Failure of Rock Slopes with Intermittent Joints: Failure Process and Stability Calculation Models

Rock slopes with intermittent joints in open-pit mines are complex geological bodies composed of intact rock and discontinuous structural planes, and their stability analysis are necessary for mine disaster prevention. In this study, a series of base friction tests were performed to determine the failure process and displacement field evolution of rock slopes with intermittent joints using the speckle technique of a noncontact measurement system. Next, stability calculation models of the slopes were established from the energy perspective using the plastic limit analysis theory, and the effects of the joint inclination angle and coalescence coefficient of rock bridges on the slope stability were evaluated. The four main conclusions are as follows: (1) The failure of rock slopes with intermittent joints shows the feature of collapse-lower traction-upper push. (2) Based on the failure modes of rock bridges in slopes, the failure of rock slopes with intermittent joints could be divided into three types: tensile coalescence (type A), shear coalescence (type B), and tensile–shear coalescence (type C). (3) Among the three slope types, the stability of the type A slope is significantly influenced by rock cohesion, whereas that of the type B slope is significantly influenced by joint cohesion. The stability of type C slope is significantly influenced by the joint inclination angle and joint friction angle. (4) The local-stable slope is unstable while the first through-tensile crack in the zone of the potential sliding body higher than the critical instability height appeared. This study guides the stability evaluation and instability prediction of jointed rock slopes in openpit mines.


Introduction
Rock slopes with intermittent joints in open-pit mines are complex geological bodies composed of intact rock and discontinuous structural planes. The discontinuous joint plays a significant role in controlling the strength and failure mode of slope rock masses [1][2][3]. The high-stress concentration of the tips of intermittent joints in open-pit mine slopes is formed under the action of long-term external factors (Figure 1), which leads to the initiation, propagation, and coalescence of cracks in rock bridges. When the original joints in the slope are interconnected through rock bridges, an overall potential sliding surface will be formed inside the slope ( Figure 2). Because the deformation and failure of rock slopes with intermittent joints are progressive and the failure mode is mostly influenced by the controlling structural plane, the deformation characteristics and failure mechanism of the slopes are complex [4,5]. An in-depth study of the propagation behavior of internal joints and the macroscopic instability mechanism of rock slopes with intermittent joints is useful for the stability evaluation and engineering treatment of jointed rock slopes in open-pit mines.
The stability of a rock slope with intermittent joints is influenced by two critical factors, i.e., the mechanical properties of rock bridges and joints and the spatial distribution of intermittent joints. The failure of rock bridges in a jointed rock mass is usually investigated through smallscale rock mechanics tests or numerical simulation methods. Several scholars have investigated the mesofailure mechanism of rock masses with joints using experimental methods. Wong et al. conducted experimental studies on the compressive strength and crack coalescence pattern of rock specimens with two or more preexisting flaws under uniaxial compression [6][7][8][9][10][11][12]. Huang et al. assessed the mechanical behaviors of rocks containing preexisting flaws under unloading or dynamic conditions using fracture mechanics principles and an experimental method [5,[13][14][15]. Li et al. analyzed the cracking processes of rocks with preexisting flaws under dynamic loads [16,17]. In addition, the numerical modeling of jointed rock-mass failure has been conducted extensively. Tang et al. analyzed the generation and propagation mechanisms of cracks in jointed rock masses and evaluated the deformation and failure processes of rocks with different intermittent joint distributions using numerical methods [18][19][20][21].
Furthermore, scholars have made valuable achievements in the macroscopic failure mechanism of rock slopes with intermittent joints. Jennings developed stability calculation methods and established the failure modes of jointed slopes in open-cast mines through theoretical analysis [22][23][24]. Huang et al. studied different step-path failure modes of rock slopes with different intermittent joints using the 2D particle flow code [25,26]. Camones et al. developed the numerical manifold method for analyzing the progressive failure of rock slopes [27,28].
Most previous studies have been conducted on the failure of rock slopes with intermittent joints through numerical simulation. However, related stability calculation model studies on the failure of rock slopes with intermittent jointsare few, and the previous studies ignored the factors that influence the stability of such slope. Because the base friction test combined with the speckle technique of the noncontact measurement system can simulate the slope failure process under gravity and the displacement field development, the method is suitable for analyzing the failure of rock slopes with intermittent joints. In this study, a series of base friction tests were first performed to reveal the failure process and displacement field evolution of rock slopes with intermittent joints. Next, stability calculation models and the theoretical failure criterion of the slopes were established based on the plastic limit analysis theory, and the factors influencing the slope stability were evaluated. Finally, the applicability of the theoretical failure criterion was verified through numerical simulation analysis using FLAC 3D . This study provides theoretical guidance for the stability evaluation and reinforcement design of jointed rock slopes in open-pit mines.

Laboratory Base Friction Tests
2.1. Principle of Base Friction and Devices. The base friction test method uses the drag forces acting along the base to represent the volume force (gravity) based on the principle that the distribution of drag forces in the drag direction is similar to that of the actual gravitational field [29]. The prefabricated simulation model is laid on the test bench, and the model bottom is in close contact with the belt ( Figure 3). As the rotating roller rotates counterclockwise, friction between the model bottom and the belt is generated by the frame constraint. If the rotation speed is uniform, the test model bottom is subjected to constant frictional stress F = f γt, where f is the friction coefficient between the model and moving belt, γ is the unit weight of the model material (N/m 3 ), and t is the thickness of the model material (m) [30].
According to Saint-Venant's principle, the frictional stress can simulate gravitational stress with the model if the model is sufficiently thin [30,31]. Hence, the failure of the model caused by the belt rotation can be equivalent to the failure of the prototype under gravity. However, similar  2 Lithosphere geometric and stress conditions need to satisfied for the test results to be valid [29,32]. Base friction tests were performed using a tester at China University of Mining and Technology, Beijing ( Figure 3). The length, width, and height of the tester were 1900, 1400, and 700 mm, respectively, and those of the red model container with a wooden frame were 1000, 920, and 50 mm, respectively. The motor speed was 1.5-12 RPM, and the seamless belt was driven using a rotating roller with a diameter of 300 mm. During the base friction tests, a GoPro HERO 7 Black motion camera with inbuilt Wi-Fi and Bluetooth and determined to be compatible with the GoPro capture app was used to capture and record the images ( Figure 4). Furthermore, an image correlation analysis software, Vic-2D, was used to determine the total displacement field of the slope, and the deformation of any monitoring point in the model could be tracked using GIPS software to obtain the history curves of displacement and time of the tracking points.

Models and Materials.
Because of the complexity of actual slopes in open-pit mines, the slope simulation models established in this study were simplified to analyze the typical failure modes of rock slopes with intermittent joints under gravity based on the noncontact measurement results of in situ joints in open-pit iron mine. The designed thickness of the model was 15 mm, which satisfied the condition of "the sufficiently thin model" in Saint-Venant's principle. Based on several test results, six models of slopes with different joint distributions were developed ( Figure 5). Each slope model had a height of 600 mm and a width of 1000 mm. The geometry factors of the models used for the tests were 100 each, and the corresponding prototype had a height and width of 60 and 100 m, respectively. In addition, the dynamic friction coefficient between the model and the belt was 0.7, and the unit weight scaling factor was 1.
Previous studies established the deformation and failure behavior of soft foundation waste dumps through base friction tests and satisfactorily replicated the failure process of waste dumps [33,34]. Therefore, the composition and pro-portion of similar materials used in this study were determined through several material proportion tests, referring to materials selected in the previous studies ( Table 1). The physicomechanical properties of the similar materials are listed in Table 2.

Results and Analysis
2.3.1. Failure Process Analysis. Figures 6-9 depict the deformation and failure patterns of the models at different test times. Here, the test time (t) refers to the accumulated time when the belt starts to run. The potential sliding body consisted of several blocks cut from a few tensile cracks originating from around the tips of the joints and propagating to the slope surface (Figures 6-9). It was found that the tensile cracks were generally perpendicular to the original joints. The detailed analysis of the failure process of each slope model is described in the subsequent paragraphs. Because of space limitations, and the failure processes of slopes 3 and 4 were similar to that of slope 5, only slope 5 is described in this paper.
(1) Slope 1: when the test time was 4.3 s, shear coalescence first occurred in the lower-part slope at rock bridge (1), and shear cracks were formed at the right tip of joint 2 and the left tip of joint 4 ( Figure 6(b)). In addition, tensile cracks appeared perpendicular to the joints at the right tips of joints 1-4 and the left tip of joint 5 and then gradually propagated along the slope surface direction. However, the tensile cracks did not extend to the slope surface. When the test time increased to 8.2 s, shear coalescence first occurred at rock bridge (2) and then at rock bridge (3), and a shear crack appeared at the right tip of joint 4 ( Figure 6(c)). Additionally, new tensile cracks were formed on the slope and intertwined in the middle part of the slope, which caused several tensile cracks to cut the potential sliding body. As the test time increased further, many tensile cracks extending to the slope surface were formed in the sliding body located on the potential sliding surface Because of the large free face of the slope toe, the sliding body at the slope toe tended to collapse under gravity and caused the middle part of the slope to slide. Next, the separation of the toe of the slope from the main slope decreased the front support force of the middle part of the slope. Finally, the middle part of the slope slid to the slope toe under the push action of the upper part of the slope. Thus far, the progressive deformation and failure mode starting from the toe of the slope and developing toward the upper part of the slope was formed.
(2) Slope 2: the failure process of slope 2 was similar to that of slope 1 (Figure 7). When the test time was 4.7 s, shear coalescence initially occurred in the lowerpart of the slope at rock bridge (1), and shear cracks appeared at the right tips of joints 2 and 3   igure 5: Model slopes with different joint distributions adopted in test (cm). Slopes 1, 3, 4, and 5 contain a low-dip joint set (inclination angle of 25°), which are parallel but not coplanar, and the bridging angle γ (the inclination angle of the ligament between adjacent tips of joints) increases from 45°, 90°, and 115°to 135°. Slope 2 contains low-and high-dip joints (inclination angles of 25°and 60°, respectively), and the bridging angle γ is 62°. Slope 5 contains three joints with different dips and rock bridges longer than those of other slopes.  Figure 7(b)). Simultaneously, tensile cracks were formed perpendicular to the joints at the right tips of joints 1 and 3, and the left tip of joint 2 gradually propagated along the slope surface direction. However, the tensile cracks did not extend to the slope surface. When the test time was 8.3 s, the tensile crack formed at the left tip of joint 2 extended to the slope surface, and the local sliding block below the tensile crack tended to slide (Figure 7(c)). As the test time increased further, shear coalescence occurred at other rock bridges, and multiple tensile cracks extending to the slope surface were formed, resulting in the gradual disintegration of the sliding body into several blocks ( (4) Slope 6: numerous tensile cracks were formed at the tips of the joints (Figure 9), and shear cracks propagated along the joint direction at the right tips of joints 1 and 2 ( Figure 9(b)). Finally, tensile-shear failure appeared at the rock bridges of slope 6 with an increase in the test time, and the progressive deformation and failure mode of slope 6 were similar to those of other models (Figure 9(d)) Based on the above findings, several key conclusions could be summarized as follows: (1) The failure of a rock slope with intermittent joints occurs progressively, and the rock bridges are cut one by one under gravity from the bottom upward. Because of the large free face of the slope toe, the sliding body at the slope toe initially tended to collapse under gravity, causing the middle part of the slope to slide. Next, the separation of the toe of the slope from the main slope reduced the front support force of the middle part of the slope. Finally, the middle part of the slope slid to the slope toe under the push action of the upper part of the slope. Thus far, the progressive deformation and failure mode of the rock slope with intermittent joints showed collapse-lower traction-upper push Figure 10 shows the variation in the vertical displacement difference of the tracking points in slopes 1 and 2 with increasing test time, which indicates the change in the dislocation distance between different slope parts. The vertical displacement difference of the tracking points in the lower part of the slope (v ab ) first increased sharply under gravity. Next, those of the middle part (v cd ) and the upper part of the slope (v ef ) also increased successively. This trend indicated that when a tensile-coalescence crack was formed near the slope toe, the sliding body at the toe tended to collapse, and the separation of the toe of the slope from the main slope could provide sufficient space for the failure of the middle and upper sliding-body parts. Therefore, the reinforcement of potential failure of a rock slope with intermittent joints should be applied to the lower part of the slope near the toe.
(2) Based on the failure modes of rock bridges in slopes, the failure of rock slopes with intermittent joints could be divided into three cases: Case 1: shear coalescence failure (type A). In this case, the rock bridge length was relatively short, and the bridging angle was generally lower than 90°. Type A slopes corresponded to slope 1 ( Figure 6(a)) and slope 2 ( Figure 6(b)) in this study. Case 2: tensile coalescence failure (type B). Here, the rock bridge length was relatively short, and the bridging angle was generally higher than 90°. Type B slopes corresponded to slopes 3-5 ( Figure 6(c)). Case 3: tensileshear coalescence failure (type C). The rock bridge length was relatively long. In this study, type C slopes corresponded to slope 6 (Figure 6(d)).

Deformation
Analysis. The speckle technique of the noncontact measurement system was used to investigate the deformation behavior and analyze the total displacement field of the slopes. Figure 11 shows the contours of the total displacement of the different models. The displacement of the sliding body above the step-path slip surface was significantly larger than that of the bedrock, and the displacement fields were somewhat partitioned into several blocks by the tensile cracks on the main body. The maximum displacement was observed near the toe of the slope, the center of the middle part of the slope, and the base of the crest. These indicated that the slope toe tended to slide first under gravity, inducing the progressive failure of the upper-sliding body due to the loss of support, which further verified the conclusion stated in Section 2.3.1.

Stability Calculation Models of Rock Slopes with Intermittent Joints
From the above test results, it was observed that the failure of rock slopes with intermittent joints started from the toe of the slope and steadily developed towards the upper part of the slope. The rock bridges in the slopes were successively cut under gravity from the bottom upward, indicating that the mechanical properties of rock bridges and joints and the spatial distribution of joints in the slope significantly influenced the slope stability. Therefore, based on the plastic limit analysis theory, the theoretical mechanical models of three slope types (types A, B, and C) were established in this study, considering the failure characteristics of rock slopes with intermittent joints under gravity. The two assumptions are as follows: (1) The intermittent joints of the slope tended toward the free face direction. And the potential sliding body tended to slide along the direction of the lowdip joints, and it satisfied the small deformation hypothesis, i.e., the potential sliding body is regarded as a rigid-plastic body with small deformation, and the geometric dimension of which is the same as that before deformation (2) For type A slopes (e.g., slopes 1 and 2), the rock bridges were fractured, owing to shear extension under gravity. For the type B slopes (e.g., slope 5), the rock bridges were fractured by tensile cracks. For the type C slope (e.g., slope 6), the rock bridges were fractured, owing to tensile-shear extension 3.1. Stability Calculation Model of A-Type Slope. Based on the slope model shown in Figure 5(a), a two-dimensional mechanical model of the type A slope was established ( Figure 12). According to the above test analysis, the rock bridges of type A slopes were fractured, owing to shear extension under gravity. As shown in Figure 12, H is the slope height, and H 0 is the height of the local surface exposure of the joint from the slope bottom. AB and BC are the joint distribution ranges along the joint trace direction and the direction perpendicular to the joint trace, respectively. a i is the length of the joint parallel to AB, d i is the projected length of an intact segment i of the rock bridge to AB, and s i is the projected length of expanded crack i to AB in the rock bridge. Δ ACD is the range of the potential sliding body after simplification. θ is the inclination angle of the slope surface, W is the weight of the overlying potential sliding body, and x − A − y is the local coordinate system.
From the geometric relationship depicted in Figure 12, the weight of the overlying potential sliding body (W) can be calculated using The coalescence coefficient of the joints along the joint trace direction (p) can be defined as follows [21,24,34]:   where n is the number of the intermittent joints, and L AB is the length of the joint distribution range along the joint trace direction.
The coalescence coefficient of rock bridges perpendicular to the joint trace direction (q) can be defined as follows [22,25]: where m is the number of expanded shear cracks at the tips of the joints, and n − 1 is the number of rock bridges. Because the sliding body analysis is based on the small deformation assumption, the sliding body size is always the same. According to the plastic limit analysis theory, assuming that the speed of the sliding body during the limit sliding is v and the direction is as shown in Figure 12, the internal energy dissipation of the sliding body during failure (P D ) could be calculated as follows [35,36].
where P D1 , P D2 , and P D3 are the internal energy dissipation values of the intermittent joints, expanded shear cracks at the joint tips, and intact rock segments at the rock bridges during the failure of type A slope, respectively. c 1 , c 2 , and c 3 are the cohesion values of the intermittent joint, expanded shear crack at the joint tip, and intact rock at the rock bridge, respectively, and φ 1 is the internal friction angle of the joints. The power generated by gravity (P G ) is the product of the weight of the sliding body and the vertical component of the velocity, and it could be calculated as follows: When the entire sliding body of type A slope tends to slide, the energy stability coefficient (E s1z ) is defined as the ratio of the internal energy dissipation to the gravity power. The energy stability coefficient (E s1z ) could be obtained using The higher the value of E s1 , the better the slope stability. When E s1 > 1, the slope is stable. The slope is in a critical state when E s1 = 1 and tends to be unstable when E s1 < 1.    Figure 5(c), a two-dimensional mechanical model of type B slope was established ( Figure 13).
As shown in Figure 13, the coalescence coefficient of rock bridges perpendicular to the joint trace direction (q′) can be obtained as follows [22,25]: According to the plastic limit analysis theory, the internal energy dissipation of the sliding body during the failure of type B slope (P D ) could be calculated as follows [35,37]: where P D1 and P D2 are the internal energy dissipation coefficients of the intermittent joints and intact rock segments at the rock bridges during the failure of type B slope, respectively, and σ t,BC is the tensile strength of the intact rock. The definitions of other parameters in Equation (8) are the same as those in Equation (4).
Therefore, when the sliding body of type B slope tended to slide as a whole, the energy stability coefficient (E s2z ) could be expressed as follows: 3.3. Stability Calculation Model of Type C Slope. Based on the slope model depicted in Figure 5(d), a two-dimensional mechanical model of type C slope was developed in this study ( Figure 14). The coalescence coefficients of the joints along the joint trace direction (p) and perpendicular to the joint trace direction (p′) can be defined as follows [22,37]: The equations for the coalescence coefficients of rock bridges along the joint trace direction (q) and perpendicular to the joint trace direction (q ′ ) can be expressed as follows [37]:  8 Lithosphere where ∑c i is the sum of the projected length of the previously expanded shear cracks to AB in the rock bridges and ∑l i is the sum of the projected length of the previously expanded tensile cracks to BC in the rock bridges.
The internal energy dissipation of the sliding body during the failure of type C slope (P D ) can be calculated as follows [35,36]: The energy stability coefficient (E s3z ) of type C slope can be obtained by
The relevant parameters above were substituted into Equations (6), (9), and (13). Variation curves of the energy stability coefficients (E sz ) of the three slope types with different rock cohesions and joint inclination angles were obtained (Figures 15-17).
The energy stability coefficients (E sz ) of the three slope types increased with increasing c 3 and decreasing α (Figure 15), indicating that c 3 and α significantly influenced the stability of rock slopes with intermittent joints. Moreover, the possibility of slope failure increased with decreasing c 3 and increasing α. When c 3 was constant and α increased from 40°to 70°, the rate of decrease in E sz of type C slope was the highest, followed by those of types A and B slopes, i.e., ΔE s3z > ΔE s1z > ΔE s2z (Figure 16). This trend signified that the stability of the type C slope was the most sensitive to the variation in α, followed by that of type A, and type B stability was the least sensitive. Furthermore, the energy stability coefficients (E sz ) of types A and C slopes sharply increased with c 3 when α was constant (lines I, II, V, and VI) ( Figure 17). However, E sz values of the type B slope were unchanged (lines III and IV), suggesting that the stabilities of types A and C slopes were significantly influenced by c 3 , whereas that of the type B slope were unaffected by c 3 . The major reason was that the rock bridges in the type B slope were in a tensile state during failure, and the coalescence mainly depended on the tensile strength of the intact rock (σ t,BC ).
The energy stability coefficients (E sz ) of the three slope types increased with c 1 and φ 1 (Figures 18 and 19), showing that c 1 and φ 1 significantly influenced the stability of rock slopes with intermittent joints, and the probability of slope failure increased with decreasing c 1 and φ 1 . When c 1 was constant and φ 1 increased from 10°to 28°, the increasing 10 Lithosphere amplitude of E sz of the type C slope was the largest and significantly higher than those of types A and B, i.e., ΔE s3z > ΔE s1z > ΔE s2z . This trend suggested that the sensitivity of type C slope stability to the variation in φ 1 was significantly higher than those types A and B slopes, and the stability of the type B slope was the least sensitive to the change in φ 1 . Furthermore, the increase rate of E sz was more rapid when φ 1 > 24°than when φ 1 < 24°( Figure 20). This difference indicated that E sz was more sensitive to the variation in φ 1 when φ 1 > 24°. When c 1 increased from 0.1 to 1.0 MPa, the increase in the amplitude of E sz of the type B slope was the highest, followed by those of types A and C slopes, i.e., Δ E s2z > ΔE s1z ≈ ΔE s3z . This behavior signified that the stability of the type B slope was more significantly influenced by the change in φ 1 than with those of types A and C slopes.
The energy stability coefficients (E sz ) of the three slope types decreased with the increase in the coalescence coeffi-cients of rock bridges along the joint trace direction (q) and perpendicular to the joint trace direction (q ′ ) ( Figure 21). These variations indicated that q and q ′ significantly influenced the stability of rock slopes with intermittent joints, and the probability of slope failure increased with increasing q and q ′ . In addition, E sz1 of the type A slope decreased with an increase in q, but it remained constant  12 Lithosphere with an increase in q′ (Figures 21(a) and 21(b)). On the contrary, E sz2 of the type B slope decreased with increasing q ′ but remained unchanged with increasing q. The main reason was that the rock bridges of the type A slope were in a shear state during failure, and the coalescence mainly depended on the shear strength of the intact rock; however, type B rock bridges were in a tensile state, and the coalescence mainly depended on the intact-rock tensile strength. Furthermore, E sz3 of the type C slope was related to both q andq′. Figure 22 shows that lines I and III change more significantly with an increase in q/q ′ than lines II and IV, and line IV shows the most gentle variation. These trends indicate that the stability sensitivities of types A and C slopes to the change in q are similar and higher than that of the type B       14 Lithosphere slope to q′. The stability of the type C slope is the least sensitive to variation in q′. As can be seen from Figure 23, lines I, II, and III change more sharply than lines IV, V, and VI. These trends suggest that the stability of the type C slope was more significantly influenced by q than q′.

Failure Criterion of Rock Slopes with Intermittent Joints.
The type A slope is used as an example. The difference function between the internal energy dissipation and gravity power was derived as follows: Furthermore, from Equation (14), the f ðHÞ equation can be reexpressed as follows: It can be observed from Equation (15) that f ðHÞ is a linear function of H. The relevant parameters in Equations (16) and (17) Figure 24. In this case, regardless of the value of H, the slope is always stable. This behavior signifies that the potential sliding body at the lower part of any through-tensile crack is stable under gravity.

4.4.2.
Type II: Local-Stable Slope. When M 1 < 0 and T Ι = N 1 − M 1 ⋅ H 0 > 0, the curve pattern of f ðHÞ assumes curve II of Figure 24. H cr is defined as the critical point of slope instability, i.e., the critical instability height (CIH). When 0 < H < H cr , the slope is stable, indicating that when the through-tensile cracks are within the height range of zero to H cr of the slope, local sliding failure will not start from these cracks. The main reason is that the sliding force of the potential sliding body at the lower part of these cracks is lower than the antisliding force generated by the rock bridges and joints, which does not satisfy the necessary and sufficient conditions for instability. When H = H cr , the slope exists in a critical state. However, when H cr < H < H 1 , the slope is unstable while the first through-tensile crack in the zone of the potential sliding body higher than the CIH appears. This behavior occurs because the sliding force of the potential sliding body at the lower part of these cracks is higher than the antisliding force provided by the rock bridges and joints, which satisfies the necessary and sufficient conditions for instability. The upper part sliding body then tends to slide, owing to the loss of the front support force. Therefore, the penetration of the first tensile crack (the initial failure position) in the zone of the potential sliding body higher than the CIH can be regarded as the failure criterion of the local-stable slope. In addition, with increasing H cr , the unstable zone (red) area in Figure 24 decreases, while the stable zone (green) area increases. These trends show that the probability of slope instability decreases with the CIH.
Let f ðHÞ = 0. Hence, the CIH of the local-stable type A slope (H cr1 ) can be determined as follows: The relevant geometric and mechanical parameters should satisfy the following conditions: where M 1 and N 1 can be calculated using Equations (16) and (17), respectively.

4.4.3.
Type III: Unstable Slope. When M 1 < 0 and T Ι = N 1 − M 1 ⋅ H 0 < 0, the f ðHÞ curve follows the curve III pattern shown in Figure 24. In this case, the slope is always unstable irrespective of the H value. This suggests that the potential sliding body at the lower part of any through-tensile crack is unstable under gravity, and the separation of the toe of the slope from the main slope decreases the front support force of the middle part of the slope. The middle part of the slope tends to slide toward the slope toe under the push action of the upper part of the slope. Similar to the type A slope, type B and C slopes can also be divided into three types based on the quantitative relationship between the correlation coefficients, i.e., stable slope (type II), local-stable slope (type I), and unstable slope (type III). Because of space limitations, the analysis will not be explained in detail in this paper. The CIH of the type B slope (H cr2 ) can be calculated as follows: The relevant geometric and mechanical parameters of the type B slope should meet the following conditions:  Figure 24: Curve patterns of f ðHÞ and stability zoning sketch of potential sliding body of local stable slope. TΙ, TΙΙ, and TΙΙΙ are the intercepts of curves 1, 2, and 3, respectively, H1 is the height of the potential sliding body, and Hcr is the critical instability height of the slope.
Similarly, the CIH of the type C slope (H cr3 ) can be calculated using the following equation: In summary, the stability discrimination process of rock slopes with intermittent joints can be depicted as follows ( Figure 25).

Conclusions
In this study, the failure behavior of rock slopes with intermittent joints under the influence of gravity was investigated through base friction tests and plastic limit theory analysis. The failure criterion was proposed. Four main conclusions are as follows: (1) The deformation and failure of rock slopes with intermittent joints showed collapse-lower traction-upper push, and they started from the lower part slope and developed toward the higher part slope. The slopes exhibited sliding failure characteristics along the direction of the low-dip joints. The displacement fields were somewhat partitioned into several blocks by the tensile cracks on the main body, and the maximum displacement occurred at the toe of the slope, the center of the middle part of the slope, and the base of the crest. The reinforcement of potential failure in a rock slope with intermittent joints is suggested to be applied to the lower part slope near the toe (2) Among the three rock slope types, the stability of the type A slope was significantly influenced by the coalescence coefficient of rock bridges along the joint trace direction and rock cohesion. The stability of the type B slope was significantly influenced by the coalescence coefficient of rock bridges perpendicular to the joint trace direction and joint cohesion but was unaffected by rock cohesion, which was different from the other slopes. The stability of the type C slope was significantly influenced by the joint inclination angle, joint friction angle, and coalescence coefficient of rock bridges along the joint trace direction (3) When M 1 > 0 and T Ι = N 1 − M 1 ⋅ H 0 > 0, the slope was always stable, regardless of the value of H. When M 1 < 0, T Ι = N 1 − M 1 ⋅ H 0 > 0, and 0 < H < H cr , the slope was stable, and when H = H cr , the slope existed in a critical state. However, when H cr < H < H 1 , the slope was unstable while the first through-tensile crack in the zone of the potential sliding body higher than the CIH appeared. When M 1 < 0 and T Ι = N 1 − M 1 ⋅ H 0 < 0, the slope was unstable while the first through-tensile crack in the slope appeared, irrespective of the H value

Data Availability
The data used to support the findings of this study are available from the corresponding author upon request. Step 1 Step 2 Step 3 Whever there is a through-tension crack in the range of H > Hcr2 Whever there is a through-tension crack in the range of H > Hcr3 Figure 25: Stability discrimination process of rock slopes with intermittent joints.
Step 1: classify the slope according to the joint distribution (types A, B, and C).
Step 2: further classify the slope according to the quantitative relationship of the correlation coefficient of the difference function (Types I, II, and III).
Step 3: evaluate the stability of the slope. 18 Lithosphere