1. Introduction
Bridges play an indispensable role in the public road network. They can improve the regional road network, thus promoting inter-regional economic development [
1]. Bridge management departments around the world have been working for a long time to develop and improve various bridge management systems (BMS), attempting to provide timely and effective bridge maintenance, repair, and rehabilitation (MR&R) through standardized and continuous technical-condition data collection and performance evaluation [
2,
3,
4,
5,
6]. Bridge management systems began to be developed extensively in the early 1990s and are now becoming increasingly mature after more than 20 years of effort [
7]. In general, a comprehensive BMS should contain four components, namely, a bridge information database, a performance deterioration model, a cost model, and a maintenance decision optimization model [
8,
9,
10]. Each part has independent functions but works in conjunction with the others. The bridge information database is used to store the attribute information, inspection history, and maintenance data of the bridge. The purpose of the performance deterioration model is to predict the future state of the bridge and its components. The cost model is utilized to estimate the cost requirements for routine maintenance and repair of bridges, and the role of the decision optimization model is to determine the best MR&R strategy [
11].
Among them, the information database and performance deterioration model are the foundation of the BMS, and they play a crucial role in assessing and predicting the technical condition of the bridge [
12,
13,
14]. Only with accurate bridge performance assessment and predictions can the development of maintenance strategies and the estimation of maintenance cost requirements be carried out successfully. In addition, unlike bridge information databases and performance deterioration models that primarily have objective attributes, the cost and decision optimization models will reflect a degree of subjectivity, as maintenance costs and strategies are often related to local bridge management goals and maintenance resource input levels [
15,
16].
The accuracy of the performance prediction of bridge components is of considerable importance; therefore, this paper used literature research and data analysis to find a suitable method to predict the service performance of bridge components. Although there are many bridge component categories, attention will only be focused on the main girders of the bridge. For this category of bridge components, the main objectives are as follows:
To analyze the deterioration of the main girder under different CRs and assess the service life of the main girder;
To seek the deterioration pattern of the performance of the main girders under different influencing factors;
To compare the deterioration of the performance of the main girders with that of the superstructure and the whole bridge.
After the background and objectives of the study are presented in this section, the remainder of the paper is organized as follows:
Section 2 presents the current state of the research through literature research;
Section 3 provides a brief explanation of the basic concepts of survival analysis, Cox regression, and Weibull distribution;
Section 4 shows the data that have been collected so far;
Section 5 discusses the estimation results obtained from the semi-parametric regression and parametric analysis used in this study; and, finally, conclusions and an outlook for future work are presented. The models developed in this paper should be useful for transport organizations to improve their maintenance strategies and operational decisions.
2. Literature Review
Many countries and regions have already established bridge management systems with relatively complete functions based on the improvement of the bridge information database, such as PONTIS and BRIDGIT in the USA, NYSDOT in New York, J-BMS in Japan, KUBA in Switzerland, OBMS in Canada, C-BMS for highway bridges and Web-BMS for urban bridges in China, etc. [
17,
18,
19,
20]. Although the data composition and accuracy of the bridge information database of these systems are not exactly the same, the basic structure is similar, and they all match the functional requirements and management objectives of the local bridge management system. For bridge deterioration models, different bridge management systems may use completely different technical solutions. For example, J-BMS and Web-BMS adopt a deterministic regression method, and PONTIS, BRIDGIT, OBMS, as well as KUBA use a stochastic model based on Markov theory [
17]. The choice of different models is usually related to the locally accumulated bridge performance database, and the prediction accuracy of bridge deterioration models often determines the success or failure of BMS [
21].
The deterministic regression method assumes that there is a certain tendency for the bridge to deteriorate, and the deterioration curve can be fitted by regression analysis. The main advantage is that the modeling process is relatively simple, and the relevant parameters are easy to update, but the disadvantage is that it cannot reflect the stochasticity and uncertainty of the bridge deterioration process. In addition, such models require high data quality, so rigorous data pre-processing is usually necessary. In turn, errors caused by subjective judgments may occur during data preprocessing [
20]. The bridge deterioration model used in the Web-BMS in Shanghai, China, is based on the bridge condition index, given by Equation (1):
where
indicates the bridge condition index, and the larger the
, the better the technical condition of the bridge;
represents the bridge age;
and
are the bridge life parameter and the curve shape parameter, respectively, both of which can be obtained by regression analysis.
In contrast, PONTIS, OBMS, and NYSDOT all adopt discrete time state-based Markov models to simulate bridge deterioration [
17]. As a special case of Markov models, Markov chains is the commonly used stochastic model [
22]. It predicts bridge performance by assigning each condition rating (CR) to the state in the Markov chain, and then calculating the probability of transition from one state to another within the scheduled time [
23]. As a first-order stochastic process, Markov chains have advantages in reflecting the randomness and uncertainty of bridge performance deterioration. However, the model based on Markov chains has two assumptions—homogeneous and memoryless—that bring some limitations to their application. Homogeneity requires that the probability of transition from one CR to another remains unchanged throughout the bridge life. Memoryless means that the future state of the bridge is only related to the present state and has nothing to do with the past [
24]. Therefore, it is difficult for such models to reflect the actual performance deterioration of the bridge.
To predict the service performance of the bridge more accurately, a great quantity of studies had further developed the deterioration model. Moses and Kleiner et al. [
25,
26] used a semi-Markov model to simulate bridge deterioration. The major difference between a semi-Markov process and a Markov process is the distribution of state durations in the process. Markov processes require state durations to be geometrically or exponentially distributed, while state durations for semi-Markov processes can be arbitrarily distributed [
27]. Therefore, the semi-Markov model is closer to the actual situation than the Markov model. However, the semi-Markov model also has some limitations in estimating transition probability. For example, the semi-Markov model cannot clearly indicate the effects of various factors on bridge deterioration. In addition, linear regression is no longer applicable due to the ordinal nature of the states (CRs of the bridge) [
28].
In view of the limitations of the above methods, Mishalani and Madanat [
29] changed from discrete time state-based models to a stochastic duration model. The duration refers to the time it takes for a bridge or a bridge component to deteriorate from one state to an adjacent state [
30]. The Weibull probability density function was used to estimate the duration of states 7 and 8 (CRs of the bridge ranged from state 0 to 9, where 0 represented the worst condition and 9 was the best). Furthermore, the effects of different factors (including traffic loads, bridge age, environmental factors, road class, structure type, and wearing surface material) on deck deterioration were also considered [
29].
Stochastic duration models can be further divided into nonparametric, parametric, and semi-parametric models [
31]. Nonparametric models can be used for survival analysis when no suitable parametric model can be fitted to the event under study. Stevens, using nonparametric models to observe the impact of different covariates on bridge survival probability, mainly investigated four factors: structure type, bridge function, number of spans, and road class. The results showed that structure type, bridge function, and road class have a great influence on bridge performance [
8]. Although nonparametric models are simple and flexible in estimating bridge performance, the relationship model between survival time and risk factors cannot be established. Parametric models can model the relationship between survival time and risk factors but require assumptions about the form of the deterioration function [
32]. In contrast, semi-parametric models can overcome these limitations. Nakat and Madanat [
33] adopted a semi-parametric Cox proportional hazards model (Cox model) with its risk function as the sample risk function, which was able to simulate the performance deterioration process that cannot be simulated by conventional parametric models. However, the accuracy of the results obtained from the Cox model was generally inferior to that of the parametric model because the Cox model used partial likelihood estimation, while the parametric model used maximum likelihood estimation [
34]. Therefore, despite the drawbacks of parametric models, they were still widely used in structural deterioration simulations. If the trend of the parametric distribution obeyed by the structural performance data can be determined in advance, parametric models could yield more accurate results than nonparametric and semi-parametric models [
35].
Depending on the form and characteristics of the data distribution, parametric models can be further classified as follows: Weibull, loglogistic, lognormal, hypertabastic, and other models [
36,
37]. Nabizadeh et al. [
38] investigated the performance deterioration pattern of the superstructure of bridges in Wisconsin using survival analysis methods and analyzed the effects of structure type, bridge age, maximum span length (MSL), and average daily traffic (ADT) on the superstructure based on a hypertabastic model. The results showed that ADT and MSL had a great impact on the reliability of the superstructure. Tabatabai et al. [
39] described the deterioration behavior of bridge decks under different influencing factors using a reliability function with a hypertabastic distribution. The results found that deck area and ADT were important factors affecting deck deterioration. Agrawal and Kawaguchi et al. [
40] used a parameter model based on the Weibull distribution to calculate the deterioration rate of bridge components through historical bridge inspection data. The results indicated that the prediction model based on Weibull distribution outperformed traditional Markov chains. Similarly, Nasrollah and Washer [
41] determined the time-in-condition ratings (TICR) of bridge superstructure components based on NBI data. The Anderson–Darling test was used to evaluate five regular distributions to determine the fitting accuracy describing the TICR probability distribution. The results showed that the Weibull distribution was well suited for calculating the TICR of the superstructure components. Manafpour et al. [
42] evaluated the transition probabilities and sojourn times for the deterioration of bridge decks using a semi-Markov model based on accelerated failure time Weibull fitted-parameters. The proposed method linked the deck deterioration with various explanatory factors, such as route type, structural system attributes, ADT, and environmental conditions. Several factors were found to be statistically significant with respect to the service life of bridge decks, including the type of rebar protection, continuous versus simply supported spans, the number of spans, overall bridge deck length, and bridge location.
Many studies have been conducted to predict the performance changes of bridges, deck systems, superstructures, and substructures, while few prediction models have been extended to specific bridge components [
43]. This may be because component-level inspection data or rating data are more difficult to obtain. However, deterioration prediction using the overall bridge ratings has the potential to overestimate the actual condition of the bridge because the overall ratings include substructure ratings that are often difficult to accurately inspect due to environmental constraints [
44,
45]. In addition, the overall bridge deterioration is obtained by a weighted average of the deterioration of individual components, whereas the maintenance of bridges is generally component-specific [
46,
47,
48]. Therefore, bridge technical condition prediction at the component level is more positive for developing targeted bridge maintenance countermeasures and optimizing maintenance capital investment. This is especially true for important bridge components with high scoring weights or components that are relatively vulnerable to damage [
49].
Taking urban bridge management in Shanghai as an example, the weight values of girder bridges are shown in
Table 1, according to the Technical Code of Maintenance for City Bridges (CJJ 99-2017) [
46]. Girder bridges are used here to represent small and medium-sized urban bridges because they account for about 91.3% of the urban bridges in Shanghai.
As can be seen from
Table 1, the substructure has the greatest weight in the overall bridge rating, but because the substructure is located underwater, inspection is often difficult to perform [
44]. Especially for the large number of small and medium-sized urban bridges, the limited maintenance resources can hardly support detailed substructure inspection of all bridges every year. Therefore, the actual management tends to overestimate the technical condition of the bridge substructure, and the use of substructure inspection data to model bridge deterioration is not reliable enough [
50]. The superstructure accounts for the second greatest weight in the overall bridge rating with a weighting of 40%. Since superstructures are usually exposed to air for a long time, they are easier to detect and have more reliable data than substructures. Therefore, it is more feasible to use superstructure data to evaluate the bridge deterioration status. Meanwhile, for girder bridges, the most important component of the superstructure is the main girder. According to the Chinese Technical Code of Maintenance for City Bridge (CJJ 99-2017) [
46], the weight of the main girder accounts for 60% of the superstructure in the technical condition evaluation. As a result, the deterioration of the main girders can be used to reflect the deterioration of the superstructure and then the deterioration of the whole bridge. This has the advantage of avoiding the problem of overestimation of deterioration prediction brought by using the overall bridge rating on the one hand and helping to control the discrete nature of the data on the other hand, as well as expanding the data set, which can improve the reliability of the deterioration model. Additionally, the main girders are also one of the main objects of bridge maintenance, and studying the deterioration behavior of the main girders is beneficial for more detailed maintenance management in the future as well.
Due to its flexibility and simplicity in fitting different types of engineering life data, the Weibull distribution has been widely used in the analysis of the time-varying reliability and service life decay behavior of infrastructure [
41,
51]. Furthermore, with the different values of the shape parameter, the Weibull distribution can be associated with different probability distributions, such as the normal distribution, the exponential distribution, and the Rayleigh distribution [
20]. Therefore, in this paper, a survival analysis model based on the Weibull distribution will be applied to model the main girder condition duration based on the inspection data of the main girders in the Shanghai urban bridge management system from 2007 to 2020. In addition, the Cox proportional hazards model will also be used to observe the influence of various factors on the survival time of main girders.
6. Conclusions and Future Work
This study investigated over 40,000 bridge main girder inspection records in the Shanghai Web-BMS from 2007 to 2020. The latest survival analysis theory was used to develop a model that fits the deterioration of main girder performance. A model based on a two-parameter Weibull distribution was used to fit the duration of reinforced concrete bridges under each CR. In addition, the Cox proportional hazards model was also used to analyze the effects of different covariates on the main girders.
The results of the parameter estimation show that the shape parameters of the Weibull distribution are all greater than 1, implying that the deterioration rate of the main girder increases with time. Based on the shape and scale parameters obtained from the Weibull distribution under each CR, the average service life of the main girder in Shanghai was predicted to be 87 years. Moreover, the COX model was used to analyze four covariates that all have different effects on the deterioration of the main girder: the area factor, structure factor, road factor, and position factor. Among them, the road factor and position factor had the most significant effects. In addition, the deterioration of the main girders was faster compared to the whole bridge and superstructure. In accordance with the analysis results of this study, bridge maintenance departments should pay more attention to the inspection and maintenance of branch roadside girders to reduce pressure on future bridge management.
Survival analysis based on bridge component data can provide useful insights into predicting the service life of bridges. In addition, when the amount of inspection data is large enough and the observation period is long enough, the method can be better applied to the prediction of the survival time of bridge components at the network level. At the same time, it can provide some basis for the optimization of bridge maintenance decisions and fund allocation. In this study, only the main girder data of the bridge superstructure was analyzed. There is a large amount of damage data for other components in the bridge management system database in Shanghai, which can be analyzed and studied for multiple components in the future to explore the patterns. Future research could also compare the deterioration patterns of bridge components in different countries, analyze the similarities and differences, and explore the reasons for them.