## 5 Probability-Based Crack Width Factors  
Nominal design surface crack widths of either \(0.3\) mm or \(0.5\) mm were calculated using the nominal values of the basic design variables for each case of slab specimen  
Figure 13: Sensitivity analysis of crack width of RC slab (first major crack in the positive moment region). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Variables**} & \multicolumn{5}{c}{**Rate of Change, (\%)**} \\ \cline{2-6}  & \(\mu-2\sigma\) & \(\mu-\sigma\) & \(\mu\) & \(\mu+\sigma\) & \(\mu+2\sigma\) \\ \hline Bottom concrete cover, \(d_{c}\) (mm) & \(-12.65\) & \(-6.61\) & \(0.00\) & \(7.23\) & \(15.15\) \\ Slab (mm) & \(8.64\) & \(4.12\) & \(0.00\) & \(-3.77\) & \(-7.24\) \\ Rebar (mm) & \(-5.08\) & \(-2.55\) & \(0.00\) & \(2.58\) & \(5.17\) \\ Concrete compressive strength, \(f_{\rm{ck}}\) (MPa) & \(1.31\) & \(0.54\) & \(0.00\) & \(-0.42\) & \(-0.76\) \\ Top concrete cover, \(d_{d}\) (mm) & \(-0.17\) & \(-0.08\) & \(0.00\) & \(0.08\) & \(0.17\) \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Sensitivity analysis of crack width of RC slab (first major crack in the positive moment region). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Location** & **Variables** & **Nominal** & **Mean/Median** & **cov/Zeta** & **PDF** \\ \hline \multirow{8}{*}{Positive moment region} & Concrete compressive strength (\(f_{ck}\)) & \(27\) MPa & \(30.4\) & \(0.222\) & Normal \\  & Rebar spacing biasness (\(\lambda_{s}\)) & - & \(0.998\) & \(0.035\) & Normal \\  & Slab thickness biasness (\(\lambda_{h}\)) & - & \(1.004\) & \(0.028\) & Normal \\  & Bottom concrete cover (\(d_{c}\)) & \(60\) mm & \(71.5\) & \(0.065\) & Normal \\  & Top concrete cover (\(d_{d}\)) & \(80\) mm & \(78.6\) & \(0.068\) & Log-Normal \\ \multirow{2}{*}{Negative moment region} & Bottom concrete cover (\(d_{c}\)) & \(80\) mm & \(78.6\) & \(0.068\) & Log-Normal \\  & Top concrete cover (\(d_{d}\)) & \(60\) mm & \(71.5\) & \(0.065\) & Normal \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Probabilistic models of basic design variables. with Equations (1) and (3). Crack widths are also calculated considering the variability inherent in the basic design variables. The probabilistic characteristics of the crack widths are evaluated through Monte Carlo simulations based on the probabilistic models of basic design variables, including the compressive strength of concrete \(f_{ck}\), rebar spacing biasness \(\lambda_{s}\), slab thickness biasness \(\lambda_{h}\), bottom concrete cover \(d_{c}\), top concrete cover \(d_{d}\), and the probabilistic models of experimental crack width biasness (\(\lambda_{c}\)). Probabilistic models of all basic design variables are presented in Table 8 and were selected based on studies from [PERSON] et al. (2015) except the probabilistic model of rebar spacing biasness (\(\lambda_{s}\)). A probabilistic model for rebar spacing biasness is developed from the information collected from various construction sites of underground RC box culverts in Korea. Probabilistic models of crack width biasness are summarized in Table 10, in which the means and covariances are estimated from the cumulative density function of the calculated-to-measured crack width ratios summarized in Figure 12.  
The thickness of the simulation slab model ranges from 300 to 500 mm. The sectional area of the bottom reinforcement in the tension side varies from 993 mm\({}^{2}\) to 2578 mm\({}^{2}\), which are widely used design values for underground RC box culverts in Korea, and the area of the upper reinforcement in the compression side is taken as half of the bottom rebar area. Different simulation sizes were assigned to various slab models based on the inventory survey, as presented in Table 11. The basic simulation size was 100 K. The slab with a thickness of 300 mm and bottom rebar area of 1324 mm\({}^{2}\) had a weight factor of 20, as shown in Table 11. Therefore, 2000 K simulations were conducted for cracks with a crack width of both 0.3 mm and 0.5 mm.