## 5 Probability-Based Crack Width Factors
For the slab with a thickness of 500 mm and bottom rebar area of 1324 mm\({}^{2}\), 100 K simulations were conducted. Therefore, in total, 8300 K simulation results were conducted to investigate each crack width model for RC slabs in underground box culverts. The Monte Carlo simulation results considering both the original (Equation (1)) and the modified (Equation (3)) equations are plotted in Figure 14 on a normal probability paper for 0.3 mm and 0.5 mm crack widths regarding the first major crack in the positive moment region (mid-span, bottom surface of the top slab). The results for the negative moment region (support area, top surface of the top slab) are plotted in Figure 15. The Monte Carlo simulation results plotted in Figures 14 and 15 both demonstrate that the modified equation has a better fitting to the normal probability distribution function compared to the original equation (Equation (1)). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Equation** & **Variables** & **Mean/Median** & **cov/Zeta** & **PDF** \\ \hline \multirow{2}{*}{Original equation (Equation (1))} & \(\lambda_{c}\) (crack width biasness) (0.3) & 1.02 & 0.184 & Normal \\  & \(\lambda_{c}\) (crack width biasness) (0.5) & 0.94 & 0.190 & Normal \\  & \(\lambda_{c}\) (crack width biasness) (0.3) & 1.03 & 0.097 & Normal \\  & \(\lambda_{c}\) (crack width biasness) (0.5) & 1.01 & 0.083 & Normal \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Probabilistic models of experimental crack width biasness (1 st major crack). The probability of having crack widths of 0.3 mm or 0.5 mm may be higher than expected due to the construction error and variability in design parameters. The mean value of the bottom concrete cover based on the information collected from the local construction sites is 71.5 mm for the design value of 60 mm, which is normally expected due to the difficulty of quality control in underground box culvert constructions. Moreover, the crack widths for the RC flexural members in the underground box culvert are sensitive to the bottom concrete cover depth. Even properly designed members can reach the target crack width of 0.3 mm or 0.5 mm at a much lower rebar stress level than the design bar stress level due to the increase in bottom concrete cover. The crack width estimations can be improved by introducing a design factor to control the occurrence probability for the target design crack width. The probability-based crack width factor, \(\phi_{\rm w}\), can be multiplied to the crack width estimation equation, Equation (3), as in the following equation for design purposes:  
\[w_{c}=\phi_{\rm w}\left[2\frac{f_{s}}{E_{s}}k\sqrt{d_{c}^{2}+\left(\frac{s}{2 }\right)^{2}}\times Y_{c}\right] \tag{6}\]  
where \(w_{c}\) is the maximum design crack width (mm) is, \(\phi_{\rm w}\) is the probability-based crack width factor, and the expressions in the parenthesis are identical to Equation (3). Based on the Monte Carlo simulation results with the modified equation (Equation (3)), the crack width factors (\(\phi_{\rm w}\)) are derived to obtain the target crack width levels of 0.3 mm and 0.5 mm depending upon the target reliability level. The safety index, \(\beta\), is directly calculated from the Monte Carlo simulation results without considering the probability  
Figure 14: Distribution of simulated crack widths: (**a**) Original equation (Equation (1)); (**b**) Modified equation (Equation (3)) for 0.3 mm crack; (**c**) Original equation (Equation (1)); (**d**) Modified equation (Equation (3)) for 0.5 mm crack widths at the positive moment region of the top slab (first major crack). distribution function. The various crack width factors (\(\phi_{\rm w}\)) are provided in Tables 12 and 13 for the positive moment region (mid-span, bottom surface) and negative moment region (support area, top surface) of the top slab for the first major crack, respectively.