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Lecture Notes in Engineering
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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