Introduction to Finite Engineeringis ideal for senior undergraduate and first-year graduate students and also as a learning resource to practicing engineers. This book provides an integrated approach to finite element methodologies. The development of finite element theory is combined with examples and exercises involving engineering applications. The steps used in the development of the theory are implemented in complete, self-contained computer programs. While the strategy and philosophy of the previous editions has been retained, the¿Fourth Edition has been updated and improved to include new material on additional topics.

** **PREFACE XIII

ABOUT THE AUTHOR XVI

** **

1 FUNDAMENTAL CONCEPTS 1

** **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

* *Special Cases, 7

1.8 Temperature Effects 8

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

* **Potential Energy *ß *, 9*

* *Rayleigh—Ritz Method, 12

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Historical References 20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 MATRIX ALGEBRA AND GAUSSIAN ELIMINATION 28

* * 2.1 Matrix Algebra 28

* *Row and Column Vectors, 29

Addition and Subtraction, 29

Multiplication by a Scalar, 29

Matrix Multiplication, 29

Transposition, 30

Differentiation and Integration, 30

Square Matrix, 31

Diagonal Matrix, 31

Identity Matrix, 31

Symmetric Matrix, 32

Upper Triangular Matrix, 32

Determinant of a Matrix, 32

Matrix Inversion, 32

Eigenvalues and Eigenvectors, 33

Positive Definite Matrix, 35

Cholesky Decomposition, 35

2.2 Gaussian Elimination 35

* *General Algorithm for Gaussian Elimination, 37

Symmetric Matrix, 40

Symmetric Banded Matrices, 40

Solution with Multiple Right Sides, 40

Gaussian Elimination with Column Reduction, 42

Skyline Solution, 44

Frontal Solution, 45

2.3 Conjugate Gradient Method for Equation Solving 45

* *Conjugate Gradient Algorithm, 46

Input Data/Output 46

Problems 47

* *Program Listings, 49

** **

3 ONE-DIMENSIONAL PROBLEMS 51

3.1 Introduction 51

3.2 Finite Element Modeling 52

* *Element Division, 52

Numbering Scheme, 53

3.3 Shape Functions and Local Coordinates 55

3.4 The Potential-Energy Approach 59

* *Element Stiffness Matrix, 60

Force Terms, 62

3.5 The Galerkin Approach 64

* *Element Stiffness, 64

Force Terms, 65

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

* *Types of Boundary Conditions, 70

Elimination Approach, 71

Penalty Approach, 76

Multipoint Constraints, 82

3.9 Quadratic Shape Functions 85

3.10 Temperature Effects 92

3.11 Problem Modeling and Boundary Conditions 96

* *Problem in Equilibrium, 96

Symmetry, 97

Two Elements with Same End Displacements, 97

Problem with a Closing Gap, 98

Input Data/Output, 98

Problems 99

* *Program Listing, 111

** **

4 TRUSSES 117

4.1 Introduction 117

4.2 Plane Trusses 118

* *Local and Global Coordinate Systems, 118

*Formulas for Calculating */ *and m, 119*

* *Element Stiffness Matrix, 120

Stress Calculations, 121

Temperature Effects, 126

4.3 Three-Dimensional Trusses 129

4.4 Assembly of Global Stiffness Matrix for the Banded and Skyline

Solutions 131

* *Assembly for Banded Solution, 131

Skyline Assembly , 132

4.5 Problem Modeling and Boundary Conditions 134

* *Inclined Support in Two Dimensions, 134

Inclined Support in Three Dimensions–Line Constraint, 134

Inclined Support in Three Dimensions–Plane Constraint, 135

Symmetry and Antisymmetry , 136

Input Data/Output, 138

Problems 139

* *Program Listing, 147

** **

5 BEAMS AND FRAMES 150

5.1 Introduction 150

* *Potential-Energy Approach, 151

Galerkin Approach, 152

5.2 Finite Element Formulation 153

* *Element Stiffness–Direct Approach, 157

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

* *Input Data/Output, 174

Problems 176

* *Program Listings, 183

** **

6 TWO-DIMENSIONAL PROBLEMS USING CONSTANT STRAIN TRIANGLES 188

6.1 Introduction 188

6.2 Finite Element Modeling 189

6.3 Constant Strain Triangle (CST) 191

* *Isoparametric Representation, 192

Potential-Energy Approach, 198

Element Stiffness, 198

Force Terms, 199

Integration Formula on a Triangle, 206

Galerkin Approach, 206

Stress Calculations, 208

Temperature Effects, 210

6.4 Problem Modeling and Boundary Conditions 212

* *Some General Comments on Dividing into Elements, 215

6.5 Patch Test and Convergence 215

* *Patch Test, 215

6.6 Orthotropic Materials 216

* *Temperature Effects, 220

Input Data/Output, 222

Problems 225

* *Program Listing, 238

** **

7 AXISYMMETRIC SOLIDS SUBJECTED TO AXISYMMETRIC LOADING 242

7.1 Introduction 242

7.2 Axisymmetric Formulation 243

7.3 Finite Element Modeling: Triangular Element 245

* *Potential-Energy Approach, 248

Body Force Term, 249

Rotating Flywheel, 249

Surface Traction, 250

Galerkin Approach, 252

Stress Calculations, 255

Temperature Effects, 256

7.4 Problem Modeling and Boundary Conditions 256

* *Cylinder Subjected to Internal Pressure, 256

Infinite Cylinder, 257

Press Fit on a Rigid Shaft, 257

Press Fit on an Elastic Shaft, 258

Belleville Spring, 259

Thermal Stress Problem, 260

Input Data/Output, 262

Problems 263

* *Program Listing, 271

** **

8 TWO-DIMENSIONAL ISOPARAMETRIC ELEMENTS

AND NUMERICAL INTEGRATION 273

8.1 Introduction 273

8.2 The Four-Node Quadrilateral 273

* *Shape Functions, 273

Element Stiffness Matrix, 276

Element Force Vectors, 279

8.3 Numerical Integration 279

* *Two-Dimensional Integrals, 283

Stiffness Integration, 283

Stress Calculations, 284

8.4 Higher Order Elements 286

* *Nine-Node Quadrilateral, 287

Eight-Node Quadrilateral, 289

Six-Node Triangle, 290

Integration on a Triangle–Symmetric Points, 291

Integration on a Triangle–Degenerate Quadrilateral, 292

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

* *Input Data/Output, 298

Problems 300

* *Program Listings, 308

** **

9 THREE-DIMENSIONAL PROBLEMS IN STRESS ANALYSIS 312

9.1 Introduction 312

9.2 Finite Element Formulation 313

* *Element Stiffness, 316

Force Terms, 317

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

* *Connectivity and Prefront Routine, 327

Element Assembly and Consideration of Specified dof, 328

Elimination of Completed dof, 328

Backsubstitution, 329

Consideration of Multipoint Constraints, 329

Input Data/Output, 330

Problems 332

* *Program Listings, 336

** **

10 SCALAR FIELD PROBLEMS 345

10.1 Introduction 345

10.2 Steady State Heat Transfer 346

* *One-Dimensional Heat Conduction, 347

One-Dimensional Heat Transfer in Thin Fins, 355

Two-Dimensional Steady-State Heat Conduction, 359

Two-Dimensional Fins, 369

Preprocessing for Program Heat2D, 370

10.3 Torsion 370

* *Triangular Element, 372

Galerkin Approach, 373

10.4 Potential Flow, Seepage, Electric and Magnetic Fields,

and Fluid Flow in Ducts 376

* *Potential Flow, 376

Seepage, 378

Electrical and Magnetic Field Problems, 379

Fluid Flow in Ducts, 381

Acoustics, 383

Boundary Conditions, 384

One-Dimensional Acoustics, 384

One-Dimensional Axial Vibrations, 386

Two-Dimensional Acoustics, 388

10.5 Conclusion *389*

* *Input Data/Output, 389

Problems 391

* *Program Listings, 402

** **

11 DYNAMIC CONSIDERATIONS 408

11.1 Introduction 408

11.2 Formulation 408

* *Solid Body with Distributed Mass, 409

11.3 Element Mass Matrices 411

11.4 Evaluation of Eigenvalues and Eigenvectors 416

* *Properties of Eigenvectors, 417

Eigenvalue—Eigenvector Evaluation, 417

Inverse Iteration Method , 420

Generalized Jacobi Method, 423

Tridiagonalization and Implicit Shift Approach, 427

Bringing Generalized Problem to Standard Form, 427

Tridiagonalization, 428

Implicit Symmetric QR Step with Wilkinson Shift

for Diagonalization, 431

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

* *Input Data/Output, 438

Problems 440

* *Program Listings, 446

** **

12 PREPROCESSING AND POSTPROCESSING 453

12.1 Introduction 453

12.2 Mesh Generation 453

* *Region and Block Representation, 453

Block Corner Nodes, Sides, and Subdivisions, 454

12.3 Postprocessing 461

* *Deformed Configuration and Mode Shape, 461

Contour Plotting, 462

Nodal Values from Known Constant Element Values

for a Triangle, 463

Least-Squares Fit for a Four-Noded Quadrilateral, 465

12.4 Conclusion 466

* *Input Data/Output, 467

Problems 468

* *Program Listings, 470

** **

**APPENDIX ** ** ***483*

* *BIBLIOGRAPHY 486

ANSWERS TO SELECTED PROBLEMS 490

INDEX 492

* *