A three-dimensional semianalytical method of determining the mechanical state of solid rock

Abstract

In coal mining, solid rock, the working, and support 'form a system with very complicated interactions. The mechanical interactions in the system result in rock-pressure phenomena which are manifested in a redistribution of the stress-strain state of the rock and the formation of loads ou the supports.The fundamental problems in rock-pressure research consisted in determining the load on the supports. A later stage In its development involved finding the optimum parameters for the system of mining (dimensions of extraction workings and pillars, depth of cut, etc.) from the viewpoint of the rock pressure factor.Present problems in rock mechanics, concerning the question of rock pressure, involve the need for three-dimensional planning of mine workings.The need to consider three-dimensional systems is due, first, to the combined working of a series of mutually interacting seams for which the sequence of working depends largely on rock pressure phenomena. Secondly, the extraction working itself, together with the surrounding development workings, forms a threedimensional system. In such a system it is sometimes difficult to find cross sections corresponding to the planar (two-dimensional) problem; in particular, it is necessary to consider the interaction of extraction and development workings the last of which are in the end cross sections of the extraction working. Furthermore, in most cases the geologic conditions and mining technology change as extraction and development operations are carried out, and this also prevents us from using planar schemes of calculation in research on rock pressure manifestations.It is obvious that three-dimensional problems must be solved in studying the mechanical state of the rock during the working of a coal seam. However, discussions of three-dimensional problems, even bymeans of the simple theory of elasticity, are few [1], owing to the mathematical difficulties and the limited computing approaches. Furthermore, solutions to three-dimensional problems in the mathematical theory of elasticity based on functional analysis or singular integral equations are general in form and difficult to apply in particular cases.In rock mechanics the consideration of three-dimensional problems began only a few years ago [2][3][4][5], but the analytical methods used in the solutions are as yet ineffective in the study of rock-pressure phenomena during the working of coal seams.One possible method of investigating the three-dimensional mechanical state of the rock is the semianalytical method of rock mechanics.The semlanalytical method involves using natural manifestations of rock pressure, in particular displacements of the rocks, as boundary conditions in analytical investigations of the state of the rock. In contrast with an earlier method [6], based on planar problems tn the theory of elasticity or creep with boundary conditions in the form of profile curves of the displacements, the three-dimensional semianalytical method considers the three-dimensional stress-strain state of the rock on the ...

Axisymmetric abutment pressure problem

“…Assuming that(z, y)~P and putting (6) into (3), we get an integral equation for the stresses in P (at the contact between the seam and the surrounding rocks): o kC1. On the right-hand side of (7) the integral over V is a known function of position, because the stresses are known in V. The constant kC~ is determined in the course of the solution from the condition (8) P+V which denotes that the total force acting on the seam P in the incremental problem is equal in magnitude and opposite in sign to the load applied to the working V. If this were not so, then from the asymptotic law of decrease of the stresses at infinity (directly proportional to the total force applied to the boundary and inversely proportional to the square of the distance from the origin) we could infer the presence of an energy source at infinity.…”

“…

The authors consider the three-dimensional problem of abutment pressure in a seam weakened by a working.It is assumed that the seam interacts with the rock without creating tangential stresses, and that being homogeneous it can be regarded as an elastic substrate.Let us introduce rectangular coordinates: The z axis is directed vertically upward and the xy plane passes along the middle of the seam.We will denote the rock-seam surface by P and the surface separating the rock from the worked-out area by V.The plane analog of this problem (in which P and V are segments of a straight line) was discussed in [1][2][3][4][5].As usual, we reduce the problem to the incremental case (i.e., we subtract the stress field corresponding to the same problem without the working), which consists in finding the solution to the equations of the static theory of elasticity for an upper half space z ~ 0 with the following boundary conditions: (4) where ~xz, ~yz, ~zz are the components of the stress tensor, w is the vertical displacement, k > 0 is the "bedding coefficient," whichis proportional to Young's modulus and inversely proportional to the thickness of the seam, y is the density of the surrounding rock, and H is the depth of the seam below the surface.Grltsko et al [6] solved the abutment pressure problem for a seam in the three-dimensional formulation by an experlmental-analytlcal method, the essence of which was that the displacements for z = 0 are determined experimentally and the stresses found from them. This avoids the difficulties due to the fact that the boundary conditions (2) and (3) are of the mixed type.

In this article we propose a different approach to the three-dimenslonal problem.

Let us reduce the incremental problem to an integral equation.

…”

Surface of abutment pressure and resistance of stowage during working of a coal seam

“…The stress increments in the solid rock, induced by mining-out work and used to calculate the abutment pressure and resistance surfaces of the stowage, are given in [2] by Eqs. (12), (15), and (17).…”

“…At the contact of the roof rocks with the coal seam and the worked-out area (z = 0), the increments of the stresses normal to the seam (Zz) and of the tangential stresses (Xz) per unit advance of the face have the form According to [2], the additional stresses induced by working of the seam are defined in terms of the stress increments due to advance of the face and working from by the equations Equations (5) are obtained in the same way as (13) in [2], and they are the result of solution of the three-dimensional problem of elasticity theory with the boundary conditions…”

“…As a result of field measurements of displacements with the form [3] w(x, y, O; l, L) =~(l, L)F~(x, l)F2(y, L), The surface of final displacements was calculated by the method in [2] using the equation As a result of field measurements of displacements with the form [3] w(x, y, O; l, L) =~(l, L)F~(x, l)F2(y, L), The surface of final displacements was calculated by the method in [2] using the equation…”