## 5 Probabilistic Flexural Strength Model of RC Slab Members
The simulation results for the ultimate moment strength are plotted on normal probability papers in Figure 19. Even though the ultimate moments show a good fit to the normal PDF, as shown in Figure 19, adaptive local PDFs are proposed for selected regions with various probable failure points of \(M_{u}\), as summarized in Table 6. The adaptive PDF is selected to fit the simulation results in the selected lower region, below (mean +\(\alpha\)), in which \(\alpha\) is the failure point location factor. The value of (mean +\(\alpha\)) is assumed to be close to the value of \(M_{u}\) at the probable failure point in the calculation of the reliability index through the advanced first-order second-moment (AFOSM) method. As described in Figure 20, the positive yield moment fits better with the 1, whereas the normal PDF is more appropriate for the negative yield moment. For the yield moment models, the failure point location factor \(\alpha\) is selected between \(-1.0\) and \(-1.5\), as summarized in Table 6, because the expected reliability levels inherent in the yield moment limits could be lower than those for the ultimate moment limits. This is shown in Figures 21 and 22. For the log-normal PDF, \(\overline{\lambda}_{R}\) is the median and \(\zeta_{R}\) is the logarithmic standard deviation. The probability models with either \(\alpha=-1.25\) or \(\alpha=-1.5\) show almost the same statistical parameters for both the positive moment and the negative moment. Figure 18: Flowchart for the evaluation of the probabilistic model of flexural strength of RC members. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Thickness, t**} & \multicolumn{4}{c}{**Bottom Rebar Area, \(A_{s}\) (\(\text{Mm}^{2}\))**} \\ \cline{2-5}  & **993.0** & **1324.0** & **1909.8** & **2578.1** \\ \hline
300 & 12 & 20 & 4 & 1 \\
350 & 4 & 16 & 8 & 1 \\
400 & 4 & 4 & 4 & 1 \\
500 & 1 & 1 & 1 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Simulation size for slab model. Figure 19: Probabilistic model of the ultimate moment strength of an RC slab. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Positive Moment**} & \multicolumn{2}{c}{**Negative Moment**} \\ \cline{2-5}
**Probable Failure Point** & \(\overline{\lambda}_{R}\) & \(V_{R}\) & \(\overline{\lambda}_{R}\) & \(V_{R}\) \\ \hline Entire & 1.39 & 0.083 & 1.44 & 0.093 \\ \(\alpha=-2.0\) & 1.39 & 0.080 & 1.44 & 0.089 \\ \(\alpha=-2.5\) & 1.38 & 0.079 & 1.45 & 0.090 \\ \(\alpha=-3.0\) & 1.38 & 0.078 & 1.45 & 0.091 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Probabilistic models of flexural moment strength with adaptive PDF. Figure 20: Probabilistic model of the yield moment strength of an RC slab. The limit state function selected for the reliability evaluation is related to the major load combinations adopted in the design of local underground RC culverts, presented in Equation (6):  
\[M-D-L_{v}-H_{v}-H_{h}<0 \tag{6}\]  
where \(M\) = flexural resistance, \(D\) = dead load effect (bias factor 0.94, cov 0.04, normal PDF), \(L_{v}\) = traffic live load effect (bias factor 2.24, cov 0.10, Gumbel's Type-I PDF), \(H_{v}\)= vertical earth load effect (bias factor 1.18, cov 0.17, normal PDF), and \(H_{h}\) = horizontal earth load effect (bias factor 1.00, cov 0.28, normal PDF) for an example case with an earth fill depth of 4 m, in which the bias factor is the ratio of the probability model mean to the nominal design value. Figure 21 summarizes the failure point location factors resulting from the reliability evaluations for various earth fill depths with the probability model (\(\alpha=-2.5\)) of \(M_{u}\), as shown in Table 6. Most location factors were found to be about \(-2.5\) for the positive ultimate moment and ranged from \(-2.5\) to \(-3.0\) for the negative ultimate moment.