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Contents

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

2.12 Solutions of Simultaneous Nonlinear Equations 121

Summary 123

References 124

Problems 124

3 Trusses 129

3.1 Definition of a Truss 129

3.2 Finite Element Formulation 130

3.3 Space Trusses 155

3.4 Overview of the ANSYS Program 157

3.5 ANSYS Workbench Environment 165

3.6 Examples Using ANSYS 165

3.7 Verification of Results 197

Summary 199

References 199

Problems 199

4 Axial Members, Beams, and Frames 209

4.1 Members Under Axial Loading 209

4.2 Beams 217

4.3 Finite Element Formulation of Beams 222

4.4 Finite Element Formulation of Frames 238

4.5 Three-Dimensional Beam Element 244

4.6 An Example Using ANSYS 246

4.7 Verification of Results 271

Summary 273

References 274

Problems 275

5 One-Dimensional Elements 287

5.1 Linear Elements 287

5.2 Quadratic Elements 291

5.3 Cubic Elements 293

5.4 Global, Local, and Natural Coordinates 296

5.5 Isoparametric Elements 298

5.6 Numerical Integration: Gauss¨CLegendre Quadrature 300

5.7 Examples of One-

Dimensional

Elements in ANSYS 305

Summary 305

References 305

Problems 305

6 Analysis of One-Dimensional Problems 312

6.1 Heat Transfer Problems 312

6.2 A Fluid Mechanics Problem 331

6.3 An Example Using ANSYS 335

6.4 Verification of Results 350

6.5 Members Under Axial Loading with Temperature Change 351

Summary 353

References 353

Problems 353

7 Two-Dimensional Elements 357

7.1 Rectangular Elements 357

7.2 Quadratic Quadrilateral Elements 361

7.3 Linear Triangular Elements 366

7.4 Quadratic Triangular Elements 371

7.5 Axisymmetric Elements 375

7.6 Isoparametric Elements 380

7.7 Two-Dimensional Integrals: Gauss¨CLegendre Quadrature 383

7.8 Examples of Two-Dimensional Elements in ANSYS 384

Summary 385

References 385

Problems 386

8 More ANSYS 393

8.1 ANSYS Program 393

8.2 ANSYS Database and Files 394

8.3 Creating a Finite Element Model with ANSYS: Preprocessing 396

8.4 h-Method Versus p-Method 410

8.5 Applying Boundary Conditions, Loads, and the Solution 410

8.6 Results of Your Finite Element Model: Postprocessing 413

8.7 Selection Options 418

8.8 Graphics Capabilities 419

8.9 Error-Estimation Procedures 421

8.10 More on ANSYS Workbench Environment 422

8.11 An Example Problem 428

Summary 441

References 442

9 Analysis of Two-Dimensional Heat Transfer Problems 443

9.1 General Conduction Problems 443

9.2 Formulation with Rectangular Elements 450

9.3 Formulation with Triangular Elements 461

9.4 Axisymmetric Formulation of Three-Dimensional Problems 480

9.5 Unsteady Heat Transfer 487

9.6 Conduction Elements Used by ANSYS 497

9.7 Examples Using ANSYS 498

9.8 Verification of Results 538

Summary 538

References 540

Problems 540

10 Analysis of Two-Dimensional Solid Mechanics Problems 552

10.1 Torsion of Members with Arbitrary Cross-Section Shape 552

10.2 Plane-Stress Formulation 568

10.3 Isoparametric Formulation: Using a Quadrilateral Element 576

10.4 Axisymmetric Formulation 583

10.5 Basic Failure Theories 585

10.6 Examples Using ANSYS 586

10.7 Verification of Results 608

Summary 608

References 610

Problems 610

11 Dynamic Problems 619

11.1 Review of Dynamics 619

11.2 Review of Vibration of Mechanical and Structural Systems 633

11.3 Lagrange¡¯s Equations 650

11.4 Finite Element Formulation of Axial Members 652

11.5 Finite Element Formulation of Beams and Frames 661

11.6 Examples Using ANSYS 675

Summar