State-of-the-art review of chloride-induced corrosion modeling in concrete structures

Abstract

The degradation of reinforced and post-tensioned concrete structures due to chloride-induced corrosion poses a significant threat to structural safety and durability. This paper offers a critical review of chloride-induced corrosion mechanisms, modeling, and structural analysis of corroded reinforced concrete (RC) structures. The review first examines chloride diffusion modeling, which governs corrosion initiation once chloride concentrations at the reinforcement exceed threshold values, and compares methods for estimating diffusion coefficients and chloride thresholds. Existing models for corrosion-induced damage, including corrosion rate, concrete cracking, reinforcement cross-section loss, and rebar–concrete bond deterioration, are then reviewed with emphasis on their assumptions and limitations. The paper further synthesizes studies on service-life prediction and structural performance—such as seismic response under corrosion effects—typically evaluated using rebar-section-loss and crack-propagation models. Special modeling considerations for post-tensioned bridge structures are also discussed. Overall, this review highlights that while corrosion modeling plays a central role in the assessment and management of RC structures, many existing approaches lack adequate validation, standardization, and robustness. Key research gaps are identified to guide future developments in corrosion modeling for RC structures.

Keywords

reinforced and post-tensioned concrete chloride-induced corrosion corrosion modeling structural performance chloride threshold diffusion modelling section loss concrete-rebar bond

Introduction

The construction industry relies heavily on reinforced concrete (RC) for durability, strength, and cost-effectiveness. However, these structures are susceptible to environmental degradation, particularly chloride-induced corrosion. Chloride ions penetrate concrete, causing corrosion of steel reinforcement in marine environments, de-icing salts, and contaminated water. This damages the structure, reducing its capacity and lifespan.

The protective passive layer on steel remains stable when concrete is highly alkaline, preventing corrosion. However, chloride ions can break down this passive layer when their concentration exceeds a critical threshold, leading to corrosion reactions involving iron, oxygen, and pore water as the electrolyte. Concrete service life can be estimated by assessing the time to chloride ingress and the time to concrete cover cracking. Time-dependent corrosion is typically described in three stages (Castaneda and Okeil, 2020), as shown in Figure 1:

• Stage I - Initiation period (Ti): The steel remains passive during this period, but concrete cover undergoes changes. The concrete absorbs chloride ions through its pores when chloride-contaminated water contacts the concrete surface. Depassivation occurs when aggressive ions surpass their critical threshold value which marks the start of the following stage.

• Stage II - Propagation period (Tc): Corrosion products formed during this stage create volumes two to three times larger than the original steel. Cracks develop in the concrete cover as the expanding corrosion products exceed the concrete’s tensile strength. The rate of deterioration increases as cracking and spalling increase environmental exposure.

• Stage III - Active corrosion state (T_s): In this stage, the reinforcement bar becomes exposed to water, oxygen, and chlorides, resulting in continued section loss and unacceptable safety levels.

Figure 1.

Three stages of time-dependent corrosion modified from (Castaneda and Okeil, 2020).

Corrosion degrades the structural performance of RC structures by reducing the cross-sectional area of steel rebar, deteriorating bond, and inducing internal stresses, thereby causing concrete cracking. Reinforced concrete’s vital role in infrastructure necessitates urgent predictive corrosion modeling. These models help engineers and decision-makers forecast corrosion, assess structural integrity, and plan maintenance, lifecycle costs, and rehabilitation.

Past reviews of chloride-induced corrosion in RC structures often focus on one or a limited number of aspects of the problem, such as chloride diffusion, corrosion initiation threshold, or specific corrosion-induced deterioration processes. In particular, corrosion modeling, structural performance, and service-life prediction have not been reviewed holistically, and post-tensioned concrete systems receive limited attention. As a result, existing reviews provide valuable but fragmented insights. This paper, however, provides a comprehensive and integrated review of chloride-induced corrosion modeling in reinforced and post-tensioned concrete structures, spanning chloride ingress and diffusion modeling, chloride threshold determination, corrosion propagation modeling, service-life prediction, and structural performance assessment. Within each aspect, representative analytical, empirical, and/or numerical models are systematically compiled and compared, and their underlying assumptions, advantages, limitations, and applicability are critically evaluated. By offering a comprehensive, unified context that systematically traces chloride-induced deterioration from corrosion initiation through propagation to structural- and system-level performance, this review advances understanding of the impact of corrosion deterioration modeling and identifies critical research gaps to guide future studies.

Chloride-induced corrosion mechanism

Chloride diffusion modeling

Chloride concentration in concrete varies over time and location and determines the initiation, stages, and degree of corrosion in RC structures. By modeling chloride diffusion, one can incorporate the evolution over time of environmental parameters into corrosion modeling, thereby quantifying the climate change impact on the life-cycle design, assessment, maintenance, and management of RC structures exposed to corrosion (Nava et al., 2023; D’Iorio et al., 2025).

The one-dimensional solution of Fick’s second law, a diffusion equation, is commonly used to model chloride penetration below:

C (x, t) = C_{s} - (C_{s} - C_{i}) \cdot \erf (\frac{x}{\sqrt{4 D_{c l} t}})

(1)

where C_s = chloride content at the concrete surface, C_i = chloride level in cement paste before environmental exposure or chloride solution immersion (as in standardized assessments), which equals 0.005% or 30 ppm unless otherwise specified, x = the distance from the surface to the center of the sampled layer (in meters), D_cl = chloride diffusion coefficient in concrete (m²/s), t = age of the core sample and erf () = error function.

However, there are three drawbacks to Fick’s second law. First, chloride-ion movement through concrete initially does not follow a linear diffusion pattern, which is assumed in Fick’s second law. Second, the classical solution considers only the diffusion coefficient (D_cl), ignoring other concrete properties, such as the water-to-cement ratio and supplementary cementitious materials. Third, Fick’s second law uses the total chloride concentration, but only free chloride ions in the pore solution cause steel corrosion. Note that chloride ions in concrete are either bound or free. Bound chlorides are chemically or physically attached within cement hydration products, trapped in gel pores or capillary voids, while free chlorides are mobile ions that can reach reinforcement and cause corrosion. Thus, modeling chloride penetration should be based on free chloride concentration.

Instead of using only Fick’s second law to determine total chloride, combining Fick’s first and second laws is able to estimate the concentration of free chloride ions (Kong et al., 2002). Fick’s first law can describe the movement of chloride ions through a unit area of saturated concrete per unit time:

j = - D_{c l} \cdot g r a d (C_{f})

(2)

where j = mass balance of chloride ions and C_f = concentration of free chlorides in grams of chloride per gram of concrete (g/g). The distribution of chloride ions over time follows Fick’s second law which is expressed as:

\frac{d C_{t}}{d t} = - d i v (j)

(3)

where C_t = overall chloride concentration which is expressed in grams of chloride per gram of concrete (g/g). The free chloride portion from the total chloride content can be isolated by inserting equation (2) into equation (3) to obtain the primary equation for chloride transport in fully saturated concrete as follows (Kong et al., 2002):

\frac{\partial C_{t}}{\partial t} = \frac{\partial C_{t}}{\partial C_{f}} \frac{\partial C_{f}}{\partial t} = d i v [D_{c l} g r a d (C_{f})]

(4)

\frac{\partial C_{f}}{\partial C_{t}} = \frac{1}{1 + \frac{A \cdot 10^{B} \cdot β_{C - S - H}}{35, 450 β_{s o l}} {(\frac{C_{f}}{35.45 β_{s o l}})}^{A - 1}}

(4a)

where ∂C_f/∂C_t = chloride binding capacity, β_sol = ratio of pore solution to the weight of concrete, which is estimated empirically, β_C-S-H = weight ratio of C-S-H gel to concrete, and A and B = two constants for the material, related to adsorbed chloride, obtained from calibration using test data. Although the second differential equation (i.e., equation (4)) differentiates between free and total chloride to better approximate the actual free chloride penetration process, Fick’s second law (i.e., equation (1)) remains a widely used model for calculating chloride concentration because of its acceptable accuracy, calculation simplicity, and lack of reliance on many input parameters that require calibration.

Multiple studies assessed corrosion mitigation methods by slowing down chloride penetration. Krauss et al. (1996) investigated epoxy-coated reinforcing bars in four structures subjected to deicing chemicals for the Minnesota Department of Transportation (MnDOT), and findings showed low w/c (water-to-cement) ratio, dense concrete overlay, or epoxy-coated reinforcements reduce chloride penetration and corrosion amount, but cracks cause localized high chloride concentrations that lead to local corrosion. Moreover, Smith and Virmani (2000) tested various epoxy-coated rebars, and emphasized that good design, proper concrete cover, corrosion inhibitors, and low-permeability concrete alone can’t prevent corrosion because concrete tends to crack but recommended using them in all reinforcement for concrete decks.

Hagen (1982, 1979) conducted research for MnDOT on protecting and repairing bridge decks, evaluated concrete overlays, sealers, membranes, and coated rebars against chloride penetration, using visual inspections, delamination testing, chloride levels, cover depth, and electrical potential measurements; results showed that membrane systems and thick overlays are effective, but membranes are less durable than overlays. Furthermore, some studies (Zhang et al., 2021a) have shown that mechanical stresses modify concrete pore structures, leading to delayed chloride penetration in compressive areas and increased penetration in tensile areas—effects that traditional Fickian diffusion models cannot predict.

Another important limitation of traditional diffusion modeling is the assumption of one-dimensional (1D) chloride transport. Biondini et al. (2004) adopted a cellular automata approach to simulate diffusion processes described by Fick’s laws in 1D, 2D, and 3D. Titi and Biondini (2016) compared analytical 1D solutions with 2D numerical diffusion models at the cross-sectional level under one-sided and multi-sided exposure conditions and showed that multidirectional chloride ingress can substantially alter predicted steel damage and structural capacity.

Concrete diffusion coefficient

The diffusion coefficient of concrete is a fundamental property that determines how quickly ions pass through the material, enabling the evaluation of concrete durability against environmental effects. Concrete diffusion coefficients can be determined by standard testing methods or other calculation approaches.

The ASTM C1556 (2004) standard in the United States requires at least 2.5 months to complete the steady-state diffusion test. The Rapid Chloride Permeability (RCP) test compared with ASTM C11202 (2009) can be completed within a few days after the standard curing period (28-day). The ASTM C1202 standard does not measure the diffusion coefficient directly. The test evaluates the total electrical charge transfer (in Coulombs) to determine concrete chloride ion permeability levels using predefined Coulomb thresholds. Stanish et al. (1997) explained that the RCP test has two drawbacks: (1) it detects all ions in the pore solution, not just chloride; (2) the high voltage raises the temperature, speeding up ion migration.

As an alternative to the RCP test, the Nordtest (1999) method is commonly adopted in Northern Europe. Similar to the RCP test, NT Build 492 is also an electrical method, that takes place between 24 and 96 hours following the completion of the standard 28-day curing period. NT Build 492 differs from other methods in that it measures the diffusion coefficient directly rather than providing a chloride permeability index. Its results are unaffected by environmental factors, making it a reliable method for predicting the service life of reinforced concrete structures (Vancura and Engineers, 2018a). The ASTM C1202 test overestimates transport in accelerated assays, but long tests (ASTM C1556 or NT Build 443) are time-consuming and may miss penetration in low-permeability concretes (Szweda et al., 2023).

Using Fick’s second law is another approach for calculating the concrete diffusion coefficient by calibrating the chloride content profile obtained from coring samples. Vancura and Engineers (2018a) implemented an iterative approach to solve Fick’s second law by adjusting C_s and D_cl to minimize the difference between the measured chloride profiles in coring samples and those predicted by Fick’s second law (error term), ultimately determining the optimal value for the concrete diffusion coefficient. Moreover, the parameter C_s has a correlation with the ambient factor of a resistivity model (Medeiros-Junior and Bem, 2020).

Prediction models have also been developed as alternatives to calculate the concrete diffusion coefficient. For example, Stanish and Thomas (2003) showed that there is a reduction in the concrete diffusion coefficient over time, which can be captured by using a decay coefficient (m), as shown below:

D_{c l} (t) = D_{r e f} {(\frac{t_{r e f}}{t})}^{m}

(5)

D_cl(t) = diffusion coefficient at time, t, and D_ref = diffusion coefficient at a reference time t_ref. Additionally, Alexander and Thomas (2015) developed a formula to estimate m considering the impact of slag and fly ash in the concrete mixture, as shown below:

m = 0.26 + 0.4 (\frac{F A}{50} + \frac{S G}{70})

(6)

where SG and FA = percentages of slag and fly ash in concrete mixture, respectively.

Table 1 lists equations from the literature for calculating the concrete diffusion coefficient. The key factors include temperature, water-to-cement ratio, activation energy, porosity, gas constant, humidity, and chloride concentration. The equations differ in their assumptions, required inputs, and applicability. Arrhenius-based formulations (e.g., Dousti et al., 2013; Li and Guo, 2021; Thomas and Bamforth, 1999) are physically motivated and most suitable when temperature effects dominate, and a reliable reference diffusion coefficient is required. Concentration-dependent models (e.g., Glass and Buenfeld, 2000a; Luping and Nilsson, 1992) account for chloride binding and are therefore more appropriate for severe exposure conditions such as marine or de-icing environments. Humidity-based relationships (e.g., Bažant, 1979; Bertolini et al., 2013) capture the strong influence of moisture on transport processes and are particularly relevant for partially saturated concrete. In contrast, mixture-based expressions linked to w/c ratio or strength (e.g., Mangat and Molloy, 1994; Zhang and Ye, 2019) are useful for comparative mixture evaluation but may have limited transferability beyond their calibration range. More comprehensive formulations (e.g., Kong et al., 2002) integrate multiple influencing factors and offer improved realism at the cost of increased data requirements. Overall, model selection should reflect the dominant exposure conditions, material characteristics, and data availability rather than reliance on a single universal formulation.

Table 1.

Summary of concrete diffusion coefficient formulas in the literature.

Reference	Model formulation	Definition of parameters	Notes
Dousti et al. (2013)	$D_{c l} = D_{r e f} \exp (- q \cdot T)$	D_cl: Concrete diffusion coefficient (m²/s) D_ref: Diffusion coefficient at a reference time (m²/s) q: Empirical constant related to temperature change T: Absolute temperature (K)	Developed based on the Arrhenius equation and used to predict the diffusion coefficient at various temperatures.
Thomas and Bamforth (1999)	$D_{c l} = D_{r e f} \exp [- ν \cdot (T_{r e f} / T)]$	ᴠ: Empirical constant T_ref: Reference temperature at reference time (K)
Li and Guo (2021)	$D_{c l} = D_{r e f} \exp [- U / (R \cdot T)]$	U: Activation energy (J/mol) R: Universal gas constant (8.314 J/mol·K)	Developed based on the Arrhenius equation and used to predict the diffusion coefficient under various environmental conditions. The activation energy $U$ is determined experimentally by fitting the Arrhenius relationship to diffusion coefficients measured at multiple temperatures.(Yuan et al., 2008)
Luping and Nilsson (1992)	$D_{c l} = D_{r e f} \exp (- b \cdot C)$	b: Empirical parameter related to changes in chloride concentration C: Chloride concentration at surface (ppm)	Widely applicable where the diffusion process is affected by concentration-dependent interactions, such as chemical binding, reduced pore space, or changes in chloride concentration at the surface (e.g., deicing salts, marine environments).
Glass and Buenfeld (2000a)	$D_{c l} = D_{r e f} / (1 + λ . C^{p})$	λ & p: Empirical parameters related to changes in chloride concentration
Hilsdorf and Kropp (1995)	$D_{c l} = D_{r e f} [1 + {(C / C_{r e f})}^{n}]$	C_ref: Reference chloride concentration at reference time (ppm) n: Empirical parameter
Bažant (1979)	$D_{c l} = D_{r e f} (1 - α \cdot R H)$	α: Empirical constant related to relative humidity RH: Relative humidity	Useful for the diffusion of corrosive ions where the diffusion processes are mainly influenced by relative humidity (e.g., severe humidity areas)
Bertolini et al. (2013)	$D_{c l} = D_{0, R H} {(R H / 100)}^{μ}$	D_0,RH: Diffusion coefficient at RH = 100% µ: Empirical parameter related to humidity
Mehta and Monteiro (2006)	$D_{c l} = D_{r e f} {(P_{r e f} / P)}^{a}$	P_ref: Reference pressure around the rebar at reference time (28 days) P: Current pressure around the rebar at the time of diffusion coefficient calculation a: Empirical parameter related to porosity	Applicable for scenarios where pressure is the main factor that influences gas or fluid diffusion through the porous concrete structure.
Khan et al. (2017)	$D_{c l} = D_{r e f} / (1 + k \sqrt{t})$	k: Empirical parameter t: Time (s)	Considering the fact that diffusion slows down over time due to physical or chemical changes in the material or environment.
Zhang and Ye (2019)	$D_{c l} = D_{r e f} {(w / c)}^{ω}$	ω: Empirical parameter related to ratio of porosity w/c: Water-cement ratio	Commonly used in concrete technology to model how the permeability and diffusivity of concrete depend on its water-to-cement ratio, and porosity.
Mangat and Molloy (1994)	$\log (D_{c l}) = {\begin{array}{l} - 3.9 {(\frac{w}{c})}^{2} + 7.2 (\frac{w}{c}) - 14 & (Normal strength concrete) \\ - 3 {(\frac{w}{c})}^{2} + 5.4 (\frac{w}{c}) - 13.7 & \begin{array}{l} (High strength concrete \\ with silica fume or slag) \end{array} \end{array}$
Kong et al. (2002)	$\begin{array}{l} D_{C l} = f_{1} (w / c, t) \cdot f_{2} (g_{i}) \cdot f_{3} (T) \cdot f_{4} (C) \\ f_{1} (w / c, t) = \frac{28 - t}{62500} + (\frac{1}{4} + \frac{28 - t}{300}) {(\frac{w}{c})}^{6.55} \\ f_{2} (g_{i}) = D_{c p} [1 + \frac{g_{i}}{(1 - g_{i}) / 3 + 1 / (D_{a g g} / D_{c p} - 1)}] \\ f_{3} (T) = \exp [\frac{U}{R} (\frac{1}{T_{r e f}} - \frac{1}{T})]; \\ f_{4} (C) = 1 - 8.333 C^{0.5} \end{array}$	g_i: Aggregate volume fraction within the concrete mix D_agg and D_cp: Diffusivity values corresponding to the aggregate and the surrounding cement paste	Comprehensively consider the most influential parameters that affect concrete durability.

Chloride limitation threshold

The corrosion process begins when chloride levels on steel exceed a threshold, breaking down the passive oxide layer (depassivation) (Paul, 2019); and this threshold is the minimum chloride amount needed to initiate rebar corrosion. Methods for determining this threshold include electrochemical monitoring, chemical analysis, empirical observation, and the assessment of factors such as chloride levels, steel type, pH, cement, moisture, and environmental conditions. Table 2 compares various methods based on material and conditions to determine the chloride threshold.

Table 2.

Summary of existing methods for determining chloride threshold.

Method		Procedure/Formula	Notes
Empirical approach (Bremner et al., 2001; Shafikhani and Chidiac, 2019; Lee and Zielske, 2014)		Prepare RC specimens • Cast RC specimens with steel reinforcement embedded at a controlled depth. Expose specimens to chloride environments • Immerse specimens in a chloride solution (e.g., NaCl) or apply simulated deicing salt exposure. • Alternatively, place specimens in marine or atmospheric chloride environments. Monitor corrosion initiation • Periodically extract powder samples from different depths near the reinforcement. • Determine chloride content using acid-soluble or water-soluble chloride tests (ASTM C1152 or ASTM C1218). • Use visual inspection, rust staining, or crack formation as qualitative indicators to determine the time of corrosion initiation. Define chloride threshold • Identify the chloride concentration at the steel-concrete interface when corrosion is first detected. • Express the threshold as % chloride by cement weight or % chloride by concrete weight.	Key principle: Visual observation of corrosion initiation in concrete specimens exposed to chloride Advantage: Realistic results considering actual concrete-steel interactions Limitation: Time-consuming, influenced by environmental variations, and subjectiveness of visual observation
Electrochemical approach (Standard, 2009; ISO, 2022)		Prepare RC specimens • Cast RC specimens with steel reinforcement embedded at a controlled depth. • Install reference electrodes in RC specimens (e.g., silver/silver chloride or saturated calomel electrode) near the steel surface. Expose specimens to chloride environments • Immerse concrete specimens in chloride solutions or subject them to cyclic wet-dry conditions. Monitor corrosion initiation • The electrochemical potential indicates corrosion initiation when it reaches between −200 mV to −350 mV (vs. Ag/AgCl or SCE) during half-cell potential testing. • The onset of corrosion happens when current density exceeds 0.1–0.2 μA/cm² based on measurements from linear polarization resistance (LPR) or electrochemical impedance spectroscopy (EIS). Define chloride threshold • Measure chloride concentration at the steel-concrete interface at the time of corrosion initiation. • Express threshold values as % chloride by cement weight or free chloride concentration in pore solution.	Key principle: Uses potential measurements and current density to detect corrosion Advantage: Fast detection, widely used in corrosion monitoring Limitation: May not always correlate with actual corrosion initiation in concrete
Chemical composition analysis	Chloride binding and free Chloride Content (Glass and Buenfeld, 2000b; Smith and Virmani, 2000)	Prepare RC specimens • Cast RC specimens using different binders (e.g., Portland cement, fly ash, slag) to study chloride binding effects. Expose samples to chloride environments • Immerse or subject samples to chloride diffusion tests (e.g., ASTM C1556). Monitor corrosion initiation • Drill powder samples from concrete at different distances from the surface. • Use water-soluble chloride testing (ASTM C1218) for free chlorides. • Use acid-soluble chloride testing (ASTM C1152) for total chlorides. • Corrosion begins when the free chloride content reaches a certain percentage of the cement weight, depending on the type of reinforcement. Define chloride threshold • Use the free chloride concentration at the steel surface at corrosion onset. • Express threshold as chloride concentration in mol/L of pore solution.	Key principle: Using only free chloride (not total chloride) in the pore solution at the steel surface to determine the corrosion initiation Advantage: More accurate than total chloride-based methods Limitation: Requires complex chloride binding models and extraction techniques
	Chloride-to-Hydroxide Ratio (Cl^-/OH^-) (Luping and Nilsson, 1993; Standard, 2000; Taerwe and Matthys, 2013)	Prepare RC specimens • Cast RC specimens using different concrete mix designs and curing the specimens under standard conditions Expose samples to chloride environments • Subject specimens to chloride exposure conditions such as sprayed or ponded salt solutions Monitor corrosion initiation • Extract pore solution from concrete specimens using vacuum extraction or high-pressure methods. • Analyze chloride concentration (Cl^-) using ion chromatography or titration. • Measure hydroxide concentration (OH^-) using pH testing and alkalinity titration. • Calculate the concentration ratio of Cl^-/OH^- • Corrosion initiation is assumed when Cl^-/OH^- ratio exceeds 0.6 to 1.0. Define chloride threshold • Determine chloride concentration at which Cl^-/OH^- reaches the critical threshold. • Express as mol/L of pore solution or % chloride by cement weight.	Key principle: Using the alkalinity of concrete to determine the corrosion initiation. Advantage: Accounts for concrete chemistry and pH effects Limitation: Requires pore solution extraction, which is complex
	Concrete quality and exposure condition (Strategic Highway Research Program 2 (SHRP2), 2019; Yang and Knight, 2018)	Prepare RC specimens • Cast RC specimens with different water-cement ratios, cement types, and supplementary materials (e.g., fly ash, slag, silica fume). Expose samples to chloride environments • Subject specimens to accelerated chloride exposure (e.g., ponding, wet-dry cycles, marine simulation). Monitor corrosion initiation • Collect data on concrete mix designs, especially water-cement ratio and cement type. • Identify cement type (OPC, blended cement) and presence of fly ash, slag, silica fume • Measure chloride profiles at different depths and exposure times. • Identify threshold by correlating chloride content at the rebar level with corrosion onset by checking visible rusting or cracking or half-cell potential measurements Define chloride threshold • Use the free chloride concentration at the steel surface at corrosion onset • Express as % chloride by cement weight (wt% Cl^-/cement)	Key principle: Using exposure data to determine chloride levels at which corrosion starts, with consideration of mix design Advantage: Useful for real-world durability assessment Limitation: Time-consuming, complexity in data interpretation
Pitting Resistance Equivalent (PRE) (ASTM G, 2000; Bertolini et al., 2013; Strategic Highway Research Program 2 (SHRP2), 2019)		Obtain stainless steel composition • Identify alloying elements (i.e., chromium, molybdenum, nitrogen) from steel reinforcement specifications. Calculate pitting resistance equivalent (PRE) • Use the formula: PRE=%Cr+3.3%Mo+A%N (%Cr= Chromium weight percentage, (%Mo = Molybdenum weight percentage, %N=Nitrogen weight percentage, A = Constant which depends on the type of reinforcement) • Reference values of Cr, Mo, N of various steel type can be found from standards such as ISO 15156, ASTM G48 Define chloride threshold • Higher PRE values correlate with higher chloride thresholds (e.g., for PRE > 40, chloride by cement weight >5%).	While PRE does not directly define the chloride threshold, it is a strong predictor of corrosion resistance in stainless steel reinforcement. A higher PRE shows higher chloride threshold

The methods summarized in Table 2 differ primarily in how corrosion initiation and the chloride threshold are defined. Empirical approaches rely on visual or physical indicators of corrosion onset and provide realistic results under real-world exposure conditions, but they are time-consuming and subject to observational error. Electrochemical methods offer faster detection and quantitative monitoring through potential and current measurements; however, their correlation with true corrosion initiation in concrete can vary with moisture, oxygen availability, and concrete resistivity. Chemical-composition–based methods, including free chloride concentration and Cl^-/OH^- ratio approaches, explicitly account for concrete chemistry and pore-solution alkalinity and are therefore more mechanistically linked to depassivation, although they require complex extraction and testing procedures. Approaches that incorporate concrete quality, exposure conditions, and rebar type improve applicability to real structures by reflecting material variability, while PRE-based methods extend threshold evaluation to stainless steel reinforcement by linking corrosion resistance to alloy composition rather than relying solely on chloride concentration. Overall, no single standardized chloride threshold applies to all concrete structures, since corrosion initiation is defined differently across empirical, electrochemical, chemical, and material-based methods. As a result, threshold values may vary, and each approach involves assumptions and limitations. Method selection should therefore be guided by the intended application and exposure conditions.

For new construction, ACI 318-95 and ACI Committee 222 (Shakouri et al., 2017) specified the chloride ion limits as a percentage of the cement weight used in the concrete mixture for various types of new structures. These limits are determined using the water-soluble test (as per ASTM C 1218 and ACI 222.1) or the acid-soluble test (as per ASTM C 1152) and are detailed in Table 3. Kamaitis (2002) also provided the chloride-ion limits as a percentage of cement weight for different types of structures, according to various codes (e.g., AFNOR 18-011, ACI 318, BS 5400, and ENV 206), as shown in Table 4.

Table 3.

Chloride limits for new construction proposed by ACI committee 222 (Shakouri et al., 2017).

Type of structure	Maximum chloride ion (Cl^-) content in concrete, percentage by weight of cement
Type of structure	Acid- soluble (ASTM C 1152)	Water- soluble (ASTM C1218)	Water-soluble (ACI 222.1)
Prestressed (pretensioned or post-tensioned)	0.08 (40%)	0.06 (40%)	0.06 (40%)
Non-prestressed, wet conditions	0.1 (50%)	0.08 (53%)	0.08 (53%)
Non- prestressed, Dry conditions	0.2 (100%)	0.15 (100%)	0.15 (100%)

Table 4.

Chloride limits for new construction in other codes (Kamaitis, 2002).

Type of structure	Maximum chloride ion (Cl-) content in concrete, percent by weight of cement in other codes
Type of structure	AFNOR 18-011	ACI 318	BS 5400	ENV 206
Concrete	1	1	1	1
Reinforced concrete	0.65	0.1-0.15	0.35	0.4
Pre- tensioned concrete	0.1	0.06	0.06	0.2
Post- tensioned concrete	0.2	0.06	0.06	0.2

The Strategic Highway Research Program 2 SHRP2 (2019) includes Appendix C, as well as research by Kamaitis (2002). Chalhoub et al. (2020), Broomfield (2023), and Vancura and Engineers (2018b) reported chloride threshold values expressed as percentages of cement weight for various reinforcement materials (including carbon steel, epoxy-coated carbon steel, galvanized carbon steel, low-carbon chromium steel, stainless-steel-clad carbon steel, austenitic stainless steel, and duplex austenitic–ferritic stainless steel). Figure 2(a)–(c) present the histograms (frequency used in various specifications and research studies) of chloride threshold values for black rebar, epoxy-coated rebar, and stainless-steel rebar, respectively. These figures highlight the variability in reported threshold values and the differences arising from material type, testing methods, and environmental assumptions across various studies. In addition, the corresponding minimum, maximum, average, and standard deviation values are summarized in Table 5.

Figure 2.

Histogram of average chloride threshold values by rebar type across references: (a) black rebar; (b) epoxy-coated rebar; and (c) stainless steel rebar.

Table 5.

Extracted chloride threshold values for different rebar types (% weight of cement).

Rebar type	Chloride threshold (% weight of cement)
Rebar type	Min	Max	Average	Std
Black rebar	0.15	2.2	0.66	0.46
Epoxy-coated rebar	0.2	3.1	1.09	0.65
Stainless steel rebar	2.2	7.1	4.19	1.53

Figure 2(a) shows that chloride threshold values for black rebar span a relatively wide range, from approximately 0.15% to 2.2% by weight of cement. However, the highest frequency of reported values is concentrated within a narrow interval of 0.5% to 1.0%, which is adopted by many codes and practical design applications. As shown in Table 5 and Figure 2(b), epoxy-coated carbon steel rebar exhibits a higher average value but a wider distribution of chloride threshold values than black rebar. The presence of the epoxy coating theoretically delays the initiation of corrosion by limiting chloride ingress to the steel surface, thereby allowing greater chloride tolerance before depassivation. Figure 2(c) and Table 5 further indicate that stainless steel reinforcement has substantially higher chloride threshold values, with reported averages exceeding 4% by weight of cement and a larger standard deviation. This behavior reflects the superior corrosion resistance of stainless steel in chloride-contaminated environments, whereas the greater scatter highlights the influence of alloy type, surface condition, and environmental conditions.

It should also be noted that some studies have proposed chloride threshold values without explicitly distinguishing between rebar types. For example, the National Cooperative Highway Research Program (Ford et al., 2011) evaluated protective strategies for concrete bridge decks using a corrosion threshold of 1.5 lb/yd³. Similarly, Hagen (1979, 1982) suggested threshold ranges of 1.1–1.6 lb/yd³ (approximately 280–410 ppm) for Minnesota bridges incorporating various protection systems. Oh et al. (2004), combining experimental testing and finite element analysis, reported total chloride threshold values ranging from 0.45% to 0.97% by weight of cement, with free chloride thresholds of approximately 0.1%, depending on cement type and mixture proportions.

Research gaps

Diffusion modeling and testing

Despite decades of research on ionic diffusion in concrete, several critical gaps remain in diffusion modeling. Existing models fail to account for multiple exposure directions, which is typical for most structures, because they consider chloride transport in only one direction from a single exposure face, which is assumed in one-dimensional Fick’s diffusion formulations reviewed in this study. This can lead to unsafe service-life predictions for structures with multiple surfaces under chloride attack. In addition, concrete diffusivity and microcracks within concrete are time-dependent under aging and loading conditions, as described earlier. However, many service-life prediction models summarized in the reviewed literature still assume a constant diffusion coefficient and a homogeneous, uncracked material, thereby oversimplifying reality.

The current modeling of chloride binding and multi-ion interactions (such as with sulfates or carbonation) relies on empirical factors or constant coefficients that do not capture the complex, time-dependent processes in concrete, as reflected in the simplified binding formulations in the reviewed studies. Additionally, Liu et al. (2024) highlighted the importance of coupling mechanical loading with carbonation-driven corrosion models, noting that stress states can modify pore structure and microcrack development, thereby altering the effective diffusion coefficient and carbonation depth. Thus, future research should develop multi-physics models that integrate transport, chemical reactions, moisture variations, and mechanical loading to more accurately simulate real-world chloride ingress and durability performance. As mentioned earlier, different standardized tests (steady-state bulk diffusion vs rapid migration) yield variable diffusion coefficients, with magnitude differences even for identical concrete samples, as reported across the reviewed experimental studies, underscoring the need for standardized testing protocols to improve consistency.

Traditional Fickian finite-element models, as commonly implemented in the diffusion-based numerical simulations reviewed herein, struggle with concrete heterogeneity and evolving properties because of their spatial uniformity and fixed-diffusion assumptions. They require extensive calibration for 2D or 3D applications. Researchers explore multi-scale and stochastic methods like lattice or cellular automata models that incorporate time-dependent diffusion and damage effects (Ma and Lin, 2022). These approaches show potential for realistic diffusion–damage interactions but are underused due to limited validation, high complexity, high computational demands, and a lack of robust software tools. Based on the reviewed numerical and computational diffusion studies, for computational tools for simulating diffusion processes, which have been extensively explored in recent years, future research should focus on experimental validation of these methods, improvements in computational efficiency through model refinement and algorithm development, and the development of standardized, user-friendly simulation tools for practical adoption.

Chloride threshold

The current models and codes determine corrosion initiation based on rebar type and water-to-cement ratio, while disregarding environmental factors and other time-varying variables (e.g., steel surface conditions and concrete quality), as evidenced by the wide ranges of threshold values reported in the reviewed experimental and field studies. This may be why the reported threshold values show large variation, leading to significant differences (of several decades) in the predicted corrosion initiation times when these values are directly incorporated into service-life prediction models. In addition, standardized testing protocols are lacking, leading to inconsistent laboratory results across the studies summarized in this review.

Future research should focus on reducing uncertainty by developing integrated models that combine chloride-transport simulations with material-property evolution and time-dependent corrosion-threshold formulations informed by the reviewed experimental evidence. A standardized testing protocol that considers comprehensive influencing factors should be developed to generate consistent, reliable data for model development, and a probabilistic approach should also be adopted to account for uncertainty in threshold determination and application. On the other hand, one could develop a new approach to predict the time to corrosion initiation without relying on the chloride threshold. The definition of corrosion initiation could shift from chloride concentration at the rebar surface to a corrosion rate threshold. The approach defines active corrosion initiation based on measurable corrosion kinetics, using a critical corrosion-rate threshold rather than chloride-content thresholds, consistent with corrosion-monitoring and electrochemical approaches discussed in the literature review.

Corrosion modeling

Corrosion rate & corrosion current density

Corrosion rate, r, describes how fast metal deteriorates, typically measured as metal oxidized per unit surface over time (e.g., µm/yr.), or as corrosion current density (i_corr) that refers to electrical flow during corrosion (in amperes/cm² or coulombs/cm²·sec) (Shafei and Shi, 2022). Corrosion current density, i_corr, and corrosion rate, r, are related through the equation below (Shafei and Shi, 2022):

r = \frac{i_{c o r r} a}{n F ρ}

(7)

where a = atomic weight; F = Faraday’s constant; n = number of electrons participating in the exchange process, is denoted by n and equals 2 for FE²⁺); and ρ = metal mass density. In corrosion testing, corrosion current density is usually measured using Linear Polarization Resistance (LPR), and equation (7) is then used to calculate the corrosion rate.

Various mathematical and empirical models have been developed to predict the corrosion rate or current density, which is summarized in Table 6. The corrosion rate and corrosion current density models summarized in Table 6 span a wide range of conceptual approaches, from simplified empirical expressions to multi-parameter and mechanistic formulations. Early empirical and resistivity-based models (e.g., Alonso et al., 1988; Molina et al., 1993) establish direct relationships between corrosion current density and easily measurable parameters such as concrete resistivity or crack development. These models are efficient for initial assessments but often overlook the combined effects of moisture, temperature, and chloride transport, reducing their reliability in changing environments. More advanced empirical models (e.g., Bastidas-Arteaga et al., 2010; Morinaga, 1988) incorporate multiple environmental and material parameters, including humidity, chloride concentration, oxygen availability, and cover depth, allowing for a more realistic representation of corrosion progression; however, their reliance on empirically fitted coefficients limits their broader applicability. Electrochemical-based and regression-driven models (e.g., DuraCrete, 1998; Pour-Ghaz et al., 2009; Song et al., 2013) explicitly link corrosion current density to measurable electrochemical quantities such as half-cell potential, anodic and cathodic current densities, or polarization parameters. These approaches are advantageous when monitoring data are available and can capture time-dependent corrosion behavior; however, they often require extensive experimental calibration and may exhibit sensitivity to measurement conditions and noise. Mechanistic formulations grounded in diffusion theory and electrochemical kinetics (e.g., Huet et al., 2007) provide a physically consistent description of corrosion by considering oxygen transport, pore structure, and reaction kinetics at the steel–concrete interface. While these models provide insight into governing mechanisms, their practical use is hindered by the need for material-specific parameters that are hard to obtain in field applications. Overall, Table 6 highlights a clear trade-off between model simplicity and physical fidelity: empirical models are suitable for rapid screening and comparative studies; multi-parameter and mechanistic approaches are better for detailed service-life prediction when ample environmental, material, and monitoring data are available.

Table 6.

Corrosion rate estimation published in the past considering influential parameters.

Researchers	Proposed models	Definition of parameters	Notes
Alonso et al. (1988)	$i_{c o r r} = \frac{3 \times 10^{4}}{ρ_{e f}}$	i_corr: Corrosion current density ρ_ef: Concrete resistivity measured under its current saturation condition	Principle: The model emerges from statistical analysis of resistivity data in relation to corrosion rates during accelerated carbonation testing. Limitation: Only considering the impact of concrete resistivity.
Molina et al. (1993)	$i_{c o r r} = \frac{Δ r}{t_{c}}$	t_c: Time to crack initiation (days) Δr: Reduction in rebar radius due to corrosion (mm)	Principle: This model is based on the decrease in rebar radius during the period until crack width reaches 0.3–0.4 mm. Limitation: The model assumes a linear relationship between corrosion rate and rebar radius reduction, neglecting complex cracking behaviors.
Morinaga (1988)	$i_{c o r r} = \frac{d_{s t}}{d^{2}} (- 0.51 - 7.6 C_{c l} + 44.97 {(w / b)}^{2} + 67.95 C_{c l} {(w / b)}^{2})$	d_st: Diameter of the reinforcing steel d: Cover depth w/b: Water to binder ratio RH: Relative humidity T: Absolute temperature (k) C_cl: Chloride content at rebar surface $C_{o_{2}}^{a i r}$ : O₂ Concentration in the air (%)	Principle: These equations are based on chloride-induced corrosion prediction, incorporating multiple influencing parameters related to concrete properties and environmental conditions. Limitation: While the second updated model includes environmental parameters, it does not account for concrete properties, like binder type and water-to-binder ratio.
Morinaga (1988)	$\begin{array}{l} i_{c o r r} = 2.59 - 0.05 T - 6.89 (R H - 0.45) - 22.87 C_{O_{2}}^{a i r} \\ - 0.99 C_{c l} + 0.14 (R H - 0.45) + 0.51 T C_{O_{2}}^{a i r} + 0.01 T C_{c l} \\ + 60.81 (R H - 0.45) C_{o_{2}}^{a i r} + 3.36 (R H - 0.45) C_{c l} + 7.32 C_{o_{2}}^{a i r} C_{c l} \end{array}$
Bastidas-Arteaga et al. (2010)	$\begin{array}{l} i_{c o r r} (t) = 102.5 + 10.09 \ln (1.69 C_{c l}) - \\ 0.0015 ρ_{e f} - \frac{39038.96}{T} + \frac{290.91}{t^{- 0.215}} \end{array}$		Principle: The model was developed from statistical analysis of data from specimens subjected to accelerated corrosion testing for five years, considering the impact of time, environmental factors, and material properties. Limitation: The equation uses fitted parameters and may not be applicable to all conditions. The Arrhenius equation assumes a constant activation energy, which may vary under different environmental conditions.
DuraCrete, (1998)	$\begin{array}{l} i_{c o r r} (t) = \frac{3 \times 10^{4}}{ρ (t)} F_{c l} F_{g a l v} F_{o x i d e} F_{o x y} \\ ρ (t) = ρ_{e f} f_{e} f_{t} {(\frac{t}{t_{0}})}^{n_{0}} \end{array}$	F_cl, F_galv , F_oxide, and F_oxy: Factors of chloride content, galvanic effects, continuous formation, aging of oxides, and availability of oxygen, respectively f_e: The factor that modifies ρ_ef based on the influence of the exposure environment f_t: The factor that affects the resistivity test method t₀: Reference time n₀: The factor for consideration of the effect of time-dependent environmental conditions	Principle: This model is an electrochemical approach for predicting corrosion rates, accounting for the effects of chloride ingress, oxygen availability, resistivity, and time-dependent concrete properties. Limitation: The empirical factors need to be calibrated experimentally, and no guidelines were provided for determining those values.
Vu and Stewart (2000)	${\begin{cases} i_{c o r r, 0} = \frac{37.8 {(1 - w / b)}^{- 1.64}}{d} & t \leq t_{i n} \\ i_{c o r r} (t) = i_{c o r r, 0} 0.85 {(t - t_{i n})}^{- 0.29} & t > t_{i n} \end{cases}$	t_in: Corrosion initiation time	Principle: This equation is influenced by factors such as concrete composition and permeability, and the use of a power-law relationship suggests that corrosion initiation depends on the interaction between the water-to-binder ratio and other environmental factors that affect chloride diffusion. Limitation: The equation lacks explicit time-dependent effects, including chloride diffusion.
Scott (2004)	$\begin{array}{l} i_{c o r r} = (1.43 \frac{c l_{90}}{f} + 0.02) \exp [(\frac{40 - d}{20}) 1.2 {(\frac{c l_{90}}{f})}^{3}] \\ f = 10^{(\| 0.5 - S \| - 0.5 + S)} \end{array}$	f: Slag correction factor cl₉₀: 90-day chloride conductivity index value S: Slag percentage weight	Principle: The equation is empirically developed and exhibits exponential sensitivity to concrete cover thickness and a nonlinear dependence on chloride concentration and material properties. Limitation: The equation does not explicitly account for time-dependent chloride diffusion, or environmental factors such as oxygen and temperature.
Pour-Ghaz et al. (2009)	$i_{c o r r} = 100 {\frac{1}{τ ρ_{e f} γ} [η T d^{k} i_{l}^{λ} + μ T v i_{L}^{ω} + θ {(T i_{L})}^{ϑ} + χ ρ^{γ} + ζ}$	ν, γ, ζ, Ƞ, ƙ, τ, ω, ϴ, χ, λ: Constants dependent on environmental and material conditions i_l: Limiting current density	Principle: This model uses regression analysis and electrochemical theory to link corrosion rate with temperature, kinetic parameter, concrete resistivity, and current density. Limitation: The equation contains multiple coefficients that must be calibrated using experimental data.
Martínez and Andrade (2009)	$i_{c o r r} = \frac{i_{c o r r}^{sing} + i_{c o r r}^{\max}}{2}$	i_corr^sing: Single measured value of corrosion rate from a specific location or time i_corr^max: Maximum expected corrosion rate in the structure	Principle: The method enables reliable corrosion-rate calculation when continuous monitoring is not feasible. Limitation: This method does not capture time-dependent changes and localized corrosion effects.
Maruya et al. (2003)	$i_{c o r r} = \frac{ρ_{e f} L_{a c}}{ϕ_{m a, c} - ϕ_{m a, a}}$	ɸ_ma,c and ɸ_ma,a: Macro-cell potentials for the cathode and anode, respectively L_ac: Spacing between the anode and cathode	Principle: The model relies on half-cell potential measurements for corrosion monitoring purposes. Limitation: This model involves multiple coefficients (Tafel slope coefficients used to calculate L_ac), which are determined experimentally from polarization measurements.
Song et al. (2013)	$\log i_{c o r r} = 8.458 - 0.508 pH + 0.5 \log i_{o, c} + 0.5 \log i_{o, a}$	i_o,c: Cathodic current density i_o,a: Anodic current density	Principle: The model operates through passive-film states to evaluate the combined effects of pH and anodic and cathodic reactions on corrosion rates. Limitation: The model relies on linear relationships between pH and anodic and cathodic current densities, as well as the logarithm of the corrosion current density, but these relationships may not hold under complex field conditions.
Gulikers (2005)	$i_{c o r r} = \frac{{F_{G}}^{- 0.8125} (98.696 \times 10^{- 3})}{{ρ_{e f}}^{0.8125}}$	F_G: Geometry factor (m^-1)	Principle: The equation calculates corrosion current density through resistivity and geometric measurements. Limitation: The model depends only on resistivity and geometry to predict corrosion behavior while ignoring other critical factors such as pH, temperature, and chloride levels.
Huet et al. (2007)	$i_{c o r r} = n_{e} F ε \sqrt{D_{O_{2, l i q u i d}} s_{a} k_{O_{2}} {s_{w}}_{r}} C_{O_{2, l i q u i d}} (x_{2})$	n_e: Valence number s_a: Normalized surface area of the oxide layer per unit volume F: Faraday’s constant s_wr: Water saturation degree of concrete ƙ_O2: Kinetic constant of oxygen reduction C_O2,liquid(x₂): Oxygen concentration in the liquid phase at depth x₂ x₂: Depth within the concrete cover where oxygen concentration is evaluated D_{O2, liquid}: Rate of diffusion of chemical species within the pore solution of concrete ε: Porosity of the concrete matrix	Principle: This model calculates corrosion current density in carbonated concrete based on a mechanistic approach, combining oxygen diffusion through the pore solution and the kinetics of oxygen reduction at the steel surface Limitation: The model assumes that oxygen transport follows Fick’s diffusion law and that oxygen concentration at depth is uniform. Moreover, it requires detailed knowledge of material-specific parameters, such as the kinetic rate constant, oxygen concentration, and the specific surface area of the oxide layer, which are difficult to obtain in the field.
Liu (1996)	$\begin{array}{l} i_{c o r r} (t) = 0.926 \exp [8.37 + 0.618 \ln (1.686 C l_{w a t e r}) \\ - \frac{3034}{T} - 1.05 \times 10^{- 4} R_{c} + 2.32 {(t - t_{i n})}^{- 0.215} + ε \end{array}$	Cl_water: Water-soluble chloride content at the steel surface R_c: Ohmic resistance of concrete ε: Model error	Principle: This empirical model estimates time-dependent corrosion current density by incorporating the effects of chloride concentration, temperature, concrete’s ohmic resistance, and elapsed corrosion time. The formula is designed to support long-term corrosion modeling and service-life assessment. Limitation: The model uses empirically derived coefficients that are valid under specific environmental and experimental conditions. It may require recalibration for different regions, materials, or exposure types. Besides, parameters such as Cl_water and R_c, are often hard to measure accurately in real structures without specialized testing or monitoring.
Mortagi and Ghosh (2022)	$i_{c o r r} (t) = \frac{0.373 \cdot {(1 - \frac{w}{c})}^{- 1.64}}{d} \cdot t^{- 0.29} \cdot f_{Δ t} (t) \cdot f_{Δ R H} (t) \cdot f_{C l} (t)$	w/c: Water-cement ratio f_Δt(t), f_ΔRH(t), f_Cl(t): Modification factors for considering temperature, relative humidity and chloride ingress	Principle: The formula estimates the time-dependent corrosion rate by integrating environmental effects, chloride ingress, and material degradation. It reflects how long-term exposure to changing climate conditions can impact the deterioration process of infrastructure, making it suitable for durability assessments and lifecycle cost analyses. Limitation: This model is empirical and based on controlled laboratory or limited field data, and experimental/field validation is needed.

Section loss

Corrosion in RC structures results in a reduction of tensile rebar diameter and yield strength over time, which impacts the structural performance (Sajedi et al., 2017). One can calculate steel loss (η_s) using iron’s atomic weight (M_Fe), reaction valence (n), corrosion current density (i_corr), and Faraday’s constant (F) via the equation below. (Guzmán et al., 2011; Guzman et al., 2012; Shafei and Shi, 2022):

η_{s} = \frac{M_{F e} i_{c o r r}}{n F}

(8)

The mass loss ratio in lab experiments compares the initial rebar weight (m₀) to that after corrosion and cleaning (m_s) (Alipour et al., 2013):

η_{s} = (m_{0} - m_{s}) / m_{0}

(9)

Table 7 summarizes different approaches for estimating reinforcement section loss, including mass loss, diameter reduction, and time-dependent corrosion models. Mass-loss models based on Faraday’s law (e.g., Guzmán et al., 2011; Law et al., 2004) relate corrosion current density to steel consumption, making them suitable for laboratory and electrochemical assessments. Diameter-based models (e.g., Andrade et al., 1993; Zhang et al., 2017) translate corrosion effects into geometric loss of reinforcement, making them more intuitive for structural analysis and crack-propagation studies, but they rely on assumptions about rust expansion and uniform penetration. Time-dependent formulations (e.g., Bažant, 1979; Morinaga, 2018; Sajedi and Huang, 2015;) incorporate corrosion rate evolution and propagation duration, enabling long-term prediction of section loss and service-life assessment; however, their accuracy depends heavily on accurate estimates of corrosion rates and exposure-dependent parameters. In summary, simpler mass-loss models work well for controlled and monitoring scenarios, while time-dependent diameter models are useful for structural performance evaluation under corrosion.

Table 7.

Some important formulas to calculate mass loss ratio due to corrosion in different research.

References	Proposed model	Definition of parameters	Notes
Guzmán et al. (2011)	$η_{s} = \frac{1}{μ_{w} μ_{v}}$	η_s: Mass loss ratio µ_w: Ratio of molecular weight of iron to the molecular weight of the corrosion products µ_v: Ratio of volume of expansive corrosion products to the volume of steel consumed	Principle: The formula calculates the mass loss ratio of corrosion products by comparing the molecular weight of steel to that of the corrosion products and the volume of steel to that of the corrosion products. The formulation relates steel mass loss to the volumetric expansion of corrosion products, providing a mechanistic basis for modeling corrosion-induced cracking in reinforced concrete. Limitation: The formula assumes uniform corrosion. Nondestructively estimating the volume of expansive corrosion products is not feasible in the field.
Law et al. (2004)	$η_{s} = (\int I_{c o r r} / C) M$	$\int I_{c o r r}$ : Total current passed C: Charge per mole of iron M: Atomic mass iron	Principle: This formula estimates the mass-loss ratio of steel reinforcement based on the total corrosion current over time. It is derived from Faraday’s law, linking electrochemical charge transfer to material loss. This model is applicable to electrochemical corrosion studies for quantifying steel degradation in RC structures. Limitation: This formula is developed based on the assumption of uniform corrosion.
Zhang et al. (2017)	$η_{s} = \frac{d_{s} + 4 δ_{0} \frac{d_{s}}{0.155} + 3 d_{s d}}{2 h \cdot d_{s t}}$	d_st: Diameter of the reinforcing steel d_s: Distance covered by the advancing corrosion front toward the adjacent unconfined edge δ₀: Representative thickness of the zone characterized by porosity d_sd: Length of rust propagation in the direction normal to d_s h: Rate of expansion caused by corrosion processes	Principle: This formula estimates the mass-loss ratio based on the propagation of the rust front and the thickness of the porous zone. It accounts for both lateral and perpendicular rust penetration, making it useful for modeling corrosion-induced cracking in concrete and the propagation of cracking in cover concrete. Limitation: This formula assumes uniform rust expansion and exact estimation of parameters like d_s, d_sd and δ₀ can be complicated and time-consuming.
Morinaga (2018)	$η_{s} = B_{c o r r} t$	B_corr: Corrosion rate parameter that varies with environmental conditions such as temperature and humidity	Principle: The formula uses corrosion rate to calculate the mass loss. The model is suitable for long-term predictions of reinforcement degradation under different exposure conditions because the corrosion rate coefficient depends on environmental factors. Limitation: The formula requires an estimate of the corrosion rate, and the corrosion rate does not account for the impact of material properties and chloride concentration.
Andrade et al. (1993)	$d (t) = d_{s t} - 0.023 i_{c o r r} t$	d(t): Rebar diameter at time t t: Time elapsed since the propagation period began	Principle: The formula calculates the reduction in rebar diameter over time due to corrosion, based on corrosion current density and exposure duration. The model uses Faraday’s law to link electrochemical charge transfer to metal loss for straightforward assessment of reinforcement deterioration. Limitation: The model makes a basic assumption about uniform rebar surface corrosion, but fails to account for pitting, which produces more destructive structural damage.
Sajedi and Huang (2015)	$\begin{array}{l} Q (t) = {\begin{cases} 0 & t \leq t_{i n} \\ 4.6 \frac{i_{c o r r} (t)}{d_{s t}} & t > t_{i n} \end{cases} \\ d (t) = \sqrt{1 - Q (t)} d_{s t} \end{array}$	i_corr(t): Corrosion current density at time t Q(t): Corrosion level at time t in terms of mass loss percentage	Principle: The formula describes the development of corrosion by assuming that no weight reduction occurs before an initiation period, after which the corrosion rate is controlled by current density. It links mass loss to rebar diameter reduction, providing a time-dependent estimation of structural deterioration. Limitation: The change in diameter at time t depends solely on the corrosion rate density and the corrosion initiation time. Given the variety of existing formulas for estimating these parameters, the results can vary significantly.
Bažant (1979)	$d (t) = \frac{p j_{r} t_{c o r r}}{ρ_{c o r r} d_{s t}} + d_{s t}$	t_corr: Steady-state corrosion or propagation period ρ_corr: Composite density coefficient accounting for both steel and corrosion products Ƥ: Perimeter of bar j_r: Instantaneous corrosion rate of rust (g/m²-s)	Principle: The formula calculates the reduction in rebar diameter based on steady-state corrosion, and it is suitable to predict long-term reinforcement deterioration in concrete structures. Limitation: The model assumes uniform corrosion, which can lead to accelerated structural deterioration. The model faces challenges in determining rust corrosion rates and estimating steady-state corrosion periods.

Corrosion cracking

Pressure needed for concrete cover cracking

Corrosion produces rust that exerts radial pressure on rebar, leading to concrete cover cracking when the pressure reaches its maximum. This pressure arises from rust-induced expansion within the rebar. Calculating this pressure is vital for modeling corrosion cracks, which affect chloride-ion diffusion and indicate corrosion progression, thereby compromising the integrity of RC structures. Additionally, steel corrosion correlates with concrete cover cracking (Zhao and Jin, 2006). Recent experimental work has also shown that sustained mechanical loading significantly influences corrosion-induced cracking behavior. Recent accelerated corrosion tests by Ma et al. (2025) on RC beams under varying loading conditions revealed that moderate sustained compression can delay cracking due to matrix densification, while high biaxial stress accelerates crack growth through lateral expansion. They also found that stirrups, by restricting expansion, increased internal strain. Based on these findings, a biaxial stress-dependent expansion model was proposed to capture the interaction between mechanical loading and corrosion-induced cracking behavior.

Thus, such pressure estimation can be used to predict both corrosion initiation and its progression. Table 8 presents some important equations for calculating the maximum pressure induced by corrosion, which can lead to cracking in the concrete cover. Empirical formulations (e.g., Torres-Acosta, 1999) provide practical expressions that incorporate reinforcement diameter, concrete cover thickness, and the length of corroding steel, making them suitable for analytical evaluation of crack initiation. Elasticity-based models derived from thick-walled cylinder theory (e.g., Basheer et al., 1996; Muñoz et al., 2007) offer a more mechanistic interpretation by explicitly accounting for concrete elastic properties and stress distribution within the cover, thereby improving physical consistency in predicting cracking onset. However, both empirical and elastic models commonly assume uniform corrosion, concentric geometry, and linear elastic behavior, which may not adequately represent localized corrosion, nonuniform rust accumulation, or material degradation in real structures during service life.

Table 8.

Some important formulas to calculate pressure-induced cover cracking due to corrosion.

References	Proposed model	Definition of parameters	Notes
Torres-Acosta (1999)	$P_{\max} = 1.54 \frac{d}{d_{s t}} {(\frac{d}{L} + 1)}^{0.72} f_{t}$	d_st: Diameter of the reinforcing steel d: Cover depth f_t: Concrete tensile strength (kg/cm²) L: Undergoing corrosion length of rebar	Principle: This empirical model estimates the critical internal pressure at which corrosion-induced tensile stress exceeds the tensile strength of concrete, using rebar diameter and the length of corroding reinforcement as key influencing factors. The formula incorporates geometric effects and material strength directly. Limitation: The model's accuracy depends heavily on precise knowledge of rebar diameter and corrosion length. However, determining the ongoing corrosion length in real structures is challenging, time-consuming, and may require invasive inspection or imaging techniques (e.g., GPR or corrosion mapping).
Muñoz et al. (2007)	$P_{\max} = \frac{E (R_{2}^{2} - R_{1}^{2}) ε_{\max}}{2 R_{2}^{2}}$	R₁ & R₂: Internal radius (steel diameter) and external radius (concrete cover) ε_max: Maximum stain in concrete caused by P_max	Principle: Based on Timoshenko’s theory (thick-walled cylinders), this formula estimates the maximum pressure from corrosion expansion in concrete. It accounts for the elastic behavior of concrete under internal stress and predicts when the induced pressure exceeds the tensile strength, leading to cracking. Limitation: The model assumes perfect concentric geometry and radial symmetry, which rarely hold in deteriorated or irregularly shaped cross-sections. It also relies on accurate measurement of internal (R₁) and external (R₂) radii, which are hard to quantify in-situ.
Basheer et al. (1996)	$P_{\max} = \frac{Δ d_{s} E_{c}}{1 + \frac{v_{c}}{2} + 1.2 v_{s} \frac{E_{c}}{E_{s}}}$	Δd_s: Rebar change diameter due to corrosion E_c & E_s: Elastic modules of concrete and steel, respectively ν_c & ν_s: Poisson’s ratio of concrete and steel, respectively	Principle: This formula estimates the internal pressure from corrosion-induced expansion based on the elastic properties of concrete and steel. It incorporates Poisson’s ratios to model stress distribution and predicts when corrosion pressure reaches a critical level, leading to cover cracking. Limitation: The model treats the system as purely elastic, which limits its validity beyond the early stages of corrosion.

Munoz et al. used a laboratory process to evaluate and validate the existing equations shown in Table 8 (Muñoz et al., 2007). Figure 3 describes the mechanical process of corrosion product expansion based on Timoshenko’s theory (thick-walled cylinders): steel is treated as a metal cylinder with initial radius ri₀, immersed in a semi-infinite concrete medium with cover depth d, and undergoing corrosion over a length L. As corrosion progresses, the original radius r₀ decreases by x, while the effective radial expansion Δr_ref due to corrosion-induced stresses in the concrete causes cracking and spalling. Munoz et al. used strain gauges to measure the maximum strain at the time of crack initiation on the sample surface. Results showed P_max aligns better with values from equations by Torres-Acosta (1999).

Figure 3.

The process of internal pressure occurring needed for cracking the cover (Muñoz et al., 2007).

Crack width

Accumulation of corrosion products near reinforcement creates tensile forces that cause cracking and expansion over time. Estimating crack width caused by corrosion is vital for assessing reinforced concrete durability. Crack development depends on factors like corrosion rate, concrete cover thickness, and environmental conditions. Table 9 presents equations from the literature for estimating crack width.

Table 9.

Some important formulas to calculate concrete crack width due to pressure generated by corrosion.

References	Proposed model	Definition of parameters	Notes
Molina et al. (1993)	$w_{c r} = 2 π x (ν_{r / s} - 1)$	w_cr: Crack width ν_r/s: Ratio between the specific volumes of rust and steel x: Corrosion penetration (μm)	Principle: The formula enables linear prediction of the effects of rust expansion on surface crack width, providing a straightforward connection between steel loss due to corrosion and concrete cracking. Limitation: The model assumes uniform volumetric expansion ratios, but these ratios are difficult to determine and vary with corrosion type and moisture content; in addition, measuring corrosion penetration requires invasive procedures.
Rodriguez et al. (2018)	$w_{c r} = 0.05 + β (x - x_{c r})$	β: Coefficient (0.01 for top rebar and 0.0125 for bottom rebar) x_cr: Corrosion penetration after cover cracking (μm)	Principle: This formula estimates the crack width by accounting for additional corrosion penetration after the cover has already cracked. It introduces a coefficient to differentiate between top and bottom rebar behavior, reflecting realistic conditions. Limitation: The model assumes a linear relationship between additional corrosion depth and crack width. Besides, the parameter β is empirical and context-specific, limiting generalizability without calibration.
Zhang et al. (2010)	$w_{c r} = 0.1916 Δ A_{S M} + 0.164$	ΔA_SM: Cross section loss	Principle: The formula estimates crack width from a linear relationship between cross-sectional loss and crack propagation, providing a straightforward method for assessing structural damage caused by steel corrosion. Limitation: The model ignores non-linear crack growth and material differences.
Vidal et al. (2004)	$w_{c r} = 0.0575 (Δ A_{s t e e l} - Δ A_{s t e e l, c r})$	ΔA_steel: Rebar cross-section loss due to corrosion (propagation stage) ΔA_steel,cr: Rebar cross-section loss at crack initiation	Principle: The model calculates crack width from the loss of reinforcement cross-section while accounting for both the initiation of cracks and their growth. Limitation: The model assumes a fixed expansion ratio and uniform corrosion, ignoring real-world variability and environmental factors like moisture and chloride ingress, which can affect crack development. Besides, obtaining accurate values for ΔA_steel, ΔA_steel,cr requires close monitoring or advanced nondestructive testing.
Andrade et al. (1993)	$w_{c r} = 15.683 {(\frac{x}{R_{0}} C T)}^{0.928}$	CT: Adjustment factor represents the combined impact of concrete tensile strength and the relationship between cover thickness and bar diameter R₀: Initial rebar radius	Principle: The model estimates crack width from the loss of reinforcement cross-section due to corrosion, accounting for crack initiation, propagation, the concrete cover-to-diameter ratio, and tensile strength. Limitation: The CT factor combines parameters like tensile strength and cover-to-diameter ratio, but its calibration varies across materials and conditions. Additionally, accurately determining the initial rebar radius (R₀) and corrosion penetration (x) in the field is challenging and requires advanced inspection methods.
Castorena-González et al. (2020)	$w_{c r}^{0.08634} = \frac{(x^{0.2} + 1.3565)}{1.8673}$		Principle: The crack width is estimated through an empirical function of corrosion penetration depth using a relationship that captures non-linear crack evolution due to corrosion conducted in a 3D finite element (FE) analysis. Limitation: While based on simulations, the formula simplifies FE outputs into an empirical expression, potentially losing accuracy across geometries or materials. It also requires the corrosion penetration depth (x), which may not be readily determined in practice.
Thoft-Christensen (2001)	$\begin{array}{l} w_{c t} (t) = {\begin{cases} 0 & t < t_{i n} \\ [d_{s t} (0) - d_{s t} (t) \frac{(α_{d} - 1) π d_{s t} (0)}{(\frac{0.5 d_{s t} (0)}{0.5 d_{s t} (0) + d} + 1) d} & t \geq t_{i n} \end{cases} \\ d_{s t} (t) = \sqrt{1 - Q (t)} d_{s t} (0) \end{array}$	d_st(0): Initial diameter of the reinforcing steel d: Cover depth Q(t): Corrosion level at time t α_d: Ratio of the density of corrosion products to that of the reinforcing steel t_in: Corrosion initiation time	Principle: The model formulates corrosion-induced cracking through a rust mass balance approach, where crack initiation occurs when the accumulated corrosion products exceed the available porous zone and elastic expansion capacity of the surrounding concrete. The critical expansion is derived using thick-walled cylinder theory, and crack width evolution is related to reinforcement cross-sectional loss over time. Limitation: A major limitation is accurately determining αd, which controls rust expansion. The model also assumes uniform corrosion and simplified geometry, possibly overlooking real-world variability and localized corrosion.

As shown in Table 9, early empirical relationships (e.g., Molina et al., 1993; Zhang et al., 2010) directly relate corrosion penetration or to reinforcement cross-section loss to crack width, offering simple, computationally efficient tools for durability assessment; however, these models typically assume uniform corrosion and linear crack growth. Extensions of these formulations (e.g., Rodriguez et al., 2018; Vidal et al., 2004) distinguish between crack initiation and propagation stages and account for additional corrosion after cover cracking, thereby improving their applicability to real exposure conditions but still relying on empirically calibrated parameters that limit generalization. Models incorporating geometric and material factors such as cover thickness, bar diameter, and concrete tensile strength (e.g., Andrade et al., 1993) provide a stronger physical interpretation of crack development, though accurate estimation of input parameters remains challenging in field applications. More advanced approaches, including finite-element-based and time-dependent formulations (e.g., Castorena-González et al., 2020; Thoft-Christensen, 2001), capture nonlinear crack evolution and the coupling between corrosion progression and mechanical response, but their complexity and data requirements restrict routine implementation. Overall, simpler empirical models are suitable for screening-level durability evaluation, whereas mechanics-based and time-dependent formulations are more appropriate for detailed service-life analysis when sufficient material and corrosion information is available.

Concrete strength

Reduction of concrete strength due to corrosion is one of the critical reasons for degradation in concrete structure performance (Shayanfar et al., 2016). Higher concrete compressive strength is typically associated with lower porosity and reduced chloride diffusivity; thus, theoretically, it could improve durability of RC structures exposed to chloride environments. Concrete compressive strength depends on two main factors: the mixture composition (water-cement ratio, aggregate type, cement type) and curing conditions. Concrete structures with higher strength generally exhibit lower porosity, resulting in reduced chloride permeability and delayed corrosion initiation, thereby enhancing the protection of embedded reinforcement in chloride-contaminated environments. However, the relationship between concrete strength and long-term structural durability considering corrosion is not straightforward and involves multiple interacting mechanisms influenced by material properties, exposure conditions, and degradation processes (Mozafarjazi and Rabiee, 2024; Vishnu and Sharma, 2012).

High-strength concrete could develop microcracks during curing or under mechanical stress. The expansion of corrosion products within the concrete matrix creates substantial tensile stresses, leading to cracking and spalling, especially when concrete has both high strength and low porosity, as high-strength concrete exhibits brittle behavior once corrosion begins. Cracks in concrete provide pathways for chloride ions to enter, thereby intensifying the corrosion reaction. Thus, the delay of corrosion initiation by stronger concrete does not necessarily lead to better long-term durability during corrosive exposure (Daniyal and Akhtar, 2020; Soraghi and Huang, 2022). Studies also show similar corrosion rates in high-strength concrete to standard concrete under tensile stresses exceeding its tensile strength (Gil-Martín et al., 2023).

On the other hand, research shows that supplementary cementitious materials (SCMs) such as fly ash, slag, and silica fume improve the durability of high-strength concrete against chloride corrosion by creating refined pore structures and reducing permeability (Shaikh and Supit, 2015). Adding these materials, along with functionally graded materials (FGM), helps minimize microcracks and enhance durability. Their long-term benefits, however, depend on a proper mix design and curing methods (Si et al., 2024; Tangtakabi et al., 2024). In summary, RC structural durability depends on concrete strength, which should be evaluated through a holistic analysis of microcrack and tensile-stress crack formation, as well as concrete permeability and SCM use.

Rebar mechanical properties

Evaluating the mechanical properties of corroded steel bars under standard loading conditions remains essential for assessing the performance of RC structures. The mechanical performance of steel bars under corrosion progressively shifts from ductile to brittle failure, accompanied by reductions in yield strength, strain-hardening capacity, and ultimate strain (Cairns et al., 2005; Ou et al., 2016; Stewart, 2009). The yield plateau disappears at approximately 16.7% corrosion (Ou et al., 2016), and fracture strain becomes equal to yield strain at about 20% corrosion (Cairns et al., 2005), which Stewart adopted as a critical corrosion threshold. Zhang et al. (2016) further reported that the ductile-to-brittle transition becomes pronounced when corrosion levels reach approximately 20–30%. While many degradation models relate strength reduction to average corrosion measures, Gu et al. (2018) demonstrated that failure behavior is strongly influenced by corrosion non-uniformity and introduced a corrosion non-uniformity factor, R, to quantify this effect. This concept has since been applied in performance- and reliability-based studies to improve structural behavior prediction over different service-life stages (Guo and Dong, 2022; Guo et al., 2020, 2021, 2023, 2024).

In addition, loading conditions can alter the impact of corrosion on rebar mechanical properties. For example, experimental studies have shown that high strain rates can enhance both the yield and ultimate strengths of corroded reinforcement under tensile loading (Zhang et al., 2016). Other studies found that the reduction in deformation capacity, strength, and fatigue life of corroded reinforcement due to corrosion could be more significant under impact or explosion loading conditions (Apostolopoulos and Papadopoulos, 2007; Zhang et al., 2012).

To quantitatively describe strength degradation, several empirical formulations have been proposed to relate the yield and ultimate strengths of corroded reinforcement to corrosion extent, typically expressed as mass loss or cross-sectional reduction. These formulations are summarized in Table 10. Overall, Table 10 indicates that the reduction in rebar strength differs between naturally and artificially corroded specimens. In addition, except for the model proposed by Gu et al. (2018), which considers the minimum remaining cross-sectional area, most models are primarily applicable to uniform corrosion conditions.

Table 10.

Formulations of yield and ultimate strength of corroded rebars from existing studies.

Resources	Formulation	Definition of parameters	Notes
Morinaga (2023)	$\begin{array}{l} f_{y c} = 36.77 - 0.589 x (%) \\ f_{u c} = 51.18 - 1.344 x (%) \end{array}$	x: Degree of corrosion, expressed as mass loss percentage (%).	Linear model based on experimental data for corroded reinforcing bars. It assumes a direct reduction in nominal yield and ultimate strengths due to corrosion-induced mass loss.
Zhang et al. (2012)	$\begin{array}{l} \frac{f_{y c}}{f_{y 0}} = \frac{1 - β_{y c} x (%)}{1 - x (%)} \\ \frac{f_{u c}}{f_{u 0}} = \frac{1 - β_{u c} x (%)}{1 - x (%)} \end{array}$	f_y0, f_u0: Yield and ultimate strengths of uncorroded rebar, respectively. β_yc, β_uc: Empirical constants for yield and ultimate loads. β_yc = 1.12, β_uc = 1.36 (Naturally corroded rebars) β_yc = 1.10, β_uc == 1.22 (Artificially corroded rebars)	The model accounts for reductions in effective cross-sectional areas and for corrosion effects on material properties.
Du et al. (2005) Cairns et al. (2005)	$\begin{array}{l} f_{y c} = f_{y 0} (1 - β_{y c} x (%)) \\ f_{u c} = f_{u 0} (1 - β_{u c} x (%)) \end{array}$	β_yc= 0.004, β_uc = 0.0061 (Artificially corroded rebars)	Linear model focusing on the mass loss impact on nominal strength, validated through experimental tests.
Ou et al. (2016)	$\begin{array}{l} \frac{f_{y c}}{f_{y 0}} = 1 - β_{y c} x (%) \\ \frac{f_{u c}}{f_{u 0}} = 1 - β_{u c} x (%) \end{array}$	β_yc = 0.0123, β_yu = 0.0125 (Naturally corroded rebars) β_yc = 0.0127, β_yu = 0.0281 (Artificially corroded rebars)	Linear model distinguishing between natural and artificial corrosion effects.
Gu et al. (2018)	$\begin{array}{l} \frac{f_{y c}}{f_{y 0}} = 1 + 0.0256 η_{\max} \\ \frac{f_{u c}}{f_{u 0}} = 1 + 0.2114 η_{\max} \end{array}$	η_max: Maximum corrosion degree, expressed as maximum cross-sectional loss percentage (%).	The model suggests a slight increase in true yield and ultimate strengths based on the minimum cross-sectional area, attributed to reduced necking effects in corroded bars. Unique in focusing on true strength at critical sections rather than nominal strength reduction.

Concrete-rebar bond behavior

Concrete and reinforcing steel function as a composite system, enabling RC structures to resist applied loads effectively, with the bond between concrete and steel playing a critical role. Such a bond is severely weakened when reinforcing steel corrodes, thereby reducing the structural performance of RC elements (Sajedi and Huang, 2015). When corrosion initiates at the rebar, the corrosion products form and expand, causing internal stress and microcracks at the rebar interface, reducing bond strength. Continued corrosion creates additional cracks, reduces concrete confinement, and causes surface spalling, thereby further degrading bond efficiency. This bond deterioration impairs stress transfer and accelerates structural decline (Feng et al., 2021). It has been shown that bond strength and initial bond stiffness increase slightly at first, then decline sharply as corrosion progresses, reflecting the rapid loss of confinement and mechanical interlock (Zhang et al., 2020). Figure 4 illustrates how corrosion causes cracking, spalling, and a weakened bond between concrete and steel (Syll and Kanakubo, 2022).

Figure 4.

The effect of corrosion on concrete-rebar bond (Gheitasi and Harris, 2015).

Bond failures include Mode I tensile (splitting), Mode II shear (pull-out), and Mixed Mode combining both. Mode I causes radial cracks around rebar, especially when confinement is insufficient. Mode II involves shear slip and crushing, common in smooth bars. When both stresses are significant, often due to corrosion or damage, a Mixed Mode causes concrete crushing and splitting, especially in corroded or deteriorated systems. (Elbusaefi, 2014). The bond failure mode at the steel–concrete interface depends on concrete cover and confinement. Increased confinement and thicker cover prevent transverse cracking and splitting, thereby avoiding pull-out failure.

Figure 5 shows the bond stress (τ_b) –slip behavior of rebars in concrete. Figure 5(a) depicts a smooth bar, where the bond is mainly due to adhesion and friction, as smooth bars lack surface ribs for mechanical interlock. Initially, cement paste adheres to the steel, providing resistance through chemical adhesion. As the load increases, this adhesion fails, leading to frictional slip. Smooth bars do not cause significant radial pressure or cracking; therefore, the bond-slip response is primarily Mode II debonding, with failure via pull-out and limited residual strength. The transition region shifts from frictional resistance near the loaded end to elastic slip at the far end. Due to their weaker bond, smooth bars achieve only 15–20% of the bond strength of deformed bars, necessitating additional anchorage, such as hooks or longer embedment (Beton, 2013; Elbusaefi, 2014; Hong and Park, 2012).

Figure 5:

Bond behavior and failure mechanisms in (a) smooth and (b) deformed bars (Elbusaefi, 2014).

Figure 5(b) shows a deformed bar’s bond behavior, where bond strength results from degradation mechanisms like crushing, shearing-off, and mechanical interlocking between the ribs and concrete. The bond-slip response has four stages: initial adhesion, rib engagement and microcracking, peak bond resistance, and post-peak degradation due to concrete failure. Initially, the bond develops through adhesion, like smooth bars. As loading continues, ribs press against concrete, increasing shear resistance. Bond stress increases with slip until it peaks, indicating that concrete crushing or shearing occurs. Degradation occurs via pull-out or splitting failure under radial tensile stresses, reflecting a mixed-mode debonding mechanism. Overall, deformed bars have higher bond strength and a more ductile, complex failure pattern than smooth bars, making them preferable in reinforced concrete. (Elbusaefi, 2014; Hong and Park, 2012).

The CEB (Beton, 2013) guidelines provide two commonly adopted models to represent the relationship between bond stress c and relative slip (s) associated with pull-out failure and splitting failure, respectively, as illustrated in Figure 6. Two key parameters define the ascending bond-slip curve, which determines the initiation of bond failure: bond strength (τ_max), representing peak stress transmitted from the reinforcement to concrete, and peak slip (s₁), indicating slip at this maximum. In splitting failure (Figure 6(a)), τ increases nonlinearly with slip to τ_max, then drops abruptly to residual stress (τ_f) due to friction, remaining constant afterward. In pull-out failure (Figure 6(b)), τ also rises nonlinearly to τ_max, stays constant between s₁ and s₂, then decreases linearly past s₂ to τ_f at s₃, covering the distance between ribs. To model bond behavior with corrosion, s₁ and τ_max are typically adjusted accordingly.

Figure 6.

Bond-slip model based on CEB (Beton, 2013). (a) Splitting; (b) pull-out.

Various formulations predict s₁ and τ_max, summarized in Table 11 by failure mode (splitting or pull-out). Some consider s₁ as a constant; others incorporate influencing factors such as bar diameter, reinforcement properties, cover depth, concrete strength, corrosion percentage, and density to account for the effects of corrosion on bond behavior. Splitting failure models predict lower s₁ values due to brittle cracking, whereas pull-out models, assuming sufficient confinement, predict higher s₁ values due to ductile shear failure (Beton, 2013). In addition, these formulations are based on different principles: some emphasize the role of transverse confinement in delaying splitting, whereas others focus on the material properties governing interface shear resistance. In particular, two formulas proposed by (Lin et al., 2017, 2019bib_lin_et_al_2019; Soraghi and Huang, 2024) explicitly incorporate the corrosion into the calculation of s₁ by establishing an inverse relationship between s₁ and the corrosion rate, recognizing that the corrosion progress reduces the effectiveness of transverse confinement and damages the interfacial concrete, leading to lower s₁ values (that is, earlier splitting or pull-out at reduced slips). Potential limitations include their empirical nature (valid only within specific ranges), the need to pre-identify the failure mode, and the neglect of factors such as bar surface condition or cyclic loading. Overall, Table 11 highlights the variation in model performance for predicting s₁ across failure modes and influencing parameters, underscoring the challenge of selecting appropriate models.

Table 11.

Summary of existing studies for peak slip (s₁).

Failure mode	Reference	Formulation of s₁	Definition of parameters	Influencing parameter
Splitting or pull-out	Kivell et al. (2015)	$s_{1} = 0.300 mm$
Splitting or pull-out	Feng et al. (2016)	$s_{1} = 0.175 mm$
Splitting	Guizani and Chaallal (2011)	s1 is proportionate to transverse reinforcement confinement		Transverse confinement
	Xu (1990)	$s_{1} = 0.0368 d_{s t}$	d_st: Diameter of reinforcing steel	d _st
	Khalaf and Huang (2016)	$s_{1} = 0.5 + 0.1 d / d_{s t} (for 1.0 < d / d_{s t} < 5.0)$	d: Cover depth	d/ d _st
	Coccia et al. (2015)	$s_{1} = 0.0035 (d - 0.2 d_{s t})$		d, d _st
	Binder (2013)	$s_{1} = {({τ_{\max, s p l i t t i n g} / 2.5 \cdot (f_{c}^{'})}^{0.5})}^{2.5}$	τ_{max, splitting:} Maximum bond stress due to splitting	τ _{max, splitting,} $f_{c}^{'}$
	Wu and Zhao (2013)	$\begin{array}{l} s_{1} = (0.7315 + K) / (5.176 + 0.33 K) \\ K = d / d_{s t} + 33 (\frac{A_{s t}}{n_{d} . d . s_{s t}}) \end{array}$	A_st: Cross-sectional area of transverse confinement bars (mm²) n_d: Number of main barss_st : Stirrups spacing (mm)K: Combined effect on bond strength	Transverse confinement quantity,K(d/d_st. A_st, n_d, d_st, s_st)
	Lin et al. (2017) Lin et al. (2019)	$\begin{array}{l} s_{1} (i_{c o r r} = 0) = 0.12 s_{r} / (1.85.8 \exp (- 1.4 K^{'})) \\ s_{1} = [5.89 i_{c o r r} + 0.12] / [216.84 i_{c o r r} + 1) (1 + 85.8 \exp (- 1.4 K^{'}))] \\ K^{'} = d / d_{s t} + 82.7 (\frac{A_{s t}}{d . s_{s t}}) \end{array}$	i_corr: Corrosion current densitys_r: Rib spacing (mm)K’: Combined effect on bond strength	S_r, i_corrTransverse confinement quantityK^’ (d/ d_st,d, A_st, s_st)
	Soraghi and Huang (2024)	$f (s_{1} \| β_{1}, β_{2}) = \frac{{(s_{1} - s_{1, \min})}^{β_{1} - 1} {(s_{1, \max} - s_{1})}^{β_{2} - 1}}{B (β_{1}, β_{2}) {(s_{1, \max} - s_{1, \min})}^{β_{1} + β_{2} + 1}}$	s_1,min ≤ s₁ ≤ s_1,maxβ₁, β₂: Beta distribution parameters	B(β₁, β₂)
	Harajli et al. (2004)	$\begin{array}{l} s_{1} = \exp (3.3 \ln ((τ_{\max, s p l i t t i n g} / τ_{\max, p u l l - o u t})) s_{1, p u l l - o u t} \\ + s_{0} \ln (τ_{\max, s p l i t t i n g} / τ_{\max, p u l l - o u t}) \end{array}$	τ_{max, pull-out}: Maximum bond strength due to pull-out s₀=0.15 mm for unconfined concretes₀=0.40 mm for confined concrete	τ _{max, splitting} /τ _{max, pull-out}
Pull-Out	Beton, (2013)	$s_{1} = 1.000 mm$
	Alsiwat and Saatcioglu (1992)	$s_{1} = \sqrt{30 / f_{c}^{'}}$		$f_{c}^{'}$
	Harajli et al. (2004) Lin et al. (2019)	$s_{1} = 0.2 \cdot s_{r}$		s _r
	Murcia-Delso and Benson Shing (2015)	$s_{1} = 0.07 d_{s t} (for 36 mm < d_{s t} < 57 mm)$		d _st
	Zhao and Zhu (2018)	$s_{1} = 0.07442 d_{s t} - 0.00093 {d_{s t}}^{2} \approx 0.1 s_{r}$		d _st , s _r
	Soraghi and Huang (2024)	$\begin{array}{l} \ln (s_{1}) = α_{0} + α_{1} \cdot (e^{- 4 Q}) + α_{2} \cdot {(\frac{d}{d_{s t}})}^{2} \cdot e^{3 Q} + \\ α_{3} \cdot \frac{d}{d_{s t}} \cdot \frac{τ_{a v g}}{\sqrt{f_{c}^{'}}} + α_{4} \cdot \frac{τ_{a v g}}{\sqrt{f_{c}^{'}}} \cdot (e^{- 4 Q}) + \\ α_{5} \cdot \frac{d}{l_{d}} \cdot {(\frac{τ_{a v g}}{\sqrt{f_{c}^{'}}} - K_{t r})}^{- 1} \cdot e^{Q} + σ \cdot ε \end{array}$	Q: Corrosion mass loss percentageα_i: Model coefficientsτ_avg: Average bond strengthK_tr: Confinement index from transverse reinforcement (stirrups)σε: Error term	Q, α_i, d/ d_st, τ_avg, $f_{c}^{'}$ _,K_tr

Table 11 reveals a clear progression from simplified constant-based expressions toward parameter-dependent models that explicitly capture confinement and corrosion effects on bond behavior. Constant s₁ models are suitable for reference or code-type applications under assumed “average” conditions, but they lack sensitivity to geometry, confinement, and deterioration mechanisms. In contrast, confinement-based formulations explicitly relate s₁ to the effectiveness of transverse reinforcement and cover geometry, making them more appropriate for splitting-controlled behavior in RC members with varying levels of confinement. Corrosion-explicit models further advance this framework by accounting for the degradation of confinement and interfacial concrete resulting from steel loss and corrosion products, thereby improving the physical consistency of deteriorated structures. However, these models typically require prior identification of the governing failure mode and calibration within defined corrosion ranges, indicating that no single formulation is universally applicable. Overall, the comparison highlights a trade-off between simplicity and physical realism, with corrosion- and confinement-aware models preferable for durability- and service-life-oriented assessments.

The maximum bond strength (τ_max) is a key quantity to assessing bond behavior but cannot be measured directly. Instead, the average bond strength (τ_avg) is typically used, defined as the maximum applied force per unit contact area between concrete and rebar. Recognizing that bond stress varies along the reinforcement length, τ_max can be estimated from τ_avg. Following the widely accepted recommendation by Perry and Thompson (1966), it is commonly assumed that τ_max ≈ 1.5 τ_avg, which provides a practical method to estimate τ_max though experimental and analytical studies.

Recognizing the significant influence of corrosion on bond degradation, numerous studies have proposed predictive models for corrosion-affected bond strength, as summarized in Table 12. Several formulations (e.g., Bhargava et al., 2007; Chung et al., 2004; Stanish et al., 1999) have been proposed to estimate the corrosion-affected bond strength by a multiplication factor formulated as a function of various influencing factors, such as corrosion mass loss percentage (Q), and geometric and material properties to the initial bond strength (τ_avg,0). However, those models need to first estimate τ_avg,0. In contrast, more comprehensive models (e.g., Bhargava et al., 2007; Murcia-Delso and Benson Shing, 2015; Perry and Thompson, 1966; Prieto et al., 2016; Sajedi and Huang, 2015; Soraghi and Huang, 2024; Zhao and Zhu, 2018) directly predict bond strength without relying on τ_avg,0 by explicitly incorporating material properties (such as concrete compressive strength f’c¸ reinforcement yield strength, f_yt), geometric properties (such as cover depth; d, rebar diameter; d_st), transverse reinforcement effects (K_tr), and bond failure mode (splitting or pull-out). These formulations enhance physical consistency and applicability for structural analysis, especially when corrosion affects confinement and failure modes. It should be noted that code-based expressions such as the CEB model (Alsiwat and Saatcioglu, 1992; Taerwe and Matthys, 2013) estimate τ_max without explicitly accounting for corrosion or steel mass loss, which limits their applicability for durability assessment of corroded structures.

Table 12.

Existing prediction models for bond strength in terms of corrosion.

Reference	Prediction model		Definition of parameters	Influencing parameter
Bhargava et al. (2007)	$τ_{a v g} = τ_{a v g, 0} . e^{- 19.8 (Q - 15)} \leq τ_{a v g, 0}$		Q: Corrosion mass loss percentageτ_avg: Average bond strengthτ_avg,0: Initial bond strength	τ _avg,0 , Q
Chung et al. (2004)	$τ_{a v g} = τ_{a v g, 0} . 0.0159 Q^{- 1.06} \leq τ_{a v g, 0}$			τ _avg,0 , Q
(Stanish et al., 1999)	$τ_{a v g} = τ_{a v g, 0} . (1 - 3.5 Q)$			τ _avg,0 , Q
Yuan and Graybeal (2015)	$τ_{a v g} = τ_{a v g, 0} \cdot (1 - 10.544 \cdot i_{c o r r} + 1.586 . (d / d_{s t}) \cdot Q)$		i_corr: Corrosion current densityd_st: Diameter of the reinforcing steeld: Cover depth	τ _avg,0 , Q, d/d _st
Prieto et al. (2016)	$\begin{array}{l} τ_{a v g} = {f_{c}^{'}}^{(2 / 3)} (m . {(\frac{1}{{d_{s t}}^{2}} + 1)}^{9.052} . {({(\frac{d_{s t}}{l_{d}})}^{2} + 1)}^{8.13} . e^{- 0.129 f_{c}^{'} / 40} {({(\frac{d + \frac{d_{s t}}{2}}{d_{s t}})}^{4} + 1)}^{0.058} \\ . {(K_{t r}^{2} + K_{t r} + 1)}^{0.498} . {(Q^{2} + 1)}^{- 0.016} - 1) \\ K_{t r} = \frac{A_{s t}}{s} \cdot d_{s t} \end{array}$		A_st: Cross-sectional area of transverse confinement bars s_st: Stirrups spacing (mm)m: Coefficient changes with bond condition and corrosion levell_d: Splice lengthK_tr: Confinement index from presence of transverse reinforcement (stirrups)	m, d, d _st , f' _c , Q,K _tr
Sajedi and Huang (2015) Soraghi and Huang (2024)	Splitting	$\begin{array}{l} Ln (\frac{τ_{a v g}}{\sqrt{f_{c}^{'}}}) = θ_{0} + θ_{1} \cdot (\frac{d}{d_{s t}} . \frac{μ + R_{r}}{1 - μ R_{r}} \cdot ψ \cdot e^{{\tilde{θ}}_{1} . Q}) + \\ θ_{2} \cdot (\frac{d_{s t}}{l_{d}}) + θ_{3} \cdot (\frac{1}{\sqrt{f_{c}^{'}}} . \frac{A_{s t} \cdot f_{y t}}{s_{s t} \cdot d_{s t}}) + σ ε \end{array}$	σε: Model error termμ: Friction coefficientΨ: Reduction factorR_r: Relative rib areaθ_i: Model coefficients b_e: Effective beam widthp_s= 0; specimens without transverse bar p_s= 1; specimens with the transverse bar f_yt: Reinforcement yield strength	Q, A _st , f _yt , d, d _st, θ _i , μ, Ψ, R _r
Sajedi and Huang (2015) Soraghi and Huang (2024)	Pull-out	$\begin{array}{l} Ln (\frac{τ_{a v g}}{\sqrt{f_{c}^{'}}}) = θ_{0} + θ_{1} \cdot (\frac{d}{d_{s t}} \cdot \frac{μ + R_{r}}{1 - μ R_{f}} . ψ . e^{{\tilde{θ}}_{1} . Q}) + θ_{2} \cdot (\frac{d_{s t}}{l_{d}}) \\ + θ_{3} \cdot (\frac{1}{\sqrt{f_{c}^{'}}} . \frac{A_{s t} . f_{y t}}{s_{s t} . d}) + θ_{4} \cdot (\frac{b_{e}}{d_{s t}} + \frac{μ + R_{r}}{1 - μ R_{f}} \cdot ψ \cdot p_{s}) + σ ε \end{array}$		Q, A _st , f _yt , d, d _st, θ _i , μ, Ψ, R _r
Taerwe and Matthys (2013)	Splitting	$τ_{\max} = 2.5 {(f_{c}^{'})}^{0.5}$	c_min: Minimum concrete cover depthc_max: Maximum concrete cover depth	K _tr , c _min , c _max , d _st , f' _c
Taerwe and Matthys (2013)	Pull-out	$τ_{\max} = 6.54 {(f_{c}^{'})}^{0.25} [{(\frac{c_{\min}}{d_{s t}})}^{0.2} \cdot {(\frac{c_{\max}}{d_{s t}})}^{0.2} \cdot {(\frac{20}{d_{s t}})}^{0.2} + 8 K_{t r}]$		K _tr , c _min , c _max , d _st , f' _c

Corrosion-induced degradation of bond significantly reduces the service life of RC structures (Sajedi et al., 2017). Additionally, bond failure mechanisms in corroded RC structures depend on the severity of corrosion and the quality of the concrete. Severe corrosion often causes splitting of the concrete cover, resulting in rapid loss of load-bearing capacity, while less severe corrosion leads to gradual bond degradation due to corrosion product accumulation at the steel–concrete interface (Soraghi and Huang, 2021). Several studies have investigated methods to improve bond performance in corrosive environments. Stainless steel, epoxy-coated reinforcement, and corrosion inhibitors can delay corrosion initiation and reduce bond degradation rates (Lee et al., 2018). Non-corrosive alternatives such as fiber-reinforced polymers (FRPs) have also been proposed, as they eliminate corrosion-related bond deterioration (James et al., 2019). However, these approaches require careful evaluation under different environmental and loading conditions.

Research gaps

The modeling of corrosion current density (i_corr) remains highly uncertain because it is sensitive to environmental fluctuations and concrete microstructure, as demonstrated by the wide variability reported in the corrosion rate studies reviewed in Corrosion rate and corrosion current density section. The majority of research models i_corr as a function of concrete resistivity or chloride concentration at rebar, while disregarding the time-dependent effects of moisture redistribution and oxygen availability discussed earlier in corrosion kinetics and transport-controlled corrosion mechanisms. Future research needs to develop dynamic models that combine environmental sensors with electrochemical data and probabilistic frameworks to model i_corr variability.

Current prediction models of steel section loss mainly depend on Faraday’s law under the assumption of uniform corrosion, as commonly adopted in the corrosion propagation models summarized in Section loss section, which is not suitable for predicting localized pitting that has a significant section loss compared to uniform corrosion. The models fail to account for how rust accumulation hinders oxygen diffusion, which leads to self-regulation of corrosion advancement, a phenomenon observed in several experimental studies reviewed herein. The suggested future work includes calibrating the current formulations with a wide range of data - different concrete materials and environmental testing conditions, and using stochastic or multi-scale methods to model localized corrosion effects, which can be verified through the rust front morphology and propagation using imaging techniques such as X-ray CT (computed tomography).

The majority of cover crack modeling is based on elastic fracture mechanics and thick-walled cylinder theory under the assumption of perfect geometries together with uniform material properties and continuous rust growth, as outlined in the analytical and numerical cracking models reviewed in Corrosion cracking section. The future work on cover cracking models should consider the propagation of multiple cracks or the effects of environmental substances entering through cracks (which impact the rust growth), and validate the predictions through strain-based measurements or acoustic emission monitoring.

The deterioration of concrete strength due to corrosion is not well understood. More research is needed to understand the complex relationships among compressive strength, permeability, microcracking development, and confinement, and how supplementary cementitious materials (SCMs) affect crack propagation resistance.

The current corrosion models determine rebar capacity reduction through area loss measurements only. Corrosion adversely affects the mechanical properties of reinforcement, including yield strength, ultimate strength, ductility, and strain capacity. These effects become increasingly pronounced when corrosion levels exceed approximately 15–20%, particularly in the presence of pitting corrosion, which induces high stress concentrations at localized pit sites. The structural analysis should consider those mechanical property changes with the pit distribution.

The process of bond deterioration remains understudied compared with corrosion modeling of other structural aspects. The current bond-slip models do not consider rebar surface condition, type of corrosion (pitting vs uniform), or bar location, despite experimental findings reviewed in Concrete rebar-bond behavior section indicating their significant impact on bond performance. While the incorporation of bond deterioration can be done through advanced FE modeling, there is still a need to develop structural analysis procedure to consider bond deterioration due to corrosion.

Structural performance with corrosion considerations

Service life prediction

RC structures degrade their resistance to chloride-induced corrosion, increasing the likelihood of bending, shear, and serviceability failures. The development of probabilistic time-dependent analysis tools for the service-life prediction of RC structures has received extensive research attention. These analyses usually follow a three-stage procedure that includes corrosion initiation, followed by corrosion propagation and then time-dependent/time-variant reliability analysis (Enright and Frangopol, 1998; Frangopol et al., 1997b; Stewart and Rosowsky, 1998a, 1998b). In the first stage, Fick’s second law of diffusion is widely used as the primary model for chloride penetration during the initiation of corrosion (Crank, 1975), with the surface chloride concentration and diffusion coefficient as the essential parameters. Rebar corrosion starts when chloride levels reach a critical threshold at rebar depth, leading to propagation that reduces rebar diameter (Ann et al., 2009; Kiesse et al., 2020; Reichert et al., 2024). This threshold and corrosion rate are key parameters studied via field monitoring and experiments. Reduced rebar diameter weakens RC structures’ resistance to bending and shear, aiding failure probability assessments under corrosion (Andrade and Alonso, 1996; Angst et al., 2009; Montemor et al., 2003).

The corrosion of reinforcement produces rust products that expand by as much as 300%. The development of tensile stresses in concrete leads to longitudinal cracking, which eventually causes spalling of the concrete cover. This serviceability failure can be evaluated by setting the time to concrete spalling (Stewart et al., 2011) or the critical amount of steel corrosion product (Akiyama et al., 2019; Biondini and Vergani, 2015) developed a 3D RC beam finite element for nonlinear analysis by modeling the damage effects of uniform and pitting corrosion in terms of reduced cross-sectional area of corroded bars, decreased ductility of reinforcing steel, deterioration of concrete strength, and spalling of concrete cover; their results demonstrated the model’s ability to replicate the effects of local corrosion damage on the overall structural response. Tran et al. (2022) demonstrated that sustained service-level loading during corrosion exposure can significantly accelerate steel loss and reduce flexural capacity. Their experiments on RC beams showed a linear relationship between load level and corrosion severity, with higher sustained bending loads resulting in up to 30% mass loss of steel and corresponding reductions in strength. These findings highlight the importance of accounting for load-corrosion interaction in service-life prediction models, particularly in evaluating residual capacity under combined mechanical and environmental deterioration.

To determine the service life, the uncertainties inherent in corrosion modeling must be quantified and accounted for. Corrosion of steel reinforcement varies spatially in RC structures due to factors like environmental exposure, concrete quality, and cover. Ignoring this variability can lead to overestimating structural reliability. Consequently, researchers are increasingly modeling corrosion variability in reliability assessments using, for example, the pitting factor (maximum pitting depth divided by average pitting depth) and the spatial variability factor (average cross-sectional area divided by minimum cross-sectional area). Another indicator is the maximum steel weight loss over 50 mm relative to the mean, which is used in finite element models to evaluate reliability considering non-uniform corrosion (Gu et al., 2018; Lim et al., 2016, 2019; Stewart, 2004, 2009; Val, 2007; Zhang et al., 2019). In addition to corrosion modeling, various physical properties and environmental factors need to be incorporated into a reliability framework to accurately forecast the structure’s service life. Strauss et al. (2013) analyzed concrete samples from bridges to update chloride distribution. Akiyama et al. (2018) used visual inspection results to update Markov chain deterioration probabilities. Srivaranun et al. (2022) updated corrosion prediction models using observed crack widths. Lu et al. (2019) developed an empirical model for steel corrosion rates, enabling reliability-based life predictions. Researchers also investigated how marine environments, CO2, and rising temperatures affect chloride-induced corrosion in RC structures (Akiyama et al., 2010; Kiesse et al., 2020; Stewart et al., 2011). Recently, Anghileri and Biondini (2025a, 2025b, 2026) adopted a Bayesian updating approach to update residual performance, safety, and reliability predictions for existing structures using experimental tests.

Table 13 summarizes the key corrosion modeling equations used to predict the service life of RC in various studies. As shown in Table 13, service-life prediction considers corrosion mechanisms by linking material degradation to structural limit states, including ultimate strength and serviceability. Early performance-based models (e.g., Frangopol et al., 1997a) couple corrosion initiation time and uniform cross-sectional loss with time-dependent flexural and shear resistance, enabling reliability analysis at the ultimate limit state, but generally assuming spatially uniform corrosion. Subsequent developments (e.g. (Stewart, 2004)) explicitly account for localized corrosion and pitting effects by modeling the reduction of the minimum remaining reinforcement area, which has been shown to govern structural capacity more accurately than average section loss. More recent probabilistic frameworks (e.g., Akiyama et al., 2019; Shi et al., 2011; Srivaranun et al., 2022) extend service-life prediction to serviceability limit states by incorporating time to corrosion initiation, time to cracking, crack propagation, and steel mass loss, often within stochastic or reliability-based frameworks. These approaches better capture corrosion uncertainty, spatial variability, and multi-stage deterioration, but need extensive calibration and data. Overall, Table 13 shows a shift from deterministic, uniform corrosion modeling to probabilistic, multi-limit-state frameworks, providing a realistic basis for the durability and long-term performance of RC structures.

Table 13.

Summary of existing formulations for modeling corrosion in service life prediction.

References	Formulation	Definition of parameters	Limit state time & corrosion process included
Frangopol et al. (1997a)	Flexural strength: $M_{r} (t) = A_{s} (t) f_{y} (d - \frac{a}{2})$ Shear strength: $V (t) = V_{c} + V_{s} (t) = V_{c} + \frac{A_{s} (t) f_{y} d}{s}$	A_s(t): Total bending reinforcement area: $A_{s} (t) = {\begin{cases} n π D_{b}^{2} / 4 for t \leq T_{1} \\ n π [D_{b} - 2 υ {(t - T_{1})}^{2}] / 4 for t > T_{1} \end{cases}$ D_b: Diameter of a rebar v: Rate of corrosionT_I: Time of corrosion initiation f_y: Yield strength of a reinforcing bar d: Effective depth a: Depth of the equivalent rectangular stress blockV_c: Shear strength of concreteA_v(t): Cross-sectional area of a stirrup:D_s: Diameter of a stirrup s: Spacing between stirrups	Ultimate limit statesTime to corrosion initiation, area loss of rebar, and yield strength of the rebar
Stewart (2004)	Flexural resistance at any element j at time t: $M_{j} (t) = M_{u l t} \frac{A_{s t} (t)}{A_{s t n o m}}$ Ultimate flexural capacity of a singly reinforced RC beam: $M_{u l t} = A_{s t} f_{y} (d - \frac{A_{s t} f_{y}}{1.7 f_{c}^{'} b})$	A_stnom: cross-sectional area of uncorroded reinforcing bar with diameter D₀ d: Effective depth b: Beam widthf_y: Yield stressf_c: Compressive strength of concreteA_st: Cross-sectional area of reinforcing bar: $A_{s t} (t) = A_{s t n o m} - \sum_{m = 1}^{n} A_{p i t_{m}} (t)$ Cross-sectional area of the pit: $A_{s} (t) = {\begin{cases} A_{1} + A_{2} p (t) \leq \frac{D_{0}}{\sqrt{2}} \\ \frac{π D_{0}^{2}}{4} - A_{1} + A_{2} \frac{D_{0}}{\sqrt{2}} < p (t) \leq D_{0} \\ \frac{π D_{0}^{2}}{4} p (t) \leq D_{0} \end{cases}$ $\begin{array}{l} A_{1} = 0.5 [θ_{1} {(\frac{D_{0}}{2})}^{2} - a \| \frac{D_{0}}{2} - \frac{p {(t)}^{2}}{D_{0}} \|]; θ_{1} = 2 \arcsin (\frac{a}{D_{0}}) \\ A_{2} = 0.5 [θ_{2} p {(t)}^{2} - a \frac{p {(t)}^{2}}{D_{0}}]; θ_{2} = 2 \arcsin (\frac{a}{2 p (t)}) \end{array}$ Maximum pit depth along a reinforcing bar: $p (t) = 0.0116 \cdot i_{c o r r} \cdot R \cdot t$ ; $a = 2 p (t) \sqrt{1 - {(\frac{p (t)}{D_{0}})}^{2}}$ i_corr: Corrosion current densityt: Time since corrosion initiation in years, and R is bounds of a Gumbel distributiona: Width of the pit	Ultimate limit statesArea loss of rebar and spatial variability of corrosion
Biondini,et al. (2006a); Biondini, et al., (2006b)(Biondini and Frangopol, 2008)	$T_{a c t u a l} = \min {(t - t_{0}) \| R \geq S, \forall t \geq t_{0}}$ or $T_{a c t u a l} = \min {(t - t_{0}) \| P_{F} \leq P_{F}^{*}, \forall t \geq t_{0}}$	T_actual: Actual service lifeR: Structural resistanceS: Structural demandP_F: Probability of failure (that is, R < S)P_F^*: An acceptable value of probability of failure t₀: Time instant at the end of the construction phase	Ultimate limit statesEffective resistance area loss for both concrete matrix and steel bars by assuming a linear relationship between the rate of damage and the mass concentration of the aggressive agent
Srivaranun et al. (2022)	Time to corrosion initiation: $t_{1} = \frac{1}{4 D_{c}} {\frac{0.1 (d + x_{G})}{e r f^{- 1} (1 - \frac{x_{4} C_{T}}{x_{5} C_{0}})}}^{2}$ Time to corrosion crack occurrence: $t_{2} = t_{1} + \frac{x_{8} Q_{c r} (c)}{x_{9} ρ_{s} V_{1}}$ Steel weight loss: $W (t) = {\begin{cases} 0 & t_{1} \geq t \\ \frac{π Φ}{(π Φ^{2}) / 4} (t - t_{1}) V_{1} x_{9} & t_{2} \geq t \geq t_{1} \\ \frac{π Φ}{(π Φ^{2}) / 4} (t - t_{1}) V_{1} x_{9} + (t - t_{2}) V_{2} x_{9}, & t > t_{2} \end{cases}$ Ratio of flexural strength of corroded beam to that of beam without corrosion: $h (Δ W) = - 1.64 \times 10^{- 4} Δ W^{2} - 9.73 \times 10^{- 3} Δ W + 1.0$	D_c: Diffusion coefficient of chloride: $\log_{D_{c}} = \log x_{7} - 6.77 {(w / c)}^{2} + 10.10 (w / c) - 3.14$ C₀: Surface chloride content on the structures d: Concrete cover erf: Error functionC_T: Critical threshold of chloride concentrationQ_cr: Critical threshold of corrosion associated with crack initiationρ_s: Steel densityV₁: Steel corrosion rate before corrosion crack occurrence $Φ$ : Diameter of the steel rebarV₂: Steel corrosion rate after corrosion cracking x₁ to x₉ are random variables and 2D random fields associated with the prediction modelΔW: Steel weight loss	Ultimate limit statesTime to corrosion initiation, time to crack initiation, and area loss of rebar
Akiyama et al. (2019)	Time to the occurrence of corrosion cracking: $T_{c r a c k e d} = T_{1} + T_{2}$ where T₁ is time to the occurrence of steel corrosion and T₂ is time from the occurrence of steel corrosion to the occurrence of crack.Performance function for steel corrosion: $\begin{array}{l} g_{d, 1} = C_{T} - χ_{1} \cdot C (c, χ_{2} D_{c}, χ_{3} C_{0}, t) \\ C (c, χ_{2} D_{c}, χ_{3} C_{0}, t) = χ_{3} C_{0} {1 - \erf (\frac{0.1 d}{2 \sqrt{χ_{2} D_{c} t}})} \end{array}$ Performance function for crack occurrence: $\begin{array}{l} g_{d, 2} = C_{T} - χ_{1} \cdot C (c, χ_{4} Q_{c r} (c) - Q_{b} (V, T_{c o}, t) \\ Q_{b} (V, T_{c o}, t) = V (t - T_{c o}) \end{array}$	χ₁ to χ₄ are model uncertainty associated with the estimation of C, D_c, the ratio of observed to predicted values, and Q_cr.C_T: Critical threshold of chloride concentration (kg/m³)C₀: Surface chloride content (kg/m³) erf: Error function d: Concrete cover (mm)D_c: Diffusion coefficient of chloride $\log_{D_{c}} = - 6.77 {(w / c)}^{2} + 10.10 (w / c) - 3.14$ W/C: Ratio of water to cement t: Time after construction (years)Q_cr: Critical threshold of corrosion amount at crack initiation $Q_{c r} = η (W_{c 1} + W_{c 2})$ η: Correction factor $\begin{array}{l} W_{c 1} = \frac{ρ_{s}}{π (γ - 1)} [α_{0} β_{0} \frac{0.22 {{(2 c + d)}^{2} + d^{2}}}{E_{c} (c + d)} f_{c}^{' 2 / 3}] \\ W_{c 2} = α_{1} β_{1} \frac{ρ_{s}}{π (γ - 1)} \frac{c + d}{5 c + 3 d} ω_{c} \end{array}$ ρ_s: Steel densityγ: Expansion rate of volume of corrosion product, determined to be 3.0α₀, β₀, α₁, and β₁ are coefficients considering the effects of concrete cover, steel bar diameter, and concrete strength d: Diameter of the steel bar f’_c: Concrete strengthE_c: Modulus of elasticity of concrete w_c: Crack width due to corrosion of the steel bar,V: Corrosion rate of the steel bar T_c0: Time in years until steel corrosion occurrence	Serviceability limit statesTime to corrosion initiation and time to crack initiation
Shi et al. (2011)	Time to corrosion damage: $T_{s p} = T_{i} + T_{1 s t} + T_{s e v}$ Time for crack to develop from crack initiation to a limit crack width: $\begin{array}{l} T_{s e v} = κ_{R} \frac{w - 0.05}{K_{c} M E_{r_{c r a c k}} r_{c r a c k}} (\frac{0.0114}{i_{c o r r} - 20}) \\ 0.25 \leq κ_{R} \leq 1, κ_{c} \geq 1, ω \leq 1.0 m m \end{array}$	T_i: Time to corrosion initiationT_1st: Time to first cracking-hairline crack of 0.05 mm width k_R: Rate of loading correction factor $κ_{R} \approx 0.95 [\exp (- \frac{0.3 i_{c o r r (\exp)}}{i_{c o r r} - 20}) - \frac{i_{c o r r (\exp)}}{2500 i_{c o r r} - 20} + 0.3]$ $i_{corr (\exp)}$ : Accelerated corrosion rate, determined as 100 μA/cm² $i_{corr - 20}$ : Corrosion current density (μA/cm²) at T = 20 °Cw: Crack width (mm)k_c: Confinement factor that represents an increase in crack propagation due to the lack of concrete confinement around external reinforcing barsME r_crackMEr_crack: Error of crack propagation model $r_{crack}$ : Rate of crack propagation (mm/h); $r_{c r a c k} = 0.0008 e^{- 1.7 ψ_{c p}}$ $ψ_{cp}$ : Cover cracking parameter: $ψ_{c p} = - \frac{d}{D f_{t}}, 0.1 \leq ψ_{c p} \leq 1$ Cover: Concrete cover (mm)D: Diameter of reinforcing bar (mm) f_t: Tensile strength of concrete (MPa)	Serviceability limit statesTime to corrosion initiation, time to crack initiation, and crack propagation

Seismic behavior

Reinforced concrete (RC) bridge structures exposed to chloride-induced corrosion experience progressive material degradation that significantly influences their seismic response. Corrosion reduces reinforcement cross-sectional area, alters mechanical properties of steel and concrete, weakens bond behavior, and may change failure mechanisms under cyclic loading. When combined with the inherent uncertainty of earthquake ground motion, these deterioration processes introduce substantial complexity into seismic performance evaluation. As a result, understanding the interaction between corrosion-induced damage and seismic demand has become a major focus in recent years, particularly in the context of life-cycle performance and reliability assessment of aging bridge infrastructure.

Numerous studies have investigated how corrosion affects seismic behavior at both structural and component levels. For example, Guo et al. (2018) examined the time-dependent seismic failure modes of coastal bridge piers by accounting for height-dependent corrosion variations. Their findings showed that plastic hinge locations may shift from the pier base toward splash and tidal zones as corrosion progresses, and that strength reduction becomes more pronounced when concrete cover cracking and bar buckling occur. Experimental investigations further validated these numerical observations, highlighting the importance of spatially varying corrosion models.

To address time-dependent deterioration within reliability frameworks, early studies primarily represented corrosion through uniform steel mass loss or stiffness degradation. Akiyama et al. (2011) incorporated probabilistic airborne chloride effects into seismic reliability assessment of bridge piers, enabling evaluation of flexural capacity reduction under corrosion. Thanapol et al. (2016) extended this approach by estimating the mean and variance of steel weight loss in plastic hinge regions using inspection data, demonstrating that improved inspection strategies reduce uncertainty in reliability predictions.

Subsequent research expanded corrosion modeling beyond global mass loss assumptions. Guo et al. (2015) evaluated time-dependent seismic demand and fragility for sea-crossing bridges considering corrosion during residual service life. Shekhar et al. (2018) compared seismic life-cycle cost analyses under varying chloride exposure conditions and proposed corrosion-deterioration modeling strategies. More refined parameterizations incorporated reinforcement section loss, yield strength deterioration, and concrete strength degradation (Guo et al., 2018; Thanapol et al., 2016), improving representation of strength and ductility loss in plastic hinge regions. In addition to considering rebar size, rebar strength, and concrete strength, which are affected by corrosion, Soraghi and Huang (2022) incorporated corroded rebar slip at the column footing into lateral displacement calculations under seismic loading by developing a simple bar stress-slip macromodel.

More recent developments emphasize localized and non-uniform corrosion mechanisms. Yuan et al. (2017) and Cui et al. (2018) incorporated pitting corrosion, post-cracking corrosion rates, and bond–slip deterioration into fragility assessment frameworks. Li et al. (2023) further addressed post-corrosion constitutive behavior, including changes in yield strength, ultimate strain, and bond performance. To facilitate practical implementation in structural analysis (El-Joukhadar et al., 2023a), proposed modifications to nonlinear FE models by introducing calibrated reduction factors for strength, stiffness, and drift based on corrosion severity. Their framework enables practical updates to seismic capacity evaluations for aging corroded columns, offering a performance-based approach aligned with experimental data.

Table 14 summarizes the corrosion-related parameters incorporated into seismic performance models of RC bridges. The listed studies show that corrosion effects are commonly represented through parameters such as steel mass loss, rebar section loss, and yield strength deterioration, which directly affect flexural capacity and ductility. Several investigations further include concrete strength degradation, reflecting the interaction between reinforcement corrosion and concrete damage. More advanced studies extend this representation by incorporating bond–slip deterioration, localized or pitting corrosion, time-dependent corrosion rates, post-corrosion constitutive behavior (yield and ultimate strength, strain), and instability-related mechanisms such as bar buckling vulnerability and bond weakening. In addition, although many studies considered the same corrosion-impacted parameters (e.g., rebar section loss or yield strength reduction), their formulations differ. Some models adopt deterministic reduction factors, whereas others employ probabilistic approaches based on inspection data or time-dependent corrosion evolution. In addition, the level of coupling between corrosion parameters and nonlinear structural analysis varies, ranging from global stiffness reduction approaches to detailed constitutive modifications at the material level. Overall, Table 14 illustrates not only the diversity of corrosion parameters considered in seismic analysis but also the variation in their mathematical implementation and structural integration across different studies.

Table 14.

Summary of corrosion modeling parameters in bridge seismic performance studies.

Reference	Modeled corrosion parameters
Akiyama et al. (2011)	Steel mass loss; Stiffness degradation; Flexural strength reduction in RC columns
Gu et al. (2018) Guo et al. (2015)	Rebar section loss; Yield strength deterioration; Concrete strength deterioration
Thanapol et al. (2016)	Steel mass depletion; Yield strength deterioration; Concrete strength deterioration
Yuan et al. (2017)	Rebar section loss; Bond-slip behavior alteration in piers
Cui et al. (2018)	Steel mass loss; Yield strength deterioration
Yu et al. (2021)	Stochastic pitting corrosion; Time-dependent corrosion rates; Localized section loss
Soaghi & Huang (2022)	Rebar section loss; Yield strength deterioration; Concrete strength deterioration; Bond-slip deterioration
Li et al. (2023)	Post-corrosion constitutive behavior (yield/ultimate strength, strain); Bond-slip deterioration
El-Joukhadar et al. (2023b)	Rebar section loss; Strength deterioration; Buckling vulnerability; Bond weakening
Li et al. (2023)	Embrittlement effects; Interfacial bond deterioration; Material strength deterioration
Guo et al. (2024)	Rebar section loss; Yield strength deterioration; Concrete strength deterioration

Life-cycle analysis

The holistic framework of life-cycle analysis (LCA) enables a comprehensive assessment of chloride-induced corrosion effects on RC structures from construction through their entire operational period. Corrosion deterioration influences not only structural performance but also inspection, maintenance, and failure costs over time. Consequently, evaluating infrastructure performance through a life-cycle perspective has become essential for managing aging RC bridges and optimizing long-term investment decisions. LCA provides a systematic framework for quantifying economic, environmental, and societal impacts associated with corrosion deterioration and maintenance interventions throughout the service life of infrastructure systems. Biondini and Frangopol (2016, 2018) provided a thorough review and survey, respectively, on the life-cycle performance of general deteriorating structural systems.

Typically, the life-cycle costs (LCCs) of RC structures include the initial cost, inspection and monitoring costs, maintenance expenses, and failure risk costs. The initial costs consist of material, design, and construction expenses that are established during project initiation (Val and Stewart, 2003). Inspection and monitoring activities are essential for detecting corrosion damage and include visual inspections (Frangopol and Liu, 2019; Kim and Frangopol, 2011), coring samples (Orbán and Gutermann, 2009; Ramanathan et al., 2021), and structural health monitoring technologies such as fiber-optic sensors (Casas and Cruz, 2003; He et al., 2022). Carsana et al. (2025) presented a diagnostic procedure that employs effective methods and criteria (including in-field nondestructive testing and chemical, physical, and microstructural analyses) to assess corrosion in existing concrete bridges. Maintenance costs exhibit significant time-dependent variation and include both direct and indirect components. Direct costs refer to activities aimed at preventing or mitigating corrosion damage, such as painting and overcoating (Liu et al., 2019; Mullard and Stewart, 2012). Indirect costs include broader social and economic impacts, such as traffic delays, increased fuel consumption, and environmental consequences including higher carbon dioxide emissions (Soliman and Frangopol, 2015). Failure risk, often expressed as life-cycle structural failure cost, is typically quantified by multiplying the probability of exceeding specific limit states by the associated failure consequences (Gong et al., 2019; Val and Stewart, 2003).

LCA also provides an effective framework for planning inspection and maintenance strategies throughout the service life of RC structures subjected to corrosion, thereby supporting budget allocation and maximizing service life. Maintenance strategies are commonly categorized as reactive (essential) and proactive (preventive) approaches (Yang et al., 2006; Zhu and Frangopol, 2013). Essential maintenance is performed after deterioration has occurred and depends on the current structural condition. Studies have compared the costs and effectiveness of various repair approaches such as patch repairs and cathodic protection (Binder, 2013; Islam and Kishi, 2010). In corrosive and seismic regions, interventions such as steel supplementation and member renewal have also been investigated (Chiu et al., 2010), while marine environments often require protective measures such as coatings or increased concrete cover (Farahani, 2019). Preventive maintenance focuses on early intervention strategies aimed at delaying corrosion deterioration, with key factors including inspection intervals (Cheng and Frangopol, 2021a, 2021bbib_cheng_and_frangopol_2021b), intervention strategies (Fernando et al., 2015; Liu and Frangopol, 2005), and repair effectiveness.

Determining optimal inspection and maintenance strategies typically involves solving an optimization problem that minimizes LCCs while maintaining required reliability levels (Frangopol et al., 1997a) first proposed a lifetime inspection and repair strategy for deteriorated RC structures, demonstrating that non-uniform inspection intervals can be more economical than uniform inspection schedules. Subsequent studies confirmed that preventive maintenance can significantly reduce the frequency and cost of essential maintenance and lower environmental impacts (Han et al., 2021; Navarro et al., 2019). The timing of initial maintenance actions has also been shown to strongly influence long-term deterioration and associated costs (García-Segura et al., 2017). More recently, optimization techniques such as genetic algorithms have been applied to determine maintenance strategies that balance safety, cost, and environmental impacts (Xie et al., 2018). In parallel, there has been increasing interest in incorporating sustainable and durable materials, such as corrosion-resistant steel (Cheng et al., 2020; Kere and Huang, 2019; Okasha et al., 2012), low-clinker cements (Cassiani et al., 2022), high performance concrete (Sajedi and Huang, 2019), and fiber-reinforced polymer reinforcements (Cadenazzi et al., 2019), to reduce long-term maintenance requirements and LCCs.

Table 15 summarizes representative formulations used for modeling corrosion processes within LCA frameworks for RC structures. The studies included in the table adopt different approaches to model the corrosion deterioration within LCA frameworks. Early formulations primarily focus on corrosion initiation time and uniform reinforcement area or mass loss to estimate expected inspection, repair, and failure costs (e.g., Frangopol et al., 1997a; Zhu and Frangopol, 2013). These models generally treat corrosion using simplified deterioration indicators and may be suitable for network-level decision-making. Subsequent studies introduce more detailed corrosion mechanisms into the LCA framework. For example, models incorporate chloride diffusion-based corrosion initiation and time-dependent reinforcement diameter reduction (García-Segura et al., 2017; Zhu and Frangopol, 2013), allowing the deterioration process to be linked directly to material degradation in structural analysis. Other formulations extend the analysis by considering corrosion-induced cracking, spalling, and serviceability limit states (Val and Stewart, 2003), which enables the evaluation of severability performance within the LCA. More recent approaches integrate additional deterioration indicators such as reinforcement yield strength degradation, bond deterioration, and corrosion-induced expansion effects (Chiu et al., 2010; Sajedi and Huang, 2019), reflecting a shift toward performance-based LCA models. Overall, Table 15 illustrates that the LCA is advancing towards integrating comprehensive deterioration mechanisms, structural performance criteria, and reliability-based decision-making frameworks.

Table 15.

Summary of existing formulations for modeling corrosion in life-cycle analysis.

References	Structure and hazard/loading type	Objective function	Corrosion process included and formulation	Definition of parameters
Frangopol et al. (1997a)	RC bridges	Minimize C_ET subject to $P_{f, l i f e} \leq P_{f, l i f e}^{*}$ Expected total cost: $C_{E T} = C_{T} + C_{P M} + C_{I N S} + C_{R E P} + C_{F}$ C_T: Initial cost of the structureC_PM: Expected cost of routine maintenanceC_INS: Expected cost of inspection and repair maintenanceC_REP: Cost of repairC_F: Expected cost of failure	Time to corrosion initiation and area loss of rebarDamage intensity, which defines the degree of existing damage due to corrosion at time t: $η (t) = {\begin{cases} 0 for 0 \leq t \leq T_{1} \\ \frac{D_{b 0} - D_{b} (t)}{D_{b 0}} for T_{1} \leq T \end{cases}$ Diameter of bending reinforcement bar at time t: $D_{b} (t) = D_{b 0} - 2 υ (t - T_{1})$	T_I: Time of corrosion initiation in yearsD_b0: Initial diameter of a bending reinforcement bar in a concrete section t: Time in years.v: corrosion rate
Zhu and Frangopol (2013)	Corroded bridges under traffic and earthquake Loads	Consequences of the bridge failure: $C_{F V} = [(C_{r e d} + C_{r u n n i n g} + C_{T L} + C_{S L} + C_{r e m}) + C_{u n d e r}] \cdot {(1 + r_{m})}^{t}$ r_m: Annual money discount rateRebuilding cost: $C_{r e b} = C_{1} W L$ C₁: Rebuilding cost per square meterW: Bridge width; L: Bridge lengthRunning cost: $C_{r u n n i n g} = C_{v e h} L_{\det} A_{D T} T_{\det}$ C_veh: Average running cost for vehicles A_DT: Average daily trafficL_det: Length of the detourT_det: Duration of the detour (days)Time-loss cost: $C_{T L} = [C_{t v a} O_{\det} (1 - \frac{T_{t r k}}{100}) + C_{t v t k} \frac{T_{t r k}}{100}] \frac{L_{\det} A_{D T} T_{\det}}{V}$ C_tva: Value of time per adultO_car: Average vehicle occupancy for carsT_trk: Average daily truck trafficC_tvtk: Value of time for truckV: Average detour speedSafety loss: $C_{S L} = (\frac{L}{D_{S}} + 1) [O_{c a r} (1 - \frac{T_{t r k}}{100}) + O_{t r k} \frac{T_{t r k}}{100}] (I C A F B)$ D_S: Safe following distance during drivingO_trk: Average vehicle occupancy for trucksICAFB is an indication of the magnitude of possible monetary compensation of the relatives of victims in the bridge failure event.Environmental losses associated with bridge failure: $C_{r e m} = C_{2} W L$ C₂: Removal cost per square meter	Time to corrosion initiation and area loss of rebarRemaining cross-sectional area of the reinforcement at time t: $A_{r} (t) = \frac{π}{4} {[d_{0} - C_{c o r r} i_{c o r r} (t - T_{i, r e b a r})]}^{2} for t > T_{i, r e b a r}$ Corrosion initiation time of reinforcing bar: $T_{i, r e b a r} = \frac{x^{2}}{4 D {[\erf c^{- 1} (\frac{C_{c r}}{C_{0}})]}^{2}}$	d₀: Initial diameter of the reinforcing bar C_corr: Corrosion coefficient i_corr: Parameter related to the rate of corrosionx: Concrete cover depthD: Chloride diffusion coefficient erfc: Complementary error functionC_cr: Critical chloride concentrationC₀: Surface chloride concentration
García-Segura et al. (2017)	Post-tensioned box-girder bridges	Life-cycle cost: $T C = T C_{m s} + C_{i n i}$ Total cost consists of economic impact (C_ms) and societal impact (C_s): $T C_{m s} = C_{m s} + C_{s}; C_{s} = C_{r u n} + C_{t i m e}$ C_run: Running cost due to detoursC_time: Time loss due to increased travel timeInitial cost: $C_{i n i} = C_{c o} \cdot V_{c o} + C_{r s} \cdot W_{r s} + C_{p s} \cdot W_{p s} + C_{f} . A_{f}$ C_co, C_rs, C_ps, and C_f: Unit cost for concrete, reinforcing steel, prestressing steel and formworkV_co: Volume of concreteW_rs, W_ps: Weight of reinforcement steel and prestressing steelA_f : Area of the formworkLife-cycle CO₂ emissions: $T E = T E_{m s} + E_{i n i}$ Total CO₂ emissions consist of environmental impact (E_ms) and societal impact (E_s): $T E_{m s} = E_{m s} + E_{s} E_{i n i} = E_{c o} \cdot V_{c o} + E_{r s} \cdot W_{r s} + E_{p s} \cdot W_{p s} + E_{f} \cdot A_{f}$ E_co, E_rs, E_ps, E_f: CO₂ emissions associated to concrete, reinforcing steel, prestressing steel and formwork	Corrosion initiation and weight loss of rebarAt time t, the chloride content at a distance x from the concrete outer surface using Fick’s second law $i_{c o r r} (t) = i_{c o r r} (1) \cdot 0.85 \cdot {(t - t_{c o r r})}^{- 0.29}$ Corrosion rate at the start of corrosion propagation: $\begin{array}{l} i_{c o r r} (1) = \frac{37.8 {(1 - w / c)}^{- 1.64}}{C_{c}} \\ D_{b} = D_{b 0} - 2 \cdot \int_{T_{i}}^{t} 0.0116 \cdot i_{c o r r} (t) d t \end{array}$	C₀: Chloride concentration on the surface erf: error functionD: Apparent diffusion coefficientD_b0: Initial reinforcement diameterD_b(t): Diameter of a reinforcement at time t;
Val and Stewart (2003)	RC structures in marine environments	Life-cycle cost: $L C C (T) = C_{D} + C_{C} + C_{Q A} + C_{I N} (T) + C_{M} (T) + E_{S F} (T)$ C_D: Design cost; C_C: Construction costC_QA: Expected cost of quality controlC_IN(T): Cost of inspectionsC_M(T): Expected cost of maintenanceE_SF(T): Expected cost of spalling during service life T $E_{S F} (T) = \sum_{i = 1}^{T / Δ t} Δ P_{f, i} \frac{C_{S F}}{{(1 + r)}^{i Δ t}}$ ΔP_f,i: Probability of a spalling incident between the (i-1)-th and ith inspectionsC_SF: Cost associated with the occurrence of spalling (i.e., repair/replacement, user losses, etc.) ; r: Discount rate	Time to corrosion initiation, time to crack initiation, and time to severe cracking and spallingProbability that cracking and spalling will occur before the time t: $P_{f} (t) = \Pr (T_{i} + T_{c r} \leq t) = \Pr (T_{i} + T_{c r 1} \leq t)$	T_i: Time to corrosion initiationT_cr1: Time to first cracking
Sajedi and Huang (2019)	RC bridges	Reliability-based design optimization: $R B D O {\begin{cases} Minimize cost (\hat{y}) \\ Subject to {\begin{cases} Probabbilistic constraint : β_{i} (x, y) - β_{T i} \geq 0 \\ Deterministic constraint : D_{j} (\hat{x}, \hat{y}) - D_{\min, j} \geq 0 \\ Ranges of design parameters : y_{k}^{t} \leq y_{k} \leq y_{k}^{u} \end{cases} \end{cases}$ Life-cycle cost: $PVC = C_{i n i t i a l} + \sum_{t = 1}^{t = T} \frac{C_{i}}{{(1 + r)}^{t}}$ C_initial: Initial cost of constructionC_i: i^th expenditure of intervene at time t that can include the costs of routine inspection, non-destructive evaluation, protection, repair, overlaying, and replacement required for the RC structure during its service lifetime T r: Discount rate	Time to corrosion initiation, area loss of rebar, yield strength of tensile rebars, and expansion of corrosion products on concrete crack widthCorrosion initiation time of the rebar: $t_{i n} = \frac{C_{b}^{2}}{4 D_{c} {[e r f^{- 1} (1 - \frac{C l_{t h}}{C l_{s}})]}^{2}}$ Time-dependent rebar diameter: $d_{b} (t) = \sqrt{1 - Q (t)} d_{b 0}$ Yield strength of tensile rebars: $f_{y} (t) = [1 - 0.005 Q (t)] f_{y 0}$ Concrete crack width: See Table 9 (Thoft-Christensen, 2001)	C_b: Concrete clear coverD_c: Chloride diffusion coefficient of an un-cracked concrete erf (∙): Gauss error functionCl_th: Chloride threshold valueCl_s: Chloride content on the surface of concreteQ(t): Corrosion level at time t in terms of percentage mass loss d_b0: Diameter of intact rebarf_y0: Yield strength of intact rebarα_d: Density ratio of the corrosion rust product to the reinforcing steel t_cr: Crack initiation time
Chiu et al. (2010)	Deteriorating RC building with high-seismic hazard	Life-cycle cost: $C_{T} = C_{I} + \sum (C_{Re p} \frac{1}{{(1 + k)}^{t_{Re p}}}) + \sum (C_{Re t} \frac{1}{{(1 + k)}^{t_{Re t}}}) + E$ C_I: Construction costC_Rep: Repair costC_Ret: Retrofitting cost t_Rep: Time of repair (year)t_Ret: Time of retrofitting (year)Deterioration risk over a specified service period: $E = \int_{0}^{T} \frac{1}{{(1 + k)}^{t}} (C_{f} P_{f} υ_{f} + C_{s} P_{s} υ_{s}) d t$ C_f: Cost of failureC_s: Cost of severe spalling or cracking v_f: Annual probability of rare earthquakes with recurrence periods of 500 yearsv_s: Annual probability of occasional earthquakes with recurrence periods of 100 yearsT: Specified service period (year) k: Discount rate	Corrosion initiation, weight loss of rebar, yield stress of reinforcing steel, and bond stress of the reinforcing steel and concreteChloride content at the surface of the reinforcing steel: $C l = (C_{0} - C_{i n i t}) (1 - \erf (\frac{x}{2 \sqrt{D_{p} t}})) + c_{i n i t}$ $\log D_{p} = - 3.9 {(w / c)}^{2} + 7.2 (w / c) - 2.5$ Rate of corrosion in propagation, acceleration and deterioration stages: $V_{c o r r} = \frac{78}{\sqrt{x}} (0.578 C l + 0.023 (w / c) - 1.52)$ Yield stress of reinforcing steel: $σ_{y} = σ_{y 0} (1 - 2.17 r / 100)$ Ultimate bond stress of the reinforcing steel and concrete: $τ_{b u} = τ_{b 0} \exp (- 0.0607 r)$ Ultimate bond stress of the reinforcing steel and concrete before corrosion: $τ_{b 0} = (1.2 + 5 p_{ω} b / d_{b}) \sqrt{σ_{b}}$	C_o: Surface chloride contentC_init: Initial chloride content in concreteW/C: Water–cement ratioD_p: Apparent diffusion coefficient erf: Error functionx: Depth of concrete covert: Time (years)σ_y0: Yield stress of the reinforcing steel before corrosion r: Corrosion of steel expressed as percentage weight lossb: Width of the memberd_b: Diameter of the main bar p_w: Stirrup ratioσ_B: Compressive strength of concrete

Post-tensioning bridges

A large number of Post-Tensioned (PT) bridges have been built to leverage their strength, faster construction, and the aesthetic benefits of a shallower superstructure. Despite these advantages, certain issues, particularly corrosion, remain persistent challenges for PT tendons, which are among the most vital structural elements of PT bridges. Instances of premature failure in PT tendons, primarily driven by recurring factors such as corrosion (see Figure 7), underscore the need for targeted, proactive maintenance strategies.

Figure 7.

Types of damage to PT Tendons. (a) corrosion and rupture of strands; (b) corrosion in anchorage; (c) corrosion and rupture of internal tendons and concrete spalling; (d) corrosion of wires visible by borescope; and (e) free water inside tendons.

PT tendons are associated with a range of damage types and mechanisms (Hurlebaus et al., 2016). Severe corrosion of external tendons causes strand ruptures (Figure 7(a)), and damage at anchorages (Figure 7(b)). Internal tendons often show corrosion and rupture only after concrete damage (Figure 7(c)), and these may be detectable only through coring or borescope inspections (Figure 7(d)). Issues like grout separation and bleeding (Figure 7(e)) have been reported, compounded by inconsistent standards across PT systems. Some tendons exhibit significant flaws due to material, design, and workmanship issues, prompting research into alternative corrosion inhibitors, such as flexible fillers, led by the Florida Department of Transportation (Cox, 2017).

In steel main tension elements of post-tensioning members, corrosion can be classified as uniform or localized. Uniform corrosion is the primary factor contributing to the degradation of structures. Extensive research has been conducted to assess and predict the corrosion rates of high-strength steel used in post-tensioned structures. Li et al. (2014) developed a model for uniform corrosion in high-strength bridge wires. Trejo et al. (2009) and Pillai et al. (2010) studied the corrosion of post-tensioning strands under various environmental conditions, presenting results on chloride concentrations, moisture levels, and stress effects. Zhang et al. (2021b) explored the corrosion performance of high-performance steel in bridge applications. Yuan et al. (2021) analyzed the temporal and spatial variability of uniform corrosion in high-strength steel wires. Kalina et al. (2011) evaluated the corrosion resistance of improved post-tensioning materials following long-term exposure in Texas. Additionally, Li et al. (2015) proposed a deterioration model for high-strength steel wires, incorporating uniform corrosion and emphasizing the need for such models in post-tensioned structures.

Corrosion testing of grout for post-tensioning ducts and stay cables to address localized corrosion began in 1992, following the publication of a report by Thompson and Michael, funded by the US Federal Highway Administration (FHWA) (Thompson et al., 1992). The testing procedure was further refined by Hamilton et al. (2000) to mitigate the corrosion of steel strands caused by off-specification grout. Subsequently, Hamilton et al. (2014) simulated prepacked grout bleeding under field conditions. Lau and Lasa (2016) reviewed the properties of post-tensioning elements, providing insights into localized corrosion issues in Florida and identifying various grout deficiencies. Meanwhile, Wang et al. (2005) investigated the corrosion of anchorage systems within post-tensioned grouted assemblies.

Table 16 offers a summary of the literature on uniform and localized corrosion of post-tensioning elements. Overall, the literature distinguishes corrosion in post-tensioned (PT) systems into two fundamentally different mechanisms: uniform corrosion affecting high-strength steel strands along their length, and localized corrosion concentrated at anchorage zones or within grout-deficient regions. Studies focusing on uniform corrosion primarily aim to characterize long-term corrosion rates and their temporal and spatial variability under different environmental exposures, emphasizing chloride concentration, moisture availability, and sustained stress effects on high-strength steel wires (e.g., Lim et al., 2016; Pillai et al., 2010; Trejo et al., 2009; Yuan et al., 2021). These models suit system-level service-life prediction but may underestimate sudden failure risks from localized damage. Recent experiments reinforce this limitation studies. Li et al. (2024) tested prestressed concrete girders with artificially induced strand corrosion and found that even ∼8% cross-sectional loss resulted in up to 30% reduction in flexural capacity and a shift from ductile concrete crushing to brittle tendon rupture. These results highlight the critical consequences of early-stage tendon corrosion and underscore the inadequacy of uniform corrosion models in predicting sudden failures in post-tensioned systems.

Table 16.

Summary of literature review on uniform and localized corrosion.

References	Subject covered
Li et al. (2015)	Uniform corrosion modeling
Trejo et al. (2009) Pillai et al. (2010)	Long-term exposure experiment- Uniform corrosion
Zhang et al. (2021b)	Corrosion performance of high-performance steel- Uniform corrosion
Yuan et al. (2021)	Temporal and spatial variability of uniform corrosion
Kalina et al. (2011)	Long-term exposure experiment- Uniform Corrosion
Thompson et al. (1992)	Corrosion testing of grout (First FHWA recommendation)-Localized corrosion
Hamilton et al. (2000)	Corrosion testing of grout- Localized corrosion
Hamilton et al. (2014)	Simulation of bleed water under field conditions- Localized corrosion
Lau and Lasa (2016)	Grout deficiency- Localized corrosion
Wang et al. (2005)	Localized corrosion in the anchorage zone

In contrast, research on localized corrosion highlights the critical role of grout quality, bleeding, segregation, and anchorage detailing in accelerating tendon deterioration, often leading to rapid strand rupture without visible external indicators (e.g., Hamilton et al., 2000; Lau and Lasa, 2016; Thompson et al., 1992; Wang et al., 2005). Localized corrosion models and experimental investigations reveal that even limited defects in grouting can significantly compromise tendon durability, underscoring the inadequacy of uniform corrosion assumptions for PT systems. As summarized in Table 16, uniform corrosion studies emphasize material behavior and exposure effects, whereas localized corrosion research focuses on construction quality, inspection techniques, and failure-prone zones such as anchorages and ducts. Consequently, Table 16 illustrates that comprehensive durability assessment of PT bridges requires combining uniform corrosion models for long-term degradation with localized corrosion evaluations to capture abrupt, brittle failure mechanisms. This distinction is essential for developing effective inspection, monitoring, and maintenance strategies tailored to the unique vulnerability of post-tensioned systems.

Typically, localized corrosion, a severe consequence of excessive bleed water, manifests within a decade of construction (Permeh and Lau, 2022). The layout and elevation of tendons in the PT system influence grout segregation and localized corrosion. Uniform corrosion involves a consistent corrosive effect across the surface, mitigated by corrosion protection barrier strands. Taeby and Mehrabi (2022) investigated both localized and uniform corrosion and proposed a deterioration model that considers factors for both types and developed equations to estimate failure risk from each and together, based on experiments and literature. In their deterministic model, the remaining cross-sectional area of steel tendons/strands can be expressed as:

A = φ_{c} \cdot A_{0} \cdot (1 - ω) \cdot g_{u}

(10)

where

φ_{c}

= the condition factor, A₀ = the intact cross-section of the tendon/strand, ω = the percentage of the tendons attacked by the localized corrosion, and g_u = deterioration function at time, t, is expressed as:

g_{u} (t) = {\begin{cases} 1 & t < t_{1} \\ {(1 - R_{U} (t - t_{1}))}^{2} & t \geq t_{1} \end{cases}

(11)

where R_U = a coefficient determined based on the environmental conditions at the bridge’s location and t₁ = time at which the corrosion protection system fails. Corrosion rates for various environments can be approximated from the existing literature; however, since rates depend on multiple factors, they should be investigated on a case-by-case basis. In addition to providing default rates, based on small-scale laboratory tests, Taeby and Mehrabi (2022); Taeby et al., (2023a) proposed a set of corrosion rates for various levels of environmental conditions. In equation (11), it is assumed that corrosion reduces the strand’s radius and that the residual materials produced by corrosion do not affect the corrosion rate. Additionally, it is presumed that corrosion is uniformly distributed along the free length of the tendon. For more details on the post-tensioning strand deterioration, references are provided here (Taeby, 2023; Taeby et al., 2023a; Taeby et al., 2023b).

Research gaps

Service life prediction

Despite substantial advances in modeling chloride-induced corrosion, a significant gap persists between corrosion process models and structural service-life prediction. This gap arises from both limitations in corrosion degradation modeling and the challenge of representing the interaction among multiple deterioration mechanisms that jointly govern structural performance. For example, for corrosion initiation, the accurate determination of chloride threshold values remains uncertain due to their sensitivity to material properties and environmental exposure (Andisheh et al., 2016) leading to large variability in predicted initiation time. Similarly, the modeling of corrosion propagation effects—particularly rust-induced cracking, spalling, and loss of confinement—remains insufficiently validated at the field scale, regardless of whether mechanical or empirical approaches are used. Furthermore, pitting corrosion represents a critical unresolved issue in service-life prediction. Most existing models represent pitting via fixed amplification factors applied to uniform corrosion rates, thereby failing to capture the stochastic nature of pit initiation, growth, and coalescence under variable material quality and exposure conditions. These simplifications obscure the translation of corrosion damage into structural limit states.

On the other hand, probabilistic methods should be used in service-life prediction to account for uncertainties; however, researchers lack sufficient, high-quality data to quantify uncertainties in essential variables, including diffusion coefficients, thresholds, and modeling errors. The lack of sufficient data leads researchers to select distribution parameters from various sources, including laboratory tests, expert judgments, and literature reviews, which may yield inconsistent results depending on the sources used. Since the service life of RC structures depends on multiple degradation processes, data-driven and statistical models that incorporate factors such as environmental exposure (humidity, temperature), chloride exposure, and structural characteristics (rebar type, cover depth, concrete quality) are effective. These models can analyze interacting factors and serve as practical tools for service life prediction.

Seismic behavior

The seismic performance of aging RC structures affected by corrosion remains inadequately characterized, particularly when corrosion-induced degradation interacts with cyclic loading demands. Time-dependent seismic fragility studies have shown that reinforcement mass loss, corrosion-induced bond degradation, and confinement deterioration collectively reduce strength, stiffness, and ductility. However, current modeling approaches adopt widely varying assumptions regarding corrosion morphology (uniform vs localized), bond–slip degradation laws, corrosion-induced cracking patterns, and post-cracking corrosion rates. These assumptions are frequently calibrated using small-scale specimens or accelerated corrosion techniques, limiting their applicability to naturally corroded structures.

A key research gap lies in the lack of validated constitutive and hysteretic models that explicitly account for corrosion-altered mechanical behavior under seismic loading. Corrosion affects not only cross-sectional area loss but also cyclic degradation mechanisms such as energy dissipation capacity, stiffness degradation, and failure modes. In addition, the spatial variability of corrosion damage, particularly pitting corrosion, introduces localized weaknesses that are rarely incorporated into seismic demand and capacity models. Another important gap concerns the updating of seismic risk and fragility functions as corrosion progresses and maintenance actions are implemented. Although inspection and repair strategies can significantly alter structural performance, their integration into time-updated, corrosion-informed seismic risk assessment frameworks remain limited. Developing standardized workflows that couple corrosion inspection data, repair actions, and seismic fragility updating represents a high-value research direction.

Life-cycle analysis

Life-cycle assessment (LCA) and life-cycle cost analysis (LCCA) of corrosion-affected RC structures are limited by the absence of standardized methodologies that consistently incorporate corrosion deterioration modeling. Existing studies employ disparate corrosion models, intervention strategies, discounting assumptions, and user cost formulations, leading to inconsistent outcomes and limited comparability. A fundamental challenge is that uncertainties in corrosion initiation and propagation timing strongly influence maintenance scheduling and cost projections, yet these uncertainties are often treated simplistically or deterministically in LCA/LCCA framework.

Furthermore, robust optimization of corrosion mitigation and maintenance strategies remains constrained by the lack of reliable deterioration and cost models capable of handling high levels of uncertainty and multiple constraints. Data limitations further compound these issues, particularly in quantifying indirect consequences of corrosion-related interventions, such as traffic disruption, environmental impacts, and cascading network effects. Addressing these gaps requires corrosion-centered probabilistic frameworks supported by standardized reporting practices and long-term field data collection.

Post-tensioning bridges

Corrosion in post-tensioned bridge systems presents unique challenges due to the interaction of environmental exposure, construction practices, material properties, and high sustained stress levels. While chloride-induced corrosion has been widely studied, other degradation mechanisms—including sulfate attack, grout incompatibility, corrosion–fatigue interaction, stress corrosion cracking, fretting corrosion, and hydrogen embrittlement—remain insufficiently integrated into existing corrosion models (Permeh and Lau, 2022; Taeby et al., 2023b). High-strength prestressing steel subjected to sustained and variable stress levels can experience coupled deterioration mechanisms, including corrosion–fatigue interaction, stress corrosion cracking, fretting corrosion, and hydrogen embrittlement, as reviewed in Post-tensioning bridges section. The combined effects of these mechanisms on tendon performance and long-term reliability are not sufficiently captured in existing modeling frameworks and require more comprehensive, coupled corrosion–mechanical formulations.

From an engineering perspective, there is a critical need for deterioration models that directly link corrosion damage in post-tensioned tendons to structural capacity retention, serviceability, and maintenance decision-making. The presence of non-cementitious barriers, such as plastic sheathing, further complicates corrosion modeling by altering moisture and ion transport paths, particularly near anchorages and deviators. Experimental studies evaluating the long-term performance of protective systems, grout or filler materials, and anchorage detailing under realistic exposure conditions remain limited. In addition, although inspection technologies for detecting corrosion in post-tensioned systems have improved, accurately quantifying corrosion severity in primary tension components remains challenging. Fully probabilistic frameworks that integrate corrosion detection, deterioration modeling, and reliability-based maintenance planning are needed to support risk-informed management of post-tensioned bridge systems.

Summary and conclusions

This paper presented a comprehensive review of modeling approaches for chloride-induced corrosion in reinforced concrete (RC) and post-tensioned (PT) concrete structures, integrating material degradation mechanisms with structural performance and long-term management considerations. The deterioration process was examined from chloride ingress and corrosion initiation through corrosion propagation, concrete cracking, bond degradation, and reinforcement mechanical deterioration, highlighting their combined influence on service life and structural safety.

Chloride ingress is commonly modeled using Fickian diffusion formulations, while more advanced approaches incorporate chloride binding, time-dependent diffusion coefficients, and environmental variability. Effective diffusion behavior is strongly influenced by concrete mix design, curing practices, and exposure conditions, and models that account for material aging and variability yield improved long-term predictions. Corrosion initiation is governed by critical chloride thresholds that vary with reinforcement type and environmental conditions; however, simplified or fixed threshold assumptions remain a major source of uncertainty in service-life estimation. Following corrosion initiation, progressive steel section loss, rust-induced cracking, and bond deterioration gradually degrade structural capacity rather than cause sudden failure. Time-dependent corrosion rate models offer more realistic predictions than constant-rate assumptions, particularly when coupled with crack development, which accelerates chloride ingress and alters transport mechanisms. These degradation processes reduce stiffness, ductility, and energy dissipation capacity, with notable implications for seismic performance and increased vulnerability to brittle failure under cyclic loading.

The review further emphasizes the importance of integrating corrosion models with probabilistic structural analysis, inspection data, and life-cycle cost assessment. Such frameworks enable reliability-based evaluation of aging infrastructure and support optimization of inspection, maintenance, and repair strategies. The effectiveness of interventions such as patching, overlays, and reinforcement replacement can be assessed within these time-dependent models, particularly when combined with monitoring and data assimilation techniques for early detection of critical deterioration stages.

Post-tensioned concrete structures require specialized consideration due to the distinct corrosion mechanisms affecting prestressing tendons, often exacerbated by grout-related deficiencies. Differentiating between uniform and localized tendon corrosion is essential for accurate risk assessment, as material variability, grout quality, and environmental exposure strongly influence failure potential. Dedicated modeling approaches for PT systems are therefore necessary to improve deterioration prediction and guide targeted maintenance planning.

Overall, corrosion modeling has progressed from simplified material-based formulations toward integrated multi-scale frameworks that link chloride transport, corrosion kinetics, mechanical degradation, structural response, and life-cycle management. While these advances provide a more realistic basis for durability-oriented design and infrastructure management, further efforts are needed to improve model validation, uncertainty quantification, and cross-stage integration. Continued development in this area will enable more reliable service-life prediction and contribute to the sustainable design and preservation of RC and PT concrete structures in aggressive environments.

Footnotes

ORCID iDs

Mehran Mozafarjazi

You Dong

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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