~
Pergamon
INTEGRITY
WITH
Eng&eering Fracture Mechanics Vol. 51, No. 3, pp. 361-380, 1995
Elsevier Science Ltd.
Printed in Great Britain.
0013-7944(94)00252-5
OF AIRCRAFT
MULTI-SITE
STRUCTURAL
FATIGUE
ELEMENTS
DAMAGE?
JAI H. PARK,,+ RIPUDAMAN SINGH,§ CHANG R. PYO¶ and SATYA N. ATLURIH
FAA Center of Excellence for Computational Modeling of Aircraft Structures, Georgia Institute of
Technology, Atlanta, GA 30332-0356, U.S.A.
Abstract--Integrity evaluation of aging structures is extremely important to ensure economic and safe
operation of the flight vehicle. A two step analytical approach has been developed to estimate the residual
strength of pressurized fuselage stiffened shell panels with multi-bay fatigue cracking. A Global Finite
Element Analysis is first carried out to obtain the load flow pattern through the damaged panel. This is
followed by the Schwartz-Neumann Alternating Method for local analysis to obtain crack tip stresses and
the relevant crack tip parameters that govern the onset of fracture. Static residual strength is evaluated
using fracture mechanics based, as well as net section yield based, criteria. The presence of holes, with
or without Multi-Site Damage (MSD), ahead of a dominant crack is found to significantly degrade the
capacity of the fuselage structure to sustain static internal pressure. An Elastic-Plastic Alternating Method
has been developed and applied, to evaluate the residual strength of fiat panels with multiple cracks. The
computational methodologies presented herein are marked improvements to the present state-of-the-art,
and are extremely efficient, both from engineering man-power, as well as computational costs, points of
view.
1. INTRODUCTION
A LARGEpercentage of in-service aircraft has exceeded its initial design life and is prone to wide
spread fatigue damage. Residual strength estimations of critical aircraft components with
detectable cracks are imperative for the safe operation of these aging airplanes. This paper presents
a part of the major effort being made to develop reliable computational tools for structural integrity
evaluation of aging, as well as new, airplanes. These computational tools are envisioned to involve
the least amounts of engineering man-power, as well as computer, resources.
Micro-cracks emanate at material/structure imperfections and discontinuities. These cracks
grow under static and fatigue loading, and coalesce to form detectable cracks. These detectable
cracks can linkup and become catastrophic, at times, within two inspection schedules. In order to
ensure safety, it is important to understand the severity of cracks and also, have an estimate of
collapse strength/residual strength of the damaged structure. This paper primarily deals with the
investigation of longitudinal cracks in stiffened fuselage shell panels, even though the proposed
methodologies apply equally well to a variety of wide spread fatigue damage situations at a variety
of locations in the pressurized fuselages of commercial planes.
In an airliner fuselage, pressurization causes stresses in the shell structure. The stiffeners
(stringers, frames and tear straps) take a part of the load, but the major fraction is taken by the
skin. Large cracks in the skin cause significant redistribution of load flow, which becomes fairly
complex due to various stiffening elements and the presence of holes/MSD ahead of a dominant
crack. Thus, the problem essentially calls for a determination of the load flow pattern through the
panel, before the crack tip stresses and other relevant crack tip parameters e.g. the stress intensity
factor (SIF), T*, etc. that govern the onset of fracture, can be determined.
The finite element alternating method has been found to provide good quantitative estimates
of SIFs at various crack tips and a qualitative insight into the mutual interactions of small, long
and even multi-bay cracks in the shell panel or at a row of fastener holes. This analysis adds to
the understanding of the residual strength of the damaged structure. Global finite element analysis
tPaper presented at the 1CAF-11, Stockholm, June 1993.
SVisiting Scholar from Chungbuk National University, Cheongju, Korea.
§Postdoctoral Fellow, member AIAA.
¶Postdoctoral Fellow.
IlInstitute Professor and Reagents" Professor of Engineering, Fellow AIAA.
361
362
J.H. PARK et
al.
of the multi-bay shell panel with cracks, generates information about the realistic load flow pattern
through the panel. The cracked portion of the shell is then isolated with the corresponding sheet
stresses and the fastener loads (if any). At this stage, for local analysis purposes, the uncracked
skin segment subjected to tractions on its boundaries (as determined from global analysis) and any
fastener loads, is modeled and a finite element analysis is performed to obtain the stress field at
the location of the cracks, which are then erased through the alternating method, using analytical
solutions for an infinite body with pressurized cracks, to determine very accurately the crack tip
stresses and the crack tip fracture parameters [1]. This sequential combination of two step FE
Methods forms an efficient computational tool for the estimation of crack tip parameters and the
corresponding static residual strength. For materials with significant ductility, an Elastic-Plastic
Alternating Method has been developed by Nikishkov and Atluri [2], which generates a very
accurate elastic-plastic stress state solution near the crack tips by using only the finite element
model of the uncracked sheet. This technique has been applied to study the Multi-Site Damage
(MSD) in A1 2024-T3 skin in the present paper. Both the techniques [1, 2] use a single finite element
mesh to study a variety of cracks, of varying lengths, in the skin. This makes the present analysis
extremely efficient.
2. PROBLEM DEFINITION
The fuselage shell panels under consideration are typical of commercial airliners. They
are stiffened along the longitudinal direction by stringers and circumferentially by frames and tear
straps. Tear straps are generally at frame locations, but can also be present at mid-frame stations
depending upon the design philosophies. A typical configuration is shown in Fig. 1. Two
shell geometries, one of each kind, have been considered. The geometrical details are presented
below.
Shell radius R
Shell skin thickness t
Distance between frames
Distance between stringers
Distance between tear straps
Width of T-straps
Thickness of T-straps t t
Frame area
Frame moment of inertia
Frame neutral axis offset
Stringer area
Stringer moment of inertia
Stringer neutral axis offset
Fuselage internal pressure
Rivet diameter D
Pitch of rivets
Material: skin, frame, stringers
Material: tear straps
Shell 1:
Shell 2:
118.5 in.
0.071 in.
20.0 in.
8.0 in.
20.0 in.
3.0 in.
0.025 in.
0.7171 infl
1.4320 in. 4
3.25 in.
0.6721 in]
0.1020 in. 4
0.68 in.
8.6 psi
0.1875 in.
1.25 in.
AI 2024-T3
Ti 8-1-1
74.0 in.
0.036 in
20.0 in.
9.25 in.
10.0 in.
2.0 in.
0.036 in.
0.160 in. 2
0.120 in. 4
3.15 in.
0.186 infl
0.040 in. 4
0.78 in.
9.0 psi
0.15625 in.
1.0 in.
A1 2024-T3
A1 2024-T3.
The lap joint configuration considered is as follows:
Length of overlap
No. of rivet rows
Pitch of rivets
No. of rivets in each bay
Rivet diameter D
Adhesive layer thickness ta
Material
3.0 in.
3
1.0 in.
20x3
0. ! 5625 in.
0.00025 in.
AI 2024-T3.
Integrity of aircraft structural elements
(a)
363
Frames
Skin
Lap.lomt ~
~'"" '
Shell panel
Stringers
Straps
(b)
Cracked Skin S e g m e n t
Fig. 1. Fuselageshell panel configuration.(a) Shell panel. (b) Cracked skin segment.
The material properties of A1 2024-T3 and Ti 8-1-1 are taken as follows:
Young's modulus E
Shear modulus G
Poisson's ratio v
Yield strength ay
Ultimate tensile strength au
1
Crack tip linkup stress 5(ay
+ au)
% Elongation
Fracture toughness Klc
Adhesive shear modulus is
Ga =
A1 2024-T3
Ti 8-1-1
10.5 x 10 3 k s i
4.2 × 10 3 k s i
0.32
47.0 ksi
64.0 ksi
55.5 ksi
18%
93.0 ksi ,,/~.
17.5 × 10 3 ksi
6.7 × 10 3 ksi
0.32
135.0 ksi
145.0 ksi
140.0 ksi
8%
1.09 x 105 psi.
Four situations of cracks are considered in this article:
(1)
(2)
(3)
(4)
a single dominant multi-bay crack in shell 1,
a single dominant multi-bay crack at a row of fastener holes in shell 1,
long cracks and MSD at lap splice in shell 2, and
a single dominant multi-bay crack at lap splice in shell 1.
In addition, a full elastic-plastic analysis of a flat panel with multiple cracks is performed, the
configurational details of which are given separately in the corresponding section.
3. ANALYTICAL APPROACH
The method of analysis adopted, is to first evaluate the load flow through the damaged panel
and then perform a refined analysis to obtain stress fields at the various crack tips. A global-local
finite element analysis procedure has been developed for this purpose.
3.1. Global analysis (FEM)
Conventional linear elastic finite element analysis of the multi-bay stiffened shell panel with
cracks, is performed as a part of the global analysis• The FEM model is briefly described below:
The fuselage skin is modeled by four-noded shell elements, with five degrees of freedom per
node. The element used is strain-based and was developed by Ashwell and Sabir [3]. Tear straps
are also modeled using the same element• The frames and stringers are modeled as two-noded, three
degrees of freedom per node, curved/straight beam elements with their shape functions degenerated
from those of shell elements. This is done to ensure compatibility within the stiffeners and the sheet.
The cracks are incorporated into the problem as unconnected nodes belonging to respective
elements. For the purpose of present global analysis, the crack tip singularity is not modeled as
364
J.H. PARK et al.
the crack will be modeled analytically in the second step, i.e. the local analysis. The fasteners are
modeled as two degrees of freedom connectors between the corresponding nodes and the fastener
stiffness is represented by the empirical relation developed by Swift [4]:
ED
with A = 5.0 and C --- 0.8 for A1 rivets. Wherever there is a crack, the stiffness along the crack
length is relieved as the fastener will not be able to bear the load in that direction. Adhesive is also
modeled as a two-noded, two degrees of freedom per node connector between the sheets. The
adhesive stiffness is modeled as
area
ga = fld ta 1- 3 / 5
t2 ) ,
(2)
aa'8ka+
where Pd is the degradation factor. The values of 0 and 1 represent total degradation and perfect
conditions, respectively. A value of 0.1 means 90% degradation. "area" Represents the bond area
being lumped at the nodal connections. Appropriate multi-point constraints have been imposed
to prevent criss-crossing of sheet nodes in the lap joint zone. The fuselage internal pressure is
applied as a uniformly distributed normal outward load on the shell panel. The four edges of the
panel are permitted to undergo only radial displacement in a cylindrical system. A typical problem
size for the configurations considered is of the order of 15 000 degrees of freedom and the computer
time is of the order of min on an HP 9000/700 series workstation.
3.2. Elastic local analysis [ E - F E A M ]
From the global analysis, the skin segment containing the cracks, holes and fasteners of
interest, is isolated with corresponding sheet stresses. The fastener holes are now modeled as
circular and the bearing loads (if any, from global analysis) are distributed as sinusoidal variations
over the periphery. The stresses due to the misfit of the rivet can also be accounted for at this stage.
This problem is solved using the Schwartz-Neumann Finite Element Alternating Method which
involves two solutions.
(1) An analytical solution to the problem of a row of cracks of arbitrary lengths with crack faces
being subjected to arbitrary tractions,
(2) Finite element solution for a strip, with/without a row of holes, but without cracks; the strip
being subjected to sheet stresses and pin bearing loads. Since the finite element solution is
only for the uncracked body and the cracks of arbitrary lengths are accounted for
analytically in step 1 above, the computational finite element mesh remains the same as the
cracks grow.
This technique and the analytical solution to the problem of multiple cracks in an infinite sheet
are presented in detail in an earlier work by Park et al. [5]. Eight-noded isoparametric elements
with two degrees of freedom per node are employed in this finite element analysis.
The crack tip stress intensity factors and the stress field are obtained directly from the FEAM
analysis. The net section stress for any ligament is obtained by taking an average over the ligament
length. To compute the plastic zone size, Irwin's formula does not seem to give a reasonable
approximation, probably due to the complexity of geometry and vicinity of other cracks/loadedholes. So the plastic zone size is estimated from the computed stress field by doubling the distance
from crack tip to the point where the stress falls to yield stress.
Critical pressure for the fuselage is that value of applied pressure differential for which either
the crack tip SIF becomes equal to K~c of 93.0 ksi x / ~ . , or the net ligament stress equals the linkup
stress of 55.5 ksi. For linear elastic analysis, this can be computed directly from the obtained values
of K~ and average ligament stress tray.
KIc
Critical pressure differential = applied pressure x - Ki
Integrity of aircraft structural elements
Critical pressure differential = applied pressure x
365
linkup stress
O'av
3.3. Elastic-plastic local analysis [EP-FEAM]
Elastic-plastic analyses for fiat panels with multiple cracks are conducted using the ElasticPlastic Finite-Element Alternating Method in conjunction with the initial stress method as recently
developed by Nikishkov and Atluri [2]. This is briefly described below:
The initial stress method [6] for the solution of elastic-plastic problems is an iterative scheme,
in which the elastic-plastic displacement and stress fields are sought as the summation of the
respective elastic solutions due to artificial volume loads. Since the initial stress method requires
only elastic solutions in the calculation procedure for an elastic-plastic problem, it is possible to
develop an elastic-plastic alternating method using the usual alternating method, which provides
an elastic solution at each iteration. Then, the algorithm for the elastic-plastic alternating method
based on stress is as follows:
Initial conditions: { q~)} =
=
[b °q =
Iterative procedure: {Aq~~} =
{ q~)} =
{a~ )} =
{t~i) } =
Approximation {t~i) } =
Analytical solution {a~i)} =
{0}
{e }
[0]
[Ko]-'{~'"-~}
{q~-')} + {Aq(oi) }
[Di[B]{ q~)}
- [n¢]{o'~)}
- [b(i)]{ U}
{a~([b")])}
{if(i)} .~. [DCP][D] l({o.~) } + y,~¢i)~t_c
,J
{~b¢')} = {P} - S,,[B]V{a 'i,} dV
Convergence criterion: IIAq~) II < ~ IIq HHere, indices " o " and "c" are used to denote quantities relating to an uncracked finite element
model and to the analytical solution for the infinite plate with the crack, respectively; { q } and {Aq }
are the displacement vector and its increment; {~, } is the finite element residual vector; {P } is the
current level of applied load; [K] is the global stiffness matrix; {a} is the current stress level; [B]
is the displacement derivative matrix; [D] is the elasticity matrix; [D e°] is the elastic-plastic
constitutive matrix; {to} is the traction vector on the crack surface; [b] is the complex coefficient
of the approximation; { U} are Chebyshev polynomials and V is a volume of the finite element
discrete model. This single step algorithm can be divided into one with several load steps. In this
case, the displacement vector { q} should be treated as an increment for the current load step. In
the present method, stresses {ac} in an infinite cracked body are computed directly from an
analytical solution at finite element integration points and are used to calculate the incremental
stresses. For computing the elastic-plastic stress increment with sufficient accuracy, it is necessary
to divide the strain increment into small sub-increments and apply a stress correction at the end
of each sub-increment. The details of the algorithm of the elastic-plastic alternating technique are
presented in Nikishkov and Atluri [2].
The elastic-plastic algorithm for multiple cracks is similar, except for the part of calculating
Chebyshev polynomials. In the present case, Chebyshev polynomials for a crack must be computed
by also accounting for the stress effects of other cracks. The complete analytical procedure for
multiple cracks is schematically shown in Fig. 2.
In the analytical model, the material is presumed to obey the plastic flow theory with a Mises
yield surface. The deformation curve is taken as
a = Cry+ ke P,
where tr is the uniaxial stress, k is the hardening coefficient given as (au - O'y)/(0.18 - try~E) and
eP is the uniaxial plastic strain.
366
J.H. PARK et
al.
lastic-Plastic Alternating Technique lot Multiple Cracks [
Ptotal
4444
Rcvcrsc Stresses at
Crack Locations.
/
/"
/
.f
/
/
eresidual
i /
j~
/
J
J
[
l
Ate Resldual Load
Vectors Zero ?
[
~
EvaluateElastic Stress
Distributions by Using
Chebyshev Polynomials.
Yes
p;a
Elastic-Plastic Stresses by
Using Initial Stress Method.
Fig. 2. Schematicof the elastic-plastic alternating method for multiplecracks.
4. SINGLE DOMINANT MULTI-BAY CRACK IN SKIN
Consider a panel of shell 1, consisting of seven frames (six bays) and six stringers (five bays).
There are seven tear straps, one at each frame location. Consider a single dominant crack aligned
longitudinally and located centrally, i.e. halfway between two stringers and symmetrical about a
frame/strap location. Swift [7] has clearly defined the design requirements for the fuselage from the
damage tolerance point of view. The fuselage stiffened shell must be able to sustain a two-bay-long
crack with broken central strap. For this requirement, consider the central tear strap also to be
cracked along with the skin. All other stiffening elements are intact.
Analyses have been carried out for various crack lengths up to 50 in., i.e. the crack spreading
in four bays. Since there is only one crack and there are no holes in the vicinity of the crack tip,
it is difficult to reasonably define an intact ligament length, thus the critical pressure is computed
only from the fracture mechanics point of view (Fig. 3).
Integrity of aircraft structural elements
Frame
--] T strap
367
I
I T strap
Frame
I
I
35.0
\
"5-
a
/
"E
0.0
0.0
I
I
I
l
10.0
I
I
i
I
8 6 psi
1½
Half Crack Length
/
"
b
0jI
•
2410
-
I
Local Analyses Region
Fig. 3. Critical pressure diagram for a single dominant multi-bay crack in the shell panel.
A point on this curve gives information about the cabin pressure required to make the crack
tip SIF critical and does not imply that the panel will fail catastrophically. The curve falls below
the applied pressure of 8.6 psi for crack lengths of 10.8 and 18.2 in. (points " a " and "b"). The
implication of this is as follows. Considering an initial half crack length of, say, 2 in. and the
pressure never exceeding 8.6 psi, the crack cannot grow under static loading, it will grow under
cyclic loading till it reaches point "a". At this stage, fast fracture will occur and the crack will
instantaneously grow up to point "b", where it will be arrested as the SIF falls below the critical
value. Further growth of the crack will be again under cyclic loading. The effect of plasticity is
not included in this calculation, and stable tearing that follows crack growth initiation is not
modeled.
5. TWO-BAY CRACK W I T H H O L E S NEAR T H E CRACK TIP
Consider a panel of shell l, consisting of seven frames (six bays) and seven stringers (six bays).
There are seven tear straps, one at each frame location. Consider a single dominant crack aligned
longitudinally along the fastener holes. The crack is located centrally and is symmetrical about a
frame/strap location. The central tear strap is broken. All other stiffening elements are intact. The
presence of holes alters the stress flow. The distance between the crack tip and the hole periphery
is found to have substantial effect on the crack tip SIF and also, for very small ligaments, the section
will yield before the crack becomes unstable. A situation of holes ahead of a two-bay crack is
analysed to study the effects of holes on the critical pressure.
Figure 4 shows the effect of holes ahead of a two-bay crack. For the sake of convenience we
will use a term "overhang", defined as the extent of crack length from the closest hole center
involved in the same crack. Zero overhang implies a hole at the crack tip and so the SIF has no
definition. For extremely small overhang, the SIF value is low and for longer overhangs, the SIF
rises sharply. The ligament stress is found to increase steadily with overhang. For any crack length
and overhang, the critical pressure is the lower of the two values corresponding to K~ = K~c and
a~v = linkup stress. Figure 4 shows the critical pressure from both considerations and interestingly,
the structure is critical from the net section failure point of view over most of the region. Again,
any point on this curve represents the pressure differential required to either make the crack
unstable, or cause tensile failure of the ligament. It is seen from this analysis that the presence of
holes (even without cracks emanating from them) ahead of a lead crack, significantly lowers the
residual strength of a stiffened panel; as surmized by Swift [8] and earlier in ref. [7].
EFM 51/3~
368
J.H. PARK et al.
Frame
-w--w- Critical
y " t ' - - ~ strap --"1
~
[
Net Section Yield
With Holes
35.0
I
o.o-r
~
I
i
I
I
2~0
|
|
I
24.0
Half Crack Length
[]
-
Local Analyses Region
Fig. 4. Critical pressure diagram for a single dominant two-bay crack with holes ahead of the crack tip.
6. L O N G CRACK AND MSD IN LAP S P L I C E
We now consider the problem of a riveted/bonded longitudinal lap joint in the shell panel.
Skin stresses have to flow through this joint. So long as the adhesive is in good condition, it
transfers most of the load across the lap, but the adhesive is found to degrade fairly fast with usage
and time, and make fasteners the major load carriers. The presence of cracks at rivet holes causes
further load redistribution. An important aspect of the situation is how one crack effects other
cracks in its close vicinity.
Consider the fuselage panel 2 with a longitudinal riveted/bonded lap joint. Let there be cracks
at the outer (critical) row of fastener holes. The adhesive is treated to be degraded, so that fasteners
share a major part of the hoop load. The fuselage panel considered for analysis, consists of three
bays in the longitudinal direction and six bays along the circumferential direction. This panel
contains four frames, seven tear straps and seven stringers. Tear straps exist at all frame and
mid-frame stations. The cracks are only considered in the central bay. Figure 5 shows 12 c a s e s
[B-M] of crack configurations that have been analysed in this section. Case A is the uncracked
situation and forms a reference. These configurations have no relationship with each other and the
long crack configurations are not necessarily arrived at through linkup of small crack situations.
In all these cases, the stiffening elements (frames, T-straps and stringers) are all intact.
6.1. Global analysis
The untracked panel is analysed as the first case, to obtain a reference and understand the
original load flow path. After the adhesive has virtually degraded (99% for the present problem)
the rivets transfer a large part of the hoop load. In this situation 83% goes through the fasteners
and only 17% of the load is taken by the frames/straps. Among the three rows of fasteners, the
outer two carry ca 36% each and the rest of the 28% is taken by the central row. In a flat panel
lap joint, normally, the center row carries a much lesser portion. The curvature seems to be a cause
of more even row-wise load distribution. The maximum sheet hoop stress is found to be 13.4 ksi.
The presence of cracks at rivet holes causes load redistribution at the joint. For various cases
considered [A-M] the load changes in straps/frames in the damaged bay are given in Table i. It
can be observed from this table that MSD, or long cracks, [B-I] do not significantly overload the
frames and straps. Even for case I, where one may say that the T-strap load has gone up from
Integrity of aircraft structural elements
i
i
A
369
0110
0
0
0
0
0
0
O
0
O
O
O
O
O
O
O
O
OIio
I
I
I
OIO
t
I
I
w
B
I
I
I
o ,o
% ~ %~=,~%o
o
O
O
O
O
O
O
O
O
O
0
0
0
0
0
0
0
0
0
OlO
I
I
I
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
010
I
I
I
010
I
I
I
I
I
I
C
0,o % ~ % ~ % ~ o
D
o Io o i ~
%~o-~s
o
E
!
I
I
010
I
I
I
0
0 I0
F
01o o-o-~ ~>-o-~ o
I
(
i
0
0
-0---0"--~.
!
I
I
G
0,0 0-%0--o-%° o
H
o 1o - o - - o - o - ~
010
O
O
O
O
O
O
O
O
O
OIio
1
!
I
0
010
I
I
I
ol, o
0
0
0
0
0
0
0
0
0
0
0
0
010
I
I
I
0',0
I
I
!
o o 0
I
I
I
I
o
o -.o~o--o--o--o-o
J
o
o oa~ %~a~O
o
o
0 ~*~:~0~'=0~¢
0 Ii 0
t
I
!
K
L
M
o
o
~
o
°
o
[
!
I
I ~ ^ ~
I
oto
~
Ic
I
I
1
I
I
.o--o--.o
-o--o
b
b
o
I
I
I
o
I
o
I
i
I
l
l
I
I
I
I
I
I
~
O
i
I
i
O
I
I
°
!
!
I
IO
T strap 4
T strap 3
j
t
t
o o,o
e
o
I
oto
T stra
5
Frame 3
Frame 2
= 0.25"
= 0.3"
= C.s"
Fig. 5. Crack configurations analysed to study the long crack and M S D interactions at lap splice.
24.5 to 84.4 lb, numerically both these values are insignificant as compared to the yield strength
of 3384 lb (2 x 0.036 × 47000 = 3384) for the strap. Some amount of overloading for the central
T-strap can be seen in cases J and K, where cracks exist across it. From cases K - M , it is evident
that if two cracks approach a tear strap from both sides and linkup at a location where there is
no frame, the strap will yield.
Table 1. Load redistribution at frames/T-straps for various cases (lb)
Case
A
B
C
D
E
F
G
H
I
J
K
L
M
Frame 2
T-strap 3
T-strap 4
T-strap 5
Frame 3
1127
1142
1142
1146
1142
1168
1153
1232
1231
1104
1284
1246
1535
24.5
34.6
34.5
34.9
32.5
47.5
40.2
86.6
84.4
-9.4
100
19.9
230
228
211
218
248
248
250
221
240
252
1510
489
4163
4558
24.5
24.1
23.9
21.8
22.0
20.4
22.6
16.6
32.3
-4.5
90.0
19.9
230
127
127
127
126
126
126
127
125
151
121
1271
1246
1535
In cases L and M, tear strap has yielded, load > 3384 lb.
370
J.H. PARK et al.
Table 2. Row-wise rivet load distribution (lb)
Row 1
Row 2
Row 3
Case
Load
%
Load
%
Load
%
Total load
A
B
C
D
E
F
G
H
I
J
K
L
M
3394
3305
3278
3280
3302
3267
3361
3223
3201
2778
2937
1669
1101
36.1
35.2
34.9
35.1
35.3
35.2
35.9
35.2
35.1
34.1
34.7
34.6
34.5
2600
2682
2696
2673
2661
2654
2627
2615
2633
2421
2485
1442
952
27.7
28.6
28.7
28.6
28.5
28.6
28.1
28.6
28.9
29.7
29.4
29.9
29.8
3405
3395
3402
3385
3382
3366
3358
3315
3272
2956
3035
1718
1141
36.2
36.2
36.3
36.3
36.2
36.2
35.9
36.2
35.9
36.2
35.9
35.6
35.7
9398
9383
9377
9339
9347
9288
9347
9154
9107
8156
8458
4831
3195
F o r all the cases, the row-wise rivet l o a d d i s t r i b u t i o n within the d a m a g e d panel is given in
T ab l e 2. It can be o b s e r v e d that as the crack length increases an d the load is shed f r o m the rivets,
the relative share o f the three rows changes marginally. Thus, as the cr ack ed r o w sheds the load,
the o t h e r two rows also do so a n d effectively, the load goes into the skin an d the frames/straps.
Physically, this a p p e a r s correct as the rivet rows are 1 in. a p a r t and with a crack o f a few inches
in the first row, the second and third rows will hardly get any load to share.
T h e rivet-wise load redistribution p a t t e r n for the critical r o w for all the cases [ A - M ] is given
in Tab l e 3. T o have an overall q u a li ta ti v e idea o f the load redistribution, the loads are also p l o t t ed
to scale in Fig. 6. R iv e t n u m b e r s 1-10 run f r o m left to right in the d a m a g e d bay. A crack diverts
the l o a d t o w a r d s its tips. So, intuitively a crack on one side o f the hole should o v e r l o a d the fastener,
an d cracks on b o t h sides will u n l o a d it. Rivets 5 an d 6 o f cases A an d B clarify this point. O n a
similar logic, the stresses in the l i g a m e n t between t w o crack tips will be high an d further increase
with crack growth. A t s o m e stage o f loading, the ligament will yield, causing a crack linkup to f o r m
a single long crack.
C o m p a r i n g the rivet load patterns for cases H an d I, we can see that even long cracks across
the tear strap do n o t affect each other. T h e rivets i n v o l v e d in long cracks take very little load, thus
significantly o v e r l o a d i n g the n e i g h b o r i n g rivets (case H). In these cases, the n e i g h b o r i n g rivets are
also f o u n d to bear some side l o a d as a c o n s e q u e n c e o f inclined load flow p a t h t h r o u g h the skin,
at the r o w o f rivets. T h e side load on rivet 8 o f case I is o f the o r d e r o f 55 lb t o w a r d s rivet 7, an d
Table 3. Rivet load distribution in the critical row (lb)
Case
Rivet
A
B
C
D
E
F
G
H
I
J
K
L
M
I
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
135
148
164
174
180
182
181
181
179
168
168
179
181
181
182
180
174
164
148
135
141
146
166
123
198
135
132
187
171
191
186
175
180
181
182
180
174
164
148
135
141
146
167
123
200
138
140
131
182
195
187
175
180
181
182
180
174
164
148
135
143
149
172
133
238
126
58
116
217
208
191
176
180
181
182
180
174
164
148
135
141
145
163
186
229
122
57
115
216
208
191
176
180
181
182
180
174
164
148
135
151
171
236
62
150
154
63
121
224
211
192
177
180
181
182
180
174
164
148
135
145
156
195
107
273
60
106
222
183
196
188
176
180
181
182
180
174
164
148
135
176
246
141
39
26
35
101
305
214
210
193
178
181
181
182
180
174
164
148
135
176
246
142
40
26
35
101
305
215
213
204
186
223
106
60
273
108
196
157
147
148
151
170
125
202
140
140
210
269
75
18
8
11
21
46
175
322
216
171
150
203
311
107
33
18
17
29
74
371
294
279
354
111
35
19
19
33
104
319
196
246
388
133
41
20
12
8
3
- 5
- 14
-14
- 5
3
8
12
20
41
133
388
246
365
120
35
17
11
9
8
3
-6
-15
-15
-6
3
8
9
11
17
35
120
365
Integrity of aircraft structural elements
371
A
I
B
I
o
C
D
o
Z,
6
E
F
G
H
J
~)~
K
Scale:
to'Z z'
I
I
I 0
I
~ 200 Ibs
Fig. 6. Rivet load redistribution pattern due to cracks at fastener holes in lap splice (Table 3).
rivet 9 in case J is 48 lb towards rivet 10. Very long cracks (over 10 rivets) are found to even reverse
the load direction on some of the centrally located rivets (cases L and M). This happens due to
inplane bending, again as a consequence of load flow diversion due to the crack. This analysis is,
of course, not correct as the tear strap has yielded. Thus, the cases L and M are not further
analysed. Looking at Table 3 one can see that case I is essentially the case when the configuration
H exists on the left side of the tear strap and the configuration G exists on the right side of the
tear strap. In this case, although the strap is overloaded, there is no mutual interaction between
the cracks and so it need not be analysed as an independent case.
6.2. Local analysis
The finite element alternating technique is employed to cases of interest, viz. B-H, J and K,
by isolating the cracked portion (row and bay) from the panel. Crack tip parameters and stress
fields are obtained from these analyses. The net section stresses and plastic zones are then
computed. Figure 7 shows the maximum net section stresses and the residual strength in terms of
cabin pressure based on the linkup stress of 55.5 ksi. Case J is intended to study the effect of a
large crack on MSD and so the local analysis is performed only on the MSD region and the
indicated maximum stress is not for the complete section, which is likely to be between rivet 9 and
10. The following observations are made from this figure:
• Case B and C - - T h e maximum net section stress is not significantly increased by MSD and
adequate residual strength still exists in the panel.
• Case C and D - - L i n k u p shoots up the maximum net section stress.
• Case B and J - - A long crack on or across the tear strap marginally increases the net section
stress in the MSD zone.
• Case D and E - - M S D near a long crack increases the ligament stress.
• Case F - H and K - - O n e or two long cracks involving five or more rivets push the maximum
stress beyond yield stress, thus substantially reducing the residual strength. The sections at the
crack tips either yield (F, H, K) or tend to yield (G).
372
J.H.
et al.
PARK
0
B
O
0
O-
-0-
O-
-O-~ -O -
0
0
C
O
0
O-
-0-
O-
- O - "q - O -
-0-
0
D
O
0
O-
--0-
O - ~--O
O
O-
0
E
O
O
0
O
0 "' "O
O
O-
0
F
O
0
0
O .....0 ~'~ -
C~
0
G
O
0
0
O- ~ 0 - ~ 0 ~ - - 0
0
0
H
O
0
-0
d
O
0
K
0
0'°---0
~
-
9
25 psi
~
0
0
0
,0
0-~0
-0-
O-
- O -'~ "O'-
O
0
0
O~
0
eZ
0--
0
.0
0
m
0
9
25 psi
Critical Pressure
Link up Stress = 55.5 ksi
Fig. 7, Maximum net section stresses and the correspondingcritical cabin pressure (psi) based net ligament
stress criterion (Table 4).
The stress intensity factors and the corresponding residual strength obtained with
K,c = 9 3 . 0 k s i v / ~ .
are shown in Fig. 8. The following observations are made from this
figure:
• Case B and C - - S h o r t and medium M S D cracks at neighboring rivets have marginal
effects, e.g. SIF at rivet 7 goes up from 14.2 to 14.5 due to a crack at rivet 8. Short/medium
cracks, one rivet hole apart, do not interact. Adequate residual strength exists in these
situations.
• Case C and D - - L i n k u p causes significant increase in the crack tip SIF, but has little effect on
the SIF at M S D in the neighborhood.
• Case D and E - - L i n k e d up cracks in close proximity affect each other marginally and do reduce
the residual strength.
• Case H and K - - I n moving from H to K, one more fastener hole is involved. Interestingly, the
SIF on one end falls and goes up on the other end. This is due to the load redistribution. The
residual strength for these cases has just fallen below the applied load o f 9.0 psi.
The maximum SIF, net section stresses and the corresponding residual strengths are listed in
Table 4.
Looking at Figs 7 and 8, and Table 4 it is quite clear that the net ligament yields before the
crack becomes unstable, under static loading conditions. Plastic zone sizes for all the cases have
been computed (Fig. 9). For M S D cases B, C and J, the plastic zones are insignificantly small. They
are found to grow as cracks link up. Table 5 gives the various K, and rp values for all the crack
cases analysed. Plastic zone size becomes significant once K, exceeds c a 50.0, which happens when
three or more rivets are involved in a single crack.
Integrity of aircraft structural elements
373
0
B
o
o
oJ
toJ
to J
o
o
C
o
o
oJ
toJ
toJ
to J
o
D
o
o
o~
E
0
o
0
o
~DJ
o
F
o
o
0-----0
O
o
0
o
0
0
H
o
o
o
o
J
o
o
O
O
K
0
0
0
o
25 psi
9
o
.4
o-1
0
o
o
0
o---o
C~j
o
o
m
I n
i
I
~Frame
0
25 psi
9
Critical
Pressure
K i c = 93.0 ksi i ~ n
Fig. 8. Stress intensity factors and the corresponding critical cabin pressure (psi) based on fracture
mechanics criterion (Table 4).
7. TWO-BAY CRACKING AT LAP SPLICE
We now consider the aspect of a multi-bay crack with, and without, an MSD ahead of it. In
this section, we consider a panel of shell 1, consisting of six bays in the longitudinal direction and
six bays along the circumferential direction, and a lap joint. This panel contains five frames, five
tear straps and seven stringers. Tear straps exist at all frame stations. Let there be a long crack
extending equally on two sides of the central broken tear strap. All the other stiffening elements
(frames, T-straps and stringers) are considered intact. We are interested in the residual strength
for various lead crack lengths and the effect of MSD ahead of the crack tip.
7.1. Single dominant crack at fastener holes
First consider the situation of a single dominant crack only. Global analysis is carried out to
obtain the rivet bearing loads and then local FEAM is applied to obtain the crack tip SIF and
net section stresses.
Table 4. Residual strengths for various cases considered (cabin pressure, psi)
Fracture mechanics
Case
B
C
D
E
F
G
H
J
K
Max K~
14.6
14.8
53.4
52.7
53.4
55.6
98.2
15.9
108.9
Net section stress
Critical pressure
Max a.~
Critical pressure
57.33
56.55
15.67
15.88
15.67
15.05
8.52
52.64
7.68
22.3
22.6
38.4
30.8
55.5
44.8
59.9
23.7
61.9
22.40
22.10
13.00
16.21
9.0
11.15
8.34
21.08
8.07
374
J . H . P A R K et al.
B
0
(D- -<)-
O-
-0-
-<)-
0
0
0
0
0
-0-
-0-
-0-
0
0
0
0
C
0
0
O-
-0-
O-
D
0
0
O-
-0-
O- e-O
0
O-e
0
0
0
0
E
0
0
0
0
0
e<)
0
O-e
0
0
0
0
F
0
0
0
0
o - e q~O
0
O~
0
0
0
0
G
0
0
0
0-t
0
0
~
0
0
0
0
0
H
0
oI-o
o
o
o
o4o
o
0
0
0
J
0
0
-0-
0--
-0-
O-
oI-o
K
o
o
o
-0-
o
0
o-1---o-
0
o-0o
I
I
1
I
I 0
0
I 0
I
I
i =
Plastic zones
smaller than
2"
= i
0.035 " are not shown
Fig. 9. Plastic zones for various crack configurations (Table 5).
In the last section, we saw that the magnitude of crack length from the hole center (overhang)
is an important factor in determining the critical pressure. For a single dominant crack, at the outer
critical row of fasteners, spreading equally on both sides of a broken tear strap, the critical pressure
diagram is shown in Fig. 10. Linkup stresses for the lead crack (ahead of which there are fastener
holes, but without MSD cracks) correspond to the net section yield between the crack tip and the
fastener hole. Up to c a 40% overhang, the shell panel is K,c critical and for the later half, it has
Table 5. SIFs and the corresponding plastic zones for all crack tips
Case
B
C
Kl
rp
10.5
13.4
13.5
12.5
14.5
14.6
0.0160
0.0269
0.0271
0.0205
0.0303
0.0300
14.4
14.2
0.0299
0.0296
10.5
13.5
0.0161
0.0272
13.6
0.0274
12.5
14.7
0.0208
0.0313
14.8
14.6
0.0311
0.0304
14.5
13.8
13.4
0.0301
0.0285
0.0279
K,
rp
11.0
14.4
14.9
13.2
53.4
50.8
0.0176
0.0318
0.0331
0.0181
0.2142
0.2010
52.7
50.3
0.2101
0.1986
53.2
53.4
0.2882
0.2089
53.4
0.2121
G
29.2
55.6
0.1090
0.2237
H
92.5
98.2
0.2923
0.3210
Case
D
E
F
Case
J
K
Kl
rp
10.6
13.8
13.7
12.6
15.2
15.0
0.0171
0.0274
0.0274
0.0214
0.0307
0.0303
15.9
15.8
0.0314
0.0316
108.9
82.5
0.3395
0.2873
Integrity of aircraft structural elements
Frame
Frame
--] T strap
35.0
[, ~/[
~
Critical
Section
,~-
.~
~
_
-
375
8.6 psi
T strap
Yieldin~1g I i i/
SIF
1 I
0.(I-0.0--
_!
10.0
Half Crack Length
Frame
20.0
~
[
Tear Strap
Fig. 10. Critical pressure diagram for a single dominant multi-bay crack at outer critical row of fastener
holes in a lap splice.
too little section strength. Interestingly, a two-bay crack is fully section strength critical.
Frames/tear straps appear to provide adequate residual strength to the panel.
7.2. Single dominant crack and MSD at fastener holes
We now explore the effect of MSD near fastener holes ahead of the dominant crack in a lap
splice. The important parameters are the lead crack length, the lead crack overhang from the
nearest fastener hole, and the number and lengths of MSD cracks near fastener holes ahead of the
lead crack. The MSD considered for the present purposes is over the three rivets immediately ahead
of the lead crack tip, as the far away MSD cracks have insignificant effect on the lead crack tip
stress field and also have lower intact ligament stress. In the MSD zone, cracks of equal length
are presumed to be present on both sides of the rivet holes. To understand the MSD effects, for
various lead crack lengths, starting from a situation where the lead crack spans over only two holes,
to a situation of a multi-bay crack involving more than 40 rivets, the following two sets of crack
configurations (overhang and MSD length) are analysed.
(1) MSD length of 0.12in., which corresponds to the maximum that can hide under the
countersunk head and stays undetected during regular economical nondestructive inspections. Two values of lead crack overhang, viz. 0.25 in. and 0.50 in., are considered.
(2) Lead crack overhang of 0.25 in., and the MSD lengths of 0.12, 0.25 and 0.35 in.
The analyses for the configurations of the two sets are performed and the residual strengths
are computed, based on linear elastic fracture mechanics, as well as net section stress criteria
(Tables 6, 7).
From Table 6, it can be seen that the small MSD cracks (0.12 in.) have marginal effect on the
residual strength of the shell panel, whether it is computed based on fracture mechanics or section
yielding criteria. Of course, for relatively large overhang values, i.e. 0.75 in., the section stress shoots
up with the presence of MSD.
J. H. PARK et al.
376
Table 6. Effect of MSD on K[, a~v and residual strength. Set 1, constant MSD crack length of 0.12 in.
LEFM
Net section stress
Half
crack
length
Kj
Critical pressure
No MSD
MSD
No MSD
1.75
3.75
5.75
7.75
9.75
11.75
13.75
15.75
17.75
18.75
19.75
20.75
21.75
22.75
27.5
41.2
53.8
64.4
72.7
79.4
84.8
88.0
81.9
68.8
47.6
28.6
30.7
36.4
28.4
42.1
54.2
64.0
71.5
77.5
82.2
83.1
79.2
65.5
43.2
25.2
28.6
35.1
28.5
19.0
14.5
12.2
10.8
9.9
9.2
8.9
9.6
11.4
16.4
27.4
25.5
21.5
2.00
4.00
6.00
8.00
10.00
12.00
14.00
16.00
18.00
19.00
20.00
21.00
22.00
23.00
27.7
41.2
53.9
64.6
73.1
80.0
85.6
88.6
81.5
67.4
47.5
31.5
32.6
37.5
28.3
41.8
54.1
64.2
72.0
78.3
83.2
84.3
78.9
64.2
43.6
28.4
30.8
36.4
28.3
19.0
14.5
12.1
10.7
9.8
9.1
8.8
9.6
I 1.6
16.5
24.8
24.0
20.9
MSD
aav
No MSD
Critical pressure
MSD
No MSD
MSD
12.4
17.4
22.4
26.9
31.0
34.4
37.1
38.5
36.0
30.5
23. I
19.7
18.3
19.1
39.4
28.6
22.2
18.5
16.1
14.5
13.5
13.0
13.8
16.1
21.I
24.6
26.7
25.7
38.5
27.4
21.3
17.7
15.4
13.9
12.9
12.4
13.3
15.6
20.7
24.2
26.1
25.0
20.6
29.1
37.5
44.9
51.2
56.5
60.6
62.2
57.2
47.1
34.3
28.5
27.6
29.9
24.6
17.6
13,7
11.4
10.0
9.0
8.4
8.1
8.8
10.7
14.4
16.9
17.8
16.7
23.2
16.4
12.7
10.6
9.3
8.4
7.9
7.7
8.3
10.1
13.9
16.7
17.3
16.0
Lead crack overhang = 0.25 in.
27.6
18.6
14.4
12.2
10.9
10.1
9.5
9.4
9.9
11.9
18.1
31.1
27.4
22.3
12.1
16.7
21.5
25.8
29.6
32.9
35.4
36.7
34.7
29.6
22.6
19.4
17.9
18.6
Lead crack overhang = 0.50 in.
T h e effect o f M S D
27.7
18.7
14.5
12.2
10.9
10.0
9.4
9.3
9.9
12.2
17.9
27.6
25.4
21.5
19.4
27.1
34.9
41.8
47.8
52.8
56.7
58.6
54.2
44.8
33.2
28.2
26.8
28.6
crack length on the critical pressure, c o m p u t e d b a s e d on linear elastic
f r a c t u r e m e c h a n i c s c r i t e r i a , is m a r g i n a l ( T a b l e 7). T h e i n c r e a s e in M S D c r a c k l e n g t h h a s t w o effects:
(1) It r e d u c e s t h e i n t a c t l i g a m e n t l e n g t h , r a i s i n g t h e n e t s e c t i o n stresses; as is v e r y c l e a r f r o m
t h e r i g h t h a l f o f T a b l e 7. T h i s i n c r e a s e s t h e l e a d c r a c k tip S I F .
(2) It relieves t h e f a s t e n e r l o a d , d i v e r t i n g t h e s h e e t s t r e s s e s a w a y f r o m t h e l e a d c r a c k tip. T h i s
l o a d f l o w r e d i s t r i b u t i o n r e d u c e s t h e l e a d c r a c k tip S I F .
Table 7. Effect of MSD crack length on residual strength. Set 2
Residual strength based on
Half
lead crack
length
1.75
3.75
5.75
7.75
9.75
11.75
13.75
15.75
17.75
18.75
19.75
20.75
21.75
22.75
LEFM
Net section stress
MSD crack length
MSD crack length
No MSD
0.12 in.
0.25 in.
0.35in.
No MSD
0.12 in.
0.25 in.
0.35 in.
29.0
19.4
14.9
12.4
II.0
10.1
9.4
9.1
9.8
11.6
16.8
28.0
26.1
22.1
28.2
19.0
14.8
12.5
11.2
10.3
9.7
9.6
10.1
12.2
18.5
31.7
27.9
22.8
28.1
18.8
14.6
12.4
ll.l
10.2
9.7
9.6
10.1
12.1
18.5
31.9
27.9
22.6
27.9
18.5
14.3
12.1
10.9
10. I
9.5
9.4
10.0
12.1
18.4
32.1
27.9
22.4
39.4
28.6
22.3
18.5
16.1
14.5
13.5
13.0
13.7
16.1
21.1
24.5
26.7
25.7
38.5
27.5
21.3
17.7
15.4
13.9
12.9
12.4
13.3
15.6
20.7
24.3
26.1
25.0
30.2
21.4
16.6
13.8
12.0
10.8
10.0
9.7
10.4
12.3
16.5
19.6
20.8
19.8
23.0
16.2
12.5
10.4
9.1
8.2
7,7
7.4
8.0
9.6
13.0
15.6
16.3
15.3
Integrity of aircraft structural elements
4o.o-I
[]
... v,so
377
.
I
20.0
10.0
0.0
I
I
I
I
I
I
I
I
I
I
I
I
1
I
1.75 3.75 5.75 7.75 9.75 11.75 13.75 15.75 17.75 18.75 19.75 20.75 21.75 22.75
Half Lead-Crack Length (in)
Fig, 11. Effect o f M S D crack length o n residual strength, based o n net section yield criterion (Table 7).
The net effect is the sum total of these two. For smaller MSD crack lengths, the load flow
redistribution dominates, causing a fall in the SIF. For longer MSD crack lengths, the net section
overloading dominates and raises the SIF. But overall, the effect of MSD on the lead crack length
SIF is marginal. However, the maximum net ligament stress increases steadily with increase in MSD
lengths as the two crack tips come closer; in this case, the plasticity dominates and crack linkup
based on the net section stress criterion is highly likely.
We see that the MSD does not significantly alter the lead crack tip SIF, but does raise the
net ligament stress when the two crack tips come closer. The use of linear elastic fracture mechanics
to study the effect of MSD infested holes ahead of a lead crack is thus questionable. It may be
concluded that the effect of plasticity and net section yield dominate the situation when holes with
MSD cracks are present ahead of a lead crack. The residual strength estimate, based on net section
yield, is significantly reduced in the presence of MSD near holes ahead of a lead crack. Figure 11
represents the fall of residual strength with MSD crack length in the form of a bar chart. Also,
judging from the results in Section 5, it is the effect of holes ahead of the lead crack that is most
dominant, whether or not these holes have MSD cracks emanating from them.
8. E L A S T I C - P L A S T I C ANALYSIS OF FLAT PANELS W I T H MSD
8.1. Problem definition
Consider an A1 2024-T3 tensile sheet of width 20 in., height 40 in. and thickness 0.04 in. with
a center crack of length " 2 a " and two MSD cracks of length " 2 b " each. Let the spacing between
the main crack and the closest MSD crack be denoted by " c " and the MSD cracks be spaced apart
by a distance " d " . The configuraton is shown in Fig. 12. Two configurations listed in Table 8 have
been analysed. Because of the double symmetry, only one quarter of the specimen needs to be
considered. The finite element mesh of the uncracked body (which is only needed in the present
Elastic-Plastic Finite-Element Alternating Method) consists of 160 quadratic, eight-noded, isoparametric elements with 40 subdivisions in the crack line direction and eight subdivisions in the
Table 8. Crack configurations (Fig. l l)
Configuration
a
b
c
d
No. 1
3.80
0.25
4.50
1.50
No. 2
2.50
0.50
3.50
2.00
378
J.H. PARK et al.
'
"
'
Ill l l lil l l l lIl l )l l l l l l l l
"
tllllllllllllltllltllllllllllllllllll
eg'°n
'
40
l l tl l l li]l l l l l l l l l l l l l
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
IIIIIIIIIIIIIIIIIIIIIIII"""'IIIIIIII
tJ= t
t
t
t
20 "
t
M e s h subdivision
-I~ ~
Fig. 12. Tensile sheet with two MSD cracks (Table 8).
load direction. Though the crack configuration changes (Table 8), a single finite element mesh of
the uncracked body is adequate in the present alternating procedure.
8.2. Results and discussion
For the two configurations considered (Table 8), both elastic [E-FEAM] and elastic-plastic
[EP-FEAM] analyses have been performed. To have a reference, an elastic analysis for a single
crack was performed, K~ obtained and the critical strength of the panel computed. This is shown
by a dashed line in Figs 13 and 14 (based on K~c = 84 ksi x/~.). Critical strengths of the two panels,
50.0
.....
a~
40.0
~ ~
£
Single crack
Based on Kic( 84 ksi- i41K)
Based on ligament yield ( 47ksi )
Based on T* ( 413 psi-in )
30.0
O1
¢-
20.0
O
~3
10.0
0.0
0.0
,~
2.0
4.c I
t~.q
8.0
Half Crack Length (inches)
I
[~}::!i!~:MS
r
J
i
10.0
i
D!g!i!iis?i~::;ii!il!iiii?iii!i!!!ii!iiiii!::i::i!iiiiiii!::i!iii?ii:i!i:i]
I
i
Fig. 13. Critical strength diagram for configuration no. 1.
Integrity of aircraft structural elements
50.0
i
~
\\
40.0
e-
379
. . . . . Singlecrack
-------0----Basedon Ktc(84 ksi- J[-ff)
.----o--- Basedon ligament yield ( 47ksi ) I
}~\ ~
Based on T* ( 413 psi-in )
I
30.0
c-
~.
20.0
No. 2
O
.=_
~3
10.0
0.0
0.0
2.0
I 410 I
610
8.0
Hall Crack Length (inches)
i
i
I
10.0
I
:::::::::::::::::::::::::::::::::::~i::~:::::~::~i~:.`!i~::!`::i::~i~::::~i~!~iii~i!~:~i:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:i Main cra,e.k ~i~{%':~!:i!~!!!!i
MSD
i! i~:!:i::::r:!i~:~i~]i!~:.ii!iiiii iiiiiiiii~ii]
::!::::;:::::::::i:::i:!:i:h;.~:;:::::::::~:i:!~!z~i:i::::;:::::::;:~:i:;;~i};i~`~:;..:~!:;:~;~`~`!~!;;.~i~t::;:;:i:!:i:i:i:;:;:;:::::::::5:::::::::~:~
Fig. 14. Critical strength diagram for configurationno. 2.
computed using three different criteria, are shown by solid lines in these figures. A point on these
curves represents the value of applied tensile stress to the panel that would cause
(1) Main crack tip SIF = 84 ksi ~/i-n. (from elastic analysis),
(2) Complete yielding of the uncracked ligament (from elastic-plastic analysis),
(3) Main crack tip T * = 413 psi-in. (from elastic-plastic analysis).
The values of K~c = 84ksi ~ / ~ . and T * = 413 psi-in., used in the present section, have been
obtained from the experimental results of Jeong [9].
The following observations can be made from these figures:
(1) Critical strength of the panel computed using elastic-plastic analysis is lower than that
computed using elastic analysis.
(2) For small main crack lengths, the panel is critical from fracture mechanics consideration,
i.e. as the applied load is increased, the crack would become unstable before any of the
sections yield under stress. When the main crack tip approaches the MSD crack tip (distance
between the tips 1.25-1.4 times the MSD length), the net section stresses go up and the intact
ligament yields before the crack becomes unstable.
(3) After first linkup, both the panels are found to be section stress critical up to the second
linkup.
The following should be noted concerning the residual strength predicted by T*:
(1) Each point on the curves based on T* represents the value of T* needed to initiate the growth
of the crack from its current length, but does not consider the actual growth of the crack
from one length to another.
(2) The effect of the plastic-wake during the actual growth is not modeled.
However, an actual simulation of the growth from one length to another is necessary to fully
understand the linkup process. The linkup criteria can be based on the crack tip values of T*
[energy flux to the crack tip region, of radius El, the CTOA (Crack Tip Opening Angle) or CTOD
(Crack Tip Opening Displacement). However, it can easily be shown that T*, CTOA and CTOD
380
J.H. PARK et al.
are all linearly related. It has been shown earlier by Atluri [10] that T * remains constant during
elastic-plastic stable crack growth. It is also well-known that C T O A remains constant during stable
growth. Thus, either T* or C T O A can be used as criteria for stable crack growth and linkup.
However, C T O A being a geometric parameter, its c o m p u t a t i o n requires a very, very refined finite
element mesh. On the other hand, T * can be calculated t h r o u g h the Equivalent D o m a i n Integral
M e t h o d developed by Nikishkov and Atluri [11], as an integral over the d o m a i n between the
far-field c o n t o u r and a c o n t o u r o f radius E near the crack tip. Thus, finite element meshes (Fig.
12) are sufficient to c o m p u t e T * . Thus, the T * based linkup criteria would be far more economical
both from engineering man-power, as well as c o m p u t e r time, points o f view. These results will be
presented shortly in a separate report.
9. C O N C L U S I O N S
Analyses o f fuselage shell panels with longitudinal cracks at various locations have been
carried out using G l o b a l - L o c a l F E M . The conclusion that can be drawn from all these analyses
is that the presence o f rivet holes (with or without M S D emanating from these holes) ahead o f a
lead crack reduces the residual strength o f the d a m a g e d panel significantly. The most important
parameter turns out to be the location o f the lead crack tip with respect to the holes and M S D
crack tips. If the net ligament is small, the section stresses shoot up, bringing the residual strength
to a very low value.
Acknowledgements--The authors are grateful for the financial support from Fedral Aviation Administration to the Center
of Excellence for Computational Modeling of Aircraft Structures, Georgia Institute of Technology. They also acknowledge
with pleasure the discussions and support from various colleagues, in particular Mr Tom Swift and P. W. Tan of FAA,
and Dr Zhao, Mr Wang, Mr Shenoy and Dr Nikishkov of Georgia Tech.
REFERENCES
[1] J. H. Park and S. N. Atluri, Fatigue growth of multiple cracks near a row of fastener holes in a fuselage lap joint.
Proc. Int. Workshop Structural Integrity o f Aging Airplanes (Edited by S. N. Atluri, C. E. Harris, A. Hoggard, N. Miller
and S. G. Sampath), pp. 91-116. Atlanta (1992). Also in Computational Mech. 13, 189-203 (1993).
[2] G. P. Nikishkov and S. N. Atluri, An analytical-numerical alternating method of elastic-plastic analysis of cracks.
Computational Mech. 13, 427-442 (1994).
[3] D. G. Ashwell and A. B. Sabir, A new cylindrical shell finite element based on simple independent shape functions.
Int. J. Mech. Sci. 14, 171-183 (1972).
[4] T. Swift, Fracture analysis of stiffened structures. A S T M STP 842, 69-107 (1984).
[5] J. H. Park, T. Ogiso and S. N. Atluri, Analysis of cracks in aging aircraft structures, with and without composite patch
repairs. Computational Mech. 10, 169-202 (1992).
[6] G. C. Nayak and O. C. Zienkiewicz, Elasto-plastic stress analysis. A generalization for various constitutive relations
including strain softening. Int. J. numer. Meth. Engng 5, 113-135 (1972).
[7] T. Swift, Damage tolerance in pressurized fuselages. 11th Plantema memorial lecture, ICAF, Canada (1987).
[8] T. Swift, Residual strength of stiffened structures. Lecture at Georgia Tech. (I0 February 1993).
[9] D. Y. Jeong, Preliminary analysis of FMI flat panel experiments. Foster-Miller and Teledyne Engineering Services,
Draft report (December 1992).
[10] S. N. Atluri, Energetic approaches and path-independent integrals in fracture mechanics, in Computational Methods
in the Mechanics o f Fracture (Edited by S. N. Atluri), Elsevier Science (1986).
[11] G. P. Nikishkov and S. N. Atluri, An equivalent domain integral method for computing crack tip integral parameters
in non-elastic, thermo-mechanical fracture. Engng Fracture Mech. 26, 851-867 (1987).
(Recewed 9 March 1994)