It is difficult to obtain and sample natural structural planes with complex surface undulations, and the consistency related to the sampling cannot be guaranteed. However, it has been proved that it is feasible to use rock structure samples prepared from cement mortar and other similar materials to replace the surface morphology and undulation of natural structural planes (Einstein et al., 1983; Plesha, 1987; Yang et al., 2001; Jafari et al., 2003; Li et al., 2008; Du et al., 2011). When using simulated structural planes to replace natural structural planes for direct shear tests, the wear degree and failure mode of simulated structural planes are similar to those of natural structural planes. The existing research results (Ladanyi and Archambault 1970; Yang et al., 2001; Jafari et al., 2003; Shen and Zhang, 2010) show that although there are differences between the sawtooth structural plane and the natural structural plane, the shear test with sawtooth structural plane is usually reasonable and can catch the surface morphology and undulations of the structural plane. Using cement mortar as the similar material, Liu et al. (2011) prepared sawtooth structural plane specimens with different initial undulant angles of 16°, 25°, 33°, 40°and 45°, respectively. Using an RJDT-A shear testing device (Figure 2), Liu et al. (2011) carried out cyclic shear tests on structural planes under four kinds normal stresses of 0.8, 1.6, 2.4, and 3.2 MPa, respectively. Due to the limitations of the RJDT-A shear testing device, the relative maximum shearing displacement was set to be ±10 mm, and the cyclic number was set to be 10 repetitions. In the shear process, the lower shear box remains stationary, and the specimen is sheared by the horizontal motion of the upper shear box. A single cyclic shear process of the rock mass structural plane is illustrated in Figure 3.

Introducing the shear strength ratio D n of the structural plane as the shear strength deterioration index, Liu et al. (2011) obtained the variation law of shear strength of structural plane. The peak shear strength of the nth cyclic shear is expressed as τ n , then the D n is defined as: D n = τ n τ 1 . (1)

The shear strength ratio D n ( 0.0 ≤ D n ≤ 1.0 ) reflects the shear strength degradation after several shear cycles: the closer D n is to 1, the lesser the influence of strength deterioration induced by cyclic shear is. On the contrary, the closer D n is to 0, the greater the influence of strength deterioration induced by cyclic shear is.

Figure 4 demonstrates the variation of shear strength ratios D n of structural planes with five kinds of initial undulant angles with cyclic shear number under σ n = 0.8   M P a . It is found that the peak shear strength of structural plane specimens with five kinds of initial undulant angles decreases with the increase cyclic shear number, though the decreasing trend is different. The D n with an initial undulant angle of 16° decreases at the lowest rate with the increase of cyclic shear number and keeps a steady decreasing trend, while the D n with an initial undulant angle of 45° decreases at the highest rate with the increase of cyclic shear number and remains basically unchanged after the 2-th cyclic shear. And the decreasing speed of D n increases with the increase of initial undulant angle. On the whole, the higher the undulations of the structural plane are, the lower the shear resisting force of the undulations is, and the occurrence of more significant shearing or wearing damage in the cycle shear is more likely. Thus, for a specimen with higher undulant angles, the shearing or wearing damage of the surface undulations is mainly in the early stage of cyclic shear, and the degree of shearing or wearing damage changes insignificantly in the subsequent cyclic shear, and for a specimen with smaller undulant angle, the undulations on the surface has strong shear resisting force, the shearing or wearing damage is mainly caused by multiple shear or cumulative wear. Thus, the D n of the specimens with smaller undulant angle decrease more slowly with the increase of the cyclic shear number.

When the shear movement of rock mass occurs, the shear resisting force mainly consists of the undulations of the structural plane and the friction of the contact sections: φ = φ b + α k (2) where φ denotes the friction angle of the structural plane, φ b denotes the basic friction angle, and α k denotes the undulant angle.

On the basis of many direct shear tests on the rough structural plane, Barton and Choubey (1977) proposed to replace the undulant angle with the dilatancy angle: α d = J R C 2 lg J C S σ n (3) where α d denotes the dilatancy angle, JRC denotes the roughness coefficient, JCS denotes the uniaxial compressive strength of the intact rock, and σ n denotes the normal stress.

Furthermore, Barton (1973) further proposed the following shear strength equation for structural plane: τ = σ n ⁡ tan ( J R C ∙ lg J C S σ n + φ b ) (4) where τ denotes the peak shear strength of structural plane.

Then, Eqs. 3, 4 can be combined as follows: τ = σ n ⁡ tan ( 2 α d + φ b ) (5) And after nth cyclic shear, the friction angle of the structural plane can be obtained: φ = 2 α n + φ n (6) where n denotes the cyclic shear number, α n and φ n denote the average dilatancy angle and the basic friction angle after the nth cyclic shear, respectively.

So, after nth cyclic shear, the peak shear strength of structural plane can be obtained: τ = σ n ⁡ tan ( 2 α n + φ n ) (7)

According to the strength characteristic test of structural plane under different shear velocity, Li et al. (2006) concluded that the peak shear strength of structural plane decreases with the increase of relative shear velocity. Thus, an exponential decay functional relationship exists between the peak strength and the relative shear velocity. According to the research of Wang and Zhang (1982) and Li et al. (2006), Liu (2017) further proposed to express the relative velocity coefficient as follows: γ ( t ) = γ r + ( 1 − γ r ) e − a | x ˙ ( t ) | (8) where x ˙ ( t ) denotes the relative shear velocity of the structural plane (m/s), γ r denotes the convergence value of the relative shear velocity correlation coefficient, and a denotes the undetermined factor. Referring to the research of Ni et al. (2013), γ r = 0.9 , a = 25 .

Therefore, combining Eqs. 7, 8, the shear strength of the structural plane introducing the relative shear velocity effect can be obtained as follows: τ = γ ( t ) ∙ σ n ⁡ tan ( 2 α n + φ n ) (9)

According to Eq. 9, the key to determining the shear strength of a structural plane during the cyclic shear is to derive variation laws of the dilatancy angle and the basic friction angle. Therefore, this section will focus on the analysis of the variation laws of the dilatancy angle and the basic friction angle during the cyclic shear.

A simplified dynamic calculation model of a bedding rock slope is illustrated in Figure 6, assuming that the potential sliding body produces unstable sliding along the rock stratum of the ith layer, di denotes the exposed length of ith rock stratum, α denotes the slope gradient, β denotes the dip angle of the rock stratum, u ¨ g denotes the seismic acceleration, and θ denotes the angle between the direction of seismic acceleration and the horizontal direction.

It should be noted that the quadrilateral area is composed of slope surface, sliding surface, slipping body top surface, and tensile crack surface of back edge of the rock slope. And the area of potential sliding body is determined by: S i = 1 2 ( ∑ n = 1 i d n ) 2 sin ⁡ α ⁡ sin ( α − β ) sin ⁡ β − 1 2 ( ∑ n = m i d n ) 2 tan ⁡ β sin ⁡   2 ( α − β ) sin ⁡   2 β . (21)

The gravity of the potential sliding body in layer i is G i , G i = γ i ( S i − S i − 1 ) (22) where γ i is the volumetric weigh of the ith rock stratum. And the total gravity of the sliding body G i t is: G i t = ∑ n = 1 i G n (23)

According to the force equilibrium analysis in dynamic analysis model, in the Y direction N = G i t ⁡ cos ⁡ β + G i t g u ¨ g ⁡ sin ( θ − β ) (24) and in the X direction T = G i t ⁡ sin ⁡ β + G i t g u ¨ g ⁡ cos ( θ − β ) (25) where T denotes the parallel (shear) reaction forces, N denotes the orthogonal (normal) reaction forces, g denotes the acceleration owing to gravity. The seismic loading can act as both driving force and resisting force. Thus, Eqs. 24, 25 are vector equations, and the u ¨ g changes with its symbol: when the seismic loading acts as the driving force, the symbol of u ¨ g is positive, and when the seismic loading acts as the resisting force, the symbol of u ¨ g is negative.

For the bedding rock landslide, the influence of cohesion (c) is mainly focused on the first cyclic shear process. Once the sliding mass generates cyclic shear under strong ground motion, the influence of cohesion (c) on shear strength is commonly ignored (Dong et al., 2020). The critical acceleration a c can be defined as the minimum seismic acceleration that leads to the potential sliding body to overcome available shear resistance and to move. If u ¨ g > a c , the potential sliding body will overcome the available shear resistance and obtain sliding displacement. According to Eqs. 9, 25, the safety factor ( F s ) of sliding body can be expressed as: F s = a v a i l a b l e   s h e a r   r e s i s t a n c e   d r i v i n g   f o r c e = τ T = γ ( t ) ∙ σ n ⁡ tan ( 2 α n + φ n ) G i t ⁡ sin ⁡ β + G i t g u ¨ g ⁡ cos ⁡ ⁡ ( θ − β ) . (26)

The safety factor F s = 1 is the critical condition for landslide instability, so assuming F S = 1 , the critical seismic acceleration is expressed as follows: a c = g [ γ ( t ) tan ( 2 α n + φ n ) cos ⁡ β − sin ⁡ β ] cos ( θ − β ) − γ ( t ) sin ( θ − β ) tan ( 2 α n + φ n ) . (27)

Based on Figure 6, the X-direction vibration balance equation can be expressed as follows: M u ¨ g ⁡ cos ( θ − β ) + M g ⁡ sin ⁡ β − γ ( t ) N ⁡ tan ( 2 α n + φ n ) = M s ¨ (28) where M is the mass of the potential sliding body, and M = G i t g . (29) And according to Eqs. 24, 28, 29 the acceleration of the potential sliding body can be expressed as follows: s ¨ = u ¨ g ⁡ cos θ − β + g ⁡ sin ⁡ β − γ t g ⁡ cos ⁡ β + u ¨ g ⁡ sin θ − β tan 2 α n + φ n = u ¨ g cos θ − β − γ t sin θ − β tan 2 α n + φ n − g γ t cos ⁡ β ⁡ tan 2 α n + φ n − sin ⁡ β (30)

After obtaining the dilatancy angle and basic friction angle after nth cyclic shear, the calculation procedures of permanent displacement are as follows:

1) The seismic acceleration u ¨ g is divided into n∆t with time-history, and the critical acceleration a c can be obtained based on Eq. (27)

Based on the experimental results of wear coefficient of the average dilatancy angle ( R s ) for specimens with the initial undulant angle of 16° (Table 1), by fitting the experimental results as Eqs. 14, 15, the correlation parameters A and B were obtained with Eq. 11, and Figures 8A,B demonstrate the fitting relationship between correlation parameters A and B and the normal stress σ n with different uniaxial compressive strengths of the intact rock σ c . Then, the fitting relationship between correlation parameters a, b, c, and d and the uniaxial compressive strength of the intact rock σ c were obtained by regression analysis, and the results are demonstrated in Figures 8C,D, with the expressions of A and B being obtained, as indicated below: A = ( 0.0039 σ n − 0.0263 ) σ c − 0.0233 σ n + 1.2222 (31) B = ( 0.0839 − 0.0074 σ n ) σ c − 0.3031 σ n + 1.3562 (32)

Based on Eqs. 31, 32, after 10-th shear cycles, the values of parameters A and B were 0.583 and 3.033, respectively. According to Eqs. 13, 20, the variations of the dilatancy angle and basic friction angle with the increase of cyclic shear number were obtained (Figure 9). The dilatancy angle and basic friction angle gradually decreased and tended to be stable with the increase of the cyclic shear number. After 10-th cyclic shear, the calculated final dilatancy angle was 7.015°. From Eq. 20, the basic friction angle after 10-th cyclic shear was 17.728°.

After obtaining the dilatancy angle and basic friction angle, based on the analysis steps of permanent displacement, the variations of critical acceleration and permanent displacement with the cyclic shear number are obtained respectively (Figure 10), where the number of cycle shear is 0, indicating that the deterioration effect of the structural plane is not considered. With the increase of the cyclic shear number, the critical acceleration decreases gradually, especially when cyclic shear number is greater than 5, the critical acceleration decreases slowly and tends to be stable. The permanent displacement increases gradually with the increase of cyclic shear number. And the permanent displacement of the slope is obtained to be 0.015 m when the deterioration effect is not considered. Before the 5-th cyclic shear, the permanent displacement increases linearly with the cyclic shear number, and then the permanent displacement increases slowly and gradually tends to be stable, and the calculated final permanent displacement is 0.420 m after 10-th cyclic shear.

The comparative calculation results of critical acceleration are presented in Table 2. The critical acceleration of the Newmark method is 0.613 g. For the Qi method, when P = 1 , the critical acceleration obtained is the same as that obtained with the Newmark method, and it can be found that the critical acceleration of the Qi method reduces with a decreasing of reduction coefficient P. This is because the shear strength gradually deteriorates during the cyclic shear process, and the ground motion load required to initiate a landslide is reduced. In the proposed method, the critical acceleration is smaller than that obtained with the Newmark method, and is between those of the Qi method with p = 0.4 and p = 0.5. It is worth noting that, the critical acceleration obtained with the Newmark method remains constant under seismic loading, while the critical acceleration reduces gradually with the deterioration of the structural plane in the proposed method and the Qi method, which is more in line with actual observations.

The calculation results of permanent displacement are shown in Table 2; Figure 11. In the proposed method, the cumulative displacement curve obtained is between those of the Qi method with p = 0.4 and p = 0.5, and in contrast the cumulative displacement is closer to the Qi method with p = 0.5. However, the permanent displacement obtained by Qi method and the proposed method is larger than that provided by the Newmark method, which illustrates that the deterioration effect of the structural plane has a significant effect on the calculation of permanent displacement, and the calculation methods that considers the deterioration effect are more conservative and reliable in terms of evaluating the stability of seismic-induced landslides.