Structural. engineering. dynamics of earthquake. s. Rajasekaran. W OODHEAD PUBLISHING LIMITED Oxford Cambridge New Delhi
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1 Structural dynamics of earthquake engineering Theory and application using MATHEMATICA and MATLAB s. Rajasekaran ocrc Press Boca Raton Boston New York Washington, DC W OODHEAD PUBLISHING LIMITED Oxford Cambridge New Delhi
2 Contents Programs available in this book Preface Acknowledgements xvii xxi xxiii Introduction to dynamics Introduetion Different types of dynamie loads Differenee between dynamie and statie problems Methodology Types of vibration Further reading Part I Structural dynamics in relation to earthquakes Free vibration of single-degree-of-freedom systems (undampedl in relation to structural dynamics during earthquakes Introduetion Formulation of the equation of motion Simple harmonie theory Newton's seeond law Simple pendulum Comparison of simple harmonie motion and uniform eireular motion Energy method Rayleigh method D'Alembert's principle Free vibration of rigid bodies without damping Program 2.1: MATLAB program to draw displacement, veloeity and aeeeleration with respeet to time
3 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 42 3 Free vibration of single-degree-of-freedom systems (under-damped) in relation to struetural dynamies 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 67 4 Foreed vibration (harmonie force) of single-degreeof-freedom systems in relation to struetural dynamies 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 93
4 Contents IX 4.12 Effectiveness of foundation Displacement isolation Vibration-measuring instruments How to evaluate damping in SDOF Response to ground acceleration Exercises Further reading 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 eva1uate amplitudes and phase angles Expression of forcing function using comp1ex variable approach Discrete fourier transform (DFT) and fast fourier transform (FFT) Gibbs phenomenon Summary Exercises Further reading Response of structures to impulse loads Introduction Impulsive loading - sine wave Program 6.1: MATLAB program to obtain maximum response for half sine cyde pulse Response to other arbitrary dynamic excitation Duhame1 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 165
5 x Contents 6.11 Summary Exercises Further reading 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 Se1ection of direct integration method Summary Exercises Further reading 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 Two-degrees-of-freedom linear system response of structures Overview Free vibration of undamped two-degrees-of-freedom system Program 9.1: MATHEMATICA program to solve coupled differential equations 273
6 Contents XI 9.4 Program 9.2: MATLAB program to solve free vibration of undamped two-degrees-of-freedom system Program 9.3: MATLAB program to solve coup1ed differential equations Coordinate coup1ing 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 Orthogona1ity relationship Norma1ization of modes Influence coefficient method Program 10.1: MATHEMATICA program far finding the solution of the characteristic equation Program 10.2: MATLAB program to find the frequencies and norma1ized mode shapes Program 10.3: MATLAB program far solving structura1 problem by the stiffness method Static condensation of stiffness matrix General viscous damping Program 10.4: MATLAB program far free vibration of MDOF with genera1ized damping Newmark's numerica1 integration 336
7 XII Contents 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 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) 410
8 Contents xiii Normal mode response to support motions Response spectrum analysis Mode acce1eration method Summary Exercises Further reading Free and forced vibration of a continuous system in relation to structural dynamics during earthquakes Introduction Vibration of astring Program 13.1: MATHEMATICA program to find displacement of astring Longitudinal vibration of a uniform rod Torsional vibration of shaft or rod Free flexura1 vibration ofbeams Program 13.2: MATHEMATICA program to find the frequency of a long beam with usua1 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 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 506
9 xiv Contents Forced vibration of a bearn Program 14.5: MATLAB program for forced vibration of a beam Vibration of a Timoshenko beam Program 14.6: MATLAB prograrn to find the frequency of a Timoshenko beam Summary Exercises Further reading 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 bearn 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 562
10 Contents xv Program 15.8: MATHEMATICA program for finding the natural frequency of a Timoshenko beam Summary Exercises References and further reading 567 Part 11 Response of structures to earthquakes 16 Earthquakes and earthquake ground motion Introduction What is an earthquake? P1ate tectonic theory Faults Earthquake be1ts in the world E1astic rebound theory Seismic waves Measuring instruments Earthquake intensity and magnitude Basic difference: magnitude versus intensity Earthquake ground motion Earthquake c1assification Asian tsunami disaster Oamage mechanisms due to earthquakes Summary Web links References and further reading Earthquake response spectra Introduction Earthquake response spectra Program 17.1: MATLAB program for drawing spectra for any specified earthquake Program 17.2: MATLAB program to draw tripartite plot Importance of response quantities Response spectrum concept Pseudo-velocity spectrum Pseudo-spectral acceleration Combined deformation, velocity and acceleration (OVA) spectrum Velestos and Newmark spectra How the response spectrum is constructed Elastic design spectrum 640
11 XVI Contents Program 17.3: MATLAB program for drawing Newmark-Hal1 design spectra Response spectrum characteristics Distinction between design and response spectra Response spectrum Site-specific response spectra Estimating the ground motion Seismic analysis and design verification Ine1astic response spectra Application of ine1astic design spectrum Ine1astic deformation Summary Exercises Further reading Earthquake analysis of linear systems Introduction Lumped mass system: shear building Modal response contribution using Chopra's method Modal analysis for r f (t) Interpretation of modal analysis Modal contribution factor Modal response and required number of modes Modal contributions Program 18.1: MATLAB program to find the ratio of dynamic shear to static shear in a multi-storey building Earthquake analysis linear systems Modal response Mu1ti-storey buildings with symmetrical plan Spectrum analysis by modal response Effective modal mass and modal height Multiple support excitation Symmetrie plan bui1dings: trans1ationa1 ground motion Summary Exercises References and further reading Building codes for aseismic design Introduction Historica1 deve10pment Coda1 provisions for seismic design Program 19.1: MATLAB program for IS1893 code Comparison of codes 763
12 Contents xvii 19.6 Design examples using IS Part Summary Exercises References and further reading Response of structures to earthquakes: approximate analysis techniques for lateral loads Introduction Simplified analysis for lateral loads Zero moment point method Approximate methods of analysis of multi-bay frames (lateral loads) Analysis of buildings simple in plan for lateral loads Summary Exercises References and further reading Response of structures to earthquakes: analysis of shear walls Introduction Shear wall frame Coupled shear walls Program 21.1: MATHEMATICA program for coupled shear wall Summary Exercises References and further reading 863 Index 864
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