Factor of safety analysis for mine pillar considering the influence of the intermediate principal stress component

Abstract

Failure of mine pillars, especially in deep underground mines, significantly threatens the safety of miners and equipment. Previous studies on mine pillar stability design have used classical constitutive models that ignore the intermediate principal stress component when determining the factor of safety. In this study, we develop and implement a three-dimensional modified Hoek–Brown (HB) constitutive model that incorporates the intermediate principal stress component into the numerical simulation tool FLAC^3D. Furthermore, we propose and apply a strength-reduction technique to determine a more accurate factor of safety for mine pillars. This novel approach provides a more comprehensive and realistic method for geomechanical analysis and pillar design, enhancing our understanding of pillar stability. Through numerical analysis, we illustrate the impact of the intermediate principal stress component on mine pillar plasticity. The factor of safety is calculated via the strength reduction method, revealing a substantial improvement from 1.7 with the classical HB model to 2.0 with the 3D HB model. Including the intermediate principal stress component reduces the evolution of plasticity in the mine pillar. For instance, the volume of plastic zones diminishes, and the factor of safety increases as the width-to-height ratio increases. Exemplary simulations show that ignoring the effect of the intermediate principal stress component, including underestimating safety levels, designing suboptimal pillar design, and misinterpreting in situ observations and measurements, can lead to severe consequences.

Keywords

Intermediate principal stress component / Numerical analysis / Factor of safety / Mine pillar / Hoek–Brown model

References

[1] R.K. Zipf Jr., C. Mark, Design methods to control violent pillar failure in room-and-pillar mines, in: L. Ozdemir, K. Hanna, K.Y. Haramy (Eds.), Proceedings of the 15th International Conference on Ground Control in Mining, Golden, CO, United States, 1996, p. 783.

[2] O. Vardar, F. Tahmasebinia, C. Zhang, I. Canbulat, S. Saydam, A review of uncontrolled pillar failures, Procedia Eng. 191 (2017) 631-637.

[3] J.N. van der Merwe, Beyond Coalbrook: what did we really learn? J. S. Afr. Inst. Min. Metall. 106 (12) (2006) 857-868.

[4] C.D. Martin, W.G. Maybee, The strength of hard-rock Pillars, Int. J. Rock Mech. Min. Sci. 37 (8) (2000) 1239-1246.

[5] Z.L. Zhou, H.Z. Zang, W.Z. Cao, X.M. Du, L. Chen, C.T. Ke, Risk assessment for the cascading failure of underground pillar sections considering interaction between pillars, Int. J. Rock Mech. Min. Sci. 124 (2019) 104142.

[6] G.S. Esterhuizen, P.L. Tyrna, M.M. Murphy, A case study of the collapse of slender pillars affected by through-going discontinuities at a limestone mine in Pennsylvania, Rock Mech. Rock Eng. 52 (12) (2019) 4941-4952.

[7] G.S. Esterhuizen, A.T. Iannacchione, J.L. Ellenberger, D.R. Dolinar, Pillar stability issues based on a survey of pillar performance in underground limestone mines, in: Proceedings of the 25th International Conference on Ground Control in Mining, Morgantown, West Virginia, USA, 2006, pp. 354-361.

[8] C. Zhang, Y.X. Zhao, P.H. Han, Q.S. Bai, Coal pillar failure analysis and instability evaluation methods: a short review and prospect, Eng. Fail. Anal. 138 (2022) 106344.

[9] D.P. Roberts, J.N. van der Merwe, I. Canbulat, E.J. Sellers, S. Coetzer, Development of a method to estimate coal pillar loading, safety in mines research advisory committee, 2002, pp. 1-90.

[10] B.A. Poulsen, Coal pillar load calculation by pressure arch theory and near field extraction ratio, Int. J. Rock Mech. Min. Sci. 47 (7) (2010) 1158-1165.

[11] J. Oke, K. Kalenchuk, Selecting the most applicable hard rock pillar design method, in: Proceedings of the 51st U.S. Rock Mechanics/Geomechanics Symposium, San Francisco, California, USA, 2017, pp. ARMA-2017-0010.

[12] Y. Yu, S.E. Chen, K.Z. Deng, H.D. Fan, Long-term stability evaluation and pillar design criterion for room-and-pillar mines, Energies 10 (10) (2017) 1644.

[13] C.Q. Li, J. Zhou, D.J. Armaghani, X.B. Li, Stability analysis of underground mine hard rock pillars via combination of finite difference methods, neural networks, and Monte Carlo simulation techniques, Undergr. Space 6 (4) (2021) 379-395.

[14] M.E. Duncan Fama, R. Trueman, M.S. Craig, Two- and three-dimensional elasto-plastic analysis for coal pillar design and its application to highwall mining, Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 32 (3) (1995) 215-225.

[15] S. Dehghan, K. Shahriar, P. Maarefvand, K. Goshtasbi, 3-D numerical modelling of Domino failure of hard rock pillars in Fetr6 Chromite Mine, Iran, and comparison with empirical methods, J. Cent. South Univ. 20 (2) (2013) 541-549.

[16] W. Sun, Q. Zhang, Y.Z. Luan, X.P. Zhang, A study of surface subsidence and coal pillar safety for strip mining in a deep mine, Environ. Earth Sci. 77 (17) (2018) 627.

[17] G.S. Esterhuizen, Extending empirical evidence through numerical modelling in rock engineering design, J. S. Afr. Inst. Min. Metall. 114 (10) (2014) 755-764.

[18] H.S. Mun, H.S. Yang, B.H. Kim, W.B. Kim, J.G. Kim, J.G. Kim, S.J. Kim, Numerical investigation on pillar stability of underground cavern associated with shape effect, in: Proceedings of the ISRM Regional Symposium-7th Asian Rock Mechanics Symposium, Seoul, Korea, 2012, pp. ISRM-ARMS7-2012-152.

[19] T. Chen, H.S. Mitri, Strategic sill pillar design for reduced hanging wall overbreak in longhole mining, Int. J. Min. Sci. Technol. 31 (5) (2021) 975-982.

[20] J. Wang, D.B. Apel, Y.Y. Pu, R. Hall, C. Wei, M. Sepehri, Numerical modeling for rock bursts: a state-of-the-art review, J. Rock Mech. Geotech. Eng. 13 (2) (2021) 457-478.

[21] F. Schmidt, J. Menéndez, H. Konietzky, Z.M. Jiang, J.M. Fernández-Oro, L. Álvarez, A. Bernardo-Sánchez, Technical feasibility of lined mining tunnels in closed coal mines as underground reservoirs of compressed air energy storage systems, J. Energy Storage 78 (2024) 110055.

[22] R. Wu, J.H. Xu, C. Li, Z.L. Wang, S. Qin, Stress distribution of mine roof with the boundary element method, Eng. Anal. Bound. Elem. 50 (2015) 39-46.

[23] A. Sarathchandran, Three Dimensional Numerical Modelling of Coal Mine Roadways under High Horizontal Stress Fields (Dissertation), University of Exeter, Exeter, UK, 2014.

[24] A. Sainoki, H.S. Mitri, Numerical investigation into pillar failure induced by time-dependent skin degradation, Int. J. Min. Sci. Technol. 27 (4) (2017) 591-597.

[25] H. Konietzky, X. Li, W. Chen, Lifetime prediction of rocks, in: B. Shen, O. Stephansson, M. Rinne (Eds.), Modelling Rock Fracturing Processes, Springer, Cham, 2020, pp. 475-529.

[26] F. Laouafa, M. Ghoreychi, Analysis of failure in a salt room and pillar mine, in: H.S. Mitri, S. Shnorhokian, M.K. Kumral, A. Sasmito, A. Sainoki (Eds.), Proceedings of the 3rd International Symposium on Mine Safety Science and Engineering, Montreal, Canada, 2016, pp. 108-113.

[27] S. Chakraborti, H. Konietzky, K. Walter, A comparative study of different approaches for factor of safety calculations by shear strength reduction technique for non-linear Hoek-Brown failure criterion, Geotech. Geol. Eng. 30 (4) (2012) 925-934.

[28] W. Minkley, W. Menzel, H. Koxnietzky, L. te Kamp, A visco-elasto-plastic softening model and its application for solving static and dynamic stability problems in potash mining, in: D. Billaux, C. Detournay, R. Hart, X. Rachez (Eds.), FLAC and Numerical Modeling in Geomechanics, first ed., CRC Press, London, 2001, pp. 21-27.

[29] K.Z. Xia, C.X. Chen, X.M. Liu, Y. Wang, X.T. Liu, J.H. Yuan, Estimating shear strength of high-level pillars supported with cemented backfilling using the Hoek-Brown strength criterion, J. Rock Mech. Geotech. Eng. 16 (2) (2024) 454-469.

[30] G. Walton, M. Diederichs, A. Punkkinen, J. Whitmore, Back analysis of a pillar monitoring experiment at 2.4 km depth in the Sudbury Basin, Canada, Int. J. Rock Mech. Min. Sci. 85 (2016) 33-51.

[31] E. Hoek, E.T. Brown, The Hoek-Brown failure criterion and GSI-2018 edition, J. Rock Mech. Geotech. Eng. 11 (3) (2019) 445-463.

[32] J.F. Labuz, A. Zang, Mohr-Coulomb failure criterion, Rock Mech. Rock Eng. 45 (6) (2012) 975-979.

[33] J. Handin, H.C. Heard, J.N. Magouirk, Effects of the intermediate principal stress on the failure of limestone, dolomite, and glass at different temperatures and strain rates, J. Geophys. Res. 72 (2) (1967) 611-640.

[34] A. Hackston, E. Rutter, The Mohr-Coulomb criterion for intact rock strength and friction-a re-evaluation and consideration of failure under polyaxial stresses, Solid Earth Discuss. 7 (4) (2015) 3843-3883.

[35] K. Mogi, Pressure dependence of rock strength and transition from brittle fracture to ductile flow, Bull. Earthq. Res. Inst. 44 (1966) 215-232.

[36] K. Mogi, Effect of the triaxial stress system on the failure of dolomite and limestone, Tectonophysics 11 (2) (1971) 111-127.

[37] K. Mogi, Effect of the intermediate principal stress on rock failure, J. Geophys. Res. 72 (20) (1967) 5117-5131.

[38] K. Mogi, Dilatancy of rocks under general triaxial stress states with special reference to earthquake precursors, J. Phys. Earth 25 (Supplement) (1978) S203-S217.

[39] M.Q. You, True-triaxial strength criteria for rock, Int. J. Rock Mech. Min. Sci. 46 (1) (2009) 115-127.

[40] H. Jiang, Y.L. Xie, A new three-dimensional Hoek-Brown strength criterion, Acta Mech. Sin. 28 (2) (2012) 393-406.

[41] M.H. Mehranpour, P.H.S.W. Kulatilake, Comparison of six major intact rock failure criteria using a particle flow approach under true-triaxial stress condition, Geomech. Geophys. Geo Energy Geo Resour. 2 (4) (2016) 203-229.

[42] L.B. Colmenares, M.D. Zoback, A statistical evaluation of intact rock failure criteria constrained by polyaxial test data for five different rocks, Int. J. Rock Mech. Min. Sci. 39 (6) (2002) 695-729.

[43] N.B. Setiawan, R.W. Zimmerman, Wellbore breakout prediction in transversely isotropic rocks using true-triaxial failure criteria, Int. J. Rock Mech. Min. Sci. 112 (2018) 313-322.

[44] L. Zhang, A generalized three-dimensional Hoek-Brown strength criterion, Rock Mech. Rock Eng. 41 (6) (2008) 893-915.

[45] A.K. Schwartzkopff, A. Sainoki, T. Bruning, M. Karakus, A conceptual three-dimensional frictional model to predict the effect of the intermediate principal stress based on the Mohr-Coulomb and Hoek-Brown failure criteria, Int. J. Rock Mech. Min. Sci. 172 (2023) 105605.

[46] Q.S. Bai, M. Friedel, H. Konietzky, Experimental investigation of stress unloading effects on rock damage and confining pressure-dependent crack initiation stress of porous sandstone under true triaxial stress environments, Rock Mech. Bull. 3 (2) (2024) 100111.

[47] X.T. Feng, X.W. Zhang, C.X. Yang, R. Kong, X.Y. Liu, S. Peng, Evaluation and reduction of the end friction effect in true triaxial tests on hard rocks, Int. J. Rock Mech. Min. Sci. 97 (2017) 144-148.

[48] X.T. Feng, J. Zhao, X.W. Zhang, R. Kong, A novel true triaxial apparatus for studying the time-dependent behaviour of hard rocks under high stress, Rock Mech. Rock Eng. 51 (9) (2018) 2653-2667.

[49] X.T. Feng, X.W. Zhang, R. Kong, G. Wang, A novel Mogi type true triaxial testing apparatus and its use to obtain complete stress-strain curves of hard rocks, Rock Mech. Rock Eng. 49 (5) (2016) 1649-1662.

[50] X.T. Feng, Y.H. Gao, X.W. Zhang, Z.F. Wang, Y. Zhang, Q. Han, Evolution of the mechanical and strength parameters of hard rocks in the true triaxial cyclic loading and unloading tests, Int. J. Rock Mech. Min. Sci. 131 (2020) 104349.

[51] X.D. Pan, J.A. Hudson, A simplified three dimensional Hoek-Brown yield criterion, in: Proceedings of the ISRM International Symposium, Madrid, Spain, 1988, pp. 95-103.

[52] S.D. Priest, Determination of shear strength and three-dimensional yield strength for the Hoek-Brown criterion, Rock Mech. Rock Eng. 38 (4) (2005) 299-327.

[53] W.Q. Cai, H.H. Zhu, W.H. Liang, L.Y. Zhang, W. Wu, A new version of the generalized Zhang-Zhu strength criterion and a discussion on its smoothness and convexity, Rock Mech. Rock Eng. 54 (8) (2021) 4265-4281.

[54] L. Zhang, A generalized three-dimensional Hoek-Brown strength criterion, Rock Mech. Rock Eng. 41 (6) (2008) 893-915.

[55] Q. Zhang, H.H. Zhu, L.Y. Zhang, Modification of a generalized three-dimensional Hoek-Brown strength criterion, Int. J. Rock Mech. Min. Sci. 59 (2013) 80-96.

[56] C.D. Chang, B. Haimson, A failure criterion for rocks based on true triaxial testing, Rock Mech. Rock Eng. 45 (6) (2012) 1007-1010.

[57] H.Z. Li, T. Guo, Y.L. Nan, B. Han, A simplified three-dimensional extension of Hoek-Brown strength criterion, J. Rock Mech. Geotech. Eng. 13 (3) (2021) 568-578.

[58] L.R. Alejano, A. Bobet, Drucker-Prager criterion, Rock Mech. Rock Eng. 45 (6) (2012) 995-999.

[59] H.H. Chen, H.H. Zhu, L.Y. Zhang, Rock slope stability analysis incorporating the effects of intermediate principal stress, Rock Mech. Rock Eng. 56 (6) (2023) 4271-4289.

[60] E. Hoek, E.T. Brown, Empirical strength criterion for rock masses, J. Geotech. Eng. Div. 106 (9) (1980) 1013-1035.

[61] A. Sainoki, D. Maina, A.K. Schwartzkopff, Y. Obara, M. Karakus, Impact of the intermediate stress component in a plastic potential function on rock mass stability around a sequentially excavated large underground cavity, Int. J. Rock Mech. Min. Sci. 127 (2020) 104223.

[62] A. Elyasi, K. Goshtasbi, Using different rock failure criteria in wellbore stability analysis, Geomech. Energy Environ. 2 (2015) 15-21.

[63] E. Hoek, P. Marinos, A brief history of the development of the Hoek-Brown failure criterion, Soils Rocks 30 (2) (2007) 85-95.

[64] V. Marinos, P. Marinos, E. Hoek, Geological Strength Index (GSI). A characterization tool for assessing engineering properties for rock masses, in: M. Romana, Á. Perucho, C. Olalla, (Eds.), Underground Works under Special Conditions-Proceedings of the Workshop (W1), Madrid, Spain, 2007, pp. 13-21.

[65] B. Haimson, C. Chang, A new true triaxial cell for testing mechanical properties of rock, and its use to determine rock strength and deformability of Westerly granite, Int. J. Rock Mech. Min. Sci. 37 (1-2) (2000) 285-296.

[66] L.H. Hu, L.Y. Yu, M.H. Ju, X.Z. Li, C.A. Tang, Effects of intermediate stress on deep rock strain bursts under true triaxial stresses, J. Rock Mech. Geotech. Eng. 15 (3) (2023) 659-682.

[67] G. Walton, A new perspective on the brittle-ductile transition of rocks, Rock Mech. Rock Eng. 54 (12) (2021) 5993-6006.

[68] K. Mogi, Fracture and flow of rocks under high triaxial compression, J. Geophys. Res. 76 (5) (1971) 1255-1269.

[69] M.D. Ingraham, K.A. Issen, D.J. Holcomb, Response of Castlegate sandstone to true triaxial states of stress, J. Geophys. Res. Solid Earth 118 (2) (2013) 536-552.

[70] X. Olivella, C. Agelet de Saracíbar Bosch, Continuum Mechanics for Engineers. Theory and Problems, second ed., 2017.

[71] I.F. Manual, Fast Lagrangian Analysis Continua-Version 7.0 User Manual, Itasca Consulting Group, Minnesota, USA, 2015.

[72] R.E. Hammah, T.E. Yacoub, B.C. Corkum, J.H. Curran, The shear strength reduction method for the generalized Hoek-Brown criterion, in: Proceedings of the 40th U.S. Symposium on Rock Mechanics (USRMS): Rock Mechanics for Energy, Mineral and Infrastructure Development in the Northern Regions, Anchorage, USA, 2005.

[73] X. Chen, J.L. Ren, D.Y. Wang, Y.N. Lyu, H. Zhang, A generalized strength reduction concept and its applications to geotechnical stability analysis, Geotech. Geol. Eng. 37 (4) (2019) 2409-2424.

[74] Q.H. Jiang, Z.F. Qi, W. Wei, C.B. Zhou, Stability assessment of a high rock slope by strength reduction finite element method, Bull. Eng. Geol. Environ. 74 (4) (2015) 1153-1162.

[75] G.S. Esterhuizen, T.S. Bajpayee, M.M. Murphy, J.L. Ellenberger, Evaluation of the strength reduction method for US coal mine entry stability analysis, in: Y. Potvin, B. Brady (Eds.), Ground Support 2013: Proceedings of the Seventh International Symposium on Ground Support in Mining and Underground Construction, Australian Centre for Geomechanics, Perth, 2013, pp. 373-385.

[76] H. Zheng, L.G. Tham, D.F. Liu, On two definitions of the factor of safety commonly used in the finite element slope stability analysis, Comput. Geotech. 33 (3) (2006) 188-195.

[77] Y.M. Cheng, T. Lansivaara, W.B. Wei, Two-dimensional slope stability analysis by limit equilibrium and strength reduction methods, Comput. Geotech. 34 (3) (2007) 137-150.

[78] F.M. Liu, Stability analysis of geotechnical slope based on strength reduction method, Geotech. Geol. Eng. 38 (4) (2020) 3653-3665.

[79] Y.L. Tu, X.R. Liu, Z.L. Zhong, Y.Y. Li, New criteria for defining slope failure using the strength reduction method, Eng. Geol. 212 (2016) 63-71.

[80] D.V. Griffiths, R.M. Marquez, Three-dimensional slope stability analysis by elasto-plastic finite elements, Géotechnique 57 (6) (2007) 537-546.

[81] W.X. Fu, Y. Liao, Non-linear shear strength reduction technique in slope stability calculation, Comput. Geotech. 37 (3) (2010) 288-298.

[82] J.Y. Shen, S.D. Priest, M. Karakus, Determination of Mohr-Coulomb shear strength parameters from generalized Hoek-Brown criterion for slope stability analysis, Rock Mech. Rock Eng. 45 (1) (2012) 123-129.

[83] G. You, M. Al Mandalawi, A. Soliman, K. Dowling, P. Dahlhaus, Finite element analysis of rock slope stability using shear strength reduction method, in: W. Frikha, S. Varaksin, A. Viana da Fonseca (Eds.), Soil Testing, Soil Stability and Ground Improvement, GeoMEast 2017, Sustainable Civil Infrastructures, Springer, Cham, 2018, pp. 227-235.

[84] X. Yang, G. Yang, T. Yu, Comparison of strength reduction method for slope stability analysis based on ABAQUS FEM and FLAC^3D FDM , Appl. Mech. Mater. 170-173 (2012) 918-922.