Parametric Monte Carlo studies of rock slopes based on the Hoek–Brown failure criterion
Introduction
The strength of a rock mass is usually represented by its cohesion (c′) and friction angle (ϕ′), with slope failures evaluated using the linear Mohr–Coulomb yield criterion. However, with investigations reporting the yield criteria for rock masses to be non-linear [1], [2], [3], [4], [5], a linear failure envelope may not be suitable for estimating the rock slope stability. The results of Li et al. [6], [7] indicated that replacing a non-linear criterion with an equivalent linear failure envelope overestimates the factor of safety by up to 80%, depending on the curve fitting procedure adopted.
In engineering practice, the limit equilibrium method (LEM), as used by Bishop [8] and Janbu et al. [9], is one of the most popular approaches for estimating slope stability. However, it is well known that the solution obtained from the limit equilibrium method is not rigorous, as neither static nor kinematic admissibility conditions are satisfied. Moreover, in order to find a solution, arbitrary assumptions must be made regarding inter slice forces for a two dimensional (2D) and inter-column forces for a three dimensional (3D) case.
The application of the reliability methods in evaluating the safety of earth slopes was initiated in the 1970s [10], [11]. In these methods, the uncertainties of material properties, strength parameters, slope geometries and potential damages, for instance, can be considered and therefore more rational and effective designs are conducted. Refinement of techniques for the assessment of soil and rock slope failures continues, with amongst many other examples Juang et al. [12], Al-Homoud and Tanash [13], Low [14], Li and Lumb [15], Cassidy et al. [16], Shinoda [17] and Silva et al. [18]. As limit equilibrium analysis is the most widely used method for estimating the safety factors of slopes, the risk evaluations of slopes are generally performed by the LEM in conjunction with the reliability analysis. The authors are unaware of any investigations which utilise limit theorems to do probabilistic assessments for slope stability.
Fortunately, attractive finite element upper and lower bound approaches have been developed by Lyamin and Sloan [19], [20] and Krabbenhoft et al. [21]. These techniques can be used to bracket the true stability solutions for geotechnical problems, and are suited to using either linear or non-linear failure criteria. Using the limit theorems can not only provide a simple and useful way of analysing the stability of geotechnical structures, but they can also avoid the shortcomings of the arbitrary assumptions underpinning the LEM. These numerical upper and lower bound methods have been used to provide chart solutions for both soil slopes [22], [23], [24], [25] and rock slopes [6], [7], [26]. This work was extended by employing the latest version of Hoek–Brown failure criterion [27] in the studies of Li et al. [6], [7], [26].
The purpose of this study is to demonstrate the use of a new form of stability number [6] for rock slope designs that has been recently developed from finite element upper and lower bound limit analysis methods, and to provide guidance for its use in probabilistic assessments.
Section snippets
Why use limit analysis method instead of limit equilibrium method
The strength of jointed rock masses is notoriously difficult to assess as rock masses are generally inhomogeneous, discontinuous media composed of rock material and naturally occurring discontinuities such as joints, fractures and bedding planes. These features make any analysis using simple theoretical solutions, such as the LEM, difficult. Without including special interface or joint elements, the displacement finite element method is not suitable for analysing rock masses with fractures and
Methodology followed in the reliability assessment
In the limit analyses, for given slope geometry (H, β) and rock mass (σci, GSI, mi), the optimised solutions of the upper bound and lower bound programs can be carried out with respect to the unit weight (γ). Therefore, the stability number, Eq. (5), can be obtained. For probabilistic analyses, the average stability numbers of upper and lower bound limit analysis solutions may provide benchmarks for slope stability assessments.
In this paper, a series of case studies of rock slopes in mines are
Uncertainty in strength distributions
Firstly, the distribution types of strength parameters are investigated in the first part of the study. As stated previously, the distribution of D is not taken into account in this paper. An average value of GSI in Table 1 is adopted as the mean for reliability analysis in the following sections. From the above discussions, this part of the study concentrates on investigating the distribution types and shapes of σci and mi using Case 1a whose input parameters are shown in Table 2.
The factor of safety against the probability of failure
In engineering practice, the magnitude of the factor of safety for a slope design should be correlated with the consequence of the failure. If the probability of failure is predicted to be too high, for a cut rock slope without reinforcement, remedial methods include the reduction of slope angle and/or slope height. These both increase the factor of safety. Because practicing engineer cannot always perform a full Monte Carlo simulation, as we performed in the previous sections, it is useful to
Conclusion remarks
This study demonstrated that the finite element limit analysis methods are not only convenient tools for slope designs, but also suitable for risk analysis. Based on the Hoek–Brown failure criterion [27], the applicability of new definition in the factor of safety proposed by Li et al. [6] has been verified by comparisons to a number of published case studies. In this paper, reliability analyses were performed using the average solutions of the numerical upper and lower limit analysis methods.
References (39)
- et al.
Stability charts for rock slopes based on the Hoek–Brown failure criterion
Int J Rock Mech Min Sci
(2008) - et al.
Seismic rock slope stability charts based on limit analysis methods
Comput Geotech
(2009) - et al.
Stability analysis of existing slopes considering uncertainty
Eng Geol
(1998) - et al.
Modeling uncertainty in stability analysis for design of embankment dams on difficult foundations
Eng Geol
(2004) Reliability analysis of rock slopes involving correlated nonnormals
Int J Rock Mech Min Sci
(2007)- et al.
Limit analysis solutions for three dimensional undrained slopes
Comput Geotech
(2009) - et al.
Effect of rock mass disturbance on the stability of rock slopes using the Hoek–Brown failure criterion
Comput Geotech
(2011) - et al.
Practical estimates of rock mass strength
Int J Rock Mech Min Sci
(1997) - et al.
Limit analysis solutions for the bearing capacity of rock masses using the generalised Hoek–Brown criterion
Int J Rock Mech Min Sci
(2006) Reliability of Hoek–Brown estimates of rock mass properties and their impact on design
Int J Rock Mech Min Sci
(1998)
Prediction of engineering properties of a selected litharenite sandstone from its petrographic characteristics using correlation and multivariate statistical techniques
Eng Geol
Engineering geological properties and durability assessment of the Cappadocian tuff
Eng Geol
Empirical strength criterion for rock masses
J Geotech Eng Div ASCE
Empirical rock failure criteria
The use of slip circle in stability analysis of slopes
Geotechnique
Cited by (66)
Probability analysis of rock slope using FORM based on a nonlinear strength criterion
2023, Geomechanics for Energy and the EnvironmentGraphical charts for onsite Continuous Slope Mass Rating (CoSMR) classification using strike parallelism and joint dip or plunge of intersection
2022, Engineering GeologyCitation Excerpt :Charts are easy to use and handy to carry in the field when dealing with engineering geological problems. Slope stability charts for the factor of safety (FOS) calculation have been previously developed by many authors based on Generalized Hoek-Brown criteria (Carranza-Torres, 2004; Li et al., 2008, 2009, 2012; Shen et al., 2013; Sun et al., 2016). F1, F2, and F3 are adjustment factors for the discontinuity-slope geometrical relationship derived using continuous functions (Eqs. 3–6)(Tomás et al., 2007).
Probabilistic evaluation of three-dimensional seismic active earth pressures using sparse polynomial chaos expansions
2021, Computers and GeotechnicsActive learning relevant vector machine for reliability analysis
2021, Applied Mathematical ModellingCitation Excerpt :However, they are much smaller than the results reported by Li et al. [45]. This is because Li et al. [45] conducted a two-dimensional stability analysis of rock slopes, which should lead to conservative results than the 3D analysis performed in this study. The numbers of calls to the original model Ncall are 95, 88, 210, and 304, respectively, for the proposed approach, which are significantly reduced compared with the sparse polynomial chaos expansion method adopted by Pan and Dias [46].
Limit equilibrium solution for the rock slope stability under the coupling effect of the shear dilatancy and strain softening
2020, International Journal of Rock Mechanics and Mining SciencesReliability analysis of a rock slope based on plastic limit analysis theory with multiple failure modes
2019, Computers and Geotechnics