# Stability Analysis Method for Rock Slope with an Irregular Shear Plane Based on Interface Model

Authors: Changqing Qi; Jiabing Qi; Liuyang Li; Jin Liu
Journal: Advances in Civil Engineering
 (2018)

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## Abstract

Landslide developed in rock mass usually has irregular shear plane. An approach for calculating distributed factor of safety of the irregular shear plane was put forward in this paper. The presented method can obtain not only the detailed stability status at any grid node of a complex shear plane but also the global safety of the slope. Thus, it is helpful to thoroughly understand the mechanism of slope failure. Comparing with the result obtained through the limit equilibrium method, the presented method was proved to be more accurate and suitable for stability analysis of rock slope with a thin shear plane. The stability of a potentially unstable rock slope was analyzed based on the presented method at the end of this paper. The detailed local stability, global stability, and the potential failure mechanism were provided.


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## Body

## 1. Introduction

Landslide is one of the most severe geohazards threatening the interests of human [1, 2]. To prevent great losses of life and property, early hazard appraisal and assessment techniques needed to be implemented. Large landslides developed in rock mountains often have complex failure mechanism [3, 4]. The distributed safety along the sliding surface is as important as global safety in thoroughly understanding the failure behaviors of the slope. Hence, a convenient and practical approach to evaluate the stability of a slope should be developed which allows for the distributed safety along the sliding surface. Quantitative stability assessment method is useful in reducing the risk of disaster and mitigating the geohazardous damage. Despite the progress of simulation techniques, the use of the factor of safety (FOS) is still the most commonly adopted method in slope stability analysis. The wildly accepted definition of FOS is the ratio of shear strength of soil to shear stress required for equilibrium [4]. Among the various approaches to determine the FOS, limit equilibrium method and strength reduction method are the ordinary used methods [5–14]. These two methods are more convenient for 2D problem and regular slip surface. Some researchers have provided methods for three-dimensional slope stability analysis [15–18], but the obtained FOS was too large to use [4]. Furthermore, these methods can only obtain the global FOS of the slope. This is not enough for understanding the behavior of slope failure. The FOS may vary on different parts of the sliding surface [19].The landslides developed in rock mass usually controlled by the combination of discontinuities. The separated part is often a 3D irregular geological body. The FOS at different parts of the separated surfaces must be addressed. A method for calculating the distributed FOS along the irregular sliding surfaces in rock slope was presented. On the basis of the definition of FOS and using the interface element model, FOS at any part of the sliding surface can be obtained. Then, the global FOS of the slope was given by the weighted average of shear strength to shear stress in the overall sliding direction, which can estimate the global stability of the slope.


## 2. Methods

The interface model of Flac3D [20] was introduced to simulate the thin and weak shear plane. An interface is a connection face between two materials that is characterized by Coulomb sliding and/or tensile and shear bonding. The interface was discretized into triangular interface element. The interface node is located at the vertex of the element and has an associated representative area (Figure 1). The Coulomb shear strength criterion was employed to simulate the behavior of the interface. Flac3D is an explicit finite difference program. The program adopts the explicit, Lagrangian calculation scheme and the mixed-discretization zoning technique in solving the dynamic equations. By assuming linear variations of the variable over finite space and time intervals, derivatives of a variable with respect to space and time are approximated by finite differences. The solution requires a number of computational timesteps to reach an equilibrium status. The normal and shear stress at any computational timestep t+Δt on the representative area of a node can be obtained by(1)σt+Δt=knun+σ0,τt+Δt=τt+ksΔust+1/2Δt+τ0,where un is the absolute normal penetration of the interface node into the target face, Δus is the incremental relative shear displacement vector, σ is the normal stress, τ is the shear stress, kn is the normal stiffness, and ks is the shear stiffness.Figure 1
Interface element of Flac3D.The normal stiffnesskn and the shear stiffness ks were suggested to ten times the equivalent stiffness of the stiffest neighboring zone [20]:(2)kn=ks=10K+4/3GΔzmin,where K and G are the bulk and shear moduli of the softer side material and Δzmin is the smallest width of an adjoining zone of the softer side in the normal direction to the interface.(3)K=E31−2μ,G=E21+u,where E is Young’s modulus and μ is Poisson’s ratio.After obtaining the normal and shear stress, the FOS at any stress node can be given as follows based on the definition of FOS [5]:(4)fs=τuτ=c+σ−pw×tan⁡φτ,where fs represents the factor of safety at the grid node, τu represents shear strength, c is the cohesion, φ is the friction angle, and pw is the pore pressure when below ground water table. fs can be the referential index for stability analysis. It can give the safety status at any grid node of the slip surface.Weighted with the area of the interface node, the global FOS of the shear plane can be given as follows:(5)Fs=∑i=1nσi−pwitan⁡ϕi⁡sin⁡δi×Ai+cAi∑i=1nτi×cos⁡δi×Ai,where Fs is the global FOS of sliding surface, n is the node on the interface, Ai is the representative area of the ith node, and δi is the angle between the shear vector and the project of the overall shear direction on the representative area of the ith node.


## 3. Method Verification

The limit equilibrium method is the widely used and accepted slope stability analysis method. It was adopted to verify our method for a unit width slope with a thin-layer shear plane. The slope is 43 m in horizontal and 32.5 m in elevation. The shear plane was set to be 5 cm in the limit equilibrium model and simulated as an interface in the numerical model. The properties of the soils are given in Table1.Table 1
Properties of soils used in the verification model.
Soil type
Unit weight (kN/m3)
Young’s modulus (GPa)
Poisson’s ratio
Cohesion (kPa)
Friction angle (°)
Unstable slope
20.5
1.0
0.35
40
30
Shear plane
20.0
0.2
0.35
15
24
Bedrock mass
24.0
10
0.3
70
35The software Slide 6.0 [21] was used, and the Bishop method was selected in limit equilibrium analysis. The noncircular surface type was chosen, and the block search method was adopted. The geometric model and the result determined by the Bishop method are shown in Figure 2. The global FOS along the given shear plane is 1.109.Figure 2
Limit equilibrium result using the Bishop method of the verification model.Figure3 shows the numerical model and the result obtained by the method presented in this paper. The shear stiffness and normal stiffness obtained from (2) are both 3.2 × 1010 Pa/m. The smallest width of the adjoining zone in the normal direction to the interface is 0.5 m. The global FOS obtained by our method is 1.101. The error between the two methods is only 0.7%. It proved that the presented method has high accuracy and is acceptable in slope stability analysis. Figure 3 also shows the distribution of the FOS along the shear plane. The FOS on the back scarp of the shear plane is lower. Actually, most area of the back scarp has a FOS lower than 1.0, indicating a potential instability status of the back scarp. The FOS on the bottom sliding surface increases from the back scarp to the toe, which gives rise to a slide resistance near the toe. This phenomenon coincides with the common mechanical characteristics of a slope with a weak shear plane [22–24].Figure 3
Result obtained by the presented method of the verification model.


## 4. Application and Implementation

The presented method was used for stability analyses of a potentially unstable rock slope in Sichuan Province, Southwest China. The potentially unstable slope, located at the flank of the Yalong River, has a total volume of about 800 million·m3. It covers upslope from 2380 m to 2730 m a.s.l. (Figure 4). The shear plane was explored by boreholes. It developed from the discontinuities of the rock mass. The studied slope will be in the reservoir of a proposed hydraulic power station. The toe will be immersed in the dammed water. So the stability and potential sliding mechanism of the slope need to be studied in detail.Figure 4
Longitudinal section of the studied slope (MW, moderately weathered sandstone; SW, slightly weathered sandstone; WT, water table).The slope can be distinguished as four geologic layers from the borehole data. The top unstable slope body, with a maximum thickness of 80.0 m, overlies a layer of heavily weathered sandstone approximately 15 m thick. A moderately weathered sandstone layer underlies the heavily weathered sandstone and extends to a depth of 140–145 m. The bottom bedrock is the fresh sandstone. The physical and mechanical properties of the relevant materials are shown in Table2. As suggested in some previous studies [25, 26], the shear strength parameters used in the analysis are opportunely reduced by a factor of 0.75 for the potentially unstable soil mass and the failure surface, and 0.85 for the bedrock, to account for the influence of moisture.Table 2
Physical and mechanical properties of relevant materials.
Soil type
Natural weight (kN/m3)
Saturated weight (kN/m3)
Young’s modulus (GPa)
Poisson’s ratio
Cohesion (kPa)
Friction angle (°)
Potential unstable slope body
20.5
22.6
0.5
0.35
40
30
Shear plane
19.0
21.0
NA
0.35
15
22.5
MW sandstone
24.0
25.5
10
0.3
70
35
SW sandstone
25.0
NA
35
0.2
300
45
Fresh sandstone
27.0
NA
45
0.2
800
50The shear plane was considered as an interface in the numerical model. The shear stiffness and normal stiffness obtained from (2) are both 2 × 109 Pa/m. The smallest width of the adjoining zone in the normal direction to the interface is 4.0 m. The three-dimensional numerical grids of the slope and the interface are shown in Figure 5.Figure 5
Numerical grid of the slope and the interface.


## 5. Result Analyses

### 5.1. Slope Stability at Present

Figure6 gives the distribution of normal stress and shear stress on the interface. It can be seen from Figure 6 that the normal stress mainly concentrates on the lower part of the interface. The maximum normal stress appears at the transition area where the slope of the shear plane changes from steep to gentle. The maximum normal stress is about 1.4 MPa. The normal stress approaches to zero near the border of the shear plane. On contrary to the distribution of normal stress, the shear stress concentrates mainly on the upper part of the shear plane, indicating a large sliding potential of the upper slope. The shear stress is also small near the border and at the areas with gentle slope angles.Figure 6
Distribution of normal stress (a) and shear stress (b) at the natural state.
(a)
(b)The FOS on the shear plane was obtained from (4). Figure 7 shows the distribution of the FOS. It indicates that there are several small areas near the border of the shear plane where the factors of safety are smaller than 1.0. Most portion of the shear plane has factors of safety ranging from 1.0 to 1.5, indicating a relatively stable condition. At the lower-middle part and the toe, there are some areas where the factors of safety are above 2.0.Figure 7
Distribution of factor of safety at the natural state.Figure8 illustrates the nodal FOS on the shear plane. It can give more comprehensive information about the stability of the slope. There are sporadic points where the factors of safety are below 1.0 (red point), indicating there will be cracks or breaks near the border of the unstable body. The FOS increases downslope along the shear plane. At the upper one-third of the shear plane, nearly all the nodal factors of safety are less than 1.05. It can be concluded that the head of the slope body has a large instability potential. The top one-third of the slope lacks sufficient ability of self-stability and poses a residual thrust force on the lower part. At the middle one-third of the shear plane, most nodal factors of safety are between 1.05 and 1.5, implying a relatively stable state. At the lower one-third of the shear plane, there are lots of points with factors of safety larger than 1.5. The middle and lower parts of slope with higher FOS act as sliding resistance and provide necessary safety capacity for the stability of the slope. The global FOS of the slope obtained from (5) is 1.23. This value proves that the entire slope is more stable at present.Figure 8
Point factor of safety at the natural state.


### 5.2. Slope Stability after Impounded

The distribution of normal and shear stresses after the proposed reservoir impounded is given in Figure9. The maximum value of the normal stress reduces a little at the vicinity of the water level when the proposed reservoir is in operation. But the shear stress below water level increases greatly comparing with the natural condition. The large shear stress near the toe increases the risk of slope failure.Figure 9
Distribution of normal stress (a) and shear stress (b) after impounded.
(a)
(b)Figure10 exhibits the distribution of FOS on the shear plane after the proposed reservoir impounded. It can be seen that there are more unstable areas (with FOS < 1.0). This phenomenon is more obvious at the toe of the slope. The factors of safety on a majority of areas are below 1.5, and only very small areas have factors of safety larger than 2.0.Figure 10
Distribution of factor of safety after impounded.Figure11 demonstrates the nodal FOS on the shear plane after reservoir impounded. It reveals that the FOS decreases at nearly all the interface nodes. At the lateral borders, more unstable points appear and the distribution of unstable points approaches inward. At the top one-third and the toe of the shear plane, nearly half of the nodes are unstable. It suggests that the top one-third slope will throw more residual thrust on the lower part, and there may be failures at the toe of the slope. There are several points with factors of safety above 1.5 and only one point with a FOS larger than 2.0. Thus, the sliding resistance of the lower part will reduce. The global FOS has a clearly reduction and is now 1.09. The slope is only slightly stable and cannot ensure the safety of the project in the view of engineering. The lower part of the slope must be reinforced.Figure 11
Point factor of safety after impounded.


## 5.1. Slope Stability at Present

Figure6 gives the distribution of normal stress and shear stress on the interface. It can be seen from Figure 6 that the normal stress mainly concentrates on the lower part of the interface. The maximum normal stress appears at the transition area where the slope of the shear plane changes from steep to gentle. The maximum normal stress is about 1.4 MPa. The normal stress approaches to zero near the border of the shear plane. On contrary to the distribution of normal stress, the shear stress concentrates mainly on the upper part of the shear plane, indicating a large sliding potential of the upper slope. The shear stress is also small near the border and at the areas with gentle slope angles.Figure 6
Distribution of normal stress (a) and shear stress (b) at the natural state.
(a)
(b)The FOS on the shear plane was obtained from (4). Figure 7 shows the distribution of the FOS. It indicates that there are several small areas near the border of the shear plane where the factors of safety are smaller than 1.0. Most portion of the shear plane has factors of safety ranging from 1.0 to 1.5, indicating a relatively stable condition. At the lower-middle part and the toe, there are some areas where the factors of safety are above 2.0.Figure 7
Distribution of factor of safety at the natural state.Figure8 illustrates the nodal FOS on the shear plane. It can give more comprehensive information about the stability of the slope. There are sporadic points where the factors of safety are below 1.0 (red point), indicating there will be cracks or breaks near the border of the unstable body. The FOS increases downslope along the shear plane. At the upper one-third of the shear plane, nearly all the nodal factors of safety are less than 1.05. It can be concluded that the head of the slope body has a large instability potential. The top one-third of the slope lacks sufficient ability of self-stability and poses a residual thrust force on the lower part. At the middle one-third of the shear plane, most nodal factors of safety are between 1.05 and 1.5, implying a relatively stable state. At the lower one-third of the shear plane, there are lots of points with factors of safety larger than 1.5. The middle and lower parts of slope with higher FOS act as sliding resistance and provide necessary safety capacity for the stability of the slope. The global FOS of the slope obtained from (5) is 1.23. This value proves that the entire slope is more stable at present.Figure 8
Point factor of safety at the natural state.


## 5.2. Slope Stability after Impounded

The distribution of normal and shear stresses after the proposed reservoir impounded is given in Figure9. The maximum value of the normal stress reduces a little at the vicinity of the water level when the proposed reservoir is in operation. But the shear stress below water level increases greatly comparing with the natural condition. The large shear stress near the toe increases the risk of slope failure.Figure 9
Distribution of normal stress (a) and shear stress (b) after impounded.
(a)
(b)Figure10 exhibits the distribution of FOS on the shear plane after the proposed reservoir impounded. It can be seen that there are more unstable areas (with FOS < 1.0). This phenomenon is more obvious at the toe of the slope. The factors of safety on a majority of areas are below 1.5, and only very small areas have factors of safety larger than 2.0.Figure 10
Distribution of factor of safety after impounded.Figure11 demonstrates the nodal FOS on the shear plane after reservoir impounded. It reveals that the FOS decreases at nearly all the interface nodes. At the lateral borders, more unstable points appear and the distribution of unstable points approaches inward. At the top one-third and the toe of the shear plane, nearly half of the nodes are unstable. It suggests that the top one-third slope will throw more residual thrust on the lower part, and there may be failures at the toe of the slope. There are several points with factors of safety above 1.5 and only one point with a FOS larger than 2.0. Thus, the sliding resistance of the lower part will reduce. The global FOS has a clearly reduction and is now 1.09. The slope is only slightly stable and cannot ensure the safety of the project in the view of engineering. The lower part of the slope must be reinforced.Figure 11
Point factor of safety after impounded.


## 6. Conclusion

A method for distributed FOS calculation of irregular three-dimensional shear plane was presented in this paper. This method is more suitable for stability assessment of rock mass slope with thin-layer shear zone. The presented method was first verified and implemented and then was applied in stability assessment of a high-focused rock slope. The research leads to the following conclusions:(1)
The presented method can give the distribution and global FOS of irregular shear plane. It is useful in sliding mechanism understanding and failure mode distinguishing, as well as global stability evaluation. Comparing with the traditional FOS analysis method, the presented method can give more direct and detailed reference for landslide disaster mitigation.(2)
The error of global FOS between the presented method and Bishop method is only 0.7% for a typical rock slope. The proposed method was proved to be accurate and suitable in stability analysis of rock slope with thin shear planes.(3)
The stability of a practical rock slope was analyzed using the present method. The global FOS of the slope is 1.23 at the natural condition. The top one-third part of studied slope has a sliding potential and poses residual thrust on the lower part. The one-third part near the toe of the slope is more stable and provides sliding resistance to the entire slope. The entire slope is currently more stable.(4)
The global and nodal factors of safety both decrease when the toe of the slope was immersed in the proposed damming water. The global factor of safety reduces to 1.09, and some areas below the water level become unstable. Considering the elevated risk of landslide hazards and the threaten to the engineering infrastructure of the proposed hydropower station, slope reinforcement measures, such as pile-anchor support, surface water control, or material cutting, should be performed.



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*Source: 8190908-2018-04-01.xml*