Introduction to Finit Elements in Engineering

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Historical References 20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

Resources
Show resources for