Special Issue Article
An improved approach by introducing copula functions for structural reliability analysis under hybrid uncertainties Zheng Liu
Advances in Mechanical Engineering 2018, Vol. 10(7) 1–10 Ó The Author(s) 2018 DOI: 10.1177/1687814018785839 journals.sagepub.com/home/ade
and Xin Liu
Abstract The structural composition of the oil platform is very complicated, and its working environment is harsh, thus conducting a large number of reliability tests is not feasible, and the field tests are also hard to accomplish. So the reliability of the oil platform cannot be analyzed and calculated by the traditional reliability method which needs a lot of test data, and new methods should be studied. In recent years, imprecise probability theory has attracted more and more attention because when unified, it can quantify hybrid uncertainty. Structural reliability analysis on the basis of imprecise probability theory has made remarkable achievements in theoretical aspects, but it is scarcely used in practical engineering domains due to the complexity in the developed methods and the unavailability of suitable or specific modeling steps for applications. In this regard, we propose a unified quantification method for statistical data, fuzzy data, incomplete information, and the like, which can handle the issue of hybrid uncertainties, and then, we construct an improved imprecise structural reliability model aiming at the practical problems by introducing copula function. To verify the existing methodology, we also consider a cantilever beam widely applied in the oil platform here for the structural reliability analysis. Keywords Structural reliability, imprecise probability theory, natural extension model, copula function, cantilever beam
Date received: 19 December 2017; accepted: 5 June 2018 Handling Editor: Jose´ Correia
Introduction Oil platform is a large structure with lots of facilities for well drilling and can be used to extract and process oil and natural gas, to store product until it can be brought to shore for refining and marketing. The oil platform is made up of the hull, spud legs, pile-boots, cantilever beams, helicopter landing, and so on. As known to all, the structural composition of the oil platform is very complicated, and its working environment is harsh, thus conducting a large number of reliability tests is not feasible, and the field tests are also hard to accomplish.1 In this case, traditional reliability method which needs a lot of test data cannot be used to analyze and calculate its reliability, thus new methods should
be studied. Uncertainties influencing the reliability of engineering structures are not only one type of uncertainty, but hybrid uncertainties, and it is of great importance to find out rational methods to quantify these uncertainties.2 As we know, a structural system throughout its life circle is faced with a variety of
School of Mechanical and Electric Engineering, Guangzhou University, Guangzhou, P.R. China Corresponding author: Zheng Liu, School of Mechanical and Electric Engineering, Guangzhou University, No. 230, GuangZhou University City Outer Ring Road, Guangzhou 510006, Guangdong, P.R. China. Email:
[email protected]
Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (http://www.creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/ open-access-at-sage).
2 stresses, and it will degenerate, damage, or fail if the stress exceeds the limit, from the view point of different failure mechanisms.3,4 The boundary between a damage state and a failure one is referred to as a limit state, and the structural reliability analysis is to study the likelihood for the structure to reach its limit state under the uncertainties.5 In practical engineering, the structural reliability is affected by hybrid uncertainties, which mainly derives from physical, model, and statistical uncertainty, and can be split into stochastic and epistemic uncertainty in accordance with their property. In the light of the theory relevant to mathematical statistics, structural reliability analysis methods can be divided into probabilistic and non-probabilistic methods. Probabilistic structural reliability analysis methods employ probability theory to quantify uncertain information. In particular, strength, stiffness, loading, geometric dimensioning as well as other characteristics related to structural reliability are characterized by a vector composed of a group of stochastic variables,2 and then, a specific probability distribution is thought over for each variable. Probabilistic structural reliability analysis methods are highly efficient when statistical data of the structure is ample enough to determine an accurate probability distribution, and these methods have been widely applied in multifarious engineering applications.6–8 Nevertheless, an increasing number of experts and authors9–14 point out that the lack of data and information remains the prevailing problem in realistic projects, and it is not convincing to ponder over any specific probability distribution for the variable when statistical data are lacking. Hence, structural reliability analysis on the condition of lack of data has aroused the concern and research of many scholars and experts. Structural reliability analysis for the lack of data is studied mainly in two lines, one concentrates on studying the reliability modeling methods for particular types of data, such as partially relevant recurrence data,9 and incomplete degradation data.10 The other line concentrates on studying the mathematical theories as well as optimization algorithms11 to make up for the lack of data, such as Bayesian theory,12,13 fuzzy set,14 Dempster–Shafer theory,15 interval analysis,16 and other non-probabilistic theories; meanwhile, optimization algorithms relevant to the non-probabilistic theories are proposed.17 All of those methods deem that the reliability measure may be decided in an indeterminant bound other than a deterministic value. Yet, either probabilistic or non-probabilistic structural reliability analysis methods, they principally consider solely one type of uncertainty, stochastic uncertainty or epistemic uncertainty, and no uniform hybrid uncertainty quantification means has been put forward. Therefore, one generalized non-probabilistic theory called imprecise probability theory has drawn more and more attention in recent years.
Advances in Mechanical Engineering Imprecise probability theory can quantify information or data with any amount, and they can also effectually merge together the classical probability, fuzzy set, interval number as well as other data types in a spontaneous extension model.18 For this reason, it supplies a feasibility to quantify hybrid uncertainties in a single model. Moreover, imprecise structural reliability analysis method can be comprehended with the help of probabilistic structural reliability theory which has comparatively integral and ripe theoretical system for engineers to apply. An imprecise structural reliability analysis method has accomplished a lot in theoretical aspects,6 but it can hardly be utilized in practical engineering due to the complicacy of imprecise structural reliability modeling. Accordingly, based on a concise introduction of imprecise probability theory, we propose a unified hybrid uncertainties quantification method and construct a relatively reformative analysis model which turns out to be much more available for practical engineering in this article. The remainder of this article is designed as follows. In section ‘‘Basic concepts of imprecise probability theory,’’ a brief introduction of certain fundamental concepts of imprecise probability theory is given. Then, the unified quantification method for hybrid uncertainties is raised, and the imprecise structural reliability model is given accordingly in section ‘‘Imprecise structural reliability analysis under the unified quantification of hybrid uncertainties.’’ In section ‘‘Improved imprecise structural reliability model by introducing copula function,’’ because of the problem statement of the extant imprecise structural reliability analysis methods, we propose a modified natural extension model by introducing copula function and show its concrete procedure for practical use. For the validation of the proposed method, we have solved an engineering example in section ‘‘The application of the proposed method on the cantilever beam.’’ Ultimately, conclusions are dawn in section ‘‘Conclusion.’’
Basic concepts of imprecise probability theory The mathematical theory of imprecise probability theory can be interpreted by behavioral interpretation. As is known to all, there exists an ‘‘event’’ in probability theory, and probability distributions and other mathematical models are established via the discussion of the ‘‘event.’’ Likewise, the notion of ‘‘gamble’’ in imprecise probability theory equals to the ‘‘event’’ in probability theory, and mathematical models can be constructed by the discussion of gambles as well. In this section, we give a few basic concepts frequently used in the field of reliability engineering. More details with respect to imprecise probability theory can be found in references.6,18
Liu and Liu
3 s:t:
Definition 1. A ‘‘gamble’’ is a bounded and realvalued function defined on domain O, such as X , g(X ), f (X ). Definition 2. Assume P(X ) is the probability that gamble X will happen. Then ½P(X ), P(X ) is the bound of P(X ), and D = P(X ) P(X ), where D expresses the inaccuracy or uncertainty of the interval ½P(X ), P(X ). Here, P(X ) is the infimum of the values that P(X ) will possibly take while P(X ) is the supremum. To represent more reliability measures and quantify more multiple uncertainties, lower and upper probabilities are expanded to upper and lower expectations. According to Peng et al.,9 most reliability measures can be expressed in mathematical expectation, such as ð RðtÞ = E I½t, + ‘Þ ð X Þ = I½t, + ‘Þ ð X Þrð X ÞdX ð1Þ R+
F ðtÞ = E I½0, t ð X Þ =
ð
I½0, t ð X Þrð X ÞdX
ð2Þ
ð
rð X Þ 0,
ð
Mean time to failures ðMTTFÞ = Eð X Þ =
ð6Þ
uij ðxi Þrð X ÞdX aij , i n, j mi
Rn+
The set P is made up of all possible probabilistic density functions satisfying constraints determined by the collected data. Obviously, the collected data can be deemed as the evidence to reduce the range of P, for along with more and more information or reliability data we can gather, the feasible region becomes smaller and smaller, thus the result becomes much more precise. Distinctly, as the primal natural extension model is extremely intricate to calculate, Kuznetsov introduces duality theorem of linear programming into the above optimization model and transfers it into a linear optimization problem.10 According to Utkin et al.19, Elishakoff20 and Zhao,21 the above optimization model can be rewritten as
R+
ð
rð X ÞdX = 1, aij
Rn+
R = sup
X rð X ÞdX
c+
c, cij , dij
R+
mi n X X
cij aij dij aij
!
i=1 j=1
ð7Þ
s:t:
ð3Þ
c+
and
mi n X X
cij dij fij ð X Þ I½0, + ‘Þ ðg ð X ÞÞ
i=1 j=1
Residual MTBFðmean time between failuresÞ = E X tI½t, + ‘Þ ð X Þ
ð4Þ
and R = inf
c, cij , dij
Definition 2.3. Natural extension is a vital concept to construct reliability models for imprecise probability. In subsistent papers, natural extension models are embodied by different equivalent optimization models.19 As we can see, x1 , . . . , xn are variables in X = (x1 , . . . , xn ), and they are in connection with the dependability of a structural system, such as load, strength, physical dimension, and environmental coefficients. Assume there are in total mi reliability data relevant to the variable xi , all of which as well as the reliability measure M can be presented in the shape of M = E(g(X )), E(uij (xi )) 2 ½aij , aij , where i = 1, . . . , n, j = 1, . . . , mi , and uij (xi ) is a confirmable function corresponding to the jth information. According to Utkin et al.,19 the primitive form of natural extension model for computing the supremum and infimum of reliability measure M can be built as follows ð M ðM Þ = max min gð X Þrð X ÞdX P
P
Rn+
ð5Þ
c+
mi n X X
cij aij dij aij
!
i=1 j=1
ð8Þ
s:t: c+
mi n X X
cij dij fij ð X Þ I½0, + ‘Þ ðgð X ÞÞ
i=1 j=1
where c 2 R, cij , dij 2 R+ , c, cij , and dij are new optiaij have mization variables. Besides, fij (X ), g(X ), ½aij , exactly the same definitions as mentioned above. The above-mentioned two natural extension models are the most wide-spread used models in imprecise structural reliability analysis, and other equivalent models are also raised based on this model for specific applications.
Imprecise structural reliability analysis under the unified quantification of hybrid uncertainties Unified quantification of hybrid uncertainties Hybrid uncertainties mainly considered in practical engineering include physical uncertainty, statistical
4
Advances in Mechanical Engineering
uncertainty, modeling uncertainty, and so on, and the above uncertainties of different kinds can be qualitatively divided into stochastic uncertainty, fuzzy uncertainty, and incompleteness. According to the imprecise probability theory, stochastic and fuzzy uncertainty as well as incompleteness can be quantified into a unified form. Through it, reliability data of different types can be employed into one model. Here, we simply give the resulting mathematical expressions. The quantification of statistical data. Assume the mean of a and its variance variable X belongs to an interval ½m, m , the mean and variance of variable to an interval ½s, s X under the framework of imprecise probability theory can be then expressed as m Eð X Þ =
ð
X rð X ÞdX m
ð9Þ
Imprecise structural reliability analysis under hybrid uncertainties Presume g(X ) is the structural system’s limit-state function, where X = (x1 , x2 , . . . , xn ) and xi is the design variables with i = 1, . . . , n. Denote F is the failure region and F = fX : g(X )\0g. In this way, the reliability R of the structure is likely to be computed as R = Prfg(X ) 0g and the failure probability F may be obtained by F = Prfg ð X Þ\0g = 1 R
m2 + s2 = E X 2 =
ð16Þ
Reliability data collected beforehand are quantified as follows aij
ð
fij ð X Þrð X ÞdX aij , i = 1, . . . , n, j = 1, . . . , mi
O
O
ð
ð15Þ
ð17Þ X 2 rð X ÞdX m 2 + s 2
ð10Þ
O
, s = s , variable X degenerates to stoWhen m = m chastic variable. The quantification of fuzzy data. Here, we take triangular fuzzy number as an example, and it can be defined as 8 ðxaÞ , a x\b > > < ðbaÞ ; x=b mð xÞ = ðcx1, Þ > , b x\c > : ðcbÞ 0, others
ð R = inf I½0, P
+ ‘Þ ðg ð X ÞÞr ð X ÞdX
O
ð R = sup I½0,
ð11Þ
P
ð18Þ + ‘Þ ðg ð X ÞÞrð X ÞdX
O
The fuzzy variable x under the framework of imprecise probability theory can be indicated as h L U i ~ , P ~ EðIA ð xÞÞ = a, EðIA ð xÞÞ = 1, A = P a a
where mi is the total number of reliability data related to xi , and fij (X ) is a known function identify with the jth reliability data because j = 1, . . . , mi , r(X ) is the joint probability density function, and O is the domain of X . On the basis of the quantification of reliability data gathered for variables, the structural reliability model is able to be established as
s:t: rð X Þ 0,
ð12Þ
where I½t1 , t2 (X ) is the indicator function defined as 1, X 2 ½t1 , t2 , and A serves as a level set. I½t1 , t2 (X ) = 0, others The quantification of incomplete information. The incomplete information can be directly quantified by upper and lower expectations. For instance, the mean time between failures of one system is in the interval ½400, 500, and its reliability at t = 300 is equal or greater than 0:9. Accordingly, it can be written as 400 EðT Þ 500, f ðT Þ = T ð13Þ 0:9 E I½300, + ‘Þ ðT Þ 1, f ðT Þ = I½300, + ‘Þ ðT Þ ð14Þ
aij
ð
ð
rð X ÞdX = 1,
O
fij ð X Þrð X ÞdX aij , i = 1, . . . , n, j = 1, . . . , mi
O
ð19Þ Hence, set P is made up of all possible probability density functions which satisfied the constraints (19), that is P = fr(X )g.
Improved imprecise structural reliability model by introducing copula function Problem statement This study on imprecise structural reliability theory which are mostly restricted to artificial example but not in practice6,22 concentrates mainly on theoretical analysis. The main reasons are elucidated as follows:
Liu and Liu 1.
2.
3.
5
The great mass of natural extension models requires reliability data to be characterized by upper and lower precisions, such as E(f (x)) 2 ½a, a, and inference models are built on the basis of this form. Nevertheless, the specific quantification methods of different data types like fuzzy data are not proposed. The most imprecise structural reliability models use joint probability density function r(X ) to quantify reliability data. However, r(X ) cannot faultlessly quantify the data bound-up with the single variable and exactly delineate the correlations among different variables, and thus, these models cannot be widely applied for the structural system reliability analysis. The existing imprecise structural reliability analysis models are restricted to numerical examples but never in practice. That is to say, there exists no minute application process to be used for reference, so it seems out of the question for engineers to operate.
The problems discussed above restrict the wide application of imprecise structural reliability analysis in practical engineering to a large extent. By keeping this in mind, we introduce copula function to the analysis of natural extension model to improve the computation accuracy, for the copula function can effectively quantify the correlation of different parameters or failure modes. In this way, we can, on one hand, use single variable reliability data to augment the data amount and reduce the imprecision D. On the other hand, we can analyze the relevance between disparate variables if necessary.
An improved natural extension model by introducing copula function The improvement of natural extension model by introducing copula function. As we know, a copula function links univariate margins to their joint probability distribution23,24 and allows us to study probability density functions and correlation of variables correspondingly. Assume r(X ) is the joint probability density function of X and ri (X ) is the marginal probability density function of Xi , according to Sklar’s theorem rð X Þ = cðF1 ðx1 Þ, . . . , Fn ðxn ÞÞ
n Y
r i ðx i Þ
ð20Þ
where Fi (xi ) is marginal probability distribution function of xi , and c(F1 (x1 ), . . . , Fn (xn )) is probability density function of copula function C(F1 (x1 ), . . . , Fn (xn )), which is cðF1 ðx1 Þ, . . . , Fn ðxn ÞÞ =
n Y
∂ C ðF1 ðx1 Þ, . . . , Fn ðxn ÞÞ ð21Þ ∂F1 ðx1 Þ, . . . , ∂Fn ðxn Þ
r i ðx i Þ =
i=1
rð X Þ cðF1 ðx1 Þ, . . . , Fn ðxn ÞÞ
ð22Þ
When all variables are fully independent and c(F1 (x1 ), . . . , Fn (xn )) = 1, we can analyze the correlation by assessing the copula function. Take equation (22) to the natural extension model and thus it can be rewritten as ð R R = inf sup I½0, P
P
+ ‘Þ ðg ð X ÞÞr ð X ÞdX
ð23Þ
O
s:t: rðX Þ 0, ð
ak
ð
rð X ÞdX = 1,
O
fk ð X Þrð X ÞdX ak , k = 1, . . . , K
O
aij
ð
fij ðxi Þ rð X Þdxi aij , i = 1, . . . , n, cðF1 ðx1 Þ, . . . , Fn ðxn ÞÞ
O
j = 1, . . . , mi ð24Þ In the above equations, g(X ) is the limit-state function, and r(X ) is the joint probability density functions of X . Fi (xi ) is the marginal probability distribution function of xi , and c(F1 (x1 ), . . . , Fn (xn )) is the copula probability density function. Besides, fk (X ) is a known function in line with the kth reliability data relevant to variables vector X , where k = 1, . . . , K, fij (xi ) is a known function in accordance with the jth reliability data related to variable xi and j = 1, . . . , mi . Consequently, the above optimization model can be rewritten as the Kuznetsov’s form R = sup
c0 +
c, cij , dij
i=1
n
If marginal probability distribution functions of Fi (xi ) are successive, and copula function C(F1 (x1 ), . . . , Fn (xn )) and its probability density function c(F1 (x1 ), . . . , Fn (xn )) are unique,25,26 we may get the following equation
K X
ðck ak dk ak Þ +
! mi n X X cij aij dij aij i=1 j=1
k=1
s:t: c0 +
mi n X X i=1 j=1
+
K X
cij dij
fij ðxi Þ cðF1 ðx1 Þ, . . . , Fn ðxn ÞÞ
ðck dk Þfk ð X Þ I½0,
+ ‘Þ ðg ð X ÞÞ
k=1
ð25Þ and
6
Advances in Mechanical Engineering
R=
inf
c, cij , dij
c0 +
K X
ðck ak dk ak Þ +
mi n X X
cij aij dij aij
!
i=1 j=1
k =1
ð26Þ
s:t: c0 +
mi n X X
cij dij
i=1 j=1
K X fij ðxi Þ + ðck dk Þfk ð X Þ I½0, cðF1 ðx1 Þ, . . . , Fn ðxn ÞÞ k =1
where u1 , u2 are marginal probability distribution functions, and a is the parameter of Gumbel copula.
where c 2 R, cij , dij , ck , dk 2 R+ . The parameter estimation of the copula density function c(F1 (x1 ), . . . , Fn (xn )). The modified natural extension and models partly reduce the imprecision D of ½R, R, they correspond more to the real conditions of practical systems. According to Wu et al.,24 different copula probability density functions c(F1 (x1 ), . . . , Fn (xn )) lead to different degrees of inaccuracy. So, the determination of copula probability density function c(F1 (x1 ), . . . , Fn (xn )) is of great importance. A majority of common copula functions have been come up with in previous papers, such as multivariate Gaussian copula, Gumbel copula, and Frank copula.19 Here, we solely introduce some common copula functions as follows: 1.
Multivariate Gaussian copula and its probability density function are defined as
C ðu1 , . . . , un ; ZÞ = Fz F1 ðu1 Þ, . . . , F1 ðun Þ
ð27Þ
and cðu1 , . . . , un ; ZÞ = jZj
12
1 T 1 exp 1 Z I 1 ð28Þ 2
where Z is a symmetric positive matrix with diagonal elements equal to 1 and jZj to be its determinant, and Fz is a standard normal distribution with correlation coefficient matrix Z. Besides, 1i = F1 (ui ), T i = 1, . . . , n, 1 = (11 , . . . , 1n ) , and I is a unit matrix. When n = 2, the multivariate Gaussian copula degenerates a bivariate normal, which is
‘
‘
2 1 s 2st + t2 pffiffiffiffiffiffiffiffiffiffiffiffiffi exp dsdt 2ð1 z2 Þ 2p 1 z2
3. Frank copula function is defined as 1 ðeuu1 1Þðeuu2 1Þ ð31Þ C ðu1 , u2 ; uÞ = ln 1 + u eu 1 where u1 , u2 are marginal probability distribution functions, and a is the parameter of Gumbel copula. In normal conditions, we may utilize parameter estimation methods to estimate the copula probability density function, such as maximum likelihood, the step-bystep, and the semi parametric estimation method.26 Here, we apply the maximum likelihood estimation method to evaluate the parameter of copula probability density function.
Application of the improved structural reliability analysis model The utilization of imprecise probability theory in the domain of reliability engineering has long been studied and has achieved a lot in theory. However, no details of application process are proposed for reference. So, in this section, we will give specific application process of the unified natural extension model for the analysis of structural reliability. The process can be operated as follows:
C ðu1 , u2 ; ZÞ = F1ððu1 Þ F1ððu2 Þ
ð29Þ
2. Gumbel copula function is defined as 1 C ðu1 , u2 ; aÞ = exp ðð log u1 Þa + ð log u2 Þa Þa , a 2 ½1, + ‘Þ
+ ‘ Þ ðg ð X ÞÞ
ð30Þ
In light of the failure criteria, we should first determine the limit-state function of g(X ) = g(x1 , x2 , . . . , xn ), where x1 , . . . , xn are variables relevant to the structural reliability, such as load, strength, physical dimension, environmental coefficients, and the like. Refer to the Table 1 and collect reliability data concerning the structure or structural parameters. Choose one certain copula function C(F1 (x1 ), . . . , Fn (xn )) and take stock of its parameters. When there are limited samples, Bayesian parameter estimation method can be selected.21 At the same time, detailed selection and parameter estimation method of copula functions can refer to Hou.26 Construct a natural extension model with the help of collected reliability data. Assume, we want to measure the lower bound of structural
Liu and Liu
7
Table 1. Reliability data collected for the very variable xi . X
Data 1
.
.
Data K
Data mi
X x1 x2 . xi . xn
E(f1 (X)) 2 ½a1 , a1 E(f11 (x1 )) 2 ½a11 , a11 E(f21 (x2 )) 2 ½a21 , a21 . E(fi1 (xi )) 2 ½ai1 , ai1 . E(fn1 (xn )) 2 ½an1 , an1
. . . . . . .
. . . . . . .
E(fK (X)) 2 ½aK , aK E(f1K (x1 )) 2 ½a1K , a1K E(f2K (x2 )) 2 ½a2K , a2K . E(fiK (xi )) 2 ½aiK , aiK . E(fnK (xn )) 2 ½anK , anK
. . . . . .
E(f1m1 (x1 )) 2 ½a1m1 , a1m1 E(f2m2 (x2 )) 2 ½a2m2 , a2m2 . E(fimi (xi )) 2 ½aimi , aimi . E(fnm1 (xn )) 2 ½anmn , anmn
Figure 1. Procedure illustration of the proposed method.
reliability, the natural extension model can be established as follows: R = sup
c0 +
c, cij , dij
K X
ðck ak dk ak Þ +
mi n X X
cij aij dij aij
!
i=1 j=1
k =1
s:t: c0 +
mi n X X i=1 j=1
+
K X
cij dij
fij ðxi Þ cðF1 ðx1 Þ, . . . , Fn ðxn ÞÞ
ðck dk Þfk ð X Þ I½0,
Figure 2. A simplified structure of the cantilever beam. Table 2. Reliability data collected for p, Mcr , and l.
+ ‘Þ ð g ð X Þ Þ
k =1
ð32Þ In these equation, X = x1 , . . . , xn are variables related to the reliability of this structure. ak and ak are the lower and upper previsions, and K is the sum of reliability date related to variables vector X . aij and aij are the lower and upper previsions, and i is the total number of variables. mi is the sum of reliability date related to variable xi , and c(F1 (x1 ), . . . , Fn (xn )) is a copula density function with parameter u. r(X ) is the joint probability density function of X , and ui = ri (xi ) is the marginal probability density function of xi . The procedure of imprecise reliability analysis can be illustrated by Figure 1.
The application of the proposed method on the cantilever beam Structural reliability analysis for the cantilever beam based on imprecise probability theory The cantilever beam is a significant component for the oil platform, and its reliability affects the reliability and
Mcr p l
j=1
j=2
E1 (Mcr ) 2 ½10, 24 kNm E1 (p) 2 ½1:9, 2:6 kN E1 (l) 2 ½5, 5:3 m
E2 (Mcr ) 2 ½12, 20 kNm E2 (p) 2 ½2:0, 2:1 kN
safety of the whole system. Here, the reliability analysis of the cantilever beam is taken as the example to illustrate the counting process of the new proposed method. The cantilever beam along with force situation as are shown in Figure 2, where l is the length of the cantilever beam, and p is force acting on its end. Denote Mcr is the maximum flexion this cantilever beam can bear, and so the limit-state function of this structure can be written as g ðMcr , p, lÞ = Mcr pl
ð33Þ
Reliability data about Mcr , p, and l we have collected are displayed in Table 2. As we all know, the flexion of a structure is affected by its length. In order to determine the correlation of Mcr , p, and l, samples shown in Table 3 are then collected.
8
Advances in Mechanical Engineering standard normal distribution with correlation coefficient matrix Z of 11 = F1 (F(Mcr )),12 = F1 (F(p)), 13 = F1 (F(l)), and 1 = (11 , 12 , 13 )T . Moreover, I is a unit matrix. Let us presume Mcr , p, and l obeys normal distribution of Mcr ;N (24:15, 0:37252 ),p;N (2:25, 0:00922 ), and l;N (4:87, 0:0422 ). Hence, we can get the equation of F(Mcr ) = F((Mcr 24:15)=0:3725),F(p) = F((p 2:25) =0:0092), and F(l) = F((p 4:87)= 0:042). So, the unified natural extension model for structural reliability can be constructed as
Table 3. A set of sample data of variables Mcr , p, and l. No.
Mcr
p
l
1 2 3 4 5 6
23kNm 22:8 kN m 24:6 kN m 24 kN m 23:4 kN m 25:1 kN m
2:4kN 2:3 kN 2:3 kN 2:2 kN 2:1 kN 2:2 kN
4:9 m 4:6 m 5:1 m 5:1 m 4:9 m 4:6 m
R = sup ðc0 + 10c11 + 12c12 + 1:9c21 + 2:0c22 + 5c31 24d11 20d12 2:6d21 2:1d22 5:3d31 Þ c, cij , dij
s:t: c0 +
(
1
1 jZj2 exp 12 1T Z1 I 1
3 X
c1j d1j Mcr +
j=1
2 X
ð36Þ
)
c2j d2j p + ðc31 d32 Þl I½0,
+ ‘Þ ðMcr
pl1 Þ
j=1
and = sup ðc0 + 24c11 + 20c12 + 2:6c21 + 2:1c22 + 5:3c31 10d11 12d12 1:9d21 2:0d22 5d31 Þ R c, cij , dij
s:t: c0 +
1 1 jZj2 exp 12 1T Z1 I 1
(
3 X
c1j d1j Mcr +
j=1
F ðMcr , p, lÞ = C ðF ðMcr Þ, F ð pÞ, F ðlÞ; ZÞ = Fz F1 ðF ðMcr ÞÞ, F1 ðF ð pÞÞ, F1 ðF ðlÞÞ ð34Þ and cðF ðMcr Þ, F ð pÞ, F ðlÞ; ZÞ = jZj
ð37Þ
)
c2j d2j p + ðc31 d32 Þl
I½0,
+ ‘Þ ðMcr
pl1 Þ
j=1
Suppose the relationship between Mcr , p, and l can be quantified by multivariate Gaussian copula, then the probability distribution function and probability density function of Mcr , p, and l respectively are
12
2 X
1 T 1 exp 1 Z I 1 2
ð35Þ where F(Mcr ), F(p), and F(l) are marginal probability distribution functions of Mcr , p, and l, and Fz is a
Thus, the structural reliability of this cantilever beam can be gained as ½0:9453, 1. From the example of a cantilever beam, we can elucidate the application process of the model raised in this article for imprecise structural reliability analysis.
Result discussion Here, a cantilever beam for the structural reliability analysis has been taken into account to verify the current methodology. By introducing copula function, we can use the proposed method to quantify variables correlation. It conforms much more to the real conditions of practical systems and will arrive at a much precise result. If we do not consider variables relationship, natural extension model of a cantilever beam can be rewritten as
R = sup ðc0 + 10c11 + 12c12 + 1:9c21 + 2:0c22 + 5c31 24d11 20d12 2:6d21 2:1d22 5:3d31 Þ c, cij , dij
s:t: c0 +
3 X j=1
c1j d1j Mcr +
2 X
ð38Þ
c2j d2j p + ðc31 d32 Þl I½0,
j=1
and
+ ‘Þ ðMcr
pl1 Þ
Liu and Liu
9
Table 4. Samples data of variables Mcr , p, and l. Sample data Mcr, (kN m) p, (kN) l, (m)
23.1; 24; 24.3; 25; 24.9; 24.6; 23.8; 24; 24.1; 24.2; 24.9; 23.6; 24.8 . 2.12; 2.15; 2.24; 2.32; 2.39; 2.4; 2.38; 2.27; 2.15; 2.23; 2.34; 2.33 . 4.70; 5.08; 5.10; 4.65; 4.60; 4.78; 4.89; 5.08; 5.05; 4.88;4.75 .
= sup ðc0 + 24c11 + 20c12 + 2:6c21 + 2:1c22 + 5:3c31 10d11 12d12 1:9d21 2:0d22 5d31 Þ R c, cij , dij
s:t: c0 +
3 X j=1
c1j d1j Mcr +
2 X
ð39Þ
c2j d2j p + ðc31 d32 Þl I½0,
+ ‘Þ ðMcr
pl1 Þ
j=1
Thus, the structural reliability of a cantilever beam can be obtained as ½0:7365, 1. It is much more inaccurate than the result of the new model, and the reason lies in that copula function can be seen as an additional condition to the constraints leading to the shrink of feasible region P. In addition, interval analysis, fuzzy mathematics, and other non-probabilistic theories have been used for safety analysis as well when reliability data are in short. In line with samples data shown in Table 4, Mcr , p, and l can be indicated as interval variables, such as Mcr 2 ½23:1 kN m, 25 kN m, p 2 ½2:1 kN, 2:4 kN and l 2 ½4:6 m, 5:1 m So, means and variances of Mcr , p, and l can be obtained as Mcrc = 24:05 kN m, Mcrr = 0:95 kN m, c = 4:85 m, pccr = 2:25 kN, prcr = 0:15 kN, lcr r lcr = 0:25 m
and the mean and variance of g(Mcr , p, l) are g c (Mcr , p, l) = 13:1 kN m, gr (Mcr , p, l) = 2:14 kN m. According to Wu et al.,24 structural reliability measure R0 can be computed by 1 R0 = min max ðh + 1Þ, 1 , 1 , 2 g c ðMcr , p, lÞ where h = r g ðMcr , p, lÞ
ð40Þ
Where h is the structural reliability computed by interval analysis method.24 So the structural reliability of this cantilever beam can be gained as R0 = 1. It can be deemed that the proposed method is a little more conservative than interval analysis method. By the above-mentioned comparisons, we can conclude that the proposed method is much more effective than original natural extension models, and it is more
conservative than interval analysis methods. In this way, it is rational for general use of structural reliability analysis. Nevertheless, it should be noted that different choice of copula functions will lead to disparate precision, and so following works will focus on the determination of copula functions.
Conclusion In practical engineering, reliability data concerning to the variables vector are rather tough to get, while the single variable’s information, on the contrary, is easy to get. This article proposes an improved imprecise structural reliability analysis method by the introduction of copula functions to the natural extension model, the big advantage is that each variable’s marginal probability density functions as well as the variable correlativity can be researched and characterized, respectively. Imprecise probability theory in reliability engineering has been studied for long, and it has made great achievements in theory, but no details about application process are proposed for reference. Thus, we give the detailed reliability modeling steps in this article for engineers to refer to; moreover, an engineering example of a cantilever beam which is widely applied in the oil platform has been taken into account to illustrate the effectiveness of the novel method proposed. Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research is partially supported by the Innovative Academic Team Project of Guangzhou Education System (grant number: 1201610013).
10
Advances in Mechanical Engineering
ORCID iD Zheng Liu
https://orcid.org/0000-0002-2879-5859
13.
References 1. Sanches RF, de Jesus AM, Correia JA, et al. A probabilistic fatigue approach for riveted joints using Monte Carlo simulation. J Constr Steel Res 2015; 110: 149–162. 2. Correia J, Apetre N, Arcari A, et al. Generalized probabilistic model allowing for various fatigue damage variables. Int J Fatigue 2017; 100: 187–194. 3. Zhu SP, Lei Q and Wang QY. Mean stress and ratcheting corrections in fatigue life prediction of metals. Fatigue Fract Eng M 2017; 40: 1343–1354. 4. Yu ZY, Zhu SP, Liu Q, et al. A new energy-critical plane damage parameter for multiaxial fatigue life prediction of turbine blades. Materials 2017; 10: 513. 5. Zhu SP, Huang HZ, Peng W, et al. Probabilistic physics of failure-based framework for fatigue life prediction of aircraft gas turbine discs under uncertainty. Reliab Eng Syst Safe 2016; 146: 1–12. 6. Zhu SP, Huang HZ, Li Y, et al. Probabilistic modeling of damage accumulation for time-dependent fatigue reliability analysis of railway axle steels. Proc IMechE, Part F: J Rail and Rapid Transit 2015; 229: 23–33. 7. Zhu SP, Liu Q, Lei Q, et al. Probabilistic fatigue life prediction and reliability assessment of a high pressure turbine disc considering load variations. Int J Damage Mech. Epub ahead of print 30 October 2017. DOI: 10.1177/1056789517737132. 8. Zhu SP, Foletti S and Beretta S. Probabilistic framework for multiaxial LCF assessment under material variability. Int J Fatigue 2017; 103: 371–385. 9. Peng WW, Balakrishnan N and Huang HZ. Reliability modelling and assessment of a heterogeneously repaired system with partially relevant recurrence data. Appl Math Modell 2018; 59: 696–712. 10. Wang C and Qiu ZP. Hybrid uncertain analysis for temperature field prediction with random, fuzzy and interval parameters. Int J Therm Sci 2015; 98: 124–134. 11. Meng DB, Li YF, Huang HZ, et al. Reliability-based multidisciplinary design optimization using subset simulation analysis and its application in the hydraulic transmission mechanism design. J Mech Des 2015; 137: 051402. 12. Peng WW, Li YF, Yang YJ, et al. Bayesian degradation analysis with inverse Gaussian process models under
14.
15.
16.
17.
18. 19.
20. 21.
22. 23.
24.
25.
26.
time-varying degradation rates. IEEE T Reliab 2017; 66: 84–96. Xing Y and Jiang P. Bayesian evaluation method for reliability growth test based on the ordered Dirichlet distribution. Syst Eng Electron 2017; 39: 1178–1181. El-Damcese MA and Ramadan DA. Analyzing system reliability using fuzzy mixture generalized linear failure rate distribution. Am J Math Stat 2015; 5: 43–51. Suo B, Cheng YS, Zeng C, et al. Reliability model of typical systems for imprecise probability using evidence theory. J Syst Simul 2013; 25: 317–321. Xiao NC, Zuo MJ and Zhou C. A new adaptive sequential sampling method to construct surrogate models for efficient reliability analysis. Reliab Eng Syst Safe 2018; 169: 330–338. Meng DB, Zhang H and Huang T. A concurrent reliability optimization procedure in the earlier design phases of complex engineering systems under epistemic uncertainties. Adv Mech Eng 2016; 8: 1–8. Walley P. Statistical reasoning with imprecise probabilities. London: Chapman & Hall, 1991. Utkin LV and Kozine IO. Different faces of the natural extension. In: Proceedings of the 2nd international symposium on imprecise probabilities and their applications, Ithaca, NY, 26–29 June 2001. Maastricht: Shaker Publishing. Elishakoff I. Discussion on a non-probabilistic concept of reliability. Struct Safe 1995; 17: 195–199. Zhao J. The study of the duality theorem for multiobjective programming. Master’s Thesis, Chongqing Normal University, Chongqing, China, 2012. Nelsen RB. An introduction to copulas. New York: Springer, 1999. Xiao NC, Duan LB and Tang ZC. Surrogate-modelbased reliability method for structural systems with dependent truncated random variables. J Risk Reliab 2017; 231: 265–274. Wu JH, Wang XJ and Zhang Y. The choice of the copula function in the correlation analysis. Stat Res 2014; 31: 99–105. Zhu SP, Huang HZ, Smith R, et al. Bayesian framework for probabilistic low cycle fatigue life prediction and uncertainty modeling of aircraft turbine disk alloys. Prob Eng Mech 2013; 34: 114–122. Hou YN. The estimation and application of copula function. Master’s Thesis, Huazhong University of Science and Technology, Wuhan, China, 2013.
