reliability-based design and inspection scheduling optimization of ...

RELIABILITY-BASED DESIGN AND INSPECTION SCHEDULING OPTIMIZATION OF AN AIRCRAFT STRUCTURE CONTAINING MULTIPLE SITE DAMAGE

Erdem Acar1, Amit A. Kale2, Mehmet A. Akgün3 1,2 Research Assistants, University of Florida, Gainesville, FL 32611-6250, USA 3 Professor, Middle East Technical University,Ankara, 06531, TURKEY Abstract:

In this paper, design and inspection schedule optimization of an aircraft structure containing multiple site damage is performed using reliability based fatigue crack growth calculations. Fatigue crack growth is modeled with Walker Equation in order to take the effect of stress ratio into account. Failure of the structure is modeled using net ligament yield criterion in which the failure is predicted to occur when the plastic zones of the cracks touch each other. The structure is chosen as an unstiffened panel with two cracked holes that represent a row of cracks. A set of two-cracked panel configurations is modeled using the finite element software, MARC. The results of the finite element analyses are used to formulate geometry factors for stress intensity calculations. The geometry factor for each crack is formulated in terms of the lengths of the two cracks by fitting quadratic functions to the stress intensity factor results. Probability of failure of a panel is calculated using First Order Reliability Method (FORM), in which the initial crack lengths, Walker Equation constants and the applied load are taken as random variables. Optimization of design and inspection schedule of panels is accomplished by minimizing the life cycle cost associated with material and manufacturing, fuel consumption and inspection under the constraint of a pre-specified level of safety.

Key words:

Multiple Site Damage, Reliability-Based Design Optimization, Inspection Scheduling, First Order Reliability Method

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1.

Erdem Acar, Amit A. Kale and Mehmet A. Akgün

INTRODUCTION

One of the main causes reducing the residual strength of aircraft structures is the simultaneous fatigue cracking at various locations, which is referred to as Multiple Site Damage (MSD). The structural elements such as wings or the fuselage develop cracks along the rows of fastener holes due to fatigue. MSD plays a significant role in the life of those components, since it reduces the overall structural integrity more compared to a single crack case. When multiple cracks are close to each other their growth rates interact. Therefore, inspection of potentially critical locations in order to ensure a pre-specified level of safety is a serious required. Periodical inspections of critical parts are accomplished with suitable non-destructive evaluation (NDE) techniques. MSD designates the occurrence of multiple cracks of arbitrary length that generally occur along rows of fastener holes in the fuselage or wings of an aircraft. Cracks caused by MSD are difficult to detect and severely reduce the residual strength, fatigue life and overall integrity of an aircraft structure. Several researchers investigated the estimation of residual strength of aircraft panels with MSD. Swift [1] proposed ligament yield criterion, an analytical method that takes plasticity and crack interaction effects into account. According to the ligament yield criterion the stress level that will cause the plastic zones of two cracks touch each other is the residual strength of a panel. K-apparent criterion developed by Mar [2] states that there exists an effective fracture toughness (K-apparent) for a thin panel which permits more yielding compared to that under the plane strain condition. Davicki [3] also used fracture mechanics techniques to predict the residual strength of panels containing MSD. Jeong and Brewer [4] proposed two criteria, average displacement criterion and average stress criterion, to predict the residual strength of a panel. They assumed that the stress in the ligament between the two cracks is uniform and equal to the ultimate strength of the material prior to the failure of the panel. Cherry et al. [5] tested unstiffened aluminum panels containing MSD to determine their residual strengths and found out that the ligament yield criterion produced the most accurate predictions. Therefore, the ligament yield criterion is used here to model the failure of the panels with MSD. Various researchers studied probabilistic analysis of MSD. Cawley et al. [6] used constituent particles as a basis for estimation of distribution of fatigue cracks in MSD analysis. Xiong et al. [7] established a lognormal distribution for crack initiation based on experimental data. Pieracci and Mengali [8] proposed a statistical model to assess the probability of

Reliability-based design and inspection scheduling optimization of an aircraft structure containing multiple site damage

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occurrence of MSD. However, these papers do not address the use of inspections in the reliability analysis of aircraft structures with MSD. Inspection of the critical locations of an aircraft structure is necessary to ensure the safety of the structure; the appropriate NDE techniques are used for this purpose. However, these inspection techniques also comprise uncertainties that should be represented by probabilistic models. Various authors suggested different probability models to represent probability of detection of the inspection techniques. Among those models, the probability model used by Palmberg et al. [9] represents the probability of detection in terms of damage size by a simple equation that is easy to apply in probabilistic models. Kale et al. [10] performed the design and inspection schedule optimization of a fuselage panel containing a single crack using a probabilistic fatigue damage model. In this paper, we consider the structural component as an unstiffened panel with two holes (two cracks), and represent a row of cracks with symmetry boundary conditions. In order to take the interaction between the cracks into account, we modeled a set of two-cracked panel configurations ® by using the finite element software, MARC . The geometry factors are then formulated as functions of the lengths of both of the cracks. The ligament yield criterion is used to define the limit state function for the fatigue reliability calculations of the panels containing MSD. The fatigue crack growth is modeled using Walker equation which is a modified form of Paris Law taking the effect of stress ratio into account. First Order Reliability Method (FORM) is utilized to perform fatigue reliability calculations and the optimization of design and inspection scheduling of the panels is performed.

2.

MULTIPLE SITE DAMAGE

Several different criteria are proposed by researchers to predict the residual strength of panels containing multiple site damage (MSD). Cherry et al. [5] investigated various criteria by testing aluminum panels with MSD and concluded that the most accurate criterion for residual strength prediction of the panels is ligament yield criterion proposed by Swift [1]. Ligament yield criterion is used here to define the failure of panels. The ligament yield criterion predicts that the failure of panels occurs when the plastic zones of two cracks meet each other. Defining R1 and R2 as the plastic zone radius of the two cracks propagating towards each other and t as the distance between the crack tips (see Fig. 1), the failure condition can be expressed as

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Erdem Acar, Amit A. Kale and Mehmet A. Akgün

t = 2 R1 + 2 R2

(2.1)

Figure 1. The panel containing two cracks The radius of the plastic zone of cracks can be approximated using Irwin’s plastic zone formula as

1 Ri = 2π

⎛ Ki ⎞ ⎟ ⎜ ⎜ σ flow ⎟ ⎠ ⎝

2

(i = 1, 2)

(2.2)

where Ri is the plastic zone radius, Ki is the stress intensity factor at the tip of the cracks, and σ flow is the flow stress defined as the average of the tensile yield and ultimate stresses. Combining Eq.s 2.1 and 2.2 and simplifying, the criterion for failure can be expressed as K 12 + K 22 = π t σ 2flow

(2.3)

The stress intensity factors can be calculated through Eq. 2.4 as

K i = β i (a1 , a 2 ) σ π ai

(i = 1, 2)

(2.4)

where the geometry factors β1 and β 2 are functions of the two crack lengths a1 and a2 due to the interaction between the cracks, and σ is the stress in the panel. The geometry factors are formulated as functions of the lengths of two cracks, a1 and a2, by modelling a set of two-cracked panel ® configurations using the finite element software, MARC . Quadratic functions are fitted to the stress intensity factor results obtained from finite element analyses, which give

Reliability-based design and inspection scheduling optimization of an aircraft structure containing multiple site damage

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β 1 ( a1 , a 2 ) = 1.272 − 34.17 a1 − 75.44 a 2 + 4994 a1 a 2 + 1027 a12 + 4556 a 22 (2.5) β 2 (a1 , a 2 ) = β 1 (a 2 , a1 )

where a1 and a2 are in meters. The effect of holes is also included in the analysis. For cracks shorter than a size of “d”, where d is a fraction of the hole radius as given below, the growth is assumed to be dominated by notch-induced stress concentrations. Otherwise, crack growth is assumed to be controlled by nominal far-field stresses [11]. Hence,

(K I )a −r d

= K t ⋅ K I (a − r ) = K I (a)

(2.6)

where d ≈ 0.13 r , Kt is the stress concentration factor and r is the hole radius. In this formulation, we refer to Fig. 1 where the crack length a includes the hole radius. For an unstiffened panel the nominal stress in the panel is calculated by the simple formula

σ=

P wt p

(2.7)

where P is the applied load, w and tp are the panel width and thickness, respectively.

3.

FATIGUE LIFE CALCULATION AND LIMIT STATE FUNCTION

In this work, panels from a wing lower surface are considered. A stress spectrum for wing lower surface of a transport aircraft is given in the paper by Chang [14]. Since the given stress values are associated with that particular aircraft, we scaled that spectrum and used it. The rainflow cycle counting method is used to count cycles in the load spectrum which is then represented with an equivalent cycle such that one cycle corresponds to one flight. The cycle equivalence method used is presented in Appendix 1.

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Erdem Acar, Amit A. Kale and Mehmet A. Akgün

For the prediction of the fatigue life of the panels Walker equation is used. Walker equation is essentially a modified form of the Paris Law and takes the effect of stress ratio into account. It is given by ⎧⎪(1 − R ) m for R > 0 da i = C (z K max,i ) p , z ≡ ⎨ dN ⎪⎩1 − µR for R ≤ 0

(i = 1,2)

(3.1)

where dai/dN is the fatigue crack growth rate, C, m and p are the material properties and K max,i = β (a1 , a 2 ) σ max,i π ai . All aluminum alloys may be assumed to have the same m value of 0.6 and µ may be taken to have a value of 0.1. The material parameter C can be expressed in terms of the parameter n. For 7075-T651 aluminum alloy the relation is (when unit of stress and crack length is MPa and meters, respectively) C = 14.09 × 10 −10 − (3.768 × 10 −10 ) p

(3.2)

The successive integration of Eq. 3.1 for crack lengths a1 and a2 from initial values to the critical values that will correspond to the failure condition gives the number of cycles to failure. For probabilistic analysis of the problem, we first formulate the limitstate of the problem from the failure condition. Here the condition

π t σ 2flow − K 12 − K 22 ≤ 0

(3.3)

defines the limit-state of the problem. The variables defining the limit-state of the problem are initial crack lengths (a1)0 and (a2)0, material parameter p, and the applied load P acting on the panel. Lognormal probability density functions (pdf) are used for these parameters as is common [7, 10]. The distribution parameters for these variables are given in Table 1. Table 1. Values of the distribution parameters for the random variables

Random variable (a1)0 (mm) (a2)0 (mm) p Pmax (MN/m)

4.

Mean 0.2 0.2 2.97 0.1175

Standard Deviation 0.07 0.07 1.05 20 % of the mean value

RELIABILITY ANALYSIS WITH NDI AND INSPECTION SCHEDULING

In order to ensure a pre-specified level of safety during the service life, structural components are subjected to non-destructive inspection (NDI).

Reliability-based design and inspection scheduling optimization of an aircraft structure containing multiple site damage

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The reliability of an inspection technique is usually described in terms of its detection. The capability of the NDI technique mainly depends on the crack size and probability of detection (POD) increases as crack size increases. An extensive investigation is reported by Lewis et al. [12] that includes 22 000 inspections conducted on 174 cracks by 107 different inspectors. Berens and Honey [13] performed a statistical evaluation by employing regression analysis to fit seven different functional forms for the POD curve to the data supplied by Lewis et al. [12] and reported that the best mathematical model for POD is

POD(a ) =

α aβ 1+ α aβ

(4.1)

where α and β are regression parameters that depends on the type of NDI technique used, and a is the crack size. This model is used by Palmberg et al. [9] in probabilistic damage tolerance calculation of a panel with regression parameters α=0.0032 mm-β and β= 3.5. In-service inspections are performed to detect cracks before they grow up to critical lengths. For inspection scheduling of components with a single crack, Harkness et al. [14] proposed an augmented form of First Order Reliability Method (FORM). The most advantageous part of their approach is that the crack size distribution is not needed to be updated. The method they proposed assumes that the components with detected cracks are repaired such that their subsequent effect on probability of failure is negligible. Although this assumption yields unconservative failure probability estimations, we decided to use and explore this method since the implementation of the method is easy and computationally less expensive. The detailed explanation of the calculation of probability of failure with NDI is given in Appendix 2. The inspections are scheduled such that when the probability of failure of a panel increases such that safety drops to the target level, an inspection is conducted to reduce the probability of failure in the next flight by reducing the uncertainty in the crack length and providing repairs and replacements for the detected cracks.

5.

SIMULTANEOUS OPTIMIZATION OF DESIGN AND INSPECTION SCHEDULING

The general form of a reliability-based optimization problem may be expressed as follows:

8

Erdem Acar, Amit A. Kale and Mehmet A. Akgün

min Cost st Pf ≤ ( Pf ) t arg et

(5.1)

where Cost is the life cycle cost associated with material and manufacturing, fuel consumption and inspection. Pf is the probability of failure and (Pf )target is the pre-specified level of safety. Using the values given in the paper by Kale et al. [10], the life cycle cost is expressed as

Cost = M C W + FC W N f + N i I C

(5.2)

where MC is the material and manufacturing cost per pound, W is the fatigue controlled structural weight in pounds, FC is fuel cost per pound per flight, Nf is the service life in number of flights, Ni is the number of inspections and IC is the inspection cost for all panels including repair and replacement costs. The values of cost factors are given in Table 2. In the optimization problem stated above, the design variable is the panel thickness that changes the probability of failure by changing the stress level and also changes the structural weight according to

W =

t Wb n p

(5.3)

tb

where Wb is the baseline weight per panel, np is the number of panels and tb is the baseline thickness. Table 2. Parameters Used in Cost and Weight Calculation Material and Manufacturing Cost per Pound (MC) Fuel Cost per Flight (FC) Inspection Cost (IC) Service Life (Nf) Baseline Weight per Panel(Wb) Number of Panels (np) Baseline Thickness (tb)

6.

$ 150 $ 0.015 $ 1,000,000 40,000 flights 11.67 lb 200 1.75 mm

RESULTS

We first investigate the effects of uncertainties in material properties and initial crack lengths on the panel design; hence the random variables are

Reliability-based design and inspection scheduling optimization of an aircraft structure containing multiple site damage

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(a1)i, (a2)i and p. When panel design is performed with a safe-life approach, no inspections are necessary. In that case the desired level of safety (LoS) for panels can be attained by simply increasing the panel thickness. The panel thickness required and the cost results for safe-life design are given in Table 3. Table 3. Safe-Life Design Thickness for three different levels of safety (random parameters are (a1)i, (a2)i and n)

Level of Safety (LoS) 10-7 10-8 10-9

Panel Thickness (mm) 1.795 1.826 1.861

Cost ($ million) 1.795 1.826 1.861

Another approach in design is the fail-safe approach where inspections are conducted instead of increasing panel thicknesses. Keeping the value of the thickness at the baseline value of 1.75 mm, we performed both uniform and non-uniform inspection schedules. The inspection schedules and the cost associated with design are given in Table 4. In uniform inspection scheduling, the choice of time for the first inspection is not unique. Different industrial companies have their own methods for the determination of first inspection time. In this paper, we used 50% of the design life for the first inspection based on the paper by Swift [16]. We fixed the time for the last inspection by considering the fact that the crack growth in MSD situations speeds up as the number of cycles increases, hence the time for the last inspection was set at the 39700th cycle. Table 4. Inspection Schedules in Terms of Cycle(Flight) Numbers with Fail-Safe Design for three different levels of safety (random parameters are (a1)i, (a2)i and n)

LoS 10-7 10-8 10-9 *

Uniform Inspection Schedule 20000, 24925, 29850, 34775, 39700 20000, 23940, 27880, 31820, 35760, 39700 20000, 23283, 26567, 29850, 33133, 36417, 39700

Non-Uniform Inspection Schedule

Cost-U* ($ million)

Cost-NU* ($ million)

38492, 39488

6.75

3.75

37896, 38855, 39513

7.75

4.75

37453, 38378, 38990, 39526

8.75

5.75

Cost-U: Cost with Uniform Scheduling, Cost-NU: Cost with Non-Uniform Scheduling

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Erdem Acar, Amit A. Kale and Mehmet A. Akgün

Even though the uniform inspection scheduling is preferred in many cases, Table 4 reveals that the number of inspections required with uniform inspection scheduling case is greater than that with the non-uniform (or optimum) inspection scheduling. Table 4 also shows that the optimum inspection times are towards the end of the service life, because the probability of failure of a structural component increases as the number of cycles increases. Another reason for the inspection times to be optimum towards the end of service life is that as the number of cycles increases the interaction between the MSD cracks increases, hence the crack growth rates of MSD cracks also increase. From the results presented in Tables 3 and 4, we can conclude that although the inspections are helpful in reducing the failure probability, the panel designs based on safe-life approach are favourable. It is seen that a small percentage of increase of the thickness increases the reliability of the panels in a large amount. The increase of design thickness is preferred over the use of inspections since inspections are costly. Next, taking also the applied load as a random variable, the design and inspection scheduling optimization of panels are performed. The coefficient of variation in the load is taken as 20%. The optimum design thickness and inspection scheduling is given in Table 5. It is seen that the safe-life design is encouraging. Table 5. Optimum Design and Inspection Scheduling (random parameters are (a1)i, (a2)i n and Pmax)

LoS 10-7 10-8 10-9

Panel Thickness (mm) 2.243 2.420 2.601

Number of Insp. 0 0 0

Cost($ million) 2.243 2.420 2.601

Finally, we calculated the panel thickness required to attain a specified level of safety keeping the number of inspections fixed in order to show the trade-off between the panel thickness required and the number of inspections. The results are shown in Table 6, where the panel thickness is determined such that the probability of failure at the end of the service life is just equal to the specified level of safety. A graphical visualization is given in Fig. 2. Again, a small change in design thickness improves the level of safety significantly.

Reliability-based design and inspection scheduling optimization of an aircraft structure containing multiple site damage

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Table 6. Trade-off Between the Thickness and the Number of Inspections Ni 0 1 2 3

Panel Thickness (mm) LoS=10-7 LoS=10-8 LoS=10-9 2.243 2.420 2.601 2.161 2.337 2.518 2.102 2.277 2.456 2.068 2.243 2.421

Cost ($ million) LoS=10-7 LoS=10-8 LoS=10-9 2.243 2.420 2.601 3.161 3.337 3.518 4.102 4.277 4.456 5.068 5.243 5.421

Figure 2. Trade-off between the thickness and the number of inspections Table 6 also illustrates that when the uncertainty in the applied load is included in the analysis, the thickness required to attain a level of safety increases. There is about 80% of increase in the design thickness compared to the case when there is no uncertainty in the applied load.

7.

CONCLUSION

Optimum design and inspection schedule optimization of a wing lower panel containing multiple site damage is performed in this study. A set of two-cracked panel configurations, in which symmetry boundary conditions are used to represent a row of cracks, are modelled to take crack interactions into account. A representative load spectrum is converted into one one wherein one equivalent cycle represents a single flight. The effect of stress ratio is included via Walker equation. A reliability analysis is performed

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Erdem Acar, Amit A. Kale and Mehmet A. Akgün

using an augmented form of the FORM method. The optimization of design and inspection scheduling is carried out by attaining a minimum life cycle cost under the constraint of a pre-specified level of safety. The use of non-uniform inspection schemes are shown to be cost effective compared to uniform inspection schemes. In non-uniform inspection schemes, it is found that the inspections are preferred to be conducted near the end of service life due to the increase of probability of failure as the number of cycles increases. The increase of crack interaction effects with the increase of crack growth is another reason for the inspection times to be preferred towards the end of service life. Trade-off between the design thickness and number of inspections is presented and it is concluded that the optimum design for this geometry and loading conditions is obtained via the use of a safe-life approach. A small increase in design thickness is preferred over inspections to have improvements in the reliability of panels due to the high cost associated with the inspections. The increase in number of inspections might have been preferred over the increase of design thickness if the costs associated with the inspections were lower. A possible approach may be the use of cheaper inspections with lower crack detection capability. This leads to the investigation of optimization of inspection scheduling using different inspection types, which is beyond the scope of this study.

ACKNOWLEDGEMENT The authors thank Prof. Raphael T. Haftka of University of Florida for his invaluable help and suggestions.

APPENDIX 1. CYCLE EQUIVALENCE METHOD This method[17,18] is based on the Walker equation for crack growth, da ⎪⎧(1 − R ) m for R > 0 z≡⎨ = C (zK max ) p , dN ⎪⎩1 − µR for R ≤ 0

(A1.1)

where K max = β σ max π a . All aluminum alloys may be assumed to have the same m value of 0.6 and µ may be taken to have a value of 0.1unless more accurate values are available. Integrating Eq. (A1.1) over N cycles of constant amplitude with a maximum value of σmax and rewriting,

e N= C

⎛ 1 ⎜ ⎜ zσ ⎝ max

⎞ ⎟ ⎟ ⎠

af

p

where e ≡

∫ (β πa ) da

a0

p

(A1.2)

Reliability-based design and inspection scheduling optimization of an aircraft structure containing multiple site damage

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e represents geometry effects only. If there are a total of n flights of a single kind in one block, I is the number of different cycle types encountered (different in terms of magnitude) in one flight, σ maxi is the maximum stress of the ith cycle type, Ri is the stress ratio of the ith cycle type and ni is the total number of cycles of the ith cycle type in n flights, then,

ni =

(

ei z i σ maxi C

)− p

ai

∫ (β πa )

where ei ≡

da

p

, i = 1,..., I

(A1.3)

ai −1

It is assumed above that all cycles of ith type have the same maximum stress and the same stress ratio Ri. In practice this implies that the cycles are classified into I groups. Hence, all cycles of type i are assumed to be applied consecutively and the total crack growth due to ni cycles of type i alone in one block is effectively taken to be (ai − ai-1) which is an approximation. In flight-by-flight analysis, each flight is represented by a single cycle with R=Req ≥0, hence there will be n cycles altogether. Then,

n=

(

e (1 − Req ) m σ max eq C

)

−p

aI

where ei ≡

∫ (β πa ) da

p

(A1.4)

a0

where σ maxeq is the maximum stress for the equivalent cycle. The value of Req is chosen as 0.1 in this study. Now,

I

e ≡ ∑ ei

(A1.5)

i =1

From Eqs. (A1.3-A1.5), the maximum stress for the equivalent cycle is

σ maxeq

⎡ ⎢1 =⎢ n ⎢⎣

I

∑ i =1

⎛ z iσ max i ni ⎜ ⎜ (1 − R ) m eq ⎝

⎞ ⎟ ⎟ ⎠

p⎤

1 p

⎥ ⎥ ⎥⎦

(A1.6)

APPENDIX 2. PROBABILITY OF FAILURE CALCULATION WITH NDI According to the method proposed by Harkness et al. [13] the failure is defined for the case of undetected cracks, hence the values of random variables that lead to the detection of cracks are out of concern. Therefore, the joint probability density function (PDF) of the random variables describing the problem should be updated. The updated joint PDF of random variables is given by fU ( r ) = PNDI (r ) f R (r )

(A2.1)

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Erdem Acar, Amit A. Kale and Mehmet A. Akgün

where f R (r ) and fU (r ) are the initial (or prior) and updated (or posterior) joint PDF, PND I ( r ) is the probability that the cracks are not detected in the first I inspections. Since it is assumed that the cracks are repaired so that the detected cracks will not contribute to the failure probability anymore, the term PND I ( r ) is just the product of probabilities that a crack is not detected in the first I inspections; I

PNDI (r ) = ∏ {1 − POD[a (r , N i )]} i =1

(A2.2)

where POD[a (r , N i )] is the probability of detecting a crack of length a, and a (r , N i ) is the crack length at the ith inspection time, Ni, defined in terms of the random variable set r. After updating the distribution of random variables, the probability of failure since the last inspection can be calculated from

∫ PND (r ) f

PfsI ( N s ) =

I

R ( r ) dr

(A2.3)

Ω fsI

where Ω fsI is the domain of random variables that leads to N I ≤ N f ≤ N . For more detailed explanation of the method, the reader is referred to the paper by Harkness et al. [14]. For the two crack problem, the formulation of probability of failure needs be modified. There are four possible events regarding the crack detection in an inspection: 1) D1 xD2 : both cracks are detected, 2) D1 xD2 : 1st crack is not detected but 2nd crack is detected, 3) D1 xD2 : 1st crack is detected but 2nd crack is not detected, and 4): D1 xD2 neither cracks is detected. The formulation for the two-crack case depends on the type of repair. Upon detection of a crack, either the component is repaired fully or a partial repair can be applied such as by use of crack arresters, is performed. For the case of partial repair, three out of four possible events, namely, D1 xD2 , D1 xD2 ,and, D1 xD2 may lead to failure. In D1 xD2 event, since neither crack is detected, the failure mechanism is multiple site damage (MSD). However, in events D1 xD2 and D1 xD2 one of the cracks is detected and repaired; hence, the failure mechanism is the propagation of a single crack under the constraint of one stopped crack. The problem gets complicated under the assumption of partial repair, therefore we assumed full repair for the sake of simplicity. On the other hand, with a full repair strategy, only D1 xD2 event leads to failure. The probability of failure after the first I inspections can then be formulated as I

PND I ( r ) = ∏

i =1

8. 1.

2.

{ [ 1 − POD[a1 (r , N i )] ] ⋅ [ 1 − POD[a 2 (r , N i )] ] }

(A2.4)

REFERENCES Swift, T., “Effect of MSD on residual Strength,” In: Symposium on Multiple Site Damage (MSD) in Aging Aircraft, Warner Robins Air Logistic Center, Robins AFB, GA, Feb. 1992. Mar, J.W. “Structural Integrity of Aging Airplane: A Perspective,” In: Atluri, S.N. Sampath, S.G. and Tong, P. editors. Structural Integrity of Aging Airplanes, Berlin, Springer, 1991.

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