Several intact or broken stringers which are continuously attached to a cracked orthotropic sheet through an adhesive are considered. The effect of orthotropy on the stress intensity factors is investigated. The stringers are assumed to be partially debonded due to high stress concentrations. The shear stress distribution between the stringers and the plate and the stress intensity factors are obtained from an integral equation which represents the continuity of displacements along the bond lines

Arin, K.

English

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SEVERAL INTACT OR BROKEN
STRINGERS ATTACHED TO AN
ORTHOTROPIC SHEET WITH A CRACK
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NASA TECHNICAL REPORTa
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DE°^P.TMENT OF MECHANICAL €NGINEERINGv, ae
AND MECHANICS
LEHIGH UNIVERSITY
BETHLEHEM, PENNSYLVANIA
r:	 S EP 1976
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A
tSEVERAL INTACT OR BROKEN STRINGERS
ATTACHED TO AN ORTHOTROPIC SHEET WITH A CRACK
K. Arin
Lehigh University
Bethlehem, Pennsylvania 18015
ABSTRACT
Several intact or broken stringers which are continuously
attached to a cracked orthotropic sheet through an adhesive
are considered. The effect of orthotropy on the stress inten-
sity factors is investigated. The stringers are assumed to be
partially debonded due to high stress concentrations. The
shear stress distribution between the stringers and the plate
and the stress intensity factors are obtained from an integral
equation which represents the continuity of displacements along
the bond lines.
- 1 -
^_. i
INTRODUCTION
Several problems have been considered recently regarding
the metal sheets stiffened by a number of stringers [1-5]. The
stringers are assumed to help decrease the probability of frac-
ture initiation and/or propagation of an existing crack. As a
result of these studies for an isotropic sheet, it has been
found that the stiffening effect of stringers depends on sev-
eral parameters, i.e., the distances between the mid point of
the crack and the stringers, the debond lengths, etc [5]. Also,
the adverse effect of stringer breakage has been shown [5].
In this paper, the effect of orthotropy will be investiga-
ted. The loading will be on the crack surfaces (intact string-
ers) as well as at infinity (broken stringers) [5]. 	 There will
be no restriction as far as the number and the locations of the
stringers are concerned. The method used here consists of ob-
taining the integral equation of the problem by the use of
Green's functions.
	
Results for orthotropic plate are compared
with the case of isotropic plate to illustrate the difference.
FORMULATION OF THE PROBLEM
In formulating the problem, the following assumptions will
be made: The adhesive will be treated as a shear spring and
the plate will be in a state of generalized plane stress.	 The
shear stresses will be considered as body forces in the plate
analysis. The input q will represent either the uniform pres-
sure on the crack surface or the uniform tension at infinity.
-2-
Using the notation described below * , the continuity of dis-
placements can be expressed as [5],
h
v p (z) - v s (z) = d u P(z), z on L
s a
Here, L denotes the union of straight lines L  defined by
x = cj , b j < y < -; j = 1,...n s
 where n s is the number of
stringers, b  is the half debond length of the jth stringer
and c  is the distance between the jth stringer and the mid-
point of the crack.
As in [5], the displacements can be expressed as
v p (z) = gk o (z) + f k p ( z,z 0 ) P(z0)dY0
L
v s (z) = f k s (z,z o ) P(z 0 )dy0 + C, z on L	 (2)
L
yo = Im(zo)
Elastic constants of the plate
Elastic modulus and cross-sectional area of
the stringer
Shear modulus of the adhesive
Thickness of the plate and the adhesive
Half crack length
Stringer width
Displacements of the plate and the stringer
at z location
Shear stress in the adhesive at z location
-3-
* (E X ,Ey,vyx,Gx )•Y
(Es,As)
ua
(h p ha)
a
ds
v p (z), vs(z)
P(z)	 .
im
r	 r
where C represents the rigid body displacements assuming a
different value on each stringer. Hence,
C = C j , j = 1,...ns for z on L^	 (3)
On the other hand, from [6], k o (z) can be expressed as in
[5]9
-k (z), for crack surface loading
k o (z) =	 (4)
k * (z) + E y, for loading at infinity
Y
y = Im(z)
and
k * (z) = Re {	 1	 [q 1 u 2 (z1	 1- 3 z	 a
u1-u2 
g 2 u l (z 2 -	 z 2 - a
	]}	 (5)
where
zm = 
C  
+ u m y, m = 1,2; z on L j	 (6)
and for orthotropic materials and the generalized plane stress,
qm = E (u— 	 vyx u m ), m = 1,2	 (7)
y m
-4-
Ali
r	 ,
i
and u 1 , u 2 are the roots of (see [6])
xy	 y	 (8)
( E xvyx = Eyvxy)
k p (z,z o ) can be determined as in the 4sotropic case, by super-
position and using the formulation set forth in [6]. Hence,
k p (z,z 0 ) can be expressed as [7],
k p (z,z o ) = Lh [ - e(z,z o ) + e(z,z o )]	 (9)
where
e(z,z o ) = Re {iq l [	
u2_ul	
^(u
( u 2 - u 1 ) c 12 J ( z l " 10 ) -
c 12 log(z l
-z 10 )] - iq2
J(z 2 ,z 20 ) - (u l -	 j(	 zu2)c22z2'20)
2-ul)cl2j(zl'z10)
(u2-u2)Z22J (zl'z20))
12(p 12 _Pl	 (("1-"2)c22
- (1'1-ul)c12
J(z 2' z 10 )) + c22log(z2-z20)])
	
(10)
z mo	 c 	 + u Illy 0 , m = 1 
,2; z on L 3	 (11 )
-5-
and
J(z,z 0) = log[z^ + z Z0 - a2]
- log[z + VZ—Z --a7]	 (12)
cl	
_ c2
c ll - c 3 ' C 12 - c3
C 4
	
_ c5
c 21 - c 6 ' c 22 - c6
cl	 u11u2 + 1 2 + 1 l (1 + u2112vyx)]
c2 = ulz1 2 + V xy + 11l(u2 +112)]
c 3 
= (111-112)(111--1)(111-112)
c 4 = 112 [ u l + il l + 11 2 (1 + 111 111vyx)]
C
5 = 112 [111111 + v Xy + 112(111+111)]
c6 = (112-111)(112-112)(112-111)
and
and
(13)
(14)
-6-
Finally, ks(z,z
0 )	 is given in [5a as:
For intact stringers
 s 
,S' Y<Yo
if z and z  are on the same
k s (z,z o ) =	 stringer
Yo
A —E- ' Y'Y 0	 (15)
s 
0	 otherwise
and C = n, y - Im(z), y o _ Im(zo).
For broken stringers
0	 y<yo
ks(z,z0)
Yo-y' Y'yo
0
if z and z  are on the same
stringer
(16)
otherwise
and C V,, j = 1,...n s are unknown constants.
Hence, in the case of broken stringers, there will be n 
additional equilibrium equations to determine C j , i.e.,
-7-
T _ 	
_-	
_.
if the load at infinity is
also transferred to the
stringer
if the end of the stringer
at infinity is stress free
ESASq
_y
{ P(zo)dyo
j
0
1,...n s	 (17)
From (1) and (2), the integral equation of the problem can
be obtained as
P(z) + f k(z,z o )P(z o )dy o + d—F 
a
a C
L 
= -T— gk o (z), z on L	 (18)
a
which will be solved together with (17) to obtain the shear
stress distribution.	 Here,
dk(z,z 0 ) - -T a [k s (z,z o ) - k p (z.z o )]	 (19)
a
STRESS INTENSITY FACTOR
The stress intensity factor will be defined as
K l 
= lim [
	 x-a ]Qy(x.o)
(20)
K2 - lim [	 x-a ]Txy(x,o)
x-► a
-8-
and given as [5], [7],
K 1
	K2/
l
+
	u2 
= q + L [e(z o ) - e(z 0)]P( zo)dy0
(21)
y0 = Im( z 0)
where
9(z 0 ) =	 1 * - 1(u2-ul)CL.'JO(z10)2 w i ha v2
(4 2 -u l )r ^JO( z10) - (1' 2 -11 2 ) C 22 J 0 ( z 2 0 )1	 (22)
and
J O (z o ) = (a * - z 0 + z20- a*2)/(a* - z 0 )	 (23)
a for right tip
a =
*
	
(24)
-a for left tip
K 1 and K2 can be found from (21) by comparing the real and
the imaginary parts of the equation. For symmetric problem
(w.r.t x axis), K 2 = 0.
-9-
!	 I	 f	 1
NUMERICAL RESULTS AND CONCLUSIONS
Numerical results will be obtained for uniformly spaced
stringers, i.e.,
c 
	
= d o
 + (J- 1 ) d l . j = 1, ... n s
	 (25)
where d l is the stringer spacing and d o is the distance of
the first stringer to the mid-point of tkhe crack. The fol-
lowing data will be used:
Boron-EDox
Plate:	 Ex = 3.5 x 10 6 psi
Ey = 3.24 x 10 7 psi
vyx = 0.23
G xy = 1.23 x 10 6 psi
h p = 0.09 in
Stringer:	 A S = 0.165 in2
E S	= 1.24 x 10 7 psi
Adhesive:	 u a = 1.65 x 10 5 psi
h a = 0.004 in
q = constant
Since there are two types of orthotropic materials [81,
the numerical results can be obtained for both types. However,
-10-
II ;
for simplicity, only the boron-epoxy considered above (a
Type I material) will be used. Aluminum plate (v - 0.30,
E - 10 7 psi) will represent the isotropic case whenever a
comparison is made.
Intact stringers
Figure 4 illustrates the K/qr variation vs. b l /a for
both orthotropic and isotropic plates. It is seen that the
trend for both materials are the same. Hence, the conclusions
for the isotropic case [5] remain the same for the orthotropic
case. Since the contribution from a third stringer is rather
insignificant, only one or two stringers are consideree. In
any case, an increasing debond length, especially beyond
b l /a > 2~3, increasesthe stress intensity factor appreciably.
Figure 5 shows that the stiffening effect of the stringers
diminish rather rapidly for d o /a > 2 for both isotropic as
well as orthotropic materials. Hence, the following conclu-
sion more or less holds true for both isotropic [5] and ortho-
tropic materials: To have a low stress intensity factor, there
should be as many perfectly bonded stringers as possible be-
tween the two crack tips.
Broken stringers
Figures 6 and 7 show the dependence of K/q r values on the
debond length b i
 and d o in the case of crack surface loading.
This is given to compare with the intact stringer problem.
An obvious observation for both isotropic as well as ortho-
-11-
Itropic materials is the sudden increase in the stress inten-
sity factors when the breakage occurs which ,s quite signifi-
cant.
However, the actual problem is the one where the loads
are applied at infinity. This case is shown in Figures 8 and
9 where it is observed that tI,e stress intensity factors in-
crease beyond unity. This indicates that breakage is very
important since it makes the structure more suscept4 l-, le to
fracture. Also note that the loads at infinity are applied
to both the plate and the stringers simultaneously.
-12-
__T
ACKNOWLEDGEMENT
The author gratefully acknowledges the support of this
work by the National Aeronautics and Space Administration under
the Grant NSG 1178.
REFERENCES
[1] C. C. Poe, Jr., "Stress Intensity Factor for a Cracked
Sheet with Riveted and Uniformly Spaced Stringers", NASA
Technical Report, NASA TR R-358, 1971.
[2] C. C. Poe, Jr., "The Effect of Bro!:en Stringers on the
Stress Intensity Factor for a Uniformly Stiffened Sheet
Containing a Crack", NASA Technical Memorandum, NASA
TMX-71947, 1973.
[3] K. Arin, "A Plate with a Crack Stiffened by a Partially
Debonded Stringer", Engineering Fracture Mechanics, Vol.
6, pp. 133-140, 1974.
[4] K. Arin, "A Note on the Effect of Lateral Bending Stiff-
ness of Stringers Attached to a Plate with a Crack", Engi-
neering Fracture Mechanics, Vol. 7, pp. 173-179, 1975.
[5] K. Arin, "A Cracked Sheet Stiffened by Several Partially
Jebonded Intact or Broken Stringers", NASA Technical Re-
port, Lehigh University, Decembe r , 1975.
[6] S. G. Lekhnitskii, "Anisotropic Plates", Gordon and
Breach Science Publishers, 1968.
[7] M. D. Snyder and T. A. Cruse, "Crack Tip Stress Intensity
Factors in Finite Anisotropic Plates", Air Force Materials
Laboratory Technical Report AFML-TR-73-209, 1973.
[8] K. Arin, "An Orthotropic Laminate Composite Containing a
Layer with a Crack", NASA Technical Report, TR 74-1,
IFSM-74-57, Lehigh University, March 1974. 	 To appear in
International Journal of Engineering Science.
-13-
h
P
(EX,Ey,vyX,
I	 `	 I
E s As)
Gam )	 -- ---- -- ---	 ^
Tbi
-a
X
q	 Aebond
cj b1. ---_— ---- --
P1 ate I
i	 dsd	
ds d 	  
ado	1	 1
j th
stringer
r
	
Plate
 dhesive Stringer
j= 1,...n
Figure 1 - Geometry of t she problem (intact stringers)
Y	
♦ #
	
t
q	 f	 t I	 rte,
hP 1 I t ha
(EX' Ey'vyX'Gxy) - -- — -
C.
4"j
b
^
-a a
^
b	
X
3
Plate
_ _ _ _— ---- — -- —
ds 
I 	
dsdy^tdo
jth
str 	 er
ll
f
=^1 nsS.4
q r
Figure	 2	 - Geometry af ;:.. ` roblem	 ( broken	 stringers)
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......... n 	 = 2, b 2 /a = 0.1	 (ref. NI)
080L-0.0	 1.0	 2.0
do /a
Figure 1 - K/qV—a vs. do/a
(broken stringer, loading on the crack surfaces,
b l /a = 1.0, d 1 /a = 0.5, d s /a = 0.2)
190
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d l /a = 0.5, b l /a = 1.0, d s /a = 0.2)
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Several intact or broken stringers attached to an orthotropic sheet with a crack

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