## 6 Conclusions  
The surface crack widths in RC flexural members for typical underground box culverts built in Korea were studied. Structural members built below the ground, such as underground RC box culverts used for electric power distribution, have more likelihood to have differences between the as-built dimensions and the dimensions provided in design drawings. Due to this variability in construction dimensions, the crack widths measured in such underground structures are more likely to have differences with the calculated values if the thicknesses of the specimens are different than what was used for design. Although crack control is a serviceability concern, if the owner chooses to have a target crack width that needs better control, crack width estimations can be improved by considering this construction variability with a newly proposed probability-based crack estimation model suggested in this study. Monte Carlo simulations were conducted to evaluate the probabilistic characteristics of the important parameters that control the crack widths in RC flexural members. These parameters include the bottom concrete cover and bar spacings provided in such flexural members. The bottom concrete cover depth in the tension surface was found to be the most sensitive design variable based on Monte Carlo simulations. It was found that the mean value of the bottom concrete cover depth, considering the construction variabilities, could increase to 20% higher than the nominal design values. With the increase in bottom concrete cover, the estimated crack widths at the surface become larger with identical bar stresses. With current design practice, it was found from simulations that 0.3 mm (12 mils) cracks can be observed with a probability of 89% in the positive moment region of the top slab, while it is 45% in the negative moment region of the top slab, which is a high potential to see such cracks. Therefore, this study additionally suggests probability-based crack width design factors that can be applied to the current crack estimation models used in design, if the target crack width must be controlled to be below 0.3 mm (12 mils) or 0.5 mm (20 mils) in underground RC box culverts. Conceptualization and methodology, S.-H.K. and S.-K.W. ; experiment and formal analysis, S.H.A.S. ; investigation, S.H.A.S. and C.S. ; writing--original draft preparation, S.H.A.S. and C.S. ; writing--review and editing, C.S. and S.-H.K. ; project administration, S.-K.W. and I.C. All authors have read and agreed to the published version of the manuscript. This research work is supported by the Korea Electric Power Research Institute (Grant R18 SA02) as well as the Korea Institute of Energy Technology Evaluation and Planning (KETEP) and the Ministry of Trade, Industry and Energy (MOTIE) of the Republic of Korea. (No. 20194030202460). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Crack Width**} & \multicolumn{2}{c}{**Safety Index, \(\beta\)**} & \multicolumn{2}{c}{**The Occurrence Probability of Design Crack**} \\ \cline{2-5}  & **0.3 mm** & **0.5 mm** & **0.3 mm** & **0.5 mm** \\ \hline
1.00 & 0.13 & 0.01 & 0.4486 & 0.4948 \\
1.25 & 1.49 & 1.64 & 0.0682 & 0.0506 \\
1.50 & 2.40 & 2.72 & 0.0083 & 0.0032 \\
1.75 & 3.04 & 3.50 & 0.0012 & 0.0002 \\
2.00 & 3.52 & 4.08 & 0.0002 & 0.00002 \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Crack width factors and safety index values for negative moment region of the top slab. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Crack Width**} & \multicolumn{2}{c}{**Safety Index, \(\beta\)**} & \multicolumn{2}{c}{**The Occurrence Probability of the Design Crack**} \\ \cline{2-5}  & **0.3 mm** & **0.5 mm** & **0.3 mm** & **0.5 mm** \\ \hline
1.00 & \(-\)1.24 & \(-\)1.27 & 0.8932 & 0.8973 \\
1.25 & 0.21 & 0.30 & 0.4159 & 0.3824 \\
1.50 & 1.18 & 1.34 & 0.1184 & 0.0896 \\
1.75 & 1.88 & 2.09 & 0.0303 & 0.0184 \\
2.00 & 2.40 & 2.65 & 0.0083 & 0.0041 \\ \hline \hline \end{tabular}
\end{table}