2.1. Description of Carbonation Process

The state function is defined to describe the carbonation risk and the resistance of concrete carbonation, as given in Eq. (1)

where R(t) and S(t) denote the carbonation resistance (concrete cover depth), and the carbonation risk (carbonation depth) with time.

The carbonation depth is usually calculated as a function of time, as given in Eq. (2), with the carbonation coefficient encompassing the concrete quality and the environmental conditions.

where x is the depth of carbonation, K for the carbonation rate coefficient and t for time of exposure to the atmosphere containing carbon dioxide.

The state defined by Eq. (1) can be used to describe the critical state that carbonation progresses up to the reinforcement if carbonation depth x in Eq. (2) is substituted for carbonation risk S(t) in Eq. (1).

2.2. Monte Carlo Simulation

As commonly adopted by many researchers, the Monte Carlo simulation is a very effective technique in modelling the risk of carbonation in as probabilistic way. Unlike the deterministic method in which the state function Eq. (1) is calculated with the specific values of cover depth and carbonation depth via Eq. (2), the probability distributions of cover depth and carbonation depth should be provided in the probabilistic method. In the present study, the Monte Carlo simulation technique was used to assess the carbonation risk. For the simulation, mean value and standard derivation of the parametric values (i.e. carbonation depth and concrete cover depth) were obtained from the underground field investigation. Then, 100,000 of random samples for each parametric value were generated in the Monte Carlo simulation. The probability for carbonation to reach the depth of the steel can be calculated from the 100,000 random trials and is defined as the ratio of the number of carbonation, which can be calculated by Eq. (1) at the depth of the steel, to the number of total trials, as given in Eq. (3).

where P_t is the probability of carbonation at the depth of the steel and n(g(t) < 0) denotes the number of carbonation at the depth of the steel out of N trials (100,000 trials in this study)

2.3. Field and indoor experiments in the existing box culverts

Box culverts for power transmission constructed in five regions in Korea, as shown in Photo 1, were investigated to estimate carbonation risk and service life due to carbonation.

The used period in each box culvert is ranged from 11 years to 29 years, in which inner surface of concrete box culvert has been exposed to carbon dioxide environment, as tabulated in Table 1. The investigation was performed in three parts inside box culvert: the upper slab and opposite wall surfaces. The concrete mix proportion for the box culverts was designed for 24 MPa of compressive strength and the cover concrete de pth to the reinforcement was designed for 50 mm. The annual concentration of carbon dioxide inside the box culvert was measured from 367 ppm to 629 ppm (G150-NDIR CO₂ Analyzer, ±1.5%), as listed in Table 1. Additionally, the concrete cover depth to the steel was measured in 84 points using an ultrasonic instrument (Iron-Seeker, ±2mm) as shown in Photo 2.

After measuring the cover depth, 21 concrete core specimens were drilled out for indoor experiment including compressive strength and carbonation depth investigation, as shown in Photo 3. In order to avoid the unintended carbonation of the core specimen in a later experiment, each specimen was sealed with plastic wrap up to the indoor test. After splitting cylinder specimen in the laboratory, carbonation depth was measured in each specimen. The measurement was done with a phenolphthalein pH indicator (i.e.<10 in the pH) to determine the carbonation depth. The carbonated concrete part was not changed in color, whereas the remaining parts turned purple by the phenolphthalein indicator, as shown in Photo 3.

2.4. Accelerated carbonation test for cracked specimen

Because of the highly permeable nature of cracks, durability of concrete can be severely reduced by the presence of cracks inside concrete. Therefore, the influence of crack on the mass transport behavior such as carbon dioxide diffusion should be clearly investigated to estimate the service life of RC structures reasonably. In fact, mass transport in crack can best be described if the crack itself is modeled as separated finite elements which have the free diffusivity for carbon dioxide in numerical analysis on the carbonation process. However, in this study, in order to experimentally consider the concrete crack effect on the service life due to carbonation of box culverts, the accelerated carbonation test for the cracked specimen was performed. Beam specimen with the same compressive strength of box culvert, a design strength 24 MPa, was newly manufactured for the accelerated carbonation test. The water cement ratio was 50% and the average compressive strength was obtained as 36.4 MPa. Potential crack, which can exist in the upper slab and the vertical wall in box culvert, was artificially produced by 4 point bending test, as shown in Photo 4. The flexural crack occurred in each beam specimen after a vertical loading controlled by a universal testing machine with a capacity of 2,000 kN. Crack width which ranged from 0.05 mm to 0.6 mm was controlled by a clip gage on the bottom face of beam specimen at a speed of 0.002 mm/s. In addition to the crack width, crack depth was also measured in each beam specimen.

After measuring crack width and depth, the cracked beam specimens were equipped in the accelerated carbonation test device during 4 weeks. Referring to KS F 2584, the accelerated carbonation test device was kept in the environment condition of 20°C of temperature, 60% of relative humidity and 5% of carbon dioxide concentration, about 100 times of atmospheric CO₂ concentration inside box culvert. After 4 weeks exposure to the accelerated carbonation environment, each beam specimen was cut along the longitudinal center line, which produced the perpendicular surface to the existing crack, as shown in Photo 5. Then, a phenolphthalein solution was applied to each cracked surface to determine the carbonation depth. The carbonation depth was separately measured in a cracked part with flexural crack and a sound part with no crack. Photo 5 illustrated the whole process of the accelerated carbonation test for the cracked beam specimen, which experienced the accelerated carbonation exposure for 4 weeks.

3.1. Probability distributions for carbonation risk estimation

In this study, five box culverts for power transmission in urban areas were examined in terms of the risk of carbonation. As discussed in section 2.3, the concrete cover depth was measured by an ultra sonic detector and compared with the designed value of cover depth to the reinforcement. Additionally, the carbonation depth was examined in 18 drilled core specimens using the phenolphthalein indicator. The concrete cover depth and carbonation depth investigated in 5 regions are given in Table 2 and Fig. 1. The average of concrete cover depth measured ranged from 54 to 72 mm, although the cover depth in the design is identical to 50 mm in five regions.

The distribution of the concrete cover depth was best fitted to the normal distribution, determined by the goodness of fit test such as the Kolmogorov-Smirnov test. The average of the measured cover depth was 54.7, 60.3, 55.1, 72.1 and 59.5 mm in Seoul A, Seoul B, Seoul C, Kwangju and Mokpo respectively, which are greater than the designed cover depth. However, the standard deviation ranged from 16.2 to 26. 1mm, which showed that the quality of construction for concrete cover was not good in casting of concrete in the box culverts construction. The distribution of carbonation depth measured in core specimen was best fitted to the normal distribution, which is identical to the cover depth distribution, as given in Fig. 1. It was seen that the average of carbonation depth accounted for 14.5, 13.3, 17.3, 10.3 and 15.9 mm in Seoul A, Seoul B, Seoul C, Kwangju and Mokpo, respectively. The standard deviation was below 10 mm in all regions. Considering the different environmental conditions in five regions and the large scatter of carbonation depth in each region, it is found that the carbonation rate in each region was severely affected by CO₂ environment and the concrete quality of each box culvert.

3.2. Service life due to carbonation

The service life due to the risk of carbonation at the depth of the steel was calculated by one of the probabilistic method, the Monte Carlo simulation. The probability of corrosion due to carbonation was calculated and dotted with an increment of 1 years of time, as depicted in Fig. 2. The probability distributions in the concrete cover depth and the rate of carbonation were taken into account by producing 100,000 samples considering the mean and standard deviation of the parametric values tabulated in Table 2. The corrosion probability and the respective service life was depicted in Fig. 2 and Table 3, respectively. It is evident that the low quality of concrete or construction always shows the high probability of corrosion due to the high rate of carbonation. For the first 50 years in all regions in Fig. 2, the probability of corrosion due to carbonation was negligible and thereafter increased gradually with time depending on the regional characteristics such as carbonation rate and cover depth. More specifically, for Seoul C, which had the longest used period in 5 regions, the risk of corrosion was negligible until 170 years and thereafter considerably increased up to about 0.5 at 300 years. On the contrary, for Kwangju, which had the smallest used period in 5 regions, the risk of corrosion gradually increased after 80 years and finally reached about 0.3 at 300 years, which is 40% less than Seoul C. However, as tabulated in Table 3, the service life of box culverts a ccounted for about 206, 221, 202, 182 and 233 years in Seoul A, Seoul B, Seoul C, Kwangju and Mokpo, respectively, assuming that the limit state for carbonation at the depth of the steel is regarded as 0.1 of probability of the corrosion initiation. Generally, the threshold probability of the carbonation risk can be assumed by engineers’ preference, but it is usual that 0.1 of the probability is considered as the limit state (^{CEB-FIP, 2006}; ^{Kwon et al., 2005}).

As naturally perceived, an increase in the concrete cover depth results in a dramatic increase in the service life if a deterministic way for estimating the service life is adopted using Eq. (2) with the average values of cover depth and carbonation rate listed in Table 2. The service life due to carbonation calculated in a deterministic way is also tabulated in Table 3. For Kwangju having the greatest cover depth 72 mm of 5 regions, the service life was obtained to 544 years inspite of having the minimum carbonation rate 3.09 mm/year^0.5. It is noticeable that the service life in a deterministic way is remarkably higher than the one in a probabilistic method. For the extreme case of Kwangju, 544 years is comparable to 188 years, which is about 33% of the former. It is evident that the large differences between two method take place presumably due to the consideration of stochastic feature such as a variance of the probability distributions of cover depth and carbonation rate. Thus, it may suggest that uncertainty of cover depth and carbonation depth which occurred in the real construction site should be appropriately considered to predict the service life more practically with only a probabilistic way.

However, in five regions investigated in this study, it is found that concrete was nearly free from carbonation because the service life due to carbonation was estimated over 100 years. The service life over 100 years will be guaranteed only if the considered box culvert is assumed to be totally sound, which means that the concrete inside box culvert does not have any cracks. As generally known by concrete engineers, it is not possible that any crack does not exist in concrete structures. Thus, it seems that the service life over 100 years estimated in five box culverts is too optimistic ignoring concrete cracking in reality.

3.3. Effect of concrete crack on service life

As mentioned earlier, the durability of RC structures can severely damaged if there exist cracks in concrete. Many researchers have been trying to investigate the influence of crack on the penetration behavior (^{Song et al., 2006}; ^{Kwon et al., 2005}; ^{Kwon et al., 2009}). However, most of existing researches are performed through the numerical analysis techniques such as a finite element method (FEM) or a finite difference method (FDM) under a steady state assumption near the concrete crack (^{Boulfiza et al., 2003}; ^{Ismail et al., 2004}; ^{Song et al., 2006}). Only limited researchers have reported two dimensional concentration profiles near the crack with the consideration of an unsteady state profile by numerical approaches, but the experimental verification for the numerical result still remains a controversial problem so far (^{Nakamura et al., 2006}).

In this study, in order to consider the potential cracks inside box culvert on the service life due to carbonation, the experimental approach is adopted as described in section 2.4. Fig. 3 shows the crack depth measured through the cracked beam specimen with respect to the crack width controlled by clip gage on the bottom face of beam specimen. The controlled crack width ranged from 0.05 mm to 0.6 mm and the corresponding crack depth was measured up to about 60 mm, as shown in Fig. 3. Owing to the heterogeneity of concrete and the unstable crack growth in flexure, it seems that there is no relationship between crack width and depth.

Fig. 4 shows the comparisons of carbonation depth between the sound face and the cracked surface of an identical beam specimen. The carbonation depth of the cracked surface are gradually increasing with crack width while the one of the sound face maintains a certain level having an average value, 15.8 mm. The greater carbonation depth for cracked surface takes place from the increased permeability in the cracked surface, which causes an easier access of carbon dioxide to the inner concrete. For cracked surface, carbonation reaction was accelerated due to the greater carbonation rate, compared to sound concrete. Even for the relatively small crack width below 0.05 mm, the microstructure between two crack faces may accelerate the diffusion property of carbon dioxide and thus cause more rapid carbonation of concrete than for the sound surface with no crack.

The accelerated carbonation reaction with respect to the crack effect can be quantitatively analyzed by a simple arithmetic calculation of the carbonation depth difference between sound and cracked surface.

Fig. 5 shows the difference of carbonation depth between two faces with respect to crack depth. According to Fig. 5, it is seen that the difference of carbonation depth is not dependent of the crack depth in beam specimen.

Eq. (4) indicates a linearly fitted result of Fig. 5, and the poor relationship between two regression variables is verified by the fact that the obtained coefficient of determination, R2, is 0.64. Alternatively, the difference of carbonation depth can be plotted with respect to crack width instead of crack depth, as shown in Fig. 6. In contrast with Fig. 5 and Eq. (4), a nice representation of carbonation depth difference with crack width is obtained and a simple linear relationship can be employed with an improved coefficient of determination, 0.91 as indicated in Eq. (5).

Thus, it is found that the additional progress of carbonation in the cracked surface is linearly dependent of crack width increase in concrete specimen. If it is assumed that crack depth and crack width are separate features of concrete cracking, it is found that crack depth itself does not affect the additional carbonation reaction in comparison with crack width.

where dx is the difference of carbonation depth between cracked and sound face, d for the crack depth in cracked beam specimen and w for the crack width in cracked beam specimen.

In order to reflect the linear dependency of an additional carbonation and crack width to the service life estimation, it is necessary to employ the relative ratio of carbonation depth rather than the arithmetic difference of carbonation depth. Fig. 7 shows the relative ratio of carbonation depth of cracked face to sound face with respect to crack width. A better linear fit result represented from Eq. (6) could be obtained in view of the greatest coefficient of determination, 0.996.

where r is the relative ratio of carbonation depth between cracked and sound face and w for the crack width (mm) in cracked beam specimen.

As special cases to be applied to the estimation of service life due to carbonation, the relative ratios of carbonation depth are graphically marked at crack width, 0.3, 0.5 and 1.0 mm, as indicated in Fig. 7. The corresponding ratios are 1.24, 1.41 and 1.81, respectively. Considering Eq. (2), where the carbonation depth is directly proportional to the carbonation rate coefficient, it is found that the ratio of carbonation depth is equivalent to the ratio of carbonation rate coefficient. Accordingly, the increased carbonation rate in cracked face can be obtained by multiplying the carbonation rate in sound face by the relative ratio in Eq. (6). With an assumption the carbonation rate in cracked face has the normal distribution like the one in sound face, average and standard deviation of the carbonation rate coefficient in cracked face can be obtained by a basic rule of probability distribution, as given in Eq. (7).

where μ and σ are average and standard deviation of carbonation rate coefficient; K_crack and K_sound are carbonation rate coefficient in cracked and sound surface, respectively.

In the same manner as the service life estimated for five box culvers in the former section, the service life considering crack effect is obtained from the Monte Carlo simulation by employing the statistical coefficients defined by Eq. (7). The probability of corrosion due to carbonation at crack width, 0.3, 1.0 mm was given in Fig. 8 and Fig. 9, respectively.

Assuming that the threshold probability for the service life definition is 0.1, the service life of box culverts considering the crack effect accounted and was tabulated in Table 3 with respect to the considered crack width

As expected, the service life dramatically decreased in all the box culverts as the crack width increased up to 1.0 mm with an equivalent increase of carbonation rate coefficient following Eq. (6). It is seen that crack with of 0.3 mm resulted in about 35% decrease in the corrosion free life, while crack width of 0.5 mm significantly decreased the service life by about 50% in all the box culverts investigated in this study. For the extreme crack width of 1.0 mm causing an apparent underground leakage of water in box culverts, the service life accounted below 70 years due to about 70% decrease of service life in crack free box culverts. Considering the uncertainty of the concrete quality and construction in the practical situation, the optimistic assurance of the service life over 100 years in box culverts does not hold any longer due to the existence of cracks with a non-negligible crack width. Although the service life estimated by considering the crack effect was limited to the box culverts investigated in this study, it is suggested that the carbonation rate and service life dependences on the crack width can provide an very effective information to repair the cracks and maintain the durability in practice for most of concrete structures.