Global sensitivity analysis for transformation of Hoek-Brown failure criterion for rock mass
Abstract
A variety of engineering activities require reliable evaluation of rock strength. For instance, the stability of rock slopes depends on structural geology of rock massif in which the slope is excavated. Hoek-Brown (HB) failure criterion applied in rock design practice introduces factors based on the properties of jointed rock. The non-linear finite element safety calculation is conveniently used for calculation safety the factor of slope stability. The Mohr-Coulomb (MC) failure (strength) criterion for soil is widely applied in geotechnical design. Therefore, the appropriate transformation from HB to the equivalent MC, employing angle of shearing resistance φ and cohesion c, is necessary. This article studies the effect of jointed rock massif properties on the transformed MC parameters by using Sobol’s global sensitivity analysis (SSA) and HB transformation equations. Statistical parameters needed for the evaluation of sensitivity analysis are processed using classical statistical methods upon the emulation of Latin Hypercube Sampling simulation methods. Developed and adapted by authors techniques are illustrated by processing real rock investigation data from survey of the trachyte massif located in the Czech Republic. The first and higher order effects of random inputs are identified using SSA. It is illustrated that the effects of inputs on the MC parameters varies significantly depending on the discontinuity distribution and height of the slope.
Keyword : sensitivity analysis, reliability, statistical analysis, Latin hypercube sampling, jointed rock, rock sample, Hoek-Brown and Mohr-Coulomb failure criteria
How to Cite
Štefaňák, J., Kala, Z., Miča, L., & Norkus, A. (2018). Global sensitivity analysis for transformation of Hoek-Brown failure criterion for rock mass. Journal of Civil Engineering and Management, 24(5), 390-398. https://doi.org/10.3846/jcem.2018.5194
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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Hoek, E.; Carranza-Torres, C.; Corkum, C. 2002. Hoek-Brown failure criterion – 2002 edition, in Proceedings of NARMS-TAC Conference, 2002, Toronto, Canada, 267–273.
Hoek, E.; Carter, T.; Diederichs, M. 2013. Quantification of the geological strength index chart, in 47th U.S. Rock Mechanics/Geomechanics Symposium, 23–26 June 2013, San Francisco, CA, USA, 13–672.
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Kala, Z. 2016a. Global interval sensitivity analysis of Hermite probability density function percentiles, International Journal of Mathematical Models and Methods in Applied Sciences 10: 373–380.
Kala, Z. 2016b. Global sensitivity analysis in stability problems of steel frame structures, Journal of Civil Engineering and Management 22(3): 417–424. http://dx.doi.org/10.3846/13923730.2015.1073618
Kala, Z.; Valeš, J. 2017a. Global sensitivity analysis of lateral-torsional buckling resistance based on finite element simulations, Engineering Structures 134: 37–47. http://dx.doi.org/10.1016/j.engstruct.2016.12.032
Kala, Z.; Valeš, J. 2017b. Sensitivity assessment and lateral-torsional buckling design of I-beams using solid finite elements, Journal of Constructional Steel Research 139: 110–122. https://doi.org/10.1016/j.jcsr.2017.09.014
Kala, Z.; Valeš, J. 2018. Imperfection sensitivity analysis of steel columns at ultimate limit state, Archives of Civil and Mechanical Engineering 18(4): 1207–1218. https://doi.org/10.1016/j.acme.2018.01.009
Keshavarz Ghorabaee, M. K.; Amiri, M.; Zavadskas, E. K.; Antucheviciene, J. 2018. A new hybrid fuzzy MCDM approach for evaluation of construction equipment with sustainability considerations, Archives of Civil and Mechanical Engineering 18(1): 32–49. https://doi.org/10.1016/j.acme.2017.04.011
Li, A. J.; Cassidy, M. J.; Wang, Y.; Merifield, R. S.; Lyamin, A. V. 2012. Parametric Monte Carlo studies of rock slopes based on the Hoek–Brown failure criterion, Computers and Geotechnics 45: 11–18. https://doi.org/10.1016/j.compgeo.2012.05.010
Liang, J.; Ding, Z.; Li, J. 2017. A probabilistic analyzed method for concrete fatigue life, Probabilistic Engineering Mechanics 49: 13–21. http://dx.doi.org/10.1016/j.probengmech.2017.08.002
Lü, Q.; Low, B. K. 2011. Probabilistic analysis of underground rock excavations using response surface method and SORM, Computers and Geotechnics 38(8): 1008–1021. https://doi.org/10.1016/j.compgeo.2011.07.003
McKey, M. D.; Conover, W. J.; Beckman, R. J. 1979. A comparison of the three methods of selecting values of input variables in the analysis of output from a computer code, Technometrics 21(2): 239–245. https://doi.org/10.2307/1268522
Olson, L.; Samson, C.; Mckinnon, S. D. 2015. 3-D laser imaging of drill core for fracture detection and rock quality designation, International Journal of Rock Mechanics and Mining Sciences 73: 156–164. https://doi.org/10.1016/j.ijrmms.2014.11.004
Palmström, A. 2002. Measurement and characterization of rock mass jointing, in V. M. Sharma, K. R. Saxena. In-situ characterization of rock mass jointing. Oslo: A. A. Balkema Publishers, 49–98.
Pejkowski, L. 2017. On the material’s sensitivity to non-proportionality of fatigue loading, Archives of Civil and Mechanical Engineering 17(3): 711–727. http://dx.doi.org/10.1016/j.acme.2016.09.010
Pells, P. J.; Bieniawski, Z. T.; Hencher, S. R.; Pells, S. E. 2017. Rock quality designation (RQD): time to rest in peace, Canadian Geotechnical Journal 54(6): 825–834. https://doi.org/10.1139/cgj-2016-0012
Saltelli, A.; Annoni, P.; Azzini, I.; Campolongo, F.; Ratto, M.; Tarantola, S. 2010. Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index, Computer Physics Communications 181: 259–270. http://dx.doi.org/10.1016/j.cpc.2009.09.018
Saltelli, A.; Chan, K.; Scott, E. M. 2004. Sensitivity analysis. Wiley series in probability and statistics. New York: John Wiley and Sons. 475 p.
Sari, M. 2012. Stochastic estimation of the Hoek-Brown strength parameters using spreadsheet models, in EUROCK 2012 – ISMR International Symposium, 28–30 May 2012, Stockholm, Sweden.
Şen, Z. 2014. Rock quality designation-fracture intensity index method for geomechanical classification, Arabian Journal of Geosciences 7(7): 2915–2922. https://doi.org/10.1007/s12517-013-0975-5
Sobol’, I. M. 1993. Sensitivity analysis for non-linear mathematical models, Mathematical Modelling and Computational Experiment 1: 407–414. Translated from Russian: I. M. Sobol’. 1990. Sensitivity estimates for nonlinear mathematical models, Matematicheskoe Modelirovanie 2: 112–118.
Sobol’, I. M. 2001. Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates, Mathematics and Computers in Simulation 55(1–3): 271–280. https://doi.org/10.1016/S0378-4754(00)00270-6
Stavropoulou, M. 2014. Discontinuity frequency and block volume distribution in rock masses, International Journal of Rock Mechanics and Mining Sciences 65: 62–74. https://doi.org/10.1016/j.ijrmms.2013.11.003
Vessia, G.; Kozubal, J.; Pula, W. 2017. High dimensional model representation for reliability analyses of complex rock–soil slope stability, Archives of Civil and Mechanical Engineering 17(4): 954–963. http://dx.doi.org/10.1016/j.acme.2017.04.005
Wang, W.; Shen, J. 2017. Comparison of existing methods and a new tensile strength based model in estimating the Hoek-Brown constant mi for intact rocks, Engineering Geology 224: 87–96. https://doi.org/10.1016/j.enggeo.2017.05.008
Závacký, M.; Štefaňák, J.; Horák, V.; Miča, L. 2017. Statistical estimate of uniaxial compressive strength of rock based on Shore hardness, in EUROCK 2017 ISMR International Symposium, 2017, Prague, Czech Republic Republic, 248–255. https://doi.org/10.1016/j.proeng.2017.05.178
Zhang, L. 2010. Estimating the strength of jointed rock masses, Rock Mechanics and Rock Engineering 43(4): 391–402. https://doi.org/10.1007/s00603-009-0065-x
Zhang, L. 2016. Determination and applications of rock quality designation (RQD), Journal of Rock Mechanics and Geotechnical Engineering 8(3): 389–397. https://doi.org/10.1016/j.jrmge.2015.11.008
Barton, N. R.; Bandis, S. C. 1990. Review of predictive capability of JRC-JCS model in engineering practice, in Proceedings of the International Symposium on Rock Joints, 1990, Rotterdam, Netherlands, 603–610.
Barton, N.; Lien, R.; Lunde, J. 1974. Engineering classification of rock masses for the design of tunnel support, Rock Mechanics 6(4): 182–239. https://doi.org/10.1007/BF01239496
Borgonovo, E.; Plischke, E. 2016. Sensitivity analysis: A review of recent advances, European Journal of Operational Research 248: 869–887. https://doi.org/10.1016/j.ejor.2015.06.032
Bozorgzadeh, N.; Yanagimura, Y.; Harrison, J. 2017. Effect of small numbers of test results on accuracy of Hoek–Brown strength parameter estimations: A statistical simulation study, Rock Mechanics and Rock Engineering 50(12): 3293–3305. https://doi.org/10.1007/s00603-017-1352-6
Brinkgreve, R. B. J.; Bakker, H. L. 1991. Non-linear finite element analysis of safety factors, in Proceedings of the 7th International Conference on Computer Methods and Advances in Geomechanics, 1991, Cairns, Australia, 1117–1122.
Cai, M. 2010. Practical estimates of tensile strength and Hoek–Brown strength parameter mi of Brittle Rocks, Rock Mechanics and Rock Engineering 43(2): 167–184. https://doi.org/10.1007/s00603-009-0053-1
Cai, M. 2011. Rock mass characterization and rock property variability considerations for tunnel and cavern design, Rock Mechanics and Rock Engineering 44: 379–399. https://doi.org/10.1007/s00603-011-0138-5
Cai, Y.; Esaki, T.; Jiang, Y. 2004. A rock bolt and rock mass interaction model, International Journal of Rock Mechanics and Mining Sciences 41(7): 1055–1067. https://doi.org/10.1016/j.ijrmms.2004.04.005
Dere, D.; Hendron, A.; Patton, F.; Cording, E. 1967. Design of surface and near surface constructions in rock, in Proceedings of the 8th U. S. Symposium on Rock Mechanics, 1967, New York, USA, 237–302.
Devkota, K.; Ham, J-E.; Kim, G-W. 2009. Characteristics of discontinuity spacing of Yeongdeok granite, Geosciences Journal 13(2): 161–165. https://doi.org/10.1007/s12303-009-0015-3
Guo, H.-S.; Feng, X.-T.; Li, S.-J.; Yang, C.-X; Yao, Z.-B. 2017. Evaluation of the integrity of deep rock masses using results of digital borehole televiewers, Rock Mechanics and Rock Engineering 50(6): 1371–1382. https://doi.org/10.1007/s00603-017-1173-7
Hoek, E. 1998. Reliability of Hoek-Brown estimates of rock mass properties and their impact on design, International Journal of Rock Mechanics and Mining Sciences 35(1): 63–68. https://doi.org/10.1016/S0148-9062(97)00314-8
Hoek, E.; Brown, E. T. 1997. Practical estimates of rock mass strength, International Journal of Rock Mechanics and Mining Sciences 34(8): 1165–1186. https://doi.org/10.1016/S1365-1609(97)80069-X
Hoek, E.; Carranza-Torres, C.; Corkum, C. 2002. Hoek-Brown failure criterion – 2002 edition, in Proceedings of NARMS-TAC Conference, 2002, Toronto, Canada, 267–273.
Hoek, E.; Carter, T.; Diederichs, M. 2013. Quantification of the geological strength index chart, in 47th U.S. Rock Mechanics/Geomechanics Symposium, 23–26 June 2013, San Francisco, CA, USA, 13–672.
Horák, V.; Závacký, M.; Štefaňák, J. 2016. Výzkumná zpráva č. HS12645008L: Laboratorní zkoušky skalních hornin. Brno: Vysoké učení technické v Brně, Fakulta stavební, AdMaS - Pokročilé stavební materiály, konstrukce a technologie (in Czech).
Hudson, J. A.; Priest, S. D. 1983. Discontinuity frequency in rock masses, International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts 20(2): 73–89. https://doi.org/10.1016/0148-9062(83)90329-7
Idris, M.; Basarir, H.; Nordlund, E.; Wettainen, T. 2013. The probabilistic estimation of rock masses properties in Malmberget mein, Sweden, The Electronic Journal of Geotechnical Engineering 18: 269–287.
Iman, R. C.; Conover, W. J. 1980. Small sample sensitivity analysis techniques for computer models with an application to risk assessment, Communications in Statistics – Theory and Methods 9(17): 1749–1842. https://doi.org/10.1080/03610928008827996
Kala, Z. 2016a. Global interval sensitivity analysis of Hermite probability density function percentiles, International Journal of Mathematical Models and Methods in Applied Sciences 10: 373–380.
Kala, Z. 2016b. Global sensitivity analysis in stability problems of steel frame structures, Journal of Civil Engineering and Management 22(3): 417–424. http://dx.doi.org/10.3846/13923730.2015.1073618
Kala, Z.; Valeš, J. 2017a. Global sensitivity analysis of lateral-torsional buckling resistance based on finite element simulations, Engineering Structures 134: 37–47. http://dx.doi.org/10.1016/j.engstruct.2016.12.032
Kala, Z.; Valeš, J. 2017b. Sensitivity assessment and lateral-torsional buckling design of I-beams using solid finite elements, Journal of Constructional Steel Research 139: 110–122. https://doi.org/10.1016/j.jcsr.2017.09.014
Kala, Z.; Valeš, J. 2018. Imperfection sensitivity analysis of steel columns at ultimate limit state, Archives of Civil and Mechanical Engineering 18(4): 1207–1218. https://doi.org/10.1016/j.acme.2018.01.009
Keshavarz Ghorabaee, M. K.; Amiri, M.; Zavadskas, E. K.; Antucheviciene, J. 2018. A new hybrid fuzzy MCDM approach for evaluation of construction equipment with sustainability considerations, Archives of Civil and Mechanical Engineering 18(1): 32–49. https://doi.org/10.1016/j.acme.2017.04.011
Li, A. J.; Cassidy, M. J.; Wang, Y.; Merifield, R. S.; Lyamin, A. V. 2012. Parametric Monte Carlo studies of rock slopes based on the Hoek–Brown failure criterion, Computers and Geotechnics 45: 11–18. https://doi.org/10.1016/j.compgeo.2012.05.010
Liang, J.; Ding, Z.; Li, J. 2017. A probabilistic analyzed method for concrete fatigue life, Probabilistic Engineering Mechanics 49: 13–21. http://dx.doi.org/10.1016/j.probengmech.2017.08.002
Lü, Q.; Low, B. K. 2011. Probabilistic analysis of underground rock excavations using response surface method and SORM, Computers and Geotechnics 38(8): 1008–1021. https://doi.org/10.1016/j.compgeo.2011.07.003
McKey, M. D.; Conover, W. J.; Beckman, R. J. 1979. A comparison of the three methods of selecting values of input variables in the analysis of output from a computer code, Technometrics 21(2): 239–245. https://doi.org/10.2307/1268522
Olson, L.; Samson, C.; Mckinnon, S. D. 2015. 3-D laser imaging of drill core for fracture detection and rock quality designation, International Journal of Rock Mechanics and Mining Sciences 73: 156–164. https://doi.org/10.1016/j.ijrmms.2014.11.004
Palmström, A. 2002. Measurement and characterization of rock mass jointing, in V. M. Sharma, K. R. Saxena. In-situ characterization of rock mass jointing. Oslo: A. A. Balkema Publishers, 49–98.
Pejkowski, L. 2017. On the material’s sensitivity to non-proportionality of fatigue loading, Archives of Civil and Mechanical Engineering 17(3): 711–727. http://dx.doi.org/10.1016/j.acme.2016.09.010
Pells, P. J.; Bieniawski, Z. T.; Hencher, S. R.; Pells, S. E. 2017. Rock quality designation (RQD): time to rest in peace, Canadian Geotechnical Journal 54(6): 825–834. https://doi.org/10.1139/cgj-2016-0012
Saltelli, A.; Annoni, P.; Azzini, I.; Campolongo, F.; Ratto, M.; Tarantola, S. 2010. Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index, Computer Physics Communications 181: 259–270. http://dx.doi.org/10.1016/j.cpc.2009.09.018
Saltelli, A.; Chan, K.; Scott, E. M. 2004. Sensitivity analysis. Wiley series in probability and statistics. New York: John Wiley and Sons. 475 p.
Sari, M. 2012. Stochastic estimation of the Hoek-Brown strength parameters using spreadsheet models, in EUROCK 2012 – ISMR International Symposium, 28–30 May 2012, Stockholm, Sweden.
Şen, Z. 2014. Rock quality designation-fracture intensity index method for geomechanical classification, Arabian Journal of Geosciences 7(7): 2915–2922. https://doi.org/10.1007/s12517-013-0975-5
Sobol’, I. M. 1993. Sensitivity analysis for non-linear mathematical models, Mathematical Modelling and Computational Experiment 1: 407–414. Translated from Russian: I. M. Sobol’. 1990. Sensitivity estimates for nonlinear mathematical models, Matematicheskoe Modelirovanie 2: 112–118.
Sobol’, I. M. 2001. Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates, Mathematics and Computers in Simulation 55(1–3): 271–280. https://doi.org/10.1016/S0378-4754(00)00270-6
Stavropoulou, M. 2014. Discontinuity frequency and block volume distribution in rock masses, International Journal of Rock Mechanics and Mining Sciences 65: 62–74. https://doi.org/10.1016/j.ijrmms.2013.11.003
Vessia, G.; Kozubal, J.; Pula, W. 2017. High dimensional model representation for reliability analyses of complex rock–soil slope stability, Archives of Civil and Mechanical Engineering 17(4): 954–963. http://dx.doi.org/10.1016/j.acme.2017.04.005
Wang, W.; Shen, J. 2017. Comparison of existing methods and a new tensile strength based model in estimating the Hoek-Brown constant mi for intact rocks, Engineering Geology 224: 87–96. https://doi.org/10.1016/j.enggeo.2017.05.008
Závacký, M.; Štefaňák, J.; Horák, V.; Miča, L. 2017. Statistical estimate of uniaxial compressive strength of rock based on Shore hardness, in EUROCK 2017 ISMR International Symposium, 2017, Prague, Czech Republic Republic, 248–255. https://doi.org/10.1016/j.proeng.2017.05.178
Zhang, L. 2010. Estimating the strength of jointed rock masses, Rock Mechanics and Rock Engineering 43(4): 391–402. https://doi.org/10.1007/s00603-009-0065-x
Zhang, L. 2016. Determination and applications of rock quality designation (RQD), Journal of Rock Mechanics and Geotechnical Engineering 8(3): 389–397. https://doi.org/10.1016/j.jrmge.2015.11.008