Multi-Scale Integrated Corrosion-Adjusted Seismic Fragility Framework for Critical Infrastructure Resilience

Abstract

This study presents a novel framework for integrating corrosion effects into critical infrastructure seismic risk assessment, focusing on reinforced concrete (RC) structures. Unlike traditional seismic fragility curves, which often overlook time-dependent degradation such as corrosion, this methodology introduces an approach incorporating corrosion-induced degradation into seismic fragility curves. This framework combines time-dependent corrosion simulation with numerical modeling, using the finite–discrete element method (FDEM) to assess the reduction in structural capacity. These results are used to adjust the seismic fragility curves, capturing the increased vulnerability due to corrosion. A key novelty of this work is the development of a comprehensive risk assessment that merges the corrosion-adjusted fragility curves with seismic hazard data to estimate long-term seismic risk, introducing a cumulative risk ratio to quantify the total risk over the structure’s lifecycle. This framework is demonstrated through a case study of a one-story RC moment frame building, evaluating its seismic risk under various corrosion scenarios and locations. The simulation results showed a good fit, with a 3% to 14% difference between the case study and simulations up to 75 years. This fitness highlights the model’s accuracy in predicting structural degradation due to corrosion. Furthermore, the findings reveal a significant increase in seismic risk, particularly in moderate and intensive corrosion environments, by 59% and 100%, respectively. These insights emphasize the critical importance of incorporating corrosion effects into seismic risk assessments, offering a more accurate and effective strategy to enhance infrastructure resilience throughout its lifecycle.

1. Introduction

Critical infrastructure (CI) comprises assets and systems that are essential for a society and its economy. CI is defined as systems and assets that are essential for the nation’s security, public health, and safety, as well as its economic stability [1]. CI includes sectors such as energy (power plants, electricity grids, oil and gas facilities), transportation (roads, bridges, railways, ports, airports), water supply and wastewater systems (pipelines, treatment plants, pumping stations, storage tanks), telecommunications, and healthcare (hospitals, emergency services). Additionally, CI encompasses financial services, government facilities, and emergency services. The interdependency among these sectors means that the failure of one system can cause cascading disruptions across the others. Over recent decades, the significance of these infrastructures has grown due to the proliferation of infrastructure systems, increased dependency across both private and public sectors, and the increasing interdependencies between various infrastructure sectors [2,3].

CI consists of complex and interconnected networks, components, and sub-components, creating a system where the failure of a single element can potentially trigger a cascade of failures [2,4,5], compromising the entire system and causing widespread damage across other sectors [6,7,8]. These characteristics underscore the critical importance of regularly maintaining and protecting these systems, especially during and after extreme events. Ensuring the resilience of CIs against various hazards, including natural disasters like earthquakes, is essential for continuous service delivery and mitigating the impact of disruptions.

Strong seismic events can cause severe damage to critical infrastructures. The impact of seismic events on CI is widespread and extends beyond the structural damage, as it can lead to extended service disruptions, compromised public safety, and significant economic loss [5,9]. To enhance the resilience of these essential systems, proactive measures, such as seismic retrofitting [10] and comprehensive maintenance strategies, are required [11].

A common tool for assessing the seismic vulnerability of CI is the use of seismic fragility curves. A fragility curve represents the probability that a structure or a component will exceed a certain damage state as a function of an earthquake’s intensity measure parameter [12]. Fragility curves are usually derived using the empirical approach or analytical approach. The empirical approach relies on the observations of post-earthquake damage to develop fragility curves [13,14,15,16,17,18], while the analytical approach uses computer simulations, like pushover and time–history analyses [19,20,21,22]. However, most of the traditional fragility curves are developed based on the structural response without explicitly accounting for actual conditions and long-term degradation processes, such as corrosion. These curves often assume that the structures are in a “like-new” or adequately maintained condition, failing to distinguish between structures in good condition and those suffering from degradation. This lack of distinction can lead to significant misinterpretation in the risk assessment process and potentially mislead the risk management strategies. By not accurately representing the varied conditions of infrastructure, particularly those degraded by factors like corrosion, fragility curves might underestimate the actual vulnerability, creating significant challenges for the effective planning and prioritization of mitigation strategies.

One of the primary components of CI is the physical building or structure that contains essential systems and operations. These structures are typically constructed using reinforced concrete (RC). RC buildings offer several advantages for CI, particularly regarding sustainability, seismic resilience, and cost-effectiveness [23,24,25]. Additionally, when properly designed and maintained, RC buildings are considered relatively suitable in coastal and marine environments due to their durability and resistance to harsh conditions [26].

However, despite the advantages of RC buildings, these structures and their elements are inherently aging and deteriorating over time. Several factors accelerate the deterioration process of RC elements, including environmental conditions [27], loading patterns and intensity [28,29], and poor maintenance [11]. This process leads to a reduction in their durability and to the reduction of their structural integrity.

One of the most significant deterioration factors is the corrosion of the steel bars within the RC elements [30,31,32]. Corrosion reduces the strength and ductility of steel bars and degrades the bond between steel and concrete [33]. Additionally, the corrosion of the steel bars increases the formation of cracks, leads to concrete cover expulsion, and decreases the concrete and steel strength [34,35]. These effects collectively result in the loss in structural capacity and heightened vulnerability to failure. It is also crucial to investigate the impact of extremely aggressive environments, such as brine attacks, as highlighted in [36].

1.1. Numerical Methods in Structural Analysis

One of the earliest and most widely used numerical analysis techniques in engineering is the finite element method (FEM). However, its effectiveness decreases when addressing problems with cracking, such as granular materials and joint rocks. To address the limitations of FEM in handling discontinuities and to incorporate fracturing phenomena, Munjiza et al. [37] developed the finite–discrete element method (FDEM), which addresses these issues. This approach integrates the FEM with the discrete element method (DEM). Furthermore, this method considers fracturing through the cohesive crack elements (CCEs) embedded within the finite element mesh to accurately model fracture processes in solids. A potential function is used to calculate the nodal forces generated by the interaction of overlapping discrete elements, which is a crucial feature of FDEM in its contact interaction algorithm [38].

This method enables the transition from a continuous to a discontinuous domain as fractures initiate and propagate. The intact material begins with a distributed micro-crack zone at the crack tip called a fracture process zone (FPZ), representing the initial damage (Figure 1a). This process then progresses to a bridging zone, where interlocking phenomena help distribute stresses across the cracks (Figure 1b). Finally, it transitions to a traction-free macro-crack zone, where the material completely separates. This sequence is marked by the Mode I crack walls beginning to separate between elements once the bond stress threshold (f_t) is reached (Figure 1c). As the separation grows, the bonding stress between the elements decreases accordingly, ultimately dropping to zero once the separation surpasses a critical threshold (O_r). Propagation under mode II fracture initiation occurs when the tangential slip at an integration point, denoted as s, attains a critical threshold s_p, which aligns with the element’s intrinsic shear strength f_s (Figure 1d). Lastly, crack elements in the FDEM can undergo mixed-mode fracture behavior (Figure 1) when Mode I and Mode II displacements interact (Figure 1e). The model continuously updates the stress and displacement fields to reflect the crack opening and the relative sliding of the faces.

We chose the FDEM model to study the capacity and fracturing behavior because it has been successfully validated, including the accurate replication of crack patterns observed in various tests [39,40,41]. Furthermore, FDEM effectively simulates the dynamic behavior and fracturing of CCEs by applying energy conservation principles. This principle involves the use of fracture energy (G_c), denoted as the energy necessary for a crack to propagate; frictional dissipation (E_f), representing the energy loss due to internal frictional forces; and kinetic energy (E_k), which accounts for the energy related to the motion of the elements. These energy components enable FDEM to model the fracturing process comprehensively, capturing the intricate interactions between the material properties, energy dissipation, and the dynamic changes within the system [42,43,44].

1.2. The Impact of Corrosion on Seismic Vulnerability

As a result of the loss in structural capacity and the increased vulnerability to failure, seismic vulnerability is correspondingly heightened. The effect of corrosion on seismic vulnerability has been investigated in the literature. For instance, Pitilakis et al. [45] investigated the seismic vulnerability assessment of RC buildings. Their study found that corrosion-induced aging and soil–structure interaction (SSI) significantly increase the seismic vulnerability of RC structures over time. Additionally, Dizaj et al. [46] developed a probabilistic framework for the seismic vulnerability analysis of corroded reinforced concrete frames. Their study highlighted that disregarding the progressive nature of damage and the changes in structural performance over time underestimates the probability of failure of corroded RC frames.

Several studies have also investigated the effect of corrosion on bridges; [47,48] found that corrosion significantly deteriorates the structural integrity of RC bridges by causing cross-sectional loss, ductility degradation, and the compressional buckling of steel rebar. A similar effect was also found in the case of prestressed concrete bridges, where the corrosion-induced degradation of tendons significantly compromises the flexural and shear capacity of bridge cross-sections [49]. Crespi et al. (2022) further demonstrated that moderate corrosion can significantly alter the seismic vulnerability of RC bridges over time, requiring more proactive maintenance and retrofitting strategies [50]. De Domenico et al. [51] found that seismic safety margins could be reduced by up to 40% due to the corrosion-induced mass loss in steel bars. Furthermore, the cumulative effects of the corrosion and repeated earthquake shocks further degrade highway bridges over time, highlighting the necessity for proactive maintenance and comprehensive seismic assessments [52].

Furthermore, the impact of corrosion was also observed in the case of shallow circular tunnels. Argyroudis et al. [53] examined the effects of soil–structure interaction (SSI) and lining corrosion on the seismic vulnerability of shallow circular tunnels, highlighting the significant impact of these factors on tunnel resilience. Their findings emphasize that corrosion must be considered together with the SSI to ensure the long-term durability and safety of tunnel structures.

As demonstrated by the literature, corrosion significantly increases the seismic vulnerability of CI by deteriorating the structural integrity of RC elements. This degradation impacts the overall structural capacity of CI, making the CI systems more vulnerable to damage during seismic events.

Furthermore, corrosion is a time-dependent process [54,55], meaning the impact of corrosion on structural capacity constantly increases as the RC elements age. This highlights the importance of ongoing, time-dependent assessment and the monitoring of RC elements to obtain a comprehensive understanding of the seismic risk to CI.

1.3. Research Objective

Understanding that corrosion has a significant impact on the seismic vulnerability of CI, this paper aims to present a conceptual framework for evaluating seismic risk, accounting for the time-dependent degradation effects of corrosion on reinforced concrete (RC) elements. This research addresses a crucial gap in current seismic vulnerability assessments, which often overlook the progressive deterioration of structural integrity due to corrosion processes. This study aims to

• Develop a methodology for simulating the progression of corrosion in RC structures over time, considering different corrosion intensities;
• Quantify the impact of corrosion-induced degradation on the seismic performance of CI components, particularly focusing on changes in structural capacity;
• Integrate corrosion effects into seismic fragility analyses by proposing adjustments to traditional fragility curves, thereby providing a more accurate representation of the structural vulnerability throughout the infrastructure’s lifecycle;
• Establish a risk assessment framework that combines adjusted seismic fragility with hazard data to estimate the expectancy of the total lifecycle risk cost, considering the corrosion-induced deterioration.

The significance of this research lies in its potential to enhance the accuracy and reliability of long-term seismic risk assessments for critical infrastructure. By incorporating the effects of corrosion, this study provides a more realistic evaluation of infrastructure vulnerability, crucial for informing maintenance strategies, improving resource allocation, and enhancing the safety and reliability of essential services, particularly in seismically active regions with corrosive environments.

2. Methodology

The goal of this research is to address the corrosion progress in the risk assessment analysis. In order to achieve this goal, the methodology consist of four main steps:

• Simulation of corrosion progress;
• Analysis of corrosion-induced degradation;
• Integrating corrosion effects into seismic fragility curves;
• Executing risk assessments for infrastructure seismic resilience.

Simulation of Corrosion Progress

Within this section, we present a comprehensive modeling framework designed to evaluate the degradation of the structural capacity for a wall subjected to extreme seismic events and the effects of corrosion, which vary based on the distance from the sea and are represented by different corrosion levels, as outlined in the following steps:

1.

The first step in the modeling framework involves selecting the corrosion level. In this study, we based our selection on the criteria provided by Raoul François [56], as shown in

Table 1. Ranges of corrosion and penetration rates with their corresponding corrosion levels, as defined by [56].

Icorr [µA/cm2]	Vcorr [mm/Year]	Corrosion Level
≤0.1	≤0.001	Passive
0.1–0.5	0.001–0.005	Minor
0.5–1	0.005–0.010	Moderate
>1	>0.010	Intensive

. The table below outlines the different levels of penetration and the corrosion ranges.

We selected three levels for simulation based on the data from Table 1: (1) minor, with i = 0.2; (2) moderate, with i = 1; and (3) intensive, with i = 3.5;

2.

Calculating the time-dependent parameters from 0 to 120 years, considering a time interval (∆t) of 20 years. This time frame represents the period required for the corrosion products to transition from passive to active. Figure 2 illustrates this time-dependent corrosion process, depicting the progressive increase in corrosion levels as a function of time. It visually represents how corrosion intensifies, from the initial passive state to more severe conditions;

3.

The next step involves calculating the confinement effect of the concrete on the corrosion products, which leads to internal pressure (P) as the corrosion progresses. Figure 2 also captures this internal pressure buildup over time, correlating it with the varying levels of corrosion rates;

4.

The internal pressure (P) and other time-dependent parameters calculated in step 3 are then input into the model;

5.

Boundary conditions are applied as follows:

a. Internal pressure (P);

b. Shear stress up to failure;

c. Confinement conditions at the y- direction (back face).

These boundary conditions represent the continuum condition, as the model simulates a cube cut-off from the wall, replicating the localized effects of corrosion and shear stress within the structural system;

6.

Steps 3 and 4 are repeated for each time interval to assess the effects of varying corrosion levels over time;

7.

All models are simulated using an automated algorithm, allowing for the systematic evaluation of the time-dependent structural degradation;

8.

After the simulations, we analyze the capacity curves, crack formation, and structural response under seismic events to evaluate the combined impact of corrosion and seismic forces on the structural performance. This analysis provides insights into how the degradation of the wall due to corrosion affects its ability to resist shear forces during extreme seismic events.

Figure 2 provides a graphical abstract of how the corrosion process evolves and affects the structural integrity at different stages. This methodology offers a more accurate and evolving representation of the structural vulnerability, providing a comprehensive long-term infrastructure resilience planning tool.

3. Framework Development

3.1. Mechanical Model

Given the significant computational costs, simulations were limited to a small “cut-off” section of the wall. A concrete cube measuring 116 × 116 × 116 mm³ was extracted for simulation and analysis from the base of the wall, where the sum of the shear forces is the highest.

For simplicity, the impact of the steel embedded in the concrete (rebar stiffness) was omitted in the model. However, radial pressure resulting from the corrosion of corroded steel was integrated into the model. This pressure, derived from Equation (18), was applied around a “cylindrical hole” at the external meshing points (see Figure 3). The pressure simulates the expansion of rust, particularly around the interface layer of the steel, which creates a significant force on the surrounding concrete, leading to internal stresses within the boundary layer. These stresses, represented as radial pressure in the simulation, can induce cracking and spalling in the concrete, ultimately weakening the wall’s structural integrity.

This model represents the seismic event as shear stress (τ) applied on the shear faces of the cube. Furthermore, the face connected to the back side of the wall is confined in the y-axis due to the continuum connection to the wall, and acts as a roller in the other directions. Finally, the remaining surface is left free, with no applied forces or continuous conditions, as shown in Figure 1. The shear stresses are applied incrementally until failure by applying velocity conditions in the shear direction (note that the self-weight of a single wall, compared to the shear forces generated by a seismic event, is negligible, and the rotation of the cube section is also relatively small based on Mohr’s circle analysis). The simulation process is repeated every ten years, during which the internal pressure is recalculated with increasing values, and degraded mechanical parameters are assigned. It is important to note that we considered the limitation that corrosive products become active after 20 years, as suggested by [57,58], and therefore, degraded parameters are only applied after this period. Since we modeled the time intervals in years discretely/discontinuously, applying shear forces continuously until failure at each time interval, it is difficult to transfer the material properties degraded by cracks from one-time interval to the next. We relied on the empirical equations presented in Equations (1)–(8) to address this limitation. While this approach provides a reasonable approximation, it highlights the potential for future work to enable the continuous transmission of material property information across time steps.

3.2. Time-Dependent Parameters

Concrete compressive strength can decrease when affected by the extent of cracking caused by corrosive products reacting on the concrete surface and the steel bar. Despite extensive research, quantifying this loss and identifying the exact degradation region through analytical methods remains a considerable challenge [59,60]. We adopted the analytical expression from Vecchio and Collins (1986) [61]. In this analysis, the time-dependent deterioration of the material’s properties is primarily driven by the corrosion penetration depth d_e (Equation (5)), which evolves as a function of time t (Equation (5)), then influencing the crack width w_c (Equation (4)). The crack width, in turn, determines the expansion of the beam thickness b_f (Equation (3)), which directly impacts the tensile strain ε₁ (Equation (2)). Finally, the modified compressive strength f_ck^∗, as shown in Equation (1), is a function of the strain and therefore indirectly dependent on the corrosion penetration depth d_e, linking it back to time t.

Moreover, other mechanical properties, such as those described in Equations (6)–(9), depend on the modified compressive strength f_ck^∗, which deteriorates over time as corrosion advances. The time-dependent degradation of f_ck^∗, influences parameters such as elastic modulus, tensile strength, and other material characteristics that evolve as corrosion weakens the structural integrity over long periods.

$$f_{ck}^{\mathrm{*}}={\displaystyle \frac{f_{ck}}{1+k_r\frac{{\epsilon }_1}{{\epsilon }_{co}}}},$$

(1)

where k_r (0.1 for medium-diameter ribbed bars) relates to bar roughness and diameter [62], ε_co (0.002) is the strain at peak compressive stress as provided by the IS 456 [63]; the material is subjected to compression, but ε₁ represents the tensile strain under this compressive load for a cracked material, given by Equation (2).

$${\epsilon }_1={\displaystyle \frac{b_f-b_o}{b_0}},$$

(2)

where b₀ denotes the width of the cross-section in the absence of corrosion cracks, and b_f denotes the thickness expanded due to corrosion-induced cracking. The approximate increase in beam thickness can be determined using Equation (3).

$$b_f-b_o=n_{bars}{w}_c,$$

(3)

where n_bars denotes the no. of bars in each layer, and $w_c$ denotes the width of the crack opening corresponding to a specific level of corrosion over the period of time Δt, as expressed by Equation (4).

$$w_c={2\pi (\upsilon }_{rs}-1)d_e,$$

(4)

where ν_rs denotes the volumetric expansion ratio of oxides compared to its intact state, typically assumed to be 2 [64]; d_e denotes the penetration depth of the corrosion, expressed by the following determination using Equation (5).

$$d_e=0.0115i_{co}t,$$

(5)

where i_co denotes the corrosion current density per cm/per year, and t denotes the time passed since the onset of corrosion propagation (years). The elastic modulus (E_c) of concrete is related to its compressive strength (f_ck) at each lifetime stage, following the theory proposed by Noguchi and Nemati [65] (Equation (6)).

$$E_c=2.1\times {10}^5{\left({\displaystyle \frac{\gamma }{2.3}}\right)}^{1.5}{\left({\displaystyle \frac{f_{ck}}{200}}\right)}^{0.5},$$

(6)

where the modulus of elasticity of concrete (E_c) is measured in kgf/cm², γ denotes the density of concrete (held constant at 2.5 ton/m³), and f_ck denotes the compressive strength characteristics, also in kgf/cm².

The ACI 318 [66], as also mentioned in [67,68], provides accurate estimates for direct tensile strength of specimens where size effect is negligible. We followed Equation (7) to estimate the tensile strength f_t that is correlated to its compressive strength (f_ck) and in this context, it is defined at each time step using the following relation:

$$f_t=0.33\lambda \sqrt{f_{ck}}(\mathrm{M}\mathrm{P}\mathrm{a} \mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{s})$$

(7)

where λ accounts for lightweight concrete and will be assumed as 1.0 in this paper.

The cohesion strength c and the normal stress σ_n in FDEM, incorporated as described in the following method [69]:

$$f_s=c+{\sigma }_n\tan \left({\theta }_i\right)$$

(8)

where θ_i denotes the friction angle, and f_s is the intrinsic shear strength. The proportional shear strength is approximately 1/2–1/3 power of the compressive strength [70].

$$f_s=\sqrt{f_{ck}}$$

(9)

3.3. Internal Pressure (P)

Various models are available in the literature for predicting rust production. Andrade et al. (1993) [71] introduced a linear relationship integrating Faraday’s law, correlating the current density of corrosion (i_corr in µA/cm²) to the decrease in the steel bar diameter. A conversion factor of 0.023 (from µA/cm² to mm per year) is used:

$$D_{rb}=D_b-0.023i_{corr}∆\mathrm{t}\left(\mathrm{m}\mathrm{m}\right)$$

(10)

where, D_rb, denotes the residual steel diameter, D_b denotes the initial steel diameter, and Δt denotes the time since the onset of corrosion.

$${∆\mathrm{V}}_s=0.023\times 2\pi D_b{i}_{corr}∆\mathrm{t}({\mathrm{m}\mathrm{m}}^3/\mathrm{m}\mathrm{m})$$

(11)

Assuming a uniform corrosion process, the radius decreases from R_b to R_rb as a result of corrosion (as described in Equation (12)). This decrease has been calculated based on the volume of consumed steel per unit length of anodic steel, ${∆\mathrm{V}}_s$ (as detailed in Equation (11)).

$$R_{rb}=\sqrt{(R_b^2-{\displaystyle \raisebox{1ex}{${∆\mathrm{V}}_s$}\!\left/ \!\raisebox{-1ex}{$\pi $}\right.}}(\mathrm{m}\mathrm{m})$$

(12)

The steel radius, including the rust layer, is given as follows:

$$R_r=R_{rb}+{\delta }_o(\mathrm{m}\mathrm{m})$$

(13)

where δ_o denotes the thickness of the oxide layer around the steel surface. The total volume of oxide produced is expressed as $∆V_r=∆V_s{\rho }_s/\left({\rho }_r{r}_m\right)$, and the thickness t_r can be calculated as follows:

$$t_r=\sqrt{(R_{rb}^2+{\displaystyle \raisebox{1ex}{${∆\mathrm{V}}_r$}\!\left/ \!\raisebox{-1ex}{$\pi $}\right.}}{-R}_{rb}\left(\mathrm{m}\mathrm{m}\right)$$

(14)

By applying an equivalent pressure around the steel, the expansion of the corroding reinforcement is simulated. The first step in this process is to calculate the effective, non-dimensional mass loss:

$$\gamma ={\displaystyle \frac{\frac{{(D_b+{\delta }_o\times 2)}^2}{D_b^2}-1}{\beta -1}}$$

(15)

The strain resulting from the unrestrained expansion of the reinforcement can be determined as follows:

$${\epsilon }_{s,free}=\sqrt{1+c(b-1)}-1$$

(16)

This represents the average deformation of the corrosion area relative to the original steel diameter and changes with the corrosion levels. The confinement effect of the concrete on the steel generates internal stress, which is calculated by multiplying the average stiffness by the strain of the corroded system. To determine the average stiffness, it can be based on the volume fractions of both the steel and the layers of corrosion:

$$E_{s,eq}={\displaystyle \frac{1+c(b-1)}{\frac{1-c}{E_s}+\frac{cb}{E_o}}}(\mathrm{G}\mathrm{P}\mathrm{a})$$

(17)

where E_s denotes the modulus of elasticity of steel (200 GPa), and E₀ refers to the modulus of the elasticity of the oxide in this work taken as (7 GPa).

The internal pressure (P) denotes the confinement effect of the concrete on steel, dependent on the degree of corrosion, calculated in the following equation:

$$P=E_{s,eq}\times {\epsilon }_{s,free}$$

(18)

The mechanical parameters throughout the structure’s design life cycle, calculated using Equations (1)–(17), are presented in Figure 4.

In Figure 4, we show the mechanical parameters that govern the degradation of the structure over time due to corrosion, recalculated every 10 years to reflect increasing corrosion severity. After 20 years, corrosion becomes active, significantly decreasing the structure’s ability to resist stresses. The modulus of elasticity (Figure 4a) decreases over time, particularly under moderate and intensive corrosion, leading to a loss in stiffness and increased vulnerability to deformation under seismic shear forces. The cohesive strength (Figure 4b), which governs the bonding between elements, reduces the structure’s capacity to resist tensile and shear forces. Similarly, tensile stress (Figure 4c) decreases as cracks form due to corrosion, with an accelerated degradation after 20 years under intensive corrosion, impacting crack propagation during seismic events.

Additionally, the friction angle (Figure 4d) increases over time, indicating a shift towards more brittle behavior, particularly under intensive corrosion where material failure becomes more abrupt during seismic loads. As depicted in Figure 4, these time-dependent degradation patterns are critical for the model’s simulations. These parameters will affect the structure’s ability to resist seismic forces, as presented in Section 3.4.

3.4. Analysis of Corrosion-Induced Degradation

After assigning the mechanical parameters to the models and applying the boundary conditions, each model was subjected to increasing shear stress until failure. Crack patterns were observed at the capacity stress level during the structure’s lifecycle, captured at each time interval on the surface where the rebar appears as circular features, illustrating the various stages of this process.

Figure 5 illustrates the progression of cracking on the concrete surface surrounding the rebar at various stages in the structure’s life cycle, with the estimated cracking relative to the total cross-sectional area increasing over time. Using the areal fracture intensity method, we quantified the extent of fracturing to assess its impact. Using the measurement technique, developed by [72], to quantify fractures in discrete fracture analysis. The intensity of the areal fracture, denoted as P₂₁, is the ratio of the total length of the fracture traces to the cross-section (see for a schematic measurement of the fracture trace).

Figure 6 shows that in the initial stages (t = 0–30 years), P₂₁ is 0.000469 mm/mm², indicating minimal cracks. As time progresses (t = 50 years), early-stage cracking increases to P₂₁, reaching 0.0154 mm/mm², showing visible cracks forming around the rebar. By the mid-stage of the structure life cycle (t = 60 years), the cracking extends further from the rebar P₂₁, increasing to 0.0255. In the advanced stage (t = 70 years), the cracks spread significantly, with P₂₁ reaching 0.0538 mm/mm². This trend continues into the severe stage (t = 90 years), where the cracking area grows to P₂₁ = 0.073 mm/mm², reflecting severe structural degradation due to ongoing corrosion and mechanical stress.

On the free surface, in the shear direction, shear cracks are observed, as depicted in Figure 7.

Figure 7 depicts the development of these shear cracks on the free surface in the shear direction, distinct from the corrosion-induced cracking visible on the cross-section where the rebar appears as a circle.

Next, we evaluated the shear capacity for each model from the resulting shear stress-displacement curves. The models represent different stages in the structure’s lifecycle, where the shear capacity at time t_i and t_i+1 was divided by the shear capacity at t = 0, thereby determining the deterioration of capacity over time, as depicted in Figure 8.

Figure 8 illustrates the degradation of shear capacity over time for a wall subjected to a seismic event, considering three different levels of corrosion: intensive, moderate, and minor. The results demonstrate a clear trend of decreasing shear capacity as corrosion advances, with the reduction rate varying significantly across corrosion levels, ranging from a 20 percent reduction to an over 80 percent loss in capacity at 120 years. As the primary purpose is to achieve material degradation while keeping the simulations feasible, we implemented several simplifications in the modeling approach. We modeled only a small section out of the wall, reducing computational expense while providing insights into shear capacity and localized corrosion effects. Although the direct impact of the reinforcing steel on the concrete’s structural behavior was ignored, the steel’s corrosive pressure action on the concrete was accounted for, recognizing its significant role in the degradation process. The simulations were conducted discontinuously, applying shear forces incrementally until failure at each time step, with material strength reduction over time based on empirical equations that approximate the degradation behavior without requiring a fully continuous model. Despite these simplifications, the results capture the overall trend of material degradation, providing sufficient information for maintenance planning and structural health monitoring, and emphasizing the importance of timely maintenance interventions to preserve structural integrity.

3.5. Integrating Corrosion Effects into Seismic Fragility Curves

In this step, the degradation of the reinforced concrete (RC) is integrated into seismic vulnerability assessments. This step is based on the degradation of the RC capacity as found in the previous step.

In this research, seismic vulnerability is assessed using seismic fragility curves. A seismic fragility function for a structure or a system is mostly formulated as a lognormal cumulative distribution function (CDF) [73,74]. In order to define this function, two parameters are required to be determined: the median capacity of the component to resist a given damage state ${\mathsf{\theta }}_{\mathrm{d}\mathrm{s}}$ and the standard deviation of this capacity ${\mathsf{\beta }}_{\mathrm{d}\mathrm{s}}$. The form of this function is shown in Equation (19).

$$\mathrm{P}\left[\mathrm{D}\mathrm{S}\ge \mathrm{d}\mathrm{s}|\mathrm{I}\mathrm{M}=\mathrm{x}\right]=\Phi \left(\frac{\ln \left(\mathrm{x}/{\mathsf{\theta }}_{\mathrm{d}\mathrm{s}}\right)}{{\mathsf{\beta }}_{\mathrm{d}\mathrm{s}}}\right);\mathrm{d}\mathrm{s}\in \{\mathrm{1,2},\dots {\mathrm{N}}_{\mathrm{D}\mathrm{S}}\}$$

(19)

where P represents the conditional probability of being at or exceeding a particular damage state (DS) for a given seismic intensity, and x is defined by the earthquake intensity measure (IM) where

$\mathrm{D}\mathrm{S}$	The damage state of a particular component, {0, 1, … ${\mathrm{N}}_{\mathrm{D}\mathrm{S}}$};
$\mathrm{d}\mathrm{s}$	A particular value of the DS;
${\mathrm{N}}_{\mathrm{D}\mathrm{S}}$	The number of possible damage states;
$\mathrm{I}\mathrm{M}$	Uncertain excitation, the ground-motion-intensity measure (i.e., PGA, PGD, or PGV);
$\mathrm{X}$	A particular value of the IM;
$\Phi$	The standard cumulative normal distribution function;
${\mathsf{\theta }}_{\mathrm{d}\mathrm{s}}$	The median capacity of the component to resist a damage state ds measured in terms of the IM;
${\mathsf{\beta }}_{\mathrm{d}\mathrm{s}}$	The logarithmic standard deviation in the uncertain capacity of the component to resist a damage state ds.

When multiple damage states are defined, they are arranged in order of severity, from the least severe to the most severe. The fragility function then represents the cumulative probability of reaching or exceeding a specific damage state.

$$\mathrm{P}\left(\mathrm{D}\mathrm{S}=\mathrm{d}{\mathrm{s}}_{\mathrm{i}}|\mathrm{I}\mathrm{M}\right)=\left\{\begin{array}{c}1-P\left(\mathrm{D}\mathrm{S}\ge \mathrm{d}{\mathrm{s}}_{\mathrm{i}}|\mathrm{I}\mathrm{M}\right)\\ P\left(\mathrm{D}\mathrm{S}\ge \mathrm{d}{\mathrm{s}}_{\mathrm{i}}|\mathrm{I}\mathrm{M}\right)-P\left(\mathrm{D}\mathrm{S}\ge \mathrm{d}{\mathrm{s}}_{\mathrm{i}+1}|\mathrm{I}\mathrm{M}\right) \\ P\left(\mathrm{D}\mathrm{S}\ge \mathrm{d}{\mathrm{s}}_{\mathrm{i}}|\mathrm{I}\mathrm{M}\right)\end{array}\right.\begin{array}{c}\mathrm{i}=0\\ 1\le \mathrm{i}\le \mathrm{n}-1\\ \mathrm{i}=\mathrm{n}\end{array}$$

(20)

• Baseline fragility curve

The baseline fragility curves provide a probabilistic measure of the structure’s seismic vulnerability without corrosion and degradation consideration. The baseline curves parameters are based on the parameters that are available in the literature and common databases such as [75,76,77]. These curves are set as a starting point before the integration degradation effects;

2.

Adjusted Fragility Curve

In this step, the baseline seismic fragility curves are adjusted to account for the impact of corrosion on the structure;

3.

Adjustment of Median Capacity ${\mathsf{\theta }}_{{\mathrm{d}\mathrm{s}}_{\mathrm{i}}}$ Due to Corrosion

The adjustment is made using the degradation factor, denoted as ${\mathrm{f}}_{\mathrm{C}\mathrm{L}}\left(\mathrm{t}\right)$, which quantifies the reduction in structural capacity due to corrosion relative to the baseline parameters. The degradation factor depends on the corrosion level (CL) and the time (t) elapsed, and it is determined using analyses conducted in prior steps of this research, where the impact of corrosion on structural elements was thoroughly evaluated (as illustrated in Figure 2). The degradation factor ${\mathrm{f}}_{\mathrm{C}\mathrm{L}}\left(\mathrm{t}\right)$ is applied to the baseline median capacity ${\mathsf{\theta }}_{{\mathrm{d}\mathrm{s}}_{\mathrm{i}}}$ to yield the adjusted median capacity ${\mathsf{\theta }}_{{\mathrm{d}\mathrm{s}}_{\mathrm{i}}}^{\prime }\left(\mathrm{t}\right)$.

$${\mathsf{\theta }}_{{\mathrm{d}\mathrm{s}}_{\mathrm{i}}}^{\prime }\left(\mathrm{t}\right)={\mathsf{\theta }}_{{\mathrm{d}\mathrm{s}}_{\mathrm{i}}}·{\mathrm{f}}_{\mathrm{C}\mathrm{L}}\left(\mathrm{t}\right)$$

(21)

where

${\mathsf{\theta }}_{{\mathrm{d}\mathrm{s}}_{\mathrm{i}}}^{’}\left(\mathrm{t}\right)$	The adjusted median capacity of the structure to resist a specific damage state i at time t, accounting for the effects of corrosion over time;
${\mathsf{\theta }}_{{\mathrm{d}\mathrm{s}}_{\mathrm{i}}}$	The baseline median capacity of the structure to resist the damage state i without the corrosion consideration;
${\mathrm{f}}_{\mathrm{C}\mathrm{L}}\left(\mathrm{t}\right)$	The degradation factor, a function of time t, quantifying the reduction in structural capacity due to corrosion, based on corrosion level (CL) and elapsed time;
$\mathrm{t}$	The time in years since the beginning of the structure’s exposure to corrosive conditions;
$\mathrm{C}\mathrm{L}$	The corrosion level, which represents the severity of the corrosion affecting the structure. The corrosion level can be categorized (e.g., minor, moderate, intensive).

To enhance the reliability of our research and substantiate the simulation outcomes, the findings regarding the degradation rate were juxtaposed with the results presented by Cui et al. [78]. Their study delineated deterioration rates for RC bridges’ fragility curves over five distinct time spans within a marine environment, which is typically considered a setting of intensive corrosion. The fragility curves reported by Cui et al. [78] encompassed durations of 0, 25, 50, 75, and 100 years, providing a robust framework for comparison given the similar environmental conditions.

The comparative analysis revealed that for all damage states across the time spans of 25, 50, and 75 years, our results aligned closely with those of Cui et al., exhibiting a deviation ranging between 3% and 14%. This similarity indicates a significant validation of our simulation techniques and methodologies, underscoring the accuracy of our degradation rate estimations within the expected margins. However, it should be noted that for the 100-year time span, our results were consistently more conservative. This discrepancy suggests that the actual vulnerability of the structures might be higher than our models predict. Such an underestimation is attributed to the simplifications considered in our modeling approach, as mentioned in Section 3.1. Acknowledging this potential underestimation is crucial for refining future models and enhancing the predictive accuracy of our research in assessing the long-term impacts of corrosion on structural integrity. This awareness allows for the continuous improvement of our methodologies, ensuring more reliable and accurate predictions in future studies.

3.6. Adjustment of Dispersion Parameter (${\beta }_{ds}$) Due to Corrosion

As part of the overall procedure for integrating corrosion effects into seismic fragility assessment, this step focuses on adjusting the dispersion parameter to account for additional uncertainties introduced by corrosion. In the development of seismic fragility curves, the dispersion parameter ${\mathsf{\beta }}_{\mathrm{d}\mathrm{s}}$ helps to capture the uncertainty associated with the structural response to seismic events. To accurately reflect the impact of corrosion on structural capacity, it is proposed to adjust this parameter as well, in order to consider the additional uncertainty that was caused by the corrosion process and this analysis. The adjustment of ${\mathsf{\beta }}_{\mathrm{d}\mathrm{s}}$ in this process is aligned and based on the principles outlined in FEMA P-58 [77], which emphasizes the importance of considering uncertainties in seismic performance assessments to ensure a more comprehensive and realistic evaluation of structural vulnerability.

The uncertainty in the structural response is heightened due to the reliance on previous studies, which included experimental research with inherent variability in material properties. These studies form the basis for the simulation models used in this research, and the variability in their results must be accounted for in the fragility analysis. In addition, corrosion leads to variability in structural performance due to several factors, including non-uniform degradation of materials and differences in construction quality. These factors contribute to increased uncertainty in the structural response, which is typically not captured by the baseline fragility curve parameters.

To incorporate this additional uncertainty, the adjusted dispersion parameter ${\mathsf{\beta }}_{\mathrm{d}\mathrm{s}}^{\prime }$ is calculated using the root–sum–square (RSS) method, combining the baseline ${\mathsf{\beta }}_{\mathrm{d}\mathrm{s}}$ with the additional uncertainty attributed to corrosion ${\mathsf{\beta }}_{\mathrm{c}\mathrm{o}\mathrm{r}}$.

$${\mathsf{\beta }}_{\mathrm{d}\mathrm{s}}^{\prime }=\sqrt{{{\mathsf{\beta }}_{\mathrm{d}\mathrm{s}}}^2+{{\mathsf{\beta }}_{\mathrm{c}\mathrm{o}\mathrm{r}}}^2}$$

(22)

Here, ${\mathsf{\beta }}_{\mathrm{d}\mathrm{s}}$ represents the original dispersion derived from the baseline fragility analysis, while ${\mathsf{\beta }}_{\mathrm{c}\mathrm{o}\mathrm{r}}$ represents the uncertainty introduced by the corrosion process. The resulting ${\mathsf{\beta }}_{\mathrm{d}\mathrm{s}}^{\prime }$ is then applied in the fragility curve equations to provide a more accurate representation of the seismic vulnerability of corroded structures. This adjustment ensures that the fragility curves account for the increased variability in structural performance, leading to a more realistic assessment of seismic risk.

It should be noted that the adjustment of ${\mathsf{\beta }}_{\mathrm{d}\mathrm{s}}$ is not mandatory, but it is suggested to account for the increased variability introduced by the proposed corrosion analysis. The value of ${\mathsf{\beta }}_{\mathrm{c}\mathrm{o}\mathrm{r}}$ represents the additional uncertainty introduced by the corrosion process. A standard value of 0.25 is recommended as suitable for most cases. However, if a more detailed analysis is required, the suggested ranges are elaborated in

Table 2. Suggested ranges for ${\mathsf{\beta }}_{\mathrm{c}\mathrm{o}\mathrm{r}}$.

Uncertainty Level	Description	${\mathsf{\beta }}_{\mathbf{c}\mathbf{o}\mathbf{r}}$ Range
Low Uncertainty	Corrosion is relatively uniform and well understood.	0 to 0.2
Moderate Uncertainty	Cases with moderate variability, where corrosion rates vary due to environmental conditions.	0.2 to 0.3
High Uncertainty	Corrosion is highly variable and unpredictable, such as in structures with mixed materials or inconsistent construction quality.	0.3 to 0.5, or more

.

3.7. Executing Risk Assessment for Infrastructure Seismic Resilience

In the final step, the proposed risk assessment process is executed. This step is followed by the process that was introduced in [11]. In this step, the seismic risk expectancy of the infrastructure is evaluated, considering the influence of corrosion. The cumulative risk expectancy over the lifespan of the infrastructure, denoted as the total risk lifecycle cost for a T-years lifespan (${\mathrm{T}\mathrm{R}\mathrm{C}\mathrm{L}}_{\mathrm{T}}$), is calculated using Equation (23). This equation provides a comprehensive assessment of the overall risk to the system due to seismic events for its design life. The ${\mathrm{T}\mathrm{R}\mathrm{C}\mathrm{L}}_{\mathrm{T}}$ calculation incorporates various potential seismic scenarios, their respective probabilities of occurrence, and the expected consequences associated with each scenario. The parameter ${\mathrm{R}}_{\mathrm{U}}$ represents the total expected consequences in the event of complete system failure, which is quantified in monetary terms (US$). This comprehensive approach to risk assessment allows for an accurate evaluation of the potential impacts of seismic events on the infrastructure.

$${\mathrm{T}\mathrm{R}\mathrm{C}\mathrm{L}}_{\mathrm{T}}=\left[{\sum }_{\mathrm{t}=1}^{\mathrm{T}}{\sum }_{\mathrm{m}=1}^{\mathrm{I}\mathrm{M}}\left({\sum }_{\mathrm{i}=1}^{\mathrm{N}}\mathrm{P}\left(\mathrm{d}{\mathrm{s}}_{\mathrm{i}}|\mathrm{I}\mathrm{M}\right)·\mathrm{D}{\mathrm{R}}_{\mathrm{d}{\mathrm{s}}_{\mathrm{i}}}\right)·\mathrm{P}{\mathrm{E}}_{\mathrm{A}}\left(\mathrm{I}\mathrm{M}\right)\right]·{\mathrm{R}}_{\mathrm{U}}$$

(23)

$${\mathrm{R}}_{\mathrm{U}}=\left({\sum \mathrm{C}}_{\mathrm{R}}+{\sum \mathrm{C}}_{\mathrm{D}}\right)·{\mathrm{C}}_{\mathrm{I}}$$

(24)

where

${\mathrm{T}\mathrm{R}\mathrm{C}\mathrm{L}}_{\mathrm{T}}$—total risk for the infrastructure design life cycle;

${\mathrm{D}\mathrm{R}}_{{\mathrm{d}\mathrm{s}}_{\mathrm{i}}}$—damage rate of damage state i;

$\mathrm{P}\left(\mathrm{d}{\mathrm{s}}_{\mathrm{i}}\right|\mathrm{I}\mathrm{M})$—probability of being in a certain damage state $\mathrm{i}$ for a given $\mathrm{I}\mathrm{M}$;

$\mathrm{T}$—design lifecycle.

$\mathrm{P}{\mathrm{E}}_{\mathrm{A}}\left(\mathrm{I}\mathrm{M}\right)$—annual rate of exceedance of a given $\mathrm{I}\mathrm{M}$;

${\mathrm{C}}_{\mathrm{R}}$—repair cost (US$);

${\mathrm{C}}_{\mathrm{D}}$—direct loss (US$);

${\mathrm{C}}_{\mathrm{I}}$—indirect loss coefficient;

${\mathrm{R}}_{\mathrm{U}}$—overall consequences (US$).

4. Case Study

In this section, the proposed methodology is demonstrated through a case study involving a one-story reinforced concrete (RC) moment frame building. This simple structure is used to illustrate how the integration of corrosion effects into seismic fragility assessment can be practically implemented. Importantly, this case study applies the deterioration patterns established in the previous sections to adjust the fragility curves of the building, demonstrating how corrosion-induced degradation affects seismic vulnerability over time. This structure is utilized based on the previous work of Urlainis and Shohet [6,79], which identified these types of building structures as critical components in critical infrastructure (CI), specifically in oil pumping stations. The baseline fragility curve parameters for this structure are derived from the HAZAS database [75]. The fragility parameters and corresponding damage rates are detailed in

Table 3. Baseline fragility curve parameters and damage ratio.

Damage State		${\mathit{\theta }}_{\mathit{i}}$	${\mathit{\beta }}_{\mathit{i}}$	Damage Ratio
DS1	Slight	0.12	0.64	0.05
DS2	Moderate	0.15	0.64	0.11
DS3	Extensive	0.27	0.64	0.55
DS4	Complete	0.45	0.64	1

.

4.1. Corrosion Impact on Fragility Curves

In the analysis of structural degradation due to corrosion, a dataset that was yielded in the previous steps was utilized. This dataset provides detailed degradation rates at specified time intervals up to 120 years, categorized by varying levels of corrosion: minor, moderate, and intensive. To estimate degradation rates at unrecorded time points, interpolation methods were utilized. The fragility parameters are computed using the method described in Section 2. Baseline fragility curves, derived from the HAZAS database, are adjusted to account for corrosion effects using the deterioration patterns established in our earlier analysis. This adjustment modifies both the median capacity and dispersion parameters of the fragility curves over time, as per Equations (21) and (22), allowing for a time-dependent assessment of seismic vulnerability that incorporates corrosion impacts.

This approach facilitates dynamically adjusting the fragility curves over time, reflecting the true progression of corrosion and its impact on seismic vulnerability. Figure 9 shows a snapshot of adjusted seismic fragility curves for a one-story RC moment frame building at different points in its lifecycle, considering an intensive level of corrosion. The baseline curves (dashed lines) represent the probability of exceedance for various damage states (DS1 to DS4) without considering corrosion effects. The solid lines show the adjusted fragility curves after accounting for corrosion effects at 50, 75, 100, and 120 years.

To provide a more comprehensive evaluation of corrosion’s impact on seismic vulnerability, fragility curves for different damage states (DS1 to DS4) under various corrosion intensities are compared across different time intervals (50, 75, 100, and 120 years). The results, as shown in the Figure 10, highlight the progressive shift in the probability of exceeding specific damage states due to corrosion over time. These curves clearly illustrate how the seismic risk increases with the severity of corrosion, offering valuable insights into the long-term structural performance under varying environmental conditions.

4.2. Risk Assessment

To evaluate the impact of corrosion on seismic risk, this case study examined three locations: Haifa, Jerusalem, and Bikat HaYarden. Using acceleration data from maps provided by the Geophysical Institute of Israel [80], a hazard curve was generated for each location. These hazard curves, representing the annual probability of exceedance for various levels of ground motion, were integral to the risk assessment. They facilitated the estimation of potential seismic impacts on the structure throughout its lifespan, thereby informing the subsequent analysis of how corrosion affects seismic risk.

Next, an analysis comparing the risk of the corroded structure against the baseline (non-corroded) condition was conducted. To achieve this, the risk ratio was defined as the ratio of the annual seismic risk of a corroded structure to its baseline risk. The risk ratio, $\mathrm{R}\mathrm{R}\left(\mathrm{t}\right)$, at a given time t is defined as

$$\mathrm{R}\mathrm{R}\left(\mathrm{t}\right)={\displaystyle \frac{{\mathrm{R}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{d}}\left(\mathrm{t}\right)}{{\mathrm{R}}_{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}}\left(\mathrm{t}\right)}}$$

(25)

where

${\mathrm{R}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{d}}\left(\mathrm{t}\right)$ is the annual seismic risk of the corroded structure at time $\mathrm{t}$;

${\mathrm{R}}_{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}}\left(\mathrm{t}\right)$ is the annual seismic risk of the baseline (non-corroded) structure at time $\mathrm{t}$.

This metric illustrates the change in seismic vulnerability due to corrosion. The progression of these ratios was plotted over a 120-year period to capture the long-term effects of corrosion on seismic vulnerability. The baseline risk was derived from the structure’s original, non-corroded condition, as outlined in the fragility curves. The influence of corrosion was then integrated into these fragility curves using the degradation factors determined from the previous analysis.

The analysis presented in Figure 11 reveals patterns of risk escalation based on corrosion severity:

• Minor corrosion: the risk ratio remains close to one throughout the 120-year period across all locations, indicating that low-level corrosion has a minimal impact on seismic risk due to gradual material degradation.
• Moderate corrosion: A noticeable increase in the risk ratio begins around 40 years, with the risk ratio doubling by 120 years in both Haifa and Bikat HaYarden, and slightly less pronounced in Jerusalem. This indicates that moderate corrosion leads to significant structural compromise over time.
• Intensive corrosion: The risk ratio escalates sharply after 40 years in all locations. This underscores the severe impact of the intensive long-term effect of the corrosion on structural vulnerability. However, it should be noted that the high-risk ratio does not indicate a high risk, but a higher risk relative to baseline.

Next, to comprehensively assess the impact of corrosion on the seismic vulnerability of the structure, a cumulative risk ratio analysis was conducted. This analysis involved aggregating the annual risk ratios over a 120-year period. The cumulative risk ratio, therefore, provides a measure of the total risk accrued over the lifecycle of the structure due to corrosion, relative to the baseline condition. The cumulative risk ratio (CRR) is calculated as the ratio of the sum of the risk ratios over the entire period:

$$\mathrm{C}\mathrm{R}\mathrm{R}={\displaystyle \frac{\sum _{t=o}^{LC}{\mathrm{R}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{d}}\left(\mathrm{t}\right)}{\sum _{t=o}^{LC}{\mathrm{R}}_{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}}\left(\mathrm{t}\right)}}$$

(26)

As illustrated in Figure 12, cumulative risk ratios are presented for three corrosion levels (minor, moderate, and intensive) across three different locations (Bikat HaYarden, Jerusalem, and Haifa). Under the minor corrosion level, the cumulative risk ratio remains close to 1.10 in all locations, indicating a minimal long-term impact on seismic vulnerability. When considering moderate corrosion, the cumulative risk ratio increases more noticeably, reaching 1.59 in Bik’at HaYarden and rising to 2.01 in Jerusalem, reflecting a significant escalation in total risk over time. The most severe impact is observed under intensive corrosion conditions, with the cumulative risk ratio escalating to 6.06 in Jerusalem, and nearly tripling the baseline risk in both Bik’at HaYarden and Haifa, where the ratios reach 2.91 and 3.26, respectively. This analysis highlights the substantial amplification of seismic risk due to progressive corrosion, particularly in regions with higher baseline seismic risks and under higher corrosion levels. Notably, the variability of risk increases significantly as the level of corrosion intensifies, underscoring the heightened uncertainty and potential for severe outcomes under higher corrosion levels. This analysis highlights the substantial amplification of seismic risk due to progressive corrosion, particularly in regions with higher baseline seismic risks and under higher corrosion levels.

4.3. Sensitivity Analysis of ${\beta }_{cor}$

To evaluate the impact of ${\beta }_{cor}$, a sensitivity analysis was conducted with values ranging from 0 to 0.5. The results show that the influence of ${\beta }_{cor}$ is minimal across all corrosion levels. Further examination of the cumulative risk ratio, which considers the entire lifecycle of the infrastructure, revealed negligible changes of up to 1%. This finding suggests that the uncertainty introduced by corrosion in our model is overshadowed by other factors affecting seismic vulnerability. The limited impact of βcor implies that efforts to precisely quantify this parameter may not significantly improve overall risk assessments in most cases.

5. Discussion

5.1. Corrosion Impact on Seismic Vulnerability

The outcomes of this study reveal a significant and progressive increase in seismic risk due to corrosion over time, particularly for moderate and intensive corrosion environments. This finding underscores the critical importance of considering corrosion effects in long-term seismic vulnerability assessments of reinforced concrete (RC) critical infrastructure. While structures subject to minor corrosion levels show only minor increases in vulnerability over a 120-year period, those exposed to moderate corrosion face doubled seismic risk by the end of this timeframe. Most alarmingly, under intensive corrosion conditions, the risk ratio escalates sharply after 40 years, reaching nearly five times the baseline risk at 120 years. This exponential increase in seismic vulnerability suggests that structures in highly corrosive environments may become critically compromised much earlier than their intended design life.

The cumulative risk ratios provide a comprehensive picture of the long-term impact of corrosion on seismic vulnerability across different locations. For minor corrosion levels, the total seismic risk over the structure’s lifecycle increases only marginally, with a 10% to 16% rise depending on the location. However, moderate corrosion levels result in a more substantial increase in the total seismic risk, ranging from 59% in Bikat HaYarden to over 100% in Jerusalem, highlighting the need for more robust maintenance strategies. Most strikingly, intensive corrosion levels dramatically increase the total seismic risk, with cumulative risk ratios reaching 2.91 in Bikat HaYarden, 3.26 in Haifa, and a significant 6.06 in Jerusalem. These findings underscore the critical role of corrosion in the risk assessment process and emphasize the necessity for tailored maintenance strategies, more frequent inspections, and potentially earlier intervention or retrofitting, particularly for structures in moderately to highly corrosive environments, to maintain acceptable levels of seismic safety throughout their design life cycle.

These findings have profound implications for the management of critical infrastructure. They suggest that current design practices and maintenance schedules, which often do not fully account for the compounding effects of corrosion on seismic vulnerability, may be inadequate for ensuring the long-term resilience of RC critical infrastructures, particularly in moderate to highly corrosive environments. The results underscore the need for more proactive and tailored approaches to infrastructure management, potentially including

• More frequent and thorough structural health monitoring, especially in corrosive environments;
• Earlier and more aggressive intervention strategies for existing structures showing signs of corrosion;
• The incorporation of time-dependent seismic vulnerability assessments in long-term infrastructure planning and budgeting.

5.2. Comparison with Existing Literature

Our findings on the progressive impact of corrosion on seismic vulnerability align closely with those reported by Cui et al. [78], particularly for time spans of up to 75 years. For these periods, our results show a deviation of only 3% to 14% from Cui et al.’s findings, providing significant validation for our simulation techniques and methodologies. This congruence underscores the reliability of our degradation rate estimations within expected margins. However, for the 100-year time span, our results consistently predict lower vulnerability compared to Cui et al. This discrepancy suggests that our model may underestimate long-term risks, particularly in marine environments. This difference highlights the need for further investigation into long-term corrosion effects and their impact on seismic vulnerability.

Our research also aligns with studies by Pitilakis et al. [76] and Dizaj et al. [46] emphasizing the significant increase in the seismic vulnerability of RC structures due to aging and corrosion. However, our study provides a more comprehensive quantification of this risk increase over time, mainly through cumulative risk ratios. This approach offers a more nuanced understanding of long-term vulnerability compared to previous studies.

The findings of our study extend the work of researchers like Deng et al. [47] and De Domenico et al. [51] on bridges by applying similar principles to general RC structures in critical infrastructure. While these previous studies focused on specific structural elements, our research provides a broader framework applicable to various RC structures, offering a more versatile tool for infrastructure risk assessment.

5.3. Implications for Infrastructure Management

The findings of this study have significant practical implications for the management and maintenance of RC structures in seismic zones. The observed increase in seismic vulnerability due to corrosion over time, particularly for moderate and intensive corrosion levels, underscores the need for more proactive and adaptive management strategies. Various protection techniques that are already in practice to mitigate corrosion in reinforced concrete structures [81] should be more commonly used. In addition to the traditional methods and strategies for managing and mitigating corrosion in RC structures, the use of machine learning (ML) presents a promising avenue for enhancing the accuracy and efficiency of corrosion assessment. Recent studies have demonstrated that ML algorithms can significantly improve the prediction of corrosion progression, structural capacity, and overall durability of RC structures [82,83,84]. Furthermore, the use of machine learning (ML) has demonstrated significant potential in engineering scenarios where data availability is limited, and decisions must be made with sparse or incomplete information [85,86].

Infrastructure managers should consider implementing more frequent inspections and condition assessments, especially for structures in corrosive environments. Furthermore, the implementation of novel technologies for the early detection and continuous monitoring of corrosion is essential for effectively preventing or mitigating corrosion-related failures [87]. The cumulative risk ratios presented in this study provide a quantitative basis for adjusting maintenance schedules and allocating resources more effectively. For instance, structures in highly corrosive environments may require major interventions or retrofitting much earlier in their lifecycle than previously anticipated.

Moreover, our results highlight the critical importance of incorporating corrosion effects into long-term seismic risk assessments. Traditional seismic risk assessments that do not account for time-dependent degradation may significantly underestimate the vulnerability of structures, particularly in the latter half of their design life. This underestimation could lead to inadequate preparedness and potentially catastrophic consequences in the event of a major seismic event. By considering corrosion effects, infrastructure managers can develop more realistic and comprehensive risk management strategies, ensuring the resilience of critical infrastructure over its entire lifecycle.

From a methodological perspective, this study contributes a novel approach to integrating corrosion effects into seismic fragility curves. The proposed framework allows for a dynamic adjustment of fragility parameters based on corrosion progression, providing a more accurate representation of structural vulnerability over time. This integration of time-dependent degradation into seismic risk assessment represents a significant advancement over traditional risk assessment models. A key strength of our approach is its ability to quantify the cumulative effects of corrosion on seismic risk over extended periods, as demonstrated by the cumulative risk ratios. This framework provides a comprehensive view of lifecycle risk, aiding researchers and practitioners in making more in-formed decisions regarding maintenance, retrofitting, and resource allocation to enhance infrastructure resilience in seismically active and corrosive environments.

5.4. Future Research and Limitation

While our study provides valuable insights, it is crucial to recognize the limitations inherent in our approach. The accuracy of our results is significantly influenced by the quality of input data related to corrosion rates and structural parameters. Future research could significantly enhance the current modeling approach by addressing several key factors, such as incorporating the stiffness of the rebar, simulating the interface layer between the rebar and concrete, and enabling continuous information transmission between time steps, as discussed in Section 3.1. In the present model, each stage was simulated discretely, with incremental shear stress applied up to failure and degraded mechanical parameters manually input based on empirical reduction functions. Future work should focus on improving the transfer of information, such as crack formation and progression, from one step to the next rather than relying solely on manually adjusted degraded mechanical properties. Additionally, incorporating the rebar’s stiffness into the model could lead to more accurate predictions of shear capacity over the structure’s lifespan. The bond strength between concrete and steel is also important for overall structural performance. As corrosion weakens the surrounding concrete, the bond strength decreases, damaging the structure’s ability to transfer loads efficiently and potentially causing localized failures or cracking, particularly under seismic loads.

Furthermore, investigating the strain rate effect, accounting for void particles, and projecting behavior from the mesoscale to the structural level would significantly enhance the model’s robustness. Future research should also aim to refine these inputs through more extensive field studies and real-world data collection. These aspects offer valuable opportunities for further refining the methodology in subsequent studies.

6. Conclusions

This research introduces an innovative framework for evaluating the seismic risks associated with reinforced concrete structures within critical infrastructure, accounting for the progressive effects of corrosion. Our findings reveal a significant and time-dependent increase in seismic vulnerability due to corrosion, particularly under moderate and intensive corrosion conditions. The cumulative risk ratios demonstrate that over a 120-year lifespan, structures in highly corrosive environments may face up to six times the baseline seismic risk, depending on the location and corrosion intensity.

A key novelty of this study is that it can serve as a tool for real-time monitoring by incorporating our approach to adjusting fragility parameters based on corrosion progression. This approach offers a more realistic and accurate representation of structural vulnerability as it evolves, enabling a comprehensive lifecycle risk assessment. The framework’s capacity to quantify the cumulative impact of corrosion on seismic risk provides a valuable tool for long-term infrastructure resilience planning. As demonstrated in the case study, applying corrosion-adjusted fragility curves allows for a more detailed understanding of risk escalation across different geographic locations and environmental conditions.

Furthermore, this study underscores the critical importance of incorporating time-dependent degradation into seismic risk assessments and the need for more proactive and tailored approaches to infrastructure management. The study’s findings call for more frequent structural health monitoring, earlier intervention strategies, and the integration of time-dependent seismic vulnerability assessments in infrastructure planning and budgeting.

In conclusion, this study provides a comprehensive evaluation of substation resilience under seismic conditions, focusing on the critical components’ operational capacity and failure probability. The development of quantitative metrics and vulnerability curves offers valuable insights into the behavior of substations during seismic events, and the proposed models contribute to a deeper understanding of post-earthquake functionality. Future research should refine these models by integrating more complex factors essential for accurately predicting substation performance in seismic events. Additionally, experimental validation using real-world data will greatly enhance the models’ accuracy and practical applicability.