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Chapter 1 Introduction to Finite Element Method1

1.1 Basic Concept of Finite Element Method1

1.2 General Description of Finite Element Method2

1.2.1 Finite Element Technique in Structure Analysis2

1.2.2 Finite Element Technique in Heat Conduction12

1.2.3 Summary15

1.3 Engineering Applications of Finite Element Analysis15

1.4 Principle of Virtual Displacements and Variational Approach16

1.4.1 Principle of Virtual Displacements(PVD)17

1.4.2 Variational Formulation21

Problem Set 123

Chapter 2 General Procedure of Finite Element Method25

2.1 Interpolation Functions26

2.2 Strain-displacement Relations29

2.3 Stress-strain Relations(Constitutive Relations)30

2.4 Governing Equations in Finite Element Analysis32

2.5 Stiffness Matrices34

2.5.1 Element Stiffness Matrix34

2.5.2 Global Stiffness Matrix35

2.6 Equivalent Nodal Force Vectors40

2.6.1 Element Equivalent Nodal Force Vector40

2.6.2 Global Equivalent Nodal Force Vector43

2.7 Imposition of Boundary Condition44

2.8 Numerical Examples48

2.9 Area Coordinates55

2.10 Six-node Triangular Elements62

2.11 Linear Rectangular Elements69

Problem Set 274

Chapter 3 Formulation of Isoparametric Finite Element Matrices77

3.1 Isoparametric Concepts78

3.2 Construction of Interpolation Functions83

3.2.1 The Pascal Triangle84

3.2.2 Lagrange Polynomials85

3.2.3 Lagrange Polynomials in Dimensionless Form87

3.3 Family of Two-dimensional Isoparametric Elements88

3.4 Formulation of Isoparametric Finite Element Matrices for Plane Elasticity97

3.4.1 Interpolation Functions97

3.4.2 Strain-displacement Transformation Matrix101

3.4.3 Constitutive Relations109

3.4.4 Element Stiffness Matrix110

3.4.5 Element Load Vector114

3.5 Isoparametric Triangular Elements in Terms of Area Coordinates123

Problem Set 3129

Chapter 4 Stress Analysis of Axisymmetric Problems132

4.1 Interpolation Functions133

4.2 Strain-displacement Relations135

4.3 Stress-strain Relations136

4.4 Element Stiffness Matrix139

4.5 Element Equivalent Nodal Force Vector140

4.6 Four-node Rectangular Ring Element147

4.7 A Numerical Example149

Problem Set 4151

Chapter 5 Analysis of Three-dimensional Problems153

5.1 Convergence Considerations153

5.2 Shape Functions for Three-dimensional Elements157

5.2.1 Shape Functions for Tetrahedron Elements157

5.2.2 Shape Functions for Three-dimensional Hexahedral Elements161

5.3 Formulation of Three-dimensional Isoparametric Element Matrices170

5.3.1 Interpolation Functions171

5.3.2 Strain-displacement Relations173

5.3.3 Constitutive Relations176

5.3.4 Element Stiffness Matrix178

5.3.5 Element Load Vector180

5.4 Formulation and Calculation of Tetrahedron Element Matrices185

5.4.1 Displacement Functions185

5.4.2 Strain-displacement Transformation187

5.4.3 Stress-strain Relations188

5.4.4 Element Stiffness Matrix188

5.4.5 Element Load Vector189

5.4.6 Degeneration of Eight-node Brick Element to Tetrahedral Element192

5.5 Numerical Examples196

Problem Set 5197

Chapter 6 Finite Element Analysis for Plates and Shells199

6.1 Introduction199

6.2 Thin Plate Elements201

6.2.1 Thin Plate Theory202

6.2.2 Interpolation Functions for a Rectangular Thin Plate Element204

6.2.3 Stiffness Matrix of a Rectangular Plate Element206

6.2.4 Equivalent Nodal Force Vector and Internal Moments208

6.3 Mindlin Plate Elements210

6.3.1 Formulation with Mindlin Model210

6.3.2 Governing Equations of Equilibrium212

6.3.3 Interpolation Functions for a Mindlin Plate Element213

6.3.4 Plate Element Stiffness Matrix and Equivalent Nodal Force Vector213

6.4 Shell Elements217

6.4.1 Geometric Description of a Curved Shell Element220

6.4.2 Interpolation Functions for a Curved Shell Element222

6.4.3 Strain-displacement Transformation Relations225

6.4.4 Transformed Elasticity and Stress Matrices228

6.4.5 Element Stiffness Matrix and Load Vectors231

6.5 Mindlin Laminated Plate Element233

6.5.1 Element Displacement and Coordinate Interpolation234

6.5.2 Strain-displacement Relations237

6.5.3 Constitutive Relations239

6.5.4 Element Stiffness Matrix243

6.5.5 Governing Equations in Finite Element Analysis244

6.5.6 Calculation of Displacements and Stresses for Composite Laminates245

6.5.7 A Numerical Example246

Problem Set 6249

Chapter 7 Finite Element Analysis in Fracture Mechanics250

7.1 Displacement and Stress Fields in the Vicinity of Crack Tip250

7.2 Finite E1ement Analysis with Conventional Elements for Determination of SIF252

7.2.1 Fine Mesh-extrapolation Method253

7.2.2 Coarse Mesh-J-integral Method254

7.2.3 Coarse Mesh-stiffness Derivative Method258

7.3 Finite Element Analysis with Singular Elements for Determination of SIF260

7.3.1 Isoparametric Element Method260

7.3.2 Global-local Finite Element Method264

Problem Set 7271

Chapter 8 Heat Transfer272

8.1 Governing Equations of Heat Transfer272

8.1.1 Rate Equations in Heat Transfer272

8.1.2 Governing Differential Equation of Temperature Field273

8.1.3 Variational Formulation of Field Problems276

8.2 Finite Element Formulation for Field Problems277

8.3 One-dimensional Heat Transfer281

8.3.1 One-dimensional Linear Element281

8.3.2 One-dimensional Quadratic Element290

8.4 Two-dimensional Heat Transfer296

8.4.1 Three-node Triangular Element298

8.4.2 Higher-order Two-dimensional Elements303

8.5 Three-dimensional Heat Transfer310

8.6 Radiation Heat Transfer317

Problem Set 8319

References322