# Analysis of Vertical Earth Pressure Acting on Box Culverts through Centrifuge Model Test
## 5 Variability Analysis of Vertical Earth Pressure Acting on Box Culvert
### Variability Analysis  
For variability analysis, the probability distribution function was analyzed under the loading and unloading conditions by conducting a goodness-of-fit test of the corresponding data. The goodness-of-fit test analyzes the difference between the values predicted using the given data and the probability distributions obtained by conducting the Anderson-Darling (AD), Kolmogorov-Smirnov (KS), and chi-squared (CS) tests, and it determines the appropriate probability distribution [39]. The AD and KS tests can be applied even when the data sample is small, but the CS test can be applied only when the sample size is adequately large [41]. In this study, because the sample size was inadequate for the CS test, the goodness-of-fit was measured by conducting the AD and KS tests, and an appropriate probability distribution was obtained. The goodness-of-fit test results are shown in Figure 14 and summarized in Tables 3 and 4. The goodness-of-fit test analyzes the optimal probability distribution suitable for the histogram of the data in question, and a low value of the statistic indicates an optimal  
Figure 12: Example of earth pressure sensor with soil particle arrangement: (**a**) before and (**b**) after calibration. Figure 13: Recalculation of data after sensor size correction. probability distribution. However, the analyzed fitted probability distribution is mainly used in other fields such as finance, accounting, and systems, and it represents a theoretical probability distribution that is difficult to implement in practice. For example, the \"Burr\" probability distribution is a log-logistic distribution that is mainly used in statistics and economics, \"Johnson SB\" is an asset model for financial portfolio management, \"Dagun (4P)\" is a personal income distribution model, and the \"Gen.Extreme value\" probability distribution is used for rainfall analysis. They have different directions for estimating uncertainty and variability in geotechnics. In the field of geotechnical engineering, sample data (physical properties, bearing capacity, etc.) are generally highly variable, unlike the data of standardized materials (such as concrete or metal). It is difficult to predict the population because the number of sample data are relatively small. Therefore, if an unusual probability distribution according to the goodness-of-fit test of the sample is applied in geotechnical engineering, uncertainty in the population estimation process may increase. Generally, in geotechnical engineering, the normal and lognormal distributions are used. This is because the designer preemptively recognizes the uncertainty that occurs in the process of population estimation by selecting a probability distribution. This is partially reflected in the safety factor or resistance factor computed in the design process. Therefore, to select the optimal probability distribution in geotechnical engineering, it is reasonable to compare the normal and the lognormal distributions. Accordingly, in this  
Figure 14: Probability density functions of normal and lognormal distribution under the (**a**) loading and (**b**) unloading conditions. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{2}{c}{**KS Test**} & \multicolumn{2}{c}{**AD Test**} \\ \hline
**Rank** & **Probability Distribution** & **Statistic** & **Rank** & **Probability Distribution** & **Statistic** \\ \hline
1 & Burr & 0.0709 & 1 & Burr & 0.1892 \\ \hline
2 & Log-logistic (3P) & 0.0731 & 2 & Gumbel Max & 0.1941 \\ \hline
3 & Pearson 6 & 0.0780 & 3 & Log-Pearson 3 & 0.1965 \\ \hline
4 & Pearson 5 & 0.0786 & 4 & Pearson 5 & 0.1966 \\ \hline
5 & Dagum & 0.0799 & 5 & Gen. Extreme value & 0.1989 \\ \hline
6 & Lognormal & 0.0809 & 13 & Lognormal & 0.2243 \\ \hline
41 & Normal & 0.1258 & 32 & Normal & 0.5341 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Result of goodness-of-fit test under loading condition. study, we compared only the normal and lognormal distributions, and we considered the fitted probability distribution as lognormal.