9780130615916

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 Third Edition has been updated and improved to include new material on additional topics.Chapter topics cover fundamental concepts, matrix algebra and gaussian elimination, one-dimensional problems, trusses, two-dimensional problems using constant strain triangles, axisymmetric solids subjected to axisymmetric loading, two-dimensional isoparametric elements and numerical integration, beams and frames, three-dimensional problems in stress analysis, scalar field problems, dynamic considerations, and preprocessing and postprocessing.For practicing engineers as a valuable learning resource.

Preface

xv

Fundamental Concepts

1

(21)

Introduction

1

(1)

Historical Background

1

(1)

Outline of Presentation

2

(1)

Stresses and Equilibrium

2

(2)

Boundary Conditions

4

(1)

Strain-Displacement Relations

4

(2)

Stress-Strain Relations

6

(2)

Special Cases

7

(1)

Temperature Effects

8

(1)

Potential Energy and Equilibrium; The Rayleigh-Ritz Method

9

(4)

Potential Energy II

9

(2)

Rayleigh-Ritz Method

11

(2)

Galerkin's Method

13

(3)

Saint Venant's Principle

16

(1)

Von Mises Stress

17

(1)

Computer Programs

17

(1)

Conclusion

18

(4)

Historical References

18

(1)

Problems

18

(4)

Matrix Algebra and Gaussian Elimination

22

(23)

Matrix Algebra

22

(7)

Row and Column Vectors

23

(1)

Addition and Subtraction

23

(1)

Multiplication by a Scalar

23

(1)

Matrix Multiplication

23

(1)

Transposition

24

(1)

Differentiation and Integration

24

(1)

Square Matrix

25

(1)

Diagonal Matrix

25

(1)

Identity Matrix

25

(1)

Symmetric Matrix

25

(1)

Upper Triangular Matrix

26

(1)

Determinant of a Matrix

26

(1)

Matrix Inversion

26

(1)

Eigenvalues and Eigenvectors

27

(1)

Positive Definite Matrix

28

(1)

Cholesky Decomposition

29

(1)

Gaussian Elimination

29

(10)

General Algorithm for Gaussian Elimination

30

(3)

Symmetric Matrix

33

(1)

Symmetric Banded Matrices

33

(2)

Solution with Multiple Right Sides

35

(1)

Gaussian Elimination with Column Reduction

36

(2)

Skyline Solution

38

(1)

Frontal Solution

39

(1)

Conjugate Gradient Method for Equation Solving

39

(6)

Conjugate Gradient Algorithm

40

(1)

Problems

41

(2)

Program Listings

43

(2)

One-Dimensional Problems

45

(58)

Introduction

45

(1)

Finite Element Modeling

46

(2)

Element Division

46

(1)

Numbering Scheme

47

(1)

Coordinates and Shape Functions

48

(4)

The Potential-Energy Approach

52

(4)

Element Stiffness Matrix

53

(1)

Force Terms

54

(2)

The Galerkin Approach

56

(2)

Element Stiffness

56

(1)

Force Terms

57

(1)

Assembly of the Global Stiffness Matrix and Load Vector

58

(3)

Properties of K

61

(1)

The Finite Element Equations; Treatment of Boundary Conditions

62

(16)

Types of Boundary Conditions

62

(1)

Elimination Approach

63

(6)

Penalty Approach

69

(5)

Multipoint Constraints

74

(4)

Quadratic Shape Functions

78

(6)

Temperature Effects

84

(19)

Input Data File

88

(1)

Problems

88

(10)

Program Listing

98

(5)

Trusses

103

(27)

Introduction

103

(1)

Plane Trusses

104

(10)

Local and Global Coordinate Systems

104

(1)

Formulas for Calculating l and m

105

(1)

Element Stiffness Matrix

106

(1)

Stress Calculations

107

(4)

Temperature Effects

111

(3)

Three-Dimensional Trusses

114

(2)

Assembly of Global Stiffness Matrix for the Banded and Skyline Solutions

116

(14)

Assembly for Banded Solution

116

(3)

Input Data File

119

(1)

Problems

120

(8)

Program Listing

128

(2)

Two-Dimensional Problems Using Constant Strain Triangles

130

(48)

Introduction

130

(1)

Finite Element Modeling

131

(2)

Constant-Strain Triangle (CST)

133

(19)

Isoparametric Representation

135

(4)

Potential-Energy Approach

139

(1)

Element Stiffness

140

(1)

Force Terms

141

(5)

Galerkin Approach

146

(2)

Stress Calculations

148

(2)

Temperature Effects

150

(2)

Problem Modeling and Boundary Conditions

152

(2)

Some General Comments on Dividing into Elements

154

(1)

Orthotropic Materials

154

(24)

Temperature Effects

157

(3)

Input Data File

160

(2)

Problems

162

(12)

Program Listing

174

(4)

Axisymmetric Solids Subjected to Axisymmetric Loading

178

(30)

Introduction

178

(1)

Axisymmetric Formulation

179

(2)

Finite Element Modeling: Triangular Element

181

(10)

Potential-Energy Approach

183

(1)

Body Force Term

184

(1)

Rotating Flywheel

185

(1)

Surface Traction

185

(2)

Galerkin Approach

187

(3)

Stress Calculations

190

(1)

Temperature Effects

191

(1)

Problem Modeling and Boundary Conditions

191

(17)

Cylinder Subjected to Internal Pressure

191

(1)

Infinite Cylinder

192

(1)

Press Fit on a Rigid Shaft

192

(1)

Press Fit on an Elastic Shaft

193

(1)

Belleville Spring

194

(1)

Thermal Stress Problem

195

(2)

Input Data File

197

(1)

Problems

198

(7)

Program Listing

205

(3)

Two-Dimensional Isoparametric Elements and Numerical Integration

208

(29)

Introduction

208

(1)

The Four-Node Quadrilateral

208

(6)

Shape Functions

208

(3)

Element Stiffness Matrix

211

(2)

Element Force Vectors

213

(1)

Numerical Integration

214

(6)

Two-Dimensional Integrals

217

(1)

Stiffness Integration

217

(1)

Stress Calculations

218

(2)

Higher Order Elements

220

(5)

Nine-Node Quadrilateral

220

(2)

Eight-Node Quadrilateral

222

(1)

Six-Node Triangle

223

(2)

Four-Node Quadrilateral for Axisymmetric Problems

225

(1)

Conjugate Gradient Implementation of the Quadrilateral Element

226

(11)

Concluding Note

227

(1)

Input Data File

228

(2)

Problems

230

(3)

Program Listings

233

(4)

Beams and Frames

237

(38)

Introduction

237

(3)

Potential-Energy Approach

238

(1)

Galerkin Approach

239

(1)

Finite Element Formulation

240

(3)

Load Vector

243

(1)

Boundary Considerations

244

(1)

Shear Force and Bending Moment

245

(2)

Beams on Elastic Supports

247

(1)

Plane Frames

248

(5)

Three-Dimensional Frames

253

(4)

Some Comments

257

(18)

Input Data File

258

(3)

Problems

261

(6)

Program Listings

267

(8)

Three-Dimensional Problems in Stress Analysis

275

(31)

Introduction

275

(1)

Finite Element Formulation

276

(4)

Element Stiffness

279

(1)

Force Terms

280

(1)

Stress Calculations

280

(1)

Mesh Preparation

281

(4)

Hexahedral Elements and Higher Order Elements

285

(2)

Problem Modeling

287

(2)

Frontal Method for Finite Element Matrices

289

(17)

Connectivity and Prefront Routine

290

(1)

Element Assembly and Consideration of Specified dof

290

(1)

Elimination of Completed dof

291

(1)

Backsubstitution

291

(1)

Consideration of Multipoint Constraints

291

(1)

Input Data File

292

(1)

Problems

293

(4)

Program Listings

297

(9)

Scalar Field Problems

306

(61)

Introduction

306

(2)

Steady State Heat Transfer

308

(23)

One-Dimensional Heat Conduction

309

(7)

One-Dimensional Heat Transfer in Thin Fins

316

(4)

Two-Dimensional Steady-State Heat Conduction

320

(9)

Two-Dimensional Fins

329

(1)

Preprocessing for Program Heat2D

330

(1)

Torsion

331

(5)

Triangular Element

332

(1)

Galerkin Approach

333

(3)

Potential Flow, Seepage, Electric and Magnetic Fields, and Fluid Flow in Ducts

336

(12)

Potential Flow

336

(2)

Seepage

338

(1)

Electrical and Magnetic Field Problems

339

(2)

Fluid Flow in Ducts

341

(2)

Acoustics

343

(1)

Boundary Conditions

344

(1)

One-Dimensional Acoustics

344

(1)

1-D Axial Vibrations

345

(3)

Two-Dimensional Acoustics

348

(1)

Conclusion

348

(19)

Input Data File

349

(1)

Problems

350

(11)

Program Listings

361

(6)

Dynamic Considerations

367

(44)

Introduction

367

(1)

Formulation

367

(3)

Solid Body with Distributed Mass

368

(2)

Element Mass Matrices

370

(5)

Evaluation of Eigenvalues and Eigenvectors

375

(16)

Properties of Eigenvectors

376

(1)

Eigenvalue-Eigenvector Evaluation

376

(6)

Generalized Jacobi Method

382

(4)

Tridiagonalization and Implicit Shift Approach

386

(1)

Bringing Generalized Problem to Standard Form

386

(1)

Tridiagonalization

387

(3)

Implicit Symmetric QR Step with Wilkinson Shift for Diagonalization

390

(1)

Interfacing with Previous Finite Element Programs and a Program for Determining Critical Speeds of Shafts

391

(1)

Guyan Reduction

392

(2)

Rigid Body Modes

394

(2)

Conclusion

396

(15)

Input Data File

397

(2)

Problems

399

(5)

Program Listings

404

(7)

Preprocessing and Postprocessing

411

(29)

Introduction

411

(1)

Mesh Generation

411

(8)

Region and Block Representation

411

(1)

Block Corner Nodes, Sides, and Subdivisions

412

(7)

Postprocessing

419

(5)

Deformed Configuration and Mode Shape

419

(1)

Contour Plotting

420

(1)

Nodal Values from Known Constant Element Values for a Triangle

421

(2)

Least Squares Fit for a Four-Noded Quadrilateral

423

(1)

Conclusion

424

(16)

Input Data File

425

(1)

Problems

425

(2)

Program Listings

427

(13)

Appendix Proof of dA = det J dξ dη

440

(3)

Bibliography

443

(4)

Answers to Selected Problems

447

(2)

Index

449

What is included with this book?

The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any access cards, study guides, lab manuals, CDs, etc.

The Used, Rental and eBook copies of this book are not guaranteed to include any supplemental materials. Typically, only the book itself is included. This is true even if the title states it includes any access cards, study guides, lab manuals, CDs, etc.

The first edition of this book appeared over 10 years ago and the second edition followed a few years later. We received positive feedback from professors who taught from the book and from students and practicing engineers who used the book. We also benefited from the feedback received from the students in our courses for the past 20 years. We have incorporated several suggestions in this edition. The underlying philosophy of the book is to provide a clear presentation of theory, modeling, and implementation into computer programs. The pedagogy of earlier editions has been retained in this edition.New material has been introduced in several chapters. Worked examples and exercise problems have been added to supplement the learning process. Exercise problems stress both fundamental understanding and practical considerations. Theory and computer programs have been added to cover acoustics, axisymmetric quadrilateral elements, conjugate gradient approach, and eigenvalue evaluation. Three additional programs have now been introduced in this edition. All the programs have been developed to work in the Windows environment. The programs have a common structure that should enable the users to follow the development easily. The programs have been provided in Visual Basic, Microsoft Excel/Visual Basic, MATLAB, together with those provided earlier in QBASIC, FORTRAN and C. The Solutions Manual has also been updated.Chapter 1 gives a brief historical background and develops the fundamental concepts. Equations of equilibrium, stress-strain relations, strain-displacement relations, and the principles of potential energy are reviewed. The concept of Galerkin's method is introduced.Properties of matrices and determinants are reviewed in Chapter 2. The Gaussian elimination method is presented, and its relationship to the solution of symmetric banded matrix equations and the skyline solution is discussed. Cholesky decomposition and conjugate gradient method are discussed.Chapter 3 develops the key concepts of finite element formulation by considering one-dimensional problems. The steps include development of shape functions, derivation of element stiffness, formation of global stiffness, treatment of boundary conditions, solution of equations, and stress calculations. Both the potential energy approach and Galerkin's formulations are presented. Consideration of temperature effects is included.Finite element formulation for plane and three-dimensional trusses is developed in Chapter 4. The assembly of global stiffness in banded and skyline forms is explained. Computer programs for both banded and skyline solutions are given.Chapter 5 introduces the finite element formulation for two-dimensional plane stress and plane strain problems using constant strain triangle (CST) elements. Problem modeling and treatment of boundary conditions are presented in detail. Formulation for orthotropic materials is provided. Chapter 6 treats the modeling aspects of axisymmetric solids subjected to axisymmetric loading. Formulation using triangular elements is presented. Several real-world problems are included in this chapter.Chapter 7 introduces the concepts of isoparametric quadrilateral and higher order elements and numerical integration using Gaussian quadrature. Formulation for axisymmetric quadrilateral element and implementation of conjugate gradient method for quadrilateral element are given.Beams and application of Hermite shape functions are presented in Chapter 8. The chapter covers two-dimensional and three-dimensional frames.Chapter 9 presents three-dimensional stress analysis. Tetrahedral and hexahedral elements are presented. The frontal method and its implementation aspects are discussed.Scalar field problems are treated in detail in Chapter 10. While Galerkin as well as energy approaches have been used in every chapter, with equal importance, only Galerkin's approach is used i

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Introduction to Finite Elements in Engineering

Introduction to Finite Elements in Engineering

0130615919

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