__Overview of Problem Description__

The goal in analyzing the effect of widespread fatigue damage (WFD) in
the discrete source damage (DSD) problem is to evaluate the ability of the structure to
complete the current mission when a partial structural failure occurs. This analysis is
aimed at one of two or more structural details that interact by providing a fail-safe
capability in the event that one or more of the structural details has failed. The
evaluation will use the conditional single-flight probability of failure, given that DSD
is present as the measure of this ability. The prototype for this analysis is the ability
of the structure to survive the sudden appearance of a two-bay crack in the fuselage or
wing skin.

The two-bay crack is a crack that spans two bays in the skin, including
the stringer or frame between the two bays. The size of two bays is considered an upper
bound on the damage that would directly result from
penetration of an engine blade thrown from the engine in an uncontained failure or
from battle damage. The concern in this damage scenario is whether the crack-stopping
structures on either side of the damage will hold through the remainder of the mission.
The conditional probability that the crack-stopping structure will fail, given that the
DSD has occurred, provides a measure of the ability of the structure to complete the
mission.

Since the flaw size distribution changes in time, the PROF DSD analysis
is calculated as a function of time. The presence of DSD only affects the structure during
the flight in which it occurs. Therefore, the same model of the growing crack size
population that is used in a standard PROF analysis can be used to assess the influence of
aging on the conditional probability of failure given DSD. The details of the crack growth
model are given in Berens, et al.[1991]. Because of its severity, DSD, will be detected
and repaired before the next flight so that a model of crack growth in the presence of DSD
is unnecessary.

__Example Input Data__

The data from the B-707 teardown inspection performed as part of the
JSTARS assessment will be used to illustrate the procedures for an analysis of the impact
of WFD on the fail safety in the presence of DSD using PROF. A detailed description of the
data and the problems associated with using the B-707 for the JSTARS was given by Lincoln
[1997]. The example presented here centers on the fail-safety capability of stringer 7 in
the lower wing skin after stringer 8 and the adjacent wing panels have failed.

Figure UD-2.1 contains a schematic of the
B-707 wing. The left half of Figure UD-2.1 shows the entire
structure and the location of stringer 8 (S8). A cross-section of the skin and stringers
is shown in the right half of Figure UD-2.1. The example will
analyze the effect of a break in stringer 8 and the adjacent skins on the large adjacent
stringer S7.

The data were collected and the structural analyses were performed by
Boeing under an Air Force contract. The data and analysis results were delivered in a
series of letter reports and in Excel spreadsheets. The data used for this example were
extracted from the spreadsheets.

**Figure UD-2.1. **Schematic of the B-707 Wing and Side View of
the Skin and Stringer Structure [Lincoln, 1997].

The structural analyses relevant to the DSD analysis include the crack
growth curve, the stress exceedance data in the presence of DSD and the residual strength
of stringer 7 in the presence of DSD. Figure UD-2.2 contains
a plot of the crack growth curve; which was determined for intact structure under normal
conditions. The DSD analysis is not concerned with crack growth in the presence of DSD
because it is assumed that the DSD will be detected and repaired before the next flight.

**Figure UD-2.2. **Crack Growth Curve for Stringer 7 with All
Structure Intact.

Figure UD-2.3 illustrates the analysis of
the peak load distribution from the exceedance data. The basis for the exceedance data is
the spectrum used to generate the crack growth curve. The stresses were transformed to
account for the damage to stringer 8 and the adjacent panels to get the empirical stress
versus exceedance probability illustrated by the points in Figure
UD-2.3. The straight line represents the Gumbel distribution that was fit to the data.

**Figure UD-2.3. ** Peak
Stress Distribution with DSD Present.

The residual strength function is plotted in FigureUD-2.4.
The shape of the stringer is responsible for the flat
region in the residual strength function. The residual strength function was derived primarily from the stress intensity
curve for the stringer. Modifications from the Irwin criterion were required at low
crack lengths and at the flat region in the middle of the curve. At low crack lengths, the
Irwin criterion would push the residual strength to infinity, so it was necessary to
truncate the residual strength function to the maximum material strength. The stress
intensity factor actually dips between 0.5 and 1.5 inches because of the shape of the
stringer. The residual strength does not, however, decrease, resulting in the flat region
in the residual strength function.

**Figure UD-2.4. **Residual Strength as a Function of Crack
Length in Stringer 7.

The analysis was performed for two different initial crack length
distributions. The crack length data were collected from an aircraft with 57,382 flight
hours. The single-flight probability of failure is unacceptably high for the distribution
seen in the teardown data. Since many of the JSTARS aircraft will have fewer hours, the
distribution was adjusted to an age of 40,000 flight hours. The two-crack length
distribution functions are illustrated in Figure UD-2.5. A
lognormal distribution was fit to the upper tail of the teardown data and the time
adjustment was made by back extrapolating the percentiles from the 57,382 distribution
using the crack growth curve.

**Figure UD-2.5. **Comparison of the Flaw Size Density
Function at 40,000 Hours with the Density Function at 57,382 Hours.

The results of two different PROF DSD analyses are plotted in Figure UD-2.6. The solid line represents the analysis using the
flaw size distribution from the 57,382-hour aircraft as the starting point. The dashed
line plots the results from using the flaw size distribution adjusted to a 40,000-hour
aircraft. The two curves show close agreement in the overlap; however, some difference
is expected since the time points at which calculations are made do not coincide from the
two analyses.

Lincoln [1997] cited 10^{-7} as the desirable overall
single-flight probability of failure and an estimated probability of DSD as 10^{-3}.
The resultant requirement for the fail-safe capability of stringer 7 is 10^{-4}.
Clearly, the aircraft at 57,382 hours does not meet this requirement. Starting at 40,000
hours, an aircraft will have approximately 16,000 hours before the conditional
single-flight probability of failure exceeds the 10^{-4} requirement.

**Figure UD-2.6. **Comparison of Single-Flight
Probability of Failure Starting from 57,382 Hours versus 40,000 Hours.

The use of the PROF DSD analysis
module has been illustrated using data from the B-707 JSTARS aircraft. The problem
of evaluating the fail safety capability of lower wing stringers in the B-707 is an
example of the prototype DSD analysis. The essential elements that made the problem
suitable for the PROF DSD module are:

a) interest in the conditional probability of failure, given
that adjacent structural elements have failed,

b) likelihood
of failure is increased by the presence of MSD,

c) prediction
of the growth of MSD cracks with time being available, and

d) analysis
of residual strength as a function of MSD crack size being available.

__References__

Berens, A.P., Hovey, P.W., and Skinn, D.A. (1991) *Risk Analysis for Aging Aircraft, Volume 1 –
Analysis*, WL-TR-91-3066, Air Force Research Laboratory, Wright-Patterson Air Force
Base, Ohio.

Lincoln, John W. (1997), “Aging Aircraft – USAF Experiences
and Actions,” __ICAF 97 – Fatigue in New
and Aging Aircraft__, R. Cook, P Poole Eds., Proceedings of the 19^{th}
Symposium of the International Committee on aeronautical fatigue, Edinburgh Scotland.