The finite element method in engineering

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The Finite Element

Method in Engineering

Fifth Edition

Singiresu S Rao

Professor and Chairman

Department of Mechanical and Aerospace Engineering

University of Miami, Coral Gables, Florida, USA

Щ Ф М Ш

ELSEVIER

AMSTERDAM • BOSTON • HEIDELBERG • LONDON

NEW YOR • OXFORD • PARIS • SAN DIEGO

SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO

Butterworth H einemann is an imprint of Elsevier



PREFACE xiii

PART 1 • Introduction

CHAPTER 1

Overview of Finite Elem ent Me thod 3

1.1 Basic Concept 3

1.2 Historical Background 4

1.3 General App licabi l i ty of the Me thod 7

1.4 Engineering Appl icat ions of the Finite Element Metho d 9

1.5 General De scription of the Finite Eleme nt Me thod 9

1.6 One-Dimensional Problems wi th Linear Interpolat ion Model 12

1.7 One-Dimensional Problems wi th Cubic Interpolat ion Model 24

1.8 Deriva tion of Finite Elem ent Equa tions Using a Direct Approa ch 28

1.9 Com mercial Finite Element Program Packages 40

1.10 Solut ions Using Finite Element Software 40

PART 2 • Basic Procedure

CHAPTER 2

Discret izat ion of the Doma in 53

2 .1 Introduct ion 53

2.2 Basic Element Shapes 53

2.3 Discret izat ion Process 56

2.4 Node Number ing Scheme 63

2.5 Autom atic Mesh Generat ion 65

CHAPTER 3

Interpolat ion Mod els 75

3 .1 Introduct ion 75

3.2 Polynom ial Form of Interp olation Functions 77

3.3 Simplex, Com plex, and Mul t ip lex Elements 78

3.4 Interpolat ion Polynomial in Terms of Nodal Degrees of Freedom 78

3.5 Select ion of the Order of the Interpolat ion Polynomial 80

3.6 Convergence Requ irements 82

3.7 Linear Interpolat ion Polynomials in Terms of Global Coordinates 85

3.8 Interpolat ion Polynomials for Vector Quan t i t ies 96

3.9 Linear Interpolat ion Polynomials in Terms of Local Coordinates 99

3.1 0 Integrat ion of Funct ions of Natural Coordinates 10 8

3. 11 Patch Tes t 10 9

CHAPTER 4

Higher Order and Isoparametr ic Elements 11 9

4 .1 In t roduct ion 12 0

4.2 Higher Order One-Dimensional Elements 12 0

4.3 Higher Order Elements in Terms of Natural Coordinates 1 2 1

4. 4 Higher Order Elemen ts in Terms of Classical Interpolat ion

Polynomials 130



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C O N T E N T S

4. 5 One-Dimensional Elements Using Classical Interpolation Polynom ials.... 13 4

4.6 Two-Dimensional (Rectangular) Elements Using Classical

Interpolation Polynomials 135

4.7 Cont inui ty Condi t ions 137

4.8 Comparat ive Study of Elements 139

4.9 Isoparametr ic Elements

140

4. 10 Numerical Integration 148

CHAPTER

Derivation of Element Matrices and Vectors 157

5.1 Introduct ion 158

5.2 Variational Approach 158

5.3 Solution of Equil ibrium Problems U sing Variation al (Rayleigh-Ritz)

Method 163

5.4 Solution of Eigenvalue Problems U sing Variationa l (Rayleigh-Ritz)

Method 167

5.5 Solution of Propagation Problem s Using Variation al (Rayleigh-Ritz)

Method 168

5. 6 Equivalence of Finite Elem ent and Varia tional (Rayleigh-Ritz)

Methods 169

5.7 Derivation of Finite Element Equations Using Variation al

(Rayleigh-Ritz) Approach 169

5.8 We ighted Residual Approach 175

5.9 Solution of Eigenvalue Problems Using Weighted Residual Meth od 182

5.10 Solut ion of Propagat ion Problems Using Weighted Residual M eth od .. . .1 83

5 .1 1 Derivation of Finite Element Equations Using We ighted R esidual

(Galerkin) Approach 184

5.1 2 Derivation of Finite Eleme nt Equa tions Using We ighted R esidual

(Least Squares) Approach 187

5.13 Strong and Weak Form Formulat ions

189

CHAPTER 6 Assem bly of Element Matrice s and Vecto rs and Derivation of

System Equat ions 199

6. 1 Coordinate Transformation 199

6.2 Assemblage of Element Equations 204

6.3 Incorpora tion of Boundary Con ditions 211

6.4 Penalty Me thod 219

6.5 Mul t ipoint Constraints—P enal ty Metho d 223

6.6 Symmetry Condi t ions—Pena l ty Method 226

6.7 Rigid Eleme nts 228

CHAPTER

7

Num erical Solution of Finite Elemen t Equations

241


7.2 Solution of Equil ibrium Problems 242

7.3 Solution of Eigenvalue Problem s 251

7.4 Solution of Propagation Problems 262

7.5 Paral lel Proces sing in Finite Elemen t Analysis 268

PART 3

•

Application to Solid Mechanics Problems

CHAPTER 8 Basic Equations and Solution Procedure 277


8.2 Basic Equations of Solid Me chan ics

277



8.3 Formu lat ions of Sol id and Structural Mech anics 29 4

8. 4 Form ulation of Finite Eleme nt Equa tions (Static Analysis) 29 9

8.5 Nature of Finite Element Solut ions 30 3

CHAPTER

9 Analys is of Trus ses, Beam s, and Frames 3 1 1

9. 1 In t roduc t ion 3 1 1

9.2 Space Truss Element 31 2

9.3 Beam Element 32 3

9.4 Space Frame Element 32 8

9.5 Cha racteris t ics of St i f fness Matr ices 33 8

CHAPTER 10

Analys is of Plates 3 55

1 0 .1 In t roduc t ion 35 5

10 .2 Tr iangu lar Membrane E lement 35 6

10 .3 Numer ica l Resu l ts wi th Membrane E lement 36 7

10 .4 Quadrat ic Triangle Element 3 69

10 .5 Rectangular Plate Element (In-plane Forces) 37 2

10 .6 Bending Behavior of Plates 3 76

10 .7 Finite Element Analys is of Plates in Bending 3 79

10 .8 Triangular Plate Bending Element 3 79

10 .9 Num erical Resul ts wi th Bending Elements 3 83

10 .10 Analysis o f Three-Dimens ional S t ruc tures Us ing P la te E lem en ts . . . .3 86

CHAPTER 1 1

Analys is of Three-Dimen sional Problems 4 0 1

11 .1 In troduct ion 4 0 1

11. 2 Tet rahedron E lement 4 0 1

11 .3 Hexahedron E lement 40 9

11 .4 Analys is of Sol ids of Revolut ion 4 13

CHAPTER

1 2 Dynamic Analys is 42 7

1 2. 1 Dynamic Equat ions o f Mot ion 42 7

12. 2 Cons is tent and Lumped Mass Matr i ces 43 0

12 .3 Con sistent Ma ss Matr ices in a Global Coordinate System 4 39

12 .4 Free Vibrat ion Analys is 4 4 0

12 .5 Dynamic Response Using Finite Element Method 45 2

12 .6 Noncon servat ive Stabi l i ty and Flutter Problems 46 0

12.7 Subs t ruc tures Method 4 6 1

PART 4 • Application to Heat Transfer Problems

CHAPTER

13 Formulat ion and Solut ion Procedure 4 73

1 3. 1 In t roduc t ion 47 3

13 .2 Basic Equat ions of Heat Transfer 4 73

13 .3 Governing Equat ion for Three-Dimensional Bodies 47 5

13 .4 Sta teme nt o f the Prob lem 47 9

13 .5 Derivat ion of Finite Element Equat ions 48 0

CHAPTER 14

One-Dim ensional Problems 4 89

1 4 .1 In t roduc tion 48 9

14 .2 Straight Uni form Fin Analys is 4 89

14 .3 Convect ion Loss from End Surface of Fin 49 2

14 .3 Tapered Fin Analys is 49 6

14 .4 Analys is of Uni form Fins Using Quadrat ic Elements 4 9 9



14. 5 Unsteady State Problems 50 2

14 .6 Heat Transfer Problems wi th Radiat ion 50 7

CHAPTER 15

Two-Dimensional Problems 51 7

15 .1 In troduc tion 51 7

15.2 So lu tion 51 7

15 .3 Unsteady State Problems 5 26

CHAPTER 16

Three-Dimensional Problems 5 3 1

16 .1 In troduc tion 5 3 1

16 .2 Axisymmetric Problems 5 3 1

16 .3 Three-Dimensional Heat Transfer Problems 53 6

16 .4 Unsteady Sta te Prob lems 5 4 1

PART 5 • Application to Fluid Me chan ics Problems

CHAPTER 17

Basic Equations of Fluid Me chan ics 5 4 9

17 .1 In troduc tion 54 9

17 .2 Basic Characteris t ics of Fluids 5 49

17 .3 Metho ds of Describing the Mot ion of a Fluid 55 0

17 .4 Cont inui ty Equat ion 5 5 1

17 .5 Equat ions of Mot ion or Mo me ntum Equat ions 55 2

17 .6 Energy, State, and Viscosi ty Equat ions 55 7

17 .7 Solut ion Procedure 55 7

17 .8 Inviscid Fluid Flow 55 9

17 .9 Irrotational Flow 5 6 0

17 .10 Velocity Potent ial 5 6 1

17 .1 1 St ream Funct ion 56 2

17 .12 Bernoul li Equat ion 5 6 4

CHAPTER 18

Inv isc id and Incompressible Flows 5 7 1


18 .2 Potent ial Function Formulat ion 57 3

18 .3 Finite Element Solut ion Using the Galerk in Approach 57 3

18 .4 Stream Funct ion Formulat ion 5 8 4

CHAPTER 19

V iscous and Non-Newtonian Flows 5 9 1


19 .2 Stream Funct ion Formulat ion (Using Variat ional Approach) 59 2

19 .3 Veloci ty-Pressure Formulat ion (Using Galerk in Approach) 59 6

19 .4 Solut ion of Navier-S tokes Equat ions 59 8

19 .5 Stream Funct ion-Vort ic ity Formulat ion 60 0

19 .6 Flow of Non-Newtonian Fluids 6 02

19.7 Other Developments 60 7

PART 6 • Solution and Applications of

Quasi-Harmonic Equations

CHAPTER 20

Solut ion of Quasi-Harmonic Equat ions 61 3

20 .1 In t roduc t ion 61 3

20 .2 Finite Element Equat ions for Steady-State Problems 61 5

20 .3 Solut ion of Poisson s Equat ion 61 5

20 .4 Transient Field Problems 6 22



PART 7 • ABAQUS and ANSYS Softw are and MATLAB® Programs

for Finite Element Analysis

CHAPTER 2 1

Finite Element Analys is Using ABAQUS 6 3 1

21 .1 In troduct ion 6 3 1

21 .2 Examples 63 2

CHAPTER 22

Finite Element Analys is Using ANSYS 66 3

2 2 .1 In t roduc tion 66 3

22 .2 GUI Layout in ANSYS 6 6 4

22 .3 Termino logy 66 4

22 .4 Finite Element Discret izat ion 66 5

22 .5 System of Uni ts 66 7

22 .6 Stages in So lu t ion 66 7

CHAPTER 23

MATLAB Program s for Finite Elem ent Ana lysis 68 3

2 3 .1 Solut ion of Linear System of Equat ions Using Choleski Method 6 84

23 .2 Incorporat ion of Boundary Cond i t ions 68 6

23 .3 Analys is of Space Trusses 68 7

23 .4 Analys is of Plates Subjected to In-plane Loads

Us ing CST E lements 6 9 1

23 .5 Ana lys is of Three-Dimensional Structures Using CST Elements 6 94

23 .6 Tem perature Distr ibut ion in One-Dimensional Fins 69 7

23 .7 Tem perature D istr ibut ion in One-Dimensional Fins Inc luding

Radiat ion Heat Transfer 69 8

23 .8 Two-D imensional Heat Transfer Analys is 69 9

2 3 .9 Con fined Fluid Flow arou nd a Cylinder Using Pote ntial

Funct ion Approach 7 0 1

23 .1 0 Tors ion Analys is of Shafts 70 2

Append ix: Green-Gauss Theorem Integ ration by Parts in Two

and Three Dimensions)

705

Index

707

The finite element method in engineering

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