NONLINEAR STRUCTURAL DYNAMICS USING FE METHODS

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1 NONLINEAR STRUCTURAL DYNAMICS USING FE METHODS Nonlinear Structural Dynamics Using FE Methods emphasizes fundamental mechanics principles and outlines a modern approach to understanding structural dynamics. The book will be useful to practicing engineers, giving them a richer understanding of their tools and thus accelerating learning on new problems. Independent workers will find access to advanced topics presented in an accessible manner. The book successfully tackles the challenge of how to present the fundamentals of structural dynamics and infuse it with finite-element (FE) methods. First, the author establishes and develops mechanics principles that are basic enough to form the foundations of FE methods. Second, the book presents specific computer procedures to implement FE methods so that general problems can be solved that is, responses can be produced given the loads, initial conditions, and so on. Finally, the book introduces methods of analysis to leverage and expand the FE solutions. is a professor in the School of Aeronautics and Astronautics at Purdue University. His main area of research is experimental mechanics, wave propagation, and structural dynamics. Special emphasis is placed on solving inverse problems such as force and system-identification problems. He is a dedicated teacher and pedagogical innovator. He is a winner of the Frocht Award for Teaching and the Hetenyi Award for Research, both from the Society for Experimental Mechanics. This is his sixth book dealing with the mechanics of structures.

3 Nonlinear Structural Dynamics Using FE Methods Purdue University

4 32 Avenue of the Americas, New York, NY , USA Cambridge University Press is part of the University of Cambridge. It furthers the University s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. Information on this title: / c 2015 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2015 Printed in the United States of America A catalog record for this publication is available from the British Library. Library of Congress Cataloging in Publication Data Doyle, James F., 1951 Nonlinear structural dynamics using FE methods /, Purdue University. pages cm Includes bibliographical references and index. ISBN (hardback) 1. Structural analysis (Engineering) Mathematics. 2. Finite element method. 3. Nonlinear theories. I. Title. TA347.F5D dc ISBN Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet Web sites referred to in this publication and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate.

5 For Linda, who made this possible. Thank you from my heart.

7 Contents Notation page xi Introduction... 1 PART I MECHANICS AND MODELS 5 1 Dynamics of Simple Elastic Systems Motion of Simple Systems Transient Excitations and Responses Forced Vibrations through Periodic Loadings Spectral Analysis of Periodic Loadings 37 Problem Set 51 2 Dynamics of Discretized Systems Principle of Virtual Work Lagrangian Dynamics System Nonlinearities 79 Problem Set 89 3 Modeling Elastic Structures Review of the Mechanics of 3D Solids Ritz Method for Dynamic Problems Finite-Element Formulation for Frames Finite-Element Formulation for Solids Modeling Large Deformations of Solids 183 Problem Set Modeling Applied Loads Some Interaction Loadings Long-Term Periodic Loadings Conservative Loads and Systems 238 vii

8 viii Contents 4.4 Classification of Problems Based on Load Types 259 Problem Set Computational Methods Solving Large Systems of Equations Direct Time Integration of Linear Systems Time Integration of Nonlinear Systems Solving Large Eigensystems 316 Problem Set 338 PART II DYNAMIC ANALYSES Modal Analysis of Large Systems Modal Matrix Modal Superposition Method Damped Motions Advanced Modal Analyses 370 Problem Set Vibration of Rods and Beams Strong Formulation of Problems Spectral Analysis of Continuous Members Distributed Elastic Constraints 414 Problem Set Vibration of Plates and Shells Flexural Behavior of Flat Plates Membrane Behavior of Thin Flat Plates Deep Beams with Shear Deformations Thin-Walled Shells Nonlinear Vibrations of Panels 475 Problem Set Wave Propagation Introduction to Wave Propagation Spectral Analysis of Wave Motions Waves in Extended Solids Relation of Wave Responses to Vibrations 530 Problem Set Stability of the Motion Some Preliminary Stability Ideas Deformation-Dependent Loadings Stability of Parametrically Excited Systems 566

9 Contents ix 10.4 Motions in the Large 581 Problem Set 596 Project-Level Problem Set 599 References 613 Index 619

11 Notation Roman letters: a radius, plate width A cross-sectional area b, b i thickness, depth, plate length, body force c o longitudinal wave speed, E/ρ c P, c S, c R primary = E /ρ, secondary = G/ρ, and Rayleigh wave speeds C,[ C ] damping, damping matrix D plate flexural rigidity, Eh 3 /12(1 ν 2 ) ê i unit vectors E, Ê, E, Ē, Young s modulus, E = E/(1 ν 2 ), Ē = E h EI beam flexural stiffness E ij Lagrangian large-strain tensor F, ˆF, F o, F i member axial force, element nodal force F generalized nodal force g i (x) element shape functions G, Ĝ shear modulus, frequency-response function h beam or rod height, plate thickness h i interpolation functions i complex 1, counter I second moment of area, I = bh 3 /12 for rectangle J o, J, J e Jacobian J n Bessel functions of the first kind k, k 1, k 2 wavenumbers K,[ k ], [ K ] stiffness, stiffness matrices L length M, M x moment M,[ m ], [ M ] mass, mass matrices P(t), ˆP,{P} applied-force history P generalized applied load q, q u, q v, q w distributed load xi

12 xii Notation r, R radial coordinate, radius r, s, t isoparametric coordinates [ R ] rotation matrix t, t i time, traction vector T time window, period, temperature T kinetic energy [ T ] transformation matrix u(t) response; velocity, strain, etc. u, v, w displacements U strain energy V member shear force, volume V potential of conservative loads W beam width, Wronskein W work x o, y o, z o original rectilinear coordinates x, y, z deformed rectilinear coordinates Greek letters: α coefficient of thermal expansion δ small quantity, variation δ ij Kronecker delta determinant, increment ɛ, ɛ ij small quantity, strain η viscosity, damping, principal coordinate θ angular coordinate λ eigenvalue μ shear modulus, complex frequency ν Poisson s ratio total potential energy ρ o, ρ mass density σ, σ ij stress φ phase φ x, φ y, φ z rotation {φ },[ ] modal vector, matrix ω angular frequency ω o,ω d natural frequency, damped natural frequency ξ damping ratio Special symbols: 2 differential operator, ( 2 / x 2 ) + ( 2 / y 2 ) [ ] square matrix, rectangular array { } vector, spectral amplitude diagonal matrix (bar) local coordinates

13 Notation xiii ˆ (hat) vector, complex quantity (dot) time derivative Subscripts: E,G,T elastic, geometric, tangent (total) stiffness matrix i,j,k continuum tensor components, (comma) partial differentiation Superscripts: o original configuration complex conjugate prime, derivative with respect to argument Abbreviations: BC, PBC DoF, SDoF DKT EoM EVP FE FFT FRF IC MRT ODE boundary condition, periodic BC degree of freedom, single DoF discrete Kirchhoff triangular FE element equation of motion eigenvalue problem finite element fast Fourier transform frequency-response function initial condition membrane with rotation triangular FE element ordinary differential equation Primary Examples of Notation Use Discrete Systems: U u I P I = 0 I = 1,2,...,N where U is strain energy, u I is generalized DoF, P I is generalized force, N is total number of DoF, and I is the enumeration of DoF. Continuum Systems: ɛ ij = u i x o j i,j = 1, 2, 3 where ɛ ij is strain tensor, u i is Cartesian strain component, x o i is Cartesian position component, and i,j is Cartesian tensor component. Atomic Systems: P α i = V u α α = 1, 2,..., N, i = 1, 2, 3 i where P α i is Cartesian force component, V is load potential, u i is Cartesian displacement, i is Cartesian tensor component, N is total number of atoms, and α is the enumeration of atoms.

14 xiv Notation Atomic EoM: M α d2ˆr α dt 2 N = ˆP αβ ˆr = 3 β α i x iê i = vector where M α is mass, ˆr α is position vector, ˆP αβ is force vector on atom α due to atom β, and N is the total number of atoms.

Foundations and Applications of Engineering Mechanics

Foundations and Applications of Engineering Mechanics 4843/24, 2nd Floor, Ansari Road, Daryaganj, Delhi - 110002, India Cambridge University Press is part of the University of Cambridge. It furthers the