Abstract
Antidip bedding rock slopes always contain a developed set of joints that dip into the free surface; however, sometimes they have some gently inclined cross joints, and the persistence ratio (k) of these cross joints exerts important influences on the stability and failure modes. In this work, a limit-equilibrium model combined with a fracture mechanics (FM) method was established to predict the stability of antidip bedding rock slopes with very low-persistent cross joints. Rock layers that might undergo toppling failure were analyzed with the theories of Mode I fracture in FM; layers with the potential for shear–sliding failure were studied by solid mechanics and theories for Mode II fracture from FM. Then, a practical slope and a model test (reported in previous studies) were used to verify the feasibility of the proposed method. The calculated results were consistent with the actual state of the slope and the experimental result, which indicated that a limit-equilibrium model combined with an FM method could be used to predict the stability of such slopes. To further approve the rationality of the method, a contrastive analysis was conducted that used two other methods. The results showed that the predicted factors of safety (FS) of the proposed method were conservative and on the safe side. The proposed method could be used by engineers to evaluate and design such rock slopes.
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Acknowledgments
The research was financially supported by the National Natural Science Foundation of China (Grant Nos. 12072358 and 12102443) and the Youth Innovation Promotion Association, CAS (No. 2022333). The authors would like to thank the anonymous reviewers and editors for their constructive suggestions, which have significantly improved the quality of this manuscript.
References
Adhikary, D. P., and A. V. Dyskin. 2007. “Modelling of progressive and instantaneous failures of foliated rock slopes.” Rock Mech. Rock Eng. 40 (4): 349–362. https://doi.org/10.1007/s00603-006-0085-8.
Adhikary, D. P., A. V. Dyskin, R. J. Jewell, and D. P. Stewart. 1997. “A study of the mechanism of flexural toppling failure of rock slopes.” Rock Mech. Rock Eng. 30: 75–93. https://doi.org/10.1007/BF01020126.
Alejano, L. R., I. Pérez-Rey, M. Múñiz-Menéndez, A. Riquelme, and G. Walton. 2021. “Considerations relevant to the stability of granite boulders.” Rock Mech. Rock Eng. 55: 2729–2745. https://doi.org/10.1007/s00603-021-02525-9.
Alejano, L. R., C. Sánchez-Alonso, I. Pérez-Rey, J. Arzúa, E. Alonso, J. González, L. Beltramone, and A. M. Ferrero. 2018. “Block toppling stability in the case of rock blocks with rounded edges.” Eng. Geol. 234: 192–203. https://doi.org/10.1016/j.enggeo.2018.01.010.
Aliha, M. R. M., M. R. Ayatollahi, D. J. Smith, and M. J. Pavier. 2010. “Geometry and size effects on fracture trajectory in a limestone rock under mixed mode loading.” Eng. Fract. Mech. 77 (11): 2200–2212. https://doi.org/10.1016/j.engfracmech.2010.03.009.
Amini, M., A. Majdi, and Ö Aydan. 2009. “Stability analysis and the stabilisation of flexural toppling failure.” Rock Mech. Rock Eng. 42 (5): 751–782. https://doi.org/10.1007/s00603-008-0020-2.
Amini, M., A. Majdi, and M. A. Veshadi. 2012. “Stability analysis of rock slopes against block-flexure toppling failure.” Rock Mech. Rock Eng. 45 (4): 519–532. https://doi.org/10.1007/s00603-012-0220-7.
ASTM. 1990. Standard test method for plane-strain fracture toughness testing of high strength metallic materials. E399-90. West Conshohocken, PA: ASTM.
Ayatollahi, M. R., and J. Akbardoost. 2014. “Size and geometry effects on rock fracture toughness: Mode I fracture.” Rock Mech. Rock Eng. 47: 677–687. https://doi.org/10.1007/s00603-013-0430-7.
Aydan, Ö, and T. Kawamoto. 1992. “The stability of slopes and underground openings against flexural toppling and their stabilisation.” Rock Mech. Rock Eng. 25 (3): 143–165. https://doi.org/10.1007/BF01019709.
Bahrami, B., M. Nejati, M. R. Ayatollahi, and T. Driesner. 2020. “Theory and experiment on true mode II fracturing of rocks.” Eng. Fract. Mech. 240: 107314. https://doi.org/10.1016/j.engfracmech.2020.107314.
Bazant, Z. P. 1984. “Size effect on blunt fracture: Concrete, rock, metal.” J. Eng. Mech. 110 (4): 518–535. https://doi.org/10.1061/(ASCE)0733-9399(1984)110:4(518).
Bobet, A. 1999. “Analytical solutions for toppling failure.” Int. J. Rock Mech. Min. Sci. 36 (7): 971–980. https://doi.org/10.1016/S0148-9062(99)00059-5.
Bowa, V. M., and Y. Xia. 2018. “Modified analytical technique for block toppling failure of rock slopes with counter-tilted failure surface.” Indian Geotech. J. 48 (1): 713–727. https://doi.org/10.1007/s40098-018-0303-9.
Cai, W. M., V. Murti, and S. Valliapan. 1990. “Slope stability analysis using fracture mechanics approach.” Theor. Appl. Fract. Mech. 12: 261–281. https://doi.org/10.1016/0167-8442(90)90063-6.
Camones, L. A. M., E. D. A. Vargas, R. P. D. Figueiredo, and R. Q. Velloso. 2013. “Application of the discrete element method for modeling of rock crack propagation and coalescence in the step-path failure mechanism.” Eng. Geol. 153: 80–94. https://doi.org/10.1016/j.enggeo.2012.11.013.
Dai, F., M. D. Wei, N. W. Xu, T. Zhao, and Y. Xu. 2015. “Numerical investigation of the progressive fracture mechanisms of four ISRM-suggested specimens for determining the mode I fracture toughness of rocks.” Comput. Geotech. 69: 424–441. https://doi.org/10.1016/j.compgeo.2015.06.011.
Ding, B. D., Z. Y. Han, G. C. Zhang, X. T. Beng, and Y. C. Yang. 2021. “Flexural toppling mechanism and stability analysis of an anti-dip rock slope.” Rock Mech. Rock Eng. 54: 3721–3735. https://doi.org/10.1007/s00603-021-02435-w.
Erdogan, F., and G. C. Sih. 1963. “On the crack extension in plates under plane loading and transverse shear.” J. Basic Eng. 85 (4): 519–527. https://doi.org/10.1115/1.3656897.
Fowell, R. J. 1995. “ISRM commission on testing methods. Suggested method for determining mode I fracture toughness using cracked chevron notched Brazilian disc (CCNBD) specimens.” Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 32 (1): 57–64. https://doi.org/10.1016/0148-9062(94)00015-U.
García-Moya, S. A., J. González-Galindo, and C. Olalla. 2019. “Underdip toppling failure mechanism: Case study retrospective analysis and its most determinant parameters.” Int. J. Geomech. 19 (6): 05019005. https://doi.org/10.1061/(ASCE)GM.1943-5622.0001408.
Ghouli, S., B. Bahrami, M. R. Ayatollahi, T. Driesner, and M. Nejati. 2021. “Introduction of a scaling factor for fracture toughness measurement of rocks using the semi-circular bend test.” Rock Mech. Rock Eng. 54: 4041–4058. https://doi.org/10.1007/s00603-021-02468-1.
Goodman, R. E., and J. W. Bray. 1976. “Toppling of rock slopes.” In Proc., ASCE Specialty Conf. on Rock Engineering for Foundations and Slopes, 201–234. Boulder, CO: American Society of Civil Engineering.
Griffith, A. A. 1921. “The phenomena of rupture and flow in solids.” Philos. Trans. R. Soc. London, Ser. A 221: 163–198. https://doi.org/10.1098/rsta.1921.0006.
Griffith, A. A. 1924. “The theory of rupture.” In Proc., 1st Int. Congress for Applied Mechanics, 55–63. Delft, Netherlands: J. Waltman, jr.
Horii, H., and S. Nemat-Nasser. 1985. “Compression-induced microcrack growth in brittle solids: Axial splitting and shear failure.” J. Geophys. Res.: Solid Earth 90 (NB4): 3105–3125. https://doi.org/10.1029/JB090iB04p03105.
Hu, X. Z., and K. Duan. 2007. “Size effect: Influence of proximity of fracture process zone to specimen boundary.” Eng. Fract. Mech. 74 (7): 1093–1100. https://doi.org/10.1016/j.engfracmech.2006.12.009.
Hu, X. Z., and K. Duan. 2008. “Size effect and quasi-brittle fracture: The role of FPZ.” Int. J. Fract. 154 (1): 3–14.
Huang, D., D. F. Cen, G. W. Ma, and R. Q. Huang. 2015. “Step-path failure of rock slopes with intermittent joints.” Landslides 12 (5): 911–926. https://doi.org/10.1007/s10346-014-0517-6.
Huang, J. A., and S. J. Wang. 1985. “An experimental investigation concerning the comprehensive fracture toughness of some brittle rocks.” Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 22 (2): 99–104. https://doi.org/10.1016/0148-9062(85)92331-9.
Jennings, J. E. 1970. “A mathematical theory for the calculation of the stability of open cast mines.” In Proc., Symp. on Theoretical Background to the Planning of Open Pit Mines, 87–102. Rotterdam, The Netherlands: A.A. Balkema.
Justo, J., J. Castro, and S. Cicero. 2020. “Notch effect and fracture load predictions of rock beams at different temperatures using the theory of critical distances.” Int. J. Rock Mech. Min. Sci. 125: 104161. https://doi.org/10.1016/j.ijrmms.2019.104161.
Kuruppu, M. D., Y. Obara, M. R. Ayatollahi, K. P. Chong, and T. Funatsu. 2014. “ISRM-suggested method for determining the mode I static fracture toughness using semi-circular bend specimen.” Rock Mech. Rock Eng. 47 (1): 267–274. https://doi.org/10.1007/s00603-013-0422-7.
Li, A., F. Dai, Y. Liu, H. B. Du, and R. C. Jiang. 2021. “Dynamic stability evaluation of underground cavern sidewalls against flexural toppling considering excavation-induced damage.” Tunnelling Underground Space Technol. 112 (4): 103903. https://doi.org/10.1016/j.tust.2021.103903.
Li, B., Q. F. Ding, N. W. Xu, F. Dai, Y. Xu, and H. L. Qu. 2022. “Characteristics of microseismic b-value associated with rock mass large deformation in underground powerhouse caverns at different stress levels.” J. Cent. South Univ. 29 (2): 693–711. https://doi.org/10.1007/s11771-022-4946-4.
Liu, C. H., M. B. Jaksa, and A. G. Meyers. 2008. “Improved analytical solution for toppling stability analysis of rock slopes.” Int. J. Rock Mech. Min. Sci. 45 (8): 1361–1372. https://doi.org/10.1016/j.ijrmms.2008.01.009.
Liu, T. T., L. Y. Ding, F. Meng, X. P. Li, and Y. Zheng. 2021. “Stability analysis of anti-dip bedding rock slopes using a limit equilibrium model combined with bi-directional evolutionary structural optimization (BESO) method.” Comput. Geotech. 134 (4): 104–116.
Majdi, A., and M. Amini. 2011. “Analysis of geo-structural defects in flexural toppling failure.” Int. J. Rock Mech. Min. Sci. 48 (2): 175–186. https://doi.org/10.1016/j.ijrmms.2010.11.007.
Mineo, S., G. Pappalardo, and S. Onorato. 2021. “Geomechanical characterization of a rock cliff hosting a cultural heritage through ground and UAV rock mass surveys for its sustainable fruition.” Sustainability 13: 924. https://doi.org/10.3390/su13020924.
Ouchterlony, F. 1988. “ISRM commission on testing methods. Suggested methods for determining fracture toughness of rock.” Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 25: 71–96.
Palaniswamy, K., and W. G. Knauss. 1972. “Propagation of a crack under general, in-plane tension.” Int. J. Fract. 8 (1): 114–117. https://doi.org/10.1007/BF00185207.
Pappalardo, G., and S. Mineo. 2015. “Rockfall hazard and risk assessment: The promontory of the pre-Hellenic village Castelmola Case, North-Eastern Sicily (Italy).” In Vol. 2 of Engineering geology for society and territory, edited by G. Lollino, 1989–1993. Cham, Switzerland: Springer.
Pérez-Rey, I., L. R. Alejano, A. Riquelme, and L. González-deSantos. 2019. “Failure mechanisms and stability analyses of granitic boulders focusing a case study in Galicia (Spain).” Int. J. Rock Mech. Min. Sci. 119: 58–71. https://doi.org/10.1016/j.ijrmms.2019.04.009.
Pérez-Rey, I., M. Muñiz-Menéndez, J. González, F. Vagnon, G. Walton, and L. R. Alejano. 2021. “Laboratory physical modelling of block toppling instability by means of tilt tests.” Eng. Geol. 282: 105994. https://doi.org/10.1016/j.enggeo.2021.105994.
Rao, Q. H., Z. Q. Sun, O. Stephansson, C. L. Li, and B. Stillborg. 2003. “Shear fracture (Mode II) of brittle rock.” Int. J. Rock Mech. Min. Sci. 40 (3): 355–375. https://doi.org/10.1016/S1365-1609(03)00003-0.
Roy, D. G., T. N. Singh, and J. Kodikara. 2017. “Influence of joint anisotropy on the fracturing behavior of a sedimentary rock.” Eng. Geol. 228: 224–237. https://doi.org/10.1016/j.enggeo.2017.08.016.
Sagaseta, C., J. M. Sánchez, and J. Cañizal. 2001. “A general analytical solution for the required anchor force in rock slopes with toppling failure.” Int. J. Rock Mech. Min. Sci. 38 (3): 421–435. https://doi.org/10.1016/S1365-1609(01)00011-9.
Sarfaraz, H. 2020. “Stability analysis of flexural toppling failure using the Sarma’s method.” Geotech. Geol. Eng. 38 (4): 3667–3682. https://doi.org/10.1007/s10706-020-01244-2.
Scavia, C. 1990. “Fracture mechanics approach to stability analysis of rock slopes.” Eng. Fract. Mech. 35: 899–910. https://doi.org/10.1016/0013-7944(90)90174-F.
Scavia, C. 1996. “The effect of scale on rock fracture toughness: A fractal approach.” Geotechnique 46 (4): 683–693. https://doi.org/10.1680/geot.1996.46.4.683.
Sih, G. C. 1974. “Strain-energy-density factor applied to mixed mode crack problems.” Int. J. Fract. 10 (3): 305–321. https://doi.org/10.1007/BF00035493.
Stead, D., J. S. Coggan, and E. Eberhardt. 2004. “Realistic simulation of rock slope failure mechanisms: The need to incorporate principles of fracture mechanics.” Int. J. Rock Mech. Min. Sci. 41 (3): 1–6.
Tada, H., P. C. Paris, and G. R. Irwin. 2003. The stress analysis of cracks handbook. 3rd ed. New York: ASME.
Wang, X. G., Z. X. Jia, and Z. Chen. 1996. “The research of stability analysis of toppling failure of jointed rock slopes.” [In Chinese.] J. Hydraul. Eng. 3: 7–12.
Wang, Y. S., X. Z. Hu, L. Liang, and W. C. Zhu. 2016. “Determination of tensile strength and fracture toughness of concrete using notched 3-p-b specimens.” Eng. Fract. Mech. 160: 67–77. https://doi.org/10.1016/j.engfracmech.2016.03.036.
Wu, J. H., Y. Ohnishi, G. H. Shi, and S. Nishiyama. 2005. “Theory of three-dimensional discontinuous deformation analysis and its application to a slope toppling at Amatoribashi, Japan.” Int. J. Geomech. 5 (3): 179–195. https://doi.org/10.1061/(ASCE)1532-3641(2005)5:3(179).
Xia, K. Z., C. X. Chen, T. L. Wang, Y. Zheng, and Y. Wang. 2022. “Estimating the geological strength index and disturbance factor in the Hoek-Brown criterion using the acoustic wave velocity in the rock mass.” Eng. Geol. 306: 106745. https://doi.org/10.1016/j.enggeo.2022.106745.
Xu, J., and Z. Li. 2019. “Crack propagation and coalescence of step-path failure in rocks.” Rock Mech. Rock Eng. 52 (2): 965–979. https://doi.org/10.1007/s00603-018-1661-4.
Zhang, C., Y. Wang, and T. Jiang. 2020. “The propagation mechanism of an oblique straight crack in a rock sample and the effect of osmotic pressure under in-plane biaxial compression.” Arabian J. Geosci. 13 (15): 736. https://doi.org/10.1007/s12517-020-05682-3.
Zhang, J., Z. Y. Chen, and X. G. Wang. 2007. “Centrifuge modeling of rock slopes susceptible to block toppling.” Rock Mech. Rock Eng. 40 (4): 363–382. https://doi.org/10.1007/s00603-006-0112-9.
Zhao, W., R. Q. Wang, and T. K. Nian. 2020. “Stability analysis of anti-dip rock slopes with flexural toppling failure based on deformation compatibility.” Rock Mech. Rock Eng. 53: 3207–3221. https://doi.org/10.1007/s00603-020-02098-z.
Zheng, Y., C. Chen, T. Liu, and Z. H. Ren. 2021a. “A new method of assessing the stability of anti-dip bedding rock slopes subjected to earthquake.” Bull. Eng. Geol. Environ. 80: 3693–3710. https://doi.org/10.1007/s10064-021-02188-4.
Zheng, Y., C. X. Chen, T. T. Liu, K. Z. Xia, and X. M. Liu. 2018. “Stability analysis of rock slopes against sliding or flexural-toppling failure.” Bull. Eng. Geol. Environ. 77: 1383–1403. https://doi.org/10.1007/s10064-017-1062-z.
Zheng, Y., C. X. Chen, F. Meng, T. T. Liu, and K. Z. Xia. 2020a. “Assessing the stability of rock slopes with respect to flexural toppling failure using a limit equilibrium model and genetic algorithm.” Comput. Geotech. 124: 103619. https://doi.org/10.1016/j.compgeo.2020.103619.
Zheng, Y., C. X. Chen, F. Meng, H. N. Zhang, K. Z. Xia, and X. B. Chen. 2020b. “Assessing the stability of rock slopes with respect to block-flexure toppling failure using a force-transfer model and genetic algorithm.” Rock Mech. Rock Eng. 53: 3433–3445. https://doi.org/10.1007/s00603-020-02122-2.
Zheng, Y., R. Wang, C. Chen, and F. Meng. 2022. “Fast stability assessment of rock slopes subjected to flexural toppling failure using adaptive moment estimation (Adam) algorithm.” Landslides 19: 2149–2158. https://doi.org/10.1007/s10346-022-01902-x.
Zheng, Y., R. Q. Wang, C. X. Chen, C. Y. Sun, Z. H. Ren, and W. Zhang. 2021b. ‘“Dynamic analysis of anti-dip bedding rock slopes reinforced by pre-stressed cables using discrete element method.” Eng. Analy. Boundary Elem. 130: 79–93. https://doi.org/10.1016/j.enganabound.2021.05.014.
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International Journal of Geomechanics
Volume 23 • Issue 5 • May 2023
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© 2023 American Society of Civil Engineers.
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Received: Jun 29, 2022
Accepted: Nov 20, 2022
Published online: Feb 28, 2023
Published in print: May 1, 2023
Discussion open until: Jul 28, 2023
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