Investigation of the second main problem of the theory of elasticity for a layer with several cylindrical cavities

V Yu Miroshnikov

doi:10.20535/2521-1943.2019.86.165291

Abstract

Background. When designing spatial structures, parts and mechanisms, underground structures and communications, it is necessary to have an idea of the stress state of such a structure.Objective. It is necessary, with high accuracy, to find displacements and stresses in the body of the layer, which has longitudinal cylindrical cavities, and also to analyze its stress-strain state. At the boundaries of the layer and at the boundaries of the cavities, displacements are given.Methods. To achieve the declared goal, an analytical-numerical method has been developed for a layer with circular endless cylindrical cavities parallel to each other and to the surfaces of the layer. The solution of the spatial problem of the theory of elasticity is obtained by the generalized Fourier method with respect to the system of Lame equations in cylindrical coordinates associated with cylinders and Cartesian coordinates associated with layer boundaries. Special formulas are applied for the transition between coordinate systems for basic solutions. The infinite systems of linear algebraic equations obtained as a result of satisfying the boundary conditions are solved by the reduction method. A numerical study of the determinant gives reason to argue that this system of equations has a unique solution. As a result, displacements and stresses at various points of the elastic body of the layer were obtained. The reduction parameters were chosen so that the accuracy of the boundary conditions reached 10-4.Results. The analysis of the stress - strain state of the layer body at different geometric locations of two cylindrical cavities in it is carried out. It turned out that with equal distance between the cavity and the layer boundary from the surface of the cylinder in question, the layer boundary has a greater effect on the stress state of the body.Conclusions. With an increase in the reduction parameter, the accuracy of fulfilling the boundary conditions increases, but the calculation time also increases.The above analysis can be used for preliminary selection of the calculation model, and the proposed method for calculation, with high accuracy, the selected calculation scheme.

Highlights

Parts and mechanisms, underground structures and communications, it is necessary to have an idea of the stress state of such a structure
V.S., Ukrainets N.A. (2015), “Application of the generalized Fourier method to the solution of the first main problem of the theory of elasticity in a half-space with a cylindrical cavity”, Visnyk Zaporizkogo natsionalnogo universytetu, vol 2, pp. 193 – 202

Summary

Постановка задачі

В пружному однорідному шарі розташовано N кругових циліндричних порожнин радіусом Rp, непересічних між собою і межами шару, де p – номер циліндра, p 1, 2, ..., N. Потрібно знайти розв’язок рівняння Ламе за умов, що на межах шару та на межах циліндричних. Усі задані вектори і функції будемо вважати швидко спадаючими до нуля на далеких відстанях від початку координат по координаті z для циліндра та по координатах x і z для меж шару. Розв’язок задачі Виберемо базисні розв’язки рівняння Ламе для зазначених систем координат у вигляді [11]:. Для переходу між системами ко ординат (рис.1) скористаємось формулами: – для переходу від розв’язків Sk,m циліндричної системи координат до розв’язків шару u k (при y>0) та u k (при y

Числові дослідження напруженого стану

Objective