## 3 Results and Discussion
### Statistical Analysis  
Statistical analysis was performed by considering the relative density of the sand (Dr), depth of the pressure transducers (z), the angle made by the line connecting the pressure transducer to the foundation center point (\(\alpha\)), and lateral stress values, which were normalized by the ultimate bearing capacity (\(\Delta\sigma\)). The experimental lateral stress increments calculated using Equation (5) are shown in Figure 13. An independent dataset was used to verify Equation (5), as shown in Figure 13. The descriptive statistical parameters for the 60 lateral stress values at different relative densities and depths in the 12 model experiments are listed in Table 6.  
The regression study examines multiple stress correlations, with the optimal coefficient of determination in the nonlinear statistical analysis obtained as R\({}^{2}\) = 0.982. Figure 13 compares the experimental and estimated stress increases. The correlation obtained from the statistical analysis is expressed as follows (Equation (6)), and the variable coefficients are presented in Table 7.  
\[\Delta\sigma=\ e^{(T:z+V\cdot(D_{r})^{0.01}+W\cdot B+X\cdot(\alpha)^{0.6}+Y \cdot(z/B)^{0.45}+Z)} \tag{6}\]  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \(\Delta\sigma\) (kPa) & z (m) & B (m) & z/B & \(\alpha\) & Dr (\%) \\ \hline Average & 0.017 & 0.30 & 0.12 & 2.71 & 36.33 & 0.51 \\ Minimum & 0.023 & 0.10 & 0.08 & 0.63 & 14.93 & 0.10 \\ Maximum & 0.092 & 0.50 & 0.16 & 6.25 & 53.13 & 0.90 \\ Standard Deviation & 0.022 & 0.14 & 0.03 & 1.55 & 13.71 & 0.30 \\ Skewness & 2.3143 & 0 & 0 & 0.7005 & \(-\)0.3622 & \(-\)0.1064 \\ Kurtosis & 4.7189 & \(-\)1.3081 & \(-\)1.5259 & \(-\)0.1588 & \(-\)1.2081 & \(-\)1.3293 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Descriptive statistical parameters of experimental data.  
Statistical analysis studies conducted to estimate the bearing capacity of foundations by taking into account the foundation geometry and soil properties (unit volume weight of sand and internal friction angle) yield results that are quite close to the existing theoretical calculations [71]. However, some studies have attempted to estimate using artificial neural networks [72; 73; 74]. However, statistical analysis studies on determining lateral stresses have not been encountered in the literature, except for a few artificial intelligence learning studies. The high R\({}^{2}\) value obtained (0.982) shows that the nonlinear regression analysis  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Coefficient** & **T (1 st)** & **V (2 nd)** & **W (3 rd)** & **X (4 th)** & **Y (5 th)** & **Z (6 th)** \\ \hline Values & \(-14.21\) & \(-112.33\) & \(-48.60\) & \(2.69\) & \(-40.94\) & \(140.20\) \\ \hline \hline \end{tabular}
\end{table}
Table 7: Variable coefficients in the equation were obtained.  
Figure 13: Comparison of experimental and predicted data.  
takes into account the combined effects of parameters such as relative density, depth, and angle on the lateral stress increase quite accurately. This result is consistent with similar nonlinear regression techniques used in previous studies, such as [PERSON] et al. (2019) [75]. These studies demonstrate that nonlinear models are an effective tool for understanding the complex structure of soil mechanics.  
However, this study has some limitations. Although the equation obtained predicts the lateral stress increments well, it would be useful to verify it with additional experimental data for different soil types and load conditions. In addition, the effects of other factors, such as the presence of water and soil size distribution, were not considered in this study. These parameters may contribute to the differences in the lateral stress behavior. Future studies can extend the analysis to include these variables and examine the effects of different boundary conditions on the lateral stress responses.