1. Introduction
Infrastructure systems, such as transportation, energy, water, wastewater, telecommunications, healthcare facilities, financial systems, educational institutions, and emergency services, are essential for the continuous performance of modern society and the economy in ordinary times and during emergencies. Infrastructures are complex systems composed of structural and nonstructural components [
1]. Damage to a single component can lead to the disruption of the entire system, and the latter implies that the vulnerability of the infrastructure depends on its layout and functional–topological relationships. Therefore, when considering the seismic performance of the system, it is crucial to consider the vulnerability at the component level.
A severe seismic event can cause significant damage to infrastructure systems, resulting in severe direct and indirect consequences, as was recently catastrophically demonstrated in the Turkey–Syria earthquake [
2,
3]. These outcomes can trigger cascading and rippling effects across various sectors, leading to economic losses, physical destruction, and human casualties [
1]. Therefore, the resilience of infrastructures after seismic events is a critical aspect of ensuring the safety and functionality of communities.
Infrastructure resilience is an increasingly important multidisciplinary field that integrates technical, social, and economic dimensions to prepare for, respond to, and recover from disasters. Cimellaro et al. presented an analytical framework [
4] in which resilience is understood as a time-dependent recovery process influenced by societal preparedness and public policies. Rasulo et al. extended this to the seismic resilience of road networks, emphasizing the critical role of bridges in maintaining network functionality [
5]. Bocchini et al. [
6] argued that resilience and sustainability are complementary, both involving lifecycle analyses and social–economic impacts. Those works suggest that resilience is not just about recovery but also about adaptive evolution in the face of disruptions, making it a key consideration for both policy and engineering decisions.
Seismic fragility curves are a common method for assessing the expected damage of various types of infrastructure due to seismic events. The fragility curve represents the probability that a component or system will reach or exceed a given damage state as a function of an earthquake’s intensity-measure (IM) parameter, such as peak ground acceleration (PGA), peak ground velocity (PGV), and peak ground displacement (PGD) [
7]. Fragility curves can be used for individual components or the entire systems [
8,
9].
The general formulation of a fragility function of a structure or system is framed as a lognormal cumulative distribution function (CDF) [
10,
11,
12]. To accurately define this function, it is essential to determine two parameters: the median capacity of the component to resist the damage state (
) and the standard deviation of the capacity (
), as presented in Equation (1).
where
stands for a conditional probability of being at or exceeding a particular damage state
for a given seismic intensity and
is defined by the earthquake-intensity measure (
).
where,
: The uncertain damage state of a particular component, {0, 1, …};
: A particular value of the DS;
: The number of possible damage states;
: Uncertain excitation, the ground-motion-intensity measure (i.e., PGA, PGD, or PGV);
: A particular value of the IM;
: The standard cumulative normal distribution function;
: The median capacity of the component to resist a damage state ds measured in terms of the IM;
: The logarithmic standard deviation of the uncertain capacity of the component to resist a damage state ds.
In instances where more than one damage state is defined, the damage states are ordered by damage severity (from the least severe to the most severe damage), and the fragility function defines the cumulative probability of being in a specified damage state. Equation (2) expresses the distribution of probabilities of exceeding different levels of damage for a given IM value.
Many studies have focused on the development and application of seismic fragility curves for different types of infrastructure, including different types of buildings [
13,
14,
15] and special structures, such as churches [
16], bridges [
17], different steel tanks [
18,
19,
20], power grids [
21], water networks [
22], transportation infrastructure [
23], oil-pumping stations [
9], and concrete dams [
24]. The fragility curves are developed regarding various factors, such as the geometry of the elements, the materials, the overall capacity of the system, and several other factors. However, one aspect that has received less attention in developing seismic fragility curves is the integration of maintenance considerations with seismic resistance.
Maintenance of infrastructures is critical to ensuring durability, functionality, and effectiveness [
25]. Maintenance activities include routine inspections, required repairs and replacements, and upgrades to maintain the structural integrity, reliability, and performance of the infrastructure system. Several studies have explored different aspects of infrastructure maintenance, including planning and scheduling [
26,
27], maintenance expenditures [
28,
29], maintenance practice challenges [
30], monitoring, and climate-related disaster planning in asset management [
31].
The maintenance level of the infrastructure is a significant factor that can impact the vulnerability of the infrastructure system to seismic events [
32,
33]. Furthermore, the functionality of the infrastructure depends on the continuous performance of each component within the system. Therefore, the system components’ maintenance level can significantly impact the seismic vulnerability of the entire system. Proper maintenance can prevent deficiencies or wear and tear, ensure the system is in a suitable condition, and better resist seismic impact. In contrast, improper maintenance can increase the probability of failure and compromise the system’s seismic resistance. In addition, most maintenance practices often do not consider the vulnerability of the components to seismic events, resulting in a gap between maintenance operations and seismic-risk-reduction strategies. Thus, maintenance activities affect seismic performance and may foster resilience to seismic events.
Shohet [
34] introduced a Building Performance Indicator (BPI) in order to quantify the performance of an entire building, relying on the performance assessment of its particular systems and components. The methodology introduces the implementation of systematic rating scales to evaluate the condition of the building’s components and combining them using lifecycle cost principles. Subsequently, the overall state of the infrastructure is assessed via the BPI, which is derived from the weighted average of the scores attributed to the various building systems and components and their LCC significance in the overall building LCC. The BPI considers several criteria, such as the actual physical performance of the systems, the frequency of failures in building systems, and the actual preventive maintenance carried out on the building structure and systems.
Several studies aimed to consider maintenance parameters regarding seismic vulnerability. Manos et al. [
35] discussed maintenance issues related to the structural integrity of stone-masonry bridges. However, no analytical process was introduced. Crespi et al. [
36] investigated the seismic performance of reinforced-concrete bridges under several corrosion scenarios, as the corrosion levels represent the maintenance status. Zanini et al. [
37] analyzed the seismic vulnerability of corroded bridges in transport networks by developing fragility curves that accounted for steel-reinforcement corrosion. Soltani et al. [
38] presented the relationship between the maintenance cost and the engineering-demand parameters (EDPs) for the case of infill walls. Tecchio et al. [
39] intended to provide seismic fragility models for two generalized classes of single-span masonry arch bridges considering the material degradation and longitudinal cracks [
39]. It was found that the seismic fragility of masonry bridges increases when the effects of degradation are considered, as the loss of material was found to be the most influential defect. Moreover, an integrated approach that includes infrastructure maintenance was presented by [
40]. The paper proposed an integrated maintenance–safety framework, demonstrating a strong correlation between maintenance and safety levels. A case study of a public facility validated the framework, emphasizing unified maintenance–safety procedures to enhance facility performance.
Various analytical frameworks and indicators have been proposed to enable risk-informed decision-making for the seismic mitigation of critical infrastructure. Wang et al. presented a methodology integrating adjusted fragility curves into risk functions to evaluate mitigation strategies quantitatively [
41]. Furthermore, Urlainis and Shohet incorporated fragility analysis with fault-tree modeling to assess risk expectancy and proposed a Risk Mitigation to Investment Ratio indicator for prioritizing retrofitting alternatives based on risk-reduction cost-effectiveness [
42]. Wei et al. developed a benefit–cost analysis approach to evaluate the economic feasibility of seismic retrofitting in moderate-seismicity regions, demonstrating its application through a case study in Tiberias, Israel [
43]. These studies exemplify different tools and techniques to appraise seismic risk and guide mitigation decisions through analytical indicators.
Moreover, it should be noted that the risk assessment for infrastructures with more than one component is a complex task. Nuti et al. developed a model to evaluate the seismic fragility of electric power network components and the overall network capability considering component damage states, power flow, and soil conditions. The analysis emphasized the importance of accurate geotechnical modeling for predicting seismic response and safety [
44]. Furthermore, Nuti et al. (2010) discussed modeling approaches for the seismic risk assessment of large-scale-infrastructure networks, including electric power, water, and transportation systems. The analysis emphasized need for network-level modeling to capture component interactions and cascading failures. Case studies demonstrated Monte Carlo simulations for the probabilistic seismic analysis of networks [
45].
Rasulo et al. presented a modeling framework combining GIS, seismic risk analysis, and traffic simulation to assess direct and indirect earthquake impacts on road networks. The methodology was demonstrated through a case study of a bridge network in Central Italy, emphasizing the importance of calibrated traffic models for quantifying postseismic network accessibility and delays [
46].
This review sheds light on the gap between the analytical models in seismic resistance and the analytical–empirical models in maintenance that can be integrated into a comprehensive synergetic framework. Therefore, this research aims to establish fragility curves that integrate seismic and maintenance factors, thereby enabling a comprehensive performance methodology.
This study proposes a comprehensive approach toward earthquake-resilient infrastructures by incorporating maintenance factors into the seismic-risk-analysis process. By incorporating the maintenance-level data into the seismic fragility curves, this paper hypothesizes that an advanced, innovative, and reliable representation of the system vulnerability will be produced. By integrating maintenance considerations with seismic fragility curves, infrastructure owners and managers can make informed decisions regarding maintenance strategies and investments to enhance the resilience of their assets.
The uniqueness of our work lies in the pioneering integration of maintenance considerations with seismic fragility curves, a feature which is distinctly absent in the existing literature. This groundbreaking consolidation allows us to present a more comprehensive, holistic view of infrastructure resilience that goes beyond immediate seismic resistance to include long-term sustainability through effective maintenance. In traditional seismic fragility models, the focus is primarily on understanding how infrastructure responds to earthquakes without considering how ongoing maintenance activities can impact this response. Our integrated approach aims to bridge this gap.
4. Sensitivity Analysis
In this section, a sensitivity analysis is conducted. Sensitivity analyses assess how varying values of independent variables, such as the BPI score and P score, influence a specific dependent variable. In our context, the dependent variable of interest is the TRLC. Therefore, the sensitivity analysis is aimed at analyzing the BPI score and the P score.
In order to gain a comprehensive understanding of the BPI score’s influence, a set of five distinct Monte Carlo simulations were conducted, each with n = 500 trials. These simulations were performed for five different mean BPI-score values: 75, 80, 85, 90, and 95. In this set of simulations, the standard deviation was set at 20% of the mean BPI value, and any generated BPI values exceeding 100 were regenerated. In total, an additional 2500 simulations were carried out.
Figure 7 illustrates the sensitivity of the total risk to variations of the mean BPI. As the mean BPI ascends, there is a discernible decline in the total risk. This inverse relationship suggests that, as the building performance index improves (i.e., increases), the associated risk is reduced. This aligns with the expectation that buildings with superior performance metrics would likely possess a reduced risk of incurring damage or failure. Then, for a deeper understanding of the relationship, and to quantify the change in risk for a unit change in BPI, a regression model was executed. The linear regression model is demonstrated in
Figure 7, exhibiting an
value of 0.982. According to the regression model, each unit increase in BPI corresponds to a decrease in risk by USD 13,971. In other words, a change of one unit in the BPI will impact 0.1% of the TRLC. Additionally,
Figure 8 presents a box plot illustrating the distribution of the TRLC for each distinct BPI value. The figure presents the median risk, and as the BPI increases, the median risk decreases, aligning with our earlier findings. The height of each box represents the interquartile range (IQR), which is the interval between the 25th and 75th percentiles. The IQR remains relatively consistent across different BPI values, indicating that the spread or variability in risk is consistent. The points outside the whiskers represent potential outliers. It is noticeable that there are some outliers in the data. This indicated the higher possibility of extreme scenarios.
Analysis of the distribution of risks for each BPI value was carried out.
Table 5 and
Figure 9 present the distribution of the TRLC for different BPI values. It can be noticed that, as the BPI increases, there is a shift to the left in the distributions, indicating a decrease in the total risk, which aligns with our earlier findings from the regression analysis. This indicates that effective maintenance significantly mitigates the seismic risk, while lack of maintenance increases the seismic risk.
To analyze the P-score value, an additional five distinct Monte Carlo simulations were carried out, each consisting of n = 400 trials (a total of an additional 2000 simulations). These simulations were implemented for five different mean P-score values. These values were determined based on variations in a single factor, MP. The MP values were set to 1, 2, 3, 4, and 5, corresponding to P scores of 17.42, 21.13, 24.83, 28.54, and 32.25, respectively.
Table 6 portrays a summary of the total risk over the lifecycle (TRLC) statistics for the various P scores, detailing the mean, standard deviation, minimum, interquartile ranges, 99th percentile, and maximum values for each simulation set.
Figure 10 presents that the linear relationship between the parameter P and the mean total risk over the lifecycle (LC) is evident. An
value of near 1.00 indicates a very strong positive correlation. As the P score increases, there is a corresponding rise in the mean total risk over the LC. In addition, an increase of one unit of the P-score value will increase the TRLC by USD 12,071.
Figure 11 provides a detailed perspective of the risk distribution patterns associated with different P values. The figure highlights the variance shift in the TRLC as P increases. For lower P values, the total risk is relatively more concentrated, as shown by the narrower interquartile range (IQR). As P grows, the spread of the risk data becomes more expansive, indicating a broader dispersion and higher variability in the TRLC. This trend is particularly visible in the lengthening of the boxplots’ whiskers and the increased number of outliers at higher P values.
Figure 12 provides a comprehensive visualization of the risk distribution for the lifecycle across a range of P scores. Each histogram represents the frequency distribution of total risks associated with a specific P value. It is noticeable that the distribution of risks shifts as the P value changes; it fits to earlier findings.
5. Results and Discussion
The results from the Monte Carlo simulation illustrated a comprehensive analysis of the risk patterns for an infrastructure project spanning a 75-year design lifecycle. These results clarify the interaction between the BPI and P scores and their cumulative effect on the total risk over the lifecycle (TRLC). Specifically, it was observed that both the BPI and the P score possess statistically significant correlations with the TRLC. This finding indicates that, as the infrastructure’s performance enhances, there is a concurrent mitigation in the associated risk.
In addition to the primary simulation, a sensitivity analysis was undertaken to delve deeper into the specific influences of the BPI score and the P score on the total risk over the lifecycle (TRLC). This rigorous analysis aimed to recognize the individual and comparative impacts of these two parameters on the overall risk dynamics of the infrastructure project.
The sensitivity analysis relating to the BPI score revealed that, as the BPI score ascends, indicating an enhanced performance index of the infrastructure, there is a concurrent reduction in the associated risk. This trend signifies the inherent balance between infrastructure performance and the potential risks associated with it. A higher BPI score is synonymous with a better-performing infrastructure, and it is intuitively understood that better performance equates to reduced risks. However, the exact quantification and relationship were established through this analysis, enabling more informed decision-making.
The sensitivity analysis concerning the P score highlighted its strong correlation with the TRLC. A notable trend observed was that higher P values, indicative of deteriorating infrastructure conditions, were associated with increased seismic risk variance. Scenarios characterized by elevated P scores intrinsically possess a wider spectrum of potential risks. This suggests that infrastructures in poorer conditions come with greater uncertainties concerning potential risks. In practical terms, for stakeholders or decision-makers, a higher P value does not only translate to an increase in risk, but also signifies a heightened unpredictability in potential outcomes. This insight underscores the importance of robust risk-mitigation strategies, especially in high-P scenarios, to cater to the broader range of potential risks.