To illustrate the use of
PROF for the calculation of the failure probability of a lap joint in the presence of
various degrees of corrosive thinning.

This sample problem illustrates the use of the PROF risk analysis
computer program for evaluating the probability of failure of a lap joint with various
degrees of corrosion and two different multi-site damage scenarios. The cracking scenarios
are from test specimens but are considered representative of corrosion in lap joints. The
analyses requires multiple runs of PROF with each given a different structural condition.
The multiple runs are then combined based on the probability of occurrence on the
conditions. Inspection intervals based on crack growth in pristine structure are compared
with those assuming the presence of corrosion and multi-site damage.

__Overview of Problem Description__

The example risk analysis of a lap joint with MSD and corrosion is based
on data from a specimen that is representative of a fuselage lap joint. The lap joint
specimens had been used in a fatigue test program by Carleton
University and the National Research Council (NRC) of Canada [Scott,1997; Eastaugh,
et al., 1995]. Crack growth predictions for the specimens were performed as part of a
program to develop an analytical corrosion damage assessment framework and the specimen
test data were used to verify the predictions. The example is presented to demonstrate the
risk analysis methodology.

The specimen, Figure UD-4.1, is constructed
of two 1 mm sheets of 2024-T3 clad aluminum with three rows of 4 mm 2117-T4 rivets
(MS20426AD5-5). The rivet pattern has 25.4 mm pitch and
row spacing with an edge margin of 9.1 mm. The test specimens were 25.4 cm wide
with eight fasteners in each row across the width.

**Figure UD-4.1.** Schematic
of Lap Joint Specimen.

Constant amplitude fatigue tests had been conducted at Carleton
University on the lap joint specimens in non-corroded
and corroded conditions with a constant amplitude far field stress of 88.9 MPa with
*R* = 0.2. Details
of the test procedure and resulting fatigue crack growth data are presented in
Eastaugh, et al. [1995]. Nine non-corroded specimens were tested to failure to provide baseline data for comparison with corrosion specimens. Only data from these non-corroded baseline specimen
tests are used in this example. Histories of crack size versus
cycles for all cracks that initiated in the top row of rivet holes were recorded during the tests and were
available for analysis. Examination of the histories showed that 95 percent or more of the joint life
was expended when the lead crack reached about 9
mm and crack growth became unstable. Further, lead cracks initiated in accordance with two
dominant scenarios. Scenario 1 is defined as a
single crack originating from one side of a
central hole. Scenario 2 is defined as approximately simultaneous, diametric cracks originating from both
sides of a central hole. Subsequent analysis showed significantly
shorter lives for the double initial cracks. Analysis also showed that assuming
both cracks were of equal size produced only 5 percent shorter lives than assuming one
crack was twice the size of the second. Consequently, the assumptions were made that:

a)
joint life is determined by the initiation and growth of lead cracks that originate by one of two scenarios,

b) cracks
are of equal size in the double crack scenario, and,

c) panel
is essentially failed when the lead crack reaches 9 mm.

Because first cracks were simultaneously discovered in different holes
in four of the nine data sets, there were a total of 13 lead cracks. Eight were from
Scenario 1 and five were from Scenario 2. For this population of structural elements, it
was assumed that probability of a randomly selected lap joint having a Scenario 1 lead
crack was 8/13 and the probability of a randomly selected lap joint having a Scenario 2
lead crack was 5/13.

Crack growth analyses were performed for both scenarios [Trego, et al.,
1998]. Stress analysis was performed using FRANC2D/L, a finite element, fracture mechanics
analysis code with crack propagation capability
[Wawryznek & Ingraffea, 1994; Swenson & James, 1997]. The resulting crack
tip stress intensity factor values as a function of
crack size were then input to the crack growth code AFGROW [Boyd, et al., 1998] for
selected degrees of corrosion severity. The no-corrosion, constant amplitude peak stress of the baseline fatigue tests and crack growth analyses was 88.9 MPa with an *R* ratio of 0.2. Predicted cyclic life from
0.25 mm to 9 mm averaged about 30 percent more
than the test data.

Corrosion severity was modeled in
terms of percent of thinning with the attendant increase in stress. To reflect
corrosion severity, crack growth predictions were made for the somewhat arbitrarily selected levels of 0, 2, 5, 8, and 10 percent
corrosive thinning by proportionate adjustments of the stress levels.

For the specimen conditions being modeled, the population of lap joint
specimens has been divided into sub-populations based
on combinations of two MSD scenarios and five corrosion severity levels. Cracking
occurred in the two dominant MSD scenarios whose influence on crack growth was exhibited
through the stress intensity factor. Corrosion severity was characterized by the metric of
uniform thickness loss whose influence on crack growth is exhibited through the
experienced stress levels. Each combination of MSD scenario and thickness loss produces a
different crack growth analysis so that each combination must be individually analyzed in
the risk analysis.

Figure UD-4.2 illustrates the partitioning
of the total population of the lap joints into the ten sub-populations. Every lap joint
must fit into one of the sets of conditions defined by MSD scenario and thickness loss. The probability that cracks will initiate
under Scenarios 1 and 2 are *p*_{1}
(=8/13) and *p*_{2} (=5/13), respectively.
The probability that a randomly selected lap joint will
have uniform thickness loss level *j* is *q*_{j}._{ }POF*(T/S*_{i},L_{j}) =* *POF_{ij}(T) is the probability of fracture as
a function of time for the combination of MSD Scenario *i* and thickness loss *j*. The calculation of the unconditional
probability of failure for a random lap joint in the fleet for each corrosion severity
level is shown in the last column. An analogous calculation could be performed across
severity levels to obtain composite failure probabilities for each MSD scenario.

Corrosion Severity |
Proportion of Joints |
Dominant MSD |
Composite over MSD |

Scenario 1 *p*_{1} |
Scenario 2 *p*_{2} |

Thickness Loss 1 |
q_{1} |
POF_{11}(T) |
POF_{21}(T) |
p_{1}POF_{11}(T)+p_{2}POF_{21}(T) |

Thickness Loss 2 |
q_{2} |
POF_{12}(T) |
POF_{22}(T) |
p_{1}POF_{12}(T)+p_{2}POF_{22}(T) |

Thickness Loss 3 |
q_{3} |
POF_{13}(T) |
POF_{23}(T) |
p_{1}POF_{13}(T)+p_{2}POF_{23}(T) |

Thickness Loss 4 |
q_{4} |
POF_{14}(T) |
POF_{24}(T) |
p_{1}POF_{14}(T)+p_{2}POF_{24}(T) |

Thickness Loss 5 |
q_{5} |
POF_{15}(T) |
POF_{25}(T) |
p_{1}POF_{15}(T)+p_{2}POF_{25}(T) |

POF_{ij}(T)
= POF(T/S_{i},L_{j}) = Probability of failure for Scenario i, Thickness
Loss j |

p_{ }=
Proportion of lap joints with crack initiating under Scenario i |

q_{j} = Proportion of lap joints with uniform thickness
loss at level j |

**Figure UD-4.2.** Conditional
Failure Probabilities for two MSD Scenarios and Five Levels of Uniform Thickness Loss.

An interpretation of the corrosion effects can be made directly from the
PROF output. If an estimate of the distribution of thickness loss in the fleet is also
available, the results of the individual runs of PROF can be combined using Equation
(UD-4.1) to provide an overall fracture probability for a randomly-selected detail.

POF(*T*) = S POF(*T/C*_{i}) · P(*C*_{i}) |
(UD-4.1) |

Further, the distribution of time to reach a fixed fracture probability
can be inferred from the percentiles associated with the corrosion severity levels. These
analyses will be demonstrated for corrosion in a representative lap joint.

It is realized that the risk analysis discussed herein does not account
for the stress levels increasing as a result of
increasing corrosion over the analysis period. At present, there are no accepted
models for the corrosion damage growth (thickness
loss) as a function of time so that the crack
growth calculations are based on the state of
corrosion at the beginning of the analysis interval.
In reality, the stresses in the spectrum should be
slowly increasing. If this effect could be accounted for in the deterministic analysis, the crack growth data
input to PROF would reflect the change. However, the peak stress distribution would need
to be made more severe at discrete increments. This added complexity could also be
introduced by adding the additional level of conditioning and performing multiple PROF
runs for each of the other ten conditions. This added level
of conditioning provides insight into the total number of different runs that might be
required to completely analyze a structure.

It might be noted that in the lap joint
example of this paper, the peak stress distribution had no effect on the failure
probability. The failure of the joint specimen was determined by reaching an unstable
crack growth state when the lead crack reached 9 mm, a size far below the critical crack
size for the applied far field stress.

** **

__PROF Input__

The risk analysis for the lap joint
corrosion example requires ten individual runs of PROF – two MSD scenarios and
five stress levels for each of the MSD scenarios. The most significant inputs for the runs
of this lap joint example are the crack growth
projections and the initial crack size distribution.
The other PROF inputs that reflect the changes between runs are the table of stress
intensity factor divided by stress *(K/s)* as a function of crack size and the distribution
of peak stresses. These were changed between runs even though they had no affect on the
results. *K/s*
came from the FRANC2D/L analysis. The peak stress distribution was estimated by a Gumbel
extreme value distribution that had a mean at the appropriate constant amplitude level and
a very small standard deviation to reflect the constant amplitude nature of the tests.
Fracture toughness for the specimen was assumed to be normally distributed, with a mean
and standard deviation of 152 and 11.4 MpaÖm,
respectively. Because the example being modeled does not include inspection and repair
cycles, reasonable, but arbitrary, data were used to define the inspection capability and
the equivalent repair flaw size distributions.

The AFGROW crack growth curves for
Scenarios 1 and 2 are presented in Figures UD-4.3 and UD-4.4, respectively. Each figure contains five crack growth
curves reflecting the five levels of corrosion severity. The shorter crack growth lives
from Scenario 2 are apparent from a comparison of these figures.

**Figure UD-4.3. **Crack Size versus Cycles for Scenario 1.

** **

**Figure UD-4.4. **Crack Size versus Cycles for Scenario 2.

The initiating flaw size distribution was generated by back calculating
from the sizes of the first observed lead cracks and their corresponding ages in the
specimen test data. The back calculation was performed in two steps. First, the
no-corrosion crack size versus cycles data of Figures UD-4.3
and UD-4.4 were used to determine the time at which each lead
crack would have reached 0.25 mm. An exponential growth model was then fit to each lead crack to estimate an equivalent crack size at
50,000 cycles. Note that the inverse of this process returns each of the observed
lead cracks to its original size and cycles.

The times to reach 0.25 mm for the cracks from the two MSD scenarios
were statistically indistinguishable. Similarly, there was no statistical difference
between the equivalent lead crack sizes from the two MSD scenarios at 50,000 cycles. The
two sets of data were pooled to obtain the initiating flaw size distribution. The
equivalent crack sizes at 50,000 cycles were fit with a mixture
of two Weibulls as shown in Figure UD-4.5. Also indicated in Figure UD-4.5 are the MSD scenarios of origin of the lead
cracks.

**Figure UD-4.5. **Weibull Mixture of Initial Crack Sizes.

__PROF
Risk Analysis Results__

Probability of failure as a function of cycles was calculated for each
of the ten combinations of cracking scenario and corrosion severity. Failure of the lap
joint specimens was defined as the lead crack exceeding 9 mm, as previously discussed. Figures UD-4.6 and UD-4.7 present
the failure probabilities as a function of experienced cycles for Scenarios 1 and 2,
respectively. The failure probabilities behave as expected with increased risk of failure
at a fixed age for Scenario 2 as compared to Scenario 1, and increasing risk of failure as
the stress level increases due to corrosion material loss. These calculations do not
account for any additional corrosive thinning after the start of the analysis.

**Figure UD-4.6. **POF versus Cycles for Scenario 1.

**Figure UD-4.7. **POF versus Cycles for Scenario 2.

As a gross check on the capability of the risk analysis methodology, Figure UD-4.8 compares the calculated probability of failure as
a function of cycles for 0% corrosion for Scenarios 1 and 2 to the observed distributions
of failure times. Superimposed on the predicted failure probabilities are the observed
cumulative distributions of the cycles to failure from the lap joints that were the basis of the analysis. The observed cumulative
distribution function was obtained by ordering the
cycles to failure and dividing the ranks of the ordered times by the sample size plus one.
That is,

F(*T*_{i})
= *i/(n+1**)* |
(UD-4.2) |

where *i* is the rank for *T*_{i}, the time at which the *i*^{th} crack exceeded 9 mm, and *n*
is the number of observed cracks that met the definition for the scenario. Sample
sizes for Scenarios 1 and 2 were eight and five, as noted earlier. The differences between
the observed and predicted probabilities of failure are most likely due to the
conservative deterministic life predictions or the extrapolation of the
crack-size-versus-cycles relation that was required to obtain the initiating distribution
of crack sizes.

**Figure UD-4.8. **POF
versus Cycles for Scenarios 1 and 2 Showing Comparison with Observed Data.

Figures UD-4.6
and UD-4.7 presented the conditional failure probabilities
given the respective cracking scenario. The unconditional failure probability for a lap
joint chosen at random from the population being
analyzed is calculated as a weighted average of the conditional probabilities where
the weighting factors are the proportion of specimens, which will initiate cracks in the
two scenarios. See Equation UD-4.1 and Figure UD-4.2. The weighting factors were estimated from the lap
joint data in which eight of the 13 lead cracks were from Scenario 1 (initial lead
crack from one side of the hole) and five of the 13 were from Scenario 2 (initial lead
crack from diametrically opposite sides of the hole).
Thus, *p*_{1} = 8/13 and *p*_{2} = 5/13. Using these factors, a
comparison of the observed and predicted cycles to failure for the composite of the
two scenarios without corrosion is shown in Figure UD-4.9.
Again, the difference between the predicted and observed distributions of cycles to
failure displays the somewhat non-conservative risks of the predicted failure
probabilities.

**Figure UD-4.9. **POF versus Cycles for Composite of Scenarios
1 and 2 Showing Comparison with Observed Data.

Figure
UD-4.10 summarizes the probabilities of failure for a randomly-selected lap joint that
can have either MSD scenario and is subject to the expected stress history for five levels
of corrosion severity. These results will be interpreted both in terms of the times to
reach a defined probability of fracture (POF) and in terms of the relative differences in
POF at a fixed number of cycles.

**Figure
UD-4.10. **POF
versus Cycles for Scenario Composites.

The cycles to reach a fixed POF for the different degrees of corrosion severity can be read from Figure UD-4.10 as indicated, for example, at POF equal
to 0.001 and 0.0001. Assume that the proportion of lap joints in the population that
contain each of the five degrees of corrosion is known. Then the distribution of the time
to reach the POF levels can also be inferred.

To illustrate, three representative
distributions of corrosion damage were assumed, as given in Table
UD-4.1. Mix 1 is symmetric about a five percent material
loss. Mix 2 is representative of a more severely corroded population. Mix 3 is
representative of a less severely corroded population and is considered to be more representative of the corrosion that would be expected
in aircraft.

**Table
UD-4.1. **Assumed Distributions of
Corrosion Damage.

Severity |
Mix 1 |
Mix 2 |
Mix 3 |

0% |
5 |
5 |
15 |

2% |
25 |
15 |
40 |

5% |
40 |
35 |
25 |

8% |
25 |
35 |
15 |

10% |
5 |
10 |
5 |

Figure UD-4.11 presents a
histogram of Mix 3. The corresponding percentage of lap joints would be expected to reach
the selected POF level in the indicated number of cycles. The histogram for cycles to reach POF = 0.0001 for severity Mix
3 is shown in Figure UD-4.12. The cumulative
distribution of time to reach the two POF levels for the three distributions of corrosion
severity are shown in Figure UD-4.13.

**Figure
UD-4.11. **Example Histogram of Levels of
Corrosion Damage – Severity Mix 3.

**Figure
UD-4.12.** Example Histogram of Cycles to POF = 0.0001
– Severity Mix 3.

**Figure
UD-4.13. **Cumulative Distribution of
Cycles to Selected POF – 3 Corrosion Severities.

At a fixed number of cycles, the failure risk of a corroded lap joint
can significantly exceed that of a non-corroded lap joint. To illustrate this difference, Figure UD-4.14 presents the ratio of failure probabilities for
each of the four degrees of corrosion severity to that of the non-corroded lap joints. The
ratios are presented as a function of the failure probability of the non-corroded lap
joint. The lap joint failure probability for the severity characterized by ten percent
thinning can be 70 times greater than that of a non-corroded lap joint. If maintenance
scheduling were based on keeping the failure probability below about 0.0001 to 0.001, a
lap joint with ten percent corrosion thinning would have a 25 to 50 times greater chance
of resulting in fracture.

**Figure
UD4-14. **Risk Ratios Normalized to No
Corrosion Condition.

** **

**MSD/Corrosion Example Summary**

This example demonstrates that it is possible to extend PROF to include
probabilistic descriptions of the factors which influence fatigue life. In particular, a
risk analysis was performed for fatigue failures in a representative lap joint in which
the crack growth calculation was influenced by corrosion thickness loss and two scenarios
of MSD. The basic approach to the analysis was to use deterministic crack growth calculations for different percentiles of the influencing
factors in the probability of failure calculations,
yielding conditional probabilities of failure. The full use of the analysis assumed
that estimates of the proportion of Scenarios 1 and 2 and an estimage of the proportion of
lapjoint with the discrete level of corrosive thinning were available, so that the
conditional failure probabilities can be combined or otherwise interpreted.

In the lap joint example of this paper, the relative frequency of the
two dominant MSD scenarios was estimated from data from a test program of the modeled
specimen. Example distributions of thickness loss were assumed to demonstrate the
calculations and interpretation. For this example, a ten percent thickness loss increased
the failure probability by a factor of as much as 70 over the no-corrosion condition.
Depending on the consequences of failure, inspection intervals based on the no-corrosion stress levels could pose
a safety issue to corroded joints. The results were also used to demonstrate the
generation of the distribution of time to a fixed risk.

** **

__References__

Boyd, K., Harter, J.A., and Krishnan (1998), *AFGROW User’s Manual Version 3.1.1*,
WL-TR-97-3053, Air Force Research Laboratory, Wright-Patterson Air Force Base, Ohio.

Eastaugh, G.F., Simpson, D.L., Straznicky, P.V., and Wakeman, R.B.
(1995), “A Special Uniaxial Coupon Test
Specimen for the Simulation of Multiple Site Fatigue Crack Growth and Link-Up in Fuselage
Skin Splices,” National Research Council of Canada and Carleton University,
AGARD-CP-568.

Scott, J.P. (1997), “Corrosion and Multiple Site Damage in Riveted
Fuselage Lap Joints,” Master’s Thesis, Carleton University.

Swenson, D. and James, M. (1997)
“FRANC2D/L: A Crack Propagation Simulator for Plane Layered Structures”, Version
1.4 User’s Guide, Kansas State University.

Trego, A., Cope, D., Johnson, P.,
and West, D. (1998) “Analytical Methodology for Assessing Corrosion and Fatigue in
Fuselage Lap Joints,” 1998 Air Force Corrosion Program Conference Proceedings, April
1998, Macon, Georgia.

Wawrzynek, P.A., and Ingraffea, A.R. (1994), “FRANC2D: A
Two-Dimensional Crack Propagation Simulator, Version 2.7, User’s Guide,” NASA
CR-4572.