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 Β© 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
Todayβs 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 todayβs real life systems needsmore detailedand complicated statistical analysis. Dependence between thecomponents is one of the intractable realistic assumptionsthat need to be carefully considered [1β4].
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 [5β7]. 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 functionsβ 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 systemβs 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. Sklarβs theory[12] clearly indicated that, given random variables X =
(π1, π2, . . . , π
π)with continuousmarginal distribution func-
tions πΉπ1
, πΉπ2
, . . . , πΉππ
and π-dimensional joint distributionfunction πΉ, there exists a unique copula functionπΆ such that,for all x = (π₯
1, π₯2, . . . , π₯
π) β Rπ,
πΉ (π₯1, π₯2, . . . , π₯
π)
= πΆ (πΉπ1
(π₯1) , πΉπ2
(π₯2) , . . . , πΉ
ππ
(π₯π)) .
(1)
According to (1), the joint probability density function ofthe random variableX = (π
1, π2, . . . , π
π) is given as follows:
π (π₯1, π₯2, . . . , π₯
π)
= π (πΉπ1
(π₯1) , πΉπ2
(π₯2) , . . . , πΉ
ππ
(π₯π))
π
β
π=1
πππ
(π₯π) ,
(2)
where πππ
(π₯π) is the marginal probability density function
(PDF) and π(u) is the joint PDF of the copula functiondenoted as follows:
π (π’1, . . . , π’
π) =
π
ππΆ
ππ’1β β β ππ’π
(π’1, . . . , π’
π) . (3)
IfπΉππ
for π = 1, 2, . . . , π have generalized inverse functionsπΉ
β1
ππ
for π = 1, 2, . . . , π, then, (1) can be rewritten as follows:
πΉ (πΉ
β1
π1
(π’1) , πΉ
β1
π2
(π’2) , . . . , πΉ
β1
ππ
(π’π))
= πΆ (π’1, π’2, . . . , π’
π) ,
(4)
where π’π= πΉππ
(π = 1, 2, . . . , π), (π’1, π’2, . . . , π’
π) = 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 π
πβ‘ πΉππ
(ππ), π = 1, 2, . . . , π, are all
uniformly distributed on (0, 1). Conversely,ππ= πΉ
β1
ππ
(ππ), π =
1, 2, . . . , π, U = (π1, π2, . . . , π
π); with this in mind, (4) is
expressed as
π (π1β€ π₯1, π2β€ π₯2, . . . , π
πβ€ π₯π) = π (π
1
β€ πΉ
β1
π1
(π’1) , π2β€ πΉ
β1
π2
(π’2) , . . . , π
πβ€ πΉ
β1
ππ
(π’π))
= π (π1β€ π’1, π2β€ π’2, . . . , π
πβ€ π’π)
= πΆ (π’1, π’2, . . . , π’
π) .
(5)
According to (1)β(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
πΆopt = argπΆπ
maxπ (Hπ‘| π·, πΌ) , (6)
where πΆopt is the optical copula function, Hπ‘is denoted
as the hypotheses that the right copula is copula πΆπ(π =
1, 2, . . . , πΏ), as the candidate copulas, πΆπis denoted by πΆ
π=
πΆπ(π’π, Vπ; ππ) (π = 1, 2, . . . , π), and π pair of quantiles (π’
π, Vπ)
are supposed as πmutually independent pairs composing thedata set π·. Besides, πΌ stands for some a priori information.According to Bayesian formula, it can be obtained that
π (Hπ‘| π·, πΌ) =
π (π· | Hπ‘, πΌ) π (H
π‘| πΌ)
π (π· | πΌ)
, (7)
where π(π· | Hπ‘, πΌ) is the likelihood, π(H
π‘| πΌ) is the a priori
copula family, andπ(π· | πΌ) is the normalization constant.Therank correlation coefficient Kendallβs π, denoted by π = π
π(π)
to be the correlation parameters for all the selected copulas,is introduced in order to give the formula for π(H
π‘| π·, πΌ):
π (Hπ‘| π·, πΌ)
= β«
πβΞ©
π (π· | Hπ‘, π, πΌ) π (H
π‘| π, πΌ) π (π | πΌ)
π (π· | πΌ)
ππ,
(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 π = β0.7581Gaussian PDF π = β0.7581
Gau
ssia
nc(u,οΏ½)
Figure 1: PDF and contour plots of Gaussian copula function.
where π(π· | Hπ‘, π, πΌ) is the likelihood with respect to π,
because of the π mutually independent pairs, and is alsodescribed as the copula density:
π (π· | Hπ‘, π, πΌ) =
π
β
π=1
πΆπ(π’π, Vπ; π
β1
π(π)) . (9)
Substituting (9) into (8), we obtain
π (Hπ‘| π·, πΌ) =
1
π (π· | πΌ)
β«
πβΞ©
π
β
π=1
πΆπ(π’π, Vπ; π
β1
π(π))
β π (Hπ‘| π, πΌ) π (π | πΌ) ππ,
(10)
where π(π· | πΌ) is the normalization constant and π(Hπ‘|
π, πΌ)π(π | πΌ) = 1/π(Ξ), where π(Ξ) is denoted as the Lebesguemeasure of Ξ = [β1, 1] in this paper; therefore, (10) can bealso expressed as follows:
π (Hπ‘| π·, πΌ) β
1
π (Ξ)
β«
Ξ©πβ©Ξ
π
β
π=1
πΆπ(π’π, Vπ; π
β1
π(π)) ππ. (11)
Obviously, the optimal copula function selection criteriacan be described as the computation of the highest weightπ
π:
ππ=
1
π (Ξ)
β«
Ξ©πβ©Ξ
π
β
π=1
ππ(π’π, Vπ; π
β1
π(π)) ππ. (12)
In especial, the βrightβ 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 1β4.
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,
πΆ (u, k; a, π) =π
β
π=1
πππΆπ(u, k; π
π) (π β€ 4) , (13)
where πΆπ(π = 1, 2, . . . , π) are, respectively, denoted as one
of the chosen copulas among the four copulas and a =
(π1, π2, . . . , π
π) is the weighting coefficient vector along with
the inner parameter vector π = (π1, π2, . . . , π
π) of the
corresponding copulas, and u, ^ are variables of the chosencopula functions, and what is more, π
1+ π2+ β β β + π
π= 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 {π
π}π, π =
1, 2, . . . , π, π = 1, 2, . . . , π, for each of the performancefunctions and obtain the sequences {πΉ
π}π= πΉ({π
π}π), π =
1, 2, . . . , π, π = 1, 2, . . . , π, of the corresponding empirical
4 Mathematical Problems in Engineering
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 π = 2.5Gumbel PDF π = 2.5
c(u,οΏ½)
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 π = 1.6352 Clayton PDF π = 1.6352
c(u,οΏ½)
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 π = β6.6800 Frank PDF π = β6.6800
c(u,οΏ½)
Figure 4: PDF and contour plots of Frank copula function.
Mathematical Problems in Engineering 5
Table 1: Basic cases of the parameters of copula functions.
Copula π = π(π) π β Ξ© π β Ξ©
Gaussian 2
π
arcsin π [β1, 1] [β1, 1]
Gumbel 1 β π
β1[0, 1] [1, +β)
Clayton 1 β
2
2 + π
(0, 1] (0, +β)
Frank 1 β
4
π
(1 β
1
π
β«
π
0
π‘
π
π‘β 1
ππ‘) [β1, 0) βͺ (0, 1] (ββ, 0) βͺ (0, +β)
distribution function by mean of Matlab software, where π 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 Kendallβs π 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:
πΉOLS = β
1
π
π
β
π=1
(πΉ ((πΉ1)
π, (πΉ2)
π, . . . , (πΉ
π)
π) β πΆ ((πΉ
1)
π, (πΉ2)
π, . . . , (πΉ
π)
π))
2
, (14)
where πΉ(β ) is the joint empirical distribution function of theempirical distribution function sequences for each perfor-mance function and πΆ((πΉ
1)π, (πΉ2)π, . . . , (πΉ
π)π) 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 πΉOLS 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
ππ(X) = π
π(X1, π2, . . . , π
π) , π = 1, 2. (15)
With (5), as a check, the probability that both failuremodes occur is denoted asπ (π1(X) β€ 0, π
2(X) β€ 0)
= π (πΉπ1
(π1(X)) β€ πΉ
π1(0) , πΉ
π2
(π2(X)) β€ πΉ
π2(0))
= π (π1β€ πΉπ1(0) , π
2β€ πΉπ2(0))
= πΆ (πΉπ1(0) , πΉ
π2(0)) = πΆ (π
ππ1
, πππ2
) .
(16)
πΉg(0, 0) = πΆ(πΉπ1
(0), πΉπ2
(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
ππ= π (π
1(X) β€ 0 βͺ π
2(X) β€ 0)
= π (π1(X) β€ 0) + π (π
2(X) β€ 0)
β π (π1(X) β€ 0, π
2(X) β€ 0)
= πππ1
+ πππ2
β πΆ(πππ1
, πππ2
) ,
(17)
where πππ1
, πππ2
, respectively, denote the failure probabilityof the two failure modes and πΆ(β ) 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,
6 Mathematical Problems in Engineering
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
ππ(X) = π
π(π1, π2, . . . , π
π) , π = 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:
ππ= π (π
1(X) β€ 0 β© π
2(X) β€ 0) = πΆ (π
ππ1
, πππ2
) , (19)
where πππ1
, πππ2
, respectively, denote the failure probabilityof the two failure modes and πΆ(β ) 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
ππ(X) = π
π(π1, π2, . . . , π
π) , π = 1, 2, . . . , π. (20)
With (5), as a check, the probability that all the failuremodes meantime occur is denoted as
π (π1(X) β€ 0, π
2(X) β€ 0, . . . , π
π(X) β€ 0)
= π (πΉπ1
(π1(X)) β€ πΉ
π1(0) , πΉ
π2
(π2(X))
β€ πΉπ2(0) , . . . , πΉ
ππ
(ππ(X)) β€ πΉ
ππ(0)) = π (π
1
β€ πΉπ1(0) , π
2β€ πΉπ2(0) , . . . , π
πβ€ πΉππ(0))
= πΆ (πΉπ1(0) , πΉ
π2(0) , . . . , πΉ
ππ(0))
= πΆ (πππ1
, πππ2
, . . . , ππππ
) .
(21)
πΉg(0, 0, . . . , 0) = πΆ(πΉπ1
(0), πΉπ2
(0), . . . , πΉππ
(0)) is statedin the fundamental theorem of Sklar. Therefore, the failureprobability of multiple-component series system (at least one
1 2 i NΒ· Β· Β·Β· Β· Β·
Figure 7: Multiple-component series system.
failure mode of the multiple components occurred) can besolved asππ= π (π
1(X) β€ 0 βͺ π
2(X) β€ 0 βͺ β β β βͺ π
π(X) β€ 0)
=
π
β
π=1
π (ππ(X) β€ 0) β β
π1<π2
π (ππ1(X) β€ 0, π
π2(X)
β€ 0) + β β β + (β1)
π+1β
π1<π2<β β β <ππ
π (ππ1(X) β€ 0, π
π2(X)
β€ 0, . . . , πππ(X) β€ 0) + β β β + (β1)
π+1π (π1(X) β€ 0,
π2(X) β€ 0, . . . , π
π(X) β€ 0) =
π
β
π=1
ππππ
β β
π1<π2
πΆ(ππππ1
, ππππ2
) + β β β + (β1)
π+1
β β
π1<i2<β β β <ππ
πΆ(ππππ1
, ππππ2
, . . . , πππππ
) + β β β
+ (β1)
π+1πΆ(πππ1
, πππ2
, . . . , ππππ
) ,
(22)
where πππ1
, πππ2
, . . . , ππππ
, respectively, denote the failureprobability of the multiple failure modes andπΆ(β ) 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
ππ(X) = π
π(π1, π2, . . . , π
π) , π = 1, 2, . . . , π. (23)
With (5), the failure probability of two-component par-allel system (all the failure modes of all the componentsmeantime occurred) can be obtained:
ππ= π (π
1(X) β€ 0, π
2(X) β€ 0, . . . , π
π(X) β€ 0)
= π (πΉπ1
(π1(X)) β€ πΉ
π1(0) , πΉ
π2
(π2(X))
β€ πΉπ2(0) , . . . , πΉ
ππ
(ππ(X)) β€ πΉ
ππ(0)) = π (π
1
β€ πΉπ1(0) , π
2β€ πΉπ2(0) , . . . , π
πβ€ πΉππ(0))
= πΆ (πΉπ1(0) , πΉ
π2(0) , . . . , πΉ
ππ(0))
= πΆ (πππ1
, πππ2
, . . . , ππππ
) ,
(24)
where πππ1
, πππ2
, . . . , ππππ
, respectively, denote the failureprobability of the π failure modes and πΆ(β ) is multiple-component parallel system copula function.
Mathematical Problems in Engineering 7
1 2 i NΒ· Β· Β·Β· Β· Β·
Figure 8: Multiple-component parallel system.
1 2 N
Β· Β· Β·
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 systemsβ failure probability can be obtained, and then,with (25), the failure probability of the series-parallel systemcan be solved:
ππs-p
= π(
π
β
π=1
πΉπ) = π(
π
β
π=1
ππ
β
π
πΉππ)
=
π
β
π=1
π (πΉπ) β β
π1<π2
{π (πΉπ1
) π (πΉπ2
)} + β β β
+ (β1)
π+1β
π1<π2<β β β <ππ
{
{
{
ππ
β
π=π1
π (πΉπ)
}
}
}
+ β β β
+ (β1)
π+1
π
β
π=1
π (πΉπ) ,
(25)
where π is the total number of the subparallel systems; ππ
is the total number of the components in the πth subparallelsystem; πΉ
π= β
ππ
π=1πΉππis the failure probability of the πth
subparallel system;πΉππ= (πππ(X) β€ 0) is the failure probability
of the πth component in the πth subparallel system; π(πΉπ), π =
1, 2, . . . , π, 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 subseriessystemsβ failure probability can be obtained, and then, with(26), the failure probability of the parallel-series system canbe solved as
ππp-s
= π(
π
β
π=1
πΉπ) =
π
β
π=1
π (πΉπ) =
π
β
π=1
π(
ππ
β
π=1
πΉππ) , (26)
whereπ is the total number of the subseries systems;ππis the
total number of the components in the πth subseries system;πΉπ= β
ππ
π=1πΉππis the failure probability of the πth subseries
system; πΉππ= (πππ(X) β€ 0) is the failure probability of the πth
component in the πth subseries system; π(πΉπ), π = 1, 2, . . . , π,
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 kNβ m (mean value) and91.783 kNβ 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 girderβs failure mode is
ππ= π (π
π1, ππ2, ππ3) = π (π
π, πΊπ, ππ) = π πβ πΊπβ ππ,
π = 1, 2, . . . , 9,
(27)
8 Mathematical Problems in Engineering
Table 2: Distribution parameters of beamsβ dead load effects in 32nd year.
Girder number 1 2 3 4 5 6 7 8 9Mean value/kNβ m 212.86 221.79 223.91 230.74 260.69 230.74 223.91 221.79 212.86Standard deviation/kNβ m 9.68 9.68 9.68 9.68 9.68 9.68 9.68 9.68 9.68
Table 3: Distribution parameters of beamsβ maximum live load effects in 32nd year.
Girder number 1 2 3 4 5 6 7 8 9Mean value/kNβ m 85.42 96.42 116.8 144.4 180.9 144.4 116.8 96.42 85.42Standard deviation/kNβ 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#
Β· Β· Β·
Figure 12: Series-parallel system.
where π πis the resistance of the πth girder, πΊ
πis the dead load
effect of the πth girder, and ππis the live load effect of the πth
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.
π π 1 β π β π π πΌ π
1#-2# 0.7438 0.1958 0.0604 β0.1868 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 β0.3490 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 13β16.
With FORM, the corresponding reliability indices to eachgirderβs failure mode are, respectively,
π½1= 5.3429,
π½2= 5.1159,
π½3= 4.8484,
π½4= 4.4417,
π½5= 3.6869.
(28)
Then, with the equation ππ= Ξ¦(βπ½), the corresponding
failure probability to each girderβs failure mode is, respec-tively,
πππ1
= 4.57 Γ 10
β8,
πππ2
= 1.56 Γ 10
β7,
πππ3
= 6.22 Γ 10
β7,
πππ4
= 4.46 Γ 10
β6,
πππ5
= 1.14 Γ 10
β4.
(29)
Mathematical Problems in Engineering 9
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:
ππ12
= πΆ (ππ1
, ππ2
) = 8.61 Γ 10
β11,
ππ23
= πΆ (ππ2
, ππ3
) = 3.12 Γ 10
β10,
ππ34
= πΆ (ππ3
, ππ4
) = 1.32 Γ 10
β9,
ππ45
= πΆ (ππ4
, ππ5
) = 1.31 Γ 10
β7,
(30)
where ππ12
is the failure probability when girder 1# andgirder 2# meantime failed, π
π23
is the failure probability whengirder 2# and girder 3# meantime failed, π
π34
is the failureprobability when girder 3# and girder 4# meantime failed,and π
π45
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
ππ= (β1)
2
8
β
π=π+1,π=1
ππππ
+ (β1)
3
7
β
π‘=π+1,π=π+1,π>π,π=1
ππππ
ππππ‘
+ β β β + (β1)
9(ππ12
ππ23
ππ34
ππ45
ππ56
ππ67
ππ78
ππ89
)
β
8
β
π=π+1,π=1
ππππ
β
7
β
π‘=π+1,π=π+1,π>π,π=1
ππππ
ππππ‘
= 2 (ππ12
+ ππ23
+ ππ34
+ ππ45
) β 4 (ππ12
ππ23
+ ππ12
ππ34
+ ππ12
ππ45
+ ππ23
ππ34
+ ππ23
ππ45
+ ππ34
ππ45
) β π
2
π12
β π
2
π23
β π
2
π34
β π
2
π45
= 2.65 Γ 10
β7,
(31)
10 Mathematical Problems in Engineering
0
0.5
1
0
0.5
10
2
4
6
8
Mix
ed co
pula
PD
F
F2F3
(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 F23
(c) Empirical distribution versus mixed copula
200 300 400 500 600 700 800200
300
400
500
600
700
800
Limit state function g2
Lim
it st
ate f
unct
iong3
(d) Scatter plot for two limit state functions
Figure 14: The subsystem 2#-3#.
where, because the structural system is symmetric, the failureprobability of the subsystems at symmetric positions is thesame. Namely, π
π12
= ππ89
, ππ23
= ππ78
, . . . , ππ45
=
ππ56
.For each subparallel system, the failure probability with-
out considering the correlation between the two componentsis
βπ12
= ππ1
ππ2
= 7.13 Γ 10
β15,
βπ23
= ππ2
ππ3
= 9.70 Γ 10
β14,
βπ34
= ππ3
ππ4
= 2.77 Γ 10
β12,
βπ45
= ππ4
ππ5
= 5.08 Γ 10
β10.
(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
ππ= (β1)
2
8
β
π=π+1,π=1
βπππ
+ (β1)
3
7
β
π‘=π+1,π=π+1,π>π,π=1
βπππ
βπππ‘
+ β β β + (β1)
9(βπ12
βπ23
βπ34
βπ45
βπ56
βπ67
βπ78
βπ89
)
β
8
β
π=π+1,π=1
βπππ
β
7
β
π‘=π+1,π=π+1,π>π,π=1
βπππ
βπππ‘
= 2 (βπ12
+ βπ23
+ βπ34
+ βπ45
) β 4 (βπ12
βπ23
+ βπ12
βπ34
+ βπ12
βπ45
+ βπ23
βπ34
+ βπ23
βπ45
+ βπ34
βπ45
) β β
2
π12
β β
2
π23
β β
2
π34
β β
2
π45
= 1.02 Γ 10
β9.
(33)
From the above solved systemβs failure probability shownin (31) and (33), it can be seen that the failure probability of the
Mathematical Problems in Engineering 11
0
0.5
1
0
0.5
10
2
4
6
Mix
ed co
pula
PD
F
F4 F3
(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 F34
(c) Empirical distribution versus mixed copula
200 300 400 500 600 700 800200
300
400
500
600
700
Limit state function g3
Lim
it st
ate f
unct
iong4
(d) Scatter plot for two limit state functions
Figure 15: The subsystem 3#-4#.
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).
12 Mathematical Problems in Engineering
F5 F40
0.5
1
0
0.5
10
0.5
1
1.5
2
Mix
ed co
pula
PD
F
(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
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
(c) Empirical distribution versus mixed copula
Limit state function g4
Lim
it st
ate f
unct
iong5
0 200 400 600 800100
200
300
400
500
600
(d) Scatter plot for two limit state functions
Figure 16: The subsystem 4#-5#.
References
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