## 3 Probabilistic Model of Basic Design Variables
### Probabilistic Models of Dimensions of Cross Sections  
The sectional dimensions (thickness and cover depths) of slab members of RC box culverts were measured from various construction sites, whereby a core was drilled through the depth of the slab. As shown in Figure 7, the bias factor of slab thickness (\(\lambda_{t}\)) was found to be 1.004 with a cov of 0.032, based on the fitting of a normal probability distribution function.  
The vertical locations of the upper and the lower reinforcements were measured using a rebar detector. The design value for cover depths is 60 mm for the inner cover depth (\(c_{s1}\)distance from the bottom side of the slab to the centroid of the lower rebar) and 80 mm for the outer cover depth (\(c_{s2}\)distance from the top side of the slab to the centroid of the upper rebar). These are the recommended cover depths for all local underground RC culverts for the electricity supply.  
The probabilistic flexural strength models herein were developed for positive moment capacity (cover depths of tensile and compressive rebars--60 mm and 80 mm) and negative moment capacity (cover depths of tensile and compressive rebars--80 mm and 60 mm). The effective depth decreases when the cover depth of the tensile rebar increases. Hence, these probabilistic models are evaluated considering the upper part of the distribution. Furthermore, probabilistic models of the cover depth of the compressive rebar are determined with the lower part of the distributions, because the moment capacity decreases as the cover depth of compressive rebar decreases.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline
**Diameter** & **D10** & **D13** & **D16** & **D19** & **D22** & **D25** & **D29** & **D32** & **Total** \\ \hline Mean, \(\overline{x}\) & 502.7 & 474.3 & 471.1 & 467.3 & 463.7 & 466.3 & 463.5 & 464.3 & 468.4 \\ cov, \(\delta_{X}\) & 0.011 & 0.045 & 0.032 & 0.043 & 0.045 & 0.029 & 0.049 & 0.035 & 0.042 \\ n & 6 & 54 & 54 & 54 & 48 & 36 & 33 & 33 & 318 \\ \hline \end{tabular}
\end{table}
Table 1: Statistical yield strength data of SD400 rebars (unit: MPa).  
Figure 6: CDF for yield strength of steel rebar (normal PDF).  
As illustrated in Figure 8, a normal PDF with a mean of 71.5 mm and an SD of 4.66 mm (cov 0.065) is proposed for the cover depth of a tensile rebar (positive moment). A normal PDF with a mean of 67.2 mm and SD of 4.35 mm (cov 0.065) is proposed for the cover depth of a compressive rebar (negative moment).  
The outer cover depth (nominal depth--80 mm) is modeled with a log-normal distribution. A log-normal PDF with a median of 75.8 mm and a \(\zeta\) (logarithmic standard deviation or zeta) of 0.144 is proposed for the cover depth of a tensile rebar (negative moment), as illustrated in Figure 9. For the cover depth of a compressive rebar (positive moment), a log-normal PDF with a median of 78.6 mm and a \(\zeta\) of 0.068 is proposed.  
[PERSON] et al. [1] proposed a unified normal distribution model for construction errors in members of all dimensions, which have been adopted in other studies [4; 5; 6; 7].  
Figure 8: CDF for inner cover depth of slab member (normal PDF).  
Figure 7: CDF for thickness of slab member (normal PDF).