Finite Element Method: Element Solutions

✍ Scribed by Yongtao Lyu

• Publisher: Springer
• Year: 2022
• Tongue: English
• Leaves: 210
• Category: Library

✦ Synopsis

This textbook is intended to be used by the senior engineering undergraduate and the graduate student. Nowadays, the finite element method has become one of the most widely used techniques in all the engineering fields, including aerospace engineering, mechanical engineering, biomedical engineering, etc. To unveil the FE technique, the textbook provides a detailed description of the finite element method, starting from the most important basic theoretical basis, e.g., the Galerkin method, the variational principle, followed by the detailed description of the various types of finite elements, including the bar, the beam, the triangular, the rectangular, the 3D elements. The primary aim of the textbook is to provide a comprehensive description of the FE solutions using different types of elements. Therefore, the properties of different elements and the solution discrepancies caused by using different elements are highlighted in the book. Thus, the textbook is very helpful for engineers to understand the behaviours of different types of elements. Additionally, the textbook can help the students and engineers write FE codes based on the theories presented in the book. Furthermore, the textbook can serve as the basis for some advanced computational mechanics courses, such as the nonlinear finite element method. 

✦ Table of Contents

Preface
Contents
About the Author
Abbreviations
Symbols
List of Figures
List of Tables
1 Introduction
1.1 Introduction
1.2 Introduction to the Finite Element Method
1.2.1 Brief History of the Finite Element Method
1.2.2 Introduction to the Commonly Used Finite Element Software
1.3 Some Basic Knowledge from the Theory of Elasticity
1.3.1 The Four Basic Assumptions
1.3.2 Some Preliminary Knowledge on Tensor Operation
1.3.3 Three Main Equations in the Theory of Elasticity
1.3.4 Types of Boundary Conditions
1.3.5 The Plane Stress and Plane Strain Problems
2 Theoretical Basis of the Finite Element Method
2.1 Introduction
2.2 Equivalent Integral Form of the Differential Equation
2.3 The Weighted Residual Method
2.4 The Variational Principle
2.4.1 Establishment of the Variational Principle for Differential Equations
2.4.2 The Ritz Method
2.5 The Principle of Virtual Work
2.5.1 The Virtual Displacement Principle
2.5.2 The Virtual Stress Principle
2.5.3 The Minimal Potential Energy Principle
2.5.4 The Minimal Complementary Energy Principle
3 Finite Element Analysis Using Bar Element
3.1 Introduction
3.2 The Finite Element Calculation Procedure
3.3 Property of the Shape Function for the bar Element
3.4 Property of the Stiffness Matrix for the bar Element
3.5 The Coordinate Transformation for bar Elements
3.6 An Example of the FE Analysis Using the bar Element
4 Finite Element Analysis Using Beam Element
4.1 Introduction
4.2 The Finite Element Calculation Procedure
4.2.1 Some Preliminary Knowledge on Beam Element
4.3 FE Analysis Procedure Using Beam Element
4.4 Calculation of the Elemental Equivalent Nodal Forces
4.5 Coordinate Transformation in the Beam Analysis
4.6 Treatment of the Boundary Conditions
5 Finite Element Analysis Using Triangular Element
5.1 Introduction
5.2 FE Analysis Procedure Using Triangular Element
5.3 Properties of the Shape Function for Triangular Element
5.4 The Area Coordinate
5.5 Properties of the Global Stiffness Matrix
5.6 Calculation of the Equivalent Nodal Forces
5.7 An Example of the FE Analysis Using Triangular Element
6 Finite Element Analysis Using Rectangular Element
6.1 Introduction
6.2 FE Analysis Procedure Using Rectangular Element
6.3 The Shape Function for Rectangular Element
6.4 Iso-Parametric Element
6.5 Numerical Integration
6.5.1 The Newton–Cotes Integration Method
6.5.2 The Gauss Integration Method
6.6 An Example of the FE Analysis Using Rectangular Element
7 Finite Element Analysis Using 3D Elements
7.1 Introduction
7.2 FE Analysis Using Tetrahedral Element
7.2.1 FE Analysis Procedure Using Tetrahedral Element
7.2.2 The Volume Coordinates and Their Properties
7.3 FE Analysis Procedure Using Hexahedral Element
8 High Order Lagrange Element
8.1 Introduction
8.2 Definition of Lagrange and Hermite Elements
8.3 The 1D High Order Lagrange Element
8.4 The 2D High Order Lagrange Element
8.4.1 The High Order Triangular Element
8.4.2 The High Order Rectangular Element
8.5 The 3D High Order Lagrange Element
8.5.1 The High Order Tetrahedral Element
8.5.2 The High Order Hexahedral Element