Finite Element Analysis

Finite Element Analysis
Mircea Rades
Contents
Preface i
Contents iii
1. Introduction 1
1.1 Object of FEA 1
1.2 Finite element displacement method 3
1.3 Historical view 4
1.4 Stages of FEA 5
2. Displacement Method 9
2.1 Equilibrium equations 9
2.2 Conditions for geometric compatibility 10
2.3 Force/elongation relations 11
2.4 Boundary conditions 12
2.5 Solving for displacements 12
2.6 Comparison of the force method and displacement method 13
3. Direct Stiffness Method 17
3.1 Stiffness matrix for a bar element 17
3.2 Transformation from local to global coordinates 19
3.2.1 Coordinate transformation 19
3.2.2 Force transformation 20
3.2.3 Element stiffness matrix in global coordinates 21
3.2.4 Properties of the element stiffness matrix 22
3.3 Link’s truss 25
3.4 Direct method 26
3.5 Compatibility of nodal displacements 28
3.6 Expanded element stiffness matrix 29
3.7 Unreduced global stiffness matrix 30
iv FINITE ELEMENT ANALYSIS
3.8 Joint force equilibrium equations 31
3.9 Reduced global stiffness matrix 33
3.10 Reactions and internal forces 35
3.11 Thermal loads and stresses 36
3.12 Node numbering 37
Exercises 41
4. Bars and shafts 47
4.1 Plane bar elements 47
4.1.1 Differential equation of equilibrium 47
4.1.2 Coordinates and shape functions 48
4.1.3 Bar not loaded between ends 49
4.1.4 Element stiffness matrix in local coordinates 51
4.1.5 Bar loaded between ends 52
4.1.6 Vector of element nodal forces 55
4.1.7 Assembly of the global stiffness matrix and load vector 56
4.1.8 Initial strain effects 59
4.2 Plane shaft elements 60
Exercises 63
5. Beams, frames and grids 79
5.1 Finite element discretization 79
5.2 Static analysis of a uniform beam 81
5.3 Uniform beam not loaded between ends 83
5.3.1 Shape functions 84
5.3.2 Stiffness matrix 86
5.3.3 Physical significance of the stiffness matrix 88
5.4 Uniform beam loaded between ends 89
5.4.1 Consistent vector of nodal forces 89
5.4.2 Higher degree interpolation functions 92
5.4.3 Bending moment and shear force 95
5.5 Basic convergence requirements 96
5.6 Frame element 97
5.6.1 Axial effects 97
5.6.2 Stiffness matrix and load vector in local coordinates 98
CONTENTS v
5.6.3 Coordinate transformation 98
5.6.4 Stiffness matrix and load vector in global coordinates 100
5.7 Assembly of the global stiffness matrix 100
5.8 Grids 111
5.9 Deep beam bending element 116
5.9.1 Static analysis of a uniform beam 117
5.9.2 Shape functions 118
5.9.3 Stiffness matrix 121
6. Linear elasticity 123
6.1 Matrix notation for loads, stresses and strain 123
6.2 Equations of equilibrium inside V 125
6.3 Equations of equilibrium on the surface S? 126
6.4 Strain-displacement relations 127
6.5 Stress-strain relations 128
6.6 Temperature effects 130
6.7 Strain energy 130
7. Energy methods 131
7.1 Principle of virtual work 131
7.1.1 Virtual displacements 131
7.1.2 Virtual work of external forces 133
7.1.3 Virtual work of internal forces 133
7.1.4 Principle of virtual displacements 134
7.1.5 Proof that PDV is equivalent to equilibrium equations 137
7.2 Principle of minimum total potential energy 139
7.2.1 Strain energy 139
7.2.2 External potential energy 140
7.2.3 Total potential energy 140
7.3 The Rayleigh-Ritz method 143
7.4 FEM – a localized version of the Rayleigh-Ritz method 148
7.4.1 FEM in Structural Mechanics 148
7.4.2 Discretization 149
7.4.3 Principle of virtual displacements 149
vi FINITE ELEMENT ANALYSIS
7.4.4 Approximating functions for the element 149
7.4.5 Compatibility between strains and nodal displacements 150
7.4.6 Element stiffness matrix and load vector 151
7.4.7 Assembly of the global stiffness matrix and load vector 151
7.4.8 Solution and back-substitution 152
8 Two-dimensional elements 153
8.1 The plane constant-strain triangle (CST) 153
8.1.1. Discretization of structure 153
8.1.2 Polynomial approximation of the displacement field 154
8.1.3 Nodal approximation of the displacement field 155
8.1.4 The matrix [ ] B 158
8.1.5 Element stiffness matrix and load vector 159
8.1.6 Remarks 160
8.2 Rectangular elements 176
8.2.1 The four-node rectangle (linear) 176
8.2.2 The eight-node rectangle (quadratic) 178
8.3 Triangular elements 180
8.3.1 Area coordinates 180
8.3.2 Linear strain triangle (LST) 182
8.3.3 Quadratic strain triangle 185
8.4 Equilibrium, convergence and compatibility 187
8.4.1 Equilibrium vs. compatibility 187
8.4.2 Convergence and compatibility 188
9 Isoparametric elements 191
9.1 Linear quadrilateral element 191
9.1.1 Natural coordinates 192
9.1.2 Shape functions 193
9.1.3 The displacement field 194
9.1.4 Mapping from natural to Cartesian coordinates 195
9.1.5 Element stiffness matrix 198
9.1.6 Element load vectors 199
9.2 Numerical integration 200
9.2.1 One dimensional Gauss quadrature 200
CONTENTS vii
9.2.2 Two dimensional Gauss quadrature 203
9.2.3 Stiffness integration 204
9.2.4 Stress calculations 207
9.3 Eight-node quadrilateral 208
9.3.1 Shape functions 209
9.3.2 Shape function derivatives 210
9.3.3 Determinant of the Jacobian matrix 211
9.3.4 Element stiffness matrix 211
9.3.5 Stress calculation 213
9.3.6 Consistent nodal forces 214
9.4 Nine-node quadrilateral 219
9.5 Six-node triangle 221
9.6 Jacobian positiveness 223
10 Plate bending 225
10.1 Thin plate theory (Kirchhoff) 225
10.2 Thick plate theory (Reissner-Mindlin) 229
10.3 Rectangular plate-bending elements 232
10.3.1 ACM element (non-conforming) 232
10.3.2 BFS element (conforming) 238
10.3.3 HTK thick rectangular element 239
10.4 Triangular plate-bending elements 244
10.4.1 Thin triangular element (non-conforming) 245
10.4.2 Thick triangular element (conforming) 248
10.4.3 Discrete Kirchhoff triangles (DKT) 250
References 257
Index 265
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