Table of Contents

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 31.2 Single-Degree-of-Freedom System 71.3 Force-Displacement Relation 81.4 Damping Force 121.5 Equation of Motion: External Force 141.6 Mass-Spring-Damper System 191.7 Equation of Motion: Earthquake Excitation 231.8 Problem Statement and Element Forces 261.9 Combining Static and Dynamic Responses 281.10 Methods of Solution of the Differential Equation 281.11 Study of SDF Systems: Organization 33Appendix 1: Stiffness Coefficients for a Flexural

Element 33

2 Free Vibration 39 2.1 Undamped Free Vibration 392.2 Viscously Damped Free Vibration 482.3 Energy in Free Vibration 562.4 Coulomb-Damped Free Vibration 573 Response to Harmonic and Periodic Excitations 65 Part A: Viscously Damped Systems: Basic Results 66 3.1 Harmonic Vibration of Undamped Systems 663.2 Harmonic Vibration with Viscous Damping 72Part B: Viscously Damped Systems: Applications 85 3.3 Response to Vibration Generator 853.4 Natural Frequency and Damping from Harmonic

Tests 87

3.5 Force Transmission and Vibration Isolation 903.6 Response to Ground Motion and Vibration

Isolation 91

3.7 Vibration-Measuring Instruments 953.8 Energy Dissipated in Viscous Damping 993.9 Equivalent Viscous Damping 103Part C: Systems with Nonviscous Damping 105 3.10 Harmonic Vibration with Rate-Independent

Damping 105

3.11 Harmonic Vibration with Coulomb Friction 109Part D: Response to Periodic Excitation 113 3.12 Fourier Series Representation 1143.13 Response to Periodic Force 114Appendix 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 1264.2 Response to Arbitrary Force 127Part B: Response to Step and Ramp Forces 129 4.3 Step Force 1294.4 Ramp or Linearly Increasing Force 1314.5 Step Force with Finite Rise Time 132Part C: Response to Pulse Excitations 135 4.6 Solution Methods 1354.7 Rectangular Pulse Force 1374.8 Half-Cycle Sine Pulse Force 1434.9 Symmetrical Triangular Pulse Force 1484.10 Effects of Pulse Shape and Approximate Analysis for

Short Pulses 151

4.11 Effects of Viscous Damping 1544.12 Response to Ground Motion 1555 Numerical Evaluation of Dynamic Response 165 5.1 Time-Stepping Methods 1655.2 Methods Based on Interpolation of Excitation 1675.3 Central Difference Method 1715.4 Newmark's Method 1745.5 Stability and Computational Error 1805.6 Nonlinear Systems: Central Difference Method 1835.7 Nonlinear Systems: Newmark's Method 1836 Earthquake Response of Linear Systems 197 6.1 Earthquake Excitation 1976.2 Equation of Motion 2036.3 Response Quantities 2046.4 Response History 2056.5 Response Spectrum Concept 2076.6 Deformation, Pseudo-velocity, and Pseudo-acceleration

Response Spectra 208

6.7 Peak Structural Response from the Response

Spectrum 217

6.8 Response Spectrum Characteristics 2226.9 Elastic Design Spectrum 2306.10 Comparison of Design and Response Spectra 2396.11 Distinction between Design and Response

Spectra 241

6.12 Velocity and Acceleration Response Spectra 242Appendix 6: El Centro, 1940 Ground Motion 2467 Earthquake Response of Inelastic Systems 257 7.1 Force-Deformation Relations 2587.2 Normalized Yield Strength, Yield Strength Reduction

Factor, and Ductility Factor 264

7.3 Equation of Motion and Controlling Parameters 2657.4 Effects of Yielding 2667.5 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 2777.8 Relative Effects of Yielding and Damping 2797.9 Dissipated Energy 2807.10 Supplemental Energy Dissipation Devices 2837.11 Inelastic Design Spectrum 2887.12 Applications of the Design Spectrum 2957.13 Comparison of Design and Response

Spectra 301

8 Generalized Single-Degree-of-Freedom Systems 305 8.1 Generalized SDF Systems 3058.2 Rigid-Body Assemblages 3078.3 Systems with Distributed Mass and Elasticity 3098.4 Lumped-Mass System: Shear Building 3218.5 Natural Vibration Frequency by Rayleigh's

Method 328

8.6 Selection of Shape Function 332Appendix 8: Inertia Forces for Rigid Bodies 336PART 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 3459.2 General Approach for Linear Systems 3509.3 Static Condensation 3679.4 Planar or Symmetric-Plan Systems: Ground

Motion 370

9.5 One-Story Unsymmetric-Plan Buildings 3759.6 Multistory Unsymmetric-Plan Buildings 3819.7 Multiple Support Excitation 3859.8 Inelastic Systems 3909.9 Problem Statement 3909.10 Element Forces 3919.11 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 40210.2 Natural Vibration Frequencies and Modes 40410.3 Modal and Spectral Matrices 40610.4 Orthogonality of Modes 40710.5 Interpretation of Modal Orthogonality 40810.6 Normalization of Modes 40810.7 Modal Expansion of Displacements 418Part B: Free Vibration Response 419 10.8 Solution of Free Vibration Equations: Undamped

Systems 419

10.9 Systems with Damping 42210.10 Solution of Free Vibration Equations: Classically

Damped Systems 423

Part C: Computation of Vibration Properties 426 10.11 Solution Methods for the Eigenvalue Problem 42610.12 Rayleigh's Quotient 42810.13 Inverse Vector Iteration Method 42810.14 Vector Iteration with Shifts: Preferred Procedure 43310.15 Transformation of k = 2m 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 44511.2 Estimating Modal Damping Ratios 450Part B: Construction of Damping Matrix 452 11.3 Damping Matrix 45211.4 Classical Damping Matrix 45311.5 Nonclassical Damping Matrix 46212 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 46512.2 Vibration Absorber or Tuned Mass Damper 468Part B: Modal Analysis 470 12.3 Modal Equations for Undamped Systems 47012.4 Modal Equations for Damped Systems 47312.5 Displacement Response 47412.6 Element Forces 47512.7 Modal Analysis: Summary 475Part C: Modal Response Contributions 480 12.8 Modal Expansion of Excitation Vectorp(t) = sp(t) 48012.9 Modal Analysis for p(t) = sp(t) 48412.10 Modal Contribution Factors 48512.11 Modal Responses and Required Number of Modes 487Part D: Special Analysis Procedures 494 12.12 Static Correction Method 49412.13 Mode Acceleration Superposition Method 49712.14 Mode Acceleration Superposition Method: Arbitrary

Excitation 498

13 Earthquake Analysis of Linear Systems 511 Part A: Response History Analysis 512 13.1 Modal Analysis 51213.2 Multistory Buildings with Symmetric Plan 51813.3 Multistory Buildings with Unsymmetric Plan 53713.4 Torsional Response of Symmetric-Plan Buildings 54813.5 Response Analysis for Multiple Support

Excitation 552

13.6 Structural Idealization and Earthquake Response 558Part B: Response Spectrum Analysis 559 13.7 Peak Response from Earthquake Response

Spectrum 559

13.8 Multistory Buildings with Symmetric Plan 56413.9 Multistory Buildings with Unsymmetric Plan 57613.10 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 61414.2 Free Vibration 61514.3 Unit Impulse Response 61614.4 Earthquake Response 617Part B: Nonclassically Damped Systems 618 14.5 Natural Vibration Frequencies and Modes 61814.6 Orthogonality of Modes 61914.7 Free Vibration 62314.8 Unit Impulse Response 62814.9 Earthquake Response 63214.10 Systems with Real-Valued Eigenvalues 63414.11 Response Spectrum Analysis 64214.12 Summary 643Appendix 14: Derivations 64415 Reduction of Degrees of Freedom 653 15.1 Kinematic Constraints 65415.2 Mass Lumping in Selected DOFs 65515.3 Rayleigh-Ritz Method 65515.4 Selection of Ritz Vectors 65915.5 Dynamic Analysis Using Ritz Vectors 66416 Numerical Evaluation of Dynamic Response 669 16.1 Time-Stepping Methods 66916.2 Linear Systems with Nonclassical Damping 67116.3 Nonlinear Systems 67717 Systems with Distributed Mass and Elasticity 693 17.1 Equation of Undamped Motion: Applied Forces 69417.2 Equation of Undamped Motion: Support

Excitation 695

17.3 Natural Vibration Frequencies and Modes 69617.4 Modal Orthogonality 70317.5 Modal Analysis of Forced Dynamic Response 70517.6 Earthquake Response History Analysis 71217.7 Earthquake Response Spectrum Analysis 71717.8 Difficulty in Analyzing Practical Systems 72018 Introduction to the Finite Element Method 725 Part A: Rayleigh-Ritz Method 725 18.1 Formulation Using Conservation of Energy 72518.2 Formulation Using Virtual Work 72918.3 Disadvantages of Rayleigh-Ritz Method 731Part B: Finite Element Method 731 18.4 Finite Element Approximation 73118.5 Analysis Procedure 73318.6 Element Degrees of Freedom and Interpolation

Functions 735

18.7 Element Stiffness Matrix 73618.8 Element Mass Matrix 73718.9 Element (Applied) Force Vector 73918.10 Comparison of Finite Element and Exact

Solutions 743

18.11 Dynamic Analysis of Structural Continua 744PART 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 A on Response 75819.3 Modal Contribution Factors 75919.4 Influence of T1 on Higher-Mode Response 76119.5 Influence of A on Higher-Mode Response 76419.6 Heightwise Variation of Higher-Mode Response 76519.7 How Many Modes to Include 76720 Earthquake Analysis and Response of Inelastic Buildings 771 Part A: Nonlinear Response History Analysis 772 20.1 Equations of Motion: Formulation and Solution 77220.2 Computing Seismic Demands: Factors

To Be Considered 773

20.3 Story Drift Demands 77720.4 Strength Demands for SDF and MDF Systems 783Part B: Approximate Analysis Procedures 784 20.5 Motivation and Basic Concept 78420.6 Uncoupled Modal Response History Analysis 78620.7 Modal Pushover Analysis 79320.8 Evaluation of Modal Pushover Analysis 79820.9 Simplified Modal Pushover Analysis

for Practical Application 803

21 Earthquake Dynamics of Base-Isolated Buildings 805 21.1 Isolation Systems 80521.2 Base-Isolated One-Story Buildings 80821.3 Effectiveness of Base Isolation 81421.4 Base-Isolated Multistory Buildings 81821.5 Applications of Base Isolation 82422 Structural Dynamics in Building Codes 831 Part A: Building Codes and Structural Dynamics 832 22.1 International Building Code (United States), 2009 83222.2 National Building Code of Canada, 2010 83522.3 Mexico Federal District Code, 2004 83722.4 Eurocode 8, 2004 84022.5 Structural Dynamics in Building Codes 842Part B: Evaluation of Building Codes 848 22.6 Base Shear 84822.7 Story Shears and Equivalent Static Forces 85222.8 Overturning Moments 85422.9 Concluding Remarks 85723 Structural Dynamics in Building Evaluation Guidelines 859 23.1 Nonlinear Dynamic Procedure: Current Practice 86023.2 SDF-System Estimate of Roof Displacement 86123.3 Estimating Deformation of Inelastic SDF Systems 86423.4 Nonlinear Static Procedures 87023.5 Concluding Remarks 876A Frequency-Domain Method of Response Analysis 879 B Notation 901 C Answers to Selected Problems 913Index 929

About the Author

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.