TEXTBOOK OF FINITE ELEMENT ANALYSIS

[Pages:340]Textbook of

Finite Element Analysis

P. Seshu

Textbook of

Finite Element Analysis

P. Seshu

Professor Department of Mechanical Engineering Indian Institute of Technology Bombay

Mumbai

NEW DELHI 2012

Rs. 250.00

TEXTBOOK OF FINITE ELEMENT ANALYSIS P. Seshu

? 2003 by PHI Learning Private Limited, New 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-978-81-203-2315-5

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

Tenth Printing

L

L

January, 2012

Published by Asoke K. Ghosh, PHI Learning Private Limited, M-97, Connaught Circus, New Delhi-110001 and Printed by Mohan Makhijani at Rekha Printers Private Limited, New Delhi-110020.

Respectfully Dedicated to My Parents and Teachers

Contents

Preface

1. Introduction

1.1 Typical Application Examples 4 1.1.1 Automotive Applications 4 1.1.2 Manufacturing Process Simulation 7 1.1.3 Electrical and Electronics Engineering Applications 8 1.1.4 Aerospace Applications 14

Summary 14

2. Finite Element Formulation Starting from Governing Differential Equations

2.1 Weighted Residual MethodUse of a Single Continuous Trial Function 16

2.2 The General Weighted Residual (WR) Statement 28 2.3 Weak (Variational) Form of the Weighted Residual Statement 33 2.4 Comparison of Differential Equation, Weighted Residual and Weak

Forms 36 2.5 Piece-wise Continuous Trial Function Solution of the Weak Form 41 2.6 One-dimensional Bar Finite Element 48 2.7 One-dimensional Heat Transfer Element 57 Summary 61 Problems 61

3. Finite Element Formulation Based on Stationarity of a Functional

3.1 Introduction 66 3.2 Functional and Differential Equation Forms 67 3.3 Principle of Stationary Total Potential (PSTP) 73

3.3.1 RayleighRitz Method 75

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115 16 65

66 88

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3.4 Piece-wise Continuous Trial FunctionsFinite Element Method 81 3.4.1 Bar Element Formulated from the Stationarity of a Functional 81 3.4.2 One-dimensional Heat Transfer Element Based on the Stationarity of a Functional 83

3.5 Meaning of Finite Element Equations 84 Summary 87 Problems 88

4. One-dimensional Finite Element Analysis

4.1 General Form of the Total Potential for 1-d 89 4.2 Generic Form of Finite Element Equations 90 4.3 The Linear Bar Finite Element 93 4.4 The Quadratic Bar Element 101

4.4.1 Determination of Shape Functions 101 4.4.2 Element Matrices 102 4.5 Beam Element 117 4.5.1 Selection of Nodal d.o.f. 117 4.5.2 Determination of Shape Functions 118 4.5.3 Element Matrices 119 4.6 Frame Element 125 4.7 One-dimensional Heat Transfer 132 Summary 138 Problems 138

89144

5. Two-dimensional Finite Element Analysis

5.1 IntroductionDimensionality of a Problem 145 5.2 Approximation of Geometry and Field Variable 148

5.2.1 Simple Three-noded Triangular Element 149 5.2.2 Four-noded Rectangular Element 152 5.2.3 Six-noded Triangular Element 153 5.3 Natural Coordinates and Coordinate Transformation 156 5.3.1 Alternate Methods of Deriving Shape Functions 157 5.3.2 Natural CoordinatesQuadrilateral Elements 159 5.3.3 Natural CoordinatesTriangular Elements 164 5.4 2-d Elements for Structural Mechanics 167 5.4.1 Generic Relations 167 5.4.2 Three-noded Triangular Element 171 5.4.3 Four-noded Rectangular Element 179 5.4.4 Compatibility of Displacements 181 5.4.5 Four-node Quadrilateral Element 183 5.4.6 Eight-node Quadrilateral Element 188 5.4.7 Nine-node Quadrilateral Element 190 5.4.8 Six-node Triangular Element 192

145231

5.5 Numerical Integration 194 5.5.1 Trapezoidal Rule 195 5.5.2 Simpsons 1/3 Rule 196 5.5.3 NewtonCotes Formula 197 5.5.4 Gauss Quadrature Formula 198 5.5.4 Gauss Quadrature in Two Dimensions 201

5.6 Incorporation of Boundary Conditions 205 5.7 Solution of Static Equilibrium Equations 206 5.8 2-d Fluid Flow 220 Summary 225 Problems 226

6. Dynamic Analysis Using Finite Elements

6.1 Introduction 232 6.2 Vibration Problems 232 6.3 Equations of Motion Based on Weak Form 235

6.3.1 Axial Vibration of a Rod 235 6.3.2 Transverse Vibration of a Beam 237 6.4 Equations of Motion Using Lagranges Approach 240 6.4.1 Formulation of Finite Element Equations 242 6.4.2 Consistent Mass Matrices for Various Elements 245 6.5 Consistent and Lumped Mass Matrices 246 6.5.1 HRZ Lumping Scheme 247 6.6 Form of Finite Element Equations for Vibration Problems 253 6.7 Some Properties of Eigenpairs 255 6.8 Solution of Eigenvalue Problems 257 6.8.1 Transformation Based Methods 258 6.8.2 Vector Iteration Methods 264 6.9 Transient Vibration Analysis 272 6.9.1 Modelling of Damping 272 6.9.2 The Mode Superposition Scheme 275 6.9.3 Direct Integration Methods 279 6.10 Thermal TransientsUnsteady Heat Transfer in a Pin-Fin 289 Summary 293 Problems 293

7. Application Examples

7.1 Finite Element Analysis of Crankshaft Torsional Vibrations 295 7.1.1 Beam Element Model of Crankshaft Assembly 296 7.1.2 Results and Discussion 299 7.1.3 Dynamic Response Analysis 301

7.2 Axisymmetric Finite Element Analysis of a Pressure Vessel 303 7.2.1 Finite Element Formulation for Axisymmetric Loads 304 7.2.2 Stress Analysis of a Pressure Vessel 305

Contents vii

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Appendix ASuggested Mini-Project Topics

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Project 1: Thermal Analysis of a Pressure Vessel 309

Project 2: Structural Dynamic Analysis of a Pressure Vessel 310

Project 3: Dynamics of a Scooter Frame 312

Project 4: Automotive Chassis Dynamics 313

Project 5: Analysis of a Turbine Disk 316

Project 6: Dynamic Analysis of a Building 317

Project 7: Thermal Analysis of an IC Engine Cylinder 318

Project 8: Stress Concentration 319

Project 9: Dynamics of a Hard Disk Drive Read/Write Head Assembly 319

Appendix BReview of Preliminaries

B1.1 Matrix Algebra 321 B1.2 Interpolation 322

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Appendix CTypical Finite Element Program

324 328

Index

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