Rajasekaran_Structural dynamics of earthquake engineering PDF

Preview

Summary

i Structural dynamics of earthquake engineering ii Related titles: Analysis and design of plated structures Volume 1: Stability (ISBN 978-1-85573-967-3) Steel-plated structures are used in a variety of marine and land-based applications, such as ships, off-shore platforms, power and chemical plants,...

Description

i

Structural dynamics of earthquake engineering

ii

Related titles: Analysis and design of plated structures Volume 1: Stability (ISBN 978-1-85573-967-3) Steel-plated structures are used in a variety of marine and land-based applications, such as ships, off-shore platforms, power and chemical plants, box-girder cranes and bridges. This first volume in a two-volume sequence considers the various types of buckling that plated structures are likely to encounter. Chapters also review buckling in a range of materials from steel to differing types of composite. The book discusses the behaviour of differing types of components used in steel-plated structures as well as curved, stiffened, corrugated, laminated and other types of plate design. Together with its companion volume, this is an essential reference in the design, construction and maintenance of plated structures. Analysis and design of plated structures Volume 2: Dynamics (ISBN 978-1-84569-116-5) This second of two volumes on plated structures reviews dynamics, particularly vibration. Chapters review the behaviour of components such as beams and plates together with materials such as steel, ceramic–metal and other composites. Together with its companion volume, this important book summarises the wealth of research on understanding the behaviour of thin-walled (plated) components. It is an invaluable resource for all those concerned with the design, construction and service life of plated structures. Multiscale materials modelling: fundamentals and applications (ISBN 978-1-84569-071-7) The survival and success of many future industries relies heavily on engineered materials and products with improved performance available at relatively low cost. This demands not only the rapid development of new/improved processing techniques but also a better understanding and control of the materials themselves, their structure and properties. The aim of multiscale modelling is to predict the behaviour of materials from their fundamental atomic structure. This emerging technique is revolutionising our understanding of material properties and how they can be altered. This important book reviews both the principles of multiscale materials modelling and the ways it can be applied to understand and improve the performance of structural materials. Details of these and other Woodhead Publishing materials books can be obtained by: • visiting our web site at www.woodheadpublishing.com • contacting Customer Services (e-mail: [email protected]; fax: +44 (0) 1223 893694; tel.: +44 (0) 1223 891358 ext. 130; address: Woodhead Publishing Limited, Abington Hall, Granta Park, Great Abington, Cambridge CB21 6AH, UK) If you would like to receive information on forthcoming titles, please send your address details to: Francis Dodds (address, tel. and fax as above; e-mail: [email protected]). Please confirm which subject areas you are interested in.

iii

Structural dynamics of earthquake engineering Theory and application using MATHEMATICA and MATLAB S. Rajasekaran

CRC Press Boca Raton Boston New York Washington, DC

WOODHEAD

PUBLISHING LIMITED

Oxford

Cambridge

New Delhi

iv Published by Woodhead Publishing Limited, Abington Hall, Granta Park, Great Abington, Cambridge CB21 6AH, UK www.woodheadpublishing.com Woodhead Publishing India Private Limited, G-2, Vardaan House, 7/28 Ansari Road, Daryaganj, New Delhi – 110002, India Published in North America by CRC Press LLC, 6000 Broken Sound Parkway, NW, Suite 300, Boca Raton, FL 33487, USA First published 2009, Woodhead Publishing Limited and CRC Press LLC © 2009, Woodhead Publishing Limited The author has asserted his moral rights. This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. Reasonable efforts have been made to publish reliable data and information, but the author and the publishers cannot assume responsibility for the validity of all materials. Neither the author nor the publishers, nor anyone else associated with this publication, shall be liable for any loss, damage or liability directly or indirectly caused or alleged to be caused by this book. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming and recording, or by any information storage or retrieval system, without permission in writing from Woodhead Publishing Limited. The consent of Woodhead Publishing Limited does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from Woodhead Publishing Limited for such copying. Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library. Library of Congress Cataloging in Publication Data A catalog record for this book is available from the Library of Congress. Woodhead Publishing ISBN 978-1-84569-518-7 (book) Woodhead Publishing ISBN 978-1-84569-573-6 (e-book) CRC Press ISBN 978-1-4398-0132-1 CRC Press order number: N10026 The publishers’ policy is to use permanent paper from mills that operate a sustainable forestry policy, and which has been manufactured from pulp which is processed using acid-free and elemental chlorine-free practices. Furthermore, the publishers ensure that the text paper and cover board used have met acceptable environmental accreditation standards. Typeset by Replika Press Pvt Ltd, India Printed by TJ International Limited, Padstow, Cornwall, UK

v

To my wife Saraswathi

vi

vii

Contents

Programs available in this book

xvii

Preface

xxi

Acknowledgements 1 1.1 1.2 1.3 1.4 1.5 1.6

Introduction to dynamics Introduction Different types of dynamic loads Difference between dynamic and static problems Methodology Types of vibration Further reading

xxiii 1 1 2 2 4 5 6

Part I Structural dynam ics in relation to earthquakes 2

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11

Free vibration of single-degree-of-freedom systems (undamped) in relation to structural dynamics during earthquakes Introduction Formulation of the equation of motion Simple harmonic theory Newton’s second law Simple pendulum Comparison of simple harmonic motion and uniform circular motion Energy method Rayleigh method D’Alembert’s principle Free vibration of rigid bodies without damping Program 2.1: MATLAB program to draw displacement, velocity and acceleration with respect to time

9 9 9 10 14 19

21 22 24 24 25 28

viii

Contents

2.12

Program 2.2: MATHEMATICA program to draw displacement, velocity and acceleration with respect to time Free vibration of structural systems Exercises Further reading

2.13 2.14 2.15 3

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 4

4.1 4.2 4.3 4.4 4.5

4.6 4.7 4.8 4.9 4.10 4.11

Free vibration of single-degree-of-freedom systems (under-damped) in relation to structural dynamics during earthquakes Introduction Damping free vibrations Logarithmic decrement Hysteresis damaping Coulomb damping Numerical method to find response due to initial conditions only Program 3.1: MATLAB program for free vibration of under-damped SDOF systems Program 3.2: MATHEMATICA program for free vibration of damped SDOF systems Summary Exercises Further reading Forced vibration (harmonic force) of single-degreeof-freedom systems in relation to structural dynamics during earthquakes Forced vibration without damping Beating phenomenon Resonance Forced vibration with damping Program 4.1: MATHEMATICA program to find displacement response of under-damped system subjected to sinusoidal loading Recurrence formula of Wilson Program 4.2: MATLAB program for finding response due to harmonic force Vector relationship in forced vibration Rotating imbalance Transmissibility (force isolation) Program 4.3: MATLAB program to compute MF, MX/me and TR

29 32 39 42

43 43 43 47 53 55

58 59 60 66 66 67

68 68 73 75 77

79 81 82 83 87 92 93

Contents

ix

4.12 4.13 4.14 4.15 4.16 4.17 4.18

Effectiveness of foundation Displacement isolation Vibration-measuring instruments How to evaluate damping in SDOF Response to ground acceleration Exercises Further reading

94 95 96 97 99 101 103

5 5.1 5.2 5.3

Response of structures to periodic dynamic loadings Introduction Fourier analysis Program 5.1: MATHEMATICA program to determine Fourier coefficients of forcing function Response to periodic excitation Program 5.2: MATHEMATICA program for finding the response to a periodic function Frequency domain analysis Alternative form of Fourier series Program 5.3: MATLAB program to evaluate amplitudes and phase angles Expression of forcing function using complex variable approach Discrete fourier transform (DFT) and fast fourier transform (FFT) Gibbs phenomenon Summary Exercises Further reading

105 105 106

Response of structures to impulse loads Introduction Impulsive loading – sine wave Program 6.1: MATLAB program to obtain maximum response for half sine cycle pulse Response to other arbitrary dynamic excitation Duhamel integral Response to arbitrary dynamic excitation Response spectrum Program 6.3: MATLAB program to find the response spectrum for any load pulse Laplace transform Program 6.4: MATHEMATICA program for Laplace transform method

136 136 136

5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 6 6.1 6.2 6.3

6.4 6.5 6.6 6.7 6.8 6.9 6.10

111 115 116 121 122 124 127 131 132 133 133 134

140 141 146 148 157 158 163 165

x

Contents

6.11 6.12 6.12

Summary Exercises Further reading

7

Dynamic response of structures using numerical methods Introduction Time stepping methods Types of time stepping method Response to base excitation Wilson’s procedure (recommended) Response of elasto-plastic SDOF system Program 7.10: MATLAB program for dynamic response for elasto-plastic SDOF system Response spectra by numerical integration Numerical method for evaluation of the Duhamel integral Selection of direct integration method Summary Exercises Further reading

7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 7.12 7.13 8

8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12 9

9.1 9.2 9.3

Generalized coordinates and energy methods in relation to structural dynamics during earthquakes Introduction Principle of virtual work Generalized SDOF system: rigid bodies Systems having distributed stiffness and distributed mass Rayleigh method Improved Rayleigh method Hamilton’s principle Lagrange’s equations Computer-generated Euler–Lagrange equations using MATHEMATICA Summary Exercises Further reading

167 167 169

171 171 172 173 211 217 224

227 231 232 236 237 237 238

240 240 240 241 243 248 250 251 253

259 262 263 264

Tw o-degrees-of-freedom linear system response of structures 266 Overview 266 Free vibration of undamped two-degrees-of-freedom system 266 Program 9.1: MATHEMATICA program to solve coupled differential equations 273

Contents

9.4 9.5 9.6 9.7 9.8

9.9 9.10 9.11 9.12 9.13

9.14 9.15 9.16 10

10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.10 10.11 10.12 10.13 10.14

Program 9.2: MATLAB program to solve free vibration of undamped two-degrees-of-freedom system Program 9.3: MATLAB program to solve coupled differential equations Coordinate coupling Simple system: two storey shear building Program 9.4: MATHEMATICA program for finding the responses of an undamped two-degrees-of-freedom system – free vibration Forced vibration of two-degrees-of-freedom undamped system Program 9.5: MATHEMATICA program for forced vibration of two-degrees-of-freedom undamped system Vibration absorber Forced response of a two-degrees-of-freedom under-damped system Program 9.6: MATLAB program for displacement response of two-degrees-of-freedom under-damped system for forced vibration Summary Exercises Further reading Free vibration of multiple degrees of freedom in relation to structural dynamics during earthquakes Introduction Modelling of a continuous system as an MDOF system Equations of motion of an MDOF system Free undamped vibration of an MDOF system Orthogonality relationship Normalization of modes Influence coefficient method Program 10.1: MATHEMATICA program for finding the solution of the characteristic equation Program 10.2: MATLAB program to find the frequencies and normalized mode shapes Program 10.3: MATLAB program for solving structural problem by the stiffness method Static condensation of stiffness matrix General viscous damping Program 10.4: MATLAB program for free vibration of MDOF with generalized damping Newmark’s numerical integration

xi

274 276 280 285

288 293 295 297 298

301 302 302 303

305 305 306 307 308 310 311 313

318 318 326 330 331 332 336

xii

Contents

10.15

Program 10.5: MATLAB program for Newmark’s method of MDOF with generalized damping Forced response of a three-degrees-of-freedom under-damped system Summary Exercises Further reading

10.16 10.17 10.18 10.19 11

11.1 11.2 11.3 11.4 11.5 11.6

11.7 11.8 11.9 11.10 11.11 11.12 12

12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 12.10

Numerical solution methods for natural frequencies and mode shapes in relation to structural dynamics during earthquakes Introduction General solution methods for eigen problems Vector iteration technique Jacobi’s method Transfer matrix method to find the fundamental frequency of a multi-storeyed building (shear frame) Program 11.1: MATHEMATICA program to find the fundamental frequency and the corresponding mode shape (transfer matrix method) Holzer method for torsional vibrations Approximate methods for finding the natural frequencies Dunkerley’s approximation Summary Exercises Further reading Time history response by mode superposition in relation to structural dynamics during earthquakes Introduction Limitations Mode displacement method for uncoupled system Modal participation factor Time history analysis Mode superposition solution for systems with classical damping Numerical evaluation of modal response Program 12.1: MATLAB program for dynamic response using modal superposition Dynamic analysis using direct integration methods Program 12.2: MATLAB program for finding dynamic response of MDOF using direct integration method (Newmark’s method)

337 338 341 342 343

344 344 344 346 359

361

364 366 369 377 380 380 381

383 383 383 384 387 388

397 401 405 410

410

Contents

12.11 12.12 12.13 12.14 12.15 12.16

Normal mode response to support motions Response spectrum analysis Mode acceleration method Summary Exercises Further reading

13

Free and forced vibration of a continuous system in relation to structural dynamics during earthquakes Introduction Vibration of a string Program 13.1: MATHEMATICA program to find displacement of a string Longitudinal vibration of a uniform rod Torsional vibration of shaft or rod Free flexural vibration of beams Program 13.2: MATHEMATICA program to find the frequency of a long beam with usual boundary conditions Orthogonality of normal modes Effect of axial force (tension or compression) Effect of rotary inertia and shear deformation Forced axial vibration of bars Beams subjected to moving loads Summary Exercises Further reading

13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 13.9 13.10 13.11 13.12 13.13 13.14 13.15 14

14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 14.9 14.10 14.11

Finite element method in relation to structural dynamics during earthquakes Introduction Dynamic analysis Torsional vibration of a shaft Axial vibration of rods Assumed modes method Program 14.1: MATLAB program for the assumed modes method Truss element Program 14.2: MATLAB program for free vibration of trusses Beam element Program 14.3: MATHEMATICA program for evaluation of stiffness matrix, and mass matrix of a beam element Program 14.4: MATLAB program to find the natural frequency of beams or rigid frames

xiii

415 416 424 427 427 430

431 431 432

435 436 441 443 449 457 458 462 465 469 474 474 475

477 477 478 478 484 485

488 490 496 499 502 506

xiv

Contents

14.12 14.13

Forced vibration of a beam Program 14.5: MATLAB program for forced vibration of a beam Vibration of a Timoshenko beam Program 14.6: MATLAB program to find the frequency of a Timoshenko beam Summary Exercises Further reading

14.14 14.15 14.16 14.17 14.18 15

15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 15.9 15.10 15.11

15.12 15.13 15.14 15.15 15.16 15.17 15.18 15.19 15.20 15.21

Differential quadrature and transformation methods for vibration problems in relation to structural dynamics during earthquakes Introduction DQ method Lagrangian interpolation Differential quadrature method formulation HDQ method Transverse vibration of pre-tensioned cable Program 15.1: MATLAB program for finding the natural frequency of lateral vibration of a pre-tensioned string Lateral vibration of uniform Euler beams Program 15.2: MATLAB program for free vibration of an Euler beam To find natural frequency and mode shape given variation of D = EI for Euler beam with axial load Program 15.3: MATLAB program for solving free vibration problem of non-prismatic beam with or without axial load Vibration of Timoshenko beam by DQ method Program 15.4: MATLAB program for free vibration analysis of Timoshenko beam DT method Transverse vibration of pre-tensioned cable Program 15.5: MATHEMATICA program for finding the natural frequency of vibration of a pre-tensioned cable Free vibration analysis of Euler beam Program 15.6: MATHEMATICA program for finding the natural frequency of vibration an Euler beam Natural frequency of Euler beam subjected to axial load Program 15.7: MATHEMATICA program for finding the natural frequency an Euler beam subjected to axial load Natural frequency of a Timoshenko beam

510 512 515 518 522 522 524

525 525 525 526 527 528 529

530 537 540 542

545 548 550 553 553 555 556 558 559 561 562

Contents

15.22 15.23 15.24 15.25

Program 15.8: MATHEMATICA program for finding the natural frequency of a Timoshenko beam Summary Exercises References and further reading

xv

563 565 566 567

Part II Response of structures to earthquakes 16 16.1 16.2 16.3 16.4 16.5 16.6 16.7 16.8 16.9 16.10 16.11 16.12 16.13 16.14 16.15 10.16 16.17

Earthquakes and earthquake ground motion Introduction What is an earthquake? Plate tectonic theory Faults Earthquake belts in the world Elastic rebound theory Seismic waves Measuring instruments Earthquake intensity and magnitude Basic difference: magnitude versus intensity Earthquake ground motion Earthquake classification Asian tsunami disaster Damage mechanisms due to earthquakes Summary Web links References and further reading

571 571 572 577 578 580 582 582 586 588 594 594 599 599 601 601 601 602