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

A residual approach to advanced topics in finite element analysis of solids and structures
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. In ten chapters, Advanced Topics in Finite Element Analysis of Structures: with Mathematica® and MATLAB® Computations covers:Essential backgroundAnalysis of elastic solidsSolids of revolutionMultifield formulations for beam elementsMultifield formulations for analysis of elastic solidsPlates and shellsIntroduction to nonlinear problemsMaterial nonlinearityGeometric nonlinearityContact problems
An associated Web site (wiley.com/go/bhatti) includes expanded computational details of some of the lengthy examples in the text. It also contains live MATLAB® files and Mathematica® notebooks to overcome the tedious nature of calculations associated with finite elements.

Spis treści:
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.
2.8.4 Quadrilateral Element with Additional Bending Shape Functions.
2.9 Computer Implementation: fe2Quad.
Problems.
3 SOLIDS OF REVOLUTION.
3.1 Equations of Elasticity in Cylindrical Coordinates.
3.2 Axisymmetric Analysis.
3.2.1 Potential Energ y.
3.2.2 Finite Element Equations.
3.2.3 Three–Node Triangular Element.
3.2.4 Mapped Quadrilateral Elements.
3.3 Unsymmetrical Loading.
3.3.1 Fourier Series Representation of Loading.
3.3.2 Finite Element Formulation for Symmetric Loading Terms.
3.3.3 Finite Element Formulation for Antisymmetric Loading Terms.
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) Rectan gular 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.
7.2.2 Load Increments.
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.
9.1.1 Deformation Gradient.
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