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

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 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 k h and geometric coeﬃcients m i and GSI.


Introduction
e 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][3][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. e mechanical mechanism of the load transfer of anchoring bar is studied by Lutz et al. [8][9][10]. e 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. e 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. en, the optimum design of mechanical model is calculated by Li-De and Cong-Xin [13] by using the strength reduction method of finite element. e 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. en, 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. e 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.

e Relevant Fracture Criterion.
e nonlinear united strength failure criterion can be expressed as [22] where σ c 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: 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. e magnitude of m i 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) 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: 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]:

Kinematic Method of Limit
Analysis. e 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. e implementation of the kinematic method of limit analysis depends on the following fundamental inequality [26]: where U is any virtual admissible velocity field. Θ ex (U) is the work performed by external loading, and Θ mr (U) is the maximum resisting work originating from the rock materials. e formula of maximum resisting work according to Saada's work [27] can be obtained as 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 F(σ) ≤ 0.

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For nonlinear twin-shear rock masses, the corresponding 3D closed-form expressions of the foregoing ψ-functions are obtained as follows [28]: where d 1 , d 2 , and d 3 represent the eigenvalues of d.
Similarly, it has been found that the ψ-functions relative to a velocity jump and defined by equation (7) take the following form: where It should be noted that the conditions tr d > 0 in equation (11) and ese conditions are necessary for the kinematic method of limit analysis expressed by equation (7) which results in nontrivial upper bound solutions.

e Analysis of Failure Mechanism.
e 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 AC is ϕ. e 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. ere exist uniform distribution external loads q in the roof of rock slope. ere 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. e inclination angle of anchoring bar is β with a horizontal direction. Five parameters are involved in such a mechanism: four angles θ 1 , θ 2 , ϕ, and β and the distance r 0 � R defining the radius of log-spiral curve for θ � θ 1 ; accordingly, this curve is defined in polar coordinates (o, r, θ) by equation r � r 0 e (θ− θ 1 )tan ϕ .

e Work Rate of External Forces and Maximum Resisting
Work Rate

e Work of External Forces in the Upper Layer of Rock Slope
(1) e work rate of gravitational force is where Θ ug is the work rate of gravitational force in the upper layer of rock slope; c 1 is the unit weight of rock mass in the upper layer of rock slope. (2) e work rate of external load q is where Θ uq is the work rate of uniform distribution load q and b is the width of roof. (3) e work rate of seismic load is where Θ uhs is the work rate of horizontal earthquake loads in the upper layer; k h is the horizontal earthquake load coefficient. (4) e work rate of roof bolt is where Θ up is the work rate of antisliding force about anchors in the upper layer; P i is the antisliding force of the ith roof bolt. (5) e maximum resisting work rate is The upper layer The lower layer Figure 1: e anchoring mechanism of rock slope.

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

e Work Rate of External Forces in the Lower Layer of Rock Slope
(1) e work rate of gravitational force is where Θ lg is the work of gravitational force; expressions of l 1 , l 2 , and l 3 can be, respectively, expressed as (2) e work rate of seismic load is where Θ lhs is the work rate of seismic load. e expressions of l 4 , l 5 , and l 6 can be, respectively, expressed as where different parameters are shown in formulas (20)-(21c). e expressions of r B and θ B can be, respectively, expressed as (3) e work rate of roof bolt is where Θ lp is the work rate of antislide force of roof bolts; P i is the antisliding force of the ith bolt. i is the number of roof bolts.
(4) e maximum resisting work rate is defined as follows.
e maximum resisting work results from the velocity jump along the log-spiral line CF, and the expression of resisting work derived from equations (10)-(12) can be depicted as follows:

e Expression of Anchoring Force.
According to the fundamental inequality (7), the expression of anchoring force n i�1 P i can be shown as follows:

e 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 f � 1, the critical total anchoring force can be expressed as

e 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: r 0 � 12 m, mi � 10, GSI � 10, and b � 5 m; It is shown in Figure 2 that the total anchoring force P increases as the velocity inclination angle increases whether Hoek-Brown criterion [30]

e Influence of Bolt Inclination Angle on the Total Anchoring
Force. e 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. e following parameters are determined: r 0 � 12 m, m i � 10, GSI � 10, and b � 5 m; , and θ B � 35 ∘ . e 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. e 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%.

e Influence of Rock Layer ickness on the Total Anchoring Force P.
e 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: r 0 � 12 m, mi � 10, GSI � 10, and b � 5 m; c 1 � 15(kN/m 3 ) and c 2 � 20(kN/m 3 ); h 1 � 8 m; h 2 � 9 m; 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, θ B � 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 (h 1 /h 2 ) on total anchoring force increase gradually with the increase of inclination angle ϕ; to illustrate, for ϕ � 100, when (h 1 /h 2 ) increases from 1/2 to 2/ 3, the value of total anchoring force P increases approximately by 11.76%; then, for ϕ � 250, when (h 1 /h 2 ) 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.

e Influence of Surcharge Load q on the Anchoring Force
P. When the influence of surcharge load q 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. e following parameters are listed: r 0 � 12 m, mi � 10, GSI � 10, and b � 5 m; c 1 � 15(kN/m 3 ) and c 2 � 20(kN/m 3 ); h 1 � 8 m; h 2 � 9 m; 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, θ B � 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 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 k h increases from 0.08 to 0.24, the total anchoring force increases from 0.5206 to 1.0322.  Table 1. It can be found in Table 1 that the total anchoring force decreases gradually with the increase of geometric coefficients m i and GSI, and the geometric parameters have a great influence on the total anchoring force P.

Conclusions
e 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. e intermediate principal stress, the anchoring inclination angle β, thickness ratio (h 1 /h 2 ) of rock layer, surcharge load q, seismic acceleration coefficients k h , and geometric coefficients GSIand m i have great influences on the total anchoring force. Finally, the following conclusions are drawn: (1) e intermediate principal stress has a great influence on the total anchoring force of bolt P, so when different criteria are selected, the influence of intermediate principal stress is not omitted. e total anchoring force P decreases gradually with the increase of inclination angle ϕ. e total anchoring force P increases gradually as the anchoring inclination angle β increases. e influence of anchoring inclination angle β on the total anchoring force decreases gradually with the increase of velocity inclination angle ϕ.
(2) e total anchoring force P increases gradually as the thickness ratio of rock layer (namely, different rock types) increases. But the effects of thickness ratio (h 1 /h 2 ) on total anchoring force increase gradually with the increase of inclination angle ϕ. e total anchoring force P decreases gradually with the increase of inclination angle ϕ and surcharge load q. (3) e total anchoring force increases gradually with the increase of horizontal seismic acceleration coefficients. e total anchoring force decreases gradually with the increase of geometric coefficients m i and GSI, and the geometric parameters have a great influence on the total anchoring force P.

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

Conflicts of Interest
e authors declare that they have no conflicts of interest.