04_19_Reliability Sensitivity Interval Probability_SMO_2011.pdf

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

RESEARCH PAPER

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: [email protected]

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)

}(1)

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

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

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)

and

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] =

G(x)


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

)

(11)

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 by

P If = P[G(

XI)

< 0]

=

G(xI )


In the normal regression analysis, all m samples are equallyweighted. However, the sample point near the limit state ismore important than others. The coefficients of the functionare determined by the weighted linear regression expressedas (Kaymaz and Mcmahon 2005)

a =(

xTDwxD)1

xTDwG (18)

where w is an m m diagonal matrix of weights which isgiven by

w (x) =

w (x1) 0w (x2)

. . .

0 w (xm)

(19)

The following expression is suitable to obtain the weight foreach sample point (Kaymaz and Mcmahon 2005)

w(x j) = exp

[

G(x j) gworst

gworst

]

(20)

gworst = maxG(x j) , j = 1, 2, , m (21)

5. Use the linearized tangent function GL (X) to replacethe original function G(X) in the reliability analysis.

The limit-state function G(X) = 0 and its linearized func-tion GL (X) = 0 in the two dimensional space are shown inFig. 2.

(X) = 0G

(X) = 0LG

1x

2x

saferegion

failureregion

( )

Contours

of f X X

0

* * *1 2( , )X x x

Fig. 2 Limit-state function and its linearized function with two basicvariables

For an illustration purpose, we assume that all randomvariables are normally distributed and mutually independentin this paper. From the weighted regression analysis, thelinearized performance function GL (X) can be written inthe same form as (15).

According to (10) and (15), the reliability index isgiven by

= GLGL

(22)

where

GL = a0 +n

j=1a jX j (23)

and

GL =

n

j=1

(a jX j

)2 (24)

From the discussion above, (10), (11), (15), and (22), (23),(24), for the linearized limit-state function GL(X) = 0,when the random variables and P-boxes are present in thesystem. The system probability of failure and reliabilityindex derived from interval-valued probabilities becomes

P If =[P f , Pf

]=

( I

)(25)

where

I =[,

](26)

and

= GLGL=

a0 +i

j=1a jX j +

n

j=i+1a jX j

i

j=1(a jX j

)2 +n

j=i+1(a jX j

)2(27)

and

= GL GL

=a0 +

i

j=1a jX j +

n

j=i+1a jX j

i

j=1

(a j X j

)2 +n

j=i+1(a jX j

)2(28)

respectively. Here, for a parameter X j N ([X j ,X j ],[ X j ,X j ]), the midpoint values of intervals [X j ,X j ],[ X j ,X j ] are expressed as X j =

X j

+X j2 , and


X j = X j

+X j2 . The interval [X j ,X j ] of parameter X j

can be expressed by [X j X j , X j + X j ], whereX j =

X j X j2 . Likewise, the expression of the stan-

dard deviation could be obtained. Now, we use two variationcoefficients which are defined as

j =X j

X j, j =

X j

X j(29)

The variation vector of XI = (XIi , Xi ) = (X I1 , X I2 , , X Ii ;Xi+1, Xi+2, , Xn) can be expressed as = [1, 2, 3, , i , 0, 0, 0, , 0] (30)and

= [1, 2, 3, , i , 0, 0, , 0] (31)respectively. From (29), (30), (31), the mean and deviationof a parameter X j N ([X j ,X j ], [ X j ,X j ]) in intervalform become

IX j =[

X j, X j

]= [X j

(1 j

), X j

(1 + j

)](32)

and

IX j =[ X j , X j

]= [X j

(1 j

), X j

(1 + j

)](33)

respectively.From (27), (28) and (32), (33), the lower bound and

upper bound of reliability indexes can be expressed as

=

GL

GL=

a0 +i

j=1a j X j

(1 j

)+n

j=i+1a jX j

i

j=1[a j X j

(1 + j

)]2 +n

j=i+1(a jX j

)2

(34)

and

= GL GL

=a0 +

i

j=1a j X j

(1 + j

)+n

j=i+1a jX j

i

j=1[a j X j

(1 j

)]2 +n

j=i+1(a jX j

)2

(35)

respectively.

4 Reliability sensitivity analysis in interval form

Reliability sensitivity analysis is used to find the rate ofchange in the probability of failure due to the changes inthe parameters, such as means and standard deviations, ofdistributions (Du 2005). Furthermore, sensitivity analysis

could be used to provide information for the reliability-based design. As discussed in previous sections, we canlinearize a non-linear limit-state function by simulation.

The reliability sensitivity of a parameter X j is usuallydefined as (Guo and Du 2009; Du 2005; Melchers andAhammed 2004)(

PfX j

, PfX j

)

(36)

where Pf is the probability of failure; X j and X j arethe mean value and standard deviation of parameter X j ,respectively.

From (9), (15), and the chain rule of partial derivative,

the reliability sensitivities

( PfX j

, PfX j

)can be expressed

as

PfX j

= P [GL (X) < 0]X j

= ()X j

= ()

X j

(37)

and

PfX j

= P [GL (X) < 0]X j

= ()X j

= ()

X j

(38)

respectively.Since all random variables are normally distributed, the

partial derivatives of ()

can be calculated as

()

=(

12

e

12 x2dx)

= 1

2e

12

2(39)

The partial derivatives of X j

and X j

can be calcu-

lated as (Du 2005)

X j=

(

GLGL

)

X j= a j

GL(40)

and

X j=

(

GLGL

)

X j= a

2j GL X j

3GL

(41)

respectively. Based on (37), (38) and (40), (41), an approx-imate approach to calculate the system reliability sensi-tivities

PfX j

and PfX j

becomes (Melchers and Ahammed

2004)

PfX j

P [GL (X) < 0]X j

= ()

X j= a j

2GLexp

[

12

(GL

GL

)2]

(42)


Table 1 Distribution details of random variables

Variable Mean Standard Variation Distribution

deviation coefficient

X1 40 5 (1, 1) Normal

X2 50 2.5 (2, 2) Normal

X3 1,000 200 (3, 3) Normal

and PfX j

P [GL (X) < 0]X j

= ()

X j= a

2j GL X j

2 3GLexp

[

12

(GL

GL

)2]

(43)

respectively.

As discussed in Section 3, when both epistemic andaleatory uncertainties are considered, reliability sensitivityof X j is mathematically an interval with its lower and upperbounds. From (25), (42) and (43), and from the interval

operation, the lower and upper bounds of reliability sensitiv-ity of each parameter can be expressed as two optimizationproblems

P IfX j

=[

min

( P IfX j

)

, max

( P IfX j

)]

(44)

min (max)

( P IfX j

)

= min (max){

a j2GL

exp

[

12

(GL

GL

)2]}

s.t.X j

X j X j ( j = 1, 2, , i) X j X j X j ( j = 1, 2, , i)

GL GL GL

GL GL GL

(45)

and

P IfX j

=[

min

( P IfX j

)

, max

( P IfX j

)]

(46)

Table 2 Results of numericalexample 1 Sensitivity type Variation coefficients Results provided by the

proposed method

P IfX1

1, 2, 3 = 1, 2, 3 = 0.05 [2.45 103, 9.66 105]1, 2, 3 = 1, 2, 3 = 0.03 [1.46 103, 2.12 104]1, 2, 3 = 1, 2, 3 = 0.01 [8.18 104, 4.31 104]1, 2, 3 = 1, 2, 3 = 0.00 5.99 104 (6.02 104)

P IfX2

1, 2, 3 = 1, 2, 3 = 0.05 [1.40 103, 5.52 105]1, 2, 3 = 1, 2, 3 = 0.03 [8.33 104, 1.21 104]1, 2, 3 = 1, 2, 3 = 0.01 [4.68 104, 2.47 104]1, 2, 3 = 1, 2, 3 = 0.00 3.42 104 (3.64 104)

P IfX3

1, 2, 3 = 1, 2, 3 = 0.05 [2.01 106, 5.10 105]1, 2, 3 = 1, 2, 3 = 0.03 [4.41 106, 3.03 105]1, 2, 3 = 1, 2, 3 = 0.01 [8.97 106, 1.70 105]1, 2, 3 = 1, 2, 3 = 0.00 1.25 105 (1.23 105)

P IfX1

1, 2, 3 = 1, 2, 3 = 0.05 [1.67 104, 7.27 103]1, 2, 3 = 1, 2, 3 = 0.03 [4.11 104, 3.89 103]1, 2, 3 = 1, 2, 3 = 0.01 [9.31 104, 1.97 103]1, 2, 3 = 1, 2, 3 = 0.00 1.36 103 (1.35 103)

P IfX2

1, 2, 3 = 1, 2, 3 = 0.05 [2.73 105, 1.19 103]1, 2, 3 = 1, 2, 3 = 0.03 [6.71 105, 6.36 104]1, 2, 3 = 1, 2, 3 = 0.01 [1.52 104, 3.21 104]1, 2, 3 = 1, 2, 3 = 0.00 2.23 104 (2.76 104)

P IfX3

1, 2, 3 = 1, 2, 3 = 0.05 [2.89 106, 1.26 104]1, 2, 3 = 1, 2, 3 = 0.03 [7.10 106, 6.73 105]1, 2, 3 = 1, 2, 3 = 0.01 [1.61 105, 3.40 105]1, 2, 3 = 1, 2, 3 = 0.00 2.36 105 (2.32 105)


0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05-2.5

-2

-1.5

-1

-0.5

0

0.5x 10

-3

Upper sensitivities of X1

Lower sensitivities of X1

Upper sensitivities of X2Lower sensitivities of X2

Upper sensitivities of X3

Lower sensitivities of X3

Fig. 3 Parameter mean sensitivities with different coefficients

min (max)

( P IfX j

)

= min (max){

a2j GL X j2 3GL

exp

[

12

(GL

GL

)2]}

s.t.X j

X j X j ( j = 1, 2, , i) X j X j X j ( j = 1, 2, , i)

GL GL GL

GL GL GL

(47)

respectively. Since the two optimization problems are verycomplicated and time consuming, for the purpose of conve-nience and simplicity, (42), (43) and the interval operation

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

1

2

3

4

5

6

7

8x 10

-3

Upper deviation sensitivities of X1Lower deviation sensitivities of X1

Upper deviation sensitivities of X2

Lower deviation sensitivities of X2

Upper deviation sensitivities of X3Lower deviation sensitivities of X3

Fig. 4 Parameter deviation sensitivities with different coefficients

Wave Generator Flexspline Circular Spline

Fig. 5 Schematic of a harmonic drive

can be further used to calculate the approximate intervalbounds of parameter sensitivities as follows

P IfX j

{

a j2 GL

exp

[

12

(GL

GL

)2]

,

a j2GL

exp

12

(GL GL

)2

(48)

and

P IfX j

a2j GL X j

2 3GLexp

12

(GL GL

)2

,

a2j GLX j2 3GL

exp

[

12

(GL

GL

)2]}

(49)

5 Numerical examples

In this section, four examples are provided to demon-strate the application of the proposed method as well asits effectiveness. All parameters are expressed by P-boxesin Example 1. Two parameters are expressed by P-boxeswhile one is expressed by precise probability distributionin Example 2. Example 3 is used to demonstrate the accu-racy of the proposed method when the limit-state function


Variable Mean Standard Variation Distribution

deviation coefficient

Th(N.m) 350 35 (1, 1) Normal

Nv(rpm) 0.1 0.01 (2, 2) Normal

K 1.3 0.1 0 Normal

T (N.m) 2,000 0 0 . . . . . .


Table 4 Results of example 2Sensitivity type Variation coefficient Results provided by the

proposed method

P IfTh

1, 2 = 1, 2 = 0.05 [2.356 103, 1.585 103]1, 2 = 1, 2 = 0.03 [2.199 103, 1.727 103]1, 2 = 1, 2 = 0.01 [2.029 103, 1.875 103]1, 2 = 1, 2 = 0.00 1.950 103 (1.801 103)

P IfNv

1, 2 = 1, 2 = 0.05 [1.457, 2.165]1, 2 = 1, 2 = 0.03 [1.587, 2.016]1, 2 = 1, 2 = 0.01 [1.722, 1.864]1, 2 = 1, 2 = 0.00 1.791 (1.744)

P IfK

1, 2 = 1, 2 = 0.05 [0.322, 0.478]1, 2 = 1, 2 = 0.03 [0.350, 0.444]1, 2 = 1, 2 = 0.01 [0.380, 0.412]1, 2 = 1, 2 = 0.00 0.396 (0.386)

P IfTh

1, 2 = 1, 2 = 0.05 [2.066 103, 4.115 103]1, 2 = 1, 2 = 0.03 [2.385 103, 3.603 103]1, 2 = 1, 2 = 0.01 [2.744 103, 3.148 103]1, 2 = 1, 2 = 0.00 2.939 103 (2.740 103)

P IfNv

1, 2 = 1, 2 = 0.05 [0.498, 0.993]1, 2 = 1, 2 = 0.03 [0.575, 0.869]1, 2 = 1, 2 = 0.01 [0.662, 0.759]1, 2 = 1, 2 = 0.00 0.709 (0.669)

P IfK

1, 2 = 1, 2 = 0.05 [0.243, 0.484]1, 2 = 1, 2 = 0.03 [0.280, 0.424]1, 2 = 1, 2 = 0.01 [0.323, 0.370]1, 2 = 1, 2 = 0.00 0.346 (0.352)

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050.2

0.4

0.6

0.8

1

Low

er a

nd u

pper

bou

nds

of m

ean

sens

itivi

ties

Variation coefficients

1.2

1.4

1.6

1.8

2

2.2

Lower mean sensitivities of Nv

Upper mean sensitivities of NvLower mean sensitivities of K

Upper mean sensitivities of K

Fig. 6 Parameter mean sensitivities with different variationcoefficients

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05-2.4

-2.3

-2.2

-2.1

Low

er a

nd u

pper

bou

nds

of m

ean

sens

itivi

ties

Variation coefficients

-1.

-2

9

-1.8

-1.7

-1.6

-1.5

-1.4x 10

-3

Lower mean sensitivities of Th

Upper mean sensitivities of Th

Fig. 7 Parameters mean sensitivities with different variationcoefficients


0 0.005 0.01 0.015 0.02 0.025Variations coefficients

Low

er a

nd u

pper

bou

nds

of d

evia

tion

sens

itivi

ties

0.03 0.035 0.04 0.045 0.050.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Lower deviation sensitivities of Nv

Upper deviation sensitivities of NvLower deviation sensitivities of K

Upper deviation sensitivities of K

Fig. 8 Parameters deviation sensitivities with different variationcoefficients

is a highly non-linear function. In Example 4, the limit-statefunction is a black-box.

5.1 Example 1: a mathematical problem

Consider a non-linear limit state function with 3 normal ran-dom variables. The limit-state function is G (X) = X1 X2 X3 = 0. The distribution details of random variables aregiven in Table 1 (Melchers and Ahammed 2004).

106 samples are used and 1194 samples fall in theconstraint domain. With weighted regression analysis, theapproximating hyper-plane GL is

GL (X) = 1299.77 + 47.13X1 + 26.94X2 0.980X3

0 0.005 0.01 0.015 0.02 0.025Variation coefficients

0.03 0.035 0.04 0.045 0.052

2.5

3

Low

er a

nd u

pper

bou

nds

of d

evia

tion

sens

itivi

ties

3.5

4

x 10-3

Lower deviation sensitivities of Th

Upper deviation sensitivities of Th

Fig. 9 Parameters deviation sensitivities with different variationcoefficients

d

ft

wt

fb

L

a P

Fig. 10 A beam

The interval-valued reliability sensitivities based on the hy-per-plane GL under different variation coefficients (1, 1)are shown in Table 2, and the values in brackets () ofTable 2 are calculated by using the MCS-based reliabilitysensitivity method with the 106 samples.

From the results in Table 2, it can be concluded thatthe reliability sensitivity of each random variable is verysensitive to its distribution parameter. 1, 2, 3 = 1,2, 3 = 0.05. This means that the variation coefficientsof all parameters are equal, that is, 1 = 2 = 3 =1 = 2 = 3 = 0.05. When we have sufficient infor-mation about all parameters of the systems, 1, 2, 3 =1, 2, 3 = 0, that is, there is no epistemic uncertaintyin the system. The sensitivity analysis results are precisevalues rather than intervals. For example, the reliabilitysensitivity of the variable X1 is (5.99 104, 1.36 103). However, in reality, it is impossible to know theparameter probability distributions precisely. If the varia-tion coefficients are 1, 2, 3 = 1, 2, 3 = 0.05, thereliability sensitivity of variable X1 is within an interval([2.45103,9.66105], [1.67104, 7.27103]).It should be noted that the variation coefficients of parame-ters may not be all equal, such as, 1 =0.05, 2 =3 =0.03.The method to handle this case is the same as that usedfor equal coefficients. For the purpose of simplicity andillustration, we assume that all variation coefficients areequal to each other. The figures of system reliability sen-sitivities are shown in Figs. 3 and 4. From Figs. 3 and 4,we know that random variables are sensitive to its distribu-tion parameters and a small change to a parameter may leadto a large change to the reliability sensitivity results. The


Variable Mean Deviation Variation Distribution

coefficient type

P 6,070 200 (1, 1) Normal

L 120 6 (2, 2) Normal

a 72 6 (3, 4) Normal

S 170,000 4,760 (4, 4) Normal

d 2.3 1/24 (5, 5) Normal

tw 0.16 1/48 (6, 6) Normal

t f 0.26 1/48 (7, 7) Normal

b f 2.3 1/24 (8, 8) Normal


variable X1 is more sensitive than the other variables in thesystem. Therefore, in reliability-based design, we need topay more attention to X1 than to the other variables.

Furthermore, in this paper, for the purpose of simplic-ity, we give an accuracy comparison between the pro-posed method and the MCS-based method for the reliability

Table 6 Results of example 3Sensitivity type Variation coefficients The proposed method

P IfP

1, , 8 = 1, , 8 = 0.01 [1.11 104, 8.19 105]1, , 8 = 1, , 8 = 0.00 9.59 105 (1.01 104)

P Ifa

1, , 8 = 1, , 8 = 0.01 [8.35 103, 1.14 102]1, , 8 = 1, , 8 = 0.00 9.78 103 (8.94 103)

P IfL

1, , 8 = 1, , 8 = 0.01 [1.40 102, 1.03 102]1, , 8 = 1, , 8 = 0.00 1.20 102 (1.15 102)

P Ifd

1, , 8 = 1, , 8 = 0.01 [0.285, 0.387]1, , 8 = 1, , 8 = 0.00 0.333 (0.352)

P Ifb f

1, , 8 = 1, , 8 = 0.01 [0.190, 0.259]1, , 8 = 1, , 8 = 0.00 0.222 (0.253)

P Iftw

1, , 8 = 1, , 8 = 0.01 [0.156, 0.212]1, , 8 = 1, , 8 = 0.00 0.183 (0.213)

P Ift f

1, , 8 = 1, , 8 = 0.01 [1.171, 1.593]1, , 8 = 1, , 8 = 0.00 1.370 (1.445)

P IfS

1, , 8 = 1, , 8 = 0.01 [4.31 106, 5.84 106]1, , 8 = 1, , 8 = 0.00 5.05 106 (5.04 106)

P IfP

1, , 8 = 1, , 8 = 0.01 [2.39 105, 3.53 105]1, , 8 = 1, , 8 = 0.00 2.92 105 (4.54 105)

P Ifa

1, , 8 = 1, , 8 = 0.01 [7.45 103, 1.10 102]1, , 8 = 1, , 8 = 0.00 9.11 103 (8.81 103)

P IfL

1, , 8 = 1, , 8 = 0.01 [1.13 102, 1.66 102]1, , 8 = 1, , 8 = 0.00 1.38 102 (1.34 102)

P Ifd

1, , 8 = 1, , 8 = 0.01 [6.02 102, 8.88 102]1, , 8 = 1, , 8 = 0.00 7.35 102 (8.69 102)

P Ifb f

1, , 8 = 1, , 8 = 0.01 [2.69 102, 3.97 102]1, , 8 = 1, , 8 = 0.00 3.28 102 (4.72 102)

P Iftw

1, , 8 = 1, , 8 = 0.01 [9.05 103, 1.34 102]1, , 8 = 1, , 8 = 0.00 1.11 102 (1.92 102)

P Ift f

1, , 8 = 1, , 8 = 0.01 [0.508, 0.750]1, , 8 = 1, , 8 = 0.00 0.621 (0.796)

P IfS

1, , 8 = 1, , 8 = 0.01 [1.58 106, 2.33 106]1, , 8 = 1, , 8 = 0.00 1.93 106 (2.00 106)


sensitivity analysis with variation coefficients equal tozeros. From Table 2, we know that the results obtained usingthe proposed method is almost identical to the results usingthe MCS-based method.

5.2 Example 2: a harmonic drive

A harmonic drive, shown in Fig. 5, is widely used in thesolar array drive mechanism and the antenna pointing mech-anism because of its high carrying capacity, light weight,small size, and etc.

The performance function of a harmonic drive for its lifeestimation is (Du 2010)

G (Th, NV , T, K , m) = 75 105

NV

(Th

K T

)3 8760 m

where m is the number of years, and Th , NV , K , and T arethe rated output torque, input speed, condition factor andnominal output torque, respectively. 8760 = (365 24) isthe total number of hours for one year. When G > 0, sys-tem is considered safe. When G < 0, system falls in thefailure domain. The distribution details of random variablesare given in Table 3. In this example, we only consider thereliability of the harmonic drive for 10 years.

5 104 samples are used and 1,640 samples fall inthe constraint domain. By applying the weighted regressionanalysis, the approximating hyper-plane GL is

GL (Nv, K , Th) = 5.8172 104 6.7030 105 Nv+ 7.2950 102Th 1.4801 105 K

The interval-valued reliability sensitivities based on thehyper-plane GL under different variation coefficients (1,1) are listed in Table 4. The values in brackets () of Table 4are calculated using the MCS-based reliability sensitivitymethod with the 106 samples.

In Example 2, the distribution parameters of Th andNv have variation coefficients which are modeled usingP-boxes. The variation coefficient of the variable K isequal to zero which is modeled using a precise probabil-ity distribution. Both epistemic and aleatory uncertaintiesare considered in this example. From the results in Table 4,we can see that the larger the variation coefficients are,the wider the interval is. The reason is that a large varia-tion coefficient represents a large uncertainty of parametersinfluence on the system. When all the variation coefficientsare zero, the distributions of all random variables can be pre-cisely determined. The sensitivity analysis results of thesevariables become precise values. The figures of systemreliability sensitivities are shown in Figs. 6, 7, 8, 9. Fromthese figures, a conclusion is reached that the system is verysensitive to the variable Nv , which is a key design variableconsidered in the reliability-based design.

(Length)

(Load)

(Width)

30

50

Fig. 11 A wrench

5.3 Example 3: a beam

As shown in Fig. 10, a beam example is used to demonstratethe accuracy of the proposed method. The performancefunction is given by (Huang and Du 2006)

Z = f (P, L , a, S, d, b f , tw, t f) = max S

where

max = Pa (L a) d2L I

and

I = b f d3 (b f tw

) (d 2t f

)3

12

The distribution details of random variables are given inTable 5. We only consider a case of Z < 50,000 in theexample.

5 104 samples are used and 2350 samples fall in theconstraint domain. With the weighted regression analysis,the approximating hyper-plane ZL is

ZL 19.220P 1960.267a + 2409.748L 66824.231d 44658.404b f 36644.644tw 274666.167t f 1.012S + 275102.329

The interval-valued reliability sensitivities based on thehyper-plane ZL under different variation coefficients (1,1) are shown in Table 6.

In this example, the limit-state function is highly non-linear. From Table 6 we know that the results calculated

Table 7 Distributions details of random variables

Variable Mean Deviation Variation Distribution

coefficient

Load 500 20 (1, 1) Normal

Length 330 15 (2, 2) Normal

Width 30 3 (3, 3) Normal

s 320 0 0


Table 8 Results of example 4Sensitivity type Variation coefficients Results provided by the proposed method

P IfX1

1, 2, 3 = 1, 2, 3 = 0.01 [6.32 104, 6.42 104]1, 2, 3 = 1, 2, 3 = 0.00 6.37 103 (5.69 103)

P IfX2

1, 2, 3 = 1, 2, 3 = 0.01 [9.73 102, 9.88 104]1, 2, 3 = 1, 2, 3 = 0.00 9.81 102 (9.01 102)

P IfX3

1, 2, 3 = 1, 2, 3 = 0.01 [3.76 103, 3.82 103]1, 2, 3 = 1, 2, 3 = 0.00 3.79 103 (4.73 103)

P IfX1

1, 2, 3 = 1, 2, 3 = 0.01 [1.25 103, 1.32 103]1, 2, 3 = 1, 2, 3 = 0.00 1.28 103 (1.58 103)

P IfX2

1, 2, 3 = 1, 2, 3 = 0.01 [5.91 102, 6.24 102]1, 2, 3 = 1, 2, 3 = 0.00 6.08 102 (6.62 102)

P IfX3

1, 2, 3 = 1, 2, 3 = 0.01 [5.89 104, 6.22 104]1, 2, 3 = 1, 2, 3 = 0.00 6.06 104 (3.63 103)

using the proposed method are not very accurate when com-pared with the results using the MCS-based method. Thevalues in brackets () are calculated using the MCS-basedmethod with 106 samples with all the variation coefficientsequal to zeros. It is observed that when the limit-state func-tion is highly non-linear, the results calculated using theproposed method are not accurate results.

5.4 Example 4: a wrench

In an example of a wrench as shown in Fig. 11, the limit-state function is given by (Wang et al. 2006)

g (max, s, Load, Width, Length) = max s = 0where max and s are the maximum stress and the ratedstress. The distribution details of random variables are givenin Table 7.

103 samples are used and 222 samples fall in the con-straint domain. With the weighted regression analysis, theapproximating hyper-plane gL is

gL 207.577 1.041X1 + 16.030X2 0.620X3where X1, X2 and X3 are used to denote random variablesLength, Width and Load, respectively.

Applying the proposed method, the interval-valued relia-bility sensitivities based on the hyper-plane gL for differentvariation coefficients (i , i ) are given in Table 8.

In this example, the limit-state function is an implicitfunction or a black-box. In order to calculate max, thefinite element analysis (FEA) method is used. The FEA forwrench is shown in Fig. 12. Generally, the results calculatedusing the proposed method are not accurate, especially forlarge-scale real engineering problems. Because FEA is timecostly, the results calculated by the MCS-based method with1,000 samples are used as reference results for accuracycomparison.

6 Conclusions

Based on the P-boxes, interval algorithm, MCS, weightedlinear aggression analysis and FORM, a new sensitivityanalysis method is proposed. In the structural reliability andsensitivity analysis, it is practically appropriate to obtain aP-box interval constraint for system random variables ratherthan precise distributions because two types of uncertain-ties, epistemic and aleatory uncertainties, exist widely inengineering practices. The results of the four examples have

Fig. 12 FEA for wrench


shown that the proposed method is effective because it pro-vides a means of reliability sensitivity analysis under eitherepistemic uncertainty, aleatory uncertainty or both of them.Generally, it is more robust than the traditional sensitiv-ity method such as the FORM-based ones because it doesnot require the MPP search. Furthermore, the proposedmethod is superior to the MCS-based method in terms ofcomputational times. In addition, the proposed method isalso applicable for the situation where the limit-state func-tion is a black box. The numerical examples indicate that asmall change to a distribution parameter may lead to a largechange to the reliability sensitivity results.

It should be noted that there are limitations to the pro-posed method based on the interval form. The proposedmethod is an approximation method that is laid on a lin-earized function instead of its original limit-state function.The linearization may cause a loss of information. Theresult of the interval bounds in the paper is an approxima-tion rather than an exact solution. Generally, the proposedmethod is not available for large-scale real engineeringproblems with highly non-linear performance functions.The more nonlinearity of a limit-state function is, the largererrors will be. Future work involves an accuracy improve-ment especially when the system failure probability is verysmall and the computational time is huge.

Acknowledgments This research was partially supported by theNational Natural Science Foundation of China under the contract num-ber 51075061 and the Specialized Research Fund for the DoctoralProgram of Higher Education of China under the contract number20090185110019.

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Reliability sensitivity analysis for structural systems in interval probability formAbstractIntroductionReliability analysis by FORMStructural reliability in interval formReliability sensitivity analysis in interval formNumerical examplesExample 1: a mathematical problemExample 2: a harmonic driveExample 3: a beamExample 4: a wrenchConclusionsReferences

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