## 4 Analysis of Crack Width
### Sensitivity Analysis of Crack Width  
The sensitivity analysis of the crack widths in the RC slabs in the underground box culverts is performed to evaluate the effects of basic design variables through Monte Carlo simulations. The numerical simulations are performed for \(\mu-2\sigma\), \(\mu-\sigma\), \(\mu\), \(\mu+\sigma\), and \(\mu+2\sigma\). The compressive strength of concrete \(f_{ck}\), rebar spacing \(s\), slab thickness \(h\), bottom concrete cover \(d_{c}\), and top concrete cover \(d_{d}\) are selected as basic variables. The statistical information for the basic design variables is given in Table 8. The results of the sensitivity analysis for the first major crack in the positive moment region are presented in Figure 13 and Table 9.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline  & \multirow{2}{*}{**Thickness**} & \multirow{2}{*}{**Rebar Stress**} & \multirow{2}{*}{\begin{tabular}{c} **Bottom** \\ **Concrete** \\ **Cover** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Rebar** \\ **Spacing** \\ **Cover** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Neutral** \\ **Axis** \\ **Factor** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **Calculated Crack** \\ **Width Using** \\ **Measured Load** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Calculated-To-** \\ **Measured Crack** \\ **Width Ratio** \\ \end{tabular} } & \multirow{2}{*}{**Ratio**} \\ \cline{2-2} \cline{8-12}
**Sp specimen** & & & & & & & & & & & & & & \\ Name & & & & & & & & & & & & & \\ \hline
**Unit** & **(mm)** & & **(MPa)** & & **(mm)** & **(mm)** & **-** & & **(mm)** & & **-** & **-** & **-** & \\ \hline Mean, \(\overline{\pi}\) & & & & & & & 0.32 & 0.33 & 0.33 & 1.01 & 1.08 & 1.14 & 1.07 & 1.13 \\ \hline Standard & & & & & & & 0.007 & 0.008 & 0.008 & 0.097 & 0.091 & 0.107 & 0.033 & 0.074 \\ deviation, \(s_{s}\) & & & & & & & 0.021 & 0.026 & 0.024 & 0.097 & 0.085 & 0.094 & 0.031 & 0.066 \\ \hline cov, \(\delta\) & & & & & & & 0.33 & 0.34 & 0.34 & 1.20 & 1.27 & 1.32 & 1.13 & 1.32 \\ \hline Minimum & & & & & & & 0.31 & 0.31 & 0.32 & 0.92 & 0.98 & 1.03 & 1.01 & 1.07 \\ \hline \hline \end{tabular}
\end{table}
Table 6: _Cont._  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c} \hline \hline  & \multirow{2}{*}{**Thickness**} & \multirow{2}{*}{**Rebar Stress**} & \multirow{2}{*}{\begin{tabular}{c} **Bottom** \\ **Concrete** \\ **Cover** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Rebar** \\ **Spacing** \\ **Cover** \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Neutral** \\ **Axis** \\ **Factor** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **Calculated Crack** \\ **Width Using** \\ **Measured Load** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Calculated-To-** \\ **Measured Crack** \\ **Width Ratio** \\ \end{tabular} } & \multirow{2}{*}{**Ratio**} \\ \cline{2-2} \cline{8-12}
**Specimen** & & & & & & & & & & & & & & & & \\ Name & & & & & & & & & & & & & & \\ \hline
**Unit** & **(mm)** & & & & & & & & & & & & & & \\  &  
As illustrated in Figure 13, crack widths are most sensitive to the bottom concrete cover (\(d_{c}\)), slab thickness (\(h\)), and rebar spacing (\(s\)), which directly affects the crack width estimations through the suggested model (Equation (3)). The crack width is changed with a rate from \(-12.65\%\) to \(15.15\%\) due to the bottom concrete cover deviation (\(\pm 2\sigma\)), and from \(8.64\%\) to \(-7.24\%\) due to the slab thickness deviation (\(\pm 2\sigma\)). The crack width changes from \(-5.08\%\) to \(5.17\%\) for rebar spacing (\(s\)). For parameters such as the compressive strength of concrete, and top concrete cover, the rate of change is between \(1.31\%\) and \(-0.76\%\), and \(-0.17\%\) and \(0.17\%\), respectively, with minimal influence on crack widths.