Probabilistic Engineering Mechanics 23 (2008) 351–358www.elsevier.com/locate/probengmech

Analysis of structural reliability under parameter uncertainties

Armen Der Kiureghian

Department of Civil and Environmental Engineering, University of California, 723 Davis Hall, Berkeley, CA 94116, United States

Received 15 July 2007; accepted 19 October 2007Available online 4 March 2008

Abstract

Formulation of structural reliability requires selection of probabilistic or physical models, which usually involve parameters to be estimatedthrough statistical inference — a process that invariably introduces uncertainties in parameter estimates. The measure of reliability thatincorporates these parameter uncertainties is termed the predictive reliability index. Methods for computing this measure and the correspondingfailure probability are introduced. A simple approximate formula is derived for the predictive reliability index, which requires a single solution ofthe reliability problem together with parameter sensitivities with respect to mean parameter values. The approach also provides measures of theuncertainties inherent in the estimates of the reliability index and the failure probability, which arise from parameter uncertainties. An illustrativeexample involving component and system problems demonstrates the influence of parameter uncertainties on the predictive reliability index andthe accuracy of the simple approximation formula.c© 2008 Elsevier Ltd. All rights reserved.

Keywords: Bayesian analysis; Parameter estimation; Parameter uncertainties; Predictive measures; Structural reliability; Systems; Uncertainties

1. Prologue

It is a privilege to be contributing to this specialissue of Probabilistic Engineering Mechanics honoring thedistinguished career and seminal contributions of Ove DalagerDitlevsen. My first impressions of Ove are from September1982, when I heard him give a series of lectures on structuralreliability at a NATO Advanced Study Institute held inBornholm, Denmark. Over the years, reading his papersand having discussions with him have profoundly influencedmy own thinking about probability theory, the nature ofuncertainties and the principles and methods of structuralreliability analysis. For that reason, I wish to start this paper bybriefly recounting from a series of communications with himon the topic of uncertainties and structural reliability duringthe summer of 1988. Provoked by a report I had written [4],which was later published in the ASCE Journal of EngineeringMechanics [5], the correspondence consisted of 12 letters,a total of 48 partly handwritten pages and 6 attachments,including detailed derivations and calculations (see Fig. 1).These communications are directly relevant to the topic of thispaper.

E-mail address: adk@ce.Berkeley.edu.

0266-8920/$ - see front matter c© 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.probengmech.2007.10.011

Fig. 1. Correspondence with Ove Ditlevsen in summer of 1988.

In [4,5], I had investigated several measures of structuralsafety relative to a set of postulated “fundamental require-ments” and advocated one measure, which included a penaltyfor uncertainty in the estimate of the reliability index arisingfrom statistical and model uncertainties. The argument wasbased on a crisp distinction of aleatory and epistemic uncer-tainties — inherent variabilities (randomness) being aleatory innature and model and statistical uncertainties being epistemic in

352 A. Der Kiureghian / Probabilistic Engineering Mechanics 23 (2008) 351–358

nature. It was argued that, since epistemic uncertainties couldbe reduced, the penalty would provide an incentive to gatheradditional data or use more refined models, thus satisfying a re-quirement of “remunerability”, which stated “The reliability in-dex shall remunerate improvements in the state of knowledge”.The main points of the discussion in the communications withOve were the nature of uncertainties and the measure of relia-bility to be used in structural design.

In his first letter in the series dated May 16, 1988, Ovestarted by “I have studied your report . . . with interest, and Ihave some comments on points where I disagree with you”.With his characteristic modesty, he added “In the following,please put in the words ‘I think that’ before all statementsformulated in absolute terms”. His first disagreement wason the nature of inherent variabilities. He wrote: “Inherentvariability is relative to a level of refinement of the model.Except for quantum mechanical phenomena inherent variabilityis reducible by more detailed modeling combined withcorresponding gathering of information. For example, this isthe case for material properties like stress–strain relationships.Thus, the inherent variability of material properties in structuralreliability is relative to the ‘usual’ level of modeling materialbehavior. The same applies to the loads. For example, theinherent variability of the wind load on a structure can inprinciple be reduced as the abilities of meteorologists to predictwind velocities from geophysical models become better andbetter. Philosophically it does not make sense to claim that windvelocities are random. Randomness is not a property of naturebut a property of the model”.

Having questioned the crisp distinction between aleatoryand epistemic uncertainties, Ove went on to say: “On apragmatic basis I agree that it is useful to distinguish betweeninherent variability and model uncertainty, but this is relativeto a given framework of engineering modeling practice....By distinguishing the intrinsic variability from uncertaintydue to estimation error and model imperfection, it is on apragmatic level clarified [as to] what can be done withina framework of usual practice by information gathering inorder to (hopefully) increase safety without changing thestructure.” The parenthetical adverb “hopefully” referred to thefact that the additional information might in fact indicate thatthe structure is less safe than estimated based on the priorinformation. He explained “by bad luck the sample average maychange with sample size so that it can happen that the reliabilityindex decreases. This event will generally be more rare than theevent of having an increase of the reliability index”.

In the report and paper, I had also defined a requirement of“completeness” as “The reliability index shall incorporate allavailable information and account for all uncertainties”. Ovewrote: “To talk about all uncertainties makes only sense relativeto some model in which the uncertainties are represented bysuitable formal elements. Otherwise a requirement containingthe word ‘all’ becomes a religious commandment”.

The fundamental point Ove was raising is that the natureof uncertainties can only be discussed within the confines of amodel. Since our correspondence in 1988, I have come to fullyappreciate this point. As we described in a recent co-authored

paper [6] “Engineering problems, including reliability, risk anddecision problems, without exception, are solved within theconfines of a model universe. This universe contains the setof physical and probabilistic models, which are employed asmathematical idealizations of reality to render a solution for theproblem at hand. The model universe may contain inherentlyuncertain quantities; furthermore, the sub-models are invariablyimperfect giving rise to additional uncertainties. Therefore, animportant part of building the model universe is the modeling ofthese uncertainties. Any discussion on the nature and characterof uncertainties should be stated within the confines of themodel universe”.

Ove’s second point of disagreement with my paper wasmy proposed “minimum-penalty reliability index”. He did notobject to the six fundamental requirements I had postulatedfor the measure of safety, except to suggest rewording ofthe completeness requirement as mentioned above. However,he insisted that what I had called “predictive reliabilityindex” satisfied all the requirements and that it was theappropriate one to use within the context of decision makingfor structural design or code development. He also pointed outthat this measure was identical to what he had earlier called“generalized reliability index” with statistical uncertainty takeninto account [7–9]. I have come to agree with this point of viewas well. In fact, one of the objectives of this paper is to formallydefine this measure of reliability and suggest several methodsfor its computation.

The importance of Ove Ditlevsen’s work is not limitedto his enormous collection of published papers and books,many of which contain seminal contributions to the theoryof structural reliability, stochastic mechanics or the principlesbehind development of probabilistic codes. These are readilyquantifiable. What is less easily quantifiable is his role as acorrective agent in the course of development of these fields.He has been a vocal critic of work by his colleagues (we haveall seen him in this role in conferences and workshops), but hiscriticism has always been honest and constructive. In a sense,he has shepherded the advancement of the field on a healthypath to its present maturity. For this and many other reasons, itis a true privilege to contribute to this issue honoring his career.

2. Objectives and scope

The focus of this paper is on the predictive reliabilityindex, or the generalized reliability index in Ove Ditlevsen’sterminology, which is a measure of structural reliability thatincorporates the influence of parameter uncertainties. Theparameters are those inherent in the probabilistic or physicalmodels employed in the formulation of the reliability model.After a brief review of the formulation of structural reliabilityproblems under parameter uncertainties, the predictive failureprobability and reliability index are formally defined andseveral methods for their computation are described. Onemethod leads to a simple approximation of the predictivereliability index, which involves little more than a singlereliability and sensitivity analysis using mean values of themodel parameters. An additional advantage of this approach

A. Der Kiureghian / Probabilistic Engineering Mechanics 23 (2008) 351–358 353

is that it also provides a measure of the uncertainty in thereliability index, which arises from parameter uncertainties. Anillustrative example demonstrates the influence of parameteruncertainties on the predictive reliability index and the accuracyof the simple approximation formula.

3. Formulation of structural reliability

The theoretical, time-invariant structural reliability problemis defined by the integral

p =

∫Ω(x)

fX(x)dx (1)

where p is the failure probability, fX(x) is the joint probabilitydensity function (PDF) of a vector of random variables X =

[X1, X2, . . . , Xn]T representing uncertain quantities such as

loads, material property constants and geometric dimensions,and Ω(x) is the failure domain of the structure in the outcomespace x = [x1, x2, . . . , xn]

T of X. For a general structuralsystem, the failure domain may be defined in the form

Ω(x) ≡

⋃k

⋂i∈Ck

gi (x) ≤ 0 (2)

where gi (x), i = 1, . . . , m, are a set of limit-state functionsformulated so that gi (x) ≤ 0 defines the failure of componenti , m denotes the number of components, and Ck is the indexset for the k-th minimum cut set, where each minimum cutset represents a minimal set of components whose joint failureconstitutes failure of the structural system. In the special casewhere each component is a cut set (failure of any componentconstitutes failure of the system), the structure is a seriessystem. Conversely, when all components collectively forma single minimum cut set, the structure constitutes a parallelsystem.

We have termed the above a theoretical formulation ofthe structural reliability problem. This is because in practiceneither the joint PDF fX(x) nor the limit-state functionsgi (x) are precisely known. Instead, idealized models of thesefunctions must be developed, typically through statisticalinference using real-world observations complemented byengineering judgment. Let fX(x|2 f ) denote the selected modelfor fX(x), in which 2 f denotes a set of parameters to beestimated from observational data. Each of the limit-statefunctions gi (x) is typically composed of several sub-models,e.g., separate capacity and demand models. However, for thesake of simplicity of the notation, here we assume each limit-state function is a single model. Thus, let gi (x, 2gi ) + Eibe the idealized model for gi (x), in which gi (x, 2gi ) isa parameterized deterministic model with 2gi denoting itsparameters and Ei = gi (x)− gi (x, 2gi ) is the residual, i.e., theerror in the model for given x and 2gi . The model parameters2gi , as well as the statistics of the model error Ei , normallyare estimated by statistical inference of laboratory or field data.It is usually reasonable to assume, possibly after an appropriatetransformation of the model and the data, that the residual hasthe normal distribution with a standard deviation Di , whichis independent of x. Furthermore, with the aim of developing

an unbiased model, the mean of Ei is set equal to zero.When multiple models are to be developed, it is additionallynecessary to estimate the cross-correlations ρi j , i 6= j , betweenpairs of model residuals. The collection of standard deviationsDi and correlation coefficients ρi j then forms the covariancematrix 6, which completely defines the joint distribution ofthe vector of model residuals E = [E1, E2, . . . , Em]

T. The setof random variables of the reliability problem now is (X, E).Furthermore, the problem involves the set 2 = (2 f , 2g, 6)

of model parameters, where (2g) is the collection of 2gi forall limit-state models. As just mentioned, these parameters needto be estimated from observational data – a process whichinvariably introduces additional uncertainties. With increasingobservational data, one can reduce uncertainties in parameterestimates; however, uncertainties in X and E cannot beinfluenced without changing the selected models. The mainfocus of this paper is on the influence of parameter uncertaintieson the reliability estimate.

Consistent with the Bayesian notion of probability, unknownmodel parameters 2 are considered as random variables. Theirdistribution is determined through the well known Bayesianupdating formula

f2(θ|z) ∝ L(z|θ) f2(θ) (3)

where f2(θ) is the prior distribution and represents informationavailable on parameters before observations are made, L(z|θ) isthe likelihood function for the set of observations z (a functionproportional to the probability of making the observations zgiven 2 = θ), and f2(θ|z) is the posterior distributionreflecting the updated information on 2 in light of observationsz. This formula can be used to estimate distribution parameters2 f as well as physical model parameters (2g, 6). In generalestimates for the two sets are statistically independent; however,statistical dependence may exist among elements of 2g and 6.Examples for the estimation of distribution parameters can befound in many textbooks [1,2]. Specific examples in structuralreliability for the estimation of physical model parameters orlimit-state function parameters can be found in [10] and [13],respectively. In the following discussion, we drop the referenceto the observations z and use the expression f2(θ) as the(updated) posterior PDF of the parameters 2.

Using the parameterized distribution model of X in (1)and including random variables E, the expression for failureprobability becomes

p(2) =

∫ε

∫Ω(x,ε,2g)

fX(x|2 f ) fE(ε|6)dxdε (4)

where fE(ε|6) is the joint normal PDF of E conditioned on thecovariance matrix 6 and

Ω(x, ε, 2g) ≡

⋃k

⋂i∈Ck

gi (x, 2gi ) + εi ≤ 0

(5)

is the parameterized failure domain in the space of expandedrandom variables (X, E). It is seen that the failure probabilityis a function of the model parameters 2 = (2 f , 2g, 6). Itfollows that, since the model parameters are uncertain, the fail-ure probability estimate is also uncertain. Consistent with the

354 A. Der Kiureghian / Probabilistic Engineering Mechanics 23 (2008) 351–358

Bayesian notion, we introduce P = p(2) as a random variablerepresenting the uncertain failure probability. We also introducethe corresponding uncertain reliability index B = β(2), whereβ(2) = Φ−1

[1 − p(2)] and Φ−1[ · ] denotes the inverse of

the standard normal cumulative probability function. These val-ues can be seen as the failure probability and reliability indexconditioned on the set of model parameters. As random vari-ables, P and B have probability distribution functions as well asstatistics, such as means and variances. In particular, we definethe PDF of P as fP (p), the PDF of B as fB(β), their meansas µP and µB , respectively, and their standard deviations asσP and σB , respectively. Note that σP and σB are measures ofthe uncertainties in the estimates of the failure probability andthe reliability index, which arise from parameter uncertainties.Since parameter uncertainties are reducible, σP and σB repre-sent measures of uncertainty, which can possibly be reduced bygathering additional observational data.

4. Predictive failure probability and reliability index

In Bayesian analysis, the distribution of a random variablethat incorporates parameter uncertainties is called the Bayesianor predictive distribution [1–3]. For the vector of randomvariables X, whose joint PDF has uncertain parameters 2 f ,the predictive distribution is defined as the expectation of theconditional distribution fX(x|2 f ) over the outcome space ofthe uncertain parameters, i.e.,

fX(x) =

∫θ f

fX(x|θ f ) f2 f (θ f )dθ f (6)

where f2 f (θ f ) is the posterior distribution of the parameters.In the same vein, we define the predictive failure probability asthe expectation of the conditional failure probability P = p(2)

over the outcome space of the uncertain parameters 2. Using(4), this can be written as

p =

∫θ

p(θ) f2(θ)dθ

=

∫θg

∫σ

∫θ f

∫ε

∫Ω(x,ε,θg)

fX(x|θ f ) fE(ε|σ ) f2 f (θ f )

× f2g,6(θg, σ )dxdεdθ f dσdθg (7)

where f2(θ) = f2 f (θ f ) f2g,6(θg, σ ) is the posterior jointPDF of the set of parameters, where statistical independencebetween 2 f and (2g, 6) is assumed. Clearly, p =

µP represents the mean of the distribution fP (p). Thecorresponding predictive reliability index is obtained from

β = Φ−1 (1 − p) . (8)

However, as shown below, β is different from the meanreliability index µB .

In a decision problem, provided the utility function isa linear function of the failure probability – as in “totalcost = initial cost + cost of failure × probability of failure” –the probability measure to be used is the predictive value in (7).In his June 1, 1988 letter, referring to [11], Ditlevsen arguedthat “the uncertainty about p f [P in the present notation]

should not have influence on decision making”. In essence,given two decision alternatives with identical predictive failureprobabilities and identical initial costs and costs of failure,the decision maker should be indifferent between the twoalternatives. This result appears somewhat counterintuitive, aswith identical costs and mean failure probabilities, one wouldexpect a preference for the case with smaller uncertaintyin the failure probability estimate. But simple derivationsin Ditlevsen’s letters show that aversion to uncertainty infailure probability is not logical. Nevertheless, knowledgeof the extent of uncertainty in the probability estimate (asmeasured by σP ) is useful for several reasons. Chief amongthese is that one then knows the extent of this uncertaintythat can possibly be removed by gathering of additional data.As we will shortly see, the failure probability estimate ismore likely to decrease than increase with increasing data.This potential benefit should be balanced by the cost ofgathering the data. Secondly, for the sake of transparency incommunicating risk, it is important that the analyst also reportsuncertainty in the estimated failure probability, not only themean value. This may help a client decide whether additionalgathering of data is worthwhile. Finally, as demonstratedin [8] by Ditlevsen and more recently in [6] and [13],epistemic uncertainties arising from the estimation of modelparameters induce statistical dependence among the estimatedstates of system components. This dependence can havesignificant influence on the predictive estimate of system failureprobability, particularly for redundant systems.

5. Computational methods

In this section we describe methods for computing thepredictive failure probability and reliability index, as well asa measure of the uncertainty in their estimates.

A straightforward way to compute predictive failureprobability in (7) is to treat all random variables and uncertainparameters together so that the set of random variables ofthe problem is (X, E, 2 f , 2g, 6). Well known computationalmethods of structural reliability, such as FORM, SORM orvarious importance sampling methods [9], can be used forthis purpose. These methods typically require a transformationof the random variables into standard normal space. For thepresent problem the conditional distribution of X given 2 f andthe conditional distribution of E given 6 are available, alongwith the posterior distributions of 2 f and (2g, 6). Because ofthe conditional form of these distributions, use of the so-called“Rosenblatt” transformation [12] is necessary for solving thereliability problem.

A second approach for computing the predictive failureprobability is to first carry out integrations on θ f and σ in (7).For this purpose we replace f2g,6(θg, σ ) by f6(σ |θg) f2g (θg)

and note that the integral of the product fX(x|θ f ) f2 f (θ f ) overθ f gives the predictive distribution of X and the integral offE(ε|σ ) f6|2g (σ |θg) over σ gives the predictive distribution ofE conditioned on 2g = θg . The result is the expression

p =

∫θg

∫ε

∫Ω(x,ε,θg)

fX(x) fE(ε|θg) f2g (θg)dxdεdθg. (9)

A. Der Kiureghian / Probabilistic Engineering Mechanics 23 (2008) 351–358 355

The set of random variables of the problem now is (X, E, 2g).This reduction in dimensionality of the problem, however, isachieved at the expense of having to derive predictive distribu-tions fX(x) of X and fE(ε|θg) of E. In many cases closed-formexpressions of these predictive distributions are available (see,e.g., [1,2]). This second formulation is useful for such cases.

The third approach is based on a nested reliabilityformulation originally suggested by Wen and Chen [14].Consider the auxiliary structural reliability problem defined bythe limit-state function

g(u, θ) = u + β(θ) (10)

where u is the outcome of a standard normal random variable Uhaving the PDF φ(u) = (2π)−1/2 exp(−u2/2) and β(θ) is theconditional reliability index of the original problem for 2 = θ.U is statistically independent of 2. The solution of this problemis identical to the predictive failure probability in (7). The proofis as follows:

Pr[g(U, 2) ≤ 0

]=

∫u+β(θ)≤0

φ(u) f2(θ)dudθ

=

∫θ

[∫−β(θ)

−∞

φ(u)du

]f2(θ)dθ

=

∫θΦ [−β(θ)] f2(θ)dθ

=

∫θ

p(θ) f2(θ)dθ

= p. (11)

It follows that the predictive failure probability can becomputed by solving the problem in (10), which in turnrequires the solution of the conditional reliability problemdefined by (4) — hence the reason for the name nestedreliability analysis. Obviously this approach is convenient if theconditional problem in (4) has a closed-form solution expressedin terms of the parameters 2. However, even when such aclosed-form solution is not available, the above formulationoffers a useful approximation for the predictive reliabilityindex. If we assume the uncertain reliability index B = β(2)

has a normal distribution, then the limit-state function in (10) isa linear function of two independent normal random variables,U and B. The solution of the corresponding reliability indexthen is simply the mean µg = 0 + µB = µB of the limit-

state function divided by its standard deviation σg =

√1 + σ 2

B .It follows that the predictive reliability index is approximatelygiven by

β ∼=µB√

1 + σ 2B

. (12)

The only approximation involved in this derivation is theassumption of normal distribution of the uncertain reliabilityindex. In many cases this assumption is not too far fromreality. It is interesting to see that the predictive reliabilityindex increases with decreasing variance of the uncertainreliability index, provided the mean reliability index is positive

and remains unchanged. Furthermore, the predictive reliabilityindex is found to be smaller than the mean reliability index inabsolute value.

The use of (12) requires knowledge of the mean and standarddeviation of the conditional reliability index. These are noteasy to compute exactly, since they require knowledge of thedistribution fB(β). However, they can be easily computed byapplying the first-order approximation method to the functionB = β(2). The result is

µB ∼= β(M2) (13)

σ 2B

∼= (∇2β)T2=M2

622 (∇2β)2=M2(14)

where M2 and 622 are the posterior mean vector andcovariance matrix of 2, respectively, and (∇2β)2=M2

issensitivity vector of the conditional reliability index withrespect to parameters 2, evaluated at the posterior meanvalues. It is seen that a single solution of the conditionalreliability problem in (4) at the posterior mean values ofthe parameters, along with parameter sensitivity analysis issufficient to approximately compute the mean and varianceof the conditional reliability index from (13), (14) andthe predictive reliability index according to the approximateformula in (12). The predictive failure probability, p, iscomputed by the inverse of (8).

As mentioned earlier, it is useful to have a measure of the un-certainty in the failure probability estimate, which arises frommodel parameter uncertainties. One way to express this uncer-tainty is to provide Bayesian credibility bounds, which expressbounds within which the failure probability lies with a specifiedprobability. Since, roughly speaking, the interval (µB ± σB)

contains 70% probability, the 70% credibility interval on theuncertain failure probability is approximately given by

〈P〉70%∼= (Φ [−(µB + σB)] ,Φ [−(µB − σB)]) . (15)

This interval is a measure of the uncertainty in the failure prob-ability estimate, which can possibly be reduced by gathering ofadditional data.

Although it is possible to derive expressions similar to (13),(14) for the mean and standard deviation of the uncertain failureprobability P = p(2), the resulting expressions usually arenot as accurate due to the strong skewness of the distributionfP (p). Furthermore, for small failure probabilities, the one-standard deviation lower credibility bound of the uncertainfailure probability often has a negative value due to the highlyskewed nature of the distribution. The expression in (15) avoidsthese shortcomings.

6. Illustrative example

Consider a structural component having the limit-statemodel

g(x) = x1 − x2 (16)

where (x1, x2) is the outcome of the vector of random variablesX = (X1, X2). Assume X1 and X2 are statistically independentnormal random variables with unknown means M1 and M2

356 A. Der Kiureghian / Probabilistic Engineering Mechanics 23 (2008) 351–358

and known variances σ 21 and σ 2

2 , respectively. Suppose theavailable information for estimating the mean values are sampleobservations of size n of X1 and X2 with respective samplemeans x1 and x2. Assuming independence of M1 and M2 andusing non-informative priors, the posterior distributions of M1and M2 are found to be independent normals with means x1and x2 and standard deviations σ1/

√n and σ2/

√n, respectively.

Thus, in this problem we have a precisely known limit-statemodel and uncertain distribution parameters 2 f = (M1, M2).Due to the linear form of the limit-state function and the normaldistribution of the random variables, the conditional reliabilityindex is easily found to be

B = β(2 f ) =M1 − M2√σ 2

1 + σ 22

. (17)

Because B is a linear function of the random variables M1 andM2, one can easily determine its mean and variance as

µB =x1 − x2√σ 2

1 + σ 22

(18)

σB =1

√n. (19)

Furthermore, since M1 and M2 are normally distributed, B hasthe normal distribution. It follows that in this case (12) gives anexact result and the predictive reliability index is given by

β =x1 − x2√(σ 2

1 + σ 22 )

1√

1 + 1/n. (20)

It is seen that, provided the sample means remain constant,the predictive reliability index increases in absolute value withincreasing sample size n and approaches the mean reliabilityindex as n approaches infinity. Of course additional data mayaffect the sample means and, therefore, the predictive reliabilityindex. However, since µB is the median reliability index andβ < µB , it is more likely that the reliability index will increasethan decrease with increasing sample size. This is consistentwith the “remunerability” requirement defined in [4,5],as mentioned in the prologue of this paper.

Now suppose the limit-state model includes a residual termε,

g(x, ε) = x1 − x2 + ε (21)

where ε denotes the outcome of a normal random variable Ehaving a zero mean and an unknown variance Σ . (Such limit-state models for seismic fragility of electrical substation equip-ment have been developed in [13].) Suppose the posterior meanof Σ is a fraction r of σ 2

1 + σ 22 , i.e., µΣ = r(σ 2

1 + σ 22 ), and

its posterior coefficient of variation is δΣ . The set of uncertainparameters now is 2 = (M1, M2,Σ ) and the conditional relia-bility index is given by

B = β(2) =M1 − M2√

σ 21 + σ 2

2 + Σ. (22)

Fig. 2. Estimates of predictive reliability index for component.

The above is no more a linear function of uncertain parameters2. Therefore, exact solutions of the mean and variance are noteasy to obtain. If the first order approximations in (13), (14) areused, the results are

µB ∼=x1 − x2√σ 2

1 + σ 22

1√

1 + r(23)

σB ∼=

√√√√ 1n(1 + r)

+(x1 − x2)2

σ 21 + σ 2

2

14(1 + r)

(rδΣ

1 + r

)2

. (24)

Using these relations in (12), the corresponding approximateestimate of predictive reliability is obtained as

β ∼=x1 − x2√σ 2

1 + σ 22

1√1 + r +

1n +

(x1−x2)2

σ 21 +σ 2

2

[rδΣ

2(1+r)

]2. (25)

As in the previous case, provided the sample means x1 and x2remain constant, the predictive reliability index increases withincreasing sample size n. Furthermore, provided r remains con-stant, the predictive reliability index increases with decreasingposterior coefficient of variation of Σ .

Fig. 2 compares the above approximation of the predictive

reliability index for (x1 − x2)/

√(σ 2

1 + σ 22 ) = 3, r =

0.2 and n = 10 and 20 as a function of the coefficientof variation δΣ with “exact” results obtained by simulationwith a 3% coefficient of variation. Also shown in thefigure are approximations of the predictive reliability indexobtained by applying first- and second-order reliabilitymethods (FORM and SORM) to the formulation in (9).

A. Der Kiureghian / Probabilistic Engineering Mechanics 23 (2008) 351–358 357

Fig. 3. Estimates of predictive reliability index for system.

Note that the approximation in (25) only involves the firstand second moments of uncertain parameters. On the otherhand, simulation and FORM/SORM results require the fulldistribution of uncertain parameters. In the present case, M1

and M2 are Gaussian distributed and Σ is assumed to belognormally distributed. It is seen that the approximation in (12)is in close agreement with the simulation results. In particular, itcorrectly predicts the decrease in the predictive reliability indexwith increasing δΣ or decreasing sample size (with fixed valuesof x1, x2 and r ). Ironically, the FORM approximation predictsan increase in the predictive reliability index with increasingcoefficient of variation of Σ , and the SORM approximationshows essentially no variation with δΣ . It is believed that thepoor performance by FORM and SORM is due to nonlinearity

in the dependence of the conditional reliability index on theuncertain parameter Σ , as is evident in (22).

Next consider a K -out-of-N system with exchangeablecomponents, each characterized by the limit-state function in(21). Such a system survives if K or more of its N componentssurvive. Conditional on the parameters 2 = (M1, M2,Σ ),random variables X1, X2 and E for different components arestatistically independent. However, all components share thesame set of uncertain parameters 2 = (M1, M2,Σ ). Asdiscussed in [6,13], this type of common parameter uncertaintyinduces statistical dependence among the estimated states ofsystem components, which can have a significant influence onthe predictive reliability of the system.

Owing to the conditional independence of the componentstates, the conditional probability of failure of the system

358 A. Der Kiureghian / Probabilistic Engineering Mechanics 23 (2008) 351–358

for given 2 = (M1, M2,Σ ) is described by the binomialcumulative distribution function

p (2) =

N∑i=N−k+1

N !

i !(N − i)![q (2)]i [1 − q (2)]N−i (26)

where

q(2) = Φ

−M1 − M2√

σ 21 + σ 2

2 + Σ

(27)

is the conditional failure probability of each component.The corresponding conditional reliability index is given byβ(2) = Φ−1 [1 − p(2)]. Using this expression, the predictivereliability index is computed by the nested reliability methodemploying the limit-state function in (10). An approximateestimate of the predictive reliability index is obtained by useof the formula in (12) together with first-order approximationsof the mean and standard deviation of the reliability index as in(13), (14). Note again that the latter analysis only requires thefirst and second moments of the parameters and computationof the conditional reliability index and its sensitivities for meanvalues of the parameters. On the other hand, computation ofthe predictive reliability index by nested reliability analysisemploying FORM, SORM or Monte Carlo simulation requiresthe full distribution of the parameters.

Fig. 3 compares approximate estimates of the predictivereliability index with FORM, SORM and “exact” simulation

results for various systems with (x1 − x2)/

√(σ 2

1 + σ 22 ) = 3,

r = 0.2, n = 10 and 20, and as functions of the coefficientof variation δΣ . Considered are systems having N = 5components with K = 1 (parallel systems), K = 3,and K = 5 (series systems). For redundant systems withK = 1 and 3, where the reliability index is higher than3, importance sampling around the design point is used. Allsimulation results have less than 3% coefficient of variation.It is observed that in all cases the predictive reliability indexof the system decreases with increasing coefficient of variationδΣ and decreasing sample size n (for fixed values of x1,x2 and r ). Furthermore, parameter uncertainties have a moreprofound effect on redundant systems (K < N ). As notedin [6,13], this is due to the effect of dependence amongestimated states of system components, which arises fromparameter uncertainties that are common to all components.The simple approximation in (12) provides close agreementwith simulation results for 1 < K and fair approximation forK = 1 (parallel systems). It is believed that the lack of closeagreement for parallel systems is due to the approximationinvolved in computing the mean and variance from (13),(14) rather than the approximation inherent in deriving (12).Overall, the expression in (12) is found to provide a consistentand reasonably accurate estimate of the predictive reliabilityindex. It is worth emphasizing once again that this equationrequires nothing more than reliability and sensitivity analyseswith posterior mean values of the parameters together withknowledge of the posterior covariance matrix of the uncertainparameters. With its simplicity, this formula provides an

attractive alternative for practical structural reliability problemsinvolving uncertain parameters, as well as for probabilisticcodified design.

7. Summary and conclusions

The problem of structural reliability under parameteruncertainties is considered. The parameters are involved inprobabilistic and physical models, which are used to formulatethe reliability problem. In practice, these parameters must beestimated by statistical inference using observational data – aprocess, which invariably yields uncertainties. The measure ofreliability that incorporates parameter uncertainties is termedthe predictive reliability index. Methods for computing thepredictive reliability index and the corresponding failureprobability are described. One method leads to a simpleapproximation formula, which involves a single computation ofthe conditional reliability index and its parameter sensitivitieswith respect to mean values of the parameters. This formulationalso provides a measure of the uncertainties in the estimatedreliability index and failure probability, which arise fromparameter uncertainties. An illustrative example demonstratesthe computation of the predictive reliability index and theaccuracy of the simple formula for component and systemreliability problems.

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