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

ISBN: 978-0-471-64807-9
608 pages
January 2006
For Instructors
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.

PREFACE.

1 ESSENTIAL BACKGROUND.

1.1 Steps in a Finite Element Solution.

1.1.1 Two-Node Uniform Bar Element.

1.2 Interpolation Functions.

1.2.1 Lagrange Interpolation for Second-Order Problems.

1.2.2 Hermite Interpolation for Fourth-Order Problems.

1.2.3 Lagrange Interpolation for Rectangular Elements.

1.2.4 Triangular Elements.

1.3 Integration by Parts.

1.3.1 Gauss’s Divergence Theorem.

1.3.2 Green-Gauss Theorem.

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

1.4 Numerical Integration Using Gauss Quadrature.

1.4.1 Gauss Quadrature for One-Dimensional Integrals.

1.4.2 Gauss Quadrature for Area Integrals.

1.4.3 Gauss Quadrature for Volume Integrals.

1.5 Mapped Elements.

1.5.1 Restrictions on Mapping of Areas.

1.5.2 Derivatives of the Assumed Solution.

1.5.3 Evaluation of Area Integrals.

1.5.4 Evaluation of Boundary Integrals.

Problems.

2 ANALYSIS OF ELASTIC SOLIDS.

2.1 Governing Equations.

2.1.1 Stresses.

2.1.2 Strains.

2.1.3 Constitutive Equations.

2.1.4 Temperature Effects and Initial Strains.

2.1.5 Stress Equilibrium Equations.

2.2 General Form of Finite Element Equations.

2.2.1 Weak Form.

2.2.2 Finite Element Equations.

2.3 Tetrahedral Element.

2.3.1 Interpolation Functions for a Tetrahedral Element.

2.3.2 Tetrahedral Element for Three-Dimensional Elasticity.

2.4 Mapped Solid Elements.

2.4.1 Interpolation Functions for an Eight-Node Solid Element.

2.4.2 Interpolation Functions for a 20-Node Solid Element.

2.4.3 Evaluation of Derivatives.

2.4.4 Integration over Volume.

2.4.5 Evaluation of Surface Integrals.

2.4.6 Evaluation of Line Integrals.

2.4.7 Complete Mathematica/MATLAB Implementations.

2.5 Stress Calculations.

2.5.1 Optimal Locations for Calculating Element Stresses.

2.5.2 Interpolation-Extrapolation of Stresses.

2.5.3 Average Nodal Stresses.

2.5.4 Iterative Improvement in Stresses.

2.6 Static Condensation.

2.7 Substructuring.

2.8 Patch Test and Incompatible Elements.

2.8.1 Convergence Requirements.

2.8.2 Extra Zero-Energy Modes.

2.8.3 Patch Test for Plane Elasticity Problems.

Problems.

3 SOLIDS OF REVOLUTION.

3.1 Equations of Elasticity in Cylindrical Coordinates.

3.2 Axisymmetric Analysis.

3.2.1 Potential Energy.

3.2.2 Finite Element Equations.

3.2.3 Three-Node Triangular Element.

Problems.

4 MULTIFIELD FORMULATIONS FOR BEAM ELEMENTS.

4.1 Euler-Bernoulli Beam Theory.

4.2 Mixed Beam Element Based on EBT.

4.3 Timoshenko Beam Theory.

4.4 Displacement-Based Beam Element for TBT.

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

4.5.1 Possible Remedies for Shear Locking.

4.6 Mixed Beam Element Based on TBT.

4.7 Four-Field Beam Element for TBT.

4.8 Linked Interpolation Beam Element for TBT.

4.9 Concluding Remarks.

Problems.

5 MULTIFIELD FORMULATIONS FOR ANALYSIS OF ELASTIC SOLIDS.

5.1 Governing Equations.

5.2 Displacement Formulation.

5.3 Stress Formulation.

5.4 Mixed Formulation.

5.5 Assumed Stress Field For Mixed Formulation.

5.5.1 Minimum Number of Stress Parameters.

5.5.2 Optimum Number of Stress Parameters.

5.5.3 Suggested Procedure for Determining Appropriate Stress Interpolation.

5.6 Analysis of Nearly Incompressible Solids.

5.6.1 Deviatoric and Volumetric Stresses and Strains.

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

5.6.3 Mixed Formulation for Nearly Incompressible Solids.

5.6.4 Finite Element Equations.

5.6.5 Assumed Pressure Solution.

5.6.6 Quadrilateral Elements for Planar Problems.

Problems.

6 PLATES AND SHELLS.

6.1 Kirchhoff Plate Theory.

6.1.1 Equilibrium Equations.

6.1.2 Stress Computations.

6.1.3 Weak Form for Displacement-Based Formulation.

6.1.4 General Form of Kirchhoff Plate Element Equations.

6.2 Rectangular Kirchhoff Plate Elements.

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

6.2.2 Patch Test for Plate Elements.

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

6.3 Triangular Kirchhoff Plate Elements.

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

6.3.2 Conforming Triangular Plate Elements.

6.4 Mixed Formulation for Kirchhoff Plates.

6.5 Mindlin Plate Theory.

6.6 Displacement-Based Finite Elements for Mindlin Plates.

6.6.1 Weak Form.

6.6.2 General Form of Mindlin Plate Element Equations.

6.6.3 Heterosis Element.

6.7 Multifield Elements for Mindlin Plates.

6.8 Analysis of Shell Structures.

6.8.1 Transformation Matrix.

6.8.2 Transformed Equations.

Problems.

7 INTRODUCTION TO NONLINEAR PROBLEMS.

7.1 Nonlinear Differential Equation.

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

7.1.2 Finite Element Solution.

7.2 Solution Procedures for Nonlinear Problems.

7.2.1 Constant Stiffness Iteration.

7.2.3 Arc-Length Method.

7.3 Linearization and Directional Derivative.

7.3.1 Examples of Linearization.

Problems.

8 MATERIAL NONLINEARITY.

8.1 Analysis of Axially Loaded Bars.

8.1.1 Weak Form.

8.1.2 Two-Node Finite Element.

8.1.3 One-Dimensional Plasticity.

8.1.4 Ramberg-Osgood Model.

8.2 Nonlinear Analysis of Trusses.

8.3 Material Nonlinearity in General Solids.

8.3.1 General Form of Finite Element Equations.

8.3.2 General Formulation for Incremental Stress-Strain Equations.

8.3.3 State Determination Procedure.

8.3.4 von Mises Yield Criterion and the Associated Hardening Models.

Problems.

9 GEOMETRIC NONLINEARITY.

9.1 Basic Continuum Mechanics Concepts.

9.1.2 Green-Lagrange Strains.

9.1.3 Cauchy and Piola-Kirchhoff Stresses.

9.2 Governing Differential Equations and Weak Forms.

9.3 Linearization of the Weak Form.

9.4 General Form of Element Tangent Matrices.

9.4.1 State Determination and Check for Convergence.

9.5 Constitutive Equations.

9.5.1 Kirchhoff Material.

9.5.2 Compressible Neo-Hookean Material.

9.6 Computations For a Planar Analysis.

9.8 Linearized Buckling Analysis.

9.9 Appendix: Double Contraction of Tensors.

9.9.1 Double Contraction of Two Second-Order Tensors.

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

Problems.

10 CONTACT PROBLEMS.

10.1 Simple Normal Contact Example.

10.1.1 Direct Solution.

10.1.2 Solution Using Normal Contact Constraint.

10.2 Contact Example Involving Friction.

10.2.1 Solution of a Beam Problem with No Frictional Resistance.

10.2.2 Frictional Constraint Function.

10.2.3 Solution of a Beam Problem with Large Frictional Resistance.

10.2.4 Solution of a Beam Problem with Small Frictional Resistance.

10.3 General Contact Problems.

10.3.1 Contact Point and Gap Calculations.

10.3.2 Forces on the Contact Surface.

10.3.3 Lagrange Multiplier Weak Form.

10.3.4 Penalty Formulation.

Problems.

BIBLIOGRAPHY.

INDEX.

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M. Asghar Bhatti, Phd, is Associate Professor in the Department of Civil and Environmental Engineering at The University of Iowa, Iowa City.
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