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PROBABILISTIC OPTIMIZATION OF A WINGBOX MODEL USING EXACTCAPACITYAPPROXIMATERESPONSEDISTRIBUTION (ECARD) By RICHARD J. PIPPY A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2008 2008 Richard J. Pippy To my parents, Richard and Evelyn, and to my brother Chris ACKNOWLEDGMENTS I would like to express my sincere gratitude to my advisor and the chair of my thesis committee, Dr. NamHo Kim, and to the cochair, Dr. Rafael T. Haftka, for their guidance, enthusiasm and continual support throughout my study. I am also grateful to committee member Dr. Peter G. Ifju for his advice and patience in reviewing this thesis. Special thanks go to Mulu Haile for his help in deriving the wingbox load conditions. I would like to thank my colleagues in the Structural and Multidisciplinary Optimization Lab at the University of Florida, in particular, Sunil Kumar, Dr. Erdem Acar, Palani Ramu, Haoyu Wang and Saad M. Mukras for their help and encouragement. TABLE OF CONTENTS page A CK N O W LED G M EN T S ................................................................. ........... ............. ..... L IST O F T A B L E S ............. ..... ............ ................................................................... . 7 LIST OF FIGURES ......................... ........ .. .... .... ............... 8 A B S T R A C T ............ ................... ............ ................................ ................ .. 9 CHAPTER 1 IN T R O D U C T IO N ............................... .................. .......................................................... .. 1 2 EXACTCAPACITYRESPONSEDISTRIBUTION (ECARD) THEORY ......................... 13 In tro d u ctio n ................... ...................1...................3.......... Characteristic Response ................................................... ............... ...... ..... 13 C o rre ctio n F a cto r ....................................................................................................16 U sin g F O R M ........................................................................................................1 8 U sin g M C S ......................................................................................................1 8 Approximate Probabilistic Optimization............................................... 19 3 ANALYTICAL EXAMPLE: APPLICATION OF ECARD TO A TENBAR TRUSS........23 In tro d u ctio n ................... ...................2...................3.......... P ro b lem D e scrip tio n ............................................................................................................... 2 3 Deterministic Optimization ............................. ............... 24 Probability of Failure Calculation Using MCS .................. ...............24 Probabilistic Optimization ..... ............................................................ ..... .... .........27 Approximate Probabilistic Optimization Using ECARD ....................................................28 4 PRACTICAL EXAMPLE: APPLICATION OF ECARD TO A WINGBOX MODEL .......36 In tro d u c tio n ....................................................................................................................... 3 6 P ro b lem D e scrip tio n ............................................................................................................... 3 6 G e o m e try ................................................................................................................... 3 6 Loading Calculations ................................. .........................................37 Deterministic Optimization .......................................................39 Probability of Failure Calculation U sing M CS ............................... ............ ..................... 40 Approximate Probabilistic Optimization Using ECARD ....................................................41 5 SUMMARY AND CONCLUSIONS .................................................52 APPENDIX CALCULATION OF MEMBER FORCES OF THE TEN BAR TRUSS .......53 L IST O F R E FE R EN C E S ....................................................... ................................... 55 B IO G R A PH IC A L SK E T C H .............................................................................. .....................57 6 LIST OF TABLES Table page 31 Param eters for the tenbar truss problem ................................................ ........ ....... 30 32 Results of deterministic optimization of the tenbar truss problem................................30 33 Probabilistic distribution types, parameters of errors and variabilities in the tenbar tru ss p ro b lem ....................................................................... .. 3 1 34 Probabilities of failure of the deterministic optimum areas .......................................31 35 Results of the probabilistic optimization of the tenbar truss ................. ............ .......32 36 Results of the ECARD optimization....................................................... ...................33 37 Results of the ECARD optimization....................................................... ..................34 41 Material Properties of 7150T77 Aluminum ...............................................................43 42 Results of deterministic optimization of the wingbox model ................. ............ .......43 43 V ariability for W ing M odel ....................................................................... ..................43 44 Probabilities of failure of the deterministic optimum design .........................................43 45 E C A R D optim ization results .................................................................. .....................43 LIST OF FIGURES Figure page 22 Calculation of the probability of failure at new design.................. .................................21 23 Calculation of the probability of failure at new design.................. .................................22 31 Geometry and loadings of the tenbar truss ....................................................................35 41 B oeing 767 w ing dim ensions............................................................................. ............44 42 B oeing 767 internal schem atic................................................ ............................... 45 43 A N SY S m odel of the w ingbox .................................................................................46 45 Sw eep of the quarterchord ............................................................................... ........ 47 46 Relationship of local lift distribution and taper ratio .....................................................48 47 Elliptical lift distribution from the root to the tip of the wingbox model .........................48 48 Equilibrium of forces on the wingbox model ........................................ .....................49 49 Pressure distribution from root to wingtip of the model..............................................49 410 Force distribution from root to wingtip of the model .......... ............ ............... 50 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science APPROXIMATE PROBABILISTIC OPTIMIZATION OF A WINGBOX MODEL USING EXACTCAPACITYRESPONSEDISTRIBUTION (ECARD) By Richard J. Pippy May 2008 Chair: NamHo Kim Cochair: Rafael T. Haftka Major: Mechanical Engineering There are two major obstacles that affect probabilistic (or reliabilitybased) structural optimization. First, uncertainties associated with errors in structural and aerodynamic modeling and quality of construction are not well characterized as statistical distributions and it has been shown that insufficient information may lead to large errors in probability calculations. Second, probabilistic optimization is computationally expensive due to multiple analyses, typically finite element analyses, for calculating probability as the structure is being redesigned. In order to overcome these obstacles, we propose an approximate probabilistic optimization method where the probabilistic calculation is confined to failure stress. This takes advantage of the fact that statistical characterization of failure stresses is required by Federal Aviation Administration regulations. The need for expensive stress distribution calculations are eliminated by condensing the stress distribution into a representative deterministic value, transforming a probabilistic optimization problem into a semideterministic optimization problem. By starting the approximate probabilistic optimization from the deterministic optimum design, a small number of iterations is expected and reliability analysis is required only once per iteration. This proposed method provides approximate sensitivity of failure probability with respect to the design variables, which is essential in risk allocation. This method is demonstrated in two examples. The first example uses a tenbar truss which demonstrates the risk allocation between the truss elements. The second example uses a wingbox model based on a Boeing 767400 aircraft which demonstrates the risk allocation between two different failure modes of stress and displacement. CHAPTER 1 INTRODUCTION There are two major barriers in front of probabilistic (or reliabilitybased) structural optimization. First, uncertainties associated with material properties, operating conditions, mathematical models, and manufacturing variability are not well characterized as statistical distributions, and insufficient information may lead to large errors in probability calculations (e.g., BenHaim and Elishakoff [1], Neal, et al. [2]). Due to this fact, many engineers are reluctant to pursue probabilistic design. The second barrier to the application of probabilistic structural optimization is computational expense. Probabilistic structural optimization is expensive because repeated stress calculations (typically FEA) are required for updating probability calculation as the structure is being redesigned. Targeting these two main barriers, we propose an approximate probabilistic optimization method that dispenses with expensive probabilistic stress calculations. In the proposed method, the probabilistic calculation is confined only to failure stress, which is often well characterized. Traditionally, reliabilitybased design optimization (RBDO) is performed based on a doubleloop optimization scheme, where the outer loop is used for design optimization while the inner loop performs a suboptimization for reliability analysis, using methods such as FirstOrder Reliability Method (FORM). Since this traditional approach is computationally expensive, even prohibitive for problems that require complex finite element analysis (FEA), alternative methods have been proposed by many researchers (e.g., Lee and Kwak [3], Kiureghian et al. [4], Tu et al. [5], Lee et al. [6], Qu and Haftka [7], Du and Chen [8] and Baabbad et al. [9]). These methods replace the probabilistic optimization with sequential deterministic optimization (often using inverse reliability measures) to reduce the computational expense. However, these approaches do not necessarily converge to the optimum design, and they do not easily lend themselves to allocating risk between failure modes in a structure where many components can fail [10]. We note, however, that most of the computational expense is associated with repeated stress calculation. So we propose an approximate probabilistic design approach that reduces the number of expensive stress calculations. That is, we approximate the probabilistic optimization that separates the uncertainties which can be evaluated inexpensively and those whose effects are expensive to evaluate. We boil down the stress distribution to a single characteristic stress by utilizing the inverse cumulative distribution of the failure stress, and we propose an inexpensive approximation of that characteristic stress. This proposed method will also improve upon a deterministic design by reallocating the safety margins between different components or failure modes. We call the proposed approximate probabilistic optimization approach ExactCapacity ApproximateResponseDistribution or ECARD. The purpose of this thesis is to further advance the version of ECARD which was originally developed by Dr. Erdem Acar and Dr. Rafael T. Haftka [11]. There is now an improved version of ECARD which was developed by Sunil Kumar, Dr. Rafael T. Haftka, and Dr. NamHo Kim. The remainder of the thesis is organized as follows. Chapter 2 discusses the theory behind the ECARD method. The application of the method to a tenbar truss problem is presented in Chapter 3 and a wingbox design in Chapter 4. Finally, the concluding remarks are listed Chapter 5. CHAPTER 2 EXACTCAPACITYRESPONSEDISTRIBUTION (ECARD) THEORY Introduction In this chapter, the approximate probabilistic optimization method, named ECARD, will be discussed. Characteristic Response In probabilistic optimization, the system constraint is often given in terms of failure probability of a performance function. We consider a specific form of performance function, given as g(x;u) = c(x;u) r(x;u) (21) where c(x; u) and r(x; u) are capacity and response, respectively. Both the capacity and response are random because they are functions of input random variables x and depend on deterministic design variables The system is considered to be failed when the response exceeds the capacity; i.e., g(x; u) < 0. We assume that the probabilistic distribution of c(x; u) is well known, while that of r(x; u) requires a large number of analyses. For example, when c(x; u) is failure stress and r(x; u) is the maximum stress of an aircraft structure, the probabilistic distribution of c(x; u) is already characterized by Federal Aviation Administration requirement. However, the probabilistic distribution of r(x; u) requires repeated computational simulations, such as finite element analysis. Since the performance function depends on two random variables, c(x; u) and r(x; u), the safety of the system can be estimated using a probability of failure, defined as Pf = Pr[g(x;u) <0] = _Fc( )fR()d (22) 00 In the above equation, Fc() is the cumulative distribution function (CDF) of capacity, and fR ({) is the probability density function (PDF) of response. The above integral can be evaluated using analytical integration, Monte Carlo simulation (MCS), or first/secondorder reliability method (FORM/SORM), among others. Smarslok et al. [18] presented a separable MCS, which is much more accurate than the traditional MCS when the performance can be separable as in Eq. 21. It is clear from Eq. 22 that accurate estimation of probability of failure requires accurate assessment of the probability distributions of both the response and capacity. When the capacity is the failure stress, the FAA requires aircraft builders to perform characterization tests in order to construct a statistical model, and then to select the allowable failure stress (Abasis or Bbasis value) based on this model. Hence, the CDF of capacity is often reasonably well characterized. On the other hand, the PDF of the response is poorly known, because it depends on the accuracy of various factors, such as material properties, operating conditions, mathematical models, and manufacturing variability. The key idea of this research is to express Pf in Eq. 22 using a characteristic value of the response, and approximate the change of Pfin terms of the change of this characteristic value. The calculation of Pf in Eq. 22 can be simplified by using the information ofFc('). From the Intermediate Value Theorem [19], there exist r* such that Eq. 22 can be rewritten as Pf = Fc (r*) fR ()d = Fc (r) (23) In the above equation, the second equality is obtained from the fact that the integral offR() is one. Equation 23 states that the effect of (the poorly characterized) probability distribution of the response can be boiled down to a single characteristic response, r*. When the probability of failure is given, the characteristic response can be calculated using the inverse transformation of Fc(r*), as r* = Fc(Pf) (24) When design variables are changed during optimization, it is possible that the distributions of both capacity and response may be changed. For the simplicity of presentation, we consider the case that the distribution of the capacity remains unchanged. We assume that the design change only affects the mean value of the response; i.e., the standard deviation remains constant. This assumption is reasonable if the design perturbation is small. In such a case, redesign changes the mean value of response from = r(px;u) to F(1 + A), where r(ux;u) is the value of response evaluated at the mean value of the input random variables. The variable A represents the relative change in response according to design change. At this point, A is unknown. Figure 21 illustrates the change in response distribution, along with the distribution of capacity. In this research, we start the probabilistic design from a known deterministic optimum. This is an important aspect of the approximate probabilistic optimization. Since the deterministic optimization uses safety margins to consider the effect of uncertainties, the deterministic optimum design is close to the probabilistic optimum design. This will satisfy the above assumption of small design change. The goal of proposed probabilistic design is then to improve upon the deterministic design by reallocating the safety margins between different components or failure modes. First, the change in design variables will change the mean of response from r to r(1 + A), while maintaining the same standard deviation. The change in the response distribution will then change the probability of failure according to Eq. 22. From Eq. 24, the characteristic response will also be changed from r* to r*(l+A*), where A* is the relative change in characteristic response. Unfortunately, this process requires calculation of the probability of failure at the new design. The novel idea of the proposed approach is to reverse this process by approximating the relation between A and A* so that the characteristic response at the new design can be calculated without performing reliability analysis. For the moment, let us assume that A* can be calculated from given A. Then, the probability of failure at the new design can be calculated from pew = Fc(r*(1l+ A*)) (25) The above probability of failure will be exact if A* is the correct relative change in characteristic response. When A* is an approximated one, the probability of failure in the above equation is approximate, and we will denote it P pPP"r. The procedure illustrated in Figure 22 does not require expensive reliability analysis. It is enough to analytically evaluate the value of the CDF at the perturbed characteristic response. Correction Factor The key idea of the proposed approximate probability distribution is that the new characteristic response can be approximated without recourse to the expensive reliability analysis. The simplest approximation, used in this research, is that the relative change in the characteristic response, A* is proportional to the relative change in response, A as A* = kA (26) where k is a proportional constant that depends on how the redesign affects the stress distribution. In fact, it is the sensitivity of the characteristic response change with respect to the response change. We call it a correction factor. The above assumption in linearity is reasonable when A is relatively small. Probabilistic optimization can be viewed as risk allocation between different failure modes or different structural members. This allocation requires the sensitivity of failure probability with respect to design variables. In the proposed approximate probabilistic optimization, this sensitivity information is presented in the correction factor. We will demonstrate that a linear relationship between A and A* works well given the assumption of translating the stress distribution, especially when the design change is relatively small. We consider a lognormally distributed capacity with mean value of [tc = 100 and coefficient of variation of 8%, and normally distributed response with coefficient of variation of 20%. From Eq. 22, the mean value of the response is chosen to be [LR = 42.49 so that the probability of failure becomes Pf = 107. For a given small value of A, a new probability of failure Pfew is calculated from Eq. 22 with the mean of the response being IR(1+A). The relative change in characteristic response A* is then obtained from Eq. 24 with pnew. Figure 23 shows the relation between A and A*. We can see that the linearity assumption is quite accurate over the range of 10% < A < 10%. The slope will be the correction factor k. Figure 24 shows the effect of the A approximation on the probability of failure. In practice, the correction factor can be calculated using a finite difference method, which requires at least two reliability calculations. We will describe the procedure using a forward finite difference method, but other method can also be used in a similar way. Let Ao = 0.0 corresponds to the current design, and Ap = 0.05 to the perturbed design. The correction factor can be calculated either using FORM or MSC. We will explain both cases. Using FORM First, the probability of failure, Pf, at the current design is calculated from FirstOrder Reliability Method (FORM) with the performance function in Eq. 21. If the response is perturbed by Ap, Eq. 21 becomes g,(x;u) = c(x;u) r(x;u)(1 + Ap) (27) Using the above equation, reliability analysis is performed to calculated the perturbed probability of failure, Pf It is noted that it is unnecessary to change design variable, u, because we directly perturb the output response. Thus, the computational cost of reliability analysis using Eq. 27 can be reduced significantly. Next, the characteristic responses are calculated from Eq. 2 4, as r* = Fc'(Pf) ,= (28) r = Fc'(P) = r(1+ A*) By comparing two terms in the above equation, the relative change in the characteristic response can be obtained as A* = 1 (29) r Then, the correction factor can be obtained from k= (210) Using MCS When MCS is employed, we generate N samples of response ri = r(xi;u),i = 1,...,N, at the current design. In view of Eq. 22, the probabilities of failure at the current and perturbed design can be calculated from N Pf FC (ri) (211) i1 N P = Fc (r( + A)) (212) i=1 The remaining procedure is identical to that of FORM. Even if Eq. 211 and Eq. 212 are two different MCS, they can be combined into one because the same sample, r,, will be used. Approximate Probabilistic Optimization The proposed approximate probabilistic optimization is composed of two stages: (1) correction factor k and initial probability of failure Pf are calculated from reliability analysis, and (2) a deterministic optimization problem is solved using the approximate probability of failure from Eq. 25. The first stage is computationally expensive, while the second stage is nothing but a semideterministic optimization. We will explain the approximate probabilistic optimization procedure, as follows. 1. Perform deterministic optimization with safety margin. The probabilistic design starts from the deterministic optimum design; i.e., initial design uo = Udet and cost function Wo = Wet. Calculate the initial probability of failure, Pf = Pdet atuo0. 2. At the current design, uo, calculate deterministic value of response, ro = r(jx;uo), using the mean value of input random variables. 3. Calculate the characteristic response, ro using the inverse CDF of the P P, and the mean and c.o.v of the response. 4. Calculate the correction factor, k, using the procedure in the previous section. 5. Obtain optimum design uopt and optimum objective function Wopt by solving the following optimization problem: min W(,x,u) U (213) s.t. Papprox < pet Where r =r(lx;u) (214) A =r 1 (215) ro A* =kA (216) r* = Fc'(Pf) (217) Pfapprox = Fc, (1 + A* ) (218) 6. Calculate the actual probability of failure, Pf, at Uopt. 7. Check convergence: Papprox PJf < e. If it converged, stop the process. Otherwise, set uo = Uopt and go to Step 2 and continue. The above semideterministic optimization process uses exact distribution of the capacity (Fc) and an approximate distribution of response (MCS or FORM). Due to this aspect, we call it ExactCapacityApproximateResponse Distribution (ECARD) method. The accuracy of ECARD to locate the true optimum depends on the magnitudes of errors involved in the approximations. As shown in Figure 23 and Figure 24, the approximation is accurate if changes of the response due to redesign are small. In addition, the accuracy in estimating the correction factor k affects the convergence rate of the proposed method. The result may be somewhat sub optimal because of the convergence condition and the approximate nature of the sensitivity of probability of failure. This issue will be discussed in detail in the following section. Changes in stress PDF 0.35  0.3  0.25  S015 a a(1+A) 0' C ' * 0 a 0.1 0.05 0 Initial Stress PDF New Stress PDF PDF of Failure Stress  Characterstic Stress  New Characterstic Stress 0 2 4 6 8 10 12 14 16 18 Stress Figure 21. Distributions of response before and after redesign. Fc (r) p new  Pf r r*(+ A*) Figure 22. Calculation of the probability of failure at new design. A* 'Ii' 0 06 0 08  0 1 0 08 0 06 0 04 0 02 0 002 004 006 008 01 A Figure 23. Calculation of the probability of failure at new design 106 10 f 107 0 1 0 08 0 06 004 002 0 002 004 006 008 A Figure 24. Calculation of the probability of failure at new design y~ approx i I I I I I CHAPTER 3 ANALYTICAL EXAMPLE: APPLICATION OF ECARD TO A TENBAR TRUSS Introduction The first demonstration example is a tenbar truss problem as shown in Figure 31. This example will demonstrate risk allocation between the different truss members. First, a brief description of the problem will be given. Then, a deterministic optimization of the problem will be presented followed by the probability of failure calculation using Monte Carlo simulations. Finally, the probabilistic and ECARD optimizations will be performed, and the efficiency and accuracy of the ECARD method will be discussed. Problem Description The problem description for the tenbar truss example was taken from Haftka and Gurdal [20] (page 237). The truss structure is under two loads, P; and P2. The design objective is to minimize the total weight of the truss, W, by varying the crosssectional areas, A1, of the members while satisfying minimum gage constraints and allowable stress constraints. Input data for the truss is listed in Table 31. Member 9 was assigned a higher failure stress value in order to make the fully stressed design nonoptimal. Aircraft design often uses a knockdown factor, Kdc, in order to conservatively estimate failure stress using Abasis or BBasis methods. The A basis (or Bbasis) failure stress is the value of a failure stress exceeded by 99% (or 90% for B basis) of the population with 95% confidence. In the conservative estimation, the allowable stress of a member is related to the mean value of the failure stress through the following equation: "allowable = KdcP/ (31) In the deterministic design process, the knockdown factor is a way of considering the uncertainty in the failure stress. Deterministic Optimization Using the safety factor and knockdown factor, the deterministic optimization problem can be formulated as 10 min W = pLA, A, i=1 N, (SFPI, SFP2, A) s.t. = ri ( allowable ),, (32) 0.1 < A, where L,, N, and A, are, respectively, the length, member force, and crosssectional area of element i. A is the vector of crosssectional areas, ci and (Gallowable)i are the stress and allowable stress of an element, respectively. For this example, Gi, corresponds to the response while (Gallowable)i to the capacity, and the loads are multiplied by a safety factor in order to consider various uncertainties involved in the truss parameters, applied load, and computational errors. The analytical solution for the member forces are given in the Appendix. The above optimization problem is solved using the "fmincon" function in MATLAB. The problem converged in 7 iterations with 97 function evaluations. Table 32 lists the results of the deterministic optimization. Note that elements 2, 5, and 6 crosssectional areas reach minimum gage while element 5 is a zero force member. At optimum design, stresses in all members except for Member 5 and 9 are at the allowable stress. Probability of Failure Calculation Using MCS In this section, the probability of failure of the truss at the deterministic optimum design is evaluated using Monte Carlo Simulation (MCS). There are two purposes in calculating the probability of failure. First, it can evaluate the level of safety of the deterministic optimum design. The effects of the knockdown factor and the factor of safety are evaluated in terms of the probability of failure. Second, it can be used for the design criterion in the probabilistic optimization. In the calculation of the probability of failure, the factor of safety and the knockdown factor will not be considered. Instead, uncertainties related to errors and variability in material properties, manufacturing tolerances, and applied loads will be considered in calculating the probability of failure. There are many uncertainties involved in the design of the tenbar truss, such as variability from material properties, loads, manufacturing, and errors from numerical calculation and modeling. Failure of an element is predicted to occur when the stress in an element is greater than its failure stress. Knowing this, the performance function can be written as g ( f ) 0t (33) where the subscript 'true' stands for the true value of the relevant quantity, which is different from its calculated (or predicted) value due to errors. Adding these errors, the equation can be rewritten as g= (e,)o(1+e)o (34) Here, efis the error in failure prediction, af is the predicted failure stress, e, is the error in stress calculation, and a is the calculated stress. The errors were formulated to where positive errors correspond to a conservative design. Therefore, the error in calculated stress is positive, while the error in predicted failure stress is negative. Even though the stress calculation is exact for the tenbar truss, the error, e,, was introduced to consider the analysis of a more complex structure where the stresses are calculated from numerical methods. The calculated stress can be written in the following form S= FEA[ P(l +ejP)P1,(1l+ep)P2,(l +eA )A] (35) where GFEA stands for calculated stresses using FEA, epi and ep2 are errors in loads P1 and P2, and eA is the vector of errors (tolerances) corresponding to ten crosssectional areas. By substituting Eq. 35 into Eq. 34, the performance function can now be rearranged in separable form (i.e., in a form that allows the use of separable MCS) for each element as (1+e) S= ("),r e) E(1+ ep l)P1,(1+ eP2 )P2,(1 + eA)A] c, (36) (i ef) where c, and r are respectively, the capacity and response. Beside errors, variabilities are introduced into the performance function through random variables af, P1, P2, and A. The probabilistic parameters of errors and variabilities and their distribution types are listed in Table 33. The probabilities of failure were calculated using separable MCS, which requires smaller number of simulations to achieve the same accuracy as crude MCS [17]. After calculating the probabilities of failure for each element, the total probability of failure of the system can be approximated as 10 PFS = (P)i (37) i=1 where PFs is the system probability of failure. Calculating the probability failure in this form is Ditlevesen's firstorder upper bound; therefore the system probability of failure is estimated conservatively. Using separable MCS with 106 samples, the probabilities of failure for each element and the system are listed in Table 34. The results show that members 2, 6, and 10's probability of failure contributes to 80% of the system probability of failure. In the deterministic design process, the uncertainty in the system is considered using safety measures, such as knockdown factor and the factor of safety. However, as is clear form Table 3 4, the effects of these safety measures are not evenly distributed between members. It appears that members 2, 6, and 10 are very sensitive to these safety measures, while other members are not. They are either at minimum gage or close to it, and yet, their probabilities of failure are relatively high compared to the other members. Thus, it is possible to move some of the weight from nonsensitive members to the sensitive ones so that the system probability of failure can be reduced further while maintaining the total weight of the truss. Or, it is also possible to reduce the weight of the truss while maintaining the same level of system probability of failure. The latter possibility will be investigated in the probabilistic optimization. Probabilistic Optimization Starting from the deterministic design, the probabilistic optimization problem can be formulated such that the weight of the structure is minimized, while maintaining the same level of system probability of failure with that of the deterministic optimum design. Thus, we have 10 min W = pLiA, A, i=1 (38) s.t. PFS < PFSt Results of the probabilistic optimization are shown in Table 35. A total of 105 samples are used for MCS. The optimization converged after 59 iterations and 728 reliability assessments. The relatively large number of reliability assessments is due to the fact that the problem has ten design variables. At each iteration, the optimization algorithm calculates sensitivity using finite different method. On the other hand, the proposed ECARD method perturbs the response directly. Thus, ECARD will be efficient when the number of response is smaller than that of design variables. The overall optimization took about 125 hours using a Dell desktop computer. In order to remove instability related to random samples, a set of input random variables are generated and repeatedly used during the optimization. Overall weight is reduced by 6% (90.47 lbs) while maintaining the same system probability of failure as that of the deterministic optimum design. This reduction is achieved by reallocating the risk from the higher risk members (2, 6, and 10) to the lower risk members. The probabilistic optimization slightly increased the crosssectional areas of members 2, 6, and 10, and decreased the crosssectional areas of the other members. While the remaining member's probabilities of failure increased slightly, members 2, 6, and 10 were reduced by an order of magnitude. This risk allocation can be achieved when the sensitivities of probability of failure and weight with respect to design variables are available. In the probabilistic optimization, these sensitivities are calculated using the finite difference method. That explains the 728 reliability assessments during the optimization. In the following section, the same optimization problem will be solved using the ECARD method, which requires a smaller amount of reliability assessments, and yet the sensitivity information can be obtained in the approximate sense. Approximate Probabilistic Optimization Using ECARD In the approximate probabilistic design, the same optimization problem is used except that the approximate probability of failure is used. Thus, the optimization problem can be written as 10 min W = pLiA, A, i= (39) s.t. Paprox < Pt where the approximate system probability of failure is the sum of each members contributions. Since there are ten members, ten characteristic responses and correction factors are calculated before the ECARD optimization. This calculation is equivalent to assessing the probability of failure twice. Then, the above optimization is deterministic because the approximate probability of failure can be evaluated without MCS. Since the approximation in the probability of failure is not accurate, the above ECARD optimization is repeated until the convergence criterion, as stated in Chapter 2, is satisfied. The accuracy of the characteristic response, which depends on the number of MCS samples, affects the number of iterations needed to reach an accurate optimum. A low number of samples may appear to reduce computational costs, but actually it reduces the confidence in the probability of failure calculation resulting in an increased number of iterations to reach the accurate optimum. The number of MCS samples must be chosen accordingly for each problem. The results of the ECARD optimization are displayed in Table 36. Using 105 MCS samples, the ECARD optimization needed only four iterations and 8 reliability assessments to reach close to the probabilistic optimum. This is a significant reduction from the 728 reliability assessments of the probabilistic optimization. The weight difference when comparing the fourth iteration to the third is 0.03%, while the approximate system probability of failure equals the deterministic system probability of failure. In addition, the errors in the member approximate probability of failure calculations are less than 2%. Since, the probability of failure for member nine is very small, its probability of failure error is not accurate and the error is ignored. As expected the ECARD optimization allocates the risk between members. The crosssectional areas of the smaller members increased while they decreased in the largest members. Table 31. Parameters for the tenbar truss problem Parameters Values Dimension, b 360 inches Safety factor, SF 1.5 Load, P1 66.67 kips Load, P2 66.67 kips Knockdown factor, Kdc 0.87 Density, r 0.1 lb/in3 Modulus of elasticity, E 104 ksi Allowable stress, Gallowable 25 ksi* 2 Minimum gage 0.1 in2 *for Element 9, allowable stress is 75 ksi Table 32. Results of deterministic optimization of the tenbar truss problem Element Adt (in) 1 7.900 2 0.100 3 8.100 4 3.900 5 0.100 6 7 8 9 10 Total 0.100 5.798 5.515 3.677 0.141 W, (lb) 284.400 3.600 291.600 140.400 3.600 3.600 295.200 280.800 187.200 7.200 1497.600 Stress (ksi) 25.0 25.0 25.0 25.0 0.0 25.0 25.0 25.0 37.5 25.0 Table 33. Probabilistic distribution types, parameters of errors and variabilities in the tenbar truss problem Distribution Uncertainties type Mean Scatter Errors e, Uniform 0 5% epi Uniform 0 + 10% ep2 Uniform 0 + 10% eA (10x1 vector) Uniform 0 + 3% ef Uniform 0 + 20% Variability P1, P2 Extreme type I 66.67 kips 10% c.o.v. A (10x1 vector) Uniform A (10x1 vector) 4% bounds 25/kdc ksi or / Lognormal 8% c.o.v. 75/kdc ksi Table 34. Probabilities of failure of the deterministic optimum areas Element Pe 1 2.13E03 2 1.06E02 3 4.80E04 4 2.19E03 5 4.04E04 6 1.07E02 7 1.69E03 8 1.89E03 9 5.47E13 10 1.07E02 Total 4.08E02 Table 35. Results of the probabilistic optimization of the tenbar truss Elements A"et (in) A, (P/ e), (Pf), 1 7.9 7.192 2.13E03 5.88E03 2 0.1 0.3243 1.06E02 3.07E03 3 8.1 7.162 4.80E04 8.26E03 4 3.9 3.701 2.19E03 2.15E03 5 0.1 0.4512 4.04E04 3.18E05 6 0.1 0.3337 1.07E02 2.14E03 7 5.798 5.1697 1.69E03 1.02E02 8 5.515 4.9782 1.89E03 3.75E03 9 3.677 3.5069 5.47E13 4.70E13 10 0.141 0.4325 1.07E02 5.46E03 Total 1497.6 lbs 1407.13 lbs 4.08E02 4.08E02 Table 36. Results of the ECARD optimization Element Determ. Des. Iter 1 Iter 2 Iter 3 Iter 4 AREAS (in2) 1 7.9000 7.4487 7.4787 7.4841 7.4849 2 0.1000 0.1000 0.1000 0.1000 0.1000 3 8.1000 7.0752 7.0406 7.0401 7.0402 4 3.9000 3.9382 3.9666 3.9710 3.9716 5 0.1000 0.1000 0.1000 0.1000 0.1000 6 0.1000 0.1000 0.1000 0.1000 0.1000 7 5.7980 5.0457 5.0440 5.0442 5.0441 8 5.5150 5.3538 5.3873 5.3941 5.3951 9 3.6770 3.8416 3.9657 3.9873 3.9908 10 0.1410 0.1314 0.1310 0.1309 0.1309 Weight (lb) 1497.60 1407.16 1415.94 1417.71 1418.00 MEAN STRESSES (ksi) 1 16.6667 17.7656 17.7047 17.6934 17.6918 2 16.6667 14.479 14.0059 13.9276 13.9147 3 16.6667 18.9868 19.0693 19.0688 19.0684 4 16.6667 16.5606 16.4537 16.4378 16.4354 5 0 4.462 4.7514 4.7913 4.7977 6 16.6667 14.479 14.0059 13.9276 13.9147 7 16.6667 18.966 18.951 18.9472 18.9468 8 16.6667 17.3456 17.2576 17.2391 17.2363 9 25 24.0093 23.2747 23.1515 23.1313 10 16.6667 15.5797 15.1199 15.0471 15.0354 Table 37. Results of the ECARD optimization Element Determ. Des. Iter 1 Iter 2 Iter 3 Iter 4 APPROXIMATE PF 1 2 3 4 5 6 7 8 9 10 SYSTEM 1 2 3 4 5 6 7 8 9 10 SYSTEM 5.65E03 2.16E03 7.44E03 1.97E03 4.04E04 2.17E03 1.23E02 3.59E03 3.09E14 5.17E03 4.08E02 5.26E03 2.11E03 7.51E03 1.77E03 1.72E03 2.09E03 1.20E02 3.22E03 2.50E15 5.21E03 4.08E02 2.13E03 1.06E02 4.80E04 2.19E03 4.04E04 1.07E02 1.69E03 1.89E03 5.47E13 1.07E02 4.08E02 2.13E03 1.06E02 4.80E04 2.19E03 4.04E04 1.07E02 1.69E03 1.89E03 5.47E13 1.07E02 4.08E02 5.21E03 2.10E03 7.51E03 1.74E03 1.86E03 2.09E03 1.20E02 3.17E03 1.67E15 5.22E03 4.08E02 5.21E03 2.13E03 7.51E03 1.74E03 1.88E03 2.11E03 1.20E02 3.17E03 1.59E15 5.27E03 4.10E02 5.20E03 2.10E03 7.50E03 1.74E03 1.88E03 2.09E03 1.20E02 3.16E03 6.66E16 5.23E03 4.08E02 5.20E03 2.11E03 7.50E03 1.74E03 1.88E03 2.09E03 1.20E02 3.16E03 1.48E15 5.24E03 4.08E02 ACTUAL PF 5.53E03 5.26E03 3.09E03 2.25E03 6.95E03 7.51E03 1.96E03 1.77E03 1.72E03 1.86E03 3.09E03 2.23E03 1.21E02 1.20E02 3.49E03 3.22E03 2.77E14 2.44E15 7.14E03 5.50E03 4.51E02 4.16E02 Figure 31. Geometry and loadings of the tenbar truss CHAPTER 4 PRACTICAL EXAMPLE: APPLICATION OF ECARD TO A WINGBOX MODEL Introduction The second demonstration example is the wingbox model problem. This example will demonstrate the risk allocation between different failure modes of stress and displacement while minimizing the total weight of the wingbox. Publically available information was used to model the wingbox as a Boeing 767 wing in a 2.5 g dive. First, a description of the problem will be given which will cover model geometry, and loading calculations. Then, a deterministic optimization of the problem will be presented followed by the probability of failure calculation using Monte Carlo simulations. Finally, the ECARD optimization will be performed, and the efficiency and accuracy of the ECARD method will be discussed. Problem Description Geometry The geometry of the wingbox was modeled using the open source Boeing 767 wing schematics shown in Figures 41 and 42. The root dimensions have a width of 170 inches with a height of 50 inch. The tip dimensions are 20 inches wide, 10 inches tall. The skin thickness at the root is 1.0 inches and decreases in steps linearly to 0.1 inches at the tip of the wing. The wing is 745 inches long from root to tip. The wingbox is modeled using ANSYS Parametric Design Language (APDL) and analyzed using ANSYS Finite Element Analysis software. Since the APDL is a parametric language, it can generate finite element models with different design values. The APDL code consists of a set of commands that instructs ANSYS to build the model, add the loading and boundary conditions, and conduct the FEA analysis. The design variables of the model are the skin thicknesses at the root and midpoint of the wingbox. The tip thickness is fixed to 0.1 in. The thickness varies linearly between the root and midpoint, and between the midpoint and the tip. Figure 43 shows the ANSYS model of the wingbox. The model contains a total of 5,105 nodes, 5,652 elements and was meshed using SHELL 63 Elastic Shell elements. The element contains four nodes, four thicknesses, and orthotropic material properties. It also has six degrees of freedom at each node, translations in the x, y, and z directions and rotations about the nodal x, y, and z axes. The boundary conditions are that all nodes at the root of the wing are constrained in all directions. Figure 44 shows the meshed wingbox model. The material properties of the model are based on 7150T77 aluminum, which were taken from Metallic Materials Properties Development and Standardization (MMPDS) [21]. These material properties are listed in Table 41. Loading Calculations The stresses and deflections of the wingbox model are calculated using mechanics of materials methods. Determination of these stresses and deflections requires simplifying assumptions on the geometry of the structure. Without losing generality, the following simplifying assumptions have been made: 1. The cross section of the wing is made of thin hollow rectangular box section and the platform is trapezoidal. The wing is modeled as a cantilever beam clamped at its root section and free at its ends. 2. The lift over the surface of the wing area is replaced by a line load (lift per unit length) elliptically distributed over a sweep of a line whose locus of points are onequarter of the chord from leading edge (c/4 line), as seen in Figure 45. Wing lift distribution is directly related to the wing geometry and determines such wing performance characteristics as induced drag, structural weight, and stalling characteristics. The distribution of the aerodynamic lift along the span of a wing is commonly regarded as elliptical and depends (among other things) on the taper ratio A. For an elliptical lift distribution shown in Figure 46, the lift, w on the total span length of both wings, L is defined by the equation of an ellipse. X2 W2 + + =1 (41) (L/2)2 w Where, Wo is the maximum load per unit length at the center of the fuselage (x = 0). The value of Wo is determined from the gross weight of the aircraft at landing. For straight and level flight, the total aerodynamic lift is equal to the area of the ellipse and must be equal to the gross weight of the aircraft. Hence, lift per wing is 1 Weight of Airplane x gforce = woL (42) 4 Now solving for wo in Equation 42 at 2.5 g, W w =3.181 g (43) L where Wg is the aircraft gross weight. The lift per unit length w at any section of the wing is, hence, given by 2w L W =  X2 (44) L 2 The lift distribution varies from 747.21 lbs/in at the root to 0 lbs/in at the wingtip. The plot of the lift distribution is seen in Figure 47. The elliptical lift distribution can now be converted into a pressure load, p on the wing and the force per unit length, F acting on the leading edge of the chord as shown in Figure 48. Equilibrium of forces in the vertical direction requires Pc+Fw=0 (45) Pc+F = w Equilibrium of the moments about the quarter chord requires c ( c c \ c c F+Pc =0 F +Pc=0 4 2 4 4 4 (46) F = Pc Solving forp, and F from Eq. 45 and Eq. 46 yields w P= 2c (47) F= 2 Using these equations, the pressure at the root was calculated to be 2.198 lbs/in2 to 0 lbs/in2 at the tip, while the force at the leading edge was calculated to be 373.61 lbs/in to 0 lbs/in at the tip. Plot of the pressure and force can be seen in Figures 49 and 410. Half the distribution obtained from Eq. 47 is applied to the top of the model while the other half is applied to the bottom of the model. This was done in order to reduce errors in the FEA analysis of the wingbox model. Deterministic Optimization The deterministic optimization problem can be formulated as # elements min W= P pV i=1 s.t. Ci < (cAbasis ), (48) TipDisplacement < (TipDisplacement) alow where Vy, ci and (GAbasis) are the volume, stress and Abasis allowable stress of the material, respectively. The design variables are the skin thicknesses at the root and midpoint of the wingbox. Maximum allowable values for the problem are the Abasis value of 7150T77 aluminum for stress and 68.2 inches for displacement. In this example, ci and the calculated tip displacement corresponds to the response while (oAbasis) and allowable tip displacement to the capacity. The loads are multiplied by a safety factor of 1.5 in order to consider various uncertainties involved in the geometric parameters, applied load, and computational errors. The above optimization problem is solved using the "fmincon" function in MATLAB. The MATLAB code will input the initial design variables into ANSYS, which will conduct the analysis. ANSYS then sends the maximum von Mises element stress, tip displacement data, and total volume of the structure back to MATLAB, which will work to find the design parameters that minimizes the structural weight, while satisfying stress and displacement constraints. The optimization problem converged in 7 iterations with 36 function evaluations. Table 42 lists the results of the deterministic optimization. It should be noted that the maximum stress occurs at the root of the wing model. A contour plot of the element stress can be viewed in Figure 411. Probability of Failure Calculation Using MCS In this section, the probability of failure of the wingbox at the deterministic optimum design is evaluated using Monte Carlo Simulation (MCS). As stated in Chapter 3, the purpose of the probability of failure is to evaluate the level of safety of the deterministic optimum design. It will also be used for the design criterion in the probabilistic optimization. In the calculation of the probability of failure, the factor of safety will not be considered. Instead, uncertainties related to variability in material properties, manufacturing tolerances, and applied loads will be considered in calculating the probability of failure. Failure of the wingbox is predicted to occur when its maximum von Mises stress and tip displacement is greater than its Abasis failure stress value and allowable displacement. Knowing this, the performance function corresponding to the failure stress mode can be written as gl = (a'ba.,s) 'FEA =1 r1 (49) where c1 and r, are the capacity and response ofgi. Similarity, the performance function corresponding to the displacement failure mode can be written as g2 = (TipDisplacement)alo)ble (TipDisplacement)F c2 r2 where c2 and r2 are the capacity and response of g2. Variabilities are introduced into the performance function through random variables oA_ basis, (Tip Displacement)allowable and a load factor. The means and standard deviations of random variables are listed in Table 43. As in the tenbar truss example, the probabilities of failure were calculated using separable MCS. After calculating the probabilities of failure, the total probability of failure of the system can be approximated as PF =Pfl +Pf2 (411) where PFis the system probability of failure, Pf] is the probability of failure of the stress failure mode, and Pp is the probability of failure of the displacement failure mode. Using separable MCS with 106 samples, the probabilities of failure for each element and the system are listed in Table 44. The results show that the probability of failure of the displacement failure mode is an about six times larger than the probability of failure of the stress failure mode. Since the stress is a local performance, it depends on the thickness of the root. On the other hand, the displacement is a global performance and its value depends on the thickness of entire wingbox. Thus, it is possible to increase the thickness of the midpoint design variable, which will reduce the probability of failure of the displacement failure mode, while simultaneously decreasing the thickness of root design variable, which will increase the probability of failure of the stress failure mode. This will make it possible to reduce the weight of the wing model while maintaining the same level of system probability of failure. Approximate Probabilistic Optimization Using ECARD Starting from the deterministic design, the approximate probabilistic optimization problem can be formulated such that the weight of the structure is minimized, while maintaining the same (410) system probability of failure with that of the deterministic optimum design. Instead of the actual probability of failure, the approximate probability of failure from ECARD is used. Thus, the optimization problem can be written as # elements min W= i pVi i=1 (412) S.t. P aPProx = PDapprox + f2approx < Fdet where the approximate system probability of failure is the sum of each approximate probability of failure mode. Before the ECARD optimization, a characteristic response and correction factors are calculated for each failure mode. The results of the ECARD optimization are displayed in Table 45. Using 106 MCS samples, the ECARD optimization needed only two iterations and 4 reliability assessments to reach close to an accurate optimum. Overall weight is reduced by 0.22% (42.7 lbs) while maintaining the same system probability of failure as that of the deterministic optimum design. This reduction is achieved by reallocating the risk from the higher risk displacement failure mode to the lower risk stress failure mode. The ECARD optimization slightly increased the midpoint thickness and decreased the root thickness. This resulted in a similar level of probabilities of failure between the displacement and stress failure modes. Table 41. Material Properties of 7150T77 Aluminum 7150T77 Aluminum Yield Strength = 80.5 ksi ABasis = 74 ksi BBasis = 79 ksi Table 42. Results of deterministic optimization of the wingbox model Root Thickness (in) Midpoint Thickness (in) Deterministic Weight (lbs) 0.78226 0.44164 19,174.40 Table 43. Variability for Wing Model Uncertainties Distribution type Mean Standard Deviation COV Load Factor Normal 1 0.1 10% Failure Stress Normal 80,500 psi 5072 psi 6.30% Displacement Normal 68.2 in 3.41 in 5% Table 44. Probabilities of failure of the deterministic optimum design Pfl PF 5.26E06 3.17E05 3.69E05 Table 45. ECARD optimization results Iteration 1 Prob Thickness (in) Mean Value Approx Pf Actual Pf Root 0.7666 Stress: 50,257 psi 1.110E05 1.141E05 Midpoint 0.4533 Displacement: 45.249 in 2.584E05 2.545E05 Total Pf: 3.694E05 3.687E05 Prob Weight: 19132.6 lbs Iteration 2 Prob Thickness (in) Mean Value Approx Pf Actual Pf Root 0.7668 Stress: 50,246 psi 1.131E05 1.130E05 Midpoint 0.4531 Displacement: 45.255 in 2.563E05 2.564E05 Total Pf: 3.694E05 3.694E05 19131.7 lbs Prob Weight: flff"AvAf ;V297 SYSTEM SCHEMATIC MANUAL ttuWM Fra Pm UJMM b(I M.M m 6 3? Figure 41. Boeing 767 wing dimensions FRONT SPAR " WING BOX STRUCTURE Figure 42. Boeing 767 internal schematic BOEING 767 TRAILING EDGE ./ OF WING BOX LEADING EDGE AREAS TYPE NMl I t lI I III I I I I 1 I r ,I i l I I i 'r I I Figure 43. ANSYS model of the wingbox ELEMENTS Figure 44. Meshed ANSYS model of the wingbox ... Sweep of c oti iiiii^ ^^^^^tc/4 line,. Ctipl ..span, b Figure 45. Sweep of the quarterchord Kllipi c lomlin j W lIrp.r rjIk s= hi: o .6 R elai Sriui ]LxCiMtprI Figure 46. Relationship of local lift distribution and taper ratio 800 700 600 mo 500  400  300  200  100  n Eq 3: Spanwise Lift Distribution vs Distance from Root to Wingtip 0 100 200 300 400 500 600 700 800 Distance across wing (in) Figure 47. Elliptical lift distribution from the root to the tip of the wingbox model FA IW c/4 p c/2 chord, c Figure 48. Equilibrium of forces on the wingbox model Pressure vs Distance from Root to Wingtip I I I I I I I rrr       "       T   T    L LL 1        ~              1111111 i ,I i TTTT Illll 0 100 200 300 400 500 600 Distance across wing (in) Figure 49. Pressure distribution from root to wingtip of the model 700 800 Force at Leading Edge vs Distance from Root to Wingtip 400 300 ,,     150 . 200    I   150 t   ; ; 100    50       0 S150  I  . , ,  0 100 200 300 400 500 6 Distance across wing (in) Figure 410. Force distribution from root to wingtip of the model 00 700 800 Figure 411. Contour plot of maximum stress on the wingbox model CHAPTER 5 SUMMARY AND CONCLUSIONS An exactcapacity approximateresponsedistribution (ECARD) probabilistic optimization method that dispenses with most of the expensive structural response calculations (typically done via finite element analysis) was proposed in this paper. ECARD was demonstrated with two examples. First, probabilistic optimization of a tenbar truss problem was performed, where risk was allocated between truss members. Then, probabilistic optimization of a wingbox was performed, where risk was allocated between the different failure modes. From the results obtained in these two demonstration problems, we reached to the following conclusions. 1. In the tenbar truss problem, ECARD converged to near optima that allocated risk between failure modes much more efficiently than the deterministic optima. The differences between the true and approximate optima were due to the errors involved in probability of failure estimations, which led to errors in the derivatives of probabilities of failure with respect to design variables that is required in risk allocation problems. 2. The approximate optimum required four inexpensive ECARD iterations and five probability of failure calculations for the tenbar truss example to locate the approximate optimum. In the wingbox example, two ECARD iterations were required and probabilities of failure of the elements are calculated three times to locate the approximate optimum. This represents substantial reduction in the number of probability calculation require for full probabilistic optimization. APPENDIX CALCULATION OF MEMBER FORCES OF THE TEN BAR TRUSS Analytical solution to tenbar truss problem is given in Elishakoff et al. [16]. The member forces satisfy the following equilibrium and compatibility equations. Note: Values with "*" are incorrect in the reference. The correct expressions are: 1 N = P, N (A1) 1 N,2 N,o (A2) 1 N3 = P 2P N8 (A3) 1 N4 = P2 N1 (A4) 1 1 N5 = P2 N8 N (A5) 1 N6 = N10 (A6) N7 = 2(P + P) + N8 (A7) N* = ba22 a12b (A8) a11a22 a12a21 N9 = 2P + N10 (A9) N0* a11b2 a21b1 N1o b (A10) a,1122 12a21 where 1 1 1 2^2 2^2 ^ 1^ all*= +++ + (A11) A, A3 A5 + A7 a 2* = a21* (A12) A5 1 1 1 1 2 i 24 ^ 1^ a22 ++ + +(A13) SA2 A4 A5 A6 Ag A]( b  /P2 P, + 2 P2 P 2F2 (P +P2 (A 14) b=4 A(A14) A A3 A (A15) [22 P 4P2(A15) 24 A 5 A I LIST OF REFERENCES [1] Ben Haim, Y., and Elishakoff, I., Convex Models of Uncertainty in Applied Mechanics, Elsevier, Amsterdam, 1990. [2] Neal, D.M., Matthews, W.T., and Vangel, M.G., "Uncertainties in Obtaining High Reliability from StressStrength Models," Proceedings of the 9th DOD/NASA/FAA Conference on Fibrous Composites in Structural Design, Vol. 1, Department of Defense, Lake Tahoe, NV, 1992, pp. 503521. [3] Lee, T.W and Kwak, B.M., "A Reliabilitybased Optimal Design Using Advanced First Order Second Moment Method," Mechanics of Structures and Machines, Vol. 15, No. 4, 1987, pp. 523542. [4] Kiureghian, A.D., Zhang, Y., and Li, C.C., "Inverse Reliability Problem," Journal of Engineering Mechanics, Vol. 120, No. 5, 1994, pp. 11541159. [5] Tu, J., Choi, K.K., and Park, Y.H., "A New Study on Reliability Based Design Optimization," ASME Journal of Mechanical Design, Vol. 121, No. 4, 1999, pp. 557 564. [6] Lee, J.O., Yang, Y.S., Ruy, W.S., "A Comparative Study on Reliabilityindex and Targetperformancebased Probabilistic Structural Design Optimization," Computers and Structures, Vol. 80, No. 34, 2002, pp. 257269. [7] Qu, X., and Haftka, R.T., "Reliabilitybased Design Optimization Using Probabilistic Sufficiency Factor," Journal of Structural and Multidisciplinary Optimization, Vol. 27, No.5, 2004, pp. 314325. [8] Du, X., and Chen, W., "Sequential Optimization and Reliability Assessment Method for Efficient Probabilistic Design," ASME Journal of Mechanical Design, Vol. 126, No. 2, 2004, pp. 225233. [9] Baabbad, M.A., Nikolaidis, E., and Kapania, R.K., "New Approach for System ReliabilityBased Design Optimization," AIAA Journal, Vol. 44, No. 5, May 2006, pp. 10871096. [10] Mogami, K., Nishiwaki, S., Izui, K, Yoshimura, M., and Kogiso, N., "Reliabilitybased Structural Optimization of Frame Structures for Multiple Failure Criteria Using Topology Optimization Techniques," in press, Structural and Multidisciplinary Optimization, 2006. [11] Acar, E., Kumar S., Pippy R.J., Kim N.H., Haftka R. T., "Approximate Probabilistic Optimization Using ExactCapacityApproximateResponseDistribution (ECARD)", submitted, 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 2006. [12] Oberkampf, W.L., DeLand, S.M., Rutherford, B.M., Diegert, K.V. and Alvin, K.F., "Estimation of Total Uncertainty in Modeling and Simulation", Sandia National Laboratory Report, SAND20000824, Albuquerque, NM, April 2000. [13] Oberkampf, W.L., Deland, S.M., Rutherford, B.M., Diegert, K.V., and Alvin, K.F., "Error and Uncertainty in Modeling and Simulation," Reliability Engineering and System Safety, Vol. 75, 2002, pp. 333357. [14] Acar, E., Kale, A. and Haftka, R.T., "Effects of Error, Variability, Testing and Safety Factors on Aircraft Safety," Proceedings of the NSF Workshop on Reliable Engineering Computing, 2004, pp. 103118. [15] Acar, E., Kale, A., and Haftka, R.T., "Comparing Effectiveness of Measures that Improve Aircraft Structural Safety," submitted, ASCE Journal of Aerospace Engineering, 2006. [16] Haftka, R.T., and Gurdal, Z., "Elements of Structural Optimization," Kluwer Academic Publishers, 3rd edition, 1992. [17] Elishakoff, I., Haftka, R.T., and Fang, J., "Structural Design Under Bounded UncertaintyOptimization with Antioptimization," Computers and Structures, Vol. 53, No. 6, 1994, pp. 14011405. [18] Smarslok, B.P., Haftka, R.T., and Kim, N.H., "Taking Advantage of Separable Limit States in Sampling Procedures," AIAA Paper 20061632, 47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Newport, RI, May 2006. [19] Kreyzig E., Advanced Engineering Mathematics, Wiley, New York, pp. 848. [20] Haftka, R.T., and Gurdal, Z., "Elements of Structural Optimization," Kluwer Academic Publishers, 3rd edition, 1992. [21] Richard C. Rice, Jana L. Jackson, John Bakuckas, and Steven Thompson, "Metallic Material Properties Development and Standardization (MMPDS)," Federal Aviation Administration, January 2003. BIOGRAPHICAL SKETCH Richard Pippy was born in Bronxville, New York, in 1972. Upon graduation from high school, he joined the United States Marine Corps, where he served for six years. After his honorable discharge, he then attended St. Petersburg College where he received an Associate of Arts degree in 2001. He then joined the University of Florida, where he earned a Bachelor of Science degree in mechanical engineering in 2005. In 2006, he returned to the University of Florida to pursue a master's degree in mechanical engineering. Under the supervision of Dr. NamHo Kim, he earned his master's degree in 2008. PAGE 1 1 APPROXIMATE PROBABILISTIC OPTIMIZATION OF A WINGBOX MODEL USING EXACT CAPACITY APPR OXIMATE RESPONSE DISTRIBUTION (ECARD) By RICHARD J. PIPPY A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FL ORIDA 2008 PAGE 2 2 2008 Richard J. Pippy PAGE 3 3 To my parents, Richard and Evelyn, and to my brother Chris PAGE 4 4 ACKNOWLEDGMENTS I would like to express my sincere gratitud e to my advisor and the chair of my thesis committee, Dr. Nam Ho Kim and to the co c hair Dr. Rafael T. Haftka for their guidance, enthusiasm and continual support throughout my study. I am also grateful to committee member Dr. Peter G. Ifju for his advice and patience in reviewing this thesis. Special thanks go to Mulu Haile for his hel p in deriving the wingbox load conditions. I would like to thank my colleagues in the Structural and Multidisciplinary Optimization Lab at the University of F lorida i n particular, Sunil Kumar, Dr. Erdem Acar, Palani Ramu, Haoyu Wang and Saad M. Mukras for their help and encouragement PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ ............... 4 LIST OF TABLES ................................ ................................ ................................ ........................... 7 LI ST OF FIGURES ................................ ................................ ................................ ......................... 8 ABSTRACT ................................ ................................ ................................ ................................ ..... 9 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .................. 11 2 EXACT CAPACITY RESPON SE DISTRIBUTION (ECARD) THEORY ......................... 13 Introduction ................................ ................................ ................................ ............................. 13 Characteristic Response ................................ ................................ ................................ .......... 13 Correction Factor ................................ ................................ ................................ .................... 16 Using FORM ................................ ................................ ................................ ................... 18 Using MCS ................................ ................................ ................................ ...................... 18 App roximate Probabilistic Optimization ................................ ................................ ................ 19 3 ANALYTICAL EXAMPLE: APPLICATION OF ECARD TO A TEN BAR TRUSS ........ 23 Introduction ................................ ................................ ................................ ............................. 23 Problem Description ................................ ................................ ................................ ............... 23 Deterministic Optimization ................................ ................................ ................................ .... 24 Probability of Failur e Calculation Using MCS ................................ ................................ ...... 24 Probabilistic Optimization ................................ ................................ ................................ ...... 27 Approximate Probabilistic Optimization Using ECARD ................................ ....................... 28 4 PRACTICAL EXAMPLE: APPLICATION OF ECARD TO A WINGBOX MODEL ....... 36 Introduction ................................ ................................ ................................ ............................. 36 Pro blem Description ................................ ................................ ................................ ............... 36 Geometry ................................ ................................ ................................ ......................... 36 Loading Calculations ................................ ................................ ................................ ....... 37 Determin istic Optimization ................................ ................................ ................................ .... 39 Probability of Failure Calculation Using MCS ................................ ................................ ...... 40 Approximate Probabilistic Optimization Using ECARD ................................ ....................... 41 5 SUMMARY AND CONCLUSIONS ................................ ................................ ..................... 52 APPENDIX CALCULATION OF MEMBER FORCES OF THE TEN BAR TRUSS ........... 53 PAGE 6 6 LIST OF REFERENCE S ................................ ................................ ................................ ............... 55 BIOGRAPHICAL SKETCH ................................ ................................ ................................ ......... 57 PAGE 7 7 LIST OF TABLES Table page 3 1 Parameters for the ten bar truss problem ................................ ................................ ........... 30 3 2 Results of deterministic optimization of the ten bar truss problem ................................ ... 30 3 3 Probabilistic distribution types, parameters of errors and variabilities in the ten bar truss problem ................................ ................................ ................................ ...................... 31 3 4 Probabilities of failure of the deterministic optimum a reas ................................ ............... 31 3 5 Results of the probabilistic optimization of the ten bar truss ................................ ............ 32 3 6 Results of the ECARD optimizatio n ................................ ................................ .................. 33 3 7 Results of the ECARD optimization ................................ ................................ .................. 34 4 1 Material Properties of 7150 T77 Aluminum ................................ ................................ ..... 43 4 2 Results of deterministic optimization of the wingbox model ................................ ............ 43 4 3 Variability for Wing Model ................................ ................................ ............................... 43 4 4 Probabilities of failure of the deterministic optimum design ................................ ............. 43 4 5 ECARD optimization results ................................ ................................ ............................. 43 PAGE 8 8 LIST OF FI GURES Figure page 2 2 Calculation of the probability of failure at new design. ................................ ..................... 21 2 3 Calculation of t he probability of failure at new design ................................ ...................... 22 3 1 Geometry and loadings of the ten bar truss ................................ ................................ ....... 35 4 1 Boeing 767 wing dimensions ................................ ................................ ............................. 44 4 2 Boeing 767 internal schematic ................................ ................................ ........................... 45 4 3 ANSYS model of the wingbox ................................ ................................ .......................... 46 4 5 Sweep of the quarter chord ................................ ................................ ................................ 47 4 6 Relationship of local lift distribution and taper ratio ................................ ......................... 48 4 7 Elliptical lift distribution from the root to the tip of the wingbox model .......................... 48 4 8 Equilibrium of forces on the wingbox model ................................ ................................ .... 49 4 9 Pressure distribution from root to wingtip of the model ................................ .................... 49 4 10 Force distribution from root to wingtip of the model ................................ ........................ 50 PAGE 9 9 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science APPROXIMATE PROBABILISTIC OPTIMIZATION OF A WINGBOX MOD EL USING EXACT CAPACITY RESPONSE DISTRIBUTION (ECARD) By Richard J. Pippy May 2008 Chair: Nam Ho Kim Co chair: Rafael T. Haftka Major: Mechanical Engineering There are two major obstacles that affect probabilistic (or reliability based) structural optimi zation. First, uncertainties associated with errors in structural and aerodynamic modeling and quality of construction are not well characterized as statistical distributions and it has been shown that insufficient information may lead to large errors in p robability calculations. Second, probabilistic optimization is computationally expensive due to multiple analyses, typically finite element analyses, for calculating probability as the structure is being redesigned. In order to overcome these obstacles, we propose an approximate probabilistic optimization method where the probabilistic calculation is confined to failure stress. This takes advantage of the fact that statistical characterization of failure stresses is required by Federal Aviation Administrati on regulations. The need for expensive stress distribution calculations are eliminated by condensing the stress distribution into a representative deterministic value, transforming a probabilistic optimization problem into a semi deterministic optimization problem. By starting the approximate probabilistic optimization from the deterministic optimum design, a small number of iterations is expected and reliability analysis is required only once per iteration. This proposed method provides approximate sensiti vity of failure probability with respect to the design PAGE 10 10 variables which is essential in risk allocation. This method is demonstrated in two examples. The first example uses a ten bar truss which demonstrate s the risk allocation between the truss elements. The second example uses a wingbox model based on a Boeing 767 400 aircraft which demonstrates the risk allocation between two different failure modes of stress and displacement PAGE 11 11 CHAPTER 1 INTRODUCTION There are two ma jor barriers in front of probabilistic (or reliability based) structural optimization First, uncertainties associated with material properties, operating conditions, mathematical models, and manufacturing variability are not well characterized as statisti cal distributions and insufficient information may lead to large errors in probabilit y calculations (e.g., Ben Haim and Elishakoff [1], Neal, et al. [2]). Due to this fact, many engineers are reluctant to pursue probabilistic design. The second barrier to the application of probabilistic structural optimization is computational expense. Probabilistic structural optimization is expensive because repeated stress calculations (typically FEA) are required for updating probability calculation as the structure i s being re designed. Targeting these two main barriers, we propose an approximate probabilistic optimization method that dispenses with expensive probabilistic stress calculations. In the proposed method, the probabilistic calculation is confined only to f ailure stress, which is often well characterized. Traditionally, reliability based design optimization (RBDO) is performed based on a double loop optimization scheme, where the outer loop is used for design optimization while the inner loop performs a sub optimization for reliability analysis, using methods such as First Order Reliability Method (FORM). Since this traditional approach is computationally expensive, even prohibitive for problems that require complex finite element analysis (FEA), alternative methods have been proposed by many researchers (e.g., Lee and Kwak [3], Kiureghian et al. [4], Tu et al. [5], Lee et al. [6], Qu and Haftka [7], Du and Chen [8] and Ba abbad et al. [9]). These methods replace the probabilistic optimization with sequential deterministic optimization (often using inverse reliability measures) to reduce the computational expense. However, these approaches do not necessarily converge to the optimum design, and they do not easily lend themselves to PAGE 12 12 allocating risk between failur e modes in a structure where many components can fail [10]. We note, however, that most of the computational expense is associated with repeated stress calculation. So we propose an approximate probabilistic design approach that reduces the number of expe nsive stress calculations. That is, we approximate the probabilistic optimization that separates the uncertainties which can be evaluated inexpensively and those whose effects are expensive to evaluate. We boil down the stress distribution to a single char acteristic stress by utilizing the inverse cumulative distribution of the failure stress, and we propose an inexpensive approximation of that characteristic stress. This proposed method will also improve upon a deterministic design by reallocating the safe ty margins between different components or failure modes. We call the proposed approximate probabilistic optimization approach Exact Capacity Approximate R esponse D istribution or ECAR D The purpose of this thesis is to further advance the version of ECARD which was originally developed by Dr. Erdem Acar and Dr. Rafael T. Haftka [11] There is now an improved version of ECARD which was developed by Sunil Kumar, Dr. Rafael T. Haftka, and Dr. Nam Ho Kim. The remainder of the thesis is organized as follows. Cha pter 2 discusses the theory behind the ECARD method. The application of the method to a ten bar truss problem is presented in Chapter 3 and a wingbox design in Chapter 4. Finally, the concluding remarks are listed Chapter 5. PAGE 13 13 CHAPTER 2 EXACT CAPACI TY RESPONSE DISTRIBUTION (ECARD) THEORY Introduction In this chapter, the approximate probabilistic optimization method named ECARD will be discussed. Characteristic Response In probabilistic optimization, the system constraint is often given in terms o f failure probability of a performance function. We consider a specific form of performance function, given as (2 1) where and are capacity and response, respectively. Both the capacity and response are random because they are functions of input random variables and depend on deterministic design variables The system is considered to be failed when the response exceeds the capacity ; i.e., g ( x ; u ) < 0 We assume that the probabilistic distribution of is well known, while that of requires a l arge number of analyses. For example, when c ( x ; u ) is failure stress and r ( x ; u ) is the maximum stress of an aircraft structure, the probabilistic distribution of c ( x ; u ) is already characterized by Federal Aviation Administration requirement. However, the probabilistic distribution of r ( x ; u ) requires repeated computational simulations, such as finite element analysis. Since the performance function depends on two random variables, and the safety of t he system can be estimated using a probability of failure, defined as (2 2) PAGE 14 14 In the above equation, is the cumulative distribution function (CDF) o f capacity, and is the probability density function (PDF) of response. The above integral can be evaluated using analytical integration, Monte Carlo simulation (MCS), or first /second order reliability method (FORM/SORM), among o thers. Smarslok et al [ 18 ] presented a separable MCS, which is much more accurate than the traditional MCS when the performance can be separable as in Eq. 2 1. It is clear from Eq. 2 2 that accurate estimation of probability of failure requires accurate a ssessment of the probability distributions of both the response and capacity. When the capacity is the failure stress, the FAA requires aircraft builders to perform characterization tests in order to construct a statistical model, and then to select the al lowable failure stress (A basis or B basis value) based on this model. Hence, the CDF of capacity is often reasonably well characterized. On the other hand, the PDF of the response is poorly known, because it depends on the accuracy of various factors, suc h as material properties, operating conditions, mathematical models, and manufacturing variability. The key idea of this research is to express P f in Eq. 2 2 using a characteristic value of the response, and approximate the change of P f in terms of the ch ange of this characteristic value. The calculation of in Eq. 2 2 can be simplified by using the information of From the Intermediate Value Theorem [ 19 ], there exist such that Eq. 2 2 can be re written as (2 3) In the above equation, the second equality is obtained from the fact that the integral of f R is one. Equation 2 3 states tha t the effect of (the poorly characterized) probability distribution of the response can be boiled down to a single characteristic response, r When the probability of PAGE 15 15 failure is given, the characteristic response can be calculated using the inverse transf ormation of F C ( r ) as (2 4) When design variables are changed during optimization, it is possible that the distributions of b oth capacity and response may be changed. For the simplicity of presentation, w e consider the case that the distribution of the capacit y remains unchanged. We assume that the design change only affects the mean value of the response; i.e., the standard dev iation remains constant. This assumption is reasonable if the design perturbation is small. In such a case, redesign changes the mean value of response from to where is the v alue the relative change in response according to design change. Figure 2 1 illustrates the change in response distribution, along with the distribution of capacity. In this research we start the probabilistic design from a known deterministic optimum. This is an important aspect of the approximate probabilistic optimization. Since the deterministic optimization uses safety margins to consider the effect of uncertainties, the deterministic optimum design is close to the probabilistic optimum design. This will satisfy the above assumption of small design change. The goal of proposed probabilistic design is then to improve upon the det erministic design by reallocating the safety margins between different components or failure modes. First, the change in design variables will change the mean of response from to while maintaining the same st andard deviation. The change in the response distribution will then change the probability of failure according to Eq. 2 2. From Eq. 2 4, the characteristic response will also be changed from r* to r* where is the relative change in characteristic PAGE 16 16 response. Unfortunately, t his process requires calculation of the probability of failure at the new design. The novel idea of the proposed approach is to reverse this process by approximating the re lation and so that the characteristic response at the new design can be calculated without performing reliability analysis For the moment, let us assume can be calculated from given Then, the probability of failure at the new desig n can be calculated from (2 5) The above probability of failure will be exact if is the correct relative change in characteristic response. W hen is an approximated one, the probability of failure in the above equation is approximate, and we will denote it The procedure illustrated in Figure 2 2 does not require expensive reliability analysis. It is enough to analyt ically evaluate the value of the CDF at the perturbed characteristic response. Correction Factor The key idea of the proposed approximate probability distribution is that the new characteristic response can be approximated without recourse to the expensive reliability analysis. The simplest approximation, used in this research, is that the relative change in the characteristic response, is proportional to the relative change in response, as (2 6) where is a proportional c onstant that depends on how the redesign af fects the stress distribution. In fact, i t is the sensitivity of the characteristic response change with respect to the r esponse change. W e call it a correction factor The above assumption in linearity is reasona ble when is relatively small. PAGE 17 17 Probabilistic optimization can be viewed as risk allocation between different failure modes or different structural members. This allocation requires the sensitivity of failure probability with respect to design variables. In the pr oposed approximate probabilistic optimization, this sensitivity information is presented in the correction factor. We will demonstrate that a linear relationship between works well given the assumption of translating the stress distribution espec ially when the design change is relatively small We consider a lognorma l l y distributed capacity with mean value of C = 100 and coefficient of variation of 8%, and normally distributed response with coefficient of variation of 20%. From Eq. 2 2, the mean value of the response is chosen to be R = 42.49 so that the probability of failure becomes P f = 10 7 For a given small value o a new probability of failure is calculated from Eq. 2 2 with the mean of the response being R ( ) The relative change in characteristic response is then obtained from Eq. 2 4 with Figure 2 3 shows the relation *. We can see that the linearity assumption is quite accurate over the range of %. The slope will be the correction factor k Figure 2 4 shows the effect of the approximation on the probability of failure. In practice, the correction factor can be calculated using a finite difference method, which requires at least two reliability c alculations. We will describe the procedure using a forward finite difference method, but other method can also be used in a similar way. Let o = 0.0 corresponds to the current design, and p = 0.05 to the perturbed design. The correction factor can be ca lculated either using FORM or MSC. We will explain both cases. PAGE 18 18 Using FORM First, the probability of failure at the current design is calculated from First Order Reliability Method (FORM) with the performance function in Eq. 2 1. If the response is perturbed by p Eq. 2 1 becomes (2 7) Using the above equation, reliability analysis is performed to calculated the perturbed probability of failure, It is noted that it is unnecessary to change design variable, because we directly perturb the output response. Thus, the computational cost of reliability analysis using Eq. 2 7 can be reduced significantly. Next, the characte ristic responses are calculated from Eq. 2 4, as (2 8) By comparing two terms in the above equation, the relative change in the characteristic res ponse can be obtained as (2 9) Then, the correction factor can be obtained from (2 10) Using MCS When MCS is employed, we generate N samples of response at the current design. In view of Eq. 2 2, the probabili ties of failure at the current and perturbed design can be calculated from PAGE 19 19 (2 11) (2 12) The remaining procedure is identical to that of FORM. Even if Eq. 2 11 and Eq. 2 12 are two different MCS, they can be combined into one because the same sample, r i will be used. Approxi mate Probabilistic Optimization The proposed approximate probabilistic optimization is composed of two stages: (1) correction factor and initial probability of failure are calculated from reliability an alysis, and (2) a deterministic optimization problem is solved using the approximate probability of failure from Eq. 2 5. The first stage is computationally expensive, while the second stage is nothing but a semi deterministic optimization. We will explain the approximate probabilistic optimization procedure, as follows. 1. Perform deterministic optimization with safety margin. The probabilistic design starts from the deterministic optimum design; i.e., initial de s ign and cost functi on Calculate the initial probability of failure at 2. At the current design calculate deterministic value of response, u sing the mean value of input random variables. 3. Calculate the characteristic response, r 0 using the inverse CDF of the P f p and the mean and c.o.v of the response. 4. Calculate the correction factor using the procedure in the p revious section. 5. Obtain optimum design and optimum objective function by solv ing the following optimization problem: (2 13) PAGE 20 20 Where (2 14) (2 15) (2 16) (2 17) (2 18) 6. Calculate the actual probability of fail ure, at 7. Check convergence: If i t converged, stop the process. Otherwise, set and go to Step 2 and continue The above semi deterministic optimiz ation process uses exact distribution of the capacity and an approximate distribution of response (MCS or FORM). Due to this aspect, we call it Exact Capacity Approximate Response Distribution (ECARD) method. The accuracy of ECAR D to locate the true optimum depends on the magnitudes of errors involved in the approximations. As shown in Figure 2 3 and Figure 2 4 the approximation is accurate if changes of the response due to redesign are small. In addition, the accuracy in estimat ing the correction factor affects the convergence rate of the proposed method. The result may be somewhat sub optimal because of the convergence condition and the approximate nature of the sensitivity of probability of failure. T his issue will be discussed in detail in the following section. PAGE 21 21 Figure 2 1 Distributions of response before and after redesign. Figure 2 2 Calculation of the probability of failure at new design. PAGE 22 22 Figure 2 3 Calculation of the probability of failure at new design Figure 2 4 Calculation of the probability of failure at new design PAGE 23 23 CHAPTER 3 ANALYTICAL EXAMPLE: APPLICATION OF ECARD TO A TEN BAR TRUSS Introduction The first demonstration example is a ten ba r truss problem as shown in Figure 3 1. This example will demonstrate risk allocation between the different truss members. First, a brief description of the problem will be given. Then, a deterministic optimization of the problem will be presented followed by the probability of failure calculation using Monte Carlo simulations. Finally, the probabilistic and ECARD optimizations will be performed and the efficiency and accuracy of the ECARD method will be discussed Problem Description The problem descrip tion for the ten bar truss example was taken from Haftka and Gurdal [ 20 ] (page 237). The truss structure is under two loads, P 1 and P 2 The design objective is to minimize the total weight of the truss, W by varying the cross sectional areas, A i of the m embers while satisfying minimum gage constraints and allowable stress constraints. Input data for the truss is listed in Table 3 1. Member 9 was assigned a higher failure stress value in order to make the fully stressed design non optimal Aircraft design often uses a knockdown factor, K dc in order to conservatively estimate failure stress using A basis or B Basis methods The A basis (or B basis) failure stress is the value of a failure stress exceeded by 99% ( or 90% for B basis) of the pop u lation with 95 % confidence. In the conservative estimation, t he allowable stress of a member is related to the mean value of the failure stress through the following equation : (3 1) In the deterministic design process, the knockdown factor is a way of considering the uncertainty in the failure stress PAGE 24 24 Deterministic Optimization Using the safety factor and knockdown factor, t he deterministic optim ization problem can be formulated as (3 2) w here L i N i and A i are, respectively, the length, member force, and cross sectional area of element i A is the vector of cross s ectional areas, i and allowable ) i are the stress and allowable stress of an element, respectively. i corresponds to the response while allowable ) i to the capacity, and the loads are multiplied by a safety factor in order to consider various uncert ainties involved in the truss parameters applied load, and computational errors The analytical solution for the member forces are given in the Appendix. The above optimization problem is solved using the d in 7 iterations with 97 function evaluations. Table 3 2 lists the results of the deterministic optimization. Note that elements 2, 5, and 6 cross sectional areas reach minimum gage while element 5 is a zero force member. At optimum design, stresses in al l members except for Member 5 and 9 are at the allowable stress. Probability of Failure Calculation Using MCS In this section, the probability of failure of the truss at the deterministic optimum design is evaluated using Monte Carlo Simulation (MCS). Ther e are two purposes in calculating the probability of failure. First, it can evaluate the level of safety of the deterministic optimum design. The effects of the knockdown factor and the factor of safety are evaluated in terms of the probability of failure. Second, it can be used for the design criterion in the probabilistic PAGE 25 25 optimization. In the calculation of the probability of failure, the factor of safety and the knockdown factor will not be considered. Instead, uncertainties related to errors and variabi lity in material properties, manufacturing tolerances, and applied loads will be considered in calculating the probability of failure. There are many uncertainties involved in the design of the ten bar truss, such as variability from material properties, loads, manufacturing, and errors from numerical calculation and modeling. Failure of an element is predicted to occur when the stress in an element is greater than its failure stress. Knowing this, the performance function can be written as (3 3) from its calculated (or predicted) value due to errors. Adding these errors, t he equation ca n be rewritten as (3 4) Here, e f is the error in failure prediction, f is the predicted fa ilure stress, e is the error in stress calculation and is the calculated stress The errors were formulated to where positive errors correspond to a conservative design. Therefore, the error in calculated stress is positive, while the error in predicte d failure stress is negative. Even though the stress calculation is exact for the ten bar truss the error, e was introduced to consider the analysis of a more complex structure where the stresses are calculated from numerical methods The calcula ted str ess can be written in the following form (3 5) where FEA stands for calculated stresses using FEA, e P1 and e P2 are errors in loads P 1 and P 2 and e A is the vector of err ors (tolerances) corresponding to ten cross sectional areas. By PAGE 26 26 substituting Eq. 3 5 into Eq. 3 4, t he performance function can now be rearranged in separable form (i.e., in a form that allows the use of separable MCS) for each element as (3 6) where and are respectively, the capacity and response. Beside errors, variabilities are introduced into the performance function through random variables f P 1 P 2 and A. The probabilistic parameters of errors and variabilities and their distribution types are listed in Table 3 3. The probabilities of failure were calculated using separable MCS, which requires smaller number of simulations to achieve the same accuracy as crude MCS [ 17 ] After calculating the probabilities of failure for each element, the total probability of failure of the system can be approximated as (3 7) where is the system probability of failure. Calculating the probability failure in this form is Ditle order upper bound; therefore the system probability of failure is estimated conservatively. Using separable MCS with 10 6 samples the probabilities of failure for each element and the system are listed in Table 3 4. The results show that members probability of failure contributes to 80% of the system prob ability of failure. In the deterministic design process, the uncertainty in the system is considered using safety measures, such as knockdown factor and the factor of safety. However, as is clear form Table 3 4, the effects of these safety measures are no t evenly distributed between members. It appears that members 2, 6, and 10 are very sensitive to these safety measures, while other members are not. They are either at minimum gage or close to it, and yet, their probabilities of failure are relatively high compared to the other members. Thus, it is possible to move some of the weight PAGE 27 27 from non sensitive members to the sensitive ones so that the system probability of failure can be reduced further while maintaining the total weight of the truss. Or, it is als o possible to reduce the weight of the truss while maintaining the same level of system probability of failure. The latter possibility will be investigated in the probabilistic optimization. Probabilistic Optimization Starting from the deterministic desig n, the probabilistic optimization problem can be formulated such that the weight of the structure is minimized, while maintaining the same level of system probability of failure with that of the deterministic optimum design. Thus, we have (3 8) Results of the probabilistic optimization are shown in Table 3 5. A total of 10 5 samples are used for MCS. The optimization converged after 59 i terations and 728 reliability assessments. The relatively large number of reliability assessments is due to the fact that the problem has ten design variables. At each iteration, the optimization algorithm calculates sensitivity using finite different meth od. On the other hand, the proposed ECARD method perturbs the response directly. Thus, ECARD will be efficient when the number of response is smaller than that of design variables. The ov erall optimization t ook about 125 hours using a Dell deskt op compute r In order to remove instability related to random samples, a set of input random variables are generated and repeatedly used during the optimization. Overall weight is reduced by 6% (90.47 lbs) while maintaining the same system probability of failure as that of the deterministic optimum design This reduction is achieved by reallocating the risk from the higher risk members (2, 6, and 10) to the lower risk members. The probabilistic optimization slightly increased the cross sectional PAGE 28 28 areas of members 2, 6 and 10, and decreased t he cross sectional areas of the other members. While the remaining probabilities of failure increased slightly, members 2, 6 and 10 were reduced by an order of magnitude. This risk allocation can be achieved when the sens itivities of probability of failure and weight with respect to design variables are available. In the probabilistic optimization, these sensitivities are calculated using the finite difference method. That explains the 728 reliability assessments during th e optimization. In the following section, the same optimization problem will be solved using the ECARD method, which requires a smaller amount of reliability assessments, and yet the sensitivity information can be obtained in the approximate sense. Approx imate Probabilistic Optimization Using ECARD In the approximate probabilistic design, the same optimization problem is used except that the approximate probability of failure is used. Thus, the optimization problem can be written as (3 9) where the approximate system probability of failure is the sum of each members contributions. Since there are ten members, ten characteristic response s and correction factors are calculated before the ECARD optimization. This calculation is equivalent to assessing the probability of failure twice. Then, the above optimization is deterministic because the approximate probability of failure c an be evaluat ed without MCS. Since the approximation in the probability of failure is not accurate, the above ECARD optimization is repeated until the convergence criterion, as stated in Chapter 2, is satisfied. The accuracy of the characteristic response, which depen ds on the number of MCS samples, affects the number of iterations needed to reach an accurate optimum. A low number of PAGE 29 29 samples may appear to reduce computational costs, but actually it reduces the confidence in the probability of failure calculation result ing in an increased number of iterations to reach the accurate optimum. The number of MCS samples must be chosen accordingly for each problem. The results of the ECARD optimization are displayed in Table 3 6. Using 10 5 MCS samples, the ECARD optimization n eeded only four iterations and 8 reliability assessments to reach close to the probabilistic optimum This is a significant reduction from the 728 reliability assessments of the probabilistic optimization The weight difference when comparing the fourth it eration to the third is 0.03%, while the approximate system probability of failure equals the deterministic system probability of failure. In addition, the errors in the member approximate probability of failure calculations are less than 2%. Since, the pr obability of failure for member nine is very small, its probability of failure error is not accurate and the error is ignored As expected the ECARD optimization allocates the risk between members. T he cross sectional areas of the smaller members increased while they decreased in the largest members. PAGE 30 30 Table 3 1. Parameters for the ten bar truss problem Parameters Values Dimension, b 360 inches Safety factor, S F 1.5 Load, P 1 66.67 kips Load, P 2 66.67 kips Knockdown factor, K dc 0.87 Density, r 0.1 lb/ in 3 Modulus of elasticity, E 10 4 ksi Allowable stress, allowable 25 ksi* Minimum gage 0.1 in 2 *for Element 9, allowable stress is 75 ksi Table 3 2. Results of deterministic optimization of the ten bar truss problem Element A i det (in) W i (lb) Str ess (ksi) 1 7.900 284.400 25.0 2 0.100 3.600 25.0 3 8.100 291.600 25.0 4 3.900 140.400 25.0 5 0.100 3.600 0.0 6 0.100 3.600 25.0 7 5.798 295.200 25.0 8 5.515 280.800 25.0 9 3.677 187.200 37.5 10 0.141 7.200 25.0 Total 1497.600  PAGE 31 31 Table 3 3. Proba bilistic distribution types, parameters of errors and variabilities in the ten bar truss problem Uncertainties Distribution type Mean Scatter Errors e Uniform 0 5% e P1 Uniform 0 10% e P2 Uniform 0 10% e A (10 1 vector) Uniform 0 3% e f Uniform 0 20% Variability P 1 P 2 Extreme type I 66.67 k ips 10% c.o.v. A (10 1 vector) Uniform A (10 1 vector) 4% bounds Lognormal 25 /k dc ksi or 8% c.o.v. 75 /k dc ksi Table 3 4. P robabilities of failure of the d eterministic optimum areas Element P f det 1 2.13E 03 2 1.06E 02 3 4.80E 04 4 2.19E 03 5 4.04E 04 6 1.07E 02 7 1.69E 03 8 1.89E 03 9 5.47E 13 10 1.07E 02 Total 4.08E 02 PAGE 32 32 Table 3 5. Results of the probabilistic optimization of the ten bar truss Elements A i det (in) A i ( P f det ) i ( P f ) i 1 7.9 7.192 2.13E 03 5.88E 03 2 0.1 0.3243 1.06E 02 3.07E 03 3 8.1 7.162 4.80E 04 8.26E 03 4 3.9 3.701 2.19E 03 2.15E 03 5 0.1 0.451 2 4.04E 04 3.18E 05 6 0.1 0.3337 1.07E 02 2.14E 03 7 5.798 5.1697 1.69E 03 1.02E 02 8 5.515 4.9782 1.89E 03 3.75E 03 9 3.677 3.5069 5.47E 13 4.70E 13 10 0.141 0.4325 1.07E 02 5.46E 03 Total 1497.6 lbs 1407.13 lbs 4.08E 02 4.08E 02 PAGE 33 33 Tabl e 3 6. Results of the ECARD optimization Element Determ. Des. Iter 1 Iter 2 Iter 3 Iter 4 AREAS (in 2 ) 1 7.9000 7.4487 7.4787 7.4841 7.4849 2 0.1000 0.1000 0.1000 0.1000 0.1000 3 8.1000 7.0752 7.0406 7.0401 7.0402 4 3.9000 3.9382 3.9666 3.9710 3.9716 5 0.1000 0.1000 0.1000 0.1000 0.1000 6 0.1000 0.1000 0.1000 0.1000 0.1000 7 5.7980 5.0457 5.0440 5.0442 5.0441 8 5.5150 5.3538 5.3873 5.3941 5.3951 9 3.6770 3.8416 3.9657 3.9873 3.9908 10 0.1410 0.1314 0.1310 0.1309 0.1309 Weight (lb) 1497.60 1407.1 6 1415.94 1417.71 1418.00 MEAN STRESSES (ksi) 1 16.6667 17.7656 17.7047 17.6934 17.6918 2 16.6667 14.479 14.0059 13.9276 13.9147 3 16.6667 18.9868 19.0693 19.0688 19.0684 4 16.6667 16.5606 16.4537 16.4378 16.4354 5 0 4.462 4.7514 4.7913 4.7 977 6 16.6667 14.479 14.0059 13.9276 13.9147 7 16.6667 18.966 18.951 18.9472 18.9468 8 16.6667 17.3456 17.2576 17.2391 17.2363 9 25 24.0093 23.2747 23.1515 23.1313 10 16.6667 15.5797 15.1199 15.0471 15.0354 PAGE 34 34 Table 3 7 Results of th e ECARD optimization Element Determ. Des. Iter 1 Iter 2 Iter 3 Iter 4 APPROXIMATE P F 1 2.13E 03 5.65E 03 5.26E 03 5.21E 03 5.20E 03 2 1.06E 02 2.16E 03 2.11E 03 2.10E 03 2.10E 03 3 4.80E 04 7.44E 03 7.51E 03 7.51E 03 7.50E 03 4 2.19E 03 1.97E 03 1.77 E 03 1.74E 03 1.74E 03 5 4.04E 04 4.04E 04 1.72E 03 1.86E 03 1.88E 03 6 1.07E 02 2.17E 03 2.09E 03 2.09E 03 2.09E 03 7 1.69E 03 1.23E 02 1.20E 02 1.20E 02 1.20E 02 8 1.89E 03 3.59E 03 3.22E 03 3.17E 03 3.16E 03 9 5.47E 13 3.09E 14 2.50E 15 1.67E 15 6. 66E 16 10 1.07E 02 5.17E 03 5.21E 03 5.22E 03 5.23E 03 SYSTEM 4.08E 02 4.08E 02 4.08E 02 4.08E 02 4.08E 02 ACTUAL P F 1 2.13E 03 5.53E 03 5.26E 03 5.21E 03 5.20E 03 2 1.06E 02 3.09E 03 2.25E 03 2.13E 03 2.11E 03 3 4.80E 04 6.95E 03 7.51E 03 7.51E 03 7 .50E 03 4 2.19E 03 1.96E 03 1.77E 03 1.74E 03 1.74E 03 5 4.04E 04 1.72E 03 1.86E 03 1.88E 03 1.88E 03 6 1.07E 02 3.09E 03 2.23E 03 2.11E 03 2.09E 03 7 1.69E 03 1.21E 02 1.20E 02 1.20E 02 1.20E 02 8 1.89E 03 3.49E 03 3.22E 03 3.17E 03 3.16E 03 9 5.47E 13 2.77E 14 2.44E 15 1.59E 15 1.48E 15 10 1.07E 02 7.14E 03 5.50E 03 5.27E 03 5.24E 03 SYSTEM 4.08E 02 4.51E 02 4.16E 02 4.10E 02 4.08E 02 PAGE 35 35 Figure 3 1. Geometry and loadings of the ten bar truss PAGE 36 36 CHAPTER 4 PRACTICAL EXAMPLE: APPLICATION OF ECARD T O A WINGBOX MODEL Introduction The second demonstration example is the wingbox model problem This example will demonstrate the risk allocation between different failure modes of stress and displacement while minimizing the total weight of the wingbox Pub lically available information was used to model the wingbox as a Boeing 767 wing in a 2.5 g dive First, a description of the problem will be given which will cover model geometry, and loading calculations Then, a deterministic optimization of the problem will be presented followed by the probability of failure calculation using Monte Carlo simulations. Finally, the ECARD optimization will be performed, and the efficiency and accuracy of the ECARD method will be discussed Problem Description Geometry The geometry of the wingbox was modeled using the open source Boeing 767 wing schematics shown in Figures 4 1 and 4 2. The root dimensions have a width of 170 inches with a height of 50 inch. The tip dimensions are 20 inches wide, 10 inches tall. The skin thi ckness at the root is 1.0 inches and decreases in steps linearly to 0.1 inches at the tip of the wing. The wing is 745 inches long from root to tip. The wingbox is modeled using ANSYS Parametric Design Language (APDL) and analyzed using ANSYS Finite Eleme nt Analysis software. Since the APDL is a parametric language, it can generate finite element models with different design values. The APDL code consists of a set of commands that instructs ANSYS to build the model, add the loading and boundary conditions, and conduct the FEA analysis. The design variables of the model are the skin thicknesses at the root and midpoint of the wingbox. The tip thickness is fixed to 0.1 in. PAGE 37 37 The thickness varies linearly between the root and midpoint, and between the midpoint and the tip. Figure 4 3 shows the ANSYS model of the wingbox. The model contains a total of 5,105 nodes, 5,652 elements and was meshed using SHELL 63 Elastic Shell elements. The element contains four nodes, four thicknesses, and orthotropic material proper ties. It also has six degrees of freedom at each node, translations in the x, y, and z directions and rotations about the nodal x, y, and z axes. The boundary conditions are that all nodes at the root of the wing are constrained in all directions. Figure 4 4 shows the meshed wingbox mode l The material properties of the model are based on 7150 T77 aluminum, which were taken from Metallic Materials Properties Development and Standardization (MMPDS) [21 ]. The se material properties are listed in Table 4 1. Loa ding C alculations The stresses and deflections of the wingbox model are calculated using mechanics of materials methods. Determination of these stresses and deflections requires simplifying assumptions on the geometry of the structure. Without losing gene rality, the following simplifying assumptions have been made: 1. The cross section of the wing is made of thin hollow rectangular box section and the platform is trapezoidal. The wing is modeled as a cantilever beam clamped at its root section and free at its ends. 2. The lift over the surface of the wing area is replaced by a line load (lift per unit length) elliptically distributed over a sweep of a line whose locus of pointes are one quarter of the chord from leading edge (c/4 line), as seen in Figure 4 5 Wing lift distribution is directly related to the wing geometry and determines such wing performance characteristics as induced drag, structural weight, and stalling characteristics. The distribution of the aerodynamic lift along the span of a wing is com monly regarded as elliptical PAGE 38 38 and depends (among other things) on the taper ratio For an elliptical lift distribution shown in Fig ure 4 6 the lift w on the total span length of both wings, L is defined by the equation of an ellipse. (4 1) Where, w o is the maximum load per unit length at the center of the fuselage ( x = 0). The value of w o is determined from the gross weight of t he aircraft at landing. For straight and level flight, the total aerodynamic lift is equal to the area of the ellipse and must be equal to the gross weight of the aircraft. Hence, lift per wing is Weight of Airplane x g force (4 2) Now solving for w o in Equation 4 2 a t 2.5 g, (4 3) where W g is the aircraft gross weight. The lift per unit length w at any section of the wing is, hence, given by (4 4) The lift distribution va ries from 747.21 lbs/in at the root to 0 lbs/in at the wingtip. The plot of the lift di stribution is seen in Figure 4 7 The elliptical lift distribution can now be converted into a pressure load, p on the wing and the force per unit length F acting on th e leading edge of the chord as shown in Figure 4 8 Equilibrium of forces in the vertical direction requires (4 5) Equilibriu m of the moments about the quarter chord requires PAGE 39 39 (4 6) Solving for p and F from Eq. 4 5 and Eq. 4 6 yields (4 7) Using these equations, the pressure at the root was calculated to be 2.198 lbs/in 2 to 0 lbs/in 2 at the tip, while the force at the leading edge was calculated to be 373 .61 lbs/in to 0 lbs/in at the tip. Plot of the pressure and force can be seen in Figures 4 9 and 4 10 Half the distribution obtained from Eq. 4 7 is applied to the top of the model while the other half is applied to the bottom of the model. This was done in order to reduce errors in the FEA analysis of the wingbox model. Deterministic Optimization The deterministic optimization problem can be formulated as (4 8) where V i i and A basis ) are the volume, stress and A basis allowable stress of the material respectively The design variables are the skin thicknesses at the root and midpoint of the wingbox. Maximum allowable values for the problem are the A basis value of 7150 T77 aluminum for stress and 68.2 inches for displacement. In i and the calculated tip displacement A basis ) and a llowable tip displacement to the capacity. T he loads are multiplied by a safety factor of 1.5 in order to consider various uncertainties involved in the geometric parameters applied load, and computational errors The PAGE 40 40 MATLAB code will input the initial design variables into ANSYS, which will conduct the analysis. ANSYS then sends the maximum von M ises element stress, tip displacement data and total volume of the structure back to MATLAB, which will work to find the design parameters that minimizes the structur al weight, while satisfying stress and displacement constraints The optimization problem converged in 7 iterations with 36 function evaluations. Table 4 2 lists the results of the deterministic optimization. It should be noted that the maximum stress occu rs at the root of the wing model. A contour plot of the element stress can be viewed in Figure 4 11. Probability of Failure Calculation Using MCS In this section, the probability of failure of the wingbox at the deterministic optimum design is evaluated u sing Monte Carlo Simulation (MCS). As stated in C hapter 3, the purpose of the probability of failure is to evaluate the level of safety of the deterministic optimum design. It will also be used for the design criterion in the probabilistic optimization. In the calculation of the probability of failure, the factor of safety will not be considered. Instead, uncertainties related to variability in material properties, manufacturing tolerances, and applied loads will be considered in calculating the probability of failure. Failure of the wingbox is predicted to occur when its maximum von Mises stress and tip displacement is greater than its A basis failure stress value and allowable displacement. Knowing this, the performance function corresponding to the failu re stress mode can be written as (4 9) where and are the capacity an d response of g 1 Similarity, the performance function corresponding to the displacement failure mode can be written as PAGE 41 41 (4 10) where and are the capa city and response of g 2 Variabilities are introduced into the performance function through random variables A basis (Tip Displacement) allowable and a load factor. The means and standard deviations of random variables are listed in Table 4 3. As in the ten bar truss example, the probabilities of failure were calculated using separable MCS. After calculating the probabili ties of failure the total probability of failure of the system can be approximated as (4 11) where P F is the syste m probability of failure, P f1 is the probability of failure of the stress failure mode, and P f2 is the probability of failure of the displacement failure mode. Using separable MCS with 10 6 samples, the probabilities of failure for each element and the syst em are listed in Table 4 4. The results show that the probability of failure of the displacement failure mode is an about six times larger than the probability of failure of the stress failure mode. Since the stress is a local performance, it depends on th e thickness of the root. On the other hand, the displacement is a global performance and its value depends on the t hickness of entire wingbox. Thu s, it is possible to increase the thickness of the midpoint design variable, which will reduce the probabili ty of failure of the displacement failure mode, while simultaneously decreasing the thickness of root design variable, which will increase the probability of failure of the stress failure mode. This will make it possible to reduce the weight of the wing mo del while maintaining the same level of system probability of failure. Approximate Probabilistic Optimization Using ECARD Starting from the deterministic design, the approximate probabilistic optimization problem can be formulated such that the weight of the structure is minimized, while maintaining the same PAGE 42 42 system probability of failure with that of the dete rministic optimum design. In stead of the actual probability of failure, the approximate probability of failure from ECARD is used. Thus, the optimizat ion problem can be written as (4 12) where the approximate system probability of failure is the sum of each approximate probability of failure mode. Before the ECARD optim ization, a characteristic response and correction factors are calculated for each failure mode. The results of the ECARD optimization are displayed in Table 4 5. Using 10 6 MCS samples, the ECARD optimization needed only two iterations and 4 reliability ass essments to reach close to an accurate optimum. Overall weight is reduced by 0.22% (42.7 lbs) while maintaining the same system probability of failure as that of the deterministic optimum design. This reduction is achieved by reallocating the risk from the higher risk displacement failure mode to the lower risk stress failure mode. The ECARD optimization slightly increased the midpoint thickness and decreased the root thickness. This resulted in a similar level of probabilities of failure between the displa cement and stress failure modes. PAGE 43 43 Table 4 1. Material Properties of 7150 T77 Aluminum 7150 T77 Aluminum Yield Strength = 80.5 ksi A Basis = 74 ksi B Basis = 79 ksi Table 4 2. Results of deterministic optimization of the wingbox model Root Th ickness (in) Mid point Thickness (in) Deterministic Weight (lbs) 0.78226 0.44164 19,174.40 Table 4 3. Variability for Wing Model Uncertainties Distribution type Mean Standard Deviation COV Load Factor Normal 1 0.1 10% Failure Stress Normal 80,500 psi 5072 psi 6.30% Displacement Normal 68.2 in 3.41 in 5% Table 4 4 P robabilities of failure of the deterministic optimum design P f1 P f2 P F 5.26 E 06 3.17 E 05 3.69 E 05 Table 4 5. ECAR D optimization results Iteration 1 Prob Thickness (in) Mean Value Approx Pf Actual Pf Root 0.7666 Stress: 50,257 psi 1.110E 05 1.141E 05 Midpoint 0.4533 Displacement: 45.249 in 2.584E 05 2.545E 05 Total Pf: 3.694E 05 3.687E 05 Prob Weight: 19132.6 lbs Iteration 2 Prob Thickness (in) M ean Value Approx Pf Actual Pf Root 0.7668 Stress: 50,246 psi 1.131E 05 1.130E 05 Midpoint 0.4531 Displacement: 45.255 in 2.563E 05 2.564E 05 Total Pf: 3.694E 05 3.694E 05 Prob Weight: 19131.7 lbs PAGE 44 44 Figure 4 1. Boeing 767 wing dimensions PAGE 45 45 F igure 4 2. Boeing 767 internal schematic PAGE 46 46 Figure 4 3. ANSYS model of the wingbox PAGE 47 47 Figure 4 4. Meshed ANSYS model of the wingbox Figure 4 5 Sweep of the quarter chord PAGE 48 48 Figure 4 6 Relationship of local lift distribution and taper ratio Figure 4 7 Elliptical lift distribution from the roo t to the tip of the wingbox model PAGE 49 49 Figure 4 8 Equilibrium of forces on the wingbox model Figure 4 9 Pressure distribution from root to wingtip of the model W F c/2 p c/4 chord, c PAGE 50 50 Figure 4 10 Force distribution from root to wingtip of the m odel PAGE 51 51 Figure 4 11. Contour plot of maximum stress on the wingbox model PAGE 52 52 CHAPTER 5 SUMMARY AND CONCLUSI ONS An exact capacity approximate response distribution (ECARD) probabilistic optimization method that dispenses with most of the expensive structural response calculations (typically done via finite element analysis) was proposed in this paper. ECAR D was demonstrated with two examples. First, probabilistic optimization of a ten bar truss problem was performed, where risk was allocated between truss mem bers. Then, probabilistic optimization of a wingbox was performed, where risk was allocated between the different failure modes. From the results obtained in these two demonstration problems, we reached to the following conclusions. 1. In the ten bar truss pr oblem ECARD converged to near optima that allocated risk between failure modes much more efficiently than the deterministic optima. The differences between the true and approximate optima were due to the errors involved in probability of failure estimatio ns, which led to errors in the derivatives of probabilities of failure with respect to design variables that is required in risk allocation problems. 2. The approximate optimum required four inexpensive ECARD iterations and five probability of failure calculations for the ten bar truss example to locate the approximate optimum. In the wingbox example, two ECAR D iterations were required and probabilities of failure of the elements are calculated three times to locate the approximate optimum. This represents substantial reduction in the number of probability calculation require for full probabilistic optimization. PAGE 53 53 APPENDIX CALCULATION OF MEMBE R FORCES OF THE TEN BAR TRUSS Analytical solution to ten bar truss problem is given in Elishakoff et al. [ 16 ] The member incorrect in the reference. The corre ct expressions are: (A 1) (A 2) (A 3) (A 4) (A 5) (A 6) (A 7 ) (A 8) (A 9) (A 10) where PAGE 54 54 (A 11) (A 12) (A 13) (A 14) (A 15) PAGE 55 55 LIST OF REFERENCE S [1] Ben Haim, Y., and Elishakof f, I., Convex Models of Uncertainty in Applied Mechanics, Elsevier, Amsterdam, 1990. [2] Reliability from Stress Conference on Fibrous Composites in Structural Design, Vol. 1, Department of Defense, Lake Tahoe, NV, 1992, pp. 503 521. [3] based Optimal Design Using Advanced First Vol. 15, No. 4, 1987, pp. 523 542. [4] Engineering Mechanics, Vol. 120, No. 5, 1994, pp. 1154 1159. [5] esign 564. [6] index and Target performance nd Structures, Vol. 80, No. 3 4, 2002, pp. 257 269. [7] based Design Optimization Using Probabilistic No.5, 2004, pp. 314 325. [8] Du, 2004, pp. 225 233. [9] Ba ystem Reliability 1087 1096. [10] based Structural Optimization of Frame Structures for Multiple Failure Cri teria Using Topology [11] Optimization Using Exact Capacity Approximate Response Distrib submitted, 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 2006. PAGE 56 56 [12] Oberkampf, W.L., DeLand, S.M., Rutherford, B.M., Diegert, K.V. and Alvin, K.F., Laboratory Report, SAND2000 0824, Albuquerque, NM, April 2000. [13] Oberkampf, W.L., Deland, S.M., Rutherford, B.M., Diegert, K.V., and Alvin, K.F., S afety, Vol. 75, 2002, pp. 333 357. [14] Computing, 2004, pp. 103 118. [15] Acar, E., Kale, A., and Haftka, R.T., "Comparing Effectiveness of Measures that Improve Aircraft Structural Safety," submitted, ASCE Journal of Aerospace Engineering, 2006. [16] Publishers, 3 rd edition, 1992. [17] Uncertainty Optimization with Anti No. 6, 1994, pp. 1401 1405. [18] Taking Advantage of Separable Limit 1632, 47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Newport, RI, May 2006. [19] Kreyzig E ., Advanced Engineering Mathematics, Wiley, Ne w York, pp. 848. [20] Publishers, 3rd edition, 1992. [21] Richard C. Rice, Jana L. Jackson, John Bakuckas, and Steven Thompson Material Properties Development and Stand Administration, January 2003. PAGE 57 57 BIOGRAPHICAL SKETCH Richard Pippy was born in Bronxville, New York in 1972. Upon graduation from high school, he joined the United States Marine Corps where he served for six years. Aft er his honorable discharge, he then attended St. Petersburg College where he received an Associate of A rts degree in 2001. He then joined the University of Florida where he earned a Bachelor of Science degree in mechanical engineering in 2005. In 2006, he returned to the University of Nam 