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# Advanced Topics in Finite Element Analysis of Structures: With Mathematica and MATLAB Computations

ISBN: 978-0-471-64807-9 January 2006 608 Pages

## Description

Starting from governing differential equations, a unique and consistently weighted residual approach is used to present advanced topics in finite element analysis of structures, such as mixed and hybrid formulations, material and geometric nonlinearities, and contact problems. This book features a hands-on approach to understanding advanced concepts of the finite element method (FEM) through integrated Mathematica and MATLAB® exercises.

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CONTENTS OF THE BOOK WEB SITE xi

PREFACE xiii

1 ESSENTIAL BACKGROUND 1

1.1 Steps in a Finite Element Solution 2

1.1.1 Two-Node Uniform Bar Element 2

1.2 Interpolation Functions 12

1.2.1 Lagrange Interpolation for Second-Order Problems 14

1.2.2 Hermite Interpolation for Fourth-Order Problems 14

1.2.3 Lagrange Interpolation for Rectangular Elements 16

1.2.4 Triangular Elements 21

1.3 Integration by Parts 23

1.3.1 Gauss’s Divergence Theorem 23

1.3.2 Green-Gauss Theorem 24

1.3.3 Green-Gauss Theorem as Integration by Parts in Two Dimensions 24

1.4 Numerical Integration Using Gauss Quadrature 25

1.4.1 Gauss Quadrature for One-Dimensional Integrals 25

1.4.2 Gauss Quadrature for Area Integrals 26

1.4.3 Gauss Quadrature for Volume Integrals 28

1.5 Mapped Elements 28

1.5.1 Restrictions on Mapping of Areas 29

1.5.2 Derivatives of the Assumed Solution 30

1.5.3 Evaluation of Area Integrals 31

1.5.4 Evaluation of Boundary Integrals 32

Problems 33

2 ANALYSIS OF ELASTIC SOLIDS 37

2.1 Governing Equations 37

2.1.1 Stresses 37

2.1.2 Strains 39

2.1.3 Constitutive Equations 40

2.1.4 Temperature Effects and Initial Strains 40

2.1.5 Stress Equilibrium Equations 41

2.2 General Form of Finite Element Equations 41

2.2.1 Weak Form 41

2.2.2 Finite Element Equations 43

2.3 Tetrahedral Element 45

2.3.1 Interpolation Functions for a Tetrahedral Element 45

2.3.2 Tetrahedral Element for Three-Dimensional Elasticity 49

2.4 Mapped Solid Elements 57

2.4.1 Interpolation Functions for an Eight-Node Solid Element 58

2.4.2 Interpolation Functions for a 20-Node Solid Element 59

2.4.3 Evaluation of Derivatives 60

2.4.4 Integration over Volume 61

2.4.5 Evaluation of Surface Integrals 63

2.4.6 Evaluation of Line Integrals 67

2.4.7 Complete Mathematica MATLAB Implementations 70

2.5 Stress Calculations 77

2.5.1 Optimal Locations for Calculating Element Stresses 77

2.5.2 Interpolation-Extrapolation of Stresses 78

2.5.3 Average Nodal Stresses 79

2.5.4 Iterative Improvement in Stresses 82

2.6 Static Condensation 84

2.7 Substructuring 85

2.8 Patch Test and Incompatible Elements 90

2.8.1 Convergence Requirements 91

2.8.2 Extra Zero-Energy Modes 91

2.8.3 Patch Test for Plane Elasticity Problems 92

Problems 115

3 SOLIDS OF REVOLUTION 120

3.1 Equations of Elasticity in Cylindrical Coordinates 120

3.2 Axisymmetric Analysis 122

3.2.1 Potential Energy 123

3.2.2 Finite Element Equations 123

3.2.3 Three-Node Triangular Element 125

Problems 166

4 MULTIFIELD FORMULATIONS FOR BEAM ELEMENTS 170

4.1 Euler-Bernoulli Beam Theory 171

4.2 Mixed Beam Element Based on EBT 173

4.3 Timoshenko Beam Theory 180

4.4 Displacement-Based Beam Element for TBT 183

4.5 Shear Locking in Displacement-Based Beam Elements for TBT 189

4.5.1 Possible Remedies for Shear Locking 190

4.6 Mixed Beam Element Based on TBT 193

4.7 Four-Field Beam Element for TBT 198

4.8 Linked Interpolation Beam Element for TBT 205

4.9 Concluding Remarks 211

Problems 212

5 MULTIFIELD FORMULATIONS FOR ANALYSIS OF ELASTIC SOLIDS 215

5.1 Governing Equations 215

5.2 Displacement Formulation 218

5.3 Stress Formulation 221

5.4 Mixed Formulation 224

5.5 Assumed Stress Field For Mixed Formulation 228

5.5.1 Minimum Number of Stress Parameters 228

5.5.2 Optimum Number of Stress Parameters 229

5.5.3 Suggested Procedure for Determining Appropriate Stress Interpolation 230

5.6 Analysis of Nearly Incompressible Solids 234

5.6.1 Deviatoric and Volumetric Stresses and Strains 236

5.6.2 Poisson Ratio Locking in the Displacement-Based Finite Elements 238

5.6.3 Mixed Formulation for Nearly Incompressible Solids 240

5.6.4 Finite Element Equations 242

5.6.5 Assumed Pressure Solution 245

5.6.6 Quadrilateral Elements for Planar Problems 246

Problems 258

6 PLATES AND SHELLS 261

6.1 Kirchhoff Plate Theory 262

6.1.1 Equilibrium Equations 266

6.1.2 Stress Computations 267

6.1.3 Weak Form for Displacement-Based Formulation 270

6.1.4 General Form of Kirchhoff Plate Element Equations 273

6.2 Rectangular Kirchhoff Plate Elements 275

6.2.1 MZC (Melosh, Zienkiewicz, and Cheung) Rectangular Plate Element 275

6.2.2 Patch Test for Plate Elements 284

6.2.3 BFS (Bogner, Fox, and Schmit) Rectangular Plate Element 291

6.3 Triangular Kirchhoff Plate Elements 299

6.3.1 BCIZ (Bazeley, Cheung, Irons, and Zienkiewicz) Triangular Plate Element 299

6.3.2 Conforming Triangular Plate Elements 305

6.4 Mixed Formulation for Kirchhoff Plates 307

6.5 Mindlin Plate Theory 311

6.6 Displacement-Based Finite Elements for Mindlin Plates 314

6.6.1 Weak Form 314

6.6.2 General Form of Mindlin Plate Element Equations 316

6.6.3 Heterosis Element 323

6.7 Multifield Elements for Mindlin Plates 325

6.8 Analysis of Shell Structures 331

6.8.1 Transformation Matrix 332

6.8.2 Transformed Equations 335

Problems 336

7 INTRODUCTION TO NONLINEAR PROBLEMS 340

7.1 Nonlinear Differential Equation 341

7.1.1 Approximate Solutions Using the Classical Form of the Galerkin Method 341

7.1.2 Finite Element Solution 344

7.2 Solution Procedures for Nonlinear Problems 353

7.2.1 Constant Stiffness Iteration 354

7.2.3 Arc-Length Method 365

7.3 Linearization and Directional Derivative 379

7.3.1 Examples of Linearization 380

Problems 383

8 MATERIAL NONLINEARITY 386

8.1 Analysis of Axially Loaded Bars 387

8.1.1 Weak Form 388

8.1.2 Two-Node Finite Element 389

8.1.3 One-Dimensional Plasticity 391

8.1.4 Ramberg-Osgood Model 414

8.2 Nonlinear Analysis of Trusses 424

8.3 Material Nonlinearity in General Solids 434

8.3.1 General Form of Finite Element Equations 434

8.3.2 General Formulation for Incremental Stress-Strain Equations 437

8.3.3 State Determination Procedure 440

8.3.4 von Mises Yield Criterion and the Associated Hardening Models 449

Problems 462

9 GEOMETRIC NONLINEARITY 466

9.1 Basic Continuum Mechanics Concepts 467

9.1.2 Green-Lagrange Strains 476

9.1.3 Cauchy and Piola-Kirchhoff Stresses 481

9.2 Governing Differential Equations and Weak Forms 482

9.3 Linearization of the Weak Form 488

9.4 General Form of Element Tangent Matrices 493

9.4.1 State Determination and Check for Convergence 496

9.5 Constitutive Equations 498

9.5.1 Kirchhoff Material 498

9.5.2 Compressible Neo-Hookean Material 499

9.6 Computations For a Planar Analysis 509

9.7.1 Linearized External Virtual Work for Pressure Loading: General Three-Dimensional Case 529

9.8 Linearized Buckling Analysis 536

9.8.1 Buckling Load for Trusses 537

9.9 Appendix: Double Contraction of Tensors 542

9.9.1 Double Contraction of Two Second-Order Tensors 542

9.9.2 Double Contraction of a Fourth-Order Tensor with a Second-Order Tensor 543

Problems 545

10 CONTACT PROBLEMS 549

10.1 Simple Normal Contact Example 549

10.1.1 Direct Solution 549

10.1.2 Solution Using Normal Contact Constraint 551

10.2 Contact Example Involving Friction 554

10.2.1 Solution of a Beam Problem with No Frictional Resistance 555

10.2.2 Frictional Constraint Function 555

10.2.3 Solution of a Beam Problem with Large Frictional Resistance 556

10.2.4 Solution of a Beam Problem with Small Frictional Resistance 557

10.3 General Contact Problems 557

10.3.1 Contact Point and Gap Calculations 558

10.3.2 Forces on the Contact Surface 564

10.3.3 Lagrange Multiplier Weak Form 565

10.3.4 Penalty Formulation 568

Problems 575

BIBLIOGRAPHY 579

INDEX 585