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Chapter 1 Introduction to Finite Element Method 1

1.1 Basic Concept of Finite Element Method 1

1.2 General Description of Finite Element Method 2

1.2.1 Finite Element Technique in Structure Analysis 2

1.2.2 Finite Element Technique in Heat Conduction 12

1.2.3 Summary 15

1.3 Engineering Applications of Finite Element Analysis 15

1.4 Principle of Virtual Displacements and Variational Approach 16

1.4.1 Principle of Virtual Displacements(PVD) 17

1.4.2 Variational Formulation 21

Problem Set 1 23

Chapter 2 General Procedure of Finite Element Method 25

2.1 Interpolation Functions 26

2.2 Strain-displacement Relations 29

2.3 Stress-strain Relations(Constitutive Relations) 30

2.4 Governing Equations in Finite Element Analysis 32

2.5 Stiffness Matrices 34

2.5.1 Element Stiffness Matrix 34

2.5.2 Global Stiffness Matrix 35

2.6 Equivalent Nodal Force Vectors 40

2.6.1 Element Equivalent Nodal Force Vector 40

2.6.2 Global Equivalent Nodal Force Vector 43

2.7 Imposition of Boundary Condition 44

2.8 Numerical Examples 48

2.9 Area Coordinates 55

2.10 Six-node Triangular Elements 62

2.11 Linear Rectangular Elements 69

Problem Set 2 74

Chapter 3 Formulation of Isoparametric Finite Element Matrices 77

3.1 Isoparametric Concepts 78

3.2 Construction of Interpolation Functions 83

3.2.1 The Pascal Triangle 84

3.2.2 Lagrange Polynomials 85

3.2.3 Lagrange Polynomials in Dimensionless Form 87

3.3 Family of Two-dimensional Isoparametric Elements 88

3.4 Formulation of Isoparametric Finite Element Matrices for Plane Elasticity 97

3.4.1 Interpolation Functions 97

3.4.2 Strain-displacement Transformation Matrix 101

3.4.3 Constitutive Relations 109

3.4.4 Element Stiffness Matrix 110

3.4.5 Element Load Vector 114

3.5 Isoparametric Triangular Elements in Terms of Area Coordinates 123

Problem Set 3 129

Chapter 4 Stress Analysis of Axisymmetric Problems 132

4.1 Interpolation Functions 133

4.2 Strain-displacement Relations 135

4.3 Stress-strain Relations 136

4.4 Element Stiffness Matrix 139

4.5 Element Equivalent Nodal Force Vector 140

4.6 Four-node Rectangular Ring Element 147

4.7 A Numerical Example 149

Problem Set 4 151

Chapter 5 Analysis of Three-dimensional Problems 153

5.1 Convergence Considerations 153

5.2 Shape Functions for Three-dimensional Elements 157

5.2.1 Shape Functions for Tetrahedron Elements 157

5.2.2 Shape Functions for Three-dimensional Hexahedral Elements 161

5.3 Formulation of Three-dimensional Isoparametric Element Matrices 170

5.3.1 Interpolation Functions 171

5.3.2 Strain-displacement Relations 173

5.3.3 Constitutive Relations 176

5.3.4 Element Stiffness Matrix 178

5.3.5 Element Load Vector 180

5.4 Formulation and Calculation of Tetrahedron Element Matrices 185

5.4.1 Displacement Functions 185

5.4.2 Strain-displacement Transformation 187

5.4.3 Stress-strain Relations 188

5.4.4 Element Stiffness Matrix 188

5.4.5 Element Load Vector 189

5.4.6 Degeneration of Eight-node Brick Element to Tetrahedral Element 192

5.5 Numerical Examples 196

Problem Set 5 197

Chapter 6 Finite Element Analysis for Plates and Shells 199

6.1 Introduction 199

6.2 Thin Plate Elements 201

6.2.1 Thin Plate Theory 202

6.2.2 Interpolation Functions for a Rectangular Thin Plate Element 204

6.2.3 Stiffness Matrix of a Rectangular Plate Element 206

6.2.4 Equivalent Nodal Force Vector and Internal Moments 208

6.3 Mindlin Plate Elements 210

6.3.1 Formulation with Mindlin Model 210

6.3.2 Governing Equations of Equilibrium 212

6.3.3 Interpolation Functions for a Mindlin Plate Element 213

6.3.4 Plate Element Stiffness Matrix and Equivalent Nodal Force Vector 213

6.4 Shell Elements 217

6.4.1 Geometric Description of a Curved Shell Element 220

6.4.2 Interpolation Functions for a Curved Shell Element 222

6.4.3 Strain-displacement Transformation Relations 225

6.4.4 Transformed Elasticity and Stress Matrices 228

6.4.5 Element Stiffness Matrix and Load Vectors 231

6.5 Mindlin Laminated Plate Element 233

6.5.1 Element Displacement and Coordinate Interpolation 234

6.5.2 Strain-displacement Relations 237

6.5.3 Constitutive Relations 239

6.5.4 Element Stiffness Matrix 243

6.5.5 Governing Equations in Finite Element Analysis 244

6.5.6 Calculation of Displacements and Stresses for Composite Laminates 245

6.5.7 A Numerical Example 246

Problem Set 6 249

Chapter 7 Finite Element Analysis in Fracture Mechanics 250

7.1 Displacement and Stress Fields in the Vicinity of Crack Tip 250

7.2 Finite E1ement Analysis with Conventional Elements for Determination of SIF 252

7.2.1 Fine Mesh-extrapolation Method 253

7.2.2 Coarse Mesh-J-integral Method 254

7.2.3 Coarse Mesh-stiffness Derivative Method 258

7.3 Finite Element Analysis with Singular Elements for Determination of SIF 260

7.3.1 Isoparametric Element Method 260

7.3.2 Global-local Finite Element Method 264

Problem Set 7 271

Chapter 8 Heat Transfer 272

8.1 Governing Equations of Heat Transfer 272

8.1.1 Rate Equations in Heat Transfer 272

8.1.2 Governing Differential Equation of Temperature Field 273

8.1.3 Variational Formulation of Field Problems 276

8.2 Finite Element Formulation for Field Problems 277

8.3 One-dimensional Heat Transfer 281

8.3.1 One-dimensional Linear Element 281

8.3.2 One-dimensional Quadratic Element 290

8.4 Two-dimensional Heat Transfer 296

8.4.1 Three-node Triangular Element 298

8.4.2 Higher-order Two-dimensional Elements 303

8.5 Three-dimensional Heat Transfer 310

8.6 Radiation Heat Transfer 317

Problem Set 8 319

References 322