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Dynamics of Structures

by: Chopra, Anil K.

Dynamics of Structures

by: Chopra, Anil K.

ISBN 13:
9780132858038
ISBN 10:
0132858037
Edition: 4th
Format: Paperback
Copyright: 12/06/2011
Publisher: Pearson
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Dynamics of Structures > ISBN13: 9780132858038

Summary

Designed for senior-level and graduate courses in Dynamics of Structures and Earthquake Engineering.

Dynamics of Structures includes many topics encompassing the theory of structural dynamics and the application of this theory regarding earthquake analysis, response, and design of structures. No prior knowledge of structural dynamics is assumed and the manner of presentation is sufficiently detailed and integrated, to make the book suitable for self-study by students and professional engineers.

Author Biography

Anil K. Chopra received his Bachelor of Science degree in Civil Engineering from Banaras Hindu University, India, in 1960, the Master of Science degree from the University of California, Berkeley, in 1963, and the Doctor of Philosophy degree, also from Berkeley, in 1966.

After serving as an Assistant Professor at the University of Minnesota, Minneapolis, he joined the faculty at the University of California, Berkeley where he has served as Assistant Professor (1967-71), Associate Professor (1971-76), Professor (1976- ), Vice Chair (1980-83) and Chair (1991-93, 1994-97) of the Structural Engineering, Mechanics and Materials program in the Department of Civil and Environmental Engineering. He has been responsible for the development and teaching of courses in structural engineering, structural dynamics, and earthquake engineering.

His research activities have included studies of structural dynamics, various problems in earthquake analysis and design of buildings, dynamic soil-structure interaction, dynamic fluid-structure interaction, and earthquake analysis and design of concrete dams. He has authored more than 300 published papers on this work, a monograph, Earthquake Dynamics of Structures, A Primer, 2005, and a textbook, Dynamics of Structures: Theory and Applications to Earthquake Engineering, 1995, 2001, and 2007.

Professor Chopra serves as a consultant on earthquake engineering problems to numerous governmental and private organizations. He is a Member of the American Society of Civil Engineers, where he has served as Chairman (1986) of the Engineering Mechanics Division Executive Committee and also Chairman (1991) of the Structural Division Executive Committee. He was a member of the Board of Directors of the Earthquake Engineering Research Institute (1990-93), the Structural Engineers Association of Northern California (1987-89), the Seismological Society of America (1982-83), and the Applied Technology Council (1972-74). He served as a member of the Steering Committee for the Eighth World Conference on Earthquake Engineering, San Francisco, 1984, and as Chairman of the National Research Council Committee on Natural Disasters (1982-83). Currently, he serves as Executive Editor of Earthquake Engineering and Structural Dynamics, the journal of the International Association for Earthquake Engineering.

Foreword xxi

Preface xxiii

Acknowledgments xxxi

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 Nonlinear Systems: Central Difference Method 183

5.7 Nonlinear Systems: Newmark’s Method 183

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: El Centro, 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 Supplemental 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 One-Story Unsymmetric-Plan Buildings 375

9.6 Multistory Unsymmetric-Plan Buildings 381

9.7 Multiple Support Excitation 385

9.8 Inelastic Systems 390

9.9 Problem Statement 390

9.10 Element Forces 391

9.11 Methods for Solving the Equations of Motion:

Overview 391

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 Systems with Damping 422

10.10 Solution of Free Vibration Equations: Classically

Damped Systems 423

Part C: Computation of Vibration Properties 426

10.11 Solution Methods for the Eigenvalue Problem 426

10.12 Rayleigh’s Quotient 428

10.13 Inverse Vector Iteration Method 428

10.14 Vector Iteration with Shifts: Preferred Procedure 433

10.15 Transformation of k φ = ω2m φ to the Standard

Form 438

11 Damping in Structures 445

Part A: Experimental Data and Recommended Modal

Damping Ratios 445

11.1 Vibration Properties of Millikan Library Building 445

11.2 Estimating Modal Damping Ratios 450

Part B: Construction of Damping Matrix 452

11.3 Damping Matrix 452

11.4 Classical Damping Matrix 453

11.5 Nonclassical Damping Matrix 462

12 Dynamic Analysis and Response of Linear Systems 465

Part A: Two-Degree-of-Freedom Systems 465

12.1 Analysis of Two-DOF Systems without Damping 465

12.2 Vibration Absorber or Tuned Mass Damper 468

Part B: Modal Analysis 470

12.3 Modal Equations for Undamped Systems 470

12.4 Modal Equations for Damped Systems 473

12.5 Displacement Response 474

12.6 Element Forces 475

12.7 Modal Analysis: Summary 475

Part C: Modal Response Contributions 480

12.8 Modal Expansion of Excitation Vector

p ( t ) = s p ( t ) 480

12.9 Modal Analysis for p ( t ) = s p ( t ) 484

12.10 Modal Contribution Factors 485

12.11 Modal Responses and Required Number of Modes 487

Part D: Special Analysis Procedures 494

12.12 Static Correction Method 494

12.13 Mode Acceleration Superposition Method 497

12.14 Mode Acceleration Superposition Method: Arbitrary

Excitation 498

13 Earthquake Analysis of Linear Systems 511

Part A: Response History Analysis 512

13.1 Modal Analysis 512

13.2 Multistory Buildings with Symmetric Plan 518

13.3 Multistory Buildings with Unsymmetric Plan 537

13.4 Torsional Response of Symmetric-Plan Buildings 548

13.5 Response Analysis for Multiple Support

Excitation 552

13.6 Structural Idealization and Earthquake Response 558

Part B: Response Spectrum Analysis 559

13.7 Peak Response from Earthquake Response

Spectrum 559

13.8 Multistory Buildings with Symmetric Plan 564

13.9 Multistory Buildings with Unsymmetric Plan 576

13.10 A Response-Spectrum-Based Envelope for

Simultaneous Responses 584

13.11 Response to Multi-Component Ground

Motion 592

14 Analysis of Nonclassically Damped Linear Systems 613

Part A: Classically Damped Systems: Reformulation 614

14.1 Natural Vibration Frequencies and Modes 614

14.2 Free Vibration 615

14.3 Unit Impulse Response 616

14.4 Earthquake Response 617

Part B: Nonclassically Damped Systems 618

14.5 Natural Vibration Frequencies and Modes 618

14.6 Orthogonality of Modes 619

14.7 Free Vibration 623

14.8 Unit Impulse Response 628

14.9 Earthquake Response 632

14.10 Systems with Real-Valued Eigenvalues 634

14.11 Response Spectrum Analysis 642

14.12 Summary 643

Appendix 14: Derivations 644

15 Reduction of Degrees of Freedom 653

15.1 Kinematic Constraints 654

15.2 Mass Lumping in Selected DOFs 655

15.3 Rayleigh—Ritz Method 655

15.4 Selection of Ritz Vectors 659

15.5 Dynamic Analysis Using Ritz Vectors 664

16 Numerical Evaluation of Dynamic Response 669

16.1 Time-Stepping Methods 669

16.2 Linear Systems with Nonclassical Damping 671

16.3 Nonlinear Systems 677

17 Systems with Distributed Mass and Elasticity 693

17.1 Equation of Undamped Motion: Applied Forces 694

17.2 Equation of Undamped Motion: Support

Excitation 695

17.3 Natural Vibration Frequencies and Modes 696

17.4 Modal Orthogonality 703

17.5 Modal Analysis of Forced Dynamic Response 705

17.6 Earthquake Response History Analysis 712

17.7 Earthquake Response Spectrum Analysis 717

17.8 Difficulty in Analyzing Practical Systems 720

18 Introduction to the Finite Element Method 725

Part A: Rayleigh—Ritz Method 725

18.1 Formulation Using Conservation of Energy 725

18.2 Formulation Using Virtual Work 729

18.3 Disadvantages of Rayleigh—Ritz Method 731

Part B: Finite Element Method 731

18.4 Finite Element Approximation 731

18.5 Analysis Procedure 733

18.6 Element Degrees of Freedom and Interpolation

Functions 735

18.7 Element Stiffness Matrix 736

18.8 Element Mass Matrix 737

18.9 Element (Applied) Force Vector 739

18.10 Comparison of Finite Element and Exact

Solutions 743

18.11 Dynamic Analysis of Structural Continua 744

PART III EARTHQUAKE RESPONSE, DESIGN, AND EVALUATION

OF MULTISTORY BUILDINGS 751

19 Earthquake Response of Linearly Elastic Buildings 753

19.1 Systems Analyzed, Design Spectrum, and Response

Quantities 753

19.2 Influence of T1 and Á on Response 758

19.3 Modal Contribution Factors 759

19.4 Influence of T1 on Higher-Mode Response 761

19.5 Influence of Á on Higher-Mode Response 764

19.6 Heightwise Variation of Higher-Mode Response 765

19.7 How Many Modes to Include 767

20 Earthquake Analysis and Response of Inelastic Buildings 771

Part A: Nonlinear Response History Analysis 772

20.1 Equations of Motion: Formulation and Solution 772

20.2 Computing Seismic Demands: Factors

To Be Considered 773

20.3 Story Drift Demands 777

20.4 Strength Demands for SDF and MDF Systems 783

Part B: Approximate Analysis Procedures 784

20.5 Motivation and Basic Concept 784

20.6 Uncoupled Modal Response History Analysis 786

20.7 Modal Pushover Analysis 793

20.8 Evaluation of Modal Pushover Analysis 798

20.9 Simplified Modal Pushover Analysis

for Practical Application 803

21 Earthquake Dynamics of Base-Isolated Buildings 805

21.1 Isolation Systems 805

21.2 Base-Isolated One-Story Buildings 808

21.3 Effectiveness of Base Isolation 814

21.4 Base-Isolated Multistory Buildings 818

21.5 Applications of Base Isolation 824

22 Structural Dynamics in Building Codes 831

Part A: Building Codes and Structural Dynamics 832

22.1 International Building Code (United States), 2009 832

22.2 National Building Code of Canada, 2010 835

22.3 Mexico Federal District Code, 2004 837

22.4 Eurocode 8, 2004 840

22.5 Structural Dynamics in Building Codes 842

Part B: Evaluation of Building Codes 848

22.6 Base Shear 848

22.7 Story Shears and Equivalent Static Forces 852

22.8 Overturning Moments 854

22.9 Concluding Remarks 857

23 Structural Dynamics in Building Evaluation Guidelines 859

23.1 Nonlinear Dynamic Procedure: Current Practice 860

23.2 SDF-System Estimate of Roof Displacement 861

23.3 Estimating Deformation of Inelastic SDF Systems 864

23.4 Nonlinear Static Procedures 870

23.5 Concluding Remarks 876

A Frequency-Domain Method of Response Analysis 879

B Notation 901

C Answers to Selected Problems 913

Index 929

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