Title: Effect of
Discontinuity States on the Risk Assessment of Corroded Fuselage Lap Joints.
To illustrate the effect that different discontinuity
states (initial and modified) have on a risk assessment of fuselage lap joints.
This problem focuses on: (1) the methods that can be used
to investigate the statistical characteristics of different discontinuity
states (DS) and (2) a risk assessment of fuselage lap joints that contain
multi-site damage (MSD) and corrosion.
A statistical test for homogeneity will be described, which can be used
to determine if a significant difference exists between the modified
discontinuity states (MDS) present in naturally corroded fuselage lap joints
and those found in artificially corroded joints. A goodness-of-fit test is applied to determine the best-fit
distribution for the pristine and corroded DS data. The use of the best-fit distributions for the pristine and
corroded DS data in a risk assessment analysis will be described. The risk assessment program PROF (PRobability
Of Fracture) is used in this problem.
Covered: Corrosion, Discontinuity States (DS),
Statistical Methods, Risk Assessment.
of Structure: Fuselage Lap Joints
Relevant Sections of Handbook: 8
Min Liao and Nicholas C. Bellinger
Company Name: National
Research Council Canada
for Aerospace Research
Road, Ottawa, Ontario Canada K1A 0R6
990-9812, Website: www.nrc.ca/iar
Contact Point: Jerzy P. Komorowski
Overview of Problem Description
A new corrosion management approach has been proposed with
the intent of anticipating, planning, and managing corrosion, which stands in
sharp contrast to the present ‘find and fix’ philosophy (Peeler, 2000). This new philosophy uses the holistic
(‘cradle-to-grave’) life assessment approach to address time and cyclic load
issues (Brooks et al., 2000), and the cornerstone of this approach is the
discontinuity state (DS) concept, which was described in detail in (Hoeppner,
1981). To characterize the different
discontinuity states present in pristine (initial discontinuity state, IDS) and
corroded (modified discontinuity state, MDS) fuselage lap joints, sections need
to be taken from representative samples of the material, polished and the
different discontinuity states that are present documented. Such a study was carried out on corroded fuselage
lap joints to document the different modified discontinuity states
present. The results from this study
are presented in Example NRC-1 in this handbook. Experiments are also required in order to determine which of the
different discontinuity states strain-energy fields (load spectrum) influence. Results from such tests are presented in
Example NRC-1 in this handbook.
Due to the random nature of the different DS values, such
as micro-porosity and inclusions (examples of IDS) and pit depth, intergranular
cracks, exfoliation and environmentally assisted cracks (examples of MDS), a
statistical analysis will be described in which a test for homogeneity and a
goodness-of-fit test will be carried out.
Using the results from the statistical analysis, the effect that the DS
distribution has on a risk assessment of fuselage lap joints will be examined.
Coupon Test and Experimental DS values
earlier, the experiments required to generate the DS data that will be
influenced by strain-energy fields are described in Example NRC-1 in this handbook. To carry out these tests, coupons were machined from pristine,
artificially and naturally corroded lap joints containing three levels of
material thinning, 0%, 2% and 5% thickness loss. The holistic life approach recognizes four distinct phases of
component life (nucleation, short crack, long crack, and final instability
(Hoeppner and Chandrasekaran, 1998)), and requires physical discontinuity
measurements and life modeling in the nucleation phase.
determine the DS values for the different corrosion levels, the majority of the
fracture surfaces from the pristine, 2% artificial and 2% natural coupons were
examined with the aid of a scanning electron microscope (SEM). For the pristine coupons the nucleation
sites were located along the non-machined edge while for the corroded coupons,
they were located along the corroded faying surface edge. Although all the nucleation sites were
semi-elliptical in shape, a semi-circular crack was used to describe the DS
with an equivalent area to the initial semi-elliptical crack. It should be pointed out that the equivalent
corrosion damage (ECD) defined in Example NRC-1 of this handbook corresponds to the MDS values in the holistic
life assessment approach.
To verify the ability of the DS concept to predict the
life of each coupon, the AFGROW crack growth program was used to predict the
number of cycles to failure, which were then compared to the particular
experimental result of the corresponding coupon (Bellinger et al., 2001). The results showed that all the predicted
cycles to failure were within 10% of the experimental results (the largest
error occurred when multiple nucleation sites were present in the
coupons). This good correlation
indicated that the DS values could be back calculated from the number of cycles
to failure of the coupons using the AFGROW program. Therefore, the DS data for the 5% corroded coupons, were
determined by back calculations.
To compare the different DS data without knowing the
best-fit distribution, each set of DS samples was ranked and their empirical
distribution functions (EDFs) plotted as shown in Figure
NRC-3.1. As can be seen from this
figure, there is distinct difference between the EDFs of the 2% and 5%
corrosion DS samples while for the pristine and 2% corrosion, the difference is
relatively small. The mean and standard
deviations of all the DS samples are presented in Table
Empirical distribution function (EDF) of DS data, (
) denotes sample size.
Table NRC-3.1. IDS and MDS values.
IDS Pristine (inch)
Artificial MDS 2% (inch)
Natural MDS 2% (inch)
Combined MDS 2% (inch)
Artificial MDS 5% (inch)
Natural MDS 5% (inch)
Combined MDS 5% (inch)
Coefficient of variation
Statistical Characteristics of DS data
Test for homogeneity of artificial and natural MDS samples
Although the sample size is small for each set of DS data
generated, combining the natural and artificial results could increase it. However, before this can be accomplished, a
test for homogeneity based on the k-sample Anderson-Darling statistic, which is
recommended in MIL-HDBK-17 and 5 (Department of
Defense, 1997, 1998) must first be carried out. This test is used to determine whether a significant difference
exists between two samples (in this case artificial and natural corrosion MDS
samples) so that they could be pooled together to get a larger sample.
Table NRC-3.2 presents the
k-sample Anderson-Darling test results for the artificial and natural MDS
samples for the 2% and 5% corrosion levels.
This table shows that the hypothesis that the artificial and natural MDS
samples, either for the 2% or 5% corrosion, are from the same population was
not rejected at a significance level (SL) of 5%. In addition, a previous study (Eastaugh et al., 2000) has shown
that the physical appearance and microscopic topography of the damage from
artificially and naturally corroded lap splices were similar. Based on these results, it was concluded
that there was no significant difference in the damage resulting from the
accelerated corrosion process as compared to the damage associated with the natural
Therefore, the two MDS samples were combined, and the mean and standard
deviation of the combined MDS data are also presented in Table
NRC-3.1 also shows the EDF of the combined MDS data for the 2% and 5%
Table NRC-3.2. k-sample Anderson-darling test
value of ADK
2% artificial and natural MDS
Two samples are from the same population
can not reject
5% artificial and natural MDS samples
Two samples are from the same
not reject at SL=5%
Best-fit distributions of DS data
Except for the 5% corrosion MDS data, the majority of the
DS data were measured from the fracture surfaces with the aid of a SEM. Generally, in a material degradation
process, failure may depend on the strength of the weakest element, or it may
depend on the largest crack-like discontinuity present in the material. Therefore, it is reasonable to assume that
the DS values used in this example are the largest values among all
discontinuities. Based on the physical
behavior of the DS, the mathematical simplicity as well as the usability in
engineering (Liao et al., 2001b), six continuous distributions, presented in Table NRC-3.3, were selected as alternative (candidate)
distributions to describe the DS data.
Table NRC-3.3. Alternative distributions.
Domain of variable
Type-I extreme value distribution (EVD) of smallest values
Gumbel (Type-I EVD of largest values)
Frechet (Type-II EVD of largest values)
In this example, all the alternative distributions were
tested to fit the pristine, combined 2%, and combined 5% corrosion MDS
data. Anderson-Darling goodness-of fit
(A-D GOF) test (Department of Defense, 1997, 1998) was used to quantitatively
examine which distribution could provide the best fit to the DS data. All the parameters for the six alternative
distributions were estimated using the maximum likelihood estimators (MLEs)
(Liao et al., 2001b). The results
the pristine IDS data, the Gumbel, Lognormal, Frechet, and Normal distributions
were highly acceptable (significance level, SL>20%), the Weibull
distribution was acceptable
(5%<SL<20%), and only the Type-I EVD of smallest values was unacceptable (SL<5%);
the combined 2% MDS data, the Frechet and Gumbel distributions were highly acceptable, and the other
alternative distributions were all unacceptable;
the combined 5% MDS data, Type-I EVD of smallest values, Weibull, and Normal
distributions were highly acceptable,
the Gumbel distribution was acceptable,
and only the Frechet was unacceptable;
the Gumbel distribution was acceptable
for all the DS data sets.
Another method to determine which distribution describes
the data best is to plot the different distributions on a probability
paper. This example plotted the six
alternative distributions and the DS data on Normal probability papers, and the
results are shown in Figures NRC-3.2 to NRC-3.4 for the pristine, combined 2%, and combined
5%, respectively. The
symmetrical ranks (Shimokawa and Liao, 1999); i.e., pi=(i-0.5)/n, were used as the plotting positions for the DS data. After carefully examining these plots, the
same conclusions from the A-D GOF test could be drawn. It should be emphasized that the
goodness-of-fit test and the probability plot are complementary to each other
and both should be used to determine the best-fit distribution.
Figure NRC-3.2. Pristine IDS data plot on Normal
Figure NRC-3.3. Combined 2% MDS data plot on Normal
Figure NRC-3.4. Combined 5% MDS data plot on Normal
Risk Analysis of Fuselage Lap Joints
MSD corrosion/fatigue test
Tests were carried out on multi-site damage (MSD) lap
splice specimens to determine the effect that corrosion has on the fatigue life
of a longitudinal fuselage lap joint (Eastaugh et al., 2000). Figure NRC-3.5
shows a schematic of the specimen, which was constructed of two 1.0 mm (0.040 inch) sheets of 2024-T3 clad aluminum with three rows of 4 mm (5/32 inch) 2117-T4 countersink rivets. Specimens were pre-corroded using
an accelerated corrosion process. Three
corrosion levels were examined: 0%, 2%, and 5% average material loss. Fatigue tests were then performed by
applying a constant amplitude loading such that the stress approximately one inch
away from the critical rivet row was 98.5 MPa
(14.3 Ksi) with a stress ratio of
0.02 and a frequency of 8 Hz.
Figure NRC-3.5. Schematic of the MSD specimen.
Examinations of the failed specimens revealed that the
majority of the crack nucleation sites were located away from the rivet hole
along the faying surface and were semi-elliptical in shape. Although different MSD scenarios were
observed in the pristine and corroded MSD tests, the onset of MSD, that is life
to visible cracks, occupied over 80% of the total life and thus was used as the
failure criteria for the risk analysis.
The probability of failure (POF) for the onset of MSD in both the
pristine and corroded specimens was predicted using the computer code, PROF (PRobability
Of Fracture) (Berens et al., 1991) (Hovey et al., 1998). PROF has been used for quantifying risk and
cost associated with inspection, replacement, and retirement of aging
aircraft. The input data was found to
play a key role in obtaining accurate POF predictions (Liao and Xiong, 2001a)
(Liao and Xiong, 2000).
Input data preparation for PROF
Initial crack size distribution (ICSD) –
To investigate the influence of the DS distributions on the POF predictions,
all the acceptable distributions were
used as ICSD in the risk analysis.
Since PROF required a tabular format for the ICSD input, i.e., (ai, F(ai)), 1000
points of (ai, F(ai))
data were generated based on the distribution function and estimated parameters
(Liao et al., 2001b).
Median crack growth
curve – The crack growth analysis of pristine and corroded lap joints was
accomplished earlier using the classic model of a corner crack at a straight
hole in AFGROW (Bellinger et al., 2001).
Stress correction factors, generated using a three dimensional finite
element analysis, were used to take into account the bending, bearing, and
corrosion pillowing that occur in non-corroded and corroded lap joints. Material
thinning for the corroded joints was also taken into account by increasing the
remote stress by the appropriate amount.
Using the methods of (Bellinger et al., 2001), the median crack growth (a-N) curves for pristine and corroded
lap joints were obtained, which are shown in Figure
NRC-3.6. One hundred points of
tabular (ai, Ni) data were used in PROF.
Calculated a-N curves for pristine and corroded MSD
length – In this example, a visible crack length of 2.54 mm (0.1 inch), i.e., the onset of MSD, was taken as the critical crack
length for both the pristine and corroded specimens. This crack length was measured from the edge of the hole as it
emerged beyond the rivet head on the outer skin of the specimen and was chosen
because it was small enough not to be influenced by an adjacent crack.
Fracture toughness distribution and geometry
factor – For the 1.0 mm (0.04 inch) sheet of 2024-T3 clad aluminum, the fracture toughness
distribution was assumed to follow a normal distribution with a mean and
standard deviation of 151.6 MPaÖm (138.0 KsiÖin) and 5.5 MPaÖm
(5.0 KsiÖin), respectively (The Boeing Company, 1998). Since this analysis defined a small critical
crack length, the fracture toughness criterion had no influence on the risk
analysis. The input of the fracture
toughness distribution was needed to run the software. For the same reason, the geometry factors
were also arbitrarily set to be small values so that the fracture toughness
criterion would not affect the calculated results.
Maximum stress distribution – The Gumble
distributions with a mean at the constant amplitude level in the test and a
small standard deviation were used for the maximum stress distribution. Using the method of (The Boeing Company,
1998), the Gumble parameters for the pristine and corroded specimens were
calculated by taking into account the material loss due to corrosion. Since the critical crack length criterion
was applied in this example, the maximum stress distribution had an
insignificant effect on the risk analysis.
Probability of detection, POD(a), and repaired crack size distribution (RCSD)
– Since this risk analysis does not involve any inspection or repair
activities, arbitrary but reasonable data were used to define the POD(a), and the RCSD.
Comparison of analytical and test results
Figures NRC-3.7 and NRC-3.8 show the results from the POF predictions for
the pristine and corroded specimens.
The experimental results, which were ranked and also plotted in these
figures, used the symmetrical ranks as the plotting positions. The experimental results, which as mentioned
earlier was the number of cycles to visible cracks, were observed from the
central four holes of the top rivet row and resulted in a sample size of 18, 7,
and 1 for the pristine, 2%, and 5% corroded specimens, respectively.
Figure NRC-3.7. POF predictions for pristine
and corroded MSD specimens using the highly
acceptable DS distributions.
Figure NRC-3.8. POF predictions for pristine and corroded
MSD specimens using the acceptable DS distributions.
Figure NRC-3.7 presents the
POF predictions for the pristine and corroded specimens using different highly acceptable (best-fit) DS
distributions. The results from Figure NRC-3.7 indicate that:
For the pristine and 2%
corroded specimens, predictions are close to the test results. Again A-D GOF tests (distribution free GOF
test (Lawless, 1982)) were carried out and indicated that the
predictions fit the test results very well (all SL>15%). All best-fit DS
distributions produced close POF results to each other, especially for the
For the 5% corroded
specimens, the prediction can’t be compared to the test result since there is
only one test datum available, however, the predicted mean life is close to
this test datum;
The POF results for the 2% corroded specimens are much
higher than those for the pristine specimens, that is, corrosion in lap joints,
even at low thickness loss levels, can result in a great increase of the
POF. From the predictions, the POF
difference between 2% and 5% is less than that between the pristine and
2%. This is consistent with the finding
that the corroded surface topography at 5% may be “smoother” than that at 2%
(Bellinger et al., 2000).
Figure NRC-3.8 presents the
POF predictions for the pristine and corroded specimens using the acceptable DS distributions and is shown
in logarithmic scale to more easily distinguish the results in the low
probability zone (<0.1). Although
the test data did not have a large enough sample size to verify the predictions
in the low probability zone, Figure NRC-3.8 does
reveal the following:
For the pristine specimens,
the different DS distributions produced significantly different POF
predictions in the low probability zone.
Assuming that 10-7 is an acceptable risk level for
maintenance scheduling (Lincoln, 2000), the Frechet DS distribution gave the
shortest time while the Weibull DS distribution gave the longest time and the
time difference was about 130,000 cycles.
For the 2% corroded specimens, the different MDS
distributions also produced different POF predictions in the low probability
zone, though the difference was not as significant as in the pristine
case. However, the maintenance schedule
could be significantly shortened due to prior corrosion even at this low
thickness loss level. In this example,
at a risk level of 10-7, the time difference between the pristine
and 2% corrosion cases was about 9,000 cycles, according to the POF predictions
produced by the Gumbel DS distribution.
Figure NRC-3.8 also shows that
the POF curves for the corroded specimens in the low probability zones are
“steeper” than the curve for the pristine results, given the same type of DS
distribution, for example the Gumbel distribution. This steeper curve would have a profound effect on the
probability of failure if corrosion was missed during routine inspections. Figure NRC-3.9
illustrates an example, in which a risk assessment was carried out to maintain a
POF level under 10-7 using the Gumbel DS distribution for both the
pristine and corroded specimens and a log-logistic distribution (Berens et al.,
1991) was assumed for the POD(a) with m=0.01
and s=0.1. To maintain the acceptable POF level for the pristine specimens,
the first inspection would have to be carried out at 170,000 cycles, while the
second inspection would be required at about 236,000 cycles. If during the first inspection, 5% corrosion
was missed, this would significantly increase the POF at the second inspection
interval by three orders of magnitude (10-4). Figure NRC-3.9
also shows that a risk assessment can aid in scheduling the maintenance
associated with corrosion. This
assessment could allow a corroded lap joint to remain in service until the next
scheduled maintenance while maintaining the acceptable POF level.
Figure NRC-3.9. Corrosion risk assessment
example: what-if scenario predictions.
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