EBOOK - The Finite Element Method - A Practical Course (G.R. Liu & S.S. Quek)

This book provides unified and detailed course material on the FEM for engineers and university students to solve primarily linear problems in mechanical and civil engineering, with the main focus on structural mechanics and heat transfer.
The aim of the book is to provide the necessary concepts, theories and techniques of the FEM for readers to be able to use a commercial FEM package comfortably to solve practical problems and structural analysis and heat transfer. Important fundamental and classical theories are introduced in a straightforward and easy to understand fashion.

Modern, state-of-the-art treatment of engineering problems in designing and analysing structural and thermal systems, including microstructural systems, are also discussed. Useful key techniques in FEMs are described in depth, and case studies are provided to demonstrate the theory, methodology, techniques and the practical applications of the FEM.
Equipped with the concepts, theories and modelling techniques described in this book, readers should be able to use a commercial FEM software package effectively to solve engineering structural problems in a professional manner.

1 Computational Modelling 1
1.1 Introduction 1
1.2 Physical Problems in Engineering 3
1.3 Computational Modelling using the FEM 4
1.4 Simulation 7
1.5 Visualization 9
2 Introduction to Mechanics for Solids and Structures 12
2.1 Introduction 12
2.2 Equations for Three-Dimensional Solids 13
2.3 Equations for Two-Dimensional Solids 19
2.4 Equations for Truss Members 22
2.5 Equations for Beams 24
2.6 Equations for Plates 28
2.7 Remarks 34
3 Fundamentals for Finite Element Method 35
3.1 Introduction 35
3.2 Strong and Weak Forms 36
3.3 Hamilton’s Principle 37
3.4 FEM Procedure 38
3.5 Static Analysis 58
3.6 Analysis of Free Vibration (Eigenvalue Analysis) 58
3.7 Transient Response 60
3.8 Remarks 64
3.9 Review Questions 65
4 FEM for Trusses 67
4.1 Introduction 67
4.2 FEM Equations 67
4.3 Worked Examples 76
4.4 High Order One-Dimensional Elements 87
4.5 Review Questions 88
5 FEM for Beams 90
5.1 Introduction 90
5.2 FEM Equations 90
5.3 Remarks 95
5.4 Worked Examples 95
5.5 Case study: Resonant Frequencies of Micro Resonant Transducer 98
5.6 Review Questions 107
6 FEM for Frames 108
6.1 Introduction 108
6.2 FEM Equations for Planar Frames 109
6.3 FEM Equations for Space Frames 114
6.4 Remarks 120
6.5 Case Study: Finite Element Analysis of a Bicycle Frame 121
6.6 Review Questions 127
7 FEM for Two-Dimensional Solids 129
7.1 Introduction 129
7.2 Linear Triangular Elements 131
7.3 Linear Rectangular Elements 141
7.4 Linear Quadrilateral Elements 148
7.5 Higher Order Elements 153
7.6 Elements with Curved Edges 160
7.7 Comments on Gauss Integration 161
7.8 Case Study: Side Drive Micro-Motor 162
7.9 Review Questions 171
8 FEM for Plates and Shells 173
8.1 Introduction 173
8.2 Plate Elements 173
8.3 Shell Elements 180
8.4 Remarks 184
8.5 Case Study: Natural Frequencies of Micro-Motor 185
8.6 Case Study: Transient Analysis of a Micro-Motor 192
8.7 Review Questions 198
9 FEM for 3D Solids 199
9.1 Introduction 199
9.2 Tetrahedron Element 200
9.3 Hexahedron Element 209
9.4 Higher Order Elements 216
9.5 Elements with Curved Surfaces 222
9.6 Case Study: Stress and Strain Analysis of a Quantum Dot
Heterostructure 223
9.7 Review Questions 232
10 Special Purpose Elements 233
10.1 Introduction 233
10.2 Crack Tip Elements 234
10.3 Methods for Infinite Domains 236
10.4 Finite Strip Elements 242
10.5 Strip Element Method (SEM) 245
11 Modelling Techniques 246
11.1 Introduction 246
11.2 CPU Time Estimation 247
11.3 Geometry Modelling 248
11.4 Meshing 250
11.5 Mesh Compatibility 254
11.6 Use of Symmetry 256
11.7 Modelling of Offsets 265
11.8 Modelling of Supports 270
11.9 Modelling of Joints 271
11.10 Other Applications of MPC Equations 274
11.11 Implementation of MPC Equations 278
11.12 Review Questions 280
12 FEM for Heat Transfer Problems 282
12.1 Field Problems 282
12.2 Weighted Residual Approach for FEM 288
12.3 1D Heat Transfer Problem 289
12.4 2D Heat Transfer Problem 303
12.5 Summary 316
12.6 Case Study: Temperature Distribution of Heated Road Surface 318
12.7 Review Questions 321
13.1 Introduction 324
13.2 Basic Building Block: Keywords and Data Lines 325
13.3 Using Sets 326
13.4 ABAQUS Input Syntax Rules 327
13.5 Defining a Finite Element Model in ABAQUS 329
13.6 General Procedures 339

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This book provides unified and detailed course material on the FEM for engineers and university students to solve primarily linear problems in mechanical and civil engineering, with the main focus on structural mechanics and heat transfer.
The aim of the book is to provide the necessary concepts, theories and techniques of the FEM for readers to be able to use a commercial FEM package comfortably to solve practical problems and structural analysis and heat transfer. Important fundamental and classical theories are introduced in a straightforward and easy to understand fashion.

Modern, state-of-the-art treatment of engineering problems in designing and analysing structural and thermal systems, including microstructural systems, are also discussed. Useful key techniques in FEMs are described in depth, and case studies are provided to demonstrate the theory, methodology, techniques and the practical applications of the FEM.
Equipped with the concepts, theories and modelling techniques described in this book, readers should be able to use a commercial FEM software package effectively to solve engineering structural problems in a professional manner.

1 Computational Modelling 1
1.1 Introduction 1
1.2 Physical Problems in Engineering 3
1.3 Computational Modelling using the FEM 4
1.4 Simulation 7
1.5 Visualization 9
2 Introduction to Mechanics for Solids and Structures 12
2.1 Introduction 12
2.2 Equations for Three-Dimensional Solids 13
2.3 Equations for Two-Dimensional Solids 19
2.4 Equations for Truss Members 22
2.5 Equations for Beams 24
2.6 Equations for Plates 28
2.7 Remarks 34
3 Fundamentals for Finite Element Method 35
3.1 Introduction 35
3.2 Strong and Weak Forms 36
3.3 Hamilton’s Principle 37
3.4 FEM Procedure 38
3.5 Static Analysis 58
3.6 Analysis of Free Vibration (Eigenvalue Analysis) 58
3.7 Transient Response 60
3.8 Remarks 64
3.9 Review Questions 65
4 FEM for Trusses 67
4.1 Introduction 67
4.2 FEM Equations 67
4.3 Worked Examples 76
4.4 High Order One-Dimensional Elements 87
4.5 Review Questions 88
5 FEM for Beams 90
5.1 Introduction 90
5.2 FEM Equations 90
5.3 Remarks 95
5.4 Worked Examples 95
5.5 Case study: Resonant Frequencies of Micro Resonant Transducer 98
5.6 Review Questions 107
6 FEM for Frames 108
6.1 Introduction 108
6.2 FEM Equations for Planar Frames 109
6.3 FEM Equations for Space Frames 114
6.4 Remarks 120
6.5 Case Study: Finite Element Analysis of a Bicycle Frame 121
6.6 Review Questions 127
7 FEM for Two-Dimensional Solids 129
7.1 Introduction 129
7.2 Linear Triangular Elements 131
7.3 Linear Rectangular Elements 141
7.4 Linear Quadrilateral Elements 148
7.5 Higher Order Elements 153
7.6 Elements with Curved Edges 160
7.7 Comments on Gauss Integration 161
7.8 Case Study: Side Drive Micro-Motor 162
7.9 Review Questions 171
8 FEM for Plates and Shells 173
8.1 Introduction 173
8.2 Plate Elements 173
8.3 Shell Elements 180
8.4 Remarks 184
8.5 Case Study: Natural Frequencies of Micro-Motor 185
8.6 Case Study: Transient Analysis of a Micro-Motor 192
8.7 Review Questions 198
9 FEM for 3D Solids 199
9.1 Introduction 199
9.2 Tetrahedron Element 200
9.3 Hexahedron Element 209
9.4 Higher Order Elements 216
9.5 Elements with Curved Surfaces 222
9.6 Case Study: Stress and Strain Analysis of a Quantum Dot
Heterostructure 223
9.7 Review Questions 232
10 Special Purpose Elements 233
10.1 Introduction 233
10.2 Crack Tip Elements 234
10.3 Methods for Infinite Domains 236
10.4 Finite Strip Elements 242
10.5 Strip Element Method (SEM) 245
11 Modelling Techniques 246
11.1 Introduction 246
11.2 CPU Time Estimation 247
11.3 Geometry Modelling 248
11.4 Meshing 250
11.5 Mesh Compatibility 254
11.6 Use of Symmetry 256
11.7 Modelling of Offsets 265
11.8 Modelling of Supports 270
11.9 Modelling of Joints 271
11.10 Other Applications of MPC Equations 274
11.11 Implementation of MPC Equations 278
11.12 Review Questions 280
12 FEM for Heat Transfer Problems 282
12.1 Field Problems 282
12.2 Weighted Residual Approach for FEM 288
12.3 1D Heat Transfer Problem 289
12.4 2D Heat Transfer Problem 303
12.5 Summary 316
12.6 Case Study: Temperature Distribution of Heated Road Surface 318
12.7 Review Questions 321
13.1 Introduction 324
13.2 Basic Building Block: Keywords and Data Lines 325
13.3 Using Sets 326
13.4 ABAQUS Input Syntax Rules 327
13.5 Defining a Finite Element Model in ABAQUS 329
13.6 General Procedures 339

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