Bridge performance degradation model based on the multi-variate bayesian dynamic linear method

Abstract

The degradation of bridge structural performance arises from the combined influence of various factors. Performance assessment and reliable prediction of bridge performance degradation through effective utilizing of detection information updates is a challenging problem. In this paper, the concept of performance indicators is redefined, employing to delineate bridge performance degradation. A bridge performance degradation model (the error ≤8%) is formulated, considering the multiple-variable Bayesian dynamic linear method (MBDLM) and revealing the coupling mechanisms among factors influencing bridge performance degradation. On this basis, the prediction performance of the model is quantitatively evaluated by three metrics: mean squared error, predictive mean squared error and mean absolute percentage error. A methodology is presented for the assessment, prediction, and maintenance reinforcement of in-service bridge structural performance degradation. This approach holds promise for future applications in safety assessments and the decision-making process for preventive maintenance of operational bridges.

Keywords

Bridge engineering performance indicators multi-variate bridge performance degradation model multi-variate bayesian dynamic linear method monitoring system dynamic performance degradation prediction

Introduction

The safety and external durability of bridge structures are crucial factors when evaluating and predicting the performance degradation of in-service bridges. The evaluation and prediction of bridge performance degradation based on initial monitoring data poses a significant challenge due to the time-dependent nature of bridge structural systems and the complex interplay of various factors (including natural factors, human factors, and traffic loads) (Fereshtehnejad et al., 2017; Wang, 2022; Zhang et al., 2021). Various methodologies have been devised to evaluate bridge performance, encompassing time-dependent reliability (Garg et al., 2022), bridge health monitoring systems (Xia et al., 2020), and time-invariant reliability (Straub et al., 2020). Nevertheless, depending exclusively on reliability metrics to evaluate bridge performance lacks the necessary precision to accurately forecast bridge performance degradation.

The assessment and degradation of the bridge structural performance can be achieved by systematically monitoring various influential factors, including natural factors and human factors. However, a critical prerequisite for this lies in the development of an effective model for bridge performance degradation. This model should be capable of offering a quantitative representation of the process of bridge performance degradation. Moreover, it ought to elucidate the complex coupling mechanisms among the diverse influencing factors. Presently, bridge performance degradation models are primarily characterized by the inclusion of physical models (Wang et al., 2021), mathematical models (Moscoso et al., 2022), and time-dependent reliability decay models (Ge et al., 2022). The physical model was utilized to scrutinize specific degradation processes manifesting in bridge components, such as steel corrosion and concrete carbonation (Sun et al., 2020). Strauss et al. (2013) proposed a methodology to assess the degradation of bridge performance influenced by chloride ion erosion. This approach involved evaluating the rust condition and estimating the remaining service life throughout the degradation process. Zambon et al. (2018) utilized concrete carbonation and chloride ion erosion as fundamental parameters for forecasting the remaining operational longevity of existing concrete bridges within the infrastructure network.

Within the realm of mathematical modeling, the BDLM framework was employed to assess the probability of structural states and the performance of in-service bridge structures by utilizing inspection information updates (Goulet, 2017; Goulet et al., 2018; Wang et al., 2022). Lei et al. (2023) constructed a policymaking framework considering environmental, economic, and safety metrics through convolutional autoencoder-structured deep-Q network. Lai et al. (2024) suggested an integrated monitoring-based optimal management method based on the partially observable Markov decision processes and Bayesian forecasting for realizing an optimal maintenance strategy within the service life. Sun et al. (2019) have proposed a method for predicting the degradation of bridge performance. This approach employs an artificial intelligence algorithm and entails extracting the initial structural state information from monitoring data. Xia et al. (2021) established a framework for predicting and assessing the performance degradation of regional bridge networks. Liu and Fan (2020a) proposed a deep-learning-based approach for modeling the rating data of bridge components. Lei et al. (2024) proposed an ensemble machine learning model (Lei et al., 2023) considering Bayesian-optimized for predicting the critical seismic demands of urban highway bridges.

The model for time-dependent reliability degradation transformed complex time-dependent reliability considerations into a more tractable conventional reliability problem. Liu et al. (2020) proposed a methodology to predict extreme stress levels in bridges, utilizing the Dynamic Linear Trend Model (DLTM), Dynamic Coupled Linear Model (DCLM), and Bayesian Probability Recursive Process. Subsequently, the authors conducted predictions and assessments of the bridge structure reliability. Liu and Fan (2020b) took into account the existence of nonlinear correlations among various control monitoring points and formulated a Multivariate Bayesian Dynamic Gaussian Copula Model (MBDGCM) to forecast the dynamic reliability of bridges. Fan et al. (2018) introduced a BDLM that includes second-order polynomials and time-varying weighting coefficients. They utilized extreme stress data gathered from the health monitoring iteratively updated to predict dynamic reliability.

The Bayesian dynamic linear method, functioning as an unsupervised algorithm, has been applied in various domains for anomaly detection. By integrating the Rao-Blackwellized Particle Filter with BDLM, abnormal data was detected using a continuous Monte Carlo approach (Nguyen et al., 2019; Nickless et al., 2018). Zhang et al. (2021) proposed a BDLM utilizing maximum likelihood estimation for anomaly detection in structural health monitoring data. Nguyen et al. (2018) proposed an anomaly detection methodology that integrates the BDLM framework with Kalman filtering theory.

In summary, scholars, both nationally and internationally, have extensively researched the bridge performance degradation and prediction. However, a substantial gap between theoretical models and practical applications persists due to the simplifying assumptions and idealized conditions inherent in theoretical models. In practical applications, the factors influencing bridge performance degradation are numerous and complex, spanning various aspects such as the environment, loads, and materials. Consequently, the disparity between theoretical models and practical applications is noteworthy, necessitating further refinement and optimization of models to enhance the accuracy of predicting bridge performance degradation. To address the limitations in existing research, this paper takes into account natural factors, human factors, and traffic loads, establishing a bridge performance degradation model for precise predictions and extending its applicability. The study aims to achieve reliable predictions of bridge performance degradation through the development of a robust bridge performance degradation model. The model facilitates the retroactive calculation of the proportion of factors contributing to bridge performance degradation based on performance indicators. This research effort aims to establish a strong theoretical framework for bridge structure maintenance.

This study proposes a bridge performance degradation model based on the multiple-variable Bayesian dynamic linear method (MBDLM) to predict and assess performance degradation of bridges. Section “Bridge Performance Degradation Model” illustrates the construction and application process of the constructed model. Section “Construction of Bridge Performance Degradation Model” first shows the definition of performance indicators and the determination of the weight of each variable. Subsequently, the MBDLM is introduced and the bridge performance degradation model is constructed. Section “Determination of the Model’s Parameters” expresses determination method of the model’s parameters. Section “Monitoring System of MBDLM” discusses the monitoring system of MBDLM. Section “The Evaluation of the Model’s Predictive Performance” illustrates the evaluation method of the model’s predictive performance. In section “Validation of an Engineering Case” shows the validation of an engineering case using the proposed model. Finally, Section “Conclusions” presents some appropriate conclusions drawn from this research.

Bridge performance degradation model

Construction of bridge performance degradation model

Contemporary bridge structures exhibit characteristics akin to “time-dependent structures”, where the dependability of individual components and the overarching system experiences alterations over time (Li et al., 2020; Luo et al., 2021). Previous models of bridge performance degradation characterized the process through changes in reliability (Fan et al., 2018; Liu et al., 2020). In our model, bridge performance indicators are employed to delineate the process of bridge performance degradation. Bridge performance deterioration is a complex process influenced by a multitude of factors. The paper addresses the modeling of bridge performance degradation, considering the aspects of structural suitability, durability, and safety. This comprehensive model incorporates four key factors (Hao, 2018): natural factors, human factors, external durability, and structural safety (see Figure 1).

Figure 1.

Schematic diagram of bridges performance degradation system.

Precipitation has the potential to induce corrosion, sedimentation, and erosion, leading to a reduction in structural stability. Strong winds can cause vibrations and structural fatigue, adversely impacting both stability and load-bearing capacity. Temperature differentials may result in thermal expansion and contraction, causing deformation and cracks, thereby influencing overall stability and structural longevity. The fatigue and vibration of bridge structures are ascribed to the volume of traffic, leading to a reduction in durability. Furthermore, the annual overloading rate contributes to structural degradation and material aging, thereby diminishing the overall performance. Elevating the strength of concrete enhances the load-bearing capacity and longevity of bridges. Corrosion is initiated by carbonation, leading to a reduction in both strength and overall structural performance. Increased concentrations of chloride ions result in corrosion, cracking, and spalling, thereby diminishing the long-term durability of the structure. Decreased corrosion potential in steel reinforcement initiates structural corrosion, cracking, and deformation, leading to a reduction in both overall strength and durability. The variations in deflection and strain coefficients act as indicators reflecting the evolutionary trends in the structural performance of the bridge, offering valuable insights for maintenance and reinforcement strategies. Insufficient impact coefficients present a risk of structural instability and damage, thereby compromising safety and durability. Moreover, a low fundamental frequency may reduce seismic performance, demanding meticulous consideration of reinforcement measures.

Due to the diverse dimensional units inherent in the detection data, an initial procedure involves the transformation of this data into dimensionless performance indicators, as indicated in equation (1). This process is conducted with the explicit aim of facilitating the analysis of bridge performance degradation. Subsequently, the grey correlation method is utilized to ascertain the individual weights of each variable, ultimately leading to the formulation of the bridge performance model. The main content of this research is on constructing a bridge performance degradation model based on the multivariate Bayesian dynamic linear method, with the performance indicators serving merely convenient in model construction.

The performance indicators were defined based on the “Code for Maintenance and Evaluation of Highway Bridges” (JTG 5120-2021) (Ministry of Transport of the People’s Republic of China, 2021). According to the regulations outlined in the code regarding the assessment grade of bridge technical conditions and state descriptions, the performance indicators are redefined based on the load capacity and state grades of the bridge. The bridge condition assessments and their respective scoring ranges are classified into five discrete categories, identified as Class 1

(90, 100]

, Class 2

(80, 90]

, Class 3

(60, 80]

, Class 4

(40, 60]

, and Class 5

(0, 40]

. Therefore, this paper categorizes performance indicators into five discrete intervals (see Table 1), wherein the values of these indicators are determined by employing interpolation methodologies.

Table 1.

Performance Indicators Level and Interval Definition.

Condition rating	State	Carrying capacity	Performance indicators interval
1	Excellent	Satisfied with the design standards	$(4, 5]$
2	Good	Reach the design indices	$(3, 4]$
3	Fair	Reduced by less than 10 %	$(2, 3]$
4	Poor	Reduce by 10%∼25%	$(1, 2]$
5	Severe	Reduce by more than 25%	$(0, 1]$

How can the detection variables that influence bridge performance degradation be transformed into performance indicators? Initially, the detection variables are classified according to the specification. The points of the highest probability within the grey-class are selected to construct respectively grey-class intervals of each variable (see Figure 2). Subsequently, each detection variable is divided into five intervals (Hao, 2018), corresponding to the state levels specified in Table 1. Then these detection variables are transformed into performance indicators through linear interpolation. Given that the detection data of the variable $i$ at time $t$ is represented as $X_{i, t} (i = 1, 2, \dots, r)$ , the conversion into the respective performance indicator transpires as outlined below.

y_{i, t} = q_{u} + \frac{1}{x_{i, u + 1} - x_{i, u}} (X_{i, t} - x_{i, u + 1})

(1)

where

y_{i, t}

is the performance indicators of variable

i

at time

t

(

t = 1, 2, \dots, T

);

x_{i, u}

and

x_{i, u + 1}

represent respectively the upper and lower bounds of the gray class interval of the variable

i

;

q_{u}

is the lower range value of the performance indicators interval.

Figure 2.

Grey class interval of (a) Natural factor; (b) Human factor; (c) External durability; and (d) Structural Safety.

Establishing a model for predicting bridge performance degradation critically depends on accurately determining the correlation coefficients and weights associated with the numerous contributing factors. This study, based on the Grey Relational Analysis theory (Guo et al., 2022), calculates correlation coefficients and weights of each factor pertinent and bridge technical conditions. After obtaining the weights of various factors affecting bridge performance degradation and the performance indicators, the bridge performance model is proposed as shown in equation (2).

y_{t} = \sum_{i} λ_{i} {y_{i, t}}^{α (t)}

(2)

where

y_{t}

is bridge technical condition at time

t

;

λ_{i}

is the weight of variable

i

;

α (t)

is a performance index which represents a variable parameter in equation (2), as

α (t) = b + η t

b

is a constant term,

η

is a regression coefficient.

The multivariate Bayesian dynamic linear model (MBDLM) is a prediction method combining subjective and objective information. The model can be regarded as a continuous learning process, and the relevant parameters of the prior distribution are constantly updated by introducing new performance index information. Consequently, the predictions of bridge performance degradation are improved to be more consistent with the real-world scenarios. Bridge performance indicators demonstrate temporal variability. To improve the accuracy of predicting bridge performance degradation under real-world conditions, a MBDLM is introduced. This method encompasses a multivariate observation equation, a state equation, and initial prior information. The multivariate observation equation, state equation, and initial prior information (Wang et al., 2020) are as follows.

y_{i, t} = F_{i, t}^{T} θ_{i, t} + v_{i, t}, v_{i, t} \sim N (0, V_{i, t})

(3)

θ_{i, t} = G_{i, t} θ_{i, t - 1} + w_{i, t}, w_{i, t} \sim N (0, W_{i, t})

(4)

(θ_{i, t - 1} | D_{i, t - 1}) \sim N (m_{i, t - 1}, C_{i, t - 1})

(5)

where

y_{i, t} = {(y_{1, t}, y_{2, t}, \dots, y_{r, t})}^{T}

is column vector data for the performance indicators value of the variable

i

at time

t

;

θ_{i, t} = {(θ_{1, t}, θ_{2, t}, \dots, θ_{r, t})}^{T}

is a column vector data for the performance indicators value of the random variable

i

at time

t

;

v_{i, t} = (v_{1, t}, v_{2, t}, \dots, v_{r, t})

is column vector data for performance indicators observation error of the random variable

i

at time

t

;

V_{i, t}

r \times r

variance matrix of performance indicators observation error;

F_{i, t}

is known

r \times r

dimensional regression unit matrix of multivariable observation equation;

N (\cdot)

is normal probability density function;

T

represents matrix transpose.

G_{i, t}

is a

r \times r

unit dimensional state transition matrix;

w_{i, t} = (w_{1, t}, w_{2, t}, \dots, w_{3, t})

is state error data of the random variable

i

at time

t

;

W_{i, t}

r \times r

dimensional variance matrix of state error.

D_{t - 1} = {y_{i, t}, D_{i, t - 1}}

is a system set composed of all the effective information for performance indicators of the random variable

i

before time

t - 1

(include time

t - 1

);

m_{i, t - 1} = {(m_{1, t - 1}, m_{2, t - 1}, \dots, m_{r, t - 1})}^{T}

is column vector data for initial state variable expectation of variable

i

at time

t - 1

;

C_{i, t - 1}

r \times r

dimensional variance matrix for the initial state of the performance indicators.

Through combing equations (3)–(5), the relationship between the performance indicators observation vector and the state parameters is obtained as follows.

\begin{array}{l} (y_{i, t} | θ_{i, t}) \sim N (F_{i, t}^{T} θ_{i, t}, V_{i, t}) \\ (θ_{i, t} | θ_{i, t - 1}) \sim N (G_{i, t} θ_{i, t - 1}, W_{i, t}) \end{array}

(6)

Following the multivariate Bayesian dynamic linear method and Bayesian update theory, probabilistic updates and predictions for the random variables of bridge performance indicators can be carried out. The process of probability recursion is outlined as follows (Wang, 2017).

(1) The posterior probability distribution of state parameters at time $t - 1$ .

(θ_{i, t - 1} | D_{i, t - 1}) \sim N (m_{i, t - 1}, C_{i, t - 1})

(7)

(2) The prior probability distribution of state parameters at time $t$ .

(θ_{i, t} | D_{i, t - 1}) \sim N (a_{i, t}, R_{i, t})

(8a)

a_{i, t} = G_{i, t} E (θ_{i, t - 1} | D_{i, t - 1}) + E (w_{i, t}) = G_{i, t} m_{i, t - 1}

(8b)

R_{i, t} = G_{i, t} V a r (θ_{i, t - 1} | D_{i, t - 1}) G_{i, t}^{T} + V a r (w_{i, t}) = G_{i, t} C_{i, t - 1} G_{i, t}^{T} + W_{i, t}

(8c)

(3) The one-step prediction distribution of the observation data at time $t$ .

(y_{i, t} | D_{i, t - 1}) \sim N (f_{i, t}, Q_{i, t})

(9a)

f_{i, t} = F_{i, t}^{T} E (θ_{i, t} | D_{i, t - 1}) + E (v_{i, t}) = F_{i, t}^{T} a_{i, t}

(9b)

Q_{t} = F_{i, t}^{T} V a r (θ_{i, t} | D_{i, t - 1}) F_{i, t} + V a r (v_{i, t}) = F_{i, t}^{T} R_{i, t} F_{i, t} + V_{i, t}

(9c)

(4) The posterior probability distribution of state parameters at time $t$ .

(θ_{i, t} | D_{i, t}) \sim N (m_{i, t}, C_{i, t})

(10)

where

m_{i, t} = a_{i, t} + A_{i, t} e_{i, t}

;

C_{i, t} = R_{i, t} - A_{i, t} A_{i, t}^{T} Q_{i, t}

;

e_{i, t} = y_{i, t} - f_{i, t}

;

A_{i, t} = R_{i, t} F_{i, t} / Q_{i, t}

Linear models are able to effectively describe the variation trend in the short term. Moreover, given the continuous correction and updating capabilities of Bayesian dynamic models, linear models are preferable when there is insufficient understanding of long-term basic trends. Due to limited insight into the long-term trends in bridge performance, the most versatile linear model is chosen to elucidate the process of bridge performance degradation. The specific values for $F_{i, t}^{T}$ , $G_{i, t}^{T}$ , and $θ_{i, t}$ are provided below.

F_{i, t}^{T} = G_{i, t}^{T} = {[\begin{array}{l} 1 & 0 & \dots & 0 \\ 0 & 1 & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & 1 \end{array}]}_{r \times r}

(11)

θ_{i, t} = [\begin{array}{l} x_{i, t} \\ ε_{i, t} \end{array}] = [\begin{array}{l} x_{1, t} & x_{2, t} & \dots & x_{r, t} \\ ε_{1, t} & ε_{2, t} & \dots & ε_{r, t} \end{array}]

(12)

where

x_{i, t}

is the mean for bridge structure performance degradation indicators observation variables of variable

i

at time

t

;

ε_{i, t}

is average degradation speed of bridge structure for variable

i

from time

t - 1

t

; it is assumed that they are all normal random variable.

The multivariate observation equation, state equation, and initial prior information pertaining to the Bayesian dynamic linear method considering multiple variables can be ascertained through the concurrent of equations (3)–(5), (11), and (12).

y_{i, t} = F_{i, t}^{T} θ_{i, t} + v_{i, t} = {[\begin{array}{l} 1 & 0 & \dots & 1 \\ 0 & 1 & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & 1 \end{array}]}_{r \times r} {[θ_{1, t}, θ_{2, t}, \dots, θ_{r, t}]}^{T} + {[v_{1, t}, v_{2, t}, \dots, v_{r, t}]}^{T}

(13)

θ_{i, t} = G_{i, t} θ_{i, t - 1} + w_{i, t} = {[\begin{array}{l} 1 & 0 & \dots & 0 \\ 0 & 1 & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & 1 \end{array}]}_{r \times r} {[θ_{1, t - 1}, θ_{2, t - 1}, \dots, θ_{r, t - 1}]}^{T} + {[w_{1, t}, w_{2, t}, \dots, w_{r, t}]}^{T}

(14)

(θ_{i, t - 1} | D_{i, t - 1}) \sim N (m_{i, t - 1}, C_{i, t - 1}), i = 1, 2, \dots, r

(15)

In order to describe the coupling mechanism between different bridge influencing factors and bridge performance. The bridge performance degradation model can be ascertained through the simultaneous integration of equations (2) and (13).

y_{t} = \sum_{i} λ_{i} {{[\begin{array}{l} 1 & 0 & \dots & 0 \\ 0 & 1 & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & 1 \end{array}]}_{r \times r} {[θ_{1, t}, θ_{2, t}, \dots, θ_{r, t}]}^{T} + {[v_{1, t}, v_{2, t}, \dots, v_{r, t}]}^{T}}^{α (t)}

(16)

Figure 3 presents the developmental framework for the proposed bridge performance degradation model that predict and assess performance degradation and help determine an appropriate maintenance schedule. A brief summary of procedures in the proposed assessment framework are included, as follows:

Figure 3.

Bridge performance degradation prediction and assessment framework.

Determination of the Model’s parameters

As described in references (Goulet et al., 2018; Liu et al., 2020; Wang et al., 2019), the main probability parameters of MBDLM are $V_{i, t}$ , $W_{i, t}$ , $m_{i, t}$ , and $C_{i, t}$ . After obtaining a large amount of test data, the data is preprocessed based on the five-point cubic smoothing method (Wahalthantri et al., 2016) to obtain the trend random data, as follows:

{\begin{cases} Y_{i, 1} = \frac{1}{70} [69 y_{i, 1} + 4 (y_{i, 2} + y_{i, 4}) - 6 y_{i, 3} - y_{i, 5}] \\ Y_{i, 2} = \frac{1}{35} [2 (y_{i, 1} + y_{i, 5}) + 27 y_{i, 2} + 12 y_{i, 3} - 8 y_{i, 4}] \\ Y_{i, t} = \frac{1}{35} [- 3 (y_{i, t - 2} + y_{i, t + 2}) + 12 (y_{i, t - 1} + y_{i, t + 1}) + 17 y_{i, t}] \\ Y_{i, T - 1} = \frac{1}{35} [2 (y_{i, T - 4} + y_{i, T}) + 27 y_{i, T - 1} + 12 y_{i, T - 2} - 8 y_{i, T - 3}] \\ Y_{i, T} = \frac{1}{70} [69 y_{i, T} + 4 (y_{i, T - 3} + y_{i, T - 1}) - 6 y_{i, T - 2} - y_{i, T - 4}] \end{cases} (\begin{array}{l} t = 1, 2, \dots, T \\ i = 1, 2, \dots, r \end{array})

(17)

where

y_{i, t}

denotes the monitoring data of variable

i

prior to the smoothing process;

Y_{i, t}

signifies the monitoring data of variable

i

posteriori to the smoothing process.

$V_{i, t}$ estimate the variance through the difference between the trend data and the test data. Due to the unobservability of state variables, the symbol $W_{i, t}$ is determined in tandem with the variance of the multivariate prior information and discount factor.

W_{i, t} = - G_{i, t} C_{i, t - 1} G_{i, t}^{T} + \frac{C_{i, t - 1}}{δ}

(18)

where

δ

is a discount factor, the value range is

[0.95, 0.98]

$m_{i, t}$ represents a mean column vector comprising the mean values of state variables at time $t$ and prior time points across different monitoring locations; $C_{i, t}$ denotes the covariance matrix encompassing all state variables at time $t$ and preceding instances.

m_{i, t - 1} = {(m_{1, t - 1}, m_{2, t - 1}, \dots, m_{r, t - 1})}^{T}

(19)

C_{i, t - 1} = (\begin{array}{c} c_{1, t - 1} & c_{1, 2, t - 1} & \dots & c_{1, r, t - 1} \\ c_{2, 1, t - 1} & c_{2, t - 1} & \dots & c_{2, r, t - 1} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ c_{r, 1, t - 1} & c_{r, 2, t - 1} & \dots & c_{r, r, t - 1} \end{array})

(20)

where

c_{i, j, t}

is the covariance of the state variable

i

and state variable

j

c_{i, j, t} = \frac{\sum_{l = 1}^{n_{2}} \sum_{k = 1}^{n_{1}} (θ_{l, j, t} \times θ_{k, i, t})}{n_{1} \times n_{2}} - m_{i, t} \times m_{j, t}

, the symbol

θ_{k, i, t}

denotes the k-th state data of the state variable

i

at time

t

and preceding instances.

Monitoring system of MBDLM

The complexity, time-dependent attributes, and uncertainties of the bridge performance degradation can possibly result in diminished predictive accuracy of the model. To enhance the future predictive precision of the model, it is essential to control and correct. Thus, the resolution of this issue entails evaluating the performance of the model through a comparative analysis with one or more alternative models.

Evaluation indicator- Bayes factor

Considering two models with identical mathematical structures, one designated for prediction $P_{0}$ (Prediction model) and the other for comparative analysis $P_{1}$ (Alternative model), their respective probability density functions are denoted as $P_{0} (y_{i, t} | D_{i, t - 1})$ and $P_{1} (y_{i, t} | D_{i, t - 1})$ , as expressed in equation (21).

{\begin{cases} P_{0} (y_{i, t} | D_{i, t - 1}) = \frac{1}{\sqrt{2 π}} e^{- \frac{1}{2} β_{t e}^{2}} \\ P_{1} (y_{i, t} | D_{i, t - 1}) = \frac{1}{\sqrt{2 π k^{2}}} e^{- \frac{1}{2 k^{2}} β_{t e}^{2}} \end{cases}

(21)

where

β_{t e}

is the standardized prediction error of the BDLM;

β_{t e} = (y_{i, t} - f_{i, t}) / \sqrt{Q_{i, t}}

;

k

is the standard deviation of the alternative model.

The MBDLM monitoring system is constructed based on a single Bayes factor to effectively eliminate abnormal detection information. The singular Bayes factor is defined as the ratio of probability density functions between the one-step predictive model and the alternative model, as depicted in equation (22).

H (t) = \frac{P_{0} (y_{i, t} | D_{i, t - 1})}{P_{1} (y_{i, t} | D_{i, t - 1})} = k e^{- \frac{β_{t e}^{2} (k^{2} - 1)}{2 k^{2}}}

(22)

Identification of outliers

Utilizing the Bayes factor to assess the predictive performance of the model, a specific criterion denoted as $β_{t e}$ is established. When the Bayes factor $H (t)$ , calculated from a particular observed value $y_{i, t}$ , is lower than $β_{t e}$ , $y_{i, t}$ is characterized as an outlier. Considering the significant presence of standardized prediction errors, the variation curves of the singular Bayes factor for $k$ values ranging from 2 to 6 are explored. The variation curve of the singular Bayes factor is presented in Figure 4. When $β_{t e} \leq 2.5$ , notable differences emerge among each alternative model. Singular Bayes factors approach approximately 0.2 for $k = 3$ or 4. When $2.5 \leq β_{t e} \leq 3.0$ , the curve exhibits a gradual reduction in slope. Conversely, when $β_{t e} \geq 3.0$ , $H (t)$ consistently remains below 0.1, the curve displays minimal variations, indicating a weak correlation and a loss of significance in the selection process. As a result, the selection of $k = 3$ and $H (t) = 0.15$ serves as the criteria for discerning anomalous performance indicators within the detection data.

Figure 4.

Variation curves of single Bayesian factor.

Automatic intervention

After the outlier is identified, it indicates that the model performance deteriorates and the model is subject to automatic intervention. This intervention strategy is similar to subjective intervention, involving both the removal of outliers and the extension and refinement of the model. The latter typically entails heightening the uncertainty pertaining to model parameters, specifically augmenting the variance of state errors.

The evaluation of the Model’s predictive performance

This paper conducts a quantitative evaluation of the predictive performance of the multivariate Bayesian dynamic linear method, which employs three metrics: mean squared error (MSE), predictive mean squared error (PMSE), and mean absolute percentage error (MAPE). Calculations are performed as follows in equations (23) to (25).

M S E = \frac{1}{n} \sum_{z = 1}^{n} {(y_{i, t} - f_{i, t}^{z})}^{2}

(23)

P M S E = \frac{1}{n - z_{0}} {\sum_{z = z_{0} + 1}^{n} (y_{i, t} - f_{i, t}^{z} | D_{i, t}^{z_{0}})}^{2}

(24)

M A P E = \frac{1}{n} | \frac{y_{i, t} - f_{i, t}^{z}}{y_{i, t}} | \times 100 %

(25)

where

n

is detected iterations;

f_{i, t}^{z}

f_{i, t}

after z update;

D_{i, t}^{z_{0}}

D_{i, t}

after z update;

z_{0}

represents the number of updates.

The MSE is used to quantitatively evaluate the predictive performance of the MBDLM based on all available performance indicators obtained from the detection data. PMSE is employed to evaluate the predictive performance of the model utilizing the updated detection data performance indicators. In contrast, MAPE is employed to provide a comprehensive assessment of the prediction accuracy of the model. Therefore, to offer a comprehensive assessment of the predictive performance of the model from various angles, this study chooses to use MSE, PMSE, and MAPE simultaneously as evaluation metrics.

Validation of an engineering case

In instances of time-dependent degradation in bridge performance, model parameters must be determined through experimentation or estimation. These parameters serve as input variables, encompassing the performance index

α (t)

(see Appendix A). Therefore, specific parameters within the model require a fitting process, since not all parameters can be obtained through a priori model. The proposed model for bridge performance degradation presented in this paper is validated by utilizing data regarding the factors influencing the bridge performance deterioration and the bridge technical condition, which is from Fufeng Bridge in Changchun City, covering the period from 2002 to 2016 (Hao, 2018). The test data are shown in Table 2. 87.5% of the data is allocated to the training and validation datasets, with the remaining 12.5% reserved for the testing dataset.

Table 2.

Detection data of the fufeng bridge from 2002 to 2016.

Aspects	Factors	Year
Aspects	Factors	2002	2004	2006	2008	2010	2012	2014	2016
Natural factor	Annual precipitation (mm)	492	545	634	718	733.5	702.3	550	775.1
	Wind condition (days)	23	12	26	18	5	16	3	11
	Temperature difference (°C)	54	53	53	51	50	54	52	51
Human factor	Daily traffic volume (thousand)	0.8448	0.9836	1.1452	1.3333	1.5528	1.7446	1.96	2.2021
Human factor	Annual overloading rate (%)	56	53	54	51	68	59	59	67
External durability	Concrete strength	0.988	0.978	0.964	0.948	0.942	0.936	0.924	0.904
	Carbonization status (mm)	0.07	0.12	0.08	0.2	0.19	0.26	0.35	0.41
	Chloride concentration (%)	0.03	0.07	0.1	0.11	0.14	0.17	0.22	0.2
	Steel corrosion potential (mV)	−71	−91	−116	−176	−126	−219	−177	−190
Structural safety	Deflection examination coefficient	0.47	0.52	0.64	0.56	0.71	0.86	0.96	0.99
	Strain calibration coefficient	0.53	0.47	0.51	0.63	0.55	0.78	0.66	0.7
	Impact coefficient	0.178	0.395	0.266	0.451	0.682	0.709	0.879	0.913
	Fundamental frequency (Hz)	1.982	1.665	1.826	1.558	1.405	1.291	0.953	0.845
Bridge performance	Bridge technical condition	92.2	87.9	84	76.5	71.2	68.6	65	56.6

Upon obtaining the bridge inspection variables and technical conditions, the initial step involves judging the relevant gray class interval. Based on the redefined concept of performance indicators, one can proceed to identify the corresponding state level and performance indicator interval. Then, equation (1) is used to convert the detection data into performance indicators (see Appendix B).

Drawing upon the obtained performance indicators of detection data and the established MBDLM, predictive performance indicators for physical factor (see Figure 5), human factor (see Figure 6), external durability (see Figure 7), and structural safety (see Figure 8) have been obtained after the eighth update by incorporating performance indicators to update the prior model.

Figure 5.

Prediction performance indicators of natural factors: (a) Annual precipitation; (b) Wind condition; and (c) Temperature difference.

Figure 6.

Prediction performance indicators of human factors: (a) Daily traffic volume; and (b) Annual overloading rate.

Figure 7.

Prediction performance indicators of external durability: (a) Concrete strength; (b) Carbonization status; (c) Chloride concentration; and (d) Potential of steel corrosion.

Figure 8.

Prediction performance indicators of the bridge structural safety: (a) Deflection test coefficient; (b) Strain calibration coefficient; (c) Impact factor; and (d) Fundamental frequency.

Figures 5 –8 show the predictive curves of the factors influencing the degradation of bridge performance. During the prediction process, the constructed monitoring system effectively removes outliers from the performance indicators of detection data. Following multiple updates, the predictive curves conspicuously towards the performance indicators of detection data, exemplifying the capability of the established MBDLM to incorporate objective detection information. By employing probabilistic inference with prior information, the predicted results reasonably capture the variations of the variables that influence the bridge performance degradation.

Figure 9 demonstrates that the proposed model aligns favorably with the experimental outcomes, which displaying a maximum error of 8%. The bridge performance degradation model, as proposed in this study, exhibits a stronger alignment with experimental data when compared to both the conventional grey model and the grey model augmented with residual correction (Hao, 2018). Additionally, it becomes obvious that the grey model unreliably predicts the initial service performance of bridges, which contrasts starkly with the actual results, thereby accentuating the accuracy and superiority of the bridge performance degradation model proposed in this paper. The performance of bridges gradually deteriorates over time in service. This phenomenon arises from the cumulative impact of traffic and environmental loads, resulting in a series of structural deteriorations, including material corrosion and degradation. As a consequence, this results in a reduction in both the load-bearing capacity and overall reliability of the structure.

Figure 9.

Verify the applicability of the bridge performance degradation model.

Results and discussions

Currently, the detection data is converted into dimensionless, continuous random variables using the refined concept of performance indicators. Variable weights are obtained based on the grey correlation method, ultimately leading to the formulation of a comprehensive bridge performance equation. The bridge performance degradation model is ultimately established through the change of performance indicators with time. This method elucidates variations in bridge performance by monitoring fluctuations in performance indicators. Consequently, the bridge performance degradation is determined by temporal progression rather than reliance on computations using reliability methodologies. Furthermore, the weights representing the factors contributing to the degradation of bridge performance were determined through the analysis of detection data. The predictive analysis is conducted on the temporal evolution of various bridge performance indicators (see Figure 5∼8).

The bridge technical condition performance indicators are forecasted using a Bayesian dynamic linear method. A prior model is constructed through equation (5) (see Figure 10). Subsequently, this model underwent update of the prior information to obtain the updated curve using Bayesian probability recursion and the real-world bridge technical condition (see Figure 11). The impact of the initial and secondary updates is not significant, as the curves closely align with the prior model. However, beginning with the third update, the updated data gradually converges toward the performance indicators of detection information. The results show that directly predicting the bridge technical condition, without accounting for the coupled effects among variables, results in occasional outcomes that are insufficient for reliably forecasting bridge performance degradation.

Figure 10.

Compared with the prior model and performance indicators.

Figure 11.

Process of model update.

This study aims to verify the predictive performance of the MBDLM by quantitatively evaluating the response of information updates on the model. Three evaluation metrics are employed, namely mean squared error, predictive mean squared error, and mean absolute percentage error. Figures 12 –15 give a quantitative evaluation of prediction performance related to natural factors, human factors, external durability, and structural safety. Over the course of 104 update iterations, 89.42% of MSE and 93.27% of PMSE data experienced a notable decrease after the update, and the minimum chloride concentration is observed. This underscores the effectiveness of information updates in enhancing the predictive performance of the MBDLM. MAPE decreased from 9.67% after the first update to 3.22% after the eight updates, indicating that the prediction accuracy has been significantly improved by 6.45%. The enhancement meets the prediction requirements, indicating a substantial increase in the model’s precision through information updates. Therefore, the information updates lead to the improvement of both the predictive performance and accuracy of the MBDLM.

Figure 12.

Validation of the MBDLM’s predictive performance based on natural factors. (a) mean square error (MSE); (b) predicted mean square error (PMSE); and (c) mean absolute percentage error (MAPE).

Figure 13.

Validation of the MBDLM’s predictive performance based on human factors. (a)MSE; (b) PMSE; and (c) MAPE.

Figure 14.

Validation of the MBDLM’s predictive performance based on external durability. (a)MSE; (b) PMSE; and (c) MAPE.

Figure 15.

Validation of the MBDLM’s predictive performance based on structural safety. (a) MSE; (b) PMSE; and (c) MAPE.

The main limitation and improvements of this study are the following three points. (1) Expanding the applicability of the model. Although this study has constructed performance degradation for predicting and assessing performance degradation, future research could explore extending the application of this model to bridges of different types and scales. This exploration would enable evaluating the model’s effectiveness and applicability across a broader range of contexts. (2) Optimizing the predictive performance of the model. Future research endeavors may involve exploring refined data collection methodologies and employing advanced model evaluation techniques. Moreover, investigation into the degradation mechanism in the target structures could ensure the scientific performance indicators. Furthermore, adjusting the weighting of existing variables holds promise for facilitating a more comprehensive assessment and prediction of bridge performance degradation. (3) Deeping the research of coupling mechanism. Despite this study elucidating the coupling mechanisms among factors influencing bridge performance degradation, the investigation into the interplay among different factors can be further explored in the future.

Conclusions

In summary, this study presents the following key contributions: Firstly, it refines the concept of performance indicators for the characterization of bridge performance degradation, followed by the construction of a bridge performance degradation model based on these indicators. Secondly, it elucidates the coupling mechanisms among factors influencing bridge performance degradation, thereby expanding the applicability of bridge performance degradation theory. Finally, a model for the bridge performance degradation and a method for predictive performance assessment have been developed. These provide a theoretical foundation for the assessment, prognosis, and maintenance improvement of the in-service structural performance degradation of bridges. The bridge performance degradation model is to describe the overall change in bridge performance based on the deterioration of performance indicators. It does not require to reflect the bridge performance deterioration through the variations in reliability. Therefore, the projected outcomes of the proposed model demonstrate a noteworthy alignment with experimental observations, surpassing the predictions of alternative models. The bridge performance degradation theory posited in this study holds promise for future applications in safety assessments and proactive maintenance decision-making processes for operational bridges.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors are grateful for support from the National Natural Science Foundation of China (52168042), the Science and Technology Plan of Gansu Province of China (22JR5RA250), and the Graduate student “Innovation Star” of the Gansu Provincial Department of Education of China (2023CXZX-460).

ORCID iD

Guojun Yang

Appendix

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