Cyclic Direct Shear Test on Rock Sample with Stepped Structural Plane

Abstract

By conducting the direct shear test on a sample with a stepped structural plane and selecting representative samples for the cyclic shear test under fixed normal stress, the effects of the number (N) of the stepped structural plane, the top length (l₁) of the stepped plane and the height (h) of the stepped structural plane on the shearing characteristics and damage morphologies of the specimens were studied. Test results show that: The shear strength of the stepped structural plane and the deterioration index of shear strength are affected by the stress concentration at the corners of the structural plane, the length of the rock bridge, the area and the aspect ratio of structural plane and the ability to resist edge failure of the specimen. When the value of l₁ and h is fixed, as N raises from 1 to 5, the shear strength reaches the minimum value at N = 5. Shear failure is transformed into direct shear failure. In the specimen with single stepped structural plane, when the value of h is fixed, as l₁ from 10 to 50 mm, the frictional sliding force between the crack surfaces will decrease firstly and then increases with the growth of number of diagonal crack. In the double stepped structural plane specimen, when the value of h is fixed, as l₁ from 10 to 50 mm, the shear strength and c increases first then decreases. When the value of l₁ is fixed, as h from 10 to 25 mm, the shear strength reaches the maximum value at h = 20 mm. The deterioration index of shear strength was inversely proportional to the number of cyclic shears, which is in turn proportional to N, h, and l₁.

1 Introduction

The mechanical properties and deformation characteristics of rock structural planes affect the stability of rock mass engineering. Therefore, many researchers have been conducted shear tests to study the mechanical characteristics of rock masses with different structural planes including the stepped rock mass structural plane which is common in nature. Thus, studying its shear mechanical properties is very necessary.

At present, a number of experiments and numerical simulation studies on the shear mechanical properties of smooth structural plane have already been conducted. The study by Yang et al. (2000) focused on successive shear mechanisms and strength degradation in shear stress displacement of rock joints influenced by progressive shearing of asperities. Gu et al. (2003) presented the results of a laboratory investigation on the shear behavior of Sydney Hawkesbury sandstone-concrete joints with unbonded interfaces. Joint roughness has been simulated using regular triangular asperities and fractal profiles. Ghazvinian et al. (2010) investigated the shear behavior of discontinuities caused by bedding planes of weakness between two different rock types with high strength difference. Asadi et al. (2012) simulated the mechanical behavior of synthetic rock fracture profiles during direct shear tests by using the two-dimensional particle flow computer code PFC2D. Bahaaddini et al. (2013) used the discrete element code PFC2D to numerically simulate the shear behavior of rock joints. Cai-Chu et al. (2014) conducted a study on new peak shear strength criterion of rock joints based on quantified surface description. Sang-Woo et al. (2014) investigated the influence of wavelength of a tooth-shaped surface asperity on the characteristics of fracture wavelength and plane interface shear strength. Using the discrete element method, Huang et al. (2014) simulated and discussed the dynamic direct shear tests on rough rock joints with 3D surface morphologies. Jahanian et al. (2015) performed a series of constant normal load direct shear tests to investigate the shear strength of artificial samples with infilled rough joint surfaces having different asperity and infill characteristics. To study the mechanisms of bursts related to structural plane, Zhou et al. (2015) selected four typical examples to illustrate the temporal and spatial characteristics of the burst and the exposed plane. Bahaaddini et al. (2016) studied the mechanisms of asperity degradation of rock joints under direct shear tests. Meng et al. (2018) conducted shear tests on tensile granite joints. Jie Liu et al. (2018) conducted a study on new peak shear strength criterion of rock joints based on quantified surface description. Zhu et al. (2018) investigated the shear process and mechanics of joints with regular and irregular triangular asperities through a series of direct shear tests, where a high-speed camera was used to record the deformation and failure of asperities. Zhang et al. (2018) carried out a study on the influence of asperity morphology on failure characteristics and shear strength properties of rock joints under direct shear rests. Fan et al. (2019) conducted a series of direct shear tests on a broken joint obtained by breaking rock cubes under constant normal load condition. Yanbing Liu et al. (2020) analyzed the structural plane in coal bed.

Scholars have not involve a lot about stepped structural plane. Kwon et al. (2010) made the shear behavior of rectangular asperities on rock joints to highlight the role of asperity characteristics related to joint shear behavior. To reasonably characterize a rock surface, Ban et al. (2018) proposed two new roughness parameters.

Under the action of cyclic loads such as earthquakes and blasting, the structural plane is often subjected to cyclic shear loads. Therefore, it is necessary to study the shear mechanical properties of the structural plane under cyclic loading. Jafari et al. (2003) evaluated the shear strength of rock joints subjected to cyclic loading. Mirzaghorbanali et al. (2014) investigated the effects of cyclic loading on the shear behavior of infilled rock joints under constant normal stiffness conditions. Niktabar et al. (2017) conducted a study on the effect of rock mass’ joint roughness on its cyclic shear behavior.

Above all, most researchers mainly focus on the mechanical properties and crack evolution law of samples with jagged structural plane. However, there are few analyses on samples with stepped structural plane. In addition, there are fewer studies on the changing laws of mechanical properties of samples of stepped structural plane under the cyclic loading. Therefore, in this paper, the effect of shearing times on the shear mechanical properties of a manually made cement specimen was investigated by subjecting it to shear test to study the normal stress, the length and height of the stepped structural plane, and the number of stepped structural plane.

2 Sample Preparation and Test Procedures

2.1 Sample Preparation

No. 325 cement, medium-sized sand, and tap water were poured into a cement mortar mold (100 mm × 100 mm × 100 mm) to simulate rock like formation. The water–cement ratio was 0.65, and the mortar ratio was 1:2. Plastic sheets were inserted into the cement mortar after 2 h and were then pulled out after 12 h to form empty cracks to simulate a structural plane. Pouring the samples out of the mold after 24 h and putting them in water and curing it for 28 days according to the standard process, the samples with stepped structure plane were completed.

2.2 Experimental Procedure

2.2.1 Uniaxial Compression Test of Specimen

Uniaxial compression test was carried out with three test specimens to obtain the uniaxial compressive strength (UCS), so as to determine the normal stress magnitude of the direct shear test. The UCS of the three samples were 15.81, 15.29, and 15.7 MPa, respectively, and the average compressive strength was 15.6 MPa.

2.2.2 Shear Test of Stepped Structure Test Samples

When a shear test was performed on a sample with a stepped structural plane, in order to have the better observation about the shear strength of the sample under different normal stresses, four different normal stresses are applied, namely, 0.1σ_c (1.56 MPa), 0.2σ_c (3.12 MPa), 0.3σ_c (4.68 MPa), and 0.4σ_c (6.24 MPa). Each shear test under normal stress was repeated with three test samples, and their average shear strength was taken. According to test standards, the tangential loading rates were set to 1 kN/s.

In one cycle, the sample under normal stress of 3.12 MPa was taken for cyclic shearing. After each cycle, the test piece was restored and circulated again for five times. At the end of each cycle, the average shear strength of the three test samples in each group was taken. According to the direct shear test standard, the tangential loading rates similarly were set to 1 kN/s.

2.3 Groups of Shear Test

In nature, stepped structural plane is the common defect of rock. The number, height and top length of the stepped structural planes largely determine the shear mechanical properties of rocks, and these mechanical parameters are also the important index for engineering stability analysis and safety evaluation.

The length of the rock bridge and the distance between the structural plane and the edge of the sample can be changed by changing the top length and the number of stepped structural plane. By changing the height, different aspect ratios of the stepped structural plane can be formed, so as to restore the complex structural plane in rock mass as much as possible. Therefore, according to the number, height and top length of different stepped structural plane, they are divided into the following groups.

2.3.1 The Different Number of Steps when the Top Length of the Stepped Structural Plane is Fixed

When the top length of the stepped structural plane is fixed at 10 mm, the number of stepped structural plane is different as shown in Fig. 1.

Fig. 1

Schematic diagram of the sample with the fixed top length and the different number of stepped structural plane (N = 1, 2, 3, 4, 5 and h = 10 mm, 15 mm, 20 mm, 25 mm)

In Fig. 1, l₁ is the top length of the stepped structural plane which is 10 mm and h is the height of the stepped structural plane, which is 10, 15, 20, and 25 mm. When the number N of stepped structural plane is 1, 2, 3, and 4, l₂ is the length of both sides at the bottom of the stepped structural plane, which is 10 mm for each step. The length from the bottom left of the structural plane to the left edge of the sample is defined as l₃ and that from the bottom left of the stepped structural plane to the right edge of the sample is defined as l₄, and l₃ = l₄. When N is 5, l₂ is the length of the lower bottom edges of both sides of the three stepped structural plane in the middle and both are 10 mm, l₅ is the length of the left bottom edge of the stepped structural plane, and l₆ is the length of the bottom right edge of the stepped structural plane, with l₅ and l₆ at 5 mm.

The specific test grouping is shown in Table 1.

Table 1 Test grouping with the fixed top length and the different number of stepped structural plane

2.3.2 Test Group with Single Stepped structural Plane

When the sample contains a single stepped structural plane, the length l₁ and height h of the top of the stepped structural plane change.

In Fig. 2, l₂ is the stepped structural plane with a top length of 10, 20, 30, 40, and 50 mm and h is the height of the stepped structural plane, which is 10, 15, 20, and 25 mm. l₂ is the length of the bottom of the left and right sides of the stepped structural plane, which is 10 mm for each step. The length from the bottom left of the leftmost stepped structural plane to the left edge of the test sample is defined as l₃ and that from the bottom left of the stepped structural plane on the right to the right edge of the sample is defined as l₄, with l₃ = l₄.

Fig. 2

Schematic diagram of the sample with the single stepped structural plane (l₁ = 10 mm, 20 mm, 30 mm, 40 mm, 50 mm and the value of h is fixed)

The test can be divided into several groups according to the values of l₁, l₂, l₃, and h, as shown in Table 2.

Table 2 Test grouping with the sample of single stepped structural plane

2.3.3 Test Group with Double Stepped Structural Plane

When the test sample contains a double stepped structural plane, the length l₁ and the height h of the top of the stepped structural plane change.

In Fig. 3, l₁ is the stepped structural plane with a top length of 10, 20 and 30 mm and h is the height of the stepped structural plane, which is 10, 15, 20, and 25 mm. l₂ is the length of the bottom of the left and right sides of the stepped structural plane, which is 10 mm for each step. The length from the bottom left of the stepped structural plane to the left edge of the test sample is defined as l₃ and that from the bottom left of the stepped structural plane on the right to the right edge of the sample is defined as l₄, with l₃ = l₄.

Fig. 3

Schematic diagram of the sample with the double stepped structural plane (l₁ = 10 mm, 20 mm, 30 mm and the value of h is fixed)

The test can be divided into several groups according to the values of l₁, l₂, l₃, and h, as shown in Table 3.

Table 3 Test grouping with the sample of double stepped structural plane

3 Shear Test Results of Specimen with the Fixed Top Length and the Different Number of the Stepped Structural Plane

3.1 Analysis of Shear Test Results After One Cycle

3.1.1 Analysis of Shear Strength After One Cycle

When l₁ = 10 mm, l₂ = 10 mm, and h = 10, 15, 20, and 25 mm, the change law of the sample’s shear strength with N is shown in Fig. 4.

Fig. 4

Variation of the shear strength of specimen with N (l₁ = 10 mm, l₂ = 10 mm, and h = 10 mm, 15 mm, 20 mm, 25 mm)

In Fig. 4, when l₁ = 10 mm, l₂ = 10 mm, and h = 10, 15, 20, and 25 mm, since the lower corners of the stepped structural plane are prone to occur the stress concentration, when N increases continuously, the number of lower corner of the stepped structural plane raises and the stress concentration is more obvious, which reduces the shear strength of the sample.

When l₁ = 10 mm, l₂ = 10 mm, and N = 1, 2, 3, 4, and 5, the variation law of the shear strength of the sample with the height of the stepped structural plane h is shown in Fig. 5.

Fig. 5

Variation of the shear strength of specimen with h (l₁ = 10 mm, l₂ = 10 mm, and N = 1, 2, 3, 4, 5)

As shown in Fig. 5, when the normal stress is 3.12 MPa and N = 1, the shear strength of the sample is proportional to h. If N = 2, the shear strength increases first and then decreases with h. In samples with multiple stepped structural planes, h cannot change the length of the rock bridge, nor can it increase the number of corner of the plane. Therefore, under the same normal stress, changing h has little effect to the shear strength. In conclusion, the shear strength of the sample is proportional to the normal stress, but under different normal stress conditions, the change of the shear strength of the sample with h is not obvious.

3.1.2 Analysis of Shear Strength Parameters After One Cycle

Using the Mohr–Coulomb criterion to analyze the shear strength of the sample, the expression is given in Eq. (1):

$$\tau = c + \sigma tg\varphi$$

(1)

In the given equation, τ is the shear strength, c is the cohesion, σ is the normal stress, and φ is the internal friction angle.

When h and l₁ are unchanged, the change with N is shown in Fig. 6.

Fig. 6

Variation of the values of c and φ with N (l₁ = 10 mm and h = 10 mm, 15 mm, 20 mm, 25 mm)

As N increases, the internal friction between the stepped structural planes is continuously increasing, at the same time, the stress concentration at the corners of the stepped structural plane is also more obvious. Before N = 3, the internal friction has a small effect on the stress concentration generated. After N = 3, the internal friction force has a greater impact on the stress concentration, hence, the cohesion tends to decrease first and then increase, which reaches the minimum value when N = 3. However, the internal friction angle increases first and then decreases, which reaches the maximum value when N = 2 and N = 3.

3.1.3 Damage Morphology After One Cycle

When h and the normal stress are the same, the change law of the sample’s failure characteristics with the N is similar, so the representative sample fracture diagram is used for description. When the h = 25 mm, l₁ = 10 mm, normal stress is 3.12 MPa and 4.68 MPa, and N = 1, 2, 3, 4, 5 the failure characteristics of the sample are shown in Figs. 7 and 8.

Fig. 7

The damage morphology of specimen when normal stress is 3.12 MPa (h = 25 mm, l₁ = 10 mm and N = 1, 2, 3, 4, 5)

Fig. 8

The damage morphology of specimen when normal stress is 4.68 MPa (h = 25 mm, l₁ = 10 mm and N = 1, 2, 3, 4, 5)

It can be seen from Fig. 7 that when l₁ = 10 mm, l₂ = 10 mm and h = 25 mm. If N is small, oblique cracks are easy to occur on the top and bottom of the structural plane, but as N increases, the sample will undergo direct shear failure and there are very few oblique cracks in the sample. If N is the same, the sample will fail as the normal stress increases. The number of cracks will also increase, and the shear strength of the sample will increase accordingly.

3.2 Analysis of Shear Test Results After Cyclic Shear

This article defines the deterioration index of shear strength:

$$R\tau = \tau fn/\tau f1$$

(2)

In the formula, $\tau fn$ is the shear strength of the sample after the n shear cycle; $\tau f1$ is the shear strength of the sample after the first shear cycle, and $R\tau$ is the shear strength deterioration index.

When l₁ = 10 mm, l₂ = 10 mm, normal stress is 3.12 MPa, and N = 1, 2, 3, 4, 5, the shear strength of samples with different heights of stepped structural plane changes with the number of cyclic shear. The law is shown in Fig. 9.

Fig. 9

Variation of the shear strength ratio of sample with n (l₁ = 10 mm, l₂ = 10 mm, and N = 1, 2, 3, 4, 5)

If the structural planes l₁, l₂, and N are the same, as the number of cyclic shears increases, the interaction between the stepped structural plane weakens, so $R\tau$ is inversely proportional to the number of cyclic shears n. If N is fixed and the number of cyclic shears is the same, $R\tau$ is proportional to the stepped structural plane h. After 5th cyclic shears, when h is continuously increasing from h = 10 mm with a height of 5 mm, the difference between the values of $R\tau$ with h = 15 mm and h = 20 mm is the largest, usually reaching 0.05 ~ 0.1.

When l₁ = 10 mm, l₂ = 10 mm, normal stress is 3.12 MPa, and h = 10, 15, 20, and 25 mm, the shear strength of samples with different height of stepped structural plane changes with the number of cyclic shears as shown in Fig. 10.

Fig. 10

Variation of the shear strength ratio of sample with n (l₁ = 10 mm, h = 10 mm, 15 mm, 20 mm, 25 mm)

If l₁, l₂, and N are the same, $R\tau$ is inversely proportional to the number of cyclic shears. If the structural plane h is the same and the number of cyclic shears is the same, with the increase of N, the internal area of the stepped structural plane and the length of the rock bridge increase, resulting in more oblique cracks and a stronger ability to resist cyclic shears, so $R\tau$ is proportional to the number N of stepped structural plane. After 5th cyclic shears, the value of $R\tau$ of the sample with N = 1 generally decreases by 60% to 70%. As N increases, the sample’s ability to resist cyclic shears increases. When N = 5, the value of $R\tau$ is only decreasing by 15% ~ 30%.

4 Shear Test Results of Single Stepped Specimens with the Different Top Length of the Stepped Structural Plane

4.1 Analysis of Shear Test Results After One Cycle

4.1.1 Analysis of Shear Strength After One Cycle

When N = 1, l₂ = 10 mm, and h = 10, 15, 20, and 25 mm, the variation of the shear strength of the sample with the top length of the stepped structural plane l₁ is shown in Fig. 11.

Fig. 11

Variation of the shear strength of specimen with l₁ (N = 1, l₂ = 10 mm, and h = 10 mm, 15 mm, 20 mm, 25 mm)

As shown in Fig. 11, the peak strength of the test sample shows a certain regularity under l₁. When h = 10 mm, 15 mm, 20 mm, at this time, the aspect ratio of the stepped structural plane is relatively small. At this time, l₁ plays a leading role in the change of the shear strength. with the increase of l₁, the length of the rock bridge increases, the number of oblique cracks generated and the interaction force between them increase, so the shear strength increases. However, if the length of the rock bridge is too long, the distance between the structural plane and the edge of the specimen is too small, so that the ability of the sample to resist edge damage is weakened, and the shear strength is reduced. Therefore, the peak shear strength of the test sample increases and then decreases and the amplitude is larger. When h = 25 mm, the aspect ratio of the stepped structural plane is relatively large. At this time, l₁ and h work together, the peak shear strength decreases first and then increases, finally, it decreases again. The shear strength reaches the minimum value when l₁ = 40 mm.

When N = 1, l₂ = 10 mm, and l₁ = 10, 20, 30, 40, and 50 mm, the variation law of the sample’s shear strength with the height of the stepped structural plane h is shown in Fig. 12.

Fig. 12

Variation of the shear strength of specimen with h (N = 1, l₂ = 10 mm, and l₁ = 10 mm, 20 mm, 30 mm, 40 mm, 50 mm)

As shown in Fig. 12, when the normal stress is 6.24 MPa and if l₁ = 20 mm, the shear strength of the sample is proportional to h. If l₁ = 30, 40, and 50 mm, the shear strength of the sample is inversely proportional to h. Under the action of different normal stresses, if h = 10 mm, the area of the stepped structural plane will be too small, but if h = 25 mm, the aspect ratio of the stepped structural surface will be too large. The above two situations will weaken the shear strength, so the peak shear strength of the sample always appears at h = 15 and 20 mm but the overall change law is not obvious.

4.1.2 Analysis of Shear Strength Parameters After One Cycle

Equation (1) defined above was used to analyze the shear strength parameters after one cycle.

When N and h are unchanged, the change of c and φ with l₁ is shown in Fig. 13.

Fig. 13

Variation of the values of c and φ with l₁ (N = 1 and h = 10 mm, 15 mm, 20 mm, 25 mm)

It can be seen from Fig. 13 that with the increase of l₁, the length of rock bridge continues to increase, the interaction of oblique cracks increases, and the sliding friction continues to increase, so the cohesion is inversely proportional to l₁, which reaches the maximum value when l₁ = 40 and 50 mm. If h = 10 and 15 mm, the aspect ratio of the structural plane is relatively small, as l₁ increases, the internal friction angle increases first and then decreases. If h = 20 and 25 mm, the aspect ratio of the structural surface is relatively large, with the increase of l₁, the internal friction angle showed a trend of decreasing first and then increasing and finally decreasing.

4.1.3 Damage Morphology After One Cycle

When N is fixed and the normal stress is the same, the failure characteristics of the sample change with the height h and the top length l₁ of the structural plane are more similar, so a representative sample fracture diagram is used for description. When l₁ = 10, 20, 30, 40, 50 mm, h = 15 and 20 mm, normal stress is 3.12 MPa, 4.68 MPa, and 6.24 MPa, the destruction characteristics of the sample are shown in Figs. 14, 15, 16, and 17.

Fig. 14

The damage morphology of specimen when normal stress is 3.12 MPa ( N = 1, h = 15 mm and l₁ = 10 mm,, 20 mm, 30 mm, 40 mm, 50 mm)

Fig. 15

The damage morphology of specimen when normal stress is 6.24 MPa ( N = 1, h = 15 mm and l₁ = 10 mm,, 20 mm, 30 mm, 40 mm, 50 mm)

Fig. 16

The damage morphology of specimen when normal stress is 3.12 MPa, ( N = 1, h = 20 mm and l₁ = 10 mm,, 20 mm, 30 mm, 40 mm, 50 mm)

Fig. 17

The damage morphology of specimen when normal stress is 4.68 MPa, ( N = 1, h = 20 mm and l₁ = 10 mm,, 20 mm, 30 mm, 40 mm, 50 mm)

It can be seen from Figs. 14, 15, 16, and 17 that when N = 1 and l₂ = 10 mm, and if l₁ is small, oblique cracks are easily generated along the top and bottom of the stepped structural plane during shear failure. However, as l₁ continues to increase until 30 mm, diagonal cracks begin to form inside the step, and at this time, the sample has the highest shear strength. If l₁ is greater than 30 mm, the diagonal cracks inside the stepped structural plane are reduced, and the number of diagonal cracks generated at the top and bottom of the stepped structural plane is reduced. When l₁ is unchanged and N = 1, the number of oblique cracks at the time of sample failure increases with an increase of h, and the failure mode is changed from direct shear failure to shear failure. When the same sample is used for direct shear under different normal stress conditions, the number of oblique cracks at the time of sample failure increases with an increase of normal stress.

4.2 Analysis of Shear Test Results After Cyclic Shear

Equation (2) is used to analyze the shear test results after cyclic shearing.

When N = 1, l₂ = 10 mm, normal stress is 3.12 MPa, and h = 10, 15, 20, and 25 mm, the shear strength of samples with h changes with the number of cyclic shear loads as shown in Fig. 18.

Fig. 18

Variation of the shear strength ratio of sample with n (l₁ = 10 mm, l₂ = 10 mm, and h = 10 mm, 15 mm, 20 mm, 25 mm)

As can be seen from Fig. 18, if the structural planes l₁, l₂, and N are the same, the shear strength is inversely proportional to the number of cyclic shears. After the first shearing, the value of $R\tau$ is mostly reduced by 30%, but after the fifth cyclic shearing, its value is only about 10% lower than that of the first shearing because the interaction force between most oblique cracks disappears, and the ability to resist cyclic loads is greatly reduced. If the number of cyclic shears is the same, with the increase of l₁, the length of the rock bridge increases, the mutual friction between the oblique cracks increases, and the ability to resist cyclic shears is stronger. Above all, the $R\tau$ is proportional to the top length l₁ of the stepped structure plane.

When N = 1, l₂ = 10 mm, normal stress is 3.12 MPa, and l₁ = 20, 30, 40, and 50 mm, the shear strength of the samples with l₁ changes with the number of cyclic shear loads as shown in Fig. 19.

Fig. 19

Variation of the shear strength ratio of sample with n (N = 1, l₂ = 10 mm, and l₁ = 10 mm, 20 mm, 30 mm, 40 mm, 50 mm)

When N = 1, l₂ = 10 mm, and l₁ = 20, 30, 40, and 50 mm under normal stress of 3.12 MPa, under the same cyclic shearing times, with the increase of h, the area of the stepped structural plane increases, the number of oblique cracks increases, and the ability to resist cyclic loads is stronger, so that $R\tau$ is proportional to h.

5 Shear Test Results of Double Stepped Specimens with the Different Top Length and Height of the Stepped Structural Plane

5.1 Analysis of Shear Test Results After One Cycle

5.1.1 Analysis of Shear Strength After One Cycle

When N = 2, l₂ = 10 mm, and h = 10, 15, 20, and 25, the variation law of the shear strength of the sample with the top length l₁ of the stepped structural plane is shown in Fig. 20.

Fig. 20

Variation of the shear strength of specimen with l₁ (N = 2, l₂ = 10 mm, and h = 10 mm, 15 mm, 20 mm, 25 mm)

In Fig. 20, when N = 2, l₂ = 10 mm, h = 10, 15, 20, and 25 mm, and as l₁ increases, the length of the rock bridge generated by the structural plane increases, and the mutual friction between the oblique cracks increases. At the same time, the distance between the structural plane and the edge of the sample reduces, and the ability to resist edge damage is weakened. Therefore, when l₁ is less than 20 mm, the increase in length length of the rock bridge has little effect on the shear strength. After l₁ is greater than 20 mm, the increase in the length of the rock bridge has an increasing effect on the shear strength. So the shear strength of the sample decreases first and then increases. The minimum value is reached when l₁ is 20 mm.

When N = 2, l₂ = 10 mm, and l₁ = 10, 20, and 30 mm, the variation law of the shear strength of the sample with the height h of the stepped structural surface is shown in Fig. 21.

Fig. 21

Variation of the shear strength of specimen with h (N = 2, l₂ = 10 mm, and l₁ = 10 mm, 20 mm, 30 mm)

As shown in Fig. 21, when N = 2, l₂ = 10 mm, l₁ = 10, 20, and 30 mm, and as h increases, the area of the structural plane gradually increases, the number of oblique cracks increases, and the shear strength increases. When h = 20 mm, the aspect ratio of the structural plane is moderate, and the number of oblique cracks is large, hence, the shear strength is the highest at this time, But when h = 25 mm, the aspect ratio of the structural plane is too large, and the shear strength is reduced. The peak value is reached when h = 20 mm.

5.1.2 Analysis of Shear Strength Parameters After One Cycle

Equation (1) defined is used to analyze the shear strength parameters after a cycle.

When the height h of the stepped structural plane is unchanged, the change of c and φ is shown in Fig. 22.

Fig. 22

Variation of the values of c and φ with l₁ (N = 2 and h = 10 mm, 15 mm, 20 mm, 25 mm)

It can be seen from Fig. 22 that with the increase of l₁, the length of the rock bridge increases, the number of oblique cracks increases, and the mutual friction between oblique cracks increases, so the cohesion is proportional to the top length l₁ of the structural surface, which reaches the maximum value when l₁ = 30 mm. As l₁ increases, the internal friction angle is inversely proportional to l₁, which reaches the minimum value when l₁ = 30 mm.

5.1.3 Damage Morphology After One Cycle

When N and the normal stress is the same, the failure characteristics of the sample change similarly with h and l₁, so a representative sample fracture diagram is used for description. When l₁ = 10, 20, 30 mm, h = 20 and 25 mm, and normal stress is 1.56 MPa, 3.12 MPa, and 6.24 MPa, the destruction characteristics of the sample are shown in Figs. 23, 24, 25, and 26.

Fig. 23

The damage morphology of specimen when normal stress is 1.56 MPa, (N = 2, h = 20 mm and l₁ = 10 mm, 20 mm, 30 mm)

Fig. 24

The damage morphology of specimen when normal stress is 3.12 MPa, (N = 2, h = 20 mm and l₁ = 10 mm, 20 mm, 30 mm)

Fig. 25

The damage morphology of specimen when normal stress is 1.56 MPa, (N = 2, h = 25 mm and l₁ = 10 mm, 20 mm, 30 mm)

Fig. 26

The damage morphology of specimen when normal stress is 6.24 MPa, (N = 2, h = 25 mm and l₁ = 10 mm, 20 mm, 30 mm)

If N = 2, l₂ = 10 mm, and if the height h or the top length l₁ of the stepped structural plane continues to increase, the diagonal cracks generated on the sample increase and gradually change from direct shear failure to shear failure. However, it is worth noting that when h is too high, shear failure occurs directly on the structural plane instead of shear failure. If h is fixed, as the normal stress increases, the oblique crack when the sample fails increases and the shear strength of the sample will increase accordingly.

5.2 Analysis of Shear Test Results After Cyclic Shear

Equation (2) is used to analyze the shear test results after cyclic shearing.

When N = 2, l₂ = 10 mm, normal stress is 3.12 MPa, and h = 10, 15, 20, and 25 mm, the shear strength of the sample with l₁ and h changes with the number of cyclic shear loads as shown in Fig. 27.

Fig. 27

Variation of the shear strength ratio of sample with n (l₁ = 10 mm, l₂ = 10 mm, and h = 10 mm, 15 mm, 20 mm, 25 mm)

When the normal stress is 3.12 MPa, N and l₁ are fixed, which decreases with the increase of the number of cyclic shear. When h is the same, with the increase of l₁, the length of the rock bridge increases, the mutual friction between the oblique cracks is continuously increased, and the ability to resist cyclic shears is stronger.

When N = 2, l₂ = 10 mm, normal stress is 3.12 MPa, and l₁ = 10, 20, and 30 mm, the shear strength of the stepped structural surface of the sample with h value changes with the number of cyclic shear cycles as shown in Fig. 28.

Fig. 28

Variation of the shear strength ratio of sample with n (N = 1, l₂ = 10 mm, and l₁ = 10 mm, 20 mm, 30 mm)

When N = 2, l₂ = 10 mm, l₁ = 20 and 30 mm, and normal stress is 3.12 MPa, under the same cyclic shear number, with the increase of h, the area of the stepped structural plane increases, resulting in more oblique cracks, and the ability to resist cyclic loads is stronger. Therefore,$R\tau$ is proportional to h.

6 Discussion and Conclusions

• 1. In the direct shear test of the same sample, as the normal stress continues to increase, the number of cracks in the specimen and the shear strength continuously increases, respectively.

• 2. When the top length of the joint plane l₁ and the height of the joint plane h is fixed, as the number of joint plane N increases from 1 to 5, the stress concentration at the corners of the stepped structural plane is more obvious, friction between cracks also reduced, therefore, the cohesion c of specimen and the shear strength reaches the minimum value as N is 5. Shear failure is transformed into direct shear failure. In the specimen with different number of joint plane, when N is 1 or 2, the cracks are likely to occur along the top and bottom of the step during shear failure, but as N increases, the sample will directly undergo shear failure and there are very few cracks in the specimen.

• 3. In the direct shear test of the specimen with a single stepped structural plane, when the value of h ≤ 20 mm, as l₁ from 10 to 50 mm, the number of oblique cracks generated on the structural plane and the interaction force between them increase. At the same time, the distance between the structural plane and the two edges of the sample is too small, so the ability to resist edge damage is weakened. As l₁ = 30 mm and 40 mm, the shear strength reaches the maximum value. And damage morphologies of rock mass specimens changed from shear failure to direct shear failure. However, when the h ≥ 20 mm, the shear strength will increase as l₁ = 30 mm, 40 mm, but the general trend will not change.

• 4. In the direct shear test of the double stepped structural plane specimen, when the value of h is fixed, as l₁ from 10 to 30 mm, the increase in the length of the rock bridge has a gradually increasing, the shear strength and c decreases first then increases and reaches the minimum value as l₁ = 20 mm. When the value of l₁ is fixed, as h from 10 to 25 mm, the area and aspect ratio of the structural plane increase, cracks on the surface are increasing, so the shear strength and c increases and then decreases and reaches the maximum value as h = 20 mm, damage morphologies also changed from shear failure to direct shear failure.

• 5. In cyclic shearing, as the number of cyclic shears increases, the degradation index of shear strength decreases, but the reduction rate continues to slow down. As N, h or l₁ increase, the deterioration index continues to increase.