Introduction

Due to the influence of the special geological structure and orogeny in the geological history, a large number of bedding slopes are widely distributed in the mountainous areas of southwest China. Because the steep bedding slope with the dip angle greater than the slope angle is relatively stable in the natural state, there is little attention to the instability mechanism of this kind of slope. However, according to the sorting and analysis of disasters around Wenchuan by some scholars in recent years1,2,3,4,5,6,7,8,9, the instability phenomenon of steep bedding slope also widely exists, such as Tangjiashan landslide in Beichuan, Ganmofang landslide in Anxian and Xiaojiaqiao landslide in Anxian, etc., these are typical representatives of this kind of landslide. Therefore, the above three landslides are selected as the cases of steep bedding landslide for analysis, and the slope generalization model is established according to the common points of these landslides, in order to further study the seismic dynamic failure mechanism of steep bedding slope through numerical simulation.

At present, research methods for the stability and dynamic response of rock slopes under seismic action include post-earthquake field investigation, physical model test, and numerical simulation10,11,12. As one of the most effective methods for studying the failure mode of slopes under earthquakes, shaking table model test has been widely applied to the dynamic response analysis of slopes with different structural types13. Dong et al.1 conducted large-scale shaking table tests on bedding rock slopes, investigated the failure modes of bedding slopes under dynamic action, and analyzed the effects of input wave shape, frequency and amplitude on the dynamic response of slopes. Reference3,14,15 revealed the instability and failure process of slopes with different structural types under strong earthquakes through shaking table model tests on two types of rock slopes, namely, anti-dipping and bedding slopes. Wang et al.16 studied the dynamic response of bedding-structured slopes via large-scale shaking table model tests. Fan et al.17 carried out large-scale shaking table tests on bedding rock slopes with weak interlayers, and explored the effects of input seismic wave characteristics and the saturation state of argillized interlayers on the dynamic response of bedding rock slopes. Che et al.18 performed large-scale shaking table tests on rock slopes with bedding joints, and studied the acceleration response characteristics of slopes and their influence on slope failure modes. Liu et al.19 and Chen et al.20 investigated the effects of weak interlayer thickness and dip angle on the dynamic response and stability of rock slopes by means of large-scale shaking table tests on slopes containing weak interlayers.

With the development of computer technology, numerical simulation technology has been widely used in seismic dynamic stability analysis of rock slope. Li et al.21 used discrete element UDEC to analyze the influence of slope height, slope, rock inclination and seismic wave parameters on the safety factor of bedding rock slopes under seismic load. Ni et al.22 used the dynamic vector method to analyze the dynamic stability of rock slopes under seismic load based on the discrete element program 3DEC. This method overcomes the defect that the traditional limit equilibrium method does not consider the dynamic characteristics of slope materials or the fluctuation characteristics of seismic load. Chen et al.23 took the landslide of Baocheng Railway 109 tunnel triggered by Wenchuan earthquake as an example, and used discrete element numerical simulation to study the instability mechanism of jointed rock slope under earthquake action. Wu et al.24 and Feng et al.14 used the discontinuous deformation analysis method to simulate and analyze the dynamic failure of the slope model, and reproduced the failure mode of the landslide after the earthquake. Yuan et al.25 used PFC2D to conduct discrete element simulation for Donghekou landslide induced by Wenchuan earthquake, and studied the formation mechanism of the landslide and the cause of the “projectile” phenomenon. Lin et al.26 studied the influence of slope angle and seismic wave amplitude on the deformation and failure characteristics of bedding slope through discrete element numerical simulation. Feng et al.14 used the discontinuous deformation analysis method (DDA) to simulate the dynamic failure of layered slope models.

The Particle flow code (PFC) based on discrete element method describes the deformation and failure of granular aggregate materials by Newton’s second law, force-displacement law and bond failure criterion. Because it only needs to define the properties of particles and bonding, rather than the overall constitutive relationship, PFC has great advantages in the study of rock mechanics. At present, many scholars have used PFC to study the failure mechanism and stability of slope rock mass, such as: He et al.27 and Zhao et al.28 have used PFC to study the progressive failure and stability of jointed rock slope. Shi et al.29 and Bian et al.30 conducted particle flow simulation studies on the failure mechanism of jointed rock slopes under earthquake action. Huang et al.31 studied the failure form of step slip of rock slope under different penetration modes of rock bridge by PFC. Yang et al.32 studied the failure characteristics of multiface slopes under earthquake action through PFC. Zhang and Li33 used PFC to study the instability process of bedding shale slopes.

Although numerous meaningful studies have been conducted by previous researchers in the above-mentioned fields, the existing research has mainly focused on the failure mechanisms of gently or moderately dipping bedding slopes, with relatively limited attention paid to steeply dipping ones. For steeply dipping bedding rock slopes where the dip angle of rock strata is greater than the slope angle, they generally maintain good stability under natural conditions. However, once instability occurs under the action of strong earthquakes, they are prone to developing into high-speed landslides with extremely severe destructive consequences. Taking advantage of the merits of model testing and particle flow simulation techniques, this study adopts large-scale shaking table model tests and the PFC2D simulation method to investigate the deformation and failure mechanisms of steeply dipping bedding rock slopes under seismic dynamic loading. Meanwhile, it analyzes the influences of rock stratum dip angle and thickness on the deformation and failure characteristics of slopes as well as the failure modes of slope locking sections.

Shaking table model test

Design and fabrication of similar physical slope models

Similar relationships and similar materials

Based on the discussion of Buckingham’s theorem and dynamic similarity conditions, the similarity relationships of various physical quantities were finally determined by considering geometric conditions, physical conditions and kinematic conditions. In the shaking table model test, ten parameters including density ρ, length L and time t were selected as control parameters. The involved parameters are shown in Table 1.

Table 1 Main similarity constants in model test.
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Based on previous research, six materials including barite powder, quartz sand, gypsum, iron ore concentrate powder, ethanol, and rosin were selected as the similar materials for this test. Among them, barite powder and quartz sand served as fine aggregate and coarse aggregate respectively, gypsum functioned as a cementing agent, and the rosin-ethanol solution acted as a bonding material. Laboratory tests showed that the mass ratio of iron ore concentrate powder, barite powder, quartz sand, gypsum, rosin, and ethanol was determined to be 7.4:66.3:31.6:7.1:1:5.7. The physical and mechanical parameters of the similar materials were as follows: density of 2.50 g/cm³, compressive strength of 0.853 MPa, tensile strength of 0.099 MPa, elastic modulus of 124.46 MPa, Poisson’s ratio of 0.12, internal friction angle of 34.8°, and cohesion of 0.294 MPa.

Slope model production

In the test, the slope angle of the model slope was designed to be 50°. Restricted by the bearing capacity of the shaking table platform, the height of the model slope was set as 1.3 m, which is equivalent to a prototype slope height of 20.8 m according to the geometric similarity coefficient given in Table 1. The dip angle of the rock strata in the shaking table test of this study was designed as 55°. The model slope was constructed by paving the prepared similar materials layer by layer. During the model preparation process, the physical and mechanical indices were guaranteed by controlling the material density, and the bedding planes were coated with mica powder. The generalized design of the slope model is illustrated in Fig. 1.

Fig. 1
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Slope model.

Loading scheme

This test was aimed at investigating the seismic failure mechanism of steeply dipping bedding rock slopes. The Wolong seismic wave and sine wave were adopted as the input seismic waves, with the input direction set as the unidirectional X-axis. In accordance with the similarity criteria, the Wolong seismic wave was compressed with a time similarity coefficient of Ct=4 (see Fig. 2), and the frequency of the input sine wave was set at 10 Hz. Subsequently, the amplitudes of both the Wolong seismic wave and the 10 Hz sine wave were gradually increased from 0.1 g to 0.6 g at an interval of 0.1 g, so as to explore the evolutionary pattern of deformation and failure of the slope under the action of seismic waves with progressively increasing amplitudes. The specific input scheme is presented in Table 2.

Fig. 2
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Time-history curve of acceleration of natural seismic wave.

Table 2 Test input scheme.
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Test equipment and process

The shaking table tests were carried out on a 3.0 m×3.0 m three-directional six-degree-of-freedom seismic simulation shaking table in the School of Civil Engineering and Architecture, Henan University. The hardware of this shaking table test system consists of the table platform, actuators, accumulators, hydraulic power supply (HPS), control cabinet, and computer. The main technical parameters of the shaking table equipment are shown in Table 3. The test procedure of the shaking table was as follows: hoisting the model onto the table platform→connecting and debugging the sensors→conducting test loading and observing cracks→recording the failure morphology of the slope and collecting test data.

Table 3 Technical parameters of vibration table.
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Analysis of test results

During the test, after each round of seismic wave excitation, the failure status of the model slope was carefully observed and photographed, and the entire test process was recorded by video. The failure characteristics and failure modes of the model slope were then analyzed accordingly. Figure 3 shows the deformation and failure status of the model slope during the test. In the test with a sine wave input of 0.3 g amplitude, minor tensile cracks appeared at the shoulder of the slope top. When the sine wave amplitude was increased to 0.4 g, the tensile cracks propagated downward along the bedding planes, forming a locking section at the slope toe, and the stability of the slope was mainly maintained by the support of this locking section (Fig. 3a). Subsequently, in the test with a 0.5 g amplitude sine wave input, the locking section at the slope toe was sheared off, and the resulting shear fracture connected with the bedding cracks to form a sliding surface, leading to the sliding failure of the slope. Finally, under the action of the 0.6 g amplitude sine wave, the slope top underwent progressive tensile fracturing layer by layer under seismic loading, while the slope suffered further sliding failure (Fig. 3b).

Fig. 3
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Deformation and failure of the model slope in the test.

Based on the analysis of the deformation and failure process and characteristics of large-scale shaking table model tests on steeply dipping bedding rock slopes, the evolutionary process of deformation and failure modes of steeply dipping bedding rock slopes under seismic dynamic actions can be divided into four stages.

Formation stage of tensile cracks at slope crest

For steeply dipping bedding rock slopes where the dip angle of rock strata is greater than the slope angle, the slope generally maintains favorable stability under natural conditions. However, the bedding planes within the slope constitute the weakest structural plane. Under the action of seismic waves, affected by the elevation effect of seismic acceleration, the rock mass near the slope shoulder undergoes the most significant stress changes, which induces the first occurrence of tensile cracks along the bedding planes, as illustrated in Fig. 4a.

Fig. 4
Fig. 4
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Failure mode diagram of steep bedding rock slope under seismic dynamic action: (a) Tensile cracks develop at slope crest, (b) Tensile cracks propagate downward, forming locked segments, (c) Sudden shear rupture of locked segment leads to sliding surface penetration, (d) Slope sliding failure.

Stage of tensile crack downward propagation and locked segment formation at slope toe

Since the bedding planes are the weakest structural components of the slope, under the action of horizontal seismic forces, both the cracks and the bedding planes exhibit tensile characteristics. As a result, the tensile cracks gradually propagate downward along the internal bedding planes of the slope. After the formation of the trailing-edge tensile fractures, a locking section of the slope is formed at the leading-edge position of the slope toe, and the stability of the slope is mainly maintained by the support of this locking section, as shown in Fig. 4b.

Stage of locked segment shear rupture and sliding surface coalescence

With the continuous application of seismic loading, stress concentration occurs at the locking section formed at the lower part of the slope. Under the combined action of intense seismic forces and the gravity of the overlying sliding mass, the accumulated stress within the rock mass of the locking section at the slope lower part exceeds the bearing capacity of the rock mass. Consequently, the locking section undergoes sudden shear failure, and the shear fracture surface of the locking section combines with the tensile cracks along the bedding planes of the slope to form a through-going sliding surface, as shown in Fig. 4c.

Translational sliding phase of slope

With the further persistence of seismic loading, the rock mass of the middle and upper sliding body loses support due to the shear failure of the locking section, and slides toward the free face along the through-going sliding surface, resulting in the overall progressive sliding failure of the landslide. Meanwhile, constrained by the loss of restraint, the rock strata inside the slope top undergo progressive tensile fracturing layer by layer under the action of seismic waves, leading to further damage of the slope, as shown in Fig. 4d. Therefore, the failure mode of steeply dipping bedding rock slopes under seismic dynamic actions is defined as tensile-shear progressive sliding failure.

Particle flow simulation of slope failure evolution process

Selection of bonding model and mesoscopic parameters

There are two types of particle bonding models in PFC: contact bonding model and parallel bonding model. Among them, the parallel bonding model has the characteristics of tensile resistance, shear and moment effects and bond failure-the deterioration of the material macroscopic stiffness, which is more suitable for simulating rock materials.

The meso-parameters in PFC are not directly related to the macro-physical and mechanical parameters of the material. The process of establishing a corresponding numerical model to make the macroscopic phenomena and physical and mechanical properties shown in the numerical test match the results of the indoor or in-situ test is called the calibration of the microscopic parameters. The elastic modulus, Poisson’s ratio, compressive strength and tensile strength of rock material are calibrated by particle flow program (simulating uniaxial, triaxial and Brazilian splitting test, Fig. 5), and the cohesion c and internal friction angle φ of rock material are calibrated by using More-Coulomb strength theory. The process of parameter calibration is cumbersome, and the “trial-and-error method” is used to change the mesoscopic parameters repeatedly until the macroscopic parameters obtained by the simulation test are very close to the main physical and mechanical parameters of the actual rock material. Table 4 shows the physical and mechanical parameters of landslide prototype rock mass. Through repeated trial calculation of triaxial model test, the corresponding micro parameters are finally determined, as shown in Table 5.

Fig. 5
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Uniaxial compression test and direct shear test models: (a) Calculation model for uniaxial compression test, (b) Calculation model for direct shear test.

Table 4 Physical and mechanical parameters of rock mass.
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Table 5 Numerical calculation of micro parameters.
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Generation of granular geological body and model scheme

In principle, the smaller the radius of the sphere selected in the PFC modeling process, the closer the generated particle aggregate is to the real rock mass, but this will generate a huge number of particles, and the speed and capacity of the computer cannot meet the requirements. After continuous trial calculation, the number of particles generated can reach the desired calculation accuracy and save a lot of calculation time when the number of particles is 2 × 104. A boundary wall with an overall size of 45 m high and 40 m wide (Fig. 6) is established, and a particle aggregate with an initial porosity of 0.1 (the porosity is not the actual porosity, but is set for discrete sphere units) and a particle size of 0.1–0.2 m is generated in the wall, which is automatically balanced under the condition of no gravity and near no friction. Due to the limitation of the wall, the contact force between the particles is very large during the generation of particles. The contact force between particles is very large due to the limitation of the wall, the particle size needs to be appropriately scaled so that the internal average stress of the particle aggregate relative to the stress field generated by the subsequent self weight can be ignored. Finally, the float is processed to generate a uniform and dense initial particle aggregate.

Fig. 6
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Particle flow model of slope.

After the initial particle aggregate is generated, the particles are bonded according to the microscopic parameters calibrated in Table 5. Then, set the acceleration of gravity and carry out the initial balance under the action of gravity (the maximum unbalance force ratio is less than 10 − 5 magnitude) to form the initial ground stress. The model boundary is simulated by Wall element. The slope top cracks and weak interlayers are simulated by PFC built-in discrete fracture network (DFN).

In PFC, there are generally four steps to generate the final model: generating particle aggregates with complete rock block properties, stress initialization, fracture generation and particle bonding setting. In order to study the influence of the rock stratum inclination and thickness on the deformation and failure mode of steep bedding rock slope, the slope angle is 50°, When the rock thickness is 3.25 m, four different dip angles of 50°, 55°, 60° and 65° are set, When the dip angle of rock stratum is 55°, four different rock stratum thicknesses of 2.75 m, 3.25 m, 3.75 m and 4.25 m are set.

Boundary conditions and seismic wave input scheme

Since the wall in the discrete element program PFC can only apply velocity but not force, and the acceleration time history curve is integrated to obtain the velocity time history curve. Therefore, the particle boundary condition is adopted to convert the seismic wave into force and load it on the particle boundary, and the input seismic wave propagation direction is from bottom to top. The first 30s of E-W wave monitored by Wolong in the 2008 Wenchuan earthquake is used as the bottom of the horizontal seismic wave input model. The input Wolong wave acceleration and velocity time history curve is shown in Fig. 7.

Fig. 7
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Wolong wave acceleration and velocity time history curve.

Analysis of numerical simulation results

Failure mode of slope under earthquake action

Taking the steep bedding slope with a dip angle of 55 ° and a thickness of 3.25 m as an example, the failure process of the slope model under different earthquake durations is shown in Fig. 8. From the simulation results, it can be seen that the steep bedding slope is not easy to slip because the dip angle of the rock stratum is greater than the slope angle, and the rock mass at the foot of the slope is the part with relatively concentrated stress in the slope. Under the action of horizontal seismic force with the duration of seismic wave less than 15s, the slope is not damaged, but only tensile cracks appear in the upper part of the rock layer of the slope and some cracks appear in the rock at the foot of the slope. When the seismic wave duration reaches 20s, there are many cracks in the rock layer. The cracks spread from top to bottom and penetrate through the bottom of the slope, and a locked segment is formed at the bottom of the slope. And the cracks at the locked segment increase, but the slope has not been subject to sliding failure. It can be seen that the failure process of steep bedding slope under earthquake can be summarized as follows: the formation of tensile crack at the rock layer, the expansion of tensile crack at the layer, the formation of locked segment at the bottom of slope, and the sudden shear of locked segment leads to sudden instability of the slope. This is consistent with the deformation and failure process of steeply inclined bedding rock slopes in the vibration table model test.

Fig. 8
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Failure process of slope model under different earthquake duration.

Influence of the strata inclination on the failure mode of the slope

Figure 9 shows the failure modes of the model slope under different rock stratum inclination. It can be seen from the simulation results that when the dip angle of the rock stratum is 50°, shear failure occurs in the locked segment of the slope under earthquake action, and the sliding surface of the slope is composed of cracks at the rock stratum layer and the fracture surface of the locked segment, as shown in Fig. 9a. Figure 9b shows the failure characteristics of the model slope when the dip angle of the rock stratum is 55 °. The locked segment of the slope also appears shear failure under the action of seismic inertia force, and the failure characteristics of the slope are the same as when the dip angle of the rock stratum is 50°. When the dip angle of the rock stratum reaches 60°, due to the large dip angle of the layer, the slope is not easy to slide and lose stability as a whole. In the process of horizontal seismic force, the upper layer of the slope is first pulled apart under the left-right swing of the seismic wave. Moreover, due to the free surface conditions of the rock stratum near the slope surface, the displacement amplitude of the left-right swing is the largest, Therefore, the layer near the slope surface is first pulled apart; After the outermost rock layer is fractured, the restraint effect on the rock layer inside the slope decreases, and with the increase of the duration of seismic wave action, the rock layer inside the slope gradually cracked layer by layer from the outside to the inside (Fig. 9c). When the dip angle of the rock stratum is 65°, the same as the simulation results when the dip angle of the rock stratum is 60°, the locked segment of the slope is not damaged under the action of earthquake, while the tension crack first appears along the layer, resulting in the tension crack of the rock mass between the rock strata, Finally, under the further action of seismic force, the slope appears layer by layer tensile failure from outside to inside (Fig. 9d).

Fig. 9
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Failure diagram of slope model under different rock stratum inclination.

It can be seen from the simulation results that the rock dip angle has a certain influence on the failure characteristics of the locked segment under dynamic action. When the dip angle of the rock stratum is 50° and 55°, the locked segment of the slope appears shear failure under the action of earthquake, and the sliding surface of the slope is composed of cracks at the layer and fracture surface of the locked segment. When the dip angle of rock stratum reaches 60° and 65°, the locked segment of the slope is not damaged. The reason for this result is that the dip angle of the rock stratum is too large. Even if there is a rock layer in the slope, the locked segment of the slope cannot be sheared under the action of seismic dynamics. At the same time, the larger dip angle of the rock stratum is more likely to cause toppling failure of the slope. Therefore, when the dip angle of the rock stratum is 60° and 65°, the failure of the slope is layer by layer tensile failure from the outside to the inside.

Influence of rock stratum thickness on slope failure mode

Figure 10 shows the failure characteristics of model slopes under different rock layer thicknesses. It can be seen from the simulation results that the thickness of the rock layer has little effect on the failure mode of the steep bedding slope, and the failure of the slope is consistent with the results of the shaking table model test. All of them are tensile cracks at the rock stratum under the action of seismic power, the tensile cracks expand downward along the layer, the locked segment is formed at the bottom of the slope, and the locked segment is cut suddenly, resulting in the sudden instability of the slope. At the same time, the fracture mode of the locked segment of the slope with different rock thickness is basically the same, which is shear failure under the combined action of seismic inertia force and overburden load. However, under the action of the same input wave amplitude, the failure status of steep bedding slope with different rock thickness is obviously different, and the sliding failure of slope with small rock thickness is more serious than that of slope with large rock thickness under seismic dynamic action. This is because the slenderness ratio of rock stratum decreases with the increase of layer thickness, which makes the ability of rock layer to resist flexural failure gradually enhanced.

Fig. 10
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Failure diagram of slope model under different rock thickness.

Conclusions

Based on shaking table model tests and the particle flow code PFC2D, this paper investigates the failure characteristics of steeply dipping bedding rock slopes and the failure modes of slope locking sections under seismic dynamic actions. Meanwhile, it analyzes the effects of rock stratum dip angle and thickness on the failure modes of steeply dipping bedding rock slopes and the fracture characteristics of locking sections. The conclusions are as follows:

  1. (1)

    Under the action of seismic waves, minor tensile cracks first appeared in the rock strata at the slope shoulder. As the amplitude of the seismic waves increased, the tensile cracks propagated downward along the rock strata and formed a locking section at the slope toe in the lower part of the slope. With a further increase in the amplitude of the seismic waves, the rock mass of the locking section underwent sudden shear failure under the combined action of seismic forces and the gravity of the overlying sliding mass. The shear fracture surface of the locking section in the lower part of the landslide combined with the tensile cracks along the bedding planes of the slope to form a through-going sliding surface, and the slope slid toward the free face along this through-going sliding surface, resulting in sliding failure. Meanwhile, due to the loss of constraint, the rock strata inside the slope top underwent progressive tensile fracturing layer by layer under the action of seismic waves, which further exacerbated the sliding failure of the slope.

  2. (2)

    The deformation and evolution process of steeply dipping bedding rock slopes under seismic dynamic actions can be divided into four stages: the emergence of tensile cracks in the rock strata at the slope top, the downward propagation of tensile cracks along the bedding planes accompanied by the formation of a locking section at the slope toe, the shear fracture of the locking section and the penetration of the sliding surface, and the sudden instability and sliding of the slope. Meanwhile, the failure mode of steeply dipping bedding rock slopes under seismic dynamic actions is tensile-shear progressive sliding failure.

  3. (3)

    The dip angle of rock stratum has an influence on the failure mode of steep bedding slope under the action of earthquake. When the dip angle of rock stratum is 50° and 55°, the failure of the slope is sliding failure. The sliding surface is composed of rock stratum layer and shear section of locked segment. The failure mode of the locked segment of the slope body is shear failure under the combined action of seismic inertial force and the overlying load. When the dip angle of rock stratum is 60° and 65°, the locked segment of the slope is not damaged, and the failure of the slope is layer by layer tensile fracture from outside to inside.

  4. (4)

    Through the numerical simulation of steep bedding slope with different rock thickness under the action of earthquake, it is found that the thickness of rock stratum has little effect on the failure mode of slope and the fracture mode of locked segment. The failure of steep bedding slope with different rock thickness is sliding failure, and the slope sliding surface is composed of rock layer and shear section of locked segment. The failure of the locked segment is also the shear failure under the combined action of seismic inertia force and overburden load, but the sliding failure of the model slope with thin rock layer is more serious than that with thick rock layer.

  5. (5)

    Through the numerical simulation analysis of steep bedding slope under seismic dynamic action, it is found that there are two failure modes of steep bedding slope: sliding instability and layer by layer tensile crack. Moreover, the occurrence of these two failure forms is due to the first tensile fracture of the rock along the rock layer surface at the slope shoulder. Therefore, the reinforcement measures for the rock layer at the slope shoulder should be emphatically considered in the reinforcement design of the steep bedding slope. This understanding has important theoretical significance and engineering guiding significance for the seismic stability evaluation, disaster prevention and reduction of steep bedding locked rock slope.

The model test in this study is a one-off large-scale physical experiment. Restricted by limitations of testing equipment, funding, and time, comparative tests with multiple groups of models of varying heights were not conducted. However, we verified the rationality of the designed model dimensions through literature research and theoretical analysis, by drawing on the published research findings of similar shaking table tests on rock slopes. Future research will further refine the scope of application of the results by conducting verification with field monitoring data.