# Prediction of Concrete Failure Time Based on Statistical Properties of Compressive Strength
## 3 Prediction for the Mean Value of Compressive Strength  
Carbonation-induced corrosion can lead to a residual capacity loss of the reinforced concrete structures [37]. Generally, carbonation depth is used to define the carbonized level of the concrete. To analyze the relation between the carbonation depth and the compressive strength of concrete, and to develop a prediction model for the mean value of compressive strength (MVCS) in which the environment factors (temperature and humidity) are considered, the ratio of compressive strength to carbonation depth was investigated. In addition, the concrete failure time calculated by this prediction model was used to compare with those calculated by time-varying reliability. Based on the existing model of carbonation depth [17], the carbonation depth is expressed as  
\[\overline{D}(t)=7.742\text{e}^{-0.05f_{cm}}\times\sqrt{t}\times\alpha_{T} \times\alpha_{RH} \tag{1}\]  
\[\alpha_{T}=\left(T/T_{0}\right)^{0.25} \tag{2}\]  
\[\alpha_{RH}=RH(100-RH)/RH_{0}(100-RH_{0}) \tag{3}\]  
\[f_{cm}=f_{c\omega k}/(1-1.6456) \tag{4}\]  
where \(\overline{D}\) is the mean value of carbonation depth in a nonstandard condition (mm); \(\alpha_{T}\) is the coefficient of temperature; \(\alpha_{\text{RH}}\) is the coefficient of humidity; \(T\) is the environmental temperature (\({}^{\circ}\)C); \(T_{0}\) is the average temperature (\({}^{\circ}\)C); \(RH\) is the environmental humidity (\(\%\)); \(RH\) 0 is the average humidity (\(\%\)); \(t\) is the service time of concrete (year); \(f_{cm}\) is the mean value of the standard value of compressive strength (N/mm\({}^{2}\)); \(f_{c\omega k}\) (equal to concrete grade) is the standard value of compressive strength (N/mm\({}^{2}\)), which is observed by converting \(f_{c}\) (design value of compressive strength (N/mm\({}^{2}\))) [38]; and \(\delta\) is the coefficient of variation, which can be obtained via the Code for Design of Concrete Structures [39]. According to the investigation of the relevant climate data, the climate changes among different years were not apparent. In order to simplify the calculation of Equation (1), \(T\) is set as the median of all mean values of the temperature of different years, and \(RH\) is the median of all mean values of the humidity of different years. Different from the prediction model of carbonation depth in the existing paper [17], \(f_{c\omega k}\) is replaced by \(f_{cm}\), which is more appropriate to calculate the ratio \(Ra(t)\), as shown in Equation (4). Relevant parameters calculating the carbonation depth are presented in Table 1.  
To analyze the relation between compressive strength of concrete and carbonation depth, the ratio of relative concrete strength to carbonation depth is developed as  
\[Ra(t)=Rs(t)/\overline{D}(t) \tag{5}\]  
\[Rs(t)=\mu_{Rs}(t)/f_{c} \tag{6}\]  
where \(Ra\) is the ratio of the relative MVCS to the mean value of carbonation depth (MVCD); \(Rs\) is the relative MVCS; and \(\mu_{Rs}\) is the MVCS of different years (N/mm\({}^{2}\)), which is obtained based on 10,317 experimental datapoints, as shown in Table 2. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Relevant Parameter** & \(f_{c}\)=18.0 N/mm\({}^{2}\) & \(f_{c}\)=21.0 N/mm\({}^{2}\) \\ \hline \(T\)(\({}^{\circ}\)C) & 15.425 & 15.425 \\ \(T\)0(\({}^{\circ}\)C) & 15.208 & 15.208 \\ \(RH\)(\(\%\)) & 62 & 62 \\ \(RH\) 0 (\(\%\)) & 69 & 69 \\ \(f_{c\omega k}\)(N/mm\({}^{2}\)) & 37.6 & 45 \\ \(\delta\) & 0.16 & 0.15 \\ \(f_{cm}\) (N/mm\({}^{2}\)) & 51 & 59.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Relevant parameters calculating the carbonation depth in Yokohama, Japan. According to Table 2, it was found that the compressive strength of concrete decreased with the increase of time. Based on Equations (1)-(6) and Tables 1 and 2, _Ra_ values of different servicing years were fitted as shown in Figure 6.  
According to Figure 6, _Ra_ decreased as time increased, showing that the compressive strength of concrete decreased with the increase of carbonation depth over time.