# The Analysis of Anchoring Mechanism of Rock Slope in Two Layers Based on the Nonlinear Twin-Shear Strength Criterion

Authors: X. B. Gu; Y. H. Wang; X. J. Ji; Y. H. Zhu
Journal: Advances in Civil Engineering
 (2021)

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## 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 boltP, so when different criteria are selected, the influence of intermediate principal stress should not be omitted. The effects of thickness ratio h1/h2 on total anchoring force increase gradually with the increase of inclination angle ϕ. The total anchoring force P decreases gradually with the increase of inclination angle ϕ and surcharge load q. The total anchoring force increases gradually with the increase of horizontal seismic acceleration coefficients kh and geometric coefficients mi and GSI.


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## Body

## 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 Section2, 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](1)F=σ1−11+ccσ2+σ3−σcm1+cσccσ2+σ3+s1/2=0,whenF≥F′,(2)F‘=11+bσ1+cσ2−σ3−σcmσ3σc+s≤0,whenF<F′,where σc is the uniaxial compressive strength of intact rock, m is constant value, and b represents the intermediate principle stress coefficients; s is a constant value relevant to the rock mass and it can be determined in the two following equations:(3)mmi=expGSI−10028−14D,(4)s=expGSI−1009−3D,where D is a disturbance coefficient, and its range is from 0.0 to 1.0.GSI is the geological strength index, and its range is from 0 to 80. The magnitude of mi 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 ofmi≈7, it is applied to calculate carbonate rock types, such as dolomite, limestone, and marble(2)
When the value ofmi≈10, it is applied to calculate lithified argillaceous rock types, such as mudstone, siltstone shale, and slate(3)
When the value ofmi≈15, it is applied to calculate arenaceous rock types, such as mudstone, siltstone shale, and slate(4)
When the value ofmi≈17, it is applied to calculate fine-grained polymineralic igneous crystalline rock types, such as andesite, dolerite, and diabase(5)
When the value ofmi≈25, it is applied to calculate coarse-grained polymineralic igneous and metamorphic rock types, such as amphibolite, gabbro, gneiss, granite, and quartz dioriteIn equations (1) and (2), when c=0, this demonstrates that the influences of the intermediate principal stress are omitted, so Hoek–Brown criterion can be expressed as follows:(5)F=σ1−σ3−σcmσ3σc+s1/2.Forc = 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]:(6)Fσ¯=σ1−12σ1+σ2−mσc2×σ2+σ3+sσc2≤0,whenF>F′,Fσ¯=12σ1+σ2−σ3−mσcσ3+sσc2≤0,whenF<F′.


### 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]: (7)ΘexU¯≤ΘmrU¯,where U¯ is any virtual admissible velocity field. ΘexU¯ is the work performed by external loading, and ΘmrU¯ 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 (8)ΘmrU¯=∫Ωψd¯¯x¯dΩ+∫Σψv¯x¯;U¯x¯dΣ,where d¯¯ is the strain rate sensor associated with U¯ at any point of rock mass volume Ω. U¯x¯ means the jump at a point x¯ through a possible velocity discontinuity surface Σ and Σ′ following its normal v¯x¯, and the ψ-functions are the support functions defined by the duality from the strength condition Fσ¯¯≤0.(9)ψd¯¯=supσ¯¯σ¯¯:d¯¯|Fσ≤0,(10)ψν;U¯=supσ¯¯U¯⋅σ¯¯⋅ν¯|Fσ≤0.For nonlinear twin-shear rock masses, the corresponding 3D closed-form expressions of the foregoingψ-functions are obtained as follows [28]:(11)ψd¯¯=−sσcmtrd¯¯+34mσcχd¯¯trd¯¯,if trd¯¯≥0,(12)χd¯¯=max0,d1+max0,d2+max0,d32,where d1, d2, and d3represent the eigenvalues of d¯¯.Similarly, it has been found that theψ-functions relative to a velocity jump and defined by equation (7) take the following form: (13)ψν¯;U¯=−sσcmU¯⋅ν¯+34mσcχν¯;U¯U¯⋅ν¯,ifU¯⋅ν¯>0,where(14)χν¯;U¯=14U¯−U¯⋅ν¯2.It should be noted that the conditionstrd¯¯>0 in equation (11) and U¯⋅ν¯>0 in equation (13) demonstrate the fact that ψd¯¯<+∞ and ψν¯;U¯<+∞, respectively. These conditions are necessary for the kinematic method of limit analysis expressed by equation (7) which results in nontrivial upper bound solutions.


## 2.1. The Relevant Fracture Criterion

The nonlinear united strength failure criterion can be expressed as [22](1)F=σ1−11+ccσ2+σ3−σcm1+cσccσ2+σ3+s1/2=0,whenF≥F′,(2)F‘=11+bσ1+cσ2−σ3−σcmσ3σc+s≤0,whenF<F′,where σc is the uniaxial compressive strength of intact rock, m is constant value, and b represents the intermediate principle stress coefficients; s is a constant value relevant to the rock mass and it can be determined in the two following equations:(3)mmi=expGSI−10028−14D,(4)s=expGSI−1009−3D,where D is a disturbance coefficient, and its range is from 0.0 to 1.0.GSI is the geological strength index, and its range is from 0 to 80. The magnitude of mi 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 ofmi≈7, it is applied to calculate carbonate rock types, such as dolomite, limestone, and marble(2)
When the value ofmi≈10, it is applied to calculate lithified argillaceous rock types, such as mudstone, siltstone shale, and slate(3)
When the value ofmi≈15, it is applied to calculate arenaceous rock types, such as mudstone, siltstone shale, and slate(4)
When the value ofmi≈17, it is applied to calculate fine-grained polymineralic igneous crystalline rock types, such as andesite, dolerite, and diabase(5)
When the value ofmi≈25, it is applied to calculate coarse-grained polymineralic igneous and metamorphic rock types, such as amphibolite, gabbro, gneiss, granite, and quartz dioriteIn equations (1) and (2), when c=0, this demonstrates that the influences of the intermediate principal stress are omitted, so Hoek–Brown criterion can be expressed as follows:(5)F=σ1−σ3−σcmσ3σc+s1/2.Forc = 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]:(6)Fσ¯=σ1−12σ1+σ2−mσc2×σ2+σ3+sσc2≤0,whenF>F′,Fσ¯=12σ1+σ2−σ3−mσcσ3+sσc2≤0,whenF<F′.


## 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]: (7)ΘexU¯≤ΘmrU¯,where U¯ is any virtual admissible velocity field. ΘexU¯ is the work performed by external loading, and ΘmrU¯ 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 (8)ΘmrU¯=∫Ωψd¯¯x¯dΩ+∫Σψv¯x¯;U¯x¯dΣ,where d¯¯ is the strain rate sensor associated with U¯ at any point of rock mass volume Ω. U¯x¯ means the jump at a point x¯ through a possible velocity discontinuity surface Σ and Σ′ following its normal v¯x¯, and the ψ-functions are the support functions defined by the duality from the strength condition Fσ¯¯≤0.(9)ψd¯¯=supσ¯¯σ¯¯:d¯¯|Fσ≤0,(10)ψν;U¯=supσ¯¯U¯⋅σ¯¯⋅ν¯|Fσ≤0.For nonlinear twin-shear rock masses, the corresponding 3D closed-form expressions of the foregoingψ-functions are obtained as follows [28]:(11)ψd¯¯=−sσcmtrd¯¯+34mσcχd¯¯trd¯¯,if trd¯¯≥0,(12)χd¯¯=max0,d1+max0,d2+max0,d32,where d1, d2, and d3represent the eigenvalues of d¯¯.Similarly, it has been found that theψ-functions relative to a velocity jump and defined by equation (7) take the following form: (13)ψν¯;U¯=−sσcmU¯⋅ν¯+34mσcχν¯;U¯U¯⋅ν¯,ifU¯⋅ν¯>0,where(14)χν¯;U¯=14U¯−U¯⋅ν¯2.It should be noted that the conditionstrd¯¯>0 in equation (11) and U¯⋅ν¯>0 in equation (13) demonstrate the fact that ψd¯¯<+∞ and ψν¯;U¯<+∞, 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 Figure1 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 AC is ϕ. 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 q 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 O with an angular velocity ω, the curve CE separating this volume from the structure that is kept motionless is a log-spiral arc and focus O, and it necessarily follows that the velocity jump at any point of line CE is inclined at angle ϕ with respect to the tangent at the same point. The inclination angle of anchoring bar is β with a horizontal direction. Five parameters are involved in such a mechanism: four angles θ1, θ2, ϕ, and β and the distance r0=R defining the radius of log-spiral curve for θ=θ1; accordingly, this curve is defined in polar coordinates o,r,θ by equation r=r0eθ−θ1tanϕ.Figure 1
The anchoring mechanism of rock slope.


### 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(15)Θug=γ1bh1Vsinβ−ϕ,whereΘug is the work rate of gravitational force in the upper layer of rock slope; γ1 is the unit weight of rock mass in the upper layer of rock slope.(2)
The work rate of external loadq is(16)Θuq=qbVsinβ−ϕ,whereΘuq is the work rate of uniform distribution load q and b is the width of roof.(3)
The work rate of seismic load is(17)Θuhs=−khγ1bh1Vcosβ−ϕ,whereΘuhs is the work rate of horizontal earthquake loads in the upper layer; kh is the horizontal earthquake load coefficient.(4)
The work rate of roof bolt is(18)Θup=−∑i=1nPiVcosβ−ϕcosβ,whereΘup is the work rate of antisliding force about anchors in the upper layer; Pi is the antisliding force of the ith roof bolt.(5)
The maximum resisting work rate is(19)Θmru=σCuh1sinαVcosϕ−sm+34m1−sinϕ2sinϕ2,whereΘmru is the maximum resisting work rate in the upper layer; σCu 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(20)Θlg=γ2r03ωl1+l2+l3,whereΘlg is the work of gravitational force; expressions of l1, l2, and l3 can be, respectively, expressed as (20a)l1=13e3tanφθ1−θ23tanφcosθ2+sinθ2−3tanφcosθ1+sinθ19tan2φ+1,(20b)l2=16sinθ12br0cosθ1−br02,(20c)l3=−13rBr03sin3θB+αcosα21sin2θB+α−1sin2θ2+α+sinαcosθB+αsinθB+αcosθ2+αsinθ2+α.(2)
The work rate of seismic load is(21)Θlhs=khγ2r03ωl4+l5+l6,whereΘlhs is the work rate of seismic load. The expressions of l4, l5, and l6 can be, respectively, expressed as (21a)l4=13etanφθ1−θ23tanφcosθ2−cosθ2−3tanφsinθ1−cosθ19tan2φ+1,(21b)l5=−13br0sin3θ1,(21c)l6=−13rBr03sin3θB+α−sinα21sin2θB+α−1sin2θ2+α+cosαcosθB+αsinθB+αcosθ2+αsinθ2+α,where different parameters are shown in formulas (20)–(21c). The expressions of rB and θB can be, respectively, expressed as (21d)rB=r01−2br0cosθ1+br02,(21e)θB=arctansinθ1cosθ1−b/r0.(3)
The work rate of roof bolt is(22)Θlp=∑i=1nPir0ωeθ−θ1tanϕsinθ−β,whereΘlp is the work rate of antislide force of roof bolts; Pi is the antisliding force of the ith bolt. i 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 lineCF, and the expression of resisting work derived from equations (10)–(12) can be depicted as follows: (23)Θmrl=12σCLωr02e2θ1−θ2tanϕ−1−sm+34m1−sinϕ2sinϕ2.


#### 3.2.3. The Expression of Anchoring Force

According to the fundamental inequality (7), the expression of anchoring force ∑i=1nPi can be shown as follows: (24)∑i=1nPi≤Θmru+Θmrl−Θug−Θuq+Θuhs−Θlhs−Θlgr0ωeθ−θ1tanϕsinθ−β−Vcosβ−θcosβ.


#### 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 (25)F=∑i=1nPi0.5γH2.According to the principal of limit analysis theory, whenf=1, the critical total anchoring force can be expressed as (26)F≤Fu=minΘmru+Θmrl−Θug−Θuq+Θuhs−Θlhs−Θlg0.5h1+h22r0ωeθ−θ1tanϕsinθ−β−Vcosβ−θcosβ1γ1+1γ2.


## 3.1. The Analysis of Failure Mechanism

The failure mechanism, depicted in Figure1 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 AC is ϕ. 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 q 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 O with an angular velocity ω, the curve CE separating this volume from the structure that is kept motionless is a log-spiral arc and focus O, and it necessarily follows that the velocity jump at any point of line CE is inclined at angle ϕ with respect to the tangent at the same point. The inclination angle of anchoring bar is β with a horizontal direction. Five parameters are involved in such a mechanism: four angles θ1, θ2, ϕ, and β and the distance r0=R defining the radius of log-spiral curve for θ=θ1; accordingly, this curve is defined in polar coordinates o,r,θ by equation r=r0eθ−θ1tanϕ.Figure 1
The anchoring mechanism of rock slope.


## 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(15)Θug=γ1bh1Vsinβ−ϕ,whereΘug is the work rate of gravitational force in the upper layer of rock slope; γ1 is the unit weight of rock mass in the upper layer of rock slope.(2)
The work rate of external loadq is(16)Θuq=qbVsinβ−ϕ,whereΘuq is the work rate of uniform distribution load q and b is the width of roof.(3)
The work rate of seismic load is(17)Θuhs=−khγ1bh1Vcosβ−ϕ,whereΘuhs is the work rate of horizontal earthquake loads in the upper layer; kh is the horizontal earthquake load coefficient.(4)
The work rate of roof bolt is(18)Θup=−∑i=1nPiVcosβ−ϕcosβ,whereΘup is the work rate of antisliding force about anchors in the upper layer; Pi is the antisliding force of the ith roof bolt.(5)
The maximum resisting work rate is(19)Θmru=σCuh1sinαVcosϕ−sm+34m1−sinϕ2sinϕ2,whereΘmru is the maximum resisting work rate in the upper layer; σCu 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(20)Θlg=γ2r03ωl1+l2+l3,whereΘlg is the work of gravitational force; expressions of l1, l2, and l3 can be, respectively, expressed as (20a)l1=13e3tanφθ1−θ23tanφcosθ2+sinθ2−3tanφcosθ1+sinθ19tan2φ+1,(20b)l2=16sinθ12br0cosθ1−br02,(20c)l3=−13rBr03sin3θB+αcosα21sin2θB+α−1sin2θ2+α+sinαcosθB+αsinθB+αcosθ2+αsinθ2+α.(2)
The work rate of seismic load is(21)Θlhs=khγ2r03ωl4+l5+l6,whereΘlhs is the work rate of seismic load. The expressions of l4, l5, and l6 can be, respectively, expressed as (21a)l4=13etanφθ1−θ23tanφcosθ2−cosθ2−3tanφsinθ1−cosθ19tan2φ+1,(21b)l5=−13br0sin3θ1,(21c)l6=−13rBr03sin3θB+α−sinα21sin2θB+α−1sin2θ2+α+cosαcosθB+αsinθB+αcosθ2+αsinθ2+α,where different parameters are shown in formulas (20)–(21c). The expressions of rB and θB can be, respectively, expressed as (21d)rB=r01−2br0cosθ1+br02,(21e)θB=arctansinθ1cosθ1−b/r0.(3)
The work rate of roof bolt is(22)Θlp=∑i=1nPir0ωeθ−θ1tanϕsinθ−β,whereΘlp is the work rate of antislide force of roof bolts; Pi is the antisliding force of the ith bolt. i 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 lineCF, and the expression of resisting work derived from equations (10)–(12) can be depicted as follows: (23)Θmrl=12σCLωr02e2θ1−θ2tanϕ−1−sm+34m1−sinϕ2sinϕ2.


### 3.2.3. The Expression of Anchoring Force

According to the fundamental inequality (7), the expression of anchoring force ∑i=1nPi can be shown as follows: (24)∑i=1nPi≤Θmru+Θmrl−Θug−Θuq+Θuhs−Θlhs−Θlgr0ωeθ−θ1tanϕsinθ−β−Vcosβ−θcosβ.


### 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 (25)F=∑i=1nPi0.5γH2.According to the principal of limit analysis theory, whenf=1, the critical total anchoring force can be expressed as (26)F≤Fu=minΘmru+Θmrl−Θug−Θuq+Θuhs−Θlhs−Θlg0.5h1+h22r0ωeθ−θ1tanϕsinθ−β−Vcosβ−θcosβ1γ1+1γ2.


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

(1)
The work rate of gravitational force is(15)Θug=γ1bh1Vsinβ−ϕ,whereΘug is the work rate of gravitational force in the upper layer of rock slope; γ1 is the unit weight of rock mass in the upper layer of rock slope.(2)
The work rate of external loadq is(16)Θuq=qbVsinβ−ϕ,whereΘuq is the work rate of uniform distribution load q and b is the width of roof.(3)
The work rate of seismic load is(17)Θuhs=−khγ1bh1Vcosβ−ϕ,whereΘuhs is the work rate of horizontal earthquake loads in the upper layer; kh is the horizontal earthquake load coefficient.(4)
The work rate of roof bolt is(18)Θup=−∑i=1nPiVcosβ−ϕcosβ,whereΘup is the work rate of antisliding force about anchors in the upper layer; Pi is the antisliding force of the ith roof bolt.(5)
The maximum resisting work rate is(19)Θmru=σCuh1sinαVcosϕ−sm+34m1−sinϕ2sinϕ2,whereΘmru is the maximum resisting work rate in the upper layer; σCu 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(20)Θlg=γ2r03ωl1+l2+l3,whereΘlg is the work of gravitational force; expressions of l1, l2, and l3 can be, respectively, expressed as (20a)l1=13e3tanφθ1−θ23tanφcosθ2+sinθ2−3tanφcosθ1+sinθ19tan2φ+1,(20b)l2=16sinθ12br0cosθ1−br02,(20c)l3=−13rBr03sin3θB+αcosα21sin2θB+α−1sin2θ2+α+sinαcosθB+αsinθB+αcosθ2+αsinθ2+α.(2)
The work rate of seismic load is(21)Θlhs=khγ2r03ωl4+l5+l6,whereΘlhs is the work rate of seismic load. The expressions of l4, l5, and l6 can be, respectively, expressed as (21a)l4=13etanφθ1−θ23tanφcosθ2−cosθ2−3tanφsinθ1−cosθ19tan2φ+1,(21b)l5=−13br0sin3θ1,(21c)l6=−13rBr03sin3θB+α−sinα21sin2θB+α−1sin2θ2+α+cosαcosθB+αsinθB+αcosθ2+αsinθ2+α,where different parameters are shown in formulas (20)–(21c). The expressions of rB and θB can be, respectively, expressed as (21d)rB=r01−2br0cosθ1+br02,(21e)θB=arctansinθ1cosθ1−b/r0.(3)
The work rate of roof bolt is(22)Θlp=∑i=1nPir0ωeθ−θ1tanϕsinθ−β,whereΘlp is the work rate of antislide force of roof bolts; Pi is the antisliding force of the ith bolt. i 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 lineCF, and the expression of resisting work derived from equations (10)–(12) can be depicted as follows: (23)Θmrl=12σCLωr02e2θ1−θ2tanϕ−1−sm+34m1−sinϕ2sinϕ2.


## 3.2.3. The Expression of Anchoring Force

According to the fundamental inequality (7), the expression of anchoring force ∑i=1nPi can be shown as follows: (24)∑i=1nPi≤Θmru+Θmrl−Θug−Θuq+Θuhs−Θlhs−Θlgr0ωeθ−θ1tanϕsinθ−β−Vcosβ−θcosβ.


## 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 (25)F=∑i=1nPi0.5γH2.According to the principal of limit analysis theory, whenf=1, the critical total anchoring force can be expressed as (26)F≤Fu=minΘmru+Θmrl−Θug−Θuq+Θuhs−Θlhs−Θlg0.5h1+h22r0ωeθ−θ1tanϕsinθ−β−Vcosβ−θcosβ1γ1+1γ2.


## 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:r0 = 12 m, mi = 10, GSI = 10, and b = 5 m; γ1=15kN/m3, γ2=20kN/m3, h1=8m, h2=9m, q = 20 kPa, ω = 0.2, V = 0.1 mm/s, σcu = 6 MPa, σcl = 10 MPa, α=20∘, kh = 0.18, θ1 = 200, θ2 = 500, θ = 300, θB=35∘, θB=35∘, and β=20∘. 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 c=1 (it means that the influence of intermediate principal stress is not considered) is higher than one of c=0 (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 P, so when different criteria are selected, the influence of intermediate principal stress is not omitted.Figure 2
The relation between intermediate principal stress and anchoring force.


### 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 forβ = 10°, 20°, and 30° and the comparison and analysis are carried out among three cases. The following parameters are determined: r0 = 12 m, mi = 10, GSI = 10, and b = 5 m; γ1=15kN/m3, γ2=20kN/m3, h1=8m, h2=9m, q = 20 kPa, ω = 0.2, V = 0.1 mm/s, σcu = 6 MPa, σcl = 10 MPa, α = 20°, kh = 0.18, θ1 = 20°, θ2 = 50°, θ = 30°, and θB=35∘. 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 ϕ; and it can also be found that the total anchoring force P increases gradually as the anchoring inclination angle β increases. The influence of anchoring inclination angle β on the total anchoring force decreases gradually with the increase of velocity inclination angle ϕ. For example, for ϕ=10°, the value of P increases approximately by 108% when the value of α increases from 10° to 30°; then, for ϕ=15°, it increases only by 59.8%.Figure 3
The relation between bolt inclination angle and anchoring force.


### 4.3. The Influence of Rock Layer Thickness on the Total Anchoring ForceP

The effects of thickness of rock layer on the total anchoring force are depicted in Figure4. 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: r0 = 12 m, mi = 10, GSI = 10, and b = 5 m; γ1=15kN/m3 and γ2=20kN/m3; h1=8m; h2=9m; q = 20 kPa, ω = 0.2, V = 0.1 mm/s, σcu = 6 MPa, σcl = 10 MPa, α = 200, kh = 0.18, θ1 = 200, θ2 = 500, θ = 300, θB=35°, and β=20°. It can be found in Figure 4 that the total anchoring force P 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 h1/h2 on total anchoring force increase gradually with the increase of inclination angle ϕ; to illustrate, for ϕ = 100, when h1/h2 increases from 1/2 to 2/3, the value of total anchoring force P increases approximately by 11.76%; then, for ϕ = 250, when h1/h2 increases from 1/2 to 2/3, the value of P increases by 27.12%. So, conclusions can be drawn, showing that the total anchoring force P increases gradually with the increase of thickness ratio, but the influence of the thickness ratio on the anchoring force P gradually increases with the increase of velocity inclination angle ϕ.Figure 4
The relation between unit weight and anchoring force.


### 4.4. The Influence of Surcharge Loadq on the Anchoring Force P

When the influence of surcharge loadq 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: r0 = 12 m, mi = 10, GSI = 10, and b = 5 m; γ1=15kN/m3 and γ2=20kN/m3; h1=8m; h2=9m; ω = 0.2, V = 0.1 mm/s, σcu = 6 MPa, σcl = 10 MPa, α  = 200, kh = 0.18, θ1 = 200, θ2 = 500, θ = 300, θB=35°, and β=20°; 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 P decreases gradually with the increase of inclination angle ϕ and surcharge load q. So, conclusions can be drawn, showing that the surcharge load q of rock slope has great influences on the total anchoring force P.Figure 5
The relation between surcharge load and anchoring force.


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

When the influences of horizontal seismic acceleration coefficientskh on the anchoring force are considered, the following parameters are listed: r0 = 12 m, mi = 10, GSI = 10, and b = 5 m; γ1=15kN/m3 and γ2=20kN/m3; h1=8m; h2=9m; q = 30 kPa, ω = 0.2, V = 0.1 mm/s, σcu = 6 MPa, σcl = 10 MPa, α  = 200, θ1 = 200, θ2 = 500, θ = 300, and θB=350. The effects of kh 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 kh increases from 0.08 to 0.24, the total anchoring force increases from 0.5206 to 1.0322.Figure 6
The relation between horizontal seismic acceleration coefficients and anchoring force.


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

When the influences of geometric coefficientsGSI and mi on the anchoring force are, respectively, regarded, the following parameters are listed: r0 = 12 m; b = 5 m; γ1=15kN/m3 and γ2=20kN/m3; h1=8m; h2=9m; q = 30 kPa, ω = 0.2, V = 0.1 mm/s, σcu = 6 MPa, σcl = 10 MPa, α = 200, kh  = 0.18, θ1 = 200, θ2 = 500, θ = 300, and θB=35∘. The effects of GSI and mi 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 miand GSI, and the geometric parameters have a great influence on the total anchoring force P.Table 1
The relation between anchoring forceP and geometric parameters GSI and mi.
GSImi=1mi=10mi=15mi=17mi=25101.42561.28261.05860.82640.6952201.27541.13460.94210.71880.5684301.13151.00230.86850.60210.5023400.95240.89330.78760.53210.3956500.75640.69920.62750.48970.3211600.60250.56240.50330.45330.2836


## 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:r0 = 12 m, mi = 10, GSI = 10, and b = 5 m; γ1=15kN/m3, γ2=20kN/m3, h1=8m, h2=9m, q = 20 kPa, ω = 0.2, V = 0.1 mm/s, σcu = 6 MPa, σcl = 10 MPa, α=20∘, kh = 0.18, θ1 = 200, θ2 = 500, θ = 300, θB=35∘, θB=35∘, and β=20∘. 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 c=1 (it means that the influence of intermediate principal stress is not considered) is higher than one of c=0 (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 P, so when different criteria are selected, the influence of intermediate principal stress is not omitted.Figure 2
The relation between intermediate principal stress and anchoring force.


## 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 forβ = 10°, 20°, and 30° and the comparison and analysis are carried out among three cases. The following parameters are determined: r0 = 12 m, mi = 10, GSI = 10, and b = 5 m; γ1=15kN/m3, γ2=20kN/m3, h1=8m, h2=9m, q = 20 kPa, ω = 0.2, V = 0.1 mm/s, σcu = 6 MPa, σcl = 10 MPa, α = 20°, kh = 0.18, θ1 = 20°, θ2 = 50°, θ = 30°, and θB=35∘. 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 ϕ; and it can also be found that the total anchoring force P increases gradually as the anchoring inclination angle β increases. The influence of anchoring inclination angle β on the total anchoring force decreases gradually with the increase of velocity inclination angle ϕ. For example, for ϕ=10°, the value of P increases approximately by 108% when the value of α increases from 10° to 30°; then, for ϕ=15°, it increases only by 59.8%.Figure 3
The relation between bolt inclination angle and anchoring force.


## 4.3. The Influence of Rock Layer Thickness on the Total Anchoring ForceP

The effects of thickness of rock layer on the total anchoring force are depicted in Figure4. 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: r0 = 12 m, mi = 10, GSI = 10, and b = 5 m; γ1=15kN/m3 and γ2=20kN/m3; h1=8m; h2=9m; q = 20 kPa, ω = 0.2, V = 0.1 mm/s, σcu = 6 MPa, σcl = 10 MPa, α = 200, kh = 0.18, θ1 = 200, θ2 = 500, θ = 300, θB=35°, and β=20°. It can be found in Figure 4 that the total anchoring force P 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 h1/h2 on total anchoring force increase gradually with the increase of inclination angle ϕ; to illustrate, for ϕ = 100, when h1/h2 increases from 1/2 to 2/3, the value of total anchoring force P increases approximately by 11.76%; then, for ϕ = 250, when h1/h2 increases from 1/2 to 2/3, the value of P increases by 27.12%. So, conclusions can be drawn, showing that the total anchoring force P increases gradually with the increase of thickness ratio, but the influence of the thickness ratio on the anchoring force P gradually increases with the increase of velocity inclination angle ϕ.Figure 4
The relation between unit weight and anchoring force.


## 4.4. The Influence of Surcharge Loadq on the Anchoring Force P

When the influence of surcharge loadq 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: r0 = 12 m, mi = 10, GSI = 10, and b = 5 m; γ1=15kN/m3 and γ2=20kN/m3; h1=8m; h2=9m; ω = 0.2, V = 0.1 mm/s, σcu = 6 MPa, σcl = 10 MPa, α  = 200, kh = 0.18, θ1 = 200, θ2 = 500, θ = 300, θB=35°, and β=20°; 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 P decreases gradually with the increase of inclination angle ϕ and surcharge load q. So, conclusions can be drawn, showing that the surcharge load q of rock slope has great influences on the total anchoring force P.Figure 5
The relation between surcharge load and anchoring force.


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

When the influences of horizontal seismic acceleration coefficientskh on the anchoring force are considered, the following parameters are listed: r0 = 12 m, mi = 10, GSI = 10, and b = 5 m; γ1=15kN/m3 and γ2=20kN/m3; h1=8m; h2=9m; q = 30 kPa, ω = 0.2, V = 0.1 mm/s, σcu = 6 MPa, σcl = 10 MPa, α  = 200, θ1 = 200, θ2 = 500, θ = 300, and θB=350. The effects of kh 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 kh increases from 0.08 to 0.24, the total anchoring force increases from 0.5206 to 1.0322.Figure 6
The relation between horizontal seismic acceleration coefficients and anchoring force.


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

When the influences of geometric coefficientsGSI and mi on the anchoring force are, respectively, regarded, the following parameters are listed: r0 = 12 m; b = 5 m; γ1=15kN/m3 and γ2=20kN/m3; h1=8m; h2=9m; q = 30 kPa, ω = 0.2, V = 0.1 mm/s, σcu = 6 MPa, σcl = 10 MPa, α = 200, kh  = 0.18, θ1 = 200, θ2 = 500, θ = 300, and θB=35∘. The effects of GSI and mi 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 miand GSI, and the geometric parameters have a great influence on the total anchoring force P.Table 1
The relation between anchoring forceP and geometric parameters GSI and mi.
GSImi=1mi=10mi=15mi=17mi=25101.42561.28261.05860.82640.6952201.27541.13460.94210.71880.5684301.13151.00230.86850.60210.5023400.95240.89330.78760.53210.3956500.75640.69920.62750.48970.3211600.60250.56240.50330.45330.2836


## 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 angleβ, thickness ratio h1/h2 of rock layer, surcharge load q, seismic acceleration coefficients kh, and geometric coefficients GSIand mi 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 boltP, so when different criteria are selected, the influence of intermediate principal stress is not omitted. The total anchoring force P decreases gradually with the increase of inclination angle ϕ. The total anchoring force P increases gradually as the anchoring inclination angle β increases. The influence of anchoring inclination angle β on the total anchoring force decreases gradually with the increase of velocity inclination angle ϕ.(2)
The total anchoring forceP increases gradually as the thickness ratio of rock layer (namely, different rock types) increases. But the effects of thickness ratio h1/h2 on total anchoring force increase gradually with the increase of inclination angle ϕ. The total anchoring force P decreases gradually with the increase of inclination angle ϕ and surcharge load q.(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 coefficientsmi and GSI, and the geometric parameters have a great influence on the total anchoring force P.



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*Source: 5565320-2021-05-03.xml*