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Finite Element Methodand ComputationalStructural Dynamics

Manish Shrikhande

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Finite Element M

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Shrikhande

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ISBN:978-81-203-4995-7

Finite Element Methodand ComputationalStructural Dynamics

Manish Shrikhande

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rimarily intended for senior undergraduate and postgraduate students of civil, mechanical and aerospace/aeronautical engineering, this text emphasises the importance of reliability in engineering

computations and understanding the process of computer aided engineering.

Written with a view to promote the correct use of finite element technology and to present a detailed studyof a set of essential computational tools for the practice of structural dynamics, this book is a ready-reckonerfor an in-depth discussion of finite element theory and estimation and control of errors in computations. It is specifically aimed at the audience with interest in vibrations and stress analysis. Several worked out examplesand exercise problems have been included to describe the various aspects of finite element theory and modelling. The exercise on error analysis will be extremely helpful in grasping the essence of posteriori error analysis and mesh refinement.

KEY FEATURES

Thorough discussion of numerical algorithms for reliable and efficient computation.

Ready-to-use finite element system and other scientific applications.

Tips for improving the quality of finite element solutions.

Companion DVD containing ready to use finite element applications.

THE AUTHOR

MANISH SHRIKHANDE, Ph.D. (IIT Kanpur), is Professor at Department of Earthquake Engineering, IndianInstitute of Technology Roorkee, and a member of Indian Society of Earthquake Technology and Earthquake Engineering Research Institute. He is a recipient of the Young Engineer Award of Indian National Academy of Engineering (INAE) and the Career Award of AICTE.

Manish shrikhande

ProfessorDepartment of Earthquake EngineeringIndian Institute of Technology Roorkee

Finite element Method and Computational structural dynamics

Delhi-1100922014

Finite element method and Computational StruCtural dynamiCS (with dVd-rom)Manish Shrikhande

© 2014 by PHI Learning Private Limited, Delhi. All rights reserved. No part of this book may be reproduced in any form, by mimeograph or any other means, without permission in writing from the publisher.

iSBn-978-81-203-4995-7

The export rights of this book are vested solely with the publisher.

Published by Asoke K. Ghosh, PHI Learning Private Limited, Rimjhim House, 111, Patparganj Industrial Estate, Delhi-110092 and Printed by Raj Press, New Delhi-110012.

Tothe loving memory of my parents

satish Gangadhar shrikhandeand

Prabha shrikhande

who taught me to persevere

List of Figures xi

List of Tables xvii

Preface xix

Part I—FInIte element method

1. mathematical modelling, differential equations and approximate Solutions 3 – 31 1.1 Mathematical Modelling in Engineering 3 1.2 Approximation via Method of Weighted Residuals 5 1.2.1 Variants of Method of Weighted Residuals 6 1.2.2 Essential and Natural Boundary Conditions 10 1.2.3 Solution of Coupled Differential Equations 12 1.3 Approximation via Variational Principles 17 1.3.1 Variational Principles and Method of Weighted Residuals 19 1.4 Convergence of the Approximate Solution 19 1.5 From Continua to Discontinua 25 1.5.1 The Finite Element Concept 26 1.5.2 Approximation over Finite Elements 27 Exercises 28 Suggested Further Reading 31

2. Finite elements of one-dimension 32 – 70 2.1 Introduction 32 2.2 Finite Elements of C 0 Continuity 32 2.2.1 Developing a Finite Element Model 35 2.2.2 Assembly of Finite Element Equations 40 2.2.3 Incorporation of Boundary Conditions 42 2.2.4 Solution of Finite Element Equations and Post-processing 46 2.2.5 Global and Local Coordinate Systems 47 2.3 Finite Elements of C 1 Continuity 49 2.3.1 Euler–Bernoulli Beam 49

Contents

v

vi Contents

2.4 Finite Element Modelling for Shear Flexible Beams 56 2.4.1 Developing a Finite Element Model 57 2.4.2 Shear Locking: Problem and Solution 60 2.5 Finite Element Modelling for Beam-Column 63 2.5.1 Elements with Arbitrary Orientation 65 2.6 Finite Element Modelling for Grillage System 66 2.7 Comments on Finite Element Approximation 66 Exercises 67 Suggested Further Reading 69

3. Finite elements of two and three dimensions 71 – 135 3.1 A Review of Three–Dimensional Elasticity 71 3.1.1 Equations of Equilibrium 71 3.1.2 Strain–Displacement Relations 73 3.1.3 St. Venant Compatibility Equations 74 3.2 Weighted Residual Formulation and the Weak Form 74 3.3 Approximations for Two-Dimensional Elasticity 77 3.3.1 Plane Strain 79 3.3.2 Plane Stress 80 3.3.3 Axisymmetry 81 3.4 Finite Elements for Two-Dimensional Domains 84 3.4.1 Triangular Elements 86 3.4.2 Rectangular Elements 93 3.5 Finite Elements for Three-Dimensional Domains 99 3.5.1 Tetrahedral Elements 101 3.5.2 Pentahedral (Wedge) Elements 106 3.5.3 Hexahedral (Brick) Elements 107 3.6 Development of Finite Element Equations 109 3.6.1 Two-Dimensional Problems 111 3.6.2 Axisymmetric Problems 131 Exercises 134 Suggested Further Reading 135

4. mapped elements 136 – 173 4.1 Finite Elements with Curved Boundaries 136 4.1.1 Parametric Mapping 137 4.2 Iso-parametric Finite Elements 139 4.2.1 Geometric Compatibility of Elements 139 4.2.2 Continuity of Primary Variables in Distorted Domains 140 4.2.3 Completeness of Interpolated Field in Distorted Domains 140 4.3 Evaluation of Integrals in Distorted Domains 141 4.3.1 Cartesian Derivatives for Rectangular and Hexahedral Domains 142 4.3.2 Cartesian Derivatives for Triangular and Tetrahedral Domains 143 4.3.3 Evaluation of Boundary Integrals for Distorted Domains 147 4.4 Element Distortion and Accuracy of Field Approximation 148 4.5 Degenerate Elements 153 4.6 Rules for Discretisation of Domain via Finite Elements 156 4.6.1 Transition Elements 156

Contents vii

4.7 Element Defects and Remedies 158 4.7.1 Parasitic Shear 159 4.8 Goodness of a Finite Element 163 4.8.1 The Eigenvalue Test 163 4.8.2 Independent Deformation Modes 164 4.8.3 The Patch Test 165 4.9 Post-processing and Stress Recovery 168 4.9.1 Stress Averaging and Contouring 169 4.10 Useful Tips for Finite Element Modelling 169 Exercises 170 Suggested Further Reading 172

5. Finite elements for plates and Shells 174 – 197 5.1 Introduction 174 5.1.1 Terminology 175 5.2 Kirchhoff’s Theory for Thin Plates 175 5.2.1 Equilibrium Equations 178 5.2.2 Weighted Residual Statement and the Weak Form 180 5.2.3 Boundary Conditions 181 5.2.4 Finite Element Model for Kirchhoff Plate Bending 182 5.3 First Order Shear Deformation Theory for Thick Plates 185 5.3.1 Equilibrium Equations 186 5.3.2 Weighted Residual Statement and the Weak Form 187 5.3.3 Finite Element Model for Reissner–Mindlin Plate Bending 187 5.4 Finite Elements for Shells 190 5.4.1 Degenerated Shell Element 190 5.5 Closure 196 Exercises 196 Suggested Further Reading 196

6. error analysis and Convergence of Finite element Solution 198 – 204 6.1 Introduction 198 6.2 A Posteriori Error Analysis 200 6.2.1 Super-convergent Patch Recovery 201 Exercises 204 Suggested Further Reading 204

7. the time dimension 205 – 221 7.1 The Nature of Time Dependent Response 205 7.2 System Properties in Time Dependent Problems 207 7.3 Free Vibration and Normal Modes 212 7.4 The Mode Superposition Method 215 7.4.1 Mode Displacement Formulation 217 7.4.2 Mode Acceleration Formulation 217 7.4.3 Inclusion of Rigid Body Modes 218 Exercises 220 Suggested Further Reading 221

viii Contents

Part II—ComPutatIonal StruCtural dynamICS

8. Solution of linear Simultaneous equations 225 – 250 8.1 Introduction 225 8.2 Direct Solvers 226 8.2.1 Gaussian Elimination 226 8.2.2 Cholesky Factorisation 233 8.3 Condition Number and Numerical Ill-Conditioning 234 8.3.1 Matrix Equilibration 235 8.4 Iterative Solvers 236 8.4.1 Jacobi and Gauss–Seidel Iterations 236 8.4.2 Method of Steepest Descent 238 8.4.3 Method of Conjugate Gradients 240 8.5 Solution of Sparse Systems 246 8.5.1 Sparse Matrix Storage 247 Suggested Further Reading 249

9. the algebraic eigenvalue problem 251 – 286 9.1 Introduction 251 9.2 Geometrical Interpretation 254 9.3 Eigenvectors as Orthogonal Basis 254 9.4 Sensitivity of Eigenvalues 255 9.5 Similarity Transforms 256 9.5.1 Givens Transformation 257 9.5.2 Householder Transformation 258 9.6 Computation of Eigensystem 261 9.6.1 Gershgorin Discs 261 9.6.2 Schur Decomposition 262 9.6.3 Method of Power Iterations 262 9.6.4 Rayleigh Quotient Iterations 267 9.6.5 Subspace Iterations 268 9.6.6 QR Iterations 269 9.6.7 Jacobi’s Method of Successive Rotations 274 9.6.8 Lanczos Iterations 278 9.6.9 QZ Iterations for Generalised Eigenvalue Problem 279 9.6.10 Sturm Sequence 281 9.7 The Quadratic Eigenvalue Problem 283 Suggested Further Reading 286

10. Singular Value decomposition 287– 297 10.1 Introduction 287 10.2 Relation to Eigenvalue Problem 288 10.3 Existence of SVD 289 10.3.1 Geometrical Interpretation of SVD 289 10.4 Stable and Efficient Computation of SVD 289

Contents ix

10.5 Applications 292 10.5.1 Least-Squares Solution 292 10.5.2 Reduced Rank Approximation and Feature Extraction 295 Suggested Further Reading 296

11. time marching: numerical Solution of initial Value problems 298 – 325 11.1 Introduction 298 11.2 Methods based on Taylor Series Expansion 299 11.2.1 Runge–Kutta Methods 301 11.2.2 Linear Multi-Step and Multi-Value Methods 307 11.3 Direct Methods for Vibration Problems 311 11.3.1 Single Step Methods 311 11.3.2 Multi-Step Methods 315 11.4 Analysis of Time Marching Schemes 317 11.4.1 Stability of Single-Step Methods 317 11.4.2 Consistency of Single-Step Methods 320 11.4.3 Stability of Multi-Step Methods 320 11.4.4 Accuracy of Time-Marching Schemes 321 Suggested Further Reading 324

12. discrete Fourier transform 326 – 343 12.1 Introduction 326 12.2 Discrete Time Data 329 12.3 Discrete Fourier Transform 332 12.3.1 DFT as a Linear Transformation 332 12.3.2 Properties of DFT 333 12.4 Fast (Finite) Fourier Transform 335 12.5 DFT Applications 337 12.5.1 Convolution and Deconvolution 337 12.5.2 Vibration Data Processing 341 Suggested Further Reading 343

13. System Identification:The InverseVibrationProblem 344–399 13.1 Introduction 344 13.2 State-Space Model 345 13.3 The z-Transform 347 13.3.1 System Transfer Function 350 13.4 Design of Experiments for System Identification 351 13.4.1 Time Domain or Frequency Domain? 352 13.5 Classical System Identification Based on Input–Output Set 353 13.5.1 Estimation of Impulse Response Function 353 13.5.2 Empirical Transfer Function Estimation 357 13.5.3 Parametric Estimate of Transfer Function 365 13.6 Output-Only Identification 366 13.6.1 Random Decrement Technique 366 13.6.2 Natural Excitation Technique 367

Finite Element Method AndComputational Structural Dynamics

Publisher : PHI Learning ISBN : 9788120349957Author : SHRIKHANDE,MANISH

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