The effect of several stringers on the stress intensity factors at the tips of a crack is considered. The stringers which were continuously attached to the plate and placed perpendicular to the crack may be partially debonded due to high stress concentrations. Since the stringers may even break under excessive loading conditions, both intact and broken stringers are considered to investigate the effect of rupture. The continuity of displacements along the bond lines leads to an integral equation which is solved to give the shear stress distribution in the adhesive and the stress intensity factors at the crack tips

Arin, K.

English

NASA Technical Reports Server

NASACR- 144952
LEHIGH UNIVERSITY
OFFICE
A CRACKED SHEET STIFFENED
BY SEVERAL PARTIALLY DEBONDED
INTACT OR BROKEN STRINGERS
RESEARCH BY
K, ARIN
NASA TECHNICAL REPORT
DECEMBER 1975
NSG 1178
DEPARTMENT OF MECHANICAL ENGINEERING
-^  - •, '.*
AND MECHANICS
LEHIGH UNIVERSITY
BETHLEHEM, PENNSYLVANIA
https://ntrs.nasa.gov/search.jsp?R=19760007431 2020-03-22T18:21:46+00:00Z
A CRACKED SHEET STIFFENED
BY SEVERAL PARTIALLY DEBONDED
INTACT OR BROKEN STRINGERS
K. ARIN
Lehigh University, Bethlehem, Pennsylvania 18015
ABSTRACT
The effect of several stringers on the stress intensity
factors at the tips of a crack is considered. The stringers
which are continuously attached to the plate and placed per-
pendicular to the crack may be partially debonded due to high
stress concentrations. Since the stringers may even break
under excessive loading conditions, both intact and broken
stringers are considered to investigate the effect of rupture.
The continuity of displacements along the bond lines leads to
an integral equation which is solved to give the shear stress
distribution in the adhesive and the stress intensity factors
at the crack tips.
Introduction
Stiffened panels, i.e. metal sheets with stringers con-
tinuously bonded through an adhesive have long been of major
interest [1], [5]. Greif and Sanders have given the solution
of a stringer perfectly bonded to a cracked sheet [1]. On the
other hand, the case with riveted stringers - both intact and
broken - has been treated by Poe [2] and [3]. Furthermore,
the problem of a cracked isotropic plate stiffened by a stringer
which may be partially debonded has been considered by Arin [4].
It has been concluded that the debonding process as well as the
stringer placing are quite important as far as the stiffening
effect of the stringer is concerned. In a separate work [5]
-1-
the effect of lateral bending stiffness of the stringer has
been investigated.
However, due to high load levels stringer breakage can
occur in addition to debonding. Also, in actual structures
several stringers are present instead of one. Hence the prob-
lem of several intact or broken stringers will be considered
in this paper. The method employed here is the same as the
one used in [4] and therefore most of the results will be used
without derivation.
For intact stringers the case where the loads are applied
on the crack surfaces will be considered (see Fig. 1). This
will also give the singular part of the solution around the
crack tips. The actual problem where the loads are applied at
infinity can be obtained by a simple superposition. However,
for broken stringers the actual problem will be treated as it
is (see Fig. 2) due to difficulties involved in superposition.
In all these cases loads will be considered uniform.
The technique used here makes it possible to consider any
number of stringers located at arbitrary locations. For the sake
of simplicity the numerical results will be given for uniformly
spaced stringers all located along the positive x axis. How-
ever, the results for the stringers located along the negative
x axis can be obtained by simply substituting for d - the
distance of the first stringer to the y axis - its value with
a negative sign.
-2-
Formulation of the Problem
The problem will.be formulated using the same notation*
as in [4]. Also, due to symmetry only the upper half of the
plate will be considered. The adhesive will be treated as a
shear spring and the shear stresses will be considered as
body forces in the plate solution (generalized plane stress).
Let q represent the uniform pressure applied on the crack sur-
faces in the case of intact stringers (see Fig. 1) and the
uniform tension applied at infinity in the case of broken
stringers (see Fig. 2). Then the continuity of displacements
can be written as [4]
v_(z) - v (z) = jHr p(z) • z on L (1)v
 s a
Here, L denotes the union of straight lines L. defined by
J
x = c., b. < y < °°; j = l,...n where n is the number of
J J ~ ^ ^
stringers, b. is the half debond length of the j th stringer.
J
For uniformly spaced stringers we have
c. = d0 + (j-l)d1 (2)
(E,v): Elastic constants of the plate, y = E/2(l+v)
K = (3-v)/(l+v) for generalized pTane stress.
(ES,A ): Elastic modulus and cross-sectional area of the
stringer.
y : Shear modulus of the adhesive.
(h*,h ): Thicknesses of the plate and the adhesive.
p
 a: Half crack length
d • Stringer width
v (z)v^(z)rDisplacements of the plate and the correspondingp
 stringer at z location.
P(z): Shear stress in the adhesive at z location.
-3-
where d-j is the stringer spacing and dQ is the distance of
the first stringer to the mid-point of the crack.
The displacements can be expressed as [4]
vp(z) = qk0(z) +. Jkp(z.z0)P(z0)dy0
L
vs(z) = jks(z,z0)P(z0)dy0 + C, z on L
L
yo = Im(zb) (3)
Note that C represents the rigid body displacement and assumes
a different constant value on each stringer. Hence if z is
L . then C = C . , j = 1 , . ...n- .
J J 5
Kernels in (3) are given as follows:
k (z): For either intact or broken stringers.
on
k (z), for crack surface loading
k 0 (z ) [ K , f
k * ( z )
 + i•  ( ) + 75— y, for loading at infinity
y = I m ( z ) (4)
and
*k (z) = -<)y - 2y Re z
P
(5 )
-4-
kp(z.z0): From [4]
where
K[1og(z-z0) + Iog(z-z0)]
.50) - e1(z,z0)
(T+K)[e4(z,z0) - e4(z,i0)] + K[e3(z,z0)
^^ 'V ' 93(z'zo)
z-z.
(6)
. J t ) = log ZZ
92(z) = log[z + /z2-az]
z-z
z-z
1 +
(z) - Kzn
>4Cz.zQ) -}^ :^0(:
-5-
IU0> • .
- 1 (7)
ks(z,zQ): Defining y = Im(z), y = Im(z )
for intact stringers
if z and z are on the
same stringer
0 otherwise (8)
and C = 0. Also note that due to symmetry the equilibrium
conditions for stringers are automatically satisfied.
For broken stringers, from the solution of a one dimen-
sional elastic body we obtain
ks(z,z0) -
* 0 , y
, y > y
s 0 .
if z and z are on the
same stringer
0 otherwise (9)
and C., j = l,...n are unknown constants. For these nj s s
additional unknowns we consider the following ng equilibrium
equations for the stringers to obtain a compatible system.
-6-
jP(z0)dy0 =
LJ
%r EsV if the load at infinity isalso transferred to the
stringer
if the end of the stringer
at infinity is stress free
j = 1 ,... n (10)
Hence from (1) and (3) the integral equation of the prob-
lem can be obtained as
P(z) + fk(z,z0)P(z0)dy0 + -j£i c = -|£i q k0(z),
, a a
z on L
(11)
which will be considered together with equations (10) and
solved for the shear stress distribution. Note that C in
(11) actually represents n . unknowns-..'
Here
(12)
The stress intensity factor will be defined as
K = lim [/ZU-a)]o (x,0)
x+a y
and given as [4]
(13)
= Im(zQ)
where
(14)
-7-
1
7 TT 1
a +I(z -)•
ao- zo
n{ fZ°" iol
[V2oj
1 + K +
: *o>
ao- zoj
(15)
and
•i:a for right tipa for left tip (16)
Numerical Results and Conclusions
The numerical results are obtained for an aluminum plate
of elastic constants v = 0.30, E = 69.0 GN/m2 (10.0 x 106 psi)
and thickness hp = 2.3 mm (0.09 in.). The cross-sectional
area of the stringer and the elastic modulus are assumed to be
As = 106 mm2 (0.165 in2) and Es = 85.5 GN/m2 (12.4 x lo6 psi)
respectively. The adhesive is supposed to have a shear
modulus of ya = 1.14 GN/m2 (0.165 x lo6 psi) and thickness ofd
ha = 0.1 mm (0.004 in.). Results are given for both left and
right crack tips. It is also assumed that q = constant.
Intact stringers: Fig. 4 shows the effect of debonding
on the stress intensity factors. With the first stringer lo-
cated at dQ = 0.5a, the stringers have little stiffening effect
for b 1 / a > 2 even if the other stringers are still perfectly
bonded to the sheet. As expected [4], the stress intensity
factor appears to be quite insensitive to the third stringer
-8-
due to its location being rather far from the crack tip.
Fig. 5 illustrates the way the stringer spacing influences
the stress intensity factors. Namely more stringers located
between the crack tips result in appreciably smaller stress
intensity factors (compare Fig. 4 and Fig. 5). In this
case, the third stringer being located away from the crack
tip, again has no significant effect. One interesting re-
sult is that as long as there is at least one perfectly
bonded stringer between the crack tips, the debonding process
occurring in any other stringer will result
in only a small drop in the stiffening effect but not a dras-
tic one. Again note that by the time b,/a = 2 most of the
first stringer's stiffening effect is already diminished.
The effect of b2, the debond length of the second stringer
is shown in Fig. 6, keeping b, and b3 constant. For similar
reasons the third stringer is unimportant as far as K factors
are concerned. Note that K/q/a" values approximately approach
to those of a single stringer with b1/a = 1.0, dQ/a = 0.5
(compare Figures 4 and 6). K/q/a vs. b2/a variation appears
to be similar to K/q/a vs. b-j/a. In both cases a considerable
loss of stiffening effect of the stringers is observed with
the increasing b, or b2- It is also possible to show the de-
pendence on b., in a similar fashion. However if the third
stringer is located away from the crack tip, K values will not
be affected significantly. For example
-9-
K le f tS ' ° '8 9 1 > Kr1ght £ °'801 for do /a = °' 5 1 n s * 3 > d1/a = ''°
> by/a = b2/a = 1.0
Kleft " °'945' Kright? °-868 for d0/a = 1 • 0 J d$/a =^0.2
for al 1 b3 values.
The effect of the location of the first stringer namely d iso
shown in Figures 7 and 8. As expected, whatever the number
of stringers, the spacings and the debond lengths are, K/q/a
values rapidly approach unity starting around d /a~2. In all
these curves it is apparent that the main factor for low K/q/a"
values is to have as many perfectly bonded stringers as possi-
ble between the two crack tips. For one stringer lowest K
occurs if the stringer is placed on the crack and approxi-
mately one fourth of the crack length away from the corres-
ponding crack tip. But for two or more stringers this de-
pends on the other parameters. However the fact that the
lowest K values will be obtained by the maximum number of
stringers critically placed on the crack sti l l remains the
same. That is why in most cases lowest K occurs for d = 0.
The dependence of K on the crack length is illustrated in
Figures 9 and 10. For all the curves, the minimum K for the
right tip occurs when dQ/a = 0.30 - 0.50, which is in agree-
ment with similar conclusions drawn previously for a single
stringer [4].
Broken Stringers; In the case of broken stringers the
actual problem , i.e. with the loads applied at infinity will
be considered directly due to the fact that superposition will
-10-
not simplify the solution. We will consider uniform loading
at infinity, i.e. q = constant (see Fig. 2). However for the
sake of comparison with the intact stringers the case where
the loads are applied on the crack surfaces will also be con-
sidered to determine the extent to which the stiffening ef-
fect of the stringers is lost due to breakage. One immediate
observation from Figures 11, 13 and 15 is that the stress in-
tensity factors are considerably higher than those of the
intact stringer results. This suggests that the loss of
stiffening effect is quite significant. Other than this,
similar trends will be observed for broken stringers. How-
ever the adverse effect of stringer breakage does not end
here. As can be seen from Figures 12, 14 and 16 the stress
intensity factors increase much further beyond unity thereby
making a stiffened structure even more susceptible to frac-
ture if the breakage occurs. This would be the same whether
the loads applied at infinity are transmitted through the
plate only or through the plate and the stringers simultan-
eously. High K/q/a values can be attributed to the p u l l i n g
effect of the broken stringers (for intact stringers pulling
works to the advantage of the structure by reducing the stress
intensity factors).
The results in Fig. 11 have similarity with Figures 4 and
5 and therefore can be interpreted in an identical manner.
Fig. 12 also illustrates the K/q/a" dependence on b-|/a in the
-11-
case of loads applied at infinity. The stress intensity
factors for the right tip now become higher than those of
the left tip. And as expected the curves corresponding to
a single stringer tend to approach unity as b, increases in-
definitely. The relation K/q/a vs. dQ/a shown in Fig. 13
for the crack surface loading can also be explained as in
Fig. 7. However K/q/a vs. dQ/a illustrated in Fig. 14 in-
dicates values significantly higher than unity for the loads
applied at infinity. One important observation is that the
K values will shoot up appreciably if one of the stringers is
placed on or very close to one of the tips and debonding in
that particular stringer is either very small or non-existent,
This phenomenon is the opposite of the one observed in the
case of intact stringers (see Fig. 7 and 8). It should also
be noted that all K/q/a" values approach unity as d0 increases,
Fig. 15 illustrates similar findings for K/q/a vs. half crack
length as in Figs. 9 and 10 with the exception of higher val-
ues for K/q/a". Same relation is given in Fig. 16 for the
loads applied at infinity. Finally, to give an idea as to
how d affects the stress intensity factors Fig. 17 shows
this relation for the parameters taken the same as in [4]
for the sake of comparison. As seen from Fig. 17, K/q/a
does not change more than approximately 2%.
-12-
Acknowledgment
The author gratefully acknowledges the support of this
work by the National Aeronautics and Space Administration
under the Grant NS6 1178. He would also like to thank
Mr. Elmo Alexander Jr. for his help in preparing the initial
drawings.
References
1. R. Greif and J. L. Sanders, Jr., "The Effect of a
Stringer on the Stress in a Cracked Sheet", Trans.
ASME, Ser. E., J. of Applied Mechanics, Vol. 32,
pp. 59-66, 1965.
2. 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.
3. C. C. Poe, Jr., "The Effect of Broken Stringers on
the Stress Intensity Factor for a Uniformly Stiffened
Sheet Containing a Crack", NASA Technical Memorandum,
NASA TMX-71947, 1973.
4. K. Arin, "A Plate with a Crack Stiffened by a Par-
tially Debonded Stringer", Engineering Fracture Mechanics,
Vol. 6, pp. 133-140, 1974.
5. K. Arin, "A Note on the Effect of Lateral Bending Stiff-
ness of Stringers Attached to a Plate with a Crack",
Engineering Fracture Mechanics, Vol. 7., pp. 173-179,
1975.
-13-
ffl
m
Fig. 1.
ttt t t t
j th
stringer
j = l,...ns
Geometry of the problem
(intact stringers) .
t t t
Debond
Plate Adhesive Stringer
t
4U I UU4 1.4 4
Fig. 2. Geometry of the problem
(broken stringers)
y0 ur-rq « y]
M t M tt „ ' ^1
—n i . • >- 5t—i *-TTTTTTTa +
s/ « _
;i
p_
ha
Plate Adhesive j thStringer
For uniformly spaced stringerrC= d-»-( j «1 )d j =1 ...p
3. Free-body diagrams
100
Q90
K 0.80
0.70
0.60
0.50
left tip
= 0
0.0 1.0
F1g. 4.
2.0 30 4.0 5.0
b,/a
K/q/a" vs. bi/a(intact stringer, dn/a = 0.5, d,/a = 1.0,
°
1.00
0.90
0.80
n = 2, b = 0
K 0.70
q/o* — — — —
 n =
 3 , = 0
0.60
0.50
0.4O
1.0 2.0 3.O 40
D.O
5.0
F1g. 5. K/q/a vs. bi/a
(intact stringer, dQ/a = d-j/a = 0.5, ds/a = 0.2)
1.00
0.90
left tip
> .Wjfiyt^rT .T« .•rr.fr.....m..-?-.. g..CTf a^  --y. .•?•—a-
0.80
K
q/o" 0.70
O.60
0.50
n$ = 2, d-|/a = 0.5
n$ = 2, d1/a = 1.0
ns = 3, d1/a = 0.5,
ns = 3, d-|/a = 1.0, = 0
0.40 j i i I i i I
0.0 0.5 1.0
Fig. 6. K/q/T vs.
(intact stringer, d0/a = 0.5, bi/a-»1.0*
d /a = 0.2)
K0.90h
0.80
0.70F
n = 3, b/a = 0.5, 6 = 00.60\-
0.50
0.40
Fig . 7 . K/q/a" vs . d Q / a(intact stringer, b-j/a = 1.0, ds/a = 0.2)
1.00
0,90
0.80
0.70
0.60
0,50
Q4O
left tip
. ng = 3, d.,/a
n =2, dn/a
. n* = 2, d|/a
-l.O,
ns = ;5,
ns = ^'
00 1.0
<t/a
20
F1g. 8. K/q/a vs. dQ /aix/ v^y \A w *r • *• U» »(intact stringer, ds/a-=0.2)
1.00
0.90
0.80
K_0.7Q
q/cT
0.60
050
0.40
030
*! \\/ / '*
»V>-; / ' x
» > V /
 x^
n$ = 3, = 0.5,
*
* './
oo
j L
| /I /
/ 5.0
a At 10.0 15.0
- l . -O . d $ / d 0 - 0 . 2 )
1.00
0.90k
0.80h
0.70k
0.60K
0.50k
0.40
0.30L
»*•
Fig. 10 K/q/a" vs. a/d0
(Intact stringer, ds/do
1.00
K
0.90H
0.801
--—n*-2, b,/a = 0.1 W
n*>2 , b, = 0 I
0.0 b/a
Fig. 11. K/q/a vs. b-i/a
(broken stringer, loading on the crack surfaces,
d!/a = 0.5, ds /a = 0.2)
d0/a =
2.00-
Flg. 12. K/q/a vs. bi/a
(broken stringer, loading at infinity,
dy/a = 0.5, ds/a = 0.2)
1.00
left tip
K
q/o"
090
- /
n = 2,
O80
0.0 1.0
d0/a
2.0
Fig. 13. K/q/T vs. dQ /a(broken stringer, loading on the crack surfaces,
= 0.5, ds/a = 0.2)
3.00r
K
q/o"
2.00-
1.00-
F1g. 14. K/q/a vs. d0/a(broken stringer, loading at infinity,
d1/a = 0.5, bi/a = 1.0, ds/a = 0.2)
1.00
Fig. 15. K/q/a" vs. a •(broken stringer, loading on the crack surfaces,
d = 0.5, DT = 1.0, ds = 0.2)
3.0
K
<V5~
,0
0
l l
" h
. i <i f.i t
- 1 F
1 t
- '
 l;
; i
11 (
// t
1 \\\ \;
^'•- Ik\-- Cs.
" ^- ^
^. vv
^X^k A
- \ / v
v 2\ ^
ns = l> d0 = 1.0
.. -.. ng = 1
•
 U2 = 0
ns = 2,
do = o
right tip
left tip
15.0
F19
'
 16
' (b?^enSWringer, loading at infinity,
di = 0.5, b] = 1.0, ds = 0.2)
100
K
0.96
right tip
a = 0.3
i i i
0.5
1.00
K
0.94
left tip
a = 0.45
d = 0.55
j i
O.O 0.5
Fig. 17. K/q/a vs. d§(b roken s t r inger , ns = 1, ( ^ = 0 . 5 , b] = 0)


A cracked sheet stiffened by several partially debonded intact or broken stringers

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