# The Finite Element Method in Engineering

## 1st Edition

### Pergamon International Library of Science, Technology, Engineering and Social Studies

**Authors:**S. S. Rao

**eBook ISBN:**9781483136929

**Imprint:**Pergamon

**Published Date:**1st January 1982

**Page Count:**652

## Description

The Finite Element Method in Engineering introduces the various aspects of finite element method as applied to engineering problems in a systematic manner. It details the development of each of the techniques and ideas from basic principles. New concepts are illustrated with simple examples wherever possible. Several Fortran computer programs are given with example applications to serve the following purposes: to enable the reader to understand the computer implementation of the theory developed; to solve specific problems; and to indicate procedure for the development of computer programs for solving any other problem in the same area. The book begins with an overview of the finite element method. This is followed by separate chapters on numerical solution of various types of finite element equations; the general procedure of finite element analysis; the development higher order and isoparametric elements; and the application of finite element method for static and dynamic solid and structural mechanics problems like frames, plates, and solid bodies. Subsequent chapters deal with the solution of one-, two-, and three-dimensional steady state and transient heat transfer problems; the finite element solution of fluid mechanics problems; and additional applications and generalization of the finite element method.

## Table of Contents

Principal Notation

1. Introduction to Finite Element Method

1.1 Basic Concept

1.2 Historical Background

1.3 General Applicability of the Method

1.3.1 One-dimensional Heat Transfer

1.3.2 One-dimensional Fluid Flow

1.3.3 Solid Bar Under Axial Load

1.4 Engineering Applications of the Finite Element Method

1.5 General Description of the Finite Element Method

1.6 Comparison of Finite Element Method with Other Methods of Analysis

1.6.1 Derivation of the Equation of Motion for the Vibration of a Beam

1.6.2 Exact Analytical Solution (Separation of Variables Technique)

1.6.3 Approximate Analytical Solution (Rayleigh's Method)

1.6.4 Approximate Analytical Solution (Galerkin Method)

1.6.5 Finite Difference Method of Numerical Solution

1.6.6 Finite Element Method of Numerical Solution (Displacement Method)

1.7 Finite Element Program Packages

References

Problems

2. Solution of Finite Element Equations

2.1 Introduction

2.2 Solution of Equilibrium Problems

2.2.1 Gaussian Elimination Method

2.2.2 Choleski Method

2.2.3 Other Methods

2.3 Solution of Eigenvalue Problems

2.3.1 Jacobi Method

2.3.2 Power Method

2.3.3 Rayleigh-Ritz Subspace Iteration Method

2.3.4 Other Methods

2.4 Solution of Propagation Problems

2.4.1 Numerical Solution of Eq. (2.56)

2.4.2 Numerical Solution of Eq. (2.58)

References

Problems

3. General Procedure of Finite Element Method

3.1 Discretization of the Domain

3.1.1 Basic Element Shapes

3.1.2 Discretization Process

3.2 Interpolation Polynomials

3.2.1 Polynomial Form of Interpolation Functions

3.2.2 Selection of the Order of the Interpolation Polynomial

3.2.3 Convergence Requirements

3.2.4 Linear Interpolation Polynomials in Terms of Global Coordinates

3.2.5 Linear Interpolation Polynomials in Terms of Local Coordinates

3.3 Formulation of Element Characteristic Matrices and Vectors

3.3.1 Direct Approach

3.3.2 Variational Approach

3.3.3 Weighted Residual Approach

3.3.4 Coordinate Transformation

3.4 Assembly of Element Matrices and Vectors and Derivation of System Equations

3.4.1 Assemblage of Element Equations

3.4.2 Computer Implementation of the Assembly Procedure

3.4.3 Incorporation of the Boundary Conditions

3.4.4 Incorporation of Boundary Conditions in the Computer Program

3.5 Solution of Finite Element (System) Equations

3.6 Computation of Element Resultants

References

Problems

4. Higher Order and Isoparametric Element Formulations

4.1 Introduction

4.2 Higher Order One-dimensional Element

4.2.1 Quadratic Element

4.2.2 Cubic Element

4.3 Higher Order Elements in Terms of Natural Coordinates

4.3.1 One-dimensional Element

4.3.2 Two-dimensional Element (Triangular Element)

4.3.3 Derivation of Nodal Interpolation Functions

4.3.4 Three-dimensional Element (Tetrahedron Element)

4.3.5 Two-dimensional Element (Quadrilateral Element)

4.3.6 Three-dimensional Element (Hexahedron Element)

4.4 Higher Order Elements in Terms of Classical Interpolation Polynomials

4.4.1 Classical Interpolation Functions

4.4.2 One-dimensional Elements

4.4.3 Two-dimensional Elements: Rectangular Elements

4.5 Continuity Conditions

4.6 Comparative Study of Elements

4.7 Isoparametric Elements

4.7.1 Definitions

4.7.2 Shape Functions in Coordinate Transformation

4.7.3 Curved-Sided Elements

4.7.4 Derivation of Element Equations

4.8 Numerical Integration

4.8.1 In One-dimensions

4.8.3 In Three-dimensions

References

Problems

5. Solid and Structural Mechanics

5.1 Introduction

5.2 Basic Equations of Solid Mechanics

5.2.1 Introduction

5.2.2 External Equilibrium Equations

5.2.3 Equations of Internal Equilibrium

5.2.4 Stress-Strain Relations (Constitutive Relations)

5.2.5 Strain-Displacement Relations

5.2.6 Boundary Conditions

5.2.7 Compatibility Equations

5.2.8 Stress-Strain Relations for Anisotropie Materials

5.2.9 Formulations of Solid and Structural Mechanics

Static Analysis

5.3 Formulation of Equilibrium Equations

5.4 Analysis of Trusses and Frames

5.4.1 Space Truss Element

5.4.2 Space Frame Element

5.4.3 Planar Frame Element

5.4.4 Beam Element

5.4.5 Computer Program for Frame Analysis (FRAME)

5.5 Analysis of Plates

5.5.1 Introduction

5.5.2 Triangular Membrane Element

5.5.3 Numerical Results with Membrane Element

5.5.4 Computer Program for Plates under Inplane Loads (CST)

5.5.5 Bending Behavior of Plates

5.5.6 Triangular Plate Bending Element

5.5.7 Numerical Results with Bending Elements

5.5.8 Analysis of Three-dimensional Structures using Plate Elements

5.5.9 Computer Program for the Analysis of Three-dimensional Structures using Plate Elements (PLATE)

5.6 Analysis of Three-dimensional Problems

5.6.1 Introduction

5.6.2 Tetrahedron Element

5.6.3 Hexahedron Element

5.6.4 Numerical Results

5.7 Analysis of Solids of Revolution

5.7.1 Introduction

5.7.2 Formulation of Elemental Equations for an Axisymmetric Ring Element

5.7.3 Numerical Results

5.7.4 Computer Program (STRESS)

Dynamic Analysis

5.8 Dynamic Equations of Motion

5.9 Consistent and Lumped Mass Matrices

5.10 Consistent Mass Matrices in Global Coordinate System

5.10.1 Consistent Mass Matrix of a Pin-Jointed (Space Truss) Element

5.10.2 Consistent Mass Matrix of a Frame Element

5.10.3 Consistent Mass Matrix of a Triangular Membrane Element

5.10.4 Consistent Mass Matrix of a Triangular Bending Element

5.10.5 Consistent Mass Matrix of a Tetrahedron Element

5.11 Free Vibration Analysis

5.12 Computer Program for Eigenvalue Analysis of Three-dimensional Structures (PLATE)

5.13 Condensation of the Eigenvalue Problem (Eigenvalue Economizer)

5.14 Dynamic Response Calculations using Finite Element Method

5.14.1 Uncoupling the Equations of Motion of an Undamped System

5.14.2 Uncoupling the Equations of Motion of a Damped System

5.14.3 Solution of a General Second Order Differential Equation

5.15 Nonconservative Stability and Flutter Problems

References

Problems

6. Heat Transfer

6.1 Introduction

6.2 Basic Equations of Heat Transfer

6.2.1 Energy Balance Equation

6.2.2 Rate Equations

6.2.3 Governing Differential Equation for Heat Conduction in Three-dimensional Bodies

6.2.4 Statement of the Problem in Differential Equation Form

6.3 Derivation of Finite Element Equations

6.3.1 Variational Approach

6.3.2 Galerkin Approach

6.4 One-dimensional Heat Transfer

6.4.1 Straight Uniform Fin Analysis

Computer Program (HEAT1)

6.4.2 Tapered Fin Analysis

6.4.3 Straight Uniform Fin Analysis Using Quadratic Elements

6.5 Two-dimensional Heat Transfer

Computer Program (HEAT2)

6.6 Axisymmetric Heat Transfer

Computer Program (HEATAX)

6.7 Three-dimensional Heat Transfer

6.8 Unsteady State Heat Transfer Problems

6.8.1 Derivation of Element Capacitance Matrices

6.8.2 Finite Difference Solution in the Time Domain

6.9 Heat Transfer Problems with Radiation

Computer Program (RADIAT)

References

Problems

7. Fluid Mechanics

7.1 Introduction

7.2 Basic Equations of Fluid Mechanics

7.2.1 Definitions

7.2.2 Flow Field

7.2.3 Continuity Equation

7.2.4 Equations of Motion or Momentum Equations

7.2.5 Energy Equation

7.2.6 State and Viscosity Equations

7.2.7 Solution Procedure

7.2.8 Inviscid Fluid Flow

7.2.9 Irrotational Flow

7.2.10 Velocity Potential

7.2.11 Stream Function

7.2.12 Bernoulli Equation

7.3 Inviscid Incompressible Flows

7.3.1 Potential Function Formulation

7.3.2 Stream Function Formulation

7.3.3 Computer Program (PHIFLO)

7.4 Flow in Porous Media

7.4.1 Governing Equations

7.4.2 Finite Element Solution

7.4.3 Steady State Unconfined Flow Through a Dam

7.4.4 Steady State Flow Towards a Well

7.5 Wave Motion of a Shallow Basin

7.5.1 Equation of Motion

7.5.2 Boundary and Initial Conditions

7.5.3 Finite Element Solution of Eq. (7.133) Using Galerkin Approach

7.5.4 Eigenvalue Solution

7.5.5 Solution of Eq. (7.161) by Mode Superposition Method

7.6 Incompressible Viscous Flow

7.6.1 Statement of the Problem

7.6.2 Stream Function Formulation (sing Variational Approach)

7.6.3 Velocity-Pressure Formulation (Using Galerkin Approach)

7.6.4 Stream Function-Vorticity Formulation

7.7 Flow of non-Newtonian Fluids

7.7.1 Governing Equations

7.7.2 Finite Element Equations Using Galerkin Method

7.7.3 Solution Procedure

References

Problems

8. Additional Applications and Generalization of the Finite Element Method

8.1 Introduction

8.2 Steady State Field Problems

8.3 Transient Field Problems

8.4 Space-Time Finite Elements

8.5 Solution of Poisson Equation

8.5.1 Derivation of the Governing Equation for the Torsion Problem

8.5.2 Finite Element Solution

8.5.3 Computer Program (TORSON)

8.6 Solution of Helmholtz Equation

8.7 Solution of Reynolds Equation

8.7.1 Hydrodynamic Lubrication Problem

8.7.2 Finite Element Equations

8.8 Least Squares Finite Element Approach

8.8.1 Solution of a General Linear Partial Differential Equation

8.8.2 Solution of Unsteady Gas Dynamic Equations

8.9 Equilibrium, Mixed and Hybrid Elements

8.10 Miscellaneous Applications

References

Problems

Appendix A: Green-Gauss Theorem (Integration by Parts in Two and Three Dimensions)

Index

## Details

- No. of pages:
- 652

- Language:
- English

- Copyright:
- © Pergamon 1982

- Published:
- 1st January 1982

- Imprint:
- Pergamon

- eBook ISBN:
- 9781483136929