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Finite Element Method with Applications in Engineering

Book Description

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

Table of Contents

  1. Cover
  2. Title Page
  3. Contents
  4. Authors Profile
  5. Dedication
  6. Preface
  7. 1 - Introduction
    1. 1.1 - Introductory Remarks
    2. 1.2 - Mathematical Modelling of Engineering Problems
    3. 1.3 - Type of Governing Equations
      1. 1.3.1 - Initial and Boundary Conditions
    4. 1.4 - Solution Methodologies
      1. 1.4.1 - Analytical Method
      2. 1.4.2 - Physical Method
      3. 1.4.3 - Computational Method
    5. 1.5 - Numerical Modelling
    6. 1.6 - Pre-Processing and Post-Processing
    7. 1.7 - Scope of the Book
    8. 1.8 - Highlights of the Book
    9. 1.9 - How to Use the Book?
    10. 1.10 - Closing Remarks
    11. References and Further Reading
  8. 2 - Approximate Methods of Analysis
    1. 2.1 - Introduction
    2. 2.2 - Aproximating Methods
    3. 2.3 - Method of Weighted Residuals
      1. 2.3.1 - Method of Point Collocation
      2. 2.3.2 - Method of Collocation by Sub-Regions
      3. 2.3.3 - Method of Least Squares
      4. 2.3.4 - Galerkin’s Method
    4. 2.4 - Rayleigh–Ritz Method
      1. 2.4.1 - Relation between FEM and Rayleigh–Ritz Method
    5. 2.5 - Further Numerical Examples
    6. 2.6 - Closing Remarks
    7. Exercise Problems
    8. References and Further Reading
  9. 3 - Finite Element Method—An Introduction
    1. 3.1 - General
    2. 3.2 - What is FEM?
    3. 3.3 - How Does FEM Work?
    4. 3.4 - A Brief History of FEM
    5. 3.5 - FEM Applications
    6. 3.6 - Merits and Demerits of FEM
    7. 3.7 - Closing Remarks
    8. Exercise Problems
    9. References and Further Reading
  10. 4 - Different Approaches in FEM
    1. 4.1 - Introduction
    2. 4.2 - General Steps of FEM
    3. 4.3 - Different Approaches Used in FEM
      1. 4.3.1 - Direct Approach
      2. 4.3.2 - Variational Approach
      3. 4.3.3 - Energy Approach
      4. 4.3.4 - Weighted Residual Approach
    4. 4.4 - Closing Remarks
    5. Exercise Problems
    6. References and Further Reading
  11. 5 - Finite Elements and Interpolation Function
    1. 5.1 - Introduction
    2. 5.2 - Interpolation Functions
      1. 5.2.1 - One-Independent Spatial Variable
      2. 5.2.2 - Two-Independent Spatial Variables
      3. 5.2.3 - Three-Independent Spatial Variables
    3. 5.3 - One-Dimensional Elements
      1. 5.3.1 - Line Element: Linear Interpolation Function
      2. 5.3.2 - Quadratic Interpolation Function
      3. 5.3.3 - Cubic Interpolation Function
      4. 5.3.4 - Lagrangian form of Interpolation Function
      5. 5.3.5 - Further Higher Order Elements in One-Dimension
    4. 5.4 - Two-Dimensional Elements
      1. 5.4.1 - Triangular Element: Linear Interpolation Function in Cartesian Co-ordinates
      2. 5.4.2 - Triangular Element—Area Co-ordinates
      3. 5.4.3 - Integration Formula for Triangular Elements
      4. 5.4.4 - Triangular Element—Quadratic Function
      5. 5.4.5 - Triangular Element—Cubic Interpolation Function
      6. 5.4.6 - Two-Dimensional Rectangular Elements
      7. 5.4.7 - Rectangular Elements—Lagrangian form in Natural and Cartesian Co-ordinates
      8. 5.4.8 - Isoparametric Elements
      9. 5.4.9 - Lagrangian Interpolation Functions for Two-Dimensional Elements
      10. 5.4.10 - Two-Dimensional Serendipity Elements
    5. 5.5 - Three-Dimensional Elements
      1. 5.5.1 - Tetrahedral Elements
      2. 5.5.2 - Tetrahedral Elements: Quadratic Interpolation Function
      3. 5.5.3 - Tetrahedral Elements: Cubic Interpolation Function
      4. 5.5.4 - Three-Dimensional Elements—Prismatic Elements
      5. 5.5.5 - Three-Dimensional Elements in Local Co-ordinates
      6. 5.5.6 - Three-Dimensional Serendipity Elements
    6. 5.6 - Closing Remarks
    7. Exercise Problems
    8. References and Further Reading
  12. 6 - One-Dimensional Finite Element Analysis
    1. 6.1 - Introduction
    2. 6.2 - Linear Spring
      1. 6.2.1 - Expressions for Equivalent Spring Constant
    3. 6.3 - Truss Element
      1. 6.3.1 - Plane Truss
      2. 6.3.2 - Element Equations by Minimizing Potential Energy
      3. 6.3.3 - Local and Global Element Equations for a Bar in the X–Y Plane
      4. 6.3.4 - Computation of Stress for a Bar in the X–Y Plane
    4. 6.4 - Space Truss
    5. 6.5 - One-Dimensional Torsion of a Circular Shaft
    6. 6.6 - One-Dimensional Steady State Heat Conduction
    7. 6.7 - One-Dimensional Flow through Porous Media
    8. 6.8 - One-Dimensional Ideal Fluid Flow through Pipes (Inviscid Fluid Flow)
    9. 6.9 - Beam Element
      1. 6.9.1 - Review of Beam Theory
      2. 6.9.2 - Finite Element Formulation of a Beam Element
      3. 6.9.3 - Illustrative Examples
    10. 6.10 - Analyses of Plane Frames and Grids
      1. 6.10.1 - Plane Frame Analysis
      2. 6.10.2 - Grid Analysis
    11. 6.11 - Further One-Dimensional Applications
      1. 6.11.1 - Flow Network Analysis
      2. 6.11.2 - Electrical Network Analysis
    12. 6.12 - Summary of Element Matrices for One-Dimensional Finite Elements
    13. 6.13 - Closing Remarks
    14. Exercise Problems
    15. References and Further Reading
  13. 7 - Two-Dimensional Finite Element Analysis
    1. 7.1 - Introduction
    2. 7.2 - Two-Dimensional Flow through Porous Media (Seepage Flow)
      1. 7.2.1 - Step-by-step Formulation for the CST Element
    3. 7.3 - Two-Dimensional Stress Analysis
      1. 7.3.1 - Review of Theory of Elasticity
      2. 7.3.2 - Application of Three-Dimensional Equations for Two-Dimensional Analysis
      3. 7.3.3 - CST Element for Plane Stress and Plane Strain Analyses
      4. 7.3.4 - Triangular Element for Axi-symmetric Analysis
      5. 7.3.5 - Some Remarks on Triangular Elements
      6. 7.3.6 - Four-Node Rectangular Element for Plane Problems
    4. 7.4 - Iso-Parametric Formulation
      1. 7.4.1 - Two-Node Iso-Parametric Line Element (Bar Element)
      2. 7.4.2 - Four-Node Iso-Parametric Element for Plane Problems (Quadrilateral Element)
    5. 7.5 - Finite Element Solution of Partial Differential Equations by Method of Weighted Residual
      1. 7.5.1 - Governing Equations and Boundary Conditions
      2. 7.5.2 - FEM Formulation
    6. 7.6 - FEM Formulation Based on Variational Principle
    7. 7.7 - Finite Element Solution of Stokes Flow Equations
      1. 7.7.1 - Problem Statement
      2. 7.7.2 - FEM Solution
    8. 7.8 - Illustrative Examples
    9. 7.9 - Closing Remarks
    10. Exercise Problems
    11. References and Further Reading
  14. 8 - Three-Dimensional Finite Element Analysis
    1. 8.1 - Introduction
    2. 8.2 - Axi-Symmetric Solids
      1. 8.2.1 - Determination of the Fourier Coefficients
      2. 8.2.2 - Isoparametric Finite Element Formulations
    3. 8.3 - Eight-Node Isoparametric Element for Three-Dimensional Stress Analysis
    4. 8.4 - Closing Remarks
    5. Exercise Problems
    6. References and Further Reading
  15. 9 - Computer Implementation of FEM
    1. 9.1 - General
    2. 9.2 - Use of Symmetry and Anti-Symmetry Conditions in Reducing a Problem
    3. 9.3 - Static Condensation
      1. 9.3.1 - Applications of Static Condensation
      2. 9.3.2 - Static Condensation Procedure
    4. 9.4 - Computer Implementation of FEM-sfeap
    5. 9.5 - Storage Schemes for Global Structural Stiffness Matrix
    6. 9.6 - Application of Boundary Conditions
    7. 9.7 - Closing Remarks
    8. Exercise Problems
    9. References and Further Reading
  16. 10 - Further Applications of Finite Element Method
    1. 10.1 - Introduction
    2. 10.2 - Finite Element Analysis of Plates
      1. 10.2.1 - Introduction
      2. 10.2.2 - Review of Plate Theories
      3. 10.2.3 - Finite Element Formulations
    3. 10.3 - Dynamics with Finite Element Method
      1. 10.3.1 - Introduction
      2. 10.3.2 - Governing Equations
      3. 10.3.3 - Mode Superposition Method
      4. 10.3.4 - Direct Time Integration Method
    4. 10.4 - Non-Linear Analysis
      1. 10.4.1 - Finite Element Formulation for Non-Linear Analysis
      2. 10.4.2 - Solution of Non-Linear Equations
      3. 10.4.3 - Illustrative Examples
    5. 10.5 - Groundwater Flow and Contaminant Transport Modelling
      1. 10.5.1 - Introduction
      2. 10.5.2 - Governing Equations and Boundary Conditions
      3. 10.5.3 - Finite Element Formulation
      4. 10.5.4 - FEM Formulation for Groundwater Flow in Unconfined Aquifer
      5. 10.5.5 - Velocity Computation within Elements
      6. 10.5.6 - FEM Formulation for Contaminant Transport
      7. 10.5.7 - Case Study
    6. 10.6 - Hydrodynamics Simulation of Shallow Water Flow
      1. 10.6.1 - Introduction
      2. 10.6.2 - Governing Equations and Boundary Conditions
      3. 10.6.3 - Finite Element Formulation
      4. 10.6.3 - Case Study
    7. 10.7 - FEM-Software and Web Resources
      1. 10.7.1 - Introduction
      2. 10.7.2 - FEM in Structural Engineering
      3. 10.7.3 - FEM in Geotechnical Engineering
      4. 10.7.4 - FEM in Fluid Mechanics
      5. 10.7.5 - FEM in Thermal and Automobile Engineering
      6. 10.7.6 - FEM in Physics
      7. 10.7.7 - Multi-Field FEM Software
      8. 10.7.8 - FEM in Other Fields
    8. 10.8 - Concluding Remarks
    9. References and Further Reading
  17. Appendix A - Review of Matrix Algebra and Matrix Calculus
  18. Appendix B - Elements of Calculus of Variations
  19. Appendix C - Example Illustrating Use of Galerkin’s Method
  20. Appendix D - Review of Gauss Quadrature Procedure for Numerical Integration
  21. Appendix E - User’s Manual for the Simplified Finite Element Analysis Program (sfeap)
  22. Appendix F - Graphical Interface for the Simplified Finite Element Analysis Program (sfeap)
  23. Appendix G - Computer Programs for One-Dimensional and Two-Dimensional Problems
  24. Notes
  25. Acknowledgements
  26. Copyright