Reliability-Based Structural Design of Aircraft Together with Future Tests

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1 Reliability-Based Structural Design o Aircrat Together with Future Tests Erdem Acar 1 TOBB University o Economics and Technology, Söğütözü, Ankara 06560, Turkey Raphael T. Hatka 2, Nam-Ho Kim 3 University o Florida, Gainesville, FL 32611, USA Merve Türinay 4 TOBB University o Economics and Technology, Söğütözü, Ankara 06560, Turkey Chanyoung Park 5 University o Florida, Gainesville, FL 32611, USA Traditional optimization changes variables that are available in the design stage to optimize objectives, such as aircrat structural reliability. However, there are many post-design measures, such as tests and structural health monitoring that reduce uncertainty andurther improve the reliability. In this paper, a new reliability-based design ramework that can include post-design uncertainty reduction variables is proposed. Among many post-design variables, this paper ocuses on the number o coupon tests and the number o structural element tests. Uncertainty in the ailure stress prediction, variability due to the inite number o coupon tests, and uncertainties in geometry and service conditions are studied in detail. The Bayesian technique is used to update the ailure stress distribution based on results o the element tests. Tradeo plots o the number o tests, weight and probability o ailure in certiication and in service are generated, and inally reliability-based design o uture tests together with aircrat structure is perormed or minimum liecycle cost. Nomenclature A = load carrying area (width times thickness) o a small part o the overall structure be e = bound o associated with ailure criterion used while predicting ailure in the structural element tests b t = bound o error in the design thickness, e t b w = bound o error in the design width, e w e e = error associated with ailure criterion used while predicting ailure in the structural element tests e = error in predicting ailure o the entire structure in certiication or proo testec e p = error in load ulation e σ = error in stress ulation e t = error in the design thickness due to construction errors e w = error in the design width due to construction errors DOC = direct operating cost E( ) = expected value (i.e., mean value) k d = knockdown actor at coupon level due to use o conservative (B-basis) material properties k = additional knockdown actor at the structural level (nominal value is taken as 0.95) n c = number o coupon tests (nominal value is taken as 50) 1 Assistant Proessor, Mechanical Engineering, AIAA Member. 2 Distinguished Proessor, Mechanical and Aerospace Engineering, AIAA Fellow. 3 Associate Proessor, Mechanical and Aerospace Engineering, AIAA Member. 4 Research Assistant, Mechanical Engineering. 5 Research Assistant, Mechanical and Aerospace Engineering. 1

2 n e = number o element tests (nominal value is taken as 3) N a = number o aircrat in a leet (taken as 1,000) N mat = number o materials or which coupon testing is done (taken as 80) N elem = number o dierent types o structural elements tested (taken as 100) p = cost saving by reducing the structural weight by one unit P = ulated design load P d = true design load based on the FAA speciications (e.g., gust load speciication) P = probability o ailure PFCT = probability o ailing certiication test σ ca = allowable stress (B-basis) rom coupon testing σ ea = allowable stress (B-basis) rom element testing σ a = allowable stress (B-basis) o the entire structure σ c = ailure stress rom coupon testing σ e = ailure stress o the structural element (σ e ) test = element ailure stress measured in tests (σ e ) = ulated (or predicted) element ailure stress (σ e ) upd = updated value o the ulated (or predicted) element ailure stress σ = ailure stress o the overall structure σ = stress in a small part o the overall structure S F = the FAA load saety actor o 1.5 t = thickness o a small part o the overall structure v t = eect o variability on the built thickness v w = eect o variability on the built width w = width o a small part o the overall structure Subscripts built-av = average built value that diers rom the design value due to errors in construction built-var = actual built value that diers rom the average built value due to variability in construction = ulated (or predicted) value that diers rom the design value due to errors in design design = design value true = true value (error ree value) I. Introduction HE saety o aircrat structures can be achieved by designing the structure against uncertainty and by taking Tsteps to reduce the uncertainty. Saety actors and knockdown actors are examples o measures used to compensate or uncertainty during the design process. Uncertainty reduction measures (URMs), on the other hand, may be employed during the design process or later on throughout the operational lietime. Examples o URMs or aircrat structural systems include structural testing, quality control, inspection, health monitoring, maintenance, and improved structural analysis and ailure modeling. In traditional reliability-based optimization, all uncertainties that are available at the design stage are considered in ulating the reliability o the structure (e.g., Res. [1-15]). However, the actual aircrat is much saer, because ater design it is customary to engage in vigorous uncertainty reduction activities using various URMs. It would be thereore beneicial to include the eects o these planned URMs in the design process [16-19]. It may be even advantageous to design the URMs together with the structure; or example, trading o the cost o more weight against the cost o additional tests. It is challenging, though, to model the eect o uture URMs in the design process. In this study, we ocus on the structural tests as an example o URMs. There are ew papers in the literature that address the eect o tests on structural saety. Jiao and Moan [20] investigated the eect o proo tests on structural saety using Bayesian updating. They showed that the proo testing reduces the uncertainty in the strength o a structure, thereby leading to substantial reduction in probability o ailure. Jiao and Eide [21] explored the eects o testing, inspection and repair on the reliability o oshore structures. Beck and Kataygiotis [22] addressed the problem o updating a probabilistic structural model using dynamic test data rom structure by utilizing Bayesian updating. Similarly, Papadimitriou et al. [23] used Bayesian updating within a probabilistic structural analysis tool to compute the updated reliability o a structure using test 2

3 data. They ound that the reliabilities computed beore and ater updating were signiicantly dierent. In a previous paper [24], we aimed to extend the work o these earlier authors in simulating all possible outcomes o uture tests, which would allow the designer to design the tests together with the structure. In [24] we ocused on the eects o structural tests on aircrat saety. We investigated in particular the eect o the number o coupon and structural element tests on the inal distribution o the ailure stress. We assumed that the mean value o the ailure stress (mean over a large number o aircrat) is obtained rom a ailure criterion (e.g., Tsai- Wu theory [25]) using the results o coupon tests. The initial uncertainty in this mean ailure stress relects the conidence o the analytical model in this prediction as well as possible error in inite number o coupon tests. The Bayesian technique is then used to update the mean ailure stress distribution considering all possible outcomes o uture element tests. In addition, there is the variability o the ailure stress rom one aircrat to another or rom one structural component to another. We assumed that this variability is the same as the variability in coupon tests, which holds only or very large number o coupon tests. In the present paper we take into account the error associated with obtaining material properties rom a inite number o coupon tests. The objective o the present paper is to perorm reliability-based design o aircrat structure together with the uture aircrat structural tests. We assume that besides satisying a constraint on the probability o ailure, the designer needs to satisy the FAA regulations or deterministic design. In order to reconcile the two requirements, we take advantage o the act that companies oten apply additional knockdown actors to design allowables beyond the FAA requirements. We use thus use a company knockdown actor as a design variable that modulates the tradeo between cost and saety. We illustrate the generation o response surace plots o the number o tests, weight and probability o ailure in certiication and in service. The paper is organized as ollows. The next section provides the motivation or design o aircrat structures together with uture tests. Section III discusses the saety measures taken during aircrat structural design. Section IV presents a simple uncertainty classiication that distinguishes uncertainties that aect an entire leet (errors) rom the uncertainties that vary rom one aircrat to another in the same leet (variability). Section V discusses modeling o errors and variability throughout the design and testing o an aircrat, and probability o ailure ulation. Finally, the reliability-based optimization results and the concluding remarks are given in the last two sections o the paper, respectively. II. Design o Aircrat Structures Together with Future Tests According to current practices, determination o the number o structural tests is based on past experience. However, since structural tests are expensive, there is an incentive to reduce the expenditures or tests without jeopardizing saety. Intuitively, the number o expensive tests such as component tests and element tests can be reduced and the number o inexpensive tests such as coupon tests can be increased. However, beore making such decisions, the eects o these tests on aircrat saety need to be assessed. In this paper, the eect o tests on structural saety is assessed by using Monte Carlo simulation (MCS). It is assumed that the mean value o the ailure stress is obtained rom a ailure criterion using the results o coupon tests. To relect our level o conidence in the prediction o the ailure criterion, an initial uncertainty in the mean ailure stress is assumed. Then, Bayesian technique is used to update the mean ailure stress distribution considering all possible outcomes o uture element tests. In addition, the variability o the ailure stress rom one airplane to another or rom one structural component to another is modeled in the simulations. The uncertainty in the ailure stress variability due to the inite number o coupon tests is also modeled in the MCS. Ater assessing the eects o tests on saety, simultaneous design o aircrat structures and the number o uture tests will be perormed or minimum liecycle cost. The design variables are chosen as the company knockdown actor and the number o coupon and element tests. Response surace models are constructed to relate the design variables to structural weight and structural saety. The response surace models are used to compute the liecycle cost (the objective unction) and the probability o ailure (the constraint). These types o models can also be used by structural designers as well as company managers or tradeo analyses and decision making. III. Saety Measures As noted earlier, the saety o aircrat structures is achieved by designing these structures to operate well in the presence o uncertainties and taking steps to reduce the uncertainties. The ollowing gives brie description o these saety measures. 3

4 A. Saety measures or designing structures under uncertainties Load Saety Factor: In transport aircrat design, FAA regulations mandate the use o a load saety actor o 1.5 (FAR [26]). That is, aircrat structures are designed to withstand 1.5 times the limit load without ailure. Conservative Material Properties: In order to account or uncertainty in material properties, FAA regulations mandate the use o conservative material properties (FAR [27]). The conservative material properties are characterized as A-basis and/or B-basis material property values. Detailed inormation on these values is provided in Volume 1, Chapter 8 o the Composite Materials Handbook [28]. In this paper, we use B-basis values. The B-basis value is determined by ulating the value o a material property exceeded by 90% o the population with 95% conidence. The basis values are determined by testing a number o coupons selected randomly rom a material batch. In this paper, the nominal number o coupon tests is taken as 50. Other measures such as redundancy are not discussed in this paper. B. Saety measures or reducing uncertainties Improvements in accuracy o structural analysis and ailure prediction o aircrat structures reduce errors and enhance the level o saety. These improvements may be due to better modeling techniques developed by researchers, more detailed inite element models made possible by aster computers, or more accurate ailure theories. Similarly, the variability in material properties can be reduced through quality control and improved manuacturing processes. Variability reduction in damage and ageing eects is accomplished through inspections and structural health monitoring. The reader is reerred to the papers by Qu et al. [16] or eects o variability reduction, Acar et al. [17] or eects o error reduction, and Acar et al. [18] or eects o reduction o both error and variability. In this paper, we ocus on error reduction through aircrat structural tests, while the other uncertainty reduction measures are let out or uture studies. Structural tests are conducted in a building block procedure (Volume I, Chapter 2 o Re. [28]). First, individual coupons are tested to estimate the mean and variability in ailure stress. The mean structural ailure is estimated based on ailure criteria (such as Tsai-Wu) and this estimate is urther improved using element tests. Then a sub-assembly is tested, ollowed by a ull-scale test o the entire structure. In this paper, we use the simpliied three-level test procedure depicted in Figure 1. The coupon tests, structural element tests and the inal certiication test are included. The irst level is the coupon tests, where coupons (i.e., material samples) are tested to estimate ailure stress. The FAA regulation FAR requires aircrat companies to perorm enough tests to establish design values o material strength properties (A-basis or B-basis value). As the number o coupon tests increases, the errors in the assessment o the material properties are reduced. However, since testing is costly, the number o coupon tests is limited to about 100 to 300 or A-basis ulation and at least 30 or B- basis value ulation. In this paper, B-basis values are used and the nominal number o coupon tests is taken as 50. At the second level o testing, structural elements are tested. The main target o element tests is to reduce errors related to ailure theories (e.g., Tsai-Wu) used in assessing the ailure load o the structural elements. In this paper, the nominal number o structural element tests is taken as 3. At the uppermost level, certiication (or proo) testing o the overall structure is conducted (FAR [29]). This inal certiication or proo testing is intended to reduce the chance o ailure in light due to errors in the structural analysis o the overall structure (e.g., errors in inite element analysis, errors in ailure mode prediction). While ailure in light oten has atal consequences, certiication ailure oten has serious inancial implications. So we measure the success o the URMs in terms o 4 Figure 1. Simpliied three-level tests

5 their eect on probability o ailure in light and in terms o their eect on probability o certiication ailure. IV. Structural Uncertainties A good analysis o dierent sources o uncertainty in engineering simulations is provided by Oberkamp et al. [30, 31]. To simpliy the analysis, we use a classiication that distinguishes between errors (uncertainties that apply equally to the entire leet o an aircrat model) and variability (uncertainties that vary or the individual aircrat) as we used in our earlier studies [32, 33]. The distinction, presented in Table 1, is important because saety measures usually target either errors or variability. While variabilities are random uncertainties that can be readily modeled probabilistically, errors are ixed or a given aircrat model (e.g., Boeing ) but they are largely unknown. Since errors are epistemic, they are oten modeled using uzzy numbers or possibility analysis [34, 35]. We model errors probabilistically by using uniorm distributions because these distributions correspond to minimum knowledge or maximum entropy. Type o Uncertainty Error (mostly epistemic) Variability (aleatory) Table 1. Uncertainty Classiication Spread Cause Remedies Departure o the average leet o an aircrat model (e.g. Boeing ) rom an ideal Departure o an individual aircrat rom leet level average Errors in predicting structural ailure, construction errors, deliberate changes Variability in tooling, manuacturing process, and lying environment Testing and simulation to improve the mathematical model and the solution Improvement o tooling and construction. Quality control Errors are uncertain at the time o the design but they are the same or all copies o a structural component on dierent airplanes o the same model, while the variabilities vary or nominally identical copies o structural components. To model errors, we assume that we have a large number o nominally identical aircrat being designed (e.g., by Airbus, Boeing, Embraer, Bombardier, etc.), with the errors being ixed or each aircrat. V. Uncertainty Modeling and Probability o Failure Calculation We use a simple example o point stress design or yield to illustrate our methodology. The loading is assumed to ollow Type I extreme distribution since we consider the maximum load over lietime. The ailure stress is assumed to ollow lognormal distribution. To model uncertainties, it is required to simulate the coupon tests, the element tests and the certiication test. At the coupon-level, we have errors in estimating material strength properties rom coupon tests, due to limited number o coupon tests. At the element-level, we have errors in structural element strength predictions due to the inaccuracy o the ailure criterion used. At the ull scale structural-level, we have errors in structural strength predictions, error in load ulation and error in construction. Similarly, we have variability in loading, geometry and ailure stress. Ater all the errors and variability are careully introduced, the probability o ailure can be computed using MCS. Details o the overall uncertainty modeling are provided in detail in Appendix A. Within this uncertainty modeling ramework, the eects o structural element tests are considered by using Bayesian updating as discussed in Appendix B. The simulation o error and variability can be easily implemented through a two-level Monte Carlo simulation [33]. At the upper level dierent aircrat companies can be simulated by assigning random errors to each, and at the lower level we simulated variability in dimensions, material properties, and loads related to manuacturing variability and variability in service conditions can be simulated. The details o the separable MCS are provided in Appendix C. The eect o element tests on ailure stress distribution is modeled using Bayesian updating. I Bayesian updating were used directly within an MCS loop o probability o ailure ulation, the computational cost would be very high. Thereore, Bayesian updating is perormed aside in a separate MCS (a brie description o the procedure is given in Table 2), beore starting with the main MCS loop (a brie sketch o the procedure is provided in Table 3). Since the ailure stress distribution may have a general shape, we used Johnson distribution to model it, 5

6 which can be represented using our quantiles. The procedure ollowed or Bayesian updating can be described briely as ollows. First, the our quantiles o the mean ailure stress are modeled as normal distributions. Then, these quantiles are used to it a Johnson distribution to the mean ailure stress. That is, the mean ailure stress is represented as a Johnson distribution, whose parameters are themselves distributions that depend on the number o element tests as well as the error in ailure stress prediction o the elements, e e. Finally, Bayesian updating is used to update the mean ailure stress distribution as in our earlier work [36]. Table 2. Monte Carlo procedure or the separate Bayesian code (computing statistical properties o quantiles) 1. Calculate the apparent values o the mean and standard deviation o the ailure stress rom inite number o coupon tests 2. Simulate the element tests (generate random numbers using the true values o the mean and standard deviation o the ailure stress) 3. Assume very large bounds or the prior distribution o the mean ailure stress and perorm Bayesian updating using the results o element tests 4. Compute the quantiles o the updated distribution 5. Perorm steps 1-4 or 20,000 times and compute the mean and standard deviation as well as the correlation coeicient between the quantiles Table 3. Monte Carlo procedure or the main code (computing probability o ailure) 1. Compute the allowable stress based on the results o coupon tests 2. Calculate the built average load carrying area using the results o coupon tests (See Appendix A) 3. Generate random numbers or the quantiles o the updated mean ailure stress (Updating was perormed in a separate Bayesian updating code) 4. Calculate the allowable stress using the quantiles (see Appendix B) 5. Revise the built average load carrying area based on the values o the allowable stress ulated rom the coupon tests (step1), and element tests (step 4). See Appendix A or revised area ulation procedure. 6. Using the revised area, compute the probability o ailure in service (P ), and in certiication (PFCT). See Appendix C or P and PFCT ulation. VI. Optimization or minimum liecycle cost As noted earlier, the main objective o the paper is to perorm reliability-based structural design o aircrat together with uture tests. The design variables o the optimization problem are chosen as the company knockdown actor k (see Appendix A, Section C), the number o coupon tests n c, and the number o element tests n e. The reliability-based design o an aircrat structure or minimum direct operating cost (DOC) can be perormed by solving the ollowing optimization problem: Find k, nc, n e (1.1) Min DOC, c, e S.t. P k, nc, ne P k n n (1.2) (1.3) nom 0.9 k 1.0, 30 n 80, 1 n 5 (1.4) c e The P term in the constraint is taken as the value o P when the design variables take their nominal values nom (i.e., k =0.95, n c =50, and n e =3). The direct operating cost DOC is related to the design variables as described in the ollowings. The cost model used in this study is based on the paper by Kaumann et al. [37], which suggested that an optimum cost-eective design inds the proper tradeo between the minimum weight and minimum manuacturing cost solution. The direct operating cost (DOC) o the aircrat structure can be deined as DOC = C man + 0W p (2) 6

7 where C man is the manuacturing cost, p 0 is the cost penalty due to excessive structural weight, W. In our study, we modiy this cost ormulation to take into account the cost o the uncertainty reduction measures (C urm ) including tests, quality control, health monitoring, etc. In addition, we dump the manuacturing cost into the cost o excess weight. That is, the DOC is reormulated as DOC = p W + Curm (3) Here, p is the total cost saving attained by reducing the structural weight by one pound. In this study, amongst uncertainty reduction measures (URMs) noted earlier, we ocus on tests. Hence, C urm is divided into two elements as C urm = C test + C urm-other (4) where C test is the cost o tests, and C urm-other is the total cost o URMs other than tests. In this study, we ocus mainly on two types o tests: the coupon tests and the element tests. Thus, C test can be re-written as C test = C coupon + C elem + C test-other (5) where C coupon is the cost o coupon tests, C elem is the cost o element tests, and C test-other is the cost o other tests such as component tests, assembly tests, certiication test, etc. Equations (2-4) can be combined to yield where DOC = p W + C coupon + C elem + Cother (6) C other = C urm-other + C test-other + C labor (7) The direct operation cost can be re-written so as to show its dependence on the knockdown actor k, the number o coupon tests n c, and the number o element tests n e as c e c e coupon c elem e DOC k, n, n = p W k, n, n + C n + C n + C (8)Here, W is the structural weight, p is the weight penalty, C c is the cost o coupon tests, C e is the cost o element tests, and C other reers to the costs that are not aected by the choice o design variables. Since DOC is used within an optimization ramework, C other term can be dropped. Details o each term in the cost equation are provided below. Weight penalty p Curran et al. [38] proposed that the economical value o weight saving is 300 $/kg. Similarly, Kim et al. [39] reerred to a recent report by the US National Materials Advisory Board [40] that estimated that a 1 lb weight reduction amounts to a total saving o $200 or a civil transport aircrat. In our study, we vary the weight penalty between $200/lb and $1000/lb and see its eect on the optimum values o the design variables. The structural weight We take the structural weight o a typical civil transport aircrat as 50,000 lbs. Since the test costs can be attributed to leet o aircrat rather than a single one, total structural weight o the leet is considered. Thereore, the weight term in Eq. (8) can be written as other Acert k, nc, ne W k, nc, ne Na 50,000 (9) A nom Here A cert is the certiied load carrying area, and A nom is the value o A cert when the design variables take their nominal values. We assume that a typical airliner has a production line o 1,000 aircrat beore it is discontinued or substantially redesigned, so N a =1,000. 7

8 Test costs The costs or the coupon tests and element tests are based on personal communications with structural engineers in Turkish Aerospace Industries, Boeing, and NASA. They are taken as $300 or each coupon in a coupon test, and $150,000 or each element tests. Accordingly the respective costs are given as C coupon (n c ) = 300 N mat n c (in $) (10) C elem (n e ) = 150,000 N elem n e (in $) (11) where the N mat is the number o dierent materials tested or a single aircrat model, and N elem is the number dierent types o structural elements tested. In this study, these values are taken as N mat =80, and N elem =100, respectively. I the design variables take their nominal values, and p=$200/lb, the DOC is 6 DOC = 200 1,000 50, , = 10, dollars (12) According to Eq. (12), even or the lowest weigh penalty, the contribution o weight to the cost dominates the cost o tests. So i we can reduce the weight by more than 0.15% by perorming an additional element test, then we should choose to do that. This relects the large number o airplanes (assumed to be 1,000 here) that beneit rom the results o the test. We will investigate these tradeos in detail in the Results section. VII. Results In this section, irst the eects o the number o coupon tests and the number o element tests on the weight and probability o ailure will be investigated. Then, the reliability-based design o aircrat will be perormed or minimum direct operating cost. A. Response surace generation or weight and reliability index in terms o the design variables Response surace models (quadratic polynomial with all terms included) are constructed to relate the number o structural tests, and company knock down actors to structural weight, and structural saety or use in the optimization. The input variables o the response surace models and their bounds are provided in Table 4. Latin hypercube design o experiments is used to generate thirty training points within the bounds given in Table 4. The built average load carrying area (surrogate or the structural weight) and the probability o ailure are computed using Monte Carlo simulations. The accuracies o the constructed response surace models are evaluated by using leave-one-out cross-validation errors. Response surace models are constructed 30 times, each time leaving out one o the training points. The dierence between the exact response at the omitted point and that predicted by each variant response surace model deines the cross-validation error. Table 5 provides the root mean square error (RMSE), the maximum absolute error (MAE), the maximum absolute error (MAXE) as well as the mean o the response. Comparison o the error metrics to the mean o response reveals that the constructed response suraces are quite accurate. Table 4. Input variables o the response surace models and their bounds Variable k n c n e Lower bound Upper bound Table 5. Evaluating accuracies o response surace models using leave-one-out cross validation errors Response Mean o response RMSE (a) MAE (b) MAXE (c) A cert Rel. Index, β (a) RMSE: root mean square error; (b) MAE: mean absolute error; (c) MAXE: maximum absolute error 8

9 Since there are three variables, we ix one variable and plot the variation o the response with respect to other two variables to provide a graphical depiction o the constructed response suraces. The response surace plots or A cert and reliability index β are given in Figs. 2 and 3. The igures show that as k reduces, both A cert and β increases. As the number o tests increases, the load carrying area reduces. As the number o coupon tests increases, the reliability index reduces since the B-basis value increases. As the number o element tests increases, the reliability index increases since the error in ailure prediction reduces. (a) n c is ixed to 50 (b) n e is ixed to 3 (c) k is ixed to 0.95 Figure 2. Constructed response suraces or the average certiied load carrying area (a) n c is ixed to 50 (b) n e is ixed to 3 (c) k is ixed to 0.95 Figure 3. Constructed response suraces or the reliability index B. The weight and the number o coupon tests tradeos The change o A cert (surrogate or structural weight) with the number o coupon tests or the same P is shown in Table 6. The number o element tests is ixed at three, and the value o k is adjusted to maintain the same P. I the number o coupon tests is reduced rom 50 to 30, the built average load carrying area is increased by 0.28%. On the other hand, i the number o coupon tests is increased rom 50 to 80, the built average load carrying area decreases by 0.24%. The last column o Table shows that when P. is maintained at its nominal value, the probability o ailure in certiication test changes only slightly. Increasing the number o tests rom 50 to 80 brings a cost penalty o ΔC = Δn c = $720 k, while 0.24% weight saving leads to a cost saving o ΔC = (0.24% 1,000 50,000) 200 = $-24 M when p=200$/lb. So, it is expected that the RBDO will lead to the setting o n c =80. Table 6. Eects o the number o coupon tests or the same probability o ailure (k = 0.95, n e =3). Note that the k values are all smaller than 1.0, so the FAA deterministic design regulations are not violated. n c k A cert %ΔA cert P ( 10-7 ) PFCT* * PFCT: Probability o ailing in certiication test 9

10 C. The weight and the number o element tests tradeos To analyze the tradeos o the weight and the number o element tests, P can be ixed and the variation o A cert with n e can be explored. The number o coupon tests is ixed at ity, and the value o k is adjusted to maintain the same P. Table 7 shows that i we want to perorm only a single element test, then we will need to put 1.43% extra weight to achieve to the same P. The cost saving by eliminating two element tests is ΔC = 150, = $30 M, while the cost penalty due to adding 1.43% extra weight is $-163 M when p=200$/lb. So, this is not a good route. On the other hand, i we increase the number o element tests rom three to our, we can save 0.34 percent weight, yielding a cost saving o $-34 M even when the weight penalty is at its minimum value o p=200 $/lb. The cost penalty due to an additional element test is $15 M, so it is advantageous to perorm our tests. However, increasing the number o element tests rom our to ive brings only additional weight saving o 0.11%, leading to cost saving o $11 M. Recalling that an additional element test costs $15 M, it is not worthy to perorm ive tests. I the penalty parameter is taken as p=1000 $/lb, on the other hand, the weight cost saving will be $30 M, so it is now worthy to perorm ive tests. Table 7. Eects o the number o element tests or the same probability o ailure when the bound o error in ailure prediction o elements is 10% (k = 0.95, n c =50). Note that the k values are all smaller than 1.0, so the FAA deterministic design regulations are not violated. n e k A cert %ΔA cert P ( 10-7 ) PFCT The eect o reducing the error in ailure stress prediction o structural elements is investigated next. Table 8 shows the variation o A cert with the number o coupon tests or the same P when the bound o error in ailure stress prediction o structural elements is reduced rom 10% to 5%. It is seen that as the error in ailure stress predictions are reduced, the tests becomes less eective so smaller weight can be saved. But still, increasing the number o element tests to our yields 0.24% weight saving, leading to a cost saving o $24 M when p=200$/lb. Recalling that an additional element test costs $15 M, we choose to perorm the orth test. Increasing the number o tests rom our to ive yields only $9 M weight cost saving, so we choose not to perorm the orth test. The variation o the load carrying area with the number o coupon and element tests or 5% and 10% bounds o error in ailure prediction o elements are provided in Fig. 4. It is seen that the tests become more eective when the error bound is large (as expected). Perorming ive element tests instead o three leads to a weight saving o 0.45% when error bound is 10%, while the weight saving reduces to 0.33% when error bound is 5%, These tradeo plots accompanied with cost inormation provide very valuable inormation to a structural engineer. Table 8. Eects o the number o element tests or the same probability o ailure when the bound o error in ailure prediction o elements is reduced to 5% (k = 0.95, n c =50). Note that the k values are all smaller than 1.0, so the FAA deterministic design regulations are not violated. n e k A cert %ΔA cert P ( 10-7 ) PFCT

11 (a) Coupon tests (b) Element tests Figure 4. Weight (or load carrying area) and number o test tradeos when the bounds o error are 5% and 10%. D. RBDO or minimum cost Using the response surace models constructed, the RBDO problem stated in Eq. (1) is solved by using mincon unction o MATLAB based on the sequential quadratic programming algorithm. The solution o the optimization problem yields real numbers or the number o tests, but should be integer numbers. To resolve this issue, the ollowing approach is ollowed. Ater the optimum solution is obtained as real numbers, the nearest two integers or both the number o coupon tests and the number o element tests are considered, which yields our combinations. For instance, i the optimum numbers o coupon and element tests are ound as n c =50.71 and n e =3.26, respectively, then the ollowing our (n c, n e ) combinations are considered: (50, 3), (51, 3), (50, 4) and (51, 4). Then, our each o these combinations, the optimization problem in Eq. (1) is reduced to a single variable optimization problem (in terms o k only), and the optimum value o k is ulated. Finally, the combination with the best perormance (i.e., with minimum liecycle cost) is declared as the optimum. The results o the optimization or minimum cost, when the bound o error in ailure prediction o elements is taken 10%, are provided in Table 9. It is seen rom Table 9 that the optimum number o coupon tests is 80 regardless the number value o penalty parameter p or excess weight, since the coupon tests are very inexpensive. It is also ound that the optimum values o the number o element tests n e and the knockdown actor k depends on the penalty parameter or excess weight. I the penalty parameter is taken as p = 200 $/lb or p = 500 $/lb, the optimum number o element tests is our. However, i the penalty parameter is taken as p = 1000 $/lb (or larger), the optimum number o element tests is ive, since each pound o weight saving is very valuable. Table 9. RBDO results or minimum cost or various values o the penalty parameter p when the bound o error in ailure prediction o elements is 10%.(P =7.96e-8) k n c n e A cert p W ($M) C c ($M) C e ($M) DOC ($M) p = 200 $/lb Nominal , ,046 Optimum , ,011 p = 500 $/lb Nominal , ,046 Optimum , ,935 p = 1000 $/lb Nominal , ,046 Optimum , ,805 Table 10 shows the results o the optimization or minimum cost, when the bound o error in ailure prediction o elements is taken 5%. Table 10 shows that the optimum number o coupon tests is 80 regardless the number value o penalty parameter p. I the penalty parameter is taken as p = 200 $/lb, the optimum number o element tests is our. 11

12 However, i the penalty parameter is increased to p = 500 $/lb, then the weight saving becomes more important, so the optimum number o element tests is increased to ive. Table 10. RBDO results or minimum cost or various values o the penalty parameter p when the bound o error in ailure prediction o elements is 5%.(P =7.79e-8) k n c n e A cert p W ($M) C c ($M) C e ($M) DOC ($M) p = 200 $/lb Nominal , ,046 Optimum , ,019 p = 500 $/lb Nominal , ,046 Optimum , ,933 p = 1000 $/lb Nominal , ,046 Optimum , ,790 VIII. Concluding remarks Most probabilistic structural design studies assume that the uncertainties are given and resources are used to compensate or them by making the structure strong enough. Instead, the resources can be allocated or uncertainty reduction measures, URMs, or reducing uncertainties, which would in turn increase the saety. Similarly, the increased saety can be traded o or reducing direct operating cost. It would be thereore beneicial to include the eects o these planned URMs in the design process under uncertainty. This paper proposed probabilistic design o aircrat structures together with uture structural tests using probabilistic methods. The eects o structural tests on saety o aircrat structures were investigated and, it was ound when the bound o error or ailure prediction o structural elements was 10% that o i the number o coupon tests was increased rom 50 to 80 maintaining the same probability o ailure, the structural weight could be reduced by 0.24%. o i the number o element tests was increased rom three to our (or ive) maintaining the same probability o ailure, the structural weight can be reduced by 0.34% (or 0.45%). o i the bounds o error in ailure stress prediction o structural elements are reduced, then the tests becomes less eective and the weight saving reduces. Reliability-based optimization o aircrat structures was perormed and it was ound that the optimization results depend heavily on the penalty or excess weight (p) and the bound o error or ailure prediction o structural elements. It was also ound that the optimum number o coupon tests was 80 (maximum considered), regardless the penalty parameter p and the bound o error, since the coupon tests were very inexpensive. Through solving the RBDO problem, the ollowings were observed. o i the bound o error or ailure prediction o structural elements was 10%, the optimum number o element tests was our when the penalty parameter was p = 500 $/lb (or smaller), and the optimum number o element tests was ive when the penalty parameter is p = 1000 $/lb (or larger). o i the bound o error or ailure prediction o structural elements was 5%, the optimum number o element tests was our when the penalty parameter was p = 200 $/lb (or smaller), and the optimum number o element tests was ive when the penalty parameter is p = 500 $/lb (or larger). The results ound or various bounds o error or ailure prediction o structural elements basically indicated that i the aircrat companies support research activities that can improve the ailure prediction theories, they can reduce the number o structural tests or the same saety and same lietime cost. 12

13 Acknowledgments This research is partially supported by TÜBİTAK, under award 109M537, and by the NASA center and the U.S. Air Force, U.S. Air Force Research Laboratory, under award FA (Victor Giurgiutiu, Program Manager) and National Science Foundation (CMMI ). The authors grateully acknowledge these supports. Reerences 1. Lincoln, J.W., Method or Computation o Structural Failure Probability or an Aircrat, ASD-TR , Wright Patterson AFB, OH, July Lincoln, J.W., Risk Assessment o an Aging Aircrat, Journal o Aircrat, Vol. 22, No. 8, August 1985, pp Shiao, M.C., Nagpal, V.K., and Chamis, C.C., Probabilistic Structural Analysis o Aerospace Components Using NESSUS, AIAA , Wirsching, P.H., Literature Review on Mechanical Reliability and Probabilistic Design, Probabilistic Structural Analysis Methods or Select Space Propulsion System Components (PSAM), NASA CR , Vol. III, Lykins, C., Thomson, D., and Pomret, C., The Air Force s Application o Probabilistics to Gas Turbine Engines, AIAA CP, Ebberle, D.H., Newlin, L.E., Sutharshana, S., and Moore, N.R., Alternative Computational Approaches or Probabilistic Fatigue Analysis, AIAA , Ushakov, A., Kuznetsov, A.A., Stewart, A., and Mishulin, I.B., Probabilistic Design o Damage Tolerant Composite Aircrat Structures, Final Report under Annex 1 to Memorandum o Cooperation AIA/CA-71 between the FAA and Central Aero-Hydrodynamic Institute (TsAGI), Mavris, D.N., Macsotai, N. I., and Roth, B., A probabilistic design methodology or commercial aircrat engine cycle selection, Paper SAE , Society o Automotive Engineers, Long, M.W., and Narciso, J.D., Probabilistic Design Methodology or Composite Aircrat Structures. DOD/FAA/AR-99/2, Final Rept., June Ushakov, A., Stewart, A., Mishulin, I., and Pankov, A., Probabilistic Design o Damage Tolerant Composite Aircrat Structures. DOD/FAA/AR-01/55, Final Rept., January Rusk, D. T., Lin, K. Y., Swartz, D. D., and Ridgeway, G. K., Bayesian Updating o Damage Size Probabilities or Aircrat Structural Lie-cycle Management, AIAA Journal o Aircrat, Vol. 39, No. 4, July 2002, pp Allen, M., and Maute, K., Reliability-based design optimization o aeroelastic structures, Structural and Multidisciplinary Optimization, Vol. 27, 2004, pp Huang, C.K., and Lin, K.Y., A Method or Reliability Assessment o Aircrat Structures Subject to Accidental Damage, AIAA , Nam, T., Soban, D.S., and Marv s, D.N., A Non-Deterministic Aircrat Sizing Method under Probabilistic Design Constraints, AIAA , Lin, K.Y., and Styuart, A.V., Probabilistic Approach to Damage Tolerance Design o Aircrat Composite Structures, Journal o Aircrat, Vol. 44, No. 4, July-August Qu, X., Hatka, R.T., Venkataraman, S., and Johnson, T.F., Deterministic and Reliability-Based Optimization o Composite Laminates or Cryogenic Environments, AIAA Journal, Vol. 41, No. 10, 2003, pp Acar, E., Hatka, R.T., Sankar, B.V. and Qui, X., "Increasing Allowable Flight Loads by Improved Structural Modeling," AIAA Journal, Vol. 44, No. 2, 2006, pp Acar, E., Hatka, R.T. and Johnson, T.F., "Tradeo o Uncertainty Reduction Mechanisms or Reducing Structural Weight," Journal o Mechanical Design, Vol. 129, No. 3, 2007, pp Li M, Williams N, Azarm S (2009) Interval uncertainty reduction and sensitivity analysis with multi-objective design optimization. J Mech Des 131(3), pp Jiao, G., and Moan, T., "Methods o reliability model updating through additional events," Structural Saety, Vol. 9, No. 2, 1990, pp Jiao, G., and Eide, O.I., "Eects o Testing, Inspection and Repair on the Reliability o Oshore Structures," Proceedings o the Seventh Specialty Conerence, Worcester, Massachusetts, August 7-9, 1996, pp Beck, J.L., and Kataygiotis, L.S., "Updating models and their uncertainties: Bayesian Statistical Framework," Journal o Engineering Mechanics, Vol. 124, No. 4, 1998, pp Papadimitriou, C., Beck, J.L., and Kataygiotis, L.S., "Updating Robust Reliability Using Structural Test Data," Probabilistic Engineering Mechanics, Vol. 16, No. 2, 2001, pp Acar, E., Hatka, R.T., Kim, N.H., and Buchi, D., "Eects o Structural Tests on Aircrat Saety," 50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conerence, Palm Springs, CA, May Zhu, H., Sankar, B.V., and Marrey, R.V., Evaluation o Failure Criteria or Fiber Composites Using Finite Element Micromechanics, Journal o Composite Materials, Vol. 32, No. 8, 1998, pp Federal Aviation Regulations, Part 25, Airworthiness Standards: Transport Category Airplanes, Sec , Factor o Saety. 27. Federal Aviation Regulations, Part 25, Airworthiness Standards: Transport Category Airplanes, Sec , Material Strength Properties and Material Design Values. 28. Composite Materials Handbook MIL-HDBK-17, Guidelines or Property Testing o Composites, ASTM Publications,

14 29. Federal Aviation Regulations, Part 25, Airworthiness Standards: Transport Category Airplanes, Sec , Proo o Structure. 30. Oberkamp, W.L., Deland, S.M., Rutherord, B.M., Diegert, K.V., and Alvin, K.F., Estimation o Total Uncertainty in Modeling and Simulation, Sandia Report, SAND , 2000, Albuquerque, NM. 31. Oberkamp, W.L., Deland, S.M., Rutherord, B.M., Diegert, K.V., and Alvin, K.F., Error and Uncertainty in Modeling and Simulation, Reliability Engineering and System Saety, Vol. 75, 2002, pp Acar, E., Kale, A., Hatka, R.T. and Stroud, W.J., "Structural Saety Measures or Airplanes," Journal o Aircrat, Vol. 43, No. 1, 2006, pp Acar, E., Kale, A., and Hatka, R.T., "Comparing Eectiveness o Measures that Improve Aircrat Structural Saety," Journal o Aerospace Engineering, Vol. 20, No. 3, 2007, pp Antonsson, E. K., and Otto, K. N., 1995, Imprecision in Engineering Design, ASME J. Mech. Des. 117 pp Nikolaidis, E., Chen, S., Cudney, H., Hatka, R. T., and Rosca, R., 2004, Comparison o Probability and Possibility or Design Against Catastrophic Failure Under Uncertainty, ASME J. Mech. Des. 126, pp An, J., Acar, E., Hatka, R.T., Kim, N.H., Iju, P.G., and Johnson, T.F., "Being Conservative with a Limited Number o Test Results," Journal o Aircrat, Vol. 45, No. 6, 2008, pp Kaumann, M., Zenkert, D., and Wennhage, P., Integrated cost/weight optimization o aircrat structures, Structural and Multidisciplinary Optimization, DOI /s , Curran, R., Rothwell, A., and Castagne, S., Numerical Method or Cost-Weight Optimization o Stringer Skin Panels, Journal o Aircrat, Vol. 43, No. 1, 2006, pp Kim, H.A., Kennedy, D., and Gürdal, Z., Special issue on optimization o aerospace structures, Structural and Multidisciplinary Optimization, Vol. 36, No. 1, 2008, pp Anon., Materials Research to Meet 21st Century Deense Needs, Committee on Materials Research or Deense Ater Next, National Research Council, ISBN: , Available rom Noh, Y., Choi, K.K., and Du, L., Reliability-based design optimization o problems with correlated input variables using a Gaussian Copula, Structural and Multidisciplinary Optimization, Vol. 38, 2009, pp Smarslok, B., Hatka, R.T., and Kim, N-H, Comparison and Eiciency Analysis o Crude and Separable Monte Carlo Simulation Methods, 47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conerence, Newport, RI, April 2006, AIAA Paper Appendix A: Details o modeling errors and variability A. Errors in estimating material strength properties rom coupon testing Coupon tests are conducted to obtain the statistical characterization o material strength properties, such as ailure stress, and their corresponding design values (A-basis or B-basis). With a inite number n c o coupon tests, the statistical characterization involves errors. Thereore, the ulated values o the mean and the standard deviation o the ailure stress will be uncertain. We assume that the ailure stress ollows normal distribution, so the ulated mean also ollows normal distribution. In addition, when n c is larger than 25, the distribution o the ulated standard deviation tends to be normal. Then, the ulated ailure stress can be expressed as c Normal c ; Std c (A1) where ulated mean and the ulated apparent standard deviation can be expressed as c Std Normal ; n c nc 3 nc nc 1 nc 1 Std c Normal Std ; Std (A3) 2 2 where and Std( ) are, respectively, the true values o the mean and standard deviation o ailure stress. Note that Eqs. (A1) (A3) describe a random variable coming rom a distribution (normal) whose parameters are also random. In this paper, this will be reerred to as a distribution o distributions. (A2) 14

15 The allowable stress at the coupon level, ca, is computed rom the ailure stress ulated at the coupon level, c, by using a knockdown actor, k d, as ca d c k (A4) The knockdown actor k d is speciied by the FAA regulations (FAR). For instance, or the B-basis value o the ailure stress, 90% o the ailure stresses (measured in coupon tests) must exceed the allowable stress with 95% conidence. The requirement o 90% probability and 95% conidence is responsible or the knockdown actor k d in Eq. (A4). For normal distribution, the knockdown actor depends on the number coupon tests and the c.o.v. o the ailure stress as k 1 k c (A5) d B c where c c is the c.o.v. o ailure stress ulated rom coupon tests, and k B is called tolerance limit actor [Re. 28]. The tolerance limit actor k B is a unction o the number o coupon tests n c as given in Re. [28] (Volume 1, Chapter 8, page 84) as 3.19 kb exp ln( nc) (A6) nc The variation o the tolerance coeicient with the number o coupon tests is depicted in Fig. A1. Note that Eq. (A6) is valid or normal distribution. Figure A1. Variation o the tolerance coeicient with the number o coupon tests B. Errors in structural element strength predictions The second level in the testing sequence is structural element testing, where structural elements are tested to validate the accuracy o the ailure criterion used (e.g., Tsai-Wu). Here, we assume that structural element tests are conducted or a speciied combination o loads corresponding to critical loading. For this load combination, the ailure surace can be boiled down to a single ailure stress where the subscript e stands or structural element tests. The mean ailure stress o the elements e can be predicted rom the mean ailure stress o the coupons c through e k (A7) e 2d c 15

16 where k 2d is the ratio between the unidirectional ailure stress and the ailure stress o a ply in a laminate under combined loading. I the ailure theory used to predict the ailure was perect, and we perormed ininite number o coupon tests, then we could predict k exactly, and the actual value would vary only due to material variability. 2d However, neither the ailure theory is perect nor ininite tests are perormed, so the ulated value o k 2d will be 1 k e k (A8) 2d e 2d where e e is the error in the ailure theory. Note that the sign in ront o the error term is negative, since we consistently ormulate the error expressions such that a positive error implies a conservative decision. Then, the ulated value o the mean ailure stress at the element level can be related to the ulated value o the mean ailure stress at the coupon level via Here, we take k 2d k e 2d 1 e c e k 2d c = 1 or simplicity. So, we have (A9) e 1 ee c (A10) The initial distribution o e is obtained by estimate o e e and using the results o coupon tests c inormation rom element tests is used by perorming Bayesian procedure to update the ailure stress distribution (see Appendix B or details). In practice, simpler procedures are oten used, such as selecting the lowest ailure stress rom element tests. Thereore, our assumption will tend to overestimate the beneicial eect o element tests. I Bayesian updating were used directly within the main MCS loop or design load carrying area determination, the computational cost would be very high. Instead, Bayesian updating is perormed outside rom the MCS loop or a range o possible test results. It is important to note that the error deinition used in the Bayesian updating code is dierent rom the error deinition used in the MCS code. In the Bayesian updating code, the error is measured rom the ulated values o the ailure stress, e, such that the true and the ulated values o the ailure stress are related through e 1 error e true 16. The. In the MCS code, on the other hand, the error is measured rom the true value o the ailure stress such that the true and the ulated values o the ailure stress are related through e 1 ee e. Thereore, while the Bayesian updating is implemented, a random error e generated in true the main MCS code is transerred to error 1 e 1 e 1 while running the Bayesian updating code. This complication relects the act that in the MCS loop we consider many possible element analysis and test results, while the engineer carrying the element tests has a unique set o computations and test results. The allowable stress based on the element test is ulated rom ea d e k (A11) Here the updated value o the mean ailure stress updated e is used, which corresponds to the most likely value o the mean ailure stress (having the maximum PDF). Combining Eqs. (A4), (A10) and (A11), we have 1 e (A12) ea e ca

17 C. Errors in structural strength predictions Due to the complexity o the overall structural system, there will be errors in ailure prediction o the overall structure that we denote as e. I we ollow the ormulation we used in expressing e in terms o c, the ulated mean ailure stress o the overall structure,, can be expressed in terms o the ulated mean ailure stress o the structural element, e, through 1 e e (A13) The allowable stress at the structural design level, a, can be related to the allowable stress computed at the element level, ea, through the ollowing relation k 1 e (A14) a ea where k is an additional knockdown actor used at the structural level as an extra precaution. Here Combining Eqs. (A12) and (A14), we can obtain 1 1 a e ac k is taken e e k (A15) D. Errors in design As noted earlier, along with the errors in ailure stress predictions, there also exist errors in design and construction. Beore starting the structural design, aerodynamic analysis needs to be perormed to determine the loads acting on the aircrat. However, the ulated design load value, P, diers rom the actual design load P d under conditions corresponding to FAA design speciications (e.g., gust-strength speciications). Since each company has dierent design practices, the error in load ulation, e p, is dierent rom one company to another. The ulated design load P is expressed in terms o the true design load P d as P (1 e ) P (A16) P d Notice here that the sign in ront o the load error term is positive while the sign in ront o the ailure stress error terms were negative. The reason or this choice, as we noted earlier, is that we consistently ormulate the error expressions such that a positive error implies a conservative decision. Besides the error in load ulation, an aircrat company may also make errors in stress ulation. We consider a small region in a structural part, characterized by a thickness t and width w, that resists the load in that region. The value o the stress in a structural part ulated by the stress analysis team, σ, can be expressed in terms o the load values ulated by the load team P, the design width w design, and the thickness t o the structural part by introducing the term e σ representing error in the stress analysis P (1 e ) (A17) w t In this paper, we assume that the aircrat companies have the capability o predicting the stresses very accurately so that the eect o e is negligible and is taken as zero. The ulated stress value is then used by a structural designer to ulate the design thickness t design. That is, the design thickness can be ormulated as design t design S 1 F P e P SF Pd w w k 1e 1ee design a design ca (A18) 17

18 Then, the design value o the load carrying area can be expressed as A t w design design design 1 e P F d 1 1 e S P e e k ca (A19) E. Errors in construction In addition to the above errors, there will also be construction errors in the geometric parameters. These construction errors represent the dierence between the values o these parameters in an average airplane (leetaverage) built by an aircrat company and the design values o these parameters. The error in width, e w, represents the deviation o the design width o the structural part, part built by the company, wbuilt av. Thus, w design, rom the average value o the width o the structural wbuilt av 1 ew wdesign (A20) Similarly, the built thickness value will dier rom its design value such that Then, the built load carrying area t 1 e t (A21) built av t design Abuilt av can be expressed using the irst equality o Eq. (21) as 1 1 Abuilt av et ew Adesign (A22) Table A1 presents nominal values or the errors assumed here. In the results section o the paper we will vary these error bounds and investigate the eects o these changes on the probability o ailure. Table A1. Distribution o error terms and their bounds Error actors Distribution Type Mean Bounds Error in load ulation, e P Uniorm 0.0 ± 10% Error in width, e w Uniorm 0.0 ± 1% Error in thickness, e t Uniorm 0.0 ± 3% Error in ailure prediction, e Uniorm 0.0 ± 10% Error in ailure prediction, e e Uniorm 0.0 ± 10% The errors here are modeled by uniorm distributions, ollowing the principle o maximum entropy. For instance, the error in the built thickness o a structural part (e t ) is deined in terms o the error bound b using et U 0,bt built av (A23) Here U indicates that the distribution is uniorm, 0 (zero) is the average value o e t, and the error bound is b =0.03. Hence, the lower bound or the thickness value is the average value minus 3% o the average and t built av the upper bound or the thickness value is the average value plus 3% o the average. Commonly available random number generators provide random numbers uniormly distributed between 0 and 1. Then, the error in the built thickness can be ulated using such random numbers r as 2 1 et r bt (A24) built av t built F. Total error, e total The expression or the built load carrying area o a structural part computed based on coupon test results, A, can be reormulated by combining Eqs. (A19) and (A22) as built av c 18

19 where e A total SFPd 1 etotal (A25) k built av c ca 1 e 1 e 1 e P t w 1e 1ee 1 (A26) Here e total represents the cumulative eect o the individual errors on the load carrying capacity o the structural part. G. Redesign based on element tests Besides updating the ailure stress distribution, element tests have an important role o leading to design changes i the design is unsae or overly conservative. That is, i very large or very small ailure stress values are obtained rom the element tests, the company may want to increase or reduce the load carrying area o the elements. We did not ind published data on redesign practices, and so we devised a common sense approach. We assumed that i the B-basis value obtained ater element tests,, is more than 5% higher than the B-basis value obtained rom coupon ea tests, ca, then the load carrying area is reduced by ca / ea ratio. I the B-basis value obtained ater element tests is more than 2% lower than the B-basis value obtained rom coupon tests, the load carrying area is increased by ca / ea amount. This lower tolerance relects the need or saety. Otherwise, no redesign was perormed. The built load carrying area can be revised by multiplying Eq. (27) by a redesign correction actor c r as where SFPd Abuilt av cr Abuilt avc 1 etotal cr (A27) k c 1 (no redesign) c r r ca 1.01 CF.. (redesign) (A28) Since redesign requires new elements to be built and tested, it is costly. Thereore, we do not have a redesign over redesigned elements. In order to protect against uncertainties in the test o the redesigned element we have an additional 1% reduction in the ulated allowable value (see the term 1.01 in Eq. (A28)). H. Variability In the previous sections, we analyzed the dierent types o errors made in the design and construction stages, representing the dierences between the leet average values o geometry, material and loading parameters and their corresponding design values. For a given design, these parameters vary rom one aircrat to another in the leet due to variability in tooling, construction, lying environment, etc. For instance, the actual value o the thickness o a structural part, tbuilt var, is deined in terms o its leet average built value, t built av, by area t built var 1 vt tbuilt av (A29) We assume that v t has a uniorm distribution with 3% bounds (see Table A2). Then, the actual load carrying A can be deined as built var Abuilt var tbuilt var wbuilt var 1 vt 1 vw Abuilt av (A30) where v w represents eect o the variability on the leet average built width. Table A2 presents the assumed distributions or variabilities. Note that the thickness error in Table A1 is uniormly distributed with bounds o ±3%. Thus the dierence between all thicknesses over the leets o all companies is up to ±6%. However, the combined eect o the uniormly distributed error and variability is not uniormly distributed. 19

20 The loading is assumed to ollow Type I asymptotic distribution since we consider the maximum load over lietime. We assume that one o the aircrat in the leet will experience the limit load over its service lie. We assume that a typical airliner has a production line o 1,000 aircrat. Thus, the distribution parameters o the loading are computed such that the probability o an aircrat experiencing limit load over its service lie is equal to 1/1000, and the coeicient o variation o the loading is 10%. The limit load is equal to 1/S F =2/3 or our problem. The distribution parameters are ound as a=28.73 and b=0.4263, when the CDF o the Type I asymptotic distribution is deined as exp exp FX x a x b (A31) Table A2. Distribution o random variables having variability Variables Distribution type Mean Scatter Actual service load, P act Type I asymptotic a=28.73 b= Actual built width, wbuilt var Uniorm wbuilt av 1% bounds Actual built thickness, tbuilt var Uniorm tbuilt av 3% bounds Failure stress, σ Normal 1.0 8% c.o.v.** v Uniorm 0 1% bounds w v Uniorm 0 3% bounds t * For the loading a and b are not the mean and scatter o the distribution **c.o.v.= coeicient o variation I. Certiication test Ater a structural part has been built with random errors in stress, load, width, allowable stress and thickness, it may ail in certiication testing part o the airplane. Recall that the structural part will not be manuactured with complete idelity to the design due to variability in the geometric properties. That is, the actual values o these parameters wbuilt var and tbuilt var will be dierent rom their leet-average values wbuilt av and tbuilt av due to variability. The structural part is then loaded with the design axial orce o S F times P, and i the stress exceeds the ailure stress o the structure σ, then the structure ails and the design is rejected; otherwise it is certiied or use. That is, the structural part is certiied i the ollowing inequality is satisied SF P 0 v v A 1 1 t w built av (A32) The details o the Monte Carlo procedure are provided in Table A3. Table A3. Monte Carlo simulation procedure or probability o ailure ulation 1. Compute the allowable stress based on coupon tests, ca 2. Calculate the built average load carrying area using the results o coupon tests, SFPd 1 Abuilt avc (1 etotal ) kw design ca 3. Generate random numbers or the quantiles o the updated mean ailure stress 4. Calculate the B-basis value using the quantiles, ea 1be 2c n a. Compute the bounds or mean ailure stress lb 1 e e c e 1be 2c n and ub 1 e e c b. Compute the PDF o the mean ailure stress using Johnson distribution with quantiles computed in Step 3, and select the mean ailure stress value with the highest PDF within the bounds as updated e. e 20

21 c. Compute B-basis value, ea 1 kb cc e 5. Compute a correction actor or the B-basis value, updated That is, i CF < 0.9, then CF = 0.9. I CF > 1.1, then CF = 1.1. ea CF. Limit the value o CF to [0.9, 1.1]. ca 6. Revise the built average load carrying area based on the value o CF. a. I CF 0.98, then redesign is needed, we will increase the load carrying area by CF. Hence, the new load 1.01 carrying area is Abuilt av Abuilt avc. Here the actor 1.01 is used to avoid a second redesign o CF elements. b. I 0.98 CF 1.05, then no redesign is needed. So, the load carrying area is Abuilt av Abuilt avc. c. I CF 1.05, then redesign is needed, we will decrease the load carrying area by CF. Hence, the new load 1.01 carrying area is Abuilt av Abuilt avc. Here again the actor 1.01 is used to avoid a second redesign o CF elements. 7. Using Abuilt av, compute the probability o ailure in service (P ), and probability o ailure in certiication test (PFCT). Appendix B: Bayesian updating o the ailure stress distribution rom the results o element tests The initial distribution o the element ailure stress is obtained by using a ailure criterion (e.g., Tsai-Wu theory) using the results o coupon tests. There will be two sources o error in this prediction. First, since a inite number o coupon tests are perormed, the mean and standard deviation o the ailure stress obtained through the coupon tests will be dierent rom the actual mean and standard deviation. We consider a typical situation relating to updating analytical predictions o strength based on tests. We assume that the analytical prediction o the ailure stress o a structural element, e, applies to the average ailure stress e o an ininite number o nominally identical structural elements. The error e e o our analytical true prediction is deined by e 1 ee e true (B1) Here we assume that the designer can estimate the bounds b e (possibly conservative) on the magnitude o the error, and we urther assume that the errors have a uniorm distribution between the bounds. Note here that it is more convenient to deine the error to be measured rom the ulated values o the ailure stress as shown in Fig B1. As in our earlier work [36], we neglect the eect o coupon tests and assumed the initial distribution o the mean ini ailure stress e uniorm within the bounds b e as ini e 2be e e 1 e i 1 b 0 othewise e (B2) 21

22 Probability density ( ) v v ( ) e e true Figure B1: Error and variability in ailure stress. The error is centered around the computed value, and is assumed to be uniormly distributed here. The variability distribution, on the other hand, is lognormal with mean equal to the true average ailure stress. test true Then, the distribution o the mean ailure stress is updated using the Bayesian updating with a given 1, ini 1, test e upd e e ini 1, test e e d e where 1, test e Normal e ;, 1, test Std e irst test result e. Note that 1, test 1,test e is the likelihood unction relecting possible variability o the density o obtaining test result e 1, test as (B3) is not a probability distribution in e ; it is the conditional probability, given that the mean value o the ailure stress is e. Subsequent tests are test handled by the same equations, using the updated distribution, as the initial one. I the Bayesian updating procedure deined above is used directly within an MCS loop or design load carrying area determination, the computational cost will be very high. In this paper, instead, the Bayesian updating is perormed aside rom the MCS loop. In this separate loop, we irst simulate the coupon tests by drawing random samples or the mean and standard deviation o the ulated ailure stress c Std c. Then, we simulate n e number o element tests, e. The element test results along with the mean and the standard deviation are used test and to deine the likelihood unction as 1, test e Normal e ;, 1, test c Std c ini distribution e in Eq. (B3). The initial in Eq. (B3) is uniormly distributed within some bounds as given in Eq. (B4). ini 1 e i 1 be e 2bec c 0 othewise We ound that applying the error bounds b e beore the Bayesian updating or ater the updating do not matter. ini Applying the error bounds beore Bayesian updating means ulating the initial distribution e (B4) rom Eq. (B4) and then using Eq. (B3). To apply the error bounds ater the Bayesian updating, however, we irst assume very 22

23 large error bounds e ini b, ulate the initial distribution e b e to the distribution obtained using Eq. (B3). rom Eq. (B4), and inally apply the error bounds Applying the error bounds ater the Bayesian updating is more useul when we want to it distributions (e.g., Johnson distribution) to the mean ailure stress obtained through Bayesian updating. I we apply the error bounds at the beginning, the distribution ater Eq. (B3) will be a truncated one and it will be diicult to it a distribution with good idelity. However, i we apply the error bounds at the end, the distribution ater Eq. (B3) will be a continuous one and we will high likely it a good distribution. So the overall procedure is as ollows. Within an MCS loop, we generate random mean and standard deviation values or the ailure stress to be obtained through coupon tests. Then, we assume large error bounds to be used in Eq. (B4), simulate element tests and use Eq. (B3) to obtain the distribution o the mean ailure stress. Then, we compute the our quantiles o the mean ailure stress distribution. Finally, we compute the mean and standard deviations o the quantiles and we model these quantiles as normal distributions. Note that the quantiles are the values o ailure stress or CDF values o [0.067, 0.309, 0.691, 0.933]. The quantiles are unctions o the number o coupon tests (n c ), number o element tests (n e ), and the error in ailure stress prediction (e e ). At irst, we wanted to build response surace approximations (RSA) or the mean and standard deviation o the quantiles in terms o n c and e e ater each element test. So we would have ten RSAs (ive or the mean and ive or the standard deviation) in terms o n c and e e. Our numerical analysis revealed, on the other hand, that n c do not have a noticeable eect on quantiles (see Tables B1 through B3 alongside Figure B2), and the eect o the error can be represented by just multiplying the quantiles with (1-e e ) term (see Table B4). As noted earlier, the quantiles are assumed to have normal distributions. Figure B3 show the histograms o the irst and second quantiles o the mean ailure stress (or 50 coupon tests ater the third element test when e e = 0) obtained through MCS with 20,000 samples. We see that the quantiles do not exactly ollow normal distributions. Table B1. The mean and standard deviation o the quantiles o the mean ailure stress ater element tests i 30 coupon tests are perormed. Mean values o the quantiles (Q 1-4 ) Standard deviation o the quantiles (Q 1-4 ) Q 1 Q 2 Q 3 4 Q std Q std Q std Q std Q test test test test test Table B2. The mean and standard deviation o the quantiles o the mean ailure stress ater element tests i 50 coupon tests are perormed. Mean values o the quantiles (Q 1-4 ) Standard deviation o the quantiles (Q 1-4 ) Q 1 Q 2 Q 3 Q std 4 Q 1 std Q 2 std Q 3 std Q 4 test test test test test

24 Table B3. The mean and standard deviation o the quantiles o the mean ailure stress ater element tests i 80 coupon tests are perormed. Mean values o the quantiles (Q 1-4 ) Standard deviation o the quantiles (Q 1-4 ) Q 1 Q 2 Q 3 Q std 4 Q 1 std Q 2 std Q 3 std Q 4 test test test test test (a) mean value o the irst quartile (a) standard deviation o the irst quartile Figure B2. Variation o the mean and standard deviation o the irst quartile o the mean ailure stress with number o coupon tests (ater the third element test) Table B4. The variation o the mean and standard deviation o the quantiles o the mean ailure stress with the error in ailure stress prediction, e e Mean values o the quantiles (Q 1-4 ) Standard deviation o the quantiles (Q 1-4 ) e Q 1 Q 2 Q 3 Q std 4 Q 1 std Q 2 std Q 3 std Q The results obtained in this separate MCS loop are used in the main MCS loop or determining the built average load carrying area. The mean and standard deviations o the quantiles are used to it a Johnson distribution to the mean ailure stress. The error bounds are then applied to the Johnson distribution and random values rom this distribution are drawn whenever element tests are simulated. Note also that the quantiles are strongly correlated to each other, so this correlation is also included in our analysis while random quantiles are generated in the main MCS loop using Gaussian copula. The reader is reerred to the work o Noh et al. [41] or urther details o reliability estimation o problems with correlated input variables using a Gaussian Copula. 24

25 (a) histogram o the irst quartile (a) histogram o the second quartile Figure B3. Histograms o the irst and the second quantiles o the mean ailure stress (ater the third element test). The continuous lines show the normal its. Appendix C. Separable Monte Carlo simulations The prediction o probability o ailure via conventional MCS requires trillions o simulations or level o 10-7 ailure probability. In order to address the computational burden, separable Monte Carlo procedure can be used. The reader is reerred to Smarslok and Hatka [42] or more inormation on the separable Monte Carlo procedure. This procedure applies when the ailure condition can be expressed as g 1 (x 1 )>g 2 (x 2 ), where x 1 and x 2 are two disjoint sets o random variables. To take advantage o this procedure, we need to ormulate the ailure condition in a separable orm, so that g 1 will depend only on variabilities and g 2 only on errors. The common ormulation o the structural ailure condition is in the orm o a stress exceeding the material limit. This orm, however, does not satisy the separability requirement. For example, the stress depends on variability in material properties as well as design area, which relects errors in the analysis process. To bring the ailure condition to the right orm, we instead ormulate it as the required cross sectional area A being larger than the built area A. So, the ailure condition can be deined in terms o the built area and the required area as: req built av A builtav req 1v 1v t A w A req (C1) where A req is the cross-sectional area required to carry the actual loading conditions or a particular copy o an aircrat model, and A req is what the built area (leet-average) needs to be in order or the particular copy to have the required area ater allowing or variability in width and thickness. Areq Pact (C2) The required area depends only on variability, while the built area depends only on errors. When certiication testing is taken into account, the built area, Abuilt av, is replaced by the certiied area, A cert, which is the same as the built area or companies that pass certiication. However, companies that ail are not included. That is, the ailure condition is written as ailure without certiication tests: A A 0 (C3a) built av ailure with certiication tests: A A 0 (C3b) cert req req 25