Corrosion-Fatigue Life Prediction Modeling for RC Structures under Coupled Carbonation and Repeated Loading

: The coupled action of concrete carbonation and repeated loading strongly inﬂuences the safety of reinforced concrete (RC) structures and substantially reduces service life. A novel corrosion-fatigue life prediction model for RC structures under coupled carbonation and repeated loading was developed. The effect of fatigue damage on concrete carbonation and carbonation-induced corrosion rate was considered, and the acceleration of fatigue damage accumulation due to reinforcement corrosion was considered in this approach. The proposed corrosion-fatigue life prediction model was illustrated by a 6 m-span RC slab in a simply supported slab bridge for the highway, and the effects of trafﬁc frequency, overloading, carbonation environment grade, and environmental temperature and relative humidity on corrosion-fatigue life were discussed. The results indicate that the proposed model can predict the corrosion-fatigue life of RC structures simply and conveniently. Trafﬁc frequency, overloading, carbonation environment grade, M temperature and relative humidity can decrease the corrosion-fatigue life of the RC slab by up to 66.86%, 58.90%, 77.45%, and 44.95%, respectively. The research is expected to provide a framework for the corrosion-fatigue life prediction of RC structures under coupled carbonation and repeated loading. 30%, and 40%, the corrosion depth is 0 mm, 0.462 mm, 0.950 mm, 1.470 mm, and 2.029 mm, respectively, and the corresponding fatigue damage caused by a heavy vehicle passage is 5.13 × 10 − 9 , 1.32 × 10 − 8 , 3.82 × 10 − 8 , 1.33 × 10 − 7 , and 1.10 × 10 − 6 , respectively. With the increase of corrosion ratio, the fatigue damage caused by a heavy vehicle passage increases signiﬁcantly.


Introduction
Concrete is the most widely utilized building construction material for the development of infrastructures due to its cost, technical properties, and durability characteristics [1,2], and reinforced concrete (RC) structures are the most common structural form in civil engineering. Many RC structures or members, such as highway and railway bridges, crane beams of industrial plants, and offshore oil production platforms not only bear static loads but also bear repeated loads such as vehicle loads, crane loads, or wave loads [3]. Repeated loading will cause fatigue damage in materials. Moreover, fatigue failure, one of the main failure modes of RC structures, will occur with the continuous accumulation of fatigue damage [4,5]. These RC structures also bear the long-term action of the environment, such as carbonation environment and chloride environment. After a period of service, rebar corrosion caused by concrete carbonation or chloride ion ingress commonly occurs in RC structures or members [6][7][8]. Reinforcement corrosion will cause the bearing capacity degradation of RC structures. Both repeated loading and environmental action will cause material deterioration in RC structures. Fatigue damage and reinforcement corrosion are the dominant reasons for the performance degradation of RC structures [9,10]. The fatigue failure mechanism of corroded RC structures is more complex than that of RC structures without corrosion. The reason is that the failure of RC structures caused by the combined action of corrosion and fatigue is not a simple superposition of these two failure factors, but a coupled process of their mutual influence [11]. CO 2 transport in fatigue-damaged concrete is accommodated by the gas transfer between pores and cracks. Considering the adsorption and reaction rate of CO 2 in concrete, the CO 2 transport in concrete can be expressed as [39]: where x is the CO 2 transport distance in concrete at service time t, C CO 2 is the CO 2 concentration around the concrete surface (kg/m 3 ), D CO 2 is the CO 2 diffusion coefficient in concrete (m 2 /s), ϕ is the porosity of the fatigue damaged concrete, s is the pore saturation of concrete, and Q is the total consumption rate of the concrete carbonation. The amount of CO 2 transport in fatigue-damaged RC structures includes the CO 2 transport in cracks and damaged but uncracked concrete. The diffusion coefficient can be expressed as [15]: where D p is the CO 2 diffusion coefficient in damaged but uncracked concrete, D c is the CO 2 diffusion coefficient in concrete cracks, and w m and l m are the average crack width and spacing of RC members under repeated loading, respectively. CO 2 transport in concrete is related to the environment temperature [40], humidity [41], and pore structure [15]. D p can be expressed as: where RH is the environmental relative humidity, U is the activation energy of carbon dioxide gas, R is the universal gas constant, T 0 is the standard temperature of the natural environment (298.15 K), T a is the environmental absolute temperature (K), and p is the factor related to fatigue damage, which can be obtained via regression of experimental data in our previous work and is expressed as follows [15]: where D b is the damage degree of RC members (0 ≤ D b ≤ 1). Jiang et al. [32] indicated the CO 2 diffusion coefficient in concrete cracks, which can be calculated as: where δ is the area fraction of aggregates in the concrete crack, and D air and D agg are the CO 2 diffusion coefficients in air and aggregates, respectively. The test results indicate that the repeated loading slightly affects the primary crack spacing. Therefore, it can be assumed that the average crack spacing of the RC members remains constant during the repeated loading process and can be computed as [42]: l m = ξ(1.9c + 0.08d eq /ρ te ) (6) where ξ is the influence coefficient of the average crack spacing under repeated loading, c is the thickness of concrete cover (mm), d eq is the equivalent diameter of the reinforcements under tension (mm), and ρ te is the reinforcement ratio of the tensile reinforcements. Based on the fatigue test results, the concrete crack width of a RC member under repeated loading is given by [15]: where w 0 is the average initial crack width of the RC members and n f is the number of repeated loading cycles. The known experimental studies have shown that the carbonation process of fatiguedamaged concrete can be calculated by Fick's first law [31]. Consequently, the carbonation depth X of the concrete at service time t can be calculated as: where K is the carbonation coefficient. Papadakis et al. [43] proposed a theoretical model for concrete carbonation through the mass-balance condition of the carbonatable substance, which can be expressed as:

Carbonation-Induced Reinforcing Bar Corrosion
Carbonation-induced reinforcing bar corrosion is usually considered uniform (general) corrosion [44]. For the uniform corrosion model, the cross-sectional loss area ratio of reinforcement ε(t) (i.e., cross-sectional corrosion ratio) can be computed as: where A s is the initial cross-sectional area of the reinforcing bar (mm 2 ), ∆A s is the crosssectional loss area of the reinforcing bar caused by uniform corrosion (mm 2 ), a(t) is the reinforcement corrosion depth (mm), and d 0 is the initial diameter of reinforcement (mm). The uniform corrosion form of the reinforcing bar is shown in Figure 1. Based on the fatigue test results, the concrete crack width of a RC member under repeated loading is given by [15]: where w0 is the average initial crack width of the RC members and nf is the number of repeated loading cycles. The known experimental studies have shown that the carbonation process of fatiguedamaged concrete can be calculated by Fick's first law [31]. Consequently, the carbonation depth X of the concrete at service time t can be calculated as:

Carbonation-Induced Reinforcing Bar Corrosion
Carbonation-induced reinforcing bar corrosion is usually considered uniform (general) corrosion [44]. For the uniform corrosion model, the cross-sectional loss area ratio of reinforcement ε(t) (i.e., cross-sectional corrosion ratio) can be computed as: where As is the initial cross-sectional area of the reinforcing bar (mm 2 ), ΔAs is the crosssectional loss area of the reinforcing bar caused by uniform corrosion (mm 2 ), a(t) is the reinforcement corrosion depth (mm), and d0 is the initial diameter of reinforcement (mm). The uniform corrosion form of the reinforcing bar is shown in Figure 1. The Chinese code GB/T 51355-2019 [45] suggests a method for carbonation-induced reinforcing bar corrosion. When reinforcing bar corrosion initiates, a certain depth of concrete cover free of carbonation still exists, which can be defined as carbonation remaining depth X0. The carbonation remaining depth X0 can be estimated by: where mef is the carbonation environmental factor; Dk is the factor associated with both c and K. The carbonation environment factor mef can be determined by carbonation environmental aggressiveness, including the four grades A, B, C, and D (Table 1). The Chinese code GB/T 51355-2019 [45] suggests a method for carbonation-induced reinforcing bar corrosion. When reinforcing bar corrosion initiates, a certain depth of concrete cover free of carbonation still exists, which can be defined as carbonation remaining depth X 0 . The carbonation remaining depth X 0 can be estimated by: where m ef is the carbonation environmental factor; D k is the factor associated with both c and K. The carbonation environment factor m ef can be determined by carbonation environmental aggressiveness, including the four grades A, B, C, and D (Table 1). For the factor D k , when c ≤ 28 mm, When c > 28 mm, When reinforcing bar corrosion initiates, the corrosion initiation time t ini can be calculated as: The reinforcing bar corrosion rate before concrete cracking can be computed by: where K cl is the reinforcement location factor, K cl is equal to 1.6 at the corner of RC members and 1.0 at other locations, T is the environmental temperature ( • C), and f cu is the concrete cubic compressive strength (MPa).
With the increase of corrosion degree, the concrete cover will crack (corrosion-induced cracking). When the concrete cracks due to corrosion, the reinforcement corrosion depth can be estimated as: The time to corrosion-induced cracking can be calculated as: Concrete will also crack when the RC members are subjected to repeated loading. Li [46] proposed the S-N curve of concrete cracking due to repeated loading, which can be represented as: f max t / f t = 1.3681 − 0.1214lgN cr (18) where f max t is the maximum tensile stress of the concrete under constant amplitude repeated loading (MPa), f t is the concrete tensile strength (MPa), and N cr is the number of loading cycles when concrete cracks due to fatigue. When using the S-N curve for concrete cracking prediction, fatigue damage in concrete under tension caused by repeated loading is accumulated according to Miner's rule [47]: where D cr is the fatigue damage of tensile concrete caused by repeated loading, D cr ≥ 1 indicates the concrete cracking, n cr,i is the number of concrete tensile stress cycles with the i-th maximum tensile stress f max t,i , and N cr,i is the number of concrete tensile stress cycles when concrete cracks due to fatigue under the i-th maximum tensile stress f max t,i . Under the constant amplitude repeated loading, the fatigue-induced cracking time of concrete cover can be obtained: where f is the loading frequency of each period. For variable amplitude repeated loading, Song et al. [38] showed the calculation method for equivalent maximum tensile stress. The concrete cracking time t cr is the smaller value of corrosion-induced cracking time t cr,c and fatigue-induced cracking time t cr,f . Thus, the concrete cracking time t cr can be calculated as: t cr = min{t cr,c , t cr,f } When fatigue-induced concrete cracking occurs, the reinforcement corrosion depth a cr,f can be computed as: (22) When concrete cracking begins, the corrosion depth a cr can be expressed as: Because concrete cracks will accelerate reinforcing bar corrosion, the corrosion rate after concrete cracking can be estimated by: According to the above models, the reinforcing bar corrosion depth can be calculated as:

Corrosion-Fatigue Life Prediction
The corrosion-fatigue failure mode of RC structures or members is generally the brittle fracture of reinforcing bars under the coupled corrosion and fatigue [48]. The S-N curve describes the fatigue resistance of materials, and the S-N curve of reinforcing bars can be expressed as: where ∆σ is the stress range of reinforcing bars (MPa), N is the number of stress cycles at reinforcement fatigue fracture under the stress range ∆σ, and C and m are the material coefficients for the reinforcing bars. Fatigue damage accumulation in the reinforcing bar caused by repeated loading is also calculated by Miner's rule, which can be determined as [47]: where D t is the cumulative fatigue damage in the reinforcing bar, and D t ≥ 1 indicates the reinforcing bar rupture; n i is the number of reinforcing bar stress cycles with the i-th stress range ∆σ i ; and N i is the number of reinforcing bar stress cycles when the reinforcing bar fatigue ruptures under the i-th stress range ∆σ i . For variable amplitude repeated loading, Li et al. [49] showed the calculation method for equivalent stress range.
As the corrosion degree increases, the fatigue stress ranges increase, and the fatigue resistance decreases [10]. Considering the time-variant of fatigue stress range and fatigue resistance, Equation (26) can be rewritten as follows: The time-variant effect of material coefficient C can be represented by: where C 0 is the initial value material coefficient C of the reinforcing bar without corrosion. φ(t) can be calculated as [50]: The time-variant reinforcement stress range can be calculated as: where ∆σ 0 is the reinforcement stress range without corrosion under repeated loading (MPa). When the RC structure is in service until the J-th period, the cumulative fatigue damage of reinforcement D t can be calculated as: where f i,j is the loading frequency of the i-th stress range ∆σ i in the j-th period. When D t reaches 1, fatigue failure will occur, and the fatigue life t r can be obtained.

Corrosion-Fatigue Life Prediction Flowchart
The above models are applied to the corrosion-fatigue life prediction of RC structures under coupled carbonation and repeated loading. A computational program for the corrosion-fatigue life prediction was developed in the MATLAB environment (version 2016a, The Mathworks, Natick, MA, USA) following the steps outlined in Figure 2. The additional details can be summarized as follows: (1) Input the structural parameters, load parameters, material behavior, and environmental factors. Then, calculate the reinforcement and concrete stress cycles due to repeated loading. (2) Determine the CO 2 transport in fatigue damaged process through Equations (1)-(7).

Model Application
To illustrate the corrosion-fatigue life prediction model of RC structures under the coupled carbonation and repeated loading, the fatigue damage and corrosion-fatigue life of an RC slab in a highway RC simply supported slab bridge were evaluated. The span of Reinforcement corrosion initiation Equations (10)- (15) Corrosion rate acceleration Equations (21)

Model Application
To illustrate the corrosion-fatigue life prediction model of RC structures under the coupled carbonation and repeated loading, the fatigue damage and corrosion-fatigue life of an RC slab in a highway RC simply supported slab bridge were evaluated. The span of the analyzed RC slab was 6 m, with a height of 0.32 m and a width of 0.91 m. The concrete strength grade of the slab was C30 (150 mm cubic compressive strength of 30 MPa at age of 28 days). Tensile reinforcement with a diameter of 18 mm was configured at the bottom of the slab, and the reinforcement strength grade was HRB335 (hot-rolled ribbed steel bar with a yield strength standard value of 335 MPa). The other reinforcements were 10 mm in diameter and HPB300 (hot-rolled plain steel bar with a yield strength standard value of 300 MPa) in strength grade. The geometric dimensions and reinforcement of the mid-span cross-section of the RC slab are shown in Figure 3. The design service life of the 6 m-span RC slab is 100 years. The thickness of the concrete cover is 50 mm, and the carbonation environment grade of the bridge is B. The annual average environmental temperature and relative humidity at the location of the slab bridge are 15 • C and 65%, respectively. Table 2 lists the material properties and environmental factors used in the corrosion-fatigue life prediction.  Table 2 lists the material properties and environmental factors used in the corrosion-fatigue life prediction.     Because the small car only has a slight influence on the fatigue performance of the bridge, to simplify the analysis, only the influences of heavy vehicles on corrosion-fatigue life are considered in this paper. Under the dead load, secondary dead load, and vehicle load, the design values of the mid-span bending moment of the RC slab are 43.33 kN·m, 16.25 kN·m, and 132.69 kN·m, respectively. Therefore, the mid-span bending moment caused by the maximum and minimum of repeated loading acting on the RC slab is 59.58 kN·m and 192.27 kN·m, respectively. The design value of traffic frequency f of heavy vehicles on the highway is 1000/day. For highway RC bridges, it is necessary to obtain the stress response of tensile reinforcement under repeated vehicle loads when predicting the corrosion-fatigue life. Because the RC slab is a flexural member, the tensile reinforcement stress of the mid-span cross-section can be calculated as: where n s is the ratio of elastic modulus of tensile reinforcement to concrete, M is the bending moment at the mid-span of the slab, y is the distance from the position of tensile reinforcement to the neutral axis of the mid-span transfer section, and I is the iner-

Corrosion-Fatigue Life Prediction
The fatigue damage of tensile reinforcing bar with different corrosion ratios caused by a heavy vehicle passage is shown in Figure 4. When the corrosion ratio of reinforcement is 0%, 10%, 20%, 30%, and 40%, the corrosion depth is 0 mm, 0.462 mm, 0.950 mm, 1.470 mm, and 2.029 mm, respectively, and the corresponding fatigue damage caused by a heavy vehicle passage is 5.13 × 10 −9 , 1.32 × 10 −8 , 3.82 × 10 −8 , 1.33 × 10 −7 , and 1.10 × 10 −6 , respectively. With the increase of corrosion ratio, the fatigue damage caused by a heavy vehicle passage increases significantly. The selection of the S-N curve is very important to the corrosion-fatigue life tion results. Eurocode [51] presents the S-N curve for reinforcing bars in RC str which can be expressed as:

Corrosion-Fatigue Life Prediction
The fatigue damage of tensile reinforcing bar with different corrosion ratios by a heavy vehicle passage is shown in Figure 4. When the corrosion ratio of reinfor is 0%, 10%, 20%, 30%, and 40%, the corrosion depth is 0 mm, 0.462 mm, 0.950 mm mm, and 2.029 mm, respectively, and the corresponding fatigue damage caus heavy vehicle passage is 5.13 × 10 −9 , 1.32 × 10 −8 , 3.82 × 10 −8 , 1.33 × 10 −7 , and 1.10 × spectively. With the increase of corrosion ratio, the fatigue damage caused by a vehicle passage increases significantly. The repeated loading frequency is the design traffic frequency of heavy v (1000/day). Under a B-grade carbonation environment, the cumulative fatigue dam tensile reinforcement in the 6 m-span RC slab is shown in Figure 5. Under the des  The repeated loading frequency is the design traffic frequency of heavy vehicles (1000/day). Under a B-grade carbonation environment, the cumulative fatigue damage of tensile reinforcement in the 6 m-span RC slab is shown in Figure 5. Under the design service condition, the corrosion-fatigue performance of the RC slab is satisfactory. When the bridge is operated to the design service life, the cumulative fatigue damage in the tensile reinforcement in the slab is 0.372. With the deepening of reinforcement corrosion and the accelerating of fatigue damage accumulation, the corrosion-fatigue failure of the RC slab will occur, and the corrosion-fatigue life of the RC slab is 162.1 years. ematics 2021, 9,3296

Effect of Traffic Frequency and Overloading
In recent years, the traffic frequency of highways in China has in which will cause a detrimental impact on the corrosion-fatigue bridges. To study the influence of traffic frequency, the corrosionslab with traffic frequencies of 500/day, 1000/day, 3000/day, 5000/ predicted. These traffic frequencies are defined by the monitoring o China [52] and the design traffic frequency. To simplify the analysis, are considered time-invariant. The carbonation environment grade perature, and relative humidity under different cases of traffic frequ the design service condition in Section 4.1. Figure 6 illustrates the cumulative fatigue damage evolution of ment in the 6 m-span RC slab under different cases of traffic frequen of traffic frequency, the fatigue damage increases, and fatigue dam celerates. When the highway bridge is operated to the design servic fatigue damage of the two cases with traffic frequency of 500/day a and 0.372, respectively. Under the three cases with traffic frequency and 7000/day, the cumulative fatigue damage reaches the limit value 54th years of bridge operation, and corrosion-fatigue failure of RC

Effect of Traffic Frequency and Overloading
In recent years, the traffic frequency of highways in China has increased significantly, which will cause a detrimental impact on the corrosion-fatigue performance of RC bridges. To study the influence of traffic frequency, the corrosion-fatigue life of the RC slab with traffic frequencies of 500/day, 1000/day, 3000/day, 5000/day, and 7000/day is predicted. These traffic frequencies are defined by the monitoring of a highway bridge in China [52] and the design traffic frequency. To simplify the analysis, the traffic frequencies are considered time-invariant. The carbonation environment grade, environmental temperature, and relative humidity under different cases of traffic frequency are the same as the design service condition in Section 4.1. Figure 6 illustrates the cumulative fatigue damage evolution of the tensile reinforcement in the 6 m-span RC slab under different cases of traffic frequency. With the increase of traffic frequency, the fatigue damage increases, and fatigue damage accumulation accelerates. When the highway bridge is operated to the design service life, the cumulative fatigue damage of the two cases with traffic frequency of 500/day and 1000/day is 0.186 and 0.372, respectively. Under the three cases with traffic frequency 3000/day, 5000/day, and 7000/day, the cumulative fatigue damage reaches the limit value in the 95th, 70th, and 54th years of bridge operation, and corrosion-fatigue failure of RC slab occurs. Under these three cases, the corrosion-fatigue life of the 6 m-span RC slab under the coupled carbonation and repeated vehicle load cannot reach the expected design service life. Therefore, it is necessary to strengthen the maintenance and repair of the bridge. fatigue damage of the two cases with traffic frequency of 500/day and 1000/day is 0.186 and 0.372, respectively. Under the three cases with traffic frequency 3000/day, 5000/day, and 7000/day, the cumulative fatigue damage reaches the limit value in the 95th, 70th, and 54th years of bridge operation, and corrosion-fatigue failure of RC slab occurs. Under these three cases, the corrosion-fatigue life of the 6 m-span RC slab under the coupled carbonation and repeated vehicle load cannot reach the expected design service life. Therefore, it is necessary to strengthen the maintenance and repair of the bridge. Overloading may occur with the increase of highway traffic volume. Previous research results have shown that overloading will substantially reduce the corrosion-fatigue life of bridges [53]. Consequently, it is necessary to investigate the effect of overloading on the corrosion-fatigue life of RC structures. To simplify the analysis, this study considers the overloading condition by increasing the design vehicle load. Four overloading cases with different overloading ratios are considered in this study, increasing the design load by 5%, 10%, 15%, and 20%, respectively. The traffic frequency, carbonation environment grade, and environmental temperature and relative humidity of each overloading case are the same as the design service condition in Section 4.1. Figure 7 plots the cumulative fatigue damage evolution of the tensile reinforcement in the 6 m-span RC slab under different overloading cases. Similarly, with the increase of the overloading ratio, the fatigue damage increases, and fatigue damage accumulation accelerates. When the highway bridge is operated to the design service life, the cumulative fatigue damage of the two cases with overloading ratios of 5% and 10% is 0.567 and 0.861, respectively. Compared with the design service condition without overloading, the fatigue damage of the two cases increases by 55.1% and 135.8%, respectively. When the overloading ratio is 15% and 20%, the cumulative fatigue damage reaches the limit value in the 87th and 68th years of bridge operation, and corrosion-fatigue failure of RC slab occurs. An overloading ratio of 20% will reduce the corrosion-fatigue life of the RC slab significantly. Therefore, overloading must be strictly prohibited during bridge operation. Overloading may occur with the increase of highway traffic volume. Previous research results have shown that overloading will substantially reduce the corrosion-fatigue life of bridges [53]. Consequently, it is necessary to investigate the effect of overloading on the corrosion-fatigue life of RC structures. To simplify the analysis, this study considers the overloading condition by increasing the design vehicle load. Four overloading cases with different overloading ratios are considered in this study, increasing the design load by 5%, 10%, 15%, and 20%, respectively. The traffic frequency, carbonation environment grade, and environmental temperature and relative humidity of each overloading case are the same as the design service condition in Section 4.1. Figure 7 plots the cumulative fatigue damage evolution of the tensile reinforcement in the 6 m-span RC slab under different overloading cases. Similarly, with the increase of the overloading ratio, the fatigue damage increases, and fatigue damage accumulation accelerates. When the highway bridge is operated to the design service life, the cumulative fatigue damage of the two cases with overloading ratios of 5% and 10% is 0.567 and 0.861, respectively. Compared with the design service condition without overloading, the fatigue damage of the two cases increases by 55.1% and 135.8%, respectively. When the overloading ratio is 15% and 20%, the cumulative fatigue damage reaches the limit value in the 87th and 68th years of bridge operation, and corrosion-fatigue failure of RC slab occurs. An overloading ratio of 20% will reduce the corrosion-fatigue life of the RC slab significantly. Therefore, overloading must be strictly prohibited during bridge operation.     Figure 8 that the corrosionfatigue life will be substantially reduced as the traffic frequency and overloading ratio increase simultaneously, and the corrosion-fatigue life of the RC slab is 13.5 to 162.1 years under different traffic frequencies and overloading cases. The corrosion-fatigue life is 93.9 years, 68.0 years, and 53.7 years, respectively, under the traffic frequency cases of 3000/day, 5000/day, and 7000/day without overloading. Compared with the design service condition, the corrosion-fatigue life of the three traffic frequency cases decrease by 42.07%, 58.02%, and 66.86%, respectively. When the traffic frequency is 1000/day and the overloading ratio is 5%, 10%, 15%, and 20%, respectively, the corrosion-fatigue life is 133.3 years, 107.6 years, 85.3 years, and 66.6 years, respectively. The four overloading cases will reduce the corrosion-fatigue life by 17.77%, 33.61%, 47.35%, and 58.90%, respectively, compared with the design service condition.

Effect of Carbonation Environment Grade
The carbonation environment grade will affect the concrete carbona rosion rate of the reinforcing bar, which, in turn, affects the corrosionstructures. To study the effect of carbonation environment grade on corr the cumulative fatigue damage and corrosion-fatigue life of the 6 m-sp different carbonation environments are evaluated. The environmental relative humidity are the same as the design service condition. Figure 9 lative fatigue damage evolution of the tensile reinforcement in the 6 m-sp different carbonation environments and different traffic frequencies.
It can be seen from Figure 9 that with the increase of carbonation en the fatigue damage increases, and fatigue damage accumulation accel bridge is operated for 100 years with the traffic frequency of 1000/day fatigue damage of tensile reinforcement is 0.187, 0.261, 0.372, 0.483, a tively, under five environmental cases of carbonation-free and carbona grades A to D. Compared with the carbonation-free environment, the fa creases by 39.31%, 98.69%, 158.00%, and 229.24% under the four differen vironment grades. As the traffic frequency increases to 7000/day, the failure of the RC slab will occur when the bridge is operated for 77th, 63t 47th years, respectively, under carbonation-free and carbonation enviro

Effect of Carbonation Environment Grade
The carbonation environment grade will affect the concrete carbonation and the corrosion rate of the reinforcing bar, which, in turn, affects the corrosion-fatigue life of RC structures. To study the effect of carbonation environment grade on corrosion-fatigue life, the cumulative fatigue damage and corrosion-fatigue life of the 6 m-span RC slab under different carbonation environments are evaluated. The environmental temperature and relative humidity are the same as the design service condition. Figure 9 shows the cumulative fatigue damage evolution of the tensile reinforcement in the 6 m-span RC slab under different carbonation environments and different traffic frequencies.
It can be seen from Figure 9 that with the increase of carbonation environment level, the fatigue damage increases, and fatigue damage accumulation accelerates. When the bridge is operated for 100 years with the traffic frequency of 1000/day, the cumulative fatigue damage of tensile reinforcement is 0.187, 0.261, 0.372, 0.483, and 0.616, respectively, under five environmental cases of carbonation-free and carbonation environment grades A to D. Compared with the carbonation-free environment, the fatigue damage increases by 39.31%, 98.69%, 158.00%, and 229.24% under the four different carbonation environment grades. As the traffic frequency increases to 7000/day, the corrosion-fatigue failure of the RC slab will occur when the bridge is operated for 77th, 63th, 55th, 55th, and 47th years, respectively, under carbonation-free and carbonation environment grades A to D. grades A to D. Compared with the carbonation-free environment, the fatigue damage increases by 39.31%, 98.69%, 158.00%, and 229.24% under the four different carbonation environment grades. As the traffic frequency increases to 7000/day, the corrosion-fatigue failure of the RC slab will occur when the bridge is operated for 77th, 63th, 55th, 55th, and 47th years, respectively, under carbonation-free and carbonation environment grades A to D.  When the traffic frequency is 1000/day to 7000/day, the fatigue life is 534.    When the traffic frequency is 1000/day to 7000/day, the fatigue life is 534. tion environment grades A to D. When the traffic frequency increases from 1000/day to 7000/day, the fatigue life decreases by 85.71% under the carbonation-free environment, and the corrosion-fatigue life decreases by 73.20%, 66.86%, 63.83%, and 61.66%, respectively, under the carbonation environment grades A to D. Under the traffic frequency of 1000/day to 7000/day, carbonation environment grades A, B, C, and D can decrease the corrosion-fatigue life by 56.48%-18.37%, 69.67%-29.64%, 74.46%-35.33%, and 77.45%-39.48%, respectively, compared with a carbonation-free environment.

Effect of Environmental Temperature and Relative Humidity
The environmental temperature and relative humidity of the bridge service environment will affect the corrosion-fatigue life of RC structures [54]. Therefore, the corrosionfatigue life of the 6 m-span RC slab under four grades of carbonation environment with various temperatures (5-25 • C) and relative humidity (55-85%) is predicted, and the prediction results are shown in Figure 11. The traffic frequency of all cases is 1000/day. It can be seen from Figure 11 that the increase of environmental temperature and relative humidity will reduce the corrosion-fatigue life.

Effect of Environmental Temperature and Relative Humidity
The environmental temperature and relative humidity of the bridge service environment will affect the corrosion-fatigue life of RC structures [54]. Therefore, the corrosionfatigue life of the 6 m-span RC slab under four grades of carbonation environment with various temperatures (5-25 °C) and relative humidity (55-85%) is predicted, and the prediction results are shown in Figure 11. The traffic frequency of all cases is 1000/day. It can be seen from Figure 11 that the increase of environmental temperature and relative humidity will reduce the corrosion-fatigue life.
For the environmental temperatures from 5 °C to 25 °C and environmental relative humidity from 55% to 85%, the corrosion-fatigue life of the RC slab is 299.9-174.7 years, 218.9-120.5 years, 185.8-103.7 years, and 163.5-95.0 years, respectively, under carbonation environment grades A to D. The service conditions with different environmental temperatures and relative humidity will reduce the corrosion-fatigue life by up to 41.76%, 44.95%, 44.16%, and 41.87%, under the four different carbonation environment grades A to D.

Conclusions
The current study proposed a novel corrosion-fatigue life prediction model for RC structures under coupled carbonation and repeated loading. The proposed corrosion-fatigue life prediction model was illustrated by a 6 m-span RC slab in a simply supported slab bridge for the highway. The fatigue damage and the corrosion-fatigue life were pre- For the environmental temperatures from 5 • C to 25 • C and environmental relative humidity from 55% to 85%, the corrosion-fatigue life of the RC slab is 299.9-174.7 years, 218.9-120.5 years, 185.8-103.7 years, and 163.5-95.0 years, respectively, under carbonation environment grades A to D. The service conditions with different environmental temperatures and relative humidity will reduce the corrosion-fatigue life by up to 41.76%, 44.95%, 44.16%, and 41.87%, under the four different carbonation environment grades A to D.

Conclusions
The current study proposed a novel corrosion-fatigue life prediction model for RC structures under coupled carbonation and repeated loading. The proposed corrosionfatigue life prediction model was illustrated by a 6 m-span RC slab in a simply supported slab bridge for the highway. The fatigue damage and the corrosion-fatigue life were predicted. The influences of traffic frequency, overloading, carbonation environment grade, and environmental temperature and relative humidity were analyzed. The following conclusions can be drawn: