## 3 Results
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(n\) & \(F_{u}\) & \(F_{u}\) & \({F_{u}}^{2}\) & \({F_{u}}^{2}\) & \({F_{u}}^{F}_{u}\) & \(k\) & \(b\) & \(k_{x}\) \\ \hline
1 & 1648 & 511 & 419,904 & 261,121 & 331,128 & & \\ \hline
2 & 702 & 571 & 492,804 & 326,041 & 400,842 & & \\ \hline
3 & 657 & 409 & 431,649 & 167,281 & 268,713 & & \\ \hline
4 & 612 & 396 & 374,544 & 156,816 & 242,352 & 1.586 & \(-\)558.755 & 0.723 \\ \hline
5 & 628 & 465 & 394,384 & 216,225 & 292,020 & & \\ \hline
6 & 624 & 441 & 389,376 & 194,481 & 275,184 & & \\ \hline \(\Sigma\) & 3871 & 2793 & 1,810,239 & 2,502,661 & 1,321,965 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Results of calculating the functional dependence for ordinary bored piles. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{3}{*}{**Pile \#**} & \multicolumn{4}{c}{**Pile Characteristics of**} \\ \cline{2-5}  & \begin{tabular}{c} **Bored Pile with** \\ **Protective Shell,** \\ **without Soil Wetting** \\ **(F\({}_{u}^{\text{p-s}}\)), kN** \\ \end{tabular}  & \begin{tabular}{c} **Ordinary Bored Pile,** \\ **without Soil Wetting** \\ **Wetting** (F\({}_{u}^{\text{p-p}}\)), kN** \\ \end{tabular}  & \begin{tabular}{c} **Bored Pile with** \\ **Wetting** (F\({}_{u}^{\text{p-s}}\)), kN** \\ \end{tabular}  & \ \begin{tabular}{c} **Ordinary Bored Pile,** \\ **Protective Shell, after** \\ **Soil Wetting** (F\({}_{u}^{\text{p-s}}\)), kN** \\ \end{tabular}  & \begin{tabular}{c} **Ordinary Bored Pile with** \\ **Pile, after Soil** \\ **Wetting** (F\({}_{u}^{\text{p-p}}\)), kN** \\ \end{tabular}  \\ \hline pile \#1 & 1564 & 648 & 1499 & 511 \\ \hline pile \#2 & 1723 & 702 & 1582 & 571 \\ \hline pile \#3 & 1647 & 657 & 1427 & 409 \\ \hline pile \#4 & 1600 & 612 & 1527 & 396 \\ \hline pile \#5 & 1645 & 628 & 1510 & 465 \\ \hline pile \#6 & 1698 & 624 & 1463 &after which the line of best fit was drawn characterizing the proportional relationship, which deviated least of all from these points. To find the linear equation of the best-fit line using the least squares method, the coordinates of the points \(F_{u}^{\prime}\) and \(F_{u}\) were drawn until they intersected with the desired straight line (Figures 10 and 11). The values of these ordinates are equal to (\(kF_{u}^{\prime}+b\)). The ordinate distance from the point \(F_{u}^{\prime}\) to the straight line is (\(kF_{u}+b-F_{u}^{\prime}\)). Thus, the regression is expressed by the following equation:  
\[\sum(F_{u}-f(F_{u}))^{2}=\sum(kF_{u}+b-F_{u}^{\prime})^{2}\to min, \tag{4}\]  
It is assumed that the best-fit line would be the smallest value of the sum of the squares of all distances. The minimum of this sum was found according to the rules of differential calculus. To determine \(k\) and \(b\), the following equations were obtained:  
\[k=\frac{n\sum_{1}^{n}F_{u}F_{u}^{\prime}-\sum_{1}^{n}F_{u}\sum_{1}^{n}F_{u}^{ \prime}}{n\sum_{1}^{n}(F_{u})^{2}-(\sum_{1}^{n}F_{u})^{2}} \tag{5}\]  
\[=\frac{\sum_{1}^{n}F_{u}^{2}\sum_{1}^{n}F_{u}^{\prime}-\sum_{1}^{n}F_{u}F_{u} ^{\prime}}{n\sum_{1}^{n}(F_{u})^{2}-(\sum_{1}^{n}F_{u})^{2}} \tag{6}\]  
If we consider that the line passes through the origin, then the linear regression is:  
\[\sum(F_{u}^{\prime}-f(F_{u}))^{2}=\sum(k_{x}F_{u}-F_{u}^{\prime})^{2}\to min, \tag{7}\]  
To find the minimum of the equation, the partial values of its derivatives are equated to zero and a system of linear equations is obtained: \(k_{x}\sum F_{u}=\sum F_{u}^{\prime}\)  
\[\sum F_{u}+k_{x}\sum F_{u}^{2}=\sum F_{u}F_{u}^{\prime}, \tag{8}\]  
The results of bearing capacity of the piles by SLT are represented by blue dots in Figures 11 and 12. The values obtained according to \(k_{x}\sum F_{u}=\sum F_{u}^{\prime}\), where \(k_{x}\) is equal to 0.911 and 0.723, respectively, are represented by crosses. The trend lines and corresponding formulas are also presented.