ELASTIC PROBLEMS LEADING TO THE BIHARMONIC EQUATION I N REGIONS OF SECTOR TYPE Hisharn Hassanein B.Sc., Alexandria University, 1969 A THESIS SUBMITTED IN PARTIXL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Department of Mathematics @ HISHAM HASSANEIN 1973 SIMON FRASER UNIVERSITY April 1973 All rights reserved. This thesis may not be reproduced in whole or in part, by photocopy or other means, without permission of the author.
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E L A S T I C PROBLEMS LEADING T O THE BIHARMONIC EQUATION
I N REGIONS O F SECTOR TYPE
H i s h a r n H a s s a n e i n
B . S c . , A l e x a n d r i a U n i v e r s i t y , 1969
A T H E S I S SUBMITTED I N PARTIXL FULFILLMENT OF
THE REQUIREMENTS FOR THE DEGREE OF
MASTER O F S C I E N C E
i n the D e p a r t m e n t
of
M a t h e m a t i c s
@ HISHAM HASSANEIN 1973
SIMON FRASER UNIVERSITY
A p r i l 1973
A l l r ights reserved. T h i s thesis m a y n o t be
reproduced i n w h o l e or i n part , by photocopy
or other m e a n s , w i t h o u t p e r m i s s i o n of the author.
APPROVAL
NAME : H i s h a m H a s s a n e i n
DEGREE : M a s t e r of Science
T I T L E OF T H E S I S : E l a s t i c p r o b l e m s leading t o the b i h a r m o n i c equation i n
regions of sector type
EXAMINING COMMITTEE :
CHAIRMAN : G. A. C. G r a h a m
R. W. Lardner Senior Supervisor
D. L. S h a r m a
M. Singh
C - - . 7 - - ( - -
D. S h a d m a n E x t e r n a l E x a m i n e r
A p r i l 19 , 1973 DATE APPROVED :
(ii)
PARTIAL COPYRIGHT LICENSE
I hereby g r a n t t o Simon F rase r Univers i ty t h e r i g h t t o lend
my t h e s i s or d i s s e r t a t i o n ( the t i t l e of which i s shown below) t o u s e r s
of t he Simon F rase r Univers i ty Library , and t o make p a r t i a l o r s i n g l e
copies only f o r such use r s o r i n response t o a reques t from the l i b r a r y
of any o the r u n i v e r s i t y , o r o the r educa t iona l i n s t i t u t i o n , on i t s own
behal f o r f o r one of i t s u s e r s . I f u r t h e r agree t h a t permission f o r
mu l t ip l e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted
by me or the Dean of Graduate S tudies . It is understood t h a t copying
o r publ ica t ion of t h i s t h e s i s f o r f i n a n c i a l ga in s h a l l no t be allowed
without my w r i t t e n permission.
Author : -- -
1
( s igna tu re )
(name )
\ I (da te )
ABSTRACT
A number of problems i n e l a s t i c i t y may be reduced t o so lv ing t h e
4 biharmonic equation V 4 = 0 i n two dimensions under appropr ia te
boundary condit ions on the function . The purpose of t h i s t h e s i s w i l l
be t o examine c e r t a i n methods o f s o l u t i o n of t h i s equation i n regions
bounded by l i n e s r a d i a t i n g from t h e o r i g i n and by a r c s of c i r c l e s centered
a t the o r i g i n . The b a s i c region of t h i s type i s t h e s e c t o r r < a ,
-w < 8 < w , where (r, 8) a r e p o l a r coordinates. We s h a l l be e s p e c i a l l y
concerned with t h e p a r t i c u l a r case when w = r, s o t h a t t h e s e c t o r becomes
a c i r c u l a r region with a crack l y i n g between the boundary and t h e cen t re
of the c i r c l e .
Af te r reviewing i n t h e f i r s t chapter t h e b a s i c equations of p lane
e l a s t i c i t y and of the theory of p l a t e bending, and showing how i n both
cases problems may be reduced t o t h e biharmonic equat ion , we proceed i n
Chapter I1 t o examine the s o l u t i o n of a number of p a r t i c u l a r problems,
most of which involve regions of s e c t o r type. I n t h i s chapter we
consider a problem previous ly examined by Williams [ 31 concerning a
cracked cy l inder wi th imposed t r a c t i o n s on t h e boundary r = a. H e
cons t ruc t s a b a s i c s e t of e igenfunct ions s a t i s f y i n g the biharmonic equation
and the homogeneous boundary condi t ions on the crack faces 0 = *IT . Unfortunately, these eigenfunctions a r e no t orthogonal which makes it
very d i f f i c u l t t o determine t h e unknown c o e f f i c i e n t s i n t h e expression
of t h e s t r e s s function.
I n an i n t e r e s t i n g paper, Gaydon and Shepherd [ 51 consider t h e problem 1
I of a semi- inf in i te r ec tangu la r s t r i p . Each of the e igenfunct ions is
expanded i n a s e r i e s of orthonormal beam funct ions , thus enabling them t o
compute numerically t h e c o e f f i c i e n t s of the s t r e s s funct ion corresponding
t o any a r b i t r a r y d i s t r i b u t i o n of t r a c t i o n on the end of the s t r i p . I n an
extens ion of t h e i r work, Gopalacharyulu [ 6 ] follows t h e same method i n
so lv ing the s e c t o r problem. He a l s o expands t h e e igenfunct ions i n a
s e r i e s o f orthonormal beam funct ions .
Xn Chapter 111, w e consider the problem of an i n f i n i t e cy l inder of
u n i t r ad ius cracked along the plane 0 = T. Ins tead of us ing the more
complicated beam eigenfunctions a s were used by Gopalacharyulu [ 6 ] , t h e
s e t of b a s i c eigenfunctions discussed i n Chapter I1 is expanded i n terms
of simple Four ier s i n e and cosine s e r i e s . An i n f i n i t e system of simul-
taneous equations is obtained from which we can compute numerically t h e
c o e f f i c i e n t s corresponding t o a r b i t r a r y t r a c t i o n s on t h e boundary r = 1.
W e have computed these c o e f f i c i e n t s numerically f o r a p a r t i c u l a r loading
a s w e l l a s the corresponding stress d i s t r i b u t i o n around the crack tip.
The stress i n t e n s i t y f a c t o r s f o r d i f f e r e n t loadings a r e a l s o computed.
Following the same method, t h e set of simultaneous equations f o r de te r -
mining the c o e f f i c i e n t s of t h e s t r e s s funct ion i s obtained i n t h e case
of an i n f i n i t e cy l inder having a crack w i t h a rounded t i p , and a l s o i n
t h e case of a semi-circular cyl inder .
ACKNOWLEDGMENT
I wish t o thank D r . R. Lardner, Mathematics Department, Simon Frase r
Univers i ty , f o r h i s invaluable he lp and continuous encouragement.
I a l s o wish t o thank M r s . A. Gerencser f o r h e r pa t i ence i n typing my
t h e s i s .
F i n a l l y , I would l i k e t o thank Simon Fraser Universi ty f o r t h e
f i n a n c i a l a s s i s t ance I received throughout my work.
TABLE OF CONTENTS
T i t l e Page
Approval Page
A b s t r a c t
Acknowledgment
Tab le o f Conten t s
L i s t o f F i g u r e s
L i s t o f T a b l e s
CHAPTER I ELASTOSTATIC PROBLEMS LEADING TO THE BIHARMONIC
EQUATION
1.1 P l a n e e l a s t o s t a t i c problems
1.1.1 P l a n e d e f o r m a t i o n s
1.1.2 G e n e r a l i z e d p l a n e stress
1.1.3 A i r y ' s stress f u n c t i o n
1.2 P u r e bend ing o f p l a t e s
1 .2 .1 C u r v a t u r e o f s l i g h t l y b e n t p l a t e s
1.2.2 R e l a t i o n s between bend ing moments and d e f l e c t i o n
1.2.3 The d i f f e r e n t i a l e q u a t i o n o f t h e d e f l e c t i o n s u r f a c e
o f l a t e r a l l y l o a d e d p l a t e s
1.2.4 Boundary c o n d i t i o n s
CHAPTER I1 THE SOLUTION OF CERTAIN ELASTOSTATIC PROBLEMS
2 .1 I n t r o d u c t i o n
2.2 The c r a c k e d c y l i n d e r
2 .2 .1 S e p a r a b l e s o l u t i o n s o f t h e b iharmonic e q u a t i o n
2.2.2 Expansion o f stresses i n terms o f t h e basic
PAGE
i
i i
iii
v
v i
i x
X
1
1
2
3
4
6
6
10
16
e i g e n f u n c t i o n s
( v i ) '
2.2.3 Radial s t r e s s v a r i a t i o n s nea r t h e crack t i p
2.2.4 Angular v a r i a t i o n s o f the p r i n c i p a l s t r e s s e s and
the d i s t o r t i o n a l s t r a i n energy dens i ty
2.3 The bending s t r e s s d i s t r i b u t i o n a t the base of a
s t a t i o n a r y crack
2.4 Plane s t r e s s e s i n a semi- inf in i te s t r i p
2.4.1 Solut ion of the biharmonic equation
2.4.2 Expansion of t h e s t r e s s funct ion i n terms of
orthogonal beam funct ions
2.4.3 Evaluat ion of the cons tants A h 1 B~
2.5 The s e c t o r problem
2.5.1 Solut ion of the biharmonic equation
2.5.2 Expansion of the a r c t r a c t i o n s i n terms of the beam
funct ions
2.5.3 Determination of %
CHAPTER I11 SOLUTION OF THE CRACKED CYLINDER AND SEMI-CIRCLE
PROBLEMS
3.1 The cracked cy l inder problem
3.1.1 S a t i s f a c t i o n of the condit ions of o v e r a l l s e l f -
equi l ibr ium
3.1.2 Separat ion of the cons tants An and Bn
3.1.3 S t r e s s i n t e n s i t y f a c t o r
3.1.4 Examples
3.2 S t r e s s d i s t r i b u t i o n around a crack w i t h a rounded t i p
3.2.1 Solut ion of t h e biharmonic equation
3.2.2 S a t i s f a c t i o n of the boundary condi t ions on the
boundaries r = R, r = 1
(vi i 1
PAGE
34
3 5
PAGE:
3.2.3 Satisfaction of conditions of overal l equilibrium 69
3.3 Stress dis tr ibut ion i n a semi-circular sector
3.3.1 Separation of An and Bn
3 . 3 . 2 Satisfaction of self-equilibrium conditions
TABLE I
TABLE I1
BIBLIOGRAPHY
( v i i i )
LIST OF F I G U R E S
FIGURE 1
2
3
4 '
5
6
7
8
9
10
11
12
13
14
15
16 (i)
16 (ii)
16 C i i i )
17
18
PAGE
7
8
10
11
11
12
14
15
16
17
23
34
36
36
4 3
6 3a
6 3b
6 3c
6 5
70
TABLE I
TABLE I1
LIST OF TABLES
PAGE
CHAPTER I
ELASTOSTATIC PROBLEMS LEADING TO THE BIHARMONIC EQUATION
In t h i s chapter we s h a l l consider two d i f ferent types of problems
tha t lead t o the biharmonic equation: plane e l a s tos ta t i c problems and
pure bending of thin plates .
Plane e l a s tos ta t i c problems
Plane e l a s tos ta t i c problems include generalized plane s t r e s s and
plane s t r a in problems. The f i r s t of these problems ar i ses when considering
a thin plate loaded by forces applied a t the boundary, p a r a l l e l t o the plane
of the plate . Therefore, i f we take the plane of the p la t e as the xy-plane,
- - the average s t r e s s components through the p la te thickness,
OXZ and
- - - 0 w i l l a l l be zero, whereas 0 , G and r are only functions of YZ YY xy
x and y. On the other hand plane s t r a i n problems ar i se when the dimen-
sion of the body i n the z-direction is very large. I f a long cyl indrical
o r prismatical body is loaded by forces which are perpendicular t o the
longitudinal elements and do not vary along the length, it may be assumed
tha t a l l cross-sections are i n the same condition. We assume t h a t the end
sections are confined between fixed smooth r ig id planes. Since there is
no axial displacement a t the ends, and, by symmetry, a t the mid-section,
it may be assumed tha t the same holds a t every cross-section.
In th i s section we s h a l l take the coordinate axes t o be xl, x2 and
x We use the Greek indices a and B for the range 1, 2. A repeated 3'
index w i l l represent the sum of a l l allowable values of tha t index.
1.1.1 Plane deformation
A body is sa id t o be i n a s t a t e of plane deformation, o r plane s t r a in ,
p a r a l l e l t o the x x -plane, i f the component u of the displacement vector 1 2 3
- u vanishes and the components u and u are functions of the coordinates
1 2
x and x but not x Thus, a s t a t e of plane deformation i s characterized 1 2 3 '
by the formulae,
The stress-displacement relat ions i n this case w i l l be
- where the d i la ta t ion V = u and G is the modulus of r ig id i ty .
a la The equilibrium equations are
where F are the components of the body force. a
I f the solutions of these equilibrium equations are t o correspond t o
\
the s t a t e of s t r e s s t h a t can ex i s t in an e l a s t i c body, the 0 must aB
sa t i s fy the Beltrani-Mitchell compatibility equation
where 0 5 oll + a22. 1
I f the components T (xl, x2) of external s t r e s se s are specif ied a
along the boundary i n the form
where the n are the components of the ex t e r io r un i t normal vector t o B
the boundary, the formulation of the problem i s complete.
\
1.1.2 Generalized Plane s t r e s s
A body is i n the s t a t e of plane s t r e s s p a r a l l e l t o the x x -plane when 1 2
the stress components o13, ~r~~ , 033 vanish.
Sokolnikoff [ l ] has shown t h a t the Beltrami-Mitchell compatibility
equation turns out t o be
- - - - 2 h ~ - - where O = all
1 + 022' X E
X i - 2 ~ ' oaB (xlI x2) and F~ (X1. x2) are
the mean values of (5 and Fa i n a cylinder of thickness 2h, and bases a@
i n the planes x = f h , i . e . 3
- Equations (1) and (2) su f f i ce t o determine the mean s t r e s se s OclB when
t h e boundary conditions on the edge are given i n the form
1.1.3 Ai ry ' s s t r e s s function
Re s h a l l consider boundary value problems i n plane e l a s t i c i t y i n which
body fo rces a r e absent . Accordingly, w e consider t h e equi l ibr ium equations
i n t h e form
w h e r e G s a t i s f y the compat ib i l i ty equation a@
and a r e given on the boundary by
where t h e T a b ) a r e known functions of the a r c parameter S on the
boundary C.
The equi l ibr ium equations imply t h e ex i s t ence of a function @ b l r x2)
such t h a t
The compat ib i l i ty equation impl ies t h a t 4 must s a t i s f y the biharmonic
equation
in the region R.
Every s o l u t i o n of t h i s equation of c l a s s c4 is c a l l e d a biharmonic
funct ion , b u t s i n c e we a r e i n t e r e s t e d i n those s t a t e s of s t r e s s f o r which
t h e 0 a r e single-valued, we need consider only biharmonic funct ions aB
with single-valued second p a r t i a l de r iva t ives .
Expressing t h e biharmonic equation and the s t r e s s e s i n terms of x
and y , we obta in
where
and
I n p lane p o l a r coordinates, these equations become
where
and
Here CT rr' '08 and (5 are the physical components of s t r e s s with respect r e t o the polar coordinates.
1.2 Pure Bending of p la tes
The c l a s s i ca l small-deflection theory of p l a t e s , developed by Lagrange,
is based on the following assumptions:
i) points which l i e on a normal t o the mid-plane of the undeflected
p l a t e l i e on a normal t o the mid-plane of the deflected p l a t e ;
ii) the s t r e s se s normal t o the mid-plane of the p l a t e , a r i s ing from
the applied loading, are negl igible i n comparison with the s t resses
i n the plane of the p la te . Thus, every transverse s ing le loading
considered i n the thin-plate theory i s merely a discont inui ty i n
the magnitude of the shearing forces. I f the e f f e c t of the surface
load becomes of spec ia l i n t e r e s t , thick-plate theory has t o be used;
iii) the slope of the deflected p l a t e i n any direct ion is small, so t h a t
i ts square may be neglected i n comparison with unity;
i v ) the mid-plane of the p l a t e remains neut ra l during bending, i .e .
any mid-plane s t r e s se s a r i s ing from the def lect ion of the p l a t e
section i n t o a non-developable surface may be ignored.
1.2.1 Curvature of s l i gh t ly bent p l a t e s
I n discussing small deflections of a p l a t e , we take the middle plane
of the p l a t e , before bending occurs , a s the xy-plane. During bending, t h e
p a r t i c l e s on this plane undergo smal l displacements w perpendicular t o
the xy-plane and form the middle su r face of the deformed p l a t e .
I n determining the curvature of the middle su r face of the p l a t e we
s h a l l be using assumption iii, namely the s lope of the tangent t o t h e
su r face i n any d i r e c t i o n can be taken equal t o the angle t h a t the tangent
makes wi th the xy-plane, and the square of the s lope is neglected compared
t o uni ty . Thus t h e curvature of the su r face i n a plane p a r a l l e l t o t h e
xz-plane (Fig. 1) i s equal t o
S imi la r ly , t h e curvature of t h e s u r f a c e in a plane p a r a l l e l t o t h e yz-plane
is approximately equal t o
- Now, f o r any d i r e c t i o n an (Fig. 2 )
- IT 7.r But, f o r any d i r e c t i o n an making an angle a with the x-axis, - - < a 5 - 2 - 2 '
a a - = - a cosa + - s i n a . an ax a~
.- Therefore, the curvature i n the an d i r e c t i o n w i l l be
1 - = r
a aw a a aw aw 'n
- 1 an an = -(- aX cosa + - a~ s i n a ) (= cosa + - a~ s i n a )
1 = - 2 1 2 rx cos a + - r s i n a - - s in2a , r . Y xy
w h e r e t h e quant 1 a2w
ity - = - is c a l l e d t h e t w i s t of t h e su r face with r axay xy
respect t o t h e x and y axes.
- I n the case of the d i r e c t i o n a t , t h e angle with the x-axis w i l l be
IT a 1- - 2 ' and the curvature i n t h e t -d i rec t ion w i l l be given by
1 1 2 1 2 - = - s in2a . s i n a + - cos a + - r r r r t x Y xy
W e note t h a t
a - - a _ - - a a t s i n a + -
ay cosa , ax
and the re fo re
1 a a aw aw - = ( - - s i n a + - cosa) (;j-;; cosa + - s i n a ) r n t ax a~ a~
1 1 Therefore, i f the q u a n t i t i e s - - 1
and - a r e known, we can g e t the r r r x Y xy
corresponding q u a n t i t i e s r e l a t e d t o any system of axes i n c l i n e d a t an angle
a t o the o r i g i n a l system, by using equations (10) - (12) . I n order t o o b t a i n the p r i n c i p a l curvatures of the s u r f a c e and t h e
corresponding p r i n c i p a l d i r e c t i o n s , we t ry t o f i n d t h e values of t h e angle
I a f o r which - is an extremum. Thus d i f f e r e n t i a t i n g equation (10) r n
with r e spec t t o a and equating t h e r e s u l t t o zero, we f i n d t h a t
IT Me denote the roo t s of equation (13) by al and a 1 2 + - . S u b s t i t u t i n g
these values o f a i n equation (10) we obta in the two p r i n c i p a l curvatures.
W e a l s o note t h a t i f a s a t i s f i e s (13) then from (12)
i .e. the t w i s t of the su r face i s zero on t h e p r i n c i p a l p lanes .
1.2.2 Relat ions between bending moments and d e f l e c t i o n -.
the
Consider a rec tangular p l a t e under uniformly d i s t r i b u t e d moments along
edges of the p l a t e (Fig. 3 ) .
I z Figure 3
The xy-plane i s taken a s t h e middle p lane of the p l a t e before bending. M X
w i l l denote the bending moment p e r u n i t l eng th a c t i n g on t h e edges p a r a l l e l
t o the y-axis and M t h e moment p e r u n i t length ac t ing on t h e edges p a r a l l e l Y
t o the x-axis. These moments a r e considered p o s i t i v e when they produce
compression i n the upper su r face of the p l a t e and t ens ion i n t h e lower. The
th ickness h of the p l a t e is assumed t o be smal l i n comparison wi th o t h e r
dimensions.
Now, t o de r ive the expressions f o r the bending moments i n terms of the
d e f l e c t i o n of the p l a t e , we consider an element c u t o u t of the p l a t e by two
p a i r s o f planes p a r a l l e l t o t h e xz and yz-planes (Fig. 4 ) .
Using assumption i, p. (61, the l a t e r a l s i d e s of the element w i l l remain
p lane during bending and w i l l r o t a t e about t h e n e u t r a l axes nn s o as t o
remain normal t o t h e de f l ec ted middle su r face of t h e p l a t e , and the re fo re
the middle su r face w i l l no t undergo any extension during bending. The
long i tud ina l s t r a i n of an element a t a d i s t ance z from t h e n e u t r a l su r face
i n t h e x-di rec t ion (Fig. 5) i s
Z e = - xx r
X Z
Simi la r ly e =- . YY
Figure 5
Using Hooke's Law, t h e normal s t r e s s e s a r e
The couples produced by these s t r e s s e s on t h e l a t e r a l s i d e s should obviously
be equal t o the ex te rna l couples Mx dy and M dx, thus Y
0 z d x d z = M d x . r" YY Y -h/2
S u b s t i t u t i n g equations (14) i n t o (15) f o r the values o f a and XX
0 we obta in YYI
where
D i s c a l l e d the f l e x u r a l r i g i d i t y o f the p l a t e .
Now we s h a l l express the moments a c t i n g on a s e c t i o n i n c l i n e d t o t h e
x and y axes i n terms o f Mx and M . I f we c u t t h e lamina abcd
Y
(Fig. 4) by a p lane p a r a l l e l t o t h e z-axis and i n t e r s e c t i n g the lamina
along a c (Fig. 5 ) , we can determine t h e normal and shea r s t r e s s e s a c t i n g
on t h i s i n c l i n e d face i n terms of
OXX and 0 . These w i l l be given
YY
by the we l l known equations
- - 2 2 'nn Gxx cos a + o s i n a, and
W 1 Grit=-(0 - 0 ) s in2a , where a 2 YY xx
is t h e angle between t h e normal n
t o the i n c l i n e d face and the x-axis. Figure 6
Considering a l l laminas, such a s acd (Fig. 61, over t h e thickness
of the p l a t e , the normal s t r e s s e s 'nn
give t h e bending moment ac t ing on
t h e inc l ined plane , t h e
h/2 0 zdz =
nn -h/2
magnitude of which p e r u n i t length along ac is
2 2 ( cos ai-0 s i n a) zdz y2 x. YY
2 2 M cos a + M s i n a
X Y
S imi lar ly
The shear ing s t r e s s e s 'nt
w i l l give a t w i s t i n g moment a c t i n g on
the inc l ined face , the magnitude of which p e r u n i t length of a c is:
Here we note t h a t the s igns of *n and Mnt a r e chosen i n such a manner
t h a t t h e i r p o s i t i v e values a re represented by vectors i n t h e p o s i t i v e
d i r e c t i o n s of t and n respect ively .
To obta in the expression f o r Mnt i n terms of the d e f l e c t i o n w,
consider the d i s t o r t i o n of a t h i n lamina efgh wi th t h e s i d e s e f and
14
eh p a r a l l e l t o the n and t d i r e c t i o n s r e spec t ive ly , and a t a d i s t ance
z from the middle p lane (Fig. 7) . During bending of the p l a t e , t h e
p o i n t s e , f , g and h undergo
smal l displacements. The components
o f the displacement of the p o i n t e
i n the n and t d i r e c t i o n s a r e
denoted by u and v respect ive ly .
Then the displacement of the adjacent
p o i n t h i n t h e n d i r e c t i o n i s u +
t he p o i n t f i n the t d i r e c t i o n is
ments, t h e shear ing s t r a i n w i l l be
au (at' d t , and t h e displacement of
av v + - an dn. Owing t o t h e s e d isplace-
and t h e corresponding shear ing s t r e s s i s
where G is the modulus of e l a s t i c i t y i n shear .
I n order t o express u and v i n terms of the d e f l e c t i o n w of the
p l a t e , consider a s e c t i o n of the middle su r face made by t h e normal p lane
through the n-axis. The angle of r o t a t i o n i n the counter-clockwise d i rec-
t i o n of an element pq, which i n i t i a l l y w a s perpendicular t o t h e xy
aw p lane , about an a x i s perpendicular t o t h e nz p lane i s equal t o - - an
(Fig. 8). Owing t o t h i s r o t a t i o n a p o i n t of the element a t a d i s t ance
z from the n e u t r a l su r face has a displacement i n the n-direct ion equal
Figure 8
S imi la r ly considering the s e c t i o n through t h e t -axis , t h e same p o i n t w i l l
have a displacement i n the t -d i rec t ion equal t o
Therefore, t h e shear s t r e s s w i l l be
and t h e corresponding t w i s t i n g moment from i ts d e f i n i t i o n i n eqn. (21)
IT From equation (21) , we n o t i c e t h a t i f a = 0 o r -
2 l i.e. when
the n and t d i r e c t i o n s coincide with the x and y axes, Mnt = 0
and t h e r e a r e only bending moments Mx
and M ac t ing on t h e sec t ions Y
perpendicular ly t o those axes a s was assumed i n Fig. 3 and i n de r iv ing
the equations of t h i s sec t ion . From equation (22) we s e e that the t w i s t
of the su r face i s p ropor t iona l t o Mnt '
and when Mnt = 0 , t h e t w i s t i s
1 zero. Hence t h e curvatures - 1
and - a r e p r i n c i p a l curvatures. r r X Y
1.2.3 The d i f f e r e n t i a l equation of the de f l ec t ion su r face of l a t e r a l l y
loaded p l a t e s
In the l a s t s e c t i o n we expressed the bending moment M n I Mt
and
M n t ' ac t ing on a s e c t i o n p a r a l l e l t o t h e z-axis and whose normal makes
an angle a with t h e x-axis , i n terms of the d e f l e c t i o n w. The x and
y-axes were considered t o be p r i n c i p a l axes. Consider now a p l a t e under
t h e ac t ion of loads normal t o i t s su r face . W e s h a l l assume t h a t , a t the
boundary, the edges of the p l a t e a r e f r e e t o move i n the p lane of t h e p l a t e .
This way the r e a c t i v e forces a t the edges w i l l be normal t o t h e p l a t e .
Together w i t h the usual assumption t h a t t h e de f l ec t ions a r e smal l compared
t o t h e th ickness of the p l a t e , the s t r a i n i n the middle p lane may be neglec-
t e d during bending.
Consider, a s was done i n Fig. 4 , an element c u t o u t of the p l a t e by
two p a i r s o f planes p a r a l l e l t o t h e xz and yz p lanes (Fig. 9 ) . We note a M
t h a t M and M a r e p o s i t i v e X Y
i f they produce compression i n
upper l aye r s and
l a y e r s , whereas
p o s i t i v e i f they
d i r e c t i o n of the
tens ion i n lower
M and M a r e xy Y X
produce r o t a t i o n i n the L --------
I \ outward normal. We i
3 \ aM* - .. denote the shear ing forces p e r u n i t aQY
M
Q +-dy YX ay
Y ay length a c t i n g on the p lanes perpendicular
t o t h e x and y-axes by Q and Q~ X
respect ive ly . Therefore Qx and Qy w i l l be given by
Figure 9
The load w i l l be considered d i s t r i b u t e d over the upper s u r f a c e of the
p l a t e , and has the i n t e n s i t y q dx dy. Fig. 10 represents the middle
p lane with the p o s i t i v e d i r e c t i o n s of the fo rces and the moments. M
M
dx
M '
Y
1
z Figure 10
For equi l ibr ium of fo rces i n the z-d i rec t ion , we have
aQx - aQ ax dx dy + - J d y d x + q d x d y = 0 ,
ay
f r o m which it follows t h a t
Taking moments about the x-axis
aQ aQx a M a M (Q, + 2 dy)dxdy + (- dx) dy + qdxdy - 2 dydx + dxdy = 0.
ay ax ay ax
18
The moments due t o t h e load q and change i n Q and Qy may be neglec ted X
because they a r e of a h igher order , s o t h a t
S imi la r ly t ak ing moments about the y-axis we
0 .
obta in
0 .
Equations (23) - (25) completely def ine t h e equi l ibr ium of t h e
element. S u b s t i t u t i n g the values of Q and Qy from (24) and (25) X
i n t o (23) we obta in
We a l s o have
I "" h/2 M 0 z dz and M = -1 0 z d z , xy xy YX YX
-h/2 -h/2
bu t s ince 0 = 0 it follows t h a t xy YX'
Therefore equation (26) reduces t o
I f the th ickness of the p l a t e i s smal l compared t o the o t h e r dimensions,
t h e e f f e c t of the s t r e s s OZZ produced by t h e load q, and the shear ing
fo rces Qx and Qy, on the bending of the p l a t e may be neglec ted , and
19
we can make use of the r e s u l t s obtained i n the l a s t s e c t i o n f o r the case
of pure bending.
Using t h i s assumption, w e can now express equation
of the de f l ec t ion w. From equations ( 1 9 ) , (20) and
and
(27) i n terms
(22) we have
S u b s t i t u t i n g these expressions i n equation (27) we g e t
which may be re-wri t ten as
4 - 2 2 2 - a + - a' Here V = V V where V = 2
i s t h e Laplacian opera to r i n the ax ay2
rec tangular coordinates. I n p o l a r coordinates ,
From equations (24) and (25) , t h e shear ing fo rces Qx and Qy
w i l l be given by
The problem of bending of p l a t e s by a l a t e r a l load q the re fo re reduces
t o the i n t e g r a t i o n of equation (29). I f the s o l u t i o n s a t i s f y i n g the given
boundary condit ions i s found, a l l t he r e l evan t q u a n t i t i e s may be computed.
They a r e l i s t e d here f o r convenience.
I n p o l a r coordinates
a = a s i n e a - - - cose - - - ax a r r 30 '
2 1
We can e a s i l y ob ta in the corresponding values f o r the second d e r i v a t i v e s
a2 a2 - - a2 2 , and - . Hence equations (31) l ead t o the fol lowing
ax ay2 ax ay
equivalent r e s u l t s i n terms o f p o l a r q u a n t i t i e s .
1.2.4 Boundary condit ions
I n t h i s s e c t i o n we s h a l l d iscuss s e v e r a l types o f boundary condi t ions
f o r s t r a i g h t boundaries. I n the case of rec tangular p l a t e s , we assume
t h a t the x and y-axes a r e taken p a r a l l e l t o the s i d e s of the p l a t e .
From the r e s u l t s f o r the rec tangular p l a t e , w e s h a l l ob ta in t h e
corresponding ones f o r boundaries of the type 0 = constant i n p o l a r
coordinates , and express them i n terms of the de f l ec t ion w of the p l a t e
using the appropriate r e l a t i o n s from (32) .
(a) Bu i l t - in edge
The de f l ec t ion w along the b u i l t - i n edge i s zero. Furthermore,
t h e tangent plane t o t h e de f l ec ted middle su r face along t h i s edge coincides
wi th the i n i t i a l p o s i t i o n of the middle p lane of the p l a t e . Assuming t h i s
b u i l t - i n edge is a t x = a, t h e boundary condit ions a r e
I n t h e case of a s e c t o r , with t h e b u i l t - i n edge along 0 = a , say , we g e t
(b) Simply supported edge
I f the edge x = a of the p l a t e i s simply supported, t h e d e f l e c t i o n
w along t h i s edge i s zero. Also t h e edge can r o t a t e f r e e l y about the
edge l i n e , i . e . M = 0. Therefore X
W
But we no t i ce t h a t
the boundary condit ions can be w r i t t e n as
x= a ax
2 2
" I a a = 0 and V w = O .
a w - 2
a w 2
- 0 and hence = 0. This implies t h a t - - ax
x=a
Again i n the case of the s e c t o r whose edge 8 = a i s simply supported we
s h a l l obta in
2 2 l a w 1 a w + , e
w 1 @=a = 0 and N e l = [ F z + - - e=a r2 ae2
2 aw a 2
2 - 0. Therefore - = But along 8 = a f - = - - a O f a d the boundary
ar ar ae2 condi t ions a re
(c) Free edge
I n case the edge x = a o f t h e p l a t e i s f r e e , we have no bending and
t w i s t i n g moments and a l s o no v e r t i c a l shear ing fo rces , s o t h a t a t f i r s t
s i g h t it appears t h a t the appropr ia te boundary condi t ions a r e
However t h e second and t h i r d condi t ions should be combined i n one condit ion
a s fol lows. Consider the t w i s t i n g couple M d produced by the hor i - xy Y
z o n t a l forces and a c t i n g on .an element of l eng th dy of the edge x = a . a
Figure 11
We replace t h i s couple by two v e r t i c a l forces of magnitude M and dy XY
a p a r t (Fig. 11). Such a replacement does no t change the magnitude of
t w i s t i n g moments and produces only l o c a l changes i n t h e s t r e s s d i s t r i b u t i o n
a t the edge of t h e p l a t e , leaving the s t r e s s condit ion of the r e s t of t h e
p l a t e unchanged. Considering two adjacent elements of the edge, t h e d i s -
t r i b u t i o n of t w i s t i n g moments M i s s t a t i c a l l y equivalent t o a d i s t r i - XY
but ion of shear ing fo rces o f i n t e n s i t y
and the re fo re the j o i n t requirement regarding M and Qx along x = a xy
be comes
I n terms of the d e f l e c t i o n w, t he necessary boundary condi t ions w i l l
For a s e c t o r wi th a f r e e edge 0 = a , t h e corresponding boundary
condi t ions a r e
Me [ = 0 , = 0 and Ve = Qe - - 0=a
o r i n terms of the def lect ion w,
and
CHAPTER I1
THE SOLUTION OF CERTAIN ELASTOSTATIC PROBLEMS
2.1 In t roduct ion
I n t h i s chapter we s h a l l i n v e s t i g a t e t h e so lu t ion of c e r t a i n e l a s to -
s t a t i c problems leading t o t h e biharmonic equation. The genera l method
of approach w i l l be separa t ion of v a r i a b l e s , leading t o the const ruct ion
of s e t s of b a s i c eigenfunctions. The s o l u t i o n s of genera l boundary value
problems f o r the types of regions considered a r e found a s l i n e a r combin-
a t i o n s of these eigenfunctions.
I n 52.2 t h e problem of th.e cracked cyl inder , previous ly inves t iga ted
by W i l l i a m s [ 3 ] , i s considered. Here t h e body c o n s i s t s of a cy l inder r < a
deformed i n plane s t r a i n ( o r p lane s t r e s s ) by means of t r a c t i o n s imposed on
t h e boundary r = a . The cy l inder conta ins a crack running from r = 0 t o
r = a on t h e p lane 8 = r , and t h e two su r faces , 8 = f ~ , of t h e crack
a r e t r a c t i o n f r e e . I n 52.3 the corresponding problem is considered f o r '
a c i r c u l a r p l a t e , cracked along the r a d i a l l i n e 8 = IT, and deformed under
bending loads. I n both of these s e c t i o n s the problem considered is solved
t o t h e e x t e n t of obta in ing genera l expansions f o r t h e s t r e s s f i e l d s i n terms
of the r e l evan t eigenfunctions. The c o e f f i c i e n t s i n these expansions w i l l
be r e l a t e d t o the t r a c t i o n s on r = a i n the nex t chapter .
I n s e c t i o n 32.4 t h e problem of p lane deformation of a semi - in f in i t e
s t r i p , -1 < y < 1, 0 < x < a, under t r a c t i o n s imposed on the end x = 0,
i s discussed. Again an eigenfunction expansion method is used, and i n this
sec t ion the c o e f f i c i e n t s i n the expansion a r e r e l a t e d t o t h e imposed
2 7
t r a c t i o n s using a method given by Gaydon and Shepherd [5]. F i n a l l y i n
92.5 the genera l s e c t o r problem (0 5 r < a , -w < 0 < w) is discussed
f o r the case when the edges 8 = &w a r e t r ac t ion- f ree and given t r a c t i o n s
a r e imposed on r = a. The c o e f f i c i e n t s i n the expansions a r e again
obtained f o r t h i s case using the method of Gopalacharyulu [6 ] r which is a
development of t h a t of Gaydon and Shepherd [ 5 ] .
2.2 The cracked cy l inder
Consider t h e p lane s t r a i n deformation of t h e c y l i n d r i c a l region
0 5 r c a , -71 c 8 < IT under t h e condi t ion t h a t the p a r t s of the boundary
8 = +IT, 0 5 r c a a r e t r ac t ion- f ree . Le t t ing $ ( r r 0) denote t h e Airy
s t r e s s funct ion , then the corresponding s t r e s s components a r e given by
( 5 ) . The condi t ions t h a t t h e crack faces , 8 = *IT , be t r a c t i o n - f r e e a r e
t h a t Gee = (5 = 0 t h e r e , o r i n o t h e r words re
I n t e g r a t i n g these equations w i t h r e spec t t o r gives the re fo re
where A , B, C a r e funct ions of 6 only. From the con t inu i ty of and
appropr ia te de r iva t ives a t the o r i g i n , it follows t h a t these cons tants a r e
i d e n t i c a l on 8 = +IT and 8 = -IT. Now the Airy stress funct ion i s
undefined up t o l i n e a r terms i n x and y : i f we add t h e funct ion
(+AX - B + Cy) t o , t he s t r e s s e s a r e unchanged, and t h e new s t r e s s
funct ion s a t i s f i e s the boundary condit ions
2.2.1 Separable s o l u t i o n s o f the biharmonic equation
Let us begin by seeking separable s o l u t i o n s of the biharmonic equation
4 V 4 = 0 s a t i s f y i n g boundary condi t ions (36) :
where R i s a function of r only and F i s a function of 8 only.
Then equation (3) w i l l be:
1 R R f (r) F + -5(2R1' - -
r r r r
a i a R' where f ( r ) = (7 + (R" + -1.
ar r
r 4
Multiplying throughout by - RF
and d i f f e r e n t i a t i n g w i t h r e s p e c t t o
r and 8 we obta in
Thus we have the two poss ib le cases:
2 F" - cons t . , s ay , (a) F -
2 i . e . F" - A F = 0
(b) ;(r2 R" - r R' + 2R) = const . , 2p say . R
The boundary condit ions (36) imply t h a t
The s o l u t i o n of (37 ) i s
It i s easy t o check t h a t the above boundary condi t ions w i l l l ead only t o
t h e t r i v i a l so lu t ion . S imi la r ly , i f h = 0 , the d i f f e r e n t i a l equation
w i l l be
and t h e s o l u t i o n is F(0) = A + B0 . Again the boundary condi t ions w i l l
g ive only the t r i v i a l so lu t ion . F i n a l l y the case when h2 i s negat ive
may a l s o be shown t o have only the t r i v i a l so lu t ion .
Therefore, we have t o cons ider E u l e r ' s equation (38). Its associa ted
i n d i c i a 1 equation i s :
2 o r m - 2m + 2 - u = 0. I f w e le t
the two roo t s w i l l be given by
and t h e r e f o r e the s o l u t i o n of equation (38) is
The cases A = 0, 1 w i l l be considered l a t e r .
I n order t o have a f i n i t e s t r a i n energy i n the neighbourhood of the
I crack-t ip r = 0 , we requ i re t h a t the s t r e s s e s a r e 0 (T) a s r + 0.
This condit ion w i l l imply t h a t B = 0 when A > 0. Therefore t h e s t r e s s
funct ion can be w r i t t e n i n t h e form
S u b s t i t u t i n g t h i s expression f o r $ i n the bihannonic equat ion , we
ob ta in
The genera l so lu t ion of equation (44) i s
~ ( 8 ) = A cos ( A + l ) 8 + B cos ( A - 1 ) 8 + c s in(A+l) 8 + D sin Ch-1) 8 .
Here, w e s h a l l d iscuss only the symmetric s t r e s s d i s t r i b u t i o n , i .e. F
w i l l be an even function of 8. (For antisymmetric stress d i s t r i b u t i o n ,
t h e method of s o l u t i o n w i l l be t h e same.) Then F may be w r i t t e n now as
~ ( 8 ) = A cos (A+1) 8 + B cos (A-1) 8 . (45)
Since we a r e considering the symmetric s o l u t i o n , only the boundary
condi t ions
w i l l be re levant . Applying the boundary condi t ions (46) on equation
(45) , we ob ta in
Here we s h a l l have two cases :
Case i) A + B # O .
This w i l l imply t h a t
coshIT = 0 and ACA+l) t B(A-1) = 0 ,
or A (1) = 2n-1 2
, n = 1 , 2 , 3 , . . . s ince A > 0. S u b s t i t u t i n g these eigen- n n
values i n (45) , we g e t
-
I 1 n+ -
2 F ( l ) (0) = a t cos(n+ T ) ~ - -
3 cos (n -
Z-) 0 n n
n- - 2 1
1 (n + y) 0 cos (n - ?) 0 -
n 1 n + -
3 2 n - - 2
J .
Case ii) A + B = 0
This implies t h a t
(2) o r h = n n = 2, 3 , 4, . . . and t h e corresponding s o l u t i o n is n
F ( ~ ) (0) = bn[cos (n-1) 0 - cos (n+ l ) 01 . n
Now we s h a l l r e tu rn t o t h e cases A = 0 , 1. For h = 0 , t h e s o l u t i o n
of (38) w i l l be
R(r ) = r [ a + b Rnr] .
The corresponding even function ~ ( 8 ) ~ from (44) , is F C ~ ) = A case +
B0 s i n 0 , and t h e condi t ions F(T) = F' (IT) = 0 imply t h a t A = B = 0.
3 2
Thus t h e r e is no eigenfunction f o r h = 0. For h = 1, w i l l be equal
t o 2, and equation (38) w i l l be
Its s o l u t i o n i s R(r) = a r 2 + b. We must take b = 0 t o avoid t o o
s i n g u l a r behaviour a t r = 0. The remaining r2 term leads t o an
e igenfunct ion of the same type a s case ii) above. Therefore
(2) on = r F ( ~ ) n (0) , A n = 1, 2, 3, . . . . Here we note t h a t . f o r
n = 1, (2 ) (0) = b l [ l - c o s ~ e l . F1
It follows t h a t the genera l even s o l u t i o n of the biharmonic equat ion
(.3) s a t i s f y i n g t h e boundary condi t ions (46) is a,
where l, (47)
2.2.2 Expansion of s t r e s s e s i n terms o f t h e b a s i c eigenfunctions - By using equations (5 ) , t h e s t r e s s e s w i l l be
Then the normal s t r e s s a t any p o i n t (r , 0) i n the considered domain
w i l l be given by
2
Simi la r ly the shea r and t a n g e n t i a l s t r e s s e s a r e given by
3
n-1 + B r [ (n-1) s i n (n-1) 8 - ( n + l ) s i n ( n + l ) 81 1 (50) n
3 1 n +-
2 3 o (r, 8) = - - 88 3
cos (n - -1 81 n -- 2
2
n-1 + B r (n+l) [ cos (n-1) 8 - cos (n+l) 81 n (51)
The constants An
and B n = 1, 2, ..., a r e determined from the n
boundary condit ions on r = a , where a i s a f ixed rad ius i n the case
of a f i n i t e domain, o r a t i n f i n i t y i n the case of an i n f i n i t e domain.
I t may be i n t e r e s t i n g a t t h i s p o i n t t o consider the term
2 B1 r [ l - cos28l appearing i n the s t r e s s funct ion (47) . This term
corresponds t o the case where A = 1. I n rec tangu la r coordinates t h i s
2 2 2 2 term may be w r i t t e n a s B1 r [l - cos281 = 2B r s i n 8 = 28 y .
1 1
The corresponding terms for the s t r e s ses , from equation (4) , are
given by
I f axx = 0 along some s t r a igh t boundaq x = -x (Fig. 1 2 ) , the constant 0
B1 w i l l be equal t o zero.
I f Y
1 Figure 1 2
2.2.3 Radial s t r e s s variations near the crack t i p
31 Equations (49) - (51) w i l l a l l be of the form r-' + O(r ) w i t h
respect t o the rad ia l variation. The loca l s t r e s s variations i n the
v ic in i ty of the base of the crack, r + 0, are dominated by the contri-
bution of the f i r s t term. I t is also noted tha t along the l ine of
propagation of the crack, 8 = 0, the shear s t r e s s is zero. Hence
a ( r , 0) and a ,, ( r , 0) are pr incipal s t resses ; we denote them by rr '5
a2 respectively, so t h a t
In other words, a t the base of the crack there ex is t s a strong tendency
toward a s t a t e of two-dimensional hydrostatic tension which consequently
may permit the e l a s t i c ana lys i s t o apply c lose t o the crack- t ip , notwith-
s tanding the square root s t r e s s s i n g u l a r i t y . I t has been suggested t h a t
t h i s f ea tu re would tend t o reduce t h e amount o r a r e a of p l a s t i c flow a t
t h e crack- t ip which might o r d i n a r i l y be expected t o e x i s t under such high
s t r e s s magnitudes and l ead the re fo re toward more of a b r i t t l e type f a i l u r e .
2.2.4 Angular v a r i a t i o n s of the p r i n c i p a l s t r e s s e s and t h e d i s t o r t i o n a l
s t r a i n energy dens i ty
From (49) - (511, the s i n g u l a r terms i n t h e s t r e s s e s a s r -+ 0 a r e
3 0 0 'rr
- A r-li[-cos - + ~ C O S -1 , 1 2 2
9 CJ - A ;'[sin 9 + s i n -1 , r e 1 2
The p r i n c i p a l s t r e s s e s a r e given by the expression
-31 6 9 - 4 ~ ~ r cos - [ 1 f s i n - ] . 2 2
The maximum p r i n c i p a l s t r e s s occurs when - = ae 0. This condit ion a u m
impl ies t h a t cos0 = s i n - 2
, i .e. t h e maximum p r i n c i p a l s t r e s s occurs m
7T a t 0 = -
m 3 ' The value of t h i s maximum s t r e s s i s
A A1
3&" 5.2 - = x max r r 4 .
Also the expression f o r t h e d i s t o r t i o n s t r a i n energy dens i ty , i . e .
t h e t o t a l s t r a i n energy l e s s t h a t due t o change i n volume, p e r u n i t volume,
is given by
From t h i s expression we obta in
and
W i l l i a m s [ 3 ] shows t h a t the two p r i n c i p a l s t r e s s e s Ol and 0 2 have
t h e angular v a r i a t i o n shown i n Fig. 13, and t h a t the d i s t o r t i o n a l s t r a i n
energy dens i ty is as shown i n Fig.
Figure 13 Figure 14
It i s i n t e r e s t i n g t o note t h a t because of the h y d r o s t a t i c tendency,
t h e maximum energy of d i s t o r t i o n does no t occur along the l i n e of crack
3 7
-1 1 d i r e c t i o n , 0 = 0 , bu t r a t h e r a t 0* = +cos (-1 - +70 deg. , where it 3
i s one-third higher.
2.3 The bending s t r e s s d i s t r i b u t i o n a t the base of a s t a t i o n a r y crack
Following the same procedure as f o r the ex tens iona l s t r e s s d i s t r i -
bu t ion , Williams [4 ] s tud ied the s t r e s s e s around a crack p o i n t owing t o
bending loads.
The problem i s formulated as follows. We have t o s a t i s f y t h e
d i f f e r e n t i a l equation
where q is the appl ied load on t h e p l a t e . For a p l a t e s u b j e c t t o edge
loading only , q = 0. We s h a l l t ake t h e edges 8 = ?nr a s f r e e edges.
Then, from (35)
Again, a s sec t ion 2.2, t h e symmetric c h a r a c t e r i s t i c so lu t ions a r e of the
form
I n order t o s a t i s f y t h e phys ica l boundary condit ion of f i n i t e s lope a t
t h e o r i g i n , X > 0. Applying the boundary condit ions (53) , w e have n
o r , denoting A by 1, n
The second of equations (53) w i l l be
Now
There f o r e
S u b s t i t u t i n g these values i n (54) we g e t
which reduces t o
{ ( A + l ) (1-V) A + [A (1-v,) - (3+V) 1 B) C O S ~ T = O . (56)
Equation (55) w i l l g ive
which reduces t o
2n-1 NOW i f cosAn = 0, o r A(') = -
2 , n = 1, 2, 3 , . .., t h i s w i l l n
imply t h a t
1 (n + 5) (1 -V)
The r e f ore w c o s ( n - ?)€I . n n ' 1
(2) On the o the r hand i f s i n A ~ = 0, o r A = n, n = 1, 2, 3, . . ., t h i s n
w i l l imply t h a t
From the above, t h e genera l s o l u t i o n of (52) s a t i s f y i n g (53) w i l l
1 1 (n +TI (1-V)
c 0 s ( n + ~ ) 8 - 3 cos (n- T) 8 4+ (14) (n - I 'n+" "-" cos (.-I) 81 1 . (581 + \rn+l [cos in+li 0 - (l-v) - ( 3+v)
This equation may a l s o be w r i t t e n i n the form
1 - 1 n+- cos (n + -) 8
2 - 1-v - 43 + (1-v) 3
n + - n - - 2 2 n--
2
cos (n+l) - + BP+' [ 1 -v cos (n-1) 8 n n(l-V) -(3+v) J I -
The s t r e s s e s a t any p o i n t ( r , 8) a r e now determined us ing (32) :
1 7 1 [ (n+?) -v (n --I 1 - 3 - - c 0 s ( n + ~ ) 8 -
5 cos (n --I 8
(n+T)-V(n- 3 2 2
where 2 1
An = an(n - 2 and Bn = bn n ( n + l ) ;
(n-3) -V (n+l) + B r -'Os (n+l)e + (n-3) -v (n+l) cos (n-1) 8
n
3 1 (n -5) (1-V)
= -2rz n= r 1 \Anrna s i n + - 2 0 + -- . 5 3 s i n ( n - 0 (n + T) -V (n - -1 2 I
E Using the r e l a t i o n G = - 2 ( 1 + V ) ' where G i s the modulus of e l a s t i c i t y
i n shea r , t h e s t r e s s e s w i l l be
1 7 1
(n+ T) -V (n -?I = - 2 ~ z r \ (n 0 - 5 3
cos (n - -1 8 'rr n= 1 3 (n+ -v (n I
7 1 (n - -v (n + -) 2
5 3 cos (n - 6 (n +z) -v (n - Z) I
+ Bnrn-'[ -cos (n+l) 0 + cos (11-11 8 1 1 I ' (60)
+ ~ ~ r ~ - ~ [-sin(nt1) 0 + (n- (n-1) 3) -V (1-V) (n+ 1) s i n (.-I) 01 ] . (61)
I n equations (59) - (61) ' w e n o t i c e t h a t the s t r e s s e s have the
same c h a r a c t e r i s t i c square r o o t s i n g u l a r i t y a s i n the case of extension
considered i n 52.2.
2.4 Plane s t r e s s e s i n a semi - in f in i t e s t r i p
I n the two problems considered s o f a r , Williams expressed the stresses
i n a s e r i e s of non-orthogonal eigenfunctions. This c r e a t e s considerable
d i f f i c u l t y when it comes t o obta in ing the c o e f f i c i e n t s i n the expansions
i n terms of the prescr ibed boundary values. I n so lv ing a s i m i l a r problem
f o r a semi- inf in i te s t r i p , Gaydon and Shepherd [ 5 ] expanded each of the
eigenfunctions i n a s e r i e s of orthogonal functions. This way it was
p o s s i b l e t o obta in the c o e f f i c i e n t s of the s t r e s s function corresponding
t o any a r b i t r a r y d i s t r i b u t i o n of t r a c t i o n on the end of t h e s t r i p d i r e c t l y
from two numerical matr ices.
2.4.1 Solut ion of t h e biharmonic equation
The problem is formulated a s fol lows. We have t o determine t h e stress
funct ion 4 which i s t h e s o l u t i o n of the biharmonic equation
and corresponds t o ze ro t r a c t i o n s on y = -+I. The t r a c t i o n s on x = 0
a r e p resc r ibed , s o t h a t 4 s a t i s f i e s t h e boundary condi t ions
We f u r t h e r assume t h a t $ + 0 as x -+ 03. Here we s h a l l consider p ( y )
and s (y) t o be even and odd funct ions of y respect ive ly .
Figure 15
F i r s t of a l l , we look f o r separable s o l u t i o n s ,
where X(x) , Y (y) a r e r e spec t ive ly funct ions of x only and y only.
Y (y) must be an even funct ion , s a t i s f y i n g the boundary condi t ions
Y (1) = Y ' (1) = 0 . Then equation (62) w i l l reduce t o
Dividing (65) by X Y and d i f f e r e n t i a t i n g wi th r e spec t t o x and y
we obta in
This implies t h a t e i t h e r
X " - = const . h2 say i)
2 = const. 11 say . ii)
2 I f Y" - p Y = 0, the even s o l u t i o n is
Y = A coshyy .
By applying the boundary condi t ions Y ( 1 ) = Y' (1) = 0, we obta in only
2 the t r i v i a l so lu t ion . I n the same way, i f 5 0 we obta in no non-
t r i v i a l so lu t ions .
Theref ore consider the d i f f e r e n t i a l equation
Its genera l so lu t ion is
Considering the requirement t h a t X -+ 0 a s x -t m, we must have A = 0.
This requirement w i l l a l s o exclude the p o s s i b i l i t y t h a t Re h = 0. Hence
S u b s t i t u t i n g (66) i n (65) we obta in the f o u r t h o rde r ordinary
d i f f e r e n t i a l equation
Its genera l s o l u t i o n is
Yly) = CA+BY) cosAy + (C+Dy) sinhy .
Considering the even s o l u t i o n , w e ob ta in
Yly) = A coshy + ~y s inhy . (67
N o w applying the boundary condi t ions Y C 1 ) = Y ' ( 1 ) = 0 on (-67) w e ge t
A and - = -tanA.
D
I f we take D = -21, t h e cons tant A w i l l be
The s t r e s s function may now be w r i t t e n as
where
yA (y) = (cos2A-1) COSAY - 2ky sinAy ,
and
a r e the roo t s of equation (68). The constants B a r e r e a l and A ~ ' A
a r e determined from the end s t r e s s d i s t r i b u t i o n s p ( y ) and s (y) , i .e.
and
Expansion of the s t r e s s funct ion i n terms of orthogonal beam functions
The functions YA(y) a r e n o t orthogonal , s o we expand each of them i n
terms of the beam funct ions Fk(y) which a r e complete and possess orthogonal
p r o p e r t i e s i n the range -1 y 1. Fk(y) s a t i s f i e s the equation
and the boundary condit ions
F k = F i = O on y = f l .
The genera l even s o l u t i o n of equation (74) is
(y) = A cosky + b coshky . Fk
Applying the boundary condit ions (75) , we obta in
tank + tanhk = 0
A coshk and - = - - B
. Therefore cosk
cosky coshky . (y) = cosk coshk 1-
Fk cosk coshk
The normalized so lu t ion w i l l be
1 3 where the norm i s { I [FLi)y]2 dy} . It is easy t o check the following
- 1 orthogonal i ty r e l a t i o n s which w i l l be u s e f u l i n the expansions
where 6 is t h e Kronecker d e l t a . mn
The functions Fk (y) a re r e a l , which s i m p l i f i e s t h e evaluat ion of
the c o e f f i c i e n t s ; furthermore, s i n c e they s a t i s f y a fourth-order d i f f e r -
e n t i a l equat ion , with the same four boundary condi t ions , a s do t h e o r i g i n a l
funct ions Yh (y) , they give r i s e t o expansions of the l a t t e r which a r e
considerably more convergent than would be obtained by Four ie r s e r i e s .
The expansion of YA(y) w i l l be
OD
Y (y) = (coszh-1) coshy - 2h s inhy = L a ~ ~ ( y ) . A i=l i h
Mult iplying both s i d e s by F and i n t e g r a t i n g from -1 t o +1, then j
We now have
and t h e s t r e s s e s w i l l be
Evaluat ion of the cons tants and -
Now i n o rde r t o eva lua te the cons tants Ah and BA, we expand the
given funct ions p (y) and s (y) i n terms of F" and F' r e spec t ive ly .
Thus
where
S imi la r ly
where
From equations (80) and (81) , w i t h x = 0, we obta in
and
These a r e s a t i s f i e d i f
and
Now
= a ' + ib' say . i A i A
Also l e t
and
Then a ' i A ' b i A can be w r i t t e n as
where
pA = ah s i n a A coshbA - b cosa s inhb X A A
q A = -b s i n a coshb A A A -aA cosaX sinhb A
rA = cosa coshb A A
sA = s i n a s inhb A A .
Equations (84) and (85) now reduce t o
and
E{A (2aX a i X +2bX b j X ) + BX(2bX a i X - 2a b i X ) 1 = -Bi . A X
I f we p u t 2a iX = CiA , -2bfh = DiX I
2a a j X + 2bX b i A = EiX , X 2b a j X - 2aX b i X = FiX I A
then
The matrices (C. ) , (DiX) , (EiX) , (F. ) a r e known, and s o (86) and l h l h
(87) provide an i n f i n i t e system of l i n e a r equations f o r t h e unknown
c o e f f i c i e n t s (AX) , (BX) . Gaydon and Shepherd [ 5 1 computed numerically an
approximate matrix M such t h a t
Therefore f o r any boundary condi t ions p (y) and s (y) , t h e cons tants
A and B can be evaluated. Knowing these cons tants , the s t r e s s e s a t X X
any p o i n t (x, y) can now be e a s i l y evaluated.
2.5 The s e c t o r problem
Gopalacharyulu [6 ] determined t h e s t r e s s f i e l d f o r the p lane defor-
mation of a s e c t o r wi th s t r e s s - f r e e r a d i a l edges and given s e l f - e q u i l i b r a t i n g
loads on the c i r c u l a r boundary. The method followed was s i m i l a r t o t h a t
given by Gaydon and Shepherd [5 ] i n so lv ing the rec tangular s t r i p problem.
The s t r e s s function s a t i s f y i n g the biharmonic equation and t h e t r a c t i o n -
f r e e condit ions on the r a d i a l edges i s i n i t i a l l y determined a s a s e r i e s
of non-orthogonal eigenfunctions. Each of these eigenfunctions is again
expanded i n a s e r i e s of orthogonal funct ions s a t i s f y i n g a fourth-order
d i f f e r e n t i a l equation and t h e same boundary condit ions.
2.5.1 Solut ion of the biharmonic equation
The s e c t o r occupies the region -w 8 w, 0 5 r < 1. The s t r e s s
funct ion 4 has t o s a t i s f y the biharmonic equation
From equation (431, the separable s o l u t i o n s a r e of t h e form
and
(Here we a r e considering again only t h e symmetric so lu t ion . ) The function
F(8) has t o s a t i s f y the boundary condit ions
which represent the t r ac t ion- f ree condi t ions along the r a d i a l edges. The
shear and normal s t r e s s e s a r e s p e c i f i e d along the c i r c u l a r boundary,
= 010) and are = T (8) .
The boundary condit ions (88) imply t h a t
where the eigenvalues \ a r e determined from the t ranscendenta l equation
( \ + l ) s i n ( \ + l ) w cos ( A - 1 ) w - (Ak-1 ) sin(Ak-1) w cos (Ak+l)w = 0 . k
This equation may be w r i t t e n as
(\+l) sin2w + 2 s i n ( - 1 ) w cos ( A + l )w = 0 . "k k
Let pk = \ + l , s o t h a t the genera l s t r e s s function @ w i l l there-
f o r e be the l i n e a r combination of the eigenfunctions @k -
where F (8) i s given by k
The eigenvalues Pk a r e themselves determined from
p sin2w + 2 s i n ( p -2)w cosp w = 0 . k k k
2.5.2 Expansion of the a r c t r a c t i o n s i n terms of the beam funct ions
The constants % a r e determined from the appl ied loads on t h e
c i r c u l a r boundary 0 (8) and T (8) , equation (89) . Due t o t h e l ack of
or thogonal i ty of the functions % (8) , each Fk(e) i s expanded i n terms
o f the orthogonal beam funct ions ( 8 ) which a r e the s o l u t i o n s of the
four th o rde r d i f f e r e n t i a l equation
and s a t i s f y t h e boundary cond i t i ons
So lu t ions o f t h e d i f f e r e n t i a l equat ion (93 ) , s a t i s f y i n g the boundary
cond i t i ons (94) a r e
where
and t h e e igenvalues ' m s a t i s f y t h e equat ion
There e x i s t o r thogona l i t y r e l a t i o n s s i m i l a r t o those given i n (78),
namely
where 6 is t h e Kronecker d e l t a . mn
NOW expanding t h e func t ions Fk (8) i n a s e r i e s of the orthogonal
functions $m(e) , we ob ta in
%m a r e the Four ier c o e f f i c i e n t s given by
"m + - cosp w cos(p -2)w tanhum w k k 1
I n de r iv ing t h i s expression we have made use of equation (96).
The s t r e s s function given i n (90) can now be w r i t t e n as
Using the expressions f o r the s t r e s s e s i n ( 5 ) , we g e t on the a r c r = 1
and
Le t
and
Then equat ions (99) and (100) may be w r i t t e n a s
Mul t ip ly ing bo th s i d e s o f (1041 by I)" ' (8) and i n t e g r a t i n g between n
-w t o +w
From which
(1) 'n - o = - ( q 4 ~rce, @;l(e) de = Kn s a y . n (105)
'n -w
To overcome t h e d i f f i c u l t y o f non-orthogonal i ty of JI and $J; i n (103) m
Gopalacharyulu [6] m u l t i p l i e d bo th s i d e s o f t h i s equat ion by (JIn - and
i n t e g r a t e d between -w t o +w t o g e t one more r e l a t i o n between the
cons t an t s Cn and Dn
Thus, we have
From (103) and (104), the coeff ic ients Cn and D are n
2.5.3 Determination of %
I n order t o determine % l e t
qc = ek + i f k
%m = gkm + i h
km
pk = % + i B k
Hence, expressions (101) and (102) w i l l be
The r i g h t hand s i d e s o f these two equations a r e known q u a n t i t i e s ,
from (107) and (108) , and the c o e f f i c i e n t s (and hence and
hkm) a r e given e x p l i c i t l y i n (98) . Hence equations (110) provide an
i n f i n i t e system of l i n e a r a l g e b r a i c equations f o r e and fk . k
Gopalacharyulu [ 6 ] obta ins approximate numerical s o l u t i o n s of t h i s
system of equations by r e t a i n i n g only a f i n i t e number of the c o e f f i c i e n t s
\ , and shows t h a t q u i t e good r e s u l t s a r e obtained by using only very
few non-zero % I s .
CHAPTER I11
SOLUTION OF THE CRACKED CYLINDER AND SEMI-CIRCLE PROBLEMS
I n t h i s chapter we s h a l l complete the s o l u t i o n of t h e cracked
cy l inder problem described i n 52.2. The method used w i l l be s impler
than t h a t of Gaydon and Shepherd [5] and Gopalacharyulu [6] i n t h a t
Four ier cosine and s i n e s e r i e s a r e used r a t h e r than beam eigenfunctions.
I t does have the disadvantage of leading t o q u i t e slow convergence of the
s e r i e s expansions employed, b u t , a s we s h a l l s e e , reasonably accura te
approximate r e s u l t s a r e obtained by r e t a i n i n g only a few terms i n t h e
expansions.
In l a t e r sec t ions of the chapter we examine t h e problem of a crack
with a rounded t i p and the semi-circle problem using e s s e n t i a l l y the same
method. Numerical s o l u t i o n s have no t been obtained however f o r these
problems .
The cracked cy l inder problem
We wish t o so lve t h e problem of t h e plane s t r a i n deformation of a
cy l inder 0 5 r < 1 under given t r a c t i o n s on r = 1 and i n t h e case when
t h e r e is a p lane crack running from t h e a x i s o f the cy l inder t o t h e boundary
on the p lane 8 = ?IT. I n 52.2 it was shown t h a t , i f the two faces of the
crack a r e t r ac t ion- f ree , t h e Airy s t r e s s funct ion has an expansion of the
form (47) and the s t r e s s components 0 (r, 8) , Ore (r, 8) and rr
C l e o ( r , 0) a re given by (48) - (51) . The constants
An and Bn have y e t t o be determined from the given
59
t ract ions on the c i rcu lar boundary r = 1. The d i f f icu l ty here i s t h a t
these s t resses are expanded i n a non-orthogonal se r i e s of functions of 8,
which does not allow us immediately t o determine the constants An
and
Bn. Therefore we s h a l l expand each of these functions i n a simple Fourier
cosine se r i e s i n the in te rva l [-IT, IT], i n the case of the normal s t r e s ses ,
and s ine ser ies i n the case of shear s t resses . Hence the s t resses on the
c i rcu lar boundary r = 1 w i l l be writ ten as
The coefficients Ok and nd a re the Fourier coefficients defined k
and r are known once the boundary t ract ions are prescribed. k
Using (49) , (51) these coefficients are a l so given as
l. where t h e r i g h t hand s i d e s have been obtained by expanding cos (n + Z) 8
3 1 3 and cos (n --) 8 as Four ier cosine s e r i e s and s i n (n + -) 8 and s i n (n - -) 8
2 2 2
as Four ier s i n e s e r i e s and then e x t r a c t i n g the c o e f f i c i e n t s of coske i n
'rr (1, 8) and of s ink8 i n a r , 1). I n these equat ions , we def ine r 8 -
Bo = 0.
3.1.1 S a t i s f a c t i o n of the condit ions of o v e r a l l se l f -equi l ibr ium
We assume t h a t the cyl inder is i n se l f -equi l ibr ium under the prescr ibed
s t r e s s e s on the c i r c u l a r boundary r = 1. The t h r e e condi t ions of e q u i l i -
brium i n a p lane a r e
IT
i 1 [[a rr (1, e)] cos8 - [ore (1, 8) ] s ine ] = o ,
iii) r are 0, 8) d8 = 0 . -IT
Since 'rr
(1, 8) and ore (1, 8) a r e even and odd funct ions of 8
r e spec t ive ly , condit ions (ii) and ( i i i ) a r e r e a d i l y s a t i s f i e d .
Condition (i) i s equ iva len t t o al = TI. From (116)
1 n - -
- - n 2 - C -1) An '1 n = l n + - [ n
and from (117)
Thus a l l t he s e l f - e q u i l i b r a t i n g condi t ions a r e s a t i s f i e d .
3.1.2 Separat ion of the cons tants and Bn An -
I n equations (115) - (117) t h e cons tants A n and Bn a r e mixed.
I t is poss ib le though t o separa te them and t h i s s i m p l i f i e s by a g r e a t
amount the determination of these cons tants . In separa t ing these cons tan t s ,
we s h a l l express Bn
i n terms of An . Thus, from (115).
Adding (116) and (117)
L e t k -t k - 2 , we ob ta in
There f o r e Bk+ 1
and Bk-l w i l l be given by
By s u b s t i t u t i n g back i n (116) we obta in
This equation may be w r i t t e n a s
From equation (116) wi th k = 2
Using equation (118), we ob ta in
Equations (121) and (122) c o n s t i t u t e an i n f i n i t e system of simultaneous
equations from which An
may be determined. Once t h e s e have been deter -
mined, the s e t of cons tants Bn can be obtained from (118) and (119).
I n determining the c o e f f i c i e n t s An, Gauss-Seidel i t e r a t i o n method has
been used.
3 . 1 . 3 S t r e s s i n t e n s i t y f a c t o r
The s t r e s s i n t e n s i t y f a c t o r i s defined by
+J KI = l i m r Gee ( r , 0) .
r t o
Using expression (51) f o r the t a n g e n t i a l s t r e s s we ob ta in :
3.1.4 Examples
i) On the boundary r = 1, we apply a cons tant normal s t r e s s o f u n i t
"0 2
- 1.0, ok = 0, k = 1, 2, ... . We assume t h a t the re magnitude, i . e . - -
is no shea r on the boundary, i n o t h e r words T = 0 , k = 1, 2, 3 , . . . . k
I n equations (121) and (122) we s h a l l r e t a i n 100 terms (n = 1, 2 , . . . , 100) and use t h e equations with k = 2, 3 , ..., 101, s o t h a t w e have 100
simultaneous equations with a 100x100 matrix of c o e f f i c i e n t s . An IBM 370
computer model 155 i s used t o determine {A n }loo n=l ' {B }loo a r e determined n n = l
from (118) and (119).
Having found {A } and { B ~ } , t h e s t r e s s d i s t r i b u t i o n may be ca lcu la ted n
from (49) - (51) . We l e t t h e r ad ius r vary wi th 0.1 i n t e r v a l s and t h e
angle 8 with f i v e degrees i n t e r v a l s i n these expressions f o r t h e s t r e s s e s ,
thus g e t t i n g the s t r e s s d i s t r i b u t i o n throughout the cyl inder . The r e s u l t i n g
s t r e s s - f i e l d s a r e drawn i n Fig. 16, which shows l i n e s o f cons tant 0 r r r 're
and 0 . I t can be seen t h a t a and 0 approach t h e i r r e spec t ive r 8 rr
given boundary values on r = 1 and 8 = f?T, while they diverge as r + 0.
LINES OF CONSTANT NORMAL STRESS Orr
. . FIGURE 16 Ci?
LINES O F CONSTANT SHEAR STFU3SS Or,
FIGURE 16 (ii)
ii) Secondly we s h a l l c a l c u l a t e t h e s t r e s s i n t e n s i t y f a c t o r KI provided
by each of the Four ier c o e f f i c i e n t s {ok, Tk} o f t h e app l i ed s t r e s s d i s t r i -
but ion . Therefore we keep t h e shea r a s zero , and take each of the O k l s
equal t o uni ty i n t u r n while keeping the remaining ones a s ze ros , i . e.
I n the case of r = 1, f o r se l f -equi l ibr ium we had al = T 1 ' hence t h i s
condit ion may be w r i t t e n as
For each r , we have computed t h e corresponding s t r e s s i n t e n s i t y f a c t o r
using (1231, and they a r e l i s t e d i n Table I. We have only considered
t h e f i r s t 25 values of r , s i n c e f o r h igher 0 and T it becomes r r
inc reas ing ly necessary t o inc lude more than 100 non-zero c o e f f i c i e n t s
I n exac t ly the same way, we t a k e t h e normal s t r e s s t o be ze ro , and
t ake each Tk equal t o one i n t u r n , wi th t h e provis ion t h a t ol = T 1 '
ri = 6il , o1 = 6 wi th i = 1, 2 , . . . , 101 . i 1
The corresponding s t r e s s i n t e n s i t y f a c t o r s a r e l i s t e d i n Table 11.
3.2 S t r e s s d i s t r i b u t i o n around a crack with a rounded t i p
I t i s more r e a l i s t i c t o allow t h e crack t o have a rounded t i p , r a t h e r
than t h e i n f i n i t e l y sharp t i p considered s o f a r . We model t h i s s i t u a t i o n
by supposing t h a t the crack extends from a cy l inder o f smal l r ad ius R
around t h e i n f i n i t e cy l inder a x i s , t o the e x t e r n a l boundary with u n i t
radius . Thus the s i n g u l a r i t y a r i s i n g a t the cy l inder a x i s i n t h e previous
problem w i l l no t show up. The region under cons idera t ion i s R < r < 1,
-IT c 8 < IT. The boundary condi t ions x=l
a r e o , , = o = O on 8 = + 7 r , r 8
'rr and O a r e p r e s c r i b e d o n r = 1 .
r 8
The s t r e s s e s expressed i n terms o f Airy ' s
s t r e s s function a r e given by (5) wi th
the funct ion @ s a t i s f y i n g the biharmonic
4 equation V 4 = 0.
Figure 17
Considering again the separable s o l u t i o n s
we look f o r symmetric s o l u t i o n s t h a t w i l l s a t i s f y t h e condi t ions of zero
t r a c t i o n on the crack faces 8 = +IT, namely
Having found t h i s s e t of s o l u t i o n s , t h e genera l problem wi th p resc r ibed
t r a c t i o n s on r = R and r = 1 is solved by t ak ing a l i n e a r combination
of these separable so lu t ions . The c o e f f i c i e n t s i n t h i s l i n e a r combination
w i l l be found i n terms of the given t r a c t i o n s .
3.2.1 Solut ion of t h e biharmonic equation
I t was determined i n 32.2 t h a t t h e s o l u t i o n s t o the biharmonic
equation s a t i s f y i n g the zero t r a c t i o n condi t ions on t h e crack faces may
be w r i t t e n as
Using equation (47) we may w r i t e t h e s t r e s s funct ion a s
where
1 n+-
1 2 .+i) j.ol;;, 0 cos
$n(r , 0) = [ ' r n + ~ ; r - 3 (124) n-- 2
-n+ 1 + (c' rn+' + D: r ) [cos (n-1) 0 - cos (n+l) 01 . n
I n de r iv ing (124) , w e have taken a s i n s e c t i o n 2.2
2n-1 A(') = - 2
, n = 1 , 2 , 3 , . . . , n
Here we note t h a t , a s i n 52.2, f o r t h e case of A = 0, we g e t only the
t r i v i a l so lu t ion . It should a l s o be noted t h a t the terms which become
s i n g u l a r a s r + 0 cannot be r e j e c t e d f o r t h e p resen t problem.
67
From the s t r e s s funct ion given by (124), we ob ta in the s t r e s s e s (5)
3 cos (n+- 1 ) 8 cos (n-Z 3 8 n-- 3
r c-n+-) B I r-n-'] [ 2 ( r r 8) = nLl -
'rr 2 n 1 n+- 3
2 n--
2
3 n--
+ [ A ' r n 2+Blr-n-:] n - (n+-) 2 cos (n+-) 1 2 8+ (n-2) 2 cos (n--) 2 8 I
where
+ ~ ' r ~ - ' + D ' r 2 2
[ n n Cn-1) cos (n-1) 8+ (n+l) cos (n+ l ) 8
1 -n-- 5 n+-
2 1 + B r - cos (n+-) 8 + cos
n 2
+ D r n
[-(n-1) cos (n-1) 8 + (n+3) cos (n+l) 8 I1 (125)
and we def ine C f D = 0 . 0 0
Simi la r ly the shear s t r e s s w i l l be given by
3.2.2 S a t i s f a c t i o n of t h e boundary condi t ions on t h e boundaries r = R, r = 1
Expanding the normal and shea r s t r e s s e s i n Four ie r cosine and s i n e
s e r i e s r e spec t ive ly , we obta in
where Uk , k = 0 , 1, 2, ..., a r e t h e Four ier c o e f f i c i e n t s def ined by
1 Uk (r) = 7 in Urr ( r , 8) cosk0 d0 .
Simi la r ly ore ( r , 0) = C T Cr) sink0 where k = l k
IT
r = Ore k, 8) sink8 d0 , k = 1, 2 , ... . k
-IT
Therefore, f o r k = 1, 2, 3 , ...
where
For k = 0 , we g e t
I n order t o complete the s o l u t i o n of t h e o r i g i n a l l y posed problem,
w e now s e t o (R) = ' r k ( ~ ) = 0 , s i n c e t h e t r a c t i o n s on the inner boundary k
r = R a r e given t o be zero; and s e t ok ( l ) and 'Ck (1) equal t o t h e
corresponding Four ier c o e f f i c i e n t s of the given normal and shea r t r a c t i o n s
on the e x t e r n a l boundary. The r e s u l t i n g system of equations can be solved
numerically a s i n s e c t i o n 3.1.
3.2.3 S a t i s f a c t i o n of condi t ions of o v e r a l l equi l ibr ium
Here it w i l l be s u f f i c i e n t t o s a t i s f y t h e condit ion = 1 on r = l ,
s ince - brr - ore = 0 on r = R . Therefore on r = 1, using equations
(127) and (128) , we obta in
and
Hence our Four ier c o e f f i c i e n t s do s a t i s f y t h e condit ions of equil ibrium.
3 . 3 S t r e s s d i s t r i b u t i o n i n a semi-c i rcular s e c t o r
I n the case of a semi-circular boundary, t h e formulation of the problem
w i l l be a s follows :
We have t o determine a s t r e s s funct ion @ which is biharmonic i n t h e region
IT IT - - < % < - I O < r < l 0
2 0 a r e p resc r ibed on r = 1, 2 r r r r e '0%' 're
IT a r e zero on 0 = f -
2 -
Figure 18
Again, t h e s t r e s s function @ i s w r i t t e n as
, 0) = r1+I ~ ( 0 )
k- r e h > 0 t o s a t i s f y boundedness of t h e s t r a i n energy dens i ty i n the
neighbourhood of the o r ig in .
The function F i n the symmetric case (equation ( 4 5 ) ) i s
F ( 0 ) = A cos ( h + l ) 8 + B cos ( A - 1 ) 8 .
I n order t o have zero t r a c t i o n s on 0 = &'IT, we must have
'JT 'IT F(-) = F' = 0 . 2
Applying these boundary condi t ions on (45) , we ob ta in
Here we have two cases ;
(i) I f A # B , t h i s implies t h a t
IT (1) s inh - = 0 o r An = 2n .
2
This l eads t o
Cii) A = B and t h i s impl ies t h a t
Therefore the s t r e s s function may now be w r i t t e n as
2n+1 cos (2n+1) 0 + cos C2n-1) 0 $k t 0 ) = n 2 1 ~ ~ ~ r I 2n+1 2n-1
On t h e boundary r = 1, the normal and shea r s t r e s s e s a r e found t o be
where A = 2n A ' and Bn = 2 (2n-1) BA. n n
I n order t o determine An and Bn, we expand (130) and (131) i n
Four ie r cosine and s i n e s e r i e s r e spec t ive ly , t h e s e r i e s having ranges
IT IT (- - -1. Hence
2' 2
O o
'rr (1, 8) = - + Z a cos2k8 ,
2 k = l k
where ak , k = 0, 1, 2 , .. . a r e t h e Four ier c o e f f i c i e n t s def ined by
Also,
where T k t k = 1, 2, 3,. . a r e given by
The expressions f o r ak and T a r e given by k
where
2 2 2 Dnk
= [4k2 - (2n+l) ][4k - (2n-1) 1 .
3.3.1 Separat ion of An and Bn
From (128), we can express B1 i n terms of An a s
8 n - 1 - - 1 (-1) An '0
2 + -
B1 n n = l 2 (2n+l) (2n-1)
and from (133) and (134), w e ob ta in
S u b s t i t u t i n g t h e expressions f o r Bk given by (135) - (137) i n equations
(132) - (133) and rearranging t h e terms, we ob ta in t h e i n f i n i t e system of
equations (138) - (139) :
"0 8 - - - C (-1," An -2n+5 CT + - -
1 2 'rr n = l (2n+3) ( l n + l ) (2n-1)
2 I
The system of equations (138) - (139) is solved i n the same way as i n
s e c t i o n 3.1. The constants Bk a r e determined from (135), (136) and
the s t r e s s e s a r e r e a d i l y evaluated.
3.3.2 S a t i s f a c t i o n of se l f -equi l ibr ium condi t ions
I n order t o s a t i s f y se l f -equi l ibr ium, we must have
This impl ies t h a t
I n posing any boundary value problem, {CTk , rk} must be chosen t o be
c o n s i s t e n t wi th t h i s condit ion.
I t i s easy t o check using the t r igonometr ic expansion
t h a t the Four ier c o e f f i c i e n t s given by (132) - (134) do s a t i s f y
condit ion (140) .
TABLE I:
TABLE I1
BIBLIOGRAPHY
[ 1 1 Sokolnikof f, I. S. , ath he ma tical Theory o f E l a s t i c i t y , McGraw-Hill, 1956.
[ 2 ] Timoshenko, S. and ~o inowsky -Kr i ege r , S . , Theory o f P l a t e s and S h e l l s , McGraw-Hill, 1959.
[ 3 ] Wi l l i ams , M. L. , J. Appl. Mech. - 24, (1957) , 109-114.
[ 4 ] Wi l l i ams , M. L. , J. Appl. Mech. - 28, (.1961), 78-82.
[ 5 ] Gaydon, F. A. and Shepherd, FT. M . , Proc. Roy. Soc. A281, (1964) , 184-206.
[ 6 ] Gopalacharyulu, S . , Quar t . J. Mech. and Appl. Math. - 22, (1969) , 305-317.