## Contents

Programs available in this book xvii

Preface xxi

Acknowledgements xxiii

1 Introduction to dynamics 1

1.1 Introduction 1

1.2 Different types of dynamic loads 2

1.3 Difference between dynamic and static problems 2

1.4 Methodology 4

1.5 Types of vibration 5

1.6 Further reading 6

Part I Structural dynamics in relation to earthquakes

2 Free vibration of single-degree-of-freedom systems

(undamped) in relation to structural dynamics during

earthquakes 9

2.1 Introduction 9

2.2 Formulation of the equation of motion 9

2.3 Simple harmonic theory 10

2.4 Newton’s second law 14

2.5 Simple pendulum 19

2.6 Comparison of simple harmonic motion and uniform

circular motion 21

2.7 Energy method 22

2.8 Rayleigh method 24

2.9 D’Alembert’s principle 24

2.10 Free vibration of rigid bodies without damping 25

2.11 Program 2.1: MATLAB program to draw displacement,

velocity and acceleration with respect to time 28

2.12 Program 2.2: MATHEMATICA program to draw

displacement, velocity and acceleration with respect

to time 29

2.13 Free vibration of structural systems 32

2.14 Exercises 39

2.15 Further reading 42

3 Free vibration of single-degree-of-freedom systems

(under-damped) in relation to structural dynamics

during earthquakes 43

3.1 Introduction 43

3.2 Damping free vibrations 43

3.3 Logarithmic decrement 47

3.4 Hysteresis damaping 53

3.5 Coulomb damping 55

3.6 Numerical method to find response due to initial

conditions only 58

3.7 Program 3.1: MATLAB program for free vibration of

under-damped SDOF systems 59

3.8 Program 3.2: MATHEMATICA program for free

vibration of damped SDOF systems 60

3.9 Summary 66

3.10 Exercises 66

3.11 Further reading 67

4 Forced vibration (harmonic force) of single-degreeof-

freedom systems in relation to structural

dynamics during earthquakes 68

4.1 Forced vibration without damping 68

4.2 Beating phenomenon 73

4.3 Resonance 75

4.4 Forced vibration with damping 77

4.5 Program 4.1: MATHEMATICA program to find

displacement response of under-damped system

subjected to sinusoidal loading 79

4.6 Recurrence formula of Wilson 81

4.7 Program 4.2: MATLAB program for finding response

due to harmonic force 82

4.8 Vector relationship in forced vibration 83

4.9 Rotating imbalance 87

4.10 Transmissibility (force isolation) 92

4.11 Program 4.3: MATLAB program to compute MF,

MX/me and TR 93

viii Contents

4.12 Effectiveness of foundation 94

4.13 Displacement isolation 95

4.14 Vibration-measuring instruments 96

4.15 How to evaluate damping in SDOF 97

4.16 Response to ground acceleration 99

4.17 Exercises 101

4.18 Further reading 103

5 Response of structures to periodic dynamic loadings 105

5.1 Introduction 105

5.2 Fourier analysis 106

5.3 Program 5.1: MATHEMATICA program to determine

Fourier coefficients of forcing function 111

5.4 Response to periodic excitation 115

5.5 Program 5.2: MATHEMATICA program for finding the

response to a periodic function 116

5.6 Frequency domain analysis 121

5.7 Alternative form of Fourier series 122

5.8 Program 5.3: MATLAB program to evaluate amplitudes

and phase angles 124

5.9 Expression of forcing function using complex variable

approach 127

5.10 Discrete fourier transform (DFT) and fast fourier

transform (FFT) 131

5.11 Gibbs phenomenon 132

5.12 Summary 133

5.13 Exercises 133

5.14 Further reading 134

6 Response of structures to impulse loads 136

6.1 Introduction 136

6.2 Impulsive loading – sine wave 136

6.3 Program 6.1: MATLAB program to obtain maximum

response for half sine cycle pulse 140

6.4 Response to other arbitrary dynamic excitation 141

6.5 Duhamel integral 146

6.6 Response to arbitrary dynamic excitation 148

6.7 Response spectrum 157

6.8 Program 6.3: MATLAB program to find the response

spectrum for any load pulse 158

6.9 Laplace transform 163

6.10 Program 6.4: MATHEMATICA program for Laplace

transform method 165

Contents ix

6.11 Summary 167

6.12 Exercises 167

6.12 Further reading 169

7 Dynamic response of structures using numerical

methods 171

7.1 Introduction 171

7.2 Time stepping methods 172

7.3 Types of time stepping method 173

7.4 Response to base excitation 211

7.5 Wilson’s procedure (recommended) 217

7.6 Response of elasto-plastic SDOF system 224

7.7 Program 7.10: MATLAB program for dynamic response

for elasto-plastic SDOF system 227

7.8 Response spectra by numerical integration 231

7.9 Numerical method for evaluation of the Duhamel integral 232

7.10 Selection of direct integration method 236

7.11 Summary 237

7.12 Exercises 237

7.13 Further reading 238

8 Generalized coordinates and energy methods in

relation to structural dynamics during earthquakes 240

8.1 Introduction 240

8.2 Principle of virtual work 240

8.3 Generalized SDOF system: rigid bodies 241

8.4 Systems having distributed stiffness and distributed mass 243

8.5 Rayleigh method 248

8.6 Improved Rayleigh method 250

8.7 Hamilton’s principle 251

8.8 Lagrange’s equations 253

8.9 Computer-generated Euler–Lagrange equations using

MATHEMATICA 259

8.10 Summary 262

8.11 Exercises 263

8.12 Further reading 264

9 Two-degrees-of-freedom linear system response of

structures 266

9.1 Overview 266

9.2 Free vibration of undamped two-degrees-of-freedom system 266

9.3 Program 9.1: MATHEMATICA program to solve coupled

differential equations 273

x Contents

9.4 Program 9.2: MATLAB program to solve free vibration of

undamped two-degrees-of-freedom system 274

9.5 Program 9.3: MATLAB program to solve coupled

differential equations 276

9.6 Coordinate coupling 280

9.7 Simple system: two storey shear building 285

9.8 Program 9.4: MATHEMATICA program for finding the

responses of an undamped two-degrees-of-freedom

system – free vibration 288

9.9 Forced vibration of two-degrees-of-freedom undamped

system 293

9.10 Program 9.5: MATHEMATICA program for forced

vibration of two-degrees-of-freedom undamped system 295

9.11 Vibration absorber 297

9.12 Forced response of a two-degrees-of-freedom

under-damped system 298

9.13 Program 9.6: MATLAB program for displacement

response of two-degrees-of-freedom under-damped

system for forced vibration 301

9.14 Summary 302

9.15 Exercises 302

9.16 Further reading 303

10 Free vibration of multiple degrees of freedom in

relation to structural dynamics during earthquakes 305

10.1 Introduction 305

10.2 Modelling of a continuous system as an MDOF system 306

10.3 Equations of motion of an MDOF system 307

10.4 Free undamped vibration of an MDOF system 308

10.5 Orthogonality relationship 310

10.6 Normalization of modes 311

10.7 Influence coefficient method 313

10.8 Program 10.1: MATHEMATICA program for finding the

solution of the characteristic equation 318

10.9 Program 10.2: MATLAB program to find the frequencies

and normalized mode shapes 318

10.10 Program 10.3: MATLAB program for solving structural

problem by the stiffness method 326

10.11 Static condensation of stiffness matrix 330

10.12 General viscous damping 331

10.13 Program 10.4: MATLAB program for free vibration of

MDOF with generalized damping 332

10.14 Newmark’s numerical integration 336

Contents xi

10.15 Program 10.5: MATLAB program for Newmark’s

method of MDOF with generalized

damping 337

10.16 Forced response of a three-degrees-of-freedom

under-damped system 338

10.17 Summary 341

10.18 Exercises 342

10.19 Further reading 343

11 Numerical solution methods for natural frequencies

and mode shapes in relation to structural dynamics

during earthquakes 344

11.1 Introduction 344

11.2 General solution methods for eigen problems 344

11.3 Vector iteration technique 346

11.4 Jacobi’s method 359

11.5 Transfer matrix method to find the fundamental frequency

of a multi-storeyed building (shear frame) 361

11.6 Program 11.1: MATHEMATICA program to find the

fundamental frequency and the corresponding mode shape

(transfer matrix method) 364

11.7 Holzer method for torsional vibrations 366

11.8 Approximate methods for finding the natural frequencies 369

11.9 Dunkerley’s approximation 377

11.10 Summary 380

11.11 Exercises 380

11.12 Further reading 381

12 Time history response by mode superposition in

relation to structural dynamics during earthquakes 383

12.1 Introduction 383

12.2 Limitations 383

12.3 Mode displacement method for uncoupled system 384

12.4 Modal participation factor 387

12.5 Time history analysis 388

12.6 Mode superposition solution for systems with classical

damping 397

12.7 Numerical evaluation of modal response 401

12.8 Program 12.1: MATLAB program for dynamic response

using modal superposition 405

12.9 Dynamic analysis using direct integration methods 410

12.10 Program 12.2: MATLAB program for finding dynamic

response of MDOF using direct integration method

(Newmark’s method) 410

xii Contents

12.11 Normal mode response to support motions 415

12.12 Response spectrum analysis 416

12.13 Mode acceleration method 424

12.14 Summary 427

12.15 Exercises 427

12.16 Further reading 430

13 Free and forced vibration of a continuous system in

relation to structural dynamics during earthquakes 431

13.1 Introduction 431

13.2 Vibration of a string 432

13.3 Program 13.1: MATHEMATICA program to find

displacement of a string 435

13.4 Longitudinal vibration of a uniform rod 436

13.5 Torsional vibration of shaft or rod 441

13.6 Free flexural vibration of beams 443

13.7 Program 13.2: MATHEMATICA program to find the

frequency of a long beam with usual boundary conditions 449

13.8 Orthogonality of normal modes 457

13.9 Effect of axial force (tension or compression) 458

13.10 Effect of rotary inertia and shear deformation 462

13.11 Forced axial vibration of bars 465

13.12 Beams subjected to moving loads 469

13.13 Summary 474

13.14 Exercises 474

13.15 Further reading 475

14 Finite element method in relation to structural

dynamics during earthquakes 477

14.1 Introduction 477

14.2 Dynamic analysis 478

14.3 Torsional vibration of a shaft 478

14.4 Axial vibration of rods 484

14.5 Assumed modes method 485

14.6 Program 14.1: MATLAB program for the assumed modes

method 488

14.7 Truss element 490

14.8 Program 14.2: MATLAB program for free vibration of

trusses 496

14.9 Beam element 499

14.10 Program 14.3: MATHEMATICA program for evaluation

of stiffness matrix, and mass matrix of a beam element 502

14.11 Program 14.4: MATLAB program to find the natural

frequency of beams or rigid frames 506

Contents xiii

14.12 Forced vibration of a beam 510

14.13 Program 14.5: MATLAB program for forced vibration of

a beam 512

14.14 Vibration of a Timoshenko beam 515

14.15 Program 14.6: MATLAB program to find the frequency

of a Timoshenko beam 518

14.16 Summary 522

14.17 Exercises 522

14.18 Further reading 524

15 Differential quadrature and transformation methods

for vibration problems in relation to structural

dynamics during earthquakes 525

15.1 Introduction 525

15.2 DQ method 525

15.3 Lagrangian interpolation 526

15.4 Differential quadrature method formulation 527

15.5 HDQ method 528

15.6 Transverse vibration of pre-tensioned cable 529

15.7 Program 15.1: MATLAB program for finding the natural

frequency of lateral vibration of a pre-tensioned string 530

15.8 Lateral vibration of uniform Euler beams 537

15.9 Program 15.2: MATLAB program for free vibration of

an Euler beam 540

15.10 To find natural frequency and mode shape given variation

of D = EI for Euler beam with axial load 542

15.11 Program 15.3: MATLAB program for solving free

vibration problem of non-prismatic beam with or without

axial load 545

15.12 Vibration of Timoshenko beam by DQ method 548

15.13 Program 15.4: MATLAB program for free vibration

analysis of Timoshenko beam 550

15.14 DT method 553

15.15 Transverse vibration of pre-tensioned cable 553

15.16 Program 15.5: MATHEMATICA program for finding the

natural frequency of vibration of a pre-tensioned cable 555

15.17 Free vibration analysis of Euler beam 556

15.18 Program 15.6: MATHEMATICA program for finding the

natural frequency of vibration an Euler beam 558

15.19 Natural frequency of Euler beam subjected to axial load 559

15.20 Program 15.7: MATHEMATICA program for finding the

natural frequency an Euler beam subjected to axial load 561

15.21 Natural frequency of a Timoshenko beam 562

xiv Contents

15.22 Program 15.8: MATHEMATICA program for finding the

natural frequency of a Timoshenko beam 563

15.23 Summary 565

15.24 Exercises 566

15.25 References and further reading 567

Part II Response of structures to earthquakes

16 Earthquakes and earthquake ground motion 571

16.1 Introduction 571

16.2 What is an earthquake? 572

16.3 Plate tectonic theory 577

16.4 Faults 578

16.5 Earthquake belts in the world 580

16.6 Elastic rebound theory 582

16.7 Seismic waves 582

16.8 Measuring instruments 586

16.9 Earthquake intensity and magnitude 588

16.10 Basic difference: magnitude versus intensity 594

16.11 Earthquake ground motion 594

16.12 Earthquake classification 599

16.13 Asian tsunami disaster 599

16.14 Damage mechanisms due to earthquakes 601

16.15 Summary 601

10.16 Web links 601

16.17 References and further reading 602

17 Earthquake response spectra 605

17.1 Introduction 605

17.2 Earthquake response spectra 607

17.3 Program 17.1: MATLAB program for drawing spectra

for any specified earthquake 607

17.4 Program 17.2: MATLAB program to draw tripartite plot 620

17.5 Importance of response quantities 623

17.6 Response spectrum concept 626

17.7 Pseudo-velocity spectrum 628

17.8 Pseudo-spectral acceleration 630

17.9 Combined deformation, velocity and acceleration (DVA)

spectrum 631

17.10 Velestos and Newmark spectra 632

17.11 How the response spectrum is constructed 633

17.12 Elastic design spectrum 640

Contents xv

17.13 Program 17.3: MATLAB program for drawing

Newmark–Hall design spectra 642

17.14 Response spectrum characteristics 647

17.15 Distinction between design and response spectra 651

17.16 Response spectrum 652

17.17 Site-specific response spectra 655

17.18 Estimating the ground motion 656

17.19 Seismic analysis and design verification 657

17.20 Inelastic response spectra 658

17.21 Application of inelastic design spectrum 661

17.22 Inelastic deformation 663

17.23 Summary 664

17.24 Exercises 664

17.25 Further reading 666

18 Earthquake analysis of linear systems 667

18.1 Introduction 667

18.2 Lumped mass system: shear building 668

18.3 Modal response contribution using Chopra’s method 676

18.4 Modal analysis for Γ f (t) 679

18.5 Interpretation of modal analysis 681

18.6 Modal contribution factor 681

18.7 Modal response and required number of modes 682

18.8 Modal contributions 683

18.9 Program 18.1: MATLAB program to find the ratio of

dynamic shear to static shear in a multi-storey building 695

18.10 Earthquake analysis linear systems 699

18.11 Modal response 700

18.12 Multi-storey buildings with symmetrical plan 706

18.13 Spectrum analysis by modal response 715

18.14 Effective modal mass and modal height 718

18.15 Multiple support excitation 723

18.16 Symmetric plan buildings: translational ground motion 726

18.17 Summary 729

18.18 Exercises 729

18.19 References and further reading 732

19 Building codes for aseismic design 734

19.1 Introduction 734

19.2 Historical development 735

19.3 Codal provisions for seismic design 737

19.4 Program 19.1: MATLAB program for IS1893 code 753

19.5 Comparison of codes 763

xvi Contents

19.6 Design examples using IS1893 2002 Part 1 765

19.7 Summary 800

19.8 Exercises 801

19.9 References and further reading 802

20 Response of structures to earthquakes: approximate

analysis techniques for lateral loads 804

20.1 Introduction 804

20.2 Simplified analysis for lateral loads 805

20.3 Zero moment point method 805

20.4 Approximate methods of analysis of multi-bay frames

(lateral loads) 814

20.5 Analysis of buildings simple in plan for lateral loads 827

20.6 Summary 831

20.7 Exercises 831

20.8 References and further reading 832

21 Response of structures to earthquakes: analysis of

shear walls 833

21.1 Introduction 833

21.2 Shear wall frame 834

21.3 Coupled shear walls 840

21.4 Program 21.1: MATHEMATICA program for coupled

shear wall 855

21.5 Summary 860

21.6 Exercises 860

21.7 References and further reading 863

Index 864