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Performance based design of degrading structures

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Title:
Performance based design of degrading structures
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English
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Chenouda, Mouchir
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University of South Florida
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Subjects / Keywords:
Seismic
Degradation
Fragility
Displacement estimates
Collapse
Dissertations, Academic -- Civil Engineering -- Doctoral -- USF
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bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )

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Abstract:
ABSTRACT: Seismic code provisions are now adopting performance-based methodologies, where structures are designed to satisfy multiple performance objectives. Most codes rely on approximate methods to predict the desired seismic demand parameters. Most of these methods are based on simple SDOF models, and do not take into account neither MDOF nor degradation effects, which are major factors influencing structural behavior under earthquake excitations. More importantly, most of these models can not predict collapse explicitly under severe seismic loads. This research presents a newly developed model that incorporates degradation effects into seismic analysis of structures. A new energy-based approach is used to define several types of degradation effects. The research presents also an evaluation of the collapse potential of degrading SDOF and MDOF structures. Collapse under severe seismic excitations, which is typically due to the formation of structures mechanisms amplified by P-Delt a effects, was modeled in this work through the degrading hysteretic structural behavior along with P-Delta effects due to gravity loads. The model was used to conduct extensive statistical dynamic analysis of different structural systems subjected to a large set of recent earthquake records. To perform this task, finite element models of a series of generic SDOF and MDOF structures were developed. The degrading hysteretic structural behavior along with P-Delta effects due to gravity loads proved to successfully replicate explicit collapse. For each structure, collapse was investigated and inelastic displacement ratios curves were developed in case collapse doesn't occur. Furthermore, seismic fragility curves for a collapse criterion were also developed. In general, seismic fragility of a system describes the probability of the system to reach or exceed different degrees of damage. Earlier work focused on developing seismic fragility curves of systems for several values of a calibrated ^damage index. This research work focuses on developing seismic fragility curves for a collapse criterion, in an explicit form. The newly developed fragility curves represent a major advancement over damage index-based fragility curves in assessing the collapse potential of structures subject to severe seismic excitations. The research findings provide necessary information for the design evaluation phase of a performance-based earthquake design process.
Thesis:
Dissertation (Ph.D.)--University of South Florida, 2006.
Bibliography:
Includes bibliographical references.
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by Mouchir Chenouda.
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Includes vita.
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Title from PDF of title page.
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Document formatted into pages; contains 201pages.

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oclc - 133465148
usfldc doi - E14-SFE0001447
usfldc handle - e14.1447
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Performance Based Design of Degrading Structures by Mouchir Chenouda A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Civil and Environmental Engineering College of Engineering University of South Florida Co-Major Professor: Ashraf Ayoub, Ph.D. Co-Major Professor: Rajan Sen, Ph.D. William Carpenter, Ph.D. Autar Kaw, Ph.D. Kandethody Ramachandran, Ph.D. Date of Approval: March 28, 2006 Keywords: seismic, degradation, fragili ty, displacement estimates, collapse Copyright 2006, Mouchir Chenouda

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DEDICATION To all my dear family members specially my wife Mirey and my son Mark.

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i TABLE OF CONTENTS LIST OF TABLES iv LIST OF FIGURES vii ABSTRACT xvi PREFACE xviii CHAPTER 1 INTRODUCTION 1.1 Introduction 1.2 Research Objectives 1.3 Thesis Organization 1 1 2 4 CHAPTER 2 LITERATURE REVIEW 2.1 Introduction 2.2 Strength-Based Building Design Codes 2.2.1 International Building Code (United States) 2.2.2 National Building Code of Canada 2.2.3 Mexico Federal District Code 2.3 Performance-Based Building Design Codes 2.3.1 Capacity Spectrum Method 2.3.2 Method of Coefficients 2.3.3 Drawback of Current Methods 2.4 Seismic Analysis Techniques 2.5 Damage Evaluation of Building Structures 2.5.1 Damage Indices 2.5.2 Classification of Damage 2.5.3 Categorization of Damage 2.6 Previous Work on Collapse Assessment 2.6.1 P Effects 2.6.2 Degrading Hysteretic Models 2.6.3 Analytical Collapse Investigations 2.6.4 Experimental Collapse Investigations 6 6 6 7 8 9 10 15 16 18 19 22 22 23 24 26 26 27 28 29

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ii CHAPTER 3 MATERIAL MODELS AND EARTHQUAKE RECORDS 3.1 Material Models 3.1.1 Degradation 3.1.1.1 Yield (Strength) Degradation 3.1.1.2 Unloading Stiffness Degradation 3.1.1.3 Accelerated Stiffness Degradation 3.1.1.4 Cap Degradation 3.1.2 Effect of Degradation on Inelastic Systems Behavior 3.1.3 Collapse of Structural Elements 3.1.4 Experimental Verifica tion of Material Models 3.2 Earthquake Records 3.2.1 Database of Earthquake Records 3.2.2 Scaling of Earthquake Records 31 31 33 34 37 38 39 40 58 60 63 64 73 CHAPTER 4 ASSESSMENT OF DEGRADED SDOF STRUCTURES 4.1 Introduction 4.2 Degradation Effect on SDOF Sy stems Under Seismic Excitations 4.3 Displacement Estimates of SDOF Degraded Structures 4.4 Incremental Dynamic Analysis and Fragility of Collapse for SDOF 4.4.1 Ductility Capacity and St rength Reduction Factor at Collapse 4.4.1.1 Short Period Structures (2 0 T sec) 4.4.1.2 Medium Period Structures (5 0 T sec) 4.4.1.3 Long Period Structures (0 1 T sec) 4.4.1.4 Long Period Structures (0 2 T sec) 4.5 Seismic Fragility Analysis 4.5.1 IDA and Fragility Relationship 4.5.2 Strength Reduction Factor ( R ) and Ratio of Yield Force to Total Weight ( ) 4.5.2.1 Short Period Structures (2 0 T sec) 4.5.2.2 Medium Period Structures (5 0 T sec) 4.5.2.3 Long Period Structures (0 1 T sec and 0 2 T sec) 4.5.3 Standard Deviation Parameter in Fragility Curves 80 80 81 87 100 103 103 106 106 107 120 121 123 123 125 126 139 CHAPTER 5 ASSESSMENT OF DEGRADED MDOF STRUCTURES 5.1 Degradation Effect on MDOF Systems Under Seismic Excitations 5.2 Effect of Higher Modes in MDOF Structures 5.3 Building Models 5.4 Selection of Representative Buildings 5.4.1 Properties of Building Models 5.4.2 DRAIN-2DX Runner / Parser 5.5 Displacement Estimates of MDOF Degraded Structures 5.6 Seismic Fragility of Collapse for MDOF Systems 5.7 Practical Use of Proposed Design Curves 143 143 153 154 157 164 166 168 174 183

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iii CHAPTER 6 SUMMARY AND CONCLUSION 6.1 Summary 6.2 Conclusion 6.3 Recommendations 185 185 186 189 REFERENCES 192 ABOUT THE AUTHOR End Page

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iv LIST OF TABLES Table 3.1 Earthquakes Having Small Magnitude and Small Distance from Fault 64 Table 3.2 Earthquakes Having Small Magnitude and Large Distance from Fault 65 Table 3.3 Earthquakes Having Large Magnitude and Small Distance from Fault 66 Table 3.4 Earthquakes Having Large Magnitude and Large Distance from Fault 67 Table 3.5 Records Details of SMSR 68 Table 3.6 Records Details of SMLR 69 Table 3.7 Records Details of LMSR 71 Table 3.8 Records Details of LMLR 72 Table 3.9 Bilinear Un-scaled T = 1s, 09 0 74 Table 3.10 Bilinear Scaled T = 1s, 09 0 74 Table 3.11 Clough Un-scaled T = 1s, 09 0 75 Table 3.12 Clough Scaled T = 1s, 09 0 75 Table 3.13 Pinching Un-scaled T = 1s, 09 0 75 Table 3.14 Pinching Scaled T = 1s, 09 0 75 Table 3.15 Bilinear Un-scaled T = 1s 76

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v Table 3.16 Bilinear Scaled T = 1s 76 Table 3.17 Clough Un-scaled T = 1s 76 Table 3.18 Clough Scaled T = 1s 76 Table 3.19 Pinching Un-scaled T = 1s 77 Table 3.20 Pinching Scaled T = 1s 77 Table 4.1 Yield Values for SDOF Systems 88 Table 5.1 One-Story Period and Damping Ratios 158 Table 5.2 Two-Story Periods and Damping Ratios 158 Table 5.3 Three-Story Peri ods and Damping Ratios 158 Table 5.4 Five-Story Periods and Damping Ratios 159 Table 5.5 Ten-Story Periods and Damping Ratios 159 Table 5.6 One-Story Model Characteristics 161 Table 5.7 Two-Story Model Characteristics 161 Table 5.8 Three-Story Model Characteristics 161 Table 5.9 Five-Story Model Characteristics 162 Table 5.10 Ten-Story Model Characteristics 162 Table 5.11 One-Story Model Spri ngs Yield Characteristics 163 Table 5.12 Two-Story Model Spri ngs Yield Characteristics 163 Table 5.13 Three-Story Model Sp rings Yield Characteristics 163 Table 5.14 Five-Story Model Sp rings Yield Characteristics 163 Table 5.15 Ten-Story Model Spri ngs Yield Characteristics 164 Table 5.16 Building Models Tota l Height and Corresponding First Modal Period 165

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vi Table 5.17 Base Shear Distributi on at Each Floor Level (NEHRP Load Pattern, k=2) 166

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vii LIST OF FIGURES Figure 2.1 Analysis Techniques for Seismic Design 19 Figure 2.2 Non-Linear Static An alysis Technique – FEMA 356 20 Figure 2.3 Non-Linear Dynamic Analysis Technique 21 Figure 3.1 Bilinear Model 32 Figure 3.2 Modified-Clough Model 32 Figure 3.3 Pinching Model 33 Figure 3.4 Strength Degradat ion for Pinching Model 36 Figure 3.5 Unloading Stiffness De gradation for Pinching Model 37 Figure 3.6 Accelerated Stiffness Degradation for Pinching Model 38 Figure 3.7 Cap Degradation for Pinching Model 39 Figure 3.8 Bilinear Model – No Degradation 42 Figure 3.9 Bilinear Model – Low Degradation 42 Figure 3.10 Bilinear Model – Moderate Degradation 43 Figure 3.11 Bilinear Model – Severe Degradation 43 Figure 3.12 Clough Model – No Degradation 44 Figure 3.13 Clough Model – Low Degradation 44 Figure 3.14 Clough Model – Moderate Degradation 45 Figure 3.15 Clough Model – Severe Degradation 45

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viii Figure 3.16 Pinching Model – No Degradation 46 Figure 3.17 Pinching Model – Low Degradation 46 Figure 3.18 Pinching Model – Moderate Degradation 47 Figure 3.19 Pinching Model – Severe Degradation 47 Figure 3.20 Bilinear Model – Strength Degradation 48 Figure 3.21 Bilinear Model – Stiffness Degradation 48 Figure 3.22 Bilinear Model – Accelerated Degradation 49 Figure 3.23 Bilinear Model – Cap Degradation 49 Figure 3.24 Bilinear Model – Strength and Stiffness Degradation 50 Figure 3.25 Bilinear Model – Accelerated and Cap Degradation 50 Figure 3.26 Bilinear Model – Strength, Stiffness and Accelerated Degradation 51 Figure 3.27 Clough Model – Strength Degradation 51 Figure 3.28 Clough Model – Stiffness Degradation 52 Figure 3.29 Clough Model – Accelerated Degradation 52 Figure 3.30 Clough Model – Cap Degradation 53 Figure 3.31 Clough Model – Strength and Accelerated Degradation 53 Figure 3.32 Clough Model – Stiffness and Cap Degradation 54 Figure 3.33 Clough Model – Stiffness, Accelerated and Cap Degradation 54 Figure 3.34 Pinching Model – Strength Degradation 55 Figure 3.35 Pinching Model – Stiffness Degradation 55 Figure 3.36 Pinching Model – Accelerated Degradation 56 Figure 3.37 Pinching Model – Cap Degradation 56

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ix Figure 3.38 Pinching Model – Stre ngth and Cap Degradation 57 Figure 3.39 Pinching Model – Stiffness and Accelerated Degradation 57 Figure 3.40 Pinching Model – Strength, Accelerated and Cap Degradation 58 Figure 3.41 Collapse – Cap Failure 59 Figure 3.42 Collapse – Degradation Failure 60 Figure 3.43 Bilinear Model (a) Experimental (b) Analytical 62 Figure 3.44 Clough Model (a) Experimental (b) Analytical 62 Figure 3.45 Pinching Model (a) Ex perimental (b) Analytical 62 Figure 3.46 Magnitude-Distance Distribution of the 80 Earthquake Records (Medina 2000) 63 Figure 3.47 Bin I (SMSR) Scaled to 7 0 T sec 78 Figure 3.48 Bin II (SMLR) Scaled to 7 0 T sec 78 Figure 3.49 Bin III (LMSR) Scaled to 7 0 T sec 79 Figure 3.50 Bin IV (LMLR) Scaled to 7 0 T sec 79 Figure 4.1 SDOF Time History for Roof Displ., 3 Floors, Bilinear and No Degradation 83 Figure 4.2 SDOF Time History for Roof Displ., 3 Floors, Bilinear, Severe Degradation 83 Figure 4.3 SDOF Force-Displacement, 3 Floors, Bilinear and No Degradation 84 Figure 4.4 SDOF Force-Displacement, 3 Floors, Bilinear and Severe Degradation 84 Figure 4.5 SDOF Time History for Roof Displ., 10 Floors, Bilinear and No Degradation 85

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x Figure 4.6 SDOF Time History for Roof Displ., 10 Floors, Bilinear and Severe Degradation 86 Figure 4.7 SDOF Force-Displacement, 10 Floors, Bilinear and No Degradation 86 Figure 4.8 SDOF Force-Displacement, 10 Floors, Bilinear and Severe Degradation 87 Figure 4.9 Effect of St rength Reduction Factor ( R ) on Yield Force (yF) 90 Figure 4.10 Bilinear Model, Median and R=4 92 Figure 4.11 Clough Model, Median and R=4 92 Figure 4.12 Pinching Model, Median and R=4 93 Figure 4.13 Bilinear Model, 84th % and R=4 93 Figure 4.14 Clough Model, 84th % and R=4 94 Figure 4.15 Pinching Model, 84th % and R=4 94 Figure 4.16 Bilinear Model, Median and R=6 95 Figure 4.17 Clough Model, Median and R=6 95 Figure 4.18 Pinching Model, Median and R=6 96 Figure 4.19 Bilinear Model, 84th % and R=6 96 Figure 4.20 Clough Model, 84th % and R=6 97 Figure 4.21 Pinching Model, 84th % and R=6 97 Figure 4.22 Bilinear Model, Median and R=8 98 Figure 4.23 Clough Model, Median and R=8 98 Figure 4.24 Pinching Model, Median and R=8 99 Figure 4.25 Bilinear Model, 84th % and R=8 99 Figure 4.26 Clough Model, 84th % and R=8 100

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xi Figure 4.27 Pinching Model, 84th % and R=8 100 Figure 4.28 Relationship of Strength Reduction Factor ( R ) and Ductility ( 103 Figure 4.29 Bilinear Model Ductility, 4 R and 2 0 T sec 108 Figure 4.30 Bilinear Model Ductility, 100 and 2 0 T sec 109 Figure 4.31 Clough Model Ductility, 4 R and 2 0 T sec 109 Figure 4.32 Clough Model Ductility, 100 and 2 0 T sec 110 Figure 4.33 Pinching Model Ductility, 4 R and 2 0 T sec 110 Figure 4.34 Pinching Model Ductility, 100 and 2 0 T sec 111 Figure 4.35 Bilinear Model Ductility, 4 R and 5 0 T sec 111 Figure 4.36 Bilinear Model Ductility, 100 and 5 0 T sec 112 Figure 4.37 Clough Model Ductility, 4 R and 5 0 T sec 112 Figure 4.38 Clough Model Ductility, 100 and 5 0 T sec 113 Figure 4.39 Pinching Model Ductility, 4 R and 5 0 T sec 113 Figure 4.40 Pinching Model Ductility, 100 and 5 0 T sec 114 Figure 4.41 Bilinear Model Ductility, 4 R and 0 1 T sec 114 Figure 4.42 Bilinear Model Ductility, 100 and 0 1 T sec 115 Figure 4.43 Clough Model Ductility, 4 R and 0 1 T sec 115 Figure 4.44 Clough Model Ductility, 100 and 0 1 T sec 116 Figure 4.45 Pinching Model Ductility, 4 R and 0 1 T sec 116 Figure 4.46 Pinching Model Ductility, 100 and 0 1 T sec 117 Figure 4.47 Bilinear Model Ductility, 4 R and 0 2 T sec 117

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xii Figure 4.48 Bilinear Model Ductility, 100 and 0 2 T sec 118 Figure 4.49 Clough Model Ductility, 4 R and 0 2 T sec 118 Figure 4.50 Clough Model Ductility, 100 and 0 2 T sec 119 Figure 4.51 Pinching Model Ductility, 4 R and 0 2 T sec 119 Figure 4.52 Pinching Model Ductility, 100 and 0 2 T sec 120 Figure 4.53 Relationship Between IDA & Fragility Curves 121 Figure 4.54 Example of Use of Eita in Fragility Curve 123 Figure 4.55 Bilinear Model Fragility, 4 R and 2 0 T sec 127 Figure 4.56 Bilinear Model Fragility, 100 and 2 0 T sec 128 Figure 4.57 Clough Model Fragility, 4 R and 2 0 T sec 128 Figure 4.58 Clough Model Fragility, 100 and 2 0 T sec 129 Figure 4.59 Pinching Model Fragility, 4 R and 2 0 T sec 129 Figure 4.60 Pinching Model Fragility, 100 and 2 0 T sec 130 Figure 4.61 Bilinear Model Fragility, 4 R and 5 0 T sec 130 Figure 4.62 Bilinear Model Fragility, 100 and 5 0 T sec 131 Figure 4.63 Clough Model Fragility, 4 R and 5 0 T sec 131 Figure 4.64 Clough Model Fragility, 100 and 5 0 T sec 132 Figure 4.65 Pinching Model Fragility, 4 R and 5 0 T sec 132 Figure 4.66 Pinching Model Fragility, 100 and 5 0 T sec 133 Figure 4.67 Bilinear Model Fragility, 4 R and 0 1 T sec 133 Figure 4.68 Bilinear Model Fragility, 100 and 0 1 T sec 134 Figure 4.69 Clough Model Fragility, 4 R and 0 1 T sec 134

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xiii Figure 4.70 Clough Model Fragility, 100 and 0 1 T sec 135 Figure 4.71 Pinching Model Fragility, 4 R and 0 1 T sec 135 Figure 4.72 Pinching Model Fragility, 100 and 0 1 T sec 136 Figure 4.73 Bilinear Model Fragility, 4 R and 0 2 T sec 136 Figure 4.74 Bilinear Model Fragility, 100 and 0 2 T sec 137 Figure 4.75 Clough Model Fragility, 4 R and 0 2 T sec 137 Figure 4.76 Clough Model Fragility, 100 and 0 2 T sec 138 Figure 4.77 Pinching Model Fragility, 4 R and 0 2 T sec 138 Figure 4.78 Pinching Model Fragility, 100 and 0 2 T sec 139 Figure 4.79 Example of Use of Sigma in Fragility Curve 139 Figure 4.80 Effect of Parameter ( ) in Fragility Curve 141 Figure 5.1 MDOF Time History for Roof Displ., 3 Floors, Bilinear and No Degradation 144 Figure 5.2 MDOF Force-Displacement, 3 Floors, Bilinear and No Degradation 145 Figure 5.3 MDOF Time History for Roof Displ., 3 Floors, Bilinear and Low Degradation 145 Figure 5.4 MDOF Force-Displacement, 3 Floors, Bilinear and Low Degradation 146 Figure 5.5 MDOF Time History for Roof Displ., 3 Floors, Bilinear and Severe Degradation 146 Figure 5.6 MDOF Force-Displacement, 3 Floors, Bilinear and Severe Degradation 147 Figure 5.7 MDOF Time History fo r Roof Displ., 3 Floors, Clough and Severe Degradation 148

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xiv Figure 5.8 MDOF Force-Displacement, 3 Floors, Clough and Severe Degradation 148 Figure 5.9 MDOF Time History for Roof Displ., 3 Floors, Pinching and Severe Degradation 149 Figure 5.10 MDOF Force-Displacem ent, 3 Floors, Pinching and Severe Degradation 149 Figure 5.11 MDOF Time History for Roof Displ., 10 Floors, Bilinear and Low Degradation 151 Figure 5.12 MDOF Force-Displacement, 10 Floors, Bilinear and Low Degradation 151 Figure 5.13 MDOF Time History for Roof Displ., 10 Floors, Bilinear and Severe Degradation 152 Figure 5.14 MDOF Force-Displacement, 10 Floors, Bilinear and Severe Degradation 152 Figure 5.15 Beam Hinge Model 155 Figure 5.16 Column Hinge Model 155 Figure 5.17 Weak Story Model 156 Figure 5.18 Node Numbering 160 Figure 5.19 Deformed Shape Under NEHRP Load Pattern 165 Figure 5.20 Drain Runner User Interface Window 167 Figure 5.21 MDOF – Bilinear Model – 8 R 171 Figure 5.22 MDOF – Bilinear Model – 4 R 172 Figure 5.23 MDOF – Clough Model – 4 R 172 Figure 5.24 MDOF – Bilinear Model – 6 R 173 Figure 5.25 MDOF – Clough Model – 6 R 173 Figure 5.26 MDOF – Bilinear Model – 8 R 174

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xv Figure 5.27 One Floor, Bilinear Model, = 0.2231 175 Figure 5.28 Two Floors, Bilinear Model, = 0.1226 176 Figure 5.29 Three Floors, Bilinear Model, = 0.1101 176 Figure 5.30 Five Floors, Bilinear Model, = 0.0881 177 Figure 5.31 Ten Floors, Bilinear Model, = 0.0622 177 Figure 5.32 One Floor, Clough Model, = 0.2231 178 Figure 5.33 Two Floors, Clough Model, = 0.1226 178 Figure 5.34 Three Floors, Clough Model, = 0.1101 179 Figure 5.35 Five Floors, Clough Model, = 0.0881 179 Figure 5.36 Ten Floors, Clough Model, = 0.0622 180 Figure 5.37 One Floor, Pinching Model, = 0.2231 180 Figure 5.38 Two Floors, Pinching Model, = 0.1226 181 Figure 5.39 Three Floors, Pinching Model, = 0.1101 181

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xvi PERFORMANCE BASED DESIGN OF DEGRADING STRUCTURES Mouchir Chenouda ABSTRACT Seismic code provisions are now adop ting performance-based methodologies, where structures are designed to satisfy multip le performance objectives. Most codes rely on approximate methods to predict the desired seismic demand parameters. Most of these methods are based on simple SDOF models, and do not take into account neither MDOF nor degradation effects, which are major f actors influencing stru ctural behavior under earthquake excitations. More importantly, most of these models can not predict collapse explicitly under severe seismic loads. This research presents a newly developed model that incorporates degradation effects into se ismic analysis of structures. A new energybased approach is used to define several types of degradation effects. The research presents also an evaluation of the collap se potential of degrading SDOF and MDOF structures. Collapse under severe seismic ex citations, which is typically due to the formation of structures mechanisms amplifie d by P-Delta effects, was modeled in this work through the degrading hyste retic structural behavior al ong with P-Delta effects due to gravity loads. The model was used to conduc t extensive sta tistical dynamic analysis of different structural systems subjected to a large set of recent ear thquake records. To perform this task, finite element models of a series of generic SDOF and MDOF

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xvii structures were developed. The degrading hysteretic structural behavior along with PDelta effects due to gravity loads proved to successfully replicate explicit collapse. For each structure, collapse was investigated a nd inelastic displacement ratios curves were developed in case collapse doesn’t occur. Furthermore, seismic fragility curves for a collapse criterion were also developed. In gene ral, seismic fragility of a system describes the probability of the system to reach or exceed different degrees of damage. Earlier work focused on developing seismic fragility curv es of systems for several values of a calibrated damage index. This research work focuses on developing seismic fragility curves for a collapse criterion, in an explicit form. The newl y developed fragility curves represent a major advancement over damage inde x-based fragility curves in assessing the collapse potential of structures subject to severe seismic excitations. The research findings provide necessary information for th e design evaluation phase of a performancebased earthquake design process.

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xviii PREFACE This dissertation owes an enormous debt to Dr. Ashraf Ayoub. I would like to express my deepest gratitude to him for always finding time to answer my endless questions, for his long-range insight in guiding my research, for his kindness and unwavering support, and for giving me the oppo rtunity to follow my Ph.D. research without any interruption. He is indeed one of the finest peopl e I have interacted with and I consider myself truly lucky to have had the opportunity to work with him. Words cannot express my sincere gratitude and appreciation to my co-advisor Dr. Rajan Sen, who is most responsible for helping me complete my doctoral program. Without his encouragement and constant gu idance, I could not have finished this dissertation. His kindness to me from my first day at USF will never be forgotten. Besides my advisors, my dissertation comm ittee members deserve recognition for their valuable time and support: Dr. William Carpenter, Dr. Autar Kaw, and Dr. Kandethody Ramachandran. Their tremendous personal and in tellectual support is much appreciated. I would also like to thank Dr Geoffrey Okogbaa for chairi ng my Ph.D. exam committee. I am greatly indebted to Dr. Amir Makar yus of Towson University in sharing with me his knowledge in statistics and probability which was important to the advance of my work. I am also grateful to Mr. Karim S ouccar, the computer guru in the Electrical Engineering Department at the University of South Florida. Mr. Souccar deserves a

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xix special mention. After hearing my complaints and frustration with the program DRAIN2DX, he helped me develop my version of a new software program that enhances the usage of DRAIN-2DX. Financial support of the Civ il and Environmental Engineering Department at the University of South Florida and the Na tional Science Foundation (Award # 0448590) is gratefully acknowledged. Special thanks also go to Dr. Sunil Saigal, chairman of the Civil and Environmental Engineering Department, Dr. Ram Pendyala, the graduate coordinator and Dr. Manjriker Gunaratne, the interim chairman for their kindness and advice throughout my stay at USF. I would like also to thank many professors in the past: Dr. Sabri Samaan, Dr. Samir Seif, Dr. Sherif Ayoub, Dr. Boulos Salama, Mr. Magdi Bebawi, and Dr. Sameh Mehanny for getting me interested in struct ural engineering design and coming to the United States. Several friends and colleagues crossed my pa th at the University of South Florida. I owe them all a great deal. Particularly, I would like to thank my friends: Chandra Khoe, Oscar Gomez and George Bolos. From the Ci vil and Environmental Engineering staff, I would like to thank Jennifer Collum, Jackie Alderman and Ingrid Hall. Special thanks are due to Van Gladfelter, my current employer and president of the Center for Innovative St ructures for his support a nd encouragement throughout the course of this work. I would like also to tha nk my friends and colleagues in the Center for being so nice and helpful to me since my first day of work: Doug Meiser, Dimitrios

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xx “Jim” Melandinos, Dennis Zabala, Andr ew Johnson, Rodney Almonte and Randy Ciarlone. There are people towards whom my gratitude can neither be expressed nor their love can be described: my w hole family members in Egypt and especially my mom, dad and my great brother and friend Michael. They endured my absence and helped me endure it too. Last but not least, my lovely dear wife Mirey. This work would not have been possible without her selflessness and strength. She gave me hope when it was rare, advice when it was sought, support when it was n eeded. No earthly words can describe her sacrifices, love and support.

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1 CHAPTER 1 INTRODUCTION 1.1 Introduction The seismic design provisions of buildi ng codes in the Unite d States are moving towards adopting the general concept of performance based design. A Performance Based Earthquake Engineering (PBEE) design pro cess is a demand/capacity procedure that incorporates multiple performance objectives. The procedure consists of four main steps. In the first step, performance objectives of a structural system at different hazard levels are defined. In the second step, a conceptual design of the structure is performed in order to meet the objectives defined in step 1. A design evaluation phase is then needed in order to evaluate the concep tual design developed in step 2. Finally a socio-economic study is needed to finalize the process. In the design evaluation phase, seismic demands of the structure need to be evaluated as accurately as possi ble at different hazard levels for demand/capacity comparison. Most codes rely on approximate methods that predict the desired demand parameters; the most comm on two are the method of coefficients and the capacity spectrum method. Most of these methods are based on simple SDOF models, and do not take into account neither MDOF nor degradation effects, which are major factors influencing structural beha vior under earthquake excitations. More importantly, most of these models can not predict collapse explicitly under severe seismic

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2 loads. This research proposes a newly deve loped model that inco rporates degradation effects into seismic analysis of structures. Th is degrading structural behavior is essential for accurate investigation of st ructural behavior and for colla pse assessment of structures subject to severe seismic excitations. A new energy-based approach is used to define several types of degradation effects for diffe rent material models. Collapse under severe seismic excitations, which is typically due to the formation of structures mechanisms amplified by P-Delta effects, was modeled in this work through the degrading hysteretic structural behavior along with P-Delta effects due to gravity loads. The new degrading model was used to conduct extensive statistica l dynamic analysis of different structural systems subjected to a large set of recent earthquake record s with the goal of predicting their maximum inelastic deformations and inve stigating their potential for collapse. To perform this task, finite element models of a series of generic SDOF and MDOF structures were developed. The structures c overed a wide range of periods, yield values, and levels of degradation. The degrading hyste retic structural behavi or of the structural elements along with P-Delta effects due to gravity loads proved to successfully replicate explicit collapse. Inelastic displacement ratios and seismic fragility curves for a collapse criterion were developed. The research findi ngs proved to provide necessary information for the design evaluation phase of a perf ormance-based earthquake design process. 1.2 Research Objectives The main goal of the research study is to investigate the beha vior of degrading structural systems and their potential for collapse under seis mic excitations. The study is essential for the design evaluation phase of a performance-based earthquake design

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3 process, particularly for collapse preventi on limit states. To accomplish the research objectives, it is necessary to develop a ne w numerical procedure for predicting maximum inelastic displacements, and for estimating co llapse of degrading structures under seismic excitations. Several constitutive material models including both static and dynamic degradation effects were deve loped. The models include a strength softening branch to model static degradation under monotonic loads. In addition, the models incorporate four types of cyclic degradation: strength degr adation, unloading stiffness degradation, accelerated stiffness degradation, and cap de gradation. The models were added to the element library of the non-linear frame anal ysis program DRAIN-2DX. The degradation parameters were calibrated versus experimental ly tested specimens of concrete, steel and timber structures. The work consisted of c onducting statistical anal ytical studies on a large ensemble of degrading structural syst ems, and using a large suite of earthquake records representing recent events. Both SDOF and MDOF systems were investigated. In addition, several other parameters were inves tigated such as yield forces, material model types, and levels of degradation. The resu lts were used to predict maximum inelastic displacements of degrading structures, and to investigate the collapse probability of structures under earthquake excitations thr ough seismic fragility analysis. The findings proved essential in providi ng the necessary background for evaluation and modification of current seismic design codes to reflect the effect of degrada tion and potential for collapse.

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4 1.3 Thesis Organization The report is organized as follows: Chapter 1 presents an introduction and brief su mmary of the research objectives and scope of work. Chapter 2 presents a discussion of previous research work on seismic analysis procedures, methods for displacement estimate s, and damage evaluation of structures subject to earthquake ex citations. A review of current seismic design guidelines is also introduced. Chapter 3 presents a detailed description of the new degrading material models. Formulation of the energy criterion used to account for the hysteretic degrading behavior is presented. Calibration of the degradation parameters versus experimentally tested specimens is conducted. Evaluation of the eff ect of degradation on the inelastic behavior of SDOF systems is performed. The chapter also presents the database of earthquake records used to conduct the analytical studi es. A discussion on th e scaling effect of records is also presented. Chapter 4 presents the statistical analytical studies conduc ted on the degrading SDOF systems. The dynamic properties of the S DOF systems and the different variables evaluated in the statis tical studies are defined. Inelasti c displacement ratio curves are developed for degrading systems and compared to non-degraded ones. Incremental dynamic analysis is performed and ductility capacities are estimated. Seismic fragility curves for a collapse criterion are also devel oped, and conclusions regarding the collapse potential of the systems are drawn.

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5 Chapter 5 presents the statistical analytical studies conduc ted on the degrading MDOF systems. A detailed description of the MDOF systems and their dynamic characteristics is presented. Description of the inelastic model a ssumptions is also presented. The effect of higher modes and P-Delta on the degrading be havior of MDOF systems is presented through plots of MDOF displa cement ratios. Seismic fragility analysis for a collapse criterion is also conducted fo r buildings with different number of stories. Conclusions regarding the degrading behavi or of MDOF systems and their collapse potential are drawn. Finally, Chapter 6 presents a summary of the work conducted, and a discussion of the main conclusions drawn. The chapter also offers recommendations for future research work in the field.

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6 CHAPTER 2 LITERATURE REVIEW 2.1 Introduction This chapter summarizes previous research work in seismic analysis and design of building structures. The extens ive research in this fiel d, which started through the pioneering work of Biot (1933), Housner ( 1941), and Veletsos and Newmark (1960) has led to the development of several seismic de sign codes for buildings that are currently used throughout the world. A brief description of the most widely used codes of practice is presented first. 2.2 Strength-Based Building Design Codes The seismic design section in most of th e current building codes necessitates that structures has to be designed using the equivale nt static load concept. These static forces assigned at each floor level are function of structure’s properties and seismic zone. The outcome of the analysis is usually shear fo rces and overturning moments used to design against seismic loads. Recently, modern code s allows for seismic analysis using linear dynamic procedures such as response spectru m analysis and respons e history analysis. Linear dynamic analysis is required for so me special cases, for instance for buildings with long periods or for i rregular buildings. Chopra (2005) mentioned that current

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7 California Building Code requires dynamic anal ysis of hospital structures. The seismic static design provisions in 3 building c odes are presented here in after (Chopra 2001 & 2005). 2.2.1 International Building Code (United States) The base shear (bV) in the 2003 edition of the Inte rnational Building Code (IBC) is specified as: W R IC Vb (2.1) Where: I Importance factor, CPeriod-dependant coefficient based on structure location and site class, R Strength reduction factor or Elastic seismic coefficient when0 1 R, and WTotal dead load of the structure. The lateral forces at each floor are distributed over the structure height using the base shear. The equation used for the jth floor is the following: N i k i i k j j b jh w h w V F1 (2.2) Where: iwWeight of the ith floor, ih Height of the ith floor above the base, kCoefficient depending on the vibration period, and NTotal number of floors.

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8 The design forces of floors and elements are calculated by subjecting the structure to the lateral forces determined from the preceding equations. 2.2.2 National Building Code of Canada The 1995 edition of the National Building C ode of Canada specifies the base shear (bV) as: W R SIFU Vb (2.3) Where: Zonal velocity ratio vary ing between 0 and 0.4, SSeismic response factor depending on f undamental natural vibration period and seismic zone, I Seismic importance factor, FFoundation factor depending on the so il category defined in the code, UOverstrength factor, R Force modification factor reflecting de sign and construction experience, and WTotal dead load of the structure. Similar to the UBC (2003), the lateral forces at each floor are distributed over the structure height using the base shear. The design forces of story shears are calcu lated by subjecting the structure to the lateral forces. The overturning moments are multiplied by reduction factor at the structure’s base and each floor.

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9 2.2.3 Mexico Federal District Code The base shear (bV) in the 1987 edition of the Mexi co Federal District Code is calculated as follows: W Q C Ve b' (2.4) Where: eCElastic seismic coefficient depending on fundamental period and seismic zone, 'QSeismic behavior factor depe nding on several factors includ ing the structural system and structural materials, and WTotal dead load of the structure. The lateral forces at each floor are distributed over the structure height using the base shear. The basic equation used for the jth floor is similar to the previous codes mentioned: N i k i i k j j b jh w h w V F1 (2.5) Where: iwWeight of the ith floor, ih Height of the ith floor above the base, kCoefficient depending on the vibration period, and NTotal number of floors. This equation is modified and separated in to two parts depending on the value of the period at the end of the constant pseudoacceleration region of the design spectrum. The design forces of floors and elements are calculated by subjecting the structure to the lateral forces previously determ ined. The overturning moments are multiplied by

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10 reduction factor at structure’s base and each floor to obtain the design values. However, the reduced moments at any floor should not be less than the product of the story shear at that elevation and the distance to the center of gravity of the building portion above the floor elevation considered. Chopra (2005) concluded from the compar ison of the three codes that the base shear is overestimated. The reduction factor used is intended to account for several factors such as the difference between de sign strength and yiel d strength, and the performance of different structural system s and materials during precedent earthquakes. 2.3 Performance-Based Building Design Codes The seismic design provisions of buildi ng codes in the Unite d States are moving towards adopting the general concept of performance based design. A Performance Based Earthquake Engineering (PBEE) design pro cess is a demand/capacity procedure that incorporates multiple performance objectives. The procedure consists of four main steps. In the first step, performance objectives of a structural system at different hazard levels are defined: Immediate occupancy, Life safety, and Collapse prevention. In the second step, a conceptual design of the structure is performed in order to meet the objectives defined in step 1. The th ird step is a design evaluation phase needed in order to evaluate the conceptual design prev iously developed in step 2. Finally a socioeconomic study is required in the fourth st ep to finalize the process. In the design

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11 evaluation phase, seismic demands of the struct ure need to be evalua ted as accurately as possible at different hazard leve ls for demand/capacity comparison. Moehle (1992) and Priestley (1996) have shown that present criteria for the seismic design of new structures and for the seismic evaluation of existing structures can be significantly improved if they are base d on the explicit consideration of lateral deformations demands as the key design parameter rather than based on lateral forces. However, implementation of displacement-base d seismic design criteria into structural engineering practice requires simplified anal ysis procedures to estimate displacement demands imposed on structures by ea rthquake ground motions (Miranda, 2001). Miranda (1999) found that recently ther e has been a growing interest in displacement-based design procedures in wh ich lateral displacement demands are used rather than lateral force demands (Moehl e 1992). During preliminary design stages of new buildings, or for a quick seismic evaluati on of existing buildings, there is a need for estimating the maximum lateral displacemen ts that can take place in the building subjected to the design earthquake ground moti on. The estimation of the maximum roof displacement and maximum interstory drift ra tio (IDR) (defined as the ratio of the maximum interstory drift to th e interstory height) is helpfu l in recognizing the required capacities, particularly the required lateral stiffness, in order to reach the desired performance level of the building. Whittaker et al., (1998) pointed out that even though the basic objective of performance-based earthquake engineering is to construct structures that respond in more reliable behavior during earthquake excitati on, many engineers rela te performance-based earthquake engineering with overall enhanced performance (i.e., damage control). The

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12 study revealed the fact that damage of stru ctural elements in a building frame can be limited if lateral displacements are controlled to predetermined values for the specific intensity of earthquake excitation. The conclu sion drawn out from this fact was that methods to calculate dependable estimates of lateral displacements are needed since the damage control is the essence of perf ormance-based earthquake engineering. Several researchers developed procedur es for estimating maximum inelastic displacements. In most of these studies, the material models used followed simple hysteretic non-degr ading rules. Few of these studies considered degradation, but still followed very simple rules. In addition, de gradation effects were not based on physical reasoning. Furthermore, none of these studies considered collapse prediction of the structures. A brief summary of earlier st udies in this field is given below. The first research work in this field is the one by Veletsos and Newmark (1960) who analyzed SDOF systems using 3 earthqu ake records. The models were assumed elasto-plastic. They concluded that in th e regions of low frequency, the maximum inelastic deformation is equal to the maximu m elastic deformation, which is known as the equal displacement rule. They also concluded that this rule doesn’t hold true for regions of high frequency, where the inelastic displ acement considerably exceeds the elastic one. Shimazaki and Sozen (1984) conducted a similar numerical study on a SDOF system using five different hysteretic models. The models used were either bilinear or of Clough type (1966), and only El Ce ntro earthquake record was used for the analysis. No degradation was considered in their study. In their work, they developed a relation between maximum inelastic displaceme nts and correspondin g maximum elastic displacements for different values of streng th and period ratios. The conclusion of their

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13 work is that for periods higher than the char acteristic period, defi ned as the transition period between the constant a cceleration and constant veloci ty regions of the response spectra, the maximum inelastic displacement equals approximately the maximum elastic displacement regardless of the hysteresis type used, confirming the equal displacement rule. For periods less than the characteristic period, the maximum inelastic displacement exceeds that of the elastic di splacement and the amount vary depending on the type of hysteretic model and on the latera l strength of the structure relative to the elastic strength. Their conclusion was confirmed la ter by Qi and Moehle (1991). Miranda (1991, 1993a and 1993b) analy zed over 30,000 SDOF systems using a large ensemble of 124 earthquake ground motions recorded on different soil types. He developed ratios of maximum inelastic to elastic displacements for 3 types of soil conditions. He also studied the limiting peri od value where the equal displacement rule applies. The material model used in his study is also elasto-plastic Lately, Miranda and Ruiz-Garcia (2002) evaluated six different methods for predicting maximum inelastic displacements. Four methods are based on equi valent linearization techniques, while two are based on multiplying maximum elastic disp lacements by modification factors. In all methods, cyclic degradation effects were not considered. Krawinkler and his co-workers (1991, 1993 and 1997) conducted similar studies to the one by Miranda. The material models used were either bilinear, Clough or of pinching type. Degradation effects were included, but in the form of strength degradation only, or stiffness degradation only. Gupta and Kunnath (1998) conducted a sim ilar study on SDOF systems subjected to 15 ground motions. They included degradation effects using a 3 parameters model.

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14 More recently, Whittaker et al. (1998 ) conducted a numerical study on SDOF systems using 20 earthquake records. They us ed the Bouc-Wen mode l (1976) in their analysis and neglected degradation effects. They developed mean and mean + 1 sigma ratio plots of maximum inelas tic to elastic displacements for different strength values. Miranda (2000) extended his earlier work, and developed displacement ratio plots for different earthquake magnitudes, epicenter distance, and soil c onditions. His study was also on non-degrading SDOF systems. Most recently, Miranda (2001) showed that maximum inelastic displacements could be related to maximum elastic displacements either through inelastic displa cement ratios or through strength reduction factors. He also showed that the second method is a first orde r approximation of the first, and that both methods yield similar results in the absence of variability. Several studies were also conducted on MDOF systems [e.g. Ayoub and Filippou (1999a, 1999b and 2000), Saiidi and Sozen (1981), Freeman (1978), Fajfar and Fischinger (1988), Qi and Moehle (1991), and Krawinkler (1991 and 1997)]. Most researchers concluded that the demand of MDOF systems could be estimated by appropriate modification of th e response of the first mode SDOF of the system. Two methods were established in that sense, the capacity spectrum method developed originally by Freeman (1978) and adopt ed by the Applied Technology Council ATC-40 (1996), and the method of coefficients deve loped by Krawinkler (1991) and used by the Federal Emergency Management Agency FE MA-356 (2000). Both methods are similar in the sense that they are ba sed on a nonlinear static push-ove r of the structure. They are different, however, in the way they es timate the maximum “target” inelastic displacement.

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15 2.3.1 Capacity Spectrum Method The capacity spectrum method is adopted by ATC-40 and is based primarily on superimposing capacity diagram plots on demand diagram plots, and estimating the target displacement with an iterative procedure using elastic dynamic analyses. The procedure consists of the following: 1. Conducting a push-over analysis to construc t a relationship between base shear and roof displacement 2. Converting the push-over curve into a cap acity diagram. The capacity diagram represents a relationship between the firs t mode spectral displacement and spectral acceleration. The first mode spectral displacement could be easily calculated as a function of the roof displacement evaluated in 1 using modal analysis, and the first mode spectral acceleration is a function of the base shear also evaluated in 1. 3. Establishing the elastic response spectrum of the earthquake reco rd of interest, and converting it from the standard period-sp ectral acceleration form into a spectral displacement-spectral acceler ation form. The resulting diagram is referred to as a demand diagram. 4. Superimposing the demand diagram evaluate d in 3 on the capacity diagram evaluated in 2. An iterative procedure using dynamic analyses of equivalent linear systems is performed to determine the displacement demand point. The displacement demand point represents the inelastic spectral displ acement of the system subject to the record of interest.

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16 5. The demand point is converted back into a target roof displacement value. The target displacement represents the maximum roof displacement due to the earthquake record of interest. The push-over diagram is th en repeated up to the specified target displacement in order to estimate all seismic demand parameters. Several modified versions were introdu ced to improve the originally developed method. Paret et al. (1996) and Bracci et al. (1997) modifi ed the proposed procedure to account for higher mode effects. WJE ( 1996), Reinhorn (1997), Fajfar (1999), and Chopra and Goel (1999) further improved the procedure by using inel astic design spectra as defined by Newmark and Hall (1982) rather than elastic spectra. In these later versions, inelastic dynamic analyses are performed but using simple bilinear nondegrading material models. 2.3.2 Method of Coefficients In the method of coefficients adopted by FEMA-356, the target displacement at a specific hazard level is calculated by mu ltiplying the maximum corresponding elastic displacement by a series of coefficients that account for inelastic behavior, higher mode effects, and dynamic second order effects. A static pushover analysis is then conducted for the structure up to the specified maxi mum displacement in order to estimate the different seismic demand parameters. Specifically, the target displacement (t ) is calculated as follow: 2 2 3 2 1 04 e a tT S c c c c (2.6)

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17 Where: 0cModification factor that account s for MDOF effects, and is equal to the first mode participation factor at the roof, 1cModification factor that accounts for the e xpected ratio of maximum inelastic to maximum elastic displacements. It is taken as 1.5 for periods less than 0.1 sec. and 1 for periods larger the than charac teristic period defined as the period associated with the transition from the constant acceleration segment to the constant velocity segment of the spectrum, 2cModification factor that acc ounts for degradation effects, and is equal to 1.2 for periods larger than the characteristic periods, 3cModification factor that acc ounts for dynamic second-order effects, and is equal to 1 for systems with hardening ratios greater than 5%, aSThe design spectral acceleration, and eTEffective fundamental period of the structure. The period and damping-dependent coefficients 0c, 1c, 2c, and 3c were evaluated using statistical studies on representative inelas tic structural system s, by comparing their behavior to the corresponding S DOF first mode elastic structur e. The selected coefficient values were based on the average values obtained from an ensemble of earthquake records whose average acceleration response spectrum matches the ATC response spectrum for soil type 1. The factor 2cwas derived by considering models th at degrade only in strength or in stiffness. It also does not accoun t for strength softening behavior.

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18 2.3.3 Drawback of Current Methods The main drawback of both methods, the capacity spectrum method and the coefficients method, is their inability to accurately estimate maximum inelastic displacements, and to predict failure of i ndividual components of the structure, which might affect the overall response and possibly failure of the entir e structure. The reason is that both models use simple numerical proc edures in estimating the maximum expected displacement during a specific earthquake excitation. In the capacity spectrum method, only static analysis is perfor med for non-degrading systems. It is known that any material degrades in strength after reaching its full capacity under static loadings, also known as strength softening, which s ubsequently causes failure. Also any material degrades in strength and stiffness under repeated cyclic lo adings, which might cause complete loss of strength and possibly dynamic material fa ilure. Since the cap acity spectrum method considers only non-degrading systems and ne glects dynamic effects, it fails to predict failure accurately. The coefficients method al so is mainly based on static analysis, but dynamic effects are introduced by a series of approximate factors determined from extensive statistical parameter studies of simple hysteresis material models. These models also do not account for strength softening, usually the main cause of failure, and consider only strength degradation under repeated dyna mic loading. The method therefore also does not predict failure of a component accurately.

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19 2.4 Seismic Analysis Techniques Both strength-based codes and performa nce-based design codes require specific analysis techniques in order to ev aluate the desired seismic demands. Analysis Techniques for Seismic Design Linear Static Linear Non-Linear Static NonLinear Linear Dynamic Non-Linear Dynamic Figure 2.1 Analysis Techniques for Seismic Design The different seismic analysis techniques ar e shown in figure 2.1, which is a simple schematic diagram showing the different methods of analysis for seismic design. The linear static method is commonly us ed in design codes. It assumes the structure is linear elastic. The method therefor e doesn’t take into account ductility effects, and can not predict collapse accurately. Furt hermore dynamic effects are accounted for.

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20 The linear dynamic method is ba sed on either time history or modal analysis. Like the linear static method, it doesn’t account for duc tility and can not ther efore predict failure. Presently, the guidelines for buildings eval uation allow the use of non-linear static and dynamic methods. The non-linear static pr ocedure is based primarily on pushover analysis using monotonic loads up to the ta rget displacement point. The procedure is shown in figure 2.2, and is adopted in both the capacity spectrum and coefficient methods, as described earlier. Results are ac curate, however, only if higher modes effects are negligible. Roof Displacement t(FEMA 356) Figure 2.2 Non-Linear Static Analysis Technique – FEMA 356 An attempt to introduce direct dynamic effects in the non-linear analysis and design of building structures was proposed by Cornell and his co-workers (2002). The

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21 process is named Incremental Dynamic Analys is (IDA) or dynamic pushover analysis. In this process, a dynamic load-deformation plot is determined by subjecting the structure to a specific earthquake history, and then scaling the earthquak e record up several times and repeating the analysis. The process has been used by several res earchers (e.g. Mehanny and Deierlein, 2001, Yun and Foutch, 2000, a nd Lee and Foutch, 2001), and is described in figure 2.3 which shows the relationship betw een the selected force parameter (Spectral Acceleration), and deformation paramete r (maximum inter-story drift IDR). Spectral Acceleration SaIDRmaxMultiple Time History Analysis Softening Behavior Hardening Behavior Elastic Response Figure 2.3 Non-Linear Dynamic Analysis Technique Although dynamic effects were included in the incremental dynamic analysis method, failure prediction was not possible since the material models used by most researchers followed very simple rules.

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22 2.5 Damage Evaluation of Building Structures Assessment of the state of a building struct ure after being subject to an earthquake excitation is an important tool that research ers use to evaluate the accuracy of a design process. Most of the researchers applied th e concept of damage index as a mean of assessment of the damage of structural systems subject to seismic shaking. The probability of the system to reach or exceed different degrees of damage, including possible collapse is a concept known as seismi c fragility. Earlier work [e.g. Singhal A., and Kiremidjian A. S. (1998), Shinozuka M. et al. (2000), Sasani M ., and Kiureghian A. D. (2001)] focused on developing seismic fragility curves of systems fo r several values of a calibrated damage index. A damage index is a factor that represents the degree of damage of the structure, and typically ranges fr om 0 to 1, with the value of 1 representing complete collapse. Collapse was therefore e xpressed implicitly as the state of the structure when its damage index approaches a value of 1. A brief summary of current damage indices is presented next. 2.5.1 Damage Indices In a study about seismic damage indices for concrete structures, Williams and Sexsmith (1995) tried to summarize most of the known methods for calculating damage indices. They noted that i ndices may be calculated from the results of a non-linear dynamic analysis, from measured response of a structure during an earthquake, or from a comparison of the physical properties of the structure before and after the earthquake. The result of their research is summarized herein below.

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23 2.5.2 Classification of Damage Park, Ang and Wen (1987) used a simple classification based on visual signs of damage to correlate damage indices with observed damage. This classification is as follows: None: localized minor cracking at worst. Minor: minor cracking throughout. Moderate: severe cracking and localized spalling. Severe: crushing of concrete an d exposure of reinforcing bars. Collapse: collapse. Although this classification is considered ve ry simple to apply, it still needs more explanation on the interpretati on of the words. For example the word “severe” does not define clearly the magnitude of cracking. Th erefore, differences in levels of damage interpreted are expected. Another different classificati on related to the ability to repair the building after being exposed to an earthquake, was proposed by Bracci et al (1989) and Stone and Taylor (1993): Undamaged or minor damage. Repairable. Irrepairable. Collapsed. This classification may be harder to apply pr actically but it serves as a decision making tool for post-earthquake evalua tion and planning of building retrofitting. The evaluation

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24 presented by this method is essentially re lated to repair cost s leaving out other consequences that may have occurred and caused for example economic damage due to loss of the structure. The Earthquake Engineering Research In stitute (1994) implements a different scale that encounters no n-structural damage, approximate duration of loss of function and risk of fatalities to building tenants: None Slight – minor damage to non-structural el ements; building reopened in less than one week. Moderate – mainly non-structural damage little or no structural damage; building closed for up to 3 months; minor risk of loss of life. Extensive – widespread structural damage ; long term closure and possibly demolition required; high risk of loss of life. Complete – collapse or very extensive, irre pairable damage; very high risk of loss of life. Williams and Sexsmith (1995) concluded th at this classification has a greater correspondence with broader consequences. The main disadvantage is that correlation of this classification with the damage indices discussed afterward is somewhat poor. 2.5.3 Categorization of Damage The numerous damage indices proposed could be categorized between local indices and global indices. The local indices deal with the damage level in individual members or joints. While global indices deal wi th the overall structur e or a large part of

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25 it. In most cases, all these indices are dime nsionless parameters ranging between 0 for an undamaged structure and 1 for a collapsed stru cture. The intermedia te values between those two numbers tend to indicate the leve l of damage. The damage of an overall structure will be best obtained by a global index which in this case will be a grouping of local indices in differe nt parts of the struct ure or by taking into c onsideration structural modal parameters. Williams and Sexsmith (1995) summarized their research in concluding that relatively few attempts were made to adjust the local indices ag ainst observed damage but they are still far from being complete. Another limitation was that, in most of those indices, focusing on damage due to bending was the principal factor leaving damage caused by shear to be doubtful. The global indi ces, which are originated directly from local indices, are usually ac quired by using a suitable combination procedure. One can argue that a more flexible, particularly re levant approach would be more appropriate since those indices consist of a prearranged, weighted average of local indices. Even though the global softening indices have the ca pacity to describe the general damage condition of a structure, they still provide ve ry little information on damage distribution across the structure. The global indices could serve as a measur e of the performance-based design. In this research, more focus is placed on the collapse prevention level of the performancebased. The following section summarizes previous work done on global collapse assessment.

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26 2.6 Previous Work on Collapse Assessment In a recent study on collapse assessment of frame structures under seismic excitations, Ibarra and Kraw inkler (2005) reviewed the previous research on global collapse. They divided the efforts put forth in this subject into P effects and degrading hysteretic models. Analytical and experimental collapse investigations were also presented. 2.6.1 PEffects Study of global collapse began by introducing P effects to structures under seismic excitations. Under large P values, the stiffness became negative leading to collapse of the system. Jennings and Husid (1968) developed a one-story frame with springs at columns bases. Height of the struct ure, ratio of the earthquake intensity to yield level of the system, and the second slope of the bilinear model were found out to be the most critical parameters affecting collapse They also stated that duration of ground motion highly affected collapse. This fi nding was based on the likelihood of collapse increasing when load path stays longer rath er than the consider ation of degradation behavior. Gravity effect on the dynamic behavior of an SDOF system and its effect on the change of the system’s period were studied by Sun et al. (1973). They showed that depending upon a suitable coefficient and th e yield displacement, the structure can endure maximum displacement without failure Bernal (1987) focused more on this coefficient and recommended amplificati on factors based on the ratio of spectral acceleration generated with and without P effects. Bernal studied elasto-plastic

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27 SDOF systems and used same stability coe fficient for all periods of interest. His conclusion was that amplification factors a nd natural period were not considerably correlated. In 1994, McRae expanded Bernal ’s work by adding structures with more complex hysteretic response when studying P effects. Bernal (1992 and 1998) studied two-di mensional moment-resisting frames and concluded that the system’s failure mechanism is very critical to the base shear capacity needed to resist failure. He used an equiva lent elasto-plastic S DOF system comprising P effects. 2.6.2 Degrading Hysteretic Models Numerous experimental studi es proved that structural parameters influencing deformation and energy-dissipa tion characteristics affect th e hysteretic behavior. Many models were generated in this aspect. Sivaselvan and Reinhor n (2000) developed a smooth hysteretic degrading model including rule s for stiffness and strength degradation but excluding negative stiffness. Song and Pi ncheira (2000) presente d cyclic strength and stiffness degradation in their model based on dissipated hysteretic energy. Ibarra and Krawinkler (2005) used the degrading models developed by Ayoub et al. (2004) for basic bilinear, Clough and pinching hysteretic m odel. Degradation is based on energy dissipation following the rules pr oposed by Rahnama and Krawinkler (1993). The same concept was used in this research. Characteristics of thes e material models are presented in depth in chapter 3.

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28 2.6.3 Analytical Collapse Investigations Takizawa and Jennings (1980) developed a structural model equivalent to a SDOF system characterized by strength degr adation. This model was used to study the maximum capacity of an RC frame under eart hquake excitations and was among the first attempts to consider P effects and material degradation in collapse assessment. Aschheim and Moehle (1992) focused on the effects of prior seismic damage on peak displacement response of SDOF systems. Prior damage was modeled by reducing the initial stiffness assuming that resi dual displacements are negligible. Mehanny and Deierlein (2000) examined coll apse of composite structures. They calculated damage indices for a given struct ure and ground motion inte nsity record using a second-order inelastic time history analysis They reanalyzed the damaged structure throughout a second order inelasti c static analysis taking into consideration the residual displacements and gravity loads. They pres umed global collapse would take place once the applied gravity loads exceeded the maximu m vertical loads the system can endure. Lee and Foutch (2001) analytical m odels included a fracturing element implemented by Shi (1997) in DRAIN-2DX program. The Incremental Dynamic Analysis (IDA) concept developed by Vamv atsikos and Cornell (2002) was used for evaluating the global drift capacity. Global dyna mic instability was defined once the local slope of the IDA curve decreased to less than 20% of the initial slope in the elastic region. The (IDA) approach was also util ized by Jalayer (2003) to estimate global dynamic instability of regular RC structure. Williamson (2003) evaluated the response of SDOF subjected to seismic excitations taking into consideration P effects and material degradation based on

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29 modifying Park and Ang (1985) damage mode l. Miranda and Sinan (2003) investigated the lateral strengths required to present failu re of bilinear SDOF systems with negative post-yield stiffness. They c oncluded that dispersion of la teral strengths increased as negative post-yield stiffness decreased and fundamental periods increased. Adam and Krawinkler (2003) evaluate d the difference in respons e of non-linear systems under different analytical formulat ions. They concluded that la rge displacements formulation generates almost same responses as sma ll displacement formulations including cases when collapse is close. 2.6.4 Experimental Collapse Investigations Numerous experiments were performed to relate collapse with shear and axial failure in columns. Yoshimura and Yamana ka (2000) carried out several tests of RC columns subjected to low axial load. They no ticed that loading procedure enforced on each specimen determining the lateral and axial deformation as well as the input energy at failure. In addition, ratio of vertical deformation increment to lateral deformation increment at failure was not affected by cha nging the loading path. They concluded that failure takes place once the lateral load decrea se to less than 10% of the maximum load. Vian and Bruneau (2001) tested a series of shake-table experiments for a SDOF steel frame structure subjected to gradua lly increasing earthquakes intensity. The experiments were used till collapse takes plac e due to geometric nonlinearities which is a form of P effects. The conclusion was that the st ability coefficient was the key factor affecting the structure’s behavi or near collapse. Vian and Bruneau’s work was extended by Kanvinde (2003). He tested more SDOF sy stems concluding that current procedures

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30 of non-linear dynamic analysis are relia ble for failure prediction in case P effects governs the commencement of failure. Full scale shear-critical RC c oncrete building columns unde r cyclic lateral loading was tested by Sezen (2002). The test was carried out till the column could no longer sustain the applied axial load. The tests proved that loss of ax ial load does not necessarily follow instantaneously after loss of latera l load capacity. Sim ilarly, Elwood (2002) concluded that shear failure does not have to be the cause of failure of the system. They discovered that for columns having lower axial loads, axial load failure takes place at fairly large drifts, despite of whether shear failure had just occurred or occurred at much smaller drift ratios. Columns having larger axial loads experience failure usually at smaller drift ratios and may take place almost right after loss of lateral load capacity. Ibarra and Krawinkler (2005) studied the dispersion of the collapse capacity due to record variability and uncertainty of syst em parameters. They concluded that softening of the post-yield stiffness and the displacem ent at which this softening commences are the two system parameters that control th e collapse capacity of a system. Cyclic deterioration was found to be an importan t but not dominant factor for collapse evaluation. Despite the large number of research studi es on collapse, the previous literature review reveals the need of de veloping a more comprehensive procedure to assess collapse criterion in an explicit form. A new model th at incorporates degradation effects into seismic analysis of MDOF structures is requ ired and is described in the next chapter.

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31 CHAPTER 3 MATERIAL MODELS AND EARTHQUAKE RECORDS 3.1 Material Models Three material models were used in this research. The models considered were: Bilinear model to represent steel elements, Modified Clough model as per Clough, R. and Johnson, S. (1966) to represent concrete elements, and Pinching model to represent wood elements. The main skeleton for bilinear, modified Clough, and pinching models is shown in figures 3.1, 3.2, and 3.3 respectively. All models consist of an elastic branch, a strain hardening branch, and a softening branch. A residual strength is assumed in all models. However, the loading-reloading rules under cycl ic loading differ from a model to another. For the bilinear model, the initial unloading is parallel to the initial slope. The reloading curve is then bounded by the positive and negati ve strain hardening branches. As shown in figure 3.1, these branches form two main asymptotes for the model. For the modified Clough model, the initial unloadi ng is parallel as well to th e initial slope. As shown in figure 3.2, the behavior under cyclic loadi ng is characterized by targeting the maximum previous displacement point. The pinching mode l behavior is similar to the modified Clough, except that reloading consists of two branches. Th e first reloading branch is

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32 directed towards a point defi ned by a reduced target force. Then, the second branch is directed towards the previous maximum peak point as shown in figure 3.3. eK maxF yF yF c c Figure 3.1 Bilinear Model eK maxF yF yF c c Figure 3.2 Modified-Clough Model

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33 pemanant ppemananmaxFk maxF yF yF c c Figure 3.3 Pinching Model 3.1.1 Degradation It is well known from experimental verification that all materi als deteriorate as a function of the loading history. Each inelas tic excursion causes damage and the damage accumulates as the number of excursions increa ses. Therefore, it is essential to include degradation effects in m odeling hysteretic behavior. There are three common methods to calcu late degradation. The first method is based on ductility. The limitation of this me thod is that for cases when ductility is constant, there is no change between the cycles and therefore degradation does not appear in the system. Second method is a combin ation of ductility and energy. The main disadvantage of this method lies in its comple xity to apply since too many parameters are required for calculating degrada tion. The third method is derived from energy only. The method has a physical interpretation since it is related to the system capacity and hence gives advantage over the previous methods.

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34 All these models consist of a strength soft ening branch, refer to as a cap, to model strength degradation under monotonic loads. An 8 parameters en ergy-based criterion model was developed by Rahnama and Krawinkler (1993) to model four special types of cyclic degradation: Yield (Strength) degradation Unloading stiffness degradation Accelerated stiffness degradation Cap degradation 3.1.1.1 Yield (Strength) Degradation Yield degradation refers to the decrease of the yield strength value as a function of the loading history. The strength degradation paramete r is energy dependent and is derived through the following equation: ) 1 (1 i str i y i yF F (3.1) Where: i yF Yield strength at the current excursioni, 1 i yF Yield strength at the previous excursion1 i, and i str Scalar parameter, ranging from 0 to 1, th at accounts for degradation effects at the current excursioni. The parameter i str itself can be defined through the following equation:

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35 strC i j j capacity i i strE E E 1 (3.2) Where: iE Hysteretic energy dissipated in the current excursioni; i j jE1 Total hysteretic energy dissipated in a ll excursions up to the current one; and capacityE Energy dissipation capacity of the element under consideration. strC Exponent defining the rate of deterioration. The term capacityE represents the resistance of the material to cyclic degradation. The structure can be considered totally de graded once the total dissipated hysteretic energy, due to cyclic loading, attains a va lue equals to the energy dissipation capacity. Usually, capacityE is calculated as a function of the strain energy up to yield through the following equation: yystr capacityF E (3.3) Where: yF and y Initial yield strength and deformation respectively and str Constant. The values of str and strC are calibrated for each material by means of experimental data.

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36 Unloading and reloading on the elastic branch do not cause any deterioration since no hysteretic energy is dissipated. He nce, deterioration can not be considered complete and the yield strength remains at its original value. A complete deterioration in either the pos itive or negative side is achieved if one of the following conditions occurs during analysis: i i j j capacityEE E ) (1 (3.4) or if the term i str is greater than 1. Figure 3.4 represents the degraded envel ope and corresponding decrease in yield force due to strength degradation. )( i yF)1( i yF )( j yF)1( j yF)2( j yF pemanant ppemanant maxF maxFk Figure 3.4 Strength Degradation for Pinching Model

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37 3.1.1.2 Unloading Stiffness Degradation Unloading stiffness degradation refers to the unloading stiffness as a function of the loading history similar to yield (s trength) degradation. The parameter i unl used in the unloading stiffness degradation is also energy dependent but differs from the one of the strength degradation in the values of C and They are referred to by uCandu The modified unloading stiffness can be cal culated through the following equation: )1(1i unl i unl i unlkk (3.5) Where: unlk Unloading stiffness. Figure 3.5 represents the effect of unloading stiffness degradation on the hysteretic loop. )1( iK)3( iK pemanant ppemanantmaxFk )( iK )2( iK )1( iK maxF Figure 3.5 Unloading Stiffness De gradation for Pinching Model

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38 3.1.1.3 Accelerated Stiffness Degradation The reloading stiffness degrades as a f unction of cumulative loading in the peakoriented models. This effect can be taken in to consideration in th e analytical hysteretic model by modifying the target point to whic h the loading is directed referred to as accelerated stiffness degradation. The acce lerated stiffness degradation parameter i acc is similar to the strength degradation and stif fness degradation except for the values of Cand They are referred to by accCandacc The displacement value of the target point can be calculated through the following equation: )1(1i acc i tar i tar (3.6) Where: tar Displacement of the target point. The effect of the accelerated stiffness de gradation on the hysteretic behavior is represented in figure 3.6. i acc j acc )( i yF )( j yF Figure 3.6 Accelerated Stiffness Degradation for Pinching Model

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39 3.1.1.4 Cap Degradation From experimental results, it is observed that the onset point of softening moves inwards as a result of cumulative damage. This is referred to as cap degradation. If the cap slope reaches the displacement axis, then co llapse of the system in one direction is assumed. The cap degradation parameter i cap is similar to the strength, stiffness and accelerated degradations except for the values of Cand They are referred to by capCandcap The onset point of softening can be modified through the following equation: )1(1i cap i cap i cap (3.7) Where: cap Displacement of the onset point of softening. The modified envelope due to cap degr adation is represented in figure 3.7. Force Displacement E n v e l o p e O r i g i n a l E n v e l o p e O r i g i n a l ) ( iC a p O r ig i n a l) ( iC a p O r ig in a l) 1 ( iC a p D e g r a d ed) 1 ( jC a p D eg r a d ed) 2 ( jC a p D eg r a d ed )( i yF )( j yF )( i cap)1( i cap )1( j cap)2( j cap )( j cap pemanant ppemanantmaxFk maxF Figure 3.7 Cap Degradation for Pinching Model

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40 3.1.2 Effect of Degradation on Inelastic Systems Behavior Figures 3.9 to 3.40 represent the effect of degradation on the various material models previously explained. Numerical simulations varied from no degradation to low, moderate and severe degradation. Static loads were imposed on each system analyzed. Figures 3.8, 3.12 and 3.16 are for a non-degraded system for bilinear, modified Clough and pinching models respectively. They all sh are a very important characteristic which is that all the load cycles result in an envel ope form defining clearly the material model used. Once degradation is introduced in any of the material models, the load cycles begin to form a decreasing loop instead of the envelope. The numbe r of loops or load cycles sustained by the system before collapse is influenced by the level of degradation specified. The more intensity the degradation level gets, the fewer load cycles the system sustains and the faster collapse occurs. Figures 3.20, 3.24 and 3.28 focus on the strength degradation only for bilinear, modified Clough and pinching respectively. The graph reveals a strength reduction in each consecutive cycle lowering the yield valu e of the reloading cycle. The unloading stiffness degradation is presented in figures 3.21, 3.25 and 3.29 for the three models. The slope of the force-displacement curve is de creased each reloading cycle. The accelerated stiffness degradation follows the same pattern as explained in the preceding section. The reloading stiffness target a further point on the force-displacement graph moving the system towards the cap and hence accelerating failure. Figures 3.22, 3.26 and 3.30 demonstrate an example for bilinear, modifi ed Clough and pinching models respectively.

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41 The effect of cap degradation is illustrated in figures 3.23, 3.27 and 3.31. Each reloading cycle moves the cap branch closer to the origin which, eventu ally, accelerates the collapse of the system. Figures 3.32 to 3.40 s how the effect of seve ral combinations of different types of degradation on the in elastic behavior of the three models.

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42 -1.5 -1 -0.5 0 0.5 1 1.5 -8-6-4-202468DisplacementForce Figure 3.8 Bilinear Model – No Degradation -1.5 -1 -0.5 0 0.5 1 1.5 -8-6-4-202468DisplacementForce Figure 3.9 Bilinear Model – Low Degradation

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43 -1.5 -1 -0.5 0 0.5 1 1.5 -8-6-4-202468DisplacementForce Figure 3.10 Bilinear Model – Moderate Degradation -1.5 -1 -0.5 0 0.5 1 1.5 -6-4-20246DisplacementForce Figure 3.11 Bilinear Model – Severe Degradation

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44 -1.5 -1 -0.5 0 0.5 1 1.5 -8-6-4-202468DisplacementForce Figure 3.12 Clough Model – No Degradation -1.5 -1 -0.5 0 0.5 1 1.5 -8-6-4-202468DisplacementForce Figure 3.13 Clough Model – Low Degradation

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45 -1.5 -1 -0.5 0 0.5 1 1.5 -8-6-4-202468DisplacementForce Figure 3.14 Clough Model – Moderate Degradation -1.5 -1 -0.5 0 0.5 1 1.5 -8-6-4-202468DisplacementForce Figure 3.15 Clough Model – Severe Degradation

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46 -1.5 -1 -0.5 0 0.5 1 1.5 -8-6-4-202468DisplacementForce Figure 3.16 Pinching Model – No Degradation -1.5 -1 -0.5 0 0.5 1 1.5 -8-6-4-202468DisplacementForce Figure 3.17 Pinching Model – Low Degradation

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47 -1.5 -1 -0.5 0 0.5 1 1.5 -8-6-4-202468DisplacementForce Figure 3.18 Pinching Model – Moderate Degradation -1.5 -1 -0.5 0 0.5 1 1.5 -8-6-4-202468DisplacementForce Figure 3.19 Pinching Model – Severe Degradation

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48 -1.5 -1 -0.5 0 0.5 1 1.5 -8-6-4-202468DisplacementForce Figure 3.20 Bilinear Model – Strength Degradation -1.5 -1 -0.5 0 0.5 1 1.5 -8-6-4-202468DisplacementForce Figure 3.21 Bilinear Model – Stiffness Degradation

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49 -1.5 -1 -0.5 0 0.5 1 1.5 -8-6-4-202468DisplacementForce Figure 3.22 Bilinear Model – Accelerated Degradation -1.5 -1 -0.5 0 0.5 1 1.5 -8-6-4-202468DisplacementForce Figure 3.23 Bilinear Model – Cap Degradation

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50 -1.5 -1 -0.5 0 0.5 1 1.5 -8-6-4-202468DisplacementForce Figure 3.24 Bilinear Model – Strength and Stiffness Degradation -1.5 -1 -0.5 0 0.5 1 1.5 -8-6-4-202468DisplacementForce Figure 3.25 Bilinear Model – Accelerated and Cap Degradation

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51 -1.5 -1 -0.5 0 0.5 1 1.5 -8-6-4-202468DisplacementForce Figure 3.26 Bilinear Model – Strength, Stiffness and Accelerated Degradation -1.5 -1 -0.5 0 0.5 1 1.5 -8-6-4-202468DisplacementForce Figure 3.27 Clough Model – Strength Degradation

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52 -1.5 -1 -0.5 0 0.5 1 1.5 -8-6-4-202468DisplacementForce Figure 3.28 Clough Model – Stiffness Degradation -1.5 -1 -0.5 0 0.5 1 1.5 -8-6-4-202468DisplacementForce Figure 3.29 Clough Model – Accelerated Degradation

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53 -1.5 -1 -0.5 0 0.5 1 1.5 -8-6-4-202468DisplacementForce Figure 3.30 Clough Model – Cap Degradation -1.5 -1 -0.5 0 0.5 1 1.5 -8-6-4-202468DisplacementForce Figure 3.31 Clough Model – Strength and Accelerated Degradation

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54 -1.5 -1 -0.5 0 0.5 1 1.5 -8-6-4-202468DisplacementForce Figure 3.32 Clough Model – St iffness and Cap Degradation -1.5 -1 -0.5 0 0.5 1 1.5 -8-6-4-202468DisplacementForce Figure 3.33 Clough Model – Stiffness, Accelerated and Cap Degradation

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55 -1.5 -1 -0.5 0 0.5 1 1.5 -8-6-4-202468DisplacementForce Figure 3.34 Pinching Model – Strength Degradation -1.5 -1 -0.5 0 0.5 1 1.5 -8-6-4-202468DisplacementForce Figure 3.35 Pinching Model – Stiffness Degradation

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56 -1.5 -1 -0.5 0 0.5 1 1.5 -8-6-4-202468DisplacementForce Figure 3.36 Pinching Model – Accelerated Degradation -1.5 -1 -0.5 0 0.5 1 1.5 -8-6-4-202468DisplacementForce Figure 3.37 Pinching Model – Cap Degradation

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57 -1.5 -1 -0.5 0 0.5 1 1.5 -8-6-4-202468DisplacementForce Figure 3.38 Pinching Model – Strength and Cap Degradation -1.5 -1 -0.5 0 0.5 1 1.5 -8-6-4-202468DisplacementForce Figure 3.39 Pinching Model – Stiffness and Accelerated Degradation

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58 -1.5 -1 -0.5 0 0.5 1 1.5 -8-6-4-202468DisplacementForce Figure 3.40 Pinching Model – Strength, Accelerated and Cap Degradation 3.1.3 Collapse of Structural Elements A structural element is considered to have experienced complete collapse if any of the following two criteria is established: The displacement has surpassed the value of the intersection point of the softening (cap) slope with the x-axis, which is known as a cap failure (figure 3.41), or The scalar parameter in any of the degradation types, has exceeded a value of 1 which, in this case, is known as cy clic degradation failure (figure 3.42).

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59 DisplacementForce Figure 3.41 Collapse – Cap Failure It is important to note that an element mi ght fail in one direction of loading (e.g. compression), while still demonstrating resistan ce in the other direction of loading (e.g. tension). A reverse loading condition can alwa ys push the element to the direction that still shows some resistance. In this case, the element can not be considered as a collapsed structure. However, in this study, complete collapse for SDOF systems is considered if any direction of loading show s no resistance. Such assumption is considered to be conservative from a design perspective. Fo r MDOF systems though, such assumption is not considered.

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60 DisplacementForce Figure 3.42 Collapse – Degradation Failure 3.1.4 Experimental Verification of Material Models Several studies were performed to calibrate the material models proposed with the actual force-displacement data obtained fr om experimental specimens. As formerly explained, each material model represents the characteristics of a specific material: steel, concrete, or wood. The goal of the ca libration procedure is to define a value representing the behavior under cy clic loading. The coefficient consists of four subcoefficients each describing a type of degradation. For simplicity, will be assumed to be equal in the four types of degradation (i.e. d a k s)

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61 Where: s Strength degradation parameter, k Stiffness degradation parameter, a Accelerated stiffness degradation parameter, and d Cap degradation parameters. To calibrate the degradation parameters for existing structures, correlation studies with different experimental specimens are conducted. The Bilinear model was used to simulate the behavior of the steel beam specimen tested by Kraw inkler et al. (1983). Using trial and error methods, it was found that a value of 100 for all degradation parameters produces an excellent correlation wi th the experimental results, as shown in figure 3.43 (a) and (b). In a recent study, Ibarra and Kr awinkler (2005) stated that 130 would be more accurate for a bilinear model. Four specimens were used applying two different loadi ng protocols. The results s howed a satisfying correlation between the experimental and th e model load-deformation graph. The same exercise was performed on a c oncrete column specimen to calibrate it with the modified Clough model. The column specimen was tested by Sezen and Moehle (2004), and the corresponding load-deformation data was obtained from the PEER Structural Performance Database. From the analytical simulations, it was found that a value of 50 produces the best results as comp ared to the experimental ones and shown in figure 3.44 (a) and (b). The same study was conducted on a timber shear wall specimen tested at UC Irvine by Pardoen et al. (2001). A pinching model with a value of 200 for all degradation parameters produced the best correlation as shown in figure 3.45 (a) and (b).

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62 Normalized Deformation (%)Load100 1.05.0(a) (b) Figure 3.43 Bilinear Model (a) Ex perimental (b) Analytical DisplacementForce DisplacementForce (a) (b) Figure 3.44 Clough Model (a) Experimental (b) Analytical -50 -40 -30 -20 -10 0 10 20 30 40 50 -80-60-40-20020406080 Deformation (mm)Load (kN) -50 -40 -30 -20 -10 0 10 20 30 40 50 -80-60-40-20020406080 Deformation (mm)Load (kN) (a) (b) Figure 3.45 Pinching Model (a) Ex perimental (b) Analytical

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63 3.2 Earthquake Records A large database set consisting of 80 earthqu ake records is used in this research. Krawinkler et al. (1999, 2001) have used these records in several earlier st udies. The set of records consist four bins representing different pairs of magnitude (M) and distance from fault (R) as shown herein below: Bin I (SMSR): small M small R; (M < 6.5) and (R < 30 km). Bin II (SMLR): small M large R; (M < 6.5) and (R > 30 km). Bin III (LMSR): large M small R; (M > 6.5) and (R < 30 km). Bin IV (LMLR): large M large R; (M > 6.5) and (R > 30 km). Each of the above mentioned bins constitutes of 20 earthquake records which were recorded in California and correspond to soil ty pes C or D (stiff soil or soft rock) as per the NEHRP soil classification. Figure (3.46) represents the magnitude-distance distribution of the 80 records according to records details tables 3.1 to 3.8. 55 6 65 7 10.020.030.040.050.060.0Distance to Fault in km (R)Magnitude (M) Figure 3.46 Magnitude-Distance Distribution of the 80 Earthquake Records (Medina 2000)

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64 3.2.1 Database of Earthquake Records The large database of records used in this study is presented herein in tables 3.1 to 3.4. The records were sorted, as previously ex plained, into four diffe rent bins. Tables 3.5 to 3.8 contained more detailed properties for each record in terms of number of points, time step, and total time to facilitate the use of these records in analysis. Table 3.1 Earthquakes Having Small Magn itude and Small Distance from Fault Earthquake Date Station Legend Imperial Valley 10/15/79 Calipatria Fire Sta. IV79cal Imperial Valley 10/15/79 Chihuahua IV79chi Imperial Valley 10/15/79 El Centro Array #1 IV79e01 Imperial Valley 10/15/79 El Centro Array #12 IV79e12 Imperial Valley 10/15/79 El Centro Array #13 IV79e13 Imperial Valley 10/15/79 Cucapah IV79qkp Imperial Valley 10/15/79 Westmoreland Fire Sta. IV79wsm Livermore 01/24/80 San Ramon Kodad Bldg. LV80kod Livermore 01/24/80 San Ramon LV80srm Morgan Hill 04/24/84 Agnews State Hospital MH84agw Morgan Hill 04/24/84 Gilroy Array #2 MH84g02 Morgan Hill 04/24/84 Gilroy Array #3 MH84g03 Morgan Hill 04/24/84 Gilroy Array #7 MH84gmr

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65 Table 3.1 (Continued) Ptmugu 02/21/73 Port Hueneme PM73phn Palm Springs 07/08/86 Palm Springs Airport PS86psa Whittier Narrows 10/01/87 Compton-Castlegate St. WN87cas Whittier Narrows 10/01/87 Carson-Catskill Ave WN87cat Whittier Narrows 10/01/87 Brea-S. Flower Ave. WN87flo Whittier Narrows 10/01/87 LA-W 70th St. WN87w70 Whittier Narrows 10/01/87 Carson-Water St. WN87wat Table 3.2 Earthquakes Having Small Magn itude and Large Distance from Fault Earthquake Date Station Legend Borrego Mountain 10/21/42 El Centro Array #9 BO42elc Coalinga 05/02/83 Parkfield Cholame 5w CO83c05 Coalinga 05/02/83 Parkfield Cholame 8w CO83c08 Imperial Valley 10/15/79 Coachella Canal #4 IV79cc4 Imperial Valley 10/15/79 Compuertas IV79cmp Imperial Valley 10/15/79 Delta IV79dlt Imperial Valley 10/15/79 N iland Fire Station IV79nil Imperial Valley 10/15/79 Plaster City IV79pls Imperial Valley 10/15/79 Victoria IV79vct Livermore 01/24/80 Tracy-Sewage Treat. Plant LV80stp Morgan Hill 04/24/84 Capitola MH84cap

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66 Table 3.2 (Continued) Morgan Hill 04/24/84 Hollister City Hall MH84hch Morgan Hill 04/24/84 San Juan Bautista MH84sjb Palm Springs 07/08/86 San Jacinto Vall Cem PS86h06 Palm Springs 07/08/86 Indio PS86ino Whittier Narrows 10/01/87 Downey-Birchdale WN87bir Whittier Narrows 10/01/87 Cent. City CC South WN87cts Whittier Narrows 10/01/87 Long Beach Harbor WN87har Whittier Narrows 10/01/87 Terminal Island-S. Seaside WN87sse Whittier Narrows 10/01/87 Northridge-Saticoy St. WN87stc Table 3.3 Earthquakes Having Large Magn itude and Small Distance from Fault Earthquake Date Station Legend Loma Prieta 10/18/89 Agnews State Hospital LP89agw Loma Prieta 10/18/89 Capitola LP89cap Loma Prieta 10/18/89 Gilroy Array #3 LP89g03 Loma Prieta 10/18/89 Gilroy Array #4 LP89g04 Loma Prieta 10/18/89 Gilroy Array #7 LP89gmr Loma Prieta 10/18/89 Hollister City Hall LP89hch Loma Prieta 10/18/89 Hollister Diff Array LP89hda Loma Prieta 10/18/89 Sunnyva le Colton Ave. LP89svl Northridge EQ 1/17/94 Canoga Park Topanga Canyon NR94cnp

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67 Table 3.3 (Continued) Northridge EQ 1/17/94 LA Faring Rd. NR94far Northridge EQ 1/17/94 LA Fletcher NR94fle Northridge EQ 1/17/94 Gle ndale Las Palmas NR94glp Northridge EQ 1/17/94 LA Hollywood Storage FF NR94hol Northridge EQ 1/17/94 La Cr escenta New York NR94nya Northridge EQ 1/17/94 Nort hridge Saticoy NR94stc San Fernando 2/09/71 LA Hollywood Store Lot SF71pel Superstition Hills 11/24/87 BRW SH87bra Superstition Hills 11/24/87 El Ce ntro Imp. CO Center SH87icc Superstition Hills 11/24/87 PLC SH87pls Superstition Hills 11/24/87 Westmo reland Fire Station SH87wsm Table 3.4 Earthquakes Having Large Magn itude and Large Distance from Fault Earthquake Date Station Legend Borrego Mountain 4/09/68 El Centro Array #9 BM68elc Loma Prieta 10/18/89 Apeel 2E Hayward Muir SCH LP89a2e Loma Prieta 10/18/89 Fremont Emerson Court LP89fms Loma Prieta 10/18/89 Halls Valley LP89hvr Loma Prieta 10/18/89 Salinas John & Work LP89sjw Loma Prieta 10/18/89 Palo Alto Slac Lab LP89slc Northridge EQ 1/17/94 Covina W Badillo NR94bad

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68 Table 3.4 (Continued) Northridge EQ 1/17/94 Comp ton Castlegate NR94cas Northridge EQ 1/17/94 LA Centinela NR94cen Northridge EQ 1/17/94 Lakewood Del Amo NR94del Northridge EQ 1/17/94 Downey NR94dwn Northridge EQ 1/17/94 Bell Gardens Jaboneria NR94jab Northridge EQ 1/17/94 Lake H ughes #1 Fire Station #78 NR94lh1 Northridge EQ 1/17/94 La wndale Osage NR94loa Northridge EQ 1/17/94 Leona Valley #2 NR94lv2 Northridge EQ 1/17/94 Palmdale Hwy 14 & Palmdale NR94php Northridge EQ 1/17/94 LA -Pico & Sentous NR94pic Northridge EQ 1/17/94 West Covina S Orange NR94sor Northridge EQ 1/17/94 Terminal Island S Seaside NR94sse Northridge EQ 1/17/94 LA E Vernon NR94ver Table 3.5 Records Details of SMSR Legend Number of Points Time Step (sec.) Total Time of the Record (sec.) Magnitude (M) Distance from Fault (R) IV79cal 7905 .00500 39.53 6.5 23.8 IV79chi 4000 .01000 40.00 6.5 28.7 IV79e01 7800 .00500 39.00 6.5 15.5 IV79e12 7800 .00500 39.00 6.5 18.2 IV79e13 7900 .00500 39.50 6.5 21.9

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69 Table 3.5 (Continued) IV79qkp 8000 .00500 40.00 6.5 23.6 IV79wsm 7990 .00500 39.95 6.5 15.1 LV80kod 4190 .00500 20.95 5.8 17.6 LV80srm 7990 .00500 39.95 5.8 21.7 MH84agw 11980 .00500 59.90 6.2 29.4 MH84g02 5990 .00500 29.95 6.2 15.1 MH84g03 7990 .00500 39.95 6.2 14.6 MH84gmr 5990 .00500 29.95 6.2 14.0 PM73phn 4630 .00500 23.15 5.8 25.0 PS86psa 6000 .00500 30.00 6.0 16.6 WN87cas 1550 .02000 31.00 6.0 16.9 WN87cat 1640 .02000 32.80 6.0 28.1 WN87flo 1380 .02000 27.60 6.0 17.9 WN87w70 1590 .02000 31.80 6.0 16.3 WN87wat 1485 .02000 29.70 6.0 24.5 Table 3.6 Records Details of SMLR Legend Number of Points Time Step (sec.) Total Time of the Record (sec.) Magnitude (M) Distance from Fault (R) BO42elc 8000 .00500 40.00 6.5 49.0 CO83c05 4000 .01000 40.00 6.4 47.3 CO83c08 3200 .01000 32.00 6.4 50.7

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70 Table 3.6 (Continued) IV79cc4 5700 .00500 28.00 6.5 49.3 IV79cmp 3600 .01000 36.00 6.5 32.6 IV79dlt 9990 .01000 99.90 6.5 43.6 IV79nil 7990 .00500 39.95 6.5 35.9 IV79pls 3740 .00500 18.70 6.5 31.7 IV79vct 8000 .00500 40.00 6.5 54.1 LV80stp 6590 .00500 32.95 5.8 37.3 MH84cap 7200 .00500 36.00 6.2 38.1 MH84hch 5665 .00500 28.33 6.2 32.5 MH84sjb 5600 .00500 28.00 6.2 30.3 PS86h06 8000 .00500 40.00 6.0 39.6 PS86ino 6000 .00500 30.00 6.0 39.6 WN87bir 1430 .02000 28.60 6.0 56.8 WN87cts 7990 .00500 39.95 6.0 31.3 WN87har 7990 .00500 39.95 6.0 34.2 WN87sse 1140 .02000 22.80 6.0 35.7 WN87stc 2000 .02000 40.00 6.0 39.8

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71 Table 3.7 Records Details of LMSR Legend Number of Points Time Step (sec.) Total Time of the Record (sec.) Magnitude (M) Distance from Fault (R) LP89agw 8000 .00500 40.00 6.9 28.2 LP89cap 7990 .00500 39.95 6.9 14.5 LP89g03 7980 .00500 39.90 6.9 14.4 LP89g04 7990 .00500 39.95 6.9 16.1 LP89gmr 7990 .00500 39.95 6.9 24.2 LP89hch 7810 .00500 39.05 6.9 28.2 LP89hda 7920 .00500 39.60 6.9 25.8 LP89svl 7850 .00500 39.25 6.9 28.8 NR94cnp 2490 .01000 24.90 6.7 15.8 NR94far 2990 .01000 29.90 6.7 23.9 NR94fle 2990 .01000 29.90 6.7 29.5 NR94glp 2990 .01000 29.90 6.7 25.4 NR94hol 2000 .02000 40.00 6.7 25.5 NR94nya 2990 .01000 29.90 6.7 22.3 NR94stc 2990 .01000 29.90 6.7 13.3 SF71pel 2800 .01000 28.00 6.6 21.2 SH87bra 2210 .01000 22.10 6.7 18.2 SH87icc 8000 .00500 40.00 6.7 13.9 SH87pls 2220 .01000 22.20 6.7 21.0 SH87wsm 8000 .00500 40.00 6.7 13.3

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72 Table 3.8 Records Details of LMLR Legend Number of Points Time Step (sec.) Total Time of the Record (sec.) Magnitude (M) Distance from Fault (R) BM68elc 4000 .01000 40.00 6.8 46.0 LP89a2e 7990 .00500 39.95 6.9 57.4 LP89fms 7900 .00500 39.50 6.9 43.4 LP89hvr 7990 .00500 39.95 6.9 31.6 LP89sjw 7990 .00500 39.95 6.9 32.6 LP89slc 7915 .00500 39.58 6.9 36.3 NR94bad 3490 .01000 34.90 6.7 56.1 NR94cas 3970 .01000 39.70 6.7 49.6 NR94cen 2990 .01000 29.90 6.7 30.9 NR94del 3530 .01000 35.30 6.7 59.3 NR94dwn 2000 .02000 40.00 6.7 47.6 NR94jab 3490 .01000 34.90 6.7 46.6 NR94lh1 1600 .02000 32.00 6.7 36.3 NR94loa 3990 .01000 39.90 6.7 42.4 NR94lv2 1600 .02000 32.00 6.7 37.7 NR94php 6000 .01000 60.00 6.7 43.6 NR94pic 4000 .01000 40.00 6.7 32.7 NR94sor 3640 .01000 36.40 6.7 54.1 NR94sse 3490 .01000 34.90 6.7 60.0 NR94ver 2990 .01000 29.90 6.7 39.3

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73 3.2.2 Scaling of Earthquake Records Cornell and his co-workers (1999) showed in an earlier study that proper scaling of earthquake records in gene ral, does not introduce any pr econception to the response. Consequently, it will reduce the necessity of th e number of analysis needed for statistical evaluation. Moreover, proper scal ing ensures that al l records used in the study fall within the same hazard level defined by codes of pract ice. Cornell’s study pr oved that scaling an ensemble of records, even if they don’t fall initially within the same hazard level, to the median spectral acceleration va lue will not change the median values of the response quantities, but reduces considerably the vari ability in results. The conclusion he reached was also applicable to scaling to any value of spectral acceleration, whether it is higher or lower than the median value. Cornell’s appro ach was based mainly on statistical analysis of non-degrading simple bili near structural systems. In a recent study, Ayoub and Mijo (2006) st udied Cornell’s approach taking into consideration the degradation effects for different material models such as bilinear, modified Clough, and pinching. First, to c onfirm the reduction in variability due to scaling of degraded structures, he calcula ted the mean (denoted by ^) and dispersion values (denoted by ) of two demand parameters; ductility ( ) and normalized hysteretic energy (NHE). Those two statistical properties were calculated for each bin for both, the set of un-scaled records, and the set of records scaled to the mean spectral acceleration value. The results obtained for bi n I for four different cases of degradation for a SDOF system with period 1 sec, and for yield value characterized by09 0 where is the ratio of the yield force to the wei ght, are represented in tables 3.9 to 3.14.

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74 These results are compared with the non-de grading case for bilinear, modified Clough, and pinching models. The cases considered by Ayoub and Mijo were: Case (1): No degradation, Case (2): Cap slope = 6% and no cyclic degradation, Case (3): Cap slope = 6% a nd all degradation parameters 150 Case (4): Cap slope = 6% a nd all degradation parameters 100 and Case (5): Cap slope = 6% a nd all degradation parameters 50 The value of 09 0 was noted to be correspondi ng to a single common strength reduction factor R value for each record in the scaled set, but to different R values for each record in the un-scaled set. Tables 3.9-14 Median and Dispersion Values of Ductility ( ) and Normalized Hysteretic Energy (NHE) Table 3.9 Bilinear Un-scaled T = 1s, 09 0 Table 3.10 Bilinear Scaled T = 1s, 09 0 ˆ ˆNHE NHE ˆ ˆNHE NHE Case (1) 2.608 0.762 6.744 1.522 Case (1) 2.796 0.323 6.872 0.519 Case (2) 2.651 0.869 6.744 1.485 Case (2) 2.799 0.326 6.907 0.515 Case (3) 2.652 0.855 6.771 1.480 Case (3) 2.791 0.328 6.995 0.328 Case (4) 2.655 0.848 6.792 1.475 Case (4) 2.79 0.328 7.006 0.524 Case (5) 2.651 0.821 6.853 1.431 Case (5) 2.784 0.328 7.106 0.533

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75 Table 3.11 Clough Un-scaled T = 1s, 09 0 Table 3.12 Clough Scaled T = 1s, 09 0 ˆ ˆNHE NHE ˆ ˆNHE NHE Case (1) 2.558 0.857 8.962 1.054 Case (1) 2.686 0.372 8.168 0.487 Case (2) 2.587 0.951 8.962 0.98 Case (2) 2.692 0.369 8.231 0.485 Case (3) 2.570 0.948 8.914 0.956 Case (3) 2.678 0.374 8.206 0.482 Case (4) 2.569 0.951 8.927 0.943 Case (4) 2.683 0.378 8.203 0.481 Case (5) 2.574 0.886 8.908 0.927 Case (5) 2.689 0.379 8.165 0.486 Table 3.13 Pinching Un-scaled T = 1s, 09 0 Table 3.14 Pinching Scaled T = 1s, 09 0 ˆ ˆNHE NHE ˆ ˆNHE NHE Case (1) 2.624 0.818 8.333 0.937 Case (1) 2.782 0.382 7.274 0.542 Case (2) 2.647 0.876 8.333 0.886 Case (2) 2.787 0.390 7.275 0.543 Case (3) 2.663 0.882 8.362 0.874 Case (3) 2.767 0.396 7.086 0.569 Case (4) 2.668 0.884 8.381 0.868 Case (4) 2.772 0.40 7.065 0.576 Case (5) 2.707 0.901 8.414 0.824 Case (5) 2.776 0.41 6.889 0.617 From Ayoub’s results, two conclusions are dr awn. First, for most of the cases, the mean value of response quantities was not in fluenced much by the scaling procedure. Typically, a difference of less than 10% betw een the scaled and unscaled response is observed. This conclusion ensures that no prejudice is introduced by the scaling procedure. Second, the dispersion in results is significantly lower for the set of scaled records, which confirms the reduc tion in variability of results.

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76 The same study was performed for the case of bin IV scaled to the median spectral acceleration of bin I. The results show ed that the mean value of ductility was not much affected and the dispersion values were also significantly lower than those of the un-scaled set of records. Tables 3.15 to 3.20 represent the results for the fail values for both the scaled and un-scaled records. These re sults also confirm that scaling process did not prejudice the fail values, but rather reduced the vari ability of results as confirmed by the low dispersion values. Tables 3.15-20 Median and Dispersion Values of Strength at Failure (fail ) Table 3.15 Bilinear Un-scaled T = 1s Table 3.16 Bilinear Scaled T = 1s ˆ f ail f ail ˆ f ail f ail Case (1) Case (1) Case (2) 0.022 0.773 Case (2) 0.0215 0.356 Case (3) 0.023 0.712 Case (3) 0.0215 0.363 Case (4) 0.026 0.616 Case (4) 0.024 0.352 Case (5) 0.0365 0.556 Case (5) 0.0335 0.299 Table 3.17 Clough Un-scaled T = 1s Table 3.18 Clough Scaled T = 1s ˆ f ail f ail ˆ f ail f ail Case (1) Case (1) Case (2) 0.0235 0.752 Case (2) 0.0235 0.440 Case (3) 0.0255 0.723 Case (3) 0.027 0.417 Case (4) 0.028 0.742 Case (4) 0.0265 0.472 Case (5) 0.034 0.629 Case (5) 0.0345 0.409

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77 Table 3.19 Pinching Un-scaled T = 1s Table 3.20 Pinching Scaled T = 1s ˆ f ail f ail ˆ f ail f ail Case (1) Case (1) Case (2) 0.020 0.886 Case (2) 0.019 0.571 Case (3) 0.0215 0.846 Case (3) 0.021 0.533 Case (4) 0.022 0.904 Case (4) 0.024 0.565 Case (5) 0.033 0.683 Case (5) 0.028 0.526 The conclusion of Cornell’s and Ayoub’s wo rk is that proper scaling can reduce significantly the variability in results, a nd hence a much smaller number of non-linear analyses is required to conduct statistical st udies. Figures 3.47 to 3.50 show normalized scaled spectral acceleration for the four bins all scaled at a period value of 0.7 sec. The solid thick line in each of these graphs re presents the median value of the records whereas the dotted thick line corresponds to the 84th percentile. It is noted that the scaled factor of all the record s is equal to 1.0 at 7 0 Tsec since this is the target period.

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78 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0020.40.60.811.21.41.61.82PeriodSa/Sa (T=0.7sec) Figure 3.47 Bin I (SMSR) Scaled to T= 0.7 sec 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0020.40.60.811.21.41.61.82PeriodSa/Sa (T=0.7sec) Figure 3.48 Bin II (SMLR) Scaled to T= 0.7 sec

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79 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0020.40.60.811.21.41.61.82PeriodSa/Sa (T=0.7sec) Figure 3.49 Bin III (LMSR) Scaled to T= 0.7 sec 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0020.40.60.811.21.41.61.82PeriodSa/Sa (T=0.7sec) Figure 3.50 Bin IV (LMLR) Scaled to T= 0.7 sec

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80 CHAPTER 4 ASSESSMENT OF DEGRADED SDOF STRUCTURES 4.1 Introduction The number of modes necessary for an accura te dynamic analysis is a function of two parameters: modal contribution factor a nd spectral ordinates associated with the modal response equation. The following equation by Chopra (2005) explained the consequence of choosing the first J modes in analysis on the er ror in the static response: J n n Jr e11 (4.1) Where: Je Error in the static response and nr Modal contribution factor. Accordingly, the modal analysis can be reduced when the magnitude of error becomes adequately small for the target response quantity. Chopra (2005) suggested that in order to attain the desired accuracy of dynamic analysis: More modes should be considered for talle r buildings having longer periods rather than shorter buildings with smaller periods, and

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81 More modes should be considered for sh ear wall building having a higher beam-tocolumn stiffness ratio rather than moment-resisting frames buildings having smaller beam-to-column stiffness ratio. Currently, the rule of thumb is to include the number of modes e qual to one tenth the number of floors to limit the static respons e error to 10%. For example, a building having five floors would require the first modal analysis only to obtain an acceptable error. A seventeen floor structure would require at least considering the first two modes. This rule is proposed based on an assumption that the buildings systems are single bay frames. The purpose of this chapter is to inves tigate the effect of degradation on the behavior of SDOF systems, to develop a new numerical procedure for predicting maximum inelastic displacements of SDOF and first mode-dominant degrading building structures, and to predict collapse under seismi c excitations. Seismic fragility curves for a collapse criterion, defined as the probability of the system to collapse are also developed for different structural systems. The fi ndings provide necessary background for the design evaluation phase of a general perfor mance-based earthquake design process. Investigation of the degradati on effect on the behavior of S DOF structures is conducted first. 4.2 Degradation Effect on SDOF Sy stems Under Seismic Excitations Figures 4.1 to 4.2 investigate the eff ect of degradation on SDOF systems. A bilinear system equivalent to a 3-story struct ure was selected. The period of the structure 294 0T and its damping ratio % 5 The degradation parameters ( ) used was equal to 0 and 50 for no and severe degrad ation respectively. The Imperial Valley

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82 earthquake (IV79e01) recorded at station El Centro 1 was used in the analysis. This record falls into the most severe bin char acterized by large magnitude and small distance from fault. The duration of this record was 40 sec in total as show n in figure 4.1. The non-degraded system doesn’t experience coll apse as shown in figure 4.1, while the degraded system experienced collapse after 8.6 sec as shown in figure 4.2. The forcedisplacement diagrams for both non-degraded and degraded cases shown in figures 4.3 and 4.4 respectively reflect the behavior of the system. Collapse, which is denoted by a ‘*’ symbol on the graph 4.4, occurred at 2.03 inches while the maximum displacement for the non-degraded system was 1.71 inches. The behavior of the non-degraded system in the force-displacement graph was bounded by the original envelope. Initially, the behavior was in the elastic and strain hardening zone. In the last few cycles, the behavior reached the cap negative slope. In the degrad ed system, though, the behavior reached the cap in the first few cycles, and was eventually driven to collapse at a displacement that equals 2.03 inches.

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83 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 0510152025303540TimeDisplacement Figure 4.1 SDOF Time History for R oof Displ., 3 Floors, Bilinear, No Degradation -2.5 -2 -1.5 -1 -0.5 0 0.5 1 0510152025303540TimeDisplacement Figure 4.2 SDOF Time History for Roof Displ., 3 Floors, Bilinear, Severe Degradation

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84 -200 -150 -100 -50 0 50 100 150 200 -2.5-2-1.5-1-05 0 0.5 1DisplacementForce Figure 4.3 SDOF Force-Displacement, 3 Floors, Bilinear and No Degradation -200 -150 -100 -50 0 50 100 150 200 -2.5-2-1.5-1-05 0 0.5 1DisplacementForce Figure 4.4 SDOF Force-Displacement, 3 Floors, Bilinear and Severe Degradation

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85 The same material model and earthquake record were used on a system representing a ten-story structur e. The period of the system was 725 0 Tand its damping ratio was % 5 The degradation parameters ( ) used was equal to 0 and 50 for no and severe degradation respectively. Th e overall behavior show n in figures 4.5 to 4.8 is similar to that of the 3-story stru cture with the exception of the displacement values. The maximum roof displacement in the case of no degradation was equal to 3.86 inches compared to 1.71 in for the 3-floor structure. Collapse occurred for a severely degraded case after 17 sec with a roof displ acement value of 4.07 inches. The 10 stories system lasted longer than the three stories le ading to more loading cycles as shown in figure 4.6. -5 -4 -3 -2 -1 0 1 2 3 0510152025303540TimeDisplacement Figure 4.5 SDOF Time History for Roof Displ., 10 Floors, Bilinear and No Degradation

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86 -5 -4 -3 -2 -1 0 1 2 3 0510152025303540TimeDisplacement Figure 4.6 SDOF Time History for Roof Displ., 10 Floors, Bilinear and Severe Degradation -80 -60 -40 -20 0 20 40 60 80 -5-4-3-2-10123DisplacementForce Figure 4.7 SDOF Force-Displacement, 10 Floors, Bilinear and No Degradation

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87 -80 -60 -40 -20 0 20 40 60 80 -5-4-3-2-10123DisplacementForce Figure 4.8 SDOF Force-Displacement, 10 Floors, Bilinear and Severe Degradation 4.3 Displacement Estimates of SDOF Degraded Structures The goal of this part of the study is to pr edict collapse of SDOF systems, and to provide an estimate for the maximum inelasti c displacements in case collapse does not occur. A large set of building structures is selected for the study. The periods of these structures are 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.2, 1.5, 1.8 and 2.0 sec. Three values for the strength reduction factor (R) were also used in this study: 4, 6, and 8. This wide range of periods and strength reduc tion factors allowed a thorough observation of the behavior of the SDOF systems. The 4 bins of earthquake records recorded in California, and described earli er in chapter 3 are used to conduct the numerical study. The material models used are bilinear, modi fied Clough, and pinchi ng described earlier. Three different degradation cases for each of the three material models are considered

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88 and compared to a corresponding non-degrad ing system. These cases represent low (150 ), moderate (100 ), and severe degradation (50 ) respectively. The cap displacement is assumed to equal 4 times the yield displacement, and its slope equals 6% of the initial slope. The residual strength is assumed to equal zero. Plots of ratio of maximum inelastic displacements to maximum elastic displacements for different period values and for the different strength reduction factors ( R ) are generated for all degradation cases. The results for the case of Bins I-IV scaled to a spectral acceleration according to USGS values LA 10/50 are shown in figures 4.10 to 4.27. Mean collapse is defined when more than 50% of the records fa iled. The last point before collapse of the system is identified with a ‘* ’ in the plots generated, a nd no corresponding point for nondegraded systems exist. Several variables in the analysis had to be determined before conducting the analysis such as eita ( ) defined as the ratio of yield for ce to weight of the system. Table 4.1 presents values of ( ) used for different st rength reduction factor ( R ) and periods. Table 4.1 Yield Values for SDOF Systems Period 1 R 4 R 6R 8R 0.1 1.1111 0.2778 0.1852 0.1389 0.2 1.1111 0.2778 0.1852 0.1389 0.3 1.1111 0.2778 0.1852 0.1389 0.4 1.1111 0.2778 0.1852 0.1389 0.5 1.1111 0.2778 0.1852 0.1389

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89 Table 4.1 (Continued) 0.6 1.1111 0.2778 0.1852 0.1389 0.7 0.9524 0.2381 0.1587 0.1191 0.8 0.8333 0.2083 0.1389 0.1042 0.9 0.7407 0.1852 0.1235 0.0926 1.0 0.6667 0.1667 0.1111 0.0833 1.2 0.5556 0.1389 0.0926 0.0695 1.5 0.4444 0.1111 0.0741 0.0556 1.8 0.3700 0.0925 0.0617 0.0463 2.0 0.3333 0.0833 0.0556 0.0417 The ratios of maximum inelastic to maxi mum elastic displacements for a strength reduction factor 4 R are shown in figures 4.10 to 4.15. Figures 4.10, 4.11 and 4.12 show the results for a bilinear, Clough and pinching model respectively for mean values. While figures 4.13, 4.14 and 4.15 show the resu lts for the same material models but for 84th percentile values. This same set of plot s is repeated again for a strength reduction factor value of6 R in figures 4.16 to 4.21. The results are also presented for strength reduction factor 8R in figures 4.22 to 4.27. Several conclusions can be extracted from those graphs to better understand the eff ect of different variables on the ratio of maximum inelastic displacement to maximum elastic displacement. From all figures, it is clear that, not only degradation did not affect the behavior of long period structures, but also in this ra nge, the equal displacement rule still applies even for degraded systems. The effect of degradation becomes apparent for short period

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90 structures (5 0T sec) for both mean and 84th percentile values. In this range, degradation increases the maximum inelastic di splacements for all three material models. This conclusion applies as well for the different strength reduction factors. For very short periods (2 0 T sec), degraded system typically co llapse at any level of degradation. The difference between mean and 84th percentile values is clearly shown when comparing ratios at periods 3 0 T sec. For example, when examining figures 4.10 and 4.13 we notice that severely degraded systems collapse only for the 84th percentile values. This finding is justified by the fact that the 84th percentile values are more stringent than median values. Higher values of strength re duction factor also influence the collapse criteria. The evaluation of plots in figures 4.11, 4.17 and 4.23 illustrates this influence. Those three graphs share a Clough model with moderate degradation and median values but differ in the value of R For 4 R collapse occurs at 1 0 T sec while T is equal to 0.2 sec for 6R and 3 0T sec for 8 R. This can be explained by examining figure 4.9 which demonstrates that increasing the value of R is equivalent to decreasing the yield force value resulting into a mo re conservative model and consequently escalating the collapse probability. Figure 4.9 Effect of Strength Reduction Factor ( R ) on Yield Force (yF)

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91 In figures 4.10, 4.11 and 4.12, for strength reduction factor 4 R and 3 0 T, the ratios of maximum inelastic to elastic di splacement for severely degraded systems equals to 1.48, 2.04 and 2.15 for bilinear, Clough and pinching models respectively. For 6 R and 5 0 T in figures 4.16, 4.17 and 4.18, the ratios equal 1.21, 1.50 and 1.51 for bilinear, Clough and pinching models. Similarly, at 8 R and 8 0 T, the ratios in figures 4.22, 4.23 and 4.24 equal to 0.97, 1.01 and 1.01. The ratio in bilinear model leans to be lower than its corresponding values in Clough and pinching models. Moreover, the Clough and pinching values are almost identical The previous observation is justified by reviewing the characteristics of the material models explained earlier in chapter 3. Difference in material models characteri stics is also noticed when examining collapse of severely degraded systems for th e different cases of st rength reduction factor. In figure 4.10 collapse occurs at 3 0 T for 4 R For the same conditions except for 6R, collapse takes place at 5 0 T as shown in figure 4.16 w ith a 66% increase in the period value. This value equals to 0.8 sec in figure 4.22 when R reaches a value of 8 denoting a 60% increase from the previous value. For Clough models in figures 4.11, 4.17 and 4.23 collapse occurs at 2 0 T, 0.3 and 0.4 for 4 R 6 and 8 respectively with 50% and 33% increase. Likewise, in figures 4.12, 4.18 and 4.24 severely degraded pinching systems collapse at a period value of 0.3, 0.4 and 0.5 for 4 R 6 and 8 respectively with 33% and 25% increase respectively. The previous results confirm the fact that degradation has a major effect on the inelastic behavior of short period structures, and on the potential of collapse of these systems.

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92 0 0.5 1 1.5 2 2.5 0.00.40.81.21.62.0Period (sec.)inelastic/ elastic No Degradation Low Degradation Moderate Degradation Severe Degradation Figure 4.10 Bilinear Model, Median and R=4 0 0.5 1 1.5 2 2.5 3 3.5 4 0.00.40.81.21.62.0Period (sec.)inelastic/ elastic No Degradation Low Degradation Moderate Degradation Severe Degradation Figure 4.11 Clough Model, Median and R=4

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93 0 0.5 1 1.5 2 2.5 3 3.5 4 0.00.40.81.21.62.0Period (sec.)inelastic/ elastic No Degradation Low Degradation Moderate Degradation Severe Degradation Figure 4.12 Pinching Model, Median and R=4 0 0.5 1 1.5 2 2.5 3 0.00.40.81.21.62.0Period (sec.)inelastic/ elastic No Degradation Low Degradation Moderate Degradation Severe Degradation Figure 4.13 Bilinear Model, 84th % and R=4

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94 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0.00.40.81.21.62.0Period (sec.)inelastic/ elastic No Degradation Low Degradation Moderate Degradation Severe Degradation Figure 4.14 Clough Model, 84th % and R=4 0 1 2 3 4 5 6 7 0.00.40.81.21.62.0Period (sec.)inelastic/ elastic No Degradation Low Degradation Moderate Degradation Severe Degradation Figure 4.15 Pinching Model, 84th % and R=4

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95 0 0.5 1 1.5 2 2.5 0.00.40.81.21.62.0Period (sec.)inelastic/ elastic No Degradation Low Degradation Moderate Degradation Severe Degradation Figure 4.16 Bilinear Model, Median and R=6 0 0.5 1 1.5 2 2.5 3 3.5 4 0.00.40.81.21.62.0Period (sec.)inelastic/ elastic No Degradation Low Degradation Moderate Degradation Severe Degradation Figure 4.17 Clough Model, Median and R=6

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96 0 0.5 1 1.5 2 2.5 3 3.5 4 0.00.40.81.21.62.0Period (sec.)inelastic/ elastic No Degradation Low Degradation Moderate Degradation Severe Degradation Figure 4.18 Pinching Model, Median and R=6 0 0.5 1 1.5 2 2.5 3 0.00.40.81.21.62.0Period (sec.)inelastic/ elastic No Degradation Low Degradation Moderate Degradation Severe Degradation Figure 4.19 Bilinear Model, 84th % and R=6

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97 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0.00.40.81.21.62.0Period (sec.)inelastic/ elastic No Degradation Low Degradation Moderate Degradation Severe Degradation Figure 4.20 Clough Model, 84th % and R=6 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0.00.40.81.21.62.0Period (sec.)inelastic/ elastic No Degradation Low Degradation Moderate Degradation Severe Degradation Figure 4.21 Pinching Model, 84th % and R=6

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98 0 0.5 1 1.5 2 2.5 0.00.40.81.21.62.0Period (sec.)inelastic/ elastic No Degradation Low Degradation Moderate Degradation Severe Degradation Figure 4.22 Bilinear Model, Median and R=8 0 0.5 1 1.5 2 2.5 3 3.5 4 0.00.40.81.21.62.0Period (sec.)inelastic/ elastic No Degradation Low Degradation Moderate Degradation Severe Degradation Figure 4.23 Clough Model, Median and R=8

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99 0 0.5 1 1.5 2 2.5 3 3.5 4 0.00.40.81.21.62.0Period (sec.)inelastic/ elastic No Degradation Low Degradation Moderate Degradation Severe Degradation Figure 4.24 Pinching Model, Median and R=8 0 0.5 1 1.5 2 2.5 3 0.00.40.81.21.62.0Period (sec.)inelastic/ elastic No Degradation Low Degradation Moderate Degradation Severe Degradation Figure 4.25 Bilinear Model, 84th % and R=8

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100 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0.00.40.81.21.62.0Period (sec.)inelastic/ elastic No Degradation Low Degradation Moderate Degradation Severe Degradation Figure 4.26 Clough Model, 84th % and R=8 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0.00.40.81.21.62.0Period (sec.)inelastic/ elastic No Degradation Low Degradation Moderate Degradation Severe Degradation Figure 4.27 Pinching Model, 84th % and R=8

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101 4.4 Incremental Dynamic Analysis and Fragility of Collapse for SDOF Vamvatsikos and Cornell (2002) define d the Incremental Dynamic Analysis (IDA) as a parametric analysis method described in several different forms to estimate more thoroughly structural performance under seismic loads. The process involves subjecting a structural model to one (or more) ground motion record(s) which are scaled to multiple levels of intensity and hence, resulting in one (or more) curve(s) of response parameters versus intensity level. The Incremental Dynamic Analysis (IDA) plots establish a relationship between seismic demand parameters and strength parameters. Examples of seismic demand parameters could be ductility ( ) or inter-story drift, whereas examples of strength paramete rs could be spectral acceleration (aS ) or strength reduction factor ( R ) commonly used in codes of pract ices. In this study, several IDA ( R ) curves are plotted for a vari ety of degrading structures. To better understand the relationship between ( R ) and ( ) shown in figure 4.28, the following relations are introduced: y e y eF F R (4.2) y m (4.3) m e m y y eR (4.4)

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102 Where: eFValue of maximum Force if the structure remains elastic, yFValue of Force if the structure is yielding, e Value of Displacement corresponding to elastic Force, y Value of Displacement corresponding to yield Force, and m Value of maximum Displacement atta ined by the yielded structure. Using the above equations, we can derive that the starting point on a R graph would be (1,1) as shown in figure 4.28. The basi s of this finding is that the least value of maximum displacement m in a yielding system would be the yield displacement value y itself. Furthermore, a case with no strengt h reduction factor means that the elastic point and the yield point coin cide and their ratio would e qual 1. Although collapse occurs at high ductility values, it is desirable to in crease the ductility to prevent brittle failure. Ductility allows for hysteric energy to be dissipated which adds more damping to the system. Points that correspond to collapse are identified with a (*) in the IDA plot. The coordinates of the collapse points ( f ail Rfail) represent the ductility capacity and the strength reduction factor at collapse.

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103 R (1,1) Collapse Point F y e mFyFe Figure 4.28 Relationship of Strength Re duction Factor (R) and Ductility ( 4.4.1 Ductility Capacity and Strength Reduction Factor at Collapse To determine the ductility capacity and stre ngth reduction factor at collapse, the median ( R ) curves of an ensemble of structur es are plotted. Th e structures are assumed to be excited with all 80 records sc aled to a common value. Four periods are selected for this study: 0.2 sec, 0.5 sec., 1 sec., and 2 sec. Damping was assumed constant and equals 5%. All three material models, bilinear, Clough and pinching models are used in the study. The results for a structure with peri od that equals 0.2 sec. are discussed first. 4.4.1.1 Short Period Structures (2 0 Tsec) Figure 4.29 shows the median values of the ( R ) curve for a bilinear model with fixed envelope values but with differe nt degradation parame ters. All degradation parameters namely for strength, unloading stiffne ss, accelerated stiffness, and cap are assumed to be equal. A value of 150 corresponds to a system with low degradation, a value of 100 corresponds to a system with mode rate degradation, and a value of 50 corresponds to a system with severe degradation. The cap displacement cap is

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104 assumed to equal 4 times the yield displacemen t. From figure 4.29, it is rather obvious that degradation had a great effect on the value of f ail The ductility capacity of a system with no degradation, which fails mainly due to softening effects, is 35 21 fail The ductility capacity of systems with low, moderate, and high degradation is 12, 9.5, and 6.6 respectively. Degradat ion can thus reduce the ductilit y capacity by a value that could be greater than 3. The stre ngth reduction factor at collapse failR is also reduced from 5.4 for the case of no degr adation to 3.4 for the case of highly degraded structures. Figure 4.30 shows the median plot for the same system with a fixed degradation value 100 but with different cap displacement values cap 1, 4, 6, 8 and The case with cap corresponds to a system with cycl ic degradation only, and the case with 1cap corresponds to a brittle-fr acture system where cap soft ening starts right after the elastic branch. The ductility capacity, ex cept for the latter case, was not affected much by the onset of softening and ranges between 9.3 and 10.9 with the corresponding failR value ranging between 4.2 and 4.6. These re sults suggest that fo r bilinear non-brittle systems, failure is most likely due to cyclic degradation effects, rather than to softening effects. For the brittle case, the ductility capac ity is dramatically reduced to a value that equals 4.38 with a corresponding failR value of 2.2. In this case, the early presence of the cap dominates the response and drives the system quickly into failure. Figures 4.31 and 4.32 show plots similar to the ones described above, but for a Clough model. The effect of degradation is clearly manifested in figure 4.31 where f ail decreases from a value of 19.1 for the non-degr aded case to a value of 5.1 for the highly degraded case. The corresponding strength reduction factor failR is also decreased from

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105 4.2 to 2.6. The effect of the presence of the cap is illustrated in figure 4.32. A system with no cap has a ductility capacity that equals 18, while a system with a cap has a ductility capacity that ranges between 6.5 and 10.3. Th ese results reveal that Clough systems fail mainly due to softening effects. Cyclic de gradation accelerates failure, but to a much lesser extent than for Bilinear models since th e hysteretic energy dissipated in this case is much less than for a Bilinear case. The final outcome is that bilinear systems are actually more ductile than Clough systems, even though they dissipate more energy. This fact is mainly due to the different failure mode of each system. The same results of a pinched model are rather interesting, and are shown in figures 4.33 and 4.34. The ductility capacity of a non-degraded system is actually lower than that of degraded systems. The pinching st ress is originally assumed to equal half the yield value. Due to degradation effects, th e pinching point moves away from the origin, and accelerated degradation dominates the resp onse. Displacements are thus increased, and so is the ductility capacity. Figure 4.33 shows that f ail for a non-degraded system equals 13, while it equals 16.42 for a sy stem with moderate degradation. The corresponding failR value decreases with degradation though. It equals 3.8 for a nondegraded system, and 3.4 for a system with moderate degradation. Degraded pinching systems are considered thus more duct ile than corresponding Bilinear and Clough systems, while non-degraded pinching systems are more brittle than Bilinear and Clough ones.

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106 4.4.1.2 Medium Period Structures (5 0 Tsec) Figure 4.35 shows the ( R ) curve for a bilinear system. Just like the previous case, degradation has a big effect on the ductility capacity. The value of f ail equals 19.9 for a non-degraded system, and drops to a value of 12.5 for moderately degraded systems, and 7.6 for severely degraded systems. The failR value is considerably higher than in the case of 2 0T sec, and equals 9.4 for non-degraded systems and 5.8 for severely degraded systems. Figure 4.36 shows the same plot for different cap displacements. As for the case of 2 0 T sec, the onset of softening did not affect the ductility capacity, except for very brittle cases, which suggests that failure, except for the latter case, is mainly due to cyclic degradation. The effect of degradation on ductility cap acities for a Clough model is not as high as for a Bilinear model as illustrated in fi gure 4.37. A severely degraded system has a ductility capacity of 11.4, while a non-degraded system has a value of 15.2. A system with low degradation has a duc tility capacity value slightly higher than a non-degraded system due to accelerated degradation effects. Figure 4.38 also shows that the presence of a cap did have a major in fluence on the value of f ail The value of failR is in the same range as for the bilinear system. The behavior of a pinched model in this case is similar to that of the Clough model as illustrated in figur es 4.39 and 4.40. Both of these models fail mainly due to softening effects, with the cy clic degradation acceler ating the failure rate. 4.4.1.3 Long Period Structures (0 1 Tsec) It is well known that structures in this region follow the equal displacement rule, where maximum inelastic displacements equal equivalent maximum elastic

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107 displacements. It is interesting to note that the previous conclusion holds true also for degrading systems in case collapse doesn’t occur. Degradation can, however, accelerate the failure of the structure. The previous conclusion could be affirmed by examining figure 4.41. The load-deformation ( R ) dynamic response for cases with or without degradation seems to follow the same tre nd confirming the previous conclusion. The ductility capacity though equals 10.4 for a nondegraded system, 7.8 for a moderately degraded system, and 5.8 for a severely degraded system. The corresponding failR values are 9.4, 7.4, and 5.8 respectively. The presen ce of the cap, however, contributes strongly to the failure mode as shown in figure 4.42. Th e earlier the onset of softening, the weaker the overall behavior and the earlier failure occurs. The failR value drops from a value of 15.8 for a system with cyclic degrada tion only to 4.2 for a brittle system. The same conclusion drawn for bilinear systems applies for Clough systems, as shown in figures 4.43 and 4.44. The only ex ception is that Clough sy stems dissipate less energy, and thus the strength level is higher than in bilinear systems. The corresponding failR value is thus considerably larger than for bilinear systems. The value for pinching systems is even larger than for Clough system s, as shown in figures 4.45 and 4.46, since their hysteretic energy di ssipation is minimal. 4.4.1.4 Long Period Structures (0 2 Tsec) The behavior of structures with period of 2s ec. is similar to that of structures with period of 1sec., as shown in figures 4.47 to 4.52. The load-deformation dynamic response for degrading and non-degrading systems follo ws the same trend, with degradation affecting only the failure point. Hysteretic energy dissipation is much less than in

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108 structures with period of 1sec. due to the fact that a smaller number of cycles are observed, which resulted in smaller failR values. For Bilinear systems, the failR value ranges between 9.8 for severely degradi ng structures and 19.0 for non-degraded structures. For Clough systems, the failR value is even higher and ranges between 11.0 and 19.4 for severely degraded and non-degr aded systems respectively. Pinching systems exhibit the highest failR value, ranging between 14.2 and 23.8 for severely degraded and non-degraded systems respectively. 0 1 2 3 4 5 6 0510152025DuctilityStrength Reduction Facto r No Degradation Low Moderate Severe Figure 4.29 Bilinear Model Ductility, 4 R and 2 0T sec

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109 0 1 2 3 4 5 024681012DuctilityStrength Reduction Facto r Cap Disp. = 1 Cap Disp. = 4 Cap Disp. = 6 Cap Disp. = 8 No Cap Figure 4.30 Bilinear Model Ductility, 100 and 2 0T sec 0 1 2 3 4 5 0510152025DuctilityStrength Reduction Facto r No Degradation Low Moderate Severe Figure 4.31 Clough Model Ductility, 4 R and 2 0T sec

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110 0 1 2 3 4 5 6 0510152025DuctilityStrength Reduction Facto r Cap Disp. = 1 Cap Disp. = 4 Cap Disp. = 6 Cap Disp. = 8 No Cap Figure 4.32 Clough Model Ductility, 100 and 2 0T sec 0 1 2 3 4 0510152025DuctilityStrength Reduction Facto r No Degradation Low Moderate Severe Figure 4.33 Pinching Model Ductility, 4 R and 2 0T sec

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111 0 1 2 3 4 5 6 0510152025DuctilityStrength Reduction Facto r Cap Disp. = 1 Cap Disp. = 4 Cap Disp. = 6 Cap Disp. = 8 No Cap Figure 4.34 Pinching Model Ductility, 100 and 2 0T sec 0 1 2 3 4 5 6 7 8 9 10 0510152025DuctilityStrength Reduction Facto r No Degradation Low Moderate Severe Figure 4.35 Bilinear Model Ductility, 4 R and 5 0T sec

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112 0 1 2 3 4 5 6 7 8 9 10 02468101214DuctilityStrength Reduction Facto r Cap Disp. = 1 Cap Disp. = 4 Cap Disp. = 6 Cap Disp. = 8 No Cap Figure 4.36 Bilinear Model Ductility, 100 and 5 0T sec 0 1 2 3 4 5 6 7 8 9 024681012141618DuctilityStrength Reduction Facto r No Degradation Low Moderate Severe Figure 4.37 Clough Model Ductility, 4 R and 5 0T sec

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113 0 2 4 6 8 10 12 14 02468101214161820DuctilityStrength Reduction Facto r Cap Disp. = 1 Cap Disp. = 4 Cap Disp. = 6 Cap Disp. = 8 No Cap Figure 4.38 Clough Model Ductility, 100 and 5 0T sec 0 1 2 3 4 5 6 7 8 9 0246810121416DuctilityStrength Reduction Facto r No Degradation Low Moderate Severe Figure 4.39 Pinching Model Ductility, 4 R and 5 0T sec

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114 0 2 4 6 8 10 12 14 16 0481216202428DuctilityStrength Reduction Facto r Cap Disp. = 1 Cap Disp. = 4 Cap Disp. = 6 Cap Disp. = 8 No Cap Figure 4.40 Pinching Model Ductility, 100 and 5 0T sec 0 1 2 3 4 5 6 7 8 9 10 024681012DuctilityStrength Reduction Facto r No Degradation Low Moderate Severe Figure 4.41 Bilinear Model Ductility, 4 R and 0 1T sec

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115 0 2 4 6 8 10 12 14 16 18 0246810121416DuctilityStrength Reduction Facto r Cap Disp. = 1 Cap Disp. = 4 Cap Disp. = 6 Cap Disp. = 8 No Cap Figure 4.42 Bilinear Model Ductility, 100 and 0 1T sec 0 2 4 6 8 10 12 14 16 04812162024DuctilityStrength Reduction Facto r No Degradation Low Moderate Severe Figure 4.43 Clough Model Ductility, 4 R and 0 1T sec

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116 0 4 8 12 16 20 24 04812162024DuctilityStrength Reduction Facto r Cap Disp. = 1 Cap Disp. = 4 Cap Disp. = 6 Cap Disp. = 8 No Cap Figure 4.44 Clough Model Ductility, 100 and 0 1T sec 0 2 4 6 8 10 12 14 16 18 04812162024DuctilityStrength Reduction Facto r No Degradation Low Moderate Severe Figure 4.45 Pinching Model Ductility, 4 R and 0 1T sec

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117 0 5 10 15 20 25 30 051015202530DuctilityStrength Reduction Facto r Cap Disp. = 1 Cap Disp. = 4 Cap Disp. = 6 Cap Disp. = 8 No Cap Figure 4.46 Pinching Model Ductility, 100 and 0 1T sec 0 2 4 6 8 10 12 14 16 18 20 04812162024DuctilityStrength Reduction Facto r No Degradation Low Moderate Severe Figure 4.47 Bilinear Model Ductility, 4 R and 0 2T sec

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118 0 4 8 12 16 20 24 02468101214161820DuctilityStrength Reduction Facto r Cap Disp. = 1 Cap Disp. = 4 Cap Disp. = 6 Cap Disp. = 8 No Cap Figure 4.48 Bilinear Model Ductility, 100 and 0 2T sec 0 5 10 15 20 25 0510152025DuctilityStrength Reduction Facto r No Degradation Low Moderate Severe Figure 4.49 Clough Model Ductility, 4 R and 0 2T sec

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119 0 4 8 12 16 20 24 02468101214161820DuctilityStrength Reduction Facto r Cap Disp. = 1 Cap Disp. = 4 Cap Disp. = 6 Cap Disp. = 8 No Cap Figure 4.50 Clough Model Ductility, 100 and 0 2T sec 0 4 8 12 16 20 24 28 04812162024DuctilityStrength Reduction Facto r No Degradation Low Moderate Severe Figure 4.51 Pinching Model Ductility, 4 R and 0 2T sec

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120 0 5 10 15 20 25 30 35 051015202530DuctilityStrength Reduction Facto r Cap Disp. = 1 Cap Disp. = 4 Cap Disp. = 6 Cap Disp. = 8 No Cap Figure 4.52 Pinching Model Ductility, 100 and 0 2T sec 4.5 Seismic Fragility Analysis Seismic fragility curves are plots that desc ribe the probability of a system to reach or exceed different degrees of damage, includ ing possible collapse. Earlier work focused on developing seismic fragility curves of systems for several values of a calibrated damage index. A damage index is a factor that represents the degree of damage of the structure, and typically ranges from 0 to 1, with the value of 1 representing complete collapse. Collapse was therefore expressed implic itly as the state of the structure when its damage index approaches a value of 1. In this study, seismic fragility curves for a collapse criterion are developed in an exp licit form using the new degraded material models. A relationship between IDA plots, such as the ones presented earlier, and fragility curves exist. The relationship is described next.

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121 4.5.1 IDA and Fragility Relationship Figure 4.53 explains the relationship betw een the Incremental Dynamic Analysis (IDA) and Fragility curves. The solid dot on the IDA curve represents the collapse point at a 50% probability, and the corresponding R value represents the mean strength reduction factor at collapse. The challenge arises when we need to find values of strength reduction factor ( R ) corresponding to a specific probabil ity of collapse other than 50% for design purposes. Fragility curves offer this advantage as they express the entire spectrum of collapse probability. Figure 4.53 Relationship Between IDA & Fragility Curves While fragility curves are typically expressed as a function of spectral accelerations for a specific yield force they could be easily ex tended to cover different values of In this case, a relationship between spectral accelerations (aS), and the global yield force has to be established to give the structural designer the flexibility to use fragility curves for any yield value. The yield force is defined as follow:

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122 The ratio of yield force (yF) to total weight of the structure (W). It is well known that the response of a sy stem to a scaled earthquake record is identical to that of the same system with a yield force reduced by the same scale value and subject to the original unscaled earthqua ke record. Consequently, the yield force is assumed to be inversely proportiona l to the spectral acceleration (aS). This relationship could be used to modify seismic fragility curves, as explained in the next example. Assume a fragility curve, show n in figure 4.54, is drawn for = 1.8. The structural designer has found th at the subject structure has = 0.09 and he needs to find the probability of collapse of this building if hit by an earthquake having a spectral acceleration (aS) of 1.2 g. In order to use this curve, the designer should modify the spectral acceleration (aS) used to get the collapse pr obability. The new value will be: g g Sified a24 09 0 8 1 2 1 ) (mod It is worth mentioning that the combination of a specific and (aS ) values correspond to a unique strength reduction factor R, which is defined as: eR (4.5) Where: e = The value of when 1 R

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123 = 1.8Actual valueProb. of Collapse1.2Sa(g) 24 Figure 4.54 Example of Use of Eita in Fragility Curve 4.5.2 Strength Reduction Factor ( R ) and Ratio of Yield Force to Total Weight ( ) In the next discussion, the fragility curv es for a collapse criterion, for the same ensemble of SDOF systems investigated earl ier are developed, and are shown in figures 4.55 to 4.78. The data are smoothed usi ng lognormal distribution functions. The 50% collapse probability point corresponds to the po int identified w ith a ‘*’ in the mean IDA plots presented earlier, as explained before The earthquake records were scaled to a common value. The yield force is assumed to equal 0.2, however plots for different values of could be easily estimated through pr oper scaling, as discussed earlier. A discussion on the behavior of the different structures investigated is presented next. 4.5.2.1 Short Period Structures (2 0 Tsec) Figure 4.55 shows the fragility curve for a collapse criterion for a bilinear model with fixed envelope values but with differe nt degradation parame ters. All degradation

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124 parameters are assumed to be equal. A value of 150 corresponds to a system with low degradation, a value of 100 corresponds to a system w ith moderate degradation, and a value of 50 corresponds to a system with severe degradation. As in the previous study, the cap displacement cap is assumed to equal 4 times the yield displacement. From the plot, it can be shown that short period structures are susceptible to failure under earthquake excitations. A system with a cap and with no cyclic degradation has a 90% probability of failure if subject to a record with 1.8g spectral acceleration. A similar system with 8 R will have the same failure probability if the spectral acceleration equals 0.9g. Cyclic degradation tends to increase the failure probability. A severe degradation system with 4 R has a 90% failure probability if the spectral acceleration equals 1.08g. Figure 4.56 shows the plot for the same system with a fixed degradation value 100 but with different cap displacement values 1 cap 4, 6, and 8. The collapse probability, except for the first case, was not a ffected much by the onset of softening. The 90% collapse probability for all cases but the first is at a spectral acceleration of 1.5g, while it is at 0.75g for the first case. As di scussed before, these re sults agree with the previously derived conclusions th at suggest that for bilinear n on-brittle systems, failure is most likely due to cyclic degradation effects, rather than to softening effects. For the brittle case, the early presence of the cap dominates the response and drives the system quickly into failure. Figures 4.57 and 4.58 show plots similar to the ones described above, but for a Clough model. Degradation affected the failure ra te of the structure; a system with a cap and no cyclic degradation has a 90% collapse probability if the sp ectral acceleration is

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125 1.5g, while systems with low and severe degr adation reach the same collapse probability if the spectral acceleration is 1.25g and 0.92g respectively. Th e effect of the presence of the cap is illustrated in figure 4.58. A system with no cap has a 90% collapse probability at a spectral acceleration of 1.8g, while syst ems with caps reach the same probability at a spectral acceleration around 1.25g irrespective of the value of the cap. These results confirm that Clough systems fail mainly due to softening effects as discussed earlier. Cyclic degradation accelerates failure, but to a much lesser extent than for Bilinear models. Also, from the preceding plots, it is concluded that Clough systems have a different failure mode than bilinear syst ems, but overall fail with a faster rate. The same results of a pinched model are show n in figures 4.59 and 4.60. The trend of the fragility curves is similar to that of Clough models. Systems with no cap tend to fail with a slower rate, while degraded systems have a collapse probability close to that of a similar Clough model. 4.5.2.2 Medium Period Structures (5 0 Tsec) Figure 4.61 shows the collapse probability for a bilinear system. In this case, degradation has a big effect on the overall collapse probability. A system with low degradation has a 90% collapse probability at a spectral acceleration of 2.92g, while a system with severe degradation has the sa me collapse probability at a 1.8g spectral acceleration. These values are considerably hi gher than those of the case of short period structures, implying that medium period struct ures have a lower collapse probability if subject to the same earthquake record. Figure 4.62 shows the same pl ot for different cap displacements. As for the case of short period st ructures, very brittle systems failed with a

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126 much faster rate (90% colla pse probability at a spectra l acceleration of 1g), while structures with different values of ca p displacements reached the same collapse probability at a spectral acceleration that ranges between 2.44g and 3.32g. The collapse probability of Clough models is also lower than the corresponding one for short period structures. A severely degraded system has a 90% collapse probability at a spectral acceleration of 2.12g, and a system with low degradation has the same probability at a spectral acceleration of 2.76g as shown in figure 4.63. Figure 4.64 also shows that the presence of a cap increa sed the 90% collapse probability from 2.6g for a case of cap displacement of 4 to 3. 08g for a case of cap displacement of 8. The collapse probability of pinched models has a slower rate than that of both bilinear and Clough models, as shown in figures 4.65 and 4.66 implying that pinched models are less likely to collapse if subject to the same earthquake record. 4.5.2.3 Long Period Structures (0 1 T sec and 0 2 T sec) Figure 4.67 shows the fragility curve for a bilinear model with different degradation parameters. The collapse rate in ge neral is slower than that of medium period structures. Degradation had a considerable effect on the collapse rate. A severely degraded system has a 90% probability of co llapse at a spectral acceleration of 2.52g, while a system with low degradation has the same probability at 4.04g. Figure 4.68 shows the effect of the cap displacement on the behavior. Similar to medium period structures, the brittle case has a fast collaps e rate, while non-brittle cases showed a much slower one. The behavior of Clough models shows a slower co llapse rate than that of bilinear models, as shown in figure 4.69. The same is true for the case of a variable cap

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127 displacement, shown in figure 4.70. The same tr end is true if we compare the behavior of pinching models, shown in figures 4.71 and 4.72, to that of Clough models. In this case, a 90% collapse probability of a pinching mode l with moderate degradation and no cap is outside the limit of the graph. The collapse rate for long period structures with period T=2 sec. is the slowest for all cases, as s hown in figures 4.73 to 4.78. A high spectral acceleration value is needed in this case in order to produce a 90% collapse probability. Degradation, in all cases, accelerated the rate of collapse. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0123456789AgProb. of Collaps e Low Degradation Moderate Degradation Severe Degradation Low Degradation Moderate Degradation" Severe Degradation Figure 4.55 Bilinear Model Fragility, 4 R and 2 0T sec

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128 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0123456789AgProb. of Collaps e Cap Disp. = 0 Cap Disp. = 4 Cap Disp. = 6 Cap Disp. = 8 Cap Disp. = 0 Cap Disp. = 4 Cap Disp. = 6 Cap Disp. = 8 Figure 4.56 Bilinear Model Fragility, 100 and 2.0T sec 0 01 02 03 0.4 05 0.6 0.7 08 09 1 0123456789AgProb. of Collaps e Low Degradation Moderate Degradation Severe Degradation Low Degradation Moderate Degradation Severe Degradation Figure 4.57 Clough Model Fragility, 4 R and 2.0T sec

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129 0 01 02 03 0.4 05 0.6 0.7 08 09 1 0123456789AgProb. of Collaps e Cap Disp. = 4 Cap Disp. = 6 Cap Disp. = 8 Cap Disp. = 4 Cap Disp. = 6 Cap Disp. = 8 Figure 4.58 Clough Model Fragility, 100 and 2.0T sec 0 01 02 03 0.4 05 0.6 0.7 08 09 1 0123456789AgProb. of Collaps e Low Degradation Moderate Degradation Severe Degradation Low Degradation Moderate Degradation Severe Degradation Figure 4.59 Pinching Model Fragility, 4 R and 2.0T sec

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130 0 01 02 03 0.4 05 0.6 0.7 08 09 1 0123456789AgProb. of Collaps e Cap Disp. = 4 Cap Disp. = 6 Cap Disp. = 8 Cap Disp. = 4 Cap Disp. = 6 Cap Disp. = 8 Figure 4.60 Pinching Model Fragility, 100 and 2.0T sec 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0123456789AgProb. of Collaps e Low Degradation Moderate Degradation Severe Degradation Low Degradation Moderate Degradation Severe Degradation Figure 4.61 Bilinear Model Fragility, 4 R and 5.0T sec

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131 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0123456789AgProb. of Collaps e Cap Disp. = 0 Cap Disp. = 4 Cap Disp. = 6 Cap Disp. = 8 Cap Disp. = 0 Cap Disp. = 4 Cap Disp. = 6 Cap Disp. = 8 Figure 4.62 Bilinear Model Fragility, 100 and 5.0T sec 0 01 02 03 0.4 05 0.6 0.7 08 09 1 0123456789AgProb. of Collaps e Low Degradation Moderate Degradation Severe Degradation Low Degradation Moderate Degradation Severe Degradation Figure 4.63 Clough Model Fragility, 4 R and 5.0T sec

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132 0 01 02 03 0.4 05 0.6 0.7 08 09 1 0123456789AgProb. of Collaps e Cap Disp. = 4 Cap Disp. = 6 Cap Disp. = 8 Cap Disp. = 4 Cap Disp. = 6 Cap Disp. = 8 Figure 4.64 Clough Model Fragility, 100 and 5.0T sec 0 01 02 03 0.4 05 0.6 0.7 08 09 1 0123456789AgProb. of Collaps e Low Degradation Moderate Degradation Severe Degradation Low Degradation Moderate Degradation Severe Degradation Figure 4.65 Pinching Model Fragility, 4 R and 5.0T sec

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133 0 01 02 03 0.4 05 0.6 0.7 08 09 1 0123456789AgProb. of Collaps e Cap Disp. = 4 Cap Disp. = 6 Cap Disp. = 8 Cap Disp. = 4 Cap Disp. = 6 Cap Disp. = 8 Figure 4.66 Pinching Model Fragility, 100 and 5.0T sec 0 01 02 03 0.4 05 0.6 0.7 08 09 1 0123456789AgProb. of Collaps e Low Degradation Moderate Degradation Severe Degradation Low Degradation Moderate Degradation Severe Degradation Figure 4.67 Bilinear Model Fragility, 4 R and 0.1T sec

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134 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0123456789AgProb. of Collaps e Cap Disp. = 0 Cap Disp. = 4 Cap Disp. = 6 Cap Disp. = 8 Cap Disp. = 0 Cap Disp. = 4 Cap Disp. = 6 Cap Disp. = 8 Figure 4.68 Bilinear Model Fragility, 100 and 0.1T sec 0 01 02 03 0.4 05 0.6 0.7 08 09 1 0123456789AgProb. of Collaps e Low Degradation Moderate Degradation Severe Degradation Low Degradation Moderate Degradation Severe Degradation Figure 4.69 Clough Model Fragility, 4 R and 0.1T sec

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135 0 01 02 03 0.4 05 0.6 0.7 08 09 1 0123456789AgProb. of Collaps e Cap Disp. = 4 Cap Disp. = 6 Cap Disp. = 8 Cap Disp. = 4 Cap Disp. = 6 Cap Disp. = 8 Figure 4.70 Clough Model Fragility, 100 and 0.1T sec 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0123456789AgProb. of Collaps e Low Degradation Moderate Degradation Severe Degradation Low Degradation Moderate Degradation Severe Degradation Figure 4.71 Pinching Model Fragility, 4 R and 0.1T sec

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136 0 01 02 03 0.4 05 0.6 0.7 08 09 1 0123456789AgProb. of Collaps e Cap Disp. = 4 Cap Disp. = 6 Cap Disp. = 8 Cap Disp. = 4 Cap Disp. = 6 Cap Disp. = 8 Figure 4.72 Pinching Model Fragility, 100 and 0.1T sec 0 01 02 03 0.4 05 0.6 0.7 08 09 1 0123456789AgProb. of Collaps e Low Degradation Moderate Degradation Severe Degradation Low Degradation Moderate Degradation Severe Degradation Figure 4.73 Bilinear Model Fragility, 4 R and 0.2T sec

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137 0 01 02 03 0.4 05 0.6 0.7 08 09 1 0123456789AgProb. of Collaps e Cap Disp. = 0 Cap Disp. = 4 Cap Disp. = 6 Cap Disp. = 8 Cap Disp. = 0 Cap Disp. = 4 Cap Disp. = 6 Cap Disp. = 8 Figure 4.74 Bilinear Model Fragility, 100 and 0.2T sec 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0123456789AgProb. of Collaps e No Degradation Low Degradation Moderate Degradation No Degradation Low Degradation Moderate Degradation Figure 4.75 Clough Model Fragility, 4 R and 0.2T sec

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138 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0123456789AgProb. of Collaps e Cap Disp. = 4 Cap Disp. = 6 Cap Disp. = 8 Cap Disp. = 4 Cap Disp. = 6 Cap Disp. = 8 Figure 4.76 Clough Model Fragility, 100 and 0.2T sec 0 01 02 03 0.4 05 0.6 0.7 08 09 1 0123456789AgProb. of Collaps e Low Degradation Moderate Degradation Severe Degradation Low Degradation Moderate Degradation Severe Degradation Figure 4.77 Pinching Model Fragility, 4 R and 0.2T sec

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139 0 01 02 03 0.4 05 0.6 0.7 08 09 1 0123456789AgProb. of Collaps e Cap Disp. = 4 Cap Disp. = 6 Cap Disp. = 8 Cap Disp. = 4 Cap Disp. = 6 Cap Disp. = 8 Figure 4.78 Pinching Model Fragility, 100 and 0.2T sec 4.5.3 Standard Deviation Parameter in Fragility Curves The fragility curves in th is research are based on the two parameter lognormal distribution function to get the S-shape cu rve. This approach was used by several researchers (Shinozuka et al. 2000) and proved to give precise results. The Probability Density Function (PDF) and the Cumulative Density Function (CDF) of the fragility curves follow the subsequent equations: 2 50 2lnln 2 12 1 )(:Tte t tfPDF (4.6) t Ttdt e t tFCDF0 lnln 2 12 50 22 1 )(:(4.7)

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140 Where: The standard deviation, and 50T The median of the results. The two parameters required for plotti ng the lognormal curve are the mean ( ) and standard deviation ( ). For the lognormal distribution, ) ( lnmedian (4.8) In our case, (50T ) represents the value of spectral acceleration (aS ) at probability of collapse of 50%. The variable ( t ) is the number of records that caused collapse to the system. The standard deviation parameter ( ) can be calculated by minimizing the sum of squared of the residual between the data a nd the lognormal model using a solver module. Figure 4.79 shows the effect of the standard deviation parameter ( ) on the fragility curve shape. Prob. of Collapse Sa(g) 50% Value of ( ) increases in this direction Figure 4.79 Example of Use of Sigma in Fragility Curve

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141 A more general 3-parameter equation of the lognormal incorporates an additional parameter ( ) which is called (shift) or (location) parameter. The 3-parameter equation is the same as for the 2-parameter except that ( t ) is replaced by ( t). No collapse can take place before ( ) or 0 in our research case. Figure 4.80 demonstrates the 3-parameter equation and the effect of ( ) on the fragility curve. Prob. of Collapse Sa(g) Fragility curve before Fragility curve applying Figure 4.80 Effect of Parameter ( ) in Fragility Curve The 3-parameter equation didn’t show significant reduction in the residual, therefore, the 2-parameter equation was found to be satisfactory to use. Another simple technique to develop frag ility curves using 2-parameter lognormal equations is to provide the designer with the median Incremental Dynamic Analysis (IDA) curve, and the corresponding 50% probabili ty of collapse, in addition to the value of standard deviation parameter ( ). This process will give flexibility to get the probability of collapse at any value. For exampl e, let us assume that it is required to get the probability of collapse at 84%. Instead of performing multiple dynamic analyses to

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142 construct the 84th percentile Incremental Dynamic Analysis (IDA) curve till collapse or to develop a complete fragility curve, one can plot the fragility curve knowing the standard deviation parameter ( ) and the probability of collapse at 50% from the regular median IDA plot.

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143 CHAPTER 5 ASSESSMENT OF DEGRADED MDOF STRUCTURES 5.1 Degradation Effect on MDOF Sy stems Under Seismic Excitations Degradation plays an importa nt role in the behavior of MDOF structures under seismic excitations. To illustrate the effect of degradation, the response of a degraded MDOF structures is evaluated and compared to a similar non-degraded one. The structure selected is a 3-story buildi ng, whose equivalent SDOF system was evaluated earlier in chapter 4. Figures 5.1 and 5.2 show the re sponse of the non-degraded building modeled as a multi degree of freedom system. The results are compared to those of the equivalent single degree of freedom system shown in figure 4.1. The record used is also the Imperial Valley 1979 recorded at station El Centro 1. The time history trend for the roof displacement of a non-degraded MDOF system in figure 5.1 is almost identical to the SDOF response shown in figure 4.1. As we introduce low intensity degradation to the structure in figure 5.3, the failure occurs after 8.8 sec with a maximum roof displacement of 2.92 inches. The momentrotation diagram for the rotational spring at colu mn base is plotted in figure 5.4. Collapse, which is still designated by the (*) symbol, was found to be at 6.44 E-03 radians. The effect of different levels of degradation becomes more obvious when studying figures 5.5 and 5.6 of a severe degrad ation case. Collapse is recorded after 8.6

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144 sec and a roof displacement of 2.47 inches marking a faster failure to the same building with the same variables except the level of de gradation. When comparing these results to its corresponding SDOF system, collapse was found to occur at the same time but with a larger roof displacement value. This signifies that the subject stru cture will experience higher straining actions requiring a higher cap acity to withstand the new displacement demands. In other words, MDOF analysis inco rporating degradation effects will result into a more accurate and safer design. Th e difference in roof displacement values between a non degraded SDOF and a severe ly degraded MDOF can reach up to 44% which is a significant value worth co nsidering in structural design. -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 0510152025303540TimeDisplacement Figure 5.1 MDOF Time History for Roof Displ., 3 Floors, Bilinear and No Degradation

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145 -8000 -6000 -4000 -2000 0 2000 4000 6000 8000 -0.007-0.006-0.005-0.004-0.003-0.002-0.00100.0010.002RotationMoment Figure 5.2 MDOF Force-Displacement, 3 Floors, Bilinear and No Degradation -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 0510152025303540TimeDisplacement Figure 5.3 MDOF Time History for Roof Displ., 3 Floors, Bilinear and Low Degradation

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146 -8000 -6000 -4000 -2000 0 2000 4000 6000 8000 -0.007-0.006-0.005-0.004-0.003-0.002-0.00100.0010.002RotationMoment Figure 5.4 MDOF Force-Displacement, 3 Floors, Bilinear and Low Degradation -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 0510152025303540TimeDisplacement Figure 5.5 MDOF Time History for Roof Displ., 3 Floors, Bilinear and Severe Degradation

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147 -8000 -6000 -4000 -2000 0 2000 4000 6000 8000 -0.007-0.006-0.005-0.004-0.003-0.002-0.00100.0010.002RotationMoment Figure 5.6 MDOF Force-Displacement, 3 Floors, Bilinear and Severe Degrad. Variation of material models used in an alysis leads also to some deviation in displacement results. Figures 5.7 and 5.9 repr esent the time history for roof displacement and figures 5.8 and 5.10 represent the moment -rotation plots at column base for Clough and pinching models respectively for a seve re degradation case. For the Clough model, collapse was recorded at 8.7 sec with 2.32 in ches of roof displacement and a rotation of 5.6 E-03 radians. Failure in the pinching m odel was more severe in its effect. Even though collapse took place at 8.7 sec too, th e displacement achieved was equal to 2.72 inches, 10% more than that of an equivalent bilinear model and 17% more than that of an equivalent Clough model. As discussed earlier, this is probably due to the accelerated degradation effect. The rotation of the colu mn base spring at co llapse in the pinching model was equal to 6.6 E-03 radians. The moment-rotation curves for Clough and pinching models reflected the material mode ls characteristics pr eviously explained.

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148 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 0510152025303540TimeDisplacement Figure 5.7 MDOF Time History for Roof Displ., 3 Floors, Clough and Severe Degradation -8000 -6000 -4000 -2000 0 2000 4000 6000 8000 -0.007-0.006-0.005-0.004-0.003-0.002-0.00100.0010.002RotationMoment Figure 5.8 MDOF Force-Displacement, 3 Floors, Clough and Severe Degradation

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149 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 0510152025303540TimeDisplacement Figure 5.9 MDOF Time History for Roof Displ., 3 Floor s, Pinching and Severe Degradation -8000 -6000 -4000 -2000 0 2000 4000 6000 8000 -0.007-0.006-0.005-0.004-0.003-0.002-0.00100.0010.002RotationMoment Figure 5.10 MDOF Force-Displacement, 3 Floors, Pinching and Severe Degradation

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150 The number of floors of the structure unde r study plays a key role in the results. Figures 5.11 and 5.13 display roof displacemen t time history plots of a 10 story building for low and severely degraded systems resp ectively. Imposing a lo w degradation to the material models, roof displacement at failu re was equal to 5.66 inches at 16.8 sec compared to 2.92 inches for a similar 3 story. Roof displacement for a severely degraded MDOF system was equal to 5.92 inches but collapse occurred after 8.8 sec only. The equivalent displacement in the thre e-story structure was 2.47 inches. Moment-rotation graphs for low and severe degradation of a ten-story structure are presented in figures 5.12 and 5.14. The lo w degradation case experienced more cycles of loading and unloading than the severely degraded one since it resisted collapse for almost twice the duration. The rotation of the base spring was equal to 7.73 E-03 and 9.43 E-03 radians for low and severe degrad ation respectively. Th e corresponding 3-story values were 6.44 E-03 and 5.61 E-03.

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151 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 0510152025303540TimeDisplacement Figure 5.11 MDOF Time History for Roof Displ., 10 Floors, Bilinear and Low Degradation -15000 -10000 -5000 0 5000 10000 15000 -0.012-0.01-0.008-0.006-0.004-0.00200.002RotationMoment Figure 5.12 MDOF Force-Displacement, 10 Floors, Bilinear and Low Degradation

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152 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 0510152025303540TimeDisplacement Figure 5.13 MDOF Time History for Roof Displ., 10 Floors, Bilinear and Severe Degradation -15000 -10000 -5000 0 5000 10000 15000 -0.012-0.01-0.008-0.006-0.004-0.00200.002RotationMoment Figure 5.14 MDOF Force-Displacement, 10 Floors, Bilinear and Severe Degradation

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153 5.2 Effect of Higher Modes in MDOF Structures Displacements of multistory buildings ma y not be always accurately estimated from analysis of an equivalent SDOF sy stem. Chopra (2005) reviewed a comparative study carried out on several buildings to observe the differenc e between the actual displacement and the one obtained from the equivalent SDOF analysis. The results revealed a large discrepancy between both cases due to th e effects of higher modes. Errors were brought to the f act that, for individual gr ound motions, the SDOF system may drastically deviate the yielding-stimulated permanent drift in the building response. The effect of higher modes has not been accounted for in most seismic codes of practice for buildings, even in the recent FE MA-356 guidelines. The coefficients adopted in these guidelines are based on analysis of equivalent SDOF systems. The effect of higher modes was introduced in earlier studies through an additional coefficient MDOFc by Nassar and Krawinkler (1991). This new coefficient, though important, was not introduced in the FEMA guidelines for simplicity. A recent effort to improve the nonlinear static seismic analysis procedur e adopted in FEMA-356 was presented by Comartin et al. (2004). The st udy, however, still focused on equivalent SDOF systems and did not consider higher m ode effects. In addition, some of the current coefficients values recommended in FEMA-356 are not conf irmed by research results. Chopra (2005) gave an example of the coefficient limitation in the FEMA-356 equation. The 1c factor, for instance, is restricted to a maximum value of 1.5, while this value is considered small when compared with dynamic response anal ysis results. Furthermore, the current procedures are still unable to determine the global MDOF collapse in an explicit form, which might be different than that of an equivalent SDOF system.

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154 The abovementioned discussion exposes the need of a more accurate procedure to provide guidance for code user of SDOF systems to accurately estimate target displacement in MDOF systems. A new factor (MDOF ) is herein introduced that accounts for higher mode effects considering the presence of P due to gravity load together with material models degradation. A numerical stud y using a large ensemble of earthquake records is conducted to study this effect for a series of building models described afterward. 5.3 Building Models The MDOF model used in this research was selected amongst three types of regular 2-dimensional single bay frames commonly used by rese archers. Nassar and Krawinkler (1991) used these models, follo wed by Seneviratna and Krawinkler (1997), then Medina and Krawinkler (2003). The reason behind using these kinds of models was to examine the basic inelastic dynamic behavior patterns. Therefore, the torsional effects of 3-dimensional structures are not encountered. Plasti c hinges are introduced to demonstrate different material models (e.g. bilinear, Cl ough and pinching). The three models are illustrated in figures 5.15 to 5.17. They differ from each other in their yield mechanism. The firs t model, which is the “beam hinge” (BH), represents structures designed following the strong column – weak beam philosophy. In this model, plastic hinges will be formed only in beams ends and columns supports.

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155 Figure 5.15 Beam Hinge Model The second model, designated as “column hinge” (CH), represents structures designed following the weak column – strong beam philosophy. In this model, plastic hinges will be formed only in columns ends between stories and at columns supports. Figure 5.16 Column Hinge Model

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156 The last model, referred to as “weak st ory” (WS), represents structures having a strength discontinuity in their first story. In this model, a nd unlike the two previous ones, the plastic hinges will be formed only in the first story. Figure 5.17 Weak Story Model The beam-hinge model was selected in th is study since its collapse scenario is quite similar to a wide range of structures in common practice. New parameters were introduced to the building model to include thei r effect, such as degradation in material models, and P effect due to gravity loads. In order to achieve the yield mechanis m described above, the relative members’ strengths were tuned so that, under the 2003 IBC equivalent static load pattern, the plastic hinges in all beams and supports form simultaneously.

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157 As for the relative member stiffness of the three models, they were also tuned so that, under the 2003 IBC equivalent static load pa ttern, the interstory dr ift in each story is identical leading to a stra ight line deflected shape. In addition, the stiffnesses are selected such that the first mode period of each structure is equal to that gi ven by the IBC code equation: 4 / 302 0nh T (5.1) Where T First mode period in sec, and nh Height of the building in feet above the base. 5.4 Selection of Representative Buildings The period formula used in this thesis was the one specified in U.S. building codes such as IBC (2003), ATC3-06 (A TC 1978), SEAOC-96 (SEAOC 1996), and NEHRP-94 (NEHRP 1994). The formula is: 4 / 3H C Tt (5.2) Where T = first mode period in sec; H = height of the building in feet above the base; and tC = coefficient equals to 0.030 and 0.035 for R.C. and steel moment resisting frames (MRF) buildings respectively. Medina and Krawinkler (2003) used another fo rmula in their study: CN T (5.3) Where

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158 T = first mode period in sec; C = coefficient equals to 0.1 and 0.2 fo r steel and R.C. MR F respectively; and N = number of stories in this building. According to Goel and Chopra (1997), the formula N T1 0 was recommended in the NEHRP-94 provisions as an alternative formula for R. C. and steel MRF buildings. But this simple formula was restricted to buildings not exceeding 12 stories in height and a minimum story height of 10 ft. Tables 5.1 to 5.5 show, for each building analyzed in this study, the modal periods along with their corresp onding damping ratios. Table 5.1 One-Story Period and Damping Ratios Period Damping Ratio 1st mode 0.129 0.0500 Table 5.2 Two-Story Periods and Damping Ratios Period Damping Ratio 1st mode 0.217 0.0500 2nd mode 0.064 0.0353 Table 5.3 Three-Story Periods and Damping Ratios Period Damping Ratio 1st mode 0.294 0.0500 2nd mode 0.099 0.0270 3rd mode 0.046 0.0430

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159 Table 5.4 Five-Story Periods and Damping Ratios Period Damping Ratio 1st mode 0.431 0.0500 2nd mode 0.160 0.0231 3rd mode 0.085 0.0235 4th mode 0.051 0.0356 5th mode 0.034 0.0582 Table 5.5 Ten-Story Periods and Damping Ratios Period Damping Ratio 1st mode 0.725 0.0500 2nd mode 0.287 0.0216 3rd mode 0.169 0.0160 4th mode 0.114 0.0162 5th mode 0.081 0.0198 6th mode 0.061 0.0265 7th mode 0.047 0.0359 8th mode 0.037 0.0479 9th mode 0.030 0.0623 10th mode 0.025 0.0790

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160 Figure 5.18 Node Numbering

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161 Figure 5.18 show the node numbering for the di fferent floor levels models used in this research. At Columns bases, 2 nodes in series are introduced to connect the regular column section to the rotational spring. The same concept is applied to the floor beam ends where rotational springs are present at each extremity. Each floor beam in these models is divided by three nodes. The gravity loads are then applied at those nodes with a ratio of 25% of the load at ri ght and left, and 50% at the mi ddle. Tables 5.6 to 5.15 show the structural characteristic s of each building analyzed. Table 5.6 One-Story Model Characteristics Floor Column Inertia Beam Spring Stiffness Base Spring Stiffness 1st 20195.5 12201460.0 6100729.8 Table 5.7 Two-Story Model Characteristics Floor Column Inertia Beam Spring Stiffness Base Spring Stiffness 1st 25614.5 15475440.0 2nd 12022.4 7263535.9 7737720.2 Table 5.8 Three-Story Model Characteristics Floor Column Inertia Beam Spring Stiffness Base Spring Stiffness 1st 29128.5 17598487.0 2nd 21443.8 12955649.0 3rd 9950.8 6011956.1 8799243.4

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162 Table 5.9 Five-Story Model Characteristics Floor Column Inertia Beam Spring Stiffness Base Spring Stiffness 1st 34835.8 21046656.0 2nd 31420.9 18983457.0 3rd 25307.1 15289718.0 4th 17785.5 10745435.0 5th 7751.9 4683419.8 10523328.0 Table 5.10 Ten-Story Model Characteristics Floor Column Inertia Beam Spring Stiffness Base Spring Stiffness 1st 45574.7 27534704.0 2nd 44731.0 27024993.0 3rd 41709.7 25199626.0 4th 39948.7 24135692.0 5th 35820.0 21641266.0 6th 31364.2 18949224.0 7th 26072.5 15752116.0 8th 19774.9 11947322.0 9th 13029.5 7871962.0 10th 5454.2 3295233.7 13767352.0

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163 Table 5.11 One-Story Model Sprin gs Yield Characteristics Floor Beam Spring Column Spring 1st 9330.0 6660.0 Table 5.12 Two-Story Model Springs Yield Characteristics Floor Beam Spring Column Spring 1st 26800.0 14900.0 2nd 11100.0 Table 5.13 Three-Story Model Sprin gs Yield Characteristics Floor Beam Spring Column Spring 1st 43900.0 23000.0 2nd 31300.0 3rd 12500.0 Table 5.14 Five-Story Model Spr ings Yield Characteristics Floor Beam Spring Column Spring 1st 61200.0 31200.0 2nd 54700.0 3rd 43800.0 4th 29400.0 5th 11100.0

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164 Table 5.15 Ten-Story Model Sprin gs Yield Characteristics Floor Beam Spring Column Spring 1st 88500.0 44500.0 2nd 86400.0 3rd 81300.0 4th 76700.0 5th 68800.0 6th 60000.0 7th 49500.0 8th 37100.0 9th 23400.0 10th 8530.0 5.4.1 Properties of Building Models As explained earlier in chapter 3, the first modal period is derived from an equation that depends on the structure hei ght. Table 5.16 presents for each building model used in this study the total height in f eet and the corresponding period. The floor height module was set to 12 feet.

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165 Table 5.16 Building Models Total Height and Corresponding First Modal Period Number of Floors Total Height (ft) Period (sec) One 12.0 0.129 Two 24.0 0.217 Three 36.0 0.294 Five 60.0 0.431 Ten 120.0 0.725 Table 5.17 displays the total base shear va lues for each of the buildings used in this study. The IBC (2003) equation was used to calculate the base shear. A distribution of the total base shear is th en applied at each floor. The distribution follows NEHRP load pattern in order to get a linear slope for the first mode of th e displaced floors under earthquake as demonstrated in figure 5.19. Figure 5.19 Deformed Shape Under NEHRP Load Pattern

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166 Table 5.17 Base Shear Distribution at Ea ch Floor Level (NEHRP Load Pattern, k=2) Force at each floor (kips) Floor # One Floor Two Floor Thr ee Floor Five Floor Ten Floor 1 220 147 110 58 23 2 293 220 117 45 3 330 175 68 4 234 90 5 292 113 6 135 7 158 8 180 9 203 10 225 Total Base Shear 220 440 660 876 1240 5.4.2 DRAIN-2DX Runner / Parser One of the major disadvantages of DRAIN2DX is the limitation of records used per single analysis. The program allows only one record at a time to be analyzed. In order to use the 80 records contained in the four bins to perform fragility and displacement estimates curves, around 50 thousands simulatio ns were required. The need to have an automated process mounted at this point a nd brought up the concept for a new software named Drain Runner.

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167 Drain Runner is a software that deals with the end user of DRAIN-2DX in a friendly interface shown in figure 5.20. The pr ogram was created using “visual basic.net” which is a subset of the “visual studio.net” p ackage. The software allows the user to pick the number of floors along with the earthquake r ecords to be used in analysis. The output is then stored in an output file sorted by earthquake records. Figure 5.20 Drain Runner User Interface Window A step counter and a progress bar features were added to the software to allow the user to monitor the current pr ogress of the analysis. An appl ication for the use of those two attributes is clear when studying a pushover analysis for one of the MDOF systems. The initial force is increased by increments defined by the user. Since DRAIN-2DX terminates the analysis at collapse, the user can check exactly the amount of force that caused failure by adding the incremen ts up to the posted step number.

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168 5.5 Displacement Estimates of MDOF Degraded Structures The new factor (MDOF ) introduced to estimate displacements of MDOF structures combines higher mode effects and P effect due to gravity along with material degradation. More importantly, the fact or is a significant tool to predict collapse in its explicit form and not as a number on a damage index scale. The following equation defines in details the new factor (MDOF ): SDOF MDOF MDOF (5.4) Where MDOF The inelastic degraded roof disp lacement of the MDOF system, and SDOF The corresponding inelastic degraded roof displacement of the equivalent SDOF system. The mean value of the coefficient (MDOF ) was derived through analytical simulations using the building models and th e ensemble of earthquake records discussed earlier. The cases considered were bilinear and modified Clough models for one, three, five and ten floors. The bilinear model was subjected to a moderate degradation, equivalent to 100 which is representative of steel buildings. In the modified Clough model, degradation was severe (50 ), which is typical of concrete buildings. Those two values resulted from the experimental and analytical calibrati on of the degradation factors for both materials. Strength reduction factors ( R ) used in the analysis were 4, 6 and 8. Collapse was represented in the plots by the (*) symbol. The dotted lines represent an estimate of the MDOF value until the collapse point. MDOF was plotted as a function of the fundamental period for the MDOF systems analyzed.

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169 Figure 5.21 shows the MDOF effect for a non-degraded system with 8R. The value of MDOF is higher for long period structures than for short period structures. Specifically, MDOF = 1.4 for 10-story buildings, and 1.15 for two-story buildings. This is expected since it is well know n that equivalent SDOF syst ems produce larger errors if used to simulate the behavior of MDOF systems with large nu mber of degrees of freedom. The plot was also constructed for sy stems with cap and low cyclic degradation with cap. In this case, degradation had a severe effect on the inelastic MDOF displacements, particularly for short period st ructures. The trend of the plot changed and became decreasing. In other words the value of MDOF for short period structures was much higher than for long period structures (2.2 and 1.5 for 2 and 10-story structures). The effect of degradation was even more evident for systems with moderate or severe degradation, as shown in fi gure 5.22 with presents the value of MDOF for a bilinear system with 4 R Two cases are presented, a system with cap only, and a system with cap and moderate cyclic degrad ation. From the plots, it is clear that degradation dominated the beha vior of short period structur es, while the trend for long period structures was similar to non-degraded cases. In addition, systems with both cap and cyclic degradation had higher MDOF values than systems with cap only. For instance, for a five story structure, MDOF was higher by 9%. Furthermore, collapse was observed for short period structures; but for systems with both cap and cyclic degradation, it o ccurred at higher MDOF values than for systems with cap degradation only.

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170 Figure 5.23 shows the mean MDOF plot for a Clough model with 4 R Two cases are shown, a system with cap only, and a system with cap and severe cyclic degradation. The effect of degradation was more severe than for similar bilinear models. The trend of the curve was dominated by degrad ation, with larger values for short period structures than for large period structures. In addition, the MDOF values for the system with cap and cyclic degradation were much higher than those of the system with cap only. The increase in the value for a five-story structure was up to 32%. Moreover, collapse was observed in the severely degraded case for structures with period less than 0.4 sec. Strength reduction factors ( R ) affected the ratio significantly. For a non-degraded ten-story bilinear structure with 4 R MDOF was equal to 1.38 (Figure 5.22). This ratio increased by 7% in figure 5.24 to reach 1.47 when 6 R. The difference was even more noticeable when 8R where MDOF was equal to 2.46 (Figure 5.26). The percentage of increase was 78% compared to 4 R and 67% compared to 6 R. In some cases with higher strength reducti on factor values as in figures 5.24 to 5.26 degradation conditions le d to collapse for all buildings considered. This explains why the degraded cases are not plotted in t hose figures. Moreover, some degraded SDOF cases did not collapse, while their corres ponding MDOF cases collapsed. Therefore it was not possible to estimate the displacement ratio MDOF In this case, analysis of equivalent SDOF systems is not considered accurate, and MDOF analysis is necessary to estimate the seismic behavior. These cases include systems with strength reduction factors 6R, or systems with severe degradation parameters 50

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171 The previous conclusions have been draw n considering mean statistical values. To further examine the potential for collapse of MDOF systems, fragility analysis for a collapse criterion of MDOF systems need to be considered. The newly developed fragility curves will cover th e entire spectrum of collapse probability, and are described in the next section. 0 0.5 1 1.5 2 2.5 3 3.5 4 00.10.2030.40.50.60.70.8Period MDOF No Degadation Low Degradation with Cap Figure 5.21 MDOF Bilinear Model 8 R

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172 0 05 1 15 2 25 3 00.10.20.30.40.50.60.70.8Period MDOF Cap Only Cap + Moderate Degradation Figure 5.22 MDOF Bilinear Model 4 R 0 05 1 15 2 25 3 00.10.20.30.40.50.60.70.8Period MDOF Cap Only Cap + Severe Degradation Figure 5.23 MDOF Clough Model 4 R

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173 0 05 1 15 2 25 3 00.10.20.30.40.50.60.70.8Period MDOF Cap Only Cap + Moderate Degradation Figure 5.24 MDOF Bilinear Model 6 R 0 05 1 15 2 25 3 00.10.20.30.40.50.60.70.8Period MDOF Cap Only Cap + Severe Degradation Figure 5.25 MDOF Clough Model 6 R

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174 0 05 1 15 2 25 3 00.10.20.30.40.50.60.70.8Period MDOF Cap Only Cap + Moderate Degradation Figure 5.26 MDOF Bilinear Model 8 R 5.6 Seismic Fragility of Collapse for MDOF Systems A wide variety of structures was consider ed to conduct the fragility analysis in order to have a thorough evaluation of th e collapse potential of MDOF systems. The variables were number of storie s, degradation level, and material models. One, two and three stories were selected to represent s hort buildings whereas five and ten stories represented relatively long buildings. The levels of degradation consid ered in this study are: low, moderate, and severe degradati on. An assumption was made, similar to the study for SDOF systems, which is that the four types of degradation: strength, stiffness, accelerated stiffness, and cap were present si multaneously when considering any level of degradation. Bilinear, modified Clough and pinching models were the three different material models considered. The pinching mode l was not used for cases of five and ten

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175 stories since pinching models represent wood structures and from a practical point of view it is unlikely to find wood structures that high. Each case from the above different combina tions would result in a single point on the fragility curve for a single earthqua ke record and a single earthquake ground acceleration. In order to get re sults as accurate as possible, the four sets of scaled earthquake records, mentioned earlier, were used in the different cases. The plots are generated for a yield factor that corresponds to a strength reduction factor 1 R The yield factor in this case is defined as the yield base shear divi ded by the total weight of the structure. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0123456AgProb. of Collaps e Low Low Moderate Moderate Severe Severe Figure 5.27 One Floor, Bilinear Model, = 0.2231

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176 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0123456AgProb. of Collaps e Low Low Moderate Moderate Severe Severe Figure 5.28 Two Floors, Bilinear Model, = 0.1226 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0123456AgProb. of Collaps e Low Low Moderate Moderate Severe Severe Figure 5.29 Three Floors, Bilinear Model, = 0.1101

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177 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0123456AgProb. of Collaps e Low Low Moderate Moderate Severe Severe Figure 5.30 Five Floors, Bilinear Model, = 0.0881 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0123456AgProb. of Collaps e Low Low Moderate Moderate Severe Severe Figure 5.31 Ten Floors, Bilinear Model, = 0.0622

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178 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0123456AgProb. of Collaps e Low Low Moderate Moderate Severe Severe Figure 5.32 One Floor, Clough Model, = 0.2231 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0123456AgProb. of Collaps e Low Low Moderate Moderate Severe Severe Figure 5.33 Two Floors, Clough Model, = 0.1226

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179 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0123456AgProb. of Collaps e Low Low Moderate Moderate Severe Severe Figure 5.34 Three Floors, Clough Model, = 0.1101 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0123456AgProb. of Collaps e Data Model Data Model Data Model Figure 5.35 Five Floors, Clough Model, = 0.0881

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180 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0123456AgProb. of Collaps e Low Low Moderate Moderate Severe Severe Figure 5.36 Ten Floors, Clough Model, = 0.0622 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0123456AgProb. of Collaps e Low Low Moderate Moderate Severe Severe Figure 5.37 One Floor, Pinching Model, = 0.2231

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181 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0123456AgProb. of Collaps e Low Low Moderate Moderate Severe Severe Figure 5.38 Two Floors, Pinching Model, = 0.1226 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0123456AgProb. of Collaps e Low Low Moderate Moderate Severe Severe Figure 5.39 Three Floors, Pinching Model, = 0.1101

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182 The fragility curves for bilinear one, two, three, five, and ten story structures are shown in figures 5.27 to 5.31. The curves ar e plotted for a yield base shear corresponding to a strength reduction factor 1 R The plots reveal that ta ll structures (five and ten story structures) have a lower probability of collapse. The 50% and 90% probability of collapse for a moderately degraded one-s tory structure occur at 1.5g and 2.3g respectively. The same values for a ten-story structure are 3.3g and 6g. From the plots it is clear that short story ductile systems have at least a 50% probability of collapse. Tall structures, however, have a lower probability of collapse, but could still exceed a 50% probability for very ductile systems. The fragility curves for the Clough mode l are shown in figure 5.32 to 5.36. As for the bilinear model, the probability of collaps e for taller structures is less than that of short structures. The 50% and 90% probability of collapse for a moderately degraded one-story structure occur at 1.15g and 2.2g resp ectively. The same values for a ten-story structure are 2.55g and 5.05g. Ductile short-st ory systems have a probability of collapse larger than 50%, while tall structures have a probability of collapse less than 50%, similar to bilinear systems. From th e plots, Clough models have a sl ower rate of collapse than bilinear models. This conclusion was also true for SDOF systems analyzed in chapter 4. The effect of the level of degradation, however, is smaller than for bilinear models. Figures 5.37 to 5.39 show the fragility curves for the pinching model. Pinching models in general have a lower probability of collapse than similar bilinear and Clough models. Ductile pinching models with 6R still have a probabili ty of collapse larger

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183 than 50%. The level of degradation for pinc hing models did not ha ve a major effect on the behavior though. The final conclusion is that short period st ructures are more susceptible to damage than long period structures. Therefore, dama ge and collapse are expected in low rise buildings more than in high rise build ings for the same ground acceleration and degradation levels. Bilinear models have a higher collapse probability followed by Clough and pinching models. The effect of the le vel of degradation is more apparent for bilinear models than for Clough or pinching models. Furthermore a mean (50%) collapse probability is expected for ductile (6R) severely or modera tely degraded short structures for all material models. This conclusion doesn’t hold true however for structures with number of stories greater than 5. 5.7 Practical Use of Proposed Design Curves To illustrate the use of the previously developed design curves, the following example is considered. Consider a five-story build ing with period T 0.431 sec and subject to earthquake record defined us ing USGS LA 10/50 spectrum. The following steps are used to evaluate the parameter ( ) needed to use the design curves: From the USGS spectrum, calculate the spec tral acceleration of th e building. For the case of a period of 0.431 sec, this value equals 1.1 g. As a designer, select the desi red strength reduction factor ( R ). In this example, R is selected to equal 4. Assuming elastic behavior, determine the ma ximum spring forces if the building is subject to the record scaled to a value of 1.1 g determined in the first step. In this case

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184 these values are: first floor beam spri ng 61200 kip-ft, second fl oor 54700 kip-ft, third floor 43800 kip-ft, fourth fl oor 29400 kip-ft, fifth floor 11100 kip-ft and column spring 31200 kip-ft. Divide the forces obtained in the previous step by the selected R value to obtain the yield forces of the springs. Fo r the case of the selected (4 R ) these values are: first floor beam spring 15300 kip-ft, second fl oor 13675 kip-ft, third floor 10950 kip-ft, fourth floor 7350 kip-ft, fi fth floor 2775 kip-ft and column spring 7800 kip-ft. Conduct a pushover analysis of the building us ing a triangular load pattern to develop the base shear – roof displacement curve. From the curve identify the value of the yield base shear and in this case the value is equal to 22.02 kips. The parameter ( ) is defined as the ratio of the yiel d base shear to the total weight of the building which is 1000 kips. In this example ( ) is equal to 0.022.

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185 CHAPTER 6 SUMMARY AND CONCLUSION 6.1 Summary The research study presents a discussion on the behavior and collapse potential of degrading structures under seismic excita tions. The study is essential for the design evaluation phase of a performance-based eart hquake design process, particularly for collapse prevention limit states. New constituti ve models for degrading structures are developed and added to the material library of the nonlinear structural analysis program DRAIN-2DX. These material models represent bilinear models for steel structures, Clough models for concrete structures, and pinching models for timber structures. All models include a strength softening branch, re ferred to as a cap, to model strength degradation under monotonic loads. An 8 para meter energy-based model was developed to model four different types of cyclic de gradation: Yield (Str ength) degradation, Unloading stiffness degradation, Accelerated stiffness degradation, and Cap degradation. Collapse is explicitly defined if the material completely loses its strength either due to severe cyclic deterioration or to strength softening. The degradation parameters were calibrated against available experimental data. A set of earthquake records is selected to conduct studies on degrading systems. An initial study proved that efficient scaling of the records can reduce considerably the variabil ity in results without introducing any bias,

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186 and thus a much smaller number of non-linear analyses are needed. The set of scaled earthquake records was used to conduct statisti cal analysis of a larg e ensemble of generic structural systems. The systems represente d both SDOF and MDOF structures with different fundamental periods of vibration. For MDOF structures gravity loads and P effects were accounted for in the model. In addition, several other parameters were investigated. These included material type, yi eld force, and levels of degradation. For each study conducted, the degrading behavior was evaluated and compared to the nondegraded behavior through several numeri cal relationships. The relationships were expressed with plots that included: mean and 84% percentile inelastic displacement ratios, mean MDOF displacement ratios, mean incremental dynamic analysis plots, and seismic fragility curves for a collapse criter ion. The potential for collapse was explicitly studied in the fragility analysis, and was invest igated in the other analytical studies. The study proved to be essential for evaluating cu rrent analysis techni ques and new seismic design codes for buildings. Se veral conclusions were draw n from the study and are explained in the next section. 6.2 Conclusion The following conclusions were drawn from the study: Scaling of earthquake records proved to be an efficient way to reduce the variability in results, and theref ore a smaller number of nonlinear analyses is needed to conduct statistical studies. A difference less than 10% between the scaled and unscaled responses is typically observ ed ensuring that no bias is introduced by the scaling procedure. In addition, the dispersion in re sults is considerably lower for the scaled

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187 set of records. This conc lusion is valid for both non-degraded and degraded, SDOF and MDOF systems. For SDOF systems, degradation had a grea t effect on the inelastic displacement ratios, especially for short period structur es where the inelastic displacements were quite larger than the corresponding displacem ents of non-degraded systems. For very short period SDOF structures, collapse is typically expected even for systems with low strength reduction factors. For lo ng period structures, the well-known equal displacement rule is preserved even for degrading systems. In this case, collapse is not expected even for systems with large strength reduction factors. For short period SDOF structures, bilinear m odels collapse due to cyclic degradation effects, due to the large energy dissipation. In this case, degrad ation strongly reduces the ductility capacity. Clough sy stems collapse mainly due to softening effects. Cyclic degradation accelerates failure, but to a much lesser extent than for bilinear models. Pinching models are strongly affect ed by accelerated degradation. In this case, the inclusion of degradation ac tually increases the ductility capacity. For medium period SDOF structures, bilinear models fail due to cyclic degradation effects, while both Clough and pinching models fail mainly due to softening effects, with the degradation accelerating the failu re rate. For bilinear models, degradation strongly affects the ductil ity capacity. For Clough and pinching models, cyclic degradation has a smaller effect on the ductility capacity, while cap degradation strongly affected the ductility capacity.

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188 For long period SDOF structures, the load-deformation dynamic response for degrading and non-degrading systems follows the same trend, confirming the equal displacement rule. Degradation in this case affects only the failure point. For short period SDOF systems, bilinear models have the fastest collapse rate, followed by Clough and pinching models. The failure mode of each system is different though. Medium period SDOF systems have a slower collapse rate than short period systems, with the bilinear model having the fastest collapse rate. The collapse rate of long period SDOF struct ures is very slow, with most systems needing very intense earthqua ke records to collapse. Degradation had a great effect on the displa cements of MDOF structures. The effect of higher modes is typically larger for l ong period non-degraded structures than for short period ones. Degradation, however, strongly affected the displacements of higher modes of short period structures, wh ile its effect on the displacements of higher modes of long period structures was less pronounced. The final outcome was that the effect of higher modes was even tually smaller for long period degraded structures than for short peri od ones. This conclusion is part icularly true for severely degraded structures, but is also vali d for systems with low degradation. The analysis of MDOF structures showed that in some cases the MDOF structure collapsed, while its equivalent SDOF system di d not collapse. In this case, analysis of equivalent SDOF systems is not consid ered accurate, and MDOF analysis is necessary to estimate the seismic behavior. These cases include systems with short

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189 periods (4 0T sec.) and strength reduction factors 6R, or systems with severe degradation parameters 50 Seismic fragility analysis of MDOF structures showed that tall structures have a much lower probability of collapse than short st ory structures (3-story and less). Short period structures are therefore more sus ceptible to damage and collapse than long period structures. Bilinear MDOF structures have also a fa ster collapse rate followed by Clough and pinching structures. The effect of the leve l of degradation is more apparent for bilinear models than for Clough or pinching models. 6.3 Recommendations While the current study was based on extensiv e statistical evaluati on of the inelastic seismic behavior of both SDOF and MDOF de grading structures a nd their potential for collapse, further work still needs to be performed in order to better understand the complex degrading behavior of structures before fully implementing it in codes of practice. The following is a list of recomm endations and ideas for possible future research work: 1. The current study focused on MDOF build ings designed according to the BeamHinge (BH) concept, where collapse mechanisms formed due to plastic hinges occurring at the beam ends and column base. The study needs to be extended to include also buildings designe d according to the Column-H inge (CH) and Weak First Story (WS) concepts.

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190 2. The current study did not account for shear degradation of members, but rather focused on their flexural degradation. It is important to include shear effects in the analysis of MDOF structures particularly for shear critical members and columns. Earlier studies showed that the loss of shear capacity for columns might cause a subsequent loss of axial capacity which might lead to partial or full collapse of the entire building. 3. The current study was conducted for two-dime nsional structures only. While the 2D assumption might be valid for regular symm etric buildings, it might not hold true for buildings with plan irregularities, where tori sonal deformations become an issue. It is therefore important to exte nd the current study to thre e-dimensional structures. 4. The current study assumed no coupling effect between the different force actions acting on a structural element. The combin ed effect of bending, shear, axial and torsional forces is a complex, yet important effect that needs to be addressed for collapse analysis of building structures. 5. The current study accounted for P-Delta e ffect due to gravity loads along with material degradation. The effect of P-Delta needs to be further explored. In other words, the effect of excluding P-Delta a nd accounting only for material degradation needs to be fully investigated. 6. The current study focused only on generic regular building structures. The study needs to be extended to evaluate the beha vior of buildings with stiffness or mass discontinuities. Furthermore, the behavior of shear wall types of buildings needs to be also investigated.

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191 7. The study assumed the building structures analyzed are subject to ordinary earthquake records recorded on stiff soil or soft rock. The study needs to be extended to consider other types of earthquake records, such as near fault and long duration records. 8. Current seismic specifications do not account for soil-structure interaction effects even though previous research studies confirmed the impor tance of this effect. The effect of the soil interaction on the collapse potential of MDOF buildings needs to be evaluated. 9. The current study focused on evaluating inel astic target roof displacements of degrading MDOF structures, in addition to the potential of the structure for collapse. Other seismic demand parameters such as inter-story drifts and maximum plastic rotations need to be also investigated for degrading st ructures. Strength parameters such as maximum base and story shears and base and stor y overturning moments need to be also studied. 10. The study needs to investigate also the be havior of non-structur al components of degrading buildings. This could be acco mplished by investigating the effect of degradation on both, floor acce lerations and velocities. 11. Finally, a thorough evaluation and modifica tions of existing seismic design codes needs to be performed based on the outcome s of this study in order to reflect the effect of degradation and potential for coll apse. Such effect is not accounted for in existing methods for seismic demand evaluation.

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ABOUT THE AUTHOR Mouchir Chenouda received a Bachelor’s De gree in Civil Engineering from Cairo University in 1997 and a Master’s of Scien ce in Construction Ma nagement in 2002. He joined the Doctoral program at the Univer sity of South Florid a in the same year. While in the Ph.D. program at the Univ ersity of South Fl orida, Mr. Chenouda started teaching as an instructor of the Mechanics of Materials Laboratory. He was awarded the Provost’s Award for Outsta nding Teaching by a Graduate Teaching Assistant in 2004. Mr. Chenouda has also coau thored two publicati ons in structural journals and made several paper presentati ons at international seismic conferences.


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Performance based design of degrading structures
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ABSTRACT: Seismic code provisions are now adopting performance-based methodologies, where structures are designed to satisfy multiple performance objectives. Most codes rely on approximate methods to predict the desired seismic demand parameters. Most of these methods are based on simple SDOF models, and do not take into account neither MDOF nor degradation effects, which are major factors influencing structural behavior under earthquake excitations. More importantly, most of these models can not predict collapse explicitly under severe seismic loads. This research presents a newly developed model that incorporates degradation effects into seismic analysis of structures. A new energy-based approach is used to define several types of degradation effects. The research presents also an evaluation of the collapse potential of degrading SDOF and MDOF structures. Collapse under severe seismic excitations, which is typically due to the formation of structures mechanisms amplified by P-Delt a effects, was modeled in this work through the degrading hysteretic structural behavior along with P-Delta effects due to gravity loads. The model was used to conduct extensive statistical dynamic analysis of different structural systems subjected to a large set of recent earthquake records. To perform this task, finite element models of a series of generic SDOF and MDOF structures were developed. The degrading hysteretic structural behavior along with P-Delta effects due to gravity loads proved to successfully replicate explicit collapse. For each structure, collapse was investigated and inelastic displacement ratios curves were developed in case collapse doesn't occur. Furthermore, seismic fragility curves for a collapse criterion were also developed. In general, seismic fragility of a system describes the probability of the system to reach or exceed different degrees of damage. Earlier work focused on developing seismic fragility curves of systems for several values of a calibrated ^damage index. This research work focuses on developing seismic fragility curves for a collapse criterion, in an explicit form. The newly developed fragility curves represent a major advancement over damage index-based fragility curves in assessing the collapse potential of structures subject to severe seismic excitations. The research findings provide necessary information for the design evaluation phase of a performance-based earthquake design process.
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Dissertation (Ph.D.)--University of South Florida, 2006.
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Co-adviser: Rajan Sen, Ph.D.
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Displacement estimates.
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