# Structural Dynamics of Earthquake Engineering Theory and Application using MATHEMATICA and MATLAB By S Rajasekaran

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## 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
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
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
under-damped SDOF systems 59
vibration of damped SDOF systems 60
3.9 Summary 66
3.10 Exercises 66
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
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
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
6 Response of structures to impulse loads 136
6.1 Introduction 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
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
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
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
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
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
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
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
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
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
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
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
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
15.12 Vibration of Timoshenko beam by DQ method 548
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
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
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
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