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9th
European Workshop on Structural Health Monitoring
July 10-13, 2018, Manchester, United Kingdom
Preliminary Validation of Deterministic and Probabilistic Risk
Assessment of Fatigue Failures Using Experimental Results
Ribelito F. Torregosa and Weiping Hu
Aerospace Division, Defence Science and Technology Group, 506 Lorimer St.,
Fishermans Bend, Victoria, 3207, Australia
Abstract
The complexity in highly technological systems such as aircraft and the economic
pressure on owning and operating such systems demand a more accurate assessment of
their safety and reliability. For aircraft structures the safety is reflected in the predicted
fatigue life and inspection intervals. In engineering practice, fatigue life may be
calculated using either a deterministic or a probabilistic method, with the former being
the dominant method historically used. Increasingly, probabilistic risk assessment
methods are being used to evaluate aircraft safety and the availability of military assets.
To provide confidence in the increasing use of the probabilistic approach to structural
integrity assessment, this paper presents a comparison of numerical and experimental
data to validate the accuracy of probabilistic assessment in comparison to the widely
used deterministic method. The validation used the fatigue lives of aluminium test
specimens from two tests, namely the experiment done at DST for variable amplitude
loading and the data from the Virkler experiment for constant amplitude load. The
initial inspection times specified in MIL-STD1530D for both the deterministic and the
probabilistic approach were used to predict the time (i.e., number of cycles) of
inspection times. As specified in the standard, the probabilistic approach requires the
initial inspection to be conducted when the probability of failure exceeds 10-7
but is less
than 10-5
, whereas the deterministic approach requires the initial inspection to be
conducted halfway through the predicted fatigue life. The predicted initial inspection
times from each method were compared to the number of load cycles at which the first
failure occurred in the experiment. Comparisons showed that the deterministic approach
was over-conservative and predicted the initial inspection time much earlier than
observed in the experiment. The probabilistic approach gave predictions which were
both safe and closer to the experimental values.
1. Introduction
Safety inspection against fatigue failure is an integral component in the structural
integrity management of aircraft fleets. In spite of the advancement in materials and
mechanical sciences, the accurate prediction of fatigue life of a given aircraft structural
component under service load is still a significant challenge. This is because fatigue is
inherently a stochastic process. Previous research showed that fatigue crack growth
rates vary even for the same material and loading conditions. Virkler et. al., [1]
conducted tests on 68 specimens of Aluminium 2024-T3 subjected to constant
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amplitude loading and Torregosa and Hu [2] conducted tests on 85 specimens of
Aluminium 7075-T7351 subjected to a variable amplitude spectrum loading. In both
tests, the results showed a high degree of variability in crack growth rates. In common
practice, fatigue life is calculated using either a deterministic method or a probabilistic
method, with the former being the dominant one due to its simplicity. In the
deterministic approach, all parameters are treated as constant. The uncertainties are
accounted for through the application of a factor of safety. In the probabilistic or risk-
based approach, the key inputs are treated as random variables, with appropriate
probability distributions. In the context of fatigue, the safety inspection times can be
determined using either approach. In a deterministic damage tolerance analysis
following MIL-STD-1530D [3], a fatigue crack is assumed to exist at a critical
structural location and is allowed to grow but the location must be inspected halfway
through the predicted failure time in order to have another chance of detecting it before
it leads to failure. Essentially, this gives a factor of safety of 2.0. The probabilistic
structural integrity assessment of military aircraft also follows the MIL-STD-1530D [3]
in determining the safety inspection time which is set at the probability of failure (PoF)
between 10-7
and 10-5
. Although both the deterministic and the probabilistic methods
have been used in the field of structural integrity management of aircraft for many
years, research comparing the predictions of the two methods can hardly be found. In
recent years due to the increasing number of ageing aircraft and with the demand for
extending the design lives of aircraft for economic reasons, the application of the
probabilistic approach in fatigue life prediction has gained more popularity. In the
future, it is envisaged that the probabilistic approach will also be applied to other
aircraft operated by the Royal Australian Air Force. Thus the prediction using this
approach needs to be verified in comparison with the deterministic approach.
2. Variability of Fatigue Crack Growth
For many years, the variability of fatigue crack propagation has been the focus of much
research. One of the earliest and probably the most widely cited document is the
investigation conducted by Virkler et al. [1], in which 68 aluminium 2024-T3 specimens
(see Figure 1) were subjected to constant amplitude loading of 23.35 kN with R ratio of
0.20. The test specimen had a width of 152 mm and thickness of 2.54 mm. The test
specimens were pre-cracked to a crack size 9.0 mm. Torregosa and Hu [2] investigated
the variability of fatigue crack growth of 7075-T7351 aluminium alloys by testing 85
middle tension specimens, shown in Figure 2, under a variable amplitude load spectrum.
A fatigue starter was introduced through a centre notch, as shown in Figure 2. The load
spectrum consisted of 19032 load turning points which were referred to as a ‘load
block’. The maximum tensile load was 60 kN. Each test specimen was subjected to the
same load spectrum applied repeatedly until the specimen failed by fracture. Crack sizes
were measured at every 2000 load turning points (i.e., 1000 cycles) using the direct
current potential drop (DCPD) method [4]. Both studies showed that even under well
controlled conditions, cracks grow stochastically, as illustrated in Figure 3 and Figure 4,
respectively. In the DST experiment, the scale of the horizontal axis is given in load
blocks. Figure 5 shows the typical crack sizes at the start of DST testing and the crack
size shortly before fracture. It should be noted that at cycle zero, the crack sizes already
exhibited variability because the measurement started after pre-cracking under constant
amplitude load equal to 70% of the peak spectrum load value.
3
Figure 1 Virkler test specimen geometry
Figure 2 DST test specimen geometry
0 100000 200000 3000000
10
20
30
40
50
Cra
ck s
ize, a
(m
m)
Cycles Figure 3 Crack growth curves for aluminium 2024-T3 mid-tension specimens under constant
amplitude loading from Virkler experiment [1]
4
0 2 4 6 8 10 12 14 16 180
2
4
6
8
10
12
14
16
18
20
Cra
ck s
ize,
a (
mm
)
Load blocks Figure 4 Crack growth curves for aluminium 7075-T7351 mid-tension specimens under variable
amplitude load from DST experiment [2]
(a)
(b)
Figure 5 PoF Test specimen showing crack size, a) at the start of the loading, b) before fracture
3. Deterministic and Probabilistic Approaches to Aircraft Fatigue
Safety Inspection Following MIL-STD1530D
For military aircraft, safety inspection follows MIL-STD1530 [3]. The standard
specifies two methods in determining the inspection times. These are the deterministic
and the probabilistic methods. The deterministic method mitigates the risk of a crack
reaching its critical size unnoticed by requiring an inspection be conducted no later than
halfway to the time when the crack is projected to fail. The idea is to have another
chance of detecting the crack during the next inspection. This is illustrated in Figure 6.
A master crack growth curve obtained from a damage tolerance analysis (DTA) is used
to project the fatigue life of a crack from its initial size. In the probabilistic method, the
inspection times are determined based on the acceptable risk of failure. For military
aircraft as specified in MIL-STD1530D, the acceptable probability of (catastrophic)
failure (PoF) during the next single flight is PoF=10-7
[3]. When the PoF exceeds 10-7
5
but is less than 10-5
, a safety inspection is required and beyond 10-5
, the risk is
considered unacceptable. This is schematically illustrated in Figure 7. Thus, in the
probabilistic method, a risk curve is constructed and the flight hours corresponding to
10-7
and 10-5
are projected to determine the inspection times.
Figure 6 Fatigue safety inspection time using the
deterministic approach
Figure 7 Fatigue safety inspection using the
probabilistic approach [3]
Given the way in which the two methods determine the safety inspection times, it is not
difficult to compare which one is more accurate and conservative if a common data set
is used. In this paper, the experimental results shown in Figure 3 and Figure 4 are used
to validate the predictions from the two methods.
3.1 In-service versus experimental failure data
Service load fatigue failure data of airframes and components are not easily available
since airframes and their components are designed not to fail. Even with retired aircraft,
failed components are hard to find. However, other data such as service load spectra and
material properties are available. When these data are used in laboratory experiments,
the resulting failure data can be a very good replacement for actual inflight data. For this
reason, the comparison of failure prediction in this paper is based on experimental
results obtained under laboratory conditions. In this investigation, each experimental
specimen failure is treated as a failed airframe component which should not be allowed
to happen in actual service.
3.2 Predicting and validating the initial inspection time using test data
In actual fleet safety management, it is common to conduct a full scale fatigue test and
use the test result as representative data for each fleet member. However, this exercise is
very expensive and time consuming. Thus other sources of information such as
teardown inspections, experiments and computational analyses are used to manage the
safety of an aircraft. The accuracy of safety predictions, though, is very hard to validate
since aircraft data of failed components are rare. In this paper, an attempt is made to
verify the accuracy of both the deterministic and the probabilistic methods. The idea is
to use the test results of specimens subjected to representative service loads experienced
by actual airframe components. Hence, each specimen is thought to represent an aircraft
6
component, and the pool of all specimens essentially represents an aircraft fleet. From
this assumption, the accuracy of predictions can be verified from the experimental
outcome. Thus if the pool of test data represents the fleet, then the safe inspection time
is the number of load cycles before the first failure among all specimens tested.
For a prediction to be considered conservative, the predicted inspection time must be
before the first failure of any specimen. When comparing two methods, the one that
predicts a life that is closer but shorter than the time the first failure occurs is considered
more accurate.
In this paper, the predictions of the initial inspection time using the deterministic and
the probabilistic methods are made according to following procedure:
3.2.1 Deterministic procedure
1) Develop the initial crack size (ICS) distribution by randomly selecting five
specimens. The crack size at 1000 (treated as time zero) load cycles is treated as
the initial crack size;
2) Randomly select a test specimen and get its crack growth curve obtained from
the test;
3) Take the mean of the ICS in step 1 as the initial crack size, and use the crack
growth curve from step 2) to project the mean to failure, to obtain the fatigue life
tcr; and
4) The safety inspection time is obtained by dividing tcr by 2.
3.2.2 Probabilistic procedure
1) Develop the initial crack size (ICS) distribution as in the deterministic
procedure;
2) Randomly select a test specimen and get its crack growth curve obtained from
the test. Use this as the master crack growth curve;
3) Develop the peak stress distribution based on the peak stresses of the load
spectrum, and model the result using the Gumbel distribution;
4) Calculate the residual strength curve and the geometry correction factors based
on the material property and geometric configuration of the test specimen;
5) Calculate the probability of failure, PoF [5, 6], using first a probabilistic fracture
toughness and then a deterministic fracture toughness, and the following
equations:
(1)
(2)
In the above equations, f(a,t) is the crack size distribution at time t, where t is the
time (in load cycles or load blocks) and is the peak stress distribution. The
terms and represent the peak
7
stress exceedance probability. This is the probability that the applied stress
exceeds the residual strengths and respectively for a particular
crack size a and fracture toughness, .
6) On the PoF curve, project the load cycles corresponding to PoF = 10-7
and PoF =
10-5
. The projected load cycles are the range in which the initial inspection time
has to be conducted.
The ICS distribution was represented by a bounded probability distribution model (i.e.,
Beta distribution) as proposed by Torregosa and Hu [7]. The minimum bound of the
ICS is set at 0.0 mm and the upper bound is the half-width of the test specimen, for the
obvious reason. The bounded distribution is considered a more realistic model for crack
size distribution than the normal or the lognormal distribution. The probability density
function of the Beta distribution with lower bound a and upper bound b is given by the
equation [8]:
(3)
(4)
The parameters of the distribution are q and r whereas is the Beta function given
by the equation;
(5)
The mean and variance of the Beta distribution are
(6)
(7)
3.3 Comparison of the accuracy of the initial inspection time predictions based on
the DST experiment
In the DST experiment, the test specimens were subjected to a variable amplitude load
spectrum. Since the peak load is 60 kN, the peak stress exceedance probability is P=1.0
for stresses less than 60 kN and P=0.0 for stresses greater than 60 kN. The initial crack
8
size (ICS) distribution is derived from the crack sizes at 1000 cycles using five
randomly selected specimens following the Beta distribution with lower bound of 0.0
mm and upper bound based on the half-width of the test specimen (see Figure 2) . The
parameters q and r are determined using equations (6) and (7) and the calculated mean
and variance of ICS for the set of 5 specimens.
Due to the use of crack sizes at load cycle 1000, the time zero in the analysis refers to
load cycle 1000. Residual strengths are calculated assuming a fracture toughness KC
=32 MPa√m. According to MIL-HDBK-5J [9] the standard deviation of fracture
toughness for 7075-T7351 aluminium alloy ranges from 1 to 8 MPa√m. Since all the
specimens were manufactured from the same aluminium plate, the minimum standard
deviation of 1.0 MPa√m was assumed in the analysis. A randomly selected crack
growth curve from one of the test specimens was used as the master crack growth curve
for both the deterministic and the probabilistic approaches.
Using these input parameters, the fatigue lives of the test specimens were calculated
using both the deterministic and probabilistic methods. Since most of the input
parameters were randomly picked from the test results, multiple analyses were
conducted (i.e., 5 trials) in order to get a range of predictions and to see the effect of
random selection. For the probabilistic approach, separate trials were conducted for
fixed KC using the mean value and for variable KC assuming a standard deviation of 1.0
MPa√m. The resulting PoF curves are shown in Figure 8 and Figure 9 respectively. In
each figure, the PoF curves of the probabilistic method are plotted in blue colour, and
correspond to the right vertical axis. The experimental crack growth curves in black
colour are plotted against the left vertical axis.
0 2 4 6 8 10 12 14 16 180
5
10
15
20
of all data
a (
mm
)
Block
Constant Kc=32
Minimum life
10-10
10-9
10-8
10-7
10-6
10-5
10-4
10-3
PoF
Figure 8 PoF curves using deterministic Kc plotted against actual crack growth curves to failure of
test specimens
Figure 8 shows the PoF curves assuming fixed value of KC of the five trials using
randomly selected data from the experimental result. Figure 9 shows the PoF curves
considering the variability of KC using randomly selected data from the experimental
result. In the two figures, a horizontal dashed line corresponding to PoF=10-7
is drawn
intersecting the PoF curves. The intersection of each PoF curve and the line
9
corresponding to PoF=10-7
, indicates the time (i.e., load blocks) for the initial
inspection. A horizontal line corresponding to PoF=10-5
is also shown in the figure to
indicate the predicted time where the risk is no longer acceptable. It can be observed
that using a fixed value of fracture toughness predicted a conservative initial inspection
in 3 out of 4 trials. Using probabilistic fracture toughness (i.e., with standard deviation)
led to predictions which were all conservative and safe. The times corresponding to
PoF=10-7
and PoF=10-5
for each PoF curve are given in Table 1. In the same table the
deterministic initial inspection time predictions (i.e., half-life) for the five trials are also
given.
0 2 4 6 8 10 12 14 16 180
5
10
15
20
of all data
a (
mm
)
Block
Variable Kc, mean =32 Stdev=1.0
Minimum life
10-10
10-9
10-8
10-7
10-6
10-5
10-4
10-3
PoF
Figure 9 PoF curves using probabilistic Kc plotted against actual crack growth curves to failure of
test specimens
Table 1 Comparison of deterministic and probabilistic initial inspection time predictions, in load
blocks, using DST Group test data
Minimum
specimen
fatigue life
(Load blocks)
Trial
Predicted inspection time Deterministic
(Load
blocks)
Probabilistic
(Fixed KC)
(Load block range)
P=10-7
, 10-5
Probabilistic
(Variable KC)
(Load block range)
P=10-7
, 10-5
12.1
1 7.7 11.5, 11.8 9.9, 10.5
2 7.6 12.4, 12.6 10.4, 11.0
3 7.3 11.1, 11.4 9.7, 10.3
4 7.8 11.2, 11.6 10.2, 10.8
5 7.5 11.6, 11.9 10.2, 10.8
3.4 Comparison of the accuracy of initial inspection time predictions between
deterministic and probabilistic approaches as validated by Virkler data
The Virkler [1] data which were obtained by testing aluminium 2024-T3 specimens
using a constant amplitude load spectrum were used to compare the predictions for the
initial inspection time using the deterministic and the probabilistic methods. Again the
initial crack size distribution was derived using crack size data at load cycle 1000 for 5
randomly chosen specimens. It should be noted that at load cycle 0, all the crack size
data were identical at 9 mm thus the need to derive the initial crack size distribution at
10
higher load cycles. In this comparison, a master crack growth curve based on the
geometry and loads used in the test was generated using CGAP [10] and shown in
Figure 10. This crack growth curve was used in both deterministic and probabilistic
methods. Since the master crack growth curve was obtained using CGAP, the need for
multiple trials is no longer necessary because for all test specimen with the same
geometry and material properties, the resulting crack growth curve from CGAP will not
vary. For this reason, the effect of varying the standard deviations of KC was
investigated instead. In the probabilistic analysis using variable fracture toughness, the
standard deviations of KC corresponding to 0.5, 0.8, 1.0 and 1.5 were used to predict the
initial inspection time. The results of the comparison are shown in Figure 11. In the
figure the four blue curves represent the risks PoF corresponding to the four different KC
standard deviations. In the figure, the PoF curves in blue from left to right correspond to
the assumed standard deviations of 1.5, 1.0, 0.8 and 0.5 respectively. The black curves
are the crack growth curves to failure of the test specimens with vertical axis scale (i.e.,
crack length) shown to the left of the graph. The red vertical line indicates the time (i.e.,
in load cycles) when the first failure occurs among the specimen tested. Two horizontal
lines were plotted to get the time where the PoF of 10-7
and 10-5
intersect the PoF
curves. The exact times (in load cycles) the two horizontal lines intersect with the PoF
curves are given in Table 2. It was noted that using a probabilistic fracture toughness
with standard deviation of 0.8 or higher resulted in a safe prediction of the initial
inspection time. In contrast, the deterministic approach gave an over-conservative
prediction.
0 100000 200000 3000000
10
20
30
40
50
60
cra
ck s
ize (
mm
)
Cycles
Figure 10 Master crack growth curve for the
Virkler data test specimen
0 100000 200000 3000000
10
20
30
40
50
Cra
ck s
ize (
mm
)
Cycles
10-10
10-9
10-8
10-7
10-6
10-5
10-4
10-3
PoF
Figure 11 PoF curves of the 4 trials using
varying Kc standard deviations
Table 2 Comparison of deterministic and probabilistic prediction using Virkler data
Minimum
specimen
fatigue life
(Cycles)
Deterministic
inspection
(Cycles)
Probabilistic
Fixed KC
Probabilistic
Mean KC = 25 MPa√m
Load cycles
corresponding to
P=10-7
, 10-5
assumed
std. dev.
Load cycles
corresponding to
P=10-7
, 10-5
222798
129700 231117, 233164
1.5 188101, 202155
1.0 210649, 215688
0.8 215851, 220942
0.5 223529, 226726
11
4. Discussion of results
4.1 Validation of Predictions verified using test specimen under variable peak
stress spectra
Based on the results of DST experiment shown in Table 1, the deterministic prediction
of the initial inspection time as halfway through the predicted fatigue life was
conservative since the first failure among test specimen occurred at 12.4 load blocks
which is higher than the deterministic prediction of 7.3 ~ 7.8 load blocks. The
deterministic prediction can be said to be overly conservative. Furthermore, the concept
of conducting inspection halfway through the fatigue life in order to have a chance of
another inspection before fatigue failure occurs as suggested by the deterministic
approach seems validated by the experimental results because all the deterministic
predictions of inspection times occurred before failure (i.e., 14.6 ~ 15.6 load blocks).
As shown in Table 1, the probabilistic approach when considering the variability of the
fracture toughness did a very good prediction on the initial inspection, predicting as
early as 9.7 load blocks and as late as 10.4 load blocks which are not only safe but also
closer to the experimental minimum fatigue life of 12.1 load blocks. This means that the
probabilistic approach was conservative and more accurate than the deterministic
approach.
4.2 Validation of Predictions Verified using Virkler data
The second verification using the Virkler data given in Table 2 showed similar trend as
in Table 1. The deterministic prediction of the initial inspection (i.e., half-life) was
overconservative at 129700 cycles in comparison to the experimental result which was
222798 cycles. In practice, conducting an inspection too early may lead to cracks not
being found due to their small size. On the other hand, the probabilistic approach
considering the variability of fracture toughness gave a very good prediction which was
closer to the experimental value. However, it is observed that the probabilistic approach
using a deterministic (fixed value) fracture toughness slightly over-predicted the first
inspection and gave a value of 231117 cycles. What can be observed in the table is that
the highest assumed standard deviation at 1.5 MPa√m gave the most conservative (i.e.,
safe) prediction among the probabilistic predictions. Furthermore the assumed Kc
standard deviation from 0.8 to 1.5 MPa√m resulted in safe and more accurate predictions
in comparison to the deterministic predictions. Similar to the previous comparison using
variable amplitude, the deterministic approach gave a prediction which was safe but
over-conservative since the predicted inspection time of 129700 cycles was too early.
The experimental fatigue life showed that inspection can be conducted as late as 222798
cycles (i.e., the minimum fatigue life of all the specimens).
5. Conclusions
Based on the results of this investigation, conclusions are as follows:
1.) Following the guidance of MIL-STD1530D, both the deterministic and
probabilistic approach in predicting the initial inspection time gave conservative
predictions as validated by test specimens, but the probabilistic approach
predicts a life closer to the experimental results;
12
2.) The initial inspection time from the deterministic approach is overly
conservative;
3.) The application of both the deterministic and probabilistic approach in
predicting the safe fatigue life and the initial inspection time may provide
increased confidence and conservatism for the safe management of fleet.
In the future, it is envisaged that the use of both deterministic and probabilistic methods
in structural integrity management has many advantages over the use of only one
method and will enable stakeholders to balance the requirements for safety and
economy.
Acknowledgements
The authors would like to thank Dr Colin Pickthall and Dr Manfred Heller for reviewing
this paper and for their valuable comments.
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