Technical note
Stability factors for rock slopes subjected to pore water pressure based on the Hoek-Brown failure criterion

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Introduction

Since the problem of slope stability is an important field in soil and rock engineering, much work has been devoted to the problem, employing different methods such as limit equilibrium using slices, limit analysis, method of characteristics, and, more recently, the finite element technique. The stability of slopes under external forces may be expressed by a stability factor that is usually defined as the ratio of the product of the unit weight and critical height to the cohesion (γHc/c), as shown in [1]. To determine the stability factor of the slope, a common technique used is the limit equilibrium method where the slope is divided into slices. In order to solve the stability problem, some assumptions regarding the location and inclination of forces between slices are made to enable the static equilibrium equations of such slices to be solved. Due to the arbitrary assumptions made regarding the interslice forces, the solutions obtained from the technique of limit equilibrium cannot be regarded as rigorous in a strict mechanical sense. Among the available solutions methods, the finite element method is certainly the most comprehensive approach to analyze the performance of slope subjected to pore water pressures. However, high numerical costs and accurate measurements of the parameters of the geomaterials are required, which are often difficult to obtain. This makes the use of the finite element method less attractive for current applications. In recent years, many slope stability analyses have been performed using the limit analysis method. This method is based on the application of the static and kinematic theorems of plasticity. By constructing a statically admissible stress field, it is possible to find the lower bound to the true limit load (or stability factor), and, theoretically, by constructing various statically admissible stress fields, the highest possible lower bound solution can be found. The upper bound theorem states that the rate of work done by actual forces is less than or equal to the rate of energy dissipation in any kinematically admissible velocity field. A kinematically admissible velocity field is compatible with the velocities at the boundary of the mass. When the unknown quantity is a force that makes the slope collapse, the application of the upper bound theorem leads to upper bounds to the true limit load (or stability factor). The lowest possible upper bound solution is sought with an optimization scheme by trying various possible kinematically admissible failure mechanisms. The true solution lies between the highest possible lower bound solution and the lowest possible upper bound solution. The magnitude of true solution depends on the geomaterial model used for the plastic deformation prior to failure. The solution, obtained by the limit analysis method, is rigorous in that the statically admissible stress field is in equilibrium with imposed loads at every point, and that the kinematically admissible velocity field is compatible with the imposed displacement. The solution is rigorous in that no additional assumption regarding the interslice force is required.

A linear Mohr–Coulomb (MC) failure criterion is widely used in geotechnical engineering. However, experimental data have shown that the strength envelopes of almost all geomaterials are nonlinear [2], [3], [4], [5], [6]. With a nonlinear failure criterion, some researchers [7], [8], [9], [10] evaluate stability of homogenous slopes, where the effect of pore water pressure is not considered.

For a linear MC failure criterion, the effects of pore water pressure have been considered in the literature focusing on the stability analysis of soil slope problems. In general, pore water pressures are obtained from ground water conditions, which are simulated by a flow net or a pore water pressure ratio. Miller and Hamilton [11] considered pore water pressures as internal forces, and used two kinds of failure mechanism to estimate slope stability. The pore water pressures were assumed to be hydrostatic below a free water surface. However, Michalowski [12] regarded the pore water pressure as external force, and employed a log-spiral rotation failure mechanism to analyze the effects of pore water pressure on slope stability. The distribution of pore water pressure was described by a coefficient ru=u/γh, where u is the pore water pressure, γ is the soil total unit weight, and h is the depth of the point on the failure surface below the soil surface.

In the present study, the kinematical theorem of limit analysis is applied to calculate the stability factors of rock slopes subjected to pore water using the nonlinear Hoek–Brown (HB) failure criterion. Pore water pressures, regarded as external forces, are calculated using a pore water pressure ratio. The generalized tangential technique, proposed by Yang [13], [14], is employed to formulate the stability factor as an optimization problem. In the approach, the work of Michalowski [12], who employed a linear MC failure criterion, is extended using the HB failure criterion, and the work of Collins et al [10], who did not consider the effects of pore water pressures on the slope stability, is extended to consider the effects of pore water pressures. Numerical results are presented for practical use.

Section snippets

Upper bound theorem with pore water pressure

Based on the upper bound theorem of limit analysis, the limit load at collapse can be determined by equating the external rate of work to the internal energy rate of dissipation in any kinematically admissible failure mechanism [20]. The lowest upper bound solution is obtained by linear or nonlinear programming algorithm. The upper bound theorem requires some assumptions, which are as follows: (a) the material is perfectly plastic with convex yield function in stress space; and (b) the material

Modified Hoek–Brown failure criterion

Due to the fact that the rock masses of the slopes have intrinsic characteristics such as joints, faults and weak bedding planes, which render the rock anisotropic, the stability analysis of rock slopes is commonly developed using distinct element method. Block theory is theoretically accepted by researchers, which can determine potentially unstable rock block. However, with the increase of joint number, which leads to an increase in block elements within the range of considered slope, this

Generalized tangential technique

If the strength of circumscribing yield surface is more than that of the actual yield surface, the limit load corresponding to the circumscribing yield surface will be an upper bound to the actual limit load corresponding to the actual yield surface, as shown by Chen [1]. A tangential line to the nonlinear HB failure criterion is shown in Fig. 1. The strength of the tangential line exceeds or equals that of the nonlinear HB failure criterion, corresponding to the same normal stress. Thus, the

Numerical results

Without considering the effect of pore water pressure, Collins et al. [10] presented numerical results with the original HB failure criterion (n=0.5). In practice, due to the variability of natural rock masses and difference of fractured degree, the exponent, n varies according to Eq. (5). The problem considered here is the stability factor of slopes subjected to pore water pressures with the modified HB failure criterion. The numerical results to the problem have been obtained by minimizing

Conclusions

The upper bound theorem of limit analysis has been applied to the stability of jointed rock slopes subjected to pore water pressure based on the nonlinear HB failure criterion. The generalized tangential technique was used to develop the upper bound solutions. Instead of employing the actual HB failure criterion, a MC linear failure criterion, which is tangential to the actual HB failure criterion, was used to calculate the rate of external work and internal energy dissipation. Equating the

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