Theory of Nonlinear Structural Analysis

Preface ix

About the Authors xi

1 Introduction 1

1.1 History of the Force Analogy Method 1

1.2 Applications of the Force Analogy Method 4

1.2.1 Structural Vibration Control 4

1.2.2 Modal Dynamic Analysis Method 6

1.2.3 Other Design and Analysis Areas 6

1.3 Background of the Force Analogy Method 6

References 14

2 Nonlinear Static Analysis 17

2.1 Plastic Rotation 17

2.2 Force Analogy Method for Static Single-Degree-of-Freedom Systems 19

2.2.1 Inelastic Displacement 19

2.2.2 Application of the FAM on SDOF System 20

2.2.3 Nonlinear Analysis Using FAM 22

2.3 Nonlinear Structural Analysis of Moment-Resisting Frames 26

2.4 Force Analogy Method for Static Multi-Degree-of-Freedom Systems 31

2.5 Nonlinear Static Examples 36

2.6 Static Condensation 52

References 61

3 Nonlinear Dynamic Analysis 63

3.1 State Space Method for Linear Dynamic Analysis 63

3.1.1 Equation of Motion 64

3.1.2 State Space Solution 66

3.1.3 Solution Procedure 68

3.2 Dynamic Analysis with Material Nonlinearity 72

3.2.1 Force Analogy Method 72

3.2.2 State Space Analysis with the Force Analogy Method 74

3.2.3 Solution Procedure 76

3.3 Nonlinear Dynamic Analysis with Static Condensation 87

3.4 Nonlinear Dynamic Examples 99

References 109

4 Flexural Member 111

4.1 Bending and Shear Behaviors 111

4.1.1 Hysteretic Models 111

4.1.2 Displacement Decomposition 113

4.1.3 Local Plastic Mechanisms 115

4.2 Inelastic Mechanisms of Flexural Members 115

4.2.1 Elastic Displacement X' 116

4.2.2 Plastic Bending Displacement X₁'' 117

4.2.3 Plastic Shear Displacement X₂'' 117

4.2.4 Combination of the Bending and Shear Behaviors 117

4.3 Nonlinear Static Analysis of Structures with Flexural Members 118

4.3.1 Force Analogy Method for Static Single-Degree-of-Freedom Systems 118

4.3.2 Force Analogy Method for Static Multi-Degree-of-Freedom Systems 129

4.4 Nonlinear Dynamic Analysis of Structures with Flexural Members 143

4.4.1 Hysteretic Behaviors of the Flexural Members 143

4.4.2 Solution Procedure of the FAM 146

References 159

5 Axial Deformation Member 161

5.1 Physical Theory Models for Axial Members 161

5.1.1 General Parameters 162

5.1.2 Displacement Decomposition 163

5.2 Sliding Hinge Mechanisms 164

5.3 Force Analogy Method for Static Axial Members 166

5.3.1 Regions O-A_a and O-F 166

5.3.2 Region F-G 166

5.3.3 Regions A_a-A and A-B 167

5.4 Force Analogy Method for Cycling Response Analysis of Axial Members 170

5.4.1 Region B-C 170

5.4.2 Region C-D 171

5.4.3 Region D'-A² 172

5.4.4 Region D-E 173

5.4.5 Region E-F 174

5.4.6 Region A_a2-A₂ 174

5.5 Application of the Force Analogy Method in Concentrically Braced Frames 178

5.5.1 Force Analogy Method for Static SDOF CBFs 178

5.5.2 Force Analogy Method for Static MDOF CBFs 182

5.5.3 Force Analogy Method for Dynamical CBFs under Earthquake Loads 188

References 194

6 Shear Member 195

6.1 Physical Theory Models of Shear Members 195

6.1.1 Flexural Behavior 195

6.1.2 Axial Behavior 197

6.1.3 Shear Behavior 197

6.2 Local Plastic Mechanisms in the FAM 198

6.2.1 Displacement Decomposition 198

6.2.2 Behavior of VSH 199

6.2.3 Behavior of HSH 200

6.3 Nonlinear Static Analysis of the Shear Wall Structures 201

6.4 Nonlinear Dynamic Analysis of RC Frame-Shear Wall Structures 222

6.4.1 Hysteretic Behaviors of the RC Shear Wall Members 222

6.4.2 Solution Procedure of the FAM 224

References 234

7 Geometric Nonlinearity 235

7.1 Classical Stiffness Matrices with Geometric Nonlinearity 236

7.1.1 The P-¿ Approach 237

7.1.2 The Geometric Stiffness Approach 238

7.2 Stability Functions 239

7.2.1 Stiffness Matrix [k_i] 240

7.2.2 Stiffness Matrix [k_i' ] 244

7.2.3 Stiffness Matrix [K_i''] 246

7.3 Force Analogy Method with Stability Functions 250

7.4 Nonlinear Dynamic Analysis Using Stability Functions 261

7.4.1 Force Analogy Method 261

7.4.2 Nonlinear Dynamic Analysis with the Force Analogy Method 262

7.4.3 State Space Analysis with Geometric and Material Nonlinearities 263

7.5 Nonlinear Dynamic Analysis with Static Condensation Using Stability Functions 272

7.6 Nonlinear Dynamic Examples 283

References 294

8 Application of the Force Analogy Method in Modal Superposition 297

8.1 Nonlinear Static Pushover Analysis in the FAM 298

8.1.1 NSPA for Mass-Normalized SDOF Systems 299

8.1.2 NSPA for Multi-Degree-of-Freedom Systems 303

8.2 Modal Decomposition in the FAM 312

8.3 Modal Response Summation 318

8.4 Nonlinear Modal Superposition Method Example 319

References 329

9 Application: Structural Vibration Control 331

9.1 Passive Control Technique 331

9.1.1 Model of Passive Energy-Dissipation Devices 331

9.1.2 Model of Framed Structures with PEDDs 333

9.1.3 Force Analogy Method for Dynamical Analysis of Multi-Degree-Freedom Systems 334

9.2 Application of the FAM in Active or Semi-Active Structural Control 336

9.2.1 Background of MBC 336

9.2.2 Force Analogy Method in Market-Based Control 342

References 349

Index 351