bridges Other bridges Irregular Regular Irregular Regular Irregular Regular 1 No seismic analysis is required 2 EDA, ISPA, and NTHA EDA and ISPA EDA and ISPA ESA EDA ESA 3 EDA, ISPA, and NTHA EDA, ISPA, and NTHA EDA and ISPA EDA EDA ESA Table A.4 CSA S6-14 minimum analysis requirements for 2%/50 and 5%/50 hazard levels Seismic performance category Lifeline bridges Major-route bridges Other bridges Irregular Regular Irregular Regular Irregular Regular 1 No seismic analysis is required 2 EDA EDA EDA ESA EDA ESA 3 EDA EDA EDA ESA EDA ESA Table A.5 CSA S6-14 minimum analysis requirements for 10%/50 hazard level Seismic ground motion probability of exceedance in 50 years (return period) Lifeline bridges Major-route bridges Other bridges Service Damage Service Damage Service Damage 10% (475 years) Immediate None Immediate Minimal Service limited Repairable 5% (975 years) Immediate Minimal Service limited* Repairable* Service disruption* Extensive* 10% (2475 years) Service limited Repairable Service disruption Extensive Life safety Probable replacement *Optional performance levels unless required by the Regulatory Authority or the Owner Table A.6 CSA S6-14 Minimum performance levels for PBD approach 176 Service Damage Immediate Bridge shall be fully serviceable for normal traffic and repair work does not cause any service disruption. Minimal damage • General: Bridge shall remain essentially elastic with minor damage that does not affect the performance level of the structure. • Concrete Structures: Concrete compressive strains shall not exceed 0.004 and reinforcing steel strains shall not exceed yield. • Steel Structures: Steel strains shall not exceed yield. Local or global buckling shall not occur. • Connections: Connections shall not be compromised. • Displacements: Pounding shall not occur. Residual displacement, settlement, translation or rotation, of the structure or foundations, including retaining and wing walls, shall be negligible, and not compromise the performance level. • Bearings and Joints: Shall not require replacement except for possible damage to joint seals. • Restrainers: No observable damage or loss of displacement capacity to restraining systems or connected elements shall occur. • Foundations: Foundation movements shall be limited to only slight misalignment of the spans or settlement of some piers or approaches that does not interfere with normal traffic, provided that no repairs are required. Limited Bridge shall be usable for emergency traffic and be repairable without requiring bridge closure. At least 50% of the lanes, but not less than one lane, shall remain operational. If damaged normal service shall be restored within a month. Repairable damage • General: There may be some inelastic behaviour and moderate damage may occur; however, primary members shall not need to be replaced, shall be repairable in place, and shall be capable of supporting the dead load plus full live load. • Concrete structures: Reinforcing steel tensile strains shall not exceed 0.015. • Steel structures: Buckling of primary members shall not occur. Secondary members may buckle provided that stability is maintained. Net area rupture of primary members at connections shall not occur. • Connections: Primary connections shall not be compromised. • Displacements: Permanent offset shall not compromise the service and repair requirements of the bridge. No residual settlement or rotation of main structure shall occur. There may be some movement of wing walls, subject to performance and reparability. • Bearings and joints: Elastomeric bearings may be replaced. If finger joints are damaged, they shall be repairable. • Restrainers: Restraining systems shall not be damaged. • Foundations: Foundation movements shall be limited to only slight misalignment of the spans or settlement of some piers or approaches that does not interfere with normal traffic, provided that repairs can bring the structure back to the original operational capacity. • Aftershocks: The structure shall retain 90% of seismic capacity for aftershocks and shall have full capacity restored by the repairs. Table A.7 CSA S6-14 Performance criteria (continued) 177 Service Damage Service disruption The bridge shall be usable for restricted emergency traffic after inspection. The bridge shall be repairable. Repairs to restore the bridge to full service might require bridge closure. Extensive damage • General: Inelastic behaviour is expected. Members might have extensive visible damage, such as spalling of concrete and buckling of braces but strength degradation shall not occur. Members shall be capable of supporting the dead plus 50% live loads, excluding impact, including P-delta effects, without collapse. • Concrete structures: There might be extensive concrete spalling, but the confined core concrete shall not crush. Reinforcing steel tensile strains shall not exceed 0.050. • Steel structures: Global buckling of gravity load supporting elements shall not occur. • Connections: There may be severe joint distortions. • Displacements: There may be permanent offsets as long as they do not prevent use by restricted emergency traffic after inspection of the bridge, nor preclude return of full service to the bridge. • Foundations: Ground lateral and vertical movements shall not exceed those that would prevent use by restricted emergency traffic after inspection or the bridge, nor preclude return of full service to the bridge. • Bearings and joints: The superstructure may have loss of bearings but shall have adequate remaining seat length and connectivity to carry emergency traffic. Joints might require replacement. • Restrainers: Restraining systems might suffer damage but shall not fail. • Aftershocks: The structure shall retain 80% seismic capacity for aftershocks and have full capacity restored by the repairs. Life safety The structure shall not collapse and it shall be possible to evacuate the bridge safely. Probable replacement • General: Bridge spans shall remain in place but the bridge might be unusable and might have to be extensively repaired or replaced. Extensive distortion of beams and column panels might occur. Fractures at moment connections might occur but shear connections shall remain intact. Members shall be capable of supporting the dead plus 30% live loads, excluding impact, but including P-delta effects, without collapse • Displacements: Permanent offsets shall be limited such that the bridge can be evacuated safely. • Foundations: Ground lateral and vertical movements are not restricted but shall not lead to collapse of the bridge superstructure Table A.7 CSA S6-14 Performance criteria (concluded) 178 Appendix B Deriving Expressions for Post-Processing Local Strain Response The following sequence, demonstrates the derivations of the proposed expressions for post-processing the strain response of regularized distributed plasticity models, in Section 5.3.5. The expressions are derived based on the assumption that the ratio of the post-peak energy up to a specified strain limit (in the post-peak region) to the total post-peak energy is similar for the reference and the regularized material models. Figure B.1 demonstrates this similarity and shows how the post peak response of the reference material model (shaded area) is scaled to the regularized model (red lines). Using similar triangles rule, the strain limits of concrete in the reference material model, denoted by εc1 can be mapped to εc2 in the regularized model: (fcc-0.2fcc'0.8fcc') = (εc1-εocεcu-εoc) (B.1) (fcc-0.2fcc'0.8fcc') = (εc2-εocε20c-PH-εoc) (B.2) (εc1-εocεcu-εoc) = (εc2-εocε20c-PH-εoc) (B.3) εc2= (ε20c-PH-εocεcu-εoc) (εc1-εoc)+εoc (B.4) The same concept can be used to map the strain limits of reinforcing steel in the reference material model denoted by εs1 to εs2 in the regularized model. A1 and A2 are the areas under the post-peak region of the reference and regularized reinforcing steel models, respectively: A1=GsLgage (B.5) A2=GsLIP (B.6) A1Lgage=A2LIP (B.7) 0.5 (fu+fy) (εu,exp-εy)Lgage=0.5 (fu+fy) (εsu-εy)LIP (B.8) 179 (εu,exp-εy)LIP=(εsu-εy)Lgage (B.9) (εs2-εy)LIP=(εs1-εy)Lgage (B.10) εs2=εy+(εs1-εy)LgageLIP (B.11) εoc εcuf’cc0.2 f’ccEccσ ε εc1 εc2 ε20c-PHf cc (a) εy εu,expEsfuσ fyfsεs1 εs2 εsu ε (b) Figure B.1 Mapping strain limits in the post-peak region of (a) confined concrete and (b) reinforcing steel 180 Appendix C Acceleration and Displacement Response Spectra of the Selected Records for NTHA in Chapter 7 and Chapter 8 C.1 Crustal Suite (Chapter 7) 181 182 183 C.2 Subcrustal Suite (Chapter 7) 184 185 186 187 C.3 Subduction Suite (Chapter 7) 188 189 190 191 C.4 Suite of Motions for Chapter 8 192 193 194 195 Appendix D PBD Aid Using the Correlation of Damage with Seismic Intensity Measures CSA S6-14 requires using a minimum of 11 ground motion records to assess the PBD of a bridge with NTHA. While using a larger number of records is necessary to reduce the dispersion in estimating mean response, it is computationally very demanding for a trial-and-error-based design process. The additional computational effort of using 11 ground motions compared to 7 or less, which was recommended by the previous editions of the code, is especially meaningful for complex bridges, where PBD is the only permissible design approach. One way to address this problem is to select a smaller number of records out of the original suite of 11 or more records, which are likely to cause more damage, and use this smaller suite in the design process instead of the original suite. It would be most convenient, if this smaller suite of records could be selected without performing any additional analysis, and solely based on ground motion intensity measures. This would be possible, if there was an adequate correlation between the intensity measures and the response parameter such that the damage could be predicted reasonably by the intensity measures. The main objective of this chapter is to find suitable intensity measures, which correlate adequately with the damage of ductile concrete bridge columns, and to use this correlation to select a smaller suite of records for NTHA required for the PBD process. D.1 Study1: Developing the PBD Aid D.1.1 Ground Motion Intensity Measure Candidates The basic task of all ground motion intensity measures is to describe the important characteristics of strong ground motions. Identifying and describing these characteristics lead to better selection and scaling of ground motion records for the purpose of time-history analysis. Traditionally, three main characteristics of amplitude, frequency content, and duration of motions are of interest, and many intensity measures have been proposed to describe one or more of these aspects. Nevertheless, due to the complex nature of ground motions, a single factor is incapable of accurately describing all important aspects of ground motions (Kramer 1995). A number of common and proposed intensity measures were examined in this study, to correlate with damage of ductile concrete bridge columns. The examined parameters are as follows: 196 1. Amplitude parameters including peak ground acceleration (PGA), peak ground velocity (PGV), and peak ground displacement (PGD). 2. Frequency content parameters including the ratio of peak velocity to peak acceleration (Vmax/Amax). 3. Mixed parameters, which represents more than one aspect of the important ground motion characteristics including Arias intensity, characteristic intensity, Housner spectral intensity, specific energy density, and RMS acceleration and velocity. 4. Proposed parameters by Fajfar et al. (1990) and Riddell and Garcia (2001). These parameters consider a combination of peak ground parameters and significant duration of earthquakes. The definitions of the parameters in the first three numbered items can be found in Kramer (1995). D.1.2 Bridge Column Models To investigate the correlation of the aforementioned intensity measures with the seismic damage of ductile reinforced concrete bridges, simplified single-column models were utilized. In reinforced concrete bridges with ductile substructure, the substructure elements such as columns, multiple-column bents, wall-type piers, etc., are designed and detailed to incorporate the seismic damage. In contrast, the superstructure and other elements are designed using the capacity design concept to stay essentially damage free. Moreover, the response of multi-span ductile concrete bridges are dominated primarily by the first mode response, as was illustrated in the previous chapters. For these reasons, it seems justified to use simplified single-column models in place of complete models for the purpose of this study. Three bridge columns with fundamental periods of 0.5s, 1.0s, and 2.0s were designed and detailed according to the CSA S6-14 FBD approach, with minimum response modification factor of R=3.0 and importance factor of I=1.0. The three periods cover a relatively wide range of periods for multi-span bridges, except for short period bridges. The columns were all 2 m in diameter, and are 8.5 m, 14 m, and 21.5 m in height, respectively. The longitudinal reinforcement ratio is 1% in all cases. The weight of the superstructure was assumed to be 200-300 kN/m distributed along the deck. Normal strength concrete with specified compressive 197 strength of 35 MPa was assumed and the grade of reinforcement steel was 400R with minimum specified yield strength of 400 MPa. The bridge columns were modelled in SeismoStruct using DBE with fibre sections. The details of modelling and utilized material models are similar to Section 8.3, and therefore are skipped here. A concentrated mass of 800 tonne was assigned to the top node of each column. This corresponds to mass of a 30 to 40-meter-long deck span supported by each column. The associated dead load from the deck was applied as a concentrated gravity force at the top of the columns. The viscous damping was modelled using Rayleigh damping with damping ratio of 3% at the first and second periods of each column. For the boundary condition, it was assumed that the columns are fixed at the base and there is no soil-structure-interaction. D.1.3 Damage Criteria Five separate damage states of yielding, cover spalling 1, serviceability limit 2, core crushing, and reduced fracture previously introduced in Table 6.3 were employed. For convenience, the strain limits corresponding to these damage states are summarized in Table D.1. It should be noted that the core crushing strain is calculated for the cross-section of the columns used in this study, and its values is different from that of Table 6.3. Damage State Strain Limit (m/m) Yielding εs > 0.0023 Cover Spalling 1 εc < -0.004 Serviceability Limit 2 εs > 0.015 Core Crushing εc < -0.01 Reduced Fracture εs > 0.05 Table D.1 Strain-based damage Criteria D.1.4 Ground Motion Records for NTHA It is assumed that the bridges under study are all located in the city of Victoria on the southern part of Vancouver Island in British Columbia. The soil site class is assumed to be class C or firm soil, which is the Canada-wide reference ground condition for uniform representation of seismic hazard across the country (Adams and Halchuk 2003). Using EZ-FRISK, probabilistic seismic hazard analysis was performed for the city of Victoria and UHS of Victoria was obtained for six hazard levels of 50%/50, 10%/50, 5%/50, 2%/50, 1%/50, and 0.5%/50 (corresponding to 72, 475, 975, 2475, 4974, and 9975-year return periods, respectively). 198 For each type of crustal, subcrustal, and subduction earthquakes, 10 records were selected for time-history analysis. The period ranges of interest were determined as 0.1-1.5 s, 0.2-2.0 s, and 0.4-4.0 s for the bridge columns with fundamental periods of 0.5 s, 1.0 s, and 2.0 s, respectively. The details of ground motion selection are similar to Section 7.7 and therefore are skipped here for brevity. The selected records are listed in Table D.2, along with the year and location of their corresponding historical event. Only one horizontal component of the records was utilized for time-history analysis. For each bridge column, the records were linearly scaled to match the 2%/50 UHS in the period ranges of interest for that column. The scale factors for linear scaling were found by minimizing the square root of the difference between the spectral acceleration response spectrum of each record with the target spectrum, over the period range of scaling. The formulas for calculating the minimum square root of error and scale factors for linear scaling can be found in the Technical Report for PEER Ground Motion Database (2010). D.1.5 Structural Analysis NTHA was performed to obtain the response of the bridge columns at six hazard levels of 50%/50, 10%/50, 5%/50, 2%/50, 1%/50, and 0.5%/50. For each hazard level, the same suite of ground motion records, which had been already selected and scaled for 2%/50 hazard level, was rescaled. The scale factor for each hazard level was calculated as Sa(T1)hazard level i /Sa(T1)2% /50, in which Sa(T1) is the spectral acceleration of the UHS for hazard level i at the fundamental period of the bridge column, T1. In total 540 nonlinear time-history analyses were conducted. For each analysis the maximum drift ratio of the top of the bridge columns was extracted as the main response parameter of interest12. The analysis was set to stop at the first occurrence of the reduced fracture damage state. The drift ratio capacities of the columns were obtained by pushover analysis as described in Section 7.10. 12 For first-mode-response dominated structures, maximum drift ratio is a common index for damage prediction (Cordova et al. 2001) 199 Type Historical Event Record Year Location No. of Records Crustal Chi-Chi CHY028-E 1999 Taiwan 1 Crustal Imperial Valley H-DLT352 1979 California, US 1 Crustal Kern County TAF021 1952 California, US 1 Crustal Kobe SKI000 1995 Japan 1 Crustal Landers ABY090 1992 California, US 1 Crustal Loma Prieta A2E090 1989 California, US 1 Crustal Northridge UCL360 1994 California, US 1 Crustal San Fernando PDL120 1971 California, US 1 Crustal Superstition Hills B-IVW090 1987 California, US 1 Crustal Tabas BOS-T1 1978 Iran 1 Subcrustal Geiyo EHM0150103241528-EW 2001 Japan 2 EHM0160103241528-EW Subcrustal Miyagi-Oki IWT0110508161146-NS 2005 Japan 4 MYG0060508161146-EW MYG0100508161146-NS MYG0170508161146-NS Subcrustal Nisqually 0720c_a-90 2001 Washington, US 4 1032j_a-58 1416a_a-125 1423c_a-148 Subduction Hokkaido HKD0770309260450-NS 1952 Japan 4 HKD0840309260450-NS HKD0950309260450-EW HKD1090309260450-NS Subduction Maule curico1002271-EW 2010 Chile 4 hualane1002271-T stgolaflorida1002271-NS stgopenalolen1002271-EW Subduction Tohoku FKS0071103111446-NS 2011 Japan 2 FKS0121103111446-EW Table D.2 Selected ground motion records for time history analysis D.1.6 Correlation of Intensity Measures with the Response Parameter To check the correlation of damage in bridge columns with the selected ground motion intensity measures, the maximum drift ratios were plotted against each ground motion intensity measure. To do so, the maximum drift ratios for the 30 records scaled at each of the six hazard levels, were plotted against the ground motion intensity measures corresponding to that hazard level. Subsequently, linear correlation coefficient between the intensity measures and the response parameter at each hazard level were calculated. A positive correlation coefficient signifies an increase in the response parameter, and thus the damage in the bridge columns, with an increase in the intensity measure. On the other hand, a zero or a small correlation coefficient indicates that 200 the response parameter is not dependent on that intensity measure. The intensity measures, which demonstrate the strongest positive correlation with the maximum drift ratios, are of interest. Examples of such plots are given in Figure D.1 and Figure D.2 for intensity measures PGV, PGD, and Vmax/Amax. The results are shown for only three hazard levels of 50%/50, 2%/50, and 0.5%/50, for the sake of clarity. The following can be observed from the figures: 1. Among the investigated intensity measures PGV, and PGD demonstrated on average higher positive linear correlation with the response parameter, followed by Vmax/Amax, the intensity measure proposed by Fajfar et al., and the Housner spectral intensity. 2. The correlation of the aforementioned intensity measures with the response parameter depends on the hazard level. For lower hazard levels, where the response of the bridge columns is essentially elastic, the linear correlation coefficients are close to zero. The linear correlation tends to increase at higher hazard levels, and with higher levels of damage. 3. The correlation of the aforementioned intensity measures with the response parameter depends on the fundamental period of the bridge columns. 4. Referring to Fig.3, it can be observed that at higher hazard levels, the linear correlation of PGV with the response parameter is significant and positive for T1=0.5s, and 1.0s, and becomes negative for T1=2.0 s. 5. PGD demonstrates a positive linear correlation with the response parameter. The correlation becomes stronger at larger periods (T1=2.0 s, and 1.0 s), and decreases for T1=0.5 s. 6. Vmax/Amax shows a positive linear correlation with the response parameter, but less strong compared to PGD, and PGV. In this case, the positive correlation tends to improve slightly at larger periods. 7. Figure D.2 shows the maximum drift ratio of the 1.0 s period column versus PGD, for each type of earthquakes, separately. It is observed that the positive linear correlation at higher hazard levels tends to be stronger for the crustal, and subduction suites compared 201 Figure D.1 Maximum drift ratios versus PGV and Vmax/Amax for the three bridge columns 0.00.51.01.52.02.53.03.54.00 50 100 150 200Maximum Drift (%)PGV (cm/s)All-T1=0.5s0.00.51.01.52.02.53.03.54.00.00 0.05 0.10 0.15 0.20 0.25Maximum Drift (%)Vmax/Amax (s)All-T1=0.5s0.00.51.01.52.02.53.03.54.04.50 50 100 150 200Maximum Drift (%)PGV (cm/s)All-T1=1.0s0.00.51.01.52.02.53.03.54.04.50.00 0.05 0.10 0.15 0.20 0.25Maximum Drift (%)Vmax/Amax (s)All-T1=1.0s0.01.02.03.04.05.06.07.00 100 200 300Maximum Drift (%)PGV (cm/s)All-T1=2.0s0.01.02.03.04.05.06.07.00.00 0.05 0.10 0.15 0.20 0.25Maximum Drift (%)Vmax/Amax (s)All-T1=2.0s202 Figure D.2 (left column) Maximum drift ratios versus PGD for the three bridge columns and (right column) maximum drift ratios versus PGD of the individual earthquake sources for the bridge column with fundamental periods of 1.0 s 0.00.51.01.52.02.53.03.54.00 50 100 150Maximum Drift (%)PGD (cm)All-T1=0.5s0.00.51.01.52.02.53.03.54.04.50 20 40 60 80 100 120 140 160Maximum Drift (%)PGD (cm)Crustal-T1=1.0s0.00.51.01.52.02.53.03.54.04.50 50 100 150 200Maximum Drift (%)PGD (cm)All-T1=1.0s0.00.51.01.52.02.53.03.54.04.50 10 20 30 40Maximum Drift (%)PGD (cm)Subcrustal-T1=1.0s0.01.02.03.04.05.06.07.00 50 100 150Maximum Drift (%)PGD (cm)All-T1=2.0s0.00.51.01.52.02.53.03.54.04.50 20 40 60 80 100 120Maximum Drift (%)PGD (cm)Subduction-T1=1.0s203 to the subcrustal suit. This can be explained considering that the response of the bridge column to the subcrustal motions is smaller than the crustal and subduction motions, at this period, and therefore less damage is incurred to the column by the subcrustal suite. It was observed that for lower levels of damage, the positive linear correlation of the response parameter with the intensity measures is weaker in comparison to higher levels of damage. It has been already shown in several studies that Sa(T1), when utilized as the intensity measure to select ground motions, can be both inefficient and insufficient for predicting the response of structures, and the extent of damage (Baker and Cornell 2005; Baker and Cornell 2006; Tothong and Luco 2007; Luco and Bazzuro 2007; Huang et al. 2009). Based on the observations of this study, and confirmed by the literature, Sa(T1) is sufficient at the lower hazard levels, where the response of the bridge columns is essentially elastic. This is the reason why at lower hazard levels, the calculated drift ratios show minimal variation. However, as the response of the bridge columns moves into the nonlinear range at higher hazard levels, Sa(T1) becomes insufficient, and the response becomes dependent on other intensity measures. This is evident from Figure D.1 and Figure D.2 where the variation of the drift ratios increases considerably at the higher hazard levels. The observed linear correlation of the response with the investigated intensity measures could be explained theoretically as well. For instance, it is well-known that the response of single-degree-of-freedom systems with very large periods tends to be close to PGD. This is why, at higher levels of damage, where period elongation has occurred, and also at larger fundamental periods, the correlation of PGD with the response of the bridge columns is stronger. D.1.7 Selection of a Smaller Suite of Records Following the main objective of this study, it is of interest to use the observed correlations to sample a smaller suite of records out of the original suite of 30 motions for NTHA. For such a suite of records, it should be examined if the predicted level of damage is comparable to the predictions of the original suite of records. To sample the records, first the records should be ranked based on the value of the chosen intensity measure (calculated at the 2%/50 hazard level). Next, the mean of the maximum drift ratios for the one third, and half of the records (10 and 20 204 records) with the highest values of the intensity measure should be calculated. Subsequently, the mean drift ratios should be compared against the drift ratio capacities of the columns to predict the damage state of the columns. Finally, the mean drift ratio and the predicted damage state should be compared to those obtained using the original suite of 30 records. Two scenarios are possible: 1. A zero or insignificant linear correlation between the intensity measure and the response parameter at lower hazard levels. In these cases, the values of the response parameter form a small cluster (as for PGV and PGD) or a horizontal line (as for Vmax/Amax), and do not depend on the investigated intensity measures (Sa(T1) is a sufficient intensity measures for these cases). As a result, selecting a smaller suite of one third or half the number of records, whether ranked based on the intensity measure or not, would yield similar mean response values as the original suite. 2. There is a positive linear correlation between the intensity measure and the response parameter, at higher hazard levels. In these cases, the records with higher values of the examined intensity measure tend to have larger responses. Therefore, the mean of the smaller suite of records selected in this way, tend to have larger mean response value as the original suite. The above scenarios were examined using PGV, PGD, and Vmax/Amax intensity measures. Typically, it is required to include records from all three types of earthquakes for time-history analysis. Therefore, it is more reasonable to choose 3 or 5 out of 10 records for each type of earthquake, instead of choosing 10 or 15 out of 30 records, disregarding the type of earthquake. Table D.3 shows the comparison between the damage predictions using the smaller suites with the original suite, for the 1.0 s period column. For this case, PGD was employed as the intensity measure. In the table, μn is the average of the maximum drift ratios for n records with the highest PGD values. For instance, μ3 is the average of the 3 records out of 10 crustal, subcrustal, or subduction records, with the highest PGD values. Referring to the table, it can be observed that in almost all cases, the smaller suites give reasonable predictions of both the mean response and damage state of the columns. In some cases, they may overestimate the mean response and the corresponding damage. Even so, performing the design checks using the smaller suites tends to be on the conservative side, and the final design would most likely meet the specified 205 performance criteria, if checked for the original suite of records. Similar observations were made for the other two columns, but for brevity they are not presented here. μn 50%/50 Damage 10%/50 Damage 5%/50 Damage 2%/50 Damage 1%/50 Damage 0.5%/50 Damage T1=1.0 s CR μ3 0.21 M 0.98 Y 2.17 SL 2.96 C 3.29 F 3.49 F μ5 0.21 M 0.95 Y 1.90 SL 2.59 C 3.03 F 3.42 F μ10 0.23 M 1.06 Y 1.83 SL 2.39 C 2.83 C 3.19 F SC μ3 0.26 M 0.86 Y 1.39 Y 1.55 Y 2.03 SL 2.28 SL μ5 0.24 M 0.91 Y 1.36 Y 1.62 S 1.97 SL 2.21 SL μ10 0.25 M 0.90 Y 1.31 Y 1.60 S 1.93 SL 2.23 SL SD μ3 0.21 M 0.86 Y 1.99 SL 3.25 F 3.45 F 3.45 F μ5 0.21 M 0.94 Y 1.74 S 2.75 C 3.06 F 3.18 F μ10 0.25 M 0.92 Y 1.57 Y 2.34 C 2.64 C 2.91 C All μ10 0.22 M 0.97 Y 1.88 SL 2.72 C 3.07 F 3.34 F μ15 0.22 M 0.96 Y 1.75 S 2.48 C 2.92 C 3.18 F μ30 0.25 M 0.96 Y 1.57 Y 2.11 SL 2.47 C 2.77 C Table D.3 Prediction of the damage to the 1.0s period bridge column at different hazard levels using smaller suites of records. PGD was utilized as the intensity measure; μn is the average of the maximum drift ratios for n records with the highest PGD values * Abbreviations: M: minimal damage, Y: yielding, S: cover spalling 1, SL: serviceability limit 2, C: core crushing, F: reduced fracture, CR: crustal, SC: subcrustal, SD: subduction D.1.8 Conclusions The linear correlation of a number of ground motion intensity measures with the seismic damage of ductile concrete bridge columns in British Columbia has been studied. The examined intensity measures were intended to be utilized to select a smaller suite of records out of the original suite of 11 or more records, for checking performance objectives during the performance-based design trial and error process. The results of nonlinear time history analysis of three bridge columns with short, medium, and long fundamental periods, demonstrated that PGV, and PGD have reasonable strong positive linear correlation with the maximum drift ratio and damage to the columns. The correlation increases at higher hazard levels, where higher levels of damage incur to the columns. Based on the primary observations of this study, it seems reasonable to employ PGV for shorter period bridge columns (0.5<T1<1.0 s), while using PGD for longer period bridges (T1>1.0 s), as the intensity measure to select a smaller suite of records out of the original suite of motions. 206 It was also demonstrated that it is possible to employ the aforementioned intensity measures to select a suite of one third or half the size of the original suite of records, which would still yield a reasonable prediction of the mean response and the corresponding damage to the bridge columns. Using a smaller number of records for the design trial and error process is very advantageous in terms of time and computational effort for performance-based design of bridges, where multiple performance-objectives at multiple hazard levels must be met simultaneously. However, it should be recognized that the conclusions of this study are based on primary observations for a limited number of bridge columns. Further studies are needed to confirm the applicability of these outcomes to performance-based design of all ductile concrete bridges. 207 D.2 Study2: Examining the PBD Aid for the Implementation Example of Chapter 5 The objective of this study is to further examine the PBD aid developed in the previous section by applying the aid to the implementation example of Chapter 5. The response of the bridge in the longitudinal direction is dominated by the first mode with period of 1.61 s and in the lateral direction by the first two modes with periods of 0.56 and 0.14 s. Following the observations of the previous section, three intensity measures will be tested for the bridge including PGD, PGV, and PGA. The reason for adding PGA to the list was the contribution of the short period mode (0.14 s) to the lateral response. D.2.1 Sampling Records To test the correlations of the selected intensity measures with the drift ratios of the bridge, the response of the bridge in the two directions are treated individually. This is partly due to the difference in the fundamental periods of the two directions and partly due to the fact that different horizontal components act in the two directions (H1 acts in the longitudinal direction and H2 in the lateral direction). Moreover, the correlation of the intensity measures with drift ratios is tested for individual crustal, subcrustal, and subduction suites, as well as all records together. Figure D.3 to Figure D.6 shows the plots of longitudinal and lateral drift ratios versus the selected intensity measures of the two horizontal components, for individual suites and all records. The calculated correlation coefficients at 2%/50 hazard level are listed in Table D.4. The values of the correlation coefficients reveal that in the longitudinal direction, PGV of the horizontal component acting in that direction has the highest positive correlation with the longitudinal drift ratio, and in the lateral direction PGA correlates best with the lateral drift ratio. PGD correlation with the longitudinal response is only strong for the subduction and subcrustal suites. This is because compared to crustal motions, subcrustal and subduction motions contribute most to the response of the modes with longer periods (compare the period range of ground motion selection for individual suites in Section 7.7). The same reasoning explains why the correlation of PGV with the longitudinal drift ratio is weakest for the crustal suite. The weak correlation of PGV with the lateral drift ratio is due to the contribution of the 0.14 s mode. This suggests that PGV may not be a suitable candidate intensity measure for periods below 0.5 s. These observations can also be made by investigating the plots of Figure D.3 to Figure D.6. 208 What these plots demonstrate aside from the strength of the intensity measure-drift ratio correlations, is that when PGA presents the highest positive correlation with the response, PGD shows otherwise, and vice versa. D.2.2 Mean Response Predictions Based on the above discussion, H1-PGV and H2-PGA were selected as the intensity measures to sample a smaller suite of records out of the original suite. In each direction, the records were ranked based on the corresponding intensity measure. Next mean drift ratios for a smaller suite of n records with the highest values of the intensity measure were calculated and compared to the mean drift ratios obtained from the original suite of records. This comparison is made in Table D.5 and Table D.6. The results of the two tables suggest that using the PBD aid, it is possible to provide a reasonable estimate of the mean response with only using 3 out of 33 records. The mean drift ratios predicted by the sampled suites tend to be conservative with one exception of the mean drift ratios predicted for the crustal suite in the longitudinal direction. Nevertheless, even in that case the underestimation of the mean response is negligible. D.2.3 Conclusions The results of this study further confirm the applicability of the developed method as an aid to facilitate PBD trial and error process. The outcomes of the study in this section and the previous section suggest using PGA, PGV, and PGD as intensity measures to sample a smaller suite of records out of a suite of 11 or more records that is required by the code for NTHA. PGA is best suited for shorter period structures with T1<0.5 s, PGV for mid-range periods of 0.5<T1<1.5 s, and PGD for longer periods of T1>2.0 s. This method works best with structures that their response is dominated with the first mode response. 209 2%/50 CR SC SD All H1 PGD with Drx 0.19 0.57 0.44 0.49 PGV with Drx 0.31 0.58 0.69 0.63 PGA with Drx -0.18 -0.46 -0.51 -0.53 H2 PGD with Dry -0.39 -0.09 -0.58 -0.38 PGV with Dry -0.28 0.17 0.17 -0.11 PGA with Dry 0.50 0.68 0.59 0.61 Table D.4 Correlation coefficients of longitudinal and lateral drift ratios at 2%/50 hazard level with PGD, PGV, and PGA of H1 and H2 components PGV μn 10%/50 5%/50 2%/50 CR μ11 3.84 2.53 1.90 μ5 3.60 2.50 1.88 μ3 3.44 2.20 1.70 SC μ11 2.66 1.97 1.46 μ5 3.29 2.45 1.75 μ3 2.73 2.07 1.61 SD μ11 4.65 3.07 2.13 μ5 5.60 3.49 2.16 μ3 5.98 3.77 2.26 All μ33 3.72 2.53 1.83 μ11 4.89 3.07 2.11 μ7 4.44 2.88 1.98 μ5 3.88 2.33 1.76 μ3 4.35 2.74 1.95 Table D.5 Mean longitudinal drift ratios (%) of the smaller suites sampled based on H1-PGV correlation PGA μn 10%/50 5%/50 2%/50 CR μ11 1.48 0.99 0.65 μ5 1.64 1.15 0.77 μ3 1.55 1.11 0.74 SC μ11 1.60 1.08 0.72 μ5 1.87 1.26 0.81 μ3 2.13 1.48 0.94 SD μ11 1.41 0.98 0.67 μ5 1.73 1.22 0.86 μ3 1.87 1.30 0.87 All μ33 1.50 1.02 0.68 μ11 1.74 1.19 0.78 μ7 1.92 1.32 0.85 μ5 2.00 1.37 0.86 μ3 1.84 1.27 0.81 Table D.6 Mean lateral drift ratios (%) of the smaller suites sampled based on H2-PGA correlation 210 Figure D.3 Maximum longitudinal and lateral drift ratios versus PGD, PGV, and PGA of H1 and H2 components of the crustal suite 0.001.002.003.004.005.006.007.008.000 50 100 150Longitudinal Drift (%)H1-PGD (cm)0.000.501.001.502.002.500 20 40 60 80 100Lateral Drift (%)H2-PGD (cm)0.001.002.003.004.005.006.007.008.000 50 100 150Longitudinal Drift (%)H1-PGV (cm/s)0.000.501.001.502.002.500 50 100 150Lateral Drift (%)H2-PGV (cm/s)0.001.002.003.004.005.006.007.008.000 1 1Longitudinal Drift (%)H1-PGA (g)0.000.501.001.502.002.500 1 1Lateral Drift (%)H2-PGA (g)211 Figure D.4 Maximum longitudinal and lateral drift ratios versus PGD, PGV, and PGA of H1 and H2 components of the subcrustal suite 0.000.501.001.502.002.503.003.504.004.505.000 10 20 30 40Longitudinal Drift (%)H1-PGD (cm)0.000.501.001.502.002.503.000 10 20 30 40Lateral Drift (%)H2-PGD (cm)0.000.501.001.502.002.503.003.504.004.505.000 20 40 60 80Longitudinal Drift (%)H1-PGV (cm/s)0.000.501.001.502.002.503.000 20 40 60 80 100Lateral Drift (%)H2-PGV (cm/s)0.000.501.001.502.002.503.003.504.004.505.000 1 1Longitudinal Drift (%)H1-PGA (g)0.000.501.001.502.002.503.000 1 1 2Lateral Drift (%)H2-PGA (g)212 Figure D.5 Maximum longitudinal and lateral drift ratios versus PGD, PGV, and PGA of H1 and H2 components of the subduction suite 0.001.002.003.004.005.006.007.008.000 50 100 150Longitudinal Drift (%)H1-PGD (cm)0.000.501.001.502.002.500 20 40 60 80 100Lateral Drift (%)H2-PGD (cm)0.001.002.003.004.005.006.007.008.000 50 100 150 200Longitudinal Drift (%)H1-PGV (cm/s)0.000.501.001.502.002.500 50 100 150Lateral Drift (%)H2-PGV (cm/s)0.001.002.003.004.005.006.007.008.000 0 0 1 1Longitudinal Drift (%)H1-PGA (g)0.000.501.001.502.002.500 1 1Lateral Drift (%)H2-PGA (g)213 Figure D.6 Maximum longitudinal and lateral drift ratios versus PGD, PGV, and PGA of H1 and H2 components of all records 0.001.002.003.004.005.006.007.008.000 50 100 150Longitudinal Drift (%)H1-PGD (cm)0.000.501.001.502.002.503.000 20 40 60 80 100Lateral Drift (%)H2-PGD (cm)0.001.002.003.004.005.006.007.008.000 50 100 150 200Longitudinal Drift (%)H1-PGV (cm/s)0.000.501.001.502.002.503.000 50 100 150Lateral Drift (%)H2-PGV (cm/s)0.001.002.003.004.005.006.007.008.000 1 1Longitudinal Drift (%)H1-PGA (g)0.000.501.001.502.002.503.000 1 1 2Lateral Drift (%)H2-PGA (g)
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Evaluating the performance-based seismic design of RC bridges according to the 2014 Canadian Highway… Ashtari, Sepideh 2018
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Title | Evaluating the performance-based seismic design of RC bridges according to the 2014 Canadian Highway Bridge Design Code |
Creator |
Ashtari, Sepideh |
Publisher | University of British Columbia |
Date Issued | 2018 |
Description | The 2014 edition of the Canadian Highway Bridge Design Code, CSA S6-14, has adopted a performance-based design approach for the seismic design of lifeline and major-route bridges in highly seismic zones. This addition offers many opportunities as well as some challenges with regards to implementing the CSA S6-14 performance-based design provisions in practice. This thesis aims to identify these challenges through a critical review of the CSA S6-14 performance-based design provisions and to address a number of them within the scope of the thesis. The motivation behind conducting the present study is to prepare a reference document for engineers to better comprehend and implement the new provisions in practice. The focus of the thesis is on the performance-based design of new reinforced concrete bridges with ductile substructures. The addressed challenges are related to CSA S6-14 performance verification framework, calibration of performance criteria, and appropriate numerical models to evaluate the established performance criteria. A deterministic and a probabilistic framework are recommended to be used with the CSA S6-14 performance-based design approach. The applications of each of the frameworks are demonstrated through two detailed case studies and the advantages and disadvantages of each framework are discussed. The performance criteria of the code are compared against the recommended criteria in the literature and other design guidelines. Moreover, the strain limits of the code are examined to predict the damage to a number of tested reinforced concrete bridge columns. A thorough comparison of the CSA S6-14 and the updated strain limits of the BC MoTI Supplement to CSA S6-14 is presented. Finally, common modelling techniques for reinforced concrete structures including distributed and concentrated plasticity models are employed to predict the response of a number of tested bridge columns. Mesh-sensitivity issues due to the localization of plastic strains at critical sections or elements of distributed plasticity models are discussed and the methods to rectify the issue are presented and compared. A simple solution is proposed to eliminate the post-processing effort that is required to verify the strain limits of the code in distributed plasticity models, for which material model regularization is used to deal with the mesh-sensitivity issue. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2018-12-06 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0375587 |
URI | http://hdl.handle.net/2429/67978 |
Degree |
Doctor of Philosophy - PhD |
Program |
Civil Engineering |
Affiliation |
Applied Science, Faculty of Civil Engineering, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2018-02 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
AggregatedSourceRepository | DSpace |
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