Finite Element Analysis

 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