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.

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 [

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 [

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 [

The strength reduction method was first proposed by Zienkiewicz et al. in 1975 [

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 [

Generally, the critical criteria for FLAC software used in strength reduction method are the running steps and the maximum nodal unbalanced force [

According to previous studies [

Moreover, when

According to the principles of the fracture mechanics [

Equation (

According to the conditions that

According to the principles of coordinate transformation, (

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:

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.

In this study, one mining rock slope in western China is used, whose width, height, and gradient are 15 m, 18 m, and ^{1/2}. The dimension of the slope is shown in Figure

Dimensions of the mining rock slope.

The material parameters of the rock slope are as follows.

The elastic modulus ^{3}. The uniaxial compressive strength is 18 MPa.

The numerical model of FLAC software for the rock slope is shown in Figure

Numerical model of rock slope.

The main process to determine the safety factor of rock slope by this new method is as follows.

The Fish function programs are embedded into FLAC software. According to the strength reduction method, the values of

Because there are many original cracks in this rock slope, based on the engineering experience, the critical safety factor for this rock slope should not be very large. Therefore, in this study, the values of strength reduction factor are taken as 1.30, 1.35, 1.40, 1.45, and 1.50. Using these values for strength reduction factor, the corresponding maximum horizontal displacement and the minimum energy dissipation rate can be obtained as shown in Figures

Results when strength reduction factor is 1.3.

Curve of maximum horizontal displacement

Curve of minimum energy dissipation rate

Results when strength reduction factor is 1.35.

Curve of maximum horizontal displacement

Curve of minimum energy dissipation rate

Results when strength reduction factor is 1.4.

Curve of maximum horizontal displacement

Curve of minimum energy dissipation rate

Results when strength reduction factor is 1.45.

Curve of maximum horizontal displacement

Curve of minimum energy dissipation rate

Results when strength reduction factor is 1.5.

Curve of maximum horizontal displacement

Curve of minimum energy dissipation rate

From the above figures, the following conclusions can be drawn.

When the strength reduction factor is 1.3, at the beginning, the maximum horizontal displacement increases very quickly. And it can reach about 20 cm in 2 seconds. After about 4 seconds, the rock slope will arrive at stability status. At this moment, the maximum horizontal displacement will be 25 cm. Moreover, at the beginning, the minimum energy dissipation rate is about 0.4. As the time increases, the minimum energy dissipation rate increases too. And the final value of minimum energy dissipation rate is 1. In other words, the rock slope is stable. Therefore, when the strength reduction factor is 1.3, the slope will not collapse; thus the safety factor should be larger than 1.3.

When the strength reduction factor is 1.35, at the beginning, the maximum horizontal displacement also increases very quickly. And it can reach about 50 cm in 1 second. After about 5 seconds, the rock slope will arrive at stability. At this moment, the maximum horizontal displacement will be 60 cm. Moreover, at the beginning, the minimum energy dissipation rate is about 0.2. As the time increases, the minimum energy dissipation rate increases too. And the final value of minimum energy dissipation rate is 1 too. In other words, the rock slope is stable. Therefore, when the strength reduction factor is 1.35, the slope will not collapse too; thus the safety factor should be larger than 1.35.

When the strength reduction factor is 1.4, although the final value of minimum energy dissipation rate also is 1, the maximum horizontal displacement will arrive at a large value, which is 1.3 m. In other words, although the slope is stable, its inner deformation is very large. Therefore, the rock slope should approach the instability.

When the strength reduction factor is 1.45, the maximum horizontal displacement increases as an exponential law. Thus, the maximum horizontal displacement cannot be stable after 8 seconds. However, the minimum energy dissipation rate decreases to zero after 7 seconds. Therefore, at this moment, the cracks in the rock slope penetrate. And the slope fails.

On the other hand when the strength reduction factor increases to 1.5, after only 4 seconds, the minimum energy dissipation rate will decrease to zero and the maximum horizontal displacement increases to larger than 3 m. Therefore, the rock slope completely fails under this condition.

The damage status of the rock slope under the condition that the values of strength reduction factor are 1.3, 1.4, and 1.45 is shown in Figure

Damage state of rock slope.

Strength reduction factor of 1.3

Strength reduction factor of 1.4

Strength reduction factor of 1.45

From Figure

Therefore, the safety factor of this slope should be between 1.4 and 1.45. In this study, the middle value of 1.42 is selected as the safety factor of this slope.

To verify the new method proposed in this study and compare its results with those of other methods, the generally used strength reduction method is applied to analyze the rock slope.

Firstly, the initial force equilibrium status of the rock slope is computed. The curve of the unbalanced force for the rock slope system is shown in Figure

Curve of unbalanced force for the rock slope system.

As for the status of force equilibrium, the horizontal stress and vertical stress in the rock slope are computed as shown in Figure

Stress distribution of the rock slope.

Horizontal stress

Vertical stress

From Figure

After the computation for the force equilibrium is complete, the safety factor of rock slope can be obtained by the strength reduction method. The computed safety factor is 1.51. The computed result is shown in Figure

Result of the generally used strength reduction method for the rock slope.

The computed safety factors by the new method is 1.42 and that by the generally used strength reduction method is 1.51. The difference of two values for the safety factors is not very large. Therefore, the new method based on the minimum energy dissipation rate is suitable to analyze the stability of the fractured rock slope. However, the safety factor by the new method is less than that by the generally used strength reduction method. In other words, the new method can obtain one more dangerous stability status than that by other methods. Therefore, the new method should be superior to other methods, such as generally used strength reduction method.

It is very important to analyze the stability of rock slope, which is a very complicated engineering problem. Nowadays, the numerical method, such as strength reduction method, is the main method to solve this problem. To overcome the shortcomings of the generally used strength reduction method, such as the expensive computation and convergence issues due to the nonlinear iterative computations, from the basic principle of rock slope failure, that is, the rock slope failure is the process of the propagation and coalescence for cracks in rock slope, one method based on the minimum energy dissipation rate is proposed. By analysis of one mining rock slope in western China, this new method is verified. From the studies, the following conclusions can be drawn:

The massive rock slope instability inherently requires the evolution of natural discontinuities through a gradual transition from a discontinuous-continuous to a fully discontinuous medium. To determine the status of the rock slope failure, it is very crucial to master the crack propagation in the rock slope.

The energy dissipation is accompanied with the crack propagation. Thus, energy dissipation can describe the fracture process of rock mass very well. Using the energy dissipation rate, the safety factor of rock slope can be determined very accurately.

When the strength reduction factor is 1.4, the cracks in the rock slope open and propagate at some extent. When the strength reduction factor increases to 1.45, the cracks in the rock slope propagate very quickly and coalesce completely. Therefore, the safety factor of this slope should be between 1.4 and 1.45.

The computed safety factor by the new method is similar to that by the generally used strength reduction method. Therefore, the new method based on the minimum energy dissipation rate is a suitable method to analyze the stability of the fractured rock slope.

The new method can obtain one more dangerous stability status of the fractured rock slope than that by generally used strength reduction method. Thus, the new method should be superior to the generally used strength reduction method.

The authors declare that they have no competing interests.

The financial supports from The Fundamental Research Funds for the Central Universities under Grant nos. 2014B17814, 2016B10214, 2014B07014, and B15020060 are all gratefully acknowledged.