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  • Struct Multidisc Optim (2011) 44:691705DOI 10.1007/s00158-011-0652-9


    Reliability sensitivity analysis for structural systemsin interval probability form

    Ning-Cong Xiao Hong-Zhong Huang Zhonglai Wang Yu Pang Liping He

    Received: 1 August 2010 / Revised: 20 March 2011 / Accepted: 30 March 2011 / Published online: 7 May 2011c Springer-Verlag 2011

    Abstract Reliability sensitivity analysis is used to find therate of change in the probability of failure (or reliability) dueto the changes in distribution parameters such as the meansand standard deviations. Most of the existing reliability sen-sitivity analysis methods assume that all the probabilitiesand distribution parameters are precisely known. That is,every statistical parameter involved is perfectly determined.However, there are two types of uncertainties, epistemicand aleatory uncertainties that may not be perfectly deter-mined in engineering practices. In this paper, both epis-temic and aleatory uncertainties are considered in reliabilitysensitivity analysis and modeled using P-boxes. The pro-posed method is based on Monte Carlo simulation (MCS),weighted regression, interval algorithm and first order reli-ability method (FORM). We linearize original non-linearlimit-state function by MCS rather than by expansion asa first order Taylor series at most probable point (MPP)because the MPP search is an iterative optimization process.Finally, we introduce an optimization model for sensitiv-ity analysis under both aleatory and epistemic uncertainties.Four numerical examples are presented to demonstrate theproposed method.

    Keywords Reliability sensitivity analysis Epistemic uncertainty Aleatory uncertainty P-box Interval form Weighted regression

    N.-C. Xiao H.-Z. Huang (B) Z. Wang Y. Pang L. HeSchool of Mechatronics Engineering, University of ElectronicScience and Technology of China, Chengdu, Sichuan, 611731, Chinae-mail:

    1 Introduction

    In reliability analysis and reliability-based design, sensitiv-ity analysis identifies the relationship between the changein reliability and the change in the characteristics of uncer-tain variables. Sensitivity analysis is also used to identifythe most significant uncertain variables that have the highestcontribution to reliability (Guo and Du 2009). Furthermore,sensitivity analysis could be used to provide informationfor the reliability-based design. If the reliability of a designdoes not meet requirements, there are several useful waysto improve the reliability, including (1) change of the meanvalues of random variables, (2) change of the variances ofrandom variables, and (3) truncation of the distributionsof random variables (Du 2005). Sensitivity analysis canprovide the information about which random variables arethe most significant and therefore, in an effective manner,should be changed in order to improve reliability. Reliabil-ity sensitivity analysis has played a key role in structuralreliability design. A number of sensitivity analysis meth-ods exist in literature. Among them, Ditlevsen and Madsen(2007) presented an expression based on the first order relia-bility method (FORM) to evaluate reliability sensitivity for astructural system with linear limit-state and normal randomvariables. De-Lataliade et al. (2002) developed a methodusing Monte Carlo simulation (MCS) for reliability sensi-tivity estimations. Ghosh et al. (2001) proposed a methodusing the first order perturbation for stochastic sensitivityanalysis. For more information, please refer to references(Mundstok and Marczak 2009; Xing et al. 2009; Au 2005;Rahman and Wei 2008; Liu et al. 2006). In the afore-mentioned literatures, most of methods assume that all thestochastic characteristics of random variables are preciselyknown. That means that every random parameter involved isperfectly determined. However, in reality, this assumption

  • 692 N.-C. Xiao et al.

    is unrealistic because both types of uncertainties, epis-temic and aleotary uncertainties, often exist in engineeringpractice (Kiureqhian and Ditlevsen 2009; Kiureghian 2008;Huang and Zhang 2009; Huang and He 2008; Zhang et al.2010a, b; Zhang and Huang 2010). Epistemic uncertainty isderived from incomplete information, less sampling data orignorance while aleatory uncertainty comes from inherentvariations (Du 2008). Approaches to describe the epis-temic uncertainty in structural reliability analysis includethe interval analysis (Merlet 2009; Kokkolaras et al. 2006),evidence theory (Christophe and Philippe 2009), possibil-ity theory (Du et al. 2006; Nikolaidis et al. 2004; Zhouand Mourelatos 2008), imprecise probability (Aughenbaughand Herrmann 2009), fuzzy theory and P-boxes models(Tanrioven et al. 2004; Karanki et al. 2009). Aleatoryuncertainty is usually modeled using the probability the-ory. Data error is inevitable due to multiple contributionsfrom machine errors, human errors or other unexpected sit-uations. For example, a parameter is random subject to anormal distribution while the mean value and standard devi-ation can not be precisely determined. Therefore, in struc-tural reliability sensitivity analysis, a unified uncertaintyanalysis method is needed to model both epistemic and alea-tory uncertainties.

    Research efforts have been made these days on reliabilitysensitivity analysis when both epistemic and aleatory uncer-tainties are present in engineering systems (Guo and Du2007, 2009). In this paper, the parameters with sufficientinformation are modeled using probability distributionswhile others are modeled by a pair of upper and lower cumu-lative distributions (the so-called P-box). P-boxes (Utkinand Destercke 2009; Tucker and Ferson 2003) are one of thesimplest and the most popular models of sets of probabilitydistributions, directly extended from cumulative distribu-tions used in the precise case. Assume that the informationabout a random variable X is represented by a lower boundF and an upper bound F , the cumulative distribution func-tion can be defined based on the P-box bound [F, F]. Thus,the lower bound F and the upper bound F distributionsdefine a set (F, F) of precise distributions such that (Utkinand Destercke 2009; Tucker and Ferson 2003)

    (F, F) = {F |X Rv, F (X) F (X) F (X)


    where Rv denotes a set of real numbers.In the interval form, F I is used to denote the P-box

    [F, F], i.e.

    F I = [F, F] = {F F F} (2)

    -5 0 5 10 150











    Fig. 1 CDF of parameter X

    The P-box of a distribution is a closed-form function orfigure. For example, X N ([4, 6] , [1, 2]) represents thatX is normally distributed but its mean and standard devi-ation value is not known precisely. However, its meanvalue is known to locate in the interval [4, 6] and its stan-dard deviation is within the interval [1, 2]. The cumulativedistribution function (CDF) of parameter X is shown inFig. 1.

    In literature, there are studies on sensitivity analysisusing P-boxes (Ferson and Tucker 2006a, b; Hall 2006).However, the proposed sensitivity analysis methods areslightly different from reliability sensitivity analysis. Fur-thermore, Monte Carlo simulation (MCS) based methodswere used widely in the reliability sensitivity analysis, forexample, De-Lataliade et al. (2002) proposed a sensitivityestimation method based on MCS. Melchers and Ahammed(2004) proposed an efficient method for parameter sensi-tivity estimation based on MCS. However, these methodscan only model aleatory uncertainty rather than both epis-temic and aleatory uncertainties. In this paper, by inte-grating the principle of P-box, interval arithmetic, FORM,MCS, weighted regression, and the works in references(De-Lataliade et al. 2002; Melchers and Ahammed 2004),we propose a unified reliability sensitivity estimationmethod under both epistemic and aleatory uncertainties.

    This paper is organized as follows. Section 2 pro-vides a brief background about the structural reliabilityand FORM. Structural reliability analysis in the intervalform is presented in Section 3. Section 4 proposes amethod for system reliability sensitivity analysis in intervalform. Four numerical examples are presented in Section 5.Brief discussion and conclusion are presented to close thepaper.

  • Reliability sensitivity analysis for structural systems in interval probability form 693

    2 Reliability analysis by FORM

    In the system reliability analysis, the system reliability Rand probability of failure Pf are defined as (Melchers 1999)

    R = P [G (X) 0] (3)


    Pf = 1 R = P [G (X) < 0] (4)

    respectively, where P[] denotes a probability, G() is aperformance function, and X = (X1, X2, , Xn) is thevector of random variables.

    fx denotes the joint probability density function (PDF)of X, the probability of failure is calculated by the integral

    Pf = P [G (X) < 0] =


  • 694 N.-C. Xiao et al.

    a parameter, the parameter could be modeled using P-boxesas X Ij N ([x j ,x j ], [ x j ,x j ]). Then these i parameterscan be denoted by XIi =

    (X I1 , X

    I2 , X

    I3 , , X Ii

    ). The other

    (n i) parameters Xi = (Xi+1, Xi+2, Xi+3, , Xn) whichwe have sufficient information about are modeled with pre-cise probability distribution such as X j N

    (X j , X j


    The parameters of system can be expressed by

    XI =(

    XIi , Xi)=(

    X I1 , XI2 , , X Ii ; Xi+1, Xi+2, , Xn



    According to (11), the corresponding performance functioncan be expressed as G(XI ). Because both epistemic andaleatory uncertainties are present in a system, the proba-bility of failure is an interval rather than a precise value.From (5), the probability of failure P If in an interval form isgiven b


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