book

Finite Element Method with Applications in Engineering

by Y. M. Desai, T. L. Eldho

January 2011

Intermediate to advanced

492 pages

12h 45m

English

Pearson India

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Cover
Title Page
Contents
Authors Profile
Dedication
Preface
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 Method1.4.2 - Physical Method1.4.3 - 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
2 - Approximate Methods of Analysis
2.1 - Introduction
2.2 - Aproximating Methods
2.3 - Method of Weighted Residuals
2.3.1 - Method of Point Collocation2.3.2 - Method of Collocation by Sub-Regions2.3.3 - Method of Least Squares2.3.4 - 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
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
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 Approach4.3.2 - Variational Approach4.3.3 - Energy Approach4.3.4 - Weighted Residual Approach
4.4 - Closing Remarks
Exercise Problems
References and Further Reading
5 - Finite Elements and Interpolation Function
5.1 - Introduction
5.2 - Interpolation Functions
5.2.1 - One-Independent Spatial Variable5.2.2 - Two-Independent Spatial Variables5.2.3 - Three-Independent Spatial Variables
5.3 - One-Dimensional Elements
5.3.1 - Line Element: Linear Interpolation Function5.3.2 - Quadratic Interpolation Function5.3.3 - Cubic Interpolation Function5.3.4 - Lagrangian form of Interpolation Function5.3.5 - Further Higher Order Elements in One-Dimension
5.4 - Two-Dimensional Elements
5.4.1 - Triangular Element: Linear Interpolation Function in Cartesian Co-ordinates5.4.2 - Triangular Element—Area Co-ordinates5.4.3 - Integration Formula for Triangular Elements5.4.4 - Triangular Element—Quadratic Function5.4.5 - Triangular Element—Cubic Interpolation Function5.4.6 - Two-Dimensional Rectangular Elements5.4.7 - Rectangular Elements—Lagrangian form in Natural and Cartesian Co-ordinates5.4.8 - Isoparametric Elements5.4.9 - Lagrangian Interpolation Functions for Two-Dimensional Elements5.4.10 - Two-Dimensional Serendipity Elements
5.5 - Three-Dimensional Elements
5.5.1 - Tetrahedral Elements5.5.2 - Tetrahedral Elements: Quadratic Interpolation Function5.5.3 - Tetrahedral Elements: Cubic Interpolation Function5.5.4 - Three-Dimensional Elements—Prismatic Elements5.5.5 - Three-Dimensional Elements in Local Co-ordinates5.5.6 - Three-Dimensional Serendipity Elements
5.6 - Closing Remarks
Exercise Problems
References and Further Reading
6 - One-Dimensional Finite Element Analysis
6.1 - Introduction
6.2 - Linear Spring
6.2.1 - Expressions for Equivalent Spring Constant
6.3 - Truss Element
6.3.1 - Plane Truss6.3.2 - Element Equations by Minimizing Potential Energy6.3.3 - Local and Global Element Equations for a Bar in the X–Y Plane6.3.4 - 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)
6.9 - Beam Element
6.9.1 - Review of Beam Theory6.9.2 - Finite Element Formulation of a Beam Element6.9.3 - Illustrative Examples
6.10 - Analyses of Plane Frames and Grids
6.10.1 - Plane Frame Analysis6.10.2 - Grid Analysis
6.11 - Further One-Dimensional Applications
6.11.1 - Flow Network Analysis6.11.2 - Electrical Network Analysis
6.12 - Summary of Element Matrices for One-Dimensional Finite Elements
6.13 - Closing Remarks
Exercise Problems
References and Further Reading
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
7.3 - Two-Dimensional Stress Analysis
7.3.1 - Review of Theory of Elasticity7.3.2 - Application of Three-Dimensional Equations for Two-Dimensional Analysis7.3.3 - CST Element for Plane Stress and Plane Strain Analyses7.3.4 - Triangular Element for Axi-symmetric Analysis7.3.5 - Some Remarks on Triangular Elements7.3.6 - 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 - Governing Equations and Boundary Conditions7.5.2 - FEM Formulation
7.6 - FEM Formulation Based on Variational Principle
7.7 - Finite Element Solution of Stokes Flow Equations
7.7.1 - Problem Statement7.7.2 - FEM Solution
7.8 - Illustrative Examples
7.9 - Closing Remarks
Exercise Problems
References and Further Reading
8 - Three-Dimensional Finite Element Analysis
8.1 - Introduction
8.2 - Axi-Symmetric Solids
8.2.1 - Determination of the Fourier Coefficients8.2.2 - 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
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 Condensation9.3.2 - 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
10 - Further Applications of Finite Element Method
10.1 - Introduction
10.2 - Finite Element Analysis of Plates
10.2.1 - Introduction10.2.2 - Review of Plate Theories10.2.3 - Finite Element Formulations
10.3 - Dynamics with Finite Element Method
10.3.1 - Introduction10.3.2 - Governing Equations10.3.3 - Mode Superposition Method10.3.4 - Direct Time Integration Method
10.4 - Non-Linear Analysis
10.4.1 - Finite Element Formulation for Non-Linear Analysis10.4.2 - Solution of Non-Linear Equations10.4.3 - Illustrative Examples
10.5 - Groundwater Flow and Contaminant Transport Modelling
10.5.1 - Introduction10.5.2 - Governing Equations and Boundary Conditions10.5.3 - Finite Element Formulation10.5.4 - FEM Formulation for Groundwater Flow in Unconfined Aquifer10.5.5 - Velocity Computation within Elements10.5.6 - FEM Formulation for Contaminant Transport10.5.7 - Case Study
10.6 - Hydrodynamics Simulation of Shallow Water Flow
10.6.1 - Introduction10.6.2 - Governing Equations and Boundary Conditions10.6.3 - Finite Element Formulation10.6.3 - Case Study
10.7 - FEM-Software and Web Resources
10.7.1 - Introduction10.7.2 - FEM in Structural Engineering10.7.3 - FEM in Geotechnical Engineering10.7.4 - FEM in Fluid Mechanics10.7.5 - FEM in Thermal and Automobile Engineering10.7.6 - FEM in Physics10.7.7 - Multi-Field FEM Software10.7.8 - 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
Notes
Acknowledgements
Copyright

Overview

The book explains the finite element method with various engineering applications to help students, teachers, engineers and researchers. It explains mathematical modeling of engineering problems and approximate methods of analysis and different approaches

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