ON THE DISTRIBUTION OF STRESSES ON THE CONTOURS OF SINGLE UNDERGROUND HORIZONTAL WORKINGS OF VARIOUS CROSS-SECTIONS SUBJECTED TO ALL-ROUND UNIFORM PRESSURE

Abstract

The formula for normal tangential stresses for single underground workings of different cross-sectional shapes, located at a given depth, with all-round tensile uniform pressure applied at the points of the contours of the workings, is derived. As a mapping function is considered a function of complex variable, which is a polynomial of natural degree n with a pole of first order in zero, allowing the construction of various families of simple closed curves, simulating configurations of the contours of underground workings. Examples are given of contours whose cross-section has the form of a straight and inverse trapezoid, triangle, vaults with vertical and sloping walls, rhombus, rectangle, square and ellipse. Based on the method proposed by the author, the coefficients of the polynomial of the seventh degree, which performs conformal mapping of the interior of a unit circle on a plane with a trapezoidal hole of a given dimensions, are calculated. The stress state of constructed workings at the points of its contour at different depths of laying, the values of the all-round tensile uniform pressure at two fixed values of the lateral expansion coefficient of the rock is investigated. The graphical representations of the stress acting on the contour of the working in question are given. The obtained results can be used to solve problems of determining the permissible depths of mine workings and calculating the values of permissible values of uniform pressure in the points of their contours. The criterion for determining the values of these values is the condition of absence of points on the contours of workings, in which the normal tangential stresses exceeding the limits of tensile and compressive strength of the host rock.