Description
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. The finite element method is based on the premise that a complex structure can be broken down into finitely many smaller pieces (elements), the behaviour of each of which is known or can be postulated. These elements might then be assembled in some sense to model the behaviour of the structure. Intuitively this premise seems reasonable, but there are many important questions that need to be answered. In order to answer them it is necessary to apply a degree of mathematical rigour to the development of finite element techniques. The approach that will be taken in this book is to develop the fundamental ideas and methodologies based on an intuitive engineering approach, and then to support them with appropriate mathematical proofs where necessary. It will rapidly become clear that the finite element method is an extremely powerful tool for the analysis of structures (and for other field problems), but that the volume of calculations required to solve all but the most trivial of them is such that the assistance of a computer is necessary.
As stated above, many questions arise concerning finite element analysis. Some of these questions are associated with the fundamental mathematical formulations, some with numerical solution techniques, and others with the practical application of the method. In order to answer these questions, the engineer/analyst needs to understand both the nature and limitations of the finite element approximation and the fundamental behaviour of the structure. Misapplication of finite element analysis programs is most likely to arise when the analyst is ignorant of engineering phenomena.
Table of Contents
Table of Content
Introduction
Book Aims and Objectives
History and Perspective
The Finite Element Mesh: Terminology
Matrix Stiffness Methods
The Simple Bar Element
Assembly of Bar Elements – The Global Stiffness Matrix
Loads and Boundary Conditions
A Solution Strategy
Numerical Examples
Error Analysis and Ill-Conditioning
Singular Equations: Rigid Body Modes and Mechanisms
Symmetry, Anti-symmetry and Asymmetry
Thermal Loads
The Finite Element Formulation – One-Dimensional Problems
The Fundamental Equations
The Shape Function
The Finite Element Equations
The Element Stiffness Matrix for a 2 Node Bar with Linear Shape Functions
The Finite Element Formulation - Two-Dimensional Problems
The Fundamental Equations
The Finite Element Formulation for a Continuum
A Triangular Element
A Quadrilateral Element
Numerical Study - Pin-Jointed Frame with a Shear Web
Restrictions on Element Formulation - Completeness and Compatibility
Computational Implementation of the Finite Element Method
Solution Methodologies - Frontal v Banded Solvers
Storage of the Stiffness Matrix
Numerical Integration - Gaussian Quadrature
Beams, Plates, Shells and Solids
Solid Elements
A Beam Element
Plates and Shells
Parametric Element Formulation
Isoparametric bar element
Isoparametric Four-Node Quadrilateral Element
Isoparametric Eight-Node Quadrilateral Element
