# Time-Varying Reliability Evaluation of Concrete Based on Carbonation Depth
## 7 Time-Varying Reliability Index Based on the TM Method  
One of the most important steps to calculate the failure probability is to analyze the integral of probability density function (PDF), leading to the difficulty in probability computation [28]; thus, approximation methods such as the moment methods are generally applied in reliability analysis. In this paper, the TM method was used to predict the time-varying reliability index. The proposed formula for calculating TM reliability is presented in this paper [38].  
The calculation steps of the time-varying TM methods are:  
\[G\left(x\right)=c-x \tag{7}\]  
\[\beta_{\text{2M}}(t)=\mu_{G}(t)/\sigma_{G}(t) \tag{8}\]  
\[\mu_{G}(t)=c-1/\lambda(t) \tag{9}\]  
\[\sigma_{G}(t)=1/\lambda(t) \tag{10}\]  
\[\alpha_{\text{3G}}=E\left[\left(\frac{X-\mu_{G}}{\sigma_{G}}\right)^{3}\right] \tag{11}\]  
\[\beta_{\text{3M}}(t)=-\frac{\alpha_{\text{3G}}}{6}-\frac{3}{\alpha_{\text{3G }}}\ln(1-\frac{1}{3}\alpha_{\text{3G}}\beta_{\text{2M}}) \tag{12}\]  
where \(G(x)\) is the performance function, \(c\) is the concrete cover, \(x\) is the random variable of carbonation depth, \(\beta_{\text{2M}}\) is the second-moment reliability index, \(\mu_{G}\) is the mean value of of \(G(x)\), \(\sigma_{G}\) is the standard deviation of \(G(x)\), \(\alpha_{\text{3G}}\) is the skewness observed by MC simulation, \(\beta_{\text{3M}}\) is the expression of the TM method.  
Figure 14 presents the time-varying TM reliability index calculated by Equations (7)-(12). The calculated results were tested by MC simulation. The calculated errors are shown in Table 4. In this example, the concrete cover was presumed to be 35 mm. Meanwhile, all skewness values were close to \(-2\). Other TM reliability values with different concrete covers can be computed using the same method.  
Figure 14: Time-varying reliability indices with concrete cover thickness \(c\) = 35 mm.  
On the basis of Figure 14 and Table 4, \(\beta_{\text{3M}}\) and MC simulation are in good agreement and therefore this formula was used for calculating the TM reliability in this paper. Obviously, from Figure 14, \(\beta_{\text{3M}}\) has a better agreement than \(\beta_{\text{2M}}\) because skewness was taken into consideration in \(\beta_{\text{3M}}\). Figure 15 shows the different TM reliability indices with a concrete cover presumed to be 25 mm. On the basis of Figures 14 and 15, \(\beta_{\text{3M}}\) can be used to precisely determine time-varying TM reliability with different presumed concrete covers.  
Since the failure probability and reliability index are equivalent methods to carry out the assessment of structural safety, the concrete failure time obtained by time-varying failure probability can be combined with the time-varying reliability index, aiming to obtain the limit reliability index of a certain concrete cover. According to Figure 13, the concrete failure time of concrete covers with thicknesses \(c=20\) mm, \(c=25\) mm, \(c=30\) mm, and \(c=35\) mm were observed to be 41.2 years, 44.1 years, 46.8 years, 49.1 years, respectively. Then, the corresponding limit reliability indices were calculated as 0.756, 0.838, 0.886, and 0.926, respectively, as shown in Figure 16. The limit reliability indices of other concrete covers can be computed using this method.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Service Age** & **Error (\(\beta_{\text{3M}}\))** & **Skewness** \\ \hline
37 & 31.8\% & 2.01 \\
39 & 1.80\% & 1.99 \\
41 & 1.15\% & 1.96 \\
43 & 1.10\% & 2.02 \\
45 & 0.23\% & 2.01 \\
47 & 0.60\% & 2.00 \\
49 & 1.23\% & 1.97 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Different time-varying reliability indices with concrete cover thickness \(c=35\) mm.  
Figure 15: Time-varying reliability indices with concrete cover thickness \(c=25\) mm.