Математичне моделювання напруженого стану ґрунтового середовища

Abstract

A mathematical model has been developed that describes the stress state of the soil environment. The basic calculation model of the soil massif is presented in the form of a dispersed medium. Its stress state is described by a discrete-continuum model under the action of all stress components. This approach made it possible to use the main provisions of the theory of elasticity to construct a system of differential equations for the equilibrium of a disperse medium in a plane formulation of the problem. It also makes it possible to use the probabilistic properties of dispersed media to solve systems of differential equations that describe the distribution of stresses in a spacer dispersive medium with a distributive property. When an external load in the form of a concentrated force is applied to the soil mass, a differential equation for the distribution of the vertical normal stress of a parabolic type was obtained. It is one of the varieties of the heat equation. To solve the differential equations for the distribution of vertical stresses, the approach of applying an exponential function is used. It allows to ensure the symmetry of the distribution of stresses with respect to the line of action of the concentrated vertical force. It also allows characterizing the structure of the soil environment itself. To determine the adequacy of the developed mathematical model of the stress state of the soil mass, an external load in the form of a force was used. This force is given as a normal distribution function over the surface of the soil mass. Based on the developed mathematical model, the distribution function of tangential, normal vertical and horizontal stresses was obtained for a normal external force load on the soil mass. A comparative analysis of the results of theoretical and experimental studies on a prototype fine sand type showed a high degree of adequacy of the developed mathematical model of the stress state of the soil environment.