# Incorporation of Structural Health Monitoring Coupled Data in Dynamic Extreme Stress Prediction of Steel Bridges Using Dynamic Coupled Linear Models

Authors: Xueping Fan; Zhipeng Shang; Guanghong Yang; Xiaoxiong Zhao; Yuefei Liu
Journal: Advances in Civil Engineering
 (2020)

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## Abstract

In this article, an approach for using structural health monitoring coupled extreme stress data in dynamic extreme stress prediction of steel bridges is presented, where the coupled extreme stress data means the extreme stress data with dynamicity, randomness, and trend. Firstly, the modeling processes about dynamic coupled linear models (DCLM) are provided based on a supposed coupled time series; furthermore, the dynamic probabilistic recursion processes about DCLM are given with Bayes method; secondly, the monitoring dynamic coupled extreme stress data is taken as a time series, historical monitoring coupled extreme stress data-based DCLM and the corresponding Bayesian probabilistic recursion processes are given for predicting bridge extreme stresses; furthermore, the monitoring mechanism is provided for monitoring the prediction precision of DCLM; finally, the monitoring coupled extreme stress data of a steel bridge is used to illustrate the proposed approach which can provide the foundations for bridge reliability prediction and assessment.


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## Body

## 1. Introduction

Structural health monitoring (SHM) is a promising technology to improve the safety of civil infrastructures and achieve their sustainable management. For steel bridges, structural safety is very important, and the monitoring extreme stress data provided by SHM systems is an important parameter for structural safety analysis and can be used for evaluating and predicting structural dynamic safety reliability.During long-term service periods, many steel bridges are subjected to many random dynamic loads like vehicle load, temperature load, and wind load; even these loads occur simultaneously. Therefore, the accumulated extreme stress data by structural health monitoring (SHM) system commonly has the characteristic of coupling including randomness, dynamicity, and tendency [1]; furthermore, based on the monitoring coupled extreme stress data which is considered as Markov chains, dynamic coupled extreme stress prediction for steel bridges is particularly important as bridges age, which could reflect bridge safety and help to guide the important decisions regarding the structural preventive maintenance. Therefore, monitoring coupled data-based extreme stress prediction and assessment of steel bridges is accordingly a grand challenge of SHM field [2, 3].In recent years, SHM has become the escalating urgent need for 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 coupled extreme stress data of steel bridges in different sampling frequency. How to make reasonable use of these data for predicting structural extreme stresses has been one of the main scientific problems in the SHM field. So far, the researches on SHM have generally experienced two stages. The first stage, falling in the mature stage, is to install an array of sensors for monitoring, collection, and recovery of SHM data during the long-term service periods [4–12]. The second stage is mainly the application of SHM data. Most studies are mainly focused on the modal parameter identification [13, 14], structural damage detection technology [10, 13, 15, 16], model updating [3], performance prediction [17–19], and reliability assessment [20–22]. The second stage shows that some researches about the application of SHM data have been conducted, but, in the analysis processes, the coupling of SHM data is not reasonably taken into account. Therefore, monitoring coupled data-based dynamic prediction of structural coupled extreme stresses remains at the initial research stage. At present, although some achievements about structural response prediction have been made, most of them have some limitations; for example, dynamic linear models and Gaussian mixed particle filter-based extreme stress prediction method of steel bridges are proposed, but, in the analysis processes, the coupling of monitoring extreme stress data is not reasonably considered [4]; a data assimilation method about Bayesian Fourier dynamic linear prediction of periodic extreme stresses for steel bridges is proposed; in the analysis processes, the modeling processes of dynamic linear models are complex and are not generally used [23]; the ARMA models [24] suitable for short-term prediction and BDLM [17, 18] without steady prediction precision are not clearly satisfying for predicting coupled performance information; the performance prediction method through decoupling the monitoring coupled data decreases the coupling effect on the prediction results to a certain extent, but the decoupling itself is a difficult problem due to the complexity of the coupled information, and it can lead to information losses and so does its work in the prediction processes [25]; the traditional nonlinear filtering methods are usually used to build the prediction models based on curve fitting of historical dynamic data [26], so they are very subjective, and monitoring load effects tend to exhibit the coupling; thus, the traditional subjective nonlinear filtering methods will affect the precision and accuracy of the prediction results; the common regression models derived from the historical monitoring data are also very subjective and are not considering the uncertainty about the monitoring data [27, 28]; the support vector regression model needs a lot of monitoring data to be trained for completing the model prediction, which consumes a lot of time [29]. In view of the above limitations, a Bayesian dynamic coupled linear prediction method of the coupled extreme stresses for steel bridges could be explored and is the topic of the present research.Based on the above issues, this paper proposes the Bayesian dynamic coupled linear prediction method of coupled extreme stresses for steel bridges based on the monitoring coupled extreme stress data. Firstly, based on a supposed coupled time series, dynamic coupled linear models (DCLM) are built, and the corresponding dynamic probabilistic recursion processes are proposed with Bayes method (see Section2); secondly, based on Section 2, monitoring coupled extreme stress data-based DCLM and the corresponding Bayesian dynamic probabilistic recursion processes are provided for predicting coupled extreme stresses in Section 3; furthermore, the monitoring mechanism about DCLM is provided for monitoring the abnormal coupled extreme stress data (see Section 4). Finally, the effectiveness of the proposed DCLM is compared with several existing methods through the monitoring coupled extreme stress data from a steel bridge in Section 5, and some conclusions are drawn in Section 6. The main flowcharts of this paper are shown in Figure 1.Figure 1
Main flowcharts of this paper.


## 2. The General Coupled Time Series-Based Dynamic Coupled Linear Models (DCLM) and the Corresponding Bayesian Probabilistic Recursion

The flowcharts of Section2 are shown in Figure 2.Figure 2
The flowcharts of Section2.According to Figure2, the detailed contents of Section 2 are described as follows.


### 2.1. DCLM Based on a General Coupled Time Series

DCLM include state equation, monitored equation, and the initial coupled state information, where the state equation shows changes of the system with time and reflects inner dynamic changes of the system and random disturbances, the monitored equation expresses the relationship between the monitored data and the current state parameters of the system, and the initial coupled state information is expressed with the mixture of a few normal probability density functions (PDFs) about the initial coupled state. In this chapter, the general coupled time series is adopted to build DCLM, and the detailed modeling methods are described as follows.It is supposed that there is a general dynamic coupled time series setDt with (n − 1)-order local trends, which includes the data at and before time t. The state set, also taken as a coupled time series approximately obtained with cubical smoothing algorithm with five-point approximation [1, 2, 17] through resampling Dt, is denoted with Dθt, where θt is the state at time t, Dθt includes all the state data at and before time t, and the ith difference data set of Dθt is Dθt,i. In general, for the general coupled time series set Dt with (n − 1)-order local trends, difference can be carried out until the time series after difference is stationary, and the number (n − 1) of difference is the order of local trends. Besides, the Augmented Dickey-Fuller (ADF) test method [30–33] can also help ascertain the stationarity of the data before and after difference. Namely, ADF test method can help determine the order of local trends.Letθt,t∈1,2,… be the state time series; there are the following difference equations:(1)Δ0θk=θk,Δ1θk=θk+1−θk,Δ2θk=Δ1θk+1−Δ1θk,…,Δiθk=Δi−1θk+1−Δi−1θk,…,Δn−1θk=Δn−2θk+1−Δn−2θk,where Δiθk,i∈1,2,…,n−1,k∈1,2,…,t is the ith order difference data about θk.Furthermore, (2) can be obtained as(2)θk+1=θk+Δ1θk,Δ1θk+1=Δ1θk+Δ2θk,…,Δi−1θk+1=Δi−1θk+Δiθk,…,Δn−2θk+1=Δn−2θk+Δn−1θk.Suppose that all the dynamic state variablesθt,i,i∈1,2,…,n are all Markov chains and are internally independent and mutually independent between each other. Taking into account the monitoring error vt about yt conditional on the state (or level) θt,1 and transition error wt,j from state θt−1,j+θt−1,j+1 to θt,j, the (n − 1)-order DCLM are defined in the following form.Monitoring equation is(3)yt=θt,1+vt,vt∼N0,e−0.0004t2V⇒yt=Fθt+vt,vt∼N0,Vt,F=100…01×n,θt=θt,1θt,2θt,3…θt,n1×nT.State equation is(4)θt,j=θt−1,j+θt−1,j+1+wt,j,wt,j∼N0,Wt,j,j∈1,2,...,n−1,θt,n=θt−1,n+wt,n,wt,n∼N0,Wt,n,⇒θt=Gθt−1+wt,wt∼N0,Wt,θt=θt,1θt,2θt,3…θt,n1×nT,wt=wt,1wt,2wt,3…wt,n1×nT,G=1100…00110…00011…0…0000…1n×n,Wt=Wt,100…00Wt,20…000Wt,j…0…0000Wt,nn×n.Through the least residual error quadratic sum method [2], initial coupled state information can be obtained with(5)pθt−1,1Dθt−1,1≈∑i1=1G1αt−1i1Nθt−1,1;μt−1i1,ϕt−1i1;pθt−1,2Dθt−1,2≈∑i2=1G2αt−1i2Nθt−1,2;μt−1i2,ϕt−1i2,...pθt−1,iDθt−1,i≈∑iβ=1Giαt−1iβNθt−1,i;μt−1iβ,ϕt−1iβ;…pθt−1,nDθt−1,n≈∑in=1Gnαt−1inNθt−1,n;μt−1in,ϕt−1in,where yt is the data at time t; vt is the monitoring error; N• is normal probability density function (PDF); V is the initial error variance obtained with the historical time series data; θt,i is the data at time t from Dθt,i; θt,1 is the state (or level) at time t, which between times t − 1 and t changes by the addition of θt−1,2 plus the noise wt,1; θt−1,2 represents a systematic change in state (or level), which itself changes by the addition of θt−2,3 plus noise; proceeding through the state parameters for j∈1,2,…,n−1, each θt,j changes systematically via the increment θt−1,j+1 and also by the addition of the noise term wt,j; the nth component θt,n changes only stochastically; T is the symbol of transpose; wt=wt,1,…,wt,nT is the state error vector and wt∼N0,Wt, where Wt is the variance matrix; vt and wt are internally independent and mutually independent between each other. Gi,i∈1,2,…,n denotes the total number of normal PDFs about mixed Gaussian PDF, αt−1iβ is the iβth mixed weight, and Nθt−1,i;μt−1iβ,ϕt−1iβ is a Gaussian PDF with the mean μt−1iβ and the variance ϕt−1iβ.


### 2.2. Decomposition Model (Dynamic Linear Model (DLM)) of DCLM

Due to the coupling of a general time series, initial coupled state information shown in (5) includes multiple mixture Gaussian PDFs; thus, there are ∏i=1nGi DLMs that can be built and used to make dynamic probabilistic recursion of the DCLM. Each dynamic linear model (DLM) can be, respectively, obtained with equations (3), (4), and (6).Monitoring equation about theJth DLM is the same as (3), where Vt is replaced by Vt,J.State equation about theJth DLM is the same as (4), where Wt is replaced by Wt,J.Initial information about theJth DLM is(6)pθt−1,1Dθt−1,1≈Nθt−1,1;μt−1i1,ϕt−1i1,i1∈1,2,…,G1;pθt−1,2Dθt−1,2≈Nθt−1,2;μt−1i2,ϕt−1i2,i2∈1,2,…,G2;...pθt−1,iDθt−1,i≈Nθt−1,i;μt−1iβ,ϕt−1iβ,iβ∈1,2,…,Gi;…;pθt−1,nDθt−1,n≈Nθt−1,n;μt−1in,ϕt−1in,in∈1,2,…,Gn,⇒θt−1,JDθt−1,J∼Nθt−1,J;Mt−1,J,Ct−1,J,J∈1,2,…,∏i=1nGi,θt−1,J=θt−1,1θt−1,2θt−1,3…θt−1,n1×n,JT;Dθt−1,J=Dθt−1,1Dθt−1,2Dθt−1,3…Dθt−1,n1×n,JTMt−1,J=μt−1i1μt−1i2μt−1i3…μt−1in1×n,JT;Ct−1,J=ϕt−1i1000…00ϕt−1i200…000ϕt−1i30…0…0000…ϕt−1in,where Wt,J can be approximately solved with(7)Wt,J=−GCt−1,JGT+Ct−1,Jδ,where GT is the transpose of G and δ is the discount factor that is usually 0.48–0.98 [2, 17].


### 2.3. Bayesian Probabilistic Recursion of DLM and DCLM Based on a General Coupled Time Series

Bayesian probabilistic recursion of DCLM means that dynamic recursive processes about DCLM are achieved with Bayes method [34, 35]. Bayesian probabilistic recursion is good at inferring and predicting past and future parameters [34, 35], and Bayesian probabilistic recursive processes about DLM and DCLM are, respectively, shown as follows.


#### 2.3.1. Bayesian Probabilistic Recursion of DLM

(1)
With (4) and (6), the prior PDF about the state θt,J at time t can be obtained with(8)θt,JDθt−1,J∼Nat,J,Rt,J,J∈1,2,…,∏i=1nGi,whereat,J=GMt−1,J,Rt,J=GCt−1,JGT+Wt,J.(2)
With (3) and (8), the one-step forward prediction PDF at time t can be solved with(9)yt,JDθt−1,J∼Nft,J,Qt,J,J∈1,2,...,∏i=1nGi,whereft,J=Fat,J can be used to predict the dynamic data of the Jth DLM; Qt,J=FRt,JFT+e−0.0004t2V. According to the definition of highest posterior density (HPD) region [34], the predicted interval of the Jth DLM with a 95% confidential interval at time t is ft,J−1.645Qt,J,ft,J+1.645Qt,J.(3)
With Bayes method [33, 34], (8) and (9), while yt,J=yt is obtained, the posterior PDF about the state at time t can be computed with(10)θt,JDθt,J∼NMt,J,Ct,J,J∈1,2,...,∏i=1nGi,whereMt,J=at,J+At,Jet,J,Ct,J=Rt,J−At,JQt,JAt,JT,At,J=Rt,JFTQt,J−1,et,J=yt,J−ft,J,et,J is the one-step forward predicted error, and yt,J is the monitoring data at time t. Qt,J−1, the reciprocal of the one-step predicted variance, is the prediction precision of the Jth DLM.With (3)-(4) and (6)–(10), the Bayesian cyclic probabilistic recursive processes of the Jth DLM can be achieved.


#### 2.3.2. Bayesian Probabilistic Recursion of DCLM

With (3)–(10) and the distantly provided coupled data, Bayesian probabilistic recursive processes of DCLM with time-variant updated weighted coefficients are derived as follows.(1)
The a posteriori PDF about the stateθt−1 at time t − 1 is(11)θt−1Dθt−1∼∑J=1Πi=1nGipt−1,JNMt−1,J,Ct−1,J,wherept−1,J=αt−1i1,J×αt−1i2,J×αt−1i3,J×⋯×αt−1in,J/∑J=1Πi=1nGiαt−1i1,J×αt−1i2,J×αt−1i3,J×⋯×αt−1in,J can be determined with (5) and (6).(2)
The prior PDF about the stateθt at time t is(12)θtDθt−1∼∑J=1Πi=1nGipt−1,JNat,J,Rt,J,(3)
The one-step prediction PDF at timet is(13)ytDθt−1∼∑J=1Πi=1nGipt−1,JNft,J,Qt,J,where the predicted data,ft=∑J=1Πi=1nGipt−1,Jft,J,∫−∞+∞yt−ft2∑J=1Πi=1nGipt−1,JNft,J,Qt,Jdyt−1, is the prediction precision of DCLM. According to the definition of highest posterior density (HPD) region [34, 35], the predicted interval of the combinatorial DCLM with a 95% confidential interval at time t is ∑J=1Πi=1nGipt−1,Jft,J−1.645∫−∞+∞yt−ft2∑J=1Πi=1nGipt−1,JNft,J,Qt,Jdyt,∑J=1Πi=1nGipt−1,Jft,J+1.645∫−∞+∞yt−ft2∑J=1Πi=1nGipt−1,JNft,J,Qt,Jdyt.(4)
With Bayes method [34, 35], while yt is obtained, the posterior PDF about the state at time t can be computed with(14)θtDθt∼∑J=1Πi=1nGipt,JNMt,J,Ct,J,wherept,J=pt−1,Jpyt,JDθt−1,J/∑J=1Πi=1nGipt−1,Jpyt,JDθt−1,J=pt−1,JNft,J,Qt,J/∑J=1Πi=1nGipt−1,JNft,J,Qt,J.


### 2.4. Error Estimation and Time Efficiency Analysis of DCLM

#### 2.4.1. Error Estimation of DCLM

In this paper, mean square error (MSE) is adopted to measure the prediction errors, the MSE is smaller, and the corresponding model is better. The formula about MSE is(15)fMSE=∑t=1Nft−yt2N,where fMSE is the formula of MSE; N is the total number of the predicted data; ft is predicted with (13); yt is the monitoring data at time t. Some relations between MSE and sampling time intervals will be discussed in Sections 2.3.2 and 2.3.3.


#### 2.4.2. Time Efficiency Analysis of DCLM

In order to test the time efficiency of DCLM, predicted coupled extreme stresses from different time intervals are considered for analysis comparison, and their MSEs are analyzed to discover different time efficiency. Hence, the every-hour, every-two-hours, and every-four-hours extreme stresses are predicted and utilized for illustrating the time efficiency of DCLM. The analysis results about the time efficiency of DCLM can be shown in Section5.


## 2.1. DCLM Based on a General Coupled Time Series

DCLM include state equation, monitored equation, and the initial coupled state information, where the state equation shows changes of the system with time and reflects inner dynamic changes of the system and random disturbances, the monitored equation expresses the relationship between the monitored data and the current state parameters of the system, and the initial coupled state information is expressed with the mixture of a few normal probability density functions (PDFs) about the initial coupled state. In this chapter, the general coupled time series is adopted to build DCLM, and the detailed modeling methods are described as follows.It is supposed that there is a general dynamic coupled time series setDt with (n − 1)-order local trends, which includes the data at and before time t. The state set, also taken as a coupled time series approximately obtained with cubical smoothing algorithm with five-point approximation [1, 2, 17] through resampling Dt, is denoted with Dθt, where θt is the state at time t, Dθt includes all the state data at and before time t, and the ith difference data set of Dθt is Dθt,i. In general, for the general coupled time series set Dt with (n − 1)-order local trends, difference can be carried out until the time series after difference is stationary, and the number (n − 1) of difference is the order of local trends. Besides, the Augmented Dickey-Fuller (ADF) test method [30–33] can also help ascertain the stationarity of the data before and after difference. Namely, ADF test method can help determine the order of local trends.Letθt,t∈1,2,… be the state time series; there are the following difference equations:(1)Δ0θk=θk,Δ1θk=θk+1−θk,Δ2θk=Δ1θk+1−Δ1θk,…,Δiθk=Δi−1θk+1−Δi−1θk,…,Δn−1θk=Δn−2θk+1−Δn−2θk,where Δiθk,i∈1,2,…,n−1,k∈1,2,…,t is the ith order difference data about θk.Furthermore, (2) can be obtained as(2)θk+1=θk+Δ1θk,Δ1θk+1=Δ1θk+Δ2θk,…,Δi−1θk+1=Δi−1θk+Δiθk,…,Δn−2θk+1=Δn−2θk+Δn−1θk.Suppose that all the dynamic state variablesθt,i,i∈1,2,…,n are all Markov chains and are internally independent and mutually independent between each other. Taking into account the monitoring error vt about yt conditional on the state (or level) θt,1 and transition error wt,j from state θt−1,j+θt−1,j+1 to θt,j, the (n − 1)-order DCLM are defined in the following form.Monitoring equation is(3)yt=θt,1+vt,vt∼N0,e−0.0004t2V⇒yt=Fθt+vt,vt∼N0,Vt,F=100…01×n,θt=θt,1θt,2θt,3…θt,n1×nT.State equation is(4)θt,j=θt−1,j+θt−1,j+1+wt,j,wt,j∼N0,Wt,j,j∈1,2,...,n−1,θt,n=θt−1,n+wt,n,wt,n∼N0,Wt,n,⇒θt=Gθt−1+wt,wt∼N0,Wt,θt=θt,1θt,2θt,3…θt,n1×nT,wt=wt,1wt,2wt,3…wt,n1×nT,G=1100…00110…00011…0…0000…1n×n,Wt=Wt,100…00Wt,20…000Wt,j…0…0000Wt,nn×n.Through the least residual error quadratic sum method [2], initial coupled state information can be obtained with(5)pθt−1,1Dθt−1,1≈∑i1=1G1αt−1i1Nθt−1,1;μt−1i1,ϕt−1i1;pθt−1,2Dθt−1,2≈∑i2=1G2αt−1i2Nθt−1,2;μt−1i2,ϕt−1i2,...pθt−1,iDθt−1,i≈∑iβ=1Giαt−1iβNθt−1,i;μt−1iβ,ϕt−1iβ;…pθt−1,nDθt−1,n≈∑in=1Gnαt−1inNθt−1,n;μt−1in,ϕt−1in,where yt is the data at time t; vt is the monitoring error; N• is normal probability density function (PDF); V is the initial error variance obtained with the historical time series data; θt,i is the data at time t from Dθt,i; θt,1 is the state (or level) at time t, which between times t − 1 and t changes by the addition of θt−1,2 plus the noise wt,1; θt−1,2 represents a systematic change in state (or level), which itself changes by the addition of θt−2,3 plus noise; proceeding through the state parameters for j∈1,2,…,n−1, each θt,j changes systematically via the increment θt−1,j+1 and also by the addition of the noise term wt,j; the nth component θt,n changes only stochastically; T is the symbol of transpose; wt=wt,1,…,wt,nT is the state error vector and wt∼N0,Wt, where Wt is the variance matrix; vt and wt are internally independent and mutually independent between each other. Gi,i∈1,2,…,n denotes the total number of normal PDFs about mixed Gaussian PDF, αt−1iβ is the iβth mixed weight, and Nθt−1,i;μt−1iβ,ϕt−1iβ is a Gaussian PDF with the mean μt−1iβ and the variance ϕt−1iβ.


## 2.2. Decomposition Model (Dynamic Linear Model (DLM)) of DCLM

Due to the coupling of a general time series, initial coupled state information shown in (5) includes multiple mixture Gaussian PDFs; thus, there are ∏i=1nGi DLMs that can be built and used to make dynamic probabilistic recursion of the DCLM. Each dynamic linear model (DLM) can be, respectively, obtained with equations (3), (4), and (6).Monitoring equation about theJth DLM is the same as (3), where Vt is replaced by Vt,J.State equation about theJth DLM is the same as (4), where Wt is replaced by Wt,J.Initial information about theJth DLM is(6)pθt−1,1Dθt−1,1≈Nθt−1,1;μt−1i1,ϕt−1i1,i1∈1,2,…,G1;pθt−1,2Dθt−1,2≈Nθt−1,2;μt−1i2,ϕt−1i2,i2∈1,2,…,G2;...pθt−1,iDθt−1,i≈Nθt−1,i;μt−1iβ,ϕt−1iβ,iβ∈1,2,…,Gi;…;pθt−1,nDθt−1,n≈Nθt−1,n;μt−1in,ϕt−1in,in∈1,2,…,Gn,⇒θt−1,JDθt−1,J∼Nθt−1,J;Mt−1,J,Ct−1,J,J∈1,2,…,∏i=1nGi,θt−1,J=θt−1,1θt−1,2θt−1,3…θt−1,n1×n,JT;Dθt−1,J=Dθt−1,1Dθt−1,2Dθt−1,3…Dθt−1,n1×n,JTMt−1,J=μt−1i1μt−1i2μt−1i3…μt−1in1×n,JT;Ct−1,J=ϕt−1i1000…00ϕt−1i200…000ϕt−1i30…0…0000…ϕt−1in,where Wt,J can be approximately solved with(7)Wt,J=−GCt−1,JGT+Ct−1,Jδ,where GT is the transpose of G and δ is the discount factor that is usually 0.48–0.98 [2, 17].


## 2.3. Bayesian Probabilistic Recursion of DLM and DCLM Based on a General Coupled Time Series

Bayesian probabilistic recursion of DCLM means that dynamic recursive processes about DCLM are achieved with Bayes method [34, 35]. Bayesian probabilistic recursion is good at inferring and predicting past and future parameters [34, 35], and Bayesian probabilistic recursive processes about DLM and DCLM are, respectively, shown as follows.


### 2.3.1. Bayesian Probabilistic Recursion of DLM

(1)
With (4) and (6), the prior PDF about the state θt,J at time t can be obtained with(8)θt,JDθt−1,J∼Nat,J,Rt,J,J∈1,2,…,∏i=1nGi,whereat,J=GMt−1,J,Rt,J=GCt−1,JGT+Wt,J.(2)
With (3) and (8), the one-step forward prediction PDF at time t can be solved with(9)yt,JDθt−1,J∼Nft,J,Qt,J,J∈1,2,...,∏i=1nGi,whereft,J=Fat,J can be used to predict the dynamic data of the Jth DLM; Qt,J=FRt,JFT+e−0.0004t2V. According to the definition of highest posterior density (HPD) region [34], the predicted interval of the Jth DLM with a 95% confidential interval at time t is ft,J−1.645Qt,J,ft,J+1.645Qt,J.(3)
With Bayes method [33, 34], (8) and (9), while yt,J=yt is obtained, the posterior PDF about the state at time t can be computed with(10)θt,JDθt,J∼NMt,J,Ct,J,J∈1,2,...,∏i=1nGi,whereMt,J=at,J+At,Jet,J,Ct,J=Rt,J−At,JQt,JAt,JT,At,J=Rt,JFTQt,J−1,et,J=yt,J−ft,J,et,J is the one-step forward predicted error, and yt,J is the monitoring data at time t. Qt,J−1, the reciprocal of the one-step predicted variance, is the prediction precision of the Jth DLM.With (3)-(4) and (6)–(10), the Bayesian cyclic probabilistic recursive processes of the Jth DLM can be achieved.


### 2.3.2. Bayesian Probabilistic Recursion of DCLM

With (3)–(10) and the distantly provided coupled data, Bayesian probabilistic recursive processes of DCLM with time-variant updated weighted coefficients are derived as follows.(1)
The a posteriori PDF about the stateθt−1 at time t − 1 is(11)θt−1Dθt−1∼∑J=1Πi=1nGipt−1,JNMt−1,J,Ct−1,J,wherept−1,J=αt−1i1,J×αt−1i2,J×αt−1i3,J×⋯×αt−1in,J/∑J=1Πi=1nGiαt−1i1,J×αt−1i2,J×αt−1i3,J×⋯×αt−1in,J can be determined with (5) and (6).(2)
The prior PDF about the stateθt at time t is(12)θtDθt−1∼∑J=1Πi=1nGipt−1,JNat,J,Rt,J,(3)
The one-step prediction PDF at timet is(13)ytDθt−1∼∑J=1Πi=1nGipt−1,JNft,J,Qt,J,where the predicted data,ft=∑J=1Πi=1nGipt−1,Jft,J,∫−∞+∞yt−ft2∑J=1Πi=1nGipt−1,JNft,J,Qt,Jdyt−1, is the prediction precision of DCLM. According to the definition of highest posterior density (HPD) region [34, 35], the predicted interval of the combinatorial DCLM with a 95% confidential interval at time t is ∑J=1Πi=1nGipt−1,Jft,J−1.645∫−∞+∞yt−ft2∑J=1Πi=1nGipt−1,JNft,J,Qt,Jdyt,∑J=1Πi=1nGipt−1,Jft,J+1.645∫−∞+∞yt−ft2∑J=1Πi=1nGipt−1,JNft,J,Qt,Jdyt.(4)
With Bayes method [34, 35], while yt is obtained, the posterior PDF about the state at time t can be computed with(14)θtDθt∼∑J=1Πi=1nGipt,JNMt,J,Ct,J,wherept,J=pt−1,Jpyt,JDθt−1,J/∑J=1Πi=1nGipt−1,Jpyt,JDθt−1,J=pt−1,JNft,J,Qt,J/∑J=1Πi=1nGipt−1,JNft,J,Qt,J.


## 2.3.1. Bayesian Probabilistic Recursion of DLM

(1)
With (4) and (6), the prior PDF about the state θt,J at time t can be obtained with(8)θt,JDθt−1,J∼Nat,J,Rt,J,J∈1,2,…,∏i=1nGi,whereat,J=GMt−1,J,Rt,J=GCt−1,JGT+Wt,J.(2)
With (3) and (8), the one-step forward prediction PDF at time t can be solved with(9)yt,JDθt−1,J∼Nft,J,Qt,J,J∈1,2,...,∏i=1nGi,whereft,J=Fat,J can be used to predict the dynamic data of the Jth DLM; Qt,J=FRt,JFT+e−0.0004t2V. According to the definition of highest posterior density (HPD) region [34], the predicted interval of the Jth DLM with a 95% confidential interval at time t is ft,J−1.645Qt,J,ft,J+1.645Qt,J.(3)
With Bayes method [33, 34], (8) and (9), while yt,J=yt is obtained, the posterior PDF about the state at time t can be computed with(10)θt,JDθt,J∼NMt,J,Ct,J,J∈1,2,...,∏i=1nGi,whereMt,J=at,J+At,Jet,J,Ct,J=Rt,J−At,JQt,JAt,JT,At,J=Rt,JFTQt,J−1,et,J=yt,J−ft,J,et,J is the one-step forward predicted error, and yt,J is the monitoring data at time t. Qt,J−1, the reciprocal of the one-step predicted variance, is the prediction precision of the Jth DLM.With (3)-(4) and (6)–(10), the Bayesian cyclic probabilistic recursive processes of the Jth DLM can be achieved.


## 2.3.2. Bayesian Probabilistic Recursion of DCLM

With (3)–(10) and the distantly provided coupled data, Bayesian probabilistic recursive processes of DCLM with time-variant updated weighted coefficients are derived as follows.(1)
The a posteriori PDF about the stateθt−1 at time t − 1 is(11)θt−1Dθt−1∼∑J=1Πi=1nGipt−1,JNMt−1,J,Ct−1,J,wherept−1,J=αt−1i1,J×αt−1i2,J×αt−1i3,J×⋯×αt−1in,J/∑J=1Πi=1nGiαt−1i1,J×αt−1i2,J×αt−1i3,J×⋯×αt−1in,J can be determined with (5) and (6).(2)
The prior PDF about the stateθt at time t is(12)θtDθt−1∼∑J=1Πi=1nGipt−1,JNat,J,Rt,J,(3)
The one-step prediction PDF at timet is(13)ytDθt−1∼∑J=1Πi=1nGipt−1,JNft,J,Qt,J,where the predicted data,ft=∑J=1Πi=1nGipt−1,Jft,J,∫−∞+∞yt−ft2∑J=1Πi=1nGipt−1,JNft,J,Qt,Jdyt−1, is the prediction precision of DCLM. According to the definition of highest posterior density (HPD) region [34, 35], the predicted interval of the combinatorial DCLM with a 95% confidential interval at time t is ∑J=1Πi=1nGipt−1,Jft,J−1.645∫−∞+∞yt−ft2∑J=1Πi=1nGipt−1,JNft,J,Qt,Jdyt,∑J=1Πi=1nGipt−1,Jft,J+1.645∫−∞+∞yt−ft2∑J=1Πi=1nGipt−1,JNft,J,Qt,Jdyt.(4)
With Bayes method [34, 35], while yt is obtained, the posterior PDF about the state at time t can be computed with(14)θtDθt∼∑J=1Πi=1nGipt,JNMt,J,Ct,J,wherept,J=pt−1,Jpyt,JDθt−1,J/∑J=1Πi=1nGipt−1,Jpyt,JDθt−1,J=pt−1,JNft,J,Qt,J/∑J=1Πi=1nGipt−1,JNft,J,Qt,J.


## 2.4. Error Estimation and Time Efficiency Analysis of DCLM

### 2.4.1. Error Estimation of DCLM

In this paper, mean square error (MSE) is adopted to measure the prediction errors, the MSE is smaller, and the corresponding model is better. The formula about MSE is(15)fMSE=∑t=1Nft−yt2N,where fMSE is the formula of MSE; N is the total number of the predicted data; ft is predicted with (13); yt is the monitoring data at time t. Some relations between MSE and sampling time intervals will be discussed in Sections 2.3.2 and 2.3.3.


### 2.4.2. Time Efficiency Analysis of DCLM

In order to test the time efficiency of DCLM, predicted coupled extreme stresses from different time intervals are considered for analysis comparison, and their MSEs are analyzed to discover different time efficiency. Hence, the every-hour, every-two-hours, and every-four-hours extreme stresses are predicted and utilized for illustrating the time efficiency of DCLM. The analysis results about the time efficiency of DCLM can be shown in Section5.


## 2.4.1. Error Estimation of DCLM

In this paper, mean square error (MSE) is adopted to measure the prediction errors, the MSE is smaller, and the corresponding model is better. The formula about MSE is(15)fMSE=∑t=1Nft−yt2N,where fMSE is the formula of MSE; N is the total number of the predicted data; ft is predicted with (13); yt is the monitoring data at time t. Some relations between MSE and sampling time intervals will be discussed in Sections 2.3.2 and 2.3.3.


## 2.4.2. Time Efficiency Analysis of DCLM

In order to test the time efficiency of DCLM, predicted coupled extreme stresses from different time intervals are considered for analysis comparison, and their MSEs are analyzed to discover different time efficiency. Hence, the every-hour, every-two-hours, and every-four-hours extreme stresses are predicted and utilized for illustrating the time efficiency of DCLM. The analysis results about the time efficiency of DCLM can be shown in Section5.


## 3. Monitoring Coupled Extreme Stress Data-Based DCLM and the Corresponding Bayesian Probabilistic Recursion

Based on Figure2, the flowcharts of monitoring coupled extreme stress data-based DCLM and the corresponding Bayesian probabilistic recursion can be obtained and shown in Figure 3.Figure 3
The flowcharts of Section3.According to Figure3, the detailed analysis processes about monitoring coupled extreme stress data-based DCLM and Bayesian probabilistic recursion are described as follows.For many actual steel bridges, the monitoring extreme stress data is coupled by multiple load effects, and the signal with local trends mainly originates from temperature load effects. Therefore, based on equations (3)–(5), the first-order DCLM is usually enough to predict the coupled extreme stresses and it can be built as follows.Monitoring equation is(16)yt=θt+vt,vt∼N0,e−0.0004t2V⇒yt=Fθt+vt,F=10,θt=θt,βtT,vt∼N0,e−0.0004t2V.State equation is(17)θt=θt−1+βt−1+wt,1,βt=βt−1+wt,2,wt=wt,1,wt,2T∼N0,Wt,⇒θt=Gθt−1+wt,wt∼N0,Wt,θt=θtβtT,wt=wt,1wt,2T,G=1101,Wt=Wt,100Wt,2.Initial information is(18)pθt−1Dθt−1≈∑i1=1g1αt−1i1Nθt−1;μt−1i1,ϕt−1i1,pβt−1Dβt−1≈∑i2=1g2αt−1i2Nβt−1;μt−1i2,ϕt−1i2,where yt is the monitoring coupled extreme stress data at time t; V is approximately estimated with the variance of the differences between the historical states and historical monitoring coupled extreme stresses; wt=wt,1,wt,2T and vt are internally independent and mutually independent; θt is the level (or state) of the coupled extreme stress data at time t; the initial state information can be approximately obtained with cubical smoothing algorithm with five-point approximation [2, 17] through resampling the monitoring coupled extreme stress data at and before time t;βt is the state change at time t; Wt is the variance matrix about wt.Initial information in (18) includes two mixture Gaussian PDFs; thus, there are g1×g2 DLMs. Each DLM has the same monitoring equation and state equation and can be, respectively, obtained with the following.Monitoring equation about theJth DLM is the same as (16), where Vt is substituted by Vt,J.State equation about theJth DLM is the same as (17), where Wt,J=Wt,100Wt,2J.Initial information about theJth DLM is(19)pθt−1Dθt−1≈Nθt−1;μt−1i1,ϕt−1i1,i1∈1,2,…,g1,pβt−1Dβt−1≈Nβt−1;μt−1i2,ϕt−1i2,i2∈1,2,…,g2,⇒θt−1,JDθt−1,J∼NMt−1,J,Ct−1,J,J∈1,2,…,∏i=12gi,θt−1,J=θt−1βt−1JT,Dθt−1,J=Dθt−1Dβt−1JT,Mt−1,J=μt−1i1μt−1i2JT,Ct−1,J=ϕt−1i100ϕt−1i2J,where Wt,J can be approximately solved with (7).Based on equations (3)–(14) and bridge monitoring coupled extreme stress data, Bayesian probabilistic cycle recursive processes of DLM and DCLM (see equations (16)–(19)) can be achieved; furthermore, bridge extreme stress prediction can be made with (13).


## 4. Monitoring Mechanism of the DCLM

Model monitoring has three purposes. The first is to identify where model prediction function declines. The second is to cope with the faults and to monitor and update the model. The third is to improve the accuracy of future prediction.In this paper, the main idea of model monitoring mechanism is to use one or more alternative models to compare and evaluate model performance.According to the research studies in [1, 34–36], model monitoring is achieved through Bayesian factors under the normal assumption. The main idea is firstly to build an alternative model and then to combine an existing probability model for constructing the formula of Bayesian factors.In this paper, the adopted probability distribution density function of the alternative model and one-step prediction model are, respectively, in the following equations:(20)p1=2πk2−0.5exp−0.5ete2k2,(21)p0=2π−0.5exp−0.5ete2,where ete=yt−∑J=1Πi=1nGipt−1,Jft,J/∫−∞+∞yt−ft2∑J=1Πi=1nGipt−1,JNft,J,Qt,Jdyt is the standard prediction error obtained with (13) and k is the standard deviation of ete.The Bayesian factor forp0⋅ versus p1⋅ based on the observed value of yt is defined as(22)Bt=p0yt|Dt−1p1yt|Dt−1,where p0yt|Dt−1 is the one-step predictive PDF of monitoring daily extreme stress data; p1yt|Dt−1 is PDF of the alternative model, namely, the routine or standard PDF; Bt is the Bayesian factor for p0yt|Dt−1 versus p1yt|Dt−1 based on the observed value of yt.Furthermore, according to (20)–(22), the Bayesian factors can be obtained with(23)Bt=kexp−0.5ete21−k−2,where ete=yt−∑i=1npt−1,ift,i/∫−∞+∞yt−∑i=1npt−1,ift,i2∑i=1npt−1,iNft,i,Qt,idyt is the standard prediction error and k is the standard deviation of ete.For integersN=1,…,t, the Bayesian factor for p0⋅ versus p1⋅ based on the sequence of N consecutive monitoring daily extreme stress data yt,yt−1,…,yt−N+1 is built as (24); namely, the built formula BtN of the cumulative Bayesian factor is(24)BtN=∏r=t−N+1tBr=p0yt,yt−1,…,yt−N+1Dt−Np1yt,yt−1,…,yt−N+1Dt−N,where BtN is the cumulative Bayesian factor, which measures the evidence provided by the recent (up to and including time t) N consecutive observations yt,yt−1,…,yt−N+1.In this paper, according to the engineering experience, the adopted monitoring criterion [1, 34–36] is as follows: if k = 3 and B (t) < 0.15, then the corresponding monitoring data is abnormal, which needs to be removed; otherwise, the monitoring data is normal.After removing the abnormal data, if the cumulative Bayesian factor is bigger, the prediction precision of DCLM is better. The uncertainty of DCLM is smaller.


## 5. Application to an Existing Steel Bridge

Fumin bridge in Tianjin city of China, taken for the actual example [23, 36–38], is an unsymmetrical single-tower and self-anchored suspension bridge with 3D curved cables, which was designed with three-span arrangements shown in Figure 4. The total length of this bridge is 340.3 m, and the width is 40 m. The deck is mainly composed of two steel box girders with a short prestressed concrete segment at the right end anchorage, which can be seen in [23, 37, 38]; the monitoring sections of the deck are not affected by the short prestressed concrete segment; the main cables in the main span are anchored to the deck, whereas those in the side span on a vertical plane are anchored to the ground. Most parts of the bridge are made of steel, and the allowable stress of the steel is 345 MPa. About this bridge, the girder system is mainly composed of upstream and downstream girders and the crossbeam, and the support conditions are indicated in [37, 38]. The detailed design information can be obtained in [23, 37, 38].Figure 4
A view of Fumin bridge [23, 36].For the girder system of Fumin bridge, according to the monitoring program [38], there are seven monitoring sections, namely, section A, section B, section G, section C, section D, section E, and section F. The positions of these sections are displayed in Figure 5, where sections A, B, and G are in the side span; sections C, D, E, and F are in the main span. These positions play a very important role in controlling the safety-based structural performance. In the measure program for the girder system [38], the sampling rate about the strain data is 2.5 Hz sampling frequency. The longitudinal hourly monitoring extreme strain data about these seven sections are monitored through an array of Fiber Bragg Grating (FBG) strain sensors, and the corresponding extreme stress data can be obtained with the monitoring extreme strain multiplied by modulus of elasticity.Figure 5
The layout of monitoring sections about Tianjin’s Fumin bridge [23, 36, 38].When the traffic flow magnitude is extremely large, the seven sections are monitored. The dynamic coupled extreme stresses mainly caused by temperature load and vehicle load are nonstationary and can be used to build DCLM.Among the seven sections, section D is the most dangerous section, so section D is adopted as the research object. Section D installed three sensors (see Figure6): FBG01012, FBG01015, and FBG01005, where the monitoring point for FBG01012 is the most dangerous point. Therefore, the hourly coupled extreme stress data of FBG01012 is applied to build DCLM for dynamically predicting hourly coupled extreme stress about section D. The stresses from the dead weight of the steel structure and the concrete deck are not included in the coupled extreme stress data. According to [23, 36], it can be known that the stress value caused by the total dead weight is 226 MPa. Considering the factor (the value is 1.15 [23, 36]) assigned to the data provided by the sensors, the maximum extreme stress value caused by the total live loads is 345−226/1.15=103.5MPa [23].Figure 6
Sensors about section D [23, 36, 38].From August 25 to September 5, 2009, the extreme stresses at the installed point of FBG01012 were monitored for 287 hours, and the hourly monitoring coupled extreme stress data collected by sensor FBG01192 is shown in Figure7.Figure 7
Monitoring extreme stress data and state data.Now, to build the DCLM with the monitoring coupled extreme stress data shown in Figure7, the ADF test based on the R programming language [30–33] is applied to determine the order of monitoring coupled extreme stress data of the previous 287 hours. The test results show that the first-order differenced sequence of monitoring coupled extreme stresses is tested to be stationary. Therefore, the local trend order of monitoring coupled data is one; furthermore, Section 3 is reasonable and available for this example.Based on the monitoring extreme stress data of the previous 287 hours, with the cubical smoothing algorithm with five-point approximation [2, 17, 36] and the difference method [39], the initial information can be both approximately obtained and shown in Figures 7 and 8. They can be both simulated with two mixture normal distributions through the least residual error quadratic sum method [2].Figure 8
State change data (or state difference data).Based on (16)–(18) and the monitoring extreme stress data of the previous 287 hours shown in Figure 7, the built DCLM are as follows.Monitoring equation is(25)yt=θt+vt,vt∼N0,0.016e−0.0004t2,⇒yt=Fθt+vt,F=10,θt=θt,βtT,vt∼N0,0.016e−0.0004t2.State equation is the same as equation (17).The initial information is(26)pθt−1Dt−1,1≈0.6048×Nθt−1;42.6755,0.3012+0.3952×Nθt−1;44.3965,0.0990,pβt−1Dt−1,2≈0.1421×Nβt−1;−0.0394,0.0003+0.8579×Nβt−1;0.0071,0.0068,where wt=wt,1,wt,2T and vt are internally independent and mutually independent; θt is the level (or state) of the extreme stress data at time t; βt is the state change at time t.Based on (17)-(18), (25)-(26), and the monitoring coupled extreme stress data shown in Figure 7, the four built DLMs are as follows.(1)
The first DLM is as follows.Monitoring equation is the same as equation (25).State equation is(27)θt=Gθt−1+wt,wt∼N0,Wt,θt=θtβtT,wt=wt,1wt,2T,G=1101,Wt=−GCt−1GT+Ct−10.8.The initial information is(28)pθt−1Dt−1,1≈Nθt−1;42.6755,0.3012,pβt−1Dt−1,2≈Nβt−1;−0.0394,0.0003,⇒Ct−1=0.3012000.0003.(2)
The second DLM is as follows.Monitoring equation is the same as equation (25).State equation is the same as equation (27).The initial information is(29)pθt−1Dt−1,1≈Nθt−1;42.6755,0.3012,pβt−1Dt−1,2≈Nβt−1;0.0071,0.0068,⇒Ct−1=0.3012000.0068.(3)
The third DLM is as follows.Monitoring equation is the same as equation (25).State equation is the same as equation (27).The initial information is(30)pθt−1Dt−1,1≈Nθt−1;44.3965,0.0990,pβt−1Dt−1,2≈Nβt−1;−0.0394,0.0003,⇒Ct−1=0.0990000.0003.(4)
The fourth DLM is as follows.Monitoring equation is the same as equation (25).State equation is the same as equation (27).The initial information is(31)pθt−1Dt−1,1≈Nθt−1;44.3965,0.0990,pβt−1Dt−1,2≈Nβt−1;0.0071,0.0068,⇒Ct−1=0.0990000.0068.Based on the monitoring coupled extreme stress data from the 287th hour to the 1149th hour, the predicted extreme stresses and prediction precision from the 288th hour to the 1150th hour are, respectively, solved with (8)–(14), (17), and (25)–(31); the results are shown in Figures 9 and 10. From Figure 9, it can be seen that predicted coupled extreme stresses well fit the changing rules of monitoring coupled extreme stress data. Furthermore, with the distant updating of monitoring coupled extreme stress data, the prediction precision gradually increases as shown in Figure 10, which signifies that it is reasonable to predict the structural dynamic coupled extreme stresses with the built DCLM. The predicted coupled extreme stresses are all smaller than 103.5 MPa, so section D is safe.Figure 9
Monitoring stresses and one-step predicted stresses.Figure 10
One-step prediction precision.In the prediction processes, the monitoring mechanism is adopted, and the single Bayesian factors are all less than 0.15, so the monitoring extreme stresses are all normal. The cumulative Bayesian factors increase, so the prediction precision of DCLM is better, and the uncertainty of DCLM is smaller.ARIMA [30–33, 40], BDLM [36], BFDLM [23], and Fourier regressive functions [23] are utilized for fair comparison with the proposed DCLM for the prediction effects of bridge coupled extreme stresses, and the predicted results are shown in Figure 11. The mean square errors (MSE) (see equations (15) and (32)) for the predicted extreme stresses of DCLM, ARIMA, BDLM, BFDLM, and Fourier regressive function are, respectively, 0.048, 0.1035, 0.0953, 0.0538, and 0.4775. ARIMA method is suitable for short-term prediction; BDLM does not have steady prediction precision; the state equation of BFDLM is complex and relies on very subjective Fourier regressive functions; so the DCLM prediction method is more effective than the other four methods for predicting coupled extreme stresses.Figure 11
Predicted results of the five methods.In this example, the built DCLM has a first-order differential approximation of local polynomials for monitoring coupled extreme stress data at each time; therefore, it can make MSE the smallest among the above five models. MSE is smaller, and the corresponding model is more reasonable, so it can be noticed that the proposed DCLM algorithm is more suitable than the other four models for coupled extreme stress prediction of steel bridges. What is more, the DCLM can be easily and widely used in extreme stress prediction; in particular, the state equation is more feasible than the other four methods.(32)fMSE=∑t=2881150ft−yt2863,where fMSE is the formula of MSE; ft is predicted with (13); yt is the monitoring coupled extreme stress data at time t.According to the description in Section2.4.2, the extreme stresses with sampling time interval of every-two-hours and every-four-hours are predicted, and their MSEs calculated with (15) are shown in Figure 12. It can be noticed from Figure 12 that MSEs increase as the time interval increases. So, the adopted time interval is one hour in this paper.Figure 12
MSEs of DCLM about three kinds of time intervals.


## 6. Conclusions

This paper first presents the Bayesian dynamic coupled prediction method for coupled extreme stresses of steel bridges based on DCLM and Bayes method, which could effectively and simply deal with the influences of coupling on bridge monitoring information. This method is mainly applied for the monitoring coupled data. In addition, the proposed DCLM has the advantages of simplicity and convenience when dealing with monitoring coupled data; namely, the highly adaptive state equation instead of the complex state curves fitted with the historical monitoring data is reasonably applied in the filtering recursion processes of DCLM based on Bayes method. The proposed method can provide the foundations for bridge reliability prediction and assessment.



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*Source: 8712907-2020-08-11.xml*