Dynamics of Structures

Contents

Foreword xxi

Preface xxiii

Preface to the Second Edition xxv

Preface to the First Edition xxvii

Acknowledgments xxxiii

 

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 12

1.5 Equation of Motion: External Force 14

1.6 Mass—Spring—Damper System 19

1.7 Equation of Motion: Earthquake Excitation 23

1.8 Problem Statement and Element Forces 26

1.9 Combining Static and Dynamic Responses 28

1.10 Methods of Solution of the Differential Equation 28

1.11 Study of SDF Systems: Organization 33

Appendix 1: Stiffness Coefficients for a Flexural Element 33

 

2 Free Vibration 39

2.1 Undamped Free Vibration 39

2.2 Viscously Damped Free Vibration 48

2.3 Energy in Free Vibration 56

2.4 Coulomb-Damped Free Vibration 57

 

3 Response to Harmonic and Periodic Excitations 65

Part A: Viscously Damped Systems: Basic Results 66

3.1 Harmonic Vibration of Undamped Systems 66

3.2 Harmonic Vibration with Viscous Damping 72

Part B: Viscously Damped Systems: Applications 85

3.3 Response to Vibration Generator 85

3.4 Natural Frequency and Damping from Harmonic Tests 87

3.5 Force Transmission and Vibration Isolation 90

3.6 Response to Ground Motion and Vibration Isolation 91

3.7 Vibration-Measuring Instruments 95

3.8 Energy Dissipated in Viscous Damping 99

3.9 Equivalent Viscous Damping 103

Part C: Systems with Nonviscous Damping 105

3.10 Harmonic Vibration with Rate-Independent Damping 105

3.11 Harmonic Vibration with Coulomb Friction 109

Part D: Response to Periodic Excitation 113

3.12 Fourier Series Representation 114

3.13 Response to Periodic Force 114

Appendix 3: Four-Way Logarithmic Graph

Paper 118

 

4 Response to Arbitrary, Step, and Pulse Excitations 125

Part A: Response to Arbitrarily Time-Varying Forces 125

4.1 Response to Unit Impulse 126

4.2 Response to Arbitrary Force 127

Part B: Response to Step and Ramp Forces 129

4.3 Step Force 129

4.4 Ramp or Linearly Increasing Force 131

4.5 Step Force with Finite Rise Time 132

Part C: Response to Pulse Excitations 135

4.6 Solution Methods 135

4.7 Rectangular Pulse Force 137

4.8 Half-Cycle Sine Pulse Force 143

4.9 Symmetrical Triangular Pulse Force 148

4.10 Effects of Pulse Shape and Approximate Analysis for Short Pulses 151

4.11 Effects of Viscous Damping 154

4.12 Response to Ground Motion 155

 

5 Numerical Evaluation of Dynamic Response 165

5.1 Time-Stepping Methods 165

5.2 Methods Based on Interpolation of Excitation 167

5.3 Central Difference Method 171

5.4 Newmark’s Method 174

5.5 Stability and Computational Error 180

5.6 Analysis of Nonlinear Response: Central Difference Method 184

5.7 Analysis of Nonlinear Response: Newmark’s Method 184

 

6 Earthquake Response of Linear Systems 197

6.1 Earthquake Excitation 197

6.2 Equation of Motion 203

6.3 Response Quantities 204

6.4 Response History 205

6.5 Response Spectrum Concept 207

6.6 Deformation, Pseudo-velocity, and Pseudo-acceleration Response Spectra 208

6.7 Peak Structural Response from the Response Spectrum 217

6.8 Response Spectrum Characteristics 222

6.9 Elastic Design Spectrum 230

6.10 Comparison of Design and Response Spectra 239

6.11 Distinction between Design and Response Spectra 241

6.12 Velocity and Acceleration Response Spectra 242

Appendix 6: ElCentro, 1940 Ground Motion 246

 

7 Earthquake Response of Inelastic Systems 257

7.1 Force—Deformation Relations 258

7.2 Normalized Yield Strength, Yield Strength Reduction Factor, and Ductility Factor 264

7.3 Equation of Motion and Controlling Parameters 265

7.4 Effects of Yielding 266

7.5 Response Spectrum for Yield Deformation and Yield Strength 273

7.6 Yield Strength and Deformation from the Response Spectrum 277

7.7 Yield Strength—Ductility Relation 277

7.8 Relative Effects of Yielding and Damping 279

7.9 Dissipated Energy 280

7.10 Energy Dissipation Devices 283

7.11 Inelastic Design Spectrum 288

7.12 Applications of the Design Spectrum 295

7.13 Comparison of Design and Response Spectra 301

 

8 Generalized Single-Degree-of-Freedom Systems 305

8.1 Generalized SDF Systems 305

8.2 Rigid-Body Assemblages 307

8.3 Systems with Distributed Mass and Elasticity 309

8.4 Lumped-Mass System: Shear Building 321

8.5 Natural Vibration Frequency by Rayleigh’s Method 328

8.6 Selection of Shape Function 332

Appendix 8: Inertia Forces for Rigid Bodies 336

 

PART II MULTI-DEGREE-OF-FREEDOM SYSTEMS 343

 

9 Equations of Motion, Problem Statement, and Solution

Methods 345

9.1 Simple System: Two-Story Shear Building 345

9.2 General Approach for Linear Systems 350

9.3 Static Condensation 367

9.4 Planar or Symmetric-Plan Systems: Ground Motion 370

9.5 Unsymmetric-Plan Buildings: Ground Motion 375

9.6 Symmetric-Plan Buildings: Torsional Excitation 383

9.7 Multiple Support Excitation 384

9.8 Inelastic Systems 389

9.9 Problem Statement 389

9.10 Element Forces 390

9.11 Methods for Solving the Equations of Motion: Overview 390

 

10 Free Vibration 401

Part A: Natural Vibration Frequencies and Modes 402

10.1 Systems without Damping 402

10.2 Natural Vibration Frequencies and Modes 404

10.3 Modal and Spectral Matrices 406

10.4 Orthogonality of Modes 407

10.5 Interpretation of Modal Orthogonality 408

10.6 Normalization of Modes 408

10.7 Modal Expansion of Displacements 418

Part B: Free Vibration Response 419

10.8 Solution of Free Vibration Equations: Undamped Systems 419

10.9 Free Vibration of Systems with Damping 422

10.10 Solution of Free Vibration Equations: Classically Damped Systems 426

Part C: Computation of Vibration Properties 428

10.11 Solution Methods for the Eigenvalue Problem 428

10.12 Rayleigh’s Quotient 430

10.13 Inverse Vector Iteration Method 430

10.14 Vector Iteration with Shifts: Preferred Procedure 435

10.15 Transformation of kφ = ω2mφ to the StandardForm 440

 

11 Damping in Structures 447

Part A: Experimental Data and Recommended Modal Damping Ratios 447

11.1 Vibration Properties of Millikan Library Building 447

11.2 Estimating Modal Damping Ratios 452

Part B: Construction of Damping Matrix 454

11.3 Damping Matrix 454

11.4 Classical Damping Matrix 455

11.5 Nonclassical Damping Matrix 463

 

12 Dynamic Analysis and Response of Linear Systems 467

Part A: Two-Degree-of-Freedom Systems 467

12.1 Analysis of Two-DOF Systems without Damping 467

12.2 Vibration Absorber or Tuned Mass Damper 470

Part B: Modal Analysis 472

12.3 Modal Equations for Undamped Systems 472

12.4 Modal Equations for Damped Systems 475

12.5 Displacement Response 476

12.6 Element Forces 477

12.7 Modal Analysis: Summary 477

Part C: Modal Response Contributions 482

12.8 Modal Expansion of Excitation Vectorp(t) = sp(t) 482

12.9 Modal Analysis for p(t) = sp(t) 486

12.10 Modal Contribution Factors 487

12.11 Modal Responses and Required Number of Modes 489

Part D: Special Analysis Procedures 496

12.12 Static Correction Method 496

12.13 Mode Acceleration Superposition Method 499

12.14 Analysis of Nonclassically Damped Systems 500

 

13 Earthquake Analysis of Linear Systems 507

Part A: Response History Analysis 508

13.1 Modal Analysis 508

13.2 Multistory Buildings with Symmetric Plan 514

13.3 Multistory Buildings with Unsymmetric Plan 533

13.4 Torsional Response of Symmetric-Plan Buildings 544

13.5 Response Analysis for Multiple Support Excitation 548

13.6 Structural Idealization and Earthquake Response 554

Part B: Response Spectrum Analysis 555

13.7 Peak Response from Earthquake Response Spectrum 555

13.8 Multistory Buildings with Symmetric Plan 560

13.9 Multistory Buildings with Unsymmetric Plan 572

 

14 Reduction of Degrees of Freedom 593

14.1 Kinematic Constraints 594

14.2 Mass Lumping in Selected DOFs 595

14.3 Rayleigh—Ritz Method 595

14.4 Selection of Ritz Vectors 599

14.5 Dynamic Analysis Using Ritz Vectors 604

 

15 Numerical Evaluation of Dynamic Response 609

15.1 Time-Stepping Methods 609

15.2 Analysis of Linear Systems with Nonclassical Damping 611

15.3 Analysis of Nonlinear Systems 618

 

16 Systems with Distributed Mass and Elasticity 629

16.1 Equation of Undamped Motion: Applied Forces 630

16.2 Equation of Undamped Motion: Support Excitation 631

16.3 Natural Vibration Frequencies and Modes 632

16.4 Modal Orthogonality 639

16.5 Modal Analysis of Forced Dynamic Response 641

16.6 Earthquake Response History Analysis 648

16.7 Earthquake Response Spectrum Analysis 653

16.8 Difficulty in Analyzing Practical Systems 656

 

17 Introduction to the Finite Element Method 661

Part A: Rayleigh—Ritz Method 661

17.1 Formulation Using Conservation of Energy 661

17.2 Formulation Using Virtual Work 665

17.3 Disadvantages of Rayleigh—Ritz Method 667

Part B: Finite Element Method 667

17.4 Finite Element Approximation 667

17.5 Analysis Procedure 669

17.6 Element Degrees of Freedom and InterpolationFunctions 671

17.7 Element Stiffness Matrix 672

17.8 Element Mass Matrix 673

17.9 Element (Applied) Force Vector 675

17.10 Comparison of Finite Element and ExactSolutions 679

17.11 Dynamic Analysis of Structural Continua 680

 

PART III EARTHQUAKE RESPONSE AND DESIGN OF

MULTISTORY BUILDINGS 687

 

18 Earthquake Response of Linearly Elastic Buildings 689

18.1 Systems Analyzed, Design Spectrum, and Response Quantities 689

18.2 Influence of T1 and ¿ on Response 694

18.3 Modal Contribution Factors 695

18.4 Influence of T1 on Higher-Mode Response 697

18.5 Influence of ¿ on Higher-Mode Response 700

18.6 Heightwise Variation of Higher-Mode Response 701

18.7 How Many Modes to Include 703

 

19 Earthquake Analysis and Response of Inelastic Buildings 707

Part A: Nonlinear Response History Analysis 708

19.1 Equations of Motion: Formulation and Solution 708

19.2 Computing Seismic Demands: Factors

To Be Considered 709

19.3 Story Drift Demands 713

19.4 Strength Demands for SDF and MDF Systems 719

Part B: Approximate Analysis Procedures 720

19.5 Motivation and Basic Concept 720

19.6 Uncoupled Modal Response History Analysis 722

19.7 Modal Pushover Analysis 729

19.8 Evaluation of Modal Pushover Analysis 734

19.9 Simplified Modal Pushover Analysis for Practical Application 739

 

20 Earthquake Dynamics of Base-Isolated Buildings 741

20.1 Isolation Systems 741

20.2 Base-Isolated One-Story Buildings 744

20.3 Effectiveness of Base Isolation 750

20.4 Base-Isolated Multistory Buildings 754

20.5 Applications of Base Isolation 760

 

21 Structural Dynamics in Building Codes 767

Part A: Building Codes and Structural Dynamics 768

21.1 International Building Code (United States), 2006 768

21.2 National Building Code of Canada, 2005 771

21.3 Mexico Federal District Code, 2004 773

21.4 Eurocode 8, 2004 775

21.5 Structural Dynamics in Building Codes 778

Part B: Evaluation of Building Codes 784

21.6 Base Shear 784

21.7 Story Shears and Equivalent Static Forces 788

21.8 Overturning Moments 790

21.9 Concluding Remarks 793

 

22 Structural Dynamics in Building Evaluation Guidelines 795

22.1 Nonlinear Dynamic Procedure: Current Practice 796

22.2 SDF-System Estimate of Roof Displacement 797

22.3 Estimating Deformation of Inelastic SDF Systems 799

22.4 Nonlinear Static Procedure 806

22.5 Concluding Remarks 812

A Frequency-Domain Method of Response Analysis 815

B Notation 837

C Answers to Selected Problems 849

Index 865