Analytical Solution for Stress Distribution around Arbitrary Stopes Using Evolutionary Complex Variable Methods
Publication: International Journal of Geomechanics
Volume 19, Issue 10
Abstract
In this study, we proposed an evolutionary complex variable method (ECVM) to find the analytical solution for the stress distribution in an infinite homogeneous, isotropic, and elastic rock mass. This ECVM was a combination of conformal mapping functions, firefly algorithm (FA), and the complex variable theory. Conformal mapping functions were determined by FA to transform arbitrary stope configurations into unit circles. The complex variable theory was then utilized to calculate two complex potential functions, resulting in stress distribution around arbitrary stope configurations solved. A case study involving the analytical solution around rectangular stopes was performed and validated by Abaqus finite-element software. The implementation of the proposed method for arbitrary stope configurations was discussed, and conformal mapping functions for several complex stope configurations were provided. The results showed that there was a good agreement between the analytical solution and numerical modeling. The difference between the analytical solution and Abaqus were mainly around stope corners, which might be because the grid size in Abaqus is not small enough. FA was found to be efficient and advantageous in the determination of conformal mapping functions. The proposed analytical solution has practical significance because it could be used for parameter sensitivity analysis, feasibility studies, and verification of numerical modeling.
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Acknowledgments
The first author was funded by the China Scholarship Council (Grant 201606420046).
References
Amiri, B., L. Hossain, J. W. Crawford, and R. T. Wigand. 2013. “Community detection in complex networks: Multi-objective enhanced firefly algorithm.” Knowledge-Based Syst. 46 (Jul): 1–11. https://doi.org/10.1016/j.knosys.2013.01.004.
Anderson, J. D., Jr. 2010. Fundamentals of aerodynamics. Noida, India: Tata McGraw-Hill Education.
Aubertin, M., L. Li, S. Arnoldi, T. Belem, B. Bussière, M. Benzaazoua, and R. Simon. 2003. “Interaction between backfill and rock mass in narrow stopes.” In Proc., SoilRock2003: 12th Panamerican Conf. on Soil Mechanics and Geotechnical Engineering and 39th US Rock Mechanics Symp., 1157–1164. Alexandria, VA: American Rock Mechanics Association.
Brady, B. H. G., and E. T. Brown. 1985. Rock Mechanics for underground mining. London: George Allen & Unwin.
Dong, X., A. Karrech, H. Basarir, M. Elchalakani, and C. Qi. 2018. “Analytical solution of energy redistribution in rectangular openings upon in-situ rock mass alteration.” Int. J. Rock Mech. Min. Sci. 106 (Jun): 74–83. https://doi.org/10.1016/j.ijrmms.2018.04.014.
Emad, M. Z., I. Vennes, H. Mitri, and C. Kelly. 2014. “Backfill practices for sublevel stoping system.” In Mine planning and equipment selection, edited by C. Drebenstedt, 391–402. Cham, Switzerland: Springer International.
Falaknaz, N., M. Aubertin, and L. Li. 2015. “Numerical investigation of the geomechanical response of adjacent backfilled stopes.” Can. Geotech. J. 52 (10): 1507–1525. https://doi.org/10.1139/cgj-2014-0056.
Fang, Q., H. Song, and D. Zhang. 2015. “Complex variable analysis for stress distribution of an underwater tunnel in an elastic half plane.” Int. J. Numer. Anal. Methods Geomech. 39 (16): 1821–1835. https://doi.org/10.1002/nag.2375.
Fister, I., I. Fister, Jr., X. S. Yang, and J. Brest. 2013. “A comprehensive review of firefly algorithms.” Swarm Evol. Comput. 13 (Dec): 34–46. https://doi.org/10.1016/j.swevo.2013.06.001.
Gram, C. 1962. Selected numerical methods. Copenhagen, Denmark: Regnecentralen.
Huo, H., and A. Bobet, G. Fernández, and J. Ramírez. 2006. “Analytical solution for deep rectangular structures subjected to far-field shear stresses.” Tunnelling Underground Space Technol. 21 (Nov): 613–625. https://doi.org/10.1016/j.tust.2005.12.135.
Jafari, M., and M. Jafari. 2019. “Thermal stress analysis of orthotropic plate containing a rectangular hole using complex variable method.” Eur. J. Mech. A Solids 73 (Jan–Feb): 212–223. https://doi.org/10.1016/j.euromechsol.2018.08.001.
Karaoglu, K., and E. Yilmaz. 2017. “Cemented paste backfill pressure monitoring and field testing.” In Paste tailings management, edited by E. Yilmaz and M. Fall, 195–214. Cham, Switzerland: Springer International.
Kargar, A. R., R. Rahmannejad, and M. A. Hajabasi. 2014. “A semi-analytical elastic solution for stress field of lined non-circular tunnels at great depth using complex variable method.” Int. J. Solids Struct. 51 (6): 1475–1482. https://doi.org/10.1016/j.ijsolstr.2013.12.038.
Krantz, S. G. 1999. Handbook of complex variables. Boston: Birkhauser.
Kreyszig, E., E. Kreyszig, and E. J. Norminton. 2006. Advanced engineering mathematics. Hoboken, NJ: John Wiley.
Li, L. 2014. “Analytical solution for determining the required strength of a side-exposed mine backfill containing a plug.” Can. Geotech. J. 51 (5): 508–519. https://doi.org/10.1139/cgj-2013-0227.
Li, L., and M. Aubertin. 2009. “Numerical investigation of the stress state in inclined backfilled stopes.” Int. J. Geomech. 9 (2): 52–62. https://doi.org/10.1061/(ASCE)1532-3641(2009)9:2(52).
Lu, A., N. Zhang, S. Wang, and X. Zhang. 2017. “Analytical solution for a lined tunnel with arbitrary cross sections excavated in orthogonal anisotropic rock mass.” Int. J. Geomech. 17 (9): 04017044. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000912.
Lu, A. Z., N. Zhang, and L. Kuang. 2014. “Analytic solutions of stress and displacement for a non-circular tunnel at great depth including support delay.” Int. J. Rock Mech. Min. Sci. 70 (Sep): 69–81. https://doi.org/10.1016/j.ijrmms.2014.04.008.
Lu, A. Z., N. Zhang, and Y. Qin. 2015. “Analytical solutions for the stress of a lined non-circular tunnel under full-slip contact conditions.” Int. J. Rock Mech. Min. Sci. 79 (Oct): 183–192. https://doi.org/10.1016/j.ijrmms.2015.08.008.
Lu, X., W. Zhou, X. Ding, X. Shi, B. Luan, and M. Li. 2019. “Ensemble learning regression for estimating unconfined compressive strength of cemented paste backfill.” IEEE Access 7: 72,125–72,133. https://doi.org/10.1109/ACCESS.2019.2918177.
Lukić, D., A. Prokić, and P. Anagnosti. 2009. “Stress–strain field around elliptic cavities in elastic continuum.” Eur. J. Mech. A Solids 28 (1): 86–93. https://doi.org/10.1016/j.euromechsol.2008.04.005.
Ma, L., and A. M. Korsunsky. 2012. “Complex variable formulation for non-slipping plane strain contact of two elastic solids in the presence of interface mismatch eigenstrain.” Int. J. Solids Struct. 49 (9): 1177–1188. https://doi.org/10.1016/j.ijsolstr.2012.02.002.
Mo, Y., Y. Ma, and Q. Zheng. 2013. “Optimal choice of parameters for firefly algorithm.” In Proc., 2013 4th Int. Conf. on Digital Manufacturing and Automation, 887–892. Piscataway, NJ: IEEE.
Moon, P., and D. E. Spencer. 1971. Field theory handbook: Including coordinate systems, differential equations and their solutions. Berlin: Springer-Verlag.
Oberhettinger, F., and W. Magnus. 2013. Anwendung der elliptischen funktionen in physik und technik. Berlin: Springer-Verlag.
Oberkampf, W. L., S. M. DeLand, B. M. Rutherford, K. V. Diegert, and K. F. Alvin. 2002. “Error and uncertainty in modeling and simulation.” Reliab. Eng. Syst. Saf. 75 (3): 333–357. https://doi.org/10.1016/S0951-8320(01)00120-X.
Obert, L., W. I. Duvall, and R. H. Merrill. 1959. Design of underground openings in competent rock. College Park, MD: Bureau of Mines.
Park, K.-H., J.-G. Lee, and A. Owatsiriwong. 2008. “Seepage force in a drained circular tunnel: An analytical approach.” Can. Geotech. J. 45 (3): 432–436. https://doi.org/10.1139/T07-113.
Pisacane, V. L., and L. E. Malvern. 1963. “Application of numerical mapping to the Muskhelishvili method in plane elasticity.” J. Appl. Mech. 30 (3): 410–414. https://doi.org/10.1115/1.3636570.
Qi, C., and A. Fourie. 2018. “A real-time back-analysis technique to infer rheological parameters from field monitoring.” Rock Mech. Rock Eng. 51 (10): 3029–3043. https://doi.org/10.1007/s00603-018-1513-2.
Qi, C., and A. Fourie. 2019. “Numerical investigation of the stress distribution in backfilled stopes considering creep behaviour of rock mass.” Rock Mech. Rock Eng., 1–19. https://doi.org/10.1007/s00603-019-01781-0.
Qi, C., A. Fourie, Q. Chen, X. Tang, Q. Zhang, and R. Gao. 2018a. “Data-driven modelling of the flocculation process on mineral processing tailings treatment.” Journal Cleaner Prod. 196 (Sep): 505–516. https://doi.org/10.1016/j.jclepro.2018.06.054.
Qi, C., A. Fourie, and X. Zhao. 2018b. “Back-analysis method for stope displacements using gradient-boosted regression tree and firefly algorithm.” J. Comput. Civ. Eng. 32 (5): 04018031. https://doi.org/10.1061/(ASCE)CP.1943-5487.0000779.
Qi, C., A. Fourie, Q. Chen, and P. Liu. 2019a. “Application of first-principles theory in ferrite phases of cemented paste backfill.” Miner. Eng. 133: 47–51. https://doi.org/10.1016/j.mineng.2019.01.011.
Qi, C., X. Tang, X. Dong, Q. Chen, A. Fourie, and E. Liu. 2019b. “Towards intelligent mining for backfill: A genetic programming-based method for strength forecasting of cemented paste backfill.” Miner. Eng. 133: 69–79. https://doi.org/10.1016/j.mineng.2019.01.004.
Rojat, F., V. Labiouse, and P. Mestat. 2015. “Improved analytical solutions for the response of underground excavations in rock masses satisfying the generalized Hoek–Brown failure criterion.” Int. J. Rock. Mech. Min. Sci. 79 (Oct): 193–204. https://doi.org/10.1016/j.ijrmms.2015.08.002.
Sahoo, J. P., and J. Kumar. 2014. “Stability of a circular tunnel in presence of pseudostatic seismic body forces.” Tunnelling Underground Space Technol. 42 (May): 264–276. https://doi.org/10.1016/j.tust.2014.03.003.
Savin, G. N. 1961. Stress concentration around holes. New York: Pergamon Press.
Silvestri, V., and C. Tabib. 1993. “Flow field around a cylindrical pile during steady penetration.” Can. Geotech. J. 30 (2): 369–376. https://doi.org/10.1139/t93-031.
Sisavath, S., X. Jing, and R. W. Zimmerman. 2001. “Laminar flow through irregularly-shaped pores in sedimentary rocks.” Transp. Porous Media 45 (1): 41–62. https://doi.org/10.1023/A:1011898612442.
Symm, G. T. 1966. “An integral equation method in conformal mapping.” Numerische Mathematik 9 (3): 250–258. https://doi.org/10.1007/BF02162088.
Thompson, B. D., W. F. Bawden, and M. W. Grabinsky. 2012. “In situ measurements of cemented paste backfill at the Cayeli Mine.” Can. Geotech. J. 49 (7): 755–772. https://doi.org/10.1139/t2012-040.
Tran, N.-H., D.-P. Do, and D. Hoxha. 2018. “A closed-form hydro-mechanical solution for deep tunnels in elastic anisotropic rock.” Eur. J. Environ. Civ. Eng. 22 (12): 1429–1445. https://doi.org/10.1080/19648189.2017.1285253.
Wang, J., D. Milne, L. Wegner, and M. Reeves. 2007. “Numerical evaluation of the effects of stress and excavation surface geometry on the zone of relaxation around open stope hanging walls.” Int. J. Rock Mech. Min. Sci. 44 (2): 289–298. https://doi.org/10.1016/j.ijrmms.2006.07.002.
Westlake, P. C., and P. A. Wilson. 2000. “New conformal mapping technique for ship sections.” Int. Shipbuild. Prog. 47 (449): 5–22.
Xiang, Y., and W. Song. 2017. “Upper-bound limit analysis of shield tunnel stability in undrained clays using complex variable solutions for different ground-loss scenarios.” Int. J. Geomech. 17 (9): 04017057. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000946.
Yang, X. S. 2008. Nature-Inspired metaheuristic algorithms. Frome, UK: Luniver Press.
Yang, X. S., and X. He. 2013. “Firefly algorithm: Recent advances and applications.” Int. J. Swarm Intell. 1 (1): 36–50. https://doi.org/10.1504/IJSI.2013.055801.
Yilmaz, E. 2018. “Stope depth effect on field behaviour and performance of cemented paste backfills.” Int. J. Min. Reclam. Environ. 32 (4): 273–296. https://doi.org/10.1080/17480930.2017.1285858.
Yilmaz, E., T. Belem, M. Benzaazoua, and B. Bussière. 2010. “Assessment of the modified CUAPS apparatus to estimate in situ properties of cemented paste backfill.” Geotech. Test. J. 33 (5): 351–362.
Zeng, X.-T., A.-Z. Lu, and N. Zhang. 2018. “Analytical stress solution for an infinite plate containing two oval holes.” Eur. J. Mech. A Solids 67 (Jan): 291–304. https://doi.org/10.1016/j.euromechsol.2017.09.011.
Zhang, Y., and H. S. Mitri. 2008. “Elastoplastic stability analysis of mine haulage drift in the vicinity of mined stopes.” Int. J. Rock Mech. Min. Sci. 45 (4): 574–593. https://doi.org/10.1016/j.ijrmms.2007.07.020.
Zhao, G., and S. Yang. 2015. “Analytical solutions for rock stress around square tunnels using complex variable theory.” Int. J. Rock Mech. Min. Sci. 80 (Dec): 302–307. https://doi.org/10.1016/j.ijrmms.2015.09.018.
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Published In
International Journal of Geomechanics
Volume 19 • Issue 10 • October 2019
Copyright
© 2019 American Society of Civil Engineers.
History
Received: Sep 28, 2018
Accepted: Apr 9, 2019
Published online: Jul 16, 2019
Published in print: Oct 1, 2019
Discussion open until: Dec 16, 2019
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