Probabilistic function evaluation system (ProFES) for reliability-based design

STRUCTURAL SAFETY

Structural Safety 28 (2006) 164–195 www.elsevier.com/locate/strusafe

Probabilistic function evaluation system (ProFES) for reliability-based design Y.-T. Wu *, Youngwon Shin, Robert H. Sues, Mark A. Cesare Applied Research Associates, Inc., Southeast Division, Raleigh, NC 27614, United States Available online 20 April 2005

Abstract ProFES is a probabilistic analysis and design software that allows users to perform a wide range of probabilistic function evaluations including finite element analysis in a graphical environment. This paper summarizes the ProFES Version 2.0 capabilities with a focus on recent ProFES enhancements in three major areas related to probabilistic design: design optimization, damage tolerance, and possibilistic-probabilistic analysis. Several efficient computational probabilistic methods developed for the recent enhancements are highlighted and demonstrated using structural application problems.
2005 Elsevier Ltd. All rights reserved. Keywords: Probabilistic analysis software; Structural reliability; Reliability; Probability; Reliability based optimization; Probabilistic damage tolerance; Possibilistic analysis

1. Introduction As competition drives manufacturing industry to produce more reliable and affordable products that can be brought to market faster, there is a greater need for designs that balance performance, reliability, and cost. To achieve the optimal designs, designers must consider manufacturing tolerances, loads, material properties, and boundary conditions, and must design with these uncertainties in mind to ensure that products are economical and reliable and safe *

Corresponding author. Tel.: +1 919 876 0018. E-mail address: [email protected] (Y.-T. Wu).

0167-4730/$ - see front matter
2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.strusafe.2005.03.006

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Fig. 1. ProFES capabilities.

during the service life under various usage environments. Recognizing that the traditional safetyfactor based design tend to be over-conservative, major industry companies are turning to probabilistic methods to systematically assess the effects of uncertainties, to predict product reliability and performance, and achieve optimal design with cost and other constraints. Since the release of first version in 1999, ProFES has been designed to facilitate rapid development of probabilistic models from existing deterministic ones. It is built on an innovative datadriven software architecture that seamlessly integrates with commercial finite element codes, including ANSYS and MSC/NASTRAN, to simplify the modeling of uncertainties [1]. Additionally, ProFES interfaces directly to CAD/CAE applications (Unigraphics, PATRAN). This advanced feature allows shape parameters to be random variables or design variables. The ProFES GUI allows the user to build probabilistic models by examining FE models, and specifying and defining random variables, response functions, limit-states, etc. For legacy codes, ProFES can interface with an executable code with text based input and output using the file-mode capability. The user simply opens the input and output files using ProFES/GUI and highlights the input fields that are to be random or design variables. The executable code is called from GUI

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automatically. Specified fields in the input file become random variables and fields in the output file become response variables. Most recently, we have developed a batch-mode version of ProFES. Thru the file-mode capability, ProFES/GUI, which controls the analysis methods, now can use the batch-mode ProFES as an external code that outputs probabilities. This framework opens up many analysis options including possibilistic–probabilistic analysis and reliability-based design optimization with probabilistic objectives and constraints. ProFES has a suite of standard probabilistic methods including first-order reliability method (FORM), second-order reliability method (SORM), first-order second-moment (FOSM), Monte Carlo, and importance sampling. In this paper we focus on three recent major ProFES enhancements related to probabilistic design: reliability-based design optimization, reliability-based damage tolerance design, and possibility-based design. ProFES capabilities are summarized in Fig. 1. 2. Reliability-based design optimization (RBDO) 2.1. RBDO overview Physics-based modeling combined with computer-aided design has increasingly become widely accepted by the design community to reduce product design and development time as well as testing requirements. To ensure high reliability and safety, uncertainties inherent to or encountered by the product during the entire life cycle must be considered and treated in the design process. The RBDO incorporates uncertainties by adopting a probability-based design optimization framework to ensure high reliability and safety. A typical formulation for reliability-based design is: Minimize : Subject to :

F ðdÞ P ½gj ðX; dÞ > 0
P Rj ; d lk

6 dk 6

d uk ;

j ¼ 1; J

ð1Þ

k ¼ 1; K

in which F(d) is an objective function such as weight or expected life cycle cost, X(i = 1, n) is a random variable vector, d is a design (or decision) variable vector with lower bound dl and upper bound du, gj(X,d) > 0 are design limit states, with each gj(X,d) > 0 corresponds to a successful event, associated with multiple failure modes, and Rj are target reliabilities (or, 1 – allowable failure probabilities). The multiple failure events can be statistically correlated due to common random variables. A more general system reliability definition requires the use of union and intersection events, e.g., R ¼ 1  P f½ðg1 6 0Þ [ ðg2 6 0Þ
\ ½ðg3 6 0Þ [ ðg4 6 0Þ
g:

ð2Þ

Another useful formulation is to maximize reliability with resource constraints. A general RBDO formulation involves deterministic and probabilistic functions of multiple objective functions and multiple equality and inequality constraints. Design variables may be deterministic, e.g., tightly controlled geometry, or associated with random variables such as the mean value of a random variable. In addition to the needs to generate probabilistic models, probabilistic analysis in RBDO generally requires a relatively larger number of deterministic analyses that may involve highly

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complicated and computationally time-intensive numerical models such as finite element and CFD models. Fortunately, with recent advances in computational mechanics and the remarkable increase in computational power including the use of parallel computing, high-fidelity numerical modeling thought to be infeasible a decade ago is now available on the engineer
s personal computer. In the area of reliability methods, along with a large amount of research work on the development of efficient reliability calculation algorithms, significant progress also has been made on the efficiency of the overall RBDO algorithms [2]. 2.2. ProFES/RBDO overview The ProFES/RBDO analysis is GUI-driven. In the RBDO framework, the objective is to maximize system performance (e.g., payload, aerodynamic efficiency, etc.), while satisfying constraints that ensure reliable operation. In general, the objective function and constraints are probabilistic. The framework described herein allows designers and analysts to solve RBDO problems in much the same way that modern design optimization problems are posed, thereby enabling all the benefits of modern design optimization but achieving more robust designs. The designs are optimal over the range of operating conditions that a system may be subjected to while providing a desired level of reliability. The ProFES/RBDO framework supports the use of both commercial and public domain optimization packages. 2.3. ProFES/RBDO method The ProFES/RBDO method implemented in ProFES is summarized in Fig. 2. The methodology involves: (1) a multi-stage design of experiments variable screening strategy to define the significant variable set; (2) probabilistic analysis using any of ProFES
s probabilistic analysis methods and interaction with commercial CAE codes; (3) linearization of the constraints at the most probable failure points (MPPs) of each random variable, for each constraint; (4) development of a second order response surface for the objective function at the mean values of all design variables; (5) non-linear programming to find the optimum design using the response surfaces (via a public domain or commercial optimizer); and (6) updating of the MPPs for the current active constraint set using new values of the design variables. As shown the process is repeated until convergence is achieved. For details of the methods and related RBDO studies, see [3–6]. A key feature of the methodology is how the probabilistic analysis is de-coupled from the optimization process. As shown, computation of most probable failure points (MPPs) is not performed inside of the optimization loop. Rather, the MPPs are computed initially, before the first execution of the optimization loop, and then updated after the optimization loop is executed. This is equivalent to using an approximate form of the constraint during the optimization process. 2.4. RBDO example: transport aircraft wing optimization [6] To demonstrate the capabilities of ProFES/RBDO, we performed reliability-based structural optimization of a full-scale transport aircraft wing and compared the RBDO-based design with the optimized initial design based on existing safety-design rules. We obtained the NASTRAN model of the Advanced Composite Technology (ACT) wing from the Boeing Company. The base

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Fig. 2. Flow chart for ProFES/RBDO methodology.

line aircraft selected for this demonstration is the D-3308-4 configuration of the proposed Boeing 190-passenger, two-class, transport aircraft. The semi-span test article consists of upper and lower cover panels, front and rear spars, ribs, and bulkheads shown in Fig. 3. The major components are the cover panels. Each contains skin, stringers, spar caps, and intercoastal clips. The design weights for this aircraft are maximum take off gross weight (MTOGW) = 180 kips and maximum landing weight (MLW) = 167.5 kips. The critical design conditions for this composite wing box were derived from the DC-10-10 and MD-90-30 aircraft loads as follows:     

2.5 g positive balance flight maneuver (upbending). 1.0 g negative balance flight maneuver (downbending). Braked roll (ground). 15 psi fuel overpressure (either inboard tank or outboard tank). 9.0 g forward emergency landing fuel inertia (either outboard tank full or both tanks full).

For the wing model, the NASTRAN finite element model is shown in Fig. 4. The NASTRAN analysis shows the maximum deflection of 39 in. (z-component displacement = 38.8 in.) occurs at

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Fig. 3. Baseline aircraft configuration and semi-span structural arrangement [6].

Fig. 4. Location of element 2767 and node 3530 in the constraints.

node 3530 at the wing tip and the maximum vonMises stress of 6770 psi at element 2767. For our illustration, we will use the vonMises stress at element 2767 for our yielding (failure) constraint and the z-component displacement at node 3530 for our deflection constraint in the optimization set up. The objective is to minimize the weight of the wing. The constraints are chosen such as to prevent material yielding and enforcing limits on the vertical displacement at wing tip. Using ProFES/RBDO, the following formulations are used: Design objective: minimize the mean of the weight of the wing Subject to reliability constraints:  b1 6 0:0; g1 ¼ btarget 1 g2 ¼ btarget  b2 6 0:0; 2

ð3Þ

where b1 ¼ U1 ½R1
and b2 ¼ U1 ½R2
;

ð4Þ

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Table 1 Description of the design variables (see Fig. 6) Var. name

Initial design

Min.

Max.

SD

Description (unit)

T1 T2

0.55 0.33

0.385 0.297

0.715 0.363

0.005 0.005

Upper skin panel thickness (in.) Lower skin panel thickness (in.)

Table 2 Description of the random variables (see Fig. 6) Var. name R1 R2 R3 R4

Mean

SD 7

1.21 · 10 0.385 0.385 0.149

1.21 · 10 0.005 0.005 0.005

5

Distribution

Description (unit)

Normal Normal Normal Normal

E11 of lower skin panels (psi) Front spar panel thickness (in.) Rear spar panel thickness (in.) Rib panel thickness (in.)

where R1 and R2 correspond to the reliability of the stress and displacement constraints, respectively. For the purposes of demonstrating the methodology, we selected nominal loads such that R1 = R2 = R* for the original design. This design served as a reference for comparison. For RBDO, the objectives were to decrease the probability of failure by 10 times, R1 = R1 = 1(1  R*)/10, or 100 times, R1 = R1 = 1(1  R*)/100, without increasing the weight. To begin the RBDO of the wing, we first set up a verification problem with only two design random variables. The design and random variables are shown in Tables 1 and 2, respectively. The design and random variables are chosen to be the thickness of the shell sections that contribute the most weight to the wing. The two design variables T1 (thickness of shell section with 123 shell elements on the upper skin panel) and T2 (thickness of shell section with 169 elements on the lower skin panel) in Table 1 are the biggest weight contributors to the wing with total weights of 251.3 and 152.4 lbs, respectively. The three random variables R2 (thickness of shell section with 82 elements on the front spar panel), R3 (thickness of shell section with 83 elements on the rear spar panel), and R4 (thickness of shell section with 120 elements on the rib panels) in Table 2 are the next biggest weight contributors to the wing in their corresponding sections with total weights of 125.0, 116.3 and 59.1 lbs, respectively. Fig. 5 shows the shell elements to which the design and random variables are assigned. Fig. 6 shows the ProFES GUI that contains the wing model, the selected elements (left lower corner), and the definition of the design and random variables in a folder-tree structure (left upper corner). After defining the variables, the user uses the analysis menu to select optimization parameters. Table 3 shows the optimization results. We tested three versions of RBDO method: loop decoupling, Full-FORM (inner standard FORM analysis in the inner loop), and Full FORM with reusing the MPP from the previous step. The loop decoupling method requires only 156 finite element runs, i.e., a factor of 4.4 fewer runs than the Full FORM approach. Reusing the MPP from the previous step in the Full FORM approach reduces the number of finite element runs by a factor of 1.7. As shown in Table 3, all three methods give approximately the same optimal designs. The optimal design increases the weight by 7.6 1bs from that of the initial design but improves the reliability.

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Fig. 5. Design variables and random variables.

Fig. 6. Model of the Boeing
s ACT wing in ProFES GUI MSC/NASTRAN model has 3804 FE nodes, 3770 FE elements (2222 Shells + 1548 Beams), 22 material properties, and 47 shell element properties (PSHELL).

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Table 3 Wing optimization results (R1 = R2 = 1(1  R*)/10, two design variables T1, T2) Initial design T1 (in.) T2 (in.) Objective (lb) Weight of five sections (lb) No. of FE analyses* *

0.55 0.33 8275.22 704.29

Optimal design Loop-decoupling

Full-FORM

Full-FORM (Reuse MPP)

0.5431 0.3533 8282.83 711.90 156

0.5439 0.3527 8282.90 711.97 686

0.5439 0.3527 8282.89 711.96 406

Includes the number of evaluations for the objective function and the constraints.

After the RBDO analysis, we performed a probabilistic analysis of the new design with both FORM and 3000 MCS (Monte Carlo Simulation) to check the new design and confirm that the locations of the maximum stress and the displacement remain the same. We then made a more challenging problem by making the thicknesses of all five sections to be random design variables and only kept the elastic modulus to be non-design random variable. Table 4 shows that the optimal design reduces the weight by 17.3 lbs while decreasing the probability of failure by a factor of 10. Next, we ran another analysis to see how much weight could be reduced if the reliability is kept the same. The results in Table 5, using the loop decoupling method, show that the optimal design Table 4 Wing optimization results (R1 = R1 = 1(1  R*)/10, five design variables T1, T2, T3, T4, T5) T1 (in.) T2 (in.) T3 (in.) T4 (in.) T5 (in.) Objective (lb) Weight of five sections (lb) No. of FE analyses* *

Initial design

Optimal design

0.55 0.33 0.385 0.385 0.149 8275.2 704.29

0.5512 0.3563 0.3465 0.3465 0.1341 8257.9 686.97 164

Includes the number of evaluations for the objective function and the constraints.

Table 5 Wing optimization results (R1 = R1 = R*, five design variables T1, T2, T3, T4, T5) T1 (in.) T2 (in.) T3 (in.) T4 (in.) T5 (in.) Objective (lb) Weight of five sections (lb) No. of FE analyses* *

Initial design

Optimal design

0.55 0.33 0.385 0.385 0.149 8275.22 704.29

0.5581 0.3328 0.3465 0.3465 0.1341 8250.20 679.27 99

Includes the number of evaluations for the objective function and the constraints.

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Table 6 Wing optimization results (R1 = R1 = 1(1  R*)/100, five design variables T1, T2, T3, T4, T5) T1 (in.) T2 (in.) T3 (in.) T4 (in.) T5 (in.) Objective (lb) Weight of five sections (lb) No. of FE analyses* *

Initial design

Optimal design

0.55 0.33 0.385 0.385 0.149 8275.22 704.29

0.5625 0.3630 0.3465 0.3466 0.1341 8266.15 695.22 165

Includes the number of evaluations for the objective and the constraints. 1.00

Initial design 0.975

0.98

0.987

Weight

0.964 0.96 0.94 0.92 0.90

Initial Factor of 10 reliability improvement

Factor of 100 improvement

Reliability

Fig. 7. Comparison of initial design with several optimal designs using ProFES/RBDO.

reduces the weight by 25.02 lbs. Finally, we ran another analysis using the loop decoupling method using R1 = R2 = 1(1  R*)/100. As shown in Table 6, the weight is reduced by 9.07 lbs. At the end of the RBDO, we checked and confirmed that the locations of the maximum stress and the displacement remain the same. Fig. 7 compares the initial design with the three optimal designs described above. The numbers in the figure show the ratio of the weight for the optimal design as compared with the initial design for various factors of improvement in reliability as compared with the initial design. The ratios shown in the figure are for the weight of the five sections in the initial design (with the total weight of 704.29 lbs) and the weight of the same five sections in the optimal designs. We should note that the weight of these five sections constitute only 8.5% of the total weight of the wing, and it would be a natural conclusion that more weight savings may be achieved if we consider more shell sections as well as the beams in the model as design variables. However, this would require considering many more proprietary design requirements and is outside the scope of this demonstration study. 3. Reliability-based damage tolerance (RBDT) 3.1. Overview For damage tolerance designs, many structural systems rely on maintenance practices to maintain reliability over the design life or extend the life beyond the original design for economical

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reasons. This section describes the ProFES reliability-based damage tolerance (RBDT) methodology that employs a systematic approach for probabilistic fracture-mechanics damage tolerance analysis with maintenance planning under various uncertainties. An efficient and accurate probabilistic method, built on a two-stage conditional importance sampling approach, is described and demonstrated. The first stage computes risk, without inspection, using the most probable point (MPP)-based importance-sampling technique combined with a new error-checking method. The second stage computes risk, with inspection, by simulating inspection and maintenance effects using the samples generated from the Stage 1 failure domain. The error-checking procedure addresses the major shortcoming in the MPP-based approximation methods and leads to a robust and efficient sampling method. This paper also presents a strategy to significantly speedup the inspection optimization computations by re-using crack growth histories for risk and risk reduction computations without additional stress and life analyses. 3.2. ProFES/RBDT framework The ProFES/RBDT framework, summarized in Fig. 8, considers a wide range of uncertainties including:  Random or uncertain parameters in material (e.g., threshold of the stress-intensity factor, modulus of elasticity).  Defect or flaw (including size, shape, and location, and the frequency of occurrence). Fracture Mechanics Damage Tolerance Principal Structural Principal Structural Element DT Model Element DT Model - FE stress model - FE stress model - FM life model - FM life model

Crack Size

Proba bilistic Analysis Driver Adaptive Resp. Surf. Most Probable Point Importance Sampling

Update distribution after maintenance

Initial Flaw 1st Insp.

2nd Insp.

Usage Load Spectra

NDI Inspection Planning 1 Stress

Material

Modeling Error

Inspection time

Cum. Probability

Initial Flaw or FOD

At Insp.

POD

Prob. of Fracture

Time (Flight hours)

Without Inspection With Inspection

Flight Hours

Time

Critical Size

0 Crack Size

Fig. 8. ProFES/RBDT framework.

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Fig. 9. ProFES/RBDT software design.

   

Loading, type of usage (with frequency of occurrence). Finite element model (including modeling error). Crack growth model (including modeling error). Maintenance (including inspection schedules, frequency of inspections, probability of detection curves, repair/replacement methods and effects).

The ProFES/RBDT software has been developed that includes the two-stage simulation method. The design of the modularized software is illustrated in Fig. 9, where the three key modules, probabilistic analysis, FE analysis, and fatigue fracture mechanics, are replaceable by other userpreferred codes, provided that the input and output of the codes follow certain file formats. We used batch-mode version of ProFES, ANSYS, and NASGRO 3.0 for this study. 3.3. Probabilistic methods for RBDT with inspection optimization 3.3.1. Limit state and probability of failure formulations Under cyclic loading, an initial flaw will grow and cause a fracture failure when the stress-intensity factor K reaches the fracture toughness Kc. The limit state is: KðX 1 ; . . . ; X n ; N s Þ ¼ K c :

ð5Þ

The stress-intensity factor is dependent on the service life, Ns , and the random variable vector X that includes all the random variables except inspection-related parameters. An alternative limit state is: gðX; N s Þ ¼ N f ðXÞ  N s ;

ð6Þ

where Nf is the life-to-failure. The probability-of-failure without inspection, as: Z Z 0    fX ðXÞ dX; pf ¼ Pr ½N f ðXÞ 6 N s
¼ N f 6N s

in which fX(X) is the joint density.

p0f ,

can be expressed ð7Þ

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For the with-inspection case, the pf after the survived defected parts being detected, measured, and repaired or replaced needs to be re-computed with an adjusted fX(X) due to the change in crack size distribution. The sum of the two (before and after inspection) integrals is the cumulative pf [8]. The difficulty in this integral approach is in computing the adjusted fX(X) after each inspection. To overcome the efficiency issue, Harkness et al. [9] proposed an approximate FORM-based approach. An exact FORM-based formulation was proposed by Madsen et al. by computing pf conditioned on two types of inspection results: non-detection based on probability of detection (POD) and detected and measured crack size modeled as a random variable [10]. Based on the conditional-probability formulation, they developed and demonstrated the FORM solution for system reliability updating. More recently, research in this area has aimed at developing more general and accurate computational framework for damage tolerance with maintenance planning under various uncertainties [11,12]. There are two important factors to consider in selecting reliability methods for RBDT: efficiency and accuracy. The efficient MPP-based methods work well for smooth, well behaved g-functions and have been widely used. Unfortunately, these MPP-based methods do not provide error estimates and may produce large errors when the g-functions are non-smooth or highly nonlinear. To address the potential error issue, but without resorting to the standard MCS, an alternative two-stage simulation approach has been developed that promises to be highly robust while significantly more efficient than the MCS [12,13]. 3.4. Two-stage simulation method with error checking [11–13] The general RBDT approach, without limiting to the number of random variables, combines the MPP-based methods, the importance sampling technique, and a two-stage simulation process. In the following, FORM is used, but SORM could be used as well. The approach consists of four steps: Step 1. The inspection-free risk p0f is computed using the standard FORM method, i.e., compute the MPP and use the hyper-tangent surface at the MPP to estimate the probability of failure e.g. [14]. Step 2. A second FORM analysis is conducted for an adjusted, more conservative, limit state defined as: gA ðX; N s Þ ¼ N f ðXÞ  A  N s ;

ð8Þ

where A > 1 is an adjustment factor that needs to be properly selected to allow for generating more conservative samples in Step 3 to check the solution from Step 1. The selection of A > 1 is to help ensure that the second failure region contains the first failure region. The value of A can be based on the FORM result to predict a slightly larger (say 20% larger) pf. Therefore, the adjusted failure probability, pAf , is greater than p0f . Note that we can use the first MPP as the initial guess to search for the second MPP to significantly reduce the computational cost for the second FORM analysis. Step 3. Generate a number of samples in the adjusted failure region using the second MPP from Step 2. These samples are used for the following analyses:

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(1). The samples with lives shorter than NS are used to compute a new p0f using: p0f ¼ pf ð2nd FORMÞ 

No: of samples with N 6 N S Total number of samples

ð9Þ

If the calculated number is close to the first FORM solution, it would provide some improved confidence that Eq. (9) is a good estimate. In addition, the first FORM MPP should be compared with the sample-based MPP to ensure that a correct MPP has been found. On the other hand, if the two numbers are significantly different, the (first) FORM solution is probably not accurate and Eq. (9) with larger A values and more samples should be used. For the cases where FORM does provide reasonably accurate solutions, Eq. (9) provides an efficient way to check and enhance the FORM solution. (2). The samples with crack growth lives shorter than NS are used to simulate the inspection and maintenance processes, and compute conditional probability of failure, pcf , i.e., the probability of failure with inspections conditioned on the population of those components with lives shorter than NS. The simulation method is illustrated in Fig. 10. All the defects that violate the limit state are grown to failure. The histories of the crack growths should be recorded for possible later use. For each sample, at each inspection, the POD is used to simulate the effect of inspection. When a defect has been detected, the defective component will be replaced by a component using an appropriate defect distribution to generate a new defect size, and the crack growth process will continue until a failure occurs, the next inspection time is reached, or the service life is reached. Step 4. Computes the probability-of-failure as follows: pf ¼ pcf  p0f :

ð10Þ

The approach is suited for problems where the FORM solution provides a reasonable approximation. It is not suitable for ill-behaved g-function or problems with multiple MPPs. A = 1 (FORM) Short Life Domain

A=1 (Exact)

MPP

X2 Long Life Domain

N f (X )

A NS ; A 1

X1, Crack Size

Fig. 10. Compute pf-with-inspection using FORM and importance sampling.

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3.5. Efficient inspection optimization method A maintenance plan generally includes inspection schedules, inspection tools, repair methods, replacement and retirement plan. This paper focuses on inspection optimization, where the objective is to determine the best timing to inspect. Theoretically it is possible to inspect and repair as frequently as needed to manage pf. In practice, the number of inspections is constrained by the maintenance cost and the equipment downtime. The conventional approach to optimizing inspection times requires calculating pf for every feasible inspection plan selected by the optimization algorithm. Ref. [12] suggests that, under practical assumptions, sampled crack growth histories may be re-used for different inspection plans without additional stress and life analyses. For a special, but important, case where only one inspection is considered, an analytical formula for risk reduction was derived in [13] and is reproduced here with additional interpretations. The assumption for using this convenient formula is that the detected and repaired/replaced parts will not fail by the service life. Assuming one inspection at t =tINSP, the POD of the entire population is: Z 1 Population POD  pPOD ¼ PODðaÞ  f ðaSurvived parts ðtInsp ÞÞ da 0

¼ E½PODðaSurvived parts Þ
;

ð11Þ

where f(aSurvived parts(tINSP)) is the probability density function of the survived defective parts at the time of the inspection. These detective parts include the parts that would not fail at t = tService. If we limit the survived pdf to those ‘‘critical parts’’, defined as those parts that have survived past t = tINSP but would fail by t = tService, if not detected, the probability of effective detection is: Probability of Effective Detection  pD Z 1 ¼ PODðaCrit: parts ðtInsp ÞÞ  f ðaCrit: parts ðtInsp ÞÞ da 0

¼ E½PODðaCrit: parts ðtInsp ÞÞ
:

ð12Þ

The probability of the critical parts from the original population is: Probability of Critical Parts  pC ¼ p0f ðtService Þ  p0f ðtInsp Þ:

ð13Þ

Therefore, the maximum reducible risk is the product of Eqs. (12) and (13): Reducible Risk  pr ¼ pC  pD ¼ ½p0f ðtService Þ  p0f ðtInsp Þ
 E½PODðaCrit: parts ðtInsp ÞÞ


ð14Þ

and the risk with inspection at t = tService is: Risk With Inspection ¼ p0f ðtService Þ  pr :

ð15Þ

Eq. (14) gives the upper bound of reducible risk because, by chance, the replaced/repaired parts can fail by tService. However, assuming a small p0f ðtService Þ, this post-repair failure probability is small and for practical purposes may be ignored. For example, when the detected parts are replaced by the as-manufactured parts, the failure probability of these parts will be smaller than p0f ðtService Þ because crack growth time restarts at inspection time.

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Eq. (14) is a product of two time-dependent terms, where the first term is the risk-reduction potential, a monotonically decreasing function, and the second term, E[POD(a)], is a monotonically increasing function. This implies that the optimal inspection time is neither at time zero, when the average E[POD(a)] is close to zero because of small initial flaws, nor towards the end of service life, when the risk-reduction opportunity is approaching zero. Eq. (14) also implies that (1) for a better POD, the optimal inspection time is earlier when the risk-reduction potential is higher, and (2) a better POD will produce more risk reduction. Note that risk reduction is proportional to E[POD(aCrit. parts(tInsp))] which can be estimated by using the sampled cracks from the failure region; therefore, by saving the crack growth histories, we can recalculate risk reduction for any inspection time without additional stress or life analysis. 3.6. RBDT demonstration examples We summarize two examples related to rotorcraft structures. To simplify the demonstrations, the detected defective parts are replaced by perfect parts with no defects. Note, however, the RBDT software can simulate repair and replacement by using user-defined defect size distributions. 3.7. Plate model using ANSYS The structure model selected to represent a rotorcraft structural part is a plate with a hole, as shown in Fig. 11. A corner crack is assumed at the hole as indicated. The load spectra is FELIX/ 28, as shown in Fig. 12, based on the main rotor blade of a military helicopter with four mission types and 140 flights [15]. The random variables are defined in Table 7. The initial crack size distribution, plotted in Fig. 13, is based on the EIFS (equivalent initial flaw size) distribution derived from a stress-life experiment [17]. We used a simplified stochastic life model defined as: ð16Þ

N ¼ C  N model ;

where Nmodel is the fracture mechanics life model and C is the life scatter random variable defined in Table 7. To demonstrate ProFES/RBDT, we developed a 1/4 model of the plate using the ANSYS software, even though an analytical stress solution was available. The model, the load spectra, and the NASAGRO model are summarized in Fig. 14. This example represents a stochastic stress and crack growth RBDT analysis with random variables in FE and crack growth models as well as POD. L = 152 mm D = 6.35 mm 25.4 mm Corner crack Thickness = 4.54 mm. Initial flaw size = 0.1 mm at 90%

Fig. 11. Plate model [15].

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Percent of max. load in spectrum

400 300 200 100 Cycles : 2755 Flight hours = 3.26 hr

0

Main rotor blade 140 flights 4 mission types 190.5 hours 161,034 cycles

-100 -200

0

500

1000

1500 15

2000

2500

3000

Number of cycles

Fig. 12. Felix/28 helicopter load spectra [15].

Table 7 Random variables for the plate model Thickness (mm) Max. load (N) Initial flaw (mm) Delta Kth Life scatter

Distribution

Mean

SD

Cov (%)

LN N User-defined LN LN

4.54 23658 0.074 156.3 1

0.023 2200 0.0224 10 0.1

0.51 9.30 30.2 6.40 10.0

Flaw Size CDF (Equivalent Initial Flaw Size) 1 0.9 0.8 0.7

CDF

0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.05

0.1

0.15

0.2

0.25

0.3

Flaw Size (mm)

Fig. 13. Equivalent initial flaw size (EIFS) distribution.

0.35

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181

Fig. 14. Plate with hole results using ANSYS FE model (Full RBDT analysis).

The selected mean thickness of 4.54 mm results in pf = 0.1, a probability large enough to allow the analysis to be done within a reasonable time frame. This probability is conditioned on having an initial flaw at the hole with a unspecified but small probability. For a service life of 750 h, the pf result with and without an inspection is shown in Fig. 15(a). A two-parameter (median and scale) log–logistic POD model used in [7] is used. Fig. 15(b) compares the POD with the CDF of crack size. The information is useful for inspection optimization. For example, the POD will be more effective at 500 h than at 400 h because the defect CDF is closer to the POD. The total number of FE and NASGRO analyses is 220. In general, the number of FE analyses is an increasing function of the number of input random variables associated with the stress, while the number of NASGRO analyses depends on the number of input random variables associated with the life. Additionally, each sampling point requires one FE analysis and one NASGRO analysis. The total CPU time using a 2 GHZ PC for the analysis is 240 min. Of that, 216 min are for the NASGRO analyses. In general, for large complicated FE models, the required CPU time for

Y.-T. Wu et al. / Structural Safety 28 (2006) 164–195 0.1

1

0.09

0.9

0.08

0.8

0.07

0.7

0.06 0.05 0.04

Crack size CDF at 400 hours

0.5

Crack size CDF at 500 hours 0.4

0.03

0.3

0.02

0.2

0.01

0.1

0 100

(a)

0.6

Without Inspection With Inspections at 400 hours With Inspections at 500 hours

CDF

Pf

182

200

300

400

500

600

700

Flight Hours

0

800

(b)

Probability of Detection

0

2

4

6

8

10

Crack Size (mm)

Fig. 15. Plate model result using ANSYS FE model and full RBDT analysis.

FE may become dominant. For the NASGRO analyses, the CPU time needed is roughly proportional to the service life. Using the samples generated from the importance-sampling method, the risk sensitivities, defined as the sensitivity of the pf with respect to each input standard deviation, are calculated for the ‘‘with’’ and ‘‘without’’ inspection cases [11,16]. The sensitivities, which are dimensionless, are then normalized so that they sum to one. Fig. 16 compares the sampling-based sensitivities with the standard FORM-based sensitivities, also normalized. The results show that both methods provide consistent sensitivities. The uncertainty in the applied load is clearly the dominant random variable. 3.7.1. Lug model Fig. 17 shows a helicopter spindle lug model [17]. The random variables are listed in Table 8. The point load is applied to the center of the pin. Since an analytical stress formula is available, no FE analysis is needed. Fig. 18 shows the NASGRO model and the applied 1-hour load spectra. The EIFS is the same as in Fig. 13. We will compare the performance of three PODs shown in Fig. 19. For this example, we use two conservative limit states and a Monte Carlo simulation to further illustrate the issues involved in error-checking. Table 9 shows the results using three A values: 1, 0.9

Risk Sensitivity

0.8

FORM-Based

0.7

Sampling-Based

0.6 0.5 0.4 0.3 0.2 0.1 0.0 Thickness Force

Defect Threshold Life Scatter

Fig. 16. Risk sensitivities for the plate model.

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183

Fig. 17. Spindle lug [17].

Table 8 Random variables for the lug model Thickness, t (mm) Max. load (N) Initial flaw size (mm) Delta Kth Life scatter

Distribution

Mean

SD

Cov (%)

LN LN User-defined LN LN

28 145000 0.074 48 1

0.14 10000 0.0224 4 0.1

0.50 6.9 30.2 8.33 10.0

1.07, and 1.33, which are associated with target lives of 750, 800, and 1000 fight hours, as well as using 1000 Monte Carlo simulation. This example requires, on average, 20 s CPU time for each NASGRO analysis (see Table 9). Based on A = 1.067, the probability in the IS (importance sampling) region is 0.081. Of the 250 samples generated in this region, there are 216 samples in the failure region with a conditional pf of 0.864. Therefore pf is 0.0699, which is close to the FORM solution, 0.0675. This, along with the fact that the angles between the FORM and the sampling MPPs are reasonably small, suggests that the IS region covers the failure region. Therefore, the importance samples are representative, assuming the FORM-based model is a good approximation. Suppose we would like to have even higher confidence, we can use a larger A value but at a cost of more samples and CPU time. Using A = 1.33, the probability in the IS (importance sampling) region is 0.151, about twice larger than for A = 1.067. Of the 500 samples generated in this region, there are 222 samples in the failure region with a conditional pf of 0.444, about half smaller than for A = 0.167. The resulting pf is 0.0699, which is the same as for A = 1.067. This means that by doubling the IS region, no additional failure region has been found. This confirms that the IS region is sufficient, again assuming that the MPP-based model is a good approximation. The above results are very close to the Monte Carlo result (0.0693) that took 60 h of CPU time. Note that 500 IS samples are equivalent to 500/0.151 = 3311 Mont Carlo samples and therefore for this example, IS provides a slightly better accuracy. However, unlike MCS, IS could miss

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Fig. 18. Spindle lug example.

failure regions. In general, a sufficient number of Monte Carlo samples should be used to ensure that IS has not missed any significant failure region. For this example, 3000 Monte Carlo samples are sufficient because we have tested and found that all the Monte Carlo samples with lives less than 750 h are in the IS region, suggesting that the IS region well covers the failure region. For complex applications with multiple MPPs, more research is needed to devise an efficient errorchecking method. Using the 222 stored simulated crack growth histories, the risk reduction curves for three PODs are generated by applying Eq. (14). The result is shown in Fig. 20. The POD-I curve is validated using the POD simulation result, shown in Fig. 21, where pf vs. inspection time is also plotted. The unsmooth curves in Figs. 20 and 21 suggest that more samples are needed to increase the resolution. The superiority of POD-I is clear from the risk reduction plot. Using POD I, there is an ‘‘inspection window’’, roughly between 300 and 650 h, with an optimal inspection time at about

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185

Three POD Curves (Log-Logistic Function) 1 POD I POD II POD III

0.9

Probability of Detection

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

1

2

3

4

5

6

7

8

9

10

Defect Size (mm)

Fig. 19. Three POD curves for the lug example.

Table 9 Lug example results A

N = Nf

No. of samp. in samp. region (n)

FORM and samp. Beta (b)

Angle from FORM MPP ()

No. of failures in samp. region (N < 750)

Prob. in samp region (Ps) [N < Nf]

Pf in samp. region (Pc) [N < 750]

pf [N < 750]

1.00 1.07 1.33 MCS

750 800 1000 750

–

1.495 1.709 1.682 1.587

0.0 33.6 22.1 17.3

– 216 222 208

– 0.081 0.157 1.000

– 0.864 0.444 0.069

0.0675 0.0699 0.0696 0.0693

250 500 3000

550 flight hours, while using POD II and III the best inspection time is around 650 h with a narrower inspection window. The results are consistent with the comments made earlier that the best inspection time is earlier for a better POD capability and a better POD capability will always produce a better optimal risk reduction. Fig. 22 shows the simulation results of pf versus flight hours with and without inspection for POD I. It shows that, without inspection, the slope of pf changes at about 550 h, which coincides with the best inspection time. 3.8. ProFES possibilistic–probabilistic analysis 3.8.1. Overview The use of probability design theories has been held back by the data and computational requirements. On the other hand, the possibilistic approach appears to have the merits of

186

Y.-T. Wu et al. / Structural Safety 28 (2006) 164–195 Reducible Risk (Pr) Given a Flaw 0.03

POD I

0.025

POD II POD III

Pr

0.02 0.015 0.01 0.005 0

0

100

200

300

400

500

600

700

800

Inspection Time (Flight Hours)

Fig. 20. Risk reduction using Eq. (14).

Reducible Risk (Pr) Given a Flaw (POD I) 0.08 Pf at Service Life Pr (Simulation) Pr (Eq.14)

0.07 0.06

Pr

0.05 0.04 0.03 0.02 0.01 0

0

100

200

300

400

500

600

700

800

Inspection Time (Flight Hours) Fig. 21. Comparisons of risk reduction using Eq. (14) and using simulation.

requiring less data and has the potential of being computationally more efficient. Therefore, it has emerged as an alternative tool in dealing with designs associated with a great amount of input and modeling uncertainty. In particular, during the conceptual and preliminary design stages, the simplicity and the efficiency will allow a larger number of design options to be quickly evaluated.

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187

Pf Given a Flaw for Various PODs 0.08 0.07 No Inspection 0.06

POD IIIat 650 hours POD II at 650 hours

Pf

0.05

POD I at 550 hours

0.04 0.03 0.02 0.01 0

0

100

200

300

400

500

600

700

800

Flight Hours Fig. 22. Risk curves for different inspection times.

Nikolaidis, Haftka and others have studied the theoretical differences between the probabilistic approach and the possibilistic approach, and their potential effects to structural design [18–20]. Cremona and Gao [21] compared the possibilistic and the probabilistic reliability theories and proposed an efficient method analogous to the most-probable-point method in structural reliability theory to compute the most-possible-point. Langley showed that a single mathematical algorithm can encompass a wide range of probabilistic and possibilistic methods [22]. In complex engineering problems, the degrees of data availability for different parameters often are non-homogeneous. Consequently some data may be better modeled using probabilistic distributions and others modeled using possibilistic distributions. This section summarizes a hybrid approach to address the uncertainty analysis and management issues by treating inherent, irreducible randomness by probability distributions and systematic, reducible errors by possibility functions [23]. Because the uncertainty can be separated into probability or possibility, this approach provides a more flexible framework for mixing the data and expert knowledge. An important benefit of the approach is to allow designers or decision makers to identity what uncertainty reduction can potentially be achieved by decreasing the uncertainties of reducible variables. 3.9. ProFES possibilistic analysis ProFES/GUI has a possibilistic analysis capability currently handling two types of membership functions for input variables: uniform and triangle. Fig. 23 shows an example of a triangular membership function. Possibilistic analysis calculates the response possibility function by searching for the extremes in the bounded domain for each possibility level.

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1

Possibility

0.8 0.6 0.4 0.2 0 X Fig. 23. An example of triangular shape possibility function.

There are three possibilistic analysis methods implemented in ProFES: 1. Bounding analysis: For each selected possibility level, compute the bounds of the response based on the extremes of the input parameters. For n variables, this requires 2n response function evaluations. This method is suitable when the bounds of the response correspond to the bounds of the inputs. 2. Response surface: This method applies Design-of-Experiment methods available in ProFES such as the first and second-order central composite designs, and Box–Benhken design. A response surface is then developed by regression analysis. Each response surface can be used for all the alpha cuts or for a single selected alpha cut. The construction of the response depends on the design of experiment method and the number of function evaluations can range from approximately n to n2 The success of this approach depends on how well the response surface fits the original function. 3. Sampling method: The method samples randomly, using proper distributions (e.g., uniform) within the bounds of each variable. The possibility that associates with the generated value is computed for each variable. For each set of random values, the response is calculated and assigned a possibility that is equal to the minimum of the individual possibilities. By repeating the process a sufficiently large number of times, a scatter plot of response possibility versus the response can be generated. By definition, the envelope of the points defines the possibility function. The sampling method guarantees to work at the expense of analysis time.

3.10. ProFES/HYBRID framework In the hybrid method, we assume nominal probabilistic distributions to input parameters with irreducible randomness as we do under a probabilistic analysis framework, but we also assign possibility functions to modeling errors and the parameters of distribution such as mean and standard deviation. The result of such an analysis is the possibilistic function of selected probabilistic

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189

measures such as reliability, mean and standard deviation. For example, the probability of failure is not single valued but has a possibility function. Thru the file-mode interface option, ProFES can conduct a possibilistic-probabilistic analysis by including the batch-mode version of ProFES to conduct inner-loop probabilistic analysis and using the GUI to select a possibilistic analysis. 3.11. Example A simple cantilever beam shown in Fig. 24 is used to demonstrate the hybrid method. The limit state is that the tip displacement of the beam, D, equals to the allowable, D0, i.e., g ¼ D0  DðE; X ; Y ; w; tÞ

4L3 DðE; X ; Y ; w; tÞ ¼ Ewt

ð17Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s 2  2 X Y þ 2 ; 2 w t

ð18Þ

where X and Y are independent random loads, and E is the modulus of elasticity (see Table 10). The width of the beam, w, and thickness, t, are set equal to 2.4484 and 3.8884 in. The modeling error, e, is introduced to account for the error in calculating the tip displacement, D, and the limit state is changed to g = D0e Æ D(E,X,Y,w,t). The uncertainty in the modeling error is considered reducible and is modeled by a triangular possibility distribution that has a nominal value of 1.0 and is bounded by [0.95, 1.05] at zero possibility level. The probability of failure curves obtained by ProFES/Hybrid are shown in Fig. 25. 3.12. Efficient hybrid method For the hybrid approach, a more efficient possibilistic analysis method has been developed and is to be added to ProFES as an analysis option. The method is based on the advanced-mean-value (AMV) concept previously developed for probabilistic analysis [24]. For monotonic functions, the AMV approach resembles the work by Akpan et al. [25], who proposed using a response-surface

Y

L =100"

X

t w

Fig. 24. Cantilever beam example for possibilistic analysis. Table 10 Random variables for the beam example Variable

Distribution

Mean

SD

X Y E

Normal Normal Normal

500 1000 2.9E7

100 100 1.45E6

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Y.-T. Wu et al. / Structural Safety 28 (2006) 164–195 1

Nominal (Poss.=1.0) Max. Extreme (Poss.=0.0) Min. Extreme (Poss.=0.0)

0.9

Probability ity of Failure l

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1

1.2

1.4

1.6

1.8

2

2.2

Do

Fig. 25. Possibility function of pf.

update approach. However, the AMV analysis framework provides an added ability to deal with non-monotonic, well-behaved functions. Assume a continuous and smooth performance function, the Taylor
s series expansion at the mean values is:  n  X oZ ðX i  lX i Þ ð19Þ Z MV ¼ ZðlX i Þ þ oX i x¼lX i¼1 which provides a mean-value based approximation. With a slightly more computational effort, the AMV method recovers the higher order error using: Z AMV ¼ Z MV þ HðX Þ  Z MV þ HðZ MV Þ;

ð20Þ

where H(ZMV) is a function of ZMV equals to the difference between ZMV and Z (Exact) at the MPP locus of Z 9MV The steps for AMV analysis are illustrated Fig. 26. An important feature of AMV is to help detect multiple MPP problems associated with non-monotonic functions. By simply replacing the most probable point with the most possible point, we have a possibilistic analysis version of AMV, as illustrated in Fig. 27 for monotonic functions. The input parameters are assumed to have triangular-shape possibility functions. The search for the response extreme usually requires the unconstrained optimization with bounded input variables. However, since ZMV is the first-order approximate function, the extreme can be obtained using ZMV when the possibility functions for the input parameters have triangular shapes. The AMV procedure provides an exact solution for monotonic functions. In addition, it may provide indications on the nonlinear effect and suggest a correction procedure. For example, if the difference between the MV and the AMV results is large, it may suggest that the function is highly nonlinear (although the result would still be exact if the function is monotonic). If the resulting possibility function has an unusual shape such as there are two or more possibility solutions

Y.-T. Wu et al. / Structural Safety 28 (2006) 164–195

Exact Model Z(X1, X2, .. Xn)

191

CDF --> AMV (m = 1)

MV

AMV AMV+

(n+1) runs (m) Z = A0+A1X1+A2X2+... AnXn

Update Ai

MPP (X*) Search for a Selected CDF

--> MV (m = 1) --> AMV+(m = 2,3..)

CDF, X*

One X* for every selected CDF

Z

. Fig. 26. AMV probabilistic analysis procedure for monotonic functions.

Possibility Exact Model Z(X1, X2, .. Xn)

1 --> AMV

(n+1) runs Z = A0+A1X1+A2X2+... AnXn

MPP (X*) Search for a Selected Possibility

AMV

AMV MV

Possibility, X*

0

MV

Fora selected possibility

Z Fig. 27. AMV possibilistic analysis procedure for monotonic functions.

for any response value, then the response function is non-monotonic. In such cases, a quick correction is to keep the maximum possibility as the correct solution and discard the other solutions. This simple correction procedure is illustrated in Fig. 28. However, if a significant correction appears needed, indicating that the function is strongly non-monotonic, other more accurate methods should be considered. 3.13. Example The probability of failure for the beam with possibilistic distributions for the mean of the input variables is analyzed. In this example, the means of E, X, and Y are assumed to have possibility

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Possibility 1

MV

--Corrected

--Original

AMV

AMV

0 Z

Fig. 28. AMV possibilistic analysis procedure for non-monotonic functions.

functions which are bounded by nominal means ±20% of their respective standard deviations. Fig. 29 shows the MV and AMV results for the limit state in Eq. (17) with displacement tolerances of 2.2 in. The AMV result matches the exact solution, as expected. The FORM analysis is performed to calculate the probability of failure. Since the sensitivities of Pf with respect to the mean of the input random variables are available as the by-products in

1 Exact Mean Value Adv. Mean Value

0.9 0.8

Possibility

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

1

2

3

4

5

6

7

Probability of Failure (x 0.001)

Fig. 29. Possibility function for pf using AMV analysis procedure (D0 = 2.2).

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193

the FORM analysis, only one FORM analysis is needed to establish the linear approximation ZMV. Therefore, the number of FORM analyses required is reduced to 1 for MV and the number of response function evaluation is 13 in a FORM analysis. The AMV procedure uses the linear approximation in MV and 8 additional FORM analyses are performed at the extreme points to correct the results. This example demonstrates that the AMV-based possibilistic analysis method is a potentially highly computationally efficient and relatively robust approximate method.

4. Summary ProFES is a probabilistic analysis and design software that allows users to perform a wide range of probabilistic function evaluations including finite element analysis in a graphical environment. This paper summarizes the ProFES 2.0 capabilities with a focus on recent ProFES enhancements in three major areas related to probabilistic design: design optimization, damage tolerance, and possibilistic-probabilistic analysis. Thru the file-mode capability, ProFES/GUI can use the batch-mode ProFES as an external code that outputs probabilities. This framework opens up many analysis options including possibilistic-probabilistic analysis and reliability-based design optimization with probabilistic objectives and constraints. Several efficient probabilistic methods developed for the recent enhancements are highlighted and demonstrated using structural application problems. We have demonstrated the ProFES/RBDO capability through a full-scale wing design application in which we have achieved a design that improves the reliability while reducing the weight. In the area of damage tolerance, the demonstration examples suggest that the two-stage importancesampling method and the optimization method, both implemented in ProFES/RBDT, are well suited for RBDT design. Probabilistic approach tends to be non-conservative while possibilistic approach tends to be too conservative. The ProFES/Hybrid capability provides a flexible framework to treat both the random (or irreducible) and the systematic (or reducible) errors and provides a more reasonable solution reflecting the amount of available data.

Acknowledgments ProFES 1.0 was funded under US Air Force contract F33615-98-2811. The contract technical monitor was Mr. Jeff Brown. In the later versions, the ProFES/RBDO work was funded under NASA contract NAS1-99081 and the ProFES/RBDT work was funded under NASA contract NASA NAS1-98025, monitored by NASA-LaRC and co-sponsored by the FAA William J. Hughes Technical Center. The possibilistic analysis capability was also funded under NAS198025. The contract technical monitor of the NASA projects was Dr. W. Jefferson Stroud. We gratefully acknowledge the financial supports from NASA and FAA, and the direction and technical guidance of the contract technical monitors.

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References [1] Cesare MA, Sues RH. ProFES probabilistic finite element system – bringing probabilistic mechanics to the desktop, presented at the AIAA 40th SDM Conference; April 1999. [2] Frangopol Dan M, Maute Kurt. Reliability-based optimization of civil and aerospace. In: Nikolaidis E, Ghiocel DM, Singhal S, editors. Engineering Design Reliability Handbook. Boca Raton: CRC Press; 2004, ISBN 0-84931180-2 [Chapter 24]. [3] Oakley DR, Sues RH, Rhodes GS. Performance optimization of multidisciplinary mechanical systems subject to uncertainties. Probabil Eng Mech 1998;13(1):15–26. [4] Xiao Q, Sues, RH, Cesare M, Rhodes GS. Computational strategies for reliability-based multi-disciplinary optimization, presented at the 13thASCE EMD Conference; June 1999. [5] Sues RH, Cesare MA. An innovative framework for reliability-based MDO. AIAA-2000-1509, In: 41st SDM conference; April 2000. [6] Mohammad A, Shin Y, Sues RH, Wu Y-T. A framework for reliability-based multi-discipline design optimization. In: 43rd AIAA/ASME/ASCE/AHS structures, structural dynamics and materials conference; April 2002. [7] Berens AP, Hovey PW, Skinn DA. Risk analysis for aging aircraft fleets. Air Force Wright Lab Report, WL-TR91-3066, vol. 1; October 1991. [8] White P, Barter S, Molent L. Probabilistic fracture prediction based on aircraft specific fatigue test data. In: 6th Joint FAA/DoD/NASA aging aircraft conference; September 2002. [9] Harkness HH, Fleming M, Moran B, Belytschko T. Fatigue reliability with in-service inspections. In: FAA/NASA international symposium on advanced structural integrity methods for airframe durability and damage tolerance; September 1994. [10] Madsen HO, Skjong RK, Talin AG, Kirkemo F. Probabilistic fatigue crack growth analysis of offshore structures, with reliability updating through inspection. Arlington (VA): SNAME, October 5–6; 1987. [11] Wu Y-T, Enright MP, Millwater HR. Probabilistic methods for design assessment of reliability with inspection. AIAA J 2002;40(5):937–46. [12] Wu Y-T, Shiao M, Shin Y, Stroud WJ. Reliability-based damage tolerance methodology for rotorcraft structures. In: Proceeding of 2004 SAE World congress, reliability and robust design in automotive engineering, Special Publication No. 1844; March 2004. [13] Wu Y-T, Shin Y. Probabilistic damage tolerance methodology for reliability design and inspection optimization. In: Proceedings of the 45th AIAA/ASME/ASCE/AHS structures, structural dynamics and materials conference; April 2004. [14] Madsen HO, Krenk S, Lind NC. Methods of structural safety. Englewood Cliffs (NJ): Prentice Hall Inc.; 1986. [15] Everett Jr. RA. Crack-growth characteristics of fixed and rotary wing aircraft. In: 6th joint FAA/DoD/NASA aging aircraft conference; September 2002. [16] Wu Y-T. Computational methods for efficient structural reliability and reliability sensitivity analysis. AIAA J 1994;32(8):1717–23. [17] Forth SC, Everett Jr. RA, Newman JA. A novel approach to rotorcraft damage tolerance, In: 6th Joint FAA/ DoD/NASA aging aircraft conference; September 2002. [18] Nikolaidis E, Haftka RT. Comparison of probabilistic and possibility-based methods for design against catastrophic failure under uncertainty, Report; 1998. [19] Nikolaidis E, Cudney HH, Chen S, Haftka RT, Rosca R. Comparison of probabilistic and possibility theory-based methods for design under catastrophic failure under uncertainty. In: 11th international conference on design theory and methodology. Las Vegas (NV): ASME; 1999. [20] Chen S, Nikolaidis E, Cudney HH, Rosca R, Haftka RT, Comparison of probabilistic and possibility-based methods for design under uncertainty. In: 40th AIAA/ASME/ASCE/AHS/ASC structures structural dynamics and materials conference, St. Louis (MO); 1999. [21] Cremona C, Gao Y. The possibilistic reliability theory: theoretical aspects and applications. Struct Saf 1997;19(2):173–201. [22] Langley RS. Unified approach to probabilistic and possibilistic analysis of uncertainty systems. J Eng Mech 2000;126(11).

Y.-T. Wu et al. / Structural Safety 28 (2006) 164–195

195

[23] Shin Y, Wu Y-T. A hybrid possibilistic–probabilistic analysis framework for uncertainty management. In: 44th AIAA/ASME/ASCE/AHS structures, structural dynamics and materials conference; April 2003. [24] Wu Y-T, Millwater HR, Cruse TA. An advanced probabilistic structural analysis method for implicit performance functions. AIAA J 1990;28(9):1663–9. [25] Akpan UO, Koko TS, Orisamolu IR. Response surface based fuzzy analysis of structural systems. In: 41st SDM conference, AIAA-2000-1634; 2000.