# Prediction of Concrete Failure Time Based on Statistical Properties of Compressive Strength
## 3 Prediction for the Mean Value of Compressive Strength
In addition, it is known that the carbonation depth increased with the increase of time. Therefore, combined with the definition of carbonation depth, it was concluded that carbonation-induced corrosion can result in the decrease of compressive strength of concrete, which agrees with the conclusion drawn from Zha [40]. The fitting curves in Figure 6 are expressed as  
\[Ra_{1}(t)=-0.02t+1.279 \tag{7}\]  
\[Ra_{2}(t)=-0.0058t+0.5445 \tag{8}\]  
where _Ra_1 is the ratio of the relative compressive strength to carbonation depth with \(f_{c}=21.0\) N/mm\({}^{2}\); and _Ra_2 is the ratio of the relative compressive strength to carbonation depth with \(f_{c}=18.0\) N/mm\({}^{2}\). By comparing the changing rate of _Ra_1(_t_) with _Ra_2(_t_), the deterioration rate of concrete with \(f_{c}=21.0\) N/mm\({}^{2}\) is faster than that of \(f_{c}=18.0\) N/mm\({}^{2}\). \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline  & \multicolumn{6}{c}{(**a) Mean Value of Compressive Strength with \(f_{c}=21.0\) N/mm\({}^{2}\)} \\ \hline Time (Year) & 35 & 36 & 37 & 38 & 39 & 40 & 41 & 42 & 43 & 44 & 45 \\ \hline \(\mu_{\text{{R}}\text{{I}}}\) (N/mm\({}^{2}\)) & 31.95 & 30.70 & 30.19 & 31.29 & 28.04 & 25.27 & 23.78 & 23.71 & 23.54 & 24.51 & 27.43 \\ \hline \multicolumn{10}{c}{(**b) Mean Value of Compressive Strength with \(f_{c}=18.0\) N/mm\({}^{2}\)} \\ \hline Time (Year) & 42 & 43 & 44 & 45 & 46 & 47 & 48 & 49 & 50 & 51 & 52 \\ \hline \(\mu_{\text{{R}}\text{{I}}}\) (N/mm\({}^{2}\)) & 22.65 & 23.00 & 24.15 & 23.34 & 23.14 & 22.96 & 22.88 & 21.54 & 22.47 & 22.19 & 19.73 \\ \hline Time (year) & 53 & 54 & & & & & & & & & & \\ \hline \(\mu_{\text{{R}}\text{{I}}\text{{I}}}\) (N/mm\({}^{2}\)) & 20.77 & 20.03 & & & & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mean value of compressive strength with (**a**) \(f_{c}=21.0\) N/mm\({}^{2}\) and (**b**) \(f_{c}=18.0\) N/mm\({}^{2}\). Figure 6: Fitting of the ratio of the relative concrete strength to the mean value of carbonation depth. On the basis of Equations (1), (5), (7) and (8), the new prediction models of \(Rs\) are proposed as  
\[Rs_{1}(t)=\overline{D}_{1}(t)\times Ra_{1}(t)=7.742\mathrm{e}^{-0.05f_{cm1}}\times \sqrt{t}\times\alpha_{T}\times\alpha_{RH}\times(-0.02t+1.279) \tag{9}\]  
\[Rs_{2}(t)=\overline{D}_{2}(t)\times Ra_{2}(t)=7.742\mathrm{e}^{-0.05f_{cm2}} \times\sqrt{t}\times\alpha_{T}\times\alpha_{RH}\times(-0.0058t+0.5445) \tag{10}\]  
where \(Rs_{1}\) is the relative compressive strength of \(f_{c}=21.0\) N/mm\({}^{2}\); \(Rs_{2}\) is the relative compressive strength of \(f_{c}=18.0\) N/mm\({}^{2}\); \(f_{cm1}\) is the mean value of the standard value of compressive strength with \(f_{c}=21.0\) N/mm\({}^{2}\); and \(f_{cm2}\) is the mean value of the standard value of compressive strength with \(f_{c}=18.0\) N/mm\({}^{2}\). Figure 7 is developed from Equations (9) and (10), showing that with an increase of service time, the compressive strength decreased, and the speed of strength deterioration increased. \(Rs(t)=1.0\) is the limit state of concrete failure in conducting concrete safety assessments (failure mode I). As shown in Figure 7, the concrete failure time of \(f_{c}=21.0\) N/mm\({}^{2}\) was observed to be 48.2 years, while that of \(f_{c}=18.0\) N/mm\({}^{2}\) was observed to be 61.8 years. However, the coefficient of variation was not considered in this model, leading to the inaccurate prediction of concrete failure time. Figures 8-10 also present the corresponding fitting curves with acceptable R-squares, indicating that the influence of \(f_{c}\) on the coefficient of variation, the skewness, and the kurtosis was negligible. Based on Figures 8-10, three eigenvalues increased with the increase of time. Besides, the coefficient of variation increased over time, resulting in dispersion expansion of the statistic. Therefore, the accuracy of the observed concrete failure time determined only by the mean value would decrease when time increases. Figure 8: Time-varying coefficient of variation. Figure 10: Time-varying kurtosis.