Table Of ContentFinite
Element
Analysis
M. Moatamedi
and
H. Khawaja
The International Society of Multiphysics
www.multiphysics.org
p,
p,
A SCIENCE PUBLISHERS BOOK
A SCIENCE PUBLISHERS BOOK
CRC Press
Taylor & Francis Group
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Preface
Finite element analysis has become the most popular technique for studying
engineering structures in detail. It is particularly useful whenever the
complexity of the geometry or of the loading is such that alternative methods
are inappropriate.
This book consists of seven chapters introducing the finite element method
as a tool for the solution of practical engineering problems.
Chapter one introduces the aims and objectives of the book, history and
perspective relating to the engineering problem, the fundamentals, numerical
methods and application of the finite element method. This chapter also
discusses the terminology associated with the finite element mesh.
Chapter two covers the matrix stiffness methods also referred to as
displacement-based finite element method. It includes the formulation of simple
bar element in local and global coordinate systems, assembly of bar elements
and global stiffness matrix, loads and boundary conditions, a solution strategy
supported by numerical examples. It also discusses error analysis and ill-
conditioning, rigid body modes and mechanisms, symmetrical, antisymmetric
and asymmetrical as well as thermal loadings.
Chapter three focuses on the finite element formulation of one-dimensional
problems. It consists of the fundamental equations, shape function, algebraic
and matrix form of finite element equations. It presents an example of element
stiffness matrix for a 2-node bar with linear shape functions.
Chapter four provides the finite element formulation of two-dimensional
problems covering the fundamental equations and finite element formulation
for continuum for triangular and quadrilateral elements. It illustrates
numerical examples and describes the restrictions on element formulations for
completeness and compatibility.
Finite Element Analysis
Chapter five explains the computational implementation of finite element
method introducing frontal and banded solution methodologies leading to the
storage of the stiffness matrix and numerical integration by Gaussian quadrature.
Chapter six sets out the most commonly used elements namely beams,
plates, shells and solids. It includes the impact of nodal degrees of freedom
and boundary conditions in numerical examples.
Chapter seven details parametric element formulations. The focus is on
isoperimetric bar element, isoperimetric four-node quadrilateral element, and
isoperimetric eight-node quadrilateral element.
The book covers the principles of finite element analysis, including the
mathematical fundamentals as required, to construct an appropriate finite
element model of a physical system, and interpret the results of the analysis.
iv
Contents
Preface iii
1. Introduction 1
1.1 Book Aims and Objectives 1
1.2 History and Perspective 1
1.2.1 The engineering problem 1
1.2.2 The finite element method 2
1.3 The Finite Element Mesh: Terminology 4
2. Matrix Stiffness Methods 6
2.1 The Simple Bar Element 6
2.1.1 Stiffness in co-ordinate system parallel to 6
element axes
2.1.2 Transformation to global co-ordinates 8
2.2 Assembly of Bar Elements—The Global Stiffness Matrix 10
2.3 Loads and Boundary Conditions 13
2.4 A Solution Strategy 13
2.5 Numerical Examples 14
2.5.1 Uniaxial system 1 14
2.5.2 Uniaxial system 2 19
2.5.3 Pin-jointed framework 1 21
2.5.4 Redundant pin-jointed framework 24
2.6 Error Analysis and Ill-conditioning 31
2.6.1 Theory 31
2.6.2 Numerical examples 32
Finite Element Analysis
2.6.2.1 Error analysis of Example 2.5.4 32
2.6.2.2 System exhibiting ill-conditioning 33
2.6.3 Sources of ill-conditioning 35
2.7 Singular Equations: Rigid Body Modes and Mechanisms 39
2.8 Symmetry, Antisymmetry and Asymmetry 43
2.8.1 Symmetrical loading 43
2.8.2 Antisymmetrical loading 45
2.8.3 Asymmetrical loading 46
2.9 Thermal Loads 46
2.9.1 Thermal loads in a bar element 47
2.9.2 Numerical example of pre-stressing of bolt 48
using thermal loads
3. The Finite Element Formulation: One-Dimensional Problems 51
3.1 The Fundamental Equations 51
3.1.1 The equilibrium equation 51
3.1.2 The principle of virtual displacements 53
3.2 The Shape Function 53
3.3 The Finite Element Equations 55
3.3.1 Algebraic form 55
3.3.2 Matrix form 57
3.4 The Element Stiffness Matrix for a 2-Node Bar with 58
Linear Shape Functions
3.4.1 Bar of constant Young’s modulus and of 59
constant cross-section
3.4.2 Bar with linear taper 59
4. The Finite Element Formulation: Two-Dimensional Problems 62
4.1 The Fundamental Equations 62
4.1.1 Elasticity of a continuum 62
4.1.2 Vector notation 64
4.1.3 The stress/strain relationships 64
4.1.4 The principle of virtual displacements for the 66
two-dimensional continuum
vi
Contents
4.2 The Finite Element Formulation for a Continuum 66
4.3 A Triangular Element 69
4.3.1 Shape functions 69
4.3.2 The element stiffness matrix 72
4.3.3 Body forces 74
4.3.4 Surface pressures and tractions 75
4.3.5 Numerical example: A single element 77
4.4 A Quadrilateral Element 81
4.4.1 Shape functions 81
4.4.2 The element stiffness matrix 83
4.4.3 An application for the element 86
4.5 Numerical Study: Pin-jointed Frame with a Shear Web 87
4.6 Restrictions on Element Formulations—Completeness 97
and Compatibility
4.6.1 Attributes of the polynomial form of shape 98
function—number of terms and differentiability
4.6.2 Completeness—constant strain and rigid body modes 98
4.6.3 Compatibility 99
4.6.4 Conforming elements 100
4.6.5 Convergence 100
4.6.6 Non-conforming elements and the patch test 100
5. Computational Implementation of the Finite Element Method 102
5.1 Solution Methodologies—Frontal v Banded 103
5.1.1 Banded solver 105
5.1.2 Frontal solver 106
5.2 Storage of the Stiffness Matrix 107
5.3 Numerical Integration—Gaussian Quadrature 108
6. Beams, Plates, Shells and Solids 111
6.1 Solid Elements 111
6.2 A Beam Element 113
6.2.1 Nodal degrees of freedom 113
6.2.2 Shape functions 115
vii
Finite Element Analysis
6.2.3 The stiffness matrix 117
6.2.4 Numerical example 1 118
6.2.5 Numerical example 2 121
6.3 Plates and Shells 123
6.3.1 Plate elements 123
6.3.2 Shell elements 124
7. Parametric Element Formulations 126
7.1 Isoparametric Bar Element 127
7.1.1 Shape functions 127
7.1.2 The element stiffness matrix 128
7.2 Isoparametric Four-Node Quadrilateral Element 129
7.2.1 Shape functions 129
7.2.2 The element stiffness matrix 131
7.2.3 A special case—the rectangular element 134
7.3 Isoparametric Eight-Node Quadrilateral Element 137
7.3.1 Shape functions 138
7.3.2 The element stiffness matrix 140
7.3.3 The general load vector: nodal equivalent loads 141
Suggested Reading 147
Index 151
viii
1
Chapter
Introduction
Finite element analysis has become the most popular technique for studying
engineering structures in detail. It is particularly useful whenever the complexity
of the geometry or of the loading is such that alternative methods are
inappropriate. This book provides an introduction to the finite element method.
1.1 Book Aims and Objectives
• To introduce finite element analysis as a tool for the solution of practical
engineering problems.
• To teach the principles of finite element analysis, including the
mathematical fundamentals as required.
• To demonstrate how to construct an appropriate finite element model of
a physical system, and how to interpret the results of the analysis.
1.2 History and Perspective
1.2.1 The engineering problem
There are three steps in the analytical solution of a physical problem:
i) Identify the variables.
ii) Formulate governing equations describing the physical system, including
any constraints and boundary conditions.
iii) Solve the equations.
A most prudent fourth step is to validate the solutions with the experimental
data.
