Tirupathi R. Chandrupatla is Professor and Chair of Mechanical Engineering at Rowan University, Glassboro, New Jersey. He received the B.S. degree from the Regional Engineering College, Warangal, which was affiliated with Osmania University, India. He received the M.S. degree in design and manufacturing from the Indian Institute of Technology, Bombay. He started his career as a design engineer with Hindustan Machine Tools, Bangalore. He then taught in the Department of Mechanical Engineering at LLT, Bombay. He pursued his graduate studies in the Department of Aerospace Engineering and Engineering Mechanics at the University of Texas at Austin and received his Ph.D. in 1977. He subsequently taught at the University of Kentucky. Prior to joining Rowan, he was a Professor of Mechanical Engineering and Manufacturing Systems Engineering at GMI Engineering & Management Institute (formerly General Motors Institute), where he taught for 16 years.

Dr. Chandrupatla has broad research interests, which include finite element analysis, design, optimization, and manufacturing engineering. He has published widely in these areas and serves as a consultant to industry. Dr. Chandrupatla is a registered Professional Engineer and also a Certified Manufacturing Engineer. He is a member of ASEE, ASME, NSPE, SAE, and SME.

Ashok D. Belegundu is a Professor of Mechanical Engineering at The Pennsylvania State University, University Park. He was on the faculty at GMI from 1982 through 1986. He received the Ph.D. degree in 1982 from the University of Iowa and the B.S. degree from the Indian Institute of Technology, Madras. He was awarded a fellowship to spend a summer in 1993 at the NASA Lewis Research Center. During 1994-1995, he obtained a grant from the UK Science and Engineering Research Council to spend his sabbatical leave at Cranfield University, Cranfield, UK.

Dr. Belegundu's teaching and research interests include linear, nonlinear, and dynamic finite elements and optimization. He has worked on several sponsored projects for government and industry. He is an associate editor of Mechanics of Structures and Machines. He is also a member of ASME and an Associate fellow of AIAA.

1.1 Introduction 1

1.2 Historical Background 1

1.3 Outline of Presentation 2

1.4 Stresses and Equilibrium 2

1.5 Boundary Conditions 4

1.6 Strain—Displacement Relations 5

1.7 Stress—Strain Relations 6

1.8 Temperature Effects 8

1.9 Potential Energy and Equilibrium: The Rayleigh—Ritz Method 9

Potential Energy ß , 9

1.10 Galerkin’s Method 14

1.11 Saint Venant’s Principle 18

1.12 Von Mises Stress 19

1.13 Principle of Superposition 19

1.14 Computer Programs 20

1.15 Conclusion 20

Historical References 20

Problems 21

2.1 Matrix Algebra 28

2.2 Gaussian Elimination 35

2.3 Conjugate Gradient Method for Equation Solving 45

Problems 47

3.1 Introduction 51

3.2 Finite Element Modeling 52

3.3 Shape Functions and Local Coordinates 55

3.4 The Potential-Energy Approach 59

3.5 The Galerkin Approach 64

3.6 Assembly of the Global Stiffness Matrix and Load Vector 66

3.7 Properties of K 69

3.8 The Finite Element Equations: Treatment

of Boundary Conditions 70

3.9 Quadratic Shape Functions 85

3.10 Temperature Effects 92

3.11 Problem Modeling and Boundary Conditions 96

Problems 99

4.1 Introduction 117

4.2 Plane Trusses 118

Formulas for Calculating / and m, 119

4.3 Three-Dimensional Trusses 129

4.4 Assembly of Global Stiffness Matrix for the Banded and Skyline

Solutions 131

4.5 Problem Modeling and Boundary Conditions 134

Problems 139

5.1 Introduction 150

5.2 Finite Element Formulation 153

5.3 Load Vector 158

5.4 Boundary Considerations 159

5.5 Shear Force and Bending Moment 160

5.6 Beams on Elastic Supports 162

5.7 Plane Frames 163

5.8 Three-Dimensional Frames 169

5.9 Problem Modeling and Boundary Conditions 173

5.10 Some Comments 174

Problems 176

6.1 Introduction 188

6.2 Finite Element Modeling 189

6.3 Constant Strain Triangle (CST) 191

6.4 Problem Modeling and Boundary Conditions 212

6.5 Patch Test and Convergence 215

6.6 Orthotropic Materials 216

Problems 225

7.1 Introduction 242

7.2 Axisymmetric Formulation 243

7.3 Finite Element Modeling: Triangular Element 245

7.4 Problem Modeling and Boundary Conditions 256

Problems 263

8.1 Introduction 273

8.2 The Four-Node Quadrilateral 273

8.3 Numerical Integration 279

8.4 Higher Order Elements 286

8.5 Four-Node Quadrilateral for Axisymmetric Problems 294

8.6 Conjugate Gradient Implementation of the Quadrilateral Element 295

8.7 Concluding Remarks and Convergence 295

8.8 References for Convergence 297

Problems 300

9.1 Introduction 312

9.2 Finite Element Formulation 313

9.3 Stress Calculations 317

9.4 Mesh Preparation 318

9.5 Hexahedral Elements and Higher Order Elements 322

9.6 Problem Modeling 324

9.7 Frontal Method for Finite Element Matrices 326

Problems 332

10.1 Introduction 345

10.2 Steady State Heat Transfer 346

10.3 Torsion 370

10.4 Potential Flow, Seepage, Electric and Magnetic Fields,

and Fluid Flow in Ducts 376

10.5 Conclusion 389

Problems 391

11.1 Introduction 408

11.2 Formulation 408

11.3 Element Mass Matrices 411

11.4 Evaluation of Eigenvalues and Eigenvectors 416

11.5 Interfacing with Previous Finite Element Programs and a Program

for Determining Critical Speeds of Shafts 432

11.6 Guyan Reduction 433

11.7 Rigid Body Modes 436

11.8 Conclusion 438

Problems 440

12.1 Introduction 453

12.2 Mesh Generation 453

12.3 Postprocessing 461

12.4 Conclusion 466

Problems 468

APPENDIX 483

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