#### Abstract

The nonlinear strength criterion is introduced into the paper firstly, and then the geometric anchoring characteristic model of rock slope separated from two layers according to rock type with the fracture surface of log-spiral curve considering nonlinear twin-shear criterion subjected to seismic loads is established. Finally, some correlated influential parameters are analyzed, and the conclusions are drawn, showing that the intermediate principal stress has great influence on the total anchoring force of bolt *P*, so when different criteria are selected, the influence of intermediate principal stress should not be omitted. The effects of thickness ratio *h*_{1}/*h*_{2} on total anchoring force increase gradually with the increase of inclination angle . The total anchoring force *P* decreases gradually with the increase of inclination angle and surcharge load *q*. The total anchoring force increases gradually with the increase of horizontal seismic acceleration coefficients *k*_{h} and geometric coefficients *m*_{i} and GSI.

#### 1. Introduction

The destabilization of slope is a complex and dangerous geological disaster phenomenon [1]; it brings about huge harms to the safety of life and property, so the investigation on the slope stability is always focused on [2–4]. Many supporting schemes are adopted in the process of slope governance, for example, antislide pile and retaining wall. Meanwhile, their many shortcomings are found in the application of civil engineering [5], such as high cost, difficult construction, and heavy work. To overcome the mentioned problems [6], the application of the anchoring bar on the slope solves the above problem, and anchoring bar plays a great role in the support of slope [7]. So, a great deal of theoretical and experimental investigations about the supporting theory of anchoring bar are performed. The mechanical mechanism of the load transfer of anchoring bar is studied by Lutz et al. [8–10]. The influential factors about the bearing capacity of anchoring bar are analyzed by Hyet and Badwen [11] using a great deal of experimental investigations. In China, the further investigation about the anchoring technology in the geotechnical engineering is performed. The fracture modes of foundation pit with the support of anchoring bar are investigated by Jin-Qing and Zheng [12] using the limit equilibrium theory and finite difference method. Then, the optimum design of mechanical model is calculated by Li-De and Cong-Xin [13] by using the strength reduction method of finite element. The distribution law of loads along the anchoring bar and the mechanism of anchoring forces are studied by Gamse and Oberguggenberger [14] by using numerical simulation and theoretical investigation. Then, the consolidation actions of anchoring bar on the layer rock slope are simulated by Chen et al. [15] using the finite difference software FLAC3D. To investigate the anchoring mechanism of rock slope, lots of fracture criteria are utilized [16]; for example, Mohr–Coulomb criterion [17] and Hoek–Brown criterion [18] are often adopted; although the obvious effects are obtained in the actual engineering, the influences of intermediate principal stress in these criteria are often omitted [19, 20], and a large deal of engineering practices demonstrated that the influence of intermediate principal stress should be valued [21], so the twin-shear strength criterion in the paper is introduced; and, to conform to the engineering practice, earthquake loads, different rock types, and fracture modes of rock slope are, respectively, considered in the paper, a new thought and method for the stability design of rock slope can be provided.

The paper is organized as follows. In Section 2, nonlinear twin-shear criteria and method of upper bound limit analysis are introduced at first. In Section 3, the mechanical model of rock slope with the anchoring bar is established, the kinematic method of limit analysis considering nonlinear twin-shear criteria is deduced, and the anchoring mechanism of rock slope is analyzed. In Section 4, results and discussions are performed. In Section 5, conclusions are drawn.

#### 2. Method and Theory

##### 2.1. The Relevant Fracture Criterion

The nonlinear united strength failure criterion can be expressed as [22]where is the uniaxial compressive strength of intact rock, is constant value, and represents the intermediate principle stress coefficients; is a constant value relevant to the rock mass and it can be determined in the two following equations:where is a disturbance coefficient, and its range is from 0.0 to 1.0. is the geological strength index, and its range is from 0 to 80. The magnitude of is obtained from compression tests on an intact rock. If no test data are available, Hoek parents the approximate values for five types of rocks as follows [23]: (1)When the value of , it is applied to calculate carbonate rock types, such as dolomite, limestone, and marble(2)When the value of , it is applied to calculate lithified argillaceous rock types, such as mudstone, siltstone shale, and slate(3)When the value of , it is applied to calculate arenaceous rock types, such as mudstone, siltstone shale, and slate(4)When the value of , it is applied to calculate fine-grained polymineralic igneous crystalline rock types, such as andesite, dolerite, and diabase(5)When the value of , it is applied to calculate coarse-grained polymineralic igneous and metamorphic rock types, such as amphibolite, gabbro, gneiss, granite, and quartz diorite

In equations (1) and (2), when , this demonstrates that the influences of the intermediate principal stress are omitted, so Hoek–Brown criterion can be expressed as follows:

For *c* = 1, this demonstrates that the influences of the intermediate principal stress are taken into consideration, and the nonlinear united strength criterion can be obtained as follows [24]:

##### 2.2. Kinematic Method of Limit Analysis

The seismic stability analysis is performed within the framework of limit analysis theory [25]. To get the upper bound solution, the construction of velocity is required in the method. The implementation of the kinematic method of limit analysis depends on the following fundamental inequality [26]: where is any virtual admissible velocity field. is the work performed by external loading, and is the maximum resisting work originating from the rock materials. The formula of maximum resisting work according to Saada’s work [27] can be obtained as where is the strain rate sensor associated with at any point of rock mass volume . means the jump at a point through a possible velocity discontinuity surface and following its normal , and the -functions are the support functions defined by the duality from the strength condition .

For nonlinear twin-shear rock masses, the corresponding 3D closed-form expressions of the foregoing -functions are obtained as follows [28]:where , , and represent the eigenvalues of .

Similarly, it has been found that the -functions relative to a velocity jump and defined by equation (7) take the following form: where

It should be noted that the conditions in equation (11) and in equation (13) demonstrate the fact that and , respectively. These conditions are necessary for the kinematic method of limit analysis expressed by equation (7) which results in nontrivial upper bound solutions.

#### 3. The Establishment of Mechanical Model

##### 3.1. The Analysis of Failure Mechanism

The failure mechanism, depicted in Figure 1 and considered in the investigation, is a direct transportation of those usually employed for homogeneous nonlinear twin-shear strengthen slope. In such model, two layers of rock mass are composed in the rock mass slope, there are, respectively, different compressive strengths and unit weight in two layers, and the earthquake loads are considered in two layers of rock mass; and the quasistatic method is adopted in the manuscript. Because horizontal wave has a great influence on the fracture of rock slope, only horizontal action of earthquake loads is considered. Its inclined angle along the fracture surface is . The failure routine is a straight line in the upper layers of rock slope, and it is assumed that the fracture surface is parallel with the rock slope surface, so quadrangle is parallel quadrangle. There exist uniform distribution external loads in the roof of rock slope. There are lots of support bars in the rock slope surface. It is assumed that fracture surface of rock slope represents a log-spiral curve in lower layer of rock slope, a volume of rock mass is rotating about a point with an angular velocity , the curve separating this volume from the structure that is kept motionless is a log-spiral arc and focus , and it necessarily follows that the velocity jump at any point of line is inclined at angle with respect to the tangent at the same point. The inclination angle of anchoring bar is with a horizontal direction. Five parameters are involved in such a mechanism: four angles , , , and and the distance defining the radius of log-spiral curve for ; accordingly, this curve is defined in polar coordinates by equation .

##### 3.2. The Work Rate of External Forces and Maximum Resisting Work Rate

###### 3.2.1. The Work of External Forces in the Upper Layer of Rock Slope

(1)The work rate of gravitational force is where is the work rate of gravitational force in the upper layer of rock slope; is the unit weight of rock mass in the upper layer of rock slope.(2)The work rate of external load *q* is where is the work rate of uniform distribution load and is the width of roof.(3)The work rate of seismic load is where is the work rate of horizontal earthquake loads in the upper layer; is the horizontal earthquake load coefficient.(4)The work rate of roof bolt is where is the work rate of antisliding force about anchors in the upper layer; is the antisliding force of the roof bolt.(5)The maximum resisting work rate is

where is the maximum resisting work rate in the upper layer; is the compressive strength of rock mass in the upper layer.

###### 3.2.2. The Work Rate of External Forces in the Lower Layer of Rock Slope

(1)The work rate of gravitational force is where is the work of gravitational force; expressions of , , and can be, respectively, expressed as (2)The work rate of seismic load is where is the work rate of seismic load. The expressions of , , and can be, respectively, expressed as where different parameters are shown in formulas (20)–(21c). The expressions of and can be, respectively, expressed as (3)The work rate of roof bolt is where is the work rate of antislide force of roof bolts; is the antisliding force of the bolt. is the number of roof bolts.(4)The maximum resisting work rate is defined as follows.

The maximum resisting work results from the velocity jump along the log-spiral line , and the expression of resisting work derived from equations (10)–(12) can be depicted as follows:

###### 3.2.3. The Expression of Anchoring Force

According to the fundamental inequality (7), the expression of anchoring force can be shown as follows:

###### 3.2.4. The Critical Anchoring Force

According to the investigation of Ling and Leshchinsky [29], the expression of the total anchoring force is defined as

According to the principal of limit analysis theory, when , the critical total anchoring force can be expressed as

#### 4. Results and Discussion

##### 4.1. The Effects of Intermediate Principal Stress on the Total Anchoring Force

To investigate the influence of intermediate principal stress on the total anchoring force, the following parameters are listed as follows: = 12 m, mi = 10, GSI = 10, and *b* = 5 m; , , , , *q* = 20 kPa, *ω* = 0.2, V = 0.1 mm/s, *σ*cu = 6 MPa, *σ*cl = 10 MPa, , kh = 0.18, *θ*1 = 200, *θ*2 = 500, *θ* = 300, , , and . It is shown in Figure 2 that the total anchoring force *P* increases as the velocity inclination angle increases whether Hoek–Brown criterion [30] or twin-shear strength criterion strengthens. However, it can be found in comparison that the magnitude for (it means that the influence of intermediate principal stress is not considered) is higher than one of (it means that the influence of intermediate principal stress is considered), and the conclusions are drawn, showing that the intermediate principal stress has a great influence on total anchoring force of bolt , so when different criteria are selected, the influence of intermediate principal stress is not omitted.

##### 4.2. The Influence of Bolt Inclination Angle on the Total Anchoring Force

The range on the inclination angle of bolt is 10–35°, so the inclination angle of anchor is, respectively, adopted for *β* = 10°, 20°, and 30° and the comparison and analysis are carried out among three cases. The following parameters are determined: = 12 m, *m*_{i} = 10, GSI = 10, and *b* = 5 m; , , , , *q* = 20 kPa, *ω* = 0.2, *V* = 0.1 mm/s, *σ*_{cu} = 6 MPa, *σ*_{cl} = 10 MPa, = 20°, *k*_{h} = 0.18, *θ*_{1} = 20°, *θ*_{2} = 50°, *θ* = 30°, and . The effects of anchoring inclination angle *α* on the total anchoring force are depicted in Figure 3. It can be found that the total anchoring force *P* decreases gradually with the increase of inclined angle ; and it can also be found that the total anchoring force *P* increases gradually as the anchoring inclination angle increases. The influence of anchoring inclination angle on the total anchoring force decreases gradually with the increase of velocity inclination angle . For example, for , the value of *P* increases approximately by 108% when the value of increases from 10° to 30°; then, for , it increases only by 59.8%.

##### 4.3. The Influence of Rock Layer Thickness on the Total Anchoring Force *P*

The effects of thickness of rock layer on the total anchoring force are depicted in Figure 4. To explain the relation, the ratio of thickness between two rock layers is, respectively, selected as 1/3, 1/2, and 2/3 in this paper, and the following parameters are listed: = 12 m, mi = 10, GSI = 10, and *b* = 5 m; and ; ; ; *q* = 20 kPa, *ω* = 0.2, *V* = 0.1 mm/s, *σ*cu = 6 MPa, *σ*cl = 10 MPa, = 200, kh = 0.18, *θ*1 = 200, *θ*2 = 500, *θ* = 300, , and . It can be found in Figure 4 that the total anchoring force increases gradually as the thickness ratio of rock layer (namely, different rock types) increases, but it can be also found that the effects of thickness ratio on total anchoring force increase gradually with the increase of inclination angle ; to illustrate, for = 100, when increases from 1/2 to 2/3, the value of total anchoring force *P* increases approximately by 11.76%; then, for = 250, when increases from 1/2 to 2/3, the value of increases by 27.12%. So, conclusions can be drawn, showing that the total anchoring force increases gradually with the increase of thickness ratio, but the influence of the thickness ratio on the anchoring force gradually increases with the increase of velocity inclination angle .

##### 4.4. The Influence of Surcharge Load *q* on the Anchoring Force

When the influence of surcharge load is taken into account, to explain the relation, the value of surcharge load *q* is, respectively, adopted as 10 MPa, 15 Mpa, and 20 MPa in this paper. The following parameters are listed: = 12 m, mi = 10, GSI = 10, and *b* = 5 m; and ; ; ; *ω* = 0.2, V = 0.1 mm/s, *σ*cu = 6 MPa, *σ*cl = 10 MPa, = 200, kh = 0.18, *θ*1 = 200, *θ*2 = 500, *θ* = 300, , and ; the effects of surcharge load *q* on the total anchoring force are depicted in Figure 5. It can be found that the total anchoring force decreases gradually with the increase of inclination angle and surcharge load . So, conclusions can be drawn, showing that the surcharge load of rock slope has great influences on the total anchoring force .

##### 4.5. The Influence of Horizontal Seismic Acceleration Coefficients on the Total Anchoring Force

When the influences of horizontal seismic acceleration coefficients on the anchoring force are considered, the following parameters are listed: = 12 m, mi = 10, GSI = 10, and *b* = 5 m; and ; ; ; *q* = 30 kPa, *ω* = 0.2, *V* = 0.1 mm/s, *σ*cu = 6 MPa, *σ*cl = 10 MPa, = 200, *θ*1 = 200, *θ*2 = 500, *θ* = 300, and . The effects of on the total anchoring force are plotted in Figure 6. It can be found in Figure 6 that the total anchoring force increases gradually with the increase of horizontal seismic acceleration coefficients; it demonstrates that the slope becomes more dangerous; for example, for *φ* = 100, when the value of increases from 0.08 to 0.24, the total anchoring force increases from 0.5206 to 1.0322.

##### 4.6. The Influence of Geometric Coefficients and on the Total Anchoring Force

When the influences of geometric coefficients and on the anchoring force are, respectively, regarded, the following parameters are listed: = 12 m; *b* = 5 m; and ; ; ; *q* = 30 kPa, *ω* = 0.2, *V* = 0.1 mm/s, *σ*cu = 6 MPa, *σ*cl = 10 MPa, = 200, = 0.18, *θ*1 = 200, *θ*2 = 500, *θ* = 300, and . The effects of and on the anchoring force are, respectively, shown in Table 1. It can be found in Table 1 that the total anchoring force decreases gradually with the increase of geometric coefficients and , and the geometric parameters have a great influence on the total anchoring force .

#### 5. Conclusions

The limit dynamic analysis method of upper bound solution based on nonlinear twin-shear criterion is adopted to analyze anchoring mechanism of two layers of failure slope in the investigation. The intermediate principal stress, the anchoring inclination angle , thickness ratio of rock layer, surcharge load , seismic acceleration coefficients , and geometric coefficients and have great influences on the total anchoring force. Finally, the following conclusions are drawn: (1)The intermediate principal stress has a great influence on the total anchoring force of bolt , so when different criteria are selected, the influence of intermediate principal stress is not omitted. The total anchoring force decreases gradually with the increase of inclination angle . The total anchoring force increases gradually as the anchoring inclination angle increases. The influence of anchoring inclination angle on the total anchoring force decreases gradually with the increase of velocity inclination angle .(2)The total anchoring force increases gradually as the thickness ratio of rock layer (namely, different rock types) increases. But the effects of thickness ratio on total anchoring force increase gradually with the increase of inclination angle . The total anchoring force decreases gradually with the increase of inclination angle and surcharge load .(3)The total anchoring force increases gradually with the increase of horizontal seismic acceleration coefficients. The total anchoring force decreases gradually with the increase of geometric coefficients and , and the geometric parameters have a great influence on the total anchoring force .

#### Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This work was supported by the first batch of Natural Science Foundation of Sichuan Provincial Department of Education (no. 17ZA0270), the cross project in Nanyang Institute of Technology (230067), the opening projection of Sichuan Province University Key Laboratory of Bridge Non-Destruction Detecting and Engineering Computing (2017QYY01), the National Natural Science Foundation of China (41672357), the Start-Up Foundation (510126), and the Sichuan Science and Technology Program (no. 2020YJ0424).