Elsevier

Computers and Geotechnics

Volume 45, September 2012, Pages 11-18
Computers and Geotechnics

Parametric Monte Carlo studies of rock slopes based on the Hoek–Brown failure criterion

https://doi.org/10.1016/j.compgeo.2012.05.010Get rights and content

Abstract

Probabilistic evaluation of slope failures is increasingly seen as the most appropriate framework for accounting for uncertainties in design. This paper performs reliability assessments for rock slopes based on the latest version of the Hoek–Brown failure criterion. The purpose of this study is to demonstrate the use of a new form of stability number 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. The analyses show that by using this newly proposed stability number, the probability of failure (Pf) obtained from case studies agrees well with the true state of the slope. In addition, this paper details a procedure to determine the magnitude of safety factor required for rock slope design.

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.

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