Risk-Based Cost–Benefit Analysis of Ultra-High-Performance Concrete Bridge Columns Under Seismic Hazards and Corrosion

Abstract

The deployment of ultra-high-performance concrete (UHPC) is a strategic response to the urgent need for advanced building materials, particularly for the repair and enhancement of aging infrastructure. Highway bridges, which are constantly subjected to high stress, heavy usage, and corrosive environments, can be ideal candidates for UHPC application. The material’s exceptional abrasion resistance and ability to withstand severe weather conditions make it a compelling choice for projects where frequent renovation or maintenance is impractical. This study presents a risk-based cost–benefit analysis (RCBA) comparing UHPC reinforced bridge columns to conventional concrete reinforced bridge columns, focusing on seismic and corrosion hazards. While UHPC has a significantly higher initial material cost than traditional concrete, a simple comparison of initial costs alone is misleading. The RCBA methodology generally evaluates life-cycle cost, including initial construction, long-term agency costs, and user costs. The central question—whether UHPC’s superior performance justifies its higher initial investment—is addressed through RCBA. The presented RCBA is formulated as the ratio of the total life-cycle cost of conventional concrete to that of UHPC. The benefit is estimated as the difference in cumulative risks between bridges with conventional concrete and UHPC bridge columns, with fragility analysis conducted under seismic and corrosion hazards. The proposed approach is illustrated using an existing bridge located in Republic of Korea.

1. Introduction

Civil infrastructure in both developed and developing countries faces the challenge of managing an aging inventory of structures, particularly highway bridges—vital economic and transportation conduits [1,2,3]. These critical assets are constantly exposed to high stress, heavy usage, and severe environmental degradation, making conventional repair cycles economically and logistically impractical over the long term. This reality drives the adoption of advanced construction materials, most notably ultra-high-performance concrete (UHPC), a material class that fundamentally redefines the performance of cementitious composites [4,5,6,7].

UHPC represents a major advancement in materials science, characterized by superior compressive and tensile strengths, exceptional durability, and remarkable ductility. Unlike conventional concrete, which typically achieves compressive strengths of 28–41 MPa, UHPC exhibits minimum compressive strengths exceeding 120 MPa. This high strength, combined with its unique strain-hardening behavior, allows it to absorb energy and resist sudden failure, making it particularly suitable for applications in seismically active regions. The exceptional performance of UHPC stems from its meticulously engineered, highly dense, and non-porous microstructure [5,8,9]. Optimized particle packing effectively eliminates interconnected capillary pores, rendering the material nearly impervious to moisture and aggressive agents such as chlorides. This characteristic provides outstanding protection to embedded steel reinforcement against chloride-induced corrosion, a leading cause of deterioration and structural vulnerability in conventional reinforced concrete structures. Furthermore, UHPC demonstrates nearly twice the abrasion resistance of ordinary concrete, further extending its service life [6,7,10].

Despite its clear performance advantages, the most significant barrier to the widespread adoption of UHPC remains its high initial material cost compared to traditional concrete [5,6]. A simple comparison of initial construction costs alone is misleading and fails to reflect the material’s true economic value. Accordingly, the central question is whether UHPC’s two primary benefits, (a) superior initial seismic capacity from its high strength and ductility, and (b) superior long-term durability and corrosion resistance, justify its substantially higher upfront investment. The most appropriate framework for evaluating this trade-off is a life-cycle cost analysis (LCCA) or, more specifically, a risk-based cost–benefit analysis (RCBA), which assesses life-cycle cost, including initial construction, long-term agency, and user costs [3,11,12,13]. The economic advantages of UHPC primarily arise from reduced maintenance requirements, extended service life, and a corresponding decrease in cumulative future risks.

The assessment of long-term structural performance, particularly under the combined effects of deterioration and extreme events, has been a central topic in structural reliability and infrastructure management research [3,12]. Numerous studies have emphasized the significance of time-dependent deterioration in evaluating the seismic vulnerability of reinforced concrete (RC) structures. Among various deterioration mechanisms, chloride-induced corrosion is widely recognized as the primary cause of degradation in conventional concrete structures [14,15,16]. The subsequent corrosion propagation is typically modeled as uniform and/or pitting corrosion, which governs the loss of reinforcement cross-sectional area over time [17].

Incorporating these deterioration processes, time-dependent seismic risk assessment requires the integration of deterioration modeling, seismic vulnerability analysis, and time-dependent risk quantification [1,15,17,18]. Seismic vulnerability is commonly characterized by a seismic fragility function, which defines the conditional probability that seismic demand exceeds structural capacity for a given ground motion intensity. Previous research has demonstrated that, for aging RC bridges, seismic fragility progressively increases with corrosion-induced strength loss, thereby elevating the associated seismic risk over time. In this context, risk is typically defined as the product of the probability of failure (i.e., exceedance of a specific damage state) and the corresponding consequence (i.e., monetary loss) of that failure. Risk analysis has been extensively applied at both project and network levels under various hazard conditions, including fatigue, corrosion, flooding, and earthquakes [12,19,20,21,22].

The application of risk-based cost–benefit analysis (RCBA) has emerged as a critical framework for evaluating and managing complex engineering projects, particularly for long-term strategic decision-making in infrastructure systems such as bridges and transportation networks. RCBA fundamentally compares the costs of an intervention (e.g., structural strengthening or the adoption of advanced materials) with the long-term benefits derived from reduced repair costs and minimized disruption due to hazards. Previous RCBA studies including [11,13] have primarily focused on retrofitting existing bridges subjected to extreme events or on seismic life-cycle cost analyses of aging structures. In all such approaches, the cumulative expected risk over a structure’s target service life serves as the key metric for quantifying the benefit of the intervention.

The primary objective of this study is to directly address this question by proposing a rigorous risk-based cost–benefit analysis (RCBA) framework to compare UHPC bridge columns with conventional reinforced concrete (RC) columns under coupled seismic and corrosion hazards. While the most appropriate framework for this trade-off is an RCBA, most previous research has focused on the deterioration of conventional concrete. The quantitative integration of UHPC’s anti-corrosion benefits into a time-dependent seismic RCBA framework, specifically to justify its high initial cost, has not been sufficiently addressed.

To fill this gap, the proposed methodology makes several key contributions. First, it formulates the RCBA as a direct ratio of benefit to additional cost, where the benefit represents the difference in cumulative expected seismic risks over a defined service life. Second, the methodology integrates time-dependent corrosion modeling, which is significant for conventional concrete but negligible for UHPC, with time-dependent seismic fragility and risk assessments. This integration allows for the direct quantification of the long-term reduction in seismic vulnerability resulting from UHPC’s superior durability. Third, the study demonstrates that the cost-effectiveness of UHPC is highly influenced by regional seismicity and target service life, providing a practical decision-support tool. The proposed approach is illustrated through a detailed case study of an existing RC bridge in Republic of Korea.

The technical foundation for this RCBA framework relies on integrating time-dependent deterioration modeling with seismic vulnerability analysis. In conventional concrete, chloride-induced corrosion is the primary cause of degradation, leading to a loss of reinforcement cross-sectional area over time. This, in turn, increases the seismic fragility and associated risk of the structure. In this context, risk is defined as the product of the probability of failure and the corresponding monetary loss, and it serves as the key metric for quantifying the long-term benefit of an intervention.

2. Ultra High-Performance Concrete

UHPC is an advanced class of cementitious composites that represents a major leap in materials science, fundamentally redefining the performance of concrete. It is distinguished by superior compressive and tensile strengths, exceptional durability, and remarkable ductility, enabling it to withstand heavy loads and resist environmental degradation far more effectively than conventional concrete [4,5,6]. The core constituents include Portland cement, very fine sand, and water. Unlike traditional concrete, UHPC contains a much higher cement content and an extremely low water-to-binder ratio [8]. Supplementary cementitious materials are essential to the UHPC matrix, with silica fume being the most critical. Its effectiveness arises from filling, nucleation, and pozzolanic effects. The deliberately low water-to-binder ratio provides the basis for UHPC’s strength and durability but also renders the fresh mix unworkable without chemical admixtures [23]. To address this, superplasticizers (high-range water-reducing admixtures) are incorporated, producing a flowable, low-viscosity system. These admixtures allow UHPC to behave as a self-consolidating material, facilitating placement in intricate forms and heavily reinforced areas with minimal effort [8].

The compressive strength of UHPC is its most recognized and defining property. Conventional concrete typically achieves 28–41 MPa, while high-performance concrete (HPC) ranges from 50 to 100 MPa. In contrast, UHPC exhibits a minimum compressive strength of 120 MPa, owing to its low water-to-binder ratio and meticulously engineered dense microstructure. Beyond compressive capacity, UHPC reinforced with fibers can achieve tensile strengths exceeding 10 MPa and, in some cases, reaching up to 19 MPa [6,23,24]. This performance is attributed to its unique strain-hardening behavior, which enables the material to absorb energy and resist sudden failure. Such characteristics are critical for structural resilience and make UHPC particularly well suited for applications in seismic regions [5,25,26].

The long-term advantage of UHPC lies in its exceptional durability, derived from its highly dense, non-porous microstructure. Optimized particle packing eliminates interconnected capillary pores, rendering the material nearly impervious to moisture, chlorides, and other aggressive agents. This extremely low permeability protects steel reinforcement from chloride-induced corrosion, one of the primary causes of deterioration in conventional concrete. In addition, UHPC demonstrates nearly twice the abrasion resistance of ordinary concrete, further enhancing its longevity [5,8].

The most significant barrier to the widespread adoption of UHPC is its high initial cost, which can make it prohibitive for projects that do not demand its specialized properties. However, the economic case for UHPC is better assessed through a lifecycle cost analysis. Its long-term value lies in reduced maintenance, extended service life, and lower material volume requirements, which can generate substantial overall savings [27,28].

3. Corrosion Initiation and Propagation

Corrosion initiates when aggressive agents, primarily chloride ions, penetrate the concrete and damage the passive layer that naturally protects the steel reinforcement. Corrosion initiation time can be modeled based on Fick’s second law [29] and the premise that the chloride concentration reaches the critical one at a depth of the concrete cover from zero at the surface [30]. Additionally, considering the effects of temperature and relative humidity at the site of the concrete structures on the chloride diffusion, the corrosion initiation time t_in is estimated as [2]

$$t_{in}={\left\{{\displaystyle \frac{{d_c}^2}{4k_e{k}_c{k}_t{D}_c(t,T,RH)t_0^m}}{\left[er{f}^{-1}(1-{\displaystyle \frac{C{l}_c}{C{l}_s}})\right]}^{-2}\right\}}^{1/(1-m)}$$

(1)

where k_e, k_c, and k_t are the environmental factor, the curing time correction factor and the testing method factor, respectively, d_c (mm) is the concrete cover depth, D_c(t, T, RH) is the adjusted chloride diffusion coefficient considering time t, temperature T, and relative humidity RH, t₀ is the reference time period, m is the aging factor, erf⁻¹ is the inverse error function, and Cl_c (% of binder mass) and Cl_s (% of binder mass) are the critical and surface chloride concentrations, respectively. The surface chloride concentrations Cl_s is computed as [14]

$$C{l}_s=A_{cs}\times w/c+{\epsilon }_{cs}$$

(2)

where A_cs is the regression parameter, w/c is the water–cement ratio, and ε_cs is the error factor.

The corrosion propagation after corrosion initiation can be estimated by the uniform and/or pitting corrosion models [3,15,17,31,32]. The remaining reinforcement cross-sectional area A_r(t) (mm²) at time t, which is based on the combined uniform and pitting corrosion models, is expressed as [31]

$$A_r(t)=\left[A_{\mathit{ru}}(t)-A_0\right](1-{\displaystyle \frac{w_p}{2D_0}})+A_{rp}(t)$$

(3)

where A_ru(t) (mm²) and A_rp(t) (mm²) are the time-dependent remaining reinforcement cross-sectional area based on the uniform and pitting corrosion models, respectively, A₀ (mm²) is the initial cross-sectional area of reinforcement, w_p is the pit width, and D₀ (mm) is the initial diameter of the reinforcement. The formulations of A_ru(t) and A_rp(t) can be found in [3].

The corrosion rate, a quantitative measure of electrochemical activity, represents the speed of corrosion propagation. This involves the formulation of the uniform and pitting corrosion propagation models. Considering the environmental conditions, the corrosion rate at time t r_corr(t) (mm/year) can be estimated as [18]

$$r_{corr}(t)=i_{corr}(t)f_T{f}_{RH}{f}_{Cl} \mathrm{for}t>t_{\mathit{in}}$$

(4)

where i_corr(t) (μA/cm²) is the corrosion current density at time t, and f_T, f_RH and f_Cl are the factors associated with temperature, relative humidity, and chloride concentration, respectively. The details on the computations of i_corr(t), f_T, f_RH and f_Cl can be found in [16].

In conventional concrete, corrosion is a major vulnerability due to its inherent porosity. However, the high density and low porosity of UHPC can delay corrosion initiation and propagation [10]. This difference can be characterized by the adjusted chloride diffusion coefficient D_c [see Equation (1)], surface chloride concentration Cl_s [see Equation (1)], critical chloride concentration Cl_c [see Equation (1)], and corrosion current density i_corr(t) [see Equation (4)] [7,10].

4. Time-Dependent Seismic Risk Assessment

Risk has been treated as one of the representative probabilistic performance indicators for managing infrastructures under various normal and abnormal deterioration conditions and natural hazards [3,17]. It is typically calculated as the product of the probability of failure (e.g., the likelihood of a bridge collapsing) and the consequences of that failure (e.g., loss of life, economic disruption). Risk can be used at both the project level (for a single asset like a specific bridge or a section of a road) and the network level (for a portfolio of assets, such as all bridges in a network). Risk analysis has been applied to a wide range of conditions including normal deterioration such as fatigue and corrosion, and unexpected events including floods, earthquakes, extreme weather, and accidents [19,20,21,22,33,34]. Based on risk analysis, managers can develop mitigation strategies and enhance the resilience of infrastructure systems. The time-dependent seismic risk assessment requires (a) a time-dependent deterioration process considering corrosion and/or fatigue, (b) a seismic vulnerability assessment, and (c) a time-dependent risk assessment integrating (a) and (b). In this study, the time-dependent deterioration process is represented by corrosion initiation and propagation in RC structures described in the previous section.

4.1. Seismic Vulnerability Assessment

The seismic vulnerability is estimated using the seismic fragility function, which indicates the conditional probability that the seismic demand exceeds the seismic capacity for a given ground motion intensity. The seismic fragility function can be expressed as [35]

$$P\left(D\ge C\left|IM\right.\right)=\Phi \left[{\displaystyle \frac{ln\left(M_d\right)-ln\left(M_c\right)}{\sqrt{{\left({\sigma }_d\right)}^2+{\left({\sigma }_c\right)}^2}}}\right]$$

(5)

where D = seismic demand; C = structural capacity; IM = ground motion intensity (i.e., peak ground acceleration, PGA); M_d = median of the seismic demand; M_c = median of the seismic capacity; σ_d = standard deviation (SD) of the seismic demand; σ_c = SD of the seismic capacity; and Φ(·) = standard normal cumulative distribution function. M_c and σ_c in Equation (5) are predefined considering the damage states.

In this study, the time-dependent seismic vulnerability assessment using Equation (5) accounts for corrosion-induced deterioration over time. The median of the seismic demand in Equation (5) (i.e., M_d) is evaluated through nonlinear time-history analysis of a three-dimensional (3D) finite element (FE) model under selected ground motions. The 3D FE analysis is performed at each time interval considering the corrosion initiation and propagation. Through regression of the outcomes of the 3D FE analysis at predefined time intervals, the time-dependent median of seismic demand M_d(t) can be expressed as [36]

$$ln\left(M_d(t)\right)=a(t)+b(t)\cdot ln(IM)$$

(6)

where a(t) and b(t) are regression coefficients at time t. The SD of the seismic demand σ_d is obtained as [37]

$${\sigma }_d(t)=\sqrt{{\displaystyle \frac{{\displaystyle \sum _{i=1}^{N_g}{\left[ln\left(S_i(t)\right)-\left(a(t)+b(t)\cdot ln(I{M}_i)\right)\right]}^2}}{N_g-2}}}$$

(7)

where N_g = number of ground motions to be considered; and S_i(t) is the seismic response associated with the ith ground motion, which is obtained from the nonlinear time-history analysis of 3D FE.

4.2. Time-Dependent Seismic Risk

The seismic risk corresponding to a specific damage state at time t, R(t)_DS,i, is defined as the occurrence probability of an earthquake causing that damage state and the associated monetary loss, which is expressed as

$$R{(t\left|PGA\right.)}_{DS,i}=C_{lss|DS,i}(t)\cdot P\left(D{S}_i\left(t\right)\left|PGA\right.\right)$$

(8)

where C_lss|DSi(t) is the monetary loss for the ith damage state DS_i at time t, and P(DS_i(t)|PGA) indicates the probability of occurrence of the ith damage state for a given PGA. This probability can be estimated using the seismic fragility function of Equation (5). Considering the multiple damage states, the total seismic risk R(t|PGA) can be estimated as [38,39]

$$R(t\left|PGA\right.)={\displaystyle \sum _{i=1}^{N_D}R{(t\left|PGA\right.)}_{DS,i}}$$

(9)

where N_D is the number of damage states to be considered. For the upper and lower bounds of PGA (i.e., denoted as x_u and x_l, respectively), the expected total seismic risk R(t) is expressed as

$$R\left(t\right)={\displaystyle \underset{x_l}{\overset{x_u}{\int }}R\left(t\left|PGA=x\right.\right)\left|P_{exd}’\left(x\right)\right|dx}$$

(10)

where |P_exd’(x)| is the first derivative of the annual probability of exceeding a specific PGA with respect to the ground acceleration variable x.

4.3. Monetary Loss

The monetary loss for the ith damage state C_lss|DSi(t) in Equation (8) can be estimated as the sum of direct and indirect monetary losses as [3].

$$C_{lss|D{S}_i}(t)=C_{lss,dir|D{S}_i}(t)+C_{lss,ind|D{S}_i}(t)$$

(11)

where C_lss,dir|DSi(t) and C_lss,ind|DSi(t) are the direct and indirect monetary losses for the ith damage state, respectively. Direct monetary loss C_lss,dir|DSi(t) is the immediate and tangible expense directly associated with repairing or replacing the damaged structure. It typically includes repair, replacement, demolition and debris removal, and emergency response. Indirect monetary loss C_lss,ind|DSi(t) represents the broader economic and societal impacts resulting from structural failure. Typical components include loss of functionality, user costs, economic losses, social costs, and environmental costs. The details on the computations of C_lss,dir|DSi(t) and C_lss,ind|DSi(t) are available in [38,40]. It should be noted that the monetary loss C_lss|DSi(t) reflects future consequences, and is affected by the annual discount rate of money [40].

5. Cost–Benefit Analysis Based on Risk

RCBA is a critical framework for evaluating and managing engineering projects and critical infrastructure, such as bridges, dams, and transportation networks. This approach is especially valuable for strategic, long-term decision-making in the face of chronic deterioration and extreme events. At its core, RCBA compares the costs of strengthening infrastructure assets with the long-term benefits of reducing repair expenses and minimizing disruptions from natural hazards [1,3,13]. While the financial benefits of strengthening can be substantial, they are also subject to uncertainty.

In this study, the benefit C_ben is defined as the difference in the cumulative risks (including initial cost) for conventional concrete and UHPC bridge columns. The cumulative expected seismic risk R_cum is the integration of the expected total seismic risk R(t) over a predefined target service life t_tg as

$$R_{cum}={\displaystyle {\int }_0^{t_{tg}}R\left(t\right)}dt$$

(12)

As a result, the expected benefit E(C_ben) and expected benefit ratio E(R_ben) from the use of UHPC can be expressed as

$$E\left(C_{ben}\right)=\left(R_{cum,CC}+C_{ini,CC}\right)-\left(R_{cum,UHPC}+C_{ini,UHPC}\right)$$

(13)

$$E\left(R_{ben}\right)={\displaystyle \frac{R_{cum,CC}+C_{ini,CC}}{R_{cum,UHPC}+C_{ini,UHPC}}}$$

(14)

The application of UHPC instead of conventional concrete for bridge columns is considered cost-effective if E(C_ben) > 0 or E(R_ben) > 1.0.

The computational flowchart illustrates a four-stage process for the RCBA, which is performed in parallel for both conventional concrete and UHPC bridge columns, as shown in Figure 1. Stage 1 involves the time-dependent deterioration modeling, where material and environmental inputs are used to calculate corrosion initiation and propagation over time, resulting in the time-dependent remaining reinforcement area. This output feeds into Stage 2, the time-dependent seismic vulnerability assessment, which uses a 3D FE model and ground motion records to generate time-dependent fragility curves. In Stage 3, the time-dependent seismic risk assessment calculates the risk for specific damage states and integrates this risk across all damage states and possible PGAs to determine the expected total seismic risk R(t). Stage 4 is the RCBA, which calculates the cumulative expected total seismic risk R_cum for both CC and UHPC by integrating the R(t) from Stage 3 over the target service life t_tg. This cumulative risk is added to the initial construction cost for each material to find their total life-cycle costs. Finally, the expected benefit ratio E(R_ben) is calculated by dividing the total life-cycle cost of conventional concrete by that of UHPC. If this ratio is greater than 1.0, the UHPC application is determined to be cost-effective. While this study provides a rigorous RCBA framework for comparing initial material choices, the presented investigations do not address the influence of preventive and essential maintenance interventions on the corrosion process, seismic risk assessment, or the RCBA.

6. Illustrative Example

The proposed approach is applied to an existing RC bridge column located in Gwangyang City, Jeollanam-do, Republic of Korea, which is subjected to corrosion and earthquakes. The bridge consists of three spans, two parallel girders, and two columns. The overall bridge configuration and its associated 3D FE model are presented in [13,17]. Since the bridge columns are considered as the most critical components against corrosion and earthquakes [13,17,41,42], the proposed approach is based on time-dependent seismic vulnerability and risk assessments associated with the use of conventional concrete and UHPC in bridge columns. Subsequently, RCBA is performed to support decision-making on whether the use of UHPC is appropriate under various seismic scenarios.

6.1. Time-Dependent Fragility Analysis

Time-dependent fragility analysis, representing seismic vulnerability, is computed using Equation (5), considering the corrosion initiation and propagation in bridge columns. The corrosion initiation and propagation are estimated using Equations (1)–(4) with the deterministic and probabilistic variables for conventional concrete and UHPC defined in Table 1. More information on modeling and associated parameters related to corrosion initiation and propagation can be found in [17]. The mean and standard deviation of the corrosion initiation time for bridge columns with conventional concrete are 11.14 years and 4.49 years, respectively. For UHPC bridge columns, the corrosion initiation time is estimated to have a mean greater than 500 years, indicating that no corrosion will occur within the next century of service. Similar results, showing corrosion initiation times exceeding 100 years, have also been reported by [9,24], among others. Figure 2 shows the expected remaining area of a single reinforcement in bridge columns with conventional concrete and UHPC. It can be seen that, once corrosion initiates, the remaining area of a single reinforcement bar in bridge columns with conventional concrete decreases over 50 years, whereas in UHPC bridge columns the reduction in reinforcement area is negligible. This is because UHPC prevents corrosion initiation and, consequently, corrosion propagation of the reinforcement.

Considering corrosion initiation and propagation for both conventional concrete and UHPC, the nonlinear time-history analysis of the 3D FE model is performed to determine the median of seismic demand M_d in Equation (5), where 30 natural earthquake records obtained from the PEER Database [43] are employed. The material properties of conventional concrete and UHPC used for the 3D FE model are provided in Table 2. From the nonlinear time-history analysis of the 3D FE model, the median and SD of seismic demand (i.e., denoted as M_d and σ_d, respectively, in Equation (5)) are obtained. The corresponding information, including PGA values and seismic magnitudes, is summarized in [17]. The four damage states (i.e., slight, moderate, extensive, and complete damage states) are considered in the seismic fragility analysis. These damage states are represented by the displacement ductility of the bridge column, defined as the ratio of the maximum top displacement of the column to the displacement at the same location when the reinforcement first yields. The displacement associated with this ductility is estimated in the longitudinal direction, parallel to the seismic input. The medians of the seismic capacities and SD (i.e., denoted as M_c and σ_c, respectively) of the bridge column displacement ductility associated with slight, moderate, extensive, and complete damage states are provided in Table 3.

Table 1. Deterministic and probabilistic variables for estimating the corrosion initiation and propagation.

Deterministic and Random Variables	Notation	Mean	Standard Deviation	Type of Distribution
Curing time correction factor	kc	1.0	-	Deterministic
Environmental factor	ke	0.265	0.045	Gamma
Testing method factor	kt	0.832	0.024	Normal
Concrete cover depth	dc (mm)	40	4	Lognormal
Aging factor	m	0.105	0.0315	Beta(9.3; 52.7; 0; 0.7)
Reference time	t0 (years)	0.0767	-	Deterministic
Temperature for estimating Dc	T (°C)	20	-	Deterministic
Relative humidity for estimating Dc	RH	0.75	-	Deterministic
Regression parameter used for estimating the surface chloride concentration Cls	Acs	7.758	0.05	Normal
Error term used for estimating the surface chloride concentration Cls	ɛcs	0	0.4	Normal
Water-cement ratio for conventional concrete	w/c	0.5	-	Deterministic
Water-cement ratio for UHPC	w/c	0.33	-	Deterministic
Critical chloride concentration for conventional concrete	Clc (mass % of binder)	0.5	-	Deterministic
Critical chloride concentration for UHPC	Clc (mass % of binder)	1.45	-	Deterministic
Initial diameter of reinforcement	D0 (mm)	32.2	-	Deterministic

Note: Beta distribution $f(x,\alpha ,\beta ,a,b)={\displaystyle \frac{{(x-a)}^{\alpha -1}{(b-x)}^{\beta -1}}{{(b-a)}^{\alpha +\beta -1}B(\alpha ,\beta )}}$ where α = 9.3, β = 52.7, a = 0, b = 0.7, and B is the beta function. Based on information provided in [2,11,14,18,44,45].

Table 1. Deterministic and probabilistic variables for estimating the corrosion initiation and propagation.

Deterministic and Random Variables	Notation	Mean	Standard Deviation	Type of Distribution
Curing time correction factor	kc	1.0	-	Deterministic
Environmental factor	ke	0.265	0.045	Gamma
Testing method factor	kt	0.832	0.024	Normal
Concrete cover depth	dc (mm)	40	4	Lognormal
Aging factor	m	0.105	0.0315	Beta(9.3; 52.7; 0; 0.7)
Reference time	t0 (years)	0.0767	-	Deterministic
Temperature for estimating Dc	T (°C)	20	-	Deterministic
Relative humidity for estimating Dc	RH	0.75	-	Deterministic
Regression parameter used for estimating the surface chloride concentration Cls	Acs	7.758	0.05	Normal
Error term used for estimating the surface chloride concentration Cls	ɛcs	0	0.4	Normal
Water-cement ratio for conventional concrete	w/c	0.5	-	Deterministic
Water-cement ratio for UHPC	w/c	0.33	-	Deterministic
Critical chloride concentration for conventional concrete	Clc (mass % of binder)	0.5	-	Deterministic
Critical chloride concentration for UHPC	Clc (mass % of binder)	1.45	-	Deterministic
Initial diameter of reinforcement	D0 (mm)	32.2	-	Deterministic

Note: Beta distribution $f(x,\alpha ,\beta ,a,b)={\displaystyle \frac{{(x-a)}^{\alpha -1}{(b-x)}^{\beta -1}}{{(b-a)}^{\alpha +\beta -1}B(\alpha ,\beta )}}$ where α = 9.3, β = 52.7, a = 0, b = 0.7, and B is the beta function. Based on information provided in [2,11,14,18,44,45].

Table 2. Mechanical properties of conventional concrete and UHPC used for the 3D FE model with machine learning.

	Parameters (Notation, Units)	Mean	Coefficient of Variation	Type of Distribution
Conventional concrete	Rebar yield strength (MPa)	475.7	0.2	Normal
	Concrete compressive strength (MPa)	41.9	0.2	Lognormal
	Ultimate strain of concrete	0.005	0.2	Lognormal
	Elastic modulus (GPa)	31.2	0.2	Normal
	Tensile fracture energy (MPa)	0.14	0.2	Normal
	Compressive fracture energy (MPa)	35.7	0.2	Normal
	Cement (kg∕m3)	424	0.2	Normal
	Coarse aggregate (kg∕m3)	758	0.2	Normal
	Fine aggregate (kg∕m3)	854	0.2	Normal
	Water (kg∕m3)	228	0.2	Normal
UHPC	Rebar yield strength (MPa)	475.7	0.2	Normal
	Concrete compressive strength (MPa)	180	0.2	Lognormal
	Ultimate strain of concrete	0.005	0.2	Lognormal
	Elastic modulus (GPa)	53.5	0.2	Normal
	Tensile fracture energy (MPa)	14	0.2	Normal
	Compressive fracture energy (MPa)	180	0.2	Normal
	Cement (kg∕m3)	712	0.2	Normal
	Silica sand (kg∕m3)	1020	0.2	Normal
	Glass powder (kg∕m3)	211	0.2	Normal
	Water (kg∕m3)	109	0.2	Normal

Note: Based on information provided in [7,13,15,22,46].

Table 2. Mechanical properties of conventional concrete and UHPC used for the 3D FE model with machine learning.

	Parameters (Notation, Units)	Mean	Coefficient of Variation	Type of Distribution
Conventional concrete	Rebar yield strength (MPa)	475.7	0.2	Normal
	Concrete compressive strength (MPa)	41.9	0.2	Lognormal
	Ultimate strain of concrete	0.005	0.2	Lognormal
	Elastic modulus (GPa)	31.2	0.2	Normal
	Tensile fracture energy (MPa)	0.14	0.2	Normal
	Compressive fracture energy (MPa)	35.7	0.2	Normal
	Cement (kg∕m3)	424	0.2	Normal
	Coarse aggregate (kg∕m3)	758	0.2	Normal
	Fine aggregate (kg∕m3)	854	0.2	Normal
	Water (kg∕m3)	228	0.2	Normal
UHPC	Rebar yield strength (MPa)	475.7	0.2	Normal
	Concrete compressive strength (MPa)	180	0.2	Lognormal
	Ultimate strain of concrete	0.005	0.2	Lognormal
	Elastic modulus (GPa)	53.5	0.2	Normal
	Tensile fracture energy (MPa)	14	0.2	Normal
	Compressive fracture energy (MPa)	180	0.2	Normal
	Cement (kg∕m3)	712	0.2	Normal
	Silica sand (kg∕m3)	1020	0.2	Normal
	Glass powder (kg∕m3)	211	0.2	Normal
	Water (kg∕m3)	109	0.2	Normal

Note: Based on information provided in [7,13,15,22,46].

Table 3. Median of the seismic capacities and SD of the bridge column displacement ductility.

Damage State	Median of the Seismic Capacities Mc		SD of Seismic Capacities σc
	Conventional Concrete	UHPC
Minor	0.039	0.11	0.59
Moderate	1.25	1.67	0.51
Major	3.00	4.75	0.64
Collapse	4.99	6.40	0.65

Note: Based on information provided in [37].

Table 3. Median of the seismic capacities and SD of the bridge column displacement ductility.

Damage State	Median of the Seismic Capacities Mc		SD of Seismic Capacities σc
	Conventional Concrete	UHPC
Minor	0.039	0.11	0.59
Moderate	1.25	1.67	0.51
Major	3.00	4.75	0.64
Collapse	4.99	6.40	0.65

Note: Based on information provided in [37].

Figure 3 and Figure 4 show the fragility curves of the bridge column constructed with conventional concrete and UHPC, respectively, at the initial state, 25 years, and 50 years. These curves represent the probability of exceeding a specific damage state with respect to PGA, considering four damage states. As shown in Figure 3 and Figure 4, the exceedance probabilities are the largest for the slight damage state and the smallest for the complete damage state among the four damage states. When conventional concrete is used, the exceedance probability for a given PGA increases from the initial state to 50 years [see Figure 3]. In contrast, when UHPC is employed, there is no significant change in the exceedance probability over time. This is because the remaining reinforcement area in conventional concrete columns decreases with time, while that in UHPC columns remains nearly constant, as shown in Figure 2.

In order to efficiently obtain seismic fragility, the regression coefficients a(t) and b(t) in Equation (6) are expressed as closed-form functions. In this study, these coefficients are determined using a machine learning (ML) approach based on XGBoost [47]. XGBoost is a nonparametric ML method that is effective for capturing complex nonlinear relationships between input and output variables [48]. In this study, a total of 3000 input and output data sets are used for ML. These consist of 500 data sets for each ten-year interval over a 50-year period. The input data sets are the random variables associated with corrosion initiation and propagation, as well as the mechanical properties listed in Table 2. The output data are the seismic demand (i.e., displacement ductility of the bridge column) obtained from the nonlinear time-history analysis of the 3D FE model. After establishing the ML model representing the relation between PGA and ln(M_d(t)), the regression coefficients a(t) and b(t) in Equation (6) can be obtained.

The output data from the nonlinear time-history analysis of the 3D FE model and the predictions from the ML model are compared for conventional concrete and UHPC bridge columns, as shown in Figure 5, to validate the accuracy of the XGBoost-based ML approach. Figure 6 shows the regression coefficients a(t) and b(t) obtained annually from ML for conventional concrete and UHPC, along with their corresponding second-order polynomial forms. It should be noted that, since the effect of corrosion on reinforcement is negligible for UHPC, the regression coefficients a(t) and b(t) remain almost constant over time (i.e., a(t) ≅ 0.83, and b(t) ≅ 1.50, as shown in Figure 6b. The SD of the seismic capacity σ_d in Equation (5) is assumed to remain constant over time. Consequently, the time-dependent seismic fragility function of Equation (5) for conventional concrete and UHPC can be expressed in closed form.

6.2. Time-Dependent Seismic Risk

The time-dependent seismic risk for given PGA and damage state (i.e., R(t|PGA)_DS,i) can be computed using Equation (8). The computation of the monetary loss for the ith damage state C_lss|DSi(t) in Equation (8) requires extensive cost estimations related to direct and indirect monetary losses as described in Equation (11). Based on the data provided in [13], the monetary loss for the ith damage state C_lss|DSi(t) is computed. It should be noted that the monetary loss C_lss|DSi(t) is considered as a projected future value [21,22], which is calculated as

$$C_{lss\left|D{S}_i\right.}\left(t\right)=C_{lss\left|D{S}_i\right.}^{PV}{\left(1+r_{dis}\right)}^t$$

(15)

where C^PV_lss|DSi is the present monetary value of consequences associated with the ith damage state, and r_dis is the annual discount rate of money (i.e., 0.03 herein). It is assumed that the initial cost for UHPC bridge columns is five times the initial cost of those made with conventional concrete.

Figure 7 compares the time-dependent seismic risks for conventional concrete bridge columns, showing how the risks associated with slight, moderate, extensive, and complete damage states increase continuously over time for PGA values of 0.1 g, 0.3 g, and 0.5 g. The seismic risk for slight damage is highest among the four damage states under PGAs of 0.1 g and 0.3 g, as shown in Figure 7a and Figure 7b, respectively. This is because its exceedance probability during 50 years is the largest [see Figure 3], despite the monetary loss for slight damage always being the lowest of the four damage states.

However, the seismic risk R(t|PGA = 0.5 g)_DS,i for complete damage becomes the highest approaching 50 years [see Figure 7c]. This is because its exceedance probability increases over 50 years more significantly compared to the exceedance probabilities for complete damage under PGAs of 0.1 g and 0.3 g, and the monetary loss for complete damage is the largest among those for the four damage states.

The seismic risks R(t|PGA)_DS,i for UHPC bridge columns under PGAs of 0.1 g, 0.3 g, and 0.5 g are presented in Figure 8a, Figure 8b and Figure 8c, respectively, to identify the effect of the PGA, and damage states on R(t|PGA)_DS,i over time. These figures show that (a) the seismic risk for slight damage is the highest among the four damage states under PGAs of 0.1 g, 0.3 g, and 0.5 g; (b) there is no significant difference between R(t|PGA = 0.1 g)_DS,i of moderate, extensive and complete damage states [see Figure 8a]; (c) the seismic risks for extensive and complete damage states under 0.3 g are similar [see Figure 8b]; and (d) under a PGA of 0.5 g, the seismic risks for the four damage states are ordered from largest to smallest as slight, moderate, complete, and extensive, respectively [see Figure 8c]. It should be noted that the seismic risk R(t|PGA = 0.1 g)_DS,i for UHPC bridge columns increases over time even though there is no difference in the seismic fragility curves for the four damage states over time, as shown in Figure 4. This is because the monetary loss C_lss|DSi(t) in the computation of seismic risk R(t|PGA = 0.1 g)_DS,i is considered as a projected future value, as indicated in Equation (15).

The total seismic risk R(t|PGA) considers four damage states simultaneously for a given PGA, as indicated in Equation (9). The total seismic risks R(t|PGA) associated with conventional concrete and UHPC bridge columns under PGAs of 0.1 g, 0.3 g and 0.5 g are compared in Figure 9a, Figure 9b and Figure 9c, respectively, to illustrate the integrated effect of damage conditions on the seismic risk. These figures show that the total seismic risk R(t|PGA) continuously increases over 50 years, and R(t|PGA) for conventional concrete bridge columns is higher than that for UHPC bridge columns over 50 years under PGAs of 0.1 g, 0.3 g and 0.5 g. Furthermore, the seismic risks R(t|PGA) for conventional concrete and UHPC bridge columns increase with increases in PGA from 0.1 g to 0.5 g.

In order to integrate the occurrence probability of PGAs with lower and upper bound in the total seismic risks, the expected total seismic risk R(t), defined in Equation (10), is used. In this illustrative example, the upper and lower bounds of PGA are set as 0.0 g and 1.0 g, respectively. The annual exceedance probability P_exd (x), for PGA = x, is determined using seismic hazard analysis with the historical earthquake data of the target location. In this study, the annual exceedance probability P_exd (x) is expressed as [13]

$$P_{exd}\left(x\right)=C_{exd}\cdot \exp \left(-15x\right)$$

(16)

where C_exd is the scale parameter representing P_exd (x). An increase in C_exd leads to a higher annual exceedance probability for a given PGA. Figure 10 compares the time-dependent expected total seismic risks R(t) associated with conventional concrete and UHPC bridge columns when the scale parameter C_exd is equal to 0.001, 0.01 and 0.1. It can be seen that R(t) increases over 50 years, and R(t) for UHPC bridge columns is less than that for conventional concrete bridge columns across C_exd values of 0.001, 0.01, and 0.1. Moreover, R(t) increases with an increase in C_exd from 0.001 to 0.1.

The relation between cumulative expected total seismic risk R_cum and target service life for RCBA t_tg is shown in Figure 11, based on the scale parameter representing annual exceedance probability C_exd equal to 0.001, 0.01, and 0.1. This figure compares R_cum for conventional concrete and UHPC bridge columns, emphasizing that the advantage of using UHPC in terms of lower R_cum grows as the target service life gets longer. The cumulative expected seismic risk R_cum is computed using the expected total seismic risk R(t) [see Figure 10], according to Equation (12). Since R(t) presented in Figure 10 is positive over 50 years, R_cum continuously increases. From Figure 11, it can be seen that R_cum for conventional concrete bridge columns is higher than that for UHPC throughout t_tg range from 0 to 50 years, when the scale parameter C_exd is considered as 0.001, 0.01, and 0.1.

Figure 12 compares the cumulative expected total risks R_cum of conventional concrete and UHPC bridge columns as a function of C_exd at t_tg = 30 years, clearly showing the difference in R_cum between the two materials. Since a larger C_exd results in a higher annual exceedance probability, the cumulative expected seismic risk R_cum increases with an increase in C_exd. R_cum for UHPC bridge columns is less than that for conventional concrete for the range of C_exd from 0 to 0.1. The difference between R_cum for conventional concrete and UHPC bridge columns increases when C_exd increases from 0 to 0.1. This means that R_cum can be reduced more significantly by using UHPC bridge columns instead of conventional concrete bridge columns for earthquakes occurring more frequently and/or having a higher PGA.

6.3. Risk-Based Cost–Benefit Analysis

The expected benefit E(C_ben) and expected benefit ratio E(R_ben) over the target service life for RCBA t_tg are illustrated in Figure 13, when the scale parameter representing annual exceedance probability C_exd are 0.01 and 0.1. This figure is presented to identify the conditions related to C_exd and t_tg that ensure the cost-effectiveness of using UHPC instead of conventional concrete for bridge columns. E(C_ben) and E(R_ben) are defined in Equation (13), where the cumulative expected seismic risks R(t) for conventional concrete and UHPC bridge columns are provided in Figure 11b,c. As mentioned previously, this study assumes that the initial cost for bridge columns made with UHPC is five times the initial cost of those made with conventional concrete (i.e., C_ini,UHPC = r_cst·C_ini,CC, where r_cst is the cost-ratio and is equal to five herein). The UHPC application is cost-effective for bridge columns if E(C_ben) is positive or if E(R_ben) exceeds 1.0. As shown in Figure 13, both E(C_ben) and E(R_ben) increase with an increase in t_tg. This is because the difference between cumulative expected seismic risks for conventional concrete and UHPC bridge columns grows over time [see Figure 11b,c], while the initial cost remains fixed.

When the scale parameter representing the annual exceedance probability C_exd of 0.01 is adopted, the UHPC application is not cost-effective for the entire range of the target service life t_tg for RCBA (from 0 to 60 years). This is evidenced by the negative E(C_ben) and E(R_ben) being less than 1.0, as shown in Figure 13a. The UHPC application for bridge columns can be cost-effective if C_exd is 0.1, and t_tg exceeds 52.89 years [see Figure 13b]. It can be concluded that the adoption of UHPC for bridge columns is most likely to be cost-effective, when two conditions are met: (a) the application is in a seismically active region (corresponding to a larger C_exd value, which reflects a higher expected seismic risk), and (b) the bridge has a long service life (corresponding to a larger t_tg value).

Figure 14 illustrates the effect of cost-ratio r_cst on both expected benefit E(C_ben) and expected benefit ratio E(R_ben), to find the threshold of the cost-ratio r_cst achieving the cost-effectiveness of UHPC bridge columns for a given C_exd. The cost-ratio r_cst represents the ratio of the initial cost of bridge columns made with UHPC to the initial cost of those made with conventional concrete. As the cost-ratio r_cst increases, both E(C_ben) and E(R_ben) decrease. As shown in Figure 14a, when the scale parameter for the annual exceedance probability C_exd is 0.01 and the target service life for RCBA t_tg is 30 years, the adoption of UHPC for bridge columns is considered cost-effective only if the cost-ratio r_cst is less than 1.14 (i.e., the initial cost for bridge columns made with UHPC should be less than 1.14 times the initial cost of those made with conventional concrete). If the scale parameter for the annual exceedance probability C_exd is increased to 0.1 from 0.01, the cost-effectiveness of UHPC improves significantly. Accordingly, the adoption of UHPC is cost-effective for a cost-ratio r_cst less than 2.46, as shown in Figure 14b.

Figure 15 shows the effects of the cost-ratio r_cst and the target service life for RCBA t_tg on the expected benefit ratio E(R_ben) for a given C_exd = 0.1. Increasing t_tg from 20 years to 50 years raises the cost-effectiveness threshold (i.e., the limiting cost-ratio r_cst) for the UHPC application. For example, when t_tg is 30 years, the UHPC application is cost-effective if the cost-ratio r_cst is less than 2.46. An increase in t_tg from 30 years to 40 years leads to an increase in the limiting cost-ratio from 2.46 to 3.34.

7. Discussion on Limitations of the Proposed Approach

It is important to acknowledge several limitations of the proposed approach, as they may influence both its practical applicability and the accuracy of the results presented in this study.

• The framework relies on several modeling assumptions and input parameters that are inherently uncertain, including the diffusion coefficient, corrosion rate, and discount rate.
• As a detailed sensitivity analysis was not performed, the relative influence of these variables on the RCBA results could not be quantitatively assessed. Therefore, future research should incorporate probabilistic or sensitivity analyses to better quantify these uncertainties and enhance the robustness of the proposed RCBA framework.
• The presented investigations did not address the influence of preventive and essential maintenance interventions on the corrosion process, seismic risk assessment, or the RCBA. However, the developed time-dependent fragility, risk assessment, and RCBA methodologies and outcomes establish a necessary foundation for performing a more extensive life-cycle service life prediction and risk assessment that formally incorporates such maintenance strategies. Future research should expand the current risk-based life-cycle framework to explicitly integrate the effects of maintenance interventions on corrosion, seismic vulnerability, and the overall RCBA.
• The corrosion-induced deterioration in conventional concrete columns leads to an increase in the exceedance probability for a given PGA over time. In contrast, the negligible corrosion in UHPC columns means there is no significant change in the seismic fragility curves over time.
• If corrosion does occur in UHPC bridge columns, the total seismic risk increases, and the cost-effectiveness of UHPC becomes lower than that without corrosion. Consequently, both the expected benefit and the benefit ratio of adopting UHPC are reduced when corrosion is considered.

8. Conclusions

This study presents an RCBA for comparing the use of UHPC bridge columns with conventional reinforced concrete bridge columns, focusing on their performance under seismic hazards and corrosion. The methodology addressed the central question of whether UHPC’s superior long-term performance justifies its significantly higher initial material cost. The approach integrates time-dependent corrosion initiation and propagation and associated seismic fragility and risk assessments, with an illustrative example applied to an existing RC bridge in Republic of Korea. The following conclusions are drawn:

• The highly dense and less porous microstructure of UHPC dramatically delays the initiation and propagation of corrosion compared to conventional concrete. The mean corrosion initiation time for the conventional concrete bridge column was estimated at 11.14 years, while for the UHPC column, it was found to be greater than 500 years, effectively preventing corrosion within a century of service. Consequently, the reduction in reinforcement area was negligible for UHPC columns over 50 years, while it decreased for conventional concrete columns once corrosion initiated.
• The total seismic risk continuously increases over time for both types of concrete, primarily because the monetary loss is considered a projected future value. However, the total seismic risk for conventional concrete bridge columns is consistently higher than that for UHPC bridge columns over 50 years under all analyzed PGAs and scale parameters for annual exceedance probability.
• The RCBA represented by the expected benefit and expected benefit ratio demonstrates that the cost-effectiveness of UHPC is highly dependent on seismic activity and target service life. The adoption of UHPC is more likely to be cost-effective in seismically active regions, which corresponds to a larger scale parameter for annual exceedance probability and, thus, a higher expected seismic risk. Both the expected benefit and the benefit ratio increase with an increase in the target service life for the RCBA because the benefit from reduced cumulative risk grows while the initial cost remains fixed.
• While the initial cost of UHPC is a significant deterrent, its superior performance, stemming from two distinct advantages, can justify its higher initial investment. It possesses both a superior initial seismic capacity and exceptional corrosion resistance that prevents deterioration, ensuring this seismic advantage is maintained and widened over the structure’s service life. This justification is strongest for structures in highly seismic regions and those with a long target service life. The proposed approach for RCBA provides a robust framework to support infrastructure management decisions, demonstrating that the lower life-cycle cost of UHPC can ultimately outweigh its increased initial construction expense.
• The findings of this study suggest that UHPC becomes increasingly cost-effective for bridge structures under long service lives and in regions with higher seismic hazard. These results have important policy implications for infrastructure planning and investment. Specifically, agencies may consider adopting UHPC for new bridge projects in areas of high seismic risk or when a long design life is targeted, as the higher initial cost can be offset by reduced maintenance needs and lower lifecycle risk. For moderate seismicity regions or projects with shorter expected service lives, conventional concrete may remain more cost-effective. Integrating these considerations into design guidelines and funding strategies can help prioritize investments in resilient and durable infrastructure.