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Lecture Notes in Engineering Edited by C. A. Brebbia and S. A. Orszag

14

A.A. Bakr

The Boundary Integral Equation Method in Axisymmetric Stress Analysis Problems

Spri nger-Verlag Berlin Heidelberq New York Tokyo

Series Editors C. A Brebbia . S. A Orszag

Consulting Editors J. Argyris . K.-J. Bathe' A S. Cakmak' J. Connor' R. McCrory C. S. Desai' K.-P. Holz . F. A Leckie' G. Pinder' A R. S. Pont J. H. Seinfeld . P. Silvester' P. Spanos' W. Wunderlich' S. Yip

Authors Bakr, AA Department of Mechanical and Computer Aided Engineering North Staffordshire Polytechnic Beaconside Stafford ST 18 OAD UK

ISBN-13:978-3-540-16030-4 001: 10.1007/978-3-642-82644-3

e-ISBN-13:978-3-642-82644-3

Library of Congress Cataloging in Publication Data

Bakr, A. A. The boundary integral equation method in axisymmetric stress analysis problems. (Lecture notes in engineering; 14) Bibliography: p. 1. Strains and stresses. 2. Boundary value problems. 3. Integral equations. I. Title. II. Series. TA417.6.B35 1985 620.1'123 85-27641 ISBN-13: 978-3-540-16030-4 (U.S.)

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.

© Springer-Verlag Berlin, Heidelberg 1986

2061/3020-543210

FOREWORD

The Boundary Integral Equation (BIE) or the Boundary Element Method

is now well established as an efficient and accurate numerical technique

for engineering problems. This book presents the application of this

technique to axisymmetric engineering problems, where the geometry and

applied loads are symmetrical about an axis of rotation. Emphasis is

placed on using isoparametric quadratic elements which exhibit excellent

modelling capabilities. Efficient numerical integration schemes are

also presented in detail.

Unlike the Finite Element Method (FEM), the BIE adaptation to

axisymmetric problems is not a straightforward modification of the two­

or three-dimensional formulations. Two approaches can be used; either

a purely axisymmetric approach based on assuming a ring of load, or,

alternatively, integrating the three-dimensional fundamental solution

of a point load around the axis of rotational symmetry. Throughout

this ~ook, both approaches are used and are shown to arrive at identi­

cal solutions.

The book starts with axisymmetric potential problems and extends

the formulation to elasticity, thermoelasticity, centrifugal and fracture

mechanics problems. The accuracy of the formulation is demonstrated

by solving several practical engineering problems and comparing the BIE

solution to analytical or other numerical methods such as the FEM. This

book provides a foundation for further research into axisymmetric prob­

lems, such as elastoplasticity, contact, time-dependent and creep prob­

lems.

I wish to express my sincere gratitude to Dr R.T. Fenner for his

constant guidance, encouragement and excellent advice throughout the

course of this work. I would also like to thank my colleagues; Drs

K.H. Lee and E.M. Remzi for their valuable discussions on the BIE method,

and Dr M.J. Abdul-Mihsein for his collaboration on Chapters 5 and 6.

Thanks are also due to Mrs E.A. Hall for her skilful and accurate typing

of this manuscript. Finally, I am indebted to my wife, Jane, for her

patience and understanding throughout this work.

Stafford, England, December 1985 A.A. Bakr

TABLE OF CONTENTS

NOTATION

CHAPTER 1 1.1

1.2

1.3

CHAPTER 2 2.1

2.2

2.3

2.4

CHAPTER 3 3.1

3.2

3.3

INTRODUCTION AND AIMS

Introduction Literature Survey - Axisymmetric Problems Layout of Notes

AXISYMMETRIC POTENTIAL PROBLEMS

Introduction Analytical Formulation 2.2.1 The axisymmetric fundamental solution 2.2.2 The boundary integral identity

2.2.3 The axisymmetric potential kernels 2.2.4 Treatment of the axis of rotational

symmetry Numerical Implementation

2.3.1 Isoparametric quadratic elements 2.3.2 2.3.3 2.3.4

Numerical integration of the kernels Calculation of the elliptic integrals

Solutions at internal points 2.3.5 Treatment of non-homogeneous problems Examples 2.4.1 Hollow cylinder

2.4.2 Hollow sphere 2.4.3

2.4.4 2.4.5

2.4.6

Effect of element curvature Compound sphere

Reactor pressure vessel Externally grooved hollow cylinder

AXISYMMETRIC ELASTICITY PROBLEMS: FORMULATION Introduction Analytical Formulation

3.2.1 Basic equations of elasticity

3.2.2 Solution of the Navier equations 3.2.3 The boundary integral identity

3.2.4 Treatment of the axis of rotational symmetry

3.2.5 Treatment of non-homogeneous problems

Numerical Implementation

3.3.1 Isoparametric quadratic elements

1

1

3

3

6

6

7

7

10

12

14

14

15 17

20

21

22

22

23

23 24

25 26

26

39

39

40

40

41

47

49

50

50 51

CHAPTER 4

4.1

4.2

3.3.2

3.3.3

3.3.4

v

Numerical integration of the kernels

Surface stresses

Solutions at internal pOints

AXISYMMETRIC ELASTICITY PROBLEMS: EXAMPLES

Introduction

Hollow Cylinder

4.3 Hollow Sphere

4.4 Thin Sections

4.5 Compound Sphere

4.6 Spherical Cavity in a Solid Cylinder

4.7 Notched Bars

4.8 Pressure Vessel with Hemispherical End Closure

4.9 Pressure Vessel Clamp

4.10 Compression of Rubber Blocks

4.11 Externally Grooved Hollow Cylinder

4.12 Plain Reducing Socket

CHAPTER 5 AXISYMMETRIC THERMOELASTICITY PROBLEMS

5.1 Introduction

5.2 Analytical Formulation

5.3 Numerical Implementation

5.4

CHAPTER 6

6.1

6.2

6.3

6.4

5.3.1

5.3.2

Isoparametric quadratic elements

Numerical integration of the kernels

5.3.3 Solutions at internal points

Examples

5.4.1

5.4.2

5.4.3

5.4.4

5.4.5

5.4.6

Hollow cylinder

Hollow sphere

Compound sphere

Comparison with other numerical methods

Reactor pressure vessel

Externally grooved hollow cylinder

AXISYMMETRIC CENTRIFUGAL LOADING PROBLEMS

Introduction

Analytical Formulation

Numerical Implementation

6.3.1 Isoparametric quadratic elements

6.3.2 Numerical integration of the kernels

Examples

6.4.1 Rotating disk of uniform thickness

6.4.2 Rotating tapered disk

52

53

55

57

57

58

59

60

61

61

62

63

63

64

65

65

99

99

99

105

105

106

106

107

107

108

109

110

111

111

120

120

120

124

124

124

125 125

125

CHAPTER 7

7.1

7.2

7.3

7.4

7.5

CHAPTER 8

REFERENCES

VI

6.4.3 Rotating disk of variable thickness

AXISYMMETRIC FRACTURE MECHANICS PROBLEMS

Introduction

Linear Elastic Fracture Mechanics

Numerical Calculation of the Stress Intensity Factor

7.3.1

7.3.2

7.3.3

The displacement method

The stress method

Energy methods

Singularity Elements

Examples

7.5.1

7.5.2

7.5.3

7.5.4

7.5.5

7.5.6

7.5.7

Circumferential crack in a round bar

Penny-shaped crack in a round bar

Internal circumferential crack in a hollow cylinder

Flat toroidal crack in a hollow cylinder

Pressurised penny-shaped crack in a solid sphere

Circumferential cracks in grooved round bars

Modelling both faces of the crack

CONCLUSIONS

APPENDIX A LIMITING PROCESS FOR THE TERM C(P)

APPENDIX B NUMERICAL COEFFICIENTS FOR THE EVALUATION OF THE ELLIPTICAL INTEGRALS

APPENDIX C NOTATION FOR AXISYMMETRIC VECTOR AND SCALAR DIFFERENTIATION

APPENDIX D COMPONENTS OF THE TRACTION KERNELS

APPENDIX E DERIVATION OF THE AXISYMMETRIC DISPLACEMENT KERNELS FROM THE THREE-DIMENSIONAL FUNDAMENTAL SOLUTION

APPENDIX F THE DIAGONAL TERMS OF MATRIX [A]

APPENDIX G DIFFERENTIALS OF THE DISPLACEMENT AND TRACTION KERNELS

APPENDIX H THE THERMOELASTIC KERNELS

APPENDIX I DIFFERENTIALS OF THE THERMOELASTIC KERNELS

126

133

133

134

136

137

138

138

140

142

142

144

146

146

147

148

149

176

181

188

190

191

192

194

197

200

208

209

NOTATION

A

A

[ A]

AItIt ' a.i

[B]

B It It ' bi

C

[ C]

c. . .(.,

Altz ' Azlt '

Bltz ' Bzlt '

[V]

d[m,c.)

E

[ E]

[E' ]

~It ' ~z eltlt e zz ' eee eltz F [F]

Azz

B zz

area in a radial plane through the axis of rota­tional symmetry

surface area of a crack

matrix containing the integrals of the traction kernels

coefficients of the sub-matrices of the matrix [A]

coefficients used to determine the elliptic inte­grals, i = 1,5

matrix containing the integrals of the displace­ment kernels

coefficients of the sub-matrices of the matrix [B]

coefficients used to determine the elliptic inte­grals, i = 1,5

parameter contributing to the leading diagonal terms of the matrix [A] in the potential problem

solution matrix multiplying the unknown variables

coefficients used to determine the elliptic inte­grals, i = 1,5

parameter contributing to the leading diagonal terms of the matrix [A] in the elasticity problem

matrix multiplying the known variables

number assigned to the c.th node of the mth ele­ment

coefficients used to determine the elliptic inte­grals, i = 1,5

Young's modulus

matrix containing the known coefficients to be solved in the potential and elasticity problems

matrix containing the known coefficients to be solved in the thermoelasticity problem

complete elliptic integral of the second kind of modulus m

percentage compression of a rubber block

unit vectors in the radial and axial directions

strains in the radial, axial and hoop directions

shear strain

body force vector

matrix containing the integrals of the thermo­elastic kernels multiplying the temperatures

components of the body force vector in the radial and axial directions

function to be integrated using the ordinary Gaussian quadrature technique

modified function to be integrated using the logarithmic Gaussian quadrature technique

G

G

[G]

[G ']

GJt ' Gz

GI GIl ' GIll

H

He.

Hn h

IlL' I z

J

J

I n

J Jt ' Jz

1f Klm'I)

Kl

KZ

5.1 KI • KIl ' KIll

Ke.l Ke.Z KILl KJtZ

Kzl KzZ

k

M

m

mllL,m lz

Ne. !!. nIL ' nz

VIII

total number of Gaussian quadrature points

Galerkin vector

matrix containing the integrals of the thermo­elastic kernels multiplying the temperature grad­ients

matrix containing the known coefficients to be solved in the centrifugal problem components of the Galerkin vector in the radial and axial directions

strain energy release rate for fracture modes I, II and III

height of a cylinder

functions remaining non-zero over the range of integration

Hankel transform of order n

ratio between the heat transfer coefficient to the thermal conductivity

integrals of the thermoelastic kernels in the radial and axial directions

Jacobian of transformation

J-contour integral

Bessel function of order n components of the Jacobian of transformation in the radial and axial directions

complete elliptic integral of the first kind of modulus m

first potential kernel multiplying the potential gradient .

second potential kernel multiplying the potential gradient

normalised stress intensity factor

stress intensity factors for fracture modes I, II and III axisymmetric centrifugal kernels

axisymmetric thermoelastic direction

axisymmetric thermoelastic direction

thermal conductivity

total number of nodes

kernels

kernels

modulus of the elliptic integrals

in the radial

in the axial

components of the unit tangential vector in the radial and axial directions

shape function associated with a nodal point e.

unit outward normal to the surface S

components of the unit outward normal in the radial and axial directions

I'

P

Pit '

Q

QI'l-l

q

R

R, R2

Ra Rp

Pz

It (p, QJ

ItQ

Itq .

T-ij

UItIt ' UltZ ' UZIt ' Uzz

~ uR

U lt '

V

V £

V

V It '

U z

Vz

IX

arbitrary boundary point load point inside the solution domain

components of the ring load vector at p in the radial and axial directions

field boundary point Legendre function of the second kind of order zero and degree I'l-! interior point in the volume V radial distance measured from the centre of a sphere

inner radius of a cylinder or sphere outer radius of a cylinder or sphere radius of a round bar or solid cylinder

fixed radial coordinate of the load point p

physical distance between points p and Q variable radial coordinate of the boundary point Q variable radial coordinate of the interior point q

surface of the volume V distance on the path r' surface of the sphere of radius £

arbitrary scalar quantity temperatures at the internal and external surfaces of a cylinder or sphere

traction kernel functions in Cartesian coordinates, -i = 1,3, j = 1,3 axisymmetric traction kernel functions

tractions in the directions tangential and normal to the surface components of the traction vector in the radial and axial directions strain energy of the body

displacement kernel functions in Cartesian co­ordinates, -i = 1,3, j = 1,3 axisymmetric displacement kernel functions

displacement vector displacement in the radial direction from the centre of a sphere components of the displacement vector in the radial and axial directions

volume of the solution domain volume of the sphere of radius £

arbitrary vector quantity

components of an arbitrary vector in the radial and axial directions

r

r I

y

<5

<5 •• -<.j

e:

v

~

p

p

01 ,°2 ' 03

°Il ' °22 ' °33

°12'°21

°e °e °nett

x

strain energy density of the body

weighting functions associated with ordinary Gaussian quadrature pOints

weighting functions associated with logarithmic Gaussian quadrature points

fixed x-coordinate of the load point p

vector of unknown quantities

variable x-coordinate of the boundary point Q fixed y-coordinate of the load point p

vector of unknown quantities

variable y-coordinate of the boundary point Q fixed axial coordinate of the load point p

variable axial coordinate of the field point Q variable axial coordinate of the interior point q

coefficient-of thermal expansion

surface path in any radial plane through the axis of rotational symmetry

path from one surface of the crack to the other inside the solution domain

common interface between two subdomains

parameter of Legendre functions of the second kind

specific surface energy of the body

Dirac delta function

Kronecker delta

radius of small sphere centred at the load point p

angular coordinate of the load point p

angular coordinate of the boundary point Q shear modulus

Poisson's ratio

local or intrinsic coordinate

density of the material distance from crack tip

principal stresses

direct stresses in local directions 1, 2 and 3

shear stresses in the local directions 1 and 2

critical stress required for crack growth

von Mises equivalent stress

nett stress acting on the cross-section at the crack plane

direct stress in the radial direction from the centre of a sphere

w

XI

direct stresses in the radial, axial and hoop directions

shear stress

direct stress in the tangential direction to the surfaces of a sphere

unknown harmonic function satisfying Laplace's equation

potentials at the inner and outer surfaces of a cylinder or sphere

potential function

fundamental solution for Laplace's equation in three-dimensional Cartesian coordinates

angular velocity