Reliability Analysis Methods Applied toLarge Scale Aircraft Structures

J. Dıaz∗, S. Hernandez

†, L. Romera

‡and A. Baldomir

§

University of La Coruna, La Coruna, 15071, Spain

The objective of this investigation is to show how it is possible the systematic applicationof uncertainty quantification and reliability analysis methods to complex structural models.In order to do that, a parametric study is carried out, aimed to compare the performanceof several reliability analysis methods applied to the evaluation of large scale aircraft struc-tures. The methods selected are the FORM and TANA3, based on the approximation ofthe limit state function, and the latin hypercube sampling and importance sampling. Re-sults show that the methods behave differently in relation to their performance, dependingon the number of random variables and also on the expected probability of failure. Limitstate approximation methods have a better response in those cases with a reduced num-ber of random variables or limit state functions. However, in those cases having a largenumber of variables, sampling methods become an alternative, with computational costsquite similar to limit state approximation methods, except in those cases with low valuesof probability of failure.

I. Introduction

Uncertainty quantification of structural response is an essential task in the design of aircraft components.Conventional methods, involving deterministic design, define safety factors to deal with the inherent uncer-tainties found in every system. However, in reliability analysis, the goal is the obtainment of the probabilityof failure when design criteria or limit states are not satisfied. The main advantage of these methods overdeterministic design is that structural safety can be defined more accurately and, at the same time, theperformace of conventional procedures is enhanced, because in each situation its specific uncertainties aretaken into account.

The concept of uncertainty identifies the fact that it is impossible to know exactly the value of a mag-nitude, but it is possible to assess its most probable value. This probabilistic information, related to thevariable, can be estimated and when the magnitude is considered from this point of view, then is qualifiedas a random variable.

The concept of reliability refers to the probability of verifying a certain condition, and so, it cannotbe established with total certainty that a design will fulfill a limit state condition. Instead, there is someprobability pf that the limit state will not be verified. This is known as the probability of failure.

In a probabilistic analysis, the uncertainties in the basic magnitudes of the structure are considereddirectly in the analysis, changing from fixed quantities to random variables. In the case of some componentwith resistance R supporting a set of external loads which provoke a structural response S, the probabilityof failure is:

pf = P (R � S) = P (R− S � 0) = P [g (r, s) � 0] (1)

∗Assistant Professor, Structural Mechanics Group, ETSI Caminos, Canales y Puertos, Member AIAA†Professor, Structural Mechanics Group, ETSI Caminos, Canales y Puertos, Associate Fellow AIAA‡Associate Professor, Structural Mechanics Group, ETSI Caminos, Canales y Puertos§Assistant Professor, Structural Mechanics Group, ETSI Caminos, Canales y Puertos

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52nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference<BR> 19th4 - 7 April 2011, Denver, Colorado

AIAA 2011-2121

Copyright © 2011 by The authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

Where g (r, s) is known as the limit state function. If fR and fS are the probability density functions ofR and S, respectively, and fRS is their joint probability density function, then:

pf = P [g (r, s) � 0] =

� ∞

−∞

� s

−∞fRS(r, s)drds (2)

And when R and S are statistically independent:

pf =

� ∞

−∞

� s

−∞fR(r)fS(s)drds (3)

The limit state function defines if a design belongs to the failure domain, where the limit state is notverified, or to the safety domain, where it is. If a is the vector of basic variables, which contains the nrandom variables of the structure, then the domains are defined as follows:

Failure domain: Fd = {a | g(a) < 0} (4)

Security domain: Sd = {a | g(a) � 0} (5)

The boundary between both domains is known as the failure surface or limit state surface, which generallyis an hypersurface of n − 1 dimensions in the n-dimensional space of basic variables. The safety margin isnow defined as a random variable, which can be identified with the value of the limit state function:

M = g(a) (6)

From the previous considerations, it can be generalized the expression (2) corresponding to the probabilityof failure, which now is formulated as:

pf = P [g (a) � 0] =

�· · ·

�

g(a)�0

fA (a) da (7)

The equation (7) is known as the fundamental equation of reliability, where fA (a) is the joint probabilitydensity function of all the basic variables involved in the response of the system.

Except for some particular cases, the integral (7) cannot be resolved analytically, because of the non-linearity of fA (a), and also due to the fact that the number of random variables usually employed is verylarge, and therefore the dimension of the problem. Also, numerical integration methods are not capable ofsolving the equation efficiently and are impractical in multi-dimensional problems. Several methods havebeen proposed to solve this situation. So far, a number of authors have contributed with works showing theapplicability of reliability analysis methodologies to different structural elements.

In that regard, Sudret et al. 1 expose an example considering the durability of cooling towers with 12independent basic variables. They use random sampling with variance reduction techniques and a samplesize of 500 elements. The finite element model employed has 12,376 elements and the limit state functionguarantees that the stress at a critical point is below a maximum value. The critical point is definedbeforehand with the help of a deterministic analysis.

Cheng and Li 2 perform the reliability analysis of a long span steel arch bridge against wind-inducedstability failure during construction stage. They employ the first order method considering uncertantiesin the wind loads, defined as static loads, selecting two independent random variables. The failure modeis defined with a single limit state function, which avoids the global buckling of the structure. The finiteelement mesh has 3046 elements and they verify the results with the Monte Carlo method using a samplesize of 5000 elements. The authors also present the reliability analysis of the same bridge in service, butwith a less complex finite element model with 741 elements3.

Pellissetti et al. 4,5 carry out the reliability analysis of a satellite structure under dynamic loads. Theyconsider a large model with 120,000 degrees of freedom and employ Monte Carlo sampling with variancereduction techniques, ranging in the number of samples from 1500 to 100,000. The limit state selected avoidsthe exceedance of an acceleration threshold in a critical point selected beforehand. The number of randomvariables is 1320, all of them statistically independent.

The previous works constitute the state of the art in the field of applied reliability analysis. In this paper,the most appropiate methods for the design of large scale aircraft components are selected and explained.Also, a practical example is considered to demonstrate the application of the methods and their performance.

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II. Reliability Analysis Methods

The reliability analysis methodologies considered in this work can be divided in two types. The first onecovers the techniques involving approximations of the limit state surface. The considered methods, basedon the Taylor series expansion of the limit state function and other approximations, are the FORM or firstorder reliability method and the TANA3 or two point adaptive nonlinear approximation. Those methodsrequire information about the value of the limit state function and its derivatives in the vicinity of the designpoint.

The second type comprehends simulation methods. This category includes Monte Carlo simulation andits modifications, aimed to reduce the elevated computational requirements associated with them. Thosemethods are the latin hypercube sampling (LHS) and importance sampling (VRT-I). In simulation methods,samplings of the random properties are generated and feeded as an input to the system, obtaining a responsepopulation where statistical data are measured. A brief description of all the aforementioned methods ispresented next.

II.A. FORM method

The FORM method (First Order Reliability Method), developed by Hasofer and Lind 6 , uses the informationof the first two statistical moments of the random variables. It assumes that the random variables are statis-tically independent and follow a normal distribution. This does not reduce the generality of the approach,because by means of a transformation it is possible the approximation of any type of distribution in thisway.

Taking into account those considerations, the standard normal properties a� can be defined from thetransformation of the original random variables to a standard Gauss distribution as:

a�i =ai − µAi

σAi

(8)

The FORM consists in the search of the most probable point of failure (MPP) in the standardised domain,in order to allow the substitution of the limit state function by its Taylor series expansion of first order atthat point:

g(a�) � g(a�f ) +∇g(a�f )T (a� − a�f ) (9)

Where a�f , the most probable point of failure, is the point of minimum distance to the origin from thelimit state surface. Geometrically, the method supposes the approximation of the limit state surface by thetangent hyperplane at the most probable point of failure. The reliability index β, which is also known asthe Hasofer and Lind index, is the minimum distance to the most probable point of failure from the origin.This index is related to the failure surface, but it is invariant with respect to the formulation of the limitstate function:

β = −a

�Tf ∇g(a�f )�

∇g(a�f )T∇g(a�f )

(10)

The probability of failure is then obtained as:

pf = 1− Φ (β) (11)

Where Φ is the cumulative distribution function of the standard normal variable. Several methods existwhich calculate the MPP position. In that regard, the alternatives suggested by Rackwitz 7 , Rackwitz andFiessler 8 and Ayyub and Haldar 9 can be recommended.

II.B. TANA3 method

Instead of using information about the limit state function and its derivatives up to a certain order, like inthe rest of the limit state approximation methods, adaptive approximations use information generated in

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several points. Thus, a more precise estimate of nonlinear limit state functions can be calculated withoutusing derivatives of higher order, which reduces the computational cost.

The TANA3 (Two point Adaptive Nonlinear Approximations) method10 is based on an exponentialapproximation which uses information of the current iteration k and also of the previous one k − 1. So, thelimit state surface can be approximated as:

g (a) � g (ak) +n�

i=1

∂g (ak)

∂ai

(ai,k)(1−ri)

ri

�arii − arii,k

�+

λ2

2

n�

i=1

�arii − arii,k

�2(12)

Where the nonlinear index ri and the parameter λ2 can be defined as:

ri = 1 +

ln

�∂g (ak−1)

∂ai

�− ln

�∂g (ak)

∂ai

�

ln (ai,k−1)− ln (ai,k)(13)

λ2 =2 [g (ak−1)− g (ak)]

n�i=1

�arii − arii,k−1

�2+

n�i=1

�arii − arii,k

�2 −2

�n�

i=1

a1−rii,k

ri

∂g (ak)

∂ai

�arii,k−1 − arii.k

��

n�i=1

�arii − arii,k−1

�2+

n�i=1

�arii − arii,k

�2 (14)

II.C. Latin Hypercube Sampling

Sampling methods provide an estimation of the reliability by means of statistical simulations of the randomvariables. The procedure consists in the selection of a large number of samples m of the random properties,according to their probability distribution, and perform deterministic analysis with those values in order toobtain the structural response for each one of the samples. By means of the processing of those results, thestatistical moments of the structural response can be obtained. The most well known sampling method isthe Monte Carlo sampling (MCS)11. The application of this method to obtain the solution of equation (7)requires the introduction in the integrand of the function υ(a):

υ(a) =

1 , if g(a) � 0

0 , if g(a) > 0(15)

Now, the domain of integration in the expression (7) includes the whole real domain and the probabilityof failure can be formulated as:

pf = P [g (a) � 0] =

�· · ·

�υ(a)fA (a) da (16)

If a Monte Carlo sampling is applied to the previous equation, so that m samples are generated, of whichmf provoke that g(a) � 0, then the probability of failure pf,e can be estimated as:

pf,e =1

m

m�

j=1

υ(aj) =mf

m(17)

The accuracy of the previous estimation increases with the number of samples m, although it is relatedwith the probability of failure too, since a very low value of pf requires a larger number of samples to achievesome results in the failure domain.

Shooman 12 suggest the following expression to estimate, with a confidence of 95 %, the probability offailure pf with an error εpf,e :

m =4(1− pf )

ε2pf,epf

(18)

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As it can be seen, the obtention of precise results with low values of probability of failure entails alarge number of simulations. Thus, the application of direct Monte Carlo simulation is restricted to simpleproblems, not requiring excessive computational resources, or to structures where either the failure criteriacan be relaxed or the accuracy of the results or both.

On the other hand, the equation (18) clearly shows that the error does not depend on the dimensionn of the problem, and so the method is suitable for those cases with a large number of random variables,where the computational cost can be compared with the required by the limit state approximation methods.Moreover, as each one of the deterministic simulations in Monte Carlo sampling is independent from theothers, the algorithm can be easily parallelized, reducing the analysis time by a factor related with thenumber of simultaneous jobs than can be executed.

The random sampling in Monte Carlo method can be improved with some techniques of sampling se-lection. One of these methods is the Latin Hypercube Sampling (LHS), proposed by McKay et al. 13 . InLHS, the domain where random variables are defined, is divided in subdomains of equal probability and thesamples are selected so the design region is evenly covered.

If m is the number of samples and n is the number of random variables, then the domain of each variableis divided in m subdomains of equal probability. The samples are selected randomly in each subdomain andonly a sample is taken per subdomain. In this way, each row and column in the hypercube of partitions hasonly a sample.

An advantage of LHS over Monte Carlo sampling is that, if the structural response is dominated by onlyone parameter, then all the levels of the response are evaluated. This is not guaranteed by direct sampling.

II.D. Importance sampling

Importance sampling method was proposed by Kahn and Marshall 14 and was applied later to reliabilitystructural analysis15. The objective in this method is to concentrate the distribution of samples in thevicinity of the most probable point of failure, instead of spreading them over the whole domain. In order todo that, an auxiliar probability density function h(a) is defined, with the purpose of generate samples in theregion which most contributes to the value of integral (16):

pf = P [g (a) � 0] =

�· · ·

�υ(a)

fA (a)

hA(a)hA(a)da (19)

According to that, the estimation of probability of failure pf,e is now as follows:

pf,e =1

m

m�

j=1

υ(aj)fA (aj)

hA(aj)(20)

The main disadvantage of this method is that the selection of hA(a) is conditioned by the shape of failuredomain.

III. Application example

This section describes the application of the uncertainty quantification procedures previously mentionedto a finite element model of an aircraft wing (figure 1). The objective is the verification of the structuralreliability of one of the wing ribs, where several cases are defined, involving different number of randomvariables.

The finite element model used for the reliability analysis is shown in figure 1. The mesh is built withshell elements modelling the outer skin of the wing box and with beam and truss elements modelling thereinforcements, including all the ribs, except for the selected one to conduct the reliability analysis (figure 2),which is modelled with a detailed mesh using shell elements.

In relation to mesh size, the total number of elements is 59,827, with 41,623 nodes and 249,738 degreesof freedom. The boundary conditions applied assume a fixed joint of the wing with the fuselage. The finiteelement code employed is MSC Nastran 2005 16 and all the analysis were conducted in linear and staticmode.

For carrying out the calculations, a high performance computer cluster of 8 nodes with 48 processors of64 bits and 192 GB of physical memory has been used. The peak performance provided by this machine is237 GFlops.

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Figure 1. Finite element model of the wing Figure 2. Position of the wing rib

III.A. Random variable characterization

In order to evaluate the structural reliability of the proposed example, several uncertainties have been con-sidered. Those uncertainties have been classified according to their type into one of the following categories:material, geometry or loads. The type of probability distribution for each one of the random variables hasbeen chosen taken into account the physical phenomena involved with them and, without loss of generality,all the variables are statistically independent.

The material properties selected as random variables have been assigned to two zones. The first zonecorresponds to the aluminum of the box, whose Young’s modulus has been selected as a random variable in5 cases. The second zone comprises the material properties of the rib, which have been described with 10uncertainty properties. All the random variables have been characterized using a Gauss distribution with anuncertainty, modelled with the coefficient of variation δ, of 10%. The mean value is the same as the nominalvalue, taken from the deterministic model.

With respect to geometrical properties, the uncertainty is assigned to the thickness of the rib, with atotal of 60 variables which follow a lognormal distribution, whose mean values are the same as the nominalones and their coefficient of variation is 10%.

In relation to loads, 336 random properties have been selected to represent them. Each one of theproperties is characterized using a Gumbel distribution with an uncertinty δ = 15%. Their mean value isthe same as the applied to the deterministic model.

Considering this classification, a number of cases can be specified, with a different number of randomvariables in each one of them. The table 1 lists those cases, along with their description and the number ofrandom variables included. The reliability analysis and the uncertainty quantification results are presentednext for each one of the cases.

Table 1. Description and number of random variables in each analysis subcase

Case Variables Description

A1 5 Wing box material properties

A2 50 Wing box material properties and loads

A3 100 Wing box material properties and loads

A4 341 Wing box material properties and loads

A5 10 Material properties of the rib

A6 70 Material and geometrical properties of the rib

A7 406 Material and geometrical properties of the rib and loads

A8 411 Material properties of the whole model, geometrical properties of the rib and loads

III.B. Deterministic analysis

Prior to the evaluation of the reliability of the structure, a deterministic analysis is conducted with the meanvalues of the random variables, in order to evaluate the critical nodes where the limit state must be verified.

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In figure 3 the contour map of resultant displacements is plotted over the deformed shape of the mesh. Themaximum displacement is located in the wing tip, with a value of 5.77 m.

Figure 3. Deformed shape of the wing

Criticalzone

Figure 4. Critical zone of the rib

On the other hand, in figure 4, a detail is shown of the most strained zone in the model, which is describedas the critical zone. The critical zone is located around one of the minor holes of the rib. and the maximumvalue of the first principal strain ε1 is 2.167× 10−3. Consequently, this will be the node more susceptible ofdeveloping a failure event and, according to that, this will be the point where the limit state of maximumstrain must be verified in each analysis subcase.

III.C. Reliability analysis

The limit state considered to assess the structural reliability evaluates the value of the first principal strainin the rib ε1,i, which must be below of a maximum value ε1,max. According to that, the following expressionmust be verified:

g (a) =ε1,max

ε1,i− 1 � 0 (21)

The previous limit state constraint must be verified in all the nodes of the rib, but in order to avoid anexcessive processing time, a critical node has been preselected, with the help of the results obtained in thedeterministic analysis with the mean values of the random variables. In this way, only a limit state functionis considered and the reliability of the structure is evaluted based on it. This methodology, employed byother authors as Sudret et al. 1 or Pellissetti et al. 4 , offers a great reduction in the analysis time.

On the other hand, it must be mentioned that in some of the cases considered, the results of probability offailure obtained are exceptionally high. The reason for that is because in those cases, the number of randomvariables simultaneously considered is particularly large, more than the required for the certification of thestructure. Besides, these properties are modelled with a great amount of uncertainty, and it is not realisticthe consideration of many of them acting at the same time, which results in reliability values below of thespecified by the design criteria. That circumstance should be understood as the consequece of an excess ofuncertainty applied to the structure and not as the result of an invalid design. In any case, this situationnot only does not distort the conclusions that can be drawn from this example, but offers information aboutthe response of the structure in extreme situations.

In the table 2 the probability of failure and the reliability index are presented for each method and caseconsidered.

The results delivered by the FORM method, which have been calculated with the Rackwitz and Fiessleralgorithm using the Nataf transformation for the non-normal variables, properly describe the behaviour ofeach subcase. So, in the case A1, that address the effect of the uncertanties in the material properties ofthe wing box, the value of the probability of failure is very low, because of the reduced number of randomvariables simultaneously considered.

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Table 2. Reliability results

Case VariablesFORM TANA3 LHS VRT-I

β pf β pf β pf β pf

A1 5 10.262 5.246× 10−25 10.262 5.218× 10−25 5.110 1.611× 10−7 10.200 9.917× 10−25

A2 50 6.593 2.158× 10−11 6.552 2.834× 10−11 4.140 1.735× 10−5 6.854 3.587× 10−12

A3 100 6.593 2.159× 10−11 6.545 2.978× 10−11 4.180 1.458× 10−5 6.760 6.878× 10−12

A4 341 3.066 1.084× 10−3 3.082 1.028× 10−3 2.736 3.113× 10−3 2.894 1.902× 10−3

A5 10 10.763 2.566× 10−27 9.775 7.214× 10−23 ∞ 0.000 10.500 4.320× 10−26

A6 70 10.482 5.235× 10−26 9.536 7.415× 10−22 ∞ 0.000 10.221 8.026× 10−25

A7 406 2.911 1.803× 10−3 2.986 1.412× 10−3 2.640 4.150× 10−3 2.595 4.729× 10−3

A8 411 2.435 7.453× 10−3 2.626 4.320× 10−3 2.359 9.165× 10−3 2.458 6.985× 10−3

In the cases A2, A3 and A4 the random variables simulating the uncertainty in the loads are graduallyapplied, increasing from 50 in the case A2 to 100 and 341 in the cases A3 and A4, respectively. Becauseof that, a logical increment is registered in the probability of failure, emphasized by the higher uncertaintypresent in the loads, with a coefficient of variation of 15%. From A1 to A2 case, the value of pf increases inseveral orders of magnitude and the reliability index decreases by 36%. However, from the case A2 to A3,which has 50 random variables more, the results do not change, which allows to conclude that the reliabilityof the structure is not affected by those new variables, or in other words, the sensitivity of the probabilityof failure and the reliability index is low with respect to those variables. A noticeable change in the resultsof reliability can be appreciated in the case A4, where β is 2.2 times lower than in the previous case and theprobability of failure increases again, as the result of a higher sensitivity with respect to the new variablesand also due to the amount of random variables acting simultaneously.

On the other hand, in the cases A5 and A6, the effect of the uncertanties in the material and geometricalproperties of the rib is studied. As the A5 case has, like the A1 case, a small number of random variables, theprobability of failure is very low again. This situation is repeated in the case A6, which in spite of having 60random variables more than the A5, has a value of pf which only increases an order of magnitude, revealinga high tolerance of the model to uncertainties in the geometry and also in the material properties.

With regards to A7 case, which adds to the A6 case all the uncertanties assigned to the loads, an expectedincrement in the probability of failure is obtained, quite similar to that happening when changing from A1 toA4, which is an analogous situation. The probability of failure is situated now in the order of 10−3, revealingthe substantial influence of the loads over the rest of uncertainties. This fact is confirmed by the results ofthe A8 case, where the 5 random variables of the uncertainty in the wing box are added to those from caseA7. The values of reliability are now in the same order of those of the previous case, which is logical, takinginto account that the number of variables is very similar.

The results provided by the two point adaptive nonlinear approximations method are quite similar tothose achieved by the FORM method. The only differences that are worth mentioning are those obtained inthe cases A5 and A6 where, according to TANA3, the reliability is lower than the FORM case, but takinginto account the small values of pf and β, that difference can not be considered as significant. So, as theresults do not differ significantlly, in order to compare the method with the rest of the alternatives, thecomputational cost delivered by each one will be considered below.

The latin hypercube sampling method has been applied to this example using a sample size of 32,768elements. This size has been selected based on previous estimations, where it has been observed that thisnumber of simulations delivers an acceptable result in most of the cases, with an analysis time not too long.Hence, this sample size represents a compromise between precision and computational cost, although in somecases the results obtained are not acceptable. That is the case when the probability of failure is very smalland then the LHS method can not be trusted. This is the situation in the cases A1, A2, A5 and A6. As allthe aforementioned cases have a really small value of pf , the sample size selected is not capable of generateenough failure events to estimate accurately the reliability using the expression eq. (17).

However, the rest of the cases perform much better and the reliability results are more precise. If theequation eq. (18) is used to delimit the accuracy of the response, it can be observed that in the case withthe lower probability of failure, the error is below of 20% with this sample size, which confirms the validity

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of the rest of resultsFinally, the results obtained by the importantance sampling method are quite similar to those delivered

by the limit state approximation methods and it is necessary the consideration of the computational costto estimate the actual performance of each analysis procedure. In order to do that, in figures 5a to 5d it isshown the analysis time for each reliability analysis method in all the considered subcases.

Results point out that when the number of random variables is reduced, as it happens in the cases A1 toA3, A5 and A6, the limit state approximation methods offer a better alternative. Besides, a distinction canbe made between the TANA3 method (figure 5b), which performs slightly better than the FORM method(figure 5a) in the case A1, but not in the case A5. On the other hand, the importance sampling delivers acompetitive cost (figure 5d), but this is not the case of LHS sampling (figure 5c), which is usually slowerthan the VRT-I and must be discarded in those situations with a reduced number of random variables andlow probability of failure.

A1

A2

A3

A4

A5

A6

A7

A8

103 104 105 106 107

Time (s)

Cas

e

00h 44m 18s

25h 54m 40s

27h 28m 34s

102h 53m 37s

01h 58m 48s

62h 42m 00s

186h 49m 48s

212h 50m 39s

(a) FORM method

A1

A2

A3

A4

A5

A6

A7

A8

103 104 105 106 107

Time (s)

Cas

e

00h 30m 45s

47h 37m 36s

55h 01m 57s

89h 14m 55s

03h 13m 24s

73h 25m 52s

134h 51m 10s

137h 46m 40s

(b) TANA3 method

A1

A2

A3

A4

A5

A6

A7

A8

103 104 105 106 107

Time (s)

Cas

e

128h 26m 09s

130h 51m 40s

128h 12m 44s

129h 07m 57s

130h 18m 54s

131h 31m 08s

128h 53m 50s

129h 46m 52s

(c) LHS method

A1

A2

A3

A4

A5

A6

A7

A8

103 104 105 106 107

Time (s)

Cas

e

05h 46m 35s

51h 33m 44s

56h 21m 07s

98h 27m 23s

06h 45m 33s

93h 23m 37s

193h 21m 54s

215h 29m 18s

(d) VRT-I method

Figure 5. Computational times required by each method

On the contrary, in those cases with a larger number of variables, like the A4, A7 and A8, the TANA3overcomes the FORM method, and that is the reason why it can not be established a definitive conclusionabout the convenience of recommending a method over the other. Also, it is advisable to perform a previousparametric analysis in each particular case to make a decision.

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In relation to sampling methods, the importance sampling surpasses the direct latin hypercube samplingin all the situations, except for the A4 case. The difference in this case is due to the number of simultaneousrandom variables, which is larger in the A4 case, and so the importance sampling is affected by the necessityof evaluate the gradients of the limit state function with respect to all the random variables. This operationis carried out using finite differences, which increases noticeably the analysis time.

On the other hand, the computational cost of LHS method is invariant with respect to the size of theproblem. This can represent a competitive advantage in certain situations, specially in those cases with alarge number of random variables or limit state functions, where the evaluations of the derivatives of thelimit state function overloads the analysis procedure.

IV. Conclusions

In this work, a review of some existing procedures of reliability analysis and their capabilities has beencarried out. A practical example has been used to illustrate the performance of the methods employed inthe study, applied to the case of large scale aircraft structures. Finally, some conclusions can be drawn.

1. The probabilistic approach to the assessment of structural reliability represents an efficient methodologyto deal with the design process of large scale structural elements. The methods are mature now andcan be confidently used in industrial applications, instead of classical deterministic methods.

2. Taking into account the high computational cost required by the reliability analysis of large scalemodels, the methods have been employed in a parallel environment using a high performance computercluster.

3. Considering the performance of each method, it is reasonable the recommendation of using limit stateapproximation methods in those cases with a reduced number of random variables or limit statefunctions, when the values of derivatives must be calculated with procedures which increase the com-putational cost. In this situation, sampling methods can not beat the performance of limit stateapproximations.

4. However, in those cases having a large number of random variables and/or limit state functions, thenumber of iterations required to evaluate the first derivatives, causes that sampling methods become analternative, with computational costs quite similar to the limit state approximation methods. Moreover,those methods are not penalized by the problem size and can be easily parallelized, increasing theirperformance.

References

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