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Original Paper

UDC 622.272:658.012.122:51.001.57© R.K. Khalkechev, Yu.M. Levkin, K.V. Khalkechev, S.U. Kuzmenko, 2024

ISSN 0041-5790 (Print) • ISSN 2412-8333 (Online) • Ugol’ – Russian Coal Journal, 2024, № 1, pp. 61-63

DOI: http://dx.doi.org/10.18796/0041-5790-2024-1-61-63




Khalkechev R.K.1, Levkin Yu.M.2, Khalkechev K.V.1, Kuzmenko S.U.3

1National University of Science and Technology “MISIS” (NUST “MISIS”), Moscow, 119049, Russian Federation

2Moscow Polytechnic University, Moscow, 105064, Russian Federation

3K.G. Razumovsky Moscow State University of Technologies and Management (the First Cossack University), Moscow, 109004, Russian Federation

Authors Information

Khalkechev R.K., Doctor of Engineering Sciences, Professor at Subdepartment Infocommunication technologies, e-mail: syrus@list.ru

Levkin Yu.M., Doctor of Engineering Sciences, Professor, Member of the Russian Union of Surveyors, e-mail: lev5353@bk.ru

Khalkechev K.V.,Doctor of Physico-Mathematical Science, Doctor of Engineering Sciences, Professor at subdepartment Geology and mine surveying, e-mail: h_kemal@mail.ru

Kuzmenko S.U., PhD (Engineering), Teacher of University College of Information Technology, e-mail: svetik-semicvetik3@yandex.ru


In the presented article, a mathematical model has been developed that makes it possible to determine the shape of the stable pillars. On the one hand these pillars provide the stability of the mine workings, and on the other, the absence of unjustified losses of mineral resources. Unlike analogues, this model takes into account the presence of crystalline and amorphous textural components in the rocks that make up the pillar. Hereupon, fuzzy tensor analysis is used to determine the value of the external stress field, not classical tensor calculus. It allows to take into account the uncertainty degree of the values characterizing the deformation properties of inhomogeneities. As a result, using the fuzzy external stress field tensor, a mathematical expression was obtained that describes the shape of the pillar. This shape enables to realize a uniform stress distribution throughout the entire volume of rocks. The shape of the pillar is determined by the modal value of the fuzzy component of the external stress field.


Mathematical model, Stability, Coal field, Pillar, External field, Fuzzy tensor, Stress field, Analog method.


1. Qiang S., Jialiang G., Feng Y. & Ruhong B. Cooperative mining technology and strata control of close coal seams and overlying coal pillars. Alexandria Engineering Journal, 2023, (73), pp. 473–485. DOI: 10.1016/j.aej.2023.04.071.

2. R?genes E., Gomes A.D.S., Farias M.M.D. & Rasmussen L.L. Pseudo-discontinuum model to simulate hard-rock mine pillars. Underground Space, 2023, (11), pp. 81–95. DOI: 10.1016/j.undsp.2022.12.002.

3. Qu X., Chen Y. & Yin D. Experimental study on progressive failure characteristics of strip coal pillar models under different roof and floor conditions. Case Studies in Construction Materials, 2023, (18), Article e02147. DOI: 10.1016/j.cscm.2023.e02147.

4. Chen L., Zhang D.S., Yao N., Wang L., Fan G.W., Wang X.F. & Zhang W. Coupling influence of inclination angle and moisture content on mechanical properties and microcrack fracture of coal specimens. Lithosphere, 2021 (Special 7), 2022, Article 6226445. DOI: 10.2113/2022/6226445.

5. Kuzin E.A. & Khalkechev K.V. Mathematical model for determining the shape of a stable pillar of a polycrystalline structure in carbon-bearing rocks. Ugol’, 2020, (2), pp. 22-25. (In Russ.). DOI: 10.18796/0041- 5790-2020-2-22-25.

6. Khalkechev R.K. Multifractal modeling theory of rock mass deformation and destruction as the basis for automation of drilling and blasting technologies in coal open-pit mine. Ugol’, 2019, (11), pp. 32-35. (In Russ.). DOI: 10.18796/0041- 5790-2019-11-32-35.

7. Khalkecheva L.K. & Khalkechev R.K. Automated monitoring system of transport berms condition for landslide danger in the form of subsidence. Ugol’, 2022, (4), pp. 50-52. (In Russ.). DOI: 10.18796/0041-5790-2022-4-50-52.

8. Khalkechev R.K. & Khalkechev K.V. Mathematical modeling of non-uniform elastic stress field of a rock mass with crystalline block structure. Gornyj zhurnal, 2016, (3), pp. 200-205. (In Russ.). DOI: 10.17580/gzh.2016.03.05.

9. Chunuev I.K., Levkin Y.M. & Bolotbekov Z. Determination of bench, dump and road sliding wedge technological parameters. Mining Science and Technology (Russian Federation), 2021, (6), pp. 31–41. DOI: 10.17073/2500-0632-2021-1-31-41.

10. Khalkechev R.K., Khalkechev K.V. & Levkin Y.M. Mathematical model of the stress field in the pillars with due account taken of the main crack in coal fields. Ugol’, 2023, (7), pp. 56-58. (In Russ.). DOI: 10.18796/0041-5790-2023-7-56-58.

11. Khalkechev R.K., Levkin Yu.M. & Khalkechev K.V. Mathematical model development of the stress field in the pillars stratified texture in coal deposits. Ugol’, 2023, (8), pp. 84-86. (In Russ.). DOI: 10.18796/0041-5790-2023-8-84-86.

12. Piegat A. Fuzzy modeling and control. Moscow, BINOM. Laboratoria Znaniy Publ., 2013, 798 p. (In Russ.).

For citation

Khalkechev R.K., Levkin Y.M., Khalkechev K.V. & Kuzmenko S.U. Fuzzy model of the shape determining of a stable pillar in coal-bearing rocks. Ugol’, 2024, (1), pp. 61-63. (In Russ.). DOI: 10.18796/0041-5790-2024-1-61-63.

Paper info

Received October 2, 2023

Reviewed November 10, 2023

Accepted December 7, 2023


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