Finite Element Method With Applications In Engineering

Description

Cover Contents Preface Acknowledgements Authors Profile Chapter 1: Introduction 1.1 Introductory Remarks 1.2 Mathematical Modelling of Engineering Problems 1.3 Type of Governing Equations 1.3.1 Initial and Boundary Conditions 1.4 Solution Methodologies 1.4.1 Analytical Method 1.4.2 Physical Method 1.4.3 Computational Method 1.5 Numerical Modelling 1.6 Pre-Processing and Post-Processing 1.7 Scope of the Book 1.8 Highlights of the Book 1.9 How to Use the Book? 1.10 Closing Remarks References and Further Reading Chapter 2: Approximate Methods of Analysis 2.1 Introduction 2.2 Aproximating Methods 2.3 Method of Weighted Residuals 2.3.1 Method of Point Collocation 2.3.2 Method of Collocation by Sub-Regions 2.3.3 Method of Least Squares 2.3.4 Galerkin’s Method 2.4 Rayleigh–Ritz Method 2.4.1 Relation Between FEM and Rayleigh–Ritz Method 2.5 Further Numerical Examples 2.6 Closing Remarks Exercise Problems References and Further Reading Chapter 3: Finite Element Method—An Introduction 3.1 General 3.2 What is FEM? 3.3 How Does FEM Work? 3.4 A Brief History of FEM 3.5 FEM Applications 3.6 Merits and Demerits of FEM 3.7 Closing Remarks Exercise Problems References and Further Reading Chapter 4: Different Approaches in FEM 4.1 Introduction 4.2 General Steps of FEM 4.3 Different Approaches Used in FEM 4.3.1 Direct Approach 4.3.2 Variational Approach 4.3.3 Energy Approach 4.3.4 Weighted Residual Approach 4.4 Closing Remarks Exercise Problems References and Further Reading Chapter 5: Finite Elements and Interpolation Functions 5.1 Introduction 5.2 Interpolation Functions 5.2.1 One-Independent Spatial Variable 5.2.2 Two-Independent Spatial Variables 5.2.3 Three-Independent Spatial Variables 5.3 One-Dimensional Elements 5.3.1 Line Element: Linear Interpolation Function 5.3.2 Quadratic Interpolation Function 5.3.3 Cubic Interpolation Function 5.3.4 Lagrangian Form of Interpolation Function 5.3.5 Further Higher Order Elements in One-Dimension 5.4 Two-Dimensional Elements 5.4.1 Triangular Element: Linear Interpolation Function in Cartesian Co-ordinates 5.4.2 Triangular Element—Area Co-ordinates 5.4.3 Integration Formula for Triangular Elements 5.4.4 Triangular Element—Quadratic Function 5.4.5 Triangular Element—Cubic Interpolation Function 5.4.6 Two-Dimensional Rectangular Elements 5.4.7 Rectangular Elements—Lagrangian Form in Natural and Cartesian Co-ordinates 5.4.8 Isoparametric Elements 5.4.9 Lagrangian Interpolation Functions for Two-Dimensional Elements 5.4.10 Two-Dimensional Serendipity Elements 5.5 Three-Dimensional Elements 5.5.1 Tetrahedral Elements 5.5.2 Tetrahedral Elements: Quadratic Interpolation Function 5.5.3 Tetrahedral Elements: Cubic Interpolation Function 5.5.4 Three-Dimensional Elements—Prismatic Elements 5.5.5 Three-Dimensional Elements in Local Co-ordinates 5.5.6 Three-Dimensional Serendipity Elements 5.6 Closing Remarks Exercise Problems References and Further Reading Chapter 6: One-Dimensional Finite Element Analysis 6.1 Introduction 6.2 Linear Spring 6.2.1 Expressions for Equivalent Spring Constant and Nodal Forces 6.3 Truss Element 6.3.1 Plane Truss 6.3.2 Element Equations by Minimizing Potential Energy 6.3.3 Local and Global Element Equations for a Bar in the X–Y Plane 6.3.4 Computation of Stress for a Bar in the X–Y Plane 6.4 Space Truss 6.5 One-Dimensional Torsion of a Circular Shaft 6.6 One-Dimensional Steady State Heat Conduction 6.7 One-Dimensional Flow Through Porous Media 6.8 One-Dimensional Ideal Fluid Flow Through Pipes (Inviscid Fluid Flow) 6.10 Analyses of Plane Frames and Grids 6.10.1 Plane Frame Analysis 6.10.2 Grid Analysis 6.9 Beam Element 6.9.1 Review of Beam Theory 6.9.2 Finite Element Formulation of a Beam Element 6.9.3 Illustrative Examples 6.11 Further One-Dimensional Applications 6.11.1 Flow Network Analysis 6.11.2 Electrical Network Analysis 6.12 Summary of Element Matrices for One-Dimensional Finite Elements 6.13 Closing Remarks Exercise Problems References and Further Reading Chapter 7: Two-Dimensional Finite Element Analysis 7.1 Introduction 7.2 Two-Dimensional Flow Through Porous Media (Seepage Flow) 7.2.1 Step-by-step Formulation for the CST Element for Two-dimensional Confined Seepage Analysis 7.3 Two-Dimensional Stress Analysis 7.3.1 Review of Theory of Elasticity 7.3.2 Application of Three-Dimensional Equations for Two-Dimensional Analysis 7.3.3 CST Element for Plane Stress and Plane Strain Analyses 7.3.4 Triangular Element for Axi-symmetric Analysis 7.3.5 Some Remarks on Triangular Elements 7.3.6 Four-Node Rectangular Element for Plane Problems 7.4 Iso-Parametric Formulation 7.4.1 Two-Node Iso-Parametric Line Element (Bar Element) 7.4.2 Four-Node Iso-Parametric Element for Plane Problems (Quadrilateral Element) 7.5 Finite Element Solution of Partial Differential Equations by Method of Weighted Residual 7.5.1 G overning Equations and Boundary Conditions 7.5.2 FEM Formulation 7.6 FEM Formulation Based on Variational Principle 7.7 Finite Element Solution of Stokes Flow Equations 7.7.1 Problem Statement 7.7.2 FEM Solution 7.8 Illustrative Examples 7.9 Closing Remarks Exercise Problems References and Further Reading Chapter 8: Three-Dimensional Finite Element Analysis 8.1 Introduction 8.2 Axi-Symmetric Solids 8.2.1 Determination of the Fourier Coefficients 8.2.2 Isoparametric Finite Element Formulations 8.3 Eight-Node Isoparametric Element for Three-Dimensional Stress Analysis 8.4 Closing Remarks Exercise Problems References and Further Reading Chapter 9: Computer Implementation of FEM 9.1 General 9.2 Use of Symmetry and Anti-Symmetry Conditions in Reducing a Problem 9.3 Static Condensation 9.3.1 Applications of Static Condensation 9.3.2 Static Condensation Procedure 9.4 Computer Implementation of FEM-sfeap 9.5 Storage Schemes for Global Structural Stiffness Matrix 9.6 Application of Boundary Conditions 9.7 Closing Remarks Exercise Problems References and Further Reading Chapter 10: Further Applications of Finite Element Method 10.1 Introduction 10.2 Finite Element Analysis of Plates 10.2.1 Introduction 10.2.2 Review of Plate Theories 10.2.3 Finite Element Formulations 10.3 Dynamics with Finite Element Method 10.3.1 Introduction 10.3.2 Governing Equations 10.3.3 Mode Superposition Method 10.3.4 Direct Time Integration Method 10.4 Non-Linear Analysis 10.4.1 Finite Element Formulation for Non-Linear Analysis 10.4.2 Solution of Non-Linear Equations 10.4.3 Illustrative Examples 10.5 Groundwater Flow and Contaminant Transport Modelling 10.5.1 Introduction 10.5.2 Governing Equations and Boundary Conditions 10.5.3 Finite Element Formulation 10.5.4 FEM Formulation for Groundwater Flow in Unconfined Aquifer 10.5.5 Velocity Computation within Elements 10.5.6 FEM Formulation for Contaminant Transport 10.5.7 Case Study 10.6 Hydrodynamics Simulation of Shallow Water Flow 10.6.1 Introduction 10.6.2 Governing Equations and Boundary Conditions 10.6.3 Finite Element Formulation 10.6.3 Case Study 10.7 FEM-Software and Web Resources 10.7.1 Introduction 10.7.2 FEM in Structural Engineering 10.7.3 FEM in Geotechnical Engineering 10.7.4 FEM in Fluid Mechanics 10.7.5 FEM in Thermal and Automobile Engineering 10.7.6 FEM in Physics 10.7.7 Multi-Field FEM Software 10.7.8 FEM in Other Fields 10.8 Concluding Remarks References and Further Reading Appendix A: Review of Matrix Algebra and Matrix Calculus Appendix B: Elements of Calculus of Variations Appendix C: Example Illustrating Use of Galerkin’s Method Appendix D: Review of Gauss Quadrature Procedure for Numerical Integration Appendix E: User’s Manual For the Simplified Finite Element Analysis Program (sfeap) Appendix F: Graphical Interface For the Simplified Finite Element Analysis Program (sfeap) Appendix G: Computer Programs for One-Dimensional and Two-Dimensional Problems Index