Chinese Journal of Engineering

Volume 2016, Article ID 3648126, 11 pages

http://dx.doi.org/10.1155/2016/3648126

## Combinatorial Bayesian Dynamic Linear Models of Bridge Monitored Data and Reliability Prediction

^{1}Key Laboratory of Mechanics on Disaster and Environment in Western China (Lanzhou University), The Ministry of Education of China, China^{2}School of Civil Engineering and Mechanics, Lanzhou University, Lanzhou 730000, China

Received 24 November 2015; Accepted 13 January 2016

Academic Editor: Ian Smith

Copyright © 2016 Xueping Fan and Yuefei Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Considering the uncertainties and randomness of the mass structural health monitored data, the objectives of this paper are to present (a) a procedure for effective incorporation of the monitored data for the reliability prediction of structural components or structures, (b) one transforming method of Bayesian dynamic linear models (BDLMs) based on 1-order polynomial function, (c) model monitoring mechanism used to look for possible abnormal data based on BDLMs, (d) combinatorial Bayesian dynamic linear models based on the multiple BDLMs and their corresponding weights of prediction precision, and (e) an effective way of taking advantage of combinatorial Bayesian dynamic linear models to incorporate the historical data and real-time data in structural time-variant reliability prediction. Finally, a numerical example is provided to illustrate the application and feasibility of the proposed procedures and concepts.

#### 1. Introduction

Long-term ambient environments, such as chemical attack from environmental stressors and continuously increasing traffic volumes, make the physical quantities of civil infrastructure be subjected to changes in both time and space; these changes would make serious impacts on the serviceability and the ultimate capacity of structures and further have serious impacts on the remaining life of an existing structure [1]. The structural performances’ degradation processes (e.g., resistance, reliability indices), which are usually considered as Markov chains, are time-variant and irreversible. The time-variant reliability indices of bridges are dependent on both the applied loads and the remaining strength of structural components or system, which can reflect the safety and serviceability of bridges, and the reliability indices can be solved with first-order second-moment (FOSM) method [2, 3]. Therefore, assessing as well as predicting the structural time-dependent reliability indices is crucial for structural safety and serviceability assessment.

Through health monitoring of bridges, the structural basic statuses, including strains, stresses, and deflections of specified structural components or structures, can be obtained. Nowadays the research on structural health monitoring (SHM) generally experiences two stages. The first stage, falling in the mature stage, is to install an array of sensors for the observation and collection of data on a bridge structure during a period of time [4–7]. The second stage is mainly the application of health monitoring information. A sound number of studies are mainly focused on the modal parameter identification, structural damage detection technology, performance prediction, reliability assessment, and other fields [8, 9]. For research of the bridge reliability prediction and assessment, some achievements [10–14] are obtained, such as the reliability assessment of long span truss bridge, structural performance prediction based on monitored extreme data, and the use of the statistics of extremes to the reliability assessment and performance prediction of monitored highway bridges. However, due to the uncertainty of the bridges’ real-time health monitored data, the research on real-timely predicting structural reliability is at the initial stage in the world.

In this paper, considering the uncertainty of mass monitored extreme data which is time-dependent monitored data in the past days, BDLMs are introduced to combine the monitored data with the structural reliability prediction. First, with the monitored data, the single BDLMs and the corresponding model monitoring mechanism are, respectively, given, then the combinatorial prediction model of monitored extreme data is firstly built based on the built single BDLMs and the corresponding weights of prediction precisions, and the prediction precisions between the combinatorial prediction model and the single BDLMs are compared. Finally the real-timely predicted reliability indices of bridge structures are obtained with FOSM method based on the proper prediction model of monitored data. The proposed models and procedures are applied to an existing bridge.

#### 2. Bayesian Dynamic Linear Models (BDLMs)

BDLMs are the predicting approaches based on a philosophy of information updating [15, 16] 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 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. According to the definition of BDLMs [15], for each time , the general dynamic linear model is characterized by the quadruple and formally defined as follows: observation equation: state equation: initial information:where is the observation data at time ; is the observation error or the observation noise; is the state variable at time ; is normal probability density function; is the variance which indicates the uncertainty of observation errors; and are both the regression coefficient of states; is the variance which indicates the model uncertainty recursive from time to time . is the state error or state noise at time ; is the monitored total time; the initial information is the probabilistic representation of the predictors’ belief about the level at time . The mean value is a point estimate of this level , and measures the associated uncertainty. Each information set comprises all the information available at time , including , the values of the variances , and the values of the observations . Thus, the only new information becoming available at time is the observational value , so .

In this paper, the BDLMs mean that the observation equation and the state equation are both linear and are shown in (1) and (2). The 1-order polynomial function model is adopted to build the state equations.

##### 2.1. Transferred State Equation Based on 1-Order Polynomial Function and Monitored Data

For the mass and random monitored extreme data, especially for monitored data at time and before time , the discretized motion equation and the fitted 1-order polynomial function, which is commonly used for the prediction of the trend data, are adopted to predict future stress data of time , so the 1-order polynomial function can be applied to properly build the BDLMs.

*(**1) 1-Order Polynomial Function of Monitored Data*. Considerwhere is the trend data (state variable) at time ; , are coefficients; is the state error indicating the model uncertainty; is the total monitored time, unit of which is day.

*(**2) State Equation Based on (4)*. The first-order differential of (4)was considered as the discretized motion equation, where is the nominal speed of the trend data , which can be obtained with (4); is an error term. For simplicity, we consider a discretization in small interval of time (), as follows:that is,where it is assumed that the random error has density and can be estimated with (13). With a further simplification, we take unitary time intervals as one day; namely, , so that (7) can be rewritten as follows:where (8) will be used to build the state equation of BDLMs, which will be shown in (10).

##### 2.2. Transferred BDLMs Based on 1-Order Polynomial Function

Based on Section 2.1, according to the definition of BDLMs [15, 17], for each time , the general and easy forms of the dynamic linear models are defined as follows: observation equation: state equation: initial information:where is the monitored data at time ; is the state parameter indicating the level of the monitored data at time ; is obtained with (4); and are, respectively, the monitored error and the state error at time , which are all zero-mean normal random variables.

For each time , the BDLMs include the following parameters: is the variance of monitored errors at time ; is the variance of state error at time ; and are, respectively, monitored errors and state errors. It is assumed that error sequences and are internally independent, mutually independent, and independent of .

With (9)–(11), the relationships between monitored data and state parameters are shown in

It can be known from (12) that the modeling processes of BDLMs can be divided into two key steps, which are shown in Figure 1. The first step is to obtain the a priori probability density function (PDF) of at time based on the state equation and the a posteriori PDF of at time ; the second step is to get the a posteriori PDF of at time based on the a priori PDF of state parameters at time and inspection/monitored data at time .