## 2 Materials and Methods  
The graph-analytical method consists of two parts: analytical and graphical parts. In the analytical part of the method, a system of several equations is solved to determine the unknown components of the stress-strain state (SSS) of the soil mass around the pile: the shear stress \(\tau\) on the lateral surface of the pile due to friction forces and the vertical stress \(\sigma\) under the tip of the pile. The analytical solution is elastic; hence, there is no failure of the soil around the pile and no exhaustion of the bearing capacity. The graphical part of the method is used to describe the non-linear behaviour of the soil. It consists in the construction of two shear stress diagrams along the lateral surface of the pile. The first is a mobilised (effective) shear stress diagram obtained on the basis of analytical equations, while the second is a limit (resource) diagram based on the selected soil strength criterion. When overlaying the diagrams on each other, there are characteristic areas where there is exhaustion of the bearing capacity, i.e., where the mobilised diagram exceeds the ultimate diagram, and, vice a versa, where the ultimate diagram is greater than the mobilised diagram, which indicates the remaining unexhausted reserve of bearing capacity of the soil. The load that exceeds the soil strength limit is redistributed laterally and downwards until the bearing capacity is completely exhausted and pile failure occurs. At the same time, the graph-analytical method allows for controlling the distribution of loads on the pile and changing the strength characteristics of the contact zone soils. The section of the pile where the bearing capacity of the soil has been exhausted is removed from further work, which gives the pile the opportunity to move relative to the soil. The analytical solution of the settlement problem of a single pile interacting with a three-layer base in axisymmetric formulation is presented below. The calculation scheme is a <<cell>> consisting of a cylinder-shaped soil volume with a pile immersed in it. The engineering geological element number is designated as EGE. When a load \(N\) is applied to the pile head, friction forces along the lateral surface of the pile \(T_{1}\), \(T_{2}\), and \(T_{3}\) and earth back pressure \(R\) under the pile tip occur. The origin is located at the level of the pile tip, with the \(z\)-axis pointing vertically upwards and the \(r\)-axis pointing horizontally. The pile radius is denoted as \(a\) and the cell radius as \(b\). The width of the cell was iteratively selected so that the condition of zero settlement at the lateral boundaries was fulfilled, i.e., at \(r=b\), the soil settlement \(S=0\). Figure 2 shows the calculation scheme of the problem with the indication of the acting forces on the pile and with the indication of the arising stresses from the action of the applied load \(N\). As a result, the following stress components need to be determined: the reactive stress \(\sigma_{0}\) under the tip pile and the four components of the shear stresses \(\tau_{0}\), \(\tau_{1I}\), \(\tau_{12}\), and  
Figure 2: Calculation diagram of the analytical problem: (**a**) showing the acting forces; (**b**) showing the acting stresses. acting along its lateral surface. The distribution of shear stresses along the length of the pile is assumed to follow the \"trapezoidal\" law, linearly decreasing with depth. For the bottom soil layer, this law has the following form:  
\[\tau(z)=\tau_{0}+\frac{\tau_{11}-\tau_{0}}{l_{1}}.z \tag{1}\]  
where \(\tau_{0}\) is the value of shear stress at the level of the pile tip; \(\tau_{l1}\) is the value of shear stress at the boundary of the bottom and middle layers of soil; \(l_{1}\) is the length of the pile located in the bottom layer of soil or the thickness of the bottom layer of soil.