## 2 Materials and Methods
Thus, the formula for determining the elastic settlement is as follows:  
\[S_{el}=S_{l1}+S_{l2}+S_{l3}+S_{compress}+S_{0} \tag{7}\]  
where \(S_{compress}\) is the settlement due to compression of the pile shaft; \(S_{0}\) is the soil settlement under the pile tip, determined by Formula (5). The settlement of the bottom layer of soil with thickness \(l_{1}\) on the lateral surface of the pile is determined by the following formula:  
\[S_{l1}=\frac{2\cdot\tau_{0}\cdot l_{1}^{2}+\tau_{l1}\cdot l_{1}^{2}}{3\cdot E _{pile}\cdot a} \tag{8}\]  
The settlement of the middle layer of soil with thickness \(l_{2}\) on the lateral surface of the pile is determined by the following formula:  
\[S_{l2}=\frac{\tau_{l1}\cdot(l_{1}+l_{2})^{2}}{E_{pile}\cdot a}+\frac{(\tau_{l 2}-\tau_{l1})\cdot\left(\frac{(l_{1}+l_{2})^{3}}{3}-l_{1}\cdot(l_{1}+l_{2})^{ 2}\right)}{E_{pile}\cdot a\cdot l_{2}} \tag{9}\]  
The settlement of the top soil layer with thickness \(l_{3}\) on the side of the pile is determined by the following formula:  
\[S_{l3}=\frac{\tau_{l2}\cdot l^{2}}{E_{pile}\cdot a}+\frac{(\tau_{l3}-\tau_{l2} )\cdot\left(\frac{\beta}{3}-l_{2}\cdot l^{2}-l_{1}\cdot l^{2}\right)}{E_{pile }\cdot a\cdot l_{3}} \tag{10}\]  
In the graphical part of the method, the mobilised tangential stress diagram is plotted using the shear stresses obtained on the basis of the analytical part of the method, and the ultimate shear stress diagram is determined on the basis of the Coulomb-Mohr strength criterion. The characteristic diagram of mobilised shear stresses and ultimate shear stresses is shown in Figure 3.  
The superposition of the mobilised and ultimate shear stresses results in two main sections. The first section occurs when the mobilised shear stress profile exceeds the ultimate shear stress profile, i.e., the bearing capacity is exhausted at this section of the pile length and the pile is shut down. The load that can no longer be carried by this section is denoted by \(\Delta T_{n1}\). The second section characterises the remaining reserve of the bearing capacity of the soil \(\Delta T_{n2}\), i.e., where the ultimate shear stress diagram exceeds the mobilised diagram. The load \(\Delta T_{n1}\) is redistributed between the lateral surface of the pile, where reserve \(\Delta T_{n2}\) remains, and the pile tip. Figure 4 shows the calculation scheme of the graph-analytical method for determining the pile settlement for a three-layer soil mass. The non-linear settlement of the soil under the pile tip is determined by the following formula [8]:  
\[S_{pl}=S_{el}\cdot\frac{p_{cr}}{p_{cr}-p_{0}} \tag{11}\]  
where \(S_{el}\) is the elastic component of the settlement; \(p_{cr}\) is the ultimate (critical) load; \(p_{0}\) is the acting load on the pile. The elastic component of the settlement \(S_{el}\) is determined from the analytical solution. The ultimate load \(p_{cr}\) is determined using the formula proposed by [PERSON] [40; 41]:  
\[p_{cr}=\omega\cdot B_{\omega}\cdot L_{\omega}\cdot\left(s_{c}\cdot c\cdot N_{c}+s _{q}\cdot\gamma\cdot d\cdot N_{q}+0,5\cdot s_{\gamma}\cdot\gamma\cdot B_{\omega} \cdot N_{\gamma}\right) \tag{12}\]  
where \(\omega\) is the shape factor of the foundation's cross-section, for the transition from a square to circular cross-section; \(B\omega\) and \(L\omega\) are given values of length and width of the pile cross-section; \(s_{c}\), \(s_{q}\), and \(s_{\gamma}\) are shape factors of the foundation's cross-section; \(Nc\), \(Nq\), and \(N\gamma\) are dimensionless constants depending on the angle of internal friction of the soil; \(c\) is the cohesion of the soil; \(\gamma\) is the unit weight of the soil within the limits of a possible soil bulge prism; \(d\) is the depth of foundation embedment, equal to the pile length \(l\). Figure 4: Calculation scheme of the problem for determining the settlement of a single pile using the graph-analytical method. Figure 3: Characteristic diagrams: (**a**) mobilised shear stresses; (**b**) ultimate shear stresses.