The results in [1][2][3] form the basis for many solutions to the problem of the stress-deformation state in rock with an extraction working. Mikhlin [1] considers a working in the absence of rock fracture. If we imagine two symmetrical workings formed from a single one [1] with sufficient width, such that the floor and roof meet smoothly (the effect of caving is eliminated) and such that w e can neglect their mutual influence, then we can consider one of them. This problem was discussed by Barenblatt and Khristianovich [2]. Both these papers assume the seam to be absolutely rigid and the tangential stresses at the seam-rock contact m be zero, O~u~-0.An elastic seam-rock system in natural interaction was discussed by us in [4,5].For solving various practical mining problems, an experimental-analytical approach has been devised for the restoration of the stresses in a half-plane [3]; the first boundary condition is based on field measurements of the normal component of the displacement, while the second is given by one of the two hypotheses u = 0, Oxy = 0 or Oxy = koyy over the whole boundary of the half-plane.This article is in a sense an extension of the above papers [1,2,3]. On the one hand, the formulations in [1, 2] can be obtained as special cases; on the other, as in [3], we can use field displacement measurements, but those obtained from a finite section I. As in our previous articles [4,5], the rock mass is modeled by the system seam -rock, but here the seam receives only the normal load in the additional stresses; i.e., it is modeled by a setof preeompressed springs. With this representation, only the Young's modulus E 2 of the seam is involved in the calculation. where gl(x) = %(x) is the normal component of the displacement, and ga(x) = O~y is the normal component of the stress (in particular, the stresses found experimentally).Thus for the lower half-plane (rock), the boundary conditions at y = 0 take the form ~---~0, --oo

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 ...

Extensive use is made of the methods of mechanics of continua to solve problems of rock mechanics.The complexity of problems of rock mechanics is due not only to the fact that the shape of individual workings is other than round (their number may be arbitrary) but also to the diverse mining-geological conditions and properties of rocks. This leads to considerable difficulties when evaluating underground structures for strength and stability.Regardless of the model of the medium, the theory of fracture and flow, an elastic solution is assumed as a first approximation.Existing schemes for evaluating mine workings represent the real solid rock as a homogeneous isotropic medium [i, 2]. They reflect to a reasonable degree the basic physical and kinematic characteristics of the behavior of rocks. The use of conformal mapping has widened the range of solvable problems [4, 5]. Boundary-value problems for a plane with a circular hole, and a half-plane, have been the most thoroughly examined. Touching on these are problems for which the function representing the real periphery on a circle or half-plane has been found.But if the representing function is irrational, to find ~(z) we obtain a singular integral equation which must be solved.Even with the most simplifying assumptions concerning contact, the modeling of a rock mass by a set of uniform particles in contact markedly complicates the problem.Obtaining a practical solution is problematic.Taking account of the overall complexity of description of real rocks, the most important trend is the experimental-analytical method of determining the stress--strain state in the vicinity of a working [6, 7]. It employs (and this is highly significant) field measurements of shifts along lines which are formed as the boundary conditions for a half-plane; as in the case of analytical solutions, we must find the functions ~(z) and ~(z), and we then describe the components of the stress and deformation tensor over the whole region, including the boundary.In practice, e.g., it is necessary to find the state of stress of the periphery of the working directly from the measured components of the boundary shifts directly, without actual determination of the complex potentials ~(z) and ~(z), or vice versa.However, kndwing the stresses and shifts at the periphery we can readily find ~(z) and ~(z), i.e., expand the solution to the region.Many methods of stress measurement in rocks are based on conversion of the measured deformations of a borehole periphery to stresses [8].Below we examine the problem of determination of the stress--strain state at the periphery of holes for the basic boundary problems without actual determination of the functions ~ (z) and ~(z).Let us consider a body S-bounded by a contour r (the complement of S-+ F with respect to the whole plane is called S+), the same stress--strain state of which is defined by various combinations of boundary conditions.We will keep to the notation in [3] throughout the remainder of this article.The boundary problems of the first and second...

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In focus is mathematical modeling of zonal disintegration of rocks around a deep-level excavation. In the framework of the elastic model of isotropic material, the author analyzes the twodimensional case of stress field in rocks around a circular cross-section excavation. The effective compressive stresses at infinity depend on the depth of the excavation occurrence. The shear stress analysis has shown that in rocks around the excavation, the increased shear stress circle zone appears at the distance R r 3 = from the excavation center. The increased stresses come before the rock mass disintegration and create conditions for next destruction circles. The author dwells on probable influence of initial hydrostatic stress on the disintegration law. In modeling, the zonal disintegration circles appear at larger distance from the excavation center than in the experiment, due to the idealization of the classical problem formulations in rock mechanics.