Technical Degradation Prediction of Bridge Components Based on Semi-Markov Degradation Model

Abstract

To overcome the limitations of traditional Markov bridge degradation prediction models, which fail to consider the interactions between different component degradation mechanisms and struggle to accurately capture the true degradation conditions of bridge components, introducing an improved version of the traditional Markov model by incorporating the Weibull distribution. This enhancement results in a semi-Markov model that offers a probability distribution for predicting the technical condition of bridge components. Taking advantage of periodic inspection data from a highway section in Shandong Province, China. With this data, the states of bridge components are defined in the semi-Markov degradation model. The improved semi-Markov model integrates a two-parameter Weibull distribution and involves determining the parameters of the Weibull distribution, transition probability matrix, and state distribution vector. The semi-Markov degradation model, in contrast to commonly used Markov degradation models, accounts for both the state and duration of each state, resulting in significantly more accurate predictions of the degradation process of bridge components, achieving a prediction accuracy of 96%. The developed semi-Markov bridge degradation model facilitates the timely detection of changes in the technical condition of bridge components by updating the transition probability matrix according to variations in the duration of each state, thereby improving the efficiency of subsequent bridge maintenance decision-making.

1 Introduction

The failure of bridge structures can result in substantial economic, social, and environmental costs [1,2,3]. Therefore, it is crucial to proactively understand the degradation patterns of bridge structural performance [4]. Long-term degradation models for bridges play a vital role in optimizing whole-life maintenance strategies. Existing models can generally be classified into mechanism-based models and statistical-based models.

Mechanism-based degradation models begin by analyzing the different factors and principles that impact the degradation of bridge structures. These models focus on the specific deterioration mechanisms of specific bridge components. Currently, there is significant research emphasis on predicting concrete carbonation, steel reinforcement corrosion, chloride ion intrusion, and other related aspects. For example, Dai [5] investigated the deterioration process of steel corrosion in concrete members to predict the life of the structure.

Statistical-based degradation models analyze historical data on structural degradation to capture the degradation patterns of bridges.By considering both the historical and current condition of the bridge, these models can predict its future state. Currently, the main prediction methods can be broadly categorized into deterministic models and stochastic models. Deterministic models establish relationships between factors influencing the degradation process and bridge condition levels using mathematical and statistical methods. The output of these models is represented by deterministic values and does not involve probabilities. These models assume that the relationship between future bridge condition levels is deterministic as time progresses [6]. Deterministic models can be classified into methods such as linear extrapolation, regression, and curve fitting [7,8,9]. For example, Zhou [10] used regression analysis to predict the technical condition of bridges. One limitation of deterministic models is that they do not consider the interaction between degradation mechanisms of different bridge components, such as the interaction between deck panels and joints [11]. On the other hand, stochastic models account for the randomness of the bridge degradation process. They are capable of capturing the stochastic nature and uncertainty associated with the degradation process. In stochastic models, the prediction of bridge condition levels is influenced by inherent uncertainties, such as traffic loads, weather conditions, material properties, exposure to deteriorating agents, etc. Stochastic models create probabilistic prediction models. The main types of stochastic models include Markov chain methods and reliability theory-based methods. These methods represent the state levels of bridge components or parts at a specific moment in the form of probability distributions rather than deterministic values. Statistical-based degradation models are more suitable for simulating the current degradation of bridges.

Stochastic models, which effectively account for the randomness of various factors, are currently the most widely used predictive models for structural performance. Prominent bridge management systems in the United States, such as PONTIS, BRIDGIT, BLCCA, and Bridge LCC, rely on Markov chains as the theoretical foundation for bridge degradation prediction models. Jiang [12], utilizing Markov chains, developed a performance prediction model for bridges based on status-level data of bridge degradation from the Indiana Department of Transportation. Pontis [13], as part of Bridge Management System (BMS) sponsored by the Federal Highway Administration, employed Markov chains to develop core component degradation models. In Markov chain degradation models, the performance level of a bridge structure is defined by a set of discrete states, and the bridge transitions from one state to another based on constant transition probabilities. Thompson [14] analyzed the California bridge dataset to quantify observed Markov transition probabilities. Similar work can be found in Puz [15], where they proposed a probabilistic model for the full lifecycle performance of structures based on homogeneous Markov processes to calculate degradation over time. Ng [16] introduced relaxation time homogeneity of Markov processes in a more general semi-Markov process stochastic model. At present, most of the bridge management systems in the United States still use Markov chain as the theoretical basis of bridge degradation prediction model to predict the change of bridge NBI rating [17].

However, Markov models have certain limitations. Firstly, they do not consider the age of the structure or components. Secondly, they assume that future states depend solely on the current state and do not account for past events such as bridge maintenance. Lastly, they may exhibit an initial rapid degradation rate. To overcome these limitations, additional model are required to describe the time distribution of each state.

To accurately understand the degradation patterns of bridge performance and enable scientific and precise bridge maintenance, this study relies on regular inspection data from a highway bridge in Shandong province. Considering the time and condition of bridge components, an improved Markov model based on the Weibull distribution is introduced to develop a semi-Markov model that closely reflects reality. This model is utilized to predict the distribution of technical condition levels for highway bridge components, providing necessary support for subsequent intelligent bridge maintenance decision-making.

2 Construction Process of Semi-Markov Bridge Component Degradation Model

Based on the theoretical research conducted on bridge performance degradation prediction in Shandong Province, the Markov model has been selected as the approach for predicting bridge performance degradation. The semi-Markov model, which is an extension of the Markov chain model, takes into account the state of the bridge and introduces the Weibull distribution to model the duration of each state. Using existing detection data and considering both time and state variables, the Markov model is enhanced by incorporating the Weibull distribution.

2.1 State Division of Semi-Markov Bridge Degradation Model

After preprocessing the technical condition data of bridge components, it is necessary to define the states of the components in the semi-Markov bridge component degradation model. Since Standards for Technical Condition Evaluation of Highway Bridges (JTGT H21-2011) does not provide a specific definition for the technical condition level of components, this chapter refers to the classification threshold table for the technical condition of bridge components in the Standards for Technical Condition Evaluation of Highway Bridges (JTGT H21-2011) and defines five semi-Markov states as follows:

• State 1: Technical condition score [95,100]

• State 2: Technical condition score [80,95)

• State 3: Technical condition score [60,80)

• State 4: Technical condition score [40,60)

• State 5: Technical condition score [0,40)

These defined states serve as a basis for modeling the degradation process of bridge components using the semi-Markov framework.

2.2 The Weibull Distribution Parameters of the Duration of Each State of the Bridge are Determined

The Standards for Technical Condition Evaluation of Highway Bridges (JTGT H21-2011) specifies that the duration of a bridge component in a particular state follows a random variable. $T_1 ,T_2 , \ldots ,T_{n - 1}$ respectively represent the random variable of time of duration which bridge components is in each state i, the duration $T_i$ in state i is specified to obey a two-parameter Weber probability distribution, meaning $T_i \ {\text{Weibull}}(b_i ,\frac{1}{a_i })$. It is further defined that the duration of a component in any given state follows a two-parameter Weibull probability distribution.

$$ F_i (t) = \Pr \left[ {T_i \le t} \right] = 1 - e^{ - \left( {a_i t} \right)^{b_i } } $$

(1)

$$ S_i (t) = \Pr \left[ {T_i > t} \right] = 1 - F_i (t) = e^{ - \left( {a_i t} \right)^{b_i } } $$

(2)

$$ f_i (t) = \frac{\delta F_i (t)}{{\delta t}} = a_i b_i \left( {a_i t} \right)^{b_i - 1} e^{ - \left( {a_i t} \right)^{b_i } } $$

(3)

In the Eq. (1), $F_i (t)$ represents the Cumulative Density Function(CDF) corresponding to the duration of time t; $S_i (t)$ represents the Survival Function (SF)，also known as the cumulative survival rate, which refers to the probability that the component remains in state i for a duration longer than t; $f_i (t)$ represents the Probability Density Function (PDF) corresponding to the duration of time t； $T_{i \to j}$ represents the time required for the process from state i to state j； $f_{i \to j} (T_{i \to j} )$, $F_{i \to j} (T_{i \to j} )$, $S_{i \to j} (T_{i \to j} )$ represent the PDF, CDF, and SF of $T_{i \to j}$, respectively.

If the proportion of components in each state is known for each year, the parameters $a_i$ and $b_i$ can be calculated by fitting the survival function curve that corresponds to the duration. This fitting process allows for the estimation of the Weibull distribution parameters. Once the parameters are determined, a transition probability matrix can be established based on the historical data. This matrix represents the probabilities of transitioning from one state to another in each time step. By utilizing this transition probability matrix, it becomes possible to predict the future annual state distribution of bridge components. This approach provides a useful framework for forecasting the deterioration and performance of bridge components based on the available historical data and the estimated Weibull distribution parameters.

2.3 Modeling of Semi-Markov Bridge Component Degradation Process

Determination of State Transition Matrix.

Under the assumption that the degradation process of bridge components is unidirectional and does not allow direct transitions from one state to another without passing through an intermediate state, a relatively simple Markov transition probability matrix can be derived. Therefore, the degradation process cannot jump from state 1 to state 3 without going through state 2. This leads to the relatively simple Markov transition probability matrix：

$$ P^{t,t + 1} = \left[ {\begin{array}{*{20}c} {p_{11}^{t,t + 1} } & {p_{12}^{t,t + 1} } & 0 & 0 & 0 \\ 0 & {p_{22}^{t,t + 1} } & {p_{23}^{t,t + 1} } & 0 & 0 \\ 0 & 0 & {p_{33}^{t,t + 1} } & {p_{34}^{t,t + 1} } & 0 \\ 0 & 0 & 0 & {p_{44}^{t,t + 1} } & {p_{45}^{t,t + 1} } \\ 0 & 0 & 0 & 0 & {p_{55}^{t,t + 1} } \\ \end{array} } \right] $$

(4)

In the Eq. (4), $P_{ij}^{t,t + 1}$ represents the transition probability from state i at time t to state j at time t + 1, with i and j ranging from 1 to 5. The matrix P captures the probabilities of transitioning between different states in the degradation process, reflecting the underlying dynamics of the bridge component degradation. In a semi-Markov process, the system is in a specific state for a random duration, and the distribution of states depends on the current state and the next state. To exclude two-state degradation, the time step (Δt) should be sufficiently small. In this chapter, we assume Δt to be 1 year. If a bridge component is in state 1 at time t, the conditional probability of transitioning to the next state in the next time step can be represented as:

$$ {\text{P}} [X(t + 1) = 2|X(t) = 1] = P^{1,2} (t) = \frac{f_1 (t)}{{S_1 (t)}} $$

(5)

In the Eq. (5), $f_1 \left( t \right)$ represents the probability density function (PDF) of the duration of the component in state 1； $S_1 \left( t \right)$ represents the survival function (SF), which is the probability that the duration exceeds t. If the process is in state 2 at time t, the conditional probability of transitioning to the next state in the next time step is:

$$ {\text{P}} [X(t + 1) = 3|X(t) = 2] = P^{2,3} (t) = \frac{{f_{1 \to 2} (t)}}{{S_{1 \to 2} (t) - S_1 (t)}} $$

(6)

In the Eq. (6), $f_{1 \to 2} (t)$ represents the PDF which representing the sum of the duration time $T_{1 \to 2}$ in states 1 and 2; $S_{1 \to 2} (t) - S_1 (t)$ represents that probability which $T_{1 \to 2} < t$ minus the probability which $T_1 < t$, which is equivalent to the condition $X(t) = 2$.

$$ {\text{P}} [X(t + \Delta t) = i + 1|{\text{X}}({\text{t}}) = {\text{i}}] = P^{i,i + 1} (t) = \frac{{f_{1 \to i} (t)\Delta t}}{{S_{1 \to i} (t) - S_{1 \to (i - 1)} (t)}}\quad (i = 1,2, \ldots ,n - 1) $$

(7)

All conditional probabilities $P^{i,i + 1} (t)$ can be calculated by the above Eqs. (4)–(7) to generate the transition probability matrix of the semi-Markov process.

The State Distribution Vector is Determined.

In a semi-Markov chain, the probability of being in state i at time t is described by the state distribution vector ${\rm{D}}(t)$, which represents the probabilities of being in each state at time t:

$$ {\text{D}}(t) = \left\{ {d_1^t ,d_2^t , \ldots d_n^t } \right\}\begin{array}{*{20}c} {\,} & {(\begin{array}{*{20}c} {d_i^{\text{t}} \ge 0,} & {\sum\limits_{i = 1}^n {d_i^{\text{t}} } = 1} \\ \end{array} )} \\ \end{array} $$

(8)

In the Eq. (8), $d_i^{\text{t}}$ is the probability of a bridge component being in state i at time t.

The state distribution vector in time (t + k) is

$$ {\text{D}}(t + k) = {\text{D}}(t){\text{P}}^{t,t + 1} {\text{P}}^{t + 1,t + 2} \ldots {\text{P}}^{t + k - 1,t + k} $$

(9)

By utilizing the transition probability matrix established in Sect. 3.4.1, the state distribution vector for the semi-Markov degradation process can be obtained at any given time t. By solving for the state distribution vector, the probabilities of each state at any specific time can be determined. This information allows for further calculations such as average duration in a particular state, state transition probabilities, and distribution of state transition times. These calculations provide valuable insights and serve as reference and basis for subsequent bridge maintenance decision-making.

3 Case Study

This paper takes the North Line of Shandong Expressway as a case study. There are a total of 8 interchanges along the route, and out of which 222 bridges have been selected Considering the completeness of historical inspection data, 40 prestressed concrete continuous girder bridges were chosen from these 222 bridges, comprising a total of 285 prestressed concrete box girder components. Inspection data from the regular inspection reports for these 40 bridges from 2008 to 2019 have been collected, processed, and analyzed. The semi-Markov degradation model has been utilized for prediction and analysis based on this data.

3.1 Semi-Markov State Division and Statistics

Based on the five semi-Markov states defined in Sect. 3.2, the component states are divided and aggregated. The percentages of components in each state for each year are calculated and summarized, as shown in Table 1.

Table 1. The percentage of each state of the component of the prestressed reinforced concrete bridges

3.2 Modeling of Semi-Markov Degradation Processes

According to Sect. 2.3, the Matlab program for constructing the semi-Markov degradation model is written, and the age of the main beam components of the prestressed reinforced concrete bridges in Table 1 and each state percentage are substituted and calculated, so that the Weibull distribution parameters of the duration of each state in Table 2 can be easily obtained.

Table 2. Weibull distribution parameters and fitting degree of each state duration

In the Table 2, $SSE_i$ represents the sum of the squared differences between the actual and predicted values, indicating the effect of random errors; $R_i$ represents the multiple determination coefficient, measuring the success of the fit in explaining the variation in the data; $RMSE_i$ represents the square root of the mean square error, which is the expected value of the squared difference between the estimated and true parameter values.

Comparing the goodness-of-fit values from Table 2, including $SSE_i$, $R_i$, and $RMSE_i$, it can be concluded that the obtained goodness-of-fit. Overall, the fit is good, with small errors and high effectiveness. Considering $RMSE_i$ as the final measure of model accuracy, it can be inferred that the accuracy has reached above 96%. Based on these fitting results, the Weibull distribution parameters can be further used to predict the cumulative survival rate and corresponding probability density function distribution for the duration of different component states of bridge elements, as shown in Figs. 1 and 2.

Fig. 1.

Prediction of the cumulative survival rate of prestressed reinforced concrete components in each state duration

Fig. 2.

Prediction of the cumulative survival rate of prestressed reinforced concrete components in each state duration

Based on the survival function of each state duration of the bridge components and the corresponding probability density function distribution, the conditional probability matrix of transformation from t to (t + 1) year to the next state can be easily obtained from Eqs. (5), (6), (7) and (8), for example, the conditional probability matrix of transformation from 4th to 5th year is as follows

$$ p^{4,5} = \left[ {\begin{array}{*{20}c} {{0}{\text{.8154}}} & {{0}{\text{.1846}}} & {0} & {0} & {0} \\ {0} & {{0}{\text{.4219}}} & {{0}{\text{.5781}}} & {0} & {0} \\ {0} & {0} & {{0}{\text{.7155}}} & {{0}{\text{.2845}}} & {0} \\ {0} & {0} & {0} & {{0}{\text{.9565}}} & {{0}{\text{.0435}}} \\ {0} & {0} & {0} & {0} & {1} \\ \end{array} } \right] $$

(10)

From the probability matrix, we can obtain the state distribution vector for any given year of prestressed concrete components along the route. Let’s assume the state distribution vector for the first year of prestressed concrete components within the route is:

$$ {\text{D}}_1 = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} {0.6667} & 0 \\ \end{array} .3333} & 0 \\ \end{array} } & 0 & 0 \\ \end{array} } \right] $$

(11)

The state distribution vector of the box girder components in the fifth year can be obtained

$$ \begin{gathered} {\text{D}}_{5} {\text{ = D}}_1 \times p^{1,2} \times p^{2,3} \times p^{3,4} \times p^{4,5} \hfill \\ \, = \left[ {\begin{array}{*{20}l} {{ 0}{\text{.2158}}} \hfill & {{0}{\text{.1074}}} \hfill & {{0}{\text{.2940}}} \hfill & {{0}{\text{.2799}}} \hfill & {{0}{\text{.0138}}} \hfill \\ \end{array} } \right] \hfill \\ \end{gathered} $$

(12)

3.3 Comparison of Semi-Markov Model and Traditional Markov Model

In this section, we will compare the accuracy of the traditional Markov model and the proposed semi-Markov model, which have different mathematical forms and parameter estimation methods. The accuracy of the models can be evaluated by comparing their predictions with actual data. This can be done using various performance measures such as mean squared error, root mean squared error, or correlation coefficients. Additionally, cross-validation techniques can be applied to assess the out-of-sample predictive performance of the models.

By conducting a comparative study of these two models, we can determine which model provides better accuracy and prediction capabilities for the given system. The Markov state transition probability matrix based on the data in Sect. 3.2:

$$ p^{4,5} = \left[ {\begin{array}{*{20}c} {{0}{\text{.4737}}} & {{0}{\text{.3860}}} & {{0}{\text{.1404}}} & {0} & {0} \\ {0} & {{0}{\text{.4286}}} & {{0}{\text{.5238}}} & {{0}{\text{.0476}}} & {0} \\ {0} & {0} & {{0}{\text{.7473}}} & {{0}{\text{.2527}}} & {0} \\ {0} & {0} & {0} & {{0}{\text{.9804}}} & {{0}{\text{.0196}}} \\ {0} & {0} & {0} & {0} & {1} \\ \end{array} } \right] $$

(13)

Assume that the state distribution vector of a prestressed reinforced concrete components in the route is

$$ {\text{D}}_1 = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} {0.6667} & 0 \\ \end{array} .3333} & 0 \\ \end{array} } & 0 & 0 \\ \end{array} } \right] $$

(14)

The state distribution vector of the components in the fifth year can be obtained

$$ {\text{D}}_{5} {\text{ = D}}_1 \times p^{1,2} \times p^{2,3} \times p^{3,4} \times p^{4,5} = \left[ {{ 0}{\text{.1559 0}}{.2187 0}{\text{.4368 0}}{.1844 0}{\text{.0043}}} \right] $$

Based on Fig. 3, it can be observed that the proposed semi-Markov model, enhanced by introducing the Weibull distribution, exhibits significant improvement in accuracy compared to the traditional Markov model. It shows a closer fit to the actual measured data and can more accurately predict the degradation trend of bridges. This provides a scientific basis for subsequent bridge maintenance activities, enabling the rational planning of maintenance schedules, optimization of repair processes and materials selection. Consequently, it helps extend the service life of bridges while reducing maintenance costs.

Fig. 3.

Comparison of semi-markov model and traditional Markov model

4 Conclusion

This study addresses the insufficient consideration of time and limited research on component-level degradation models in current bridge condition prediction methods. Based on the assessment scores of highway bridge component conditions and considering the impact of maintenance on these scores, a semi-Markov model is constructed to predict the probability distribution of the condition levels of highway bridge components over time. The main conclusions are as follows:

• (1) The study divides the assessment scores of bridge component conditions into multiple discrete states, referring to the boundary values for assessing degradation states in specifications. The Weibull distribution parameters for the duration of each state are fitted based on the realistic data. This establishes a Weibull-based semi-Markov model for predicting the condition of bridge components.

• (2) The proposed model can update the transition probability matrix based on changes in state durations and predict the probability distribution of bridge component condition levels for any future year. It can timely identify changes in component conditions, provide references for subsequent bridge maintenance decisions, and improve maintenance efficiency.

• (3) Compared to the commonly used Markov models, the proposed approach is more realistic and comprehensive in considering the durations of different states. However, since the model requires fitting parameters for the Weibull distribution of state durations, a larger amount of regular bridge inspection data is needed to further improve the model’s accuracy.