Research Article Time-Independent Reliability Analysis of ...

Research ArticleTime-Independent Reliability Analysis of BridgeSystem Based on Mixed Copula Models

Yuefei Liu1,2 and Xueping Fan1,2

1Key Laboratory of Mechanics on Disaster and Environment in Western China, Lanzhou University,The Ministry of Education of China, Lanzhou, Gansu 730000, China2School of Civil Engineering and Mechanics, Lanzhou University, Lanzhou 730000, China

Correspondence should be addressed to Yuefei Liu; [email protected] and Xueping Fan; [email protected]

Received 25 December 2015; Accepted 19 May 2016

Academic Editor: Egidijus R. Vaidogas

Copyright © 2016 Y. Liu and X. Fan. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The actual structural systems have many failure modes. Due to the same random sources owned by the performance functions ofthese failuremodes, there usually exist some nonlinear correlations between the various failuremodes. How to handle the nonlinearcorrelations is one of the main scientific problems in the field of structural system reliability. In this paper, for the two-componentsystems and multiple-component systems with multiple failure modes, the mixed copula models for time-independent reliabilityanalysis of series systems, parallel systems, series-parallel systems, and parallel-series systems are presented.These obtained mixedcopulamodels, considering the nonlinear correlation between failuremodes, are obtainedwith the chosen optimal copula functionswith the Bayesian selection criteria and Monte Carlo Sampling (MCS) method. And a numerical example is provided to illustratethe feasibility and application of the built mixed models for structural system reliability.

1. Introduction

Today’s structural systems are becoming more complex andmore sophisticated. Therefore, the evaluation of structuralsystem reliability is becoming harder. This means that thederivations based on classical assumptions are no longersatisfactory for the analysis of systems in terms of reliability.The evaluation of today’s real life systems needsmore detailedand complicated statistical analysis. Dependence between thecomponents is one of the intractable realistic assumptionsthat need to be carefully considered [1–4].

Especially for the actual bridge system, there exist manyfailure modes such as flexural failure and shearing failure.As the limit state functions (performance function) of thesefailure modes may have the same random sources, it is notmutually exclusive yet among failure modes [5–7]. Naturally,how to model the correlation among failure modes in aspectof the reliability analysis of structural systems is one ofthe most significant topics. The classic Pearson correlationcoefficient is mainly used for characterizing the correlationamong failure modes of structural system before; however, it

has a few disadvantages.The copulas, unlike the Pearson cor-relation coefficient only applied for describing linear correla-tion, offer a flexible tool for deriving nonlinear dependence,especially tail dependence among failure modes. The aim ofthis study is to introduce the copula function as a usefultool for modeling the dependence among failure modes ofbridge system. More recently, the copula theory has beenprimarily used in mechanical engineering [8] and hydraulicengineering [9] but little used in bridge engineering.

In this paper, firstly, several commonly used ellipticalcopulas andArchimedean copulas were introduced, and theirapplication in the correlation analysis was also describedin detail. And then based on the introduced copulas, withthe aid of copula Bayesian selection criteria [7, 10, 11], aflexible mixed copula model is constructed, by means oflinear weighted model, to model the nonlinear dependenceamong failure modes. For the two-component systems andmultiple-component systemswithmultiple failuremodes, theperformance function value of failure modes is chosen asthe copula functions’ analytical variable to construct mixedcopula model. With Monte Carlo Sampling method, the

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016, Article ID 2720614, 13 pageshttp://dx.doi.org/10.1155/2016/2720614

2 Mathematical Problems in Engineering

unknown parameters of copula functions can be approx-imately determined; therefore, the mixed copula function,considering the correlation between failure modes, is built.And then use the built mixed copula model to solve thestructural system’s reliability. Finally, a numerical example isprovided to illustrate the feasibility and application of thebuilt mixed copula models.

2. Mixed Copula Models

The mixed copula models are commonly built throughdifferent combinations of several copula functions, whichrespectively possess various characteristics in aspect of thedependence modeling. For the complex and ever-changingcorrelations among random variables, they are not adequateformodeling the dependence bymeans of only a single copulafunction. Therefore, it is necessary to build a flexible mixedcopulamodel to describe the complex dependence structures.Moreover, it is essential to select several appropriate copulafunctions from the existing copulas before building themixedcopula model. The Bayesian selection criteria are just chosenfor constructing the mixed copula model, as this method isindependent of the parameter estimation andmay be appliedto all known copula classes [7, 10].

2.1. Basic Theory about Copula Functions. Sklar’s theory[12] clearly indicated that, given random variables X =

(𝑋1, 𝑋2, . . . , 𝑋

𝑛)with continuousmarginal distribution func-

tions 𝐹𝑋1

, 𝐹𝑋2

, . . . , 𝐹𝑋𝑛

and 𝑛-dimensional joint distributionfunction 𝐹, there exists a unique copula function𝐶 such that,for all x = (𝑥

1, 𝑥2, . . . , 𝑥

𝑛) ∈ R𝑛,

𝐹 (𝑥1, 𝑥2, . . . , 𝑥

𝑛)

= 𝐶 (𝐹𝑋1

(𝑥1) , 𝐹𝑋2

(𝑥2) , . . . , 𝐹

𝑋𝑛

(𝑥𝑛)) .

(1)

According to (1), the joint probability density function ofthe random variableX = (𝑋

1, 𝑋2, . . . , 𝑋

𝑛) is given as follows:

𝑝 (𝑥1, 𝑥2, . . . , 𝑥

𝑛)

= 𝑐 (𝐹𝑋1

(𝑥1) , 𝐹𝑋2

(𝑥2) , . . . , 𝐹

𝑋𝑛

(𝑥𝑛))

𝑛

∏

𝑖=1

𝑝𝑋𝑖

(𝑥𝑖) ,

(2)

where 𝑝𝑋𝑖

(𝑥𝑖) is the marginal probability density function

(PDF) and 𝑐(u) is the joint PDF of the copula functiondenoted as follows:

𝑐 (𝑢1, . . . , 𝑢

𝑛) =

𝜕

𝑛𝐶

𝜕𝑢1⋅ ⋅ ⋅ 𝜕𝑢𝑛

(𝑢1, . . . , 𝑢

𝑛) . (3)

If𝐹𝑋𝑖

for 𝑖 = 1, 2, . . . , 𝑛 have generalized inverse functions𝐹

−1

𝑋𝑖

for 𝑖 = 1, 2, . . . , 𝑛, then, (1) can be rewritten as follows:

𝐹 (𝐹

−1

𝑋1

(𝑢1) , 𝐹

−1

𝑋2

(𝑢2) , . . . , 𝐹

−1

𝑋𝑛

(𝑢𝑛))

= 𝐶 (𝑢1, 𝑢2, . . . , 𝑢

𝑛) ,

(4)

where 𝑢𝑖= 𝐹𝑋𝑖

(𝑖 = 1, 2, . . . , 𝑛), (𝑢1, 𝑢2, . . . , 𝑢

𝑛) = u.

Equation (4) reveals how to construct the copula functionof a multivariate distribution with given marginal distribu-tions. It follows from the probability integral transform thatthe random variables 𝑈

𝑖≡ 𝐹𝑋𝑖

(𝑋𝑖), 𝑖 = 1, 2, . . . , 𝑛, are all

uniformly distributed on (0, 1). Conversely,𝑋𝑖= 𝐹

−1

𝑋𝑖

(𝑈𝑖), 𝑖 =

1, 2, . . . , 𝑛, U = (𝑈1, 𝑈2, . . . , 𝑈

𝑛); with this in mind, (4) is

expressed as

𝑃 (𝑋1≤ 𝑥1, 𝑋2≤ 𝑥2, . . . , 𝑋

𝑛≤ 𝑥𝑛) = 𝑃 (𝑋

1

≤ 𝐹

−1

𝑋1

(𝑢1) , 𝑋2≤ 𝐹

−1

𝑋2

(𝑢2) , . . . , 𝑋

𝑛≤ 𝐹

−1

𝑋𝑛

(𝑢𝑛))

= 𝑃 (𝑈1≤ 𝑢1, 𝑈2≤ 𝑢2, . . . , 𝑈

𝑛≤ 𝑢𝑛)

= 𝐶 (𝑢1, 𝑢2, . . . , 𝑢

𝑛) .

(5)

According to (1)–(5), the copula function organicallycombines each marginal distribution function with the mul-tivariate joint distribution functions; therefore, it not onlyconsiders the dependence among random variables but alsosimplifies the probability modeling process for multivariaterandom variables.

2.2. Bayesian Selection Criteria of Copula Functions. Huardet al. [10] proposed the Bayesian copula selection methodapplied to the bivariate copula functions. Bayesian copulaselection method was deduced by the Bayesian hypothesistesting to choose the best copula. However, we apply thismethod to choose not just the best one but also the bettercopulas as the candidate copulas for constructing the mixedcopula model.

For the Bayesian copula selection criteria, it is essentialto propose the optical copula function selection criteriadescribed by

𝐶opt = arg𝐶𝑙

max𝑃 (H𝑡| 𝐷, 𝐼) , (6)

where 𝐶opt is the optical copula function, H𝑡is denoted

as the hypotheses that the right copula is copula 𝐶𝑙(𝑙 =

1, 2, . . . , 𝐿), as the candidate copulas, 𝐶𝑙is denoted by 𝐶

𝑙=

𝐶𝑙(𝑢𝑖, V𝑖; 𝜃𝑖) (𝑖 = 1, 2, . . . , 𝑛), and 𝑛 pair of quantiles (𝑢

𝑖, V𝑖)

are supposed as 𝑛mutually independent pairs composing thedata set 𝐷. Besides, 𝐼 stands for some a priori information.According to Bayesian formula, it can be obtained that

𝑃 (H𝑡| 𝐷, 𝐼) =

𝑃 (𝐷 | H𝑡, 𝐼) 𝑃 (H

𝑡| 𝐼)

𝑃 (𝐷 | 𝐼)

, (7)

where 𝑃(𝐷 | H𝑡, 𝐼) is the likelihood, 𝑃(H

𝑡| 𝐼) is the a priori

copula family, and𝑃(𝐷 | 𝐼) is the normalization constant.Therank correlation coefficient Kendall’s 𝜏, denoted by 𝜏 = 𝑔

𝑙(𝜃)

to be the correlation parameters for all the selected copulas,is introduced in order to give the formula for 𝑃(H

𝑡| 𝐷, 𝐼):

𝑃 (H𝑡| 𝐷, 𝐼)

= ∫

𝜏∈Ω

𝑃 (𝐷 | H𝑡, 𝜏, 𝐼) 𝑃 (H

𝑡| 𝜏, 𝐼) 𝑃 (𝜏 | 𝐼)

𝑃 (𝐷 | 𝐼)

𝑑𝜏,

(8)

Mathematical Problems in Engineering 3

0 0.2 0.4 0.6 0.8 10

0.5

10

2

4

6

8

10

u

v

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1Gaussian PDF 𝜃 = −0.7581Gaussian PDF 𝜃 = −0.7581

Gau

ssia

nc(u,�)

Figure 1: PDF and contour plots of Gaussian copula function.

where 𝑃(𝐷 | H𝑡, 𝜏, 𝐼) is the likelihood with respect to 𝜏,

because of the 𝑛 mutually independent pairs, and is alsodescribed as the copula density:

𝑃 (𝐷 | H𝑡, 𝜏, 𝐼) =

𝑛

∏

𝑙=1

𝐶𝑙(𝑢𝑖, V𝑖; 𝑔

−1

𝑙(𝜏)) . (9)

Substituting (9) into (8), we obtain

𝑃 (H𝑡| 𝐷, 𝐼) =

1

𝑃 (𝐷 | 𝐼)

∫

𝜏∈Ω

𝑛

∏

𝑙=1


−1

𝑙(𝜏))

⋅ 𝑃 (H𝑡| 𝜏, 𝐼) 𝑃 (𝜏 | 𝐼) 𝑑𝜏,

(10)

where 𝑃(𝐷 | 𝐼) is the normalization constant and 𝑃(H𝑡|

𝜏, 𝐼)𝑃(𝜏 | 𝐼) = 1/𝜆(Λ), where 𝜆(Λ) is denoted as the Lebesguemeasure of Λ = [−1, 1] in this paper; therefore, (10) can bealso expressed as follows:

𝑃 (H𝑡| 𝐷, 𝐼) ∝

1

𝜆 (Λ)

∫

Ω𝑙∩Λ

𝑛

∏

𝑙=1


−1

𝑙(𝜏)) 𝑑𝜏. (11)

Obviously, the optimal copula function selection criteriacan be described as the computation of the highest weight𝑊

𝑙:

𝑊𝑙=

1

𝜆 (Λ)

∫

Ω𝑙∩Λ

𝑛

∏

𝑖=1

𝑐𝑙(𝑢𝑖, V𝑖; 𝑔

−1

𝑙(𝜏)) 𝑑𝜏. (12)

In especial, the “right” copulas include not just the bestcopula with the highest weight but also the better copula withhigher weight for the mixed copula modeling in this paper.

Two particular classes of copulas that proved to be usefulin dependence modeling are the elliptical and Archimedeanclasses [7, 13]. The four copulas (one of the elliptical copulas,Gaussian copula, and three of the Archimedean copulas,Gumbel copula, Clayton copula, and Frank copula) areemployed as the candidates of the Bayesian copula selectioncriteria. Finally, the chosen ones are used for constructing themixed copula model. Among the chosen copulas, Gaussiancopulas and Frank copulas are able to capture symmetric

dependence structures among random variables. Differentfrom them, Gumbel copulas and Clayton copulas exhibitasymmetric dependence and Gumbel copulas are especiallyemployed for describing upper tail dependence structures,while Clayton copulas are employed for that of the lower tail.And, then, the probability density function graphs and theircontour plots are depicted as in Figures 1–4.

2.3. Modeling Method of Mixed Copula Models. There existcomplex nonlinear correlations among failure modes; there-fore, just one copula is not enough to characterize depen-dence structures among failure modes. With the aid ofa weighted combination of the selected copula functions,a mixed copula is approximately constructed in order todescribe the complex dependence structures among failuremodes [7, 14]; namely,

𝐶 (u, k; a, 𝜃) =𝑁

∑

𝑖=1

𝑎𝑖𝐶𝑖(u, k; 𝜃

𝑖) (𝑁 ≤ 4) , (13)

where 𝐶𝑖(𝑖 = 1, 2, . . . , 𝑁) are, respectively, denoted as one

of the chosen copulas among the four copulas and a =

(𝑎1, 𝑎2, . . . , 𝑎

𝑁) is the weighting coefficient vector along with

the inner parameter vector 𝜃 = (𝜃1, 𝜃2, . . . , 𝜃

𝑁) of the

corresponding copulas, and u, ^ are variables of the chosencopula functions, and what is more, 𝑎

1+ 𝑎2+ ⋅ ⋅ ⋅ + 𝑎

𝑁= 1.

The copula function and Monte Carlo Simulation (MCS)method are contributed to the mixed copula model analysis.The specific steps can be stated as follows.

2.3.1. Monte Carlo Sampling (MCS). According to the dis-tribution types of random variables for the performancefunctions of failure modes, the samples for the randomvariables can be obtained with MCS method. After sub-stituting the samples into the corresponding performancefunctions, we can get the random sequence {𝑔

𝑗}𝑖, 𝑖 =

1, 2, . . . , 𝑛, 𝑗 = 1, 2, . . . , 𝑁, for each of the performancefunctions and obtain the sequences {𝐹

𝑗}𝑖= 𝐹({𝑔

𝑗}𝑖), 𝑖 =

1, 2, . . . , 𝑛, 𝑗 = 1, 2, . . . , 𝑁, of the corresponding empirical


0 0.2 0.4 0.6 0.8 10

0.5

10

5

10

15

20

25

u

v

Gum

bel

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1Gumbel PDF 𝜃 = 2.5Gumbel PDF 𝜃 = 2.5

c(u,�)

Figure 2: PDF and contour plots of Gumbel copula function.

0 0.2 0.4 0.6 0.8 10

0.5

10

5

10

15

20

25

u

v

Clay

ton

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1Clayton PDF 𝜃 = 1.6352 Clayton PDF 𝜃 = 1.6352

c(u,�)

Figure 3: PDF and contour plots of Clayton copula function.

0 0.2 0.4 0.6 0.8 100.5

10

1

2

3

4

5

6

u

v

Fran

k

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1Frank PDF 𝜃 = −6.6800 Frank PDF 𝜃 = −6.6800

c(u,�)

Figure 4: PDF and contour plots of Frank copula function.


Table 1: Basic cases of the parameters of copula functions.

Copula 𝜏 = 𝑔(𝜃) 𝜏 ∈ Ω 𝜃 ∈ Ω

Gaussian 2

𝜋

arcsin 𝜃 [−1, 1] [−1, 1]

Gumbel 1 − 𝜃

−1[0, 1] [1, +∞)

Clayton 1 −

2

2 + 𝜃

(0, 1] (0, +∞)

Frank 1 −

4

𝜃

(1 −

1

𝜃

∫

𝜃

0

𝑡

𝑒

𝑡− 1

𝑑𝑡) [−1, 0) ∪ (0, 1] (−∞, 0) ∪ (0, +∞)

distribution function by mean of Matlab software, where 𝑔 isthe performance function and F is the empirical distributionfunction.

2.3.2. Copula Function Selection. With the scatter plot ofsamples, we can judge approximate distribution features ofthe samples. Furthermore, it is extremely vital to determinewhich classes of copula functions to choose as the candidatecopulas with Bayesian copula selection criteria. In this paper,the copula classes we chose are Gaussian copulas, Gumbel

copulas, and Clayton copulas as well as Frank copulas, whoseparameter domain along with the definition and the domainof Kendall’s 𝜏 are listed in Table 1. The next is to build themixed copula function through the selected copulas and (13).

2.3.3. Parameter Estimation for the Mixed Copula. Applyingthe least residual error quadratic summethodOLS, one of thecurve fitting criteria, we can obtain parameters of the mixedcopula function. And the formula of the OLS is representedas follows:

𝐹OLS = √

1

𝑛

𝑛

∑

𝑖=1

(𝐹 ((𝐹1)

𝑖, (𝐹2)

𝑖, . . . , (𝐹

𝑁)

𝑖) − 𝐶 ((𝐹

1)

𝑖, (𝐹2)

𝑖, . . . , (𝐹

𝑁)

𝑖))

2

, (14)

where 𝐹(⋅) is the joint empirical distribution function of theempirical distribution function sequences for each perfor-mance function and 𝐶((𝐹

1)𝑖, (𝐹2)𝑖, . . . , (𝐹

𝑁)𝑖) is the mixed

copula function value. The unknown parameter value for themixed copula function can be obtained by the optimizationcomputation with the rule of OLS, while the optimizedparameters must be determined to make sure that the valueof 𝐹OLS is the minimum.

2.3.4. Goodness-of-Fit Estimation for the Mixed Copula.ThroughMCSmethod, the values of joint empirical distribu-tion functions and the corresponding mixed copula functionare obtained, and, then, the scatter plot between them isdrawn. Finally, the goodness of fit for the mixed copula canbe determined through the scatter plot.

3. Mixed Copula Model Expressionsabout Joint Failure Probabilityof Structural System

3.1. Mixed Copula Model Expression about Joint FailureProbability of Two-Component Series System. For the two-component series systemwhich is shown in Figure 5, supposethat the performance function of the component failuremodeis

𝑔𝑖(X) = 𝑔

𝑖(X1, 𝑋2, . . . , 𝑋

𝑛) , 𝑖 = 1, 2. (15)

With (5), as a check, the probability that both failuremodes occur is denoted as𝑃 (𝑔1(X) ≤ 0, 𝑔

2(X) ≤ 0)

= 𝑃 (𝐹𝑔1

(𝑔1(X)) ≤ 𝐹

𝑔1(0) , 𝐹

𝑔2

(𝑔2(X)) ≤ 𝐹

𝑔2(0))

= 𝑃 (𝑈1≤ 𝐹𝑔1(0) , 𝑈

2≤ 𝐹𝑔2(0))

= 𝐶 (𝐹𝑔1(0) , 𝐹

𝑔2(0)) = 𝐶 (𝑝

𝑓𝑔1

, 𝑝𝑓𝑔2

) .

(16)

𝐹g(0, 0) = 𝐶(𝐹𝑔1

(0), 𝐹𝑔2

(0)) is stated in the fundamentaltheorem of Sklar. Therefore, the failure probability of two-component series system (at least one failuremode of the twocomponents occurred) can be solved as

𝑝𝑓= 𝑃 (𝑔

1(X) ≤ 0 ∪ 𝑔

2(X) ≤ 0)

= 𝑃 (𝑔1(X) ≤ 0) + 𝑃 (𝑔

2(X) ≤ 0)

− 𝑃 (𝑔1(X) ≤ 0, 𝑔

2(X) ≤ 0)

= 𝑝𝑓𝑔1

+ 𝑝𝑓𝑔2

− 𝐶(𝑝𝑓𝑔1

, 𝑝𝑓𝑔2

) ,

(17)

where 𝑝𝑓𝑔1

, 𝑝𝑓𝑔2

, respectively, denote the failure probabilityof the two failure modes and 𝐶(⋅) is two-component seriessystem copula function.

3.2. Mixed Copula Model Expression about Joint FailureProbability of Two-Component Parallel System. For the two-component parallel system which is shown in Figure 6,


1 2

Figure 5: Two-component series system.

1 2

Figure 6: Two-component parallel system.

suppose that the performance function of the componentfailure mode is

𝑔𝑖(X) = 𝑔

𝑖(𝑋1, 𝑋2, . . . , 𝑋

𝑛) , 𝑖 = 1, 2. (18)

With (5) and (16), the failure probability of two-component parallel system (two failure modes of the twocomponents meantime occurred) can be obtained:


1(X) ≤ 0 ∩ 𝑔

2(X) ≤ 0) = 𝐶 (𝑝

𝑓𝑔1

, 𝑝𝑓𝑔2

) , (19)

where 𝑝𝑓𝑔1

, 𝑝𝑓𝑔2

, respectively, denote the failure probabilityof the two failure modes and 𝐶(⋅) is two-component parallelsystem copula function.

3.3.MixedCopulaModel Expression about Joint Failure Proba-bility of Multiple-Component Series System. For the multiple-component series systemwhich is shown in Figure 7, supposethat the performance function of the component failuremodeis

𝑔𝑖(X) = 𝑔

𝑖(𝑋1, 𝑋2, . . . , 𝑋

𝑛) , 𝑖 = 1, 2, . . . , 𝑁. (20)

With (5), as a check, the probability that all the failuremodes meantime occur is denoted as

𝑃 (𝑔1(X) ≤ 0, 𝑔

2(X) ≤ 0, . . . , 𝑔

𝑁(X) ≤ 0)

= 𝑃 (𝐹𝑔1

(𝑔1(X)) ≤ 𝐹

𝑔1(0) , 𝐹

𝑔2

(𝑔2(X))

≤ 𝐹𝑔2(0) , . . . , 𝐹

𝑔𝑁

(𝑔𝑁(X)) ≤ 𝐹

𝑔𝑁(0)) = 𝑃 (𝑈

1

≤ 𝐹𝑔1(0) , 𝑈

2≤ 𝐹𝑔2(0) , . . . , 𝑈

𝑁≤ 𝐹𝑔𝑁(0))

= 𝐶 (𝐹𝑔1(0) , 𝐹

𝑔2(0) , . . . , 𝐹

𝑔𝑁(0))

= 𝐶 (𝑝𝑓𝑔1

, 𝑝𝑓𝑔2

, . . . , 𝑝𝑓𝑔𝑁

) .

(21)

𝐹g(0, 0, . . . , 0) = 𝐶(𝐹𝑔1

(0), 𝐹𝑔2

(0), . . . , 𝐹𝑔𝑁

(0)) is statedin the fundamental theorem of Sklar. Therefore, the failureprobability of multiple-component series system (at least one

1 2 i N· · ·· · ·

Figure 7: Multiple-component series system.

failure mode of the multiple components occurred) can besolved as𝑝𝑓= 𝑃 (𝑔

1(X) ≤ 0 ∪ 𝑔

2(X) ≤ 0 ∪ ⋅ ⋅ ⋅ ∪ 𝑔

𝑁(X) ≤ 0)

=

𝑁

∑

𝑖=1

𝑃 (𝑔𝑖(X) ≤ 0) − ∑

𝑖1<𝑖2

𝑃 (𝑔𝑖1(X) ≤ 0, 𝑔

𝑖2(X)

≤ 0) + ⋅ ⋅ ⋅ + (−1)

𝑛+1∑

𝑖1<𝑖2<⋅⋅⋅<𝑖𝑛

𝑃 (𝑔𝑖1(X) ≤ 0, 𝑔

𝑖2(X)

≤ 0, . . . , 𝑔𝑖𝑛(X) ≤ 0) + ⋅ ⋅ ⋅ + (−1)

𝑁+1𝑃 (𝑔1(X) ≤ 0,

𝑔2(X) ≤ 0, . . . , 𝑔

𝑁(X) ≤ 0) =

𝑛

∑

𝑖=1

𝑝𝑓𝑔𝑖

− ∑

𝑖1<𝑖2

𝐶(𝑝𝑓𝑔𝑖1

, 𝑝𝑓𝑔𝑖2

) + ⋅ ⋅ ⋅ + (−1)

𝑛+1

⋅ ∑

𝑖1<i2<⋅⋅⋅<𝑖𝑛

𝐶(𝑝𝑓𝑔𝑖1

, 𝑝𝑓𝑔𝑖2

, . . . , 𝑝𝑓𝑔𝑖𝑛

) + ⋅ ⋅ ⋅

+ (−1)

𝑁+1𝐶(𝑝𝑓𝑔1

, 𝑝𝑓𝑔2

, . . . , 𝑝𝑓𝑔𝑁

) ,

(22)

where 𝑝𝑓𝑔1

, 𝑝𝑓𝑔2

, . . . , 𝑝𝑓𝑔𝑁

, respectively, denote the failureprobability of the multiple failure modes and𝐶(⋅) is multiple-component series system copula function.

3.4. Mixed Copula Model Expression about Joint FailureProbability of Multiple-Component Parallel System. For themultiple-component parallel system which is shown inFigure 8, suppose that the performance function of thecomponent failure mode is

𝑔𝑖(X) = 𝑔

𝑖(𝑋1, 𝑋2, . . . , 𝑋

𝑛) , 𝑖 = 1, 2, . . . , 𝑁. (23)

With (5), the failure probability of two-component par-allel system (all the failure modes of all the componentsmeantime occurred) can be obtained:


1(X) ≤ 0, 𝑔

2(X) ≤ 0, . . . , 𝑔

𝑁(X) ≤ 0)

= 𝑃 (𝐹𝑔1

(𝑔1(X)) ≤ 𝐹

𝑔1(0) , 𝐹

𝑔2

(𝑔2(X))

≤ 𝐹𝑔2(0) , . . . , 𝐹

𝑔𝑁

(𝑔𝑁(X)) ≤ 𝐹

𝑔𝑁(0)) = 𝑃 (𝑈

1

≤ 𝐹𝑔1(0) , 𝑈

2≤ 𝐹𝑔2(0) , . . . , 𝑈

𝑁≤ 𝐹𝑔𝑁(0))

= 𝐶 (𝐹𝑔1(0) , 𝐹

𝑔2(0) , . . . , 𝐹

𝑔𝑁(0))

= 𝐶 (𝑝𝑓𝑔1

, 𝑝𝑓𝑔2

, . . . , 𝑝𝑓𝑔𝑁

) ,

(24)

where 𝑝𝑓𝑔1

, 𝑝𝑓𝑔2

, . . . , 𝑝𝑓𝑔𝑁

, respectively, denote the failureprobability of the 𝑁 failure modes and 𝐶(⋅) is multiple-component parallel system copula function.


1 2 i N· · ·· · ·

Figure 8: Multiple-component parallel system.

1 2 N

· · ·

Figure 9: Series-parallel system.

3.5. Mixed Copula Model Expression about Joint FailureProbability of Series-Parallel System. For series-parallel sys-tem shown in Figure 9, in this paper, only the correlationbetween internal components of each subparallel system isconsidered,while the correlation between subparallel systemsis not considered and considered to bemutually independent.Therefore, with (24), all the copula models of all the subpar-allel systems’ failure probability can be obtained, and then,with (25), the failure probability of the series-parallel systemcan be solved:

𝑝𝑓s-p

= 𝑃(

𝑁

⋃

𝑖=1

𝐹𝑖) = 𝑃(

𝑁

⋃

𝑖=1

𝑁𝑖

⋂

𝑗

𝐹𝑖𝑗)

=

𝑁

∑

𝑖=1

𝑃 (𝐹𝑖) − ∑

𝑖1<𝑖2

{𝑃 (𝐹𝑖1

) 𝑃 (𝐹𝑖2

)} + ⋅ ⋅ ⋅

+ (−1)

𝑛+1∑

𝑖1<𝑖2<⋅⋅⋅<𝑖𝑛

{

{

{

𝑖𝑛

∏

𝑘=𝑖1

𝑃 (𝐹𝑘)

}

}

}

+ ⋅ ⋅ ⋅

+ (−1)

𝑁+1

𝑁

∏

𝑖=1

𝑃 (𝐹𝑖) ,

(25)

where 𝑁 is the total number of the subparallel systems; 𝑁𝑖

is the total number of the components in the 𝑖th subparallelsystem; 𝐹

𝑖= ⋂

𝑁𝑖

𝑗=1𝐹𝑖𝑗is the failure probability of the 𝑖th

subparallel system;𝐹𝑖𝑗= (𝑔𝑖𝑗(X) ≤ 0) is the failure probability

of the 𝑗th component in the 𝑖th subparallel system; 𝑃(𝐹𝑖), 𝑖 =

1, 2, . . . , 𝑁, can be solved with (24), which are the failureprobability of the subparallel system considering correlationbetween internal components of each subparallel system.

3.6. Mixed Copula Model Expression about Joint Failure Prob-ability of Parallel-Series System. For parallel-series systemshown in Figure 10, in this paper, only the correlationbetween internal components of each subseries system isconsidered, while the correlation between subseries systemsis not considered and considered to bemutually independent.

1

2

N

......

......

Figure 10: Parallel-series system.

Therefore, with (22), all the copula models of all the subseriessystems’ failure probability can be obtained, and then, with(26), the failure probability of the parallel-series system canbe solved as

𝑝𝑓p-s

= 𝑃(

𝑁

⋂

𝑖=1

𝐹𝑖) =

𝑁

∏

𝑖=1

𝑃 (𝐹𝑖) =

𝑁

∏

𝑖=1

𝑃(

𝑁𝑖

⋃

𝑗=1

𝐹𝑖𝑗) , (26)

where𝑁 is the total number of the subseries systems;𝑁𝑖is the

total number of the components in the 𝑖th subseries system;𝐹𝑖= ⋂

𝑁𝑖

𝑗=1𝐹𝑖𝑗is the failure probability of the 𝑖th subseries

system; 𝐹𝑖𝑗= (𝑔𝑖𝑗(X) ≤ 0) is the failure probability of the 𝑗th

component in the 𝑖th subseries system; 𝑃(𝐹𝑖), 𝑖 = 1, 2, . . . , 𝑁,

can be solved with (22), which are the failure probability ofthe subseries system considering correlation between internalcomponents of each subseries system.

In this paper, firstly with First-Order Reliability Method(FORM), the structural reliability index and the correspond-ing failure probability of each failure mode can be solved,and then with the constructed mix copula functions, suchas (17), (19), (22), (24), (25), and (26), the failure probabilityof structural system considering correlation between failuremodes can be obtained.

4. Numerical Example: SystemReliability Analysis of Simply SupportedCored Slab Bridge

For the simply supported cored slab bridge shown in Fig-ure 11, the total span is 13m, the computed span is 12.6m,the clear width of bridge deck is 7m, the width of footwayon both sides of bridge deck is 1m, and the whole bridgeis composed of nine concrete cored slabs [15]. The designreference period of this bridge is 100 years. And this bridgehas been served for 32 years. At the 32nd year, the resistanceof each girder follows normal distribution; the distributionparameters are, respectively, 796.04 kN⋅m (mean value) and91.783 kN⋅m (standard deviation).

Based on Figure 11 and reference [16], the failure criterionof bridge system is as follows: if any two adjacent girders bothfailed, then the whole bridge system failed. According to thefailure criterion, the bridge system is a series-parallel system,which is shown in Figure 12.

The performance function of each girder’s failure mode is

𝑍𝑖= 𝑔 (𝑋

𝑖1, 𝑋𝑖2, 𝑋𝑖3) = 𝑔 (𝑅

𝑖, 𝐺𝑖, 𝑄𝑖) = 𝑅𝑖− 𝐺𝑖− 𝑄𝑖,

𝑖 = 1, 2, . . . , 9,

(27)


Table 2: Distribution parameters of beams’ dead load effects in 32nd year.

Girder number 1 2 3 4 5 6 7 8 9Mean value/kN⋅m 212.86 221.79 223.91 230.74 260.69 230.74 223.91 221.79 212.86Standard deviation/kN⋅m 9.68 9.68 9.68 9.68 9.68 9.68 9.68 9.68 9.68

Table 3: Distribution parameters of beams’ maximum live load effects in 32nd year.

Girder number 1 2 3 4 5 6 7 8 9Mean value/kN⋅m 85.42 96.42 116.8 144.4 180.9 144.4 116.8 96.42 85.42Standard deviation/kN⋅m 12.71 14.35 17.38 21.49 26.92 21.49 17.38 14.35 12.71

1# 2# 3# 4# 5# 6# 7# 8# 9#

900

Figure 11: Cross section of bridge (unit: cm).

1#

2# 3# 4# 5#

2# 3# 4# 8#

9#

· · ·

Figure 12: Series-parallel system.

where 𝑅𝑖is the resistance of the 𝑖th girder, 𝐺

𝑖is the dead load

effect of the 𝑖th girder, and 𝑄𝑖is the live load effect of the 𝑖th

girder. At the 32nd year, the distribution parameters aboutdead load effects are listed in Table 2, where the standarddeviation is not changed, because, for load effect, the variationof variables is very small. And the distribution parametersabout vehicle load effects are listed in Table 3, which occurdue to symmetrical variable load and is not applicated to theunsymmetrical variable load.

Based on the reliability analysis method of two-component parallel system considering the correlationbetween failures modes described in Section 3.2, thereliability analysis processes of bridge system are as follows.

Because the bridge system is symmetrical, the subpar-allel systems (1#-2#, 2#-3#, 3#-4#, and 4#-5#) are used toanalyze the reliability and failure probability of the bridgesystem.

Based on Tables 2 and 3 and (25), the scatter plot betweentwo random sampling sequences of the corresponding limitstate functions for the two failure modes of each subparallelsystem (1#-2#, 2#-3#, 3#-4#, and 4#-5#) can be obtained.Then, according to the characteristics of the obtained scatterplots, the candidate copula functions, which can approxi-mately describe distribution features of the samples, can beselected, and then with Bayesian selection criteria describedin Section 2.2, the suitable copula functions, from the selectedcandidate copula functions, can be obtained which can beused to build the mixed copula model. The parameters ofthe built mixed copula function for each subparallel systemare listed in Table 4. Finally, the mixed copula modes, for

Table 4: Parameters of each of the mixed copulas for four subsys-tems.

𝑎 𝑏 1 − 𝑎 − 𝑏 𝜌 𝛼 𝜃

1#-2# 0.7438 0.1958 0.0604 −0.1868 2.4872 4.42782#-3# 0.8521 0.1479 0 0.1014 2.4207 4.40783#-4# 0.9381 0.0619 0 0.1502 2.4611 4.39844#-5# 0.7690 0.2310 0 −0.3490 2.4814 4.4192

each subparallel system, are built. Further, PDF plots andcontour plots for themixed copulas, the scatter plots betweenempirical distributions and mixed copula functions, andscatter plots for two limit state functions are presented,respectively, in Figures 13–16.

With FORM, the corresponding reliability indices to eachgirder’s failure mode are, respectively,

𝛽1= 5.3429,

𝛽2= 5.1159,

𝛽3= 4.8484,

𝛽4= 4.4417,

𝛽5= 3.6869.

(28)

Then, with the equation 𝑝𝑓= Φ(−𝛽), the corresponding

failure probability to each girder’s failure mode is, respec-tively,

𝑝𝑓𝑔1

= 4.57 × 10

−8,

𝑝𝑓𝑔2

= 1.56 × 10

−7,

𝑝𝑓𝑔3

= 6.22 × 10

−7,

𝑝𝑓𝑔4

= 4.46 × 10

−6,

𝑝𝑓𝑔5

= 1.14 × 10

−4.

(29)


0

0.5

1

0

0.5

10

5

10

15

Mix

ed co

pula

PD

F

F2F1

(a) PDF plot for the mixed copula

Mixed copula PDF contour

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

(b) Contour plot

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Mix

ed co

pula

func

tion

Joint empirical distribution function F12

(c) Empirical distribution versus mixed copula

200 400 600 800 1000200

300

400

500

600

700

800

Limit state function g1

Lim

it st

ate f

unct

iong2

(d) Scatter plot for two limit state functions

Figure 13: The subsystem 1#-2#.

With (19) and Table 4, the following can be obtained:

𝑝𝑓12

= 𝐶 (𝑝𝑓1

, 𝑝𝑓2

) = 8.61 × 10

−11,

𝑝𝑓23

= 𝐶 (𝑝𝑓2

, 𝑝𝑓3

) = 3.12 × 10

−10,

𝑝𝑓34

= 𝐶 (𝑝𝑓3

, 𝑝𝑓4

) = 1.32 × 10

−9,

𝑝𝑓45

= 𝐶 (𝑝𝑓4

, 𝑝𝑓5

) = 1.31 × 10

−7,

(30)

where 𝑝𝑓12

is the failure probability when girder 1# andgirder 2# meantime failed, 𝑝

𝑓23

is the failure probability whengirder 2# and girder 3# meantime failed, 𝑝

𝑓34

is the failureprobability when girder 3# and girder 4# meantime failed,and 𝑝

𝑓45

is the failure probability when girder 4# and girder5# meantime failed.

Suppose that failure modes between subsystems aremutually independent, and then the failure probability ofstructural system is approximately

𝑝𝑓= (−1)

2

8

∑

𝑗=𝑖+1,𝑖=1

𝑝𝑓𝑖𝑗

+ (−1)

3

7

∑

𝑡=𝑘+1,𝑗=𝑖+1,𝑘>𝑖,𝑖=1

𝑝𝑓𝑖𝑗

𝑝𝑓𝑘𝑡

+ ⋅ ⋅ ⋅ + (−1)

9(𝑝𝑓12

𝑝𝑓23

𝑝𝑓34

𝑝𝑓45

𝑝𝑓56

𝑝𝑓67

𝑝𝑓78

𝑝𝑓89

)

≈

8

∑

𝑗=𝑖+1,𝑖=1

𝑝𝑓𝑖𝑗

−

7

∑

𝑡=𝑘+1,𝑗=𝑖+1,𝑘>𝑖,𝑖=1

𝑝𝑓𝑖𝑗

𝑝𝑓𝑘𝑡

= 2 (𝑝𝑓12

+ 𝑝𝑓23

+ 𝑝𝑓34

+ 𝑝𝑓45

) − 4 (𝑝𝑓12

𝑝𝑓23

+ 𝑝𝑓12

𝑝𝑓34

+ 𝑝𝑓12

𝑝𝑓45

+ 𝑝𝑓23

𝑝𝑓34

+ 𝑝𝑓23

𝑝𝑓45

+ 𝑝𝑓34

𝑝𝑓45

) − 𝑝

2

𝑓12

− 𝑝

2

𝑓23

− 𝑝

2

𝑓34

− 𝑝

2

𝑓45

= 2.65 × 10

−7,

(31)


0

0.5

1

0

0.5

10

2

4

6

8

Mix

ed co

pula

PD

F

F2F3



0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

(b) Contour plot

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Mix

ed co

pula

func

tion



200 300 400 500 600 700 800200

300

400

500

600

700

800


Lim

it st

ate f

unct

iong3



where, because the structural system is symmetric, the failureprobability of the subsystems at symmetric positions is thesame. Namely, 𝑝

𝑓12

= 𝑝𝑓89

, 𝑝𝑓23

= 𝑝𝑓78

, . . . , 𝑝𝑓45

=

𝑝𝑓56

.For each subparallel system, the failure probability with-

out considering the correlation between the two componentsis

ℎ𝑓12

= 𝑝𝑓1

𝑝𝑓2

= 7.13 × 10

−15,

ℎ𝑓23

= 𝑝𝑓2

𝑝𝑓3

= 9.70 × 10

−14,

ℎ𝑓34

= 𝑝𝑓3

𝑝𝑓4

= 2.77 × 10

−12,

ℎ𝑓45

= 𝑝𝑓4

𝑝𝑓5

= 5.08 × 10

−10.

(32)

Suppose that failure modes between subsystems aremutually independent, and the failure modes between the

two components of each subsystem are also mutually inde-pendent; then, the failure probability of structural system [17]is approximately

𝑝𝑓= (−1)

2

8

∑

𝑗=𝑖+1,𝑖=1

ℎ𝑓𝑖𝑗

+ (−1)

3

7

∑

𝑡=𝑘+1,𝑗=𝑖+1,𝑘>𝑖,𝑖=1

ℎ𝑓𝑖𝑗

ℎ𝑓𝑘𝑡

+ ⋅ ⋅ ⋅ + (−1)

9(ℎ𝑓12

ℎ𝑓23

ℎ𝑓34

ℎ𝑓45

ℎ𝑓56

ℎ𝑓67

ℎ𝑓78

ℎ𝑓89

)

≈

8

∑

𝑗=𝑖+1,𝑖=1

ℎ𝑓𝑖𝑗

−

7

∑

𝑡=𝑘+1,𝑗=𝑖+1,𝑘>𝑖,𝑖=1

ℎ𝑓𝑖𝑗

ℎ𝑓𝑘𝑡

= 2 (ℎ𝑓12

+ ℎ𝑓23

+ ℎ𝑓34

+ ℎ𝑓45

) − 4 (ℎ𝑓12

ℎ𝑓23

+ ℎ𝑓12

ℎ𝑓34

+ ℎ𝑓12

ℎ𝑓45

+ ℎ𝑓23

ℎ𝑓34

+ ℎ𝑓23

ℎ𝑓45

+ ℎ𝑓34

ℎ𝑓45

) − ℎ

2

𝑓12

− ℎ

2

𝑓23

− ℎ

2

𝑓34

− ℎ

2

𝑓45

= 1.02 × 10

−9.

(33)

From the above solved system’s failure probability shownin (31) and (33), it can be seen that the failure probability of the


0

0.5

1

0

0.5

10

2

4

6

Mix

ed co

pula

PD

F

F4 F3



0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

(b) Contour plot

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Mix

ed co

pula

func

tion



200 300 400 500 600 700 800200

300

400

500

600

700


Lim

it st

ate f

unct

iong4



series-parallel system considering the correlation betweentwo adjacent girders is larger than the failure probabilitywithout considering the correlation between two adjacentgirders, which showed that the series-parallel system con-sidering the correlation between two adjacent girders moreeasily failed. Further, it is illustrated that considering thecorrelation between two adjacent girders of each subparallelsystem is essential and applicable for solving the reliability ofthe series-parallel system.

5. Conclusions

For the two-component systems and multiple-componentsystems with multiple failure modes, this paper presents themixed copula models for reliability analysis of series sys-tems, parallel systems, series-parallel systems, and parallel-series systems. The mixed copula model is obtained withthe chosen optimal copula functions with the Bayesianmethod. Through a numerical example, it is illustrated

that the calculated failure probability when consideringthe correlation between failure modes is larger than thatwithout considering the correlation between failure modes.It is verified that the solved failure probability is conser-vative without considering the correlation between failuremodes.

This paper provided a new method for characterizing thecorrelation between failure modes and solving the reliabilityof the system considering correlation between failure modes.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work was supported by the Fundamental ResearchFunds for the Central Universities (lzujbky-2015-300,lzujbky-2015-301).


F5 F40

0.5

1

0

0.5

10

0.5

1

1.5

2

Mix

ed co

pula

PD

F



0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

(b) Contour plot

Joint empirical distribution functionF45

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Mix

ed co

pula

func

tion



Lim

it st

ate f

unct

iong5

0 200 400 600 800100

200

300

400

500

600



References

[1] S. Eryilmaz, “Multivariate copula based dynamic reliabilitymodeling with application to weighted-k-out-of-n systems ofdependent components,” Structural Safety, vol. 51, pp. 23–28,2014.

[2] A. S. Balu and B. N. Rao, “Multicut-high dimensional modelrepresentation for structural reliability bounds estimationunder mixed uncertainties,” Computer-Aided Civil and Infras-tructure Engineering, vol. 27, no. 6, pp. 419–438, 2012.

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