Abstract
When designing spatial structures, it is necessary to know the stressed-strained state of a body. These problems include the calculation schemes, in which there is a half-space with cylindrical cavities, at the boundaries of which the contact type conditions are assigned. The segment of such problems is not enough researched and requires further attention.The analytical and numerical algorithm for solving a special problem of the theory of elasticity for a half-space with cylindrical cavities was proposed in this paper. Radial displacements and tangential stresses are assigned at the boundaries of cavities, and one of the two types of the boundary conditions – displacement or stress – is assigned at the half-space boundary. Calculations revealed the stressed-strained state of the half-space.Under the fixed geometrical conditions, a numerical analysis of the three variants of the problem, when displacements are assigned at the half-space boundary, and of the tree variants of the problem, when stresses are assigned at the half-space boundary, was conducted. A comparative analysis of the variants with different boundary conditions was carried out.It was found that at the boundary conditions of the contact type, assigned at the boundaries of cylindrical cavities, if the assigned displacement function and the assigned function of stresses are the same, the boundary conditions at the half-space boundary in the form of stresses have more influence of the stressed state that boundary conditions in the form of displacements. It was also established that at different kinds of the assigned boundary conditions (stresses or displacements), stresses o and on the surface of applying such conditions change for the opposite, that is, from stretching to compressing and vice versa.The presented analysis can be used during designing the structures, in the calculation schemes of which there is a half-space boundary with boundary conditions of the contact type, assigned on it, and cylindrical cavities, on the surfaces of which displacements and stresses are assigned.
Highlights
In the design of various structures, underground facilities and communications, there is a need in assessing stresses in a half-space with cavities
We explore the problem, where displacement or stress are assigned at the boundaries of a cavity, and the conditions of a contact type are assigned at the boundaries of a half-space in the form of radial displacement and tangential efforts
If we find Uρ and stresses τρφ, τρz, for the right side of equation (4) on the surface of each cylindrical cavity and take into consideration boundary conditions (1), we will obtain the system of equations for coefficients
Summary
In the design of various structures, underground facilities and communications, there is a need in assessing stresses in a half-space with cavities. It is necessary to have a method for calculation of the problems of the theory of elasticity, which makes it possible to find a stressed-strained state of a half-space with cylindrical cavities. We explore the problem, where displacement or stress are assigned at the boundaries of a cavity, and the conditions of a contact type are assigned at the boundaries of a half-space in the form of radial displacement and tangential efforts. In addition to the proposed algorithm of calculation, an analysis of the stressed state, which enables prediction of the weak points at the stage of design, is presented in this paper
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