Introductory Guide To Finite Element Analysis, An

Description:
'An Introductory Guide to Finite Element Analysis' is written specifically for engineers and scientists who want to understand the fundamental theory of finite element (FE) analysis and to learn how to analyse practical problems using FE software. Based on lecture notes compiled by the author whilst running successful 'continuing professional development' courses on finite element methods over the past 12 years, this excellent book provides answers to the fundamental questions a newcomer to FEA, or a busy practising engineer, might want to ask without using unnecessarily complex mathematical explanations. This book provides the theoretical background to understanding linear finite element analysis, presents a number of examples to illustrate how practical problems can be analysed using FE software, and provides a firm foundation for readers interested in further, more advanced, non-linear applications.

Table of Contents:
Series Editor's Foreword v Author's Preface xi Notation xiii Chapter 1 Introduction and Background 1 1.1 Numerical methods in continuum mechanics 1 1.1.1 Finite element (FE) method 1 1.1.2 Boundary element (BE) method 2 1.1.3 Finite difference (FD) method 3 1.2 Definition of stress 3 1.3 Stress - strain relationships (Hooke's law) 5 1.4 Strain - displacement definitions 6 1.5 Equilibrium equations 6 1.6 Compatibility equations 8 1.7 Principle of minimum total potential energy 9 1.7.1 Stable and unstable problems 9 1.7.2 Strain energy 9 1.7.3 Work done by external forces 10 1.7.4 Total potential energy (TPE) 11 1.8 Matrix definitions 11 1.8.1 Matrix multiplication 12 1.8.2 Transpose of a matrix 12 1.8.3 Symmetric matrix 12 1.8.4 Inverse of a matrix 13 1.8.5 Strain energy matrix example 13 1.9 Layout of the book 14 Chapter 2 Structural Analysis Using Pin-Jointed Elements 17 2.1 A simple one-dimensional tension element 17 2.2 Pin-jointed structures (truss elements) 21 2.3 A structural analysis example 28 2.4 Summary of key points 35 Chapter 3 Continuum Elements (Two-Dimensional, Axisymmetric, Three-Dimensional Elements) 37 3.1 Overview of the FE formulation 37 3.2 Two-dimensional and axisymmetric assumptions 37 3.2.1 Two-dimensional problems 37 3.2.2 Axisymmetric problems 39 3.3 Two-dimensional triangular continuum elements 39 3.4 Axisymmetric (ring) continuum elements 45 3.5 Three-dimensional continuum elements 47 3.6 Summary of key points 48 Chapter 4 Energy and Variational Approaches 51 4.1 Introduction 51 4.2 Ritz method (variational approach) 51 4.2.1 Solution steps in the Ritz method 52 4.2.2 Essential and natural boundary conditions 52 4.2.3 Convergence criteria 53 4.3 Ritz example: Cantilever beam example 53 4.4 Galerkin method (weighted residuals) 58 4.5 A flexible cable example 61 4.5.1 The trial solution for the displacement function 61 4.5.2 Galerkin method 61 4.5.3 Equivalence of the Ritz and Galerkin solutions 62 4.5.4 Advantages of the Galerkin method 63 4.5.5 Equivalence of FE formulation and the Ritz/Galerkin methods 64 4.6 Summary of key points 64 Chapter 5 Higher Order Quadratic Elements 65 5.1 Properties of the shape functions 65 5.2 Isoparametric mapping 68 5.3 Transformation of variables (the Jacobian) 71 5.4 The stiffness matrix for higher order elements 72 5.5 Numerical integration using Gaussian quadrature 73 5.6 Summary of key points 76 Chapter 6 Beam, Plate, and Shell Elements 77 6.1 Structural elements 77 6.1.1 Beam elements 78