AN INTRODUCTION TO THE FINITE ELEMENT METHOD

Transcription

1 AN INTRODUCTION TO THE FINITE ELEMENT METHOD Second Edition J. N. Reddv Oscar S. Wyatt Chair in Mechanical Engineering Texas A&M University College Station, Texas McGraw-Hill, Inc. New York St. Louis San Francisco Auckland Bogota Caracas Lisbon London Madrid Mexico Milan Montreal New Delhi Paris San Juan Singapore Sydney Tokyo Toronto

2 CONTENTS Preface to the Second Edition Preface to the First Edition Part 1 Preliminaries Introduction General Comments Historical Background The Basic Concept of the Finite Element Method General Comments Approximation of the Circumference of a Circle Approximate Determination of the Center of Mass Solution of Differential Equation Some Remarks The Present Study Summary 15 References for Additional Reading 16 Integral Formulations and Variational Methods Need for Weighted-Integral Forms Some Mathematical Concepts and Formulae Boundary, Initial, and Eigenvalue Problems Integral Relations Functionals The Variational Symbol Weak Formulation of Boundary Value Problems Introduction Weighted-Integral and Weak Formulations Linear and Bilinear Forms and Quadratic Functionals Examples 35 ix

3 X CONTENTS 2.4 Variational Methods of Approximation Introduction The Rayleigh-Ritz Method The Method of Weighted Residuais Summary 57 Problems 59 References for Additional Reading 63 Part 2 Finite Element Analysis of One-Dimensional Problems 3 Second-Order Boundary Value Problems Introduction Basic Steps of Finite Element Analysis Model Boundary Value Problem Discretization of the Domain Derivation of Element Equations Connectivity of Elements Imposition of Boundary Conditions Solution of Equations Postprocessing of the Solution Radially Symmetrie Problems Applications keat Transfer Fluid Mechanics Solid Mechanics Summary 127 Problems 128 References for Additional Reading 141 Bending of Beams Introduction The Euler-Bernoulli Beam Element Governing Equation Discretization of the Domain Derivation of Element Equations Assembly of Element Equations Imposition of Boundary Conditions Solution Postprocessing of the Solution Examples Plane Truss and Euler-Bernoulli Frame Elements The Timoshenko Beam and Frame Elements Governing Equations Weak Form Finite Element Model Inclusion of Constraint Equations Summary

4 CONTENTS xi Problems 192 References for Additional Reading Finite Element Error Analysis Approximation Errors Various Measures of Errors Convergence of Solution Accuracy of the Solution Summary 207 Problems 207 References for Additional Reading Eigenvalue and Time-Dependent Problems Eigenvalue Problems Introduction Formulation of Eigenvalue Problems Finite Element Models Applications Time-Dependent Problems Introduction Semidiscrete Finite Element Models Time Approximations Mass Lumping Applications Summary 241 Problems 241 References for Additional Reading Numerical Integration and Computer Implementation Isoparametric Formulations and Numerical Integration Background Natural Coordinates Approximation of Geometry Isoparametric Formulations Numerical Integration Computer Implementation Introductory Comments General Outline Preprocessor Calculation of Element Matrices (Processor) Assembly of Element Equations (Processor) Imposition of Boundary Conditions (Processor) Solution of Equations and Postprocessing Applications of the Computer Program FEM1DV General Comments Illustrative Examples Summary 286 Problems 287 References for Additional Reading 291

5 XU CONTENTS Part 3 Finite Element Analysis of Two-Dimensional Problems 8 Single-Variable Problems Introduction Boundary Value Problems The Model Equation Finite Element Discretization Weak Form Finite Element Model Interpolation Functions Evaluation of Element Matrices and Vectors Assembly of Element Equations Postprocessing Axisymmetric Problems AnExample Some Comments on Mesh Generation and Imposition of Boundary Conditions Discretization of a Domain Generation of Finite Element Data Imposition of Boundary Conditions Applications Heat Transfer Fluid Mechanics Solid Mechanics Eigenvalue and Time-Dependent Problems Introduction Parabolic Equations Hyperbolic Equations Summary 384 Problems 386 References for Additional Reading Interpolation Functions, Numerical Integration, and Modeling Considerations Library of Elements and Interpolation Functions Introduction Triangulär Elements Rectangular Elements The Serendipity Elements Numerical Integration Preliminary Comments Coordinate Transformations Integration over a Master Rectangular Element Integration over a Master Triangulär Element Modeling Considerations Preliminary Comments Element Geometries Mesh Generation Load Representation Summary 448

6 CONTENTS Xlll Problems 448 References for Additional Reading Plane Elasticity Introduction Governing Equations Assumptions of Plane Elasticity Basic Equations Weak Formulations Preliminary Comments Principle of Virtual Displacements in Matrix Form Weak Form of the Governing Differential Equations Finite Element Model Matrix Form of the Model Weak Form Model Eigenvalue and Transient Problems Evaluation of Integrals Assembly and Boundary and Initial Conditions Examples Summary 476 Problems 476 References for Additional Reading Flows of Viscous Incompressible Fluids Preliminary Comments Governing Equations Velocity-Pressure Finite Element Model Penalty-Finite Element Model Penalty Function Method Formulation of the Flow Problem as a Constrained Problem Lagrange Multiplier Formulation Penalty Function Formulation Computational Aspects Examples Summary 502 Problems References for Additional Reading Bending of Elastic Plates Introduction Classical Plate Model Displacement Field Virtual Work Statement Finite Element Model Shear Deformable Plate Model Displacement Field Virtual Work Statement Finite Element Model Shear Locking and Reduced Integration Eigenvalue and Time-Dependent Problems 520

7 XIV CONTENTS 12.5 Examples Summary 529 Problems 529 References for Additional Reading Computer Implementation Introduction Preprocessor Element Computations: Processor Applications of the Computer Program FEM2DV Introduction Description of Mesh Genrators Applications (Illustrative Examples) Summary 563 Problems 570 References for Additional Reading 575 Part 4 Advanced Topics 14 Weighted-Residual Finite Element Models, and Finite Element Models of Nonlinear and Three-Dimensional Problems Introduction Alternative Formulations Introductory Comments Weighted-Residual Finite Element Models Mixed Formulations Nonlinear Problems General Comments Large-Deflection Bending of (Euler-Bernoulli) Beams Solution Methods for Nonlinear Algebraic Equations The 2-D Navier-Stokes Equations Three-Dimensional Problems Summary 601 Problems 602 References for Additional Reading 606 Appendixes 1 Fortran Listing of FEM1DV Fortran Listing of FEM2DV2 640 Index 679