## Description

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

**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.**

*Dynamics of Structures*## 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 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 **=

*ω*2

**m**

**to the Standard**

*φ*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 *T*1 and *Á *on Response 758

**19.3 **Modal Contribution Factors 759

**19.4 **Influence of *T*1 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*## Purchase Info ?

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