The complex combinations of potential cracking within and
between structural elements and the unknown
state of the fatigue damage in aging aircraft essentially preclude the use of
deterministic crack growth calculations for estimating the onset of
WFD. Accordingly, structural risk
analyses are being used to quantify structural capability. Current practice is to express structural
risk in terms of the single-flight probability of failure as a function of
experienced flights or flight hours from a reference age. According to Lincoln , in the USAF,
the acceptable upper bound on the single-flight failure probability is 10-7. This degree of risk implies that less than
one failure would be expected in any given fleet.
When an airframe enters service, estimates-of-failure
probability would be based on the growth of monolithic cracks at the most
severe, known critical locations and would be extremely small. Such estimates would be made on the basis of
a probabilistic characterization of initial quality. Currently, the equivalent initial flaw size distribution is used
to model the crack sizes at the critical locations. In the probabilistic approach to maintenance scheduling,
inspections would be planned at intervals that keep the failure probabilities
of the monolithic structures below 10-7. In the aging aircraft scenarios, crack size distributions are
obtained for the critical locations in the complete
load path as the basis for the estimates of failure probability. Structural failure probability is
then calculated as the conditional probability of inadequate strength given the
condition of the elements in the load path.
For example, assume there is a 10-3 probability of a discrete
source damage event, such as a sudden fatigue crack linkup across two bays in a
fuselage lap joint. To maintain an
overall catastrophic failure probability less than 10-7, the
probability of failure in this damaged state must be less than 10-4. In this example, loss of fail safety can be
said to occur at the number of flight hours when the WFD reaches the state at
which the probability of surviving the discrete event exceeds 10-4. The number of flight hours to reach such a
state of fatigue cracking has been suggested as the definition of the onset of WFD
There are several approaches that can be used to calculate
single flight failure probability, but the USAF has available a computer
program named PRobability Of Fracture (PROF) for risk
analysis in aging aircraft. PROF is a
computer program that runs in the Windows environment on a personal computer
and was specifically written to interface with the data that is available as a
result of ASIP. See Berens, et al.
 and Hovey, et al.  for complete descriptions of the development of
the program and its update to the Windows environment. Figure 8.2.1 is a
schematic of the program for calculating probability of failure as a function
of flight hours for a monolithic crack.
The figure illustrates the types of data required to perform an analysis
and the probability of failure (POF) output that is calculated as a function of
flight hours. Another calculation
module in PROF calculates probability of failure due to a discrete source
Figure 8.2.1 Schematic of the PROF Computer Program
Under ASIP, crack life predictions (a versus T) are available
for every known critical location. This
implies the availability of:
flight by flight stress spectrum, from which the distribution of maximum stress
per flight can be obtained;
intensity factors as a function of crack size, a versus K/s; and,
toughness, Kcr, from which
a distribution of fracture toughness can be inferred.
The initiating crack size distribution can be obtained from
inspection feedback, tear-down inspections, or equivalent initial flaw
sizes. Probability of detection as a
function of crack size, POD(a), is
from a characterization of the capability of the non-destructive inspection
system used during the safety inspections.
The starting point of a PROF analysis can be representative of
any arbitrary number of hours in the life of the fleet. PROF uses the deterministic a versus T curve to project the percentiles of the initiating crack size
distribution as a function of flight hours.
At defined flight hour increments, the single-flight probability of
fracture is calculated from the distributions of crack size, maximum stress per
flight, and fracture toughness. That
is, the single-flight fracture probability is the probability that the maximum
stress intensity factor (combination of the distributions of maximum stress per
flight and crack sizes) during the flight exceeds the critical stress intensity
At a maintenance cycle, the distribution of crack sizes is
changed in accordance with the POD(a) function
and the equivalent repair crack size distribution. It is assumed that all detected cracks are repaired and
the equivalent repair crack size distribution accounts for the repaired
cracks. PROF produces files of both the
pre- and post-inspection crack size distributions. The availability of these distributions allows changing the
analysis conditions at inspection times set by the analyst.
The a versus T, a versus K/s, and crack size distributions are input to PROF in tabular
form. Fracture toughness is modeled by
a normal distribution and requires values for the mean and standard
deviation. Maximum stress per flight is
modeled by the Gumbel extreme value distribution and the parameters of the
distribution can be obtained from a fit of either a flight- by-flight stress
spectrum, or an exceedance curve of all of the stresses in the spectrum. The POD(a)
function is modeled by a cumulative lognormal distribution with parameters m
and s. Fifty percent of the cracks of size m
would be detected. The parameter, s,
determines the flatness of the POD(a)
function with smaller values implying steeper POD(a) functions.
The module for the calculation of failure probability given
discrete source damage also requires an evaluation of the residual strength in
the presence of partial structural failure.
Procedures for determining residual strength in the presence of discrete
source damage for a number of representative aircraft skin structures can be
found in Swift .
Sensitivity studies have been performed on the application of
PROF in representative problems [Berens, et al., 1991]. These studies have indicated that, although
the absolute magnitudes of the fracture probabilities are strongly dependent on
the input, relative magnitudes tend to remain consistent when factors are
varied one at a time. Because of the
indefinite nature of some of the input data,
particularly the crack size information, absolute magnitudes of the fracture probabilities
are suspect. However, it is believed
that relative differences resulting from consistent variations in the
better-defined input factors are meaningful.
A single run of PROF analyzes the growth of a crack for a
single geometry, including crack type and shape. The analysis would apply to the population of structural details
that both have this geometry and are subject to an equivalent stress
spectrum. The output includes fracture
probabilities for a single structural detail, for a single aircraft when there
are multiple equivalent details, and for the entire fleet. The inspection intervals are set by the
analyst, including the possibility for an immediate inspection at time zero.
More complex problems can be analyzed by combining the results
of multiple runs. First, intermediate
output can be used to initiate new runs for changed conditions. Examples of such analyses would include the
introduction of corrosive thinning of the material, the effect of over-sizing
holes during repairs, and the effects of changing usage. The results from multiple runs for different
details can also be combined to model more complex scenarios. Examples of such scenarios include the analyses
of multi-element and multi-site damage.
There are four examples of the application of risk analysis in
sustainment scenarios in the Sample Problem Section of the Handbook. The sample problems addressed are:
Problem No. UDRI-2 – Structural Risk Assessment for a Discrete Source Damage Threat to the Fail Safety Capability of Stringer 7 in a
Boeing 707 JSTARS Airframe.
b) Problem NO. UDRI-3 – Structural Risk
Assessment for the Multiple-Element Damage Scenario at WS 405 of a C-141
No. UDRI-4 – Comparative
Risk Assessment of the Thinning Effect of Corrosion on a Representative Lap
d) Problem No. NRC-3 – Effect of
Discontinuity States on the Risk Assessment of Corroded Fuselage Lap Joints.