Advanced Probabilistic Structural Analysis Method for Implicit Performance Functions

VOL. 28, NO. 9, SEPTEMBER 1990 AIAA JOURNAL 1663

Advanced Probabilistic Structural Analysis Methodfor Implicit Performance Functions

Y.-T. Wu,* H. R. Millwater,* and T. A. CrusefSouthwest Research Institute, San Antonio, Texas 78228 0510

In probabilistic structural analysis, the performance or response functions usually are implicitly defined and mustbe solved by numerical analysis methods such as finite-element methods. In such cases, the commonly usedprobabilistic analysis tool is the mean-based second-moment method, which provides only the first two statisticalmoments. This paper presents an advanced mean-based method, which is capable of establishing the full probabilitydistributions to provide additional information for reliability design. The method requires slightly more computa-tions than the mean-based second-moment method but is highly efficient relative to the other alternative methods.Several examples are presented to demonstrate the method. In particular, the examples show that the newmean-based method can be used to solve problems involving nonmonotonic functions that result in truncateddistributions.

NomenclatureF = cumulative distribution function/ = probability density function/„ = natural frequencyH = higher-order termsn = number of random variablesP[-] = probability ofPf = probability of failureu = standard normal variableX = design random variablex = real number associated with XZ = response function or performance functionZt = linear approximation of Zz,z0 = real number associated with ZH = mean valuea = standard deviationv = Poisson's ratioO = standard normal cumulative distribution function

Superscript* = most probable point

Introduction

The need to address the uncertainties in a design has longbeen recognized. Traditionally, designers use safety fac-

tors to provide confidence. However, the safety factor ap-proach is questionable because it usually does not take intoaccount the underlying probability distributions. For exam-ple, consider the \JL ± 3<r design approach. Assuming that theunderlying distribution is Gaussian, then the probability ofexceeding the three standard deviation bounds is 0.0027. Inpractice, however, non-Gaussian distributions are not uncom-mon, and the true probabilities cannot be determined withoutthe knowledge of the underlying distributions. For consistentreliability design, therefore, it is important to determine theprobability distributions. Unfortunately, the evaluations of

Received May 8, 1989; revision received Oct. 16, 1989. Copyright© 1990 by the American Institute of Aeronautics and Astronautics,Inc. All rights reserved.

*Senior Research Engineer, Division of Engineering and MaterialSciences. Member AIAA

tDirector, Division of Engineering and Material Sciences. FellowAIAA.

the probability distributions for structural performance func-tions generally involve complicated numerical computationsthat prohibit the use of the standard Monte Carlo simulationmethod.

Because of the preceding problem, a commonly usedmethod in probabilistic structural analysis is the mean-basedsecond-moment method, which is used to obtain the meanand the standard deviation of the unknown distribution.1'2However, without the knowledge of higher moments, the firsttwo statistical moments can only provide a probability bound,which is, according to Chebyshev's inequality,

P{\X-n\>Ko}<—2 (1)

where K is a real number. This probability bound is known tobe generally too conservative. For example, for K = 3, theprobability bound is 0.11, which is far greater than the value0.0027 for a normally distributed random variable. In general,more than two moments will be required to "define" adistribution.

Recently, numerical methods have been proposed to com-pute higher statistical moments and to establish the cumula-tive distribution function (CDF) by curve fitting.3'4 Theseefforts recognize the need for defining the full distribution.However, these numerical moment methods generally mustcompute the performance function (e.g., the structural re-sponse) at well-designed calculation points (e.g., quadraturepoints) and tend to be inefficient as the number of randomvariables is large. A more effective procedure is thereforeneeded to establish the CDF, one that can be used for implicitperformance functions where the performance or response ofthe system to changes in random variables is not explicit.

A probabilistic structural analysis computer program,NESSUS, is being developed as part of a Probabilistic Struc-tural Analysis Methods (PSAM) program funded byNASA.5'6 To compute the structural response CDF effec-tively, a new probabilistic analysis tool called the advancedmean-value, first-order method7 (AMVFO) has been devel-oped. The major feature of this method is its capability toapproximate the CDF with few extra computations, relativeto the mean-based second-moment method. In this paper,since only the first-order method is presented, the AMVFOmethod will be abbreviated as the AMV method.

This paper summarizes the previous AMV method and itslimitations to certain nonlinear performance functions and

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1664 WU, MILLWATER, AND CRUSE AIAA JOURNAL

presents a generalized AMV method that is applicable tomore general nonlinear performance functions. In this paper,the generalized AMV method will be applied to several exam-ples to demonstrate the effectiveness of the method in generat-ing the CDFs for strongly nonmonotonic performancefunctions.

Review of Probabilistic Structural Analysis MethodsIn structural reliability analysis, a "performance function"

or "limit state function" g(X) is often formulated in terms ofa vector of basic design factors X = (Xl ,X2,...,Xn) in which Xfare random variables. In this paper, it is assumed that X is avector of mutually independent variables. Dependent randomvariables can be treated using transformations.7-8'11

The limit state which separates the design space into "fail-ure" and "safe" regions is g(X) = 0. The probability of failureis

Pf=P[g<0] (2)

An exact solution of pf requires the integration of a multipleintegral denoted as

Pf= (3)

where fx(x) is the joint probability density function of X, andQ is the failure region. The solution of this multiple integralis, in general, extremely complicated. Alternatively, a MonteCarlo solution provides a convenient but usually time-con-suming approximation. For practical purposes, efficient, ap-proximate analysis tools are needed. Current approaches,based on analytical approximation methods, include a fastprobability integration algorithm9 and the first- and second-order reliability methods.8

For each limit state, the most probable point is defined asfollows. First, transform the nonnormal variables X to stan-dard, normal variables u using the following transformation:

x = Fxl[<b(u)] (4)

On the limit-state surface, g(u) = 0, the most probable point isthe point that defines the minimum distance from the origin,u = 0 to the limit-state surface. This point can be found by anoptimization scheme or other iteration algorithms such as theRackwitz-Fiessler algorithm.8'10

Define Z(X) as a structural performance function; the CDFof Z(X) can then be formulated as

P(Z<z0)=Fz(z0) =1 '•JZ<Z0

(x) dx (5)

By varying z0, a series of limit states can be formed. Amost-probable-point-locus can be defined by connecting themost probable points.

In theory, it is possible to repeatedly apply the previousstructural reliability methods to compute the CDF. In reality,however, many problems may occur. The problems includeinefficiency and convergence instability due to search al-gorithms and multiple minimum distance points. Thus, thereis a need to develop a more efficient and robust procedure toestablish the CDF.

Mean-Based MethodsAssume that the Z function is "smooth" or can be

smoothed, and Taylor's series expansion of Z exists at themean values. The Z function can be expressed as

Z(A) = • (X, - j H(X)

(6)

where the derivatives are evaluated at the mean values; Zl isa random variable representing the sum of the first-orderterms, and H(X) represents the higher-order terms. The mean-based methods are defined as the methods based on the meanvalues expansion.

Mean-Value First-Order MethodBy retaining only the first-order terms, the mean and the

standard deviation of Z are:

(7)

(8)

These two statistical moments can be computed easily afterthe coefficients a, are computed. In structural analysis, thereare several ways of obtaining a, including the direct differenti-ation method and the adjoint method.12 The NESSUS codeutilizes iterative perturbation algorithms13 to efficiently com-pute the Z functions around an x and uses the solutions toestimate at. In general, the coefficients at can be computed bynumerical differentiation and the minimum required numberof Z-function evaluations is (n + 1).

Equations (7) and (8) constitute the mean-based, first-order, second-moment method. However, when the probabil-ity distributions, not just the first two moments, of Xt are fullydefined, the CDF of the first-order terms Z\ is also fullydefined. Since the Zl function is linear and explicit, its CDFcan be computed effectively using many methods, includingthe structural reliability analysis methods mentioned earlier.Thus, the mean-value first-order (MVFO, abbreviated asMV) solution defines the CDF of Z1? not just the twomoments.

For nonlinear Z functions, the MV solution is, in general,not sufficiently accurate. For simple problems, it is possible touse higher-order expansions to improve the accuracy. Forexample, a mean-value second-order (MVSO) solution can beobtained by retaining second-order terms in the series expan-sion. However, for problems involving implicit Z functionsand large «, the higher-order approach becomes difficult andinefficient. The AMV method described below provides analternative to improve the MV solution with minimum addi-tional Z-function evaluations.

Advanced Mean-Value First-Order Method7'14

The AMV method takes the MV solution one step furtherto compute the CDF. The key to the AMV method is thereduction of the truncation error by replacing the higher-order terms H(X) by a simplified function H(Zl) dependenton Zl. Ideally, the H(Z{) function should be based on theexact most-probable-point locus (MPPL) of the Z function tooptimize the truncation error. The AMV procedure simplifiesthis procedure by using the MPPL of Zl. As a result of thisapproximation, the truncation error is not optimum; however,because the Z-function correction points are generally "close"to the exact most probable points, the AMV solution providesreasonably good CDF estimations.

The step wise AMV (first-order) procedure can be summa-rized as follows:

1) Obtain the Z{ function based on perturbations aboutthe mean values.

2) Compute the CDF of Zl at selected z0 points using thefast probability integration method (other approximationtechniques such as the first- and second-order reliability meth-ods can also be used).

3) Select a number of CDF values that cover a sufficientlywide probability range.

4) For each CDF value, identify the most probable point

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SEPTEMBER 1990 ADVANCED PROBABILISTIC STRUCTURAL ANALYSIS METHOD 1665

5) Recompute Z(jc*) to replace z0 for the same CDF in theprevious step.

The preceding steps require the construction of the Z\ func-tion only once for all the CDF levels. Assuming that a numer-ical differentiation scheme is used to define the Z\ function,the required number of the Z function evaluations is (n +1+ ra), where n is the number of random variables and m thenumber of CDF levels.

Iteration AlgorithmsThe accuracy of the preceding AMV method can be further

improved by using the exact MPPL to define the H function.Based on the AMV results, two iteration algorithms, one forspecified probability level and the other for specified Z levelhave been proposed7 to improve the CDF estimates. Forcompleteness, the suggested algorithms for specified probabil-ity level, which will be used in the demonstration examples, issummarized in the following steps:

1) Construct the Zl function, initially mean-based, andsearch for z0 such that P[Zl < z0] = probability goal.

2) Use most probable point of Z\ = z0 and recompute Z.3) Obtain the new Z\ function around the most probable

point of Z\ - z0.4) Repeat the above steps unitl z0 converges.In general, the above steps require the construction of the

Zl function several times. Therefore, for complicated Z func-tions which require extensive computations, efficient sensitiv-ity computation schemes are preferred in updating the Zlfunction.

To illustrate the AMV (first-order) method and the itera-tion algorithm, consider an example where the Z function isthe first bending natural frequency of a cantilever beam. Thisfrequency can be approximated as

/„= 0.5602 (9)

where E is the modulus of elasticity, t is the thickness, p is thematerial density, and L is the length.

By numerical differentiation, at the mean values of therandom variables, the following first-order approximation canbe obtained using the results of five Z-function [i.e., Eq. (9)]evaluations.

fn\ =00 (10)

Based on Eq. (10), an MV CDF solution can be obtained asshown in Fig. 1. Note that the CDF solution is plotted on anormal probability paper.

The preceding MV solution is exact if/„ is a linear functionof the four random variables. However, since Eq. (9) is anonlinear function, Eq. (10) is subjected to error in theregions away from the mean values. For each selected proba-

I

50%

16%

2%

0.1%

0.003%

1 MV2 AMV

— 3 First Iteration

200 240 280 440

Fig. 1

320 360 400Frequency (Rad/Sec)

AMV method and iteration procedure.

480

Response, Z

Fig. 2 Modified AMV solution for concave Z function.

0.0013%

Response, Z

Fig. 3 Modified AMV solution for convex Z function.

bility, the AMV solution is obtained by calculating/„ at themost probable point (£V*,/>*,L*) obtained using Eq. (10).This requires one function evaluation offn for each selectedprobability.

In Fig. 1, three points are shown at a selected probabilitylevel (CDF = 0.003%). Point 1 is the MV solution, point 2 isthe AMV solution, and point 3 is the AMV solution using thedescribed iteration algorithm.

Generalized Advanced Mean-Value MethodThe previous AMV (first-order) method and the associated

iteration algorithms have been applied to a set of selectedproblems to validate the NESSUS code.15 In those testedexamples, the AMV method provided satisfactory solutions.In fact, the errors in the AMV probability solutions wereoften much less than the errors due to the finite-elementmodeling. However, recently it has been found that this AMVmethod has one weakness analogous to the "multiple mini-mum distance points" problem in structural reliability analy-sis. No general solution procedure was available to solve thisproblem except the Monte Carlo method. However, when theZ function is a strongly nonmonotonic function of x, theAMV-based CDF solution may suggest that the Z function isa strongly nonmonotonic function of Zl. Based on thisinformation, modified AMV solutions can be developed.

Based on its formulation, the previous AMV solution pro-cedure inherently assumes one H value for each Z value.Thus, the method is suitable if there exists only one significantmost probable point for a Z value. It may be suspected thatthe CDF result may be in great error if there is more than oneZl solution regions for some Z values. This could happenwhen the performance function is strongly nonmonotonic.

To alleviate the problem, a generalized AMV method canbe formulated by assuming that Z is a function of randomvariable Zl. Since the AMV solution provides the CDF of Zl(the MV solution) and also the relationship between Z and Zt

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1666 WU, MILLWATER, AND CRUSE AIAA JOURNAL

(the AMV solution), the CDF of Z is uniquely defined. Thisformulation, of course, is approximate because the relation-ship between Z and Z\ is only approximate. However, thisapproximation does not require any extra Z-function calcula-tions and is able to provide the information on the character-istic of the Z function (e.g., monotonic or nonmonotonic) andto suggest a CDF modification procedure.

In general, a more accurate solution would require first theidentification of all the significant most probable points andthen the modification of the probability solution by assumingmultiple-limit states. The system reliability analysis methods,particularly the reliability bounds theory,16 may be applied toestimate the probability. For practical purposes, however,such an analysis procedure may be too complicated to use. Inthe following, a simple procedure for computing an approxi-mate solution is proposed.

For the cases where the Z function is a concave- or aconvex-like function of Z t, a simple modification procedurebased on the generalized AMV method can be summarized asfollows: 1) For a concave function, the CDF plot wouldidentify a minimum value of z, zmin. For z > zmin, two CDFs,F{ and F2, can be identified for each z. Let Fl>F2; themodified CDF, based on the theory of functions of onerandom variable, is simply (Fl — F2). 2) For a convex func-tion, a zmax can be identified. For z < zmax, the modified CDFis [ 1 - (Fl - F2)]. The procedure to modify the AMV solutionis illustrated in Figs. 2 and 3 for concave and convex Zfunctions, respectively. Note that these two figures representthe cases where there are two Z{ solutions for some Z values;i.e., they are analogous to the two most-probable-point cases.

Applying the theory of functions of one random variable, itis straightforward to extend the procedure to the situationswhere there are more than two Zl solutions for a Z value.

Table 1 Variables for example 1

99.997%

99.865%

97.725%

84.134%

50.000%0.5 0.54 0.58 0.62 0.66 0.7 0.74

RMS of Tip Displacement (in.)(Thousands)

Fig. 4 NESSUS validation problem.

0.78

-20 -1083 (Degrees)

Fig. 5 Frequency as a function of material orientation angle.

Variable Mean covaDistribution

ELt£

P

6.895E + 4 MPa0.508 m0.0249 m0.056.45E - 4 m2/s3-rad22.26 kg/m3

0.030.010.010.010.100.02

LognormalNormalNormalNormalLognormalNormal

aCoefficient of variation.

Table 2 Variables for example 2

Variable Mean Standard deviation

e,02

03

ExEyGV*vyLtP

-5deg2deg3deg1.267E + 5MPa1.267E + 5MPa1.285E + 5MPa0.3860.3860.254 m0.0254 m7 16.9 kg/m3

3.87 deg3.87 deg3.87 deg00000000

However, it should be noted that, when a function is ex-tremely nonlinear (e.g., a sinusoidal wave shape), the numberof CDF points required to fully describe the nonlinear shapemay be high; therefore, care should be exercised to generatesufficient AMV solution points to accurately define the shape.When a highly nonlinear shape is detected, it may be neces-sary to perform a Monte Carlo study to supplement the AMVsolution. A procedure that applies an importance samplingtechnique to confirm or improve the AMV solution is cur-rently under development.

Numerical ExamplesThree examples are selected to demonstrate the AMV

method. The first example shows the typical AMV solution inwhich the CDF does not appear to have truncation limits. Inthis case, no modification to the AMV solution is needed. Theremaining two examples involve strong nonmonotonic perfor-mance functions that can be solved using the generalizedAMV method as just described.Example 1: Random Vibration

This problem was developed to test the NESSUS randomvibration capabilities. In this example, a cantilever beam issubjected to random base excitation. The random variablesinclude the modulus of elasticity E, material density p, damp-ing factor £, length L, thickness t, and constant accelerationpower spectral density (PSD) level WA. The PSD is modeledas a truncated white noise with cutoff frequency properlyselected such that the random loading would excite, approxi-mately, only the first mode. The random variables are definedin Table 1.

Using a single-degree-of-freedom model, the tip root-mean-square (rms) displacement can be approximated as

rms = ( ID

The NESSUS probabilistic solution is shown in Fig. 4. Inthis figure, the MV solution employs the mean-based sensitiv-ity analysis result to establish the approximate, linear Z\function [see Eq.(6)]; the AMV solution uses the informationfrom the MV solution, and the recomputation of the responseat the approximate most probable point. Note that no modifi-cation to the AMV solution is needed because the CDF curveis clearly monotonic.

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SEPTEMBER 1990 ADVANCED PROBABILISTIC STRUCTURAL ANALYSIS METHOD 1667

+ Monte Carlo (1000 Samples)

MV

99.997%

99.87%

^ 97.7%

I 84.1%

g 50%

'•ia 15.9%

U 2.28%

0.13%

0.003%240 242 244 246 248 250 252 254 256 258 260

Frequency (Hz)

Fig. 6 Modified AMY solution for example 2 (case 1: one randomangle).

240 242 244 246 248 250 252 254 256 258 260Frequency (Hz)

Fig. 7 Probability density function of the natural frequency.

To improve the result further, the iteration algorithms justdiscussed can be applied. A first iteration solution for a CDFvalue is presented in Fig. 4, which shows excellent agreementbetween the NESSUS solution and the "exact" solution basedon the Monte Carlo simulation (100,000 samples). Note thatthere was a 3.9% difference between the NESSUS result andthe analytical result at the 50% probability level. This differ-ence has been used to adjust the exact solution.

Based on Fig. 4, it is obvious that the AMV solutionimproves the MV solution significantly; whereas the firstiteration solution improves the AMV solution slightly. Sincethe AMV accuracy generally deteriorates at the extreme tailregions of the distribution, it is suggested that the iterationsshould be performed at the critical tail regions.

Example 2: Material Orientation EffectA recent probabilistic analysis of a single crystal turbine

blade has shown that the AMV solution of the first bendingfrequency CDF is truncated; i.e., the frequency has a lowerbound. To study the truncation phenomena, a simple beammodel, as shown in Fig. 5, was developed to simulate theturbine blade. The beam is made of a single crystal material,and the material orientation is defined by three angles. Thedata are shown in Table 2 in which Ex and Ey are the"off-axis" moduli.

Two cases are considered in this study. The first caseconsiders that only one material orientation angle is randomand is normally distributed. This allows a simple closed formequation for more detailed study. The second case considersthree normally distributed random orientation angles, and theNESSUS code is applied to solve the problem.

Case 1: One Random AngleThe first-mode frequency is

/„ = 0.5602 (12)

where El is the "on-axis" (or material symmetry axis) mod-ulus, which can be shown17 to be a function of the off-axismoduli and the random angle 03 as

EG(c4 + s4 - 2c2s2vy)G + c2s2E (13)

where E = Ex = Ey, c = cos(03), and s = sin(03). By substitut-ing Eq. (13) into Eq. (12), the relationship between/, and 03is shown in Fig. 5, which shows that fn is a nonmonotonicfunction in the probability significant region. Furthermore,the frequency has a minimum.

By numerical differentiation, the MV solution was obtainedas

fnl = 254.96 + 0.42396(03 - 3) (14)

Based on Eq. (14),/wl is normally distributed. Therefore, theCDF of/„! is linear on a normal probability paper, as shownin Fig. 6. The AMV solution is shown in Fig. 6 and appearsto be parabolic with a minimum of 03. The parabolic shapeindicates that/,, is a nonmonotonic function of 03; therefore,a modification to the AMV solution is needed.

By applying the generalized AMV procedure, the modifiedCDF becomes truncated at the left tail. To verify the result, a

99.87%

97.7%

I 84.1%

I50%

15.9%

2.28%

0.13%

1 MV2 AMV3 MVSO+ Monte Carlo (2000)

230 250Frequency (Hz)

270

Fig. 8 MVSO and modified AMV solutions for example 2 (case 2:three random angles).

60

50

"

10

240 242 244 246 248 250 252 254 256 258 260Natural Frequency (Hz)

Fig. 9 Monte Carlo simulation of dynamic magnification factor (1000samples).

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1668 WU, MILLWATER, AND CRUSE AIAA JOURNAL

Table 3 Variables for example 3

Variable Mean Standard deviation

fn

fe

{

250 Hz250 Hz0.01

2.5 HzOHz0

0.003%0 20 40

Dynamic Magnification Factor, D

Fig. 10 Modified AMV solution for example 3.

Monte Carlo simulation was performed, and the empiricalCDF, as shown in Fig. 6, agrees with the AMV solution.Thus, this simple case clearly shows how a concave perfor-mance function would produce a truncated distribution. Theprobability density function is shown in Fig. 7 to show howthe distribution is skewed.Case 2: Three Random Angles

In case 1, there exists only one independent random vari-able. In such a case, the generalized AMV solution is exact. Inthe present case, three random variables are assumed, asshown in Table 2. For multiple random variables, the AMVmethod offers approximate solutions.

For three random angles, it is difficult to obtain a closedform solution for/,. Therefore, the NESSUS code was usedto solve the problem numerically.

Following a similar solution procedure used in case 1, theMV and the AMV solutions are obtained, as shown in Fig. 8.The MV solution used four Z-function evaluations and theAMV solution used an additional 6 function evaluations. Toconfirm the results, a Monte Carlo simulation with 2000samples was performed, with results plotted in Fig. 8.

The result of case 2 is different from case 1 because of thetwo additional random angles. However, the major character-istic remains the same; i.e., the distribution is truncated at theleft tail.

It is noted that the error is relatively large at the centralregion (i.e., CDF w 0.5). The reason is because/„ is stronglynonlinear at this region. The error can be reduced by anMVSO analysis. The MVSO solution, presented in Fig. 8,agrees with the simulation solution very well in the centralregion. However, the error appears to be significant at theright-tail region. This error would need to be improved by theAMV procedure or, perhaps, an AMV second-order proce-dure, which would be a natural extensioin of the first-ordermethod. Because of the additions of the second-order terms,the AMV second-order method has the potential to be moreaccurate than the AMV first-order method. However, therequired number of function evaluations will be greater. Fu-ture research will be needed to explore the AMV second-ordermethod.

It is interesting to compare the MV solution with the AMVsolution using the n ± 3<r design criteria. For the MV solu-tion, because all three input random variables are normally

distributed, fni is normally distributed; therefore, the ju ± 3crprobability bounds are (0.0013, 0.9987). The correspondingfrequencies are 234.0, 260.8 Hz. Keeping the same probabilitylevels, the AMV-based bounds are 245.7, 278.3 Hz. Thus theMV solution significantly underestimates the bounds. Also,the actual probabilities corresponding to the n + 3<r boundsare (0,0.946), as opposed to (0.0013, 0.9987). This clearlydemonstrates the importance of obtaining the entire CDFresult.Example 3: Dynamic Magnification Factor

This example was selected to represent a situation wherethe response CDF is truncated at the right tail.

Consider the first-mode dynamic characteristic of a turbineblade model. The dynamic magnification factor D is

D = 1(15)

where fe is the exciting frequency, fn is the natural frequency,and f is the damping factor. The variables are defined inTable 3, which shows that fe and £ are deterministic while fnis random. As a worst-case analysis, it is assumed that/, isequal to the mean value of/„.

In this problem, D is a convex function of/„. For each Dthere are two possible/„ values. Random samples of fn weregenerated and used to calculate D using Eq. (15). Figure 9shows the 1000 samples of D as a function of/„.

Following the same procedure as described in the precedingexamples, the Monte Carlo simulation, MV, AMV, and themodified AMV solutions are shown in Fig. 10. This problemdemonstrates how a convex performance function results in atruncated distribution and how this truncated distribution canbe identified quickly based on the generalized AMV concept.

SummaryThe CDF analysis provides a more useful reliability design

tool than the conventional mean-based second-moment analy-sis. This paper presented a generalized AMV method, which iscapable of performing a CDF analysis in a robust and highlyefficient manner. Several examples were used to demonstratethe methodology. The examples included monotonic as wellas convex and concave performance functions. It was shownwhy a performance function CDF is truncated and how atruncated distribution can be identified and computed usingthe proposed procedure. The extension to the more generalnonlinear functions has also been discussed in the paper. Insummary, the generalized AMV method appears to be idealto solve problems involving implicit, complex performancefunctions.

AcknowledgmentsThis research was supported by the NASA Lewis Research

Center under Contract NAS3-24389, "Probabilistic StructuralAnalysis Methods for Select Space Propulsion System Com-ponents." The authors appreciate the support of C. C.Chamis, NASA Program Manager. The generalized AMVmethod, developed by the first author, was motivated by theiteration problems encountered during a turbine blade proba-bilistic analysis study performed at Rocketdyne. The sugges-tions made and information given by K. R. Rajagopal ofRocketdyne to use a simplified single crystal cantilever beammodel is gratefully acknowledged.

References'Wong, F. S., "First-Order, Second-Moment Methods," Computers

and Structures, Vol. 20, No. 4, 1985, pp. 779-791.2Hasofer, A. M., and Lind, N. C., "Exact and Invariant Second-

moment Code Format," Journal of Engineering Mechanics, American

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Society of Civil Engineers, Vol. 100, No. EM1, Feb. 1974, pp.111-121.

3Li, K. S., and Lumb, P., "Reliability Integration and CurveFitting," Structural Safety, Vol. 3, Oct. 1985, pp. 29-36.

4Zhou, J., and Nowak, A. S., "Integration Formulas to EvaluateFunctions of Random Variables," Structural Safety, Vol. 5, Dec.1988, pp. 267-284.

5Millwater, H., Palmer, K., Fink, P., "NESSUS/EXPERT—AnExpert System for Probabilistic Structural Analysis Methods," Pro-ceedings of the 29th AIAA/ASME/ASCE/AHS Structures, StructuralDynamics and Materials Conference, Pt. 3, AIAA, Washington, DC,1988, pp. 1283-1288.

6Cruse, T. A., Wu, Y.-T., Dias, J. B., and Rajagopal, K. R.,"Probabilistic Structural Analysis Methods and Applications," Com-puters and Structures, Vol. 30, No. 1/2, 1988, 163-170.

7Wu, Y.-T., Burnside, O. H., and Cruse, T. A., "ProbabilisticMethods for Structural Response Analysis," Computational Mechan-ics of Probabilistic and Reliability Analysis, edited by W. K. Liu andT. Belytschko, Elmepress International, Aug. 1989, pp. 181-195.

8Madsen, H. O., Krenk, S., and Lind, N. C, Methods of StructuralSafety, Prentice Hall, Englewood Cliffs, NJ, 1986.

9Wu, Y.-T., and Wirsching, P. H., "New Algorithm for StructuralReliability Estimation," Journal of Engineering Mechanics, ASCE,Vol. 113, No. 9, Sept. 1987, pp. 1319-1336.

10Rackwitz, R., and Fiessler, B., "Structural Reliability Under

Combined Load Sequences," Journal of Computers and Structures,Vol. 9, 1978, pp. 489-494.

HHohenbichler, M., and Rackwitz, R., "Non-normal DependentVectors in Structural Safety," Journal of Engineering Mechanics Divi-sion, ASCE, Vol. 100, No. EM6, Dec. 1981, pp. 1227-1238.

12Haug, E. J., Choi, K. K., and Komkov, V., Design SensitivityAnalysis of Structural Systems, Academic, New York, 1986.

13Dias, J., Nagtegaal, J., and Nakazawa S., "Iterative PerturbationAlgorithms in Probabilistic Finite Element Analysis," ComputationalMechanics of Probabilistic and Reliability Analysis, edited by W. K.Liu and T. Belytschko, Elmepress International, Virginia, 1989, pp.211-230.

14Wu, Y.-T., Burnside, O. H., and Dominguez, J., "Efficient Prob-abilistic Fracture Mechanics Analysis," Proceedings of the FourthInternational Conference on Numerical Methods in Fracture Mechan-ics, Pineridge Press, Swansea, United Kingdom, 1987, pp. 85-100.

15Wu, Y.-T., and Burnside, O. H., "Validation of the NESSUSProbabilistic Analysis Computer Program," Proceedings of the 29thAIAA/ASME/ASCE/AHS Structures, Structural Dynamics and Ma-terials Conference, Pt. 3, AIAA, Washington, DC, 1988, pp. 1267-1274.

16Ang, A. H.-S., and Tang, W. H., Probability Concepts in Engi-neering Planning and Design, Vol. 2, Wiley, New York, 1984.

17Tsai, S. W., and Hahn, H. T., Introduction to Composite Materi-als, Technomic Publishing, PA, 1980.

Recommended Reading from the AIAA t^AMM AProgress in Astronautics and Aeronautics Series . .

Dynamics of Flames and ReactiveSystems and Dynamics of Shock Waves,Explosions, and DetonationsJ. R. Bo wen, N. Manson, A. K. Oppenheim, and R. I. Soloukhin, editors

The dynamics of explosions is concerned principally with the interrelationship betweenthe rate processes of energy deposition in a compressible medium and its concurrentnonsteady flow as it occurs typically in explosion phenomena. Dynamics of reactivesystems is a broader term referring to the processes of coupling between thedynamics of fluid flow and molecular transformations in reactive media occurring inany combustion system. Dynamics of Flames and Reactive Systems covers premixedflames, diffusion flames, turbulent combustion, constant volume combustion, spraycombustion nonequilibrium flows, and combustion diagnostics. Dynamics of ShockWaves, Explosions and Detonations covers detonations in gaseous mixtures, detona-tions in two-phase systems, condensed explosives, explosions and interactions.

Dynamics of Flames andReactive Systems1985 766 pp. illus., HardbackISBN 0-915928-92-2AIAA Members $54.95Nonmembers $84.95Order Number V-95ORDER: c/o TASCO, 9 Jay Gould Ct., P.O. Box 753

Waldorf, MD 20604 Phone (301) 645-5643Dept. 415 • FAX (301) 843-0159

Sales Tax: CA residents, 7%; DC, 6%. Add $4.50 for shipping and handling. Orders under $50.00 must be prepaid. Foreign orders mustbe prepaid. Please allow 4 weeks for delivery. Prices are subject to change without notice. Returns will be accepted within 15 days.

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1985 595 pp., illus. HardbackISBN 0-915928-91-4

AIAA Members $49.95Nonmembers $79.95Order Number V-94

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Journal 39:4, 667-677. [Citation] [PDF] [PDF Plus]120. Xiaoping Du, Wei ChenConcurrent subsystem uncertainty analysis in multidisciplinary design . [Citation] [PDF] [PDF

Plus]121. Dimitri N. Mavris, Daniel A. DeLaurentis. 2000. A probabilistic approach for examining aircraft concept feasibility and

viability. Aircraft Design 3, 79-101. [CrossRef]122. Xiaoping Du, Wei Chen. 2000. Towards a Better Understanding of Modeling Feasibility Robustness in Engineering

Design. Journal of Mechanical Design 122:4, 385. [CrossRef]123. Leslie Banks-Sills, R. Eliasi. 1999. Fatigue life analysis of a cannon barrel. Engineering Failure Analysis 6, 371-385.

[CrossRef]124. J. Tu, K. K. Choi, Y. H. Park. 1999. A New Study on Reliability-Based Design Optimization. Journal of Mechanical Design

121, 557. [CrossRef]125. Y.-T. WuMethods for efficient probabilistic analysis of systems with large number of random variables . [Citation] [PDF]

[PDF Plus]

Dow

nloa

ded

by R

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HE

N o

n A

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t 7, 2

015

| http

://ar

c.ai

aa.o

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DO

I: 1

0.25

14/3

.252

66

126. J.N. Majerus, M.L. Mimnagh, J.A. Jannone, D.A. Tenney, S.P. Lamphear. 1998. Blending hierarchical economic decisionmatrices (EDM) with FE and stochastic modeling. II. Detailing EDM. Finite Elements in Analysis and Design 30, 219-234.[CrossRef]

127. Xiaoming Yu, Kuang-Hua Chang, Kyung K. Choi. 1998. Probabilistic Structural Durability Prediction. AIAA Journal 36:4,628-637. [Citation] [PDF] [PDF Plus]

128. Paul H Wirsching. 1998. Fatigue reliability. Progress in Structural Engineering and Materials 1:10.1002/pse.v1:2, 200-206.[CrossRef]

129. G. Orient, J. Depsky, S. Schwartz, D. O'Connor, D. O'Connor, G. Orient, J. Depsky, S. SchwartzProbabilistic castingporosity prediction methodology . [Citation] [PDF] [PDF Plus]

130. Xiaodong Luo, R.V. Grandhi. 1997. Astros for reliability-based multidisciplinary structural analysis and optimization.Computers & Structures 62, 737-745. [CrossRef]

131. R. G. Tryon, T. A. Cruse. 1997. Probabilistic Mesomechanical Fatigue Crack Nucleation Model. Journal of EngineeringMaterials and Technology 119, 65. [CrossRef]

132. Kuang-Hua Chang, Xiaoming Yu, Kyung ChoiProbabilistic structural stability prediction . [Citation] [PDF] [PDF Plus]133. H.S. Rajagopalan, R.V. Grandhi. 1996. Reliability based structural analysis and optimization in X window environment.

Computers & Structures 60, 1-10. [CrossRef]134. Liping Wang, R.V. Grandhi. 1996. Safety index calculation using intervening variables for structural reliability analysis.

Computers & Structures 59, 1139-1148. [CrossRef]135. Robert Tryon, Thomas VruseReliability-based mesomechanical crack nucleation model . [Citation] [PDF] [PDF Plus]136. Wing Kam Liu, Yijung Chen, Ted Belytschko, Yuan Jie Lua. 1996. Three reliability methods for fatigue crack growth.

Engineering Fracture Mechanics 53, 733-752. [CrossRef]137. Robert G. Tryon, Thomas A. Cruse, Sankaran Mahadevan. 1996. Development of a reliability-based fatigue life model

for gas turbine engine structures. Engineering Fracture Mechanics 53, 807-828. [CrossRef]138. H. R. Millwater, Y.-T. Wu, J. W. Cardinal, G. G. Chell. 1996. Application of Advanced Probabilistic Fracture Mechanics

to Life Evaluation of Turbine Rotor Blade Attachments. Journal of Engineering for Gas Turbines and Power 118, 394.[CrossRef]

139. L. Wang, R. V. Grandhi. 1995. Intervening variables and constraint approximations in safety index and failure probabilitycalculations. Structural Optimization 10, 2-8. [CrossRef]

140. Liping Wang, Ramana V. Grandhi, Dale A. Hopkins. 1995. Structural reliability optimization using an efficient safety indexcalculation procedure. International Journal for Numerical Methods in Engineering 38:10.1002/nme.v38:10, 1721-1738.[CrossRef]

141. I Orisamolu, Q LiuFinite element reliability solution of stochastic eigenvalue problems . [Citation] [PDF] [PDF Plus]142. Swamy Chandu, Harini Rajagopalan, Ramana Grandhi, Dale HopkinsReliability based structural analysis and optimization

in X-window environment . [Citation] [PDF] [PDF Plus]143. A.E. Mansour, P.H. Wirsching. 1995. Sensitivity factors and their application to marine structures. Marine Structures 8,

229-255. [CrossRef]144. Swamy V.L. Chandu, Ramana V. Grandhi. 1995. General purpose procedure for reliability based structural optimization

under parametric uncertainties. Advances in Engineering Software 23, 7-14. [CrossRef]145. M.V. Reddy, R.V. Grandhi, D.A. Hopkins. 1994. Reliability based structural optimization: A simplified safety index

approach. Computers & Structures 53, 1407-1418. [CrossRef]146. T. A. Cruse, S. Mahadevan, Q. Huang, S. Mehta. 1994. Mechanical system reliability and risk assessment. AIAA Journal

32:11, 2249-2259. [Citation] [PDF] [PDF Plus]147. Liping Wang, Ramana GrandhiIntervening variables and constraint approximations in safety index and failure probability

calculations . [Citation] [PDF] [PDF Plus]148. Y.-T. Wu. 1994. Computational methods for efficient structural reliability and reliability sensitivity analysis. AIAA Journal

32:8, 1717-1723. [Citation] [PDF] [PDF Plus]149. Wang Liping, R.V. Grandhi. 1994. Efficient safety index calculation for structural reliability analysis. Computers &

Structures 52, 103-111. [CrossRef]150. H. Millwater, Y.-T. Wu, J. CardinalProbabilistic structural analysis of fatigue and fracture . [Citation] [PDF] [PDF Plus]151. Yuan Jie Lua, Wing Kam Liu, Ted Belytschko. 1993. Curvilinear fatigue crack reliability analysis by stochastic

boundary element method. International Journal for Numerical Methods in Engineering 36:10.1002/nme.v36:22, 3841-3858.[CrossRef]

Dow

nloa

ded

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c.ai

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.252

66

152. MAHIDHAR REDDY, RAMANA GRANDHI, DALE HOPKINSRELIABILITY BASED STRUCTURALOPTIMIZATION: A SIMPLIFIED SAFETY INDEX APPROACH . [Citation] [PDF] [PDF Plus]

153. Jon C Helton. 1993. Uncertainty and sensitivity analysis techniques for use in performance assessment for radioactive wastedisposal. Reliability Engineering & System Safety 42, 327-367. [CrossRef]

154. P. B. Thanedar, S. Kodiyalam. 1992. Structural optimization using probabilistic constraints. Structural Optimization 4,236-240. [CrossRef]

155. D.S. Riha, H.R. Millwater, B.H. Thacker. 1992. Probabilistic structural analysis using a general purpose finite elementprogram. Finite Elements in Analysis and Design 11, 201-211. [CrossRef]

156. T. CRUSE, Q. HUANG, S. MEHTA, S. MAHADEVANSystem reliability and risk assessment . [Citation] [PDF] [PDFPlus]

157. T. TORNG, Y.-T. WU, H. MILLWATERStructural system reliability calculation using a probabilistic fault tree analysismethod . [Citation] [PDF] [PDF Plus]

158. PRAMOD THANEDAR, SRINIVAS KODIYALAMStructural optimization using probabilistic constraints . [Citation][PDF] [PDF Plus]

159. T. TORNG, Y.-T. WUDevelopment of a probabilistic analysis methodology for structural reliability estimation .[Citation] [PDF] [PDF Plus]

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://ar

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