Rock slope stability analysis under Hoek–Brown failure criterion with different flow rules

Bulletin of Engineering Geology and the Environment (2024) 83:181 https://doi.org/10.1007/s10064-024-03667-0 ORIGINAL PAPER Rock slope stability analysis under Hoek–Brown failure criterion with different flow rules Svetlana Melentijević1 · Zoran Berisavljević2 · Dusan Berisavljević2 · Claudio Olalla Marañón3 Received: 22 April 2023 / Accepted: 1 April 2024 / Published online: 19 April 2024 © The Author(s) 2024 Abstract The stability analysis of homogeneous rock slope following the Hoek–Brown failure criterion under the hypothesis of different flow rules is performed based on limit equilibrium and finite element methods. The applied failure criterion is the generalized Hoek–Brown that can be introduced as a shear/normal function in analysis applying different flow rules. The results are compared with those obtained by the application of equivalent shear strength parameters of the Mohr–Coulomb criterion, considering that this is still the most widely used criterion in rock slope stability analysis and is still the base for the shear strength reduction method applied in finite element modelling. Different proposals for estimating the equivalent strength parameters based on confining stress level are evaluated. The limitation of stress-dependent linear Mohr–Coulomb parameters is emphasized by analysing the vertical cut problem, for which, depending on the chosen stress level, different critical heights are obtained for the same material. Sensitivity analysis of geotechnical parameters used as input for failure criterion is performed to determine their influence on slope stability. Probabilistic analysis is conducted to determine the probability of failure when different flow rules are applied. If slope stability analysis is performed with an assumption of associative flow rule, the probability of failure is within the acceptable limits for the considered case study, while employing non-associative flow rule, the probability of failure is rather high. The chart is presented that could be readily used to estimate the combination of σci, GSI, and mi values that produce failure for the analysed case study. Keywords Circular failure surface · Hoek and Brown failure criterion · Flow rule · Shear/normal function · Stress level · Probabilistic and sensitivity analysis Abbreviations c′ Cohesion mi Rock mass constant mb Reduced value of the intact rock mass constant mi mdil Dilation parameter s and a Rock mass material constants * Svetlana Melentijević 1 Department of Geodynamics, Stratigraphy and Paleonthology, Faculty of Geological Science, Universidad Complutense de Madrid, c/ José Antonio Nováis 12, Ciudad Universitaria, 28040 Madrid, Spain 2 Department of Geotechnics, Faculty of Mining and Geology, University of Belgrade, 7 Djusina Street, 11000 Belgrade, Serbia 3 Department of Geotechnics, ETSI Caminos Canales y Puertos, Universidad Politécnica de Madrid, c/ Profesor Aranguren 3, 28040 Madrid, Spain p, q Lambe’s variables p*, q* Dimensionless form of Lambe’s variables CoF Consequence of failure CoV Coefficient of variation D Disturbance factor Erm Deformation modulus of the rock mass Ei Elastic modulus of intact rock FS Factor of safety GSI Geological strength index H Slope height H* Dimensionless height HL Lower bound solution for critical height of vertical cut HU Upper bound solution for critical height of vertical cut K0 Initial value of lateral earth pressure coefficient Pf Probability of failure RI Reliability index SD Standard deviation of FS Vol.:(0123456789) 181 Page 2 of 18 𝛽 Strength modulus used to scale the original HB failure criterion 𝛽a Strength modulus used to scale the generalized HB failure criterion 𝛿 Angle of slope inclination 𝜙′ Instantaneous friction angle 𝛾 Unit weight ′ 𝜎1 Major principal stresses at failure ′ 𝜎3 Minor principal stresses at failure 𝜎′3max Upper limit of the confining stress 𝜎ci Uniaxial compressive strength of the intact rock 𝜎′cm Rock mass compressive strength 𝜎t Tensile strength of the rock mass 𝜏 Tangential stress exerted on the failure surface 𝜎 Normal stress exerted on the failure surface 𝜏 ∗ Dimensionless form of the tangential stress 𝜎n ∗ Dimensionless form of the normal stress 𝜓 Dilatancy angle χ and k Parameters of a general equation of linear type of the flow rule 𝜁 Coefficient of toughness of the original HB failure criterion 𝜁a Coefficient of toughness of the generalized HB failure criterion μ Mean value of the FS Introduction The estimation of the factor of safety (FS) value in rock slope stability is one of the most important tasks for the design of different geotechnical works, i.e. infrastructures, mining, dams, etc. The most widely used failure criterion in the study of the behaviour of rock masses is the generalized non-linear empirical Hoek–Brown (HB) criterion (Hoek et al. 2002; Hoek and Brown 2019). This failure criterion is widely accepted for the study of homogeneous and isotropic types of rock media as an equivalent continuum, which includes weak rock mass and heavily fractured rock mass where the governing failure pattern is along a rotational failure surface rather than through intact rock and joints. In the state of the art of rock slope stability analysis, both by limit equilibrium method (LEM) and finite element method (FEM), the Mohr–Coulomb (MC) linear failure criterion is still the most widely used one considering that the material strength is expressed in terms of normal and shear stresses rather than in terms of principal stresses. In the past, most of the commercial numerical software did not use the HB failure criterion, and for this reason, the need arose to know the equivalent MC shear strength parameters (cohesion and friction angle) of the rock mass deduced from the HB criterion that should depend on the Bulletin of Engineering Geology and the Environment (2024) 83:181 range of confining stress ( 𝜎′3max ) governing the results of analysis. Nowadays, modern commercial software has introduced the HB criterion; therefore, it seems unnecessary to continue using the equivalent MC parameters, although the conversion of the parameters is still used for many geotechnical applications. For the equivalent parameters to be valid, it is important to choose the appropriate stress range for the failure mechanism depending on the geotechnical problem at hand. There are a lot of proposals for the estimation of equivalent MC parameters for tunnelling problems (Hoek et al. 2002; Sofianos 2003; Sofianos and Nomikos 2006; Jiménez et al. 2008), rock slope stability (Hoek et al. 2002; Yang and Yin 2010; Li et al. 2008; Wei et al. 2018, Rafiei Renani and Martin 2020), strip footing on rock mass (Yang and Yin 2010), etc. Anyway, the majority of the proposed solutions for determination of equivalent MC parameters provide great discrepancies due ( ) to different estimations of confining stress level 𝜎′3max . The plasticity flow rule describes the relation (...truncated)