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PART Ⅰ SINGLE-DEGREE-OF-FREEDOM SYSTEMS 1

1 Equations of Motion，Problem Statement，and Solution Methods 3

1.1 Simple Structures 3

1.2 Single-Degree-of-Freedom System 7

1.3 Force-Displacement Relation 8

1.4 Damping Force 13

1.5 Equation of Motion：External Force 14

1.6 Mass-Spring-Damper System 18

1.7 Equation of Motion：Earthquake Excitation 20

1.8 Problem Statement and Element Forces 23

1.9 Combining Static and Dynamic Responses 25

1.10 Methods of Solution of the Differential Equation 25

1.11 Study of SDF Systems：Organization 29

Appendix 1：Stiffness Coefficients for a Flexural Element 30

2 Free Vibration 35

2.1 Undamped Free Vibration 35

2.2 Viscously Damped Free Vibration 44

2.3 Energy in Free Vibration 52

2.4 Coulomb-Damped Free Vibration 53

3 Response to Harmonic and Periodic Excitations 61

Part A：Viscously Damped Systems：Basic Results 62

3.1 Harmonic Vibration of Undamped Systems 62

3.2 Harmonic Vibration with Viscous Damping 68

Part B：Viscously Damped Systems：Applications 80

3.3 Response to Vibration Generator 80

3.4 Natural Frequency and Damping from Harmonic Tests 83

3.5 Force Transmission and Vibration Isolation 85

3.6 Response to Ground Motion and Vibration Isolation 87

3.7 Vibration-Measuring Instruments 91

3.8 Energy Dissipated in Viscous Damping 94

3.9 Equivalent Viscous Damping 98

Part C：Systems with Nonviscous Damping 100

3.10 Harmonic Vibration with Rate-Independent Damping 100

3.11 Harmonic Vibration with Coulomb Friction 104

Part D：Response to Periodic Excitation 108

3.12 Fourier Series Representation 109

3.13 Response to Periodic Force 109

Appendix 3：Four-Way Logarithmic Graph Paper 113

4 Response to Arbitrary，Step，and Pulse Excitations 119

Part A：Response to Arbitrarily Time-Varying Forces 119

4.1 Response to Unit Impulse 120

4.2 Response to Arbitrary Force 121

Part B：Response to Step and Ramp Forces 123

4.3 Step Force 123

4.4 Ramp or Linearly Increasing Force 125

4.5 Step Force with Finite Rise Time 126

Part C：Response to Pulse Excitations 129

4.6 Solution Methods 129

4.7 Rectangular Pulse Force 131

4.8 Half-Cycle Sine Pulse Force 137

4.9 Symmetrical Triangular Pulse Force 142

4.10 Effects of Pulse Shape and Approximate Analysis for Short Pulses 144

4.11 Effects of Viscous Damping 147

4.12 Response to Ground Motion 149

5 Numerical Evaluation of Dynamic Response 155

5.1 Time-Stepping Methods 155

5.2 Methods Based on Interpolation of Excitation 157

5.3 Central Difference Method 161

5.4 Newmark’s Method 164

5.5 Stability and Computational Error 170

5.6 Analysis of Nonlinear Response：Central Difference Method 174

5.7 Analysis of Nonlinear Response：Newmark’s Method 174

6 Earthquake Response of Linear Systems 187

6.1 Earthquake Excitation 187

6.2 Equation of Motion 193

6.3 Response Quantities 194

6.4 Response History 195

6.5 Response Spectrum Concept 197

6.6 Deformation，Pseudo-velocity，and Pseudo-acceleration Response Spectra 198

6.7 Peak Structural Response from the Response Spectrum 206

6.8 Response Spectrum Characteristics 211

6.9 Elastic Design Spectrum 217

6.10 Comparison of Design and Response Spectra 225

6.11 Distinction between Design and Response Spectra 227

6.12 Velocity and Acceleration Response Spectra 228

Appendix 6：El Centro，1940 Ground Motion 232

7 Earthquake Response of Inelastic Systems 241

7.1 Force-Deformation Relations 242

7.2 Normalized Yield Strength，Yield Reduction Factor，and Ductility Factor 248

7.3 Equation of Motion and Controlling Parameters 249

7.4 Effects of Yielding 250

7.5 Response Spectrum for Yield Deformation and Yield Strength 257

7.6 Design Strength and Deformation from the Response Spectrum 261

7.7 Design Yield Strength 261

7.8 Relative Effects of Yielding and Damping 263

7.9 Dissipated Energy 264

7.10 Inelastic Design Spectrum 269

7.11 Comparison of Design and Response Spectra 274

8 Generalized Single-Degree-of-Freedom Systems 277

8.1 Generalized SDF Systems 277

8.2 Rigid-Body Assemblages 279

8.3 Systems with Distributed Mass and Elasticity 281

8.4 Lumped-Mass System：Shear Building 292

8.5 Natural Vibration Frequency by Rayleigh’s Method 298

8.6 Selection of Shape Function 302

Appendix 8：Inertia Forces for Rigid Bodies 306

PART Ⅱ MULTI-DEGREE-OF-FREEDOM SYSTEMS 311

9 Equations of Motion，Problem Statement，and Solution Methods 313

9.1 Simple System：Two-Story Shear Building 313

9.2 General Approach for Linear Systems 318

9.3 Static Condensation 334

9.4 Planar or Symmetric-Plan Systems：Ground Motion 337

9.5 Unsymmetric-Plan Buildings：Ground Motion 342

9.6 Symmetric-Plan Buildings：Torsional Excitation 350

9.7 Multiple Support Excitation 351

9.8 Inelastic Systems 355

9.9 Problem Statement 356

9.10 Element Forces 356

9.11 Methods for Solving the Equations of Motion：Overview 357

10 Free Vibration 365

Part A：Natural Vibration Frequencies and Modes 366

10.1 Systems without Damping 366

10.2 Natural Vibration Frequencies and Modes 368

10.3 Modal and Spectral Matrices 370

10.4 Orthogonality of Modes 371

10.5 Interpretation of Modal Orthogonality 372

10.6 Normalization of Modes 372

10.7 Modal Expansion of Displacements 382

Part B：Free Vibration Response 383

10.8 Solution of Free Vibration Equations：Undamped Systems 383

10.9 Free Vibration of Systems with Damping 386

10.10 Solution of Free Vibration Equations：Classically Damped Systems 390

Part C：Computation of Vibration Properties 392

10.11 Solution Methods for the Eigenvalue Problem 392

10.12 Rayleigh’s Quotient 394

10.13 Inverse Vector Iteration Method 394

10.14 Vector Iteration with Shifts：Preferred Procedure 399

10.15 Transformation of kφ=ω2mφ to the Standard Form 404

11 Damping in Structures 409

Part A：Experimental Data and Recommended Modal Damping Ratios 409

11.1 Vibration Properties of Millikan Library Building 409

11.2 Estimating Modal Damping Ratios 414

Part B：Construction of Damping Matrix 416

11.3 Damping Matrix 416

11.4 Classical Damping Matrix 417

11.5 Nonclassical Damping Matrix 425

12 Dynamic Analysis and Response of Linear Systems 429

Part A：Two-Degree-of-Freedom Systems 429

12.1 Analysis of Two-DOF Systems without Damping 429

12.2 Vibration Absorber or Tuned Mass Damper 432

Part B：Modal Analysis 434

12.3 Modal Equations for Undamped Systems 434

12.4 Modal Equations for Damped Systems 436

12.5 Displacement Response 438

12.6 Element Forces 438

12.7 Modal Analysis：Summary 439

Part C：Modal Response Contributions 444

12.8 Modal Expansion of Excitation Vector p（t） = sp（t） 444

12.9 Modal Analysis for p（t） = sp（t） 447

12.10 Modal Contribution Factors 448

12.11 Modal Contributions to Response 449

Part D：Special Analysis Procedures 455

12.12 Static Correction Method 455

12.13 Mode Acceleration Superposition Method 458

12.14 Analysis of Nonclassically Damped Systems 459

13 Earthquake Analysis of Linear Systems 467

Part A：Response History Analysis 468

13.1 Modal Analysis 468

13.2 Multistory Buildings with Symmetric Plan 474

13.3 Multistory Buildings with Unsymmetric Plan 492

13.4 Torsional Response of Symmetric-Plan Buildings 503

13.5 Response Analysis for Multiple Support Excitation 508

13.6 Structural Idealization and Earthquake Response 513

Part B：Response Spectrum Analysis 514

13.7 Peak Response from Earthquake Response Spectrum 514

13.8 Multistory Buildings with Symmetric Plan 519

13.9 Multistory Buildings with Unsymmetric Plan 532

14 Reduction of Degrees of Freedom 549

14.1 Kinematic Constraints 550

14.2 Static Condensation 551

14.3 Rayleigh-Ritz Method 551

14.4 Selection of Ritz Vectors 554

14.5 Dynamic Analysis Using Ritz Vectors 560

15 Numerical Evaluation of Dynamic Response 565

15.1 Time-Stepping Methods 565

15.2 Analysis of Linear Systems with Nonclassical Damping 567

15.3 Analysis of Nonlinear Systems 574

16 Systems with Distributed Mass and Elasticity 585

16.1 Equation of Undamped Motion：Applied Forces 586

16.2 Equation of Undamped Motion：Support Excitation 587

16.3 Natural Vibration Frequencies and Modes 588

16.4 Modal Orthogonality 595

16.5 Modal Analysis of Forced Dynamic Response 596

16.6 Earthquake Response History Analysis 600

16.7 Earthquake Response Spectrum Andlysis 604

16.8 Difficulty in Analyzing Practical Systems 607

17 Introduction to the Finite Element Method 613

Part A：Rayleigh-Ritz Method 613

17.1 Formulation Using Conservation of Energy 613

17.2 Formulation Using Virtual Work 617

17.3 Disadvantages of Rayleigh-Ritz Method 618

Part B：Finite Element Method 619

17.4 Finite Element Approximation 619

17.5 Analysis Procedure 621

17.6 Element Degrees of Freedom and Interpolation Functions 622

17.7 Element Stiffness Matrix 624

17.8 Element Mass Matrix 625

17.9 Element（Applied） Force Vector 626

17.10 Comparison of Finite Element and Exact Solutions 630

17.11 Dynamic Analysis of Structural Continua 632

PART Ⅲ EARTHQUAKE RESPONSE AND DESIGN OF MULTISTORY BUILDINGS 639

18 Earthquake Response of Linearly Elastic Buildings 641

18.1 Systems Analyzed，Design Spectrum，and Response Quantities 641

18.2 Influence of T1 and p on Response 646

18.3 Modal Contribution Factors 647

18.4 Influence of T1 on Higher-Mode Response 649

18.5 Influence of p on Higher-Mode Response 652

18.6 Heightwise Variation of Higher-Mode Response 653

18.7 How Many Modes to Include 655

19 Earthquake Response of Inelastic Buildings 659

19.1 Allowable Ductility and Ductility Demand 660

19.2 Buildings with “Weak” or “Soft” First Story 665

19.3 Buildings Designed for Code Force Distribution 670

19.4 Limited Scope 680

20 Earthquake Dynamics of Base-Isolated Buildings 683

20.1 Isolation Systems 683

20.2 Base-Isolated One-Story Buildings 686

20.3 Effectiveness of Base Isolation 691

20.4 Base-Isolated Multistory Buildings 695

20.5 Applications of Base Isolation 701

21 Structural Dynamics in Building Codes 703

Part A：Building Codes and Structural Dynamics 704

21.1 Uniform Building Code（United States），1994 704

21.2 National Building Code of Canada，1995 707

21.3 Mexico Federal District Code，1987 711

21.4 Structural Dynamics in Building Codes 713

Part B：Evaluation of Building Codes 720

21.5 Base Shear 720

21.6 Story Shears and Equivalent Static Forces 723

21.7 Overturning Moments 726

21.8 Concluding Remarks 728

A Notation 731

B Answers to Selected Problems 743

Index 753