In structural health monitoring (SHM) field, the structural stress prediction and assessment are the research bottleneck. To reasonably and dynamically predict structural extreme stress based on the timevariant monitored data, the objectives of this paper are to present (a) cubic functionbased Bayesian dynamic linear models (BDLM) about monitored extreme stress, (b) choosing method of optimum probability distribution functions about initial stress state, (c) monitoring mechanism of the optimum BDLM, and (d) an effective way of taking advantage of BDLM to incorporate the timevariant monitored data into structural extreme stress prediction. The monitored data of an existing bridge is adopted to illustrate the feasibility and application of the proposed models and procedures.
Nowadays, most of the service bridges are close or above their planned lifetime at home and abroad. The bridge stress can reflect the structural basic status. Therefore, it is crucial to predict the structural dynamic stress for the safety and serviceability assessment of critical infrastructure components for bridge systems. Therefore, how to build the dynamic prediction model of structural stress becomes very important.
For bridges, SHM is a very useful tool which can provide the structural basic status, including strain, stress, and deflection of specified structural components or structures. How to properly handle with the continuously provided monitored data is one of the main difficulties in the SHM field. Up to now, a sound number of studies about the data handling are mainly focused on the modal parameter identification, structural damage detection technology, data modeling, and so on [
In this paper, considering the uncertainty of structural dynamic monitored data and the diversity of initial stress state probability distributions, the optimum Bayesian dynamic linear prediction method about monitored extreme stress including nonlinear effects is given with cubic function, and the detailed contents are described in detail as follows.
Firstly, build cubic function of monitored extreme stress.
Secondly, Bayesian dynamic linear model is established based on the built cubic function.
Thirdly, considering the diversity of probability distribution functions (PDF) about initial state information, the optimum PDF is chosen with KullbackLeibler (KL) information distance. And then the optimum BDLM and the corresponding model monitoring mechanism are, respectively, given.
Finally, an actual engineering is provided to illustrate the feasibility and application of the proposed model and method in this paper.
Cubic functionbased Bayesian dynamic linear prediction approach of bridge extreme stress is shown in Figure
The building processes of the optimum Bayesian dynamic linear model.
BDLM is a predicting approach based on a philosophy of information updating [
In this paper, the cubic function is adopted to build the BDLM, the prediction precision of which is better than the BDLM built based on linear function model and quadratic function model about the SHM data. The cubic functionbased BDLM is described in detail as follows.
BDLM is presented as a special case of a general statespace model, being linear and Gaussian. So the BDLM satisfies the assumptions of a statespace model, while the basic assumptions [
(a) State variables, monitored errors, and state errors all follow normal/Gaussian distributions.
(b) (
(c) Conditionally on
For the longterm health monitoring extreme stress data, the fitted cubic function
Further, with (
Monitored equation is
State equation is
Initial state information is
For BDLM,
With (
It is known from (
The probability recursive processes of BDLM.
For the BDLM, the probability parameters need to be solved include
In this paper,
Considering the diversity of PDF about the state variable including
The extension of common PDF is better than theoretical PDF, and more reasonably used for BDLM, so the KL information distance is adopted to determine the optimum common PDF about the state variable.
If the initial state variable follows the other common PDF except Gaussian PDF, then the distribution can be approximately transferred into mixed Gaussian PDF. Take state variable
(1) With kernel density estimation method, the actual PDF
(2) Since any set of data can be fitted by a few normal distributions, namely.
(3) The weights and distribution parameters of the fitted normal distributions can be obtained with the least residual error quadratic sum method OLS, namely,
With (
The transferred common Gaussian PDF
BDLM are applicable to the prediction of the future state parameters, which can be recursive and updated like wellknown Kalman filter [
According to the definition of highest posterior density (HPD) region [
Model monitoring has three purposes. The first is to identify where model prediction function declines and in which form the model fault occurred. 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 monitoring mechanism is to construct Bayesian factors [
The single Bayesian factor for
For integers
Further through simplifying (
Then the specific expression of the Bayesian factor is obtained as
Monitoring rules of the Bayesian dynamic model is when
The I39 northbound bridge [
Map view of the I39 northbound bridge (adapted from [
I39 northbound bridge, instrumentation plan of the strain gage, CH15 (adapted from [
The monitored data of the sensor CH15 displayed the variability of the stresses caused by traffic, temperature, shrinkage, creep, and structural changes. The stresses from the dead weight of the steel structure and the concrete deck are not included in the measured data. The monitored extreme stresses are shown in Table
Dynamic monitored extreme stresses [
Time (day)  Stress 


25.23 

21.67 

19.53 

20.50 

24.44 

22.66 

25.95 

27.65 

25.26 

21.40 

21.48 

20.06 

20.60 

22.56 

23.54 

20.94 

19.16 

22.47 

23.37 

20.99 

24.15 

21.22 

22.02 

24.80 

22.51 

21.57 

21.67 

19.16 

21.67 

23.99 

21.05 

23.35 

22.66 

24.61 

25.77 

26.54 

22.83 

21.05 

24.44 

28.80 

20.24 

29.97 

24.17 

23.72 

26.85 

25.32 

26.93 

25.06 

23.01 

22.02 

23.90 

20.10 

25.24 

25.77 

27.11 

23.72 

22.65 

24.89 

27.56 

25.86 

24.61 

22.11 

21.22 

25.15 

24.64 

23.18 

21.94 

21.82 

22.66 

21.57 

29.16 

21.57 

22.92 

21.94 

21.14 

20.41 

22.76 

22.38 

25.21 

24.98 

23.82 

26.44 
Now, the monitored extreme stress data of the forward seventy days is adopted to predict the extreme stress of the later twelve days based on the built BDLM shown in (
In this existing example, the state data, which is shown in Figure
RMSE about the regressive polynomial functions.
Functions  RMSE 

Linear polynomial  2.305 
Quadratic polynomial  2.303 
Cubic polynomial  2.167 
Monitored extreme stress data and resampled extreme stress data.
Monitored data and its regression analysis.
In order to solve the distribution parameters of initial state information, the monitored extreme stress data of the forward seventy days is resampled with cubical smoothing algorithm with fivepoint approximation; then the resampled data
Initial state data and initial state difference data.
Through KolmogorovSmirnov (KS) test for the initial state data and initial state difference data, with (
With (
Monitored equation is
State equation is
Initial state information is
With (
Monitored and predicted extreme stress data.
Prediction precision of BDLM.
In Figure
Based on (
Single Bayesian factors of BDLM.
Cumulative Bayesian factors of BDLM.
The paper describes a new approach for incorporating monitored data into structural extreme stress prediction. Namely, based on the monitored extreme stress data including nonlinear effects, the cubic functionbased BDLM is utilized in this paper. This kind of model is easy to build and can be widely used in the bridge performance prediction.
The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
This work was supported by the National Natural Science Foundation of China (Project nos. 51608243 and 51702142), the Natural Science Foundation of Gansu Province of China (Project no. 1606RJYA246), the Fundamental Research Funds for the Central Universities (Project nos. lzujbky2015300, lzujbky2015301, and lzujbky2017k17) for the research, authorship, and/or publication of this article.