Advances in Materials Science and Engineering

Volume 2016 (2016), Article ID 2030238, 10 pages

http://dx.doi.org/10.1155/2016/2030238

## Numerical Study on Stability of Rock Slope Based on Energy Method

Key Laboratory of Ministry of Education for Geomechanics and Embankment Engineering, College of Civil and Transportation Engineering, Hohai University, Nanjing 210098, China

Received 26 April 2016; Revised 29 June 2016; Accepted 30 June 2016

Academic Editor: Hossein Moayedi

Copyright © 2016 Wei Gao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

To solve the main shortcoming of numerical method for analysis of the stability of rock slope, such as the selection the convergence condition for the strength reduction method, one method based on the minimum energy dissipation rate is proposed. In the new method, the basic principle of fractured rock slope failure, that is, the process of the propagation and coalescence for cracks in rock slope, is considered. Through analysis of one mining rock slope in western China, this new method is verified and compared with the generally used strength reduction method. The results show that the new method based on the minimum energy dissipation rate can be used to analyze the stability of the fractured rock slope and its result is very good. Moreover, the new method can obtain less safety factor for the rock slope than those by other methods. Therefore, the new method based on the minimum energy dissipation rate is a good method to analyze the stability of the fractured rock slope and should be superior to other generally used methods.

#### 1. Introduction

The rock slopes can be found in many engineering applications, such as highways and constructions in mountainous area, open pit mines, and hydropower projects. It is very important to maintain the stability of those rock slopes for the engineering safety. Nowadays, there are two main methods for analysis of the stability of rock slope, which are theoretical method and numerical method [1]. As the main ones of the theoretical method, the limit equilibrium-based methods remain the most popular option in rock slope engineering. These conventional methods, however, are limited to simple slope geometries, basic loading condition and rigid body assumption. Practical rock slope stability problems are intricate in the aspects of topological geometry, material anisotropy, nonlinear behavior, in situ stress, and the presence of several coupled processes. Therefore, numerical method helps to forward the approximate solutions, which would have never been possible using the conventional theoretical methods. However, the strength reduction method is the main one of the numerical method. Nowadays, there are many studies using the strength reduction method for analysis of the stability of rock slope.

For example, the loosened block rock slope of Jinyang Grand Buddha in Taiyuan of China is analyzed by the strength reduction method using Fast Lagrangian Analysis of Continua (FLAC) software [2]. Using FLAC software, Jiang [3] analyzed the stability of the bedding rock slope by strength reduction method with the ubiquitous-joint criterion. Moreover, the relationships of the bedding plane inclination angle and the safety factor are discussed. To analyze the jointed rock slopes, the strength reduction method based on the discrete element method is proposed [4]. Moreover, the sensitivity analyses are performed to better understand the critical mechanisms that lead to slope failure and to discriminate between the respective roles played by intact rock and planes of weakness at the onset of failure. To contrast techniques and highlight their advantages and drawbacks, Alejano et al. [1] analyze six theoretical examples using a limit equilibrium method and a numerical method, Universal Distinct Element Code (UDEC), combined with the strength reduction method. The first five examples concern fully joint-controlled rock slopes, for which only rigid blocks are needed, whereas the sixth example refers to a partially joint-controlled rock slope. To deal with the stability of highway cut in Eastern Slovakia, the strength reduction technique based on finite element analysis is used. In this study, the Mohr-Coulomb model and jointed rock model are all applied [5]. In the studies of Liu et al. [6], a new approach for determining the safety factor and the corresponding critical slip surface of a layered rock slope subjected to seismic excitations is presented, through a case study based on the combination of the strength reduction technique and distinct element method. Moreover, according to this proposed method, the seismic safety factor and the critical slip surface of the slope are estimated and compared with those obtained by the pseudostatic approach, combined with the limit equilibrium method. Chen et al. [7] present a comparative study of methods used for the analysis of the stability of rock slopes for hydropower projects. The comparison concerns the application of the limit equilibrium method, the block element method, and the finite element method on the stability analysis of two hydropower engineering projects in China. In those numerical studies, the strength reduction method is used. Nowadays, the linear Mohr-Coulomb criterion is generally used in the strength reduction method. But it is widely accepted that the shear strength of the rock mass is a nonlinear function of stress level. To analyze the stability of rock slope, one nonlinear Hoek-Brown shear strength reduction technique is proposed [8]. Moreover, the implementation of the nonlinear shear strength reduction method is described detailedly and a rock slope example is selected to verify this new method.

Although the strength reduction method is widely used, it has some shortcomings, such as the expensive computation and convergence issues due to the nonlinear iterative computations [9]. The selection of the convergence condition especially is a very hard work. To solve this problem, one method based on the minimum energy dissipation rate is proposed. At last, one rock slope example is used to verify the proposed method.

#### 2. Strength Reduction Method Based on Minimum Energy Dissipation Rate

The strength reduction method was first proposed by Zienkiewicz et al. in 1975 [10]. Its definition of the safety factor for a slope is often considered to be the ratio of the actual shear strength to the lowest shear strength of a rock or soil material that is required to maintain the slope in equilibrium [11]. The strength reduction method usually assumes the Mohr-Coulomb strength for slope materials. In this method, the gravitational acceleration is assumed as one constant. And, the parameters of shear strength of a rock or soil material ( and ) are reduced by one reduction factor . Thus, new parameters ( and ) can be obtained. The new parameters are reduced by another reduction factor again. Until the slope is in the limit equilibrium state, in other words, when the reduction factor increases a little, the slope will collapse. At this moment, the corresponding reduction factor is the safety factor of the slope. In other words, this method involves successive reductions (by some trial factors) in the shear strength of slope materials until failure occurs. The critical reduction factor at which collapse occurs may be regarded as the safety factor of the slope. The equations for this method are as follows [11]:where and are shear stress of slope materials before and after reduction, respectively; and are cohesion of slope materials before and after reduction, respectively; and are internal friction angle of slope materials before and after reduction, respectively; is normal stress of slope materials; is the reduction factor.

A critical problem in the determination of the safety factor of a slope using the strength reduction method is the definition of the critical (or limit equilibrium) state. Nowadays, there are three main types of criteria for determining the critical state of the slope. The first one is the deformation characteristics of the slope, such as failure shear strain developed from the toe to the top of the slope [12]; the second one is stress distribution features, such as the connectivity of the plastic zone through the slope from the toe to the crest [13]; and the third one is the nonconvergence option of solutions in numerical modeling, which is widely taken as an indicator of the collapse of the slope [11]. However, different criteria for identifying the critical state of slope in numerical modeling may lead to significant differences in the results.

Generally, the critical criteria for FLAC software used in strength reduction method are the running steps and the maximum nodal unbalanced force [14]. Nevertheless, the maximum nodal unbalanced force is used widely. The nodal unbalanced force is the sum of forces acting on a node from its neighbouring elements. If a node is in equilibrium state, these forces should sum to zero. However, the maximum nodal unbalanced force should be given by the user. Thus, the safety factor determined by this method is affected by user’s experience very largely. And the computing accuracy for the safety factor will be poor in most cases. To solve this problem, one method based on the minimum energy dissipation rate is proposed. In this method, the energy dissipation rate is applied to determine the status of the rock slope failure. The previous studies [15–17] show that rock slope failure is frequently controlled by a complex combination of discontinuities that facilitate kinematic release. The entire processes of the progressive slide surface development for rock slope related to crack initiation, propagation, coalescence, and degradation to eventual catastrophic failure are successfully captured. In other words, massive rock slope instability inherently requires the evolution of natural discontinuities through a gradual transition from a discontinuous-continuous to a fully discontinuous medium [17]. Therefore, to determine the status of the rock slope failure, it is very crucial to master the crack propagation in the rock slope. However, the energy dissipation is accompanied with the crack propagation [18, 19]. Energy dissipation can describe the fracture process of rock mass very well [19, 20]. Thus, using the energy dissipation rate, the safety factor of rock slope can be determined very accurately.

According to previous studies [19], the energy dissipation rate at any point nearby the crack tip whose coordinate is at any time during the fracture process of rock can be described as follows: where are polar coordinates when the crack tip is taking as the coordinate origin, is the time parameter of the rock failure process, is the name stress parameter near the crack tip, is the unrecoverable strain-rate of the point at time , is the nominal elastic modulus, is nominal Poisson’s ratio, is the damage variable at time , and the other parameters are the stresses of the point .

Moreover, when , , and when , . In other words, when , the energy consumption process of the point finishes, and then the point is completely damaged.

According to the principles of the fracture mechanics [21], the stress field nearby the crack tip iswhere , , and are stress intensity factors for crack modes I, II and III, respectively.

Equation (3) is substituted into (2); the following equation can be obtained:According to the least energy consumption principle [22], for one point nearby crack tip, the crack propagation orientation must be the orientation along which the energy dissipation rate is minimum. Thus, the orientation angle of crack propagation must satisfy the following conditions:Based on (4) and (5), the orientation angle of crack propagation along which the energy dissipation rate is minimum can be obtained. The minimum energy dissipation rate is one material parameter which can be determined by the test of rock material fracture. Thus, there exist the following equation:According to (6), along the orientation whose angle is , when the energy dissipation rate at the point nearby crack tip reaches the critical value , the crack propagation begins.

According to the conditions that , , , and , the minimum energy dissipation rate iswhere is the fracture toughness for crack mode I, which is a material constant.

According to the principles of coordinate transformation, (7) can be transformed aswhere is the maximum horizontal displacement.

To compute the minimum energy dissipation rate of rock slope failure, the maximum horizontal displacement must be computed beforehand. Therefore, the maximum horizontal displacement for rock slope can be obtained by the following Fish function of FLAC software: *def find_max_disp* *p_gp_gp_head* *maxdisp_value = *0.0 *maxdisp_gpid = *0 *loop while p_gp # null* *disp_gp = sqrt *(*gp_xdisp *(*p_gp*) ∧ 2) *if disp_gp > maxdisp_value* *maxdisp_value = disp_gp* *maxdisp_gpid = gp_id *(*p_gp*) *end if* *p_gp = gp_next *(*p_gp*) *end loop* *end* *find_max_disp* *print maxdisp_value maxdisp_gpid*With the obtained maximum horizontal displacement and (8), the minimum energy dissipation rate can be obtained be the following Fish function of FLAC software: *def Y* *p_z* =* zone_head* *d_ μ =*

*d_*

*π*=*d_*

*=*

*loop while p_z # null*

*r = disp_x/sin*∧ 2

*θ**/*

*π*r*Endloop*

*end*where , , and are constants that can be determined from the practical engineering conditions.

Based on the definition of safety factor by the minimum energy dissipation rate, the safety factor can be determined as follows.

Firstly, the minimum energy dissipation rate of the crack propagation is computed by an initial value of the reduction factor. When the values of the reduction factor increase, the minimum energy dissipation rate will decrease. Until the minimum energy dissipation rate approaches zero, the corresponding value of the reduction factor will be the critical value of the safety factor. In other words, when the minimum energy dissipation rate approaches zero, the crack will penetrate and coalesce; thus, the rock slope collapses.

Therefore, in this method, the energy dissipation rate is used to determine the slope stability status. Because the energy dissipation rate can estimate the moment of the crack propagation and coalescence for rock slope, the new method to determine the rock slope stability status based on the energy dissipation rate will be more precise than other methods.

#### 3. Engineering Example

In this study, one mining rock slope in western China is used, whose width, height, and gradient are 15 m, 18 m, and , respectively. According to the geological condition, there exist five original cracks in the rock slope, whose lengths are all 100 mm and inclination is all . The fracture toughness for all cracks is MPa·m^{1/2}. The dimension of the slope is shown in Figure 1, whose caption is dimensions of the mining rock slope.