Finite Element Method : Element Solutions

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. Preface 5 Contents 7 About the Author 10 Abbreviations 11 Symbols 12 List of Figures 16 List of Tables 19 1 Introduction 20 1.1 Introduction 20 1.2 Introduction to the Finite Element Method 20 1.2.1 Brief History of the Finite Element Method 20 1.2.2 Introduction to the Commonly Used Finite Element Software 22 1.3 Some Basic Knowledge from the Theory of Elasticity 23 1.3.1 The Four Basic Assumptions 23 1.3.2 Some Preliminary Knowledge on Tensor Operation 25 1.3.3 Three Main Equations in the Theory of Elasticity 26 1.3.4 Types of Boundary Conditions 33 1.3.5 The Plane Stress and Plane Strain Problems 34 2 Theoretical Basis of the Finite Element Method 38 2.1 Introduction 38 2.2 Equivalent Integral Form of the Differential Equation 39 2.3 The Weighted Residual Method 41 2.4 The Variational Principle 54 2.4.1 Establishment of the Variational Principle for Differential Equations 54 2.4.2 The Ritz Method 56 2.5 The Principle of Virtual Work 59 2.5.1 The Virtual Displacement Principle 60 2.5.2 The Virtual Stress Principle 61 2.5.3 The Minimal Potential Energy Principle 62 2.5.4 The Minimal Complementary Energy Principle 62 3 Finite Element Analysis Using Bar Element 64 3.1 Introduction 64 3.2 The Finite Element Calculation Procedure 64 3.3 Property of the Shape Function for the bar Element 70 3.4 Property of the Stiffness Matrix for the bar Element 72 3.5 The Coordinate Transformation for bar Elements 75 3.6 An Example of the FE Analysis Using the bar Element 78 4 Finite Element Analysis Using Beam Element 83 4.1 Introduction 83 4.2 The Finite Element Calculation Procedure 83 4.2.1 Some Preliminary Knowledge on Beam Element 84 4.3 FE Analysis Procedure Using Beam Element 87 4.4 Calculation of the Elemental Equivalent Nodal Forces 93 4.5 Coordinate Transformation in the Beam Analysis 100 4.6 Treatment of the Boundary Conditions 106 5 Finite Element Analysis Using Triangular Element 111 5.1 Introduction 111 5.2 FE Analysis Procedure Using Triangular Element 111 5.3 Properties of the Shape Function for Triangular Element 116 5.4 The Area Coordinate 120 5.5 Properties of the Global Stiffness Matrix 124 5.6 Calculation of the Equivalent Nodal Forces 128 5.7 An Example of the FE Analysis Using Triangular Element 132 6 Finite Element Analysis Using Rectangular Element 137 6.1 Introduction 137 6.2 FE Analysis Procedure Using Rectangular Element 137 6.3 The Shape Function for Rectangular Element 145 6.4 Iso-Parametric Element 150 6.5 Numerical Integration 158 6.5.1 The Newton–Cotes Integration Method 159 6.5.2 The Gauss Integration Method 162 6.6 An Example of the FE Analysis Using Rectangular Element 166 7 Finite Element Analysis Using 3D Elements 176 7.1 Introduction 176 7.2 FE Analysis Using Tetrahedral Element 176 7.2.1 FE Analysis Procedure Using Tetrahedral Element 176 7.2.2 The Volume Coordinates and Their Properties 181 7.3 FE Analysis Procedure Using Hexahedral Element 183 8 High Order Lagrange Element 187 8.1 Introduction 187 8.2 Definition of Lagrange and Hermite Elements 187 8.3 The 1D High Order Lagrange Element 188 8.4 The 2D High Order Lagrange Element 192 8.4.1 The High Order Triangular Element 192 8.4.2 The High Order Rectangular Element 197 8.5 The 3D High Order Lagrange Element 202 8.5.1 The High Order Tetrahedral Element 203 8.5.2 The High Order Hexahedral Element 207