Finite Element Method With Applications In Engineering

489 Pages·2011·14.859 MB·English

by  Y. M. Desai, T. I Eldho, A. H. Shah

Finite Element Method
with Applications
in Engineering
A01_DESAI2182_01_FM.indd 1 12/22/10 6:19:54 PM
This page is intentionally left blank.
A01_DESAI2182_01_FM.indd 2 12/22/10 6:19:54 PM
Finite Element Method
with Applications
in Engineering
Y. M. Desai
Professor, Department of Civil Engineering, IIT Bombay
T. I. Eldho
Professor, Department of Civil Engineering, IIT Bombay
A. H. Shah
Professor Emeritus, Department of Civil Engineering,
University of Manitoba, Canada
A01_DESAI2182_01_FM.indd 3 12/22/10 6:19:57 PM
Copyright © 2011 Dorling Kindersley (India) Pvt. Ltd.
Licensees of Pearson Education in South Asia
No part of this eBook may be used or reproduced in any manner whatsoever without the publisher’s
prior written consent.
This eBook may or may not include all assets that were part of the print version. The publisher
reserves the right to remove any material present in this eBook at any time.
ISBN 9788131724644
eISBN 9789332500839
Head Office: A-8(A), Sector 62, Knowledge Boulevard, 7th Floor, NOIDA 201 309, India
Registered Office: 11 Local Shopping Centre, Panchsheel Park, New Delhi 110 017, India
A01_DESAI2182_01_FM.indd 4 12/22/10 6:19:58 PM
Dedication
To my parents (late) Lilyben and Mangubhai
My wife Nilima and daughter Adveta
Y. M. Desai
To my parents (late) Iype and Marium
My wife Dr. Manjush and sons Iype and Basil
T. I. Eldho
To my wife Ranjan and daughters Ketki and Seema
A. H. Shah
A01_DESAI2182_01_FM.indd 5 12/22/10 6:19:58 PM
This page is intentionally left blank.
A01_DESAI2182_01_FM.indd 6 12/22/10 6:19:58 PM
CONTENTS
Preface xiii
Acknowledgements xv
Authors Profile xvii
1 Introduction 1
1.1 Introductory Remarks 1
1.2 Mathematical Modelling of Engineering Problems 1
1.3 Type of Governing Equations 4
1.3.1 Initial and Boundary Conditions 5
1.4 Solution Methodologies 6
1.4.1 Analytical Method 7
1.4.2 Physical Method 7
1.4.3 Computational Method 7
1.5 Numerical Modelling 7
1.6 Pre-Processing and Post-Processing 8
1.7 Scope of the Book 10
1.8 Highlights of the Book 10
1.9 How to Use the Book? 11
1.10 Closing Remarks 11
References and Further Reading 12
2 Approximate Methods of Analysis 13
2.1 Introduction 13
2.2 Aproximating Methods 14
2.3 Method of Weighted Residuals 14
2.3.1 Method of Point Collocation 15
2.3.2 Method of Collocation by Sub-Regions 15
2.3.3 Method of Least Squares 16
2.3.4 Galerkin’s Method 17
2.4 Rayleigh–Ritz Method 18
2.4.1 Relation Between FEM and Rayleigh–Ritz Method 20
2.5 Further Numerical Examples 20
2.6 Closing Remarks 25
Exercise Problems 25
References and Further Reading 26
3 Finite Element Method—An Introduction 27
3.1 General 27
3.2 What is FEM? 27
A01_DESAI2182_01_FM.indd 7 12/22/10 6:19:58 PM
viii | Contents
3.3 How Does FEM Work? 28
3.4 A Brief History of FEM 30
3.5 FEM Applications 31
3.6 Merits and Demerits of FEM 32
3.7 Closing Remarks 33
Exercise Problems 33
References and Further Reading 33
4 Different Approaches in FEM 35
4.1 Introduction 35
4.2 General Steps of FEM 35
4.3 Different Approaches Used in FEM 41
4.3.1 Direct Approach 41
4.3.2 Variational Approach 43
4.3.3 Energy Approach 46
4.3.4 Weighted Residual Approach 47
4.4 Closing Remarks 50
Exercise Problems 50
References and Further Reading 51
5 Finite Elements and Interpolation Functions 53
5.1 Introduction 53
5.2 Interpolation Functions 54
5.2.1 One-Independent Spatial Variable 54
5.2.2 Two-Independent Spatial Variables 55
5.2.3 Three-Independent Spatial Variables 55
5.3 One-Dimensional Elements 55
5.3.1 Line Element: Linear Interpolation Function 55
5.3.2 Quadratic Interpolation Function 58
5.3.3 Cubic Interpolation Function 60
5.3.4 Lagrangian Form of Interpolation Function 61
5.3.5 Further Higher Order Elements in
One-Dimension 63
5.4 Two-Dimensional Elements 68
5.4.1 Triangular Element: Linear Interpolation Function
in Cartesian Co-ordinates 68
5.4.2 Triangular Element—Area Co-ordinates 71
5.4.3 Integration Formula for Triangular Elements 72
5.4.4 Triangular Element—Quadratic Function 72
5.4.5 Triangular Element—Cubic Interpolation Function 74
5.4.6 Two-Dimensional Rectangular Elements 75
5.4.7 Rectangular Elements—Lagrangian Form in Natural
and Cartesian Co-ordinates 77
5.4.8 Isoparametric Elements 80
A01_DESAI2182_01_FM.indd 8 12/22/10 6:19:58 PM
Contents | ix
5.4.9 Lagrangian Interpolation Functions for
Two-Dimensional Elements 82
5.4.10 Two-Dimensional Serendipity Elements 84
5.5 Three-Dimensional Elements 88
5.5.1 Tetrahedral Elements 89
5.5.2 Tetrahedral Elements: Quadratic Interpolation Function 93
5.5.3 Tetrahedral Elements: Cubic Interpolation Function 93
5.5.4 Three-Dimensional Elements—Prismatic Elements 93
5.5.5 Three-Dimensional Elements in Local Co-ordinates 95
5.5.6 Three-Dimensional Serendipity Elements 96
5.6 Closing Remarks 97
Exercise Problems 97
References and Further Reading 98
6 One-Dimensional Finite Element Analysis 99
6.1 Introduction 99
6.2 Linear Spring 99
6.2.1 Expressions for Equivalent Spring Constant
and Nodal Forces 109
6.3 Truss Element 111
6.3.1 Plane Truss 111
6.3.2 Element Equations by Minimizing Potential
Energy 114
6.3.3 Local and Global Element Equations for a Bar
in the X–Y Plane 119
6.3.4 Computation of Stress for a Bar in the X–Y Plane 122
6.4 Space Truss 125
6.5 One-Dimensional Torsion of a Circular Shaft 130
6.6 One-Dimensional Steady State Heat Conduction 133
6.7 One-Dimensional Flow Through Porous Media 136
6.8 One-Dimensional Ideal Fluid Flow Through Pipes
(Inviscid Fluid Flow) 139
6.9 Beam Element 139
6.9.1 Review of Beam Theory 140
6.9.2 Finite Element Formulation of a Beam Element 142
6.9.3 Illustrative Examples 147
6.10 Analyses of Plane Frames and Grids 156
6.10.1 Plane Frame Analysis 156
6.10.2 Grid Analysis 166
6.11 Further One-Dimensional Applications 171
6.11.1 Flow Network Analysis 171
6.11.2 Electrical Network Analysis 178
6.12 Summary of Element Matrices for One-Dimensional
Finite Elements 181
A01_DESAI2182_01_FM.indd 9 12/22/10 6:19:58 PM