INTEGRITY ANALYSES OF SURFACE - FLAWED AIRCRAFT
ATTACHMENT LUGS: A NEW, INEXPENSIVE, 3-D ALTERNATING METHOD
82-0742
T. N i s h i o k a and S.N. A t l u r i
C e n t e r f o r t h e Advancement o f
Computational Mechanics
School o f C i v i l E n g i n e e r i n g
Georgia I n s t i t u t e o f Technology
A t l a n t a , GA 30332
Abstract
,--A new a l t e r n a t i n g method f o r t h e a n a l y s i s o f
a q u a r t e r - e l l i p t i c a l c o m e r c r a c k i s developed. The
completely g e n e r a l a n a l y t i c a l s o l u t i o n f o r a n embedded c r a c k , i n an i n f i n i t e s o l i d , s u b j e c t t o a r b i t r a r y c r a c k - f a c e t r a c t i o n s , i s implemented i n t h e
p r e s e n t a l t e r n a t i n g method.
The p r e s e n t f i n i t e e l e ment a l t e m a t i n g n e t h o d r e s u l t s i n a n i n e x p e n s i v e p r o c e d u r e f o r r o u t i n e e v a l u a t i o n of a c c u r a t e s t r e s s i n t e n s i t y f a c t o r s f o r flawed s t r u c t u r a l components.
The p r e s e n t a l t e r n a t i n g method i s a p p l i e d t o
t h e a n a l y s e s o f v a r i o u s s h a p e s of q u a r t e r - e l l i p t i c a l
c o r n e r c r a c k s ( i ) i n a b r i c k s u b j e c t t o remote t e n s i o n , ( i i ) emanating from a h o l e i n f i n i t e - t h i c k n e s s
p l a t e s , s u b j e c t t o remote t e n s i o n as w e l l a s b e a r i n g
p r e s s u r e , and ( i i i ) emanating from a p i n h o l e i n
a i r c r a f t attachment lugs s u b j e c t t o simulated pin
l o a d i n g . The r e s u l t s f o r t h e problem ( i ) and ( i i )
a r e compared w i t h t h o s a v a i l a b l e i n l i t e r a t u r e .
For
t h e problem ( i i i ) t h e s t r e s s i n t e n s i t y f a c t o r s , and
t h e i r p a r a m e t r i c v a r i a t i o n s f o r t h e c o r n e r c r a c k s of
various shapes a r e presented.
1.
Introduction
A knowledge o f a c c u r a t e s t r e s s i n t e n s i t y f a c t o r s i s e s s e n t i a l f o r a p r o p e r i n t e g r i t y a n a l y s e s of
flawed s t r u c t u r e s .
C o m e r c r a c k s a t h o l e s , such a s
i n a i r c r a f t a t t a c h m e n t l u g s , have r e c e i v e d much a t t e n t i o n due t o t h e f a c t t h a t t h e y a r e among t h e most
common f l a w s i n a i r c r a f t s t r u c t u r a l components.
Analyses of c o m e r c r a c k i n a i r c r a f t a t t a c h m e n t . ~ , s
a r e , needless t o say, t h r e e dimensional i n nature,
w h i l e i n most s t u d i e s , s o f a r , two-dimensional a n a l y s e s have been employed a s i n Refs. [ 1 , 2 ] .
For t h e t h r e e - d i m e n s i o n a l a n a l y s e s o f c o m e r
c r a c k s a t h o l e s , A t l u r i and K a t h i r e s a n [ 3 ] used a
h y b r i d 3-D c r a c k element f o r d i r e c t l y e v a l u a t i n g t h e
stress i n t e n s i t y f a c t o r s a l o n g t h e crack-border.
Hechmer and Bloom [ 4 ] , Raju and Newman [ 5 ] used 3-D
s i n g u l a r i t y wedge e l e m e n t s , i n which t h e s t r e s s i n t e n s i t y f a c t o r s were i n d i r e c t l y e x t r a c t e d from computed r e s u l t s such a s t h e nodal displacements o r
nodal forces.
Smith and K u l l g r e n [ 6 ] used a f i n i t e
element a l t e r n a t i n g method i n which t h e a n a l y t i c a l
s o l u t i o n f o r an e l l i p t i c a l c r a c k i n an i n f i n t e s o l i d ,
subject t o a cubic pressure d i s t r i b u t i o n [7] wasused
t o obtain t h e s t r e s s intensity factors. Heliot,
Labbens, and P e l l i s s i e r - T a n o n [ 8 ] used t h e boundary
i n t e g r a l e q u a t i o n method. On t h e o t h e r hand, f o r
t h e t h r e e - d i m e n s i o n a l a n a l y s e s of c o r n e r c r a c k s i n
a t t a c h m e n t l u g s , v e r y few s o l u t i o n s a r e a v a i l a b l e
in literature.
A r e c e n t comprehensive s t u d y [ 9 ] r e v e a l e d t h a t
w h i l e t h e "3-D h y b r i d c r a c k element" method [ 3 , 1 0 , 1 1 ]
y l c l d s b e t t e r a c c u r a c i e s t h a n t h e a l t e r n a t i n g method,
[12,13] t h e a l t e r n a t i n g method y e t remained a potent i a l l y cheaper t e c h n i q u e i f i t can b e improved. One
o f t h e major impediments t o o b t a i n i n g a c c u r a t e s o l u t i o n s through t h e a l t e r n a t i n g technique has b e e n t h a t
Copyrighl @ American lnstitule of Aeronautics and
Astronautics, Inc., 1982. All rights reserved.
t h e s o l u t i o n f o r a n embedded e l l i p t i c a l c r a c k i n a n
implementation o f t h e a l t e r n a t i n g method, h a s been
l i m i t e d o n l y t o a c u b i c normal p r e s s u r e v a r i a t i o n
on c r a c k s u r f a c e [ 7 ] .
R e c e n t l y , a major improvement o f t h e a l t e r n a t i n g t e c h n i q u e h a s been made by t h e p r e s e n t a u t h o r s
[14,15].
I n t h e new a l t e r n a t i n g method [ 1 4 , 1 5 ] , t h e
complete, g e n e r a l a n a l y t i c a l s o l u t i o n [15,16] f o r a n
embedded e l l i p t i c a l c r a c k i n a n i n f i n i t e s o l i d , subj e c t t o a r b i t r a r y t r a c t i o n s (normal a s w e l l a s
s h e a r ) on t h e c r a c k s u r f a c e was implemented i n conI t was
j u n c t i o n w i t h t h e f i n i t e element method.
demonstrated t h a t t h e new f i n i t e element a l t e r n a t i n g method y i e l d e d a c c u r a t e s o l u t i o n s of s t r e s s
i n t e n s i t y f a c t o r s and i s a p p r o x i m a t e l y one o r d e r
o f magnitude l e s s e x p e n s i v e i n computing c o s t s a s
compared t o t h o s e w i t h t h e h y b r i d f i n i t e element
method [ 3 , 1 0 , 1 1 ] , and o t h e r t e c h n i q u e s c u r r e n t l y
reported i n l i t e r a t u r e .
I n t h e p r e s e n t p a p e r , u s i n g t h e new f i n i t e
element a l t e r n a t i n g method 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 presented f o r q u a r t e r - e l l i p t i c a l corner cracks
of various shapes ( i ) i n a f i n i t e - t h i c k n e s s p l a t e
( b r i c k ) s u b j e c t t o remote t e n s i o n , ( i i ) a t t h e edge
of a h o l e i n f i n i t e - t h i c k n e s s p l a t e s s u b j e c t t o
remote t e n s i o n a s w e l l a s b e a r i n g p r e s s u r e l o a d i n g ,
and ( i i ) a t t h e edge of a p i n h o l e i n a i r c r a f t a t tachment l u g s s u b j e c t t o s i m u l a t e d p i n l o a d i n g .
The p r e s e n t r e s u l t s f o r t h e problems ( i ) and ( i i )
a r e compared w i t h o t h e r r e s u l t s a v a i l a b l e . For t h e
problem ( i i i ) t h e s t r e s s i n t e n s i t y f a c t o r s , and
t h e i r parametric v a r i a t i o n s f o r t h e c o m e r cracks
of v a r i o u s s h a p e s a r e p r e s e n t e d .
2.
Analytical Solution f o r an E l l i p t i c a l
Crack i n a n I n f i n i t e S o l i d w i t h A r b i t r a r y Crack-Face T r a c t i o n s
I n t h i s s e c t i o n , o n l y t h e Mode I problem i s
c o n s i d e r e d . The complete, g e n e r a l s o l u t i o n i n c l u d i n g t h e Modes I1 and I11 i s g i v e n i n Refs.
[15,16].
Suppose t h a t xl and x2 a r e c a r t e s i a n
c o o r d i n a t e s i n t h e p l a n e of t h e e l l i p t i c a l c r a c k
and x3 i s normal t o t h e c r a c k p l a n e s u c h t h a t
d e s c r i b e s t h e b o r d e r of t h e e l l i p t i c a l c r a c k of a s p e c t r a t i o ( a / a ) . The e l l i p t i c a l c o o r d i n a t e s
1 2
5, (,=1,2,3) a r e d e f i n e d a s t h e r o o t s of t h e c u b i c
equation:
L e t t h e normal t r a c t i o n a l o n g t h e c r a c k s u r f a c e b e e x p r e s s e d i n t h e form:
S a t i s f y i n g t h e boundary c o n d i t i o n on t h e c r a c k
s u r f a c e , t h e r e l a t i o n between t h e c o e f f i c i e n t s A
and C can b e summarized i n a m a t r i x form:
where A ' s a r e undetermined c o e f f i c i e n t s and t h e
p a r a m e t e r s i and j s p e c i f y t h e symmetries of t h e
load with respect t o t h e axes of t h e e l l i p s e , x
1
and x
2'
The s o l u t i o n c o r r e s p o n d i n g t o t h e l o a d exp r e s s e d by Eq. ( 3 ) can b e assumed in t e r m s of t h e
p o t e n t i a l function:
where
{A} = [B] {C}
Nxl
NxN Nxl
The d e t a i l e d c o m p l e t e e x p r e s s i o n of components o f
[B] i s g i v e n i n Ref. [ 1 5 ] .
F o r a c o m p l e t e polynomial l o a d i n g e x p r e s s e d by
Eq. ( 3 ) , t h e maximum d e g r e e of polynomial (MXDOP)
and t h e t o t a l number of c o e f f i c i e n t s N can b e exp r e s s e d , r e s p e c t i v e l y , by MXDOP=ZM+l and N=(M+l)
(2M+3)3.
F o r an i n c o m p l e t e polynomial l o a d i n g i n
which t h e symmetries of problem a r e a c c o u n t e d f o r ,
t h e maximum d e g r e e o f polynomial and t h e number o f
c o e f f i c i e n t s depend on t h e p a r a m e t e r M b u t a l s o t h e
p a r a m e t e r s i and j i n Eqs. ( 3 ) and ( 4 ) .
Once t h e c o e f f i c i e n t s C a r e d e t e r m i n e d by
s o l v i n g Eq. (10) f o r a g i v e n l o a d i n g on t h e c r a c k
surface, t h e s t r e s s i n t e n s i t y f a c t o r s corresponding
t o t h i s l o a d can b e e v a l u a t e d from t h e f o l l o w i n g
equation [15,16]:
and
and C ' s a r e a l s o undetermined c o e f f i c i e n t s . The
components of d i s p l a c e m e n t u and s t r e s s 0 . . i n
i
1J
t e r m s of f 3 a r e g i v e n by:
where 0 i s t h e e l l i p t i c a n g l e measured from xl a x i s ,
and
2
2
A = a sin 8
1
3.
+
2
2
a cos e
2
(12)
F i n i t e Element A l t e r n a t i n g Method
The a l t e r n a t i n g method f o r e l l i p t i c a l c r a c k
problem was o r i g i n a l l y developed by Shah and
Kobayashi [ 1 2 , 1 3 ] .
I n t h e i r method [ 1 2 , 1 3 ] , t h e
s o l u t i o n f o r an e l l i p t i c a l c r a c k , s u b j e c t t o a
c u b i c polynomial p r e s s u r e d i s t r i b u t i o n , i n a n
i n f i n i t e s o l i d was implemented.
S u b s e q u e n t l y Smith
e t . a l . [ 6 ] i n t r o d u c e d t h e f i n i t e element t e c h n i q u e i n t o t h e a l t e r n a t i n g method, employing t h e
same s o l u t i o n [ 7 ] used by Shah and Kobayashi [ 1 2 ,
131. The l i m i t a t i o n t o a c u b i c polynomial p r e s s u r e was o n e of t h e major impediments t o o b t a i n i n g
a c c u r a t e s o l u t i o n s through t h e a l t e r n a t i n g technique.
and
The p r e s e n t a l t e r n a t i n g method u s e s two b a s i c
s o l u t i o n s a s follows [14,15]:
where v and v a r e t h e s h e a r modulus and P o i s s o n ' s
ratio.
For l a t e r c o n v e n i e n c e , t h e s t r e s s e s g i v e n
by Eq. ( 8 ) t h r o u g h Eqs. (4-6) a r e e x p r e s s e d i n a
m a t r i x form:
Io}
6x1
=
[PI { C }
6xN Nxl
where [PI i s t h e f u n c t i o n of t h e c o o r d i n a t e s
(x1,x2,x3) and N i s t h e t o t a l number of c o e f f i c i e n t s
A o r C.
S o l u t i o n 1: The complete, g e n e r a l a n a l y t i c a l s o l u t i o n f o r an e l l i p t i c a l crack subject t o a r b i t r a r y '
l o a d i n g s on t h e c r a c k s u r f a c e , i n a n i n f i n i t e
s o l i d , a s e x p l a i n e d in t h e p r e v i o u s s e c t i o n and i n
Ref. [ 1 5 ] .
S o l u t i o n 2. A g e n e r a l n u m e r i c a l s o l u t i o n t e c h n i q u e
s u c h a s t h e f i n i t e element method o r t h e boundary
element method.
I n t h e present paper t h e f i n i t e
element method i s used t o g e n e r a t e S o l u t i o n 2 bec a u s e of i t s s i m p l i c i t y . Use of f i n i t e element
method e n a b l e s t h e a l t e r n a t i n g method t o b e a p p l i e d
t o more complex s t r u c t u r a l components.
The s t e p s r e q u i r e d i n t h e p r e s e n t a l t e r n a t i n g
method a r e e x p l a i n e d i n t h e f o l l o w i n g ( a l s o s e e
Table 1):
t h e s t r e s s boundary c o n d i t i o n on t h e e x t e r n a l
s u r f a c e s of t h e body, r e v e r s e t h e r e s i d u a l
s t r e s s e s and c a l c u l a t e e q u i v a l e n t n o d a l f o r c e s .
These n o d a l f o r c e s { Q } can b e e x p r e s s e d i n
t e r m s of t h e c o e f f i c i e n t s C:
( 1 ) S o l v e t h e uncracked body u n d e r t h e g i v e n ext e r n a l l o a d s by u s i n g f i n i t e element method.
The uncracked body h a s t h e same geometry a s
t h e g i v e n problem e x c e p t f o r t h e c r a c k . To
save computational t i m e i n solving t h e f i n i t e
element e q u a t i o n s r e p e a t e d l y , a n e f f i c i e n t
e q u a t i o n s s o l v e r OPTBLOK [ 1 7 ] which h a s a
r e s o l u t i o n f a c i l i t y was implemented a s exp l a i n e d i n Ref. [ 1 5 ] . I n OPTBLOK, t h e reduct i o n of s t i f f n e s s m a t r i x i s done o n l y o n c e a l though t h e r e d u c t i o n of l o a d v e c t o r and b a c k
s u b s t i t u t i o n may b e r e p e a t e d f o r any number
of i t e r a t i o n s , w i t h o n l y a s m a l l a d d i t i o n a l
computational time.
where [N] i s t h e m a t r i x of i s o p a r a m e t r i c e l e ment s h a p e f u n c t i o n , [ n ] i s t h e m a t r i x of t h e
normal d i r e c t i o n c o s i n e s , and [PI i s t h e b a s i s
f u n c t i o n m a t r i x f o r s t r e s s e s and d e f i n e d i n
Eq. ( 9 ) . I n o r d e r t o s a v e c o m p u t a t i o n a l t i m e ,
t h e m a t r i c e s [GIm a r e c a l c u l a t e d p r i o r t o t h e
) Using t h e f i n i t e e l e m e n t s o l u t i o n , we compute
t h e s t r e s s e s a t t h e l o c a t i o n of o r i g i n a l c r a c k
i n t h e uncracked s o l i d .
s t a r t of i t e r a t i o n p r o c e s s shown i n T a b l e 1.
Although t h e m a t r i x [PI h a s t h e s i n g u l a r i t y
o f o r d e r 1/& a t t h e c r a c k f r o n t , t h e magnitude
of t h e m a t r i x [PI ( o r s t r e s s ) d e c a y s r a p i d l y
w i t h t h e d i s t a n c e from t h e c r a c k f r o n t . Thus,
t h e m a t r i c e s [GIm a r e c a l c u l a t e d o n l y a t t h e
) Compare t h e r e s i d u a l s t r e s s e s c a l c u l a t e d i n
S t e p ( 2 ) w i t h a p e r m i s s i b l e s t r e s s magnitude.
U s u a l l y t h e p e r m i s s i b l e s t r e s s magnitude
i s chosen a s o n e p e r c e n t o f t h e maximum ext e r n a l applied s t r e s s .
s u r f a c e e l e m e n t s which s a t i s f y t h e f o l l o w i n g
condition:
A l t e r n a t i v e l y t h e convergency of t h e a n a l y s i s
i s a l s o checked w i t h a norm of s t r e s s i n t e n s i t y factor:
i s t h e d i s t a n c e of t h e c l o s e s t nomin
d a l p o i n t of e a c h s u r f a c e e l e m e n t , from t h e
c e n t e r of t h e e l l i p t i c a l c r a c k a s shown i n
F i g . 1.
where y
i n which L p o i n t s a r e chosen a l o n g t h e c r a c k
f r o n t . The change i n t h e norm of s t r e s s i n t e n s i t y f a c t o r f o r e a c h c y c l e of i t e r a t i o n
i s a l s o monitored.
F o r most c a s e s , t h e change
i n t h e norm between t h e 2nd and 3 r d i t e r a t i o n s
becomes l e s s t h a n o n e p e r c e n t .
( 4 ) To s a t i s f y t h e s t r e s s boundary c o n d i t i o n , on
t h e crack surface, reverse t h e residual
s t r e s s e s . Then d e t e r m i n e t h e c o e f f i c i e n t s
A in Eq. ( 3 ) f o r t h e a p p l i e d stress on t h e
c r a c k s u r f a c e , by u s i n g t h e f o l l o w i n g l e a s t
square f i t t i n g
.
R
where u33 i s t h e r e v e r s e d r e s i d u a l s t r e s s c a l c u l a t e d by t h e f i n i t e element method, S
C
is
t h e r e g i o n o f t h e c r a c k , and I i s t h e f u n c t i o n a l t o b e minimized.
The more d e t a i l e d proc e d u r e i n t h i s s t e p i s g i v e n i n Ref. [ 1 5 ] .
(5) Determine t h e c o e f f i c i e n t s i n Eq. ( 4 ) f o r t h e
p o t e n t i a l f u n c t i o n by s o l v i n g Eq. (10)
({cI=[BI-~{AI).
( 6 ) Calculate the s t r e s s intensity factor f o r the
c u r r e n t i t e r a t i o n by s u b s t i t u t i n g c o e f f i c i e n t s
(11).
C i n Eq.
( 7 ) C a l c u l a t e t h e r e s i d u a l s t r e s s e s on e x t e r n a l
s u r f a c e s of t h e body d u e t o t h e a p p l i e d s t r e s s
on t h e c r a c k s u r f a c e i n S t e p ( 4 ) . To s a t i s f y
(8)
Consider t h e n o d a l f o r c e s i n S t e p (7) a s ext e r n a l a p p l i e d l o a d s a c t i n g on t h e uncracked
body.
Repeat a l l s t e p s i n t h e i t e r a t i o n proc e s s u n t i l t h e r e s i d u a l s t r e s s e s on t h e c r a c k
s u r f a c e become n e g l i g i b l e ( S t e p 3 ) . To obt a i n t h e f i n a l s o l u t i o n , add t h e s t r e s s i n t e n s i t y f a c t o r s of a l l i t e r a t i o n s .
S i n c e t h e a n a l y t i c a l s o l u t i o n f o r an e l l i p t i c a l c r a c k i n an i n f i n i t e s o l i d i s implemented a s
S o l u t i o n 1, i t i s n e c e s s a r y t o d e f i n e t h e r e s i d u a l
s t r e s s e s over t h e e n t i r e crack plane including t h e
f i c t i t i o u s p o r t i o n of t h e c r a c k which l i e s o u t s i d e
of t h e f i n i t e body. Morever, it i s w e l l known t h a t
a c c u r a c y of t h e l e a s t s q u a r e s f i t t i n g i n s i d e of t h e
f i t t i n g region can be increased with t h e increasing
number of polynomial t e r m s ; however t h e f i t t i n g
c u r v e may change d r a s t i c a l l y i n t h e r e g i o n o u t s i d e
of t h e f i t t i n g . For t h e s e r e a s o n s , i n Ref. [ 5 ]
n u m e r i c a l e x p e r i m e n t a t i o n was done f o r a r r i v i n g
a t an optimum p r e s s u r e d i s t r i b u t i o n on t h e c r a c k
s u r f a c e extended i n t o t h e f i c t i t i o u s r e g i o n .
For
a s e m i - e l l i p t i c a l c r a c k which l i e s i n t h e r e g i o n
it was concluded t h a t t h e
of -a <x < a and O<x < a
1- 1
2- 2 '
f i c t i 4 b u s p r e s s u r e which, f o r t h e r e g i o n of
-a <x <O, remains c o n s t a n t i n t h e x2 d i r e c t i o n b u t
2, 2v a r i e s i n t h e xl d i r e c t i o n g i v e s t h e b e s t r e s u l t
among t h e s e v e r a l n u m e r i c a l e x p e r i m e n t s performed
i n Ref. [ 1 5 ] . The p r o c e d u r e of t h e f i c t i t i o u s
pressure d i s t r i b u t i o n f o r a semi-elliptical surf a c e c r a c k was s u c c e s s f u l l y u s e d on t h e a n a l y s e s
of s u r f a c e c r a c k s , i n f i n i t e t h i c k n e s s p l a t e s
s u b j e c t t o remote t e n s i o n a s w e l l a s remote bendi n g [ 1 5 ] , and i n p r e s s u r e v e s s e l s [ 1 8 ] .
I n t h e p r e s e n t p a p e r , t a k i n g a c c o u n t of t h e
c o n c l u s i o n drawn i n Ref. [ 1 5 ] t h e f i c t i t i o u s
p r e s s u r e d i s t r i b u t i o n shown i n F i g . 2 i s employed
f o r t h e analysis of a q u a r t e r - e l l i p t i c a l c o m e r
crack.
For t h e f i r s t q u a d r a n t ( x x > 0 ) , t h e r e 1' 2s i d u a l s t r e s s can b e c a l c u l a t e d by
the finite
element method and i s a f u n c t i o n o f t h e c o o r d i n a t e s
x1 and x2:
For t h e o t h e r q u a d r a n t s , t h e f i c t i t i o u s r e s i d u a l
s t r e s s i s defined a s
where a
-
U 3 3 ( 0 , ~ 2 ) f o r t h e second quadrant
(x1-I0, x 2 g
R
u3.
-1
R
033(0,0)
f o r t h e t h i r d quadrant
(x,
, x 2 g
R
( " 3 3 ( ~ 1 , ~ ) f o r t h e f o u r t h quadrant
Yx > o , x <O)
(18)
124.
R e s u l t s and D i s c u s s i o n s
The 20-noded i s o p a r a m e t r i c e l e m e n t s were used
i n t h e present study.
In t h e previous s t u d i e s [14,
15,183 t h e 3 x 3 ~ 3p r o d u c t Gauss i n t e g r a t i o n r u l e
was u s e d t o e v a l u a t e t h e s t i f f n e s s m a t r i c e s of t h e
20-noded i s o p a r a m e t r i c e l e m e n t s .
In t h e present
s t u d y t h e p r o d u c t Gauss i n t e g r a t i o n r u l e was r e p l a c e d by t h e 14 p o i n t s non-product r u l e f o r t h r e e d i m e n s i o n a l i n t e g r a t i o n [ 1 9 , 2 0 ] . By t e s t i n g b o t h
t h e i n t e g r a t i o n r u l e s f o r t h e problem of a q u a r t e r e l l i p t i c a l c o r n e r c r a c k , i t was found t h a t b o t h t h e
r u l e s gave almost i d e n t i c a l s t r e s s i n t e n s i t y fact o r solutions.
I n f a c t , t h e s t r e s s i n t e n s i t y fact o r s o l u t i o n o b t a i n e d by t h e 14 p o i n t s non-product
r u l e varies within
0.4% o f t h e s o l u t i o n o b t a i n e d
by t h e 27 p o i n t s p r o d u c t Gauss r u l e .
Therefore,
t h e a l l n u m e r i c a l s o l u t i o n s shown i n t h e f o l l o w i n g
s e c t i o n s were o b t a i n e d by u s i n g t h e 14 p o i n t s nonproduct i n t e g r a t i o n r u l e .
T h i s r u l e can b e e x p r e s s e d by
+
i
i s a r e f e r e n c e s t r e s s m a g n i t u d e , E(k) i s
t h e c o m p l e t e e l l i p t i c i n ~ e g r a lo f s e c o n d k i n d ,
2
2 2
2
k = ( a -a ) / a and A i s d e f i n e d by Eq. ( 1 2 ) . The
1
1 2
denominator o f t h e r i g h t h a n d s i d e o f Eq. (20) c o r responds t o t h e exact s t r e s s i n t e n s i t y f a c t o r f o r
t h e e l l i p t i c a l crack subject t o t h e constant press u r e ui on t h e c r a c k s u r f a c e , i n a n i n f i n i t e s o l i d .
The r e f e r e n c e s t r e s s a depends on t y p e o f t h e
i
problem c o n s i d e r e d .
4.1.
-
Quarter-Elliptical
Corner Crack i n B r i c k
We c o n s i d e r a b r i c k c o n t a i n i n g a q u a r t e r - e l l i p t i c a l c o r n e r c r a c k o f a s p e c t r a t i o a / a =0.4,
2 1
and s u b j e c t t o remote t e n s i o n o a t t h e e n d s of t h e
T
brick.
The g e o m e t r i e s of t h i s problem a n d t h e
f i n i t e e l e m e n t breakdown f o r t h e u n c r a c k e d b r i c k
a r e shown i n F i g . 3. Due t o t h e symmetry w i t h r e s p e c t t o t h e x 3 d i r e c t i o n , o n l y t h e u p p e r h a l f of
It s h o u l d
t h e b r i c k was modeled by f i n i t e e l e m e n t s .
b e n o t e d t h a t t h e f i n i t e element method i s used t o
a n a l y s e t h e u n c r a c k e d body, a l t h o u g h t h e mesh p a t t e r n follows t h e o r i g i n a l crack shape. Thus, a l l
t h e d i s p l a c e m e n t s u3 on t h e p l a n e o f x3=0 a r e cons t r a i n e d d u e t o t h e symmetry. The f i n i t e element
mesh shown i n F i g . 3 c o n s i s t s o f 80 twenty-noded
i s o p a r a m e t r i c e l e m e n t s w i t h 1377 d e g r e e s of f r e e dom ( b e f o r e i m p o s i t i o n of t h e boundary c o n d i t i o n s ) .
The m a t r i c e s [GI g i v e n i n E q . (16) a r e c a l c u l a t e d
m
on t h e s u r f a c e e l e m e n t s o f x = O and x = O s a t i s f y i n g
1
2
t h e c o n d i t i o n ( 1 7 ) , #'mi$
5a,, p r i o r t o t h e s t a r t
of i t e r a t i o n process.
It i s n o t e d t h a t a l l s u r f a c e e l e m e n t s on x =W, x =H, and x =L a r e excluded
1
2
3
i n t h e c a l c u l a t i o n of [GI,, s i n c e t h e s e b o u n d a r i e s
a r e f a r enough from t h e c r a c k .
+
=
B6[$(-b6,0,0)
+
c8[$(-c8,-c8,-c8)
$(b6,0,0)
+
...
$(c8,-c8,-c
8
6 terms]
)..8 t e r m s ]
(19)
where
The above numbers were o b t a i n e d by Punch 1201. A l l
n u m e r i c a l a n a l y s e s were performed by u s i n g t h e CDC
CYBER 74 a t G e o r g i a I n s t i t u t e o f Technology.
A l l problems c o n s i d e r e d h e r e c o n c e r n t h e l i n e a r
e l a s t i c Mode I problems o f q u a r t e r - e l l i p t i c a l c o r n e r
c r a c k s . To q u a n t i f y t h e e f f e c t s of a f i n i t e body,
crack aspect r a t i o , e t c . , a magnification f a c t o r
( n o r m a l i z e d s t r e s s i n t e n s i t y f a c t o r ) F . d e f i n e d by
t h e following equation i s used
The v a r i a t i o n o f t h e m a g n i f i c a t i o n f a c t o r F
T
( n o r m a l i z e d s t r e s s i n t e n s i t y f a c t o r ) i s shown i n
F i g . 4 . The m a g n i f i c a t i o n f a c t o r s were e v a l u a t e d
by u s i n g Eq. (20) w i t h t h e r e f e r e n c e s t r e s s o . = o
1 T'
I n t h i s c a s e t h e v a l u e E(k)=1.1507 was u s e d f o r
a / a =0.4.
I n t h e p r e s e n t a n a l y s i s twenty-one
2 1
1211 t e r m s of t h e f i f t h o r d e r p o l y n o m i a l (MXDOP=5:
M=2, i = 0 , 1 ; j = O , l ) i n Eq. (3) were u s e d f o r t h e
f i t t i n g of t h e r e s i d u a l s t r e s s i n S t e p ( 4 ) . F i g u r e 4 a l s o shows t h e r e s u l t w i t h t h e c u b i c polyThe
nomial fitting(PiXDOP=3: M=l; i = 0 , 1 ; j = 0 , 1 ) .
p r e s e n t r e s u l t s a r e compared w i t h t h e r e s u l t s from
Newman and R a j u [ 2 1 ] . As s e e n from t h e f i g u r e t h e
p r e s e n t r e s u l t w i t h ?TXDOP=5 i s i n a n e x c e l l e n t
agreement w i t h t h o s e o f Newman and Raju [ 2 1 ] w h i l e
t h e r e s u l t w i t h MXDOP=3 d i f f e r s from o t h e r r e s u l t s .
The s t r e s s i n t e n s i t y f a c t o r v a r i a t i o n a f t e r
e a c h i t e r a t i o n and t h e r e s i d u a l s t r e s s removed
from t h e c r a c k s u r f a c e i n e a c h i t e r a t i o n a r e shown
i n F i g s . 5a and 5b, r e s p e c t i v e l y . As s e e n from
t h e f i g u r e s , t h e i n c r e m e n t of t h e m a g n i f i c a t i o n
f a c t o r f o r each i t e r a t i o n c o r r e l a t e s w i t h t h e res i d u a l s t r e s s removed from t h e c r a c k s u r f a c e . The
magnitude o f r e s i d u a l s t r e s s d e c r e a s e s monotonica l l y w i t h t h e i n c r e a s i n g number o f i t e r a t i o n s .
7
The increment o f t h e norm f o r s t r e s s i n t e n s i t y
f a c t o r v a r i a t i o n d e f i n e d by Eq. (13) f o r t h e 4 t h
i t e r a t i o n ( f i n a l ) was o n l y 0.2%. The CPU t i m e f o r
t h i s a n a l y s i s was 990 s e c o n d s u s i n g t h e CYBER 74.
4.2.
-
Q u a r t e r - E l l i p t i c a l C o m e r Cracks Emanating
a Hole i n Finite-Thickness P l a t e s
The c o n f i g u r a t i o n of t h e specimen c o n s i d e r e d
h e r e i s shown i n F i g . 6. Two s y m m e t r i c a l q u a r t e r e l l i p t i c a l c o m e r c r a c k s emanating t h e h o l e a r e
c o n s i d e r e d . The d e f i n i t i o n of t h e problem i s ident i c a l t o t h a t i n Ref. [ 5 ] . The g e o m e t r i e s f o r t h i s
problem a r e summarized a s f o l l o w s a / a = 2 . 0 ,
1 2
t = 0 . 2 , 0 . 5 and 0 . 8
a l / a 2 = 2.0,
a / t = 0.2,
1
Ref. [ 5 ] e x c e p t n e a r t h e h o l e s u r f a c e (8=0°)Contrary t o t h i s , f o r t h e bearing pressure loading
b o t h t h e r e s u l t s show s i g n i f i c a n t d e f f e r e n c e e x c e p t
i n t h e case of a /t=0.8.
1
To u n d e r s t a n d f u r t h e r t h e s o l u t i o n s of t h e s e
problems, t h e two d i m e n s i o n a l e l a s t i c i t y s o l u t i o n s
f o r a h o l e i n an i n f i n i t e p l a t e s u b j e c t t o t h e same
l o a d i n g s a r e examined.
These s o l u t i o n s a r e g i v e n
by t h e f o l l o w i n g e q u a t i o n s ;
f o r t h e remote t e n s i o n [ 2 2 ]
0.5 and 0 . 8
u
OT
=-(I+<)
$4
2
2
R.
+ -O2T
4
R.
(1+3$)
R
c o s 2$
(22b)
For e a c h c r a c k geometry, two s e p a r a t e l o a d i n g s
were a p p l i e d t o t h e p l a t e a s shown i n F i g s . 6b
and 6c. The a p p l i e d normal s t r e s s a
on t h e
RR
h o l e boundary i n F i g . 6c i s g i v e n by
-,,
RR
= - -
3P
4Rit
2
cos $
and f o r t h e b e a r i n g p r e s s u r e l o a d i n g
3
uRR = -
where P i s t h e t o t a l f o r c e a c t i n g on t h e perpenIt i s n o t e d
d i c u l a r d i r e c t i o n t o t h e crack plane.
t h a t t h e o r i g i n of p o l a r c o o r d i n a t e s (R,$) i s l o cated a t t h e center of t h e hole.
The t y p i c a l f i n i t e element model u s e d f o r t h e
uncracked p l a t e w i t h a h o l e i s shown i n F i g . 7 ,
which c o n s i s t o f 80 f i n i t e e l e m e n t s w i t h 1377 deg r e e s of freedom ( b e f o r e i m p o s i t i o n of boundary
c o n d i t i o n ) . Due t o t h e symmetries, o n l y o n e
q u a r t e r of t h e p l a t e was used i n t h e a n l a y s i s .
The m a t r i c e s [GIm a r e c a l c u l a t e d on t h e s u r f a c e
e l e m e n t s of x = 0 , t and R=R s a t i s f y i n g t h e con1
i
d i t i o n (17), y
Sa, p r i o r t o t h e s t a r t o f i t e r min
r
a t i o n p r o c e s s . The s u r f a c e s of x =W-R
x =L can
2
i' 3
b e excluded i n t h e c a l c u l a t i o n of [GI s i n c e
m
t h e s e s u r f a c e s a r e f a r enough, i . e . , ymin&5ar
.<
S i n c e t h e g l o b a l s t i f f n e s s m a t r i x and t h e m a t r i c e s
[GIm a r e t h e same, t h e two t y p e s of l o a d i n g shown
i n F i g s . 6b a n d 6 c a r e s o l v e d a t o n c e f o r e a c h
c r a c k geometry.
Thus, a s e x p l a i n e d e a r l i e r , t h e
r e d u c t i o n o f s t i f f n e s s m a t r i x was done o n l y o n c e ,
w h i l e t h e r e d u c t i o n of l o a d v e c t o r and t h e back
s u b s t i t u t i o n were r e p e a t e d f o r t h e two l o a d i n g
cases a s well a s f o r t h e i t e r a t i o n process i n
t h e a l t e r n a t i n g t e c h n i q u e . The CPU t i m e f o r e a c h
c r a c k geometry w i t h two l o a d i n g c a s e s was a p p r o x i m a t e l y 1200 s e c o n d s u s i n g t h e CYBER 74.
"
The m a g n i f i c a t i o n f a c t o r s f o r remote t e n s i o n
and b e a r i n g p r e s s u r e l o a d i n g a r e shown i n F i g s .
8 a and 8b, r e s p e c t i v e l y . To e v a l u a t e t h e magnif i c a t i o n f a c t o r s t h e r e f e r e n c e s t r e s s was chosen
a s o.=oT f o r remote t e n s i o n and o.=o = P / 2 R . f o r
1 P
b e a r i n g p r e s s u r e l o a d i n g . The c o m p l e t e e l l i p t i c
i n t e g r a l of second kind i n Eq. ( 2 0 ) i s e q u a l t o
1 . 2 1 1 f o r a / a =2.0.
The p r e s e n t r e s u l t s a r e
1 2
compared w i t h t h o s e o b t a i n e d by Raju and Newman
[ 5 ] . F o r t h e remote t e n s i o n problem t h e p r e s e n t
r e s u l t s a r e i n e x c e l l e n t agreement w i t h t h o s e from
a
= - -
R8
UP
:
Up
(Ri/R)
2
2 2
1 1 - ( ~ - R ~ /) Rc o s 2$}(23a)
2
2 2
( R i / ~ ) (l-Ri/R ) s i n 28
(23c)
The above s o l u t i o n was d e r i v e d by u s i n g B i c k l e y ' s
p r o c e d u r e [ 2 3 ] . The d i s t r i b u t i o n s of t h e circumferential stress a
a l o n g t h e x2 a x i s ($=oO) a r e
shown i n F i g . 9a ar% 9b. A s s e e n from t h e f i g u r e
t h e s t r e s s c o n c e n t r a t i o n f a c t o r s a r e 3.0 f o r t h e
remote t e n s i o n , and 1 . 5 f o r t h e b e a r i n g p r e s s u r e
l o a d i n g of t h e form o =-3P/(4R.t). cos2$.
In
RR
F i g s . 9 a and 9b, t h e r e g i o n s covered by t h e c r a c k s
f o r t h e c a s e s of a l / t = 0 . 2 , 0.5 and 0 . 8 a r e a l s o
shown. Comparing F i g s . 8 and 9 good c o r r e l a t i o n can
b e n o t i c e d between t h e p r e s e n t m a g n i f i c a t i o n f a c t o r s and t h e s t r e s s d i s t r i b u t i o n f o r b o t h t h e l o a d i n g
cases.
For example, f o r a l l t h e c a s e s c o n s i d e r e d h e r e ,
t h e a v e r a g e v a l u e of t h e s t r e s s i n e a c h c r a c k r e g i o n
p r e d i c t s e x c e l l e n t l y t h e a v e r a g e v a l u e o f t h e magn i f i c a t i o n f a c t o r v a r i a t i o n f o r e a c h c r a c k geometry.
Morever, t h e s t r e s s d i s t r i b u t i o n o f t h e b e a r i n g p r e s s u r e c a s e d e c a y s f a s t e r t h a n t h a t o f t h e remote
t e n s i o n , a s t h e d i s t a n c e from t h e h o l e s u r f a c e i n c r e a s e s . T h i s e f f e c t can a l s o b e s e e n i n t h e v a r i a t i o n s of t h e p r e s e n t m a g n i f i c a t i o n f a c t o r s from
t h e h o l e s u r f a c e (8=0°) t o t h e p l a t e s u r f a c e (0=90°)
a s compared i n F i g s . 9a and 9b. From t h e e v i d e n c e
mentioned above, i t is b e l i e v e d t h a t t h e p r e s e n t
r e s u l t s a r e accurate.
The e f f e c t of t h e v a r i a t i o n of b e a r i n g p r e s s u r e
i s a l s o examined. Another t y p e of p r e s s u r e d i s t r i b u t i o n i s g i v e n by
The c i r c u m f e r e n t i a l s t r e s s d i s t r i b u t i o n o b t a i n e d
by B i c k l e y s o l u t i o n [ 2 3 ] i s a l s o shown i n F i g . 9b.
This cosine pressure loading gives a higher s t r e s s
c o n s e n t r a t i o n f a c t o r a s compared t o t h a t o b t a i n e d
from t h e c o s i n e s q u a r e l o a d i n g g i v e n by Eq. (21)
( s e e F i g . 9 b ) . The m a g n i f i c a t i o n f a c t o r s f o r b o t h
t h e p r e s s u r e l o a d i n g s a r e compared i n F i g . 1 0 . As
can b e e x p e c t e d from t h e e l a s t i c i t y s o l u t i o n s , t h e
magnification f a c t o r f o r t h e c o s i n e p r e s s u r e loadi n g i s s l i g h t l y h i g h e r a t t h e h o l e s u r f a c e (8=0°),
and s l i g h t l y lower a t t h e p l a t e s u r f a c e (8=900)
t h a n t h a t from c o s i n e s q u a r e l o a d i n g .
The p r e s e n t r e s u l t s shown i n F i g s . 8 , 9 were
o b t a i n e d f o r two symmetric q u a r t e r - e l l i p t i c a l c o r n e r c r a c k s . The s t r e s s i n t e n s i t y f a c t o r f o r a
s i n g l e q u a r t e r - e l l i p t i c a l c o r n e r c r a c k can b e conv e r t e d from t h e r e s u l t s f o r two symmetric c r a c k s b y
u s i n g f o r m u l a developed by Shah [ 2 4 ] :
4.3.
2Rit
+-
two
+-
(25)
Q u a r t e r - E l l i p t i c a l C o m e r Cracks
i n A i r c r a f t Attachment Lugs
Young's modulus ~ = 1 Mpsi
0 ~ and P o i s s o n ' s r a t i o
=0.33. To s i m u l a t e p i n l o a d i n g , t h e c o s i n e b e a r i n g p r e s s u r e d e f i n e d by Eq. (24) a c t i n g on o n l y a
h a l f o f t h e boundary Oc$zn, a s shown i n F i g . 11
i s c o n s i d e r e d . The a n l a y s i s was made f o r n i n e
crack geometries a s follows
a / a h = 0.5,
P
where a
=
0.2,
Thus t h e
Mode I s t r e s s i n t e n s i t y f a c t o r i s dominant and
o t h e r modes a r e n e g l i g i b l e i n t h i s c a s e .
To e v a l u a t e t h e m a g n i f i c a t i o n f a c t o r s f o r
t h i s problem t h e r e f e r e n c e s t r e s s was chosen a s
o =o =P/2R.t.
The c o m p l e t e e l l i p t i c i n t e g r a l
i
P
1
of second k i n d E(k) i n Eq. (20) i s g i v e n by 1.2111
f o r a / a = 0 . 5 and 2 . 0 , and 1.4429 f o r a / a =1.2.
P h
P h
F i g u r e s 13-15 show t h e m a g n i f i c a t i o n f a c t o r s a s a
f u n c t i o n of e l l i p t i c a l a n g l e f o r t h e a s p e c t r a t i o s
o f a / a =0.5, 1 . 2 and 2 . 0 r e s p e c t i v e l y . The e l P h
l i p t i c a l a n g l e i s a l w a y s measurer: from t h e h o l e
surface i n t h e s e cases.
F i g u r e s 16-18 show t h e
m a g n i f i c a t i o n f a c t o r s a s a f u n c t i o n of t h e c r a c k
length a t t h e p l a t e surface, a , f o r t h e crack
d e p t h of a / t = 0 . 2 , 0.5 and 0.8!
The m a g n i f i c a t i o n
h
f a c t o r s increase a s t h e crack length a decreases,
due t o t h e f a c t t h a t t h e s t r e s s c o n c e n t r a t i o n exi s t s around t h e p i n h o l e .
cracks
The geometry of t h e l u g w i t h two symmetric
q u a r t e r - e l l i p t i c a l c o r n e r c r a c k s i s shown i n F i g .
11. The l u g m a t e r i a l i s 7075-76 Aluminum w i t h
a,,/t
0 . 5 and 0.1% of t h e normal s t r e s s a33.
=,/T
2R.t
single
crack
t h e s h e a r s t r e s s e s u31 and u~~ were r e s p e c t i v e l y ,
1 . 2 , and 2 . 0
5.
0 . 5 , and 0 . 8
and ah d e n o t e c r a c k l e n g t h s a t t h e s u r f a c e s
P
of t h e p l a t e and h o l d r e s p e c t i v e l y .
The s t r e s s i n t e n s i t y f a c t o r s f o r a l l t h e c r a c k
g e o m e t r i e s a r e summarized i n T a b l e 2. The s t r e s s
i n t e n s i t y f a c t o r s were n o r m a l i z e d by t h e s t r e s s
i n t e n s i t y f a c t o r f o r t h e c r a c k s i z e of 2R i n a n
i
i n f i n i t e two-dimensional p l a t e w i t h t h e p r e s s u r e
u on t h e c r a c k s u r f a c e . As s e e n from t h e t a b l e
tKe s t r e s s i n t e n s i t y f a c t o r i n c r e a s e s w i t h t h e
i n c r e a s i n g s i z e of c r a c k . The s t r e s s i n t e n s i t y
f a c t o r s f o r a s i n g l e q u a r t e r - e l l i p t i c a l corner
c r a c k i n t h e l u g can a l s o be approximated from t h e
r e s u l t s f o r two-symmetric c o r n e r c r a c k s by u s i n g
Eq. ( 2 5 ) . The CPU t i m e f o r t h e a n a l y s e s was
a p p r o x i m a t e l y 1800 s e c o n d s u s i n g t h e CYBER 74.
Thus, a = a
D
2
and ah=al f o r a / a = 0 . 5 , and a =al and a =a
P h
P
h 2
a / a =1.2 and 2.0.
P h
The t y p i c a l f i n i t e element model used f o r t h e
uncracked l u g i s shown i n F i g . 1 2 . T h i s model cons i s t s of 140 twenty-noded i s o p a r a m e t r i c e l e m e n t s
w i t h 2250 d e g r e e s of freedom ( b e f o r e i m p o s i t i o n
o f boundary c o n d i t i o n ) .
Due t o t h e symmetry a
h a l f of t h e l u g was used i n t h e a n a l y s i s . The d i s p l a c e m e n t s were imposed a s u =O and x =-L, and u
3
3
1
= O a t x =-R
The m a t r i c e s [GIm were c a l c u l a t e d
1
i'
on t h e s u r f a c e of x = O , t , R=Ro ( x 3 3 ) , and x =R -R
2
l o 1
<5a
( x <O) s a t i s f y i n g t h e c o n d i t i o n ( 1 7 ) , y
3
min
1'
( s e e Fig. 12).
F i r s t , o n l y s t r e s s a n a l y s e s of t h e uncracked
l u g shown i n F i g . 12 were performed t o examine t h e
e f f e c t o f t h e l u g l e n g t h , c h a n g i n g L=5R t o 6R
i
i'
The a v e r a g e v a l u e of normal s t r e s s u33 a t t h e
- -
o r i g i n a l c r a c k l o c a t i o n d i f f e r s o n l y 0.02% a s t h e
l u g l e n g t h changes from 5R t o 6Ri.
Thus, t h e
i
f o l l o w i n g a n a l y s e s were done w i t h L=5R..
I n add i t i o n , t h e magnitude of s h e a r s t r e s s e B which produce t h e Mode I1 and Mode 111 s t r e s s i n t e n s i t y
f a c t o r s was a l s o examined. The a v e r a g e v a l u e s of
Concluding Remarks
The a l t e r n a t i n g method, i n c o n j u n c t i o n w i t h
t h e f i n i t e element method and t h e c o m p l e t e , g e n e r a l
a n a l y t i c a l s o l u t i o n f o r an e l l i p t i c a l c r a c k i n an
i n f i n i t e s o l i d , was developed f o r t h e a n a l y s e s o f
q u a r t e r - e l l i p t i c a l c o r n e r c r a c k s emanating a h o l e .
The p r e s e n t f i n i t e element a l t e r n a t i n g method l e a d s
t o a t l e a s t an o r d e r of magnitude l e s s e x p e n s i v e
p r o c e d u r e f o r r o u t i n e e v a l u a t i o n of a c c u r a t e s t r e s s
i n t e n s i t y f a c t o r s i n t h r e e - d i m e n s i o n a l complex
s t r u c t u r a l components, a s compared t o o t h e r t e c h niques currently reported i n l i t e r a t u r e .
E x c e l l e n t c o r r e l a t i o n was found between t h e
p r e s e n t s o l u t i o n s and t h o s e o b t a i n e d by Raju and
Newman [ 5 , 2 1 ] f o r a q u a r t e r - e l l i p t i c a l c o r n e r
c r a c k emanating a h o l e i n t h e p l a t e s u b j e c t t o
remote t e n s i o n . Also i t was found t h a t t h e magn i f i c a t i o n f a c t o r s f o r c o r n e r c r a c k s emanating
a h o l e c o r r e l a t e q u a l i t a t i v e l y t o s t r e s s concent r a t i o n around t h e h o l e .
The s t r e s s i n t e n s i t y f a c t o r s f o r v a r i o u s s h a p e s
of two symmetric q u a r t e r - e l l i p t i c a l c o r n e r c r a c k s
emanating t h e p i n h o l e o f a i r c r a f t a t t a c h m e n t l u g s
were a l s o determined by t h e p r e s e n t f i n i t e element
a l t e r n a t i n g method. The s t r e s s i n t e n s i t y f a c t o r s
f o r a s i n g L e c o r n e r c r a c k c a n b e approximated from
t h e p r e s e n t s o l u t i o n s f o r two symmetric c o m e r
c r a c k s by u s i n g S h a h ' s c o n v e r s i o n f o r m u l a [ 2 4 ] .
It was a l s o demonstrated i n t h e p r e s e n t s t u d y
t h a t t h e s t r e s s i n t e n s i t y f a c t o r solution obtained
by t h e a l t e r n a t i n g t e c h n i q e u can b e improved
s i g n i f i c a n t l y when t h e d e g r e e o f p o l y n o m i a l s i n t h e
a p p l i e d s t r e s s f o r t h e a n a l y t i c a l s o l u t i o n is increased.
Acknowledgements
c
T h i s w o r k was s u p p o r t e d b y AFOSR u n d e r g r a n t
81-0057 t o G e o r g i a I n s t i t u t e o f T e c h n o l o g y . T h e
a u t h o r s g r a t e f u l l y acknowledge t h i s s u p p o r t a s
w e l l as t h e e n c o u r a g e m e n t r e c e i v e d f r o m Dr. Anthony
Amos.
T h e a u t h o r s t h a n k M s . M a r g a r e t e Eiternan f o r
h e r c a r e f u l a s s i s t a n c e i n t h e preparation of t h i s
manuscript.
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.
Table 1
Plow Chart f o r F i n i t e Element
- Alternating
Technique
Solve t h e uncracked body under e x t e r n a l loads
by using f i n i t e element method (FEW
Step
1
Step
Step 3
+
Using FPI s o l u t i o n s compute s t r e s s e s a t
t h e l o c a t i o n of t h e crack
I Are
+
1
t h e s t r e s s e s i n s t e p ( 2 ) n e g l i g i b l e ? / Yes
+
'
No
Determine c o e f f i c i e n t s A i n t h e applied s t r e s s e s by
f i t t i n g crack f a c e s t r e s s e s i n s t e p ( 2 )
t
Determine c o e f f i c i e n t s C i n
the t e n t i a f u t o n
1
Step
+
I
1
Calculate t h e k-factors f o r t h e
current i t e r a t i o n
crack.
1
+
I
Calculate r e s i d u a l s t r e s s e s on e x t e r n a l
s u r f a c e s of t h e body due t o t h e loaded
Reverse them and c a l c u l a t e equivalent nodal f o r c e s .
+
F i g . 1:
onsider t h e nodal f o r c e s in s t e p (7) a s e x t e r n a l
applied loads a c t i n g on t h e uncracked body
Quarter-Elliptical
Crack a n d D i s t a n c e
o f S u r f a c e Element
I
(Add t h e k-factor s o l u t i o n s of a11
iterations
Table 2
The S t r e s s I n t e n s i t y F a c t o r s (K /
I "'P
i n a n Attachment Lug
r",
angle*
*)
0a2
0.5
0.8
0.2
q)
f o r ~ u a r t e r - ~ l l i p t i c aClo r n e r Cracks
0.5
0.8
The e l l i p t i c a l a n g l e measured from t h e h o l e s u r f a c e
0.2
0.5
0.8
F i g . 2:
Residual S t r e s s D i s t r i b u t i o n
o v e r t h e E n t i r e Crack S u r f a c e
Fig.
-
1.5
f
E 1
F i n i t e Element Breakdown f o r an Uncracked B r i c k
CORNER CRACK
a2/a
0.4
"
u
--- MXDOP = 3
u
3:
Q
NEWMAN
Y
R A J U (21)
IJl
---4th
0th-3rd i t e r a t i o n s
i t e r a t i o n (final)
u
LT"
b"
0
\*
0.5
-
I
W
a:
I-
cn
-1
z
Q
cn
W
cr
0
30
60
E L L I P T I C A L ANGLE 8 ( D E G R E E S )
F i g . 4:
Magnification Factors f o r
a Q u a r t e r - E l l i p t i c a l Corner Crack
1 2 3 L . 5
Cycle of Iterations
0
90
ELLIPTICAL ANGLE 8 (DEGREES)
(a)
Fig.
5:
Convergency o f t h e S o l u t i o n Obtained by t h e A l t e r n a t i n g
Technique; ( a ) S u c c e s s i v e I t e r a t i o n f o r S t r e s s I n t e n s i t y
F a c t o r , and ( b ) V a r i a t i o n of R e s i d u a l S t r e s s on t h e Crack
Surface
0-
T
TENSION
(b)
Fig. 6:
Fig. 7:
BEARING PRESSURE
(c )
Two Symmetric Quarter-Elliptical Corner Cracks Emanating
a Hole in a Finite-Thickness Plate; (a) Plate Configuration, (b) Remote Tension, and (c) Bearing Pressure Loading
Finite Element Breakdown for an Uncracked Plate with a Hole
BEARING PRESSURE
TENSION
- PRESENT
RAJU & NEWMAN ( 5 )
Fig. 8:
R A J U & NEWMAN (5)
Magnification Factors for Two Symmetric Quarter-Elliptical
Corner Cracks Emanating a Hole in a Plate; (a) Remote
Tension, and (b) Bearing Pressure Loading
\
TENSION
BEARING PRESSURE
a,/t= 0 8
Fig. 9:
Distribution of the Circumferential Stress along the x
axis; (a) Remote Tension, and (b) Bearing Pressure
Loading
,TO
0
BEARING PRESSURE
-5
= - 3 ~ / ( 4 F $t 1 .
RR
---- S R -2P/(rRi
~
t ) . cos $
Q
4
0
30
60
90
E L L I P T I C A L ANGLE 8 ( D E G R E E S )
Fig.10:
Comparison of Magnification Factors with
Different Typesof Bearing Pressure Loading
Fig.12:
Fig.11:
Typical Finite Element Mesh for the Uncracked Lug
Configuration of an Attachment Lug with Two Synunetric
Quarter-Elliptical Cracks Emanating the Pin Hole
HOLE SURFACE
(1
F
(dl
P L A T E S U R F A C E (9&l
[
L
O0 0
.3 0
60
90
E L L I P T I C A L ANGLE ( D E G R E E S )
E L L I P T I C A L ANGLE ( D E G R E E S )
Fig.13:
PLATE SURFACE (Qd
Magnification Factors versus the Elliptical Angle
Measured from the Hole Surface; a /a =0.5
P h
Fig.14:
Magnification Factors versus the Elliptical Angle
Measured from the Hole Surface: a /ah=1.2
P
!
YY
1
4#
HOLE SURFACE (04
3
PLATE S U R F A C E ( 9 0 3
0.0
30
60
90
E L L I P T I C A L ANGLE ( D E G R E E S )
Fig.15:
Magnification Factors versus the Elliptical Angle
Measured from the Hole Surface; a /a =2.0
P h
2.0
I0
k
-
0
H O L E SURFACE
I
-El-
9=&"
4- P L A T E SURFACE
ASPECT RATW) a p
h
Magnification Factors for ah/t=0.2 versus Crack Aspect
Ratio
Q
HOLE SURFACE
-0-
0 9 = 45"
a
HOLE SURFACE
8=45"
PLATE SURFACE
PLATE SURFACE
0.0
ASPECT RATIO a /a
Magnification Factors for a /t=0.5versus
h
Ratio
1 .O
1.5
ASPECT RATIO a a
d
P h
Fig.17:
05
2.0
h
Crack Aspect
Fig.18:
M a g n i f i c a t i o n F a c t o r s f o r %/t=0.8
Ratio
v e r s u s Crack Aspect