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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.

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 [

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 [

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.

BDLMs are the predicting approaches based on a philosophy of information updating [

observation equation:

state equation:

initial information:

In this paper, the BDLMs mean that the observation equation and the state equation are both linear and are shown in (

For the mass and random monitored extreme data, especially for monitored data at time

Based on Section

observation equation:

state equation:

initial information:

For each time

With (

It can be known from (

The modeling process of BDLMs.

In this paper, the monitored interval period of extreme stress data is one day;

BDLMs are presented as a special case of a general state-space model, being linear and Gaussian. So the BDLMs satisfy the assumptions of a state-space model. While the basic assumptions [

State variables, observation errors, and state errors all follow normal distributions.

(

namely,

Conditionally on (

The recursive relation between state variables and inspection/monitored variables is shown in (

Suppose that there are

observation equation:

state equation:

Normal a priori distribution about the initial information is as follows:

If the initial state data follows the lognormal distribution, then the state data can be transformed into a quasinormal distribution

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

(1) With estimation method of kernel density, the actual distribution function

(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,

BDLMs are applicable to the prediction of the future state parameters, which can be recursive and updated like well-known Kalman filter [

(1) The a posteriori distribution at time

(2) The a priori distribution at time

(3) One-step prediction distribution at time

According to the definition of highest a posterior density (HPD) region [

(4) The a posteriori distribution at time

(5) Predicted probability distribution based on the arithmetic mean at time

(6) The combinatorial prediction probability distribution at time

(7) Comparison of the predicted precisions between combinatorial prediction model and the prediction model based on the arithmetic means:

Model monitoring has three purposes. The first is to identify where model prediction function declines and which form the model fault occurred in. 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 [

In this paper, the adopted probability distribution density function of the alternative model and one-step prediction model are, respectively, (

The Bayesian factor for

Further, according to (

For integers

With (

Curves of Bayes factors versus one-step predicted errors.

After removing the abnormal data, in the changing curves of cumulative Bayesian factors, the prediction precision of the Bayesian dynamic model can be shown. Namely, if the cumulative Bayesian factor is bigger, the prediction precision of the Bayesian dynamic model is better. The uncertainty of the Bayesian dynamic models is smaller.

In this paper, the FOSM method [

Suppose there are random variables

With FOSM method, the formula of the reliability indices can be obtained with

In this paper, a five-span continuous steel plate girder bridge is taken as an example. The total length of the bridge is 188.81 m. The explicit details about the aim and results of the monitoring program for the whole bridge are given in [

The reliability index

For the real-time monitored reliability indices, the monitored data is one by one, so

The I-39 Northbound Bridge, which was described in Section

Real-time monitored extreme stresses.

Time |
Stress |
Time |
Stress |
Time |
Stress |
Time |
Stress |
---|---|---|---|---|---|---|---|

1 | 25.23 | 22 | 21.22 | 43 | 24.17 | 64 | 25.15 |

2 | 21.67 | 23 | 22.02 | 44 | 23.72 | 65 | 24.64 |

3 | 19.53 | 24 | 34.80 | 45 | 26.85 | 66 | 23.18 |

4 | 20.50 | 25 | 30.51 | 46 | 30.32 | 67 | 21.94 |

5 | 24.44 | 26 | 21.57 | 47 | 31.93 | 68 | 18.82 |

6 | 22.66 | 27 | 31.67 | 48 | 25.06 | 69 | 22.66 |

7 | 25.95 | 28 | 29.16 | 49 | 23.01 | 70 | 21.57 |

8 | 32.65 | 29 | 21.67 | 50 | 22.02 | 71 | 29.16 |

9 | 39.26 | 30 | 23.99 | 51 | 33.90 | 72 | 21.57 |

10 | 21.40 | 31 | 21.05 | 52 | 18.10 | 73 | 32.92 |

11 | 31.48 | 32 | 29.35 | 53 | 25.24 | 74 | 21.94 |

12 | 30.06 | 33 | 22.66 | 54 | 25.77 | 75 | 21.14 |

13 | 20.60 | 34 | 24.61 | 55 | 17.11 | 76 | 20.41 |

14 | 22.56 | 35 | 25.77 | 56 | 23.72 | 77 | 16.76 |

15 | 23.54 | 36 | 28.54 | 57 | 12.65 | 78 | 22.38 |

16 | 16.94 | 37 | 22.83 | 58 | 24.89 | 79 | 27.21 |

17 | 29.16 | 38 | 21.05 | 59 | 27.56 | 80 | 19.98 |

18 | 22.47 | 39 | 24.44 | 60 | 25.86 | 81 | 18.82 |

19 | 23.37 | 40 | 28.80 | 61 | 24.61 | 82 | 29.44 |

20 | 28.99 | 41 | 20.24 | 62 | 22.11 | 83 | 20.41 |

21 | 30.15 | 42 | 29.97 | 63 | 21.22 |

Curves of monitored extreme stresses.

In this existing example, the state equation, obtained with (

For obtaining the distribution parameters of initial information, the monitored extreme data of the 83 days is smoothly processed, and then the initial information of monitored data is approximately obtained, which is shown in Figure

Curves of initial information and the monitored extreme stress data.

Through Kolmogorov-Smirnov (K-S) test for the initial information, the initial a priori PDF is lognormal PDF or normal PDF shown in Figure

PDF curves fitted with the initial monitored extreme stress data (PDF: probability density function).

Based on the monitored data, with (

observation equation:

state equation:

initial information:

Equation (

Initial information follows normal distribution, and then the BDLMs are built to predict the monitored extreme data.

Initial information follows lognormal distribution; firstly the lognormal distribution must be transformed into a quasinormal distribution [

The arithmetic mean of the one-step predicted mean values, respectively, obtained with Cases

The fourth case is to build combinatorial BDLMs with BDLMs obtained with Cases

In this paper, the Bayesian factors are adopted to seek the abnormal data, and the monitored results are shown in Figures

Curves of time-dependent Bayes factors (the data of the 9th day is abnormal).

Curves of time-dependent Bayes factors after eliminating the abnormal extreme data (the data of the 9th day is deleted).

Curves of cumulative Bayesian factors.

The predicted extreme stresses and prediction precision (the reciprocal of predicted variances) of the above four cases are, respectively, shown in Figures

Predicted curves of extreme data when initial information follows normal probability distribution (Case

Predicted curves of extreme data when initial information follows lognormal probability distribution (Case

Predicted curves of extreme data based on the arithmetic mean of the two distributions (Case

Predicted curves of extreme data based on the combinatorial model (Case

Comparison among predicted data of the four cases.

Prediction precision comparisons among the predicted stresses of the four cases.

From Figures

In Figure

Curves of reliability indices based on the combinatorial model of monitored extreme data.

In this paper, based on the everyday monitored extreme stresses of bridge, the structural reliability indices are predicted with combinatorial BDLMs and FOSM method. And the following conclusions can be reached:

The BDLMs, which are used to seek the abnormal data of the mass monitoring information, are obtained with 1-order polynomial function based on the past information.

The monitored extreme stresses-based combinatorial BDLMs are firstly built. The predicted extreme stresses and the predicted ranges of the above four cases are almost the same, but as far as the prediction precision is concerned, the combinatorial BDLMs have the best prediction precision.

Based on the combinatorial BDLMs of monitored extreme stresses, structural reliability indices are predicted. Compared with deterministic monitored extreme stresses-based reliability indices, this paper considered the randomness and uncertainty of monitored data, so the predicted reliability indices are smaller. But the predicted smaller reliability indices may better reflect the actual state of the bridge. Thus, the smaller reliability indices may be more reasonably used to assess the structural safety and serviceability.

In this paper, the proposed reliability prediction method is easy and may be widely used in the structural health monitoring. BDLMs are possible to include subjective judgments with the observed data in order to obtain a more informed and accurate prediction. The numerical applications presented, using the monitored extreme data of an existing bridge, illustrate the application and feasibility of the proposed approaches and concepts.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the Fundamental Research Funds for the Central Universities (lzujbky-2015-300, lzujbky-2015-301).