Advanced Topics in Finite Element Analysis of Structures

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

• CONTENTS OF THE BOOK WEB SITE xiPREFACE xiii1 ESSENTIAL BACKGROUND 11.1 Steps in a Finite Element Solution 21.1.1 Two-Node Uniform Bar Element 21.2 Interpolation Functions 121.2.1 Lagrange Interpolation for Second-Order Problems 141.2.2 Hermite Interpolation for Fourth-Order Problems 141.2.3 Lagrange Interpolation for Rectangular Elements 161.2.4 Triangular Elements 211.3 Integration by Parts 231.3.1 Gauss’s Divergence Theorem 231.3.2 Green-Gauss Theorem 241.3.3 Green-Gauss Theorem as Integration by Parts in Two Dimensions 241.4 Numerical Integration Using Gauss Quadrature 251.4.1 Gauss Quadrature for One-Dimensional Integrals 251.4.2 Gauss Quadrature for Area Integrals 261.4.3 Gauss Quadrature for Volume Integrals 281.5 Mapped Elements 281.5.1 Restrictions on Mapping of Areas 291.5.2 Derivatives of the Assumed Solution 301.5.3 Evaluation of Area Integrals 311.5.4 Evaluation of Boundary Integrals 32Problems 332 ANALYSIS OF ELASTIC SOLIDS 372.1 Governing Equations 372.1.1 Stresses 372.1.2 Strains 392.1.3 Constitutive Equations 402.1.4 Temperature Effects and Initial Strains 402.1.5 Stress Equilibrium Equations 412.2 General Form of Finite Element Equations 412.2.1 Weak Form 412.2.2 Finite Element Equations 432.3 Tetrahedral Element 452.3.1 Interpolation Functions for a Tetrahedral Element 452.3.2 Tetrahedral Element for Three-Dimensional Elasticity 492.4 Mapped Solid Elements 572.4.1 Interpolation Functions for an Eight-Node Solid Element 582.4.2 Interpolation Functions for a 20-Node Solid Element 592.4.3 Evaluation of Derivatives 602.4.4 Integration over Volume 612.4.5 Evaluation of Surface Integrals 632.4.6 Evaluation of Line Integrals 672.4.7 Complete Mathematica MATLAB Implementations 702.5 Stress Calculations 772.5.1 Optimal Locations for Calculating Element Stresses 772.5.2 Interpolation-Extrapolation of Stresses 782.5.3 Average Nodal Stresses 792.5.4 Iterative Improvement in Stresses 822.6 Static Condensation 842.7 Substructuring 852.8 Patch Test and Incompatible Elements 902.8.1 Convergence Requirements 912.8.2 Extra Zero-Energy Modes 912.8.3 Patch Test for Plane Elasticity Problems 922.8.4 Quadrilateral Element with Additional Bending Shape Functions 982.9 Computer Implementation: fe2Quad 102Problems 1153 SOLIDS OF REVOLUTION 1203.1 Equations of Elasticity in Cylindrical Coordinates 1203.2 Axisymmetric Analysis 1223.2.1 Potential Energy 1233.2.2 Finite Element Equations 1233.2.3 Three-Node Triangular Element 1253.2.4 Mapped Quadrilateral Elements 1463.3 Unsymmetrical Loading 1543.3.1 Fourier Series Representation of Loading 1543.3.2 Finite Element Formulation for Symmetric Loading Terms 1573.3.3 Finite Element Formulation for Antisymmetric Loading Terms 161Problems 1664 MULTIFIELD FORMULATIONS FOR BEAM ELEMENTS 1704.1 Euler-Bernoulli Beam Theory 1714.2 Mixed Beam Element Based on EBT 1734.3 Timoshenko Beam Theory 1804.4 Displacement-Based Beam Element for TBT 1834.5 Shear Locking in Displacement-Based Beam Elements for TBT 1894.5.1 Possible Remedies for Shear Locking 1904.6 Mixed Beam Element Based on TBT 1934.7 Four-Field Beam Element for TBT 1984.8 Linked Interpolation Beam Element for TBT 2054.9 Concluding Remarks 211Problems 2125 MULTIFIELD FORMULATIONS FOR ANALYSIS OF ELASTIC SOLIDS 2155.1 Governing Equations 2155.2 Displacement Formulation 2185.3 Stress Formulation 2215.4 Mixed Formulation 2245.5 Assumed Stress Field For Mixed Formulation 2285.5.1 Minimum Number of Stress Parameters 2285.5.2 Optimum Number of Stress Parameters 2295.5.3 Suggested Procedure for Determining Appropriate Stress Interpolation 2305.6 Analysis of Nearly Incompressible Solids 2345.6.1 Deviatoric and Volumetric Stresses and Strains 2365.6.2 Poisson Ratio Locking in the Displacement-Based Finite Elements 2385.6.3 Mixed Formulation for Nearly Incompressible Solids 2405.6.4 Finite Element Equations 2425.6.5 Assumed Pressure Solution 2455.6.6 Quadrilateral Elements for Planar Problems 246Problems 2586 PLATES AND SHELLS 2616.1 Kirchhoff Plate Theory 2626.1.1 Equilibrium Equations 2666.1.2 Stress Computations 2676.1.3 Weak Form for Displacement-Based Formulation 2706.1.4 General Form of Kirchhoff Plate Element Equations 2736.2 Rectangular Kirchhoff Plate Elements 2756.2.1 MZC (Melosh, Zienkiewicz, and Cheung) Rectangular Plate Element 2756.2.2 Patch Test for Plate Elements 2846.2.3 BFS (Bogner, Fox, and Schmit) Rectangular Plate Element 2916.3 Triangular Kirchhoff Plate Elements 2996.3.1 BCIZ (Bazeley, Cheung, Irons, and Zienkiewicz) Triangular Plate Element 2996.3.2 Conforming Triangular Plate Elements 3056.4 Mixed Formulation for Kirchhoff Plates 3076.5 Mindlin Plate Theory 3116.6 Displacement-Based Finite Elements for Mindlin Plates 3146.6.1 Weak Form 3146.6.2 General Form of Mindlin Plate Element Equations 3166.6.3 Heterosis Element 3236.7 Multifield Elements for Mindlin Plates 3256.8 Analysis of Shell Structures 3316.8.1 Transformation Matrix 3326.8.2 Transformed Equations 335Problems 3367 INTRODUCTION TO NONLINEAR PROBLEMS 3407.1 Nonlinear Differential Equation 3417.1.1 Approximate Solutions Using the Classical Form of the Galerkin Method 3417.1.2 Finite Element Solution 3447.2 Solution Procedures for Nonlinear Problems 3537.2.1 Constant Stiffness Iteration 3547.2.2 Load Increments 3597.2.3 Arc-Length Method 3657.3 Linearization and Directional Derivative 3797.3.1 Examples of Linearization 380Problems 3838 MATERIAL NONLINEARITY 3868.1 Analysis of Axially Loaded Bars 3878.1.1 Weak Form 3888.1.2 Two-Node Finite Element 3898.1.3 One-Dimensional Plasticity 3918.1.4 Ramberg-Osgood Model 4148.2 Nonlinear Analysis of Trusses 4248.3 Material Nonlinearity in General Solids 4348.3.1 General Form of Finite Element Equations 4348.3.2 General Formulation for Incremental Stress-Strain Equations 4378.3.3 State Determination Procedure 4408.3.4 von Mises Yield Criterion and the Associated Hardening Models 449Problems 4629 GEOMETRIC NONLINEARITY 4669.1 Basic Continuum Mechanics Concepts 4679.1.1 Deformation Gradient 4679.1.2 Green-Lagrange Strains 4769.1.3 Cauchy and Piola-Kirchhoff Stresses 4819.2 Governing Differential Equations and Weak Forms 4829.3 Linearization of the Weak Form 4889.4 General Form of Element Tangent Matrices 4939.4.1 State Determination and Check for Convergence 4969.5 Constitutive Equations 4989.5.1 Kirchhoff Material 4989.5.2 Compressible Neo-Hookean Material 4999.6 Computations For a Planar Analysis 5099.7 Deformation-Dependent Loading 5299.7.1 Linearized External Virtual Work for Pressure Loading: General Three-Dimensional Case 5299.7.2 Linearized External Virtual Work for Pressure Loading: Planar Case 5349.8 Linearized Buckling Analysis 5369.8.1 Buckling Load for Trusses 5379.9 Appendix: Double Contraction of Tensors 5429.9.1 Double Contraction of Two Second-Order Tensors 5429.9.2 Double Contraction of a Fourth-Order Tensor with a Second-Order Tensor 543Problems 54510 CONTACT PROBLEMS 54910.1 Simple Normal Contact Example 54910.1.1 Direct Solution 54910.1.2 Solution Using Normal Contact Constraint 55110.2 Contact Example Involving Friction 55410.2.1 Solution of a Beam Problem with No Frictional Resistance 55510.2.2 Frictional Constraint Function 55510.2.3 Solution of a Beam Problem with Large Frictional Resistance 55610.2.4 Solution of a Beam Problem with Small Frictional Resistance 55710.3 General Contact Problems 55710.3.1 Contact Point and Gap Calculations 55810.3.2 Forces on the Contact Surface 56410.3.3 Lagrange Multiplier Weak Form 56510.3.4 Penalty Formulation 568Problems 575BIBLIOGRAPHY 579INDEX 585