Results from the simulations, conducted with the case study, are described and discussed below. Durability state limit (DSL) analysis, regarding the reinforcement depassivation and the initiation of reinforcement degradation, was conducted considering a deterministic application of Equations (1) and (2), and the probabilistic approach.
4.1. Deterministic Analysis
Figure 6 shows the diffusion depths of CO
2 and Cl
− into the concrete. The process of concrete carbonation was considered for all cities, while chloride diffusion was considered only for the cities located in coastal regions, Florianópolis and Fortaleza.
For the carbonation analysis, the exposure environments that generate the maximum and minimum carbonation depths in Equation (1) were the indoor-sheltered from rain (
KCE = 1.3) and the outdoor-exposed to rain (
KCE = 0.65), respectively. Simulations were conducted for beams with cross-sections of 22 × 40 cm (
Figure 3), and concrete produced with Portland cement CEM III/A type, and with no content additions. To perform the simulations, the CO
2 concentration and concrete compressive strength were defined with the mean values indicated in
Table 6. The relative humidity was defined with mean values indicated in
Table 5, and the other input parameters were:
Kc = 30.5;
Kfc = 1.7;
Kad = 0.32;
KCO2 = 15.5; and
KRH = 1300.0.
To perform the simulations of structures located in Florianópolis and Fortaleza, the beams are modeled with a cross-section of 22 × 40 cm (
Figure 3), and concrete is produced with Portland cement CEM III/A type, with no content additions. The Cl
− concentration and concrete compressive strength were defined with the mean values presented in
Table 6. The relative humidity and temperature were defined with mean values indicated in
Table 5, and the other input parameters were
K1 = 1.21 and
K2 = 1.0.
Based on
Figure 6, there is no occurrence of reinforcement depassivation by carbonation for concrete structures located outdoors, exposed to rain environments. For structures located indoors, sheltered from rain environments, depassivation occurs in all cases. This analysis considers that the diffusion depth of CO
2 or chlorides, before the age of 50, is greater than the cover indicated in the Brazilian design standard for RC structures (NBR 6118 [
47]) located in the simulated environments. When comparing all the cities, the CO
2 diffusion is higher in the cities of São Paulo and Brasilia, and this can be explained by the average values of relative humidity in these cities (
Table 5), which are between 60 and 70%, the range of humidity that enhances the diffusion of CO
2 in concrete [
64].
Considering structures located in coastal cities, depassivation occurs due to the diffusion of chlorides. In Florianópolis and Fortaleza, as shown in
Figure 6, the depth of chloride diffusion at the end of 50 years is greater than the concrete cover thickness (40 mm). When comparing the CO
2 and chlorides diffusion depth of these cities, at the end of the analysis, they are two to four times higher than the carbonation depth, showing that the chlorides ingress is more meaningful and aggressive, as reported by Taffese and Sistonen [
55]. However, for Zhu et al. [
40], in cases where there is a combined action of chlorides and CO
2, the initiation of corrosion must be analyzed considering the two mechanisms simultaneously. There must also be considered a synergistic effect between the aggressive agents, which could reduce the structure’s service life by up to 40% [
52].
Figure 6 also indicates that, for the durability requirements of the Brazilian standard (NBR 15575 [
65]), the specified concrete cover and compressive strength only abide by depending on the exposure conditions. Thus, the aggressiveness of the environment must be carefully defined in the design stage, so the structural element achieves its design life.
4.2. Probabilistic Analysis
The first step of the probabilistic analysis is to establish the number of samples and simulations that would provide reliable results. According to Liberati et al. [
35], the probability of failure of civil structures is between 10
−3 and 10
−6, so it is required about 10
3 to 10
5 limit state simulations. As an option to reduce computational efforts, it was decided to carry out a convergence study.
In
Figure 7, two convergence analyses are presented, referring to the variability of the reliability index with the number of samples used in the simulations. The analyses were conducted considering the process of carbonation in the city of São Paulo (
Figure 7a) and chloride diffusion in the city of Florianópolis (
Figure 7b). The samples were generated with the Importance Sampling technique [
53].
Figure 7a,b shows that the number of samples between 30,000 and 50,000 is sufficient to converge the reliability index and the probability of failure of the structure. Once the number of samples is defined, it is possible to estimate the probability of depassivation considering the simulated scenarios.
The results presented in
Figure 8 show that the reinforcement depassivation probability due to CO
2 diffusion is approximately zero for the first ten years, for all the simulated conditions of rain exposure. When analyzing the process of carbonation in an external environment and exposed to rain, the depassivation probability after 50 years is less than 10%, which attends to the durability limit state, and it is an acceptable probability for the reinforcement depassivation of concrete structures [
66]. The durability limit state marks the onset of durability failure as the reinforcement depassivation in a RC structure [
67]. According to ISO 13823:2008 [
68], the durability limit state is reached when the reliability index exceeds the value of 1.6, which is equivalent to a failure probability of 0.2.
For indoor concrete structures, considering the carbonation process and an element sheltered from the rain, the reinforcement depassivation probability reached 20% at the age of 16–23 years, depending on the city. São Paulo was the city with the earliest probability of depassivation reaching 20%, at the age of 16 years. It is a city where the relative humidity has great variation, thus structures may always be under drying/wetting cycles. Possan et al. [
67] observed that the diffusion of CO
2 is highly affected by internal humidity, and the carbonation chemical reaction would be better represented by internal moisture instead of air relative humidity. Thus, to reach a more reliable analysis, it would be interesting to use a diffusion model that considers the internal humidity of concrete. However, using this parameter as input to the model would make it impractical, since it is not simple to collect the material’s internal humidity.
In chloride-induced depassivation analyses, it was found that the aggressiveness of the environment generates depassivation probability curves with steeper slopes in the first few years, with diffusion slowing down over time. Al-Alaily et al. [
41] observed through probabilistic analyses, using Monte Carlo simulation, that the level of aggressiveness is primarily defined by the concrete’s compressive strength, i.e., its quality, and that in concrete without additions, the probability curve has high growth rates in the first few years. The authors [
41] showed that if metakaolin is added to concrete, the depassivation probability decreases and the corrosion initiation is postponed.
When comparing the results obtained in the deterministic and probabilistic analyses, it is observed that the depassivation limit state, in corrosion induced by carbonation and in the indoor environment sheltered from rain, is reached at age 20 to 30 years in all environments evaluated probabilistically, while in the deterministic analysis, depassivation occurs between 35–45 years. In coastal cities, the durability limit state is reached at 30 years in the deterministic analysis and at 10 years in the probabilistic analysis. These results point out that purely deterministic approaches are not able to properly evaluate the depassivation of concrete structures because it is necessary to consider the randomness of the variables.
To evaluate the influence of the input parameters, a sensitivity analysis of the random variables considered in the probabilistic approach is shown in
Figure 9. While
Figure 9a is related to the input parameters in the analysis with CO
2 diffusion in concrete structures in Manaus, Brasília, and São Paulo,
Figure 9b is related to the input parameters in the simulation of Cl
− diffusion in Fortaleza and Florianópolis.
Figure 9 shows that the compressive strength, in the depassivation probability due to the carbonation, had the same importance level of the chloride content on the surface, when the depassivation due to Cl
− diffusion is analyzed. Thus, concrete structures located in urban environments are more sensitive to the quality of concrete when subject to carbonation. When the chloride content is predominant concerning the content of CO
2 in the atmosphere, the chloride content on the surface indicates the level of environmental aggressiveness, and concrete quality parameters, such as concrete cover, and compressive strength, must be asserted to ensure the durability limit state.
Figure 10a,b shows the depassivation probability curves for beams produced with different types of Portland cement.
Figure 10a shows the results of simulations conducted with an indoor environment sheltered from rain, in São Paulo city.
Figure 10b shows the results of simulations conducted with structures located in Florianópolis for depassivation due to chloride diffusion.
Figure 10a shows that beams produced with types of cement CEM III/A or CEM IV/A and CEM IV/B have higher probability of depassivation by carbonation and less durability, the inverse of depassivation due to chloride diffusion (
Figure 10b). Jiang et al. [
69] and Possan et al. [
20] reported the existence of a negative influence of using additions in the carbonation process due to the reduction of the alkaline reserve of the concrete when produced with cement types CEM III/A or CEM IV/A and CEM IV/B, which have high slag contents (from 35 to 70%) and pozzolana (from 15 to 50%) in their compositions, respectively. However, Andrade [
33] reported that cement types CEM III/A and CEM IV/A and CEM IV/B are more resistant to chloride’s ingress. Rasheduzzafar et al. [
70] and Helene [
13] reported that chloride ions combine with tricalcium aluminate (C3A) to produce Friedel’s Salt, which reduces the number of chlorides in the pore solution, thus hindering its diffusion in the cementitious matrix.
Besides, although CEM II/A-S and CEMII/B-S E and CEM II/A-V cement types have additions in their composition (slag and pozzolan, respectively), the carbonation depths were lower than those obtained in beams produced with CEM III/A or CEM IV/A and CEM IV/B because they present significantly lower addition content, thus with a low reserve of alkaline ions but enough to improve the overall quality of concrete. The same analysis applies to the greater depths of chloride diffusion observed in the types CEM II/B-S and CEM II/A-V, and lesser depths in CEM III/A, CEM IV/A, and CEM IV/B.
The cement types that lead to greater durability against carbonation process were CEM I and CEM II/A-L and against chloride’s ingress were the types CEM III/A and CEM IV/A and CEM IV/B.
Figure 11 shows the probability of reinforcement depassivation related to the beam’s simulations produced with CEM III/A and CEM I cement, with different concrete compressive strengths (30, 40, and 50 MPa), in two environmental scenarios, in the cities of São Paulo and Florianópolis.
The depassivation probability due to the ingress of CO
2 in structures located in São Paulo (
Figure 11a) with 30 MPa, and a concrete cover of 30 mm, is about 98% for CEM III/A. It clearly does not guarantee the durability in this scenario. Even with CEM I, the probability of depassivation is about 44%. To extend the durability limit state of a structure with a concrete cover of 30 mm, it would be recommended to produce concrete with CEM I and at least 40 MPa, reducing the probability of depassivation to less than 20%.
The depassivation probability due to chloride diffusion in concrete structures located in Florianópolis (
Figure 11b) with 30 MPa and a concrete cover of 40 mm is about 86% for cement CEM I and 75% for CEM III/A. Just enhancing the quality of concrete by improving its compressive strength to 50 MPa would reduce the probability of depassivation to less than 25%.
Figure 12 illustrates the effect of the concrete cover thickness on the reinforcement depassivation, using contour maps referring to the variation in probability of depassivation for structures with different ages (10, 30, and 50 years), considering beams produced with cement CEM III/A and compressive strength of 30 MPa.
As before, the first
Figure 12a refers to a concrete beam subjected to carbon dioxide ingress in São Paulo in an indoor environment sheltered from the rain. The second
Figure 12b refers to a beam subject to chloride diffusion in Florianópolis. The environmental parameters employed in the simulation were defined in
Table 5 and
Table 6. The probability of depassivation is higher for chloride’s ingress, even with a cover greater than 40 mm.
The results shown in
Figure 11 and
Figure 12 confirm those indicated in
Figure 9, in which the concrete cover has lower influence than the compressive strength on corrosion induced by carbonation, whilst in corrosion induced by chlorides, the compressive strength has a higher level of influence compared to the concrete cover. Coverings smaller than 40 mm generate higher depassivation probabilities in environments governed by the action of CO
2, while coverings greater than 40 mm have higher depassivation probabilities when the structures are in coastal environments. For instance, when considering a concrete cover of 30 mm, the probability of depassivation due to the carbonation process is 97%, while there is a probability of 92% when analyzing the depassivation by Cl
− diffusion. When evaluating a cover of 35 mm, the probability of depassivation due to the carbonation process is 58%, and the probability of depassivation due to the action of chlorides is 66%. To ensure protection, the concrete cover must be increased. According to Palm et al. [
71], an association of concrete cover and good practices for construction must be designed in every project to increase service life and durability.