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Safety Factor on Rock Slopes with Tensile Cracks Using Numerical and Limit Equilibrium Models

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Abstract

Through the research it describes an analytic methodology, which allows to determine the minimum safety factor depending on the depth of tensile crack and the inclination of surface failure on the most critical condition, considering at the same time, surcharge, seismic effect and water pressure. Therefore, it studies the stability of rock slopes considering that the potential of failure surface it is constituted by two blocks with different inclinations. The superior block is limited by a tensile crack that is represented by a fracture without displacement. On the other hand, on the inferior block, geometry is formed by a potential slide plane of α inclination with the horizontal axis, in which are acting shear stresses. Fracture on superior block is characterized by a normal-tensile stresses field that act over the crack whose presence originates when the rock loses its original cohesion. Finally, comparisons are made through examples with the limit equilibrium method and finite elements method, where it determines the safety factor on dry state, water and seism, being all of them too similar. Besides, the methodology compares the track depth and the distance between the intersection point of tensile crack and the edge of the slope face, results shows that the analytic methodology is very conservative and throws the less values of this distances.

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References

  • Abd AH, Utili S (2017) Design of geosynthetic-reinforced slopes in cohesive backfills. Geotext Geomembr 45(6):627–641

    Google Scholar 

  • Agheshlui H, Sedaghat MH, Azizmohammadi S (2019) A comparative study of stress influence on fracture apertures in fragmented rocks. J Rock Mech Geotech Eng 11:38–45

    Google Scholar 

  • Aydin A, Johnson AM (1978) Development of faults as zones of deformation bands and as slip surfaces in sandstone. Pure Appl Geophys 116:931–942

    Google Scholar 

  • Aydin A, Johnson AM (1983) Analysis of faulting in porous sandstones. J Struct Geol 5:19–31

    Google Scholar 

  • Barton N, Choubey V (1976) The shear strength of rock joints in theory and practice. Rock Mech 10:1–54

    Google Scholar 

  • Belandria N, Úcar R, León F and Hassani F (2020) Stability analysis of slopes with planar failure using variational calculus and numerical methods. Front Struct Civ Eng 1–12

  • Bieniawski T (1974) Estimating the strength of rock materials. J S Afr Inst Min Metall 74(8):312–320

    Google Scholar 

  • Bieniawski T (1989) Engineering rock mechanics classifications, vol 251. Wiley, Hoboken

    Google Scholar 

  • Bishop AW (1955) The use of the slip circle in the stability analysis of slops. Géotechnique 5(1):7–17

    Google Scholar 

  • Bobet A, Einstein HH (1998) Fracture coalescence in rock-type materials under uniaxial and biaxial compression. Int J Rock Mech Min Sci Geomech 35(7):863–889

    Google Scholar 

  • Brace W, Byerlee J (1967) Recent experimental studies of brittle fracture of rocks. In: Failure and breakage of rock. AIME, pp 57–81

  • Chen G, Kemeny J, Harpalani S (1992) Fracture propagation and coalescence in marble plates with pre-cut notches under compression. In: Symposium on fractured and jointed rock mass, Lake Tahoe, CA, pp 443–448

  • Coulomb CA (1776) Essai sur une application des regles des maximis et minimis a quelquels problemesde statique relatifs, a la architecture. Mem Acad Roy Div Sav 7:343–387

  • Dai ZH, Liu ZW, Liu CY, He MC (2008) Numerical analysis of soil slope stability considering tension and shear failures. Chin J Rock Mech Eng 27(2):375–382

    Google Scholar 

  • De Melo Moura D, Bobet A (2019) Influence of flaw geometry on crack coalescence across an interface in a rock model material. In: 53rd US Rock mechanics/geomechanics symposium held in New York, NY, USA, 23–26 June, Paper 19–388

  • De Melo Moura D, Bobet A (2019) Cracking across a smooth interface in a rock-model material. In: ISRM 14th international congress of rock mechanics, Iguassu Falls, Brazil, September 13–18

  • Fellenius W (1936) Calculation of stability of earth dams. In: Transactions of the 2nd congress on large dams, 4, Washington D.C., pp 445–463

  • Hoek E, Bieniawski ZT (1984) Brittle fracture propagation in rock under compression. Int J Fract 26:276–294

    Google Scholar 

  • Hoek E, Bray JW (1977) Rock slope engineering, 2nd edn. The Institution of Mining and Metallurgy, London

    Google Scholar 

  • Hoek E, Bray JW (1981) Rock slope engineering, 3rd edn. Institution of Mining and Metallurgy, London, p 358

    Google Scholar 

  • Hoek E, Brown ET (1980) Empirical strength criterion for rock masses. J Geotech Eng Div ASCE 106(GT9):1013–1035

    Google Scholar 

  • Hoek E, Brown ET (1988) The Hoek–Brown failure criterion: a 1988 update. In: Proceedings of the 15th Canadian rock mechanics symposium, University of Toronto, pp 31–38

  • Hoek E, Brown ET (1989) The Hoek–Brown failure criterion: a 1988 updated in Rock engineering for undergrounds excavations. In: Curran JC (ed) Proceedings of the 15th Canadian rock mechanics symposium, Toronto. Department of Civil Engineering, University of Toronto, pp 31–38

  • Hoek E, Brown ET (1997) Practical estimates of rock mass strength. Int J Rock Mech Min Sci 34(8):1165–1186

    Google Scholar 

  • Hoek E, Brown ET (2019) The Hoek–Brown failure criterion and GSI. J Rock Mech Geotech Eng 11(3):445–463

    Google Scholar 

  • Hoek E, Carranza-Torres C, Corkum B (2002) Hoek–Brown failure criterion: 2002 edition. In: Proceedings of the 5th North American rock mechanics symposium and 17th tunnelling association of Canada conference: NARMS-TAC, July 7–10. University of Toronto, pp 267–271

  • Ingraffea AR, Heuze FE (1980) Finite element models for rock fracture mechanics. Int J Numer Anal Methods Geomech 4:25–43

    Google Scholar 

  • Jiefan H, Ganglin C, Yonghong Z, Ren W (1990) An experimental study of the strain field development prior to failure of a marble plate under compression. Tectonophysics 175:269–284

    Google Scholar 

  • Johnson AM, Johnson KM, Durdella J, Sozen M, Gur T (2002) An emendation of elastic rebound theory: main rupture and adjacent belt of right-lateral distortion detected by Viaduct at Kaynasli, Turkey, 12 November 1999 Duzce Earthquake. J Seismolog 6:329–346

    Google Scholar 

  • Klingbeil NW, Beuth JL (1997) Interfacial fracture testing of deposited metal layers under four-point bending. Eng Fract Mech 56(1):113–126

    Google Scholar 

  • Lajtai EZ (1974) Brittle fracture in compression. Int J Fract 10(4):525–536

    Google Scholar 

  • Michalowski RL (2013) Stability assessment of slopes with cracks using limit analysis. Can Geotech J 50(10):1011–1021

    Google Scholar 

  • Modiriasari A, Bobet A, Pyrak-Nolte LJ (2020) Seismic wave conversion caused by Shear crack initiation and growth. Rock Mech Rock Eng 53:2805–2818

    Google Scholar 

  • Mogi K (1971) Fracture and flow of rocks under triaxial compression. J Geophys Res 76(5):1255–1269

    Google Scholar 

  • Mogi K (2007) Experimental rock mechanics, vol 361. Taylor and Francis, Abingdon

    Google Scholar 

  • Morgenstern NR, Price VE (1965) The analysis of the stability of general slip surface. Géotechnique 15(1):63–79

    Google Scholar 

  • Murrell SA (1965) The effect of triaxial stress system on the strength of rocks at atmospheric temperatures. Geophys J R Astron Soc 10:231–281

    Google Scholar 

  • Olson JE (1993) Joint pattern development: effects of subcritical crack growth and mechanical crack interaction. J Geophys Res 98(B7):12251–12265

    Google Scholar 

  • Pariseau W (2007) Design analysis in rock mechanics, vol 560. Taylor and Francis, Abingdon

    Google Scholar 

  • Peng S, Johnson AM (1972) Crack growth and faulting in cylindrical specimens of chelmsford granite. Int J Rock Mech Min Sci 9:37–86

    Google Scholar 

  • Petit J, Barquins M (1988) Can natural faults propagate under mode II Conditions? Tectonics 7(6):1243–1256

    Google Scholar 

  • Pollard DD, Zeller S, Olson J, Thomas A (1990) Understanding the process of jointing in brittle rock masses. In: Proceedings: 31st US symposium on rock mechanics, A. A. Balkema, Rotterdam, pp 447–454

  • Polyakov A, Zhabin A, Averin E, Polyakov A (2019) Generalized equation for calculating rock cutting efficiency by pulsed water jets. J Rock Mech Geotech Eng 11:867–873

    Google Scholar 

  • Priest S (2005) Determination of shear strength and three-dimensional yield strength for the Hoek–Brown criterion. Rock Mech Rock Eng 38(4):299–327

    Google Scholar 

  • Puell F, De La Fuente P, Arenillas M (2004) La linealización del criterio de Hoek y Brown en el análisis de la estabilidad de taludes en roca. Revista de Obras Públicas, Julio – Agosto 3(446):37–44

    Google Scholar 

  • Ramamurthy T (1986) Stability of rock mass. Indian Geotech J 16(1):74

    Google Scholar 

  • Reyes O, Einstein HH (1991) Failure mechanism of fractured rock—a fracture coalescence model. In: Proceedings 7th international congress of rock mechanics vol 1, pp 333–340

  • Sarma SK (1973) Stability analysis of embankments and slopes. Géotechnique 23(3):423–433

    Google Scholar 

  • Shen B, Stephansson O, Einstein HH, Ghahreman B (1995) Coalescence of fractures under shear stress experiments. J Geophys Res 100:5975–5990

    Google Scholar 

  • Shukla SK, Khandelwal S, Verma VN, Sivakugan N (2009) Effect of surcharge on the stability of anchored rock slope with water filled tension crack under seismic loading condition. Geotech Geol Eng 27:529–538

    Google Scholar 

  • Spencer E (1967) A method of analysis of the stability of embankments assuming parallel inter-slice forces. Géotechnique 17(1):11–26

    Google Scholar 

  • Spencer E (1968) Effect of tensile on the stability of embankments. J Soil Mech Found Div ASCE 94(5):1159–1173

    Google Scholar 

  • Taghavi M, Dovoudi MH, Amiri-Tokaldany E, Darby SE (2010) An analytical method to estimate failure plane angle and tensile crack depth for use in riverbank stability analyses. Geomorphology 123(1–2):74–83

    Google Scholar 

  • Takeuchi K (1991) Mixed-mode fracture initiation in granular brittle materials. M.S. Thesis, Massachusetts Institute of Technology, Cambridge

  • Tang L, Zhao Z, Luo Z, Sun Y (2019) What is the role of tensile cracks in cohesive slopes? J Rock Mech Geotech Eng. https://doi.org/10.1016/j.jrmge.2018.09.007

    Article  Google Scholar 

  • Terzaghi K (1943) Theoretical soil mechanics. Wiley, New York

    Google Scholar 

  • Ucar R (2003) Una metodología reciente para determinar la resistencia al corte en macizos rocosos. Ingeniería del Terreno, E.T.S.I Minas. Universidad Politécnica de Madrid. Editor Carlos López. Madrid. vol 3, pp 99–117

  • Ucar R (2011) Una metodología reciente para determinar la resistencia al corte en macizos rocosos. In: Proceedings of PanAm CGS geotechnical conference, Toronto, Canada vol 3, pp 2694–2700

  • Ukritchon B, Keawsawasvong S (2018) Discussion on ‘‘Seismic displacement along a log-spiral failure| surface with crack using rock Hoek-Brown failure criterion’’. Soil Dyn Earthq Eng 110:141–144

    Google Scholar 

  • Utili S (2013) Investigation by limit analysis on the stability of slopes with cracks. Géotechnique 63(2):140–154

    Google Scholar 

  • Wang CH (1997) Fracture of interface cracks under combined loading. Eng Fract Mech 56(1):77–86

    Google Scholar 

  • Wong RHC, Chau KT, Tang CA, Lin P (2001) Analysis of crack coalescence in rock-like materials containing three flaws-Part I: experimental approach. Int J Rock Mech Min Sci 38:909–924

    Google Scholar 

  • Wu H, Pollard DD (1992) Propagation of a set of opening-mode fractures in layered brittle materials under uniaxial strain cycling. J Geophys Res 97(B3):3381–3396

    Google Scholar 

  • Xu D, Wu A, Li C (2019a) A linearly-independent higher-order extended numerical manifold method and its application to multiple crack growth simulation. J Rock Mech Geotech Eng 11:1256–1263

    Google Scholar 

  • Xu Y, Yao W, Xia K (2019b) Numerical study on tensile failures of heterogeneous rocks. J Rock Mech Geotech Eng 12:50–58

    Google Scholar 

  • Yu MH (2002) Advances in strength theories for materials under complex stress state in the 20 Century. Appl Mech Rev 55(3):169–218

    Google Scholar 

  • Yu MH (2003) Unified strength theory and its applications, vol 412. Springer, Berlin

    Google Scholar 

  • Zhang G, Wang R, Qian JY, Zhang JM, Qian JG (2012) Effect study of cracks on behavior of soil slope under rainfall conditions. Soils Found 52(4):634–643

    Google Scholar 

  • Zhao LH, Cheng X, Zhang Y, Li L, Li DL (2016) Stability analysis of seismic slopes with cracks. Comput Geotech 77:77–90

    Google Scholar 

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Authors and Affiliations

  1. Geological Engineering School, University of Los Andes, Mérida, Venezuela

    Norly Belandria, Roberto Úcar & Alfredo Corredor

  2. Department of Mining and Material Engineering, McGill University, Montreal, Canada

    Ferri Hassani

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  1. Norly Belandria
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  2. Roberto Úcar
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  4. Ferri Hassani
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Correspondence to Norly Belandria.

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Belandria, N., Úcar, R., Corredor, A. et al. Safety Factor on Rock Slopes with Tensile Cracks Using Numerical and Limit Equilibrium Models. Geotech Geol Eng 39, 2287–2300 (2021). https://doi.org/10.1007/s10706-020-01624-8

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