## 1 Introduction  
Experimental studies of piles using field methods show that piles are designed with a large reserve for both bearing capacity and settlement. First of all, this is due to the fact that, according to the regulatory documentation in force in the Russian Federation, piles are designed in the elastic stage of the soil [1, 2, 3, 4]. In the second place, some studies suggest that the underutilisation of bearing capacity may also be due to the fact that the soils under the tip of long piles have a higher bearing capacity than assumed in the design [1, 5]. Despite this, the authors in [5, 6, 7] have proposed methods for calculating pile settlement that take into account the non-linear behaviour of soils only along the lateral surface of the pile, while the authors in [8, 9], on the contrary, take into account the non-linear behaviour of soils only under the pile tip. However, no calculation method has been proposed in engineering practice that takes into account the non-linear behaviour of the soil both along the lateral surface of the pile and under the tip of the pile. Exceptions are numerical calculation methods, e.g., [10], which allow elastic, non-linear, viscous behaviour of the soil to be taken into account, but the final result and its interpretation will depend on the user. Secondly, the design of pile foundations does not take into account the load distribution on the pile between its lateral surface and the tip. However, this process has a significant influence on the formation of the stress-strain state of the soil mass and on the pile settlement. Based on a study of the experimental results, it can be noted that initially, most of the load is transferred to the soil through the side surface of the pile and the rest of the load is transferred through the tip of the pile. With increasing loads on the pile head, there is a gradual depletion in the bearing capacity: first, at the level of the pile head and at the contact point with the weak soil layers; then, along the entire lateral surface; and finally, failure under the pile tip, which leads to total failure [11; 12]. For example, in [13], at a load of 100 kN, almost the entire load on the soil is transmitted through the lateral surface of the pile (Figure 1). A further increase in the loads occurs mainly only on the lateral surface, and at the maximum applied load of 1100 kN, there is an increase in the pile tip load, but it does not exceed 30%. Similar results can be seen for different pile lengths with different cross-sectional shapes [14; 15; 16; 17; 18]. However, this factor was only considered by the authors in [5], and they considered the kinematic inclusion of the pile in that work. When loads are applied, the lateral surface area at the level of the head is included in the work, and then, gradually deeper soil layers are involved in the work, gradually reaching the tip of the pile and involving the soil beneath it in the work. This load distribution is an approximation as it will also depend on the stiffness of the soil. Thirdly, most of the theoretical and numerical solutions are based on the laws of continuum mechanics, which leads to their joint deformation, i.e., the settlement of the pile is equal to the settlement of the adjacent soil. But in reality, when the bearing capacity limit of the soil is reached, failure occurs, which leads to pile failure. The works by [PERSON]. [19; 20] considered the possibility of pile detachment from the ground using the mechanics of discrete bodies; however, such methods are more voluminous and difficult to calculate than methods based on the mechanics of continuous bodies, which is undoubtedly their disadvantage, especially in engineering practice. Fourthly, a new stress-strain state appears at the contact point of the structures with the soil, and the properties of the soils in the contact zone will differ from the properties of the same soils in their natural mass.