## 5 Probabilistic Flexural Strength Model of RC Slab Members  
The probabilistic characteristics of the flexural strength of RC members are examined through Monte Carlo simulations based on probabilistic models of design variables (\(f_{ck},f_{y}\), \(\lambda_{t},c_{s1},c_{s2}\)) and probabilistic models of experimental strength biasness (\(\lambda_{y}\) and \(\lambda_{u}\)). The Monte Carlo method is a numerical process of repeatedly calculating a mathematical operator including random variables with prescribed probability distributions to obtain the solution to complex probabilistic problems [19]. Probabilistic models of variables are summarized in Table 4.  
The thicknesses of the simulation slab models range from 300 mm to 500 mm. The section areas of the reinforcements in the tension side vary from 993 mm\({}^{2}\) to 2578 mm\({}^{2}\) (\(\rho\): 0.0022-0.0095), which are widely used design values in local underground RC box culverts, and the area of the reinforcements in the compression side is taken generally as half of the bottom rebar area. The positive moment capacity at the middle of the top slab and the negative moment capacity at the end of the top slab in the box section are investigated in this study. The middle section and the end section have different nominal cover depths (60 mm and 80 mm), which result in different effective depths. Different simulation sizes are assigned to the different slab models based on the inventory survey, presented in Table 5. The basic simulation size is 100 K. The slab with a thickness of 300 mm and bottom rebar area of 1324.0 mm\({}^{2}\) has a weight factor of 20 in Table 5. Therefore, 2000 K simulations are generated for the positive moment capacity. For the slab with a thickness of 500 mm and bottom rebar area of 1324.0 mm\({}^{2}\), 100 K ultimate moments are simulated. In total, 8300 K simulation results are used to investigate the ultimate flexural strength model for the positive moment in the RC box slab. The same procedure is repeated for the negative moment model. The evaluation process for the probabilistic model of flexural strength for RC members is illustrated in Figure 18. The ultimate moment strength is calculated considering both upper rebars and lower rebars that are effective in their load-carrying capacity. The yield moment strength is estimated considering only the lower rebar because the effect of the \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Variables** & **Mean/Median** & **Cov/Zeta** & **PDF** \\ \hline \(\lambda_{u}\) (\(M_{u}\)) & 1.15 & 0.043 & Normal \\ \(\lambda_{y}\) (\(M_{y}\)) & 0.99 & 0.046 & Normal \\ \(f_{ck}\) (27 MPa) & 30.4 MPa & 0.222 & Normal \\ \(f_{y}\) (400 MPa) & 468.7 MPa & 0.043 & Normal \\ \(\lambda_{t}\) (slab thickness biasness) & 1.004 & 0.032 & Normal \\ \(c_{s1}\) (cover depth of bottom rebar/tensile rebar) & 71.5 mm & 0.065 & Normal \\ \(c_{s2}\) (cover depth of upper rebar/compressive rebar) & 78.6 mm & 0.068 & LN \\ Negative moment & \(c_{s2}\) (cover depth of upper rebar/tensile rebar) & 75.8 mm & 0.144 & LN \\ \(c_{s1}\) (cover depth of bottom rebar/compressive rebar) & 67.2 mm & 0.065 & Normal \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Probabilistic models of variables. upper rebar is negligible until the yield moment is developed. The moment strengths (ultimate \(M_{u}\) and yield \(M_{y}\)) are normalized with the nominal moment strength (\(M_{n}\)), calculated with the nominal design values of all the design variables, in which only the rebars on the tension side are considered (Equation (1)), as in most design practices.