FUNDAMENTAL FINITE ELEMENT ANALYSIS AND APPLICATIONS
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1 FUNDAMENTAL FINITE ELEMENT ANALYSIS AND APPLICATIONS With Mathematica and MATLAB Computations M. ASGHAR BHATTI WILEY JOHN WILEY & SONS, INC.
2 CONTENTS OF THE BOOK WEB SITE PREFACE xi xiii 1 FINITE ELEMENT METHOD: THE BIG PICTURE Discretization and Element Equations / Plane Trass Element / Triangular Element for Two-Dimensional Heat Flow / General Remarks on Finite Element Discretization / Triangular Element for Two-Dimensional Stress Analysis / Assembly of Element Equations / Boundary Conditions and Nodal Solution / Essential Boundary Conditions by Rearranging Equations / Essential Boundary Conditions by Modifying Equations / Approximate Treatment of Essential Boundary Conditions / Computation of Reactions to Verify Overall Equilibrium / Element Solutions and Model Validity / Plane Truss Element / Triangular Element for Two-Dimensional Heat Flow / Triangular Element for Two-Dimensional Stress Analysis / Solution of Linear Equations / Solution Using Choleski Decomposition / Conjugate Gradient Method / 62
3 Vi CONTENTS 1.6 Multipoint Constraints / Solution Using Lagrange Multipliers / Solution Using Penalty Function / Units / 83 MATHEMATICAL FOUNDATION OF THE FINITE ELEMENT METHOD Axial Deformation of Bars / Differential Equation for Axial Deformations / Exact Solutions of Some Axial Deformation Problems / Axial Deformation of Bars Using Galerkin Method / Weak Form for Axial Deformations / Uniform Bar Subjected to Linearly Varying Axial Load / Tapered Bar Subjected to Linearly Varying Axial Load / One-Dimensional BVP Using Galerkin Method / Overall Solution Procedure Using Galerkin Method / Higher Order Boundary Value Problems / Rayleigh-Ritz Method / Potential Energy for Axial Deformation of Bars / Overall Solution Procedure Using the Rayleigh-Ritz Method / Uniform Bar Subjected to Linearly Varying Axial Load / Tapered Bar Subjected to Linearly Varying Axial Load / Comments on Galerkin and Rayleigh-Ritz Methods / Admissible Assumed Solution / Solution Convergence the Completeness Requirement / Galerkin versus Rayleigh-Ritz / Finite Element Form of Assumed Solutions / Linear Interpolation Functions for Second-Order Problems / Lagrange Interpolation / Galerkin Weighting Functions in Finite Element Form / Hermite Interpolation for Fourth-Order Problems / Finite Element Solution of Axial Deformation Problems / Two-Node Uniform Bar Element for Axial Deformations / Numerical Examples / 155 ONE-DIMENSIONAL BOUNDARY VALUE PROBLEM ,-Selected Applications of ID BVP / Steady-State Heat Conduction / Heat How through Thin Fins / 175
4 Vii Viscous Fluid Flow between Parallel Plates Lubrication Problem / Slider Bearing / Axial Deformation of Bars / Elastic Buckling of Long Slender Bars / Finite Element Formulation for Second-Order ID BVP / Complete Solution Procedure / Steady-State Heat Conduction / Steady-State Heat Conduction and Convection / Viscous Fluid Flow Between Parallel Plates / Elastic Buckling of Bars / Solution of Second-Order ID BVP / A Closer Look at the Interelement Derivative Terms / TRUSSES, BEAMS, AND FRAMES Plane Trasses / Space Trasses / Temperature Changes and Initial Strains in Trasses / Spring Elements / Transverse Deformation of Beams / Differential Equation for Beam Bending / Boundary Conditions for Beams / Shear Stresses in Beams / Potential Energy for Beam Bending / Transverse Deformation of a Uniform Beam / Transverse Deformation of a Tapered Beam Fixed at Both Ends / Two-Node Beam Element / Cubic Assumed Solution / Element Equations Using Rayleigh-Ritz Method / Uniform Beams Subjected to Distributed Loads / Plane Frames / Space Frames / Element Equations in Local Coordinate System / Local-to-Global Transformation / Element Solution / Frames in Multistory Buildings / 293
5 Viii CONTENTS TWO-DIMENSIONAL ELEMENTS Selected Applications of the 2D BVP / Two-Dimensional Potential Flow / Steady-State Heat Flow / Bars Subjected to Torsion / Waveguides in Electromagnetics / Integration by Parts in Higher Dimensions / Finite Element Equations Using the Galerkin Method / Rectangular Finite Elements / Four-Node Rectangular Element / Eight-Node Rectangular Element / Lagrange Interpolation for Rectangular Elements / Triangular Finite Elements / Three-Node Triangular Element / Higher Order Triangular Elements / 371 MAPPED ELEMENTS Integration Using Change of Variables / One-Dimensional Integrals / Two-Dimensional Area Integrals / Three-Dimensional Volume Integrals / Mapping Quadrilaterals Using Interpolation Functions / Mapping Lines / Mapping Quadrilateral Areas / Mapped Mesh Generation / Numerical Integration Using Gauss Quadrature / Gauss Quadrature for One-Dimensional Integrals / Gauss Quadrature for Area Integrals / Gauss Quadrature for Volume Integrals / Finite Element Computations Involving Mapped Elements / Assumed Solution / Derivatives of the Assumed Solution / Evaluation of Area Integrals / Evaluation of Boundary Integrals / Complete Mathematica and MATLAB Solutions of 2D BVP Involving Mapped Elements / Triangular Elements by Collapsing Quadrilaterals / Infinite Elements / One-Dimensional BVP / Two-Dimensional BVP / 458
6 7 ANALYSIS OF ELASTIC SOLIDS Fundamental Concepts in Elasticity / Stresses / Stress Failure Criteria / Strains / Constitutive Equations / Temperature Effects and Initial Strains / Governing Differential Equations / Stress Equilibrium Equations / Governing Differential Equations in Terms of Displacements / General Form of Finite Element Equations / Potential Energy Functional / Weak Form / Finite Element Equations / Finite Element Equations in the Presence of Initial Strains / Plane Stress and Plane Strain / Plane Stress Problem / Plane Strain Problem / Finite Element Equations / Three-Node Triangular Element / Mapped Quadrilateral Elements / Planar Finite Element Models / Pressure Vessels / Rotating Disks and Flywheels / Residual Stresses Due to Welding / Crack Tip Singularity / TRANSIENT PROBLEMS Transient Field Problems / Finite Element Equations / 5^ Triangular Element / Transient Heat Flow / Elastic Solids Subjected to Dynamic Loads / Finite Element Equations / Mass Matrices for Common Structural Elements / Free-Vibration Analysis / Transient Response Examples / 573
7 9 p-formulation p-formulation for Second-Order ID BVP / Assumed Solution Using Legendre Polynomials / Element Equations / Numerical Examples / ^-Formulation for Second-Order 2D BVP / p-mode Assumed Solution / Finite Element Equations / Assembly of Element Equations / Incorporating Essential Boundary Conditions / Applications / 624 A USE OF COMMERCIAL FEA SOFTWARE 641 A.I ANSYS Applications / 642 A. 1.1 General Steps / 643 A. 1.2 Trass Analysis / 648 A. 1.3 Steady-State Heat Flow / 651 A. 1.4 Plane Stress Analysis / 655 A.2 Optimizing Design Using ANSYS / 659 A.2.1 General Steps / 659 A.2.2 Heat Flow Example / 660 A.3 ABAQUS Applications / 663 A.3.1 Execution Procedure / 663 A.3.2 Trass Analysis / 665 A.3.3 Steady-State Heat Flow / 666 A.3.4 Plane Stress Analysis / 671 B VARIATIONAL FORM FOR BOUNDARY VALUE PROBLEMS 676 B.I Basic Concept of Variation of a Function / 676 B.2 Derivation of Equivalent Variational Form / 679 B.3 Boundary Value Problem Corresponding to a Given Functional / 683 BIBLIOGRAPHY 687 INDEX 695
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