Introduction to Finite Elements in Engineering

PREFACE XIII

ABOUT THE AUTHOR XVI

 

1 FUNDAMENTAL CONCEPTS 1



1.1 Introduction 1

1.2 Historical Background 1

1.3 Outline of Presentation 2

1.4 Stresses and Equilibrium 2

1.5 Boundary Conditions 4

1.6 Strain—Displacement Relations 5

1.7 Stress—Strain Relations 6

Special Cases, 7

1.8 Temperature Effects 8

1.9 Potential Energy and Equilibrium: The Rayleigh—Ritz Method 9

Potential Energy ß , 9

Rayleigh—Ritz Method, 12

1.10 Galerkin’s Method 14

1.11 Saint Venant’s Principle 18

1.12 Von Mises Stress 19

1.13 Principle of Superposition 19

1.14 Computer Programs 20

1.15 Conclusion 20

Historical References 20

Problems 21

 

2 MATRIX ALGEBRA AND GAUSSIAN ELIMINATION 28

2.1 Matrix Algebra 28

Row and Column Vectors, 29

Addition and Subtraction, 29

Multiplication by a Scalar, 29

Matrix Multiplication, 29

Transposition, 30

Differentiation and Integration, 30

Square Matrix, 31

Diagonal Matrix, 31

Identity Matrix, 31

Symmetric Matrix, 32

Upper Triangular Matrix, 32

Determinant of a Matrix, 32

Matrix Inversion, 32

Eigenvalues and Eigenvectors, 33

Positive Definite Matrix, 35

Cholesky Decomposition, 35

2.2 Gaussian Elimination 35

General Algorithm for Gaussian Elimination, 37

Symmetric Matrix, 40

Symmetric Banded Matrices, 40

Solution with Multiple Right Sides, 40

Gaussian Elimination with Column Reduction, 42

Skyline Solution, 44

Frontal Solution, 45

2.3 Conjugate Gradient Method for Equation Solving 45

Conjugate Gradient Algorithm, 46

Input Data/Output 46

Problems 47

Program Listings, 49

 

3 ONE-DIMENSIONAL PROBLEMS 51

3.1 Introduction 51

3.2 Finite Element Modeling 52

Element Division, 52

Numbering Scheme, 53

3.3 Shape Functions and Local Coordinates 55

3.4 The Potential-Energy Approach 59

Element Stiffness Matrix, 60

Force Terms, 62

3.5 The Galerkin Approach 64

Element Stiffness, 64

Force Terms, 65

3.6 Assembly of the Global Stiffness Matrix and Load Vector 66

3.7 Properties of K 69

3.8 The Finite Element Equations: Treatment

of Boundary Conditions 70

Types of Boundary Conditions, 70

Elimination Approach, 71

Penalty Approach, 76

Multipoint Constraints, 82

3.9 Quadratic Shape Functions 85

3.10 Temperature Effects 92

3.11 Problem Modeling and Boundary Conditions 96

Problem in Equilibrium, 96

Symmetry, 97

Two Elements with Same End Displacements, 97

Problem with a Closing Gap, 98

Input Data/Output, 98

Problems 99

Program Listing, 111

 

4 TRUSSES 117

4.1 Introduction 117

4.2 Plane Trusses 118

Local and Global Coordinate Systems, 118

Formulas for Calculating / and m, 119

Element Stiffness Matrix, 120

Stress Calculations, 121

Temperature Effects, 126

4.3 Three-Dimensional Trusses 129

4.4 Assembly of Global Stiffness Matrix for the Banded and Skyline

Solutions 131

Assembly for Banded Solution, 131

Skyline Assembly , 132

4.5 Problem Modeling and Boundary Conditions 134

Inclined Support in Two Dimensions, 134

Inclined Support in Three Dimensions–Line Constraint, 134

Inclined Support in Three Dimensions–Plane Constraint, 135

Symmetry and Antisymmetry , 136

Input Data/Output, 138

Problems 139

Program Listing, 147

 

5 BEAMS AND FRAMES 150

5.1 Introduction