Influence of Spatial Variability on the Uniaxial Compressive Responses of Rock Pillar Based on 3D Random Field
Publication: ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering
Volume 7, Issue 3
Abstract
Knowledge of the uniaxial compressive responses of rock mass is crucial for evaluating the stability of rock pillar in underground engineering. The inherent spatial variabilities of rock mass properties are generally ignored in the traditional stability evaluation of rock pillars. In this study, probabilistic analysis of rock pillar was carried out to investigate the compressive responses of spatially variable rock pillars with three-dimensional (3D) random fields. Three-dimensional random fields of rock mass properties were generated using the Cholesky decomposition method, which was implemented into the using the Fish code. The failure mechanism of spatially variable rock pillar and influence of spatial variability of rock mass properties on uniaxial compressive strength and Young’s modulus were investigated on the basis of the Hoek–Brown criterion with Latin hypercube sampling (LHS). Results indicated that the spatial variability of rock mass properties significantly influences the uniaxial compressive strength and Young’s modulus of rock pillar. The expansion of the plastic region has obvious directivity, which is closely related to the direction of the random field. The vertical scale of fluctuation has a greater effect on the mean value of the uniaxial compressive strength and Young’s modulus of rock pillar than the horizontal scale of fluctuation.
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Data Availability Statement
Some or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request. The items are listed below:
1.
The calculation program to generate 3D random fields.
2.
The files to simulate the rock pillar.
3.
The data calculated by 3D random fields in the study.
Acknowledgments
This work was supported by the Chongqing Construction Science and Technology Plan Project (2019–0045), the program of China Scholarships Council (No. 201906050026), as well as the Science and Technology Research Program of Chongqing Municipal Education Commission (Grant No. KJZD–K201900102).
References
Bieniawski, Z. T. 1968. “In situ strength and deformation characteristics of coal.” Eng. Geol. 2 (5): 325–340. https://doi.org/10.1016/0013-7952(68)90011-2.
Boyd, D. L., G. Walton, and W. Trainor–Guitton. 2019. “Quantifying spatial uncertainty in rock through geostatistical integration of borehole data and a geologist’s cross–section.” Eng. Geol. 260 (Oct): 105246. https://doi.org/10.1016/j.enggeo.2019.105246.
Brady, B. H. G., and E. Brown. 2004. Rock mechanics for underground mining. New York: Kluwer Academic Publishers.
Chen, F., L. Wang, and W. Zhang. 2019. “Reliability assessment on stability of tunnelling perpendicularly beneath an existing tunnel considering spatial variabilities of rock mass properties.” Tunnelling Underground Space Technol. 88 (Jun): 276–289. https://doi.org/10.1016/j.tust.2019.03.013.
Chen, F., R. Zhang, Y. Wang, H. Liu, B. Thomas, and W. Zhang. 2020. “Probabilistic stability analyses of slope reinforced with piles in spatially variable soils.” Int. J. Approximate Reasoning 122 (Jul): 66–79. https://doi.org/10.1016/j.ijar.2020.04.006.
Das, A. J., P. K. Mandal, P. S. Paul, and R. K. Sinha. 2019. “Generalised analytical models for the strength of the inclined as well as the flat coal pillars using rock mass failure criterion.” Rock Mech. Rock Eng. 52 (10): 3921–3946. https://doi.org/10.1007/s00603-019-01788-7.
Deng, J., Z. Q. Yue, L. G. Tham, and H. H. Zhu. 2003. “Pillar design by combining finite element methods, neural networks and reliability: A case study of the Feng Huangshan copper mine, China.” Int. J. Rock Mech. Min. Sci. 40 (4): 585–599. https://doi.org/10.1016/S1365-1609(03)00042-X.
Ellefmo, S. L., and J. Eidsvik. 2009. “Local and spatial joint frequency uncertainty and its application to rock mass characterization.” Rock Mech. Rock Eng. 42 (4): 667. https://doi.org/10.1007/s00603-008-0009-x.
Goh, A. T., W. G. Zhang, and K. S. Wong. 2019. “Deterministic and reliability analysis of basal heave stability for excavation in spatial variable soils.” Comput. Geotech. 108 (Apr): 152–160. https://doi.org/10.1016/j.compgeo.2018.12.015.
Griffiths, D. V., G. A. Fenton, and C. B. Lemons. 2002. “Probabilistic analysis of underground pillar stability.” Int. J. Numer. Anal. Methods Geomech. 26 (8): 775–791. https://doi.org/10.1002/nag.222.
Griffiths, D. V., J. Huang, and G. A. Fenton. 2009. “Influence of spatial variability on slope reliability using 2–D random fields.” J. Geotech. Geoenviron. Eng. 135 (10): 1367–1378. https://doi.org/10.1061/(ASCE)GT.1943-5606.0000099.
Griffiths, D. V., J. Huang, and G. A. Fenton. 2011. “Probabilistic infinite slope analysis.” Comput. Geotech. 38 (4): 577–584. https://doi.org/10.1016/j.compgeo.2011.03.006.
Griffiths, D. V., and R. M. Marquez. 2007. “Three–dimensional slope stability analysis by elasto–plastic finite elements.” Géotechnique 57 (6): 537–546. https://doi.org/10.1680/geot.2007.57.6.537.
Hoek, E., Brown Brown. 1997. “Practical estimates of rock mass strength.” Int. J. Rock Mech. Min. Sci. 34 (8): 1165–1186. https://doi.org/10.1016/S1365-1609(97)80069-X.
Hoek, E., and E. T. Brown. 2018. “The Hoek–Brown failure criterion and GSI–2018 edition.” J. Rock Mech. Geotech. Eng. 11 (3): 445–463. https://doi.org/10.1016/j.jrmge.2018.08.001.
Hoek, E., C. Carranza-Torres, and B. Corkum. 2002. “Hoek–Brown failure criterion—2002 edition.” Proc. NARMS–Tac 1 (1): 267–273. https://doi.org/10.1016/S1365-1609(97)80069-X.
Idris, M. A., D. Saiang, and E. Nordlund. 2015. “Stochastic assessment of pillar stability at Laisvall mine using artificial neural network.” Tunnelling Underground Space Technol. 49 (Jun): 307–319. https://doi.org/10.1016/j.tust.2015.05.003.
Jessu, K. V., A. J. Spearing, and M. Sharifzadeh. 2018. “Laboratory and numerical investigation on strength performance of inclined pillars.” Energies 11 (11): 3229. https://doi.org/10.3390/en11113229.
Ji, J., C. Zhang, Y. Gao, and J. Kodikara. 2018. “Effect of 2D spatial variability on slope reliability: A simplified FORM analysis.” Geosci. Front. 9 (6): 1631–1638. https://doi.org/10.1016/j.gsf.2017.08.004.
Jiang, S. H., D. Q. Li, Z. J. Cao, C. B. Zhou, and K. K. Phoon. 2015. “Efficient system reliability analysis of slope stability in spatially variable soils using Monte Carlo simulation.” J. Geotech. Geoenviron. Eng. 141 (2): 04014096. https://doi.org/10.1061/(ASCE)GT.1943-5606.0001227.
Jiang, S. H., D. Q. Li, L. M. Zhang, and C. B. Zhou. 2014. “Slope reliability analysis considering spatially variable shear strength parameters using a non–intrusive stochastic finite element method.” Eng. Geol. 168 (Jan): 120–128. https://doi.org/10.1016/j.enggeo.2013.11.006.
Li, D., C. C. Li, and X. Li. 2011. “Influence of sample height–to–width ratios on failure mode for rectangular prism samples of hard rock loaded in uniaxial compression.” Rock Mech. Rock Eng. 44 (3): 253–267. https://doi.org/10.1007/s00603-010-0127-0.
Li, D. Q., S. H. Jiang, Z. J. Cao, W. Zhou, C. B. Zhou, and L. M. Zhang. 2015. “A multiple response-surface method for slope reliability analysis considering spatial variability of soil properties.” Eng. Geol. 187 (Mar): 60–72. https://doi.org/10.1016/j.enggeo.2014.12.003.
Li, D. Q., T. Xiao, L. M. Zhang, and Z. J. Cao. 2019. “Stepwise covariance matrix decomposition for efficient simulation of multivariate large–scale three–dimensional random fields.” Appl. Math. Modell. 68 (Apr): 169–181. https://doi.org/10.1016/j.apm.2018.11.011.
Li, W., J. Bai, S. Peng, X. Wang, and Y. Xu. 2015. “Numerical modeling for yield pillar design: A case study.” Rock Mech. Rock Eng. 48 (1): 305–318. https://doi.org/10.1007/s00603-013-0539-8.
Li, Y. J., M. A. Hicks, and P. J. Vardon. 2016. “Uncertainty reduction and sampling efficiency in slope designs using 3D conditional random fields.” Comput. Geotech. 79 (Oct): 159–172. https://doi.org/10.1016/j.compgeo.2016.05.027.
Liu, H., and X. Qi. 2018. “Random field characterization of uniaxial compressive strength and elastic modulus for intact rocks.” Geosci. Front. 9 (6): 1609–1618. https://doi.org/10.1016/j.gsf.2017.11.014.
Luo, Z., S. Atamturktur, Y. Cai, and C. H. Juang. 2012. “Simplified approach for reliability–based design against basal–heave failure in braced excavations considering spatial effect.” J. Geotech. Geoenviron. Eng. 138 (4): 441–450. https://doi.org/10.1061/(ASCE)GT.1943-5606.0000621.
Marinos, V., P. Marinos, and E. Hoek. 2005. “The geological strength index: applications and limitations.” Bull. Eng. Geol. Environ. 64 (1): 55–65. https://doi.org/10.1007/s10064-004-0270-5.
Mayer, J. M., and D. Stead. 2017. “A comparison of traditional, step–path, and geostatistical techniques in the stability analysis of a large open pit.” Rock Mech. Rock Eng. 50 (4): 927–949. https://doi.org/10.1007/s00603-016-1148-0.
Mortazavi, A., F. P. Hassani, and M. Shabani. 2009. “A numerical investigation of rock pillar failure mechanism in underground openings.” Comput. Geotech. 36 (5): 691–697. https://doi.org/10.1016/j.compgeo.2008.11.004.
Najafi, M., S. M. E. Jalali, F. Sereshki, and A. Y. Bafghi. 2016. “Probabilistic analysis of stability of chain pillars in Tabas coal mine in Iran using Monte Carlo simulation.” J. Min. Environ. 7 (1): 25–35. https://doi.org/10.22044/JME.2016.465.
Öztürk, C. A., and E. Nasuf. 2002. “Geostatistical assessment of rock zones for tunneling.” Tunnelling Underground Space Technol. 17 (3): 275–285. https://doi.org/10.1016/S0886-7798(02)00023-8.
Phoon, K. K., and F. H. Kulhawy. 1999. “Characterization of geotechnical variability.” Can. Geotech. J. 36 (4): 612–624. https://doi.org/10.1139/t99-038.
Pinheiro, M., J. Vallejos, T. Miranda, and X. Emery. 2016. “Geostatistical simulation to map the spatial heterogeneity of geomechanical parameters: A case study with rock mass rating.” Eng. Geol. 205 (Apr): 93–103. https://doi.org/10.1016/j.enggeo.2016.03.003.
Qi, X. H., and D. Q. Li. 2018. “Effect of spatial variability of shear strength parameters on critical slip surfaces of slopes.” Eng. Geol. 239 (May): 41–49. https://doi.org/10.1016/j.enggeo.2018.03.007.
Renani, H. R., C. D. Martin, P. Varona, and L. Lorig. 2019. “Stability analysis of slopes with spatially variable strength properties.” Rock Mech. Rock Eng. 52 (10): 3791–3808. https://doi.org/10.1007/s00603-019-01828-2.
Suchomel, R., and D. Mašı. 2010. “Comparison of different probabilistic methods for predicting stability of a slope in spatially variable c–φ soil.” Comput. Geotech. 37 (1–2): 132–140. https://doi.org/10.1016/j.compgeo.2009.08.005.
Vásárhelyi, B. 2009. “A possible method for estimating the Poisson’s rate values of the rock masses.” Acta Geod. Geophys. Hung. 44 (3): 313–322. https://doi.org/10.1556/AGeod.44.2009.3.4.
Wang, B., L. Liu, Y. Li, and Q. Jiang. 2020. “Reliability analysis of slopes considering spatial variability of soil properties based on efficiently identified representative slip surfaces.” J. Rock Mech. Geotech. Eng. 12 (3): 642–655. https://doi.org/10.1016/j.jrmge.2019.12.003.
Wang, J., W. Tan, S. Feng, and R. Zhou. 2000. “Reliability analysis of an open pit coal mine slope.” Int. J. Rock Mech. Min. Sci. 37 (4): 715–721. https://doi.org/10.1016/S1365-1609(00)00005-8.
Wang, L., C. Wu, Y. Li, H. Liu, W. Zhang, and X. Chen. 2019. “Probabilistic risk assessment of unsaturated slope failure considering spatial variability of hydraulic parameters.” KSCE J. Civ. Eng. 23 (12): 5032–5040. https://doi.org/10.1007/s12205-019-0884-6.
Wang, L., C. Wu, L. Tang, W. Zhang, S. Lacasse, H. Liu, and L. Gao. 2020. “Efficient reliability analysis of earth dam slope stability using extreme gradient boosting method.” Acta Geotech. 15 (11): 3135–3150. https://doi.org/10.1007/s11440-020-00962-4.
Zhang, W., and A. T. Goh. 2015. “Numerical study of pillar stresses and interaction effects for twin rock caverns.” Int. J. Numer. Anal. Methods Geomech. 39 (2): 193–206. https://doi.org/10.1002/nag.2306.
Zhang, W., and A. T. Goh. 2017. “Reliability assessment of ultimate limit state of twin caverns.” Geomech. Geoeng. 12 (1): 48–59. https://doi.org/10.1080/17486025.2016.1162331.
Zhang, W., L. Han, X. Gu, L. Wang, F. Chen, and H. Liu. Forthcoming. “Tunneling and deep excavations in spatially variable soil and rock masses: A short review.” In Underground space. https://doi.org/10.1016/j.undsp.2020.03.003.
Zheng, F., Y. Y. Jiao, and N. Sitar. 2018. “Generalized contact model for polyhedra in three-dimensional discontinuous deformation analysis.” Int. J. Numer. Anal. Methods Geomech. 42 (13): 1471–1492. https://doi.org/10.1002/nag.2798.
Zheng, F., Y. F. Leung, J. B. Zhu, and Y. Y. Jiao. 2019. “Modified predictor-corrector solution approach for efficient discontinuous deformation analysis of jointed rock masses.” Int. J. Numer. Anal. Methods Geomech. 43 (2): 599–624. https://doi.org/10.1002/nag.2881.
Zhou, S., T. Rabczuk, and X. Zhuang. 2018a. “Phase field modeling of quasi–static and dynamic crack propagation: COMSOL implementation and case studies.” Adv. Eng. Software 122 (Aug): 31–49. https://doi.org/10.1016/j.advengsoft.2018.03.012.
Zhou, S., X. Zhuang, and T. Rabczuk. 2018b. “A phase–field modeling approach of fracture propagation in poroelastic media.” Eng. Geol. 240 (Jun): 189–203.
Zhou, S., X. Zhuang, and T. Rabczuk. 2019a. “Phase field modeling of brittle compressive–shear fractures in rock–like materials: A new driving force and a hybrid formulation.” Comput. Methods Appl. Mech. Eng. 355 (Oct): 729–752. https://doi.org/10.1016/j.cma.2019.06.021.
Zhou, S., X. Zhuang, and T. Rabczuk. 2019b. “Phase-field modeling of fluid-driven dynamic cracking in porous media.” Comput. Methods Appl. Mech. Eng. 350 (Jun): 169–198. https://doi.org/10.1016/j.cma.2019.03.001.
Zhou, S., X. Zhuang, H. Zhu, and T. Rabczuk. 2018c. “Phase field modelling of crack propagation, branching and coalescence in rocks.” Theor. Appl. Fract. Mech. 96 (Aug): 174–192. https://doi.org/10.1016/j.tafmec.2018.04.011.
Zhou, Z., H. Zang, W. Cao, X. Du, L. Chen, and C. Ke. 2019. “Risk assessment for the cascading failure of underground pillar sections considering interaction between pillars.” Int. J. Rock Mech. Min. Sci. 124 (Dec): 104142. https://doi.org/10.1016/j.ijrmms.2019.104142.
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ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering
Volume 7 • Issue 3 • September 2021
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© 2021 American Society of Civil Engineers.
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Received: Feb 19, 2021
Accepted: Apr 20, 2021
Published online: Jul 7, 2021
Published in print: Sep 1, 2021
Discussion open until: Dec 7, 2021
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