Life-Cycle Maintenance Cost Model for Concrete Bridges Using Markovian Deterioration Curves

Abstract

Long-term deterioration of concrete bridges is a natural process that requires prudent maintenance actions throughout the bridge’s life-cycle. Nowadays, there is an ongoing effort to simulate such processes into practical models. One primary element for the model’s accuracy is the datasets used for its development. The gap between underestimated or overestimated and actual values could be narrowed by utilizing real-world datasets on bridge deterioration and rehabilitation obtained from systematic inspections over time in similar environments. Therefore, the present study aims to develop an empirical probabilistic model for precisely predicting the bridge’s future performance and suitably implementing maintenance strategies that facilitate sustainable management during bridge service life based on real data. Actual records of 72 concrete bridges from motorways in Northern Greece were collected, documenting different detected defect types, condition states, and associated maintenance costs over time. Two discrete-time Markov-chain models for the bridge’s superstructure and substructure were produced, allowing for the prediction of maintenance costs that align with the given structural condition throughout its operational life. A Chi-square test demonstrated the model’s applicability to similar datasets. This enables bridge managers to obtain a comprehensive overview of the bridge’s longitudinal performance and maintenance expenditures and adopt economically sustainable solutions for the bridge’s management.

1. Introduction

Transport infrastructure projects indisputably contribute to countries’ economies and development. In particular, bridges are vital links in the transportation network and the overall development of a nation’s economic activities. Nevertheless, a significant portion of the ageing bridges deteriorate steadily, owing to combined degradation factors, such as increased traffic volumes, extreme weather conditions, etc. [1,2]. Hence, precise deterioration prediction as well as thorough maintenance planning should be constantly prioritized to keep the bridges in an acceptable condition throughout their service life.

The gradual degradation process of bridge systems integrates various uncertainties deriving from material, damage mechanisms, climate, as well as design and maintenance risk parameters [3,4]. An accurate deterioration model is a fundamental management tool since it lays the foundation for the successful upcoming maintenance and cost estimation modules [5]. Compared to the deterministic models that export single point values for the future performance of bridge elements, the probabilistic methods are regarded as more suitable for capturing the inherent uncertainties of condition data, resulting in the development of more reliable infrastructure performance models [4,6,7]. Among the probabilistic methods used for infrastructure assets’ performance modeling, the Markov-chain models are considered the most adopted method [8]. Owing to its stochastic nature, the inherent uncertainty and randomness of the bridge’s performance can be captured adequately. For this reason, many existing bridge management systems (BMSs), including Pontis and BRIDGIT, have adopted the Markov-chain process to accurately model the bridge’s progressive deterioration [9,10,11].

Maintenance planning is a crucial aspect of infrastructure asset management that demands investigation from multiple perspectives, such as economic, engineering, environmental, etc. [12]. Due to bridge’s constructability and inspection issues as well as their exposure to intense climate conditions, bridges’ life-cycle economic expenses can be significantly high [13]. Moreover, the investigation of the most beneficial maintenance interventions remains a challenging issue for project managers because it is subject to multiple conflicting objectives, such as restricted maintenance and inspection budgets or optimization of bridge performance [14].

Bridge managers regularly undertake repair actions, either time-based or condition-based, to maintain the structure’s optimal performance. However, it is claimed that, in some cases, rehabilitation decisions lead to unnecessary over-safety levels and redundant consumption of financial resources. On top of that, the budgetary allocations are usually restricted, and administrative authorities should be quite cautious regarding the prioritization of projects requiring rehabilitation actions. To eliminate such instances, the concept is to restore the serviceability and safety of bridge assets up to sufficient levels over the life-cycle of the system with the limited available funds. The selection of the most cost-effective solution by forecasting bridge performance based on inspection data is a current issue that needs to be addressed.

The collection of on-site inspection datasets significantly impacts the accuracy of subsequent asset deterioration prediction models [15]. However, due to confidential issues and sensitivity in cost data provision by management companies, the extant papers of the literature body rarely deal with life-cycle maintenance cost analysis based on real datasets. The present study aims to address this issue by collecting and processing actual condition and cost datasets from 72 concrete bridges that serve the Greek road network. Therefore, the objective of the present study is to investigate the probabilistic deterioration of bridge components based on real inspection condition data and estimate the expected life-cycle maintenance costs based on actual rehabilitation actions for capturing the comprehensive bridge behavior and promoting long-term sustainable solutions against short-term ones.

2. Systematic Literature Review

Longitudinal degradation and maintenance of concrete bridge assets stimulates the great concern of project managers and stakeholders due to various social, economic, and political implications that arise. Primarily, a systematic literature review (SLR) was conducted to holistically investigate the current state in the field of bridge life-cycle deterioration and maintenance activities. The SLR was performed via the Web of Science (WoS) Core Collection bibliographic database for the timespan between 2000 and 2025. The SLR focused exclusively on studies with real-world deterioration and cost datasets that address life-cycle deterioration and maintenance using probabilistic parametric models. Initially, 59 peer-reviewed papers were extracted from the WoS database, while the final eligible papers were seven (7) after applying specific exclusion criteria that excluded papers relevant to (i) seismic loads and life-cycle costs; (ii) bridge maintenance optimization; (iii) other construction materials than concrete, such as timber and masonry bridges; (iv) life-cycle costs of materials; (v) deterioration models without cost estimation; (vi) deterministic approaches (e.g., parametric regressions, machine learning models, etc.).

An overview of the key descriptive attributes of the literature on bridge life-cycle maintenance cost estimation reviewed in this study is presented in

Table 1. Overview of the probabilistic studies on bridge maintenance cost estimation.

Sources	Bridge Components			Data Size	Applied Probabilistic Methods
	Superstructure	Substructure	Whole Bridge
Shim and Lee (2015) [16]	✓			53	Monte-Carlo simulation
Van Noortwijk & Klatter (2004) [17]			✓	N/A	Weibull distribution
Vagdatli et al. (2023) [4]	✓			60	Dynamic Bayesian Network
Vagdatli et al. (2024) [7]	✓	✓		78	Dynamic Bayesian Network
Yianni et al. (2017) [18]	✓			4434	Petri-net model
Gadiraju et al. (2023) [19]	✓	✓		15,350	Markov-chain model
Lee et al. (2019) [20]			✓	84	Bayesian updating

. More precisely, Shim and Lee [16] conducted a Monte-Carlo simulation to develop a probabilistic analysis for annual maintenance costs for bridge decks by utilizing a dataset comprising 53 bridge decks from the state of Wyoming. Van Noortwijk and Klatter [17] proposed a probabilistic model to calculate expected maintenance or repair costs based on the lifetime data fitted to a Weibull distribution. Vagdatli et al. [4] developed a probabilistic model for bridge life-cycle maintenance cost estimation using a Dynamic Bayesian Network. A first-order Markov-chain model formulated by a dataset of 60 bridges in Greece was implemented to demonstrate the performance and expected maintenance costs of expansion joints over their service life. Later on, Vagdatli et al. [7] introduced a Dynamic Bayesian Network to estimate the life-cycle costs of bridges, based on the actual records of 78 concrete bridges in Northern Greece. For demonstrating purposes, a semi-Markovian process and survival analysis were applied to assess the deterioration progress and expected annual maintenance costs of expansion joints throughout their 10-year lifespan.

Furthermore, Yianni et al. [18] utilized a Petri-Net modeling approach to integrate a system encompassing deterioration, inspection, and maintenance modules for investigating the behavior of railway bridge main girders over 100 years. The study used real-world data from 4434 bridges in the UK, collected during inspections from 1998 to 2014. Gadiraju et al. [19] presented a deep reinforcement learning model for enhancing bridge maintenance schedules in Nebraska state by using a dataset of 15,350 bridges from the Nebraska NBI. Three different bridge components—the deck, superstructure, and substructure—shared the same transition probability matrices, obtained through Markovian computations, to demonstrate bridge deterioration and cumulative maintenance costs over 20 years. Lee et al. [20] suggested updating the deterioration state of prestressed concrete bridges through Bayesian inference to estimate maintenance costs based on the bridge’s current structural condition. A dataset of 84 Korean bridges between 1995 and 2018 was used to incorporate existing information in the proposed model.

Evidently, the majority of existing studies in the extant literature body concerning the probabilistic bridge life-cycle maintenance costs focus on the bridge’s deck system elements. Additionally, large datasets usually have not been performed under the same specifications, such as the consistent pricelists, construction standards, and administrative authorities, rendering it challenging to extract homogeneous results from them. Therefore, the present study aims to introduce a data-driven probabilistic model for life-cycle deterioration prediction and maintenance cost analysis for bridge superstructure and substructure components based on homogeneous datasets collected from 72 concrete bridges in Northern Greece.

3. Methodology

The process for the proposed framework development involves the following steps: (a) data collection and processing; (b) prediction of bridge components longitudinal deterioration; and (c) life-cycle maintenance cost analysis with real datasets. A general flowchart of the developed methodology is illustrated in Figure 1, while a detailed explanation of each step is provided in the following sections.

3.1. Markovian Stochastic Model

The Markov-chain approach is regarded as the most common stochastic procedure utilized for the deterioration prediction and asset management of infrastructure projects, such as bridges and pavements [21,22]. Markov chains are probabilistic methods that can be parameterized by empirical transition probabilities calculation between discrete states [23].

The classic Markovian model follows three assumptions [24]: (a) the process is discrete in time; (b) the behavior of the studied system is represented by a finite state space, known as condition states (CS), ${\{S}_i=S_1, S_2,S_N\}$, which are discrete random variables; and (c) the model satisfies the “Markov property”. The latter theorem indicates that the conditional probability, $P_{ij}$, of a given future condition state $j$ depends only on the current state $i$ and not on previous ones, known as the first-order Markov process [25]. In general, for a given sequence of time points $t$, the conditional probabilities follow the subsequent rule:

$$P_{ij}=prob[X\left(t+1\right)=j|X\left(t\right)=i]$$

(1)

The Markov-chain process consists of three main elements [25]: the condition probability vector, the time step of the process, and the transition matrix:

a. 

The condition probability vector is a row vector presenting the condition of bridge components utilizing the probabilities of stating in each condition state, $CR=\{a_1;a_2;\dots ,a_n\}$. The sum of all $a_i$ is equal to one, with all the elements being positive values. For instance, the initial condition vector for any new rehabilitated bridge is represented with the following form:

$$CR\left(0\right)=\{1;0;0;\dots ,0\}$$

(2)

a. 

The time step is the time interval among two consecutive states. Typically, this step is identical with the frequency of real data collection.

b. 

The Transition Probability Matrix (TPM), usually denoted as $P$, describes the bridge’s deterioration over time. A typical TPM is shown in Equation (3) along with its restrictions described in Equations (4) and (5):

$$P=\left[\begin{array}{cccc}{p}_{11}& p_{12}& \cdots & p_{1n}\\ p_{21}& p_{22}& \cdots & p_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ p_{n1}& p_{n2}& \cdots & p_{nn}\end{array}\right]$$

(3)

$$0\le p_{ij}\le 1, \mathrm{for} \mathrm{all} i \mathrm{and} j, \mathrm{where} i,j=\mathrm{0,1},2...,n$$

(4)

$$\sum _{j\in I}{p}_{ij}=1 i∊I$$

(5)

Each component $p_{ij}$ represents the transition probabilities from one condition state $i$ at time $t$ to another condition state $j$ at time $t+1.$ In cases where the nature deterioration of the presented system is examined and maintenance actions are not considered, only the upper diagonal of the TPM receive values, where $i<j$. Therefore, the general form of Equation (3) is converted into Equation (6):

$$P=\left[\begin{array}{cccc}{p}_{11}& p_{12}& \cdots & p_{1n}\\ 0& 0& \cdots & p_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & 1\end{array}\right]$$

(6)

Once the TPM is formulated, the predicted condition vector of the next state is estimated by multiplying the current state by the TPM. In the case of time-homogeneous Markov-chain models, which is the case in the present study, the transition rates are exponentially distributed, remain constant throughout all the analysis years, and are applied to any $t$ to $t+1$ transition [26], as presented in Equation (7).

$$A_t=A_{t-1}\times P=A_0\ast P^t$$

(7)

where $A_t$ is the state vector at any time $t$; $A_0$ is the state vector at time $t=0$.

Ultimately, the overall expected bridge component deterioration can be estimated by Equation (8):

$${CR}_t=CR\left(0\right)\times {P_{ij}}^t\times CR$$

(8)

where ${P_{ij}}^t$ is the TPM an any given time $t$; $CR$ is the bridge’s condition vector that remains constant $CR=\left\{\mathrm{9,8},\mathrm{7,6}\dots , 1\right\}$; and $CR\left(0\right)$ is the initial condition vector, which, in the present study, equals to $CR\left(0\right)=\{1, 0, 0, \dots , 0\}$, since it is considered that the bridges degraded from an intact level and for a period of 100 years.

3.2. Transition Probability Matrix Computation

The reliability of a Markov-chain model is highly related to the accuracy of the computed transition probabilities [15]. Two techniques are the most commonly used for estimating the matrix’s transition probabilities: collection of historical real data or experts’ judgement [25]. Ιt is acknowledged that the calculation of the transition probabilities is ideally accomplished by using the bridge’s available condition data derived from uniform time intervals over a long time period [27]. In the case of on-site data collection, the most appropriate processing method is the maximum likelihood estimator for the deterioration probabilities $P_{ij}$ calculation [28], as depicted in Equation (9):

$$P_{ij}={\displaystyle \frac{N_{ij}}{N_i}}$$

(9)

where $P_{ij}$ = the element in row $i$ and column $j$ of TPM; $N_{ij}$ = the portion of assets shifting from one condition state $i$ into another worse $j$ during one time step; $N_i$ = the total portion of assets was in state $i$ before any transition take place.

Thus, the TPMs of the present study were calculated by using actual values concerning the condition of the bridge’s components to simulate more realistic deterioration outcomes. The real inspection data of 72 concrete bridges serving the Greek motorways were collected over a period of 15 to 25 years, covering bridge operation from 1999 to 2024.

Since the Markov-chain process relies only on two consecutive years to calculate the TMP, only nine transition probabilities are necessary, each for every condition state. However, in the present study, a set of transition probabilities is available for each pair of successive years, owing to the longitudinal data collected. As observed, these values are not uniformly distributed. Hence, the average value of the computed transition probabilities was calculated for each CS over the entire analysis period. As can be seen in Equations (10) and (11), transition probabilities outside the valid range should be excluded, while the final transition probabilities, ${p_{ii}}^{valid}$, represent the average of the valid values for each condition state. This approach allows for fully utilizing the available dataset and ensures a more thorough data analysis and a more informed decision-making process.

$${PR}_i=\left\{\begin{array}{c}\overline{p_{ii}}-s_i\\ \overline{p_{ii}}+s_i\end{array}\right. i\in [9, 8, 7, \dots 1]$$

(10)

$${p_{ii}}^{valid}=\left(\overline{p_{ii}}\right) \forall p_{ii}\in {PR}_i$$

(11)

where ${PR}_i$ is the valid range of transition probabilities for each state $i$; $p_{ii}$ represents the computed transition probabilities throughout all the years of the inspection dataset for each state $i$; $\overline{p_{ii}}$ and $s_i$ are the average value and standard deviation of $p_{ii}$ values, respectively; ${p_{ii}}^{valid}$ represents the final transition probabilities for each state $i$ removing invalid values.

3.3. Validation Process of Markovian Stochastic Model

Ultimately, the Markovian models should be validated for ensuring their applicability in the light of new datasets from similar environments. In this study, quantitative validation methods were utilized to compare the predicted stochastic models with observed datasets. For this reason, the collected datasets, analyzed in Section 3.4, were divided into two distinct datasets, the training and the validation dataset; specifically, 80% of the data were utilized to train the Markovian models, and 20% was used to validate them for generalization purposes.

The Chi-square goodness-of-fit test was conducted using appropriate statistical packages in the Python 3.8.3 version programming language. According to classical binary testing theory [29], two hypothesis tests, the null hypothesis ($H_0$) and the alternative hypothesis ($H_1$), should be checked for determining the model’s accuracy:

H₀: 

The predicted conditions are not significantly different from the observed conditions.

H₁: 

The predicted conditions are significantly different from the observed conditions.

For accepting the $H_0$ hypothesis, the probability of the estimated Chi-square ($p-value$) should be greater than a given chosen significance level ($a$).

3.4. Data Collection

The main key aspects considered in the present study are summarized below:

a. 

Bridge and Component type selection: The Markovian model was calibrated using actual datasets collected from 72 existing concrete bridges serving the Greek national road network in Northern Greece. The dataset is grouped into three categories: (a) bridges with pier height greater than 15 m, (b) overpasses, and (c) underpasses. Among the bridge’s components, the superstructure and substructure were regarded as the most critical ones; hence, they were selected for the deterioration and life-cycle analysis.

Table A1. Overview of structural information of actual datasets used in the analysis.

Serial Number	Bridge’s ID	Construction Year	Span Number and Arrangement (m)	Superstructure Material
1	01.07Β.08	2009	36.30/37.50 × 7/36.30	Prestressed concrete
2	01.08.07Β	2009	36.30/37.50 × 6/36.30
3	04.08Β.09	2007	60/8 × 100/60
4	04.09.08Β	2007	60/8 × 100/60
5	07.31.32	2001	3 × 34.50/3 × 42.00/2 × 34.50/22.75
6	07.32.31	2001	22.75/34.50/3 × 43.50/34.50
7	06.42.43	2000	19.40/19.05
8	01.43	1999	32.01/31.71
9	04.43	1999	27.58/26.78
10	04.24.24A	2002	27.85/27.85/27.85
11	04.24A.24	2002	27.85/27.85/27.85
12	09.24.24A	2001	29.62/30.76/29.62
13	09.24A.24	2001	29.62/30.76/29.62
14	02.24A.25	2002	19.67/20.66/19.67
15	02.25.24A	2002	19.67/20.66/19.67
16	05.24A.25	2001	27.1/27.9/27.1
17	05.25.24A	2001	27.1/27.9/27.1
18	09.24A.25	2002	25.0/25.0
19	09.25.24A	2002	25.0/25.0
20	12.24A.25	2001	19.67/20.66/19.67
21	12.25.24A	2001	19.67/20.66/19.67
22	16.24A.25	2002	35.00/45.00/35.00
23	16.25.24A	2002	35.00/45.00/35.00
24	18.24A.25	2001	38.0/47.0 × 2/38.0
25	18.25.24A	2001	38.0/47.0 × 2/38.0
…	…	…	…
…	…	…	…	Reinforced concrete
50	04.24A.25	2003	10.5
51	04.25.24A	2003	10.5
52	06.24A.25	2003	10.5
53	06.25.24A	2003	10.5
54	07.24A.25	2000	13
55	07.25.24A	2000	13
56	08.24A.25	2003	10.5
57	08.25.24A	2003	10.5
58	10.24A.25	2000	10.5
59	10.25.24A	2000	10.5
60	11.24A.25	2000	10.5
61	11.25.24A	2000	10.5
62	13.24A.25	2000	10.5
63	13.25.24A	2000	10.5
64	14.24A.25	2002	14.00
65	14.25.24A	2002	14.00
66	15.24A.25	2000	10.5
68	15.25.24A	2000	10.5
69	17.24A.25	2000	10.5
70	17.25.24A	2000	10.5
71	19.24A.25	2001	10.5
72	19.25.24A	2001	10.5

and

Table A2. Indicative actual condition states of bridge superstructure and substructure over time.

Serial Number	Bridge’s ID	5th Year	10th Year	15th Year	20th Year	25th Year
1	01.07Β.08	8/9	7/6	6/5	-/-	-/-
2	01.08.07Β	8/9	7/6	6/5	-/-	-/-
3	04.08Β.09	7/9	7/8	5/6	-/-	-/-
4	04.09.08Β	7/9	7/8	5/6	-/-	-/-
5	07.31.32	8/9	8/8	8/7	7/7	-/-
6	07.32.31	8/9	8/8	8/7	7/7	-/-
7	06.42.43	9/9	8/8	8/7	7/6	7/5
8	01.43	8/8	7/8	7/6	6/4	6/4
9	04.43	9/9	8/8	7/7	7/7	6/7
10	04.24.24A	9/9	9/9	8/8	8/8	-/-
11	04.24A.24	9/9	9/9	8/8	8/8	-/-
12	09.24.24A	7/8	7/7	7/7	7/7	-/-
13	09.24A.24	7/8	7/7	7/7	7/7	-/-
14	02.24A.25	7/8	7/7	7/7	7/7	-/-
15	02.25.24A	7/8	7/7	7/7	7/7	-/-
16	05.24A.25	9/9	9/9	8/8	7/8	7/8
17	05.25.24A	9/9	9/9	8/8	7/8	7/8
18	09.24A.25	8/9	8/8	7/7	6/7	-/-
19	09.25.24A	8/9	8/8	7/7	6/7	-/-
20	12.24A.25	8/8	8/8	7/7	7/7	6/7
21	12.25.24A	8/8	8/8	7/7	7/7	6/7
22	16.24A.25	8/9	8/8	8/8	7/7	-/-
23	16.25.24A	8/9	8/8	8/8	7/7	-/-
24	18.24A.25	8/8	8/8	8/8	8/8	7/7
25	18.25.24A	8/8	8/8	8/8	8/8	7/7
…	…	…	…	…	…	…
50	04.24A.25	9/9	9/8	8/8	8/7	-/-
51	04.25.24A	9/9	9/8	8/8	8/7	-/-
52	06.24A.25	9/9	8/8	8/8	8/7	-/-
53	06.25.24A	9/9	8/8	8/8	8/7	-/-
54	07.24A.25	9/9	8/8	8/8	8/8	7/7
55	07.25.24A	9/9	8/8	8/8	8/8	7/7
56	08.24A.25	9/9	8/8	8/8	7/7	-/-
57	08.25.24A	9/9	8/8	8/8	7/7	-/-
58	10.24A.25	8/8	8/8	8/8	7/7	7/7
59	10.25.24A	8/8	8/8	8/8	7/7	7/7
60	11.24A.25	8/8	8/8	8/8	8/8	7/8
61	11.25.24A	8/8	8/8	8/8	8/8	7/8
62	13.24A.25	9/9	8/8	8/8	8/8	7/7
63	13.25.24A	9/9	8/8	8/8	8/8	7/7
64	14.24A.25	9/9	9/8	8/8	8/8	-/-
65	14.25.24A	9/9	9/8	8/8	7/8	-/-
66	15.24A.25	8/8	8/8	8/8	7/8	7/8
68	15.25.24A	8/8	8/8	8/8	8/8	7/8
69	17.24A.25	9/9	9/9	8/8	8/8	7/8
70	17.25.24A	9/9	9/9	8/8	8/8	7/8
71	19.24A.25	9/9	9/9	9/9	8/8	-/-
72	19.25.24A	9/9	9/9	9/9	8/8	-/-
	Data size	72/72	72/72	72/72	64/64	35/35

in Appendix A display the main structural information of the collected datasets along with the indicative condition states for superstructure and substructure observed during the bridge’s service life.

b. 

Analysis period: The real dataset was collected over a 25-year time period from bridges operating from 1999 to 2024. The bridge performance model was calculated based on actual data collected from its initial operation and projected for 100 years of its total lifespan. Since all the available data were used, a comprehensive insight into the bridge components’ deterioration can be obtained and illustrated in the formulated TPM.

c. 

Defect identification: Based on detailed visual inspections, the different types of bridges’ defects and the percentage of material defects for the superstructure and substructure elements were detected. More precisely, the most common registered damages for the collected bridges account for staining, efflorescence, cracks, spalling, exposed reinforcement and reinforcement corrosion.

d. 

Condition States: The number and ranking of CSs were adopted by the National Bridge Inventory (NBI) and are illustrated in

Table 2. Definition of bridge condition states according to NBI classification.

NBI Scale	Condition	Description
9	Excellent	New condition; no remarkable deficiencies
8	Very good	No repair needed
7	Good	Some minor problems, minor maintenance needed
6	Satisfactory	Some minor deterioration, major maintenance needed
5	Fair	Minor section loss, cracking, spalling, or scouring for minor rehabilitation; minor rehabilitation needed
4	Poor	Advanced section loss, deterioration, spalling, or scouring; major rehabilitation needed
3	Serious	Section loss, deterioration, spalling, or scouring that have seriously affected the primary structural components
2	Critical	Advanced deterioration of primary structural elements for urgent rehabilitation; bridge may be closed until corrective action is taken
1	Imminent failure	Major deterioration or loss of section; bridge may be closed to traffic, but corrective action can put it back to light service

. The classification of the bridge’s components into the individual CSs was conducted based on the aforementioned identified damages. Since there were no data on the last condition states, a by-6 TPM was considered representative of the studied case and was developed for the suggested Markov-chain probabilistic models. To obtain a complete and consistent dataset that would serve as an adequate basis for the deterioration model prediction, the data were screened to filter out any records with incomplete data.

e. 

Time step: A one-year time cycle was adopted to resemble the annual data collection regarding the bridge condition.

f. 

Maintenance improvements: For the initial phase of the model, only the data accounting for the natural deterioration of the bridge’s components was considered. Consequently, any bridge component that demonstrated an improvement in the condition value compared to the previous year was not considered; hence, all the elements under the main diagonal of TPM are equal to zero, as defined in Equation (6). In the second step of this research, where maintenance improvements take place, the TPM is modified to incorporate any possible maintenance action.

3.5. Maintenance Policies and Related Costs

In ageing systems, assets are either repaired preventively at scheduled time intervals or replaced correctively in case of failure based on emerging needs [30]. Figure 2 indicates all the possible cases of failure rates, (${\mathrm{f}}_{\mathrm{i}},\mathrm{i}=\mathrm{1,2},\dots ,\mathrm{N}$), and repair rates, (${\mathrm{r}}_{\mathrm{i}},\mathrm{i}=\mathrm{1,2},\dots ,\mathrm{N}$), that might occur in degraded systems, indicated by red and blue colors arrows, respectively. Assuming that the current time step is $t$ and the time interval among two consecutive steps is $\Delta t$, then the transition probabilities between the examined states for natural decay or preventive maintenance are shown in

Table 3. General form of transition probabilities among condition states for natural decay.

$\mathit{t}$	$\mathit{t}\mathbf{+}\mathsf{\Delta }\mathit{t}$
	1	2
1	$e^{-(f_1+f_4+f_5)\Delta t}$	$f_1\ast (1-e^{-\left(f_1+f_4+f_5\right)\Delta t})/(f_1+f_4+f_5)$
2	0	$e^{-(f_2+f_6)\Delta t}$
$\mathbf{N}\mathbf{-}\mathbf{1}$	0	0
$\mathbf{N}$	0	0
	$\mathbf{N}\mathbf{-}\mathbf{1}$	$\mathbf{N}$
1	$f_4\ast (1-e^{-\left(f_1+f_4+f_5\right)\Delta t})/(f_1+f_4+f_5)$	$f_5\ast (1-e^{-\left(f_1+f_4+f_5\right)\Delta t})/(f_1+f_4+f_5)$
2	0	$e^{-(f_2+f_6)\Delta t}$
$\mathbf{N}\mathbf{-}\mathbf{1}$	$e^{-\left(f_3\right)\Delta t}$	$1-e^{-\left(f_3\right)\Delta t}$
$\mathbf{N}$	0	1

and

Table 4. General form of transition probabilities among condition states with repair actions.

$\mathit{t}$	$\mathit{t}\mathbf{+}\mathsf{\Delta }\mathit{t}$
	1	2
1	$e^{-(f_1+f_4+f_5)\Delta t}$	$f_1\ast (1-e^{-\left(f_1+f_4+f_5\right)\Delta t})/(f_1+f_4+f_5)$
2	$r_1\ast (1-e^{-\left(r_1+f_2+f_6\right)\Delta t})/(r_1+f_2+f_6)$	$e^{-(f_2+f_6+r_1)\Delta t}$
$\mathbf{N}\mathbf{-}\mathbf{1}$	$r_2\ast (1-e^{-\left(r_2+r_6+f_3\right)\Delta t})/(r_2+r_6+f_3)$	$r_6\ast (1-e^{-\left(r_2+r_6+f_3\right)\Delta t})/(r_2+r_6+f_3)$
$\mathbf{N}$	$r_3\ast (1-e^{-\left(r_3+r_4+r_5\right)\Delta t})/(r_3+r_4+r_5)$	$r_5\ast (1-e^{-\left(r_3+r_4+r_5\right)\Delta t})/(r_3+r_4+r_5)$
	$\mathbf{N}\mathbf{-}\mathbf{1}$	$\mathbf{N}$
1	$f_4\ast (1-e^{-\left(f_1+f_4+f_5\right)\Delta t})/(f_1+f_4+f_5)$	$f_5\ast (1-e^{-\left(f_1+f_4+f_5\right)\Delta t})/(f_1+f_4+f_5)$
2	$f_2\ast (1-e^{-\left(f_2+f_6+r_1\right)\Delta t})/(f_2+f_6+r_1)$	$f_6\ast (1-e^{-\left(f_2+f_6+r_1\right)\Delta t})/(f_2+f_6+r_1)$
$\mathbf{N}\mathbf{-}\mathbf{1}$	$e^{-(f_3+r_6+r_2)\Delta t}$	$f_3\ast (1-e^{-\left(f_3+r_6+r_2\right)\Delta t})/(f_3+r_6+r_2)$
$\mathbf{N}$	$r_3\ast (1-e^{-\left(r_3+r_4+r_5\right)\Delta t})/(r_3+r_4+r_5)$	$e^{-(r_3+r_4+r_5)\Delta t}$

, respectively.

More precisely, it is clarified that in the present study, the only feasible failure transition occurs between two successive CSs, as depicted with continuous arrows in Figure 2 and calculated with relative probabilities in Table 3. Moreover, Table 4 presents all the possible solutions for imperfect and perfect maintenance actions. Various maintenance strategies, $M_z=\{M_1,M_2,\dots ,M_Z\}$, can be applicable for retaining the bridge performance at its optimum level. The appropriate maintenance actions can be determined based on the developed Markov-chain prediction of bridge components performance and according to the CSs; hence, a condition-based maintenance model is constructed. For all the maintenance works identified as necessary for improving the bridge’s condition state, the actual procurement and real costs were collected and used in the analysis calculations. The full specification of the implemented maintenance actions along with their associated actual costs and the CSs applied to are summarized in

Table 5. Maintenance actions and cost estimation breakdown.

Maintenance Works Category	Applied Condition State	Maintenance Action	Symbol	Actual Unit Costs *
Prerequisite auxiliary works	8, 7, 6, 5, 4	Supply of a special under bridge inspection platform vehicle for inspection or maintenance of the bridge underside at heights > 15 m	${\mathit{M}}_{\mathbf{1}}$	$550 \mathit{€}/\mathbf{u}\mathbf{n}\mathbf{i}\mathbf{t}$
		Use of a special motorized work platform for inspection or maintenance of the underside of bridges at heights > 15 m	${\mathit{M}}_{\mathbf{2}}$	$1100 \mathit{€}/\mathbf{u}\mathbf{n}\mathbf{i}\mathbf{t}$
		High-grade tubular iron scaffolding	${\mathit{M}}_{\mathbf{3}}$	$9 \mathit{€}/{\mathbf{m}}^{\mathbf{2}}$
Minor works	8, 7	Protective coating for concrete surfaces, siloxane/silane based, permeable to water vapor and impermeable to water and CO2, according to ELOT EN	${\mathit{M}}_{\mathbf{4}}$	$14.4 \mathit{€}/{\mathbf{m}}^{\mathbf{2}}$
	7, 6, 5	Application of high-pressure water jetting on concrete surfaces	${\mathit{M}}_{\mathbf{5}}$	$3.60 \mathit{€}/{\mathbf{m}}^{\mathbf{2}}$
	6, 5, 4	Corrosion inhibitor application on reinforced concrete elements	${\mathit{M}}_{\mathbf{6}}$	$13.3 \mathit{€}/{\mathbf{m}}^{\mathbf{2}}$
Major works	6, 5, 4	Coating concrete surface with a cement-based sealer to extract existing moisture and protect the concrete from water	${\mathit{M}}_{\mathbf{7}}$	$6 \mathit{€}/{\mathbf{m}}^{\mathbf{2}}$
	6, 5, 4	High-strength, non-shrinking thixotropic repair mortar in an average thickness of 5 cm	${\mathit{M}}_{\mathbf{8}}$	$45.6 \mathit{€}/{\mathbf{m}}^{\mathbf{3}}$
	5, 4	Filling of small width cracks (0.3–3.00 mm) in concrete structures by injection of epoxy resin	${\mathit{M}}_{\mathbf{9}}$	$20.6 \mathit{€}/\mathbf{m}$
	5, 4	Application of very high-pressure water jetting on surfaces of reinforced or prestressed concrete to expose reinforcing bars	${\mathit{M}}_{\mathbf{10}}$	$400 \mathit{€}/{\mathbf{m}}^{\mathbf{3}}$

* Current market prices in 2024.

. These activities are combined suitably according to the detected damage each time for completing one full maintenance activity for each CS.

The uncertainties related to structural and external causes lead to structural bridge failures that are distributed as random events with time-dependent probabilities of occurrence. Therefore, the expected monetary value concept is applied to the proposed life-cycle maintenance model for bridge components computation. The expected life-cycle maintenance costs are estimated by using Equation (12), without taking into account the time value of money:

$$E\left[C_{M\left(t\right)}\right]=\sum _{s=1}^{S-1}{P}_{m,s\left(t\right)}\times c_{m,s}+\sum _{s=S}^S{P}_{f,s\left(t\right)}\times c_{f,s}$$

(12)

where $s$ = the current condition state; $S$ = the overall number of condition states of bridge’s components; $P_{m,s\left(t\right)}$ = the probability of a preventive maintenance action $m$ occurring at a given time $t$ in states $s=\{\mathrm{1,2},\dots , N-1\}$; $P_{f,s\left(t\right)}$ = the probability of corrective maintenance action $f$ occurring at a given time $t$ in the final failure state $s=N$; $c_{m,s}$ = the undiscounted unit cost related to the preventive maintenance action in states $s=\{\mathrm{1,2},\dots , N-1\}$; $c_{f,s}$ = the undiscounted unit cost related to the corrective maintenance strategy in the final failure state.

4. Deterioration and Maintenance Prediction Using Markov-Chain Model

The entire calculation of the TPMs for the bridge’s superstructure and substructure components was fulfilled via Excel spreadsheets by using the training dataset and applying Equations (3)–(11) as analyzed in Section 3.1 and Section 3.2. Eventually, Equations (13) and (14) depict the transition probabilities of remaining to the same CS or transiting to the next CS after one-year step for the bridge’s superstructure and substructure, respectively.

$$P_{sup}=\left[\begin{array}{cccccc}0.844& 0.156& 0& 0& 0& 0\\ 0& 0.931& 0.069& 0& 0& 0\\ 0& 0& 0.972& 0.028& 0& 0\\ 0& 0& 0& 0.90& 0.10& 0\\ 0& 0& 0& 0& 0.91& 0.09\\ 0& 0& 0& 0& 0& 1\end{array}\right]$$

(13)

Equation (13) Transition Probability Matrix for a one-year time step of superstructure.

$$P_{sub}=\left[\begin{array}{cccccc}0.854& 0.146& 0& 0& 0& 0\\ 0& 0.908& 0.092& 0& 0& 0\\ 0& 0& 0.984& 0.016& 0& 0\\ 0& 0& 0& 0.687& 0.313& 0\\ 0& 0& 0& 0& 0.94& 0.06\\ 0& 0& 0& 0& 0& 1\end{array}\right]$$

(14)

Equation (14) Transition Probability Matrix for a one-year time step of substructure.

Considering the exported TPMs as uniform for the entire analysis period, the deterioration prediction results under natural decay emerged for each time step $t$, as shown in Figure 3 and Figure 4, while the expected CRs for the bridge’s components were calculated via Equation (8) and are presented in Figure 5 and

Table A3. CR data points of superstructure for a single bridge in 100 years.

Superstructure
Years	Condition Rating	Years	Condition Rating	Years	Condition Rating	Years	Condition Rating
0	9000	26	6935	52	5756	78	4933
1	8844	27	6885	53	5717	79	4910
2	8701	28	6840	54	5679	80	4886
3	8571	29	6809	55	5641	81	4864
4	8450	30	6736	56	5603	82	4841
5	8339	31	6687	57	5567	83	4820
6	8237	32	6639	58	5531	84	4798
7	8141	33	6591	59	5495	85	4778
8	8051	34	6543	60	5460	86	4757
9	7967	35	6496	61	5426	87	4738
10	7887	36	6449	62	5392	88	4718
11	7812	37	6402	63	5359	89	4699
12	7740	38	6356	64	5326	90	4681
13	7671	39	6310	65	5295	91	4663
14	7605	40	6265	66	5263	92	4646
15	7542	41	6220	67	5233	93	4629
16	7480	42	6175	68	5202	94	4612
17	7421	43	6131	69	5173	95	4596
18	7363	44	6088	70	5144	96	4580
19	7306	45	6044	71	5116	97	4564
20	7250	46	6002	72	5088	98	4549
21	7196	47	5959	73	5061	99	4535
22	7142	48	5918	74	5034	100	4520
23	7090	49	5876	75	5008
24	7037	50	5836	76	4983
25	6986	51	5796	77	4958

and

Table A4. CR data points of substructure for a single bridge in 100 years.

Substructure
Years	Condition Rating	Years	Condition Rating	Years	Condition Rating	Years	Condition Rating
0	9000	26	6909	52	5993	78	5347
1	8854	27	6865	53	5964	79	5326
2	8716	28	6822	54	5935	80	5306
3	8586	29	6781	55	5907	81	5286
4	8463	30	6741	56	5879	82	5266
5	8347	31	6700	57	5851	83	5246
6	8238	32	6661	58	5824	84	5227
7	8135	33	6623	59	5797	85	5208
8	8038	34	6585	60	5770	86	5190
9	7947	35	6548	61	5744	87	5171
10	7860	36	6512	62	5718	88	5153
11	7779	37	6476	63	5692	89	5135
12	7701	38	6440	64	5667	90	5118
13	7628	39	6406	65	5642	91	5100
14	7558	40	6371	66	5618	92	5083
15	7491	41	6337	67	5593	93	5066
16	7427	42	6304	68	5569	94	5050
17	7366	43	6271	69	5546	95	5033
18	7308	44	6239	70	5522	96	5017
19	7252	45	6206	71	5499	97	5002
20	7198	46	6175	72	5477	98	4986
21	7146	47	6143	73	5454	99	4971
22	7095	48	6113	74	5432	100	4955
23	7047	49	6082	75	5410
24	6999	50	6052	76	5389
25	6953	51	6022	77	5368

in Appendix B. A perfect condition probability vector was considered as an initial vector for the year 0 when the bridge initiated its operation. Afterwards, the bridge’s degradation commences with approximately the 55% to be in the State “8” by the tenth year for the superstructure component. Roughly the same percentage is shifted to the State “7” in the thirtieth year of the analysis. After the fiftieth year of the bridge’s life-cycle, the highest proportion of the cumulative percentage of transition probabilities is detected in the lower CSs from “6” to “4”. Finally, at the end of the time period, 75% of superstructures reach the lower examined state of “4” (Figure 3). Regarding the deterioration of the bridge’s substructure, a percentage of 60% reaches the “4” CS at the 100th year of the analysis period, indicating, as expected, a slower deterioration compared to the superstructure component (Figure 4).

Moreover, the crucial issue of the execution time of maintenance actions throughout the bridge’s service life should be addressed. The managing authority of bridges under consideration has set the State “5” as a condition threshold to ensure smooth operations and user safety (Figure 5). Hence, by comparing the exported results of Figure 3 and Figure 4 with the specified condition threshold, useful policy recommendations regarding the extent and the time of major maintenance actions can be derived. For instance, for both superstructure and substructure, the peak of the deterioration curves for State “5” occur during the sixth decade of bridge service life, with approximately 17% of the bridge fleet falling into this condition. However, State “4” roughly starts between the 15th and 20th years of the analysis period with an upward trend until the end of the analysis period. Consequently, the exported deterioration curves can serve as a consulting tool for bridge managers in order to obtain an overall sight of bridge degradation and to examine and prioritize suitable maintenance actions under budget constraints each time.

The results of the statistical goodness-of-fit test for superstructure and substructure deterioration curves are summarized in

Table 6. Chi-square goodness of fit results for superstructure and substructure deterioration models.

Superstructure Markovian Deterioration Model
	Training Dataset	Validation Dataset
Degree of freedom	25	25
Significance level ($a$)	0.05	0.05
$p-value$ of Chi-square test	0.487	0.356
Substructure Markovian deterioration model
	Training dataset	Validation dataset
Degree of freedom	25	25
Significance level ($a$)	0.05	0.05
$p-value$ of Chi-square test	0.651	0.508

. For this study, the significance level was set at $a=0.05$. As can be seen, the null hypothesis $H_0$ is confirmed for both training and validation datasets, since $p-value>a=0.05$. Therefore, it can be inferred that the Markovian models can effectively predict the condition of bridge’s superstructure and substructure within an acceptable range of differences.

Additionally, the deterioration curves for the bridge’s superstructure and substructure exported from the present study were compared with the corresponding ones found by widely known and reputable agencies that adhere to FHWA practices, such as those in Nebraska and New York [31,32]. As observed in Figure 6, slight differences have appeared between the Greece and Nebraska models, exhibiting similar deterioration trends for the bridge’s superstructure (Figure 6a) and substructure (Figure 6b). Also, New York’s deterioration curve decreases at the same rate as the other two models, despite its initial deterioration state being State 7 for superstructure.

Subsequently, the maintenance costs were estimated based on the aforementioned longitudinal transition probabilities over a 100-year period of the bridge’s service life. It is mentioned that for the model’s demonstrative purposes and taking into account the most commonly implemented maintenance strategy whereby the system returns to its initial state from any worse state, the repair rates with the continuous arrows of Figure 2 are considered with a 100% probability of occurrence. The maintenance actions shown in Table 5 were employed for restoring the bridge asset to its initial condition. Considering the quantities of the damaged areas of either the bridge’s superstructure or substructure and by applying appropriate multiplications with the unit costs of maintenance works of Table 5, the final repair prices were converted to $€/{\mathrm{m}}^2$ and are illustrated in Figure 7 as representative costs for each CS and each of the group of studied deteriorated bridges.

It is stated that the derived unit costs for substructure components appeared notable differences among the studied categories and their classification into three sub-categories was deemed necessary. Primarily, the difference in actual costs for the bridge and underpass substructure components is due to the types of prerequisite auxiliary works applied for the implementation of the maintenance works. Bridges with piers > 15 m typically utilize the types of $M_1$ and $M_2$, whereas underpasses make use of the $M_3$ type according to Table 5. Furthermore, it is clarified that the classification of the bridge into condition states is based on the severity and intensity of any given damage, rather than the extent of deterioration. Thus, considering the extent of the occurred damage, the cost outcomes might be slightly different from the ones derived from this study.

Ultimately, the obtained transition probabilities for each CS (Figure 3 and Figure 4) were multiplied by the actual unit maintenance costs corresponding to each CS (Figure 7) to yield the expected costs of superstructure and substructure components over the bridge’s lifetime. Figure 8, Figure 9, Figure 10 and Figure 11 provide a comprehensive overview of the expected maintenance costs throughout a 100-year bridge life-cycle. By knowing the expected maintenance costs in each year of the bridge’s remaining service life and comparing them through the formulation of different maintenance alternatives, the project managers can have a solid base for planning future expenditures in a more cost-effective way. More precisely, as expected, the cost trend aligns with the corresponding condition states. As can be seen in the cumulative cost diagram of Figure 8, if the maintenance of the superstructure is performed in the first half of the analysis period, the costs are reduced by half compared to those incurred during the second fifty-year life. Accordingly, in the case of the bridge’s substructure (Figure 9, Figure 10 and Figure 11), the costs of the first fifty years were estimated at two-thirds of the costs of the second half of the life-cycle. This can be explained by the fact that the superstructure deteriorates faster than the bridge’s substructure. As a result, by the end of the analysis period, a greater proportion of the bridge’s superstructure is classified in state “4” compared to the substructure. Consequently, restoring the superstructure to its initial condition will incur higher costs.

5. Conclusions

Bridge management actions are an incremental part of its smooth and seamless operation during its service life. These repair actions must be delivered within the bounds of available resources. For promoting feasible solutions, bridge managers should conduct life-cycle modeling procedures, which encompasses the overall expenditures of the asset. Due to various uncertain factors that impact the bridge’s operation, the probabilistic methods are considered more suitable for accurately modeling bridge performance. Therefore, the main goal of the current study is to develop a data-driven performance and maintenance cost model for the bridge’s life-cycle, to provide insights into the field of overall bridge management under uncertain circumstances, as well as highlight the significance of the usage of actual inspection and cost datasets in the accuracy of such models.

There are three primary steps in the entire process. Firstly, actual datasets regarding the bridge’s defects, condition states, as well as maintenance actions and costs were collected for further processing from 72 concrete bridges serving the Greek national highways. Secondly, considering the detected defects and the corresponding bridge’s conditions states, two discrete-time Markov-chain models were developed for the performance prediction of superstructure and substructure components over a 100-year period, with a one-year time step. Finally, the unit costs for restoring both examined bridge components to their initial condition were calculated, which allowed for the estimation of expected life-cycle maintenance costs. Ultimately, the probabilistic performance curves of the superstructure and substructure, along with the predicted life-cycle maintenance costs presented in this study, can assist asset managers in promoting more sustainable rehabilitation strategies, both structurally and economically. Since these costs have been projected over 100 years, they offer a comprehensive overview of the bridge’s total life-cycle expenses. This assessment highlights cost-effective solutions, enabling experts to eliminate unnecessary early maintenance actions and prevent severe damages that could be devastating in both economic and safety terms. Consequently, the exported deterioration curves and maintenance costs of the current study can serve as a baseline for implementing management policies for bridges constructed and operated under the same or similar conditions, such as similar construction specifications, geographical areas, and administrative authorities for ensuring reliable results. Additionally, several maintenance scenarios can be investigated using the exported life-cycle costs to identify the most optimal scenario while considering financial and safety constraints.

A practical limitation of this study is that the sample bridges have been in service for up to twenty-five years. As a result, the data collected only reflect the initial years of the bridges’ operation life. Therefore, future research will focus on implementing updating methods, such as Bayesian methods, that update the model in the light of new datasets, for even more realistic results. Moreover, with the passage of an even longer bridge lifespan that will allow for the collection of a sufficient volume of condition and maintenance datasets, non-homogeneous techniques, such as semi-Markov models, can be applied to capture the temporal variability of bridge degradation. Furthermore, bridge constitutes a multi-component system with different deterioration rates among its components. Hence, it would be fruitful to investigate more bridge components, such as bearings, deck etc., for exporting representative empirical probabilistic curves for each deterioration type.