SIMULATION OF THE SUPPORT-ENCLOSING ROCK MASS INTERACTION FOR DEEP MINING

Purpose. To develop analytical model for a support-enclosing rock interaction to determine parameters for operational stability of deep mine workings while decreasing metal consumption and increasing efficient use of resources. Methods. Involving various strength degradation functions and variations of physical and mechanical properties of rocks, mathematical modeling is used to consider the ranges of force action of a support on the enclosing rock mass of deep mine workings. Findings. Analytical dependence of a support effect on the rock border displacement as well as on the changes in cross section of the mine working has been obtained. Effective interval of the support force resistance to block limit zones of the rock mass deformations has been substantiated. Innovative approach relying on the priority of the support working capacity as well as its forming characteristics has been proposed. The results of the studies help regulate the use of available supports, and the development of new designs meeting the increased geomechanical requirements of deep mining. Originality. It has been determined for the first time that 150 – 250 kN/m2 interval of a support resistance is the most efficient and achievable; while mining deepening (more than 1000 m), a support resistance achieves 350 – 400 kN/m2. Higher values are not practical. Practical implications. The results of the studies help regulate the use of available supports, and the development of new designs meeting the increased geomechanical requirements of deep mining and to determine the required parameters of both force and deformational characteristics of supports making.


INTRODUCTION
The development of "support -rock mass" interaction system is driven by a requirement for the reliable forecasting of stability of mine workings as well as for the allowable limits determining reaction pressure of a support on a value of rock displacements (Baranowski & Lugovoi, 2008;Elmo & Stead, 2010;Gaidachuk, Koshel', & Lugovoi, 2011).
Convergence of calculation data with experiments concerning the use of the supports having related working characteristics are the criterion of the model accordance with actual conditions (Carranza-Torres & Fairhurst, 2000;Kononenko, Petlovanyi, & Zubko, 2015).Consideration of real values of the support efficiency along with its stable resistance is the essential objective.It is expedient to solve the problem in the context of al-lowable limits of the support actual effect on the value of finite displacements and estimation of the results -in the context of initial assumptions effect on the actual nature of the process physics.
The most important moment is the mechanical strength of the rock mass within a limit zone which consideration makes it possible to single out the three groups of calculation schemes: 1) according to its properties, a medium of the limit zone is identified with ideally flowing one; in this context, rock properties are taken up as similar for any point of the limit zone; 2) according to its properties, a medium of the limit zone corresponds to flowing medium with adhesion; its properties are also similar for any limit zone; however, its adhesion value differs from zero; 3) adhesion within the limit zone is taken up as a variable, and its value for each point of the limit zone is determined by means of strength degradation function.
While selecting a type of deformational strength degradation, the assumed function is quasi-similar to out-oflimit deformation diagram leg in the context of complicated stress state (σ 3 ≠ 0) and, being undefinable, it should satisfy the two basic requirements: be continuous, and correspond to border conditions within external boundary of the limit zone as well as within its internal boundary (Stovpnyk, Borodai, & Kravets, 2011;Stovpnyk, Han, Zahoruiko, & Shaidetska, 2017).
As additional conditions, it is assumed that in the context of low values of wall pressure within boundary share of the limit zone (σ r → Р), disintegration of material and its behaviour are followed by volumetric deformation, and deformational strength degradation with brittle failure prevailing (Stovpnyk & Osypov, 2017).
The effect of the increase in the support resistance (р = σ 3 ) is implemented as a result of degree dilatancy degree reduction (εν) as well as mobilization of the residual strength of disintegrated part of rock mass neighbouring the border.It is possible to subdivide the available geomechanical solutions as follows: -at the site of deep single mine working located beyond the area of mining influence as well as within a zone of front abutment pressure (up to the longwall "window"), a mode of interacting deformation is implemented in the context of rather heavy dependence of displacement value on the support working resistance; final balance the "support -neighbouring part of the rock mass" is achieved when the support resistance value is proper; -effect of mining operations disturbs external limit balance on the line of limit disintegration zone and undisturbed rock mass being under the limit state one; due to the system balance disturbance, radius of the limit zone increases together with displacements within the mine working border as well as their velocity; -generally, a new state of limit balance within a zone of front abutment pressure of a longwall is not achieved as the process is temporal; thus, the required support strengthening just decreases displacements down to a value providing necessary conditions of the mine working operations until the moment of its abandonment after the longwall advance; rate of advance determines a degree of crosssection reduction during the influence of geodynamic zone.
It should be noted that the idea of limit balance does not imply absolute attenuation of displacement; thus, it can be characterized when displacement velocity drops from its initial values being several dozens of millimeters a day down to less than 0.2 -0.5 mm/day.Within a zone of front abutment pressure, the displacements may near the values if only the zone obtains its final extension, i.e. after the longwall stoppage.
Consideration of the approaches makes it possible to apply the following (Hudson & Harrison, 2000;Jing, 2003;He, Xie, Peng, & Jiang, 2005;Jaeger, Cook, & Zimmerman, 2009;Brady & Brown, 2013;Wittke, 2014): -in the context of extended mine working it is expedient to consider weightless plane as an initial state.The plane is weakened by a cut which form corresponds to a cross-section of a mine working with remote stresses being equal to stresses within undisturbed rock mass in the central part of a future mine working (А.I.Dinnik, А.V. Morgaevski, G.I. Savin); -in the context of deep mine workings initial stress state may be assumed as hydrostatic (К.V. Ruppeneit, Yu.М.Liberman, М.А.Dolgikh, Р. Mindlin and others); -within the areas of the highest stress concentration in the neighbourhood of a mine working border, active stresses may exceed rock strength limit; in this context, brittle failure is a predominant type of rock deformation being typical for coal deposits (G.N.Kuznetsov, А.D. Panov, К.V. Ruppeneit, Yu.М.Liberman and others); -in the function of limit balance of rock which deformations are followed by their brittle failure, it is possible to be limited to the consideration of rectilinear inclined compound curve since in the context of initial hydrostatic stress state, normal stresses acting within the mine working boundary are always situated behind the circle of uniaxial compression; upon that, consideration of differences in locations of compound curves for disturbed rocks and undisturbed ones is mandatory (А.Labass, К.V. Ruppeneit, Yu.М.Liberman and others); -in the context of calculations concerning a mine working which form of cross section differs from circular one to simplify data, the substitution for equivalent circular cannot introduce significant error into final results; maximum difference of the latter is not more than 10% (К.V. Ruppeneit, М.А.Dolgikh); -currently, it is possible to consider the effect of anisotropy of real rocks on the displacement and distribution of stresses around a mine working (S.G.Lekhnitski, A.N. Zorin, M.I.Rozovski); however, if analytical studies are meant, the idea to consider anisotropy of real rock masses in terms of empirical coefficients is the most reasonable; moreover, since analytical studies of such a type should be performed together with a field experiment, its numerical value may be obtained according to the observation results.
Thus, there is abundance of experimental data for substantiated selection of a type of functions of deformational strength degradation within different points of limit zone and to reduce the number of functions, taken for the analysis in terms of the specified calculation technique.

CALCULATION MODEL OF A BORDER DISPLACEMENT AND DETERMINATION OF PRESSURE ACTING ON THE SUPPORT OF A MINE WORKING
Since volumetric deformation (i.e.dilatation in the process of pseudoplastic deformations) is the basic factor determining final value of displacements in terms of rock border of a mine working, the use of a known condition of plastic potential is necessary: where: λ -arbitrary parameter; f r -plasticity function describing rock behaviour Since, effect of a support resistance on the value of final displacements of a mine working border of cross section is in large part determined by a type of a function of distribution (strength degradation), the development of the mathematical model should involve the three relevant functions: exponential, rational, and hyperbolical.
Consider rock mass assuming it as homogenous isotropic medium where both elastic and nonelastic deformations arise after the development of a mine working (Fig. 1).

Design model for elastic and nonelastic deformations calculation
Mainly, nonelastic deformations are represented in the form of pseudoplastic brittle deformations.Round mine working is located at rather large distance h from the surface and stress state is of hydrostatic type (Poisson's ratio is ν = 0.5), i.e. axisymmetric plane problem is under consideration.
Introduce following symbols: Mark a zone of pseudoplastic deformations А within the specified plane А (А is a zone between С 2 and С 3 circles); a zone of elastic deformations В (В is a zone located beyond the С 3 circle); zone D (D is a zone between С 1 and С 2 circles) characterizing the cross section decrease by the time when final balance of "supportrock mass" approaches.
Support displacements are determined with the help of the value: where: r 0 -the radius of a circle of a mine working to which uniformly distributed pressure P is applied.
Find a change in relative area of the failure zone, i.e. area of zone D classified as a part of a circle area with r 0 radius.It is obvious that since area of S circle with R radius is equal to πR 2 , then area of zone D is equal to π (r 0 2 -(r 0 -U) 2 ), i.e.: (3)

Determination of stresses
Plasticity equation within A zone is: where: σ r -radial stresses; σ θ -tangential stresses; Balance equation takes place for σ r , σ θ values: Specify radial deformations and tangential deformations by means of ε r and ε θ respectively.Then, compatibility condition of deformations within A zone is: It should be pointed out that at infinity (if r → + ∞), normal stress components are equal to each other; τ θ = 0: Interpretation of formula ( 4) is that strength properties of material (rock) may be described with the help of rectilinear compound curve of stress circles: τ = σ tg φ 1 + K, where φ 1 is friction angle.
Write down formula (4) in another way: Then, using properties of zone А and a condition of plastic potential, following formulas for ε r and ε θ deformation can be written down in the form of: In view of ratio two from (8) for ε r from (9) we get: While integrating equations ( 5) and ( 6) in which specifications (8) are introduced, we will use boundary conditions: ( 1 1 ) where: P -the support resistance with р = const characteristic.
In Figure 1, r L means a value characterizing radius of circle С 3 separating elastic zone В and failure zone А.
In view of continuity it is necessary to assume that: For a function f(r) shown in Figure 1 by means of separated line, following equalities hold true: .
( 1 4 ) In addition, we consider implying: ( ) Insert expression (7) in formula (4): ( 1 6 ) Equation ( 16) is a differential first-order equation.Hence, we will seek its solution with the help of a method of variation of arbitrary constant., we obtain: The above formula is required to seek a solution for nonhomogeneous equation ( 16).While replacing σ r from equation ( 17) to equation ( 16) we obtain: Equation ( 18) gives ratio for C(σ r ): ( ) ( ) where: С -arbitrary constant.
Basing upon ( 17) and ( 19) find general solution for equation ( 16): Using boundary condition (12), we obtain final formula from (20): If availability of integral in formula ( 21) is taken into consideration, then function σ r (r) is a function of upper limit of an integral as well.

Determination of deformations
Inserting expressions ( 9) and (10) in differential equation ( 6) we get: where: It is not difficult to demonstrate by means of direct differentiation that solution of the equation ( 22) is: -movement, and formula (23), determine: ( 2 4 ) Apply boundary condition taking place if r = r L to determine a constant С * .
Displacements within zone B are identified with the help of the expression: Hence, the С * constant in (24) formula can be found from the equality condition of radial movements ( 24) and ( 25).
Formulate the results of the transformations under the condition that zone A (failure zone) arises: ( ) ( ) where: ν -Poisson's ratio; Е -Young's modulus.Formulate the problem using the abovementioned approach.Assume that function f(r) and parameters ν, E, R st , Р, ϕ 1 , r 0 have been set.Then, the problem solution involves two stages.
Stage one: in accordance with formulas ( 21) and ( 15) we have the equality: with the help of which, r L value characterizing zones A and В. Since, left side contains integral then it is possible to determine r L using numerical procedure.
Stage two: after determination of r L all values of the parameters are inserted in formula (26); thus, the movement is being determined: ( ) ( ) ( 2 8 ) Stage three: while using formula (3), we have a graph of s s Δ function depending upon pressure: ( ) where: U -a function of P. Besides, such a numerical approach makes it possible to determine qualitative dependences U(r), U(r L ) etc.

RESULTS OF THE NUMERICAL MODELING
To perform a more comprehensive mechanical and mathematical modeling, consider following classes of function f(r) relative to (r -r 0 ) value.

Class one of f(r) functions
Define f(r) function in exponential form: where: β 0 -a constant being determined with the help of the condition f(r L ) = 1, ( ) Thus, formula (29) is: where: n -a whole number.
It follows from formulas ( 27) and (31) that: where: ( ) ( ) (34) Equation ( 32) is the equation to identify r L .It is convenient to solve it by means of graphoanalytical method that is a method of g 1 (r) = g 2 (r) functions construction.
Table 1 shows results of the calculations.

Class two of f(r) functions
Assume that f(r) function is of rational type as a ratio of polynomials: In accordance with f(r L ) = 1, function ( 35) is: With the help of formula (36) rewrite equality (27) as follows: where: Like in the previous case, equation (37) with symbols (38) is solved by means of graphoanalytical method.Table 2 explains results of the calculations.

Class three of f(r) functions
Define f(r) in the class of hyperbolic functions: It is possible to determine β 0 while applying f(r) value within r L point: f(r L ) = 1.Then the formula is a consequence of ( 39): r L max parameter may be selected additionally (e.g.2r L , 3r L …r L ).
Value r L is being determined with the help of the equation: Numerical experiment for ( 41) and ( 42) has been performed when r L max = 2r 0 , k = 1.
Table 3 demonstrates results concerning calculations of parameters r L , U, and Δs/s.
Analysis of Tables 1, 2, and 3 obtained with the help of numerical experiment shows that more detailed studies are required when ϕ 1 = 30° as well as in the context of the increase in the range of pressure values up to 55 t/m 2 .Table 4 shows calculation results for all distribution functions f(r) under study (cases 1, 2, and 3). Figure 2 demonstrates generally the dependence of U, U/r 0 , and Δs/s parameters of the value of assumed resistance Relying upon the allowable values of rock displacements corresponding to a value of pliability of supports as well as to residual (required) area of the mine working cross section, such interval of the support resistance as 150 -250 kN/m 2 (being 60 -120 kN/m 2 in the context of standard arched supports) is technically the most efficient and achievable one.Depending upon the mining deepening (Н > 1000 m), the efficient interval Р may expand up to 350 -400 kN/m 2 ; further expansion is unreasonable.
Analysis the results of the model concerning the interaction between force parameter of a support and displacements of neighbouring rock mass involves the ne-cessity to mention sufficient convergence of the obtained theoretical results and the results of full-scale experiments providing an opportunity to expect the development of a new reference document specifying the problems of practical implementation of both available supports and the development of new ones corresponding to complicated geomechanical conditions.
Mining deepening predetermined the necessity in the innovative scientific approach developed by us.The approach is based upon the significance of a performance parameter of a support (Q) as well as characteristics forming it -working resistance (Р r ) and structural pliability (∆) owing to common physical content of the process as an activity providing the formation of "supportrock mass" system: Q = P r ×∆.In this context, a parameter of limiting bearing capability of a support (P s ) is the determinant of the possibility to increase working support (P r ) as well as the efficiency of the design (Q).

CONCLUSIONS
1. Irrespective of approaches of mathematical modeling connected with the selection of distribution function f(r), sufficient convergence of numerical results of changes in r L and U values in the context of variability of a support resistance has been obtained; the fact is illustrative of the adequacy of actually covered process concerning the "support -rock mass" system interaction.
2. Dependence of changes in r L and U values is the most suitable for both experimental and practical data if n = 2, ϕ 1 = 30°.
3. A zone of plastic deformation is characterized by the fact that its dimension (r L -r 0 ) in all modeling cases despite the properties of material, remains to be within a meter or two being important physically and useful practically.
4. The stated approach as well as the statement of the modeling problem made it possible to obtain general solution in the context of different types of strength degradation function giving more accurate results to approach the practice.
5. The obtained results help formulate reasonably the initial requirements for both power and kinematic parameters of supports belonging to a new technical level.Support becomes operationally adequate when its resistance is within 150 -250 kN/m 2 .Depending upon the mining deepening (Н > 1000 m), the efficient interval Р may expand up to 350 -400 kN/m 2 ; further expansion is unreasonable.
6. Innovative approach based upon the use of a support performance parameter as well as forming characteristics (i.e.working resistance and structural pliability) has been proposed.The approach makes it possible to design new structures for supports in mine workings.