Mathematical modelling of the safety pillar stability in layered rocks during underground mining
D.Zh. Akmatov1,2, R.V. Shevchuk1,2,3, A.A. Zarubin1,2, D.D. Veis1,2, M.V. Podloznaya2
1 Geophysical Center of the Russian Academy of Sciences, Moscow, Russian Federation
2 National University of Science and Technology "MISIS", Moscow, Russian Federation
3 Schmidt Institute of Physics of the Earth, Russian Academy of Sciences, Moscow, Russian Federation
Russian Mining Industry №3/ 2026 p. 72-78
Abstract: In order to determine the stable shape of a pillar in layered rocks, a set of mathematical models was developed, with each model corresponding to an individual layer of the rock mass. The stable shape of the entire pillar is obtained by combining the stable shapes of its individual layers. In particular, the pillar is assumed to have a layered structure and a circular crosssection subjected to external stresses. The magnitude of the applied stress is determined, among other factors, by the lengths of the fractures in the upper layer of the rock mass. The pillar stability is achieved by ensuring a uniform macroscopic stress distribution over the cross-section, i.e., by eliminating stress jumps between the layers. This condition is equivalent to requiring that the stresses per unit area of the cross-section are equal throughout the entire height of the pillar. The developed models are reduced to the boundary-value problems for differential equations describing the shapes of the curves corresponding to the stable state of each layer. Solving these equations helps to shape functions for the individual layers, which together provide a geometric description of the stable lateral surface of the entire pillar. Numerical modelling of the stress–strain state of the layered rock mass for the calculated pillar geometry confirmed the efficiency of the proposed approach, showing reduced stress concentrations in the central part and a redistribution of the load toward stronger layers.
Keywords: mathematical modeling, pillar, layered rocks, pillar stability, macroscopic stress field, rock pressure, effective elastic moduli
Acknowledgments: This work was performed as part of the State Assignment for the Geophysical Center of the RAS, adopted by the Ministry of Science and Higher Education of the Russian Federation.
For citation: Akmatov D.Zh., Shevchuk R.V., Zarubin A.A., Veis D.D., Podloznaya M.V. Mathematical modelling of the safety pillar stability in layered rocks during underground mining. Russian Mining Industry. 2026;(3):72–78. https://doi. org/10.30686/1609-9192-2026-3-72-78
Article info
Received: 11.02.2026
Revised: 24.03.2026
Accepted: 31.03.2026
Information about the authors
Dastan Zh. Akmatov – Cand. Sci. (Eng.), Senior Researcher, Laboratory of Geodynamics, Geophysical Center of the Russian Academy of Sciences, Moscow, Russian Federation; Senior Lecturer, Department of Geology and Surveying, College of Mining, National University of Science and Technology "MISIS", Moscow, Russian Federation; e-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.
Roman V. Shevchuk – Cand. Sci. (Eng.), Senior Researcher, Laboratory of Geodynamics, Geophysical Center of the Russian Academy of Sciences, Moscow, Russian Federation; Senior Lecturer, Department of Geology and Surveying, College of Mining, National University of Science and Technology "MISIS", Moscow, Russian Federation; Researcher, Geoinformatics Laboratory, Schmidt Institute of Physics of the Earth, Russian Academy of Sciences, Moscow, Russian Federation; e-mail: Shevchuk.002@ mail.ru
Artem A. Zarubin – Engineer, Laboratory of Geodynamics, Geophysical Center of the Russian Academy of Sciences, Moscow, Russian Federation; Student, National University of Science and Technology "MISIS", Moscow, Russian Federation; e-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.
Daniel D. Veis – Engineer, Laboratory of Geoinformatics and Arctic Big Data, Geophysical Center of the Russian Academy of Sciences, Moscow, Russian Federation; Student, National University of Science and Technology "MISIS", Moscow, Russian Federation; e-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.
Margarita V. Podloznaya – Student, National University of Science and Technology "MISIS", Moscow, Russian Federation; e-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.
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