To illustrate the use of PROF for the calculation of the probability of
load path failure given a representative three element load path.
This sample problem illustrates the use of the PROF risk analysis
computer program for evaluating the probability of failure of the chordwise joint at WS407
of the C-141 airframe given the structural status of the adjacent beam cap and splice
fitting. The failure probability of the chordwise joint as a function of flight hours from
a reference time is calculated using representative crack growth data, stress
distributions, and crack size distributions from inspections of C-141 airframes. Since the
failure probability of the joint is conditioned on the failed or intact status of the beam
cap and splice fitting, the probability of failure of these elements must also be
calculated and the results combined.
Overview of Problem Description
In the multi-element damage (MED) scenario, two or more structural
elements bridge the same load path and the damage states of the elements can interact. In
this scenario, failure of selected combinations of elements may not lead to system
failure, but the effects of the failures may well lead to changes in the fracture
mechanics (loads or geometry factors) of the remaining elements. Thus, the probability of
system failure changes when the non-critical elements
fail. To evaluate the failure risks of the complete structure, the functional
interaction of the structural elements must also be
taken into account. PROF can provide a reasonable approximation to this potentially complex
calculation.
A fault tree type of analysis is
first performed to identify all of the interactive states that have an affect on the
conditions leading to system failure. This step is performed external to PROF and may
prove to require extensive stress and fracture mechanics analyses. These states will represent structural conditions that can be
modeled by deterministic crack growth analysis.
PROF can then be used to calculate the conditional probability of failure, given the
potential combinations of failed and intact states of the elements. The
unconditional failure probability of the complete structure is a weighted average of the conditional probabilities in
which the weights are the probabilities of being in each of the states, i.e., the
probability that selected elements will have failed.
It is apparent that there are, potentially, a very large number of
possible combinations of structural elements that would
need to be considered in the analysis of a complex structure. From the viewpoint of
structural interaction, it is judged that three or four elements will generally suffice.
For two elements, there are only two basic combinations: the structure will fail if either
element fails (the elements are in series), or the structure will not fail if one of the
elements fails (the elements are in parallel). Note in
the latter case, that the crack growth properties of either element will change
upon failure of the other. Even this simple multi-element structure would require four
PROF runs to be combined. If there are three interacting elements, there are a total of
five basic combinations of series and parallel arrangements, and many more potential
analysis combinations that could require PROF runs.
Problem Statement
Failure occurs at WS405 in the C-141
airframe when the chordwise joint fractures. Since the stress levels and crack
growth behavior in the chordwise joint are dependent on the intact or failed status of
both the splice fitting and the beam cap, the risk analysis for WS405 must combine
conditional fracture probabilities for the relevant combinations of the states of the
structural details. The probability of failure at this wing station under routine
operations was previously calculated by Lockheed Aeronautical Systems Company (LASC) for a
single inspection interval at 31,000 spectrum hours using a Monte Carlo analysis [Cochran,
et al., 1991]. The data were re-analyzed to demonstrate using PROF to calculate the
failure risks for the same scenario.
The input required by PROF was
provided by LASC from their evaluation of the failure risks at WS405. The input data that
were used in the analyses are presented in discussed in detail in Berens [1993] and
Cochran et al. [1991].
LASC performed extensive finite element analyses of the chordwise joint,
splice fitting and beam cap at WS405 of the C-141 airframe. The intact or fractured status
of the beam cap affects the stress levels in both the splice fitting and the chordwise
joint. The intact or fractured status of the splice
fitting also affects the stress levels in the chordwise joint. Thus, different crack
size versus flight hour relations and different maximum stress per flight distributions
are needed for the various combinations of intact and fractured beam caps and splice
fittings.
Since structural failure at WS405 of the C-141 airframe occurs when the
chordwise joint fractures, LASC established a fault tree, Figure
UD-3.1, which isolated the fracture events that need to be evaluated in the
calculation of the probability of failure of WS405 [Cochran, et al., 1991]. The fault tree
of Figure UD-3.1 was restructured to demonstrate that the
WS405 failure probability can be modeled as a weighted average of the probability of
fracture of the chordwise joint, given the intact or failed status of the splice fitting
and the beam cap. The weighing factors are the probabilities of the intact or fractured
status of the splice fitting and the beam cap. The chordwise joint fracture can also be visualized in terms of the Venn diagram of Figure UD-3.2 in which
the event is partitioned four mutually-exclusive sub-events.
Figure UD-3.1. WS405
Fault Tree.
Figure UD-3.2. WS405
Venn Diagram.
Probabilistic
Approach
The probability of failure at WS405
(POF) is given by:
POF = P{CSF,SFTAC,BCTAC}
+ P{CSF,SFTAC,BCF}
+ P{CSF,SFF,BCTAC} + P{CSF,SFF,BCF}
= P{CSF ô SFTAC,BCTAC} · P{SFTAC} ·P{BCTAC}
+ P{CSF ô SFTAC,BCF}
· P{SFTAC} ·P{BCF}
+ P{CSF ô SFF,BCTAC}
· P{SFF} ·P{BCTAC}
+ P{CSF ô SFF,BCF}
· P{SFF} ·P{BCF}) |
(UD-3.1) |
where
CSF |
= chordwise joint fracture |
SFTAC |
= splice fitting intact |
SFF |
= splice fitting fractured |
BCTAC |
= beam cap intact |
BCF |
= beam cap fractured |
P{A,B,C} |
=
Probability of events A and B and C
= P{AôB,C} · P{B} · P{C} |
P{AôB,C} |
= Conditional probability of event A given the events B and C |
Note that because of the effect of
the failed or intact effect of the beam cap on the splice fitting that
P{SFF} = P{SF |
BCTAC} • P{BCTAC} + P{SF|BCF} • P{BCF} |
(UD-3.2) |
Further,
P{SFTAC} = 1 – P{SFF} |
(UD-3.3) |
P{BCTAC} = 1 – P{BCF}. |
Time histories of the conditional
probability of chordwise joint fracture given the intact or failed status of the splice
fitting and beam cap were calculated using PROF (with the appropriate a versus T
and maximum stress per flight distribution). Similarly, the time histories of the probability of the splice fitting and beam cap being in an
intact or failed status were also calculated using PROF. These numbers were combined to calculate the
unconditional probability of WS405 failure.
Selected WS405
Risk Analysis Results
PROF computed the single flight probability of fracture at ten
approximately equally spaced times throughout each usage interval. The usage intervals
were specified in terms of spectrum hours from the zero reference time (31,000 spectrum
hours in this example) and define the times at which the inspection and repair actions are
taken. In this risk evaluation at WS405 of the C-141, the analyses were performed over two
usage intervals of 328-hour duration. The reported analyses were run assuming an
inspection at the start of the analysis (Reference time T = 0 or 31000 spectrum hours).
PROF also calculates interval
probability of fracture, but only at the end of a usage interval. For the structural elements and conditions of this
example, the probability of fracture was dominated by cracks reaching unstable size (about
1 in.) as opposed to an encounter of a maximum stress in a flight. That is, the
probability of fracture was determined primarily from
the distributions of crack sizes. As a result, the single flight and interval
probabilities of fracture were equal (to three significant figures) for the
chordwise joint and the beam cap. The interval probabilities of fracture for the splice
fitting were about five percent greater than the single flight fracture probabilities.
Therefore, in this application, the single-flight fracture probabilities were used for the
probabilities of intact and fractured status of the
splice fitting and beam cap, Equation UD-3.1, in calculating the unconditional
probability of failure at the ten times in a usage interval. This assumption is expected
to occur in problems of interest because of the relatively small failure probabilities of
risks in any realistic problem.
Sample results from the WS405 analysis are as follows. Figure UD-3.3 presents the probability of fracture as a function
of spectrum hours for the splice fittings and the beam caps. This analysis assumed that
maintenance (inspection and repair of detected cracks and failures) was performed at T = 0 (31,000
spectrum hours) and a subsequent maintenance was performed at 328 hours. The figure
displays the relatively high fracture probabilities for the splice fittings, even after
the maintenance cycle. In the original data, approximately 75 percent of the beam caps were in a failed crack size state and these were repaired
before the failure probability calculations were started. The inspection capability
assumed in the analysis was not sufficient to find and repair the cracks in the splice
fittings. The effect of the failed beam cap on the fracture probability of the splice
fitting was relatively minor in comparison to other effects.
Figure UD-3.3. Failure
Probabilities of Splice Fitting and Beam Cap.
Figure
UD-3.4 presents the conditional probability of failure of the chordwise joint, given
the intact or fractured status of the splice fitting and beam cap. The
unconditional failure probability is a weighted average of these conditional
probabilities, with the weights being determined by the proportion of intact and failed
splice fittings and beam caps. Figure UD-3.5 displays the
chordwise joint (system) unconditional failure probability along with the conditional
failure probabilities. With the inspection at time zero, the intact or failed status of
the splice fitting and beam cap had relatively minor effect on the failure probability of
the system. Figure UD-3.6 compares system probabilities of
failure for the analyses with and without an inspection at time zero. The effect of the
maintenance action decreases the failure risks by about a factor of five.
Figure UD-3.4. Conditional
Failure Probabilities of Chordwise Joint.
Figure UD-3.5. Unconditional Probability of Failure of Chordwise Joint.
Figure UD-3.6. Unconditional Probability of Failure
of Chordwise Joint –
With and Without Initial
Inspection/Repair.
Summary for
Multi-Element Damage Example
The computer code PRobability Of Fracture, PROF,
was used to evaluate the probability of failure at WS405 of the C-141 aircraft. Failure
occurs at this location when the chordwise joint fails.
The stress levels experienced by the chordwise joint are dependent on the failed or intact
status of the splice fitting and the beam cap. This multi-element analysis was calculated
in terms of the failure probability of the chordwise joint, given the status of the splice
fitting and the beam cap, and the probabilities of the
condition of the splice fitting and beam cap. The probability of failure at WS405
was calculated for a set of conditions comparable to those used in an independent analysis
performed at LASC. For these conditions, the probability of a failure at WS405 in one wing
was less than 2 • 10^{-4} during a period of 656 hours of operational usage
with an inspection/repair cycle at 328 hours.
References
Berens, Alan P. (1993), “Risk Analysis for C-141, WS405,”
UDR-TR-93-20, University of Dayton Research Institute, Dayton, OH, 45469-0120.
Cochran, J.B., Bell, R.P., Alford,
R.E., and Hammond, D.O. (1991), “C-141 WS405 Risk Assessment," Proceedings of the 1991 USAF Structural Integrity
Program Conference, San Antonio, Texas, December, 1991.