Determination of an allowable value of internal uniform pressure on the underground horisontal working contour with a trapezoidal form of its cross-section

The paper presents the study results of the stress state at the contour points of an underground horizontal working carried out by the methods of the complex variable theory. This study is the basis for solving the problem of determining an allowable value of the uniform pressure transmitted to the contour of the working, with predetermined values of the depth of its laying. The condition proposed by the authors of the paper is used as a criterion of strength (stability), according to which the working’s contour strength is ensured if at any point the amount of the tangential normal stress does not exceed the tensile and compressive strength of the host soil (rock), i.e. σt ≤ σe ≤ σc . As a result of the solution, an assessment of the stress distribution on the contour of an underground horizontal working has been made. The working has a cross-section in the form of a curved trapezoid, which is under the action of uniform all-round stretching of a given intensity with a varying depth and various values of the host mass lateral expansion coefficient. The allowable values of the tensile internal pressure on the contour of an underground horizontal working of a trapezoidal cross-section have been determined at the given values of its depth and a constant coefficient value of the lateral expansion of the host mass μ = 0.25. The materials presented in the paper can be used when choosing the maximum or minimum allowable depth of the working used as a storage facility for liquid or gaseous hydrocarbons.


Introduction
One of the most urgent problems of geomechanics is the use of underground cavities as storage facilities for recovered reserves of liquid and gaseous minerals. This problem is directly related to studying the strength of underground, in particular, horizontal, workings [1][2][3][4].
It is known that after the exhaustion of mineral deposit reserves being developed by the underground method, there are many kilometers of underground mine networks left, which can be used to solve the problem mentioned above.
The use of such workings as storage facilities for gaseous hydrocarbons requires to analyze the stress state of the rock mass in the vicinity of the working and at the points of its contour by the elasticity theory methods. The most effective methods in this situation are the methods of the complex variable theory. In order to use them, it is necessary to specify the function of a complex variable that performs a conformal mapping of the interior or exterior of a unit circle onto an infinite simplyconnected area, whose boundary is a simple closed curve imitating the contour of an underground horizontal working of the required configuration [5; 6].
The construction of mapping functions, as a rule, is a rather difficult problem and when solving it, one has to abandon complex analytical expressions, replacing them with the simple ones, for example,  [7][8][9].
Let us consider the mapping function used in [10; 11], which has the form of where A, B, C, D are real numbers. This function applies a conformal mapping of the interior of the unit circle 1   to the exterior of infinite simply connected areas whose boundaries is a family of simple closed curves.
Mapping function (1) has already been used in a number of works [12][13][14] by the authors of this paper when solving problems related to the study of stresses at the contour points of underground workings and determining their allowable depth.

Goal setting and investigation objectives
Let us consider an underground horizontal working in the elastically isotropic rock mass. The form of its cross-section is determined by the mapping function (1). We assume that the working is laid at a sufficiently large depth H . What is more, along its contour (from inside), there is all-round uniform pressure of intensity p , which makes it possible to use the working as an underground storage facility of gaseous hydrocarbons, for example.
The present study is aimed at determining an allowable value of uniform pressure transmitted to the contour at preset depth values based on the stress state analysis on the contour of the working. As a strength criterion we use the one proposed by the authors, according to which the strength of the contour will be ensured if the value of the tangential normal stress at any point does not exceed the rock strength of the host rock under tension and compression, i.e.

Problem solution.
Let us consider the working whose cross-section represents a curvilinear trapezoid with dimensions 4  3.2 m.
Using an approach proposed in [15], we calculate the coefficients of mapping function (1  Following [14], we notice that the formula describing a stress state at contour points, whose geometrical structure is defined by mapping function (1) under the condition of the confining uniform pressure with specified intensity p , applied to the contour, had the following form Where  is rock volume-weight;  is lateral thrust coefficient; H is depth, p is the value of uniform pressure, applied to the contour of the hole. What is more, according to [4], we assume that at p >0, the contour of the working sustained compression of the constant value of p, and at p <0, it sustains tension of the same intensity.
Formula (2) is derived under the condition that depth H is sufficiently great. Following [4], we suppose that where R max is the maximum linear size of the contour. Finding zeros of the tangential normal stress is reduced to the solution of equation where  Taking into account the results of [14], we note that the extreme values of function () could be derived from the following equations where Let us move onto considering the problem which is connected with the study of the stress state at the points of the trapezoid cross-section contour given in figure 1. When solving the problem, we use two values of the lateral thrust coefficient:  1 =0.25 and  2 =1. The first value corresponds to the Poisson ratio, which, on average, is equal to  =0.20 for rocks [3]. The second value corresponds to the Poisson ratio  =0.5 and assumes the hydrostatical distribution of stresses in rocks, which is taken when determining stresses at rather large depths [1].
So, let us consider the trapezoid cross-section working (figure 1), corresponding to mapping function (1) (4) and (5) Then equation (7) will be as follows Then equation (7) where R c is ultimate tensile strength of rock, g is free-fall acceleration as pressure function p(). . We will look for the minimum value of function p()near the value of the argument at which the pressure function reaches its minimum at