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 (Wiley, 1995) and Advanced Mechanics of Materials (2nd Edtion, Prentice Hall, 1999, with Warren C. Young). David S. Malkus received his Ph.D. from Boston University in 1976. He spent two years at the National Bureau of Standards and seven years in the Mathematics Department of Illinois Institute of Technology. He is now Professor of Engineering Mechanics at the Univrersity of Wisconsin-Madison. His research interests concern application of the finite element method to problems of structural and continuum mechanics, in particular the flow of non-Newtonian fluids. He is a member of the Rheology Research Center (University of Wisconsin-Madison) and the Society of Rheology. Michael E. Plesha received his B.S. from the University of Illinois at Chicago, and his M.S. and Ph.D. degrees from Northwestern University, the Ph.D. degree in 1983. He has been a faculty member in the Department of Engineering Physics at the University of Wisconsin-Madison since 1983 where he is Professor of Engineering Mechanics. His research areas include constitutive modeling and finite element analysis of contact-friction problems, transient finite element analysis, and discrete element methods. Robert J. Witt received his Ph.D. in Nuclear Engineering from the Massachusetts Institute of Technology in 1987. He is now an associate professor in the Department of Engineering Physics at the University of Wisconsin-Madison. His research interests are in computational methods of fluid and solid mechanics, with particular application to nuclear systems.

Notation

Introduction

(18)

Finite Element Analysis

(2)

Problem Classification, Modeling, and Discretization

(2)

Interpolation. Elements, Nodes, and D.O.F.

(3)

Example Applications. History of FEA

(3)

Solving a Problem by FEA

(4)

Learning and Using FEA

(4)

Analytical Problems

(2)

One-Dimensional Elements and Computational Procedures

(59)

Introduction

(1)

Bar Element

(4)

Beam Element

(5)

Bar and Beam Elements of Arbitrary Orientation

(3)

Assembly of Elements

(4)

Properties of Stiffness Matrices

(4)

Boundary Conditions

(2)

Exploiting Sparsity. Solving Equations

(4)

Mechanical Loads. Stresses

(6)

Thermal Loads. Stresses

(2)

Structural Symmetry

(3)

Review. Remarks Regarding Modeling

(2)

An Application

(19)

Analytical Problems

(12)

Computational Problems

(4)

Basic Elements

(58)

Preliminaries

(5)

Interpolation and Shape Functions

(5)

Formulas for Element Matricies

(3)

Linear Triangle (CST)

(4)

Quadratic Triangle (LST)

(1)

Bilinear Rectangle (Q4)

(4)

Quadratic Rectangle (Q8, Q9)

100

(2)

Rectangular Solid Elements

102

(2)

Choice of Interpolation Functions

104

(2)

Improved Triangles and Quadrilaterals

106

(5)

Nodal Loads

111

(4)

Stress Calculation

115

(3)

Nature of a Finite Element Solution

118

(1)

Example: A Simple Stress Concentration Problem

119

(2)

An Application with High Stress Gradient

121

(15)

Analytical Problems

124

(8)

Computational Problems

132

(4)

Formulation Techniques: Variational Methods

136

(43)

Introduction

136

(1)

Principle of Stationary Potential Energy

137

(3)

Problems Having Many D.O.F.

140

(2)

Potential Energy of an Elastic Body

142

(4)

The Rayleigh-Ritz Method

146

(3)

Comments Regarding the Rayleigh-Ritz Method

149

(2)

Strong Form and Weak Form

151

(5)

Finite Element Form of the Rayleigh-Ritz Method

156

(5)

Convergence of Finite Element Solutions

161

(4)

Additional Formulations. Hybrid Elements

165

(14)

Analytical Problems

171

(8)

Formulation Techniques: Galerkin and Other Weighted Residual Methods

179

(23)

Galerkin Method

179

(3)

Methods of Weighted Residuals (MWR)

182

(4)

Galerkin Finite Element Method in One Dimension

186

(5)

Integration by Parts

191

(2)

Galerkin Finite Element Method in Two Dimensions

193

(2)

A Mixed Formulation

195

(7)

Analytical Problems

198

(4)

Isoparametric Elements

202

(57)

Introduction

202

(3)

Bilinear Quadrilateral (Q4)

205

(4)

Quadrature: [k] Obtained by Numerical Integration

209

(4)

Quadratic Quadrilaterals (Q8, Q9)

213

(4)

Hexahedral Isoparametric Elements

217

(2)

Incompatible Modes. Nodeless D.O.F.

219

(2)

Static Condensation

221

(2)

Choices in Numerical Integration

223

(4)

Load Considerations

227

(3)

Stress Calculation

230

(4)

Effect of Element Geometry

234

(3)

Validity of Isoparametric Elements

237

(1)

Patch Test

238

(2)

A 2D Application

240

(4)

A 3D Application

244

(15)

Analytical Problems

247

(8)

Computational Problems

255

(4)

Isoparametric Triangles and Tetrahedra

259

(12)

Reference Coordinates. Shape Functions

259

(3)

Element Characteristic Matrices

262

(2)

Analytical Integration. Area and Volume Coordinates

264

(2)

Numerical Integration

266

(5)

Analytical Problems

268

(3)

Coordinate Transformation and Selected Analysis Options

271

(29)

Transformation: Introduction and Vector Forms

271

(2)

Strain, Stress, and Material Property Transformation

273

(2)

Transformation of the Characteristic Matrix

275

(1)

Changing the Directions of Restraints

276

(2)

Connecting Dissimilar Elements. Rigid Elements

278

(4)

Higher Derivatives as Nodal D.O.F.

282

(1)

Fracture Mechanics. Singularity Elements

283

(3)

Elastic Foundations. Infinite Media

286

(6)

Structural Modification. Reanalysis

292

(1)

Tests of Element Quality

293

(7)

Analytical Problems

295

(4)

Computational Problems

299

(1)

Error, Error Estimation, and Convergence

300

(36)

Sources of Error

300

(2)

Ill-Conditioning

302

(4)

The Condition Number

306

(2)

Diagonal Decay Test

308

(1)

Residuals

309

(1)

Discretization Error. Convergence Rate

310

(5)

Multimesh Extrapolation

315

(3)

Mesh Revision Methods

318

(2)

Gradient (Stress) Recovery and Smoothing

320

(6)

A-Posteriori Error Estimate

326

(3)

Adaptive Meshing

329

(7)

Analytical Problems

331

(4)

Computational Problems

335

(1)

Modeling Considerations and Software Use

336

(37)

Introduction

336

(1)

Physical Behavior Versus Element Behavior

337

(3)

Element Shapes and Interconnection

340

(2)

Test Cases and Pilot Studies

342

(2)

Material Properties

344

(3)

Loads and Reactions

347

(1)

Connections in Structures

348

(4)

Boundary Conditions

352

(2)

Repetitive Symmetry

354

(2)

Stress Concentrations. Submodels

356

(2)

Substructures

358

(2)

Planning an Analysis

360

(3)

Common Mistakes

363

(2)

Checking the Model

365

(1)

Critique of Computed Results

366

(3)

Design Optimization

369

(1)

Software

370

(1)

Concluding Remarks

371

(2)

Analytical Problems

371

(1)

Computational Problems

372

(1)

Finite Elements in Structural Dynamics and Vibrations

373

(81)

Introduction

373

(1)

Dynamic Equations. Mass and Damping Matrices

374

(3)

Mass Matrices: Consistent, Diagonal, and Other

377

(6)

Natural Frequencies and Modes

383

(5)

Damping

388

(2)

Reduction of the Number of D.O.F.

390

(4)

Response History: Modal Methods

394

(4)

Response History: Ritz Vectors

398

(2)

Component Mode Synthesis (CMS)

400

(5)

Harmonic Response

405

(2)

Response History: Direct Integration Methods

407

(2)

Explicit Direct Integration

409

(7)

Implicit Direct Integration

416

(5)

Direct Integration: Stability and Accuracy Analysis

421

(5)

Analysis by Response Spectra

426

(3)

Remarks. Modeling Considerations

429

(7)

An Application: Vibration and Harmonic Response

436

(3)

An Application: Response History

439

(15)

Analytical Problems

444

(7)

Computational Problems

451

(3)

Heat Transfer and Selected Fluid Problems

454

(35)

Heat Transfer: Introduction

454

(5)

Finite Element Formulation

459

(3)

Radiation. Nonlinear Heat Transfer Problems

462

(2)

Transient Thermal Analysis

464

(3)

Modeling Considerations. Remarks

467

(2)

An Application

469

(5)

Acoustic Frequencies and Modes

474

(3)

Fluid-Structure Interaction

477

(3)

Plane Incompressible Irrotational Flow

480

(9)

Analytical Problems

482

(4)

Computational Problems

486

(3)

Constraints: Penalty Forms, Locking, and Constraint Counting

489

(19)

Explicit Constraints. Transformation Equations

489

(3)

Lagrange Multipliers to Enforce Constraints

492

(1)

Penalty Functions to Enforce Constraints

493

(2)

Implicit Penalty Constraints and Locking

495

(4)

Constraint Counting

499

(3)

Remarks About Techniques for Incompressible Media

502

(6)

Analytical Problems

504

(4)

Solids of Revolution

508

(22)

Introduction. Elasticity Relations for Axial Symmetry

508

(2)

Axisymmetric Solid Elements

510

(2)

An Application

512

(4)

Loads Without Axial Symmetry: Introduction

516

(5)

Loads Without Axial Symmetry: Some Details of FEA

521

(9)

Analytical Problems

524

(3)

Computational Problems

527

(3)

Plate Bending

530

(31)

Introduction. Plate Behavior

530

(6)

C1 (Kirchhoff) Plate Elements

536

(6)

C0 (Mindlin) Plate Elements

542

(5)

Mindlin Beam. More Devices for C0 Plate Elements

547

(4)

Boundary Conditions. Test Problems

551

(2)

An Application

553

(8)

Analytical Problems

556

(3)

Computational Problems

559

(2)

Shells

561

(34)

Introduction

561

(2)

Circular Arches and Arch Elements

563

(7)

Shells of Revolution

570

(4)

General Shells: Three- and Four-Node Elements

574

(4)

General Shells: Curved Isoparametric Elements

578

(5)

Test Cases. Remarks

583

(3)

An Axisymmetric Shell Application

586

(9)

Analytical Problems

588

(3)

Computational Problems

591

(4)

Nonlinearity: An Introduction

595

(44)

Nonlinear Problems

595

(1)

Some Solution Methods

596

(6)

Plasticity: Introduction

602

(4)

Plasticity: General Formulation for Small Strains

606

(3)

Plasticity: Formulation for Von Mises Theory

609

(3)

Plasticity: Some Computational Procedures

612

(4)

Nonlinear Dynamic Problems

616

(3)

Problems of Gaps and Contact

619

(2)

Geometric Nonlinearity

621

(5)

Modeling Considerations. Remarks

626

(13)

Analytical Problems

630

(6)

Computational Problems

636

(3)

Stress Stiffness and Buckling

639

(24)

Introduction. Energy Considerations

639

(3)

Bar and Beam Elements

642

(3)

Plate Elements

645

(1)

A General Formulation

646

(2)

Calculation of Buckling Loads

648

(2)

Remarks on Stress Stiffness and Its Uses

650

(3)

Remarks and Examples

653

(10)

Analytical Problems

656

(5)

Computational Problems

661

(2)

Appendix A MATRICES: SELECTED DEFINITIONS AND MANIPULATIONS

663

(5)

Appendix B SIMULTANEOUS ALGEBRAIC EQUATIONS

668

(7)

B.1 Overview

668

(1)

B.2 Direct Solvers

668

(3)

B.3 Iterative Solvers

671

(4)

Appendix C EIGENVALUES AND EIGENVECTORS

675

(7)

C.1 Overview

675

(1)

C.2 The Standard Eigenproblem

675

(1)

C.3 The General Eigenproblem

676

(3)

C.4 Solution Algorithms

679

(3)

References

682

(29)

Index

711

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Concepts and Applications of Finite Element Analysis

9780471356059

0471356050

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