An improved approach by introducing copula functions for structural reliability analysis under hybrid uncertainties

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.

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 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 authors^9–14 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 algorithms ¹¹ 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.

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

Definition 1

A “gamble” is a bounded and real-valued function defined on domain $\Omega$ , such as $X,g(X),f(X)$ .

Definition 2

Assume $P(X)$ is the probability that gamble $X$ will happen. Then $[\underset{¯}{P}(X),\overline{P}(X)]$ is the bound of $P(X)$ , and $\Delta =\overline{P}(X)-\underset{¯}{P}(X)$ , where $\Delta$ expresses the inaccuracy or uncertainty of the interval $[\underset{¯}{P}(X),\overline{P}(X)]$ . Here, $\underset{¯}{P}(X)$ is the infimum of the values that $P(X)$ will possibly take while $\overline{P}(X)$ 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

$$R\left(t\right)=\mathrm{E}\left(I_{\left[t,+\infty \right)}\left(X\right)\right)={\displaystyle \underset{R^{+}}{\int }{I}_{\left[t,+\infty \right)}\left(X\right)}\rho \left(X\right)\mathrm{d}X$$

(1)

$$F\left(t\right)=\mathrm{E}\left(I_{[0,t]}\left(X\right)\right)=\underset{R^{+}}{\int }{I}_{[0,t]}\left(X\right)\rho \left(X\right)\mathrm{d}X$$

(2)

$$\mathrm{Mean}\mathrm{time}\mathrm{to}\mathrm{failures}\left(\mathrm{MTTF}\right)=\mathrm{E}\left(X\right)=\underset{R^{+}}{\int }X\rho \left(X\right)\mathrm{d}X$$

(3)

and

$$\mathrm{Residual}\mathrm{MTBF}\left(\mathrm{mean}\mathrm{time}\mathrm{between}\mathrm{failures}\right)=\mathrm{E}\left(X-t|I_{\left[t,+\infty \right)}\left(X\right)\right)$$

(4)

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, $x_1,\dots ,x_n$ are variables in $X=(x_1,\dots ,x_n)$ , 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 $m_i$ reliability data relevant to the variable $x_i$ , all of which as well as the reliability measure $M$ can be presented in the shape of $M=\mathrm{E}(g(X))$ , $\mathrm{E}({\phi }_{\mathrm{ij}}(x_i))\in [{\underset{¯}{a}}_{\mathrm{ij}},{\overline{a}}_{\mathrm{ij}}]$ , where $i=1,\dots ,n$ , $j=1,\dots ,m_i$ , and ${\phi }_{\mathrm{ij}}(x_i)$ is a confirmable function corresponding to the $j\mathrm{th}$ 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

$$\overline{M}\left(\underset{¯}{M}\right)=\underset{\mathrm{P}}{max}\left(\underset{\mathrm{P}}{min}\right)\underset{{\mathrm{R}}_{+}^n}{\int }g\left(X\right)\rho \left(X\right)\mathrm{d}X$$

(5)

$$\begin{array}{c}s.t.\hfill \\ \rho \left(X\right)\ge 0,\underset{{\mathrm{R}}_{+}^n}{\int }\rho \left(X\right)\mathrm{d}X=1,{\underset{¯}{a}}_{\mathrm{ij}}\hfill \\ \le \underset{{\mathrm{R}}_{+}^n}{\int }{\phi }_{\mathrm{ij}}\left(x_i\right)\rho \left(X\right)\mathrm{d}X\le {\overline{a}}_{\mathrm{ij}},i\le n,j\le m_i\hfill \end{array}$$

(6)

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 , Elishakoff ²⁰ and Zhao, ²¹ the above optimization model can be rewritten as

$$\begin{array}{c}\underset{¯}{R}=\underset{c,c_{\mathrm{ij}},d_{\mathrm{ij}}}{sup}(c+\sum _{i=1}^n\sum _{j=1}^{m_i}(c_{\mathrm{ij}}{\underset{¯}{a}}_{\mathrm{ij}}-d_{\mathrm{ij}}{\overline{a}}_{\mathrm{ij}}))\hfill \\ s.t.\hfill \\ c+\sum _{i=1}^n\sum _{j=1}^{m_i}(c_{\mathrm{ij}}-d_{\mathrm{ij}})f_{\mathrm{ij}}\left(X\right)\le I_{[0,+\infty )}\left(g\left(X\right)\right)\hfill \end{array}$$

(7)

and

$$\begin{array}{c}\overline{R}=\underset{c,c_{\mathrm{ij}},d_{\mathrm{ij}}}{inf}(c+\sum _{i=1}^n\sum _{j=1}^{m_i}(c_{\mathrm{ij}}{\overline{a}}_{\mathrm{ij}}-d_{\mathrm{ij}}{\underset{¯}{a}}_{\mathrm{ij}}))\hfill \\ s.t.\hfill \\ c+\sum _{i=1}^n\sum _{j=1}^{m_i}(c_{\mathrm{ij}}-d_{\mathrm{ij}})f_{\mathrm{ij}}\left(X\right)\ge I_{[0,+\infty )}\left(g\left(X\right)\right)\hfill \end{array}$$

(8)

where $c\in R,c_{\mathrm{ij}},d_{\mathrm{ij}}\in R^{+}$ , $c,c_{\mathrm{ij}},\mathrm{and}{d}_{\mathrm{ij}}$ are new optimization variables. Besides, $f_{\mathrm{ij}}(X)$ , $g(X)$ , $[{\underset{¯}{a}}_{\mathrm{ij}},{\overline{a}}_{\mathrm{ij}}]$ have 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 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 variable $X$ belongs to an interval $[\underset{¯}{\mu },\overline{\mu }]$ and its variance to an interval $[\underset{¯}{\sigma },\overline{\sigma }]$ , the mean and variance of variable $X$ under the framework of imprecise probability theory can be then expressed as

$$\underset{¯}{\mu }\le \mathrm{E}\left(X\right)=\underset{\Omega }{\int }X\rho \left(X\right)\mathrm{d}X\le \overline{\mu }$$

(9)

$${\underset{¯}{\mu }}^2+{\underset{¯}{\sigma }}^2=\mathrm{E}\left(X^2\right)=\underset{\Omega }{\int }{X}^2\rho \left(X\right)\mathrm{d}X\le {\overline{\mu }}^2+{\overline{\sigma }}^2$$

(10)

When $\underset{¯}{\mu }=\overline{\mu },\underset{¯}{\sigma }=\overline{\sigma }$ , variable $X$ degenerates to stochastic variable.

The quantification of fuzzy data

Here, we take triangular fuzzy number as an example, and it can be defined as

$${\mu }^{~}\left(x\right)=\{\begin{array}{c}\frac{(x-a)}{(b-a)},a\le x<b\hfill \\ \multicolumn{1}{c}{1,x=b}\\ \multicolumn{1}{c}{\frac{(c-x)}{(c-b)},b\le x<c}\\ \multicolumn{1}{c}{0,\mathrm{others}}\end{array}$$

(11)

The fuzzy variable $x$ under the framework of imprecise probability theory can be indicated as

$$\underset{¯}{\mathrm{E}}\left(I_A\left(x\right)\right)=\alpha ,\overline{\mathrm{E}}\left(I_A\left(x\right)\right)=1,A=[{\left(\tilde{P}\right)}_{\alpha }^L,{\left(\tilde{P}\right)}_{\alpha }^U]$$

(12)

where $I_{[t_1,t_2]}(X)$ is the indicator function defined as $I_{[t_1,t_2]}(X)=\{\begin{array}{c}1,X\in [t_1,t_2]\hfill \\ \multicolumn{1}{c}{0,\mathrm{others}}\end{array}$ , and $A$ serves as $\alpha$ level set.

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 $[400,500]$ , and its reliability at $t=300$ is equal or greater than $0.9$ . Accordingly, it can be written as

$$400\le \mathrm{E}\left(T\right)\le 500,f\left(T\right)=T$$

(13)

$$0.9\le \mathrm{E}\left(I_{[300,+\infty )}\left(T\right)\right)\le 1,f\left(T\right)=I_{[300,+\infty )}\left(T\right)$$

(14)

Imprecise structural reliability analysis under hybrid uncertainties

Presume $g(X)$ is the structural system’s limit-state function, where $X=(x_1,x_2,\dots ,x_n)$ and $x_i$ is the design variables with $i=1,\dots ,n$ . Denote $\Phi$ is the failure region and $\Phi =\{X:g(X)<0\}$ . In this way, the reliability $R$ of the structure is likely to be computed as

$$R=\mathrm{Pr}\{g(X)\ge 0\}$$

(15)

and the failure probability $F$ may be obtained by

$$F=\mathrm{Pr}\{g\left(X\right)<0\}=1-R$$

(16)

Reliability data collected beforehand are quantified as follows

$${\underset{¯}{a}}_{\mathrm{ij}}\le \underset{\Omega }{\int }{f}_{\mathrm{ij}}\left(X\right)\rho \left(X\right)\mathrm{d}X\le {\overline{a}}_{\mathrm{ij}},i=1,\dots ,n,j=1,\dots ,m_i$$

(17)

where $m_i$ is the total number of reliability data related to $x_i$ , and $f_{\mathrm{ij}}(X)$ is a known function identify with the $j\mathrm{th}$ reliability data because $j=1,\dots ,m_i$ , $\rho (X)$ is the joint probability density function, and $\Omega$ 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

$$\begin{array}{c}\underset{¯}{R}=\underset{\mathrm{P}}{inf}\underset{\Omega }{\int }{I}_{[0,+\infty )}\left(g\left(X\right)\right)\rho \left(X\right)\mathrm{d}X\hfill \\ \overline{R}=\underset{\mathrm{P}}{sup}\underset{\Omega }{\int }{I}_{[0,+\infty )}\left(g\left(X\right)\right)\rho \left(X\right)\mathrm{d}X\hfill \end{array}$$

(18)

$$\begin{array}{c}s.t.\hfill \\ \rho \left(X\right)\ge 0,\underset{\Omega }{\int }\rho \left(X\right)\mathrm{d}X=1,\hfill \\ {\underset{¯}{a}}_{\mathrm{ij}}\le \underset{\Omega }{\int }{f}_{\mathrm{ij}}\left(X\right)\rho \left(X\right)\mathrm{d}X\le {\overline{a}}_{\mathrm{ij}},i=1,\dots ,n,j=1,\dots ,m_i\hfill \end{array}$$

(19)

Hence, set P is made up of all possible probability density functions which satisfied the constraints (19), that is $P=\{\rho (X)\}$ .

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 practice^6,22 concentrates mainly on theoretical analysis. The main reasons are elucidated as follows:

1. The great mass of natural extension models requires reliability data to be characterized by upper and lower precisions, such as $\mathrm{E}(f(x))\in [\underset{¯}{a},\overline{a}]$ , 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.

2. The most imprecise structural reliability models use joint probability density function $\rho (X)$ to quantify reliability data. However, $\rho (X)$ 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.

3. 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 $\Delta$ . 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 distribution^23,24 and allows us to study probability density functions and correlation of variables correspondingly.

Assume $\rho (X)$ is the joint probability density function of $X$ and ${\rho }_i(X)$ is the marginal probability density function of $X_i$ , according to Sklar’s theorem

$$\rho \left(X\right)=c(F_1\left(x_1\right),\dots ,F_n\left(x_n\right))·\underset{i=1}{\overset{n}{\Pi }}{\rho }_i\left(x_i\right)$$

(20)

where $F_i(x_i)$ is marginal probability distribution function of $x_i$ , and $c(F_1(x_1),\dots ,F_n(x_n))$ is probability density function of copula function $C(F_1(x_1),\dots ,F_n(x_n))$ , which is

$$c(F_1\left(x_1\right),\dots ,F_n\left(x_n\right))=\frac{{\partial }^{n}C(F_1\left(x_1\right),\dots ,F_n\left(x_n\right))}{\partial F_1\left(x_1\right),\dots ,\partial F_n\left(x_n\right)}$$

(21)

If marginal probability distribution functions of $F_i(x_i)$ are successive, and copula function $C(F_1(x_1),\dots ,F_n(x_n))$ and its probability density function $c(F_1(x_1),\dots ,F_n(x_n))$ are unique,^25,26 we may get the following equation

$$\underset{i=1}{\overset{n}{\Pi }}{\rho }_i\left(x_i\right)=\frac{\rho \left(X\right)}{c(F_1\left(x_1\right),\dots ,F_n\left(x_n\right))}$$

(22)

When all variables are fully independent and $c(F_1(x_1),\dots ,F_n(x_n))=1$ , 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

$$\underset{¯}{R}\left(\overline{R}\right)=\underset{\mathrm{P}}{inf}\left(\underset{\mathrm{P}}{sup}\right)\underset{\Omega }{\int }{I}_{[0,+\infty )}\left(g\left(X\right)\right)\rho \left(X\right)\mathrm{d}X$$

(23)

$$\begin{array}{c}s.t.\hfill \\ \rho \left(X\right)\ge 0,\underset{\Omega }{\int }\rho \left(X\right)\mathrm{d}X=1,\hfill \\ {\underset{¯}{a}}_k\le \underset{\Omega }{\int }{f}_k\left(X\right)\rho \left(X\right)\mathrm{d}X\le {\overline{a}}_k,k=1,\dots ,K\hfill \\ {\underset{¯}{a}}_{\mathrm{ij}}\le \underset{\Omega }{\int }\frac{f_{\mathrm{ij}}\left(x_i\right)}{c(F_1\left(x_1\right),\dots ,F_n\left(x_n\right))}\rho \left(X\right)\mathrm{d}{x}_i\le {\overline{a}}_{\mathrm{ij}},i=1,\dots ,n,\hfill \\ j=1,\dots ,m_i\hfill \end{array}$$

(24)

In the above equations, $g(X)$ is the limit-state function, and $\rho (X)$ is the joint probability density functions of $X$ . $F_i(x_i)$ is the marginal probability distribution function of $x_i$ , and $c(F_1(x_1),\dots ,F_n(x_n))$ is the copula probability density function. Besides, $f_k(X)$ is a known function in line with the $k\mathrm{th}$ reliability data relevant to variables vector $X$ , where $k=1,\dots ,K$ , $f_{\mathrm{ij}}(x_i)$ is a known function in accordance with the $j\mathrm{th}$ reliability data related to variable $x_i$ and $j=1,\dots ,m_i$ .

Consequently, the above optimization model can be rewritten as the Kuznetsov’s form

$$\begin{array}{c}\underset{¯}{R}=\underset{c,c_{\mathrm{ij}},d_{\mathrm{ij}}}{sup}(c_0+\sum _{k=1}^K(c_k{\underset{¯}{a}}_k-d_k{\overline{a}}_k)+\sum _{i=1}^n\sum _{j=1}^{m_i}(c_{\mathrm{ij}}{\underset{¯}{a}}_{\mathrm{ij}}-d_{\mathrm{ij}}{\overline{a}}_{\mathrm{ij}}))\hfill \\ s.t.\hfill \\ c_0+\sum _{i=1}^n\sum _{j=1}^{m_i}(c_{\mathrm{ij}}-d_{\mathrm{ij}})\frac{f_{\mathrm{ij}}\left(x_i\right)}{c(F_1\left(x_1\right),\dots ,F_n\left(x_n\right))}\hfill \\ +\sum _{k=1}^K(c_k-d_k)f_k\left(X\right)\le I_{[0,+\infty )}\left(g\left(X\right)\right)\hfill \end{array}$$

(25)

and

$$\begin{array}{c}\overline{R}=\underset{c,c_{\mathrm{ij}},d_{\mathrm{ij}}}{inf}(c_0+\sum _{k=1}^K(c_k{\overline{a}}_k-d_k{\underset{¯}{a}}_k)+\sum _{i=1}^n\sum _{j=1}^{m_i}(c_{\mathrm{ij}}{\overline{a}}_{\mathrm{ij}}-d_{\mathrm{ij}}{\underset{¯}{a}}_{\mathrm{ij}}))\hfill \\ s.t.\hfill \\ c_0+\sum _{i=1}^n\sum _{j=1}^{m_i}(c_{\mathrm{ij}}-d_{\mathrm{ij}})\frac{f_{\mathrm{ij}}\left(x_i\right)}{c(F_1\left(x_1\right),\dots ,F_n\left(x_n\right))}+\sum _{k=1}^K(c_k-d_k)f_k\left(X\right)\ge I_{[0,+\infty )}\left(g\left(X\right)\right)\hfill \end{array}$$

(26)

where $c\in R,c_{\mathrm{ij}},d_{\mathrm{ij}},c_k,d_k\in R^{+}$ .

The parameter estimation of the copula density function $c(F_1(x_1),\dots ,F_n(x_n))$

The modified natural extension models partly reduce the imprecision $\Delta$ of $[\underset{¯}{R},\overline{R}]$ , and they correspond more to the real conditions of practical systems. According to Wu et al., ²⁴ different copula probability density functions $c(F_1(x_1),\dots ,F_n(x_n))$ lead to different degrees of inaccuracy. So, the determination of copula probability density function $c(F_1(x_1),\dots ,F_n(x_n))$ 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

$$C(u_1,\dots ,u_n;\mathrm{Z})={\Phi }_z({\Phi }^{-1}\left(u_1\right),\dots ,{\Phi }^{-1}\left(u_n\right))$$

(27)

and

$$c(u_1,\dots ,u_n;\mathrm{Z})={\left|\mathrm{Z}\right|}^{-\frac{1}{2}}\exp (-\frac{1}{2}{\varsigma }^T({\mathrm{Z}}^{-1}-I)\varsigma )$$

(28)

where $\mathrm{Z}$ is a symmetric positive matrix with diagonal elements equal to 1 and $\left|\mathrm{Z}\right|$ to be its determinant, and ${\Phi }_z$ is a standard normal distribution with correlation coefficient matrix $\mathrm{Z}$ . Besides, ${\varsigma }_i={\Phi }^{-1}(u_i),i=1,\dots ,n$ , $\varsigma =({\varsigma }_1,\dots ,{\varsigma }_n{)}^T$ , and $I$ is a unit matrix.

When $n=2$ , the multivariate Gaussian copula degenerates a bivariate normal, which is

$$C\left(u_1,u_2;\mathrm{Z}\right)={\displaystyle \underset{-\infty }{\overset{{\Phi }^{-1}\left(u_1\right)}{\int }}{\displaystyle \underset{-\infty }{\overset{{\Phi }^{-1}\left(u_2\right)}{\int }}\frac{1}{2\pi \sqrt{1-z^2}}\exp \left[-\frac{s^2-2st+t^2}{2\left(1-z^2\right)}\right]}}\mathrm{d}s\mathrm{d}t$$

(29)

• 2. Gumbel copula function is defined as

$$C\left(u_1,u_2;\alpha \right)=\exp \left(-{\left({\left(-\log u_1\right)}^{\alpha }+{\left(-\log u_2\right)}^{\alpha }\right)}^{\frac{1}{\alpha }}\right),\alpha \in \left[1,+\infty \right)$$

(30)

where $u_1,u_2$ are marginal probability distribution functions, and $\alpha$ is the parameter of Gumbel copula.

• 3. Frank copula function is defined as

$$C(u_1,u_2;\theta )=\frac{1}{\theta }\ln (1+\frac{(e^{-\theta u_1}-1)(e^{-\theta u_2}-1)}{e^{-\theta }-1})$$

(31)

where $u_1,u_2$ are marginal probability distribution functions, and $\alpha$ 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-by-step, 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:

• In light of the failure criteria, we should first determine the limit-state function of $g(X)=g(x_1,x_2,\dots ,x_n)$ , where $x_1,\dots ,x_n$ 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 $C(F_1(x_1),\dots ,F_n(x_n))$ 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 reliability, the natural extension model can be established as follows:

$$\begin{array}{c}\underset{¯}{R}=\underset{c,c_{\mathrm{ij}},d_{\mathrm{ij}}}{sup}(c_0+\sum _{k=1}^K(c_k{\underset{¯}{a}}_k-d_k{\overline{a}}_k)+\sum _{i=1}^n\sum _{j=1}^{m_i}(c_{\mathrm{ij}}{\underset{¯}{a}}_{\mathrm{ij}}-d_{\mathrm{ij}}{\overline{a}}_{\mathrm{ij}}))\hfill \\ s.t.\hfill \\ c_0+\sum _{i=1}^n\sum _{j=1}^{m_i}(c_{\mathrm{ij}}-d_{\mathrm{ij}})\frac{f_{\mathrm{ij}}\left(x_i\right)}{c(F_1\left(x_1\right),\dots ,F_n\left(x_n\right))}\hfill \\ +\sum _{k=1}^K(c_k-d_k)f_k\left(X\right)\le I_{[0,+\infty )}\left(g\left(X\right)\right)\hfill \end{array}$$

(32)

Table 1.

Reliability data collected for the very variable $x_i$ .

$X$	Data 1	…	…	Data K		Data $m_i$
$X$	$\mathrm{E}(f_1(X))\in [{\underset{¯}{a}}_1,{\overline{a}}_1]$	…	…	$\mathrm{E}(f_K(X))\in [{\underset{¯}{a}}_K,{\overline{a}}_K]$
$x_1$	$\mathrm{E}(f_{11}(x_1))\in [{\underset{¯}{a}}_{11},{\overline{a}}_{11}]$	…	…	$\mathrm{E}(f_{1K}(x_1))\in [{\underset{¯}{a}}_{1K},{\overline{a}}_{1K}]$	…	$\mathrm{E}(f_{1m_1}(x_1))\in [{\underset{¯}{a}}_{1m_1},{\overline{a}}_{1m_1}]$
$x_2$	$\mathrm{E}(f_{21}(x_2))\in [{\underset{¯}{a}}_{21},{\overline{a}}_{21}]$	…	…	$\mathrm{E}(f_{2K}(x_2))\in [{\underset{¯}{a}}_{2K},{\overline{a}}_{2K}]$	…	$\mathrm{E}(f_{2m_2}(x_2))\in [{\underset{¯}{a}}_{2m_2},{\overline{a}}_{2m_2}]$
…	…	…	…	…	…	…
$x_i$	$\mathrm{E}(f_{i1}(x_i))\in [{\underset{¯}{a}}_{i1},{\overline{a}}_{i1}]$	…	…	$\mathrm{E}(f_{\mathrm{iK}}(x_i))\in [{\underset{¯}{a}}_{\mathrm{iK}},{\overline{a}}_{\mathrm{iK}}]$	…	$\mathrm{E}(f_{i{m}_i}(x_i))\in [{\underset{¯}{a}}_{i{m}_i},{\overline{a}}_{i{m}_i}]$
…	…	…	…	…	…	…
$x_n$	$\mathrm{E}(f_{n1}(x_n))\in [{\underset{¯}{a}}_{n1},{\overline{a}}_{n1}]$	…	…	$\mathrm{E}(f_{\mathrm{nK}}(x_n))\in [{\underset{¯}{a}}_{\mathrm{nK}},{\overline{a}}_{\mathrm{nK}}]$	…	$\mathrm{E}(f_{n{m}_1}(x_n))\in [{\underset{¯}{a}}_{n{m}_n},{\overline{a}}_{n{m}_n}]$

In these equation, $X=x_1,\dots ,x_n$ are variables related to the reliability of this structure. ${\underset{¯}{a}}_k$ and ${\overline{a}}_k$ are the lower and upper previsions, and $K$ is the sum of reliability date related to variables vector $X$ . ${\underset{¯}{a}}_{\mathrm{ij}}$ and ${\overline{a}}_{\mathrm{ij}}$ are the lower and upper previsions, and $i$ is the total number of variables. $m_i$ is the sum of reliability date related to variable $x_i$ , and $c(F_1(x_1),\dots ,F_n(x_n))$ is a copula density function with parameter $\theta$ . $\rho (X)$ is the joint probability density function of $X$ , and $u_i={\rho }_i(x_i)$ is the marginal probability density function of $x_i$ .

The procedure of imprecise reliability analysis can be illustrated by Figure 1.

Figure 1.

Procedure illustration of the proposed method.

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 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.

Figure 2.

A simplified structure of the cantilever beam.

Denote $M_{\mathrm{cr}}$ is the maximum flexion this cantilever beam can bear, and so the limit-state function of this structure can be written as

$$g(M_{\mathrm{cr}},p,l)=M_{\mathrm{cr}}-\mathrm{pl}$$

(33)

Reliability data about $M_{\mathrm{cr}},p,\mathrm{and}l$ we have collected are displayed in Table 2.

Table 2.

Reliability data collected for $p,M_{\mathrm{cr}},\mathrm{and}l$ .

	$j=1$	$j=2$
$M_{\mathrm{cr}}$	${\mathrm{E}}_1(M_{\mathrm{cr}})\in [10,24]\mathrm{kN}\mathrm{m}$	${\mathrm{E}}_2(M_{\mathrm{cr}})\in [12,20]\mathrm{kN}\mathrm{m}$
$p$	${\mathrm{E}}_1(p)\in [1.9,2.6]\mathrm{kN}$	${\mathrm{E}}_2(p)\in [2.0,2.1]\mathrm{kN}$
$l$	${\mathrm{E}}_1(l)\in [5,5.3]\mathrm{m}$

As we all know, the flexion of a structure is affected by its length. In order to determine the correlation of $M_{\mathrm{cr}},p$ , and $l$ , samples shown in Table 3 are then collected.

Table 3.

A set of sample data of variables $M_{\mathrm{cr}},p,\mathrm{and}l$ .

No.	$M_{\mathrm{cr}}$	$p$	$l$
1	$23\mathrm{kN}\mathrm{m}$	$2.4\mathrm{kN}$	$4.9\mathrm{m}$
2	$22.8\mathrm{kN}\mathrm{m}$	$2.3\mathrm{kN}$	$4.6\mathrm{m}$
3	$24.6\mathrm{kN}\mathrm{m}$	$2.3\mathrm{kN}$	$5.1\mathrm{m}$
4	$24\mathrm{kN}\mathrm{m}$	$2.2\mathrm{kN}$	$5.1\mathrm{m}$
5	$23.4\mathrm{kN}\mathrm{m}$	$2.1\mathrm{kN}$	$4.9\mathrm{m}$
6	$25.1\mathrm{kN}\mathrm{m}$	$2.2\mathrm{kN}$	$4.6\mathrm{m}$

Suppose the relationship between $M_{\mathrm{cr}},p,$ and $l$ can be quantified by multivariate Gaussian copula, then the probability distribution function and probability density function of $M_{\mathrm{cr}},p,$ and $l$ respectively are

$$F\left(M_{\mathrm{cr}},p,l\right)=C\left(F\left(M_{\mathrm{cr}}\right),F\left(p\right),F\left(l\right);\mathrm{Z}\right)={\Phi }_z\left({\Phi }^{-1}\left(F\left(M_{\mathrm{cr}}\right)\right),{\Phi }^{-1}\left(F\left(p\right)\right),{\Phi }^{-1}\left(F\left(l\right)\right)\right)$$

(34)

and

$$c(F\left(M_{\mathrm{cr}}\right),F\left(p\right),F\left(l\right);\mathrm{Z})={\left|\mathrm{Z}\right|}^{-\frac{1}{2}}\exp (-\frac{1}{2}{\varsigma }^T({\mathrm{Z}}^{-1}-I)\varsigma )$$

(35)

where $F(M_{\mathrm{cr}}),F(p)$ , and $F(l)$ are marginal probability distribution functions of $M_{\mathrm{cr}},p,$ and $l$ , and ${\Phi }_z$ is a standard normal distribution with correlation coefficient matrix $\mathrm{Z}$ of ${\varsigma }_1={\Phi }^{-1}(F(M_{\mathrm{cr}}))$ , ${\varsigma }_2={\Phi }^{-1}(F(p))$ , ${\varsigma }_3={\Phi }^{-1}(F(l))$ , and $\varsigma =({\varsigma }_1,{\varsigma }_2,{\varsigma }_3{)}^T$ . Moreover, $I$ is a unit matrix.

Let us presume $M_{\mathrm{cr}},p,$ and $l$ obeys normal distribution of $M_{\mathrm{cr}}~N(24.15,0.{3725}^2)$ , $p~N(2.25,0.{0092}^2)$ , and $l~N(4.87,0.{042}^2)$ . Hence, we can get the equation of $F(M_{\mathrm{cr}})=\Phi ((M_{\mathrm{cr}}-24.15)/0.3725)$ , $F(p)=\Phi ((p-2.25)/0.0092)$ , and $F(l)=\Phi ((p-4.87)/0.042)$ . So, the unified natural extension model for structural reliability can be constructed as

$$\begin{array}{c}\underset{¯}{R}=\underset{c,c_{\mathrm{ij}},d_{\mathrm{ij}}}{sup}(c_0+10c_{11}+12c_{12}+1.9c_{21}+2.0c_{22}+5c_{31}-24d_{11}-20d_{12}-2.6d_{21}-2.1d_{22}-5.3d_{31})\hfill \\ s.t.\hfill \\ c_0+\frac{1}{{\left|\mathrm{Z}\right|}^{-\frac{1}{2}}\exp (-\frac{1}{2}{\varsigma }^T({\mathrm{Z}}^{-1}-I)\varsigma )}\{\sum _{j=1}^3(c_{1j}-d_{1j})M_{\mathrm{cr}}+\sum _{j=1}^2(c_{2j}-d_{2j})p+(c_{31}-d_{32})l\}\le I_{[0,+\infty )}(M_{\mathrm{cr}}-p{l}_1)\hfill \end{array}$$

(36)

and

$$\begin{array}{c}\overline{R}=\underset{c,c_{\mathrm{ij}},d_{\mathrm{ij}}}{sup}(c_0+24c_{11}+20c_{12}+2.6c_{21}+2.1c_{22}+5.3c_{31}-10d_{11}-12d_{12}-1.9d_{21}-2.0d_{22}-5d_{31})\hfill \\ s.t.\hfill \\ c_0+\frac{1}{{\left|\mathrm{Z}\right|}^{-\frac{1}{2}}\exp (-\frac{1}{2}{\varsigma }^T({\mathrm{Z}}^{-1}-I)\varsigma )}\{\sum _{j=1}^3(c_{1j}-d_{1j})M_{\mathrm{cr}}+\sum _{j=1}^2(c_{2j}-d_{2j})p+(c_{31}-d_{32})l\}\ge I_{[0,+\infty )}(M_{\mathrm{cr}}-p{l}_1)\hfill \end{array}$$

(37)

Thus, the structural reliability of this cantilever beam can be gained as $[0.9453,1]$ . 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

$$\begin{array}{c}\underset{¯}{R}=\underset{c,c_{\mathrm{ij}},d_{\mathrm{ij}}}{sup}(c_0+10c_{11}+12c_{12}+1.9c_{21}+2.0c_{22}+5c_{31}-24d_{11}-20d_{12}-2.6d_{21}-2.1d_{22}-5.3d_{31})\hfill \\ s.t.\hfill \\ c_0+\sum _{j=1}^3(c_{1j}-d_{1j})M_{\mathrm{cr}}+\sum _{j=1}^2(c_{2j}-d_{2j})p+(c_{31}-d_{32})l\le I_{[0,+\infty )}(M_{\mathrm{cr}}-p{l}_1)\hfill \end{array}$$

(38)

and

$$\begin{array}{c}\overline{R}=\underset{c,c_{\mathrm{ij}},d_{\mathrm{ij}}}{sup}(c_0+24c_{11}+20c_{12}+2.6c_{21}+2.1c_{22}+5.3c_{31}-10d_{11}-12d_{12}-1.9d_{21}-2.0d_{22}-5d_{31})\hfill \\ s.t.\hfill \\ c_0+\sum _{j=1}^3(c_{1j}-d_{1j})M_{\mathrm{cr}}+\sum _{j=1}^2(c_{2j}-d_{2j})p+(c_{31}-d_{32})l\ge I_{[0,+\infty )}(M_{\mathrm{cr}}-p{l}_1)\hfill \end{array}$$

(39)

Thus, the structural reliability of a cantilever beam can be obtained as $[0.7365,1]$ . 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, $M_{\mathrm{cr}},p,$ and $l$ can be indicated as interval variables, such as

Table 4.

Samples data of variables $M_{\mathrm{cr}},p,\mathrm{and}l$ .

	Sample data
M cr , (kN m)	23.1; 24; 24.3; 25; 24.9; 24.6; 23.8; 24; 24.1; 24.2; 24.9; 23.6; 24.8 …
p , (kN)	2.12; 2.15; 2.24; 2.32; 2.39; 2.4; 2.38; 2.27; 2.15; 2.23; 2.34; 2.33 …
l , (m)	4.70; 5.08; 5.10; 4.65; 4.60; 4.78; 4.89; 5.08; 5.05; 4.88;4.75 …

$$M_{\mathrm{cr}}\in \left[23.1\mathrm{kN}\mathrm{m},25\mathrm{kN}\mathrm{m}\right],p\in \left[2.1\mathrm{kN},2.4\mathrm{kN}\right]\mathrm{and}l\in \left[4.6\mathrm{m},5.1\mathrm{m}\right]$$

So, means and variances of $M_{\mathrm{cr}},p,$ and $l$ can be obtained as

$$M_{\mathrm{cr}}^{\mathrm{c}}=24.05\mathrm{kN}\mathrm{m},M_{\mathrm{cr}}^{\mathrm{r}}=0.95\mathrm{kN}\mathrm{m},p_{\mathrm{cr}}^{\mathrm{c}}=2.25\mathrm{kN},p_{\mathrm{cr}}^{\mathrm{r}}=0.15\mathrm{kN},l_{\mathrm{cr}}^{\mathrm{c}}=4.85\mathrm{m},l_{\mathrm{cr}}^{\mathrm{r}}=0.25\mathrm{m}$$

and the mean and variance of $g(M_{\mathrm{cr}},p,l)$ are $g^{\mathrm{c}}(M_{\mathrm{cr}},p,l)=13.1\mathrm{kN}\mathrm{m},g^{\mathrm{r}}(M_{\mathrm{cr}},p,l)=2.14\mathrm{kN}\mathrm{m}$ .

According to Wu et al., ²⁴ structural reliability measure $R\prime$ can be computed by

$$R^{\prime }=\min \left\{\max \left[\frac{1}{2}\left(\eta +1\right),1\right],1\right\},\mathrm{where}\eta =\frac{g^{\mathrm{c}}\left(M_{\mathrm{cr}},p,l\right)}{g^{\mathrm{r}}\left(M_{\mathrm{cr}},p,l\right)}$$

(40)

Where $\eta$ is the structural reliability computed by interval analysis method. ²⁴ So the structural reliability of this cantilever beam can be gained as $R\prime =1$ . 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.