This article presented a new data fusion approach for reasonably predicting dynamic serviceability reliability of the long-span bridge girder. Firstly, multivariate Bayesian dynamic linear model (MBDLM) considering dynamic correlation among the multiple variables is provided to predict dynamic extreme deflections; secondly, with the proposed MBDLM, the dynamic correlation coefficients between any two performance functions can be predicted; finally, based on MBDLM and Gaussian copula technique, a new data fusion method is given to predict the serviceability reliability of the long-span bridge girder, and the monitoring extreme deflection data from an actual bridge is provided to illustrated the feasibility and application of the proposed method.

Structural Health Monitoring (SHM) is a promising technology to improve the serviceability of civil infrastructures and achieve the sustainable management. For the long-span bridges, the monitoring extreme deflection data provided by SHM systems is an important parameter for structural serviceability analysis and can be used for evaluating and predicting structural dynamic serviceability reliability.

In recent years, SHM has become the escalating urgent need for the modern bridge engineering and grew into a hot topic on both investments and researches around the world. With the innovation of sensing data acquisition, SHM systems are comprehensively deployed and used for obtaining the extreme deflection data of the long-span bridge bridges in different sampling frequency. How to make reasonable use of these data for predicting the dynamic serviceability reliability of the long-span bridge girder, has been still at the initial research stage, but it has become one of the main scientific problems in the SHM field [

Long-span bridge serviceability reliability prediction can be made with reliability analysis methods (e.g., first order second moment reliability method [

For the long-span bridges, dynamic correlation occurs among the performance functions corresponding to the deformation failure modes at different control monitoring points for the long-span bridge girder [

In view of the above existing problems, this article takes the long-span bridge girder as the research object, firstly, MBDLM is built and adopted to predict the dynamic extreme deflections. Secondly, with the predicted covariance matrix of the MBDLM, the correlation coefficients among the predicted deflections can be accurately obtained, further, the correlation coefficients among the multiple performance functions can be accurately predicted. Finally, based on the MBDLM and the Gaussian copula technique, the new data fusion approach is proposed to predict the dynamic serviceability reliability of the long-span bridge girder, and an actual bridge is provided to illustrated the feasibility of the proposed approach. The flowcharts of this paper are shown in

MBDLM is a predicting approach based on a philosophy of information updating [

(1) The monitoring variables

(2) The state variables, monitoring variables and the corresponding errors all approximately follow normal distribution.

(3) Different state variables are dependent between each other.

(4) Monitoring errors and state errors are internally and mutually dependent.

In this paper, the extreme deflection data is the maximum of the monitoring deflection data in each hour. Based on the historical monitoring extreme deflection data, MDLM (Multiple dynamic linear model) is built as follows [

Multivariate observation equation:

where,

Multivariate state equation:

where,

Initial multivariate state information:

where, ^{th} monitoring variable at and before time

Bayesian probabilistic recursion processes of MDLM can be obtained with Bayes method. The detailed steps are shown as follows [

(1) The state posteriori distribution at time

For the column initial state mean

(2) The state priori distribution at time

where,

(3) One-step prediction distribution at time

where,

(4) The state posteriori distribution at time

where,

For MBDLM, the main probability parameters include _{t+1}, _{t+1}, _{t} and _{t}; the interval period of model updating is one hour; _{t+1} can be approximately estimated with variance matrix about the differences between the state data and the monitoring extreme data at the different monitoring points, where the state data can be obtained through resampling the monitoring extreme data with cubical smoothing algorithm with five-point approximation [_{t+1} can be solved with

where,

_{t} is the mean value vector composed of mean value about the data for each state variable at and before time _{t} is the variance matrix of all the state variables at and before time

^{th} state variable, ^{th} state variable, ^{th} state variable and the ^{th} state variable, ^{th} state data about the ^{th} state variable at and before time

In ^{th} predicted monitoring variable, ^{th} predicted monitoring variable ^{th} predicted monitoring variable

The deformation failure mode at the monitoring point is: If the monitoring deflection is more than the allowable deflection, then the monitoring point failed. The predicted performance functions of deformation failure modes at the multiple monitoring points are

where,

Further, the dynamic prediction of correlation coefficients

where, ^{th} predicted performance function ^{th} predicted performance function ^{th} predicted variable ^{th} predicted variable

The long-span bridge girder exists

When the ^{th} deformation failure mode occurred, with first order second moment (FOSM) method [

where,

where,

When the

where, ^{th} deformation failure mode,

When the

Further, based on

Based on

Finally, the predicted serviceability reliability indices

where,

With

The Zhaoqing West River Bridge over West River was built in 2003 in Guangdong City, China. It is a typical continuous rigid frame bridge, the girders of which are all reinforced concrete box-girders. The main spans include five spans

The extreme deflection at each point is defined as the maximum of the monitoring deflections at each point in each hour. The extreme deflections at the three monitoring points (A, B and C) for the 4# box-girder are monitored for 200 h, which ensure that the probability statistic characteristics of the monitoring extreme deflections at each monitoring point can be extracted correctly. The monitoring extreme deflections and the smoothly-processed state data at Points A, B and C are shown in

In _{t+1}, _{t+1}, _{t} and _{t}) can be estimated with Section 2.3. Through Kolmogorov-Smirnov (K-S) test [

Multivariate observation equation:

where,

Multivariate state equation:

where,

Initial multivariate state information:

where,

Based on the monitoring extreme deflection data &

The predicted performance functions of the deformation failure modes at the monitoring points A, B and C are, respectively,

where, ^{th} monitoring point.

The deformation failure modes at the monitoring points A, B and C are serial. Based on

where, ^{th} monitoring point.

Further, based on

The predicted correlation coefficients between performance functions corresponding to two deformation failure modes can be computed with

Based on

With the

From

This paper presented a new data fusion approach for the dynamic serviceability reliability prediction of the long-span bridge girder through combining MBDLM and Gaussian copula model based on monitoring extreme deflection data. The proposed method considered the dynamic correlation coefficients among the multiple deformation failure modes.

Through the illustration of the actual long-span bridge girder, the results show that there exists dynamic correlation among multiple failure modes (

The authors would like to thank the Editor and the anonymous reviewers for their constructive comments and valuable suggestions to improve the quality of the article.