Sensitivity of probability-of-failure estimates with respect to probability of detection curve parameters

International Journal of Pressure Vessels and Piping 92 (2012) 84e95

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Sensitivity of probability-of-failure estimates with respect to probability of detection curve parameters J. Garza, H. Millwater* University of Texas at San Antonio, Mechanical Engineering, 1 UTSA circle, EB 3.04.50, San Antonio, TX 78249, USA

a r t i c l e i n f o

a b s t r a c t

Article history: Received 18 June 2010 Received in revised form 7 November 2011 Accepted 11 November 2011

A methodology has been developed and demonstrated that can be used to compute the sensitivity of the probability-of-failure (POF) with respect to the parameters of inspection processes that are simulated using probability of detection (POD) curves. The formulation is such that the probabilistic sensitivities can be obtained at negligible cost using sampling methods by reusing the samples used to compute the POF. As a result, the methodology can be implemented for negligible cost in a post-processing nonintrusive manner thereby facilitating implementation with existing or commercial codes. The formulation is generic and not limited to any specific random variables, fracture mechanics formulation, or any specific POD curve as long as the POD is modeled parametrically. Sensitivity estimates for the cases of different POD curves at multiple inspections, and the same POD curves at multiple inspections have been derived. Several numerical examples are presented and show excellent agreement with finite difference estimates with significant computational savings.
2011 Elsevier Ltd. All rights reserved.

Keywords: Probabilistic fracture mechanics Probability-of-detection Nondestructive evaluation Nondestructive inspection Inspection sensitivity

1. Introduction Nondestructive evaluation (NDE), also known as Nondestructive Testing (NDT) or Nondestructive Inspection (NDI), plays a vital role in fracture control plans. Methods such as visual, dye penetrant, ultrasonics, radiography, and eddy current are among the common inspection techniques used to ensure structural integrity [1]. The type of inspection and the times of inspection must be carefully selected to ensure safety with reasonable cost. There are a number of industries that have a long history of application of NDE methods for structural integrity. For example, applications include nuclear [2e4], petroleum [5], aircraft structures [6e8], gas turbines [9,10], and offshore structures [11] to name a few. Rummel et al. [12] provide a summary of a number of issues related to the application of NDE methods to systems. The efficacy of an inspection process is characterized through the POD curve [13]. The POD defines the probability of detecting a defect as a function of the size of the defect. This concept is well known and POD curves for a particular inspection process, material, etc. are developed through statistical experiments using seeded samples of various sizes, multiple inspectors, etc [8,14]. For ^ vs. a ” opportunities for a number of example, the “hit-miss” or “a

* Corresponding author. Tel.: þ1 210 458 4481; fax: þ1 210 458 6504. E-mail address: [email protected] (H. Millwater). 0308-0161/$ e see front matter
2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijpvp.2011.11.009

inspections and a number of operators are tabulated and analyzed statistically to determine the percentiles, e.g., 50 and 95% of the POD curve [8,15]. The percentiles are often curve fit to parametric forms such as log-logistic, lognormal, log-odds, etc. Although to date the development of a POD curve is largely based on lab experiments, computational methods are becoming more prevalent in “Model Assisted POD”, MAPOD, methods [16]. Simulation of the inspection process is usually incorporated within a probabilistic fracture mechanics analysis also known as a risk assessment. The analysis in general considers random variables of crack size and aspect ratio, material properties, loading, inspection efficacy (represented by a POD curve) and inspection times. The output conveys an estimate of the POF of the structure as a function of time or cycles with and without inspection. The inspection process and frequency is often varied toward developing an optimized inspection schedule [17,18]. The simulation of inspection requires the assignment of a POD curve. Typically, the selection of a POD curve is based on an established catalog of POD curves pertinent to the material and inspection method under consideration. For example, POD curves for aluminum, titanium and stainless steel materials and certain geometries are cataloged in [14]. The curve fits are the best available information based upon statistical analysis of the experimental data but there always exists statistical uncertainty in the numerical values obtained. Application of existing cataloged POD curves is convenient since no new testing is required; however, there is

J. Garza, H. Millwater / International Journal of Pressure Vessels and Piping 92 (2012) 84e95

Nomenclature POF Pf(t) t tf tf0 tq x y fX(x) I(x,t) g(x,t) ai ac C m

mode I stress intensity factor plane strain fracture toughness stress intensity factor range crack size at time tq total number of Monte Carlo samples number of Monte Carlo samples that reach inspection q and are not detected during inspection q q inspection number q parameters of a POD curve PODq(q,a(y,tq)) probability of detection curve for inspection q CPODq(q,a(y,tq)) complementary PODq, equals 1 minus PODq vPf ðtÞ sensitivity of the POF with respect to the parameters q vq of a POD curve vPODq 1 Uq(q,a(y,tq))  vq CPODq KI KIc DK a(y,tq) N Nq

probability-of-failure POF at time t time cycles-to-failure specified number of cycles time of inspection q all random variables all random variables that affect crack size a joint PDF of random variables indicator function that denotes structural failure limit state function initial crack size critical crack size Paris law constant Paris law constant

always some uncertainty about the applicability of the labdeveloped POD curves to a field inspection where issues such as access, visual acuity, cleanliness and others arise. Also, sometimes due to cost and schedule constraints, POD curves are developed for a new inspection procedure using “transfer functions” from an existing set of POD curves from a related scenario [15,16]. Brausch et al. [15] discuss the application of an “inspectability factor” that encapsulates human factors challenges. These factors, which range from 1 to 2, should be used to adjust the expected aNDI (largest crack length that may be assumed to be missed during an inspection). In summary, as discussed above, there are a number of reasons to doubt the “exact” values for the parameters of a POD curve for any particular application. In addition, it may be useful to conduct “what-if” scenarios to quickly assess the value of changes in the inspection method in reducing the POF or reducing the cost of inspections. As a result, it is useful to have a quantitative estimate how the POF varies as a function of the parameters of a particular POD curve. These questions can be easily estimated once the sensitivities are computed and this information provides an estimate as to how much variation in the POF can be expected due to the mismatch of the environment for development of the POD curve versus its field application. Given the value of sensitivities and the difficulty with current parameter studies and finite difference approaches that require reanalyses, a more formal investigation into sensitivity methods of the POF with respect to the parameters of a POD curve is warranted and presented here. The paper is organized as follows. The methodology to determine the sensitivities for a single inspection is developed in detail in Section 2 followed by a summary of the results for multiple inspections. Variance estimates of the sensitivities are presented in Section 3. Two numerical examples are presented in Section 4 followed by the Conclusions in Section 5.

equal to zero, i.e., g(x,t) ¼ tf(x)tf0 where tf is the cycles-to-failure, defined when KI(t)  KIc. KIc is the fracture toughness of the material and KI is the mode I stress intensity factor. The fatigue crack growth model is implicit in determining the cycles-to-failure tf. Defining an indicator function such that

 Iðx; tÞ ¼

gðx; tÞ>0 safe gðx; tÞ  0 fail

ZN Pf ðtÞ ¼

Iðx; tÞfx ðxÞdx

(1)

gðx;tÞ0

where Pf denotes the POF, t is the time in cycles, x is a vector of random variables, fx(x) is the joint density function of the random variables, and g(x,t) is a limit state function used to define failure. If failure occurs before the observation time, tf0, g(x,t) is less than or

(3)

N

and approximated using sampling as

Pf ðtÞ ¼

N 1 X Iðxi ; tÞ N i¼1

(4)

where N denotes the number of samples and xi a vector of the random variables for realization i. To characterize an inspection process, the probability of detecting a crack of size a is defined by PODq(qq,a(y,tq)), where tq is the time of inspection q, a is the crack size, y is a vector of random variables that affect a, and qq is a vector of parameters that are used to describe PODq. The vector of random variables that affect crack size y is a subset of the total vector of random variables x. For example, typical random variables that affect a include the initial crack size, loading, crack growth parameters, and geometry correction factor, whereas fracture toughness is contained in x but not y unless fracture toughness is an explicit component of the crack growth equation. The POF after inspection can be determined for a single inspection at t1 as

ð1  POD1 ðq1 ; aðy; t1 ÞÞÞfx ðxÞdx t>t1

Pf ðtÞ ¼

Z

(2)

the POF can be written as

The POF for fatigue analysis without inspection is evaluated as

fx ðxÞdx

0 1

Z

2. Methodology for probability-of-failure sensitivities

Pf ðtÞ ¼

85

(5)

gðx;tÞ0

which can be written in terms of the complementary POD, CPODq ðqq ; aðy; tq ÞÞ ¼ 1  PODq ðqq ; aðy; tq ÞÞ, and indicator function as

ZN Pf ðtÞ ¼ N

Iðx; tÞCPOD1 ðq1 ; aðy; t1 ÞÞfx ðxÞdx

t>t1

(6)

86

J. Garza, H. Millwater / International Journal of Pressure Vessels and Piping 92 (2012) 84e95

The POF can be estimated using sampling as

8 N > 1 X > > Iðxi ; tÞ > < N i¼1 Pf ðtÞz N1 > X > > P ðt Þ þ 1 > Iðxi ; tÞ : f 1 N

t  t1 (7) t>t1

reused to compute the sensitivities. As such, the sensitivities will be obtained for negligible computational cost. Two distinct cases occur when computing sensitivities for multiple inspections: a) different POD curves are applied at each inspection time, and b) the same POD curves are applied at each inspection time. Sensitivities for both cases are derived below.

i¼1

where N denotes the number of samples, N1 is the number of samples that have not failed before t1 nor detected at t1, xi denotes a vector of the random variables for realization i, and Pf(t1) denotes the POF just before inspection 1. In calculating Pf, any defects that are detected are removed from the population. Fig. 1 shows an example of the POF versus cycles without inspection and with inspections at 10,000 and 20,000 cycles. For multiple inspections, the POF for Q inspections can be written

ZN Pf ðtÞ ¼

Iðx; tÞ

Q Y

  CPODq qq ; a y; tq



fx ðxÞdx t>tq

(8)

q¼1

N

and estimated using sampling as shown for two inspections as

8 N > 1 X > > Iðxi ; tÞ > > > N > > i¼1 > > N1 < 1 X Pf ðtÞz Pf ðt1 Þ þ Iðxi ; tÞ > N i¼1 > > > > N2 > > 1 X > > Iðxi ; tÞ > : Pf ðt2 Þ þ N i¼1

N

This can be written in terms of a new parameter U as

8 0 t t1 > > vPf ðtÞ < ZN ¼ Iðx;tÞU1 ðq1 ;aðy;t1 ÞÞCPOD1 ðq1 ;aðy;t1 ÞÞfx ðxÞdx t>t1 > vq 1 > : (11)

t1 < t  t2

(9)

where the term U is defined generically for any inspection time as





Uq qq ;a y;tq t>t2



   vCPODq qq ;a y;tq 1    vq q CPODq qq ;a y;tq    vPODq qq ;a y;tq 1    ¼ vqq CPODq qq ;a y;tq ¼

(12) Eq. (11) can be written in terms of the expectation operator as

vPf ðtÞ ¼ vq 1

The sensitivity of the POF with respect to the parameters of A POD curve can be determined by differentiation Eq. (6) for a single inspection and Eq. (8) for multiple inspections. A key efficiency consideration is to develop the methodology such that the same Monte Carlo samples that are used to compute the POF can be



0 Ef CPOD1 ½Iðx; tÞU1 ðq1 ; aðy; t1 ÞÞ

t  t1 t>t1

(13)

The expected value can be approximated using sampling as

8 > <

0 vPf ðtÞ N1 z 1 X Iðxi ; tÞU1 ðq1 ; aðyi ; t1 ÞÞ > vq 1 :N i¼1

t  t1 t>t1

(14)

where q1 represents the vector of parameters of POD1, N1 denotes the samples that reach inspection 1, i.e., have not failed before t1 nor detected, a(yi,t1) denotes the crack size at time t1, yi represents the random variables that affect crack size for realization i, xi represents all the random variables for realization i, and N denotes the total number of samples in the simulation. From Eq. (14), the N1 1 X ½. However, this expected value is estimated as Ef CPOD1 ½z N1 i ¼ 1

Inspection at 10,000 and 20,000 Cycles 1 No Inspection With Inspection

0.8

Probability−of−Failure

(10)

N

2.1. Sensitivity with respect to POD parameters

0.7 0.6

equation will give the sensitivity conditioned on the samples that reached inspection 1 and are not detected. In order to provide the sensitivity with respect to the total samples N, the conditional N1 N 1 X ½. probability estimate N1/N is used such that 1 Ef CPOD1 ½z N N i¼1

0.5 0.4 0.3 0.2 0.1 0

8 0 t  t1 < ZN vPf ðtÞ vCPOD1 ðq1 ; aðy; t1 ÞÞ ¼ Iðx; tÞ fx ðxÞdx t>t1 : vq 1 vq1

0 < t  t1

where Nq is the number of samples that reach inspection q and were not detected by any previous inspection. Extension of Eq. (9) for more than two inspections is self-evident.

0.9

2.1.1. Single inspection case The sensitivity of the POF with respect to q1 (the parameters of the POD curve used at inspection 1) for a single inspection can be obtained by taking the partial derivative of Eq. (6) with respect to q1. For a single inspection,

0

1

2

3

4

5

6

7

Cycles Fig. 1. POF with and without inspection example.

8

9

10 4

x 10

Some comments on the governing equation for sensitivity analysis, Eq. (14), are in order. First, the form of the equation is very simple and is analogous to the equation to estimate the POF. Second, and most importantly, the same samples x, used to estimate the POF are reused to estimate the sensitivity; therefore, the sensitivities are estimated for negligible computational cost. Third, U is only computed at the time of inspection t1, it is only a function

J. Garza, H. Millwater / International Journal of Pressure Vessels and Piping 92 (2012) 84e95

of the random variables that affect crack size, and it will only be accumulated for failed samples. The time dependent nature of the sensitivity is embodied in the indicator function, I(x,t), which is a function of all random variables and all time. When a failure occurs for t > t1, the random variable values for that realization are used to compute U at the time of inspection. That is, the crack size used to evaluate U is the size of the crack at the time of inspection for the realization under consideration. It is clear that the magnitude of the sensitivity will increase monotonically with time if U is always positive or negative for any realization of random variables since U is always summed. However, if U changes sign as a function of t, the sensitivities may increase or decrease with time as demonstrated below. 2.1.2. Multiple inspection sensitivities The above methodology can be extended to multiple inspections. Results for two inspections are given below; the extension to more inspections is self-evident. Two scenarios must be considered: different POD curves used at each inspection, and the same POD curve used at both inspections. The derivations below address the time interval subsequent to the time of the second inspection, i.e., t > t2, since the equation for the sensitivity of the POF with respect to the parameters of the first inspection, Eq. (14), is unchanged by application of a second inspection. Note, however, that the numerical values of vPf/vq1, may change since subsequent inspections may remove cracks that would have failed otherwise and, therefore, contributed to vPf/vq1. 2.1.2.1. Different POD curves. In this case the POD’s are different at each inspection, POD1 s POD2, and therefore q1 s q2. The derivation of the sensitivity of the POF with respect to the parameters that govern the 2nd inspection is given in Appendix A, the results are summarized here. The sensitivity vPf/vq2 can be approximated using sampling as

t  t2 t>t2

(15)

where N2 denotes the samples that have not failed before t2 nor detected by the first or second inspections. 2.1.2.2. Same POD curves. If the same POD curve is used at each inspection then POD1 ¼ POD2 and q1 ¼ q2 ¼ q. The derivation is given in Appendix B. The sensitivity vPf/vq2 can be approximated using sampling as N2 vPf ðtÞ 1X ¼ Iðx; tÞfUðq; aðyi ; t1 ÞÞ þ Uðq; aðyi ; t2 ÞÞg t>t2 N i¼1 vq

PODðaÞ ¼

1 1 ln½a  m pffiffiffi þ Erf 2 2 2s

87

(17)

where a denotes the crack size at the time of inspection, m and s are parameters of the POD curve, and Erf denotes the error function. Um and Us per Eq. (12) are

# " ðln½a  mÞ2 Exp  pffiffiffi 2s2 2  Um ¼  pffiffiffiffi  ln½a  m s p pffiffiffiffiffiffi Erf 1 2s

Us ¼

(18)

ln½a  m

s

Um

(19)

Fig. 2 shows a plot of Um and Us for values of m ¼ 9, s ¼ 0.8. From Eq. (18) and the figure, it is clear that Um is monotonically increasing with respect to crack size a and is always positive. Therefore, the sensitivity, vPf/vm, will increase in magnitude monotonically with time and be always positive (an increase in m, which shifts the POD curve to the right, will increase the POF). However, as seen from Eq. (19) and Fig. 2, Us can change sign depending upon the relationship of crack size to m as shown in Eq. (19) by the ln[a]  m term. Cracks that have failed that are short cracks at the time of inspection will result in a negative Us, whereas long cracks at the time of inspection that will result in a positive Us. Therefore, it is problem dependent whether vPf/vs will monotonically increase with respect to time or not; the sign of the sensitivity depends upon the form of the POD curve, the crack sizes at the time of inspection for cracks that cause failure and the relative proportion of positive to negative values of Us.

3. Variance estimates The sensitivity estimates computed using sampling are random variables and their accuracy depends upon the problem under investigation and the number of samples used. An estimate for the variance (or standard deviation) of the sensitivities can be computed be adopting and modifying the variance estimate in [19] as Nq  2     2 1  1 X V vPf =vq z 2  vPf =vq Iðxi ; tÞU q; a yi ; tq N N i¼1

(20)

4 3.5

(16)

Eq. (16) is similar to the sensitivity equation obtained for different POD curves. The only difference is that if the same POD curve is used, the U’s from all previous inspections are summed.

3 2.5 2

Ωq

8 0 > vPf ðtÞ < X N2 z 1 Iðxi ; tÞU2 ðq2 ; aðyi ; t2 ÞÞ > vq2 :N i¼1



1.5

Ω

μ

2.2. Properties of U Eqs. (14) and (16) show that the partial derivative Pf(t)/vq will increase monotonically with time if U is always positive or negative for any realization of random variables but may be nonmonotonic if U changes sign as a function of time. Therefore, it is instructive to observe the properties of U for some standard forms. A typical form of the POD curve is a cumulative lognormal distribution of the form

1 Ω

0.5

σ

0 −0.5

0

1

2

3

Crack Size (m) Fig. 2. Plot of U’s for lognormal POD, m1 ¼ 9, s1 ¼ 0.8.

4 −4

x 10

88

J. Garza, H. Millwater / International Journal of Pressure Vessels and Piping 92 (2012) 84e95

The sensitivities follow a normal distribution; therefore, the 95% confidence bounds can be computed as

pffiffiffiffi pffiffiffiffi vPf =vq  1:96 V  vPf =vq  vPf =vq þ 1:96 V

Probability of Detection Curves 1 POD2

0.9

(21)

0.8

The variance and confidence bounds can be obtained from a single analysis.

0.7 POD1

0.6

Two examples are presented to demonstrate and verify the methodology. In both cases, the probabilistic sensitivities computed using the equations derived here are compared against finite difference estimates. Finite difference estimates require multiple probabilistic analyses with a separate small perturbation and analysis for each parameter in the POD curves. In addition, the number of samples required to accurately compute small differences in the POF is typically very large; usually an order of magnitude or more larger than the number needed to compute the POF and to compute the sensitivities using the methods presented here.

POD

4. Numerical examples

0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Crack Size (m)

1 −3

x 10

Fig. 3. POD curves, lognormal format, m1 ¼ 9, s1 ¼ 0.8 and m2 ¼ 11, s2 ¼ 0.8.

4.1. Closed-form example An academic example is presented to demonstrate the methodology. The problem is fictitious, but serves to demonstrate the elements of the method using an easy to understand problem. The problem consists of an edge through crack growing in a semiinfinite rectangular plate under constant amplitude loading, Ds ¼ 675 MPa and the R ratio is zero, i.e., R ¼ smin/smax ¼ 0. The initial crack size and material properties are representative of titanium, however, the POD curve is contrived. Four random variables are considered: initial crack size (lognormal), Paris crack growth constants log10(C) and m (correlated normal) and fracture toughness (normal). The problem parameters are given in Table 1. The random variable vector x consists of x˛ai,C,m,KIC, and the y vector of random variables consists of y˛ai,C,m. For this crack geometry using the Paris crack growth law, the crack size as a function of cycles a, the critical crack size ac, the cycles-to-failure tf, the crack size aq at any time of inspection tq, and the limit state g(x,t) can be determined in closed form as shown below.

da=dt ¼ CðDKÞm

aq ¼

(27)

i

gðx; tÞ ¼ tf  tf 0

(28)

where tf0 is a specified number of cycles. The POD curve is modeled as a lognormal distribution as shown in Eq. (17) with Um and Us given by Eqs. (18) and (19). Inspections are simulated at 10,000 and 20,000 cycles using the lognormal POD with m1 ¼ 9, s 1¼ 0.8 for inspection 1 and m2 ¼ 11, s2 ¼ 0.8 for inspection 2. Fig. 3 plots the POD curves for the two inspections. In this example, q1 ¼ ½m1 ; s1  and q2 ¼ ½m2 ; s2 . 4.1.1. Single inspection A single inspection is simulated at 10,000 cycles using parameters m1 ¼ 9, s1 ¼ 0.8. The POF as a function of cycles with and without inspection is shown in Fig. 4. The sensitivity of the POF

(22)

Inspection at 10,000 Cycles 1

(23)

pffiffiffiffiffiffiffiffiffi KIc ¼ 1:12smax paC

(24)

pffiffiffi2   ac ¼ KIc = 1:12smax p

(25)

2  pffiffiffim ðm  2ÞC 1:12Ds p

1

1  ðm2Þ=2 ðm2Þ=2 ai ac

! (26)

No Inspection With Inspection

0.9 0.8

Probability−of−Failure

pffiffiffiffiffiffi DK ¼ 1:12Ds pa

tf ¼

 pffiffiffiffim !2=ð2mÞ ðm  2ÞC 1:12Ds p  tq ðm2Þ=2 2 a 1

0.7 0.6 0.5 0.4 0.3 0.2

Table 1 Example 1 random variables. Random variable pffiffiffiffiffi KIC ðMPa mÞ ai (mm) Crack growth, log10(C) Crack growth, m

0.1

Mean

St. Dev.

Distribution type

55 15.1 11.8 3.81

5.5 8.48 0.157 0.146

Normal Lognormal Correlated normal rCm ¼ 0.9751

0

0

1

2

3

4

5

Cycles

6

7

8

9

10 4

x 10

Fig. 4. Single inspection case: POF without inspection and with an inspection at 10,000 cycles.

J. Garza, H. Millwater / International Journal of Pressure Vessels and Piping 92 (2012) 84e95

vPf/vs is accounted for by the sign change in Us, as shown in Eq. (19). Crack sizes that fail soon after inspections are large cracks and, therefore, Us is positive for these cracks. Conversely, crack sizes that fail later are small at the time of inspection which lead to a negative value of Us. At about 18,000 cycles, the positive and negative values of Us balance and the sensitivity is approximately zero. For subsequent times, the sum of Us becomes negative indicating that an increase in s1 will cause a decrease in the POF at later times.

POF Sensitivity wrt POD Parameter μ

1

0.14

0.12

MCS Simulations Finite Difference

∂Pf / ∂μ

1

0.1

0.08

0.06

0.04

0.02

0

0

1

2

3

4

5

6

7

8

9

Cycles

10 4

x 10

Fig. 5. Single inspection case: POF sensitivity with respect to m1.

with respect to m1 and s1 as a function of cycles is given in Figs. 5 and 6, respectively, with the solid (red) line denoting the results using the equations developed here and finite difference results denoted by the dotted (black) points. One million samples were used for the calculation. The results indicate a very good agreement with the finite difference solutions verifying the accuracy of the methodology. Fig. 5 shows that the sensitivity with respect to m1 is always positive; an increase in m1 will increase the POF. This is because the POD becomes less sensitive as the mean is increased, e.g., compare the POD curves in Fig. 3. Fig. 6 shows that the sensitivity with respect to s1 is first positive up to about 18,000 cycles, then negative. The implications are that for a short time period, approximately 8000 cycles, the POF increases when the standard deviation of the POD curve increases, then the POF decreases subsequently. Thus, one would need to sacrifice some short-term increase in risk in order to gain significant reduction in POF over the long term. This change in sign can be explained qualitatively as follows. If Us increases, the POD curve is better at detecting smaller cracks (ln(a) < m) but worse at detecting larger cracks (ln(a) > m). This relationship and the change in sign of POF Sensitivity wrt POD Parameter σ1 0.02 MCS Simulations Finite Difference

0

∂Pf / ∂σ1

−0.02

−0.04

−0.06

−0.08

−0.1

0

1

2

3

4

5

6

7

8

Cycles Fig. 6. Single inspection case: POF sensitivity with respect to s1.

9

89

10 4

x 10

Fig. 7. Single inspection case: histogram of U’s and crack size.

90

J. Garza, H. Millwater / International Journal of Pressure Vessels and Piping 92 (2012) 84e95 0.14

Coefficient of Variation 0.4

1 Sample Data Equation Estimate

0.12

c.o.v. ∂P / ∂μ 1 f

f

∂P / ∂μ 1

0.08

0.06

∂Pf / ∂σ1

0.2

0

c.o.v. ∂Pf / ∂σ1

1E3 1E4 1E5 1E6

0.1

0.04

0.02 ∂P / ∂μ f

0 0

1

2

3

4

5

6

7

8

9

0

10 4

Cycles

0

1

2

4

x 10

vPf ð12; 000Þ=vs is positive. For tf0 ¼ 15,000, both negative and positive Us values exist. For tf0 ¼ 50,000, clearly the large majority of Us values are negative, hence, a negative.

vPf ð50; 000Þ=vs It is also obvious that the cracks sizes at the time of inspection become smaller as time increases. Convergence of the sensitivity with respect to the number of samples is shown in Figs. 8 and 9. The results indicate that good convergence is achieved at approximately 10,000 samples. Using the variance estimate give in Eq. (20) and the estimated sensitivities, the coefficient of variances (COV ¼ standard deviation/mean) is compared from a single analysis against the result obtained by

−1 10

8

4

Cycles

Fig. 8. Single inspection case: Convergence of vPf(t)/vm1 with respect to number of samples.

Fig. 7 shows histograms of values of crack sizes of Us (blue bars) and the crack sizes (red points) at the time of inspection for failures that occur after 10,000 cycles but before 12,000, 15,000 and 50,000 cycles, respectively. The histograms clearly show that failure at early times (tf0 ¼ 12,000 cycles) is due to large crack sizes leading to positive Us values. The net result is that

6

x 10

Fig. 10. Coefficient of variation of sensitivities.

rerunning the sensitivity equation (1000) times, each time with a different sequence of random numbers. The comparisons are presented in Fig. 10. The results show a good correspondence, thus verifying the variance equation. 4.1.2. Two inspections Two inspections are simulated at 10,000 and 20,000 cycles with m1 ¼ 9, s1 ¼ 0.8 for inspection 1 and m2 ¼ 11, s2 ¼ 0.8 for inspection 2. Inspection 2 is significantly more sensitive than inspection 1, see Fig. 3. Analyses using both “different” and the “same” POD curves used for the inspections were considered. 4.1.2.1. Different POD curves. The POF as a function of the number of cycles with and without inspection is shown in Fig. 1. The sensitivities for m1, s1, m2, s2 and finite difference estimates using one million samples are shown in Figs. 11 and 12, respectively. The sensitivities are in good agreement with finite difference results. The sensitivity with respect to m1 or m2 is monotonic and always positive, indicating that a decrease in m always decreases the POF. This is to be expected as a decrease in m shifts the POD curve to the left and increases its sensitivity. POF Sensitivity wrt POD 0.15

0.02 1E3 1E4 1E5 1E6

0

0.1

1 f

∂P / ∂σ

μ2

∂P / ∂μ f q

−0.02

−0.04

μ1

0.05 −0.06

−0.08

0 −0.1

0

1

2

3

4

5

Cycles

6

7

8

9

10 4

x 10

Fig. 9. Single inspection case: Convergence of vPf(t)/vs1 with respect to number of sample.

0

1

2

3

4

5

Cycles

6

7

8

9

10 4

x 10

Fig. 11. Multiple inspections, different POD curves: POF sensitivities with respect to m1 and m2.

J. Garza, H. Millwater / International Journal of Pressure Vessels and Piping 92 (2012) 84e95

comprises one flight. The RPM values were then used to scale a local stress reference value using a reference RPM value as

POF Sensitivity wrt POD 0.08 0.06

sðxÞApplied ¼

σ

0.04

91

RPM RPMref

!2

sðxÞref

(29)

2

f

∂P / ∂σ

q

0.02 0 −0.02 σ1

−0.04 −0.06 −0.08 0

1

2

3

4

5

Cycles

6

7

8

9

10 4

x 10

Fig. 12. Multiple inspections, different POD curves: POF sensitivities with respect to s1 and s2.

As with the single inspection case, the sensitivity with respect to

s1 is first positive than negative. The reason is the same as in the single inspection case: an increase in s improves the detection of smaller cracks but decreases the ability to detect larger cracks. In the case of s2, the sensitivity rises then decreases but remains positive; thus an increase in s2 increases the POF for all t. These results indicate that it may not always be intuitive as to even the sign of the effect on the POF due to a change in a POD parameter. Fig. 13 shows a histogram of Us values (blue bars) and the crack sizes (red points) at the time of the second inspection for failures that occur after 20,000 cycles but before 30,000, 40,000 and 80,000, respectively. At 30,000 cycles, values for Us are all positive. At 40,000 cycles, some negative values of Us occur but the sum total is still positive. At 80,000 cycles, a large percentage of negative Us values occur but still not enough to ensure a negative value of vPf/vs2. Most likely, if the simulation was continued further, at some time vPf/vs2 would become negative however the change in vPf/vs2 with respect to time is very slow, see Fig. 12. Convergence of the sensitivities with respect to the number of samples is shown in Figs. 14 and 15, respectively. Convergence occurs after approximately ten thousand samples. 4.1.2.2. Same POD curves. The POF as a function of the number of cycles with and without inspection is shown in Fig. 16. When the same POD curve is used at both inspections m1 ¼ m2 ¼ 9 and s1 ¼ s2 ¼ 0.8. The sensitivities for m, s, and finite difference estimates are shown in Figs. 17 and 18, respectively. The sensitivities are in good agreement with finite difference results. As before, the sensitivities with respect to s show initially a positive value then a negative value as time increases. Convergence of the sensitivities with respect to the number of samples is shown in Figs. 19 and 20, respectively. Convergence is achieved at approximately 10,000 samples. 4.2. Inspection of an engine disk This example computes the sensitivity of the POF with respect to the POD parameters of a defect located at a bolt hole in a gas turbine engine disk. The properties used are for titanium and are shown in Table 1. The loading consisted of 70 load pairs derived from a rainflow analysis of revolutions per minute (RPM) data from a flight data recorder, see Fig. 21. Application of these loads

Fig. 13. Multiple inspections, different POD curves: histogram of U’s and crack sizes for 2 inspections.

92

J. Garza, H. Millwater / International Journal of Pressure Vessels and Piping 92 (2012) 84e95

POF Sensitivity wrt POD Parameter μ1

0.16 μ

0.3

2

MCS Simulations Finite Difference

0.14 μ

0.25

1

0.12

0.2

f

0.08

1

1E3 1E4 1E5 1E6

∂P / ∂μ

f

∂P / ∂μ q

0.1

0.06

0.1

0.04 0.02 0

0.15

0.05 0

1

2

3

4

5

6

7

8

9

Cycles

10

0

4

x 10

Fig. 14. Multiple inspections, different POD curves: Convergence of vPf(t)/vm1 and vPf(t)/vm2.

Fig. 15. Multiple inspections, different POD curves: Convergence of vPf(t)/vs1 and vPf(t)/vs2.

0

1

2

4

5

6

7

8

9

Cycles

10 4

x 10

Fig. 17. Multiple inspections, same POD curves: POF sensitivity with respect to m1 ¼ m2 ¼ 9.

where s(x)ref denotes a stress dieout away from the hole and x denotes the distance into the disk from the bolt hole. The stress dieout was determined from finite element analysis and is shown in Fig. 22. The analysis applied the stress history for multiple flights until fracture and repeated the process for each Monte Carlo sample to determine the POF. A semi-circular surface crack was grown from the edge of the hole using the variable amplitude stress field described by Eq. (29) until fracture as determined by the stress intensity factor exceeding the fracture toughness. The POF up to 20,000 cycles was computed. A POD curve representative of an eddy current inspection (m1 ¼ 7.16, s1 ¼ 0.416) was applied at 16,000 cycles using the lognormal model, Eq. (17). Fig. 23 shows the POF with and without inspection as a function of the cycles with 95% confidence intervals. The probabilistic sensitivities vPf/vm, and vPf/vs are shown in Figs. 24 and 25. Included in the figures are the 95% confidence bounds obtained from Eq. (21) and the finite difference results; the finite difference results are clearly contained within the confidence bounds. 100,000 POF Sensitivity wrt POD Parameter σ1

Inspection at 10,000 and 20,000 Cycles 1

0.02

No Inspection With Inspection

0.9

3

MCS Simulations Finite Difference

0 −0.02

0.7

−0.04 1

0.6

∂P / ∂σ

0.5

f

Probability−of−Failure

0.8

0.4

−0.06 −0.08

0.3 −0.1

0.2

−0.12

0.1 0 0

1

2

3

4

5

Cycles

6

7

8

9

10 4

x 10

Fig. 16. Multiple inspections, same POD curves: POF without inspection and with two inspections at 10,000 and 20,000 cycles.

−0.14

0

1

2

3

4

5

Cycles

6

7

8

9

10 4

x 10

Fig. 18. Multiple inspections, same POD curves: POF sensitivity with respect to s1 ¼ s2 ¼ 0.8.

J. Garza, H. Millwater / International Journal of Pressure Vessels and Piping 92 (2012) 84e95

0.25

93

Stress Dieout

1 0.95 1E3 1E4 1E5 1E6

0.15

0.9

Normalized Stress

∂P / ∂μ f 1

0.2

0.1

0.05

0.85 0.8 0.75 0.7 0.65 0.6

0

0

1

2

3

4

5

6

7

8

9

Cycles

0.55

10 4

x 10

0.5

Fig. 19. Multiple inspections, same POD curves: Convergence of vPf(t)/vm1.

0

0.05

0.1

0.15

0.2

0.25

Distance from Hole (m) Fig. 22. Stress dieout from hole.

0.02 1E3 1E4 1E5 1E6

0

−3

6

−0.02

Inspection at 16,000 Cycles

x 10

No Inspection With Inspection 95%LB 95%UB

5

Probability−of−Failure

f

∂P / ∂σ

1

−0.04 −0.06 −0.08 −0.1

4

3

2

−0.12 −0.14

0

1

2

3

4

5 Cycles

6

7

8

9

10

1

4

x 10

Fig. 20. Multiple inspections, same POD curves: Convergence of vPf(t)/vs1.

0 1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

Cycles

4

Fig. 23. Single inspection case for hole in disk: POF without inspection and with an inspection at 16,000 cycles.

Variable Amplitude Spectrum 900 800

POF Sensitivity wrt POD Parameter μ1

−3

3

x 10

MCS Simulations 95%LB 95%UB Finite Difference

700

2.5

600

2

500

∂Pf / ∂μ1

Stress [MPa]

2 x 10

400 300

1.5

1 200

0.5

100 0

0 0

20

40

60

80

100

Load Steps Fig. 21. Variable amplitude loading.

120

140

1

1.1

1.2

1.3

1.4

1.5

Cycles

1.6

1.7

1.8

1.9

2 4

x 10

Fig. 24. Single inspection case for hole in disk: POF sensitivity with respect to m1.

94

J. Garza, H. Millwater / International Journal of Pressure Vessels and Piping 92 (2012) 84e95

POF Sensitivity wrt POD Parameter σ1

−4

x 10

8

MCS Simulations 95%LB 95%UB Finite Difference

7 6

∂Pf / ∂σ1

samples were used for the sensitivities using the equations developed here and 4.5 M samples were needed to obtain convergence in the finite difference results. As expected, the sensitivity with respect to m1 was always positive. The sensitivity with respect to s1 is positive over the cycle range of interest although the sensitivities are clearly approaching a negative value as demonstrate in the first example. Figs. 26 and 27 show the convergence of the sensitivities as a function of the number of samples. The results have clearly converged at 100,000 samples.

5 4 3

5. Conclusions 2 1 0 1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2 4

Cycles

x 10

Fig. 25. Single inspection case for hole in disk: POF sensitivity with respect to s1.

−3

x 10

3

1E4 1E5 1E6

2.5

f

∂P / ∂μ

1

2

1.5

1

0.5

0 1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

Cycles

2 x 10

x 10

To demonstrate implementing multiple inspections, consider two inspections, POD1(q1,a(y,t1)) and POD2(q2,a(y,t2)) implemented at times t1 and t2, respectively. The derivations below address the time interval t > t2 since the form of the sensitivity for 0 < t < t2 (single inspection case), is already given by Eq. (14). In this case the POD’s are different at each inspection, POD1 s POD2, and therefore q1 s q2. The equation that defines the sensitivity with respect to the parameters q1 is unchanged from the single inspection case, Eq. (14), for any number of subsequent inspections. Note, however, that the numerical evaluation of vPf/vq1 may change since subsequent inspections may remove crack sizes that would have failed otherwise and, therefore, contributed to vPf/vq1. The POF after the second inspection (t > t2) is

1E4 1E5 1E6

7 6 5 1

This research was supported by the Materials and Manufacturing Directorate, Air Force Research Laboratory, Wright-Patterson AFB, under Contract no. FA8650-07-C-5060, Patrick Golden AFRL/RXLMN program manager. Appendix A. Different POD curves

−4

∂Pf / ∂σ

Acknowledgment

4

Fig. 26. Single inspection case for hole in disk: Convergence of vPf(t)/vm1 with respect to number of samples.

8

NDE methods play a critical role in fracture control plans. The efficacy of the NDE methods is embodied in the POD curve that relates the probability of detecting a defect versus the defect size. A methodology has been developed and demonstrated that will compute the sensitivity of the POF with respect to the parameters of the POD curve. These sensitivities provide a convenient and low cost method to assess the sign and magnitude of potential changes in the POF with respect to variations in the parameters describing the POD curve. The formulation is generic and not limited to any specific random variables, fracture mechanics formulation, or any specific POD curve as long as the POD is modeled parametrically. Variance estimates were also derived to assess the quality of the sensitivities. The key to the methodology is that the same samples used to compute the POF are reused to compute the sensitivities. As such, the methodology can be implemented in a post-processing nonintrusive manner thereby facilitating implementation with existing or commercial codes; the methodology only needs the crack sizes at the times of inspection and the cycles-to-failure for each realization. The resulting equations are remarkably simple and involve an evaluation and summation of a U function that is derived from the POD curve. Several numerical examples demonstrated the accuracy of the methodology as compared against much costlier finite difference results.

4 3 2 1 0 −1 −2 1

1.1

1.2

1.3

1.4

1.5

Cycles

1.6

1.7

1.8

1.9

2 4

x 10

Fig. 27. Single inspection case for hole in disk: Convergence of vPf(t)/vs1 with respect to number of samples.

Pf ðtÞ¼

ZN Iðx;tÞCPOD1 ðq1 ;aðy;t1 ÞÞCPOD2 ðq2 ;aðy;t2 ÞÞfx ðxÞdx t>t2 N

(A1)

J. Garza, H. Millwater / International Journal of Pressure Vessels and Piping 92 (2012) 84e95

The sensitivity of the POF with respect to q2 can be derived by differentiating Eq. (A1) yielding

vPf ðtÞ ¼ vq2

ZN N

t>t2

(A2)

t>t2

(A3)

t  t2 t>t2

(A4)

where N2 denotes the samples that have not failed before t2 nor detected by the first or second inspections, and N is the total number of samples. Similar to the numerical evaluation of vPf/vq1, the numerical value of vPf/vq2 may change since subsequent inspections may remove crack sizes that would have failed otherwise and, therefore, contributed to vPf/vq2. Generalizing, the sensitivity of the POF with respect to qq can be written

8 0 > vPf ðtÞ < X Nq    z 1 Iðxi ; tÞUq qq ; a yi ; tq > vq q :N i¼1

t  tq t>tq

(A5)

where Nq denotes the samples that reach inspection q (have not failed before tq nor are detected by any previous inspections). Remarkably, this equation is the same form as Eq. (14) for a single inspection. The summation in Eq. (A5) is divided by N not Nq to account for the conditional probability of the expected value operator and to provide the sensitivity with respect to the original samples N. Appendix B. Same POD curves The POF after the second inspection is

ZN Pf ðtÞ ¼

Iðx; tÞCPODðq; aðy; t1 ÞÞCPODðq; aðy; t2 ÞÞfx ðxÞdx t>t2

-N (B1) The POF sensitivity with respect to parameter q can be calculated by taking the partial derivative of Eq. (B1) yielding

vPf ðtÞ ¼ vq

ZN Iðx;tÞ N

v½CPODðq;aðy;t1 ÞÞCPODðq;aðy;t2 ÞÞ fx ðxÞdx t>t2 vq (B2)

Using the product rule

vPf ðtÞ ¼ vq

ZN N

N



Uðq; aðy; t1 ÞÞþ CPODðq; aðy; t1 ÞÞ Uðq; aðy; t2 ÞÞ

¼ Ef CPOD1 CPOD2 ½Iðx; tÞfUðq; aðy; t1 ÞÞ þ Uðq; aðy; t2 ÞÞg t>t2

(B4) (B5)

N2 vPf ðtÞ 1 X z Iðx; tÞfUðq; aðyi ; t1 ÞÞ þ Uðq; aðyi ; t2 ÞÞg t>t2 N vq i¼1

(B6)

For two inspections, the sensitivity vPf/vq is estimated as,

The expected value operator can be approximated as

8 0 > vPf ðtÞ < X N2 z 1 Iðxi ; tÞU2 ðq2 ; aðyi ; t2 ÞÞ > vq2 :N i¼1

Iðx; tÞ

The expected value operator can be approximated using sampling to yield

where U is given by Eq. (12). Applying the expectation operator to Eq. (A2) yields,

vPf ðtÞ ¼ Ef CPOD1 CPOD2 ½Iðx; tÞU2 ðq2 ; aðy; t2 ÞÞ vq2



ZN

 CPODðq; aðy; t2 ÞÞfx ðxÞdx

Iðx; tÞU2 ðq2 ; aðy; t2 ÞÞCPOD1 ðq1 ; aðy; t1 ÞÞ

 CPOD2 ðq2 ; aðy; t2 ÞÞfx ðxÞ dx

vPf ðtÞ ¼ vq

95

8 9 vCPODðq;aðy;t1 ÞÞ > < = CPODðq;aðy;t2 ÞÞþ > vq Iðx;tÞ fx ðxÞdx > : vCPODðq;aðy;t2 ÞÞ CPODðq;aðy;t ÞÞ > ; 1 vq (B3)

Multiplying by ðCPODðt1 Þ$CPODðt2 ÞÞ=ðCPODðt1 Þ$CPODðt2 ÞÞ, using the definition of U, and factoring yields

8 0 tt1 > > > N1 > 1X > > Iðx;tÞUðq;aðyi ;t1 ÞÞ t1 N2 vq > > 1X > > > Iðx;tÞfUðq;aðyi ;t1 ÞÞþUðq;aðyi ;t2 ÞÞg t>t2 : N

(B7)

i¼1

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