Advanced topics in finite element analysis of structures : with Mathematica and MATLAB computations

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.

Author Notes

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

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

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

1.4 Numerical Integration Using Gauss Quadrature

1.5 Mapped Elements

Problems

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 Twenty 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 The 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

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

3.2.4 Mapped quadrilateral elements

3.3 Unsymmetrical Loading

Problems

Multi-Field Formulations for Beam Elements

4.1 Euler-Bernoulli Beam Theory (EBT)

4.2 Mixed Beam Element Based on EBT

4.3 Timoshenko Beam Theory (TBT)

4.4 Displacement Based Element for Timoshenko Beam6

4.5 Shear Locking in Displacement Based Elements for Timoshenko Beam

4.6 Mixed Beam Element Based on Timoshenko Beam Theory

4.7 A Four Field Timoshenko Beam Element

4.8 Timoshenko Beam Element Using Linked Interpolation

4.9 Problems

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

5.6 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

Plates and Shells

6.1 Kirchhoff Plate Theory

Stress resultants (shear and moment intensities

Stress resultants on an arbitrarily oriented plane

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 Rectangular Plate Element (Melosh, Zienkiewicz, and Cheung

6.2.2 The Patch Test for Plate Elements

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

6.3 Triangular Kirchhoff Plate Elements

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

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

Problems

Introduction

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Author Notes

Table of Contents