Introduction to Finite Elements in Engineering F O U R T H
E D I T I O N
TIRUPATHI R. CHANDRUPATLA Rowan University Glassboro, New Jersey
ASHOK D. BELEGUNDU The Pennsylvania State University University Park, Pennsylvania
International Edition contributions by T. RAMESH National Institute of Technology Technology Tiruchirapalli, India
CHAITALI RAY Bengal Engineering and Science University Shibpur, India
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ISBN-10: 0-273-76368-7 ISBN-13: 978-0-273-76368-0
To our parents
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Contents
PREFACE
13
ABOUT THE AUTHORS
16
1 FUNDAMENTAL CONCEPTS
1.1 1.2 1.3 1.4 1.5 1.6 1.7
Introduction Introduct ion 17 Historical Histori cal Background 17 Outline of Presentation Presentat ion 18 Stresses and Equilibrium Equilib rium 18 Boundary Boundar y Conditions Conditio ns 20 Strain–Displaceme Strain–Di splacement nt Relations Stress–Strain Stress–St rain Relations 22
17
21
Special Cases, 23
1.8 1.9
Temperature Effects 24 Potential Energy and Equilibrium: Equilibr ium: The Rayleigh–Ritz Rayleigh–R itz Method
25
Potential Energy, , 25 Rayleigh–Ritz Method, 28
1.10 1.11 1.12 1.13 1.14 1.15
Galerkin Galerki n’s Method 30 Saint Venant’s Principle Princip le 34 Von Mises Stress 35 Principle of Superposition 35 Computer Programs 36 Conclusion Conclus ion 36 Historical References 36 Problems 37
2 MATRIX MATRIX ALGEBRA AND GAUSSIAN GAUSSIAN ELIMINATION ELIMINATION
2.1
44
Matrix Algebra 44 Row and Column Vectors, 45
5
6
Contents Addition and Subtraction, 45 Multiplication by a Scalar, 45 Matrix Multiplication, 45 Transposition, 46 Differentiation and Integration, 46 Square Matrix, 47 Diagonal Matrix, 47 Identity Matrix, 47 Symmetric Matrix, Mat rix, 48 Upper Triangular Triangular Matrix, 48 Determinant of a Matrix, 48 Matrix Inversion, 48 Eigenvalues and Eigenvectors, Ei genvectors, 49 Positive Definite Matrix, 51 Cholesky Decomposition, 51
2.2
Gaussian Elimination 51 General Algorithm for Gaussian Elimination, 53 Symmetric Matrix, Mat rix, 56 Symmetric Banded Matrices, 56 Solution with Multiple Right Ri ght Sides, 58 Gaussian Elimination with Column Reduction, 58 Skyline Solution, 60 Frontal Solution, 61
2.3
Conjugate Gradient Method for Equation Solving 61 Conjugate Gradient Algorithm, 62 Input Data/Output, 62
Problems 63 Program Listings, 65
3 ONE-DIMENSIONAL PROBLEMS
3.1 3.2
Introduction 67 Finite Element Modeling 68 Element Division, 68 Numbering Scheme, 69
3.3 3.4
Shape Functions and Local Coordinates 71 The Potential-Energy Potential-Energy Approach 75 Element Stiffness Matrix, 76 Force Terms, 78
3.5
The Galerkin Approach 80 Element Stiffness, 80 Force Terms, 81
3.6 3.7 3.8
Assembly of the Global Stiffness Matrix and Load Vector 82 Properties of K 85 The Finite Element Equations: Equation s: Treatment of Boundary Conditions 86
67
Contents
7
Types of Boundary Conditions, 86 Elimination Approach, 87 Penalty Approach, 92 Multipoint Constraints, 98
3.9 3.10 3.11
Quadratic Shape Functions 101 Temperature Effects 108 Problem Modeling and Boundary Conditions 112 Problem in Equilibrium, 112 Symmetry, 113 Two Elements with Same End Displacements, 113 Problem with a Closing Gap, 114 114 Input Data/Output, 114 114
Problems
115
Program Listing, 127
4 TRUSSES
4.1 4.2
133
Introduction 133 Plane Trusses 134 Local and Global Coordinate Systems, 134 Formulas for Calculating and m, 135 Element Stiffness Matrix, 136 Stress Calculations, Calculati ons, 137 Temperature Effects, 142
4.3 4.4
Three-Dimensional Three-Dimensio nal Trusses 145 Assembly of Global Stiffness Matrix for the Banded and Skyline Solutions 147 Assembly for Banded Solution, 147 147 Skyline Assembly , 148 148
4.5
Problem Modeling and Boundary Conditions 150 150 Inclined Support in Two Two Dimensions, 150 Inclined Support in Three Dimensions—Line Constraint, 150 Inclined Support in Three Dimensions—Plane Constraint, 151 Symmetry and Antisymmetry , 152 Input Data/Output, 154
Problems
155
Program Listing, 163
5 BEAMS AND FRAMES
5.1
Introduction 166 Potential-Energy Approach, 167 Galerkin Approach, 168
5.2
Finite Element Formulation 169 Element Stiffness—Direct Approach, Approach, 173
5.3
Load Vector
174 174
166