Gaussian copula–Bayesian dynamic linear model–based time-dependent reliability prediction of bridge structures considering nonlinear correlation between failure modes

Abstract

Bridge structures in service are subjected to long-term ambient environments and continuously increasing traffic demands; therefore, the physical quantities of the existing bridge structures are subjected to changes in both time and space. Through health monitoring for bridges, the data of the load effects of bridge structures, including strain, stress, deflection, and so on, of the specified structural components or structures, can be obtained. The novel monitoring systems installed in bridge structures contain sensors providing a large amount of monitored data. Proper processing of the continuously provided monitored data is one of the main difficulties in the field of structural health monitoring for time-dependent reliability prediction of structural components and/or structures. Under the actions of the common random sources of the time-dependent input load data, the time-dependent nonlinear correlation will exist among the time-dependent output load effect data. The Bayesian dynamic linear models are introduced to predict the time-dependent output variables and model the time-dependent nonlinear correlation coefficients between them. Then, the Gaussian copula–Bayesian dynamic linear models are built based on the Gaussian copula theory and the time-dependent correlation coefficients. The models can better and more feasibly predict the future reliability of bridge structures. Finally, an actual application example is provided to illustrate the feasibility and application of the built Gaussian copula–Bayesian dynamic linear models for structural reliability prediction.

Introduction

The time-dependent structural reliability formula is often expressed in terms of a vector of basic random variables characterizing uncertainties in quantities such as time-variant resistances (time-variant allowed load effect caused by structure degradation) and time-dependent load effects which are usually correlated with each other. For example, peak and permanent displacements of a system subjected to earthquake loading are correlated with each other. ¹ Uncertainty in structural reliability analysis^2–6 and the dependence modeling^7–9 have caused more attentions by the planners at home and abroad. But how to make reasonable uncertainty model is rarely studied in the analysis of structural time-dependent reliability.

To predict structural time-variant reliability reasonably, the complete probability information such as the time-dependent joint cumulative distribution function (CDF) and the time-variant joint probability density function (PDF) of the random time-variant vector should be known. ¹⁰ In engineering practice, however, the joint PDF cannot be often obtained due to the limited data. In most practical applications, only the marginal PDFs and the covariance matrix are available, so the incomplete statistical data are quite commonly available.^11–13 Furthermore, the marginal distributions are often nonnormal distributions, and it is accepted that the modeling and simulation of the non-Gaussian random vectors are challenging problems. ¹⁴ However, the nonnormal distributions could be transferred into the combination of a few normal distributions based on such limited information. So, the joint PDFs could be approximately defined. Consequently, an evaluation of the failure probability is possible through modeling and constructing the approximated multivariate distribution based on incomplete statistical data.

It is evident that the time-dependent reliability of an individual component cannot represent the time-variant reliability of the entire structural system. ¹⁵ So, it is of practical interest to reasonably determine the time-dependent reliability of the entire structural system. In addition, the time-dependent nonlinear correlation between failure modes of the entire structural system should be considered. And the copula function can handle with the correlation modeling, and the application of copulas in modeling temporal dependence of time series data appears to be a recent phenomenon. ¹⁶ Dynamic or time-varying copula models for bivariate time series data have been studied by Patton, ¹⁷ Dias and Embrachts, ¹⁸ and Van den Goorbergh et al., ¹⁹ and Joe ⁷ studied Markov time series using copulas with constant parameter, but little is used in structural system reliability analysis. With these in mind, time-dependent Gaussian copulas for modeling the time-dependent joint distributions on the time-dependent system reliability could be explored and are the topic of the present research.

The objective of this study is to predict the time-dependent reliability of bridge structures considering time-variant nonlinear correlation between failure modes based on the built Gaussian copula–Bayesian dynamic linear models (BDLMs), to achieve this goal; this article is organized as follows: In section “Gaussian copula theory,” the Gaussian copula theory is described; especially, the concepts of Pearson correlation coefficient is introduced briefly. In section “Gaussian copula-BDLMs,” one BDLM and its probability recursion processes are provided to predict the relevant information, and the predicted time-variant correlation coefficient formulas between different BDLMs are presented for constructing the time-dependent Gaussian copula function. In section “The modeling processes of Gaussian copula-BDLMs,” one kind of correlation coefficient-based Gaussian copula-BDLMs and their modeling processes are given in detail. In section “Reliability prediction of structural system considering the nonlinear correlation between failure modes,” the built Gaussian copula-BDLMs are used to predict the structural time-dependent reliability. One numerical example is provided to illustrate the feasibility and application of the built models in section “A numerical example: reliability prediction of Yitong River Bridge.”

Gaussian copula theory

Sklar’s theory ²⁰ clearly indicated that given random variables $\mathbf{X}=(X_1,X_2,\dots ,X_n)$ with continuous marginal distribution function $F_{X_1},F_{X_2},\dots ,F_{X_n}$ and n-dimensional joint distribution function F, there exists a unique copula function C such that for all $\mathbf{x}=(x_1,x_2,\dots ,x_n)\in {\mathbf{R}}^n$

$$F(x_1,x_2,\dots ,x_n)=C(F_{X_1}\left(x_1\right),F_{X_2}\left(x_2\right),\dots ,F_{X_n}\left(x_n\right))$$

(1)

According to equation (1), the joint PDF of the random variable $\mathbf{X}=(X_1,X_2,\dots ,X_n)$ is given as

$$p\left(x_1,x_2,\dots ,x_n\right)=c\left(F_{X_1}\left(x_1\right),F_{X_2}\left(x_2\right),\dots ,F_{X_n}\left(x_n\right)\right){\displaystyle \prod _{i=1}^n{p}_{X_i}\left(x_i\right)}$$

(2)

where $p_{X_i}(x_i)$ is the marginal PDF and $c(\mathbf{u})$ is the joint PDF of the copula function denoted as

$$c(u_1,\dots ,u_n)=\frac{{\partial }^{n}C}{\partial u_1\cdots \partial u_n}(u_1,\dots ,u_n)$$

(3)

If $F_{X_i}$ for $i=1,2,\dots ,n$ , have generalized inverse functions $F_{X_i}^{-1}$ for $i=1,2,\dots ,n$ , then equation (1) can be rewritten as

$$F(F_{X_1}^{-1}\left(u_1\right),F_{X_2}^{-1}\left(u_2\right),\dots ,F_{X_n}^{-1}\left(u_n\right))=C(u_1,u_2,\dots ,u_n)$$

(4)

where $u_i=F_{X_i}(i=1,2,\dots ,n)$ and $(u_1,u_2,\dots ,u_n)=\mathbf{u}$ .

Equation (4) reveals how to construct the copula function of a multivariate distribution with given marginal distributions. It follows from the probability integral transform that the random variables $U_i\equiv F_{X_i}(X_i),i=1,2,\dots ,n$ , are all uniformly distributed on (0, 1). Conversely, $X_i=F_{X_i}^{-1}(U_i),i=1,2,\dots ,n$ , and $\mathbf{U}=(U_1,U_2,\dots ,U_n)$ ; with this in mind, equation (4) is expressed as

$$\begin{array}{c}P(X_1\le x_1,X_2\le x_2,\dots ,X_n\le x_n)\hfill \\ =P(X_1\le F_{X_1}^{-1}\left(u_1\right),X_2\le F_{X_2}^{-1}\left(u_2\right),\dots ,X_n\le F_{X_n}^{-1}\left(u_n\right))\hfill \\ =P(U_1\le u_1,U_2\le u_2,\dots ,U_n\le u_n)\hfill \\ =C(u_1,u_2,\dots ,u_n)\hfill \end{array}$$

(5)

According to equations (1)–(5), the copula function organically combines each marginal distribution functions with the multivariate joint distribution functions; therefore, it not only considers the dependence among random variables but also simplifies the probability modeling process for multivariate random variables.

According to the construction method of the elliptical copulas, Gaussian copula function, belonging to the frequently used elliptical copula family, is defined as the joint normal CDF of multiple standard normal variables. For two-dimensional copula function, the corresponding CDF C and PDF c are, respectively

$$\begin{array}{c}C(u_1,u_2;\rho )={\Phi }_G({\Phi }^{-1}\left(u_1\right),{\Phi }^{-1}\left(u_2\right);\rho )\hfill \\ =\underset{-\infty }{\overset{{\Phi }^{-1}\left(u_1\right)}{\int }}\underset{-\infty }{\overset{{\Phi }^{-1}\left(u_2\right)}{\int }}\frac{1}{2\pi {(1-{\rho }^2)}^{1/2}}\left(\frac{-(s^2-2\rho \mathrm{st}+t^2)}{2(1-{\rho }^2)}\right)\mathrm{dsdt}\hfill \end{array}$$

(6)

$$\begin{array}{l}c\left(u_1,u_2;\rho \right)=\frac{1}{\sqrt{1-{\rho }^2}}\\ \exp \left(-\frac{{\Phi }^{-1}{\left(u_1\right)}^2+{\Phi }^{-1}{\left(u_2\right)}^2-2\rho {\Phi }^{-1}\left(u_1\right){\Phi }^{-1}\left(u_2\right)}{2\left(1-{\rho }^2\right)}\right)·\\ \exp \left(-\frac{{\Phi }^{-1}{\left(u_1\right)}^2{\Phi }^{-1}{\left(u_2\right)}^2}{2}\right)\end{array}$$

(7)

where $u_1=F_1(x_1)$ , $u_2=F_2(x_2)$ , $F_1(x_1)$ , and $F_2(x_2)$ are, respectively, the marginal CDF of $x_1$ and $x_2$ ; $\Phi$ is standard normal CDF; ${\Phi }^{-1}$ is the inverse standard normal distribution; and $\rho$ is a correlation parameter, restricted to the interval from −1 to 1. ²¹

With equations (6) and (7), the joint CDF and joint PDF of $x_1$ and $x_2$ can be, respectively, obtained as

$$\begin{array}{cc}\multicolumn{1}{c}{F(x_1,x_2)}& =C(u_1,u_2;\rho )=C(F_1\left(x_1\right),F_2\left(x_2\right);\rho )\hfill \\ \multicolumn{1}{c}{}& ={\Phi }_G({\Phi }^{-1}\left(F_1\left(x_1\right)\right),{\Phi }^{-1}\left(F_2\left(x_2\right)\right);\rho )\hfill \end{array}$$

(8)

$$\begin{array}{cc}\multicolumn{1}{c}{f(x_1,x_2)}& =c(u_1,u_2;\rho )f_1\left(x_1\right)f_2\left(x_2\right)\hfill \\ \multicolumn{1}{c}{}& =c(F_1\left(x_1\right),F_2\left(x_2\right);\rho )f_1\left(x_1\right)f_2\left(x_2\right)\hfill \end{array}$$

(9)

As far as the Gaussian copula function is concerned, the main problem is to determine the correlation parameter $\rho$ , which is solved with the usual Pearson product–moment correlation coefficient ²² in this article.

Pearson correlation coefficient

The Pearson product–moment correlation is also called linear correlation. ²³ Let $x_1$ and $x_2$ be two real valued random variables with finite variances. The Pearson correlation coefficient between $x_1$ and $x_2$ , denoted as $\rho (1,2)$ , is defined as

$$\begin{array}{cc}\multicolumn{1}{c}{\rho (1,2)}& =\frac{\mathrm{cov}(x_1,x_2)}{\sqrt{D\left(x_1\right)}\sqrt{D\left(x_2\right)}}=\frac{\mathrm{cov}(x_1,x_2)}{{\sigma }_{x_1}{\sigma }_{x_2}}\hfill \\ \multicolumn{1}{c}{}& =\underset{-\infty }{\overset{+\infty }{\int }}\underset{-\infty }{\overset{+\infty }{\int }}\left(\frac{x_1-{\mu }_{x_1}}{{\sigma }_{x_1}}\right)\left(\frac{x_2-{\mu }_{x_2}}{{\sigma }_{x_2}}\right)f(x_1,x_2)d{x}_{1}d{x}_2\hfill \end{array}$$

(10)

where $\mathrm{cov}(x_1,x_2)$ is the covariance between $x_1$ and $x_2$ ; $D(x_1)$ and $D(x_2)$ are, respectively, the variances of $x_1$ and $x_2$ ; ${\mu }_{x_1}$ and ${\mu }_{x_2}$ are, respectively, the mean values of $x_1$ and $x_2$ ; ${\sigma }_{x_1}$ and ${\sigma }_{x_2}$ are, respectively, the standard deviations of $x_1$ and $x_2$ ; and $f(x_1,x_2)$ is the joint PDF of $x_1$ and $x_2$ .

Based on the sampled data of $x_1$ and $x_2$ , the linear correlation coefficient is also denoted as

$$\rho (1,2)=\frac{\sum _{i=1}^N(x_{1i}-{\overline{x}}_1)(x_{2i}-{\overline{x}}_2)}{\sqrt{\sum _{i=1}^N{(x_{1i}-{\overline{x}}_1)}^2}\sqrt{\sum _{i=1}^N{(x_{2i}-{\overline{x}}_2)}^2}}$$

(11)

where $(x_{1i},x_{2i})$ is a pair sample about $x_1$ and $x_2$ ; N is the total pair number; and ${\overline{x}}_1$ and ${\overline{x}}_2$ are, respectively, the mean of the sample about $x_1$ and $x_2$ . If two random variables are completely dependent, then $\rho (1,2)=\pm 1$ , otherwise, $\rho (1,2)\in (-1,1)$ .

With equation (10), the relation ²⁴ between linear correlation coefficient and correlation parameters of Gaussian copula function can be expressed with

$$\begin{array}{cc}\multicolumn{1}{c}{\rho (1,2)}& =\underset{-\infty }{\overset{+\infty }{\int }}\underset{-\infty }{\overset{+\infty }{\int }}\left(\frac{x_1-{\mu }_{x_1}}{{\sigma }_{x_1}}\right)\left(\frac{x_2-{\mu }_{x_2}}{{\sigma }_{x_2}}\right)f(x_1,x_2)d{x}_{1}d{x}_2\hfill \\ \multicolumn{1}{c}{}& =\underset{-\infty }{\overset{+\infty }{\int }}\underset{-\infty }{\overset{+\infty }{\int }}\left(\frac{x_1-{\mu }_{x_1}}{{\sigma }_{x_1}}\right)\left(\frac{x_2-{\mu }_{x_2}}{{\sigma }_{x_2}}\right)c(F_1\left(x_1\right),F_2\left(x_2\right);\rho )f_1\left(x_1\right)f_2\left(x_2\right)d{x}_{1}d{x}_2\hfill \\ \multicolumn{1}{c}{}& =\underset{-\infty }{\overset{+\infty }{\int }}\underset{-\infty }{\overset{+\infty }{\int }}\left(\frac{x_1-{\mu }_{x_1}}{{\sigma }_{x_1}}\right)\left(\frac{x_2-{\mu }_{x_2}}{{\sigma }_{x_2}}\right)\frac{f_1\left(x_1\right)f_2\left(x_2\right)}{\sqrt{1-{\rho }^2}}\exp \left(f\left(\rho \right)\right)\exp \left(\frac{{\Phi }^{-1}{\left(u_1\right)}^2{\Phi }^{-1}{\left(u_2\right)}^2}{2}\right)d{x}_{1}d{x}_2\hfill \end{array}$$

(12)

where $f(\rho )=-({\Phi }^{-1}(u_1{)}^2+{\Phi }^{-1}(u_2{)}^2-2\rho {\Phi }^{-1}(u_1){\Phi }^{-1}(u_2))/(2(1-{\rho }^2))$ .

Based on equations (10) and (11), the linear correlation coefficients between $x_1$ and $x_2$ can be easily obtained, then the correlation parameters $\rho$ of Gaussian copula function can be computed with equation (12). Considering the computation complexity of equation (12), the following method is adopted to compute the correlation parameters of Gaussian copula function.

According to equations (10) and (11), if the marginal distributions of $x_1$ and $x_2$ are both normal distributions, combining the concept of Gaussian copula function, the correlation parameters $\rho$ can be obtained

$$\begin{array}{cc}\multicolumn{1}{c}{\rho }& =\rho ({\Phi }^{-1}\left(u_1\right),{\Phi }^{-1}\left(u_2\right))\hfill \\ \multicolumn{1}{c}{}& =\rho ({\Phi }^{-1}\left(F_1\left(x_1\right)\right),{\Phi }^{-1}\left(F_2\left(x_2\right)\right))\hfill \\ \multicolumn{1}{c}{}& =\rho (x_1,x_2)\hfill \\ \multicolumn{1}{c}{}& =\rho (1,2)\hfill \end{array}$$

(13)

With equation (13), the correlation parameters of Gaussian copula function can be solved. Due to the special characteristics of Gaussian copula function, namely, the marginal distributions of $x_1$ and $x_2$ are both normal distributions, so the Gaussian copula function about $x_1$ and $x_2$ can be easily obtained.

Gaussian copulas are able to capture symmetric dependence structures (including positive correlation and negative correlation) among random variables, which are shown in Figures 1 and 2.

Figure 1.

(a) The PDF plot for a Gaussian copula PDF, (b) the contour plot, and (c) the PDF plot on the antidiagonal line $x_2=-x_1+1$ .

Figure 2.

(a) The PDF plot for a Gaussian copula PDF, (b) the contour plot, and (c) the PDF plot on the diagonal line x₁ = x_2.

Gaussian copula function has been widely applied in static copula theory, but the time-dependent correlation modeling for the dynamic time series cannot be well handled with the static copula theory. In order to reasonably assess the structural time-variant reliability, the dynamic Gaussian copula function with time-dependent correlation parameters, namely, Gaussian copula-BDLMs, is built in this article, which can analyze the time-variant nonlinear correlation between different time series.

Gaussian copula-BDLMs

BDLMs

BDLMs are the predicting approaches based on a philosophy of information updating ²⁵ which define a dynamic model system of time series processes that can incorporate all useful monitored information into the model to update the prediction model. The BDLMs include a state equation, an observation equation, the initial information, and the time-dependent probability recursion processes based on Bayesian method. The state equation, which is very critical to determine, shows changes of the system with time and reflects inner dynamic changes of the system and random disturbances. The observation equation expresses the relationship between the measured data and the current state parameters of the system.

The BDLMs mean that the observation equation and the state equation are both linear. First-order autoregression (AR(1)) model is adopted to build the state equations in this article.

BDLMs based on AR(1) model

The time series model, AR(1) model, ²⁶ can be used to predict the future data of time t only based on the monitored data of time t, so that the AR(1) model can be applied to simply build the BDLMs.

• 1. AR(1) model

$$\left(\begin{array}{c}{y}_{1,t+1}\\ y_{2,t+1}\end{array}\right)=\left(\begin{array}{c}{\lambda }_1{y}_{1,t}\\ {\lambda }_2{y}_{2,t}\end{array}\right)+\left(\begin{array}{c}{\nu }_{1,t+1}\\ {\nu }_{2,t+1}\end{array}\right),\left(\begin{array}{c}{\nu }_{1,t+1}\\ {\nu }_{2,t+1}\end{array}\right)~\mathrm{N}\left(\mathbf{0},\left(\begin{array}{c}{V}_{1,t+1}\\ V_{2,t+1}\end{array}\right)\right),t=1,2,\dots ,T$$

(14)

where $y_{i,t+1}(i=1,2)$ is the observation data of the ith random variable at time t + 1, ${\nu }_{i,t+1}$ is the observation error of the ith AR(1) model, T is the total monitored time, and ${\lambda }_i$ is the regression coefficient of the ith AR(1) model.

• 2. State equation based on equation (14): Based on equation (14), the state variable keeps in line with the trend changes of monitored data; therefore, the transformed approximate state equation of monitored data is

$$\left(\begin{array}{c}{\theta }_{1,t+1}\\ {\theta }_{2,t+1}\end{array}\right)=\left(\begin{array}{c}{\lambda }_1{\theta }_{1,t}\\ {\lambda }_2{\theta }_{2,t}\end{array}\right)+\left(\begin{array}{c}{\omega }_{1,t+1}\\ {\omega }_{2,t+1}\end{array}\right),\left(\begin{array}{c}{\omega }_{1,t+1}\\ {\omega }_{2,t+1}\end{array}\right)~\mathrm{N}\left(0,\left(\begin{array}{c}{W}_{1,t+1}\\ W_{2,t+1}\end{array}\right)\right),t=1,2,\dots ,T$$

(15)

where ${\lambda }_i$ can be approximately obtained with equation (14) and ${\omega }_{i,t+1}$ is the state error of the ith AR(1) model.

• 3. Transferred BDLMs based on AR(1) model: According to the definition of BDLMs, ²⁵ based on AR(1) model, for each time t, the general and easy form of the dynamic linear models is defined as follows:

• State equation

  $$\left(\begin{array}{c}{\theta }_{1,t+1}\\ {\theta }_{2,t+1}\end{array}\right)=\left(\begin{array}{c}{\lambda }_1{\theta }_{1,t}\\ {\lambda }_2{\theta }_{2,t}\end{array}\right)+\left(\begin{array}{c}{\omega }_{1,t+1}\\ {\omega }_{2,t+1}\end{array}\right),\left(\begin{array}{c}{\omega }_{1,t+1}\\ {\omega }_{2,t+1}\end{array}\right)~\mathrm{N}\left(0,\left(\begin{array}{c}{W}_{1,t+1}\\ W_{2,t+1}\end{array}\right)\right)$$

  (17)

• Initial information

  $$\left(\begin{array}{c}{\theta }_{1,t}|{\mathbf{D}}_{1,t}\hfill \\ \multicolumn{1}{c}{{\theta }_{2,t}|{\mathbf{D}}_{2,t}}\end{array}\right)~\mathrm{N}(\left(\begin{array}{c}{m}_{1,t}\hfill \\ \multicolumn{1}{c}{m_{2,t}}\end{array}\right),\left(\begin{array}{c}{C}_{1,t}\hfill \\ \multicolumn{1}{c}{C_{2,t}}\end{array}\right)),t=1,2,\dots ,T$$

  (18)

where ${\mathbf{D}}_{i,t+1}(i=1,2)$ is the information set of the ith monitored variable at and before time t and ${\mathbf{D}}_{i,t+1}=\{y_{i,t+1},{\mathbf{D}}_{i,t}\}$ . It is assumed that error sequences ( ${\nu }_{i,t+1}$ and ${\omega }_{i,t+1}$ ) are internally independent, mutually independent, and independent of ${\theta }_{i,t+1}$ .

Probability recursion of BDLMs

The probability recursion processes of BDLMs based on AR(1) model

• 1. The state posteriori distribution at time t: For the mean m_i,t and the variance C_i,t, there is

$$({\theta }_{i,t}|{\mathit{D}}_{i,t})~\mathrm{N}[m_{i,t},C_{i,t}]$$

(19)

• 2. The state priori distribution at time t + 1

$$({\theta }_{i,t+1}|{\mathit{D}}_{i,t})~\mathrm{N}[a_{i,t+1},R_{i,t+1}]$$

(20)

where $a_{i,t+1}={\lambda }_i{m}_{i,t}\mathrm{and}{R}_{i,t+1}={\lambda }_i{C}_{i,t}\lambda {\prime }_i+W_{i,t+1}$ .

• 3. One-step prediction distribution at time t + 1

$$y_{i,t+1}|{\mathbf{D}}_{i,t}~\mathrm{N}({\mu }_{i,t+1},{\sigma }_{i,t+1}^2)$$

(21)

where ${\mu }_{i,t+1}=E\left(y_{i,t+1}|{\mathbf{D}}_{i,t}\right)\mathrm{and}{\sigma }_{i,t+1}^2=\mathrm{var}\left(y_{i,t+1}|{\mathbf{D}}_{i,t}\right)=R_{i,t+1}+V_{i,t+1}$ .

• 4. The state posteriori distribution at time t + 1

$${\theta }_{i,t+1}|{\mathbf{D}}_{i,t+1}~\mathrm{N}(m_{i,t+1},C_{i,t+1})$$

(22)

where $C_{i,t+1}=\mathrm{var}({\theta }_{i,t+1}|{\mathbf{D}}_{i,t+1})=R_{i,t+1}-A_{i,t+1}{A}_{i,t+1}^T{\sigma }_{i,t+1}^2$ , $m_{i,t+1}=E({\theta }_{i,t+1}|{\mathbf{D}}_{i,t+1})=a_{i,t+1}+A_{i,t+1}{e}_{i,t+1}$ , and $A_{i,t+1}=R_{i,t+1}({\sigma }_{i,t+1}^2{)}^{-1},e_{i,t+1}=y_{i,t+1}-f_{i,t+1}$ (one-step prediction error).

Determination of the main probability parameters of BDLMs

For the BDLMs, the main probability parameters are V_t+1, W_t+1, m_t, and C_t. The method of determining the main probability parameters is described in the following.

In this article, the interval period of model updating is 1 day; V_t+1 is estimated with the variance of differences between fitted trend data and monitored data. According to the researches,^25,27W_t+1 can be solved with equation (23)

$$W_{t+1}=\frac{-G_{t+1}{C}_t{G\prime }_{t+1}+C_t}{\delta }$$

(23)

where $G{\prime }_{t+1}$ is the transpose of $G_{t+1}$ and $\delta$ is the discount factor, which is usually 0.95–0.98 given by engineering experience of the authors.

$m_t$ is the mean value of state variable at and before time t; $C_t$ is the variance of state variable at and before time t. They can be both estimated with the smoothly processed or resampled monitored data at time t and before time t.

If the initial state data follow the lognormal distribution, then the state data can be transformed into a quasi-normal distribution ²⁸ with equations (24) and (25), the distribution parameters are, respectively, $\mu \prime$ (mean value) and $\sigma \prime$ (standard deviation)

$$\sigma \prime =\frac{\varphi (-1.645)}{g\left(x_0\right)}$$

(24)

$$\mu \prime =x_0+1.645\sigma \prime$$

(25)

where g(·) is the actual fitted PDF of the sample data (lognormal PDF), G(·) is the actual probability distribution function (lognormal probability distribution function), and G(x₀) = 0.05.

If the initial state data follow the other distributions, then the distribution can be approximately obtained as follows:

• 1. With estimation method of kernel density, the actual distribution function $G({\theta }_{t-1})$ of the initial state data is approximately $g({\theta }_{t-1})$ , namely

  $$G\left({\theta }_{t-1}\right)\approx g\left({\theta }_{t-1}\right)$$

  (26)

• 2. Since any set of data can be fitted by a few normal distributions, namely

$$G\left({\theta }_{t-1}\right)\approx \sum _{i=1}^n{p}_i\Phi \left(\frac{{\theta }_{t-1}-{\mu }_i}{{\sigma }_i}\right)$$

(27)

where $\sum _{i=1}^n{p}_i=1$ , $p_i\ge 0$ , and $\Phi (·)$ denotes the cumulative probability distribution functions of standard normal distribution.

• 3. The weights and distribution parameters of the fitted normal distributions can be obtained with the least residual error quadratic sum method ordinary least squares (OLS), namely

$$\mathrm{OLS}=\sqrt{\frac{1}{n}\sum _{j=1}^k{(g\left({\theta }_{t-1}\right)-\left(\sum _{i=1}^n{p}_i\Phi \left(\frac{{\theta }_{t-1}-{\mu }_i}{{\sigma }_i}\right)\right))}^2}$$

(28)

where $p_i$ is the weight. The values of unknown parameters for the fitted distributions can be possessed by the optimization computation with the rule of OLS. Furthermore, the optimized parameters must be determined to make sure that the value of OLS is the minimum.

Therefore, the distribution parameters (mean and variance) of state at time t are, respectively

$$m_t=\sum _{i=1}^N{p}_i{\mu }_i$$

(29)

$$C_t=\sum _{i=1}^N{p}_i^2{\sigma }_i^2$$

(30)

Correlation parameters of Gaussian copula-BDLMs

This article takes the two-dimensional random variables as the example, based on the BDLMs, the time-variant correlation parameters of Gaussian copula function is computed through Pearson linear correlation coefficients:

• 1. Time-variant correlation parameters of Gaussian copula function based on Pearson linear correlation coefficients: For the corresponding BDLMs to two different monitored variables $y_{1,t+1}$ and $y_{2,t+1}$ , the Pearson linear correlation coefficients between the two predicted random variables can be approximately expressed as

$${\rho }_{t+1}\left(1,2\right)=\frac{\mathrm{cov}\left(y_{1,t+1},y_{2,t+1}|{\mathbf{D}}_t\right)}{{\sigma }_{1,t+1}{\sigma }_{2,t+1}}\approx \frac{\mathrm{cov}\left({\mathbf{D}}_{1,t},{\mathbf{D}}_{2,t}\right)}{{\sigma }_{1,t+1}{\sigma }_{2,t+1}}$$

(31)

Because two predicted random variables both follow normal distributions, with equation (13), time-variant correlation parameter ${\rho }_{t+1}$ of Gaussian copula function for two predicted random variables at time t + 1, based on Pearson linear correlation coefficients, is

$${\rho }_{t+1}={\rho }_{t+1}(1,2)$$

(32)

For different BDLMs, the marginal distributions of the predicted random variables $y_{1,t}$ and $y_{2,t}$ are different, so the case about correlation parameters of Gaussian copula function was provided in the following.

Correlation parameters of Gaussian copula function considering the correlation between the predicted random variables based on the AR(1)-BDLMs

• 1. Time-variant correlation parameters of Gaussian copula function based on Pearson linear correlation coefficients: Suppose that the predicted random variables $y_{1,t+1}$ and $y_{2,t+1}$ are both obtained with the AR(1)-based BDLMs. Then, with equations (21) and (31), the Pearson linear correlation coefficient between them is

$${\rho }_{t+1}\left(1,2\right)=\frac{\mathrm{cov}\left(y_{1,t+1},y_{2,t+1}|{\mathbf{D}}_t\right)}{{\sigma }_{1,t+1}{\sigma }_{2,t+1}}\approx \frac{\mathrm{cov}\left({\mathbf{D}}_{1,t},{\mathbf{D}}_{2,t}\right)}{\sqrt{{\lambda }_1{C}_{1,t}{{\lambda }^{\prime }}_1+W_{1,t+1}+V_{1,t+1}}\sqrt{{\lambda }_2{C}_{2,t}{{\lambda }^{\prime }}_2+W_{2,t+1}+V_{2,t+1}}}$$

(33)

Furthermore, the corresponding correlation parameters of Gaussian copula function at time t + 1 is

$${\rho }_{t+1}={\rho }_{t+1}(1,2)$$

(34)

The modeling processes of Gaussian copula-BDLMs

The modeling processes of Gaussian copula-BDLMs based on the Pearson correlated coefficients

With equations (33) and (34), the correlation parameters of Gaussian copula function at time t + 1 is

$${\rho }_{t+1}={\rho }_{t+1}(1,2)$$

(35)

Then, based on the BDLMs of two predicted random variables $y_{1,t+1}$ and $y_{2,t+1}$ , and the Pearson linear correlation coefficients, the modeling processes of Gaussian copula-BDLMs are as follows:

$y_{1,t+1}$ follows normal distribution $\mathrm{N}({\mu }_{1,t+1},{\sigma }_{1,t+1}^2)$ , and $y_{2,t+1}$ also follows normal distribution $\mathrm{N}({\mu }_{2,t+1},{\sigma }_{2,t+1}^2)$ , namely

$$\begin{array}{c}F\left(y_{1,t+1}\right)=\mathrm{N}({\mu }_{1,t+1},{\sigma }_{1,t+1}^2)=\Phi \left(\frac{y_{1,t+1}-{\mu }_{1,t+1}}{{\sigma }_{1,t+1}}\right)\hfill \\ G\left(y_{2,t+1}\right)=\mathrm{N}({\mu }_{2,t+1},{\sigma }_{2,t+1}^2)=\Phi \left(\frac{y_{2,t+1}-{\mu }_{2,t+1}}{{\sigma }_{2,t+1}}\right)\hfill \end{array}$$

(36)

Let

$$u_{t+1}=F\left(y_{1,t+1}\right),v_{t+1}=G\left(y_{2,t+1}\right)$$

(37)

Then, the Gaussian copula-BDLMs can be expressed as

$$\begin{array}{l}\left(y_{1,t+1},y_{2,t+1}\right)~C\left(F\left(y_{1,t+1}\right),G\left(y_{2,t+1}\right)\right)=C\left(u,v\right)\\ ={\Phi }_G\left(\Phi \left(\frac{y_{1,t+1}-{\mu }_{1,t+1}}{{\sigma }_{1,t+1}}\right),\Phi \left(\frac{y_{2,t+1}-{\mu }_{2,t+1}}{{\sigma }_{2,t+1}}\right);{\rho }_{t+1}\right)\end{array}$$

(38)

where ${\Phi }_G(·)$ is Gaussian copula function and ${\rho }_{t+1}$ is the time-variant correlation parameters of Gaussian copula functions based on BDLMs.

Reliability prediction of structural system considering the nonlinear correlation between failure modes

Some structure has two failure modes, and the corresponding performance functions are

$$g_{i,t+1}(X_1,X_2,\dots ,X_n)=\left[\sigma \right]-y_{i,t+1},i=1,2$$

(39)

where $[\sigma ]$ is the allowed data and $y_{i,t+1}$ is the predicted data with BDLMs at time t + 1.

The failure probability, when the two failure modes meantime occurred, is

$$\begin{array}{c}P(g_{1,t+1}\left(\mathbf{X}\right)\le 0,g_{2,t+1}\left(\mathbf{X}\right)\le 0)\hfill \\ =P(F\left(g_{1,t+1}\left(\mathbf{X}\right)\right)\le F\left(0\right),G\left(g_{2,t+1}\left(\mathbf{X}\right)\right)\le G\left(0\right))\hfill \\ =P(U_{t+1}\le F\left(0\right),V_{t+1}\le G\left(0\right))=C(F\left(0\right),G\left(0\right);{\rho }_{t+1})\hfill \\ =C(p_{f_{g_{1,t+1}}},p_{f_{g_{2,t+1}}};{\rho }_{t+1})={\Phi }_G(p_{f_{g_{1,t+1}}},p_{f_{g_{2,t+1}}};{\rho }_{t+1})\hfill \end{array}$$

(40)

where

$$\begin{array}{c}{U}_{t+1}=\Phi \left(\frac{\left[\sigma \right]-y_{1,t+1}-{\mu }_{1,t+1}}{{\sigma }_{1,t+1}}\right),F\left(0\right)=\Phi \left(\frac{-{\mu }_{1,t+1}}{{\sigma }_{1,t+1}}\right)\hfill \\ V_{t+1}=\Phi \left(\frac{\left[\sigma \right]-y_{2,t+1}-{\mu }_{2,t+1}}{{\sigma }_{2,t+1}}\right),G\left(0\right)=\Phi \left(\frac{-{\mu }_{2,t+1}}{{\sigma }_{2,t+1}}\right)\hfill \end{array}$$

(41)

${\rho }_{t+1}$ can be obtained by the following method:

First, with equation (13), the following can be solved

$$\begin{array}{cc}\multicolumn{1}{c}{{\rho }_{t+1}}& =\rho ({\Phi }^{-1}\left(u_{t+1}\right),{\Phi }^{-1}\left(v_{t+1}\right))\hfill \\ \multicolumn{1}{c}{}& =\rho ({\Phi }^{-1}\left(F_1\left(g_{1,t+1}\right)\right),{\Phi }^{-1}\left(F_2\left(g_{2,t+1}\right)\right))\hfill \\ \multicolumn{1}{c}{}& =\rho (g_{1,t+1},g_{2,t+1})=\rho (\left[\sigma \right]-y_{1,t+1},\left[\sigma \right]-y_{2,t+1})\hfill \end{array}$$

(42)

Then, with equations (33) and (34), the correlation parameters ${\rho }_{t+1}$ of Gaussian copula function at time t + 1 can be computed. Finally, with equation (40), the failure probability of two-dimensional series system is

$$\begin{array}{c}{p}_{f,t+1}=P\left(g_{1,t+1}\left(\mathbf{X}\right)\le 0\cup g_{2,t+1}\left(\mathbf{X}\right)\le 0\right)\\ =P\left(g_{1,t+1}\left(\mathbf{X}\right)\le 0\right)+P\left(g_{2,t+1}\left(\mathbf{X}\right)\le 0\right)-P\left(g_{1,t+1}\left(\mathbf{X}\right)\le 0,g_{2,t+1}\left(\mathbf{X}\right)\le 0\right)\\ =p_{f_{g_{1,t+1}}}+p_{f_{g_{2,t+1}}}-C\left(p_{f_{g_{1,t+1}}},p_{f_{g_{2,t+1}}};{\rho }_{t+1}\right)\\ =p_{f_{g_{1,t+1}}}+p_{f_{g_{2,t+1}}}-{\Phi }_G\left(p_{f_{g_{1,t+1}}},p_{f_{g_{2,t+1}}};{\rho }_{t+1}\right)\end{array}$$

(43)

where $p_{f_{g_{1,t+1}}}\mathrm{and}{p}_{f_{g_{2,t+1}}}$ are, respectively, the failure of the two failure modes and C is copula function.

Failure probability of multiple-dimensional series system can be solved based on equation (43), by computing the time-variant correlation parameter matrix and the marginal distributions of all the predicted random variables based on BDLMs.

A numerical example: reliability prediction of Yitong River Bridge

Yitong River Bridge over Yitong River was built in 2009 in the city of Changchun, China. It is a flying swallow profiled concrete-filled steel tube arch bridge which is at no. 102 National Highway in Changchun. The main bridge of Yitong River Bridge is 260 m, and the main span is 158 m, which is shown in Figures 3–5. Details of this bridge are provided in Qu and Xie ²⁹ and Zhao. ³⁰ The monitoring program for this bridge included the assessment of the stain and deflection of specified structural components.

Figure 3.

Yitong River bridge (Changchun city from Jilin Province in China).

Figure 4.

The layout of Yitong River Bridge.

Figure 5.

Schematic diagram of Yitong River Bridge.

The everyday extreme deflections of the middle parts for upper stream and lower stream middle spans are monitored for 45 days, which are shown in Table 1. The distribution parameters of the allowed deflection about the middle parts of upper stream and lower stream main spans are, respectively, $L/600$ (mean value) and 0.115 (coefficient of variation), where L is the length of the main span.

Table 1.

Monitored and predicted information for mid-span deflection of upper stream and lower stream main spans.

Mid-span deflection of upper stream main span				Mid-span deflection of lower stream main span				Copula time-dependent correlation parameters
Time (day)	Monitored deflection (mm)	Predicted mean (mm)	Predicted variance (mm2)	Time (day)	Monitored deflection (mm)	Predicted mean (mm)	Predicted variance (mm2)
1	45.72	45.97	4.58	1	44.67	46.14	3.61
2	45.10	45.83	3.44	2	45.15	45.45	2.71	−1.00
3	46.69	45.57	3.06	3	45.56	45.39	2.41	0.57
4	47.17	45.85	2.87	4	44.01	45.49	2.26	−0.37
5	47.20	46.11	2.75	5	46.62	45.23	2.17	0.21
6	45.01	46.29	2.67	6	42.96	45.52	2.11	0.52
7	48.08	46.08	2.62	7	47.31	45.18	2.06	0.71
8	43.46	46.33	2.58	8	42.56	45.52	2.03	0.81
9	47.19	45.98	2.55	9	47.08	45.21	2.01	0.82
10	43.38	46.10	2.52	10	43.30	45.47	1.99	0.83
11	45.34	45.81	2.50	11	45.61	45.30	1.97	0.80
12	45.50	45.75	2.48	12	45.26	45.38	1.96	0.80
13	44.74	45.72	2.47	13	43.10	45.41	1.94	0.80
14	46.74	45.62	2.46	14	47.89	45.28	1.93	0.79
15	42.92	45.70	2.45	15	45.61	45.53	1.93	0.64
16	44.35	45.48	2.44	16	46.27	45.58	1.92	0.59
17	46.86	45.39	2.43	17	46.06	45.68	1.91	0.60
18	45.91	45.47	2.42	18	46.20	45.75	1.91	0.60
19	46.52	45.49	2.41	19	46.39	45.83	1.90	0.61
20	47.36	45.54	2.41	20	46.57	45.91	1.90	0.62
21	48.22	45.63	2.40	21	46.86	46.00	1.89	0.64
22	48.21	45.76	2.40	22	47.12	46.10	1.89	0.67
23	49.17	45.88	2.39	23	47.99	46.20	1.88	0.71
24	49.09	46.04	2.39	24	48.08	46.34	1.88	0.74
25	48.98	46.18	2.39	25	46.94	46.48	1.88	0.74
26	47.75	46.30	2.38	26	46.67	46.55	1.88	0.74
27	46.84	46.36	2.38	27	46.45	46.61	1.87	0.74
28	45.18	46.36	2.38	28	46.18	46.65	1.87	0.73
29	44.09	46.30	2.37	29	45.93	46.68	1.87	0.71
30	45.50	46.19	2.37	30	45.64	46.70	1.87	0.71
31	45.92	46.14	2.37	31	45.40	46.70	1.87	0.71
32	46.01	46.12	2.37	32	45.12	46.69	1.86	0.71
33	44.97	46.10	2.36	33	45.31	46.68	1.86	0.71
34	45.74	46.04	2.36	34	45.60	46.67	1.86	0.71
35	45.04	46.01	2.36	35	45.92	46.68	1.86	0.70
36	45.64	45.97	2.36	36	46.26	46.70	1.86	0.70
37	45.26	45.94	2.36	37	46.60	46.73	1.86	0.68
38	45.24	45.90	2.36	38	46.94	46.78	1.85	0.66
39	45.67	45.86	2.35	39	47.29	46.83	1.85	0.64
40	44.74	45.84	2.35	40	47.62	46.90	1.85	0.59
41	46.03	45.79	2.35	41	47.68	46.97	1.85	0.58
42	44.30	45.79	2.35	42	47.50	47.05	1.85	0.53
43	46.20	45.72	2.35	43	47.20	47.11	1.85	0.53
44	44.24	45.72	2.35	44	46.92	47.17	1.85	0.50
45	45.93	45.66	2.35	45	46.65	47.21	1.85	0.50

Note: Figure 10 and Figure 11 are plotted based on the shaded data in Table 1 and Table 2.

The performance functions about the middle deflections of the upper stream and lower stream main spans are, respectively

$$\begin{array}{c}{Z}_{y_{1,t}}=\left[\sigma \right]-\gamma y_{1,t}=\left[\sigma \right]-1.15y_{1,t}\hfill \\ Z_{y_{2,t}}=\left[\sigma \right]-\gamma y_{2,t}=\left[\sigma \right]-1.15y_{2,t}\hfill \end{array}$$

(44)

where $[\sigma ]$ follows normal distribution and the distribution parameters of which are 98.75 mm (mean value) and 11.36 mm (standard deviation). The upper stream and lower stream main spans are connected in series, so the time-dependent structural failure probability is

$$p_{f,t+1}=p_{Z_{y_{1,t}}}+p_{Z_{y_{2,t}}}-{\Phi }_G(p_{Z_{y_{1,t}}},p_{Z_{y_{2,t}}};{\rho }_{t+1})$$

(45)

$V_t$ can be approximately estimated with former monitored data, and $\delta =0.98\mathrm{and}{W}_t=-C_{t-1}+C_{t-1}/\delta$ according to the actual engineering experience of the authors. Through Kolmogorov–Smirnov (K-S) test for the former information, the state variable, observational error, and state error all follow normal distributions. The built BDLMs are

• Observation equation

  $$\left(\begin{array}{c}{y}_{1,t}\hfill \\ \multicolumn{1}{c}{y_{2,t}}\end{array}\right)=\left(\begin{array}{c}{\theta }_{1,t}\hfill \\ \multicolumn{1}{c}{{\theta }_{2,t}}\end{array}\right)+\left(\begin{array}{c}{\nu }_{1,t}\hfill \\ \multicolumn{1}{c}{{\nu }_{2,t}}\end{array}\right),\left(\begin{array}{c}{\nu }_{1,t}\hfill \\ \multicolumn{1}{c}{{\nu }_{2,t}}\end{array}\right)~\mathrm{N}(0,\left(\begin{array}{c}{V}_{1,t}\hfill \\ \multicolumn{1}{c}{V_{2,t}}\end{array}\right))$$

  (46)

• State equation

  $$\left(\begin{array}{c}{\theta }_{1,t}\\ {\theta }_{2,t}\end{array}\right)=\left(\begin{array}{c}{\phi }_1{\theta }_{1,t-1}\\ {\phi }_2{\theta }_{2,t-1}\end{array}\right)+\left(\begin{array}{c}{\omega }_{1,t}\\ {\omega }_{2,t}\end{array}\right),\left(\begin{array}{c}{\omega }_{1,t}\\ {\omega }_{2,t}\end{array}\right)~\mathrm{N}\left(\mathbf{0},\left(\begin{array}{c}{W}_{1,t}\\ W_{2,t}\end{array}\right)\right),\left(\begin{array}{c}{W}_{1,t}\\ W_{2,t}\end{array}\right)=\left(\begin{array}{c}-{\phi }_1^2{C}_{1,t-1}\\ -{\phi }_2^2{C}_{2,t-1}\end{array}\right)+\left(\begin{array}{c}\frac{C_{1,t-1}}{\delta }\\ \frac{C_{2,t-1}}{\delta }\end{array}\right)$$

  (47)

• Initial information

  $$\left(\begin{array}{c}{\theta }_{1,t-1}|{\mathbf{D}}_{1,t-1}\\ {\theta }_{2,t-1}|{\mathbf{D}}_{2,t-1}\end{array}\right)~\mathrm{N}\left(\left(\begin{array}{c}{m}_{1,t-1}\\ m_{2,t-1}\end{array}\right),\left(\begin{array}{c}{C}_{1,t-1}\\ C_{2,t-1}\end{array}\right)\right),t=1,2,\dots ,T$$

  (48)

where ${\phi }_1=0.98$ , ${\phi }_2=0.96$ , and $\delta =0.98$ .

Based on section “Gaussian copula-BDLMs,” the monitored deflection data, predicted deflection data, and the time-dependent correlation parameters of the built Gaussian copula functions are, respectively, shown in Table 2 and Figures 6–9.

Table 2.

Reliability indices and failure probability of upper stream and lower stream main spans.

Upper stream main span			Lower stream main span			Copula time-variant correlation parameters
Time (day)	Reliability indices	Failure probability (10−5)	Time (day)	Reliability indices	Failure probability (10−5)
1	4.5422	0.2783	1	4.5493	0.2691
2	4.5799	0.2326	2	4.6300	0.1828	−1.00
3	4.6112	0.2001	3	4.6415	0.1729	0.57
4	4.5916	0.2199	4	4.6369	0.1768	−0.37
5	4.5716	0.2420	5	4.6614	0.1570	0.21
6	4.5578	0.2584	6	4.6374	0.1764	0.52
7	4.5770	0.2358	7	4.6679	0.1521	0.71
8	4.5558	0.2610	8	4.6395	0.1747	0.81
9	4.5877	0.2241	9	4.6667	0.1530	0.82
10	4.5778	0.2349	10	4.6451	0.1700	0.83
11	4.6032	0.2080	11	4.6601	0.1580	0.80
12	4.6085	0.2028	12	4.6536	0.1631	0.80
13	4.6119	0.1995	13	4.6504	0.1656	0.80
14	4.6202	0.1917	14	4.6628	0.1560	0.79
15	4.6143	0.1973	15	4.6412	0.1732	0.64
16	4.6331	0.1801	16	4.6365	0.1772	0.59
17	4.6413	0.1731	17	4.6282	0.1844	0.60
18	4.6342	0.1791	18	4.6219	0.1901	0.60
19	4.6332	0.1800	19	4.6153	0.1963	0.61
20	4.6291	0.1836	20	4.6081	0.2032	0.62
21	4.6212	0.1908	21	4.6006	0.2106	0.64
22	4.6099	0.2014	22	4.5922	0.2193	0.67
23	4.5997	0.2116	23	4.5832	0.2290	0.71
24	4.5860	0.2259	24	4.5708	0.2429	0.74
25	4.5739	0.2394	25	4.5589	0.2571	0.74
26	4.5632	0.2519	26	4.5527	0.2648	0.74
27	4.5584	0.2577	27	4.5480	0.2708	0.74
28	4.5577	0.2585	28	4.5443	0.2755	0.73
29	4.5636	0.2514	29	4.5419	0.2787	0.71
30	4.5733	0.2400	30	4.5406	0.2805	0.71
31	4.5771	0.2357	31	4.5402	0.2810	0.71
32	4.5791	0.2334	32	4.5407	0.2803	0.71
33	4.5808	0.2316	33	4.5421	0.2785	0.71
34	4.5860	0.2260	34	4.5426	0.2778	0.71
35	4.5882	0.2235	35	4.5420	0.2785	0.70
36	4.5927	0.2188	36	4.5403	0.2808	0.70
37	4.5950	0.2164	37	4.5375	0.2846	0.68
38	4.5984	0.2128	38	4.5336	0.2899	0.66
39	4.6017	0.2095	39	4.5288	0.2966	0.64
40	4.6036	0.2077	40	4.5231	0.3047	0.59
41	4.6082	0.2031	41	4.5165	0.3143	0.58
42	4.6087	0.2026	42	4.5101	0.3240	0.53
43	4.6143	0.1972	43	4.5044	0.3327	0.53
44	4.6141	0.1974	44	4.4999	0.3400	0.50
45	4.6196	0.1922	45	4.4963	0.3457	0.50

Note: Figure 10 and Figure 11 are plotted based on the shaded data in Table 1 and Table 2.

Figure 6.

Monitored and predicted upper stream deflection.

Figure 7.

Monitored and predicted lower stream deflection.

Figure 8.

Prediction precision of upper stream and lower stream deflections.

Figure 9.

Linear correlation coefficients between performance functions.

The correlation between the failure modes of the middle parts for upper stream and lower stream middle spans is time dependent and nonlinear, and Figures 9 and 10 can reflect the rules.

Figure 10.

PDF and contour plots of time-dependent Gaussian copula functions.

Based on the built performance functions: equation (44), with first-order second moment (FOSM) method,^11,31 the time-dependent reliability indices, and failure probability of mid-span parts of upper stream and lower stream main spans can be computed, as shown in Table 2.

Based on Table 2, with equation (45), the structural failure probability at special time can be obtained, as shown in Table 3 and Figure 11, which can be obtained that the series-system failure probability considering the correlation between failure modes is smaller than the one without considering the correlation between failure modes. Therefore, the obtained series-system failure probability without considering the correlation between failure modes is conservative.

Table 3.

Structural system failure probability on special days.

Day	3rd	4th	5th	6th	13th
Dependent	3.7078 × 10−6	3.9670 × 10−6	3.9898 × 10−6	4.3324 × 10−6	3.4548 × 10−6
Independent	3.7300 × 10−6	3.9670 × 10−6	3.9900 × 10−6	4.3480 × 10−6	3.6510 × 10−6
Day	21st	29th	35th	40th	45th
Dependent	3.9645 × 10−6	5.1663 × 10−6	4.9056 × 10−6	5.8039 × 10−6	5.3629 × 10−6
Independent	4.0140 × 10−6	5.3010 × 10−6	5.0200 × 10−6	5.1240 × 10−6	5.3790 × 10−6

Figure 11.

Structural system failure probability on special days.

Conclusion

This article provided a new model: Gaussian copula-BDLMs, which can characterize the nonlinear correlation between failure modes and solve the reliability of the system considering nonlinear correlation between failure modes. Based on the built Gaussian copula-BDLMs, the time-dependent reliability of the series system (Yitong River Bridge main span), consisted of upper stream and lower stream middle spans, is analyzed. The results show that because there exists nonlinear correlation between the middle deflections of the upper stream and lower stream middle spans, the series-system failure probability considering the nonlinear correlation between failure modes is smaller than the one without considering the nonlinear correlation between failure modes. Therefore, it is essential and important to consider the time-dependent correlation between failure modes for obtaining the more accurate and reasonable structural system reliability.