Resilience Assessment for Corroded Reinforced Concrete Bridge Piers Against Vessel Impact

Abstract

The resilience concept is well established in engineering, but the quantitative studies of vessel impact resilience for bridge structures remain limited. This paper presents an integrated framework for assessing vessel impact resilience under combined rebar corrosion and vessel collision effects. First, a corroded reinforced concrete bridge is considered for nonlinear static analysis to quantify initial corrosion damage and for nonlinear dynamic analysis to evaluate post-impact function loss. Then, recovery for each damage state is modeled by using both negative exponential and triangular recovery functions to estimate restoration times and to obtain a vessel impact resilience index. The results show that increasing corrosion severity markedly reduces resilience capacity. Furthermore, resilience indices obtained from the negative exponential function generally exceed those from the triangular function, and this improvement becomes more significant at lower resilience levels. Resilience indices calculated by using negative exponential and triangular recovery functions show negligible differences when the concrete bridge is in the uncorroded initial state and the vessel impact velocity is below 1.5 m/s. However, as reinforcement corrosion increases, the maximum discrepancy between these two recovery functions also increases, reaching a value of 67% at a corrosion level of 15.0%. From the numerical results obtained from a case study, it is important to select an appropriate recovery model when assessing vessel impact resilience. For rapid initial restoration followed by slower long-term recovery, the negative exponential model yields greater resilience gains compared to the triangular model. The proposed method thus provides an effective tool for engineers and decision makers to evaluate and improve the vessel impact resilience of aging bridges under the combined corrosion and impact effects. This proposes a quantitative metric for resilience-based condition assessment and maintenance planning.

1. Introduction

For coastal bridges, vessel impact generates one of the extreme loads capable of causing severe structural damage or collapse, leading to safety incidents and disrupting transportation systems. The report by the World Association for Waterborne Infrastructure in 2018 documented 35 bridge collapses worldwide due to vessel impact between 1960 and 2015. In 2019, at least two barges struck the San Jacinto Bridge, approximately 15 miles east of the San Jacinto River, causing significant structural damage to its piers [1]. Irhayyim et al. [2] identified flood scour and vessel collision as the major causes of bridge failure and reported that vessel impact-induced collapse accounts for 12 percent of such incidents, which is expected to rise with increasing maritime traffic. Therefore, it is critically important to investigate both the degradation of structural performance following vessel impact and the capacity for restoring the bridge to its initial condition.

Current research on structural resilience focuses predominantly on either a structure’s capacity to withstand external hazards or its efficiency in post-event recovery. Resilience is commonly evaluated across four attributes: redundancy, robustness, rapidity, and resourcefulness [3,4]. Despite growing interest, resilience assessments in engineering remain limited, with most studies focusing on seismic resilience. Aroquipa et al. [5] introduced a simplified probabilistic risk assessment method that quantifies building seismic capacity by incorporating mean annual repair time. Huang et al. [6] developed a probabilistic resilience assessment framework to account for corrosive environments under seismic loading. They applied this framework to evaluate aging reinforced concrete bridge piers across various failure modes. He et al. [7] adapted quantitative seismic resilience methods for slope engineering, proposing a preliminary evaluation framework for slope seismic resilience. Chen et al. [8] established a paradigmatic resilience model for metro systems by integrating individual structural performance, inter-structural interactions, and post-disaster recovery processes. Therefore, these studies underscore both the diversity of approaches and the need for broader applications of resilience assessments across engineering disciplines.

Although a well-established theoretical framework exists for evaluating bridge resilience under seismic loading, research on bridge resilience under multi-hazard coupling remains limited, particularly regarding the mechanisms of life-cycle performance degradation and the resilience enhancement strategies for hazard modes such as flooding, tsunamis, hurricanes, impacts, and blasts. Qeshta et al. [9] conducted an assessment of coastal bridge resilience to extreme waves by proposing a method within the classical performance-based earthquake engineering (PBEE) framework. Qiu et al. [10] examined the resilience of coastal reinforced concrete bridges under multiple hazards, in order to assess recovery capacity in terms of repair time, repair cost, and carbon footprint. Yan et al. [11] reviewed advancements in blast resilience for critical infrastructure, providing the critical scientific challenges in enhancing structural resilience against explosive loading.

Coastal bridges endure seawater erosion and wave action all the time, which induces surface cracking in concrete and accelerates chloride ingress into reinforced concrete structural members. Once chloride concentration at the rebar surface exceeds a critical threshold, the passive oxide layer breaks down and initiates reinforcement corrosion [12]. Studies have shown that corrosion reduces rebar yield strength and effective cross-sectional area [13], while the volumetric expansion of corrosion products is several times the lost steel volume and causes concrete cover cracking and subsequent deterioration of structural performance [14,15]. Kagermanov [16] developed a nonlinear finite element model to conduct sensitivity analyses of various failure modes, accounting for reduced rebar section and yield strength, concrete cracking, reduced concrete area, and degraded bond–slip behavior under corrosive conditions. Wang [17] employed finite element simulations to investigate the effects of non-uniform corrosion and stirrup confinement on cracking patterns in reinforced concrete beams. Ma et al. [18] analyzed seismic performance changes in bridge piers subjected to different corrosion durations using finite element methods. Although significant progress has been made in assessing resilience under isolated corrosion effects and coupled corrosion and seismic loading, the quantitative evaluations of vessel impact resilience for corrosion-vulnerable coastal piers remain limited.

This study investigates the degradation of vessel impact resilience indices in reinforced concrete structures subjected to both environmental corrosion and external impact loading. Corrosion-induced degradation is modeled by accounting for reductions in rebar cross-sectional area, yield strength, bond–slip behavior, and concrete strength. A simplified finite element model of a bridge is constructed, and the impact load is applied using quasi-static analysis. Nonlinear dynamic analysis is then conducted to evaluate the effects of vessel collision velocity and tonnage on structural performance. Then, a limit state function is given by using the response surface method, and a Monte Carlo simulation is performed to derive failure probability curves under different corrosion levels and impact velocities. A resilience assessment framework is applied, which incorporates repair time and a recovery function by using two types of recovery functions, i.e., exponential and triangular models, to study differences in restoration behavior. Finally, the vessel impact resilience indices are calculated for varying corrosion levels, and the influence of both corrosion severity and recovery assumptions on structural resilience performance is investigated.

2. Corrosion-Induced Degradation Model for Reinforced Concrete Materials

Coastal bridges are subjected to long-term marine wind wave actions, which accelerate chloride ingress and subsequently lead to internal reinforcement corrosion and surface cracking of concrete due to the volumetric expansion of corrosion products, leading to a reduction in the overall structural performance. This study considers four main aspects of material degradation: (1) reduction in the effective cross-sectional area of the reinforcement; (2) decrease in rebar yield and ultimate strength; (3) reduction in the bond strength between rebar and concrete; and (4) loss of concrete tensile strength caused by corrosion-induced cracking.

Considering that chloride-induced corrosion of reinforcement leads to expansion and cracking of the concrete cover, the corrosion level ${\eta }_s$ is obtained based on the mass loss of reinforcement before and after corrosion, given as

$${\eta }_s=1-m_s/m_0$$

(1)

where $m_0$ is the initial mass of uncorroded reinforcement and $m_s$ is the residual mass of reinforcement after removing corrosion products. The effective cross-sectional area $A_s$ of corroded reinforcement is then derived from

$$A_s=\left(1-{\eta }_s\right)A_0$$

(2)

where $A_0$ denotes the effective cross-sectional area of the uncorroded reinforcement.

Corrosion leads to degradation in the mechanical properties of reinforcement, including yield strength, ultimate strength, and ultimate elongation. In this study, the constitutive model for corroded reinforcement given in [19] is adopted. This model effectively captures the mechanical behavior of reinforcement subjected to natural atmospheric corrosion in exposed environments. The yield strength $f_{ys}$ and ultimate strength $f_{us}$ of the corroded reinforcement are determined from

$$f_{ys}={\displaystyle \frac{1-1.231{\eta }_s}{1-{\eta }_s}}{f}_{y0}$$

(3)

$$f_{us}={\displaystyle \frac{1-1.245{\eta }_s}{1-{\eta }_s}}{f}_{u0}$$

(4)

where $f_{y0}$ and $f_{u0}$ represent the yield and ultimate strengths of uncorroded reinforcement, respectively. In addition, corrosion reduces the ultimate strain of reinforcement due to changes in cross-sectional dimensions and the development of geometric non-uniformity, which induces stress concentrations. The ultimate strain ${\epsilon }_{sus}$ of the corroded reinforcement is estimated as

$${\epsilon }_{sus}=e^{-2.093{\eta }_s}{\epsilon }_{su0}$$

(5)

where ${\epsilon }_{su0}$ represents the ultimate strain of the uncorroded reinforcement.

The bond–slip constitutive model is adopted by using the τ-s (bond stress–slip) curve from Yu et al. [20]. Corrosion-induced bond strength degradation is simulated through a reduction coefficient. The bond strength reduction factor follows the model proposed by Jiang et al. [21]. The bond–slip relationship $\tau \left({\eta }_{\mathrm{s}}\right)$ for the corroded rebar concrete interfaces is defined as

$$\tau \left({\eta }_s\right)=\beta {\tau }_0\left({\eta }_s\right)$$

(6)

where ${\tau }_0\left({\eta }_{\mathrm{s}}\right)$ denotes the original bond–slip relationship for uncorroded reinforcement, and β is the corrosion-dependent bond strength reduction coefficient, calculated by the equation given in [21], expressed here as

$$\beta =\left\{\begin{array}{l}1\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left({\eta }_s\le 1.5\%\right)\\ 1.192e^{-11.7{\eta }_s}\hspace{1em}\left({\eta }_s>1.5\%\right)\end{array}\right.$$

(7)

Corrosion products from rebar not only degrade bond strength but also cause concrete cover cracking due to rust expansion, reducing the mechanical properties of the cover concrete. This study introduces a reduction factor to simulate strength degradation of cover concrete [22]. The compressive strength $f_s^{\ast }$ of cover concrete after reinforcement corrosion is given as

$$f_s^{\ast }={\displaystyle \frac{f_s}{1+K{\epsilon }_1/{\epsilon }_{s0}}}$$

(8)

where $f_s$ is the compressive strength of cover concrete before corrosion; $K$= 0.1 is the reduction coefficient; ${\epsilon }_{s0}$ is the compressive strain at peak stress of uncorroded concrete; and ${\epsilon }_1$ is the average tensile strain induced by corrosion-induced cracking, which correlates with the volumetric expansion of corrosion products. The tensile strain ${\epsilon }_1$ is obtained from

$${\epsilon }_1={\displaystyle \frac{\pi n_l{\varphi }_l\left(1-\sqrt{1-{\eta }_s}\right)}{p_{cp}}}$$

(9)

where $n_l$ is the number of longitudinal reinforcement bars within the cross-section of the structural member; ${\varphi }_l$ is the diameter of uncorroded longitudinal reinforcement; and $p_{cp}$ is the perimeter of the cross-section of the structure.

Due to the confinement effect provided by stirrups, the tensile and compressive strength of core concrete is increased. However, corrosion leads to a reduction in the stirrup confinement efficiency. In this study, the constitutive model for corroded stirrup confined concrete proposed by Vu [23], based on the Mander model and applicable to circular sections, is employed to calculate the core concrete strength $f_{cc}^{\prime }$, given as

$$f_{cc}^{\prime }=\left(1-\alpha {\eta }_s\right)f_{co}^{\prime }\left(-1.254+2.254\sqrt{1+{\displaystyle \frac{7.94f_l^{\prime }}{f_{co}^{\prime }}}}-2{\displaystyle \frac{f_l^{\prime }}{f_{co}^{\prime }}}\right)$$

(10)

where $\alpha$ is a stress modification factor obtained through regression analysis of experimental data. For circular cross-sections, $\alpha$ is taken as 0.51; $f_{co}^{\prime }$ is the compressive strength of unconfined concrete; and $f_l^{\prime }$ is the effective lateral confining stress. Due to corrosion-induced reductions in stirrup cross-sectional area, yield strength, and the bond strength between stirrups and concrete, the lateral confinement provided by stirrups is significantly weakened. The effective lateral confining stress $f_l^{\prime }$ is obtained as

$$f_l^{\prime }={\displaystyle \frac{1}{2}}{k}_e{\rho }_{sc}{f}_{y\mathrm{s}}$$

(11)

where the confinement effectiveness coefficient is defined as $k_e=A_e/A_{cc}$, in where $A_e$ is the area of the effectively confined concrete core; and $A_{\mathrm{cc}}$ is the total area of concrete within the cross-section that is confined by stirrups; $f_{y\mathrm{s}}$ denotes the yield strength of the corroded transverse reinforcement; and ${\rho }_{sc}$ is the volumetric transverse reinforcement ratio for the corroded confined concrete, calculated from

$${\rho }_{sc}=\left(1-{\eta }_s\right){\rho }_s$$

(12)

where ${\rho }_s$ represents the volumetric transverse reinforcement ratio of the uncorroded confined concrete.

3. Structural Vessel Impact Resilience Assessment

3.1. Vessel Impact Fragility Analysis

Structural fragility is defined as the conditional probability that a structure exceeds a prescribed limit state under varying hazard intensities. This metric quantitatively characterizes the relationship between hazard intensity and damage severity. The probability that the structural dynamic response under vessel impact exceeds the predefined limit state can be expressed as

$$I_i=I\left(N\ge L{S}_i\left|V=v\right.\right)$$

(13)

where $I_i$ denotes the conditional probability of structural failure in damage state i; $N$ is the compressive ultimate capacity of the structure; $V$ represents the vessel impact velocity considered as a random variable; $v$ is the actual value of that velocity; and $LS$ represents the prescribed limit state that the structural response may exceed.

Since bridge piers are critical elements that must stably support and efficiently transfer vertical loads from the superstructure, their vertical load-carrying capacity serves as a key performance metric after damage. Consequently, the ultimate compressive capacity of the pier is adopted as the limit state criterion. Following the loss ratio classification in [24], four limit states, i.e., $D_1$, $D_2$, $D_3$, and $D_4$, are defined, corresponding to residual capacity levels of 10%, 20%, 40%, and 70% of the original ultimate compressive strength, respectively. Here, a damage $D=1-N_{\mathrm{r}}/N$ is introduced, where $N_{\mathrm{r}}$ is the residual load-carrying capacity after damage, and $N$ is the original ultimate compressive capacity. The ranges for each damage state are summarized in

Table 1. Damage state range values.

Damage State	Loss Range of Ultimate Compressive Capacity
Intact	<10%
Slight Damage	$10\% \le D$ < 20%
Moderate Damage	$20\% \le D$ < 40%
Severe Damage	$40\% \le D$ < 70%
Collapse	≥70%

.

Fragility analysis of bridge vessel impact requires the systematic collection of structural dynamic response data under multiple scenarios. Given the high computational cost of traditional finite element methods in full probabilistic analyses, this study introduces the response surface method to construct explicit surrogate models for high-dimensional dynamic responses, significantly reducing simulation effort. Then, the probabilistic distribution of concrete component resistance is coupled with a statistical model of vessel mass. Large-scale Monte Carlo sampling is then employed to quantify the exceedance probabilities of limit states for each damage state, ultimately generating the probabilistic fragility curve of the bridge pier under vessel impact loading. The workflow is illustrated in Figure 1.

3.2. Vessel Impact Resilience Assessment Method

3.2.1. Method for Quantifying the Vessel Impact Resilience Index

Structural resilience, as a key quantitative metric for the ability of an engineering system to withstand hazard impacts and recover functionality, fundamentally involves the time-varying recovery of system functionality following a disruptive event. This recovery is typically represented by a functional recovery curve, which traces the dynamic trajectory of structural functionality from the initial damage state induced by the hazard, through progressive restoration to acceptable service levels, and ultimately back to the original baseline performance.

This paper focuses on evaluating the resilience of the corroded reinforced concrete structures under vessel impact. To analyze how corrosion influences the time-varying characteristics of impact resilience, the established seismic resilience assessment framework is adopted to quantify the evolution of system functionality after vessel collision. The time-varying system functionality of the corroded reinforced concrete structures in seismic events is illustrated in Figure 2.

The normalized system functionality $Q\left(t\right)$ as a function of time t represents the structural performance function, which is evaluated in terms of the structural resistance to impact. For reinforced concrete structures, the vessel impact resilience $R$ is quantified by [25]

$$R={\displaystyle {\int }_{t_{OE}}^{t_{OE}+T_{\mathrm{RE}}}\frac{Q(t)}{T_{\mathrm{RE}}}}\mathrm{d}t$$

(14)

where $t$ is time; $t_{OE}$ denotes the time of the impact event; and $T_{RE}$ is the repair completion time following vessel collision. The functionality function $Q\left(t\right)$ is defined as

$$Q(t)=1-L(h)\cdot H(t)\cdot f_{\mathrm{rec}}\left(t\right)$$

(15)

in which $L(h)$ is the loss function of the concrete under corrosion at the time of impact; $f_{\mathrm{rec}}\left(t\right)$ is the recovery function; and $H\left(t\right)$ is the Heaviside step function. Since bridge piers may experience degradation before the impact event due to pre-existing corrosion, $Q\left(t\right)$ is redefined following Cimellaro’s approach to include initial loss. The modified functionality function now becomes

$$Q(t)=1-L_c({\eta }_s)-L_h({\eta }_s,h)\cdot H(t)\cdot f_{\mathrm{rec}}\left(t\right)$$

(16)

where $L_h({\eta }_s,h)$ is the impact induced loss function under corrosion state $L_c({\eta }_s)$ is the initial loss function of the corroded pier, which is quantified from nonlinear static analysis by using the residual ultimate compressive capacity as the remaining functionality metric, given as

$$L_c({\eta }_s)=1-{\displaystyle \frac{N_a\left(t\right)}{N_a\left(0\right)}}$$

(17)

in which $N_a\left(0\right)$ is the maximum vertical capacity of the uncorroded pier; and $N_a\left(t\right)$ is the time-varying vertical capacity.

The impact induced loss function $L_h({\eta }_s,h)$ is calculated from the probabilistic fragility curves for each damage state by multiplying the failure probability by the corresponding damage ratio, expressed as

$$L_h({\eta }_s,h)={\displaystyle \sum _{j=1}^5\left[r_j\cdot P_j\left(LS=j\right)\right]}$$

(18)

where j ranges 1 to 5 and corresponds to the five damage states from intact to collapse; $r_j$ is the loss ratio of the structure in damage state j; and $P_j$ is the probability of damage state j, obtained from the fragility model as

$$P_j=\left\{\begin{array}{l}1-I_1\left(LS=j\right),\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} j=1\\ I_{j-1}\left(LS=j-1\right)-I_j\left(LS=j\right),\hspace{1em}j=2,3,4\\ I_4\left(LS=j\right),\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}j=5\end{array}\right.$$

(19)

3.2.2. Recovery Time

Within the bridge vessel impact resilience assessment framework, recovery time is a core quantitative metric defined as the duration required for a structure in a given post-impact damage state to be restored to a predetermined serviceability level. Recovery time typically correlates with damage severity, and the more severe the damage, the longer the maintenance and repair period. To parameterize recovery time for this study, values from [26] are adopted and rationalized for five damage states. Therefore, recovery times in days for the intact, slight damage, moderate damage, severe damage, and collapse states are set to 0 d, 10 d, 20 d, 60 d, and 75 d, respectively.

3.2.3. Recovery Function

Functional recovery functions describe how a bridge’s performance or functionality gradually restores over time following a disruptive event. The repair process depends on many factors, including material allocation, funding, workforce deployment, repair strategies, seasonal conditions, etc. Based on the recovery functions in [25], two forms are considered, i.e., the triangular recovery function $f_{rec}^{tri}\left(t\right)$ and the negative exponential recovery function $f_{rec}^{neg}$, defined as

$$f_{rec}^{tri}\left(t\right)={\displaystyle \frac{a}{2}}\cdot \left(1+\cos \left(\pi b\cdot {\displaystyle \frac{t-t_{OE}}{T_{RE}}}\right)\right)$$

(20)

$$f_{rec}^{neg}\left(t\right)=a\cdot \exp \left(-b\cdot {\displaystyle \frac{t-t_{OE}}{T_{RE}}}\right)$$

(21)

where a and b are constants to be determined by the damage state and the target recovery level. For the triangular recovery function, both a and b are suggested to be taken as 1; for the negative exponential recovery function, a is suggested to be 1 and b to be ln(200) according to the results in [27]. When vessel impact accidents occur, scenarios with pre-allocated resources show accelerated initial recovery rates due to immediate resource influx, followed by progressive deceleration. Such cases employ the negative exponential recovery function. Conversely, scenarios with limited social response capacity demonstrate gradual acceleration after self-organized recovery mobilization. Here the triangular recovery function is adopted in the calculations.

4. Case Study

4.1. Finite Element Modeling of Bridge Piers

In the dynamic analysis of bridge piers subjected to vessel impact, the short duration of impact loading generates the structural dynamic response primarily within the pier region. Conducting such transient dynamic simulations with a fully detailed full-bridge finite element model could result in significant computational expense. Therefore, a simplified modeling strategy is adopted in this study, in which the bridge pier model is primarily considered for the full-bridge system. To reasonably capture the inertial effects induced by the superstructure during impact, the superstructure is idealized as a rigid mass block. Following the assumptions in [28], the gravitational load of this mass block is set to 10% of the pier’s vertical compressive capacity. The finite element model of the pier is illustrated in Figure 3. The pier height is H = 15 m, with a diameter of D = 1.4 m. The concrete compressive strength is 26.8 MPa. Grade HRB400 steel is used for both longitudinal and transverse reinforcement. The longitudinal bars have a diameter of 28 mm, with a reinforcement ratio of 1.12%. The stirrups are 16 mm in diameter and are spaced at 200 mm intervals. The soil–structure interaction at the pier base is neglected. A fixed boundary condition is applied at the bottom of the pier, while only the vertical degree of freedom is released at the top.

4.2. Materials

In this study, finite element analyses were performed by using the software Abaqus 2017, and the concrete plastic damage model was adopted to simulate the nonlinear mechanical behavior of concrete under complex stress states [29]. Concrete is discretized in space using eight-node linear brick elements with reduced integration algorithm (C3D8R), and the damage evolution is simulated based on the energy equivalence principle. The elastoplastic behavior of the reinforcing steel is represented by a constitutive model, and its mechanical response is simulated using two-node linear beam elements (B31) with first-order shear deformable formulation. The concrete material parameters employed in the finite element model are summarized in

Table 2. Concrete and rebar material parameters for the bridge pier.

Parameter	Density ρ ($\mathbf{Kg}\cdot {\mathbf{m}}^{\mathbf{-}\mathbf{3}}$)	Young’s Modulus E (MPa)	Poisson’s Ratio υ	Eccentricity e	Biaxial Compression Strength Ratio${\mathit{f}}_{\mathit{b}\mathbf{0}}\mathbf{/}{\mathit{f}}_{\mathit{c}\mathbf{0}}$
Value	2400	32,500	0.2	0.1	1.16
Parameter	Yield Surface Eccentricity Ratio ${\mathit{K}}_{\mathbf{1}}$	Dilation Angle φ (°)	Viscosity Parameter µ (Second)	Concrete Strength ${\mathit{f}}_{\mathit{c}}$ (MPa)	Rebar Strength ${\mathit{f}}_{\mathit{y}}$ (MPa)
Value	0.66667	30°	$1\times {10}^{-5}$	26.8	400

.

The bond slip behavior between reinforcement and concrete is modeled by introducing connector elements between the nodes of the longitudinal reinforcement and the corresponding concrete nodes, with appropriate interfacial properties assigned. Since the slip primarily occurs perpendicular to the cross-section, the connection type is set to the Cartesian translation, in which the two directions orthogonal to the reinforcement axis are defined as rigidly connected. The nonlinear axial force along the reinforcement direction is computed as the product of the nodal surface area and the axial stress at the reinforcement nodes.

Based on the reinforced concrete corrosion degradation model given in Section 2, mechanical parameters for the degradation of the material properties of the concrete pier were obtained and are shown in Figure 4.

As shown in Figure 4, as the corrosion level increases, the strength degradation of the cover concrete is significantly faster than that of the core concrete. This is because the expansive pressure of corrosion products initially acts on the cover concrete layer. Then, cracks are generated along the reinforcement that directly damage the cover concrete integrity, and larger spalled areas appear in the cover concrete as reinforcement corrosion progresses [23]. Furthermore, the lack of lateral confinement in the cover concrete region allows cracks to propagate directionally, whereas hoop reinforcement in the core concrete region constrains damage accumulation.

4.3. Vessel Impact Force Analysis

Three primary methods are available for estimating vessel impact forces, i.e., equivalent static analysis, simplified dynamic computational models, and refined finite element simulations. Due to its simplicity and ease of use, this study employs the equivalent static analysis; the impact force for a barge on the pier can be estimated from the following two methods:

According to the Guide Specifications and Commentary for Vessel Collision Design of Highway Bridges [30], the barge impact force $P_B$ is calculated as follows

$$P_B=\left\{\begin{array}{l}4112a_B\begin{array}{cc}& \end{array}\begin{array}{cc}& \end{array}\hspace{1em} \left(a_B>0.34\right)\\ 1349+110a_B\begin{array}{cc}& \end{array}\left(a_B>0.34\right)\end{array}\right.$$

(22)

where $a_B$ is the bow damage dimension of the barge, which depends on the impact kinetic energy of the vessel and is calculated as

$$a_B=10.2\left(\sqrt{1+{\displaystyle \frac{KE}{5672}}-1}\right)$$

(23)

in which the impact energy $KE$ is proportional to the mass of the vessel and the square of its relative velocity, and also accounts for the added mass effect between hull and water, expressed as

$$KE={\displaystyle \frac{C_{H}W{V}^2}{29.2}}$$

(24)

in which $C_H$ is the hydrodynamic added mass coefficient, taken as 1.25; $V$ is the vessel speed (ft/s); and $W$ is the vessel mass (ton).

According to the equivalent static force method for barge collision proposed in the vessel collision design of highway bridges [31], the design impact force $P_B$ for a barge is given by

$$P_B=0.0115\cdot M^{0.70}\cdot V_a$$

(25)

where $M$ is the full load mass (ton) and $V_a$ is the vessel impact velocity (m/s).

The barge impact force predicted by the AASHTO is substantially higher than that from the vessel collision design of highway bridges for vessels below 3000 t. Since this study models a barge whose mass is much smaller than that of a sea-going ship, and because the barge impact force obtained by the vessel collision design of highway bridges is more in line with the vessel impact conditions studied in this study, which is considered more appropriate for the present analysis.

4.4. Computational Scenarios

Simulation of vessel pier impact requires selecting appropriate vessel parameters. In this study, the primary variables are vessel impact velocity and vessel mass, with the impact height assumed at mid-pier elevation. Focusing on Class III and IV waterways, the representative barge masses of 1000 t and 500 t can be considered in [32]. For impact velocity, statistics indicate an average barge collision speed of approximately 2 m/s [33], while the AASHTO Guide specifies a minimum design speed of 0.514 m/s. Accordingly, this study adopts a velocity range of 0.5 m/s to 3.0 m/s.

From this analysis, the key structural response parameters are corrosion level, vessel impact velocity, and vessel mass. To reduce computational cost and improve efficiency, an explicit surrogate model is constructed through the response surface method. A three-factor, three-level Box–Behnken design is employed, with corrosion level, vessel impact velocity, and vessel mass as independent variables and residual load-carrying capacity as the dependent variable. The experimental factors and levels are summarized in

Table 3. Response surface experiment factors and levels for response surface method.

Levels	Parameters
	Corrosion Level (%)	Tonnage (Ton)	Velocity (m/s)
−1	0.0	100	0.50
0	7.5	400	1.75
1	15.0	700	3.00

.

4.5. Finite Element Simulation Damage Analysis

4.5.1. Nonlinear Static Analysis

A series of axial compression simulations was conducted on the corroded bridge piers to investigate the residual vertical load-carrying capacity of the corroded reinforced concrete structures at varying corrosion levels. Nonlinear static analysis is employed, with the pier base fully fixed to prevent numerical instabilities upon load application. A smooth step load curve is used to apply vertical load. The reaction force at the pier crest increased with load application until reaching a peak, which is recorded as the pier’s residual capacity. By using the corroded reinforced concrete model, axial compression simulations with smooth step loading are performed for corrosion levels of 0.0%, 2.5%, 5.0%, 7.5%, 10.0%, 12.5%, and 15.0%. The reaction forces corresponding to axial displacements are obtained to generate axial load–displacement curves for each corrosion level, as shown in Figure 5.

In the uncorroded initial state, the residual axial capacity of the pier can be calculated from the ACI expression for axially loaded reinforced concrete members [34], given as

$$N_u\le 0.85\cdot f_{c0}^{\prime }\left(A-A_s^{\prime }\right)+f_s{A}_s^{\prime }$$

(26)

where $f_{c0}^{\prime }$ is the concrete compressive strength; $f_s$ is the reinforcement compressive strength; $A$ is the gross cross-sectional area; and $A_s^{\prime }$ is the area of longitudinal reinforcement. To account for the confinement enhancement provided by the corroded stirrups, the nominal capacity given by this formula is multiplied by the enhancement factor of 1.138 derived from Vu’s circular corroded stirrup confinement model in Equation (10).

The discrepancy between the code-based calculation and the finite element simulation results is less than 5%, confirming the validity of the finite element modeling strategy. Specifically, the initial residual bearing capacity of the pier is calculated as 47.31 MPa according to the code formula, while the finite element simulation yields 46.72 MPa for the initial maximum bearing capacity. Three experimental datasets on the degradation of concrete column axial compressive capacity related to corrosion level from various sources [35,36,37] were also collected for the model validation. The comparison between experimental data and the results obtained from the present models is shown in Figure 6. From Figure 6, the present results match well the experimental data available, therefore confirming the effectiveness of the present model.

As corrosion progresses, the residual capacities obtained from finite element simulations at corrosion levels of 0.0%, 2.5%, 5.0%, 7.5%, 10.0%, 12.5%, and 15.0% are converted into residual functionality indices from Equation (17). The resulting functionality values are plotted in Figure 5, with values of 1.0000, 0.9568, 0.8990, 0.8626, 0.8476, 0.8241, and 0.7931, respectively.

4.5.2. Nonlinear Dynamic Analysis

The combination of elevated corrosion levels and increased vessel impact kinetic energy significantly amplifies the dynamic response of the pier, indicated by higher peak stresses in critical regions. When impact occurs at the mid-height of the pier, the shear stress at the impact location reaches its maximum value and exceeds that at the pier base or cap. This high shear stress concentration leads to a predominantly shear mode failure in the impact region. The structure is considered to have reached a stabilized state when its axial reaction and vertical displacement curves stabilize. By starting from this equilibrium, a smooth step vertical load representing the equivalent static load level is incrementally increased until structural failure occurs. The reaction force reaches a peak just before failure, then rapidly decays as damage accumulates.

A quantitative comparison of the responses across corrosion levels is presented in Figure 7, based on the mid-height concrete displacement on the tension side and the mid-height reinforcement stress in the compression zone at the stabilized stage under equivalent static loading.

Under the same corrosion conditions, as the vessel impact kinetic energy increases, the dynamic response of the pier increases. In Figure 7, at lower energy thresholds, the horizontal displacement and equivalent stress at the impact section stabilize, with only limited fluctuations. However, as impact energy continues to rise, both displacement and stress increase sharply at the same time, with the increasing rate as energy levels grow. For a given impact energy, higher corrosion levels similarly amplify these response parameters, and the more severe the corrosion damage, the greater the amplification in horizontal displacement and equivalent stress.

After the structure stabilizes, a vertical load is incrementally applied until failure. The stress damage patterns at the time of failure vary with corrosion state, and greater corrosion severity produces more pronounced damage. Damage contours of tensile cracking in concrete and equivalent stress failure in reinforcement under the same impact conditions at different corrosion levels are presented in Figure 8.

Once the structure stabilizes, an axial load is applied until failure. As shown in Figure 8, higher impact energies induce tensile cracking in the pier’s cross-section within the Y–Z plane. Under the same impact energy, elevated corrosion levels accelerate the overall damage, and the existing cracks widen while new microcracks increase significantly. Increased corrosion severity intensifies localized damage at both the pier base and cap and leads to progressive expansion of the damaged zones.

The coupled reaction force reaches its maximum just before failure and then rapidly decays. Figure 9 presents the degradation of load-carrying capacity across corrosion levels under varying impact energies. From the results, under the same corrosion conditions, increases in vessel impact kinetic energy lead to a progressive decline in the residual load-carrying capacity of the pier. Moreover, as corrosion damage accumulates, the magnitude of capacity loss under the same impact energy grows significantly. The material degradation associated with higher corrosion levels further amplifies the structural response to impact energy, resulting in steeper decay slopes of residual capacity as both corrosion levels and impact energy increase.

4.6. Response Surface-Based Surrogate Model

Based on the nonlinear dynamic analyses, the residual load-carrying capacities under various scenarios are obtained, and the peak absolute value of each capacity curve is selected as the response variable. These response values are then collected across all scenarios and used to develop a surrogate model via a three-factor design. These three factors are corrosion level, vessel mass, and vessel impact velocity, and the objective function is the residual capacity $N_c$ of the pier shaft. The scenario design matrix and resulting responses are presented in Figure 10.

With the corrosion level, vessel mass, and vessel impact velocity as the predictor variables and the residual load-carrying capacity of the pier shaft as the response, the data from Figure 10 are analyzed through an analysis of variance on the regression equation derived from the Box–Behnken sampling matrix. The results are summarized in

Table 4. Analysis of variance for the response surface regression model with three factors.

Source of Variation	Sum of Squares	Degrees of Freedom	Mean Square	F-Value	p-Value
Model	468.37	9	52.04	136.92	<0.0001
Corrosion level (A)	188.67	1	188.67	496.37	<0.0001
Tonnage (B)	62.44	1	62.44	164.28	<0.0001
Velocity (C)	105.85	1	105.85	278.49	<0.0001
AB	2.10	1	2.10	5.53	0.0509
AC	0.22	1	0.22	0.59	0.4662
BC	62.02	1	62.02	163.16	<0.0001
A2	12.53	1	12.53	32.96	0.0007
B2	0.67	1	0.67	1.77	0.2248
C2	35.72	1	35.72	93.97	<0.0001
Residual	2.66	7	0.38
Lack of Fit	2.66	3	0.89
Pure Error	0.00	4	0.00
Total	471.03	16

.

The coefficient of determination R-squared of 0.9944 and the adjusted R-squared of 0.9871 approach unity, indicating that the model shows over 98% of the variance in the response with high fitting accuracy. The predicted R-squared of 0.9096 differs from the adjusted R-squared by 0.0775, which is less than 0.2, demonstrating no evidence of overfitting and good agreement between predicted and experimental data. The signal-to-noise ratio far exceeds the threshold of 4, confirming that the model effectively distinguishes signal from noise.

From the results in Table 4, the p-value of the model is less than 0.001, indicating a highly significant fit and excellent model adequacy. According to the significance tests in the analysis of variance, any term with a p-value below 0.05 is considered significant and thus a primary contributor to model accuracy. The interaction terms AB, AC, and the quadratic term B² are identified as nonsignificant. Therefore, a polynomial regression retaining only the significant terms is performed to construct the response surface model. The residual load-carrying capacity $N_c$ is then expressed as

$$\begin{array}{c}{N}_c=38.52997-0.934278\cdot A+0.015035\cdot B+8.004\cdot C-0.0105\cdot B\cdot C\\ +0.030667\cdot A^2-1.864\cdot C^2\end{array}$$

(27)

The fitted response surfaces clearly illustrate the effects of each factor on the response. The response surfaces corresponding to different corrosion levels and vessel masses are shown in Figure 11.

4.7. Fragility Analysis

In this study, a rigorously defined structural performance metric is employed, specifically the residual vertical load-carrying capacity of the bridge pier, as the primary damage indicator to quantify the extent of damage following a vessel collision event. The prediction of this critical capacity metric across a diverse range of potential impact scenarios is efficiently obtained through the implementation of a response surface surrogate model. This sophisticated model serves as a computationally tractable approximation derived from more complex underlying structural analyses. To explicitly incorporate the inherent uncertainties associated with both the impacting vessel and the structural resistance characteristics, key parameters within the fragility analysis framework are considered as random variables. Focusing specifically on the prevalent Class III and IV vessels operating within the relevant waterways, the vessel mass is modeled with uncertainty. It is assumed to follow a normal distribution, with the distribution parameters calibrated based on the existing maximum design masses for these vessel classes on the specified waterways.

In this study, a coefficient of variation (COV) of 0.30 is adopted to reflect the anticipated variability in operational loading conditions. Furthermore, for each analytically defined damage state, e.g., minor spalling, severe cracking, or partial collapse, the sectional resistance corresponding to the onset of that state is calculated. This resistance is determined as the product of the initial resistance of the pier, undamaged load-carrying capacity, and a specific damage ratio associated with the damage state threshold. The uncertainty inherent in this structural resistance capacity is characterized by its COV, and its value is directly adopted from the statistically robust findings reported by Nowak et al. [38]. The extensive reliability database indicates a characteristic COV range of 0.080 to 0.085 for flexural structural members with longitudinal reinforcement ratios between 0.6% and 1.6%, which are representative of typical bridge pier construction.

The fragility function, formally denoted as $P_f$ is quantitatively defined as the conditional probability that the structural response induced by a vessel impact reaches or exceeds a predefined limit state $L_i$ with a specific damage state severity, given a specific vessel impact speed $V$. The estimation of this probability function is rigorously performed using Monte Carlo simulation techniques. The computational procedure involves systematically evaluating a large number of potential scenarios. For each discretely sampled vessel speed $V$ within the range of interest, a substantial number of Monte Carlo trials are executed. Within each individual trial, values for all the designated random variables are generated.

The impact-induced structural demand and the corresponding sectional resistance capacity for the limit state $L_i$ are then computed for that specific set of sampled parameters. The number of trials where the calculated demand meets or exceeds the calculated capacity for state $L_i$ is recorded. The fragility probability $P_f$ at speed $V$ is subsequently estimated as the simple ratio of this exceedance count to the total number of trials conducted at the given speed. This estimation process is repeated comprehensively across the entire spectrum of relevant vessel impact speeds. Finally, the derived fragility probabilities $P_f$ are graphically represented by plotting them against the corresponding vessel speed $V$ for each distinct limit state $L_i$. Fragility curves are then generated as functions of impact velocity to show the probability of reaching or exceeding each damage severity level. The complete set of resulting fragility curves, illustrating the vulnerability of the bridge pier to vessel collision under the proposed probabilistic framework, is presented in Figure 12.

Analysis of the vessel impact fragility curves presented in Figure 12 for distinct damage states reveals critical trends. Under the slight damage state, the probability of failure $P_f$ demonstrates asymptotic convergence towards 1.0, as both the corrosion level and the impact velocity increase concurrently. Conversely, for the collapse state, the failure probability $P_f$ remains proximate to zero across the vast majority of evaluated scenario combinations. The elevated corrosion level causes a substantial increase in the conditional failure probability for a given impact condition, with the magnitude of this increase showing a positive correlation with the severity of corrosion. Furthermore, the interaction between increasing impact velocity and growing corrosion significantly increases the rate of failure probability. Specifically, as the impact velocity continues to rise in conjunction with higher levels of corrosion, the resultant rate of growth in the computed failure probability undergoes a significant acceleration.

4.8. Vessel Impact Resilience Evaluation

The quantification of vessel impact resilience for the corroded bridge piers follows a rigorous computational methodology. Initial damage states induced by corrosion are obtained through nonlinear static analysis, which accurately characterizes the reduction in structural integrity due to material degradation prior to impact. Subsequently, nonlinear dynamic simulations are employed to determine performance degradation across diverse impact scenarios, providing the complex interaction between vessel collision dynamics and pre-existing corrosion damage. These computational results are systematically integrated with damage state-related recovery times and mathematical recovery function models, e.g., triangular and negative exponential formulations, to quantify the comprehensive vessel impact resilience metric, as represented in Figure 13.

Examination of Figure 5 and Equation (16) reveals that corrosion severity has a deterministic relationship with initial post-impact functionality loss. Specifically, more severe corrosion states correlate directly with greater immediate reductions in resilience, agreeing well with the damage levels quantified through static structural analysis. From Figure 13, as impact velocity increases, impact loading increases, and the vessel impact resilience undergoes a systematic decrease. Under the same impact loading conditions, higher corrosion levels consistently yield reduced resilience indices. However, the influence of corrosion severity on the resilience metric diminishes progressively. For instance, as the corrosion level increases from 0.0% to 7.5%, a substantially larger resilience reduction appears from 7.5% to 15%. This nonlinear behavior reflects the influence of corrosion progression on residual load-carrying capacity degradation.

For different recovery models, resilience indices obtained from the negative exponential recovery function, which models scenarios with immediate resource deployment, consistently exceed those computed by using the triangular recovery function. The latter assumes the constrained initial resources followed by gradual mobilization of repair capabilities, resulting in systematically lower resilience estimates. This divergence underscores the critical operational significance of pre-positioned emergency resources for optimizing post-impact functional recovery.

Under conditions of corrosion levels of 0.0%, 2.5%, 5.0%, 7.5%, 10.0%, 12.5%, and 15.0%, the vessel impact resilience of the pier is evaluated under different repair strategies. By using the proposed method and assuming ample resources for rapid restoration with the negative exponential recovery function, the impact resilience of the pier can be significantly improved, as shown in Figure 14.

The quantitative relationship between the resilience index $R$ and the choice of recovery function model has a significant influence on the magnitude of $R$ itself. When $R$ approaches its theoretical maximum value of 1.0, representing near complete functional recovery, the selection of the recovery function model, whether negative exponential or triangular, has negligible influence on the obtained resilience metric. However, as $R$ decreases, indicating more severe post-impact functional loss, a systematic and progressively widening divergence appears between the resilience indices obtained from the two recovery models. Specifically, resilience values obtained from the negative exponential recovery function consistently exceed those calculated by using the triangular recovery function. This difference reaches its maximum magnitude, quantified at up to 67% higher resilience increase, under the conditions corresponding to the lowest $R$ values.

This empirical comparison shows a critical operational principle, i.e., the pre-positioning and optimization of post-impact recovery resources, including strategically located emergency material stockpiles, pre-contracted rapid repair technologies, and mobilized technical response teams, fundamentally affect the restoration scheme. Using a negative exponential recovery function, which more accurately reflects the scenarios characterized by immediate resource availability and accelerated initial recovery phases, provides an objective quantitative measure. The results demonstrate how optimized resource allocation directly enhances a bridge’s functional recovery trajectory following an impact event, particularly under high damage scenarios where conventional recovery models underestimate achievable resilience. The proposed model thus serves as an effective tool for infrastructure resilience planning and resource prioritization.

5. Conclusions

This study proposes a quantitative framework for evaluating the vessel impact resilience of the corroded reinforced concrete bridges. The proposed method employs a computationally efficient and simplified finite element model explicitly integrating critical corrosion-induced degradation mechanisms. These mechanisms consider the reduction in effective reinforcement cross-sectional area, decreased yield strength of steel rebars, deterioration of bond slip behavior at the rebar–concrete interface, and the progressive weakening of the concrete cover surrounding the reinforcement. This study uses equivalent static analysis, a validated simplification for impact scenarios, to simulate vessel collisions across a comprehensive scenarios of vessel masses and impact velocities.

From the finite element simulations, a probabilistic framework is implemented. Within this framework, a response surface methodology is applied to replace a closed-form limit state equation characterizing the structural performance under impact loads. Extensive Monte Carlo sampling techniques are then employed for the given limit state equation to compute robust failure probabilities. These probabilities are evaluated across varying degrees of corrosion levels and impact velocities. The critical resilience metric is then obtained by integrating post-impact functional recovery, modeled through two distinct mathematical representations, i.e., a linear triangular recovery function and a negative exponential recovery function. From the results obtained from the case study, the following conclusions are noted:

• The study confirms a statistically significant inverse relationship between the residual load-carrying capacity of the bridge structure and the corrosion level. Notably, the rate of capacity degradation shows a nonlinear relationship, demonstrating a significant decrease at elevated corrosion levels, compared to initial corrosion stages. Concurrently, the functional loss index has a positive correlation with increasing corrosion level. The progression of functional loss is characterized by an initially rapid increase phase, transitioning towards a more gradual asymptotic increase as cumulative damage grows under higher corrosion states.
• Reinforcement corrosion increases structural vulnerability to vessel impact. The computed failure probabilities demonstrate a substantial increase with rising corrosion level. Crucially, as the kinetic energy of the impacting vessel increases, structures with higher corrosion levels have failure probabilities asymptotically approaching unity. This trend shows the paramount importance of implementing a proactive, scheduled maintenance scheme, particularly for bridges with extended service lives, which are inherently more susceptible to corrosion progression.
• The vessel impact resilience index indicates a clear and quantifiable decline as corrosion severity increases. Furthermore, the choice of recovery function significantly influences the calculated resilience metric. The negative exponential recovery model yields higher resilience indices compared to the triangular model across the investigated parameter space. The difference between the two recovery models could reach a substantial 67% relative improvement under the conditions corresponding to low resilience levels. The results highlight the critical operational significance of strategically pre-positioning post-event recovery resources to maximize the efficiency and speed of functional restoration following an impact event, particularly for corroding structures.