Preface xi

Acknowledgments xv

**1 Introduction 1**

1.1 Engineering Problems 2

1.2 Numerical Methods 5

1.3 A Brief History of the Finite Element Method and ANSYS 6

1.4 Basic Steps in the Finite Element Method 6

1.5 Direct Formulation 8

1.6 Minimum Total Potential Energy Formulation 37

1.7 Weighted Residual Formulations 43

1.8 Verification of Results 48

1.9 Understanding the Problem 49

Summary 54

References 54

Problems 54

**2 Matrix Algebra 66**

2.1 Basic Definitions 66

2.2 Matrix Addition or Subtraction 69

2.3 Matrix Multiplication 69

2.4 Partitioning of a Matrix 73

2.5 Transpose of a Matrix 77

2.6 Determinant of a Matrix 81

2.7 Solutions of Simultaneous Linear Equations 86

2.8 Inverse of a Matrix 94

2.9 Eigenvalues and Eigenvectors 98

2.10 Using MATLAB to Manipulate Matrices 102

2.11 Using Excel to Manipulate Matrices 106

Summary 120

References 121

Problems 121

**3 Trusses 125**

3.1 Definition of a Truss 125

3.2 Finite Element Formulation 126

3.3 Space Trusses 151

3.4 Overview of the ANSYS Program 153

3.5 Examples Using ANSYS 161

3.6 Verification of Results 193

Summary 195

References 195

Problems 195

**4 Axial members, Beams, and Frames 205**

4.1 Members Under Axial Loading 205

4.2 Beams 213

4.3 Finite Element Formulation of Beams 218

4.4 Finite Element Formulation of Frames 234

4.5 Three- Dimensional Beam Element 240

4.6 An Example Using ANSYS 242

4.7 Verification of Results 267

Summary 269

References 270

Problems 271

**5 One- Dimensional Elements 283**

5.1 Linear Elements 283

5.2 Quadratic Elements 287

5.3 Cubic Elements 289

5.4 Global, Local, and Natural Coordinates 292

5.5 Isoparametric Elements 294

5.6 Numerical Integration: Gauss–Legendre Quadrature 296

5.7 Examples of One- Dimensional Elements in ANSYS 301

Summary 301

References 301

Problems 301

**6 Analysis of One- Dimensional Problems 308**

6.1 Heat Transfer Problems 308

6.2 A Fluid Mechanics Problem 327

6.3 An Example Using ANSYS 331

6.4 Verification of Results 346

Summary 347

References 347

Problems 348

** ** 7 Two- Dimensional Elements 351

** ** 7.1 Rectangular Elements 351

7.2 Quadratic Quadrilateral Elements 355

7.3 Linear Triangular Elements 360

7.4 Quadratic Triangular Elements 365

7.5 Axisymmetric Elements 369

7.6 Isoparametric Elements 374

7.7 Two- Dimensional Integrals: Gauss—Legendre Quadrature 377

7.8 Examples of Two- Dimensional Elements in ANSYS 378

Summary 379

References 379

Problems 380

** ** 8 More Ansys 387

** ** 8.1 ANSYS Program 387

8.2 ANSYS Database and Files 388

8.3 Creating a Finite Element Model with ANSYS: Preprocessing 390

8.4 h- Method Versus p- Method 404

8.5 Applying Boundary Conditions, Loads, and the Solution 404

8.6 Results of Your Finite Element Model: Postprocessing 407

8.7 Selection Options 412

8.8 Graphics Capabilities 413

8.9 Error- Estimation Procedures 415

8.10 An Example Problem 417

Summary 431

References 432

** ** 9 Analysis of Two- Dimensional Heat Transfer Problems 433

** ** 9.1 General Conduction Problems 433

9.2 Formulation with Rectangular Elements 440

9.3 Formulation with Triangular Elements 451

9.4 Axisymmetric Formulation of Three- Dimensional Problems 470

9.5 Unsteady Heat Transfer 477

9.6 Conduction Elements used by ANSYS 487

9.7 Examples Using ANSYS 488

9.8 Verification of Results 528

Summary 528

References 530

Problems 530

** ** 10 Analysis of Two- Dimensional Solid Mechanics Problems 542

** ** 10.1 Torsion of Members with Arbitrary Cross- Section Shape 542

10.2 Plane- Stress Formulation 558

10.3 Isoparametric Formulation: Using a Quadrilateral Element 566

10.4 Axisymmetric Formulation 573

10.5 Basic Failure Theories 575

10.6 Examples Using ANSYS 576

10.7 Verification of Results 598

Summary 598

References 600

Problems 600

** ** 11 Dynamic Problems 609

** ** 11.1 Review of Dynamics 609

11.2 Review of Vibration of Mechanical and Structural Systems 623

11.3 Lagrange’s Equations 640

11.4 Finite Element Formulation of Axial Members 642

11.5 Finite Element Formulation of Beams and Frames 651

11.6 Examples Using ANSYS 665

Summary 684

References 684

Problems 684

** ** 12 Analysis of Fluid Mechanics Problems 691

** ** 12.1 Direct Formulation of Flow Through Pipes 691

12.2 Ideal Fluid Flow 703

12.3 Groundwater Flow 709

12.4 Examples Using ANSYS 712

12.5 Verification of Results 733

Summary 734

References 735

Problems 736

** ** 13 Three- Dimensional Elements 741

** ** 13.1 The Four- Node Tetrahedral Element 741

13.2 Analysis of Three- Dimensional Solid Problems Using Four- Node

13.3 The Eight- Node Brick Element 749

13.4 The Ten- Node Tetrahedral Element 751

13.5 The Twenty- Node Brick Element 752

13.6 Examples of Three- Dimensional Elements in ANSYS 754

13.7 Basic Solid- Modeling Ideas 758

13.8 A Thermal Example Using ANSYS 769

13.9 A Structural Example Using ANSYS 786

Summary 799

References 799

Problems 799

Tetrahedral Elements 744

** ** 14 Design and Material Selection 808

** ** 14.1 Engineering Design Process 809

14.2 Material Selection 812

14.3 Electrical, Mechanical, and Thermophysical Properties of Materials 813

14.4 Common Solid Engineering Materials 815

14.5 Some Common Fluid Materials 822

Summary 824

References 824

Problems 824

** ** 15 Design Optimization 826

** ** 15.1 Introduction to Design Optimization 826

15.2 The Parametric Design Language of ANSYS 830

15.3 Examples of Batch Files 832

Summary 843

References 844

Problems 844

Appendix A mechanical Properties of some materials 845

Appendix B thermophysical Properties of some materials 848

Appendix C Properties of Common line and Area shapes 849

Appendix D Geometrical Properties of structural steel shapes 852

Appendix e Conversion Factors 856

Appendix F An introduction to mAtlAB 858

index 893