Introduction to Finite Element Analysis and Design

Introduces the basic concepts of FEM in an easy-to-use format so that students and professionals can use the method efficiently and interpret results properly

Finite element method (FEM) is a powerful tool for solving engineering problems both in solid structural mechanics and fluid mechanics. This book presents all of the theoretical aspects of FEM that students of engineering will need. It eliminates overlong math equations in favour of basic concepts, and reviews of the mathematics and mechanics of materials in order to illustrate the concepts of FEM. It introduces these concepts by including examples using six different commercial programs online.

The all-new, second edition of Introduction to Finite Element Analysis and Design provides many more exercise problems than the first edition. It includes a significant amount of material in modelling issues by using several practical examples from engineering applications. The book features new coverage of buckling of beams and frames and extends heat transfer analyses from 1D (in the previous edition) to 2D. It also covers 3D solid element and its application, as well as 2D. Additionally, readers will find an increase in coverage of finite element analysis of dynamic problems. There is also a companion website with examples that are concurrent with the most recent version of the commercial programs.

• Offers elaborate explanations of basic finite element procedures
• Delivers clear explanations of the capabilities and limitations of finite element analysis
• Includes application examples and tutorials for commercial finite element software, such as MATLAB, ANSYS, ABAQUS and NASTRAN
• Provides numerous examples and exercise problems
• Comes with a complete solution manual and results of several engineering design projects

NAM-HO KIM, PHD is a professor in the Department of Mechanical & Aerospace Engineering at the University of Florida, USA. His research interests are in computational mechanics and design optimization, in particular, nonlinear solid mechanics and design under uncertainty.

BHAVANI V. SANKAR, PHD is a professor in the Department of Mechanical & Aerospace Engineering at the University of Florida, USA. His research interests are in mechanics of composite materials and structures, in particular, micromechanics and fracture mechanics.

ASHOK V. KUMAR, PHD is an Associate Professor in the Department of Mechanical & Aerospace Engineering at the University of Florida, USA. His research focus is in the area of computational methods and design optimization.

Preface ix

About the Companion Website xi

1 Direct Method – Springs, Bars, and Truss Elements 1

1.1 Illustration of the Direct Method 2

1.2 Uniaxial Bar Element 7

1.3 Plane Truss Elements 15

1.4 Three-Dimensional Truss Elements (Space Truss) 27

1.5 Thermal Stresses 32

1.6 Finite Element Modeling Practice for Truss 39

1.7 Projects 45

1.8 Exercises 49

2 Weighted Residual Methods for One-Dimensional Problems 63

2.1 Exact vs. Approximate Solution 63

2.2 Galerkin Method 67

2.3 Higher-Order Differential Equations 72

2.4 Finite Element Approximation 75

2.5 Energy Methods 89

2.6 Exercises 99

3 Finite Element Analysis of Beams and Frames 107

3.1 Review of Elementary Beam Theory 107

3.2 Rayleigh-Ritz Method 112

3.3 Finite Element Formulation for Beams 117

3.4 Plane Frame Elements 136

3.5 Buckling of Beams 142

3.6 Buckling of Frames 154

3.7 Finite Element Modeling Practice for Beams 157

3.8 Project 162

3.9 Exercises 163

4 Finite Elements for Heat Transfer Problems 175

4.1 Introduction 175

4.2 Fourier Heat Conduction Equation 176

4.3 Finite Element Analysis – Direct Method 178

4.4 Galerkin’s Method for Heat Conduction Problems 184

4.5 Convection Boundary Conditions 191

4.6 Two-Dimensional Heat Transfer 198

4.7 3-Node Triangular Elements for Two-Dimensional Heat Transfer 204

4.8 Finite Element Modeling Practice for 2-D Heat Transfer 213

4.9 Exercises 215

5 Review of Solid Mechanics 221

5.1 Introduction 221

5.2 Stress 222

5.3 Strain 234

5.4 Stress–Strain Relationship 240

5.5 Boundary Value Problems 244

5.6 Principle of Minimum Potential Energy for Plane Solids 249

5.7 Failure Theories 250

5.8 Safety Factor 256

5.9 Exercises 259

6 Finite Elements for Two-Dimensional Solid Mechanics 269

6.1 Introduction 269

6.2 Types of Two-Dimensional Problems 269

6.3 Constant Strain Triangular (CST) Element 272

6.4 Four–Node Rectangular Element 286

6.5 Axisymmetric Element 296

6.6 Finite Element Modeling Practice for Solids 300

6.7 Project 305

6.8 Exercises 306

7 Isoparametric Finite Elements 315

7.1 Introduction 315

7.2 One-Dimensional Isoparametric Elements 316

7.3 Two-Dimensional Isoparametric Quadrilateral Element 326

7.4 Numerical Integration 337

7.5 Higher-Order Quadrilateral Elements 343

7.6 Isoparametric Triangular Elements 349

7.7 Three-Dimensional Isoparametric Elements 355

7.8 Finite Element Modeling Practice for Isoparametric Elements 359

7.9 Projects 368

7.10 Exercises 369

8 Finite Element Analysis for Dynamic Problems 377

8.1 Introduction 377

8.2 Dynamic Equation of Motion and Mass Matrix 378

8.3 Natural Vibration: Natural Frequencies and Mode Shapes 384

8.4 Forced Vibration: Direct Integration Approach 392

8.5 Method of Mode Superposition 404

8.6 Dynamic Analysis with Structural Damping 410

8.7 Finite Element Modeling Practice for Dynamic Problems 414

8.8 Exercises 423

9 Finite Element Procedure and Modeling 427

9.1 Introduction 427

9.2 Finite Element Analysis Procedures 427

9.3 Finite Element Modeling Issues 446

9.4 Error Analysis and Convergence 460

9.5 Project 466

9.6 Exercises 467

10 Structural Design Using Finite Elements 473

10.1 Introduction 473

10.2 Conservatism in Structural Design 474

10.3 Intuitive Design: Fully Stressed Design 480

10.4 Design Parameterization 484

10.5 Parametric Study – Sensitivity Analysis 486

10.6 Structural Optimization 491

10.7 Projects 505

10.8 Exercises 507

Appendix Mathematical Preliminaries 511

A.1 Vectors and Matrices 511

A.2 Vector-Matrix Calculus 514

A.3 Matrix Equations and Solution 518

A.4 Eigenvalues and Eigenvectors 524

A.5 Quadratic Forms 528

A.6 Maxima and Minima of Functions 529

A.7 Exercises 530

Index 533

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