Finite Element Modeling for Stress Analysis

Oriented toward those who will use finite elements (FE) rather than toward theoreticians and computer programmers. Emphasizes the behavior of FE and how to use the FE method successfully. Includes several examples of FE analysis-each one features a critique of the accuracy of the solutions. Contains end-of-chapter exercises and extensive advice about FE modeling.

Robert D. Cooke received his Ph.D. in Theoretical and Applied Mechancis from the University of Illinois in 1963. Since then he has been at the University of Wisconsin-Madison, where he is now a professor in the Department of Engineering Physics. His interests include stress analysis and finite element methods. In addition to the present book, he is author of Finite Element Modeling for Stress Analysis and Advanced Mechanics of Materials.

Notation

xi

Introduction

1

(16)

The Finite Element Method

1

(2)

Elements and Nodes

3

(3)

Modeling the Problem and Checking Results

6

(3)

Discretization and Other Approximations

9

(1)

Responsibility of the User

10

(2)

Elementary Matrix Algebra

12

(5)

Analytical Problems

13

(4)

Bars and Beams. Linear Static Analysis

17

(24)

Introduction

17

(1)

Stiffness Matrix Formulation: Bar Element

18

(2)

Stiffness Matrix Formulation: Beam Element

20

(5)

Properties of k and K. Avoiding Singularity

25

(1)

Mechanical Loads. Stresses

26

(4)

Thermal Loads. Stresses

30

(3)

An Application

33

(8)

Analytical Problems

36

(2)

Computational Problems

38

(3)

Plane Problems

41

(34)

Introduction

41

(5)

Constant Strain Triangle (CST)

46

(1)

Linear Strain Triangle (LST)

47

(1)

Bilinear Quadrilateral (Q4)

48

(2)

Quadratic Quadrilateral (Q8)

50

(1)

Improved Bilinear Quadrilateral (Q6)

51

(3)

Elements with ``Drilling'' d.o.f.

54

(1)

Elements of More General Shape

55

(1)

Loads

56

(3)

Stress Calculation. Other Remarks

59

(3)

Comparative Examples

62

(1)

An Application

63

(12)

Analytical Problems

67

(3)

Computational Problems

70

(5)

Isoparametric Elements and Solution Techniques

75

(30)

Node Numbering and Matrix Sparsity

75

(2)

Equation Solving

77

(1)

Transformations

78

(2)

Isoparametric Elements: Formulation

80

(2)

Gauss Quadrature and Isoparametric Elements

82

(2)

Choice of Quadrature Rule. Instabilities

84

(2)

Stress Calculation and Gauss Points

86

(1)

Nature of Finite Element Solution

86

(2)

Convergence Requirements. Patch Test

88

(1)

Infinite Media and Infinite Elements

89

(1)

Substructures

90

(2)

Symmetry

92

(3)

Constraints

95

(10)

Analytical Problems

98

(5)

Computational Problems

103

(2)

Modeling, Errors, and Accuracy in Linear Analysis

105

(40)

Modeling in General

105

(1)

Structure Behavior and Element Behavior

106

(4)

Element Tests and Element Shapes

110

(3)

Test Cases and Pilot Studies

113

(1)

Material Properties

114

(2)

Loads

116

(2)

Connections

118

(4)

Boundary Conditions

122

(2)

Planning the Analysis

124

(2)

Numerical Error: Sources and Detection

126

(4)

Common Mistakes

130

(1)

Checking the Model

131

(2)

Critique of FE Results

133

(3)

Stress Concentrations. Submodeling

136

(2)

Convergence with Mesh Refinement

138

(3)

Error Measures and Adaptivity

141

(4)

Solids and Solids of Revolution

145

(26)

Introduction

145

(4)

Elements for 3D Solids

149

(3)

A 3D Application

152

(3)

Axisymmetric Solid Elements

155

(2)

An Axisymmetric Application

157

(4)

Nonaxisymmetric Loads

161

(10)

Analytical Problems

164

(2)

Computational Problems

166

(5)

Plates and Shells

171

(34)

Plate Displacements, Strains and Stresses

171

(4)

Finite Elements for Plates

175

(6)

A Plate Application

181

(3)

Shells and Shell Theory

184

(2)

Finite Elements for Shells

186

(4)

An Axisymmetric Shell Application

190

(2)

A General Shell Application

192

(13)

Analytical Problems

196

(2)

Computational Problems

198

(7)

Thermal Analysis

205

(22)

Introduction. Some Basic Equations

205

(3)

Finite Elements in Thermal Analysis

208

(3)

Radiation and Other Nonlinearities

211

(1)

Thermal Transients

212

(1)

Modeling Considerations

213

(2)

An Application

215

(12)

Analytical Problems

220

(2)

Computational Problems

222

(5)

Vibration and Dynamics

227

(48)

Introduction

227

(1)

Basic Equations. Vibration

227

(3)

Mass Matrices

230

(2)

Undamped Free Vibration

232

(2)

Damping

234

(1)

Reduction

235

(2)

Model Equations

237

(2)

Harmonic Response Analysis

239

(2)

Dynamic Response Analysis

241

(3)

Response Spectrum Analysis

244

(2)

Remarks. Modeling Considerations

246

(12)

An Application: Vibration and Harmonic Response

258

(3)

An Application: Dynamic Response

261

(14)

Analytical Problems

265

(3)

Computational Problems

268

(7)

Nonlinearity in Stress Analysis

275

(34)

Introduction

275

(2)

Solution Algorithms. Convergence Criteria

277

(3)

Stress Stiffening

280

(3)

Geometric Nonlinearity and Buckling

283

(3)

Material Nonlinearity

286

(3)

Problems of Gaps and Contact

289

(3)

Remarks. Modeling Considerations

292

(3)

Applications

295

(14)

Analytical Problems

299

(2)

Computational Problems

301

(8)

References

309

(6)

Index

315

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Finite Element Modeling for Stress Analysis

Summary

Author Biography

Table of Contents

Supplemental Materials