Efficient conditional failure analysis: applicationto an aircraft engine component

LUC HUYSE* and MICHAEL P. ENRIGHT

Southwest Research Institute, 6220 Culebra Rd, San Antonio, TX 78238, USA

(Received 16 May 2005; accepted in revised form 16 November 2005)

Although many engineering systems and components have a very low probability of

failure, the consequences of failures can be catastrophic. The low failure probability

precludes the safety certification of such systems or components on the basis of repeated

experiments. Therefore the certification must necessarily be done on the basis of

adequately validated computational models. Conditional failure analysis (CFA) can be

used to provide insight into the failure of a component and the most likely range of values

of the input random variables associated with failure. It not only provides valuable

information to the engineer for improvement of component design, but also helps to

validate the mathematical models and probabilistic input distributions associated with

reliability computations. The CFA technique outlined in this paper is based on

generation of conditional failure samples obtained during computation of component

reliability. It provides a graphical description of the failure region that can be used for

failure analysis. The technique is illustrated for a practical design problem involving the

fatigue crack growth limit state of a gas turbine engine component. The results can be

applied to failure analysis and model validation of general aerospace components.

Keywords: Conditional failure; Failure analysis; Importance sampling; Monte Carlo

simulation; Probabilistic methods; Response surface method

1. Introduction

The design of engineering systems and components is

affected by uncertainties (e.g. incomplete understanding of

the physics, lack of information, inherently variable opera-

ting conditions, among many others). When a probabilistic

evaluation is performed, the effects of uncertainty must be

accounted regardless of the source of the uncertainties. If

the random variables x are characterized by a probability

density function (PDF), the failure probability pF can be

computed as an integral over the failure domain:

pF ¼Zn-fold

failure

� � �Z

fXðxÞdx ¼Zn-fold

gðxÞ< 0

� � �Z

fXðxÞdx ð1Þ

where g(x)5 0 is the failure domain.

From a design perspective it is important to identify the

range of values of the individual random variables that lead

to failure (herein referred to as conditional failure analysis

or CFA) so that they can be avoided where possible. In CFA

the original PDF of the random variables is redefined in

terms of failure F, where F describes the failure domain

{x j (g(x)5 0)}. In this context, the conditional failure den-

sity fX(x j F) of a random variable X (i.e. probability density

function of x given that F has occurred) can be written as:

fXðx j FÞ ¼ fX½x j gðxÞ < 0� ð2Þ

A typical reliability assessment identifies the risk associated

with a specific limit state or set of limit states with a focus

on the most likely combination of variables that lead to

failure. CFA provides additional insight regarding the phy-

sics of the problem. For example, computational models

*Corresponding author. Email: lhuyse@swri.org

Structure and Infrastructure Engineering, Vol. 2, Nos. 3 – 4, September –December 2006, 221 – 230

Structure and Infrastructure EngineeringISSN 1573-2479 print/ISSN 1744-8980 online ª 2006 Taylor & Francis

http://www.tandf.co.uk/journalsDOI: 10.1080/15732470600590374

typically involve approximations that are valid only for a

particular range of input parameters. The high-density region

of a joint conditional failure density fX(x j F) represents thecombinations of input parameters X for which the

computational model must adequately predict the physical

behavior. It can be compared to the densities of the input

variables to ensure that none of the model assumptions are

violated. This helps to validate that the physics are

adequately represented in the computational model.

CFA also provides information useful for selecting the

PDF that best fit the experimental data. Typically, an

explicit PDF is selected based on a goodness-of-fit test,

such as Kolmogorov – Smirnov (Chakravarti et al. 1967) or

its tail-improved Anderson –Darling version (Stephens

1974), the Shapiro –Wilks test (Shapiro and Wilks 1965),

or the chi-square test (Snedecor and Cochran 1989), among

others. A good fit often cannot be achieved over the entire

range, and the fit results may be influenced by data values

that do not contribute to failure. Since the PDF is used for

failure probability computations, the accuracy of the fit is

most critical in the neighborhood of the most probable

failure point (MPP). The MPP may or may not be located

in the tail of the fitted distribution. The conditional density

provides more detail regarding the range of values over

which a good fit must be achieved.

In this paper, several approaches for generating condi-

tional densities are discussed. A CFA technique is pres-

ented, in which the conditional densities are generated

during the computation of component reliability using a

hybrid approach (Wu et al. 2002) consisting of numerical

integration combined with the response surface method

(Myers and Montgomery 2002). The technique is illustrated

for a practical design problem involving the fatigue crack

growth limit state of a commercial aircraft gas turbine

engine component. It provides the additional information

needed for statistical analysis of the failure region asso-

ciated with each random variable. The results can be ap-

plied to failure analysis and model validation of general

aerospace components.

2. Approaches for generating conditional probability

distributions

Typical quantities of interest for CFA include the condi-

tional mean E(x j F), conditional variance Var(x j F), andx*, the values of the input variables x at the MPP:

Eðx j FÞ ¼Z

fXðx j FÞ x dx ¼ZgðxÞ<0

x fXðxÞ dx ð3Þ

Varðx j FÞ ¼Z½x� Eðx j FÞ�2 fXðx j FÞdx ð4Þ

x� ¼ argfmax ½fXðx j FÞ�g ¼ arg maxgðxÞ<0

½ fXðxÞ�� �

ð5Þ

A number of techniques can be used to generate

conditional densities. For example, direct Monte Carlo

simulation can be used using the following procedure: (1)

generate a sample from the original (unconditional) density

function f(x), (2) evaluate the limit state g(x), (3) if

g(x)5 0, accept the sample (i.e. use in evaluation of the

conditional metric of interest), otherwise, reject the sample.

However, this method is very inefficient because approxi-

mately 1/pF samples are required to generate a single

sample from the conditional density function. Since

additional samples are required for a statistical analysis

of failure (e.g. estimation of second or higher order

moments, correlations, etc.), this approach requires many

more samples than are ordinarily required for Monte

Carlo-based failure probability computations.

Second moment methods can be used to identify the

MPP. The MPP (or, equivalently, the Point of Maximum

Likelihood) identifies the most likely combination of

random variables associated with failure. However, no

expressions are available to easily compute the conditional

statistics:

EðhðxÞ j FÞ ¼Z

fXðx j FÞ hðxÞ dx ¼ZgðxÞ<0

fXðxÞ hðxÞ dx

ð6Þwhere h(x) represents the importance sampling density

(ISD).

The location of the MPP can, however, be used as a

starting point to construct approximations to the condi-

tional density f(x j F). For example, the asymptotic ISD

(Maes et al. 1993) is fairly straightforward to construct

from the MPP location and main curvature direction of the

limit state function at the MPP. The Asymptotic Impor-

tance Sampling estimator typically has a small variance,

which makes it a good candidate to evaluate the

conditional expectations in equation (6) typically required

for failure analysis.

Importance sampling (IS) can also be used to character-

ize conditional densities. The efficiency of IS depends on the

selection of an ISD. The IS method leads to an unbiased

estimate provided that the ISD is non-zero over the entire

failure domain (Madsen et al. 1986):

pF ¼Z

fXðx j FÞ dx ¼ZgðxÞ<0

fðxÞ dx ¼ZgðxÞ<0

wðxÞ hðxÞ dx

ð7Þ

where w(x)¼ f(x)/h(x) is the location dependent scaling

factor associated with the ISD. The variance of the

IS-based failure probability estimate is given by (Au and

Beck 2002):

VarðpFÞ ¼ZgðxÞ<0

ðwðxÞ � pFÞ2hðxÞdx ð8Þ

222 L. Huyse and M. P. Enright

Equation (8) shows that the optimal choice for the ISD is

given by the conditional density f(x j F) when w(x)¼ pF for

all x, and the variance Var(pF) is zero. The conditional

density is obtained by adjusting w(x) to reduce Var(pF) to a

small value. Au and Beck (2002) recently demonstrated that

the conditional failure density is obtained as the asympto-

tically density resulting of iterative Markov Chain Monte

Carlo simulations, based on an arbitrary ISD. This

removes the need to perform the potentially computation-

ally expensive limit state analysis.

In this paper an approach is presented where the ISD is

given by the exact conditional density. The failure

probability is computed using numerical integration and

the conditional densities are generated as a by-product of

the integration. If the random variables are independent,

equation (1) can be expressed as:

pF ¼Zn-fold

failure

� � �Z

fx1ðx1Þ � fx2ðx2Þ � � � fxnðxnÞdx1 � dx2 � � � dxn

ð9Þ

It is possible to solve the limit state function g(x) for one of

the random variables, say X1. In most applications this

cannot be done in closed form. However, if an accurate

response surface can be generated for the problem, X1 can

be solved numerically. If the failure domain can be

described in terms of X1, equation (9) can be rewritten as:

pF ¼Zðn�1Þ-fold

� � �Z

Fx1 ½x1ðx2; x3; . . . ; xnÞ j F�

� fx2ðx2Þ � � � fxnðxnÞdx2 � � � dxn ð10Þ

This process can be repeated for the remaining random

variables, X2, . . . ,Xn. The cumulative distribution function

(CDF) for the n-th random variable FXn(xn j F) is the

conditional CDF of Xn given failure F. Conditional samples

for Xn, say xn,i, can be generated directly from FXn(xn j F).

Conditional samples for the (n – 1)-th random variable

are generated using FXn71(xn71 j F) for the range of values

xn j F. This process of back-substitution is repeated for

all remaining random variables.

The accuracy of the conditional density estimate is

dependent on the techniques used for numerical integration

and identification of the failure domain. This approach is

practically limited to problems with relatively few variables

and a well-defined failure domain.

3. Application to aircraft gas turbine engine component

Over the past several years, a probabilistic methodology

has been under development for crack growth life predic-

tion of high-energy rotary component in commercial

aircraft gas turbine engines (Leverant et al. 2000). The

methodology was recommended by the Federal Aviation

Administration (FAA 2001) to address the occurrence of

rare metallurgical defects that can lead to uncontained

engine failures (NTSB 1990). It has been implemented in a

probabilistic damage tolerance computer program called

DARWIN1 (Leverant et al. 2004, Enright et al. 2005,

McClung et al. 2006).

This example demonstrates the use of conditional

densities (generated as part of the importance sampling

method in DARWIN) to predict the failure behavior of a

gas turbine engine component (figure 1). The conditional

densities of the primary (non-inspection related) random

variables are shown in table 1, including a comparison of

the statistical descriptors for the original and conditional

densities. The use of the conditional densities for failure

analysis (i.e. CFA) is illustrated, providing insight for

design and maintenance planning decisions.

Figure 1. Application example: commercial aircraft gas

turbine engine component.

Table 1. Summary statistics for the original and conditionaldensities associated with gas turbine engine random variables.

Random variable

Statistical

descriptor

Original

density

Conditional

density

Initial defect area, d mean (mm2) 16.0 125.0

log(d) COV 42.5% 30%

Stress scatter factor, B median 1.0 1.63

Stress scatter factor, B COV 20.0% 16.0%

Life scatter factor, S median 1.0 0.928

Life scatter factor, S COV 20.0% 20.0%

S and B correlation 0.0 0.09

d and S correlation 0.0 0.15

d and B correlation 0.0 70.72

Efficient conditional failure analysis 223

3.1 Limit state

A fatigue crack growth failure limit state is considered, in

which it is assumed that material failure (i.e. fracture)

occurs when the maximum stress intensity factor K exceeds

the fracture toughness KIc . If an initial defect is present, the

probability of fracture pF can be expressed as:

pF ¼ PðK � KIcÞ ¼ PðNfail � NserviceÞ ð11Þ

3.2 Random variables

The uncertainty associated with fatigue crack growth

computations is modeled using three general random

variables related to variability in initial defect size, applied

stress, and material properties (Leverant et al. 2000). In this

application each of the random variables is treated as an

independent random variable. The initial defect size is

based on an exceedance model developed by the Aerospace

Industries Association (AIA 1997). The model is converted

to a CDF as follows (Wu et al. 2002):

CDFðdiÞ ¼excðdmin Þ � excðdÞ

excðdmin Þ � excðdmax Þð12Þ

where exc(.) denotes the exceedance probability (given by

the complementary CDF).

The exceedance and CDF curves of the initial defect size

distribution used in this application are shown in figure 2.

In equation (12) it is assumed that the CDF is bounded at

either end of the exceedance curve (the validity of this

assumption is addressed later in the paper).

The uncertainty associated with applied stress is modeled

as a log-normally distributed stress scatter factor B. In

practice this uncertainty may be relatively small with a

coefficient of variation (COV) of 10% or less. In this

application the COV is set to 20% to better explore the

influence of this variable (the median value is set to 1.0).

The applied stress is equal to the product of the stress

scatter factor and finite element stress results sFE:

sapplied ¼ b� sFE ð13Þ

The influence of materials-related variability on fatigue

crack growth life is modeled using a log-normally

distributed life scatter factor S. In this application example

it is assumed to have a median value of 1.0 and a 20%

COV. The fatigue crack growth life Nfail is equal to the

product of the life scatter factor and the deterministic crack

growth life NFCG:

Nfail ¼ s�NFCG ð14Þ

For this example, fatigue crack growth is modeled as (Paris

and Erdogan 1963):

da

dN¼ CðDKÞm ð15Þ

where a is the crack area, C and m are empirical regression

coefficients associated with the Paris Law. The life scatter

can also be considered as an error band applied to the

regression curve of the dd/dN data. If life scatter is

included, equation (15) can be expressed as:

da

dN¼ 1

s� CðDKÞm ð16Þ

3.3 Crack growth life response surface

Figure 3 shows how the median predicted life NFCG

decreases with increasing initial defect area d and stress

scatter value b. This life is computed using the Flight_Life

fracture mechanics code which is an integral part of the

DARWIN software. For this crack growth computation a

Paris Law (equation (15)) was used and no redistribution of

stresses due to the crack growth was considered. For each

load step only a single FE analysis on an uncracked

impeller model is required to obtain the stresses. The

computational cost of setting up this response surface can

be considerably reduced if it can be truncated in regions

that do not influence risk. The computational cost of

predicting the fatigue crack growth life NFCG is directly

proportional to the median life value itself. The likeli-

hood of failure becomes negligible if the median life

NFCG�Ntarget. In DARWIN the response surface is

truncated at life values five standard deviations above the

median life scatter value. In addition, since the predicted

life increases monotonically with decreasing values of the

initial defect size and stress scatter factor, the response

surface can be truncated for relatively small defect (d5 dt)

Figure 2. Exceedance values and corresponding cumulative

distribution associated with initial defect size.

224 L. Huyse and M. P. Enright

or stress scatter (b5 bt) values. Practical experience has

indicated that organizing the computations in this manner

reduces the computational cost of response surface

preparation by 30 – 60%.

3.4 Numerical integration over original densities

The probability of fracture can be expressed as an integral

over the failure domain in terms of the three independent

random variables described in the previous section:

pF ¼ZZZ

failure

fdðdÞfBðbÞfSðsÞdddbds ð17Þ

To solve this integral, the failure domain must be identified.

Introduce a critical defect size d* defined as the smallest

defect that will cause fracture within a specified service life

Nservice. Since the failure domain is limited to the defects

that exceed the critical defect size (i.e. d4 d*), equation

(17) can be rewritten as:

pF ¼ZZð1� FD½d�ðb; sÞ�ÞfBðbÞfSðsÞdbds ð18Þ

where

FD½d�ðb; sÞ� ¼ Pr½d � d�ðb; sÞ� ð19Þ

and d*(b,s) is the critical defect size d* associated with

specific values of the life and stress scatter factors. The

values d*(b,s) are obtained directly from a response surface

of predicted life values (figure 3) by setting the predicted

life equal to Ntarget/s. The corresponding exceedance

probability Pr(d4 d*(b,s)) is computed using the initial

defect CDF, as shown in figure 4. Introducing the stress

scatter and life scatter random variables, the probability

density of the integrand of equation (18) (i.e.

Pr(d� d*)fB(b)fs(s)) is obtained as shown in figure 5. The

failure probability at Nservice is equal to the integral of the

PDF shown in this figure. The conditional densities are

obtained based on the failure region identified during

evaluation of equation (19) and are subsequently used for

importance sampling to predict the influence of inspection

on the probability of fracture (see Wu et al. 2002, Huyse

and Enright 2003 for further details on the importance

sampling method in DARWIN).

3.5 Generation of conditional densities

The conditional PDFs can be developed during the

numerical integration with little additional effort. This is

a key advantage over the use of Monte Carlo methods

that may require many samples. The process is outlined

below.

1. Integrate Pr(d� d*)fB(b)fs(s) (see figure 5) over the

stress scatter factor B. This results in a conditional

likelihood curve for the life scatter factor S.

2. Obtain a conditional life scatter factor sample value sifrom this conditional density.

3. Obtain a sample stress scatter factor bi from the

likelihood curve Pr(d� d* j si) fB(b)fs(si). In effect, the

conditional likelihood curve for the stress scatter factor

B is obtained by setting S¼ si in figure 5.

Figure 4. Exceedance probability Pr(d4 d*) associated

with life and stress scatter factor values.

Figure 3. Response surface of predicted life as function of

the initial defect size and stress scatter factor values.

Efficient conditional failure analysis 225

The exceedance probability Pr(d4 d* j bi, si) correspondingto the sample values si and bi is shown in figure 4. The

critical defect size d* can be derived from the exceedance

probability value Pr(d4 d* j bi, si) through the initial defect

distribution (figure 1). A sample initial defect size diis subsequently generated from the conditional PDF

fD(d j d4 d*), which is a truncated and rescaled version

of the original initial defect size distribution and includes

only defect values that exceed the critical defect (i.e.

d4 d*).

3.6 Failure analysis of conditional densities

The joint density is dependent on several random variables

and is difficult to visualize in two-dimensional space.

Original and conditional joint PDFs for each pair of

random variables (i.e. stress scatter and initial defect size,

stress scatter and life scatter factor, life scatter factor and

initial defect size) were therefore constructed, shown in

figure 6. It is important to note that these conditional

densities are readily obtained as a by-product of the

numerical integration scheme in DARWIN and do not

require sampling. A significant number of samples (about

1000/pF or 100 000/pF samples – depending on the desired

accuracy of the conditional histograms) would be required if

these densities were generated using Monte Carlo sampling.

The conditional densities in figure 6d and f indicate that

failure is most likely to occur when the initial defect area is

approximately 0.1 mm2 (note: the initial defect size is

plotted on a logarithmic scale). Smaller defects are plentiful

(see figure 6a and c) but do not cause failure before the

specified service life of 20 000 cycles. Although failure is

virtually guaranteed when large defects are present, they do

not occur frequently enough to have a significant influence

on risk. It is interesting to note that the joint conditional

likelihood values are very small near the truncated tails of

the defect distribution. It can be concluded that, for this

example, the tails of the initial defect distribution do not

have a significant influence on the total risk. Therefore,

although the defect distribution is truncated, this assump-

tion does not appear to have an influence on the

conditional density and associated failure probability

predictions.

The high-density regions associated with the life scatter

factor are nearly identical for the original (figure 6a and b)

and conditional (figure 6d and e) densities. This indicates

that, in this example, the failure risk is not very sensitive to

the uncertainty in the fatigue crack growth model.

For the stress scatter factor, the high-density region in

the conditional density is located near stress values that are

approximately 60% greater than for these associated with

the original density (compare figure 6b and c with 6e and f,

respectively). This indicates that failure is more likely to

occur for large stress scatter values.

Summary statistics for the original and conditional

densities associated with the three random variables are

indicated in table 1. The results in table 1 indicate that

failure is more likely to occur if the median stress scatter

factor is about 60% larger than average and the geometric

mean initial defect size is about eight times larger than

average (based on the original densities). The conditional

density associated with the life scatter factor is very similar

to the original density, suggesting that it may not have as

much influence on the failure as the other random

variables. It is interesting to note that the COV associated

with the logarithm of initial defect size conditional density

Figure 5. Conditional failure density associated with life and stress scatter factor values: (a) three-dimensional plot, and (b)

contour plot.

226 L. Huyse and M. P. Enright

Figure 6. Joint original and conditional probability densities: (a,b,c) original densities; (d,e,f) conditional densities.

Efficient conditional failure analysis 227

is much smaller than the COV of the original density. This

is entirely due to a shift in the mean since the standard

deviation has actually increased. This illustrates the overall

reduction in variability associated with the conditional

initial defect density.

The three random variables are uncorrelated. However,

the conditional variables x j F are correlated since they are

all related by the failure event F. The strong negative

correlation between conditional stress scatter factor and

conditional initial defect size (table 1) suggests that failure

is not likely to occur for small initial defects unless the

stress scatter factor is relatively high. However, large initial

defects can lead to failure even when stress is relatively

small. The likelihood of failure due to a combination of

large defect and small stress scatter is about equal to the

likelihood of failure caused by a combination of small

defects and high stress scatter values (see figure 6f). Some

correlation also exists among the remaining conditional

variables.

3.7 Application of CFA to selection of manufacturing

inspection method

In this section an illustration of how the failure densities

may be used in practice is presented. Nondestructive

inspection methods can be used to reduce the probability

of fracture of engine disks. Assume that five different crack

inspection techniques (IT 1 – 5) with log-normal PDF are

available (see figure 7). The median probability of detection

(POD) value decreases by an order of magnitude for each

IT, and the amount of scatter is identical for all ITs. These

inspections are applied before the component is placed in

service. The influence of these inspections on the cumula-

tive probability of fracture of a typical engine disk (Huyse

and Enright 2003) is shown in figure 8. It can be observed

that the largest changes in the probability of fracture occur

when the technique is changed from IT-2 to IT-3, and from

IT-3 to IT-4. Techniques IT-1 and IT-2 have only a small

influence on probability of fracture reduction.

The most effective inspection techniques detect cracks

that can lead to fracture during the service life of the

component. CFA can be used to identify the techniques

that focus on the cracks that can lead to fracture. This is

illustrated by superimposing the ITs from figure 7 on the

densities from figure 6a and d, shown in figure 9. When the

ranges of detection for the each of the ITs are overlaid on

top of the original densities (figure 9a), it appears that only

techniques IT-5 and IT-6 significantly influence the prob-

ability of fracture results. However, since many of the

smaller cracks shown in this figure do not lead to failure

within the service life, this result is misleading. Overlaying

the ranges of techniques IT-4 and IT-5 with the conditional

densities (figure 9b), it can be observed that a significant

portion of the cracks that lead to fracture can be detected.

This is consistent with the cumulative probability simula-

tion results shown in figure 8, and demonstrates how the

conditional densities can be used for maintenance planning

decisions. It is important to note that the results shown in

figure 9 are based on conditional densities only. Since no

simulations are required to quantify the influence of the

various inspections on the probability of fracture, the

results can be achieved much more efficiently.

4. Summary

The paper demonstrates a method for generating condi-

tional failure densities when only a small number of

random variables are present. The following observations

can be made regarding the CFA technique presented in

this paper:

1. It provides the engineer with valuable information

regarding the most likely combination of input

Figure 7. Probability of detection curves associated with

several inspection techniques.

Figure 8. Failure probability per flight cycle associated with

several inspection techniques.

228 L. Huyse and M. P. Enright

variables associated with failure (similar to the most

probable failure point associated with the first-order

reliability method).

2. It indicates the most important range of values for each

of the input random variables and can be used to guide

modeling and validation experiments for these vari-

ables.

3. It can be used as a measure of the sensitivity of a

random variable to the probability of fracture (i.e. the

degree to which these densities differ indicates the

relative sensitivity).

4. The conditional densities can be used to efficiently

gauge the effectiveness of in-service or maintenance

inspections. In this case no simulations are required to

select the most effective inspection.

The technique provides the additional information needed

for statistical analysis of the failure regions associated with

the random variables. The results can be applied to failure

analysis and model validation of general aerospace

components.

Acknowledgments

This work was supported by the Federal Aviation

Administration under Cooperative Agreement 95-G-041

and Grant 99-G-016. The authors wish to thank Joe Wilson

(FAA Technical Center project manager) and Tim

Mouzakis (FAA Engine and Propeller Directorate) for

their continued support. The ongoing contributions of the

Industry Steering Committee (Darryl Lehmann, Pratt &

Whitney; Jon Tschopp, General Electric; Ahsan Jameel,

Honeywell; Jon Dubke, Rolls-Royce) are also gratefully

acknowledged. The authors would also like to acknowledge

Y.-T. (Justin) Wu and Harry R. Millwater Jr for their

pioneering work in this field.

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Appendix – nomenclature

a crack size

b, B stress scatter factor value

C empirical coefficient in the Paris Law

d defect size

d* critical defect size

dmin minimum defect size

dmax maximum defect size

exc(d) number of defects of size greater than or

equal to d

E(x j F) conditional expectation (mean value) of x

given failure

fX(x) PDF of random variable X

fX j F(x j F) conditional PDF of x given failure

F failure domain

FX(x) CDF of random variable X

FX j F(x j F) conditional CDF of random variable X

given failure

g(x) limit state function

h(x) importance sampling density

K stress intensity factor

KIcfracture toughness

m empirical coefficient in the Paris Law

n number of random variables

Nservice service life (in flight cycles)

Ntarget target number of flight cycles

Nfail number of flight cycles to failure

NFCG deterministic (i.e. median) fatigue crack

growth life

pF failure probability

s, S life scatter factor value

Var(X) variance of the random variable X

X random variables (vector if bold)

x sample value of random variable X

x* value of X at most probable failure point

(MPP)

w(x) location-dependent scaling factor for the

Importance Sampling Density

sapplied actual, applied stress level

sFE stress level predicted by FE model

230 L. Huyse and M. P. Enright