Experimental Study on the Effect of Single Structural Plane on the Stability of Shallow Tunnel Surrounding Rock

Abstract

There are a large number of structural planes distributed in the surrounding rock of a tunnel, and this is one of the key factors causing a tunnel’s instability. Due to different geological and historical conditions, the distribution characteristics and the occurrence of structural planes in the rock mass also have significant differences. In engineering, it is common to encounter structural planes that cut across the tunnel section and have a significant impact on the stability. The occurrence of structural planes is a key factor controlling the mechanical behavior of the surrounding rock. Based on this, laboratory uniaxial compression tests were carried out by constructing a small tunnel physical model with single structural planes of different inclination angles. A related numerical simulation analysis was also carried out. This research indicates that: (1) Under the influence of a single structural plane, the dip direction with 30°–60° is the most dangerous situation, and when the dip angle of the structural plane is between 38 and 88°, it will slip along the structural plane. (2) According to the mechanical mechanism, there are three types of cracks: tensile cracks, shear cracks, and tensile shear cracks. According to the deformation characteristics, there are four types: tension, friction, bending, and shear. (3) There is a certain correlation between the strength of a single discontinuity rock mass and that of a multi-jointed rock mass. When the dip angle of the joints’ combination is 45°+60°, the rock mass is in its weakest state.

1. Introduction

Discontinuous interfaces are widely distributed in all types of geological environments, including rock mass structural planes, bedding, faults, and slip zones, and are the main factor controlling the properties of rock that contains cohesion and friction angle. The discontinuous interfaces are inevitably encountered in engineering because of its wide distribution and properties.

The anisotropy of a rock mass has always been an important and difficult issue in rock and soil research. Most research on the anisotropy of interface strength focuses on the discontinuity. The anisotropy of rock and soil mass is mainly controlled by the orientation of the interface, and the anisotropy can be effectively characterized by the strength of the rock and soil mass at interfaces with different dip angles [1,2,3,4]. Furthermore, as a typical engineering geological model, rock and soil mass contact in tunnel engineering has a more obvious response to anisotropy because of the differences between the two materials which contain the morphology of the interface and the plasticity behavior. The interfaces affecting tunnel engineering mainly include the bedding, structural surface, and sliding zone. As a particular type of interface, the slide belt is divided into the primary slide belt and secondary slide belt. The primary slip zone mainly includes the rock mass structural plane, bedding, and soil–rock interface, and is closely related to the topographic structure. The secondary slip zone is the interface formed under the action of human engineering activities or geological forces, and the slide band is curved into the soil and straight into the rock. When the tunnel passes through an interface with different orientations, different forms of deformation and failure may occur, resulting in the instability of the surrounding rock and a series of failures in the tunnel chamber [5]. The tunnel deformation is closely related to the movement mode of the interface, which is controlled by the tunnel’s orientation. Therefore, the orientation of the strata is an important factor affecting stability (Figure 1). Presently, the methods used in tunnel–landslide system research mainly include numerical simulation [6,7,8], field monitoring [9,10,11], and laboratory testing [12,13,14]. Subsequently, the influence of landslides on tunnels was investigated through a series of mechanical analyses [15,16]. This is a very broad research area, and the spatial location that includes the landslide and tunnel, the stratigraphic structure and lithology, the landslide starting mechanism, and the tunnel support are important considerations with wide coverage by many studies.

Many factors, such as the rock and soil characteristics, shape of the interface, water content state, and stress history, affect the strength of the interface. From the perspective of influencing factors, the interface is actually a special structural surface of a rock mass. Therefore, the study on the mechanical properties of the structural surface is also applicable to the interface; the difference lies in that there are different materials on both sides of the interface. Jaeger proposed the shear failure condition of the structural plane to investigate the strength characteristics [17]. Patton investigated the strength characteristics of the structural plane based on the earliest Mohr–Coulomb strength criterion, and proposed the rock mass structure strength criterion [18]. Barton proposed the concept of structural plane roughness based on Patton’s results [19]. The structural plane empirical strength criterion was proposed based on the strength of the sidewall of the structural plane and the friction angle of the rock. The strength criterion was constantly supplemented and improved [20,21,22,23,24,25,26] in the next years. Subsequent studies on the structural plane strength were mostly based on the empirical strength criterion proposed by Barton [27,28,29,30]. Additionally, Maksimović conducted substantial research on the dilation behavior in the shear process [31,32,33]. The non-linear strength criterion of the structural plane was proposed and developed, and is a relatively complete method. As mentioned above, the strength criteria for rock mass discontinuities also have certain applicability to interfaces. Currently, however, research on the strength criteria for a packed structural plane or soil–rock interface is lacking. Yang proposed an anisotropic shear strength criterion for the soil–rock interface based on energy conservation [34]. Additionally, in tunnel engineering, the interface strength is an inevitable element to the analysis of the stability of the surrounding rock, which is related to the stability analysis of the surrounding rock [35,36,37]. Therefore, it is very important to combine the strength criterion of the interface with the stability analysis of the surrounding rock.

In terms of the shear behavior of rock mass discontinuity, Yang et al. [38] studied the creep shear mechanical behavior of regular jugged discontinuity. Zhang et al. [39] studied the shear mechanics characteristics under different normal loads with regular jugged discontinuity as the object. In addition, many scholars have studied the effects of structural plane morphology, roughness, and filling on shear strength. However, anisotropy is a very important property of a rock mass, and the geometric relationship between the structural plane and the rock mass is the key factor to determine the anisotropy of the rock mass. Therefore, studying the geometric relationship between the structural plane and the rock mass is of great significance for an in-depth understanding of anisotropy. Mu et al. [40] carried out the block shear test of a 45° dip plane, which provided the basic idea for this study. Ali et al. [41] studied the shear behavior of the packed structural plane by using PFC2D software, and the structural plane inclination was also 45°. The object of these studies was a single structural plane with the shear test as the main test method. However, the anisotropy of a rock mass cannot isolate the discontinuity, so it is necessary to analyze the rock mass with the discontinuity through compression test. The bedding development of sedimentary rocks contains natural structural planes, which are typical transverse isotropic materials. The analysis of rock anisotropy can be realized by controlling the compression direction of different angles [42]. However, in practical engineering, the rock mass structural plane is often distributed in two or more pieces, and it is difficult to carry out further research by laboratory test and simple calculation. Discrete element numerical simulation software can effectively solve this problem. Therefore, numerical simulation becomes an effective means in multi-joint analysis [43]. Tunnel engineering is buried in an underground rock mass, and its stability is closely related to the property of the rock mass. A physical model test is an effective research method, through which many meaningful results can be obtained [44]. However, few of these studies focused on the impact of the structure on the tunnel structure, so it is necessary to conduct in-depth research and analysis.

Based on the above-mentioned research background, this study investigated the mechanical behavior of a tunnel–interface system under the influence of the dip angle. By constructing a simplified physical model, uniaxial compression tests were carried out and the deformation of the model during the test was monitored in real time using the particle capture method [45,46,47,48]. Finally, two types of physical and mechanical models were abstracted: the anisotropic mechanical model of the interface and the arching mechanical model of the surrounding rock. The results obtained by this study have reference value for related research and provide the theoretical basis for the subsequent study of surrounding rock support methods.

2. Materials and Methods

2.1. Test Material

Iron crystal sand colloidal geotechnical material was used as the test simulation material, and the main components contained iron ore powder, barite powder, quartz sand, rosin, alcohol, and gypsum powder [49]. To ensure uniformity, all samples were kept in a humidity box for 14 days at the constant temperature of 25 °C and 35% humidity. Interface molds with seven different angles (0°, 15°, 30°, 45°, 60°, 75°, and 90°) were printed using a 3D printer. The mold pedestal thickness was 0.5 cm with the size of 10 cm × 10 cm × 10 cm (Figure 2). In the process of sample preparation, the air bubbles were fully discharged by vibration at a certain frequency. The sample surface was leveled and placed at room temperature. After eight hours, the sample was removed and placed inside a humidity box at constant temperature for curing. The roughness characteristics are important; therefore, to ensure that each sample had the same roughness, the surface was polished smooth (JRC = 1) using 600 mesh sandpaper.

In this study, a shallow-buried tunnel was considered as the object of investigation, and the thickness of the cover soil was determined to be less than twice the tunnel span. Therefore, in this study, the tunnel span was 3 cm and set in the middle of a cube. The distance between the tunnel roof and the surface was 3 cm. In shallow-buried tunnels, the lateral failure is mainly manifested by tensile failure under vertical load, because the binding force of lateral plane is small and the tensile strength of rock mass is much lower than that of compressive strength. In order to study the characteristics of lateral deformation and tensile failure of rock mass under the influence of tunnel, no lateral constraint was set in this test.

The sample numbers and initial compression wave velocities are listed in

Table 1. Sample information.

Sample Number	Dip Angle α	Figure	Initial Compressive Wave Velocity Vp (m/s) (±50)
o-1	/		2942
o-2			2856
o-3			2922
0-1	0°		2444
0-2			2356
0-3			2532
15-1	15°		2155
15-2			2048
15-3			2102
30-1	30°		1942
30-2			2014
30-3			1988
45-1	45°		2195
45-2			2012
45-3			2128
60-1	60°		2013
60-2			2022
60-3			1997
75-1	75°		1900
75-2			1854
75-3			1922
90-1	90°		1794
90-2			1802
90-3			1688

. According to the initial wave velocity, the interface samples with different inclination angles have obvious anisotropic characteristics, and the wave velocity decreases as the dip angle increases. The compressional wave velocity represents the resistance of the specimen to compression deformation, and a positive correlation exists between them [50]. The strength of the sample can be preliminarily assessed by a wave velocity test.

2.2. Methods and Apparatus

In this study, uniaxial compression tests were conducted according to the standard for test methods of engineering rock mass to test the strength of the samples. The uniaxial compressive strength (UCS) tests were carried out subsequently. The instrument used in this study was the MTS universal material testing machine, which can apply a maximum load of 600 kN (Figure 3a). The test was carried out in compression mode with displacement control at the rate of 0.005 mm/s, with three samples in each group.

Speckle interferometry was used to monitor the specimen surface deformation. A layer of white primer was sprayed onto the surface of the specimen, and the scattered points were subsequently sprayed with black paint. Images were captured by a camera at different moments, and the deformation of the speckle at different moments was calculated and analyzed by vector calculation (Figure 3c).

The compressive wave velocities of the samples were measured before and after the mechanical test, and the development of internal cracks was characterized by the wave velocities and spectral characteristics. Before the acoustic wave test, Vaseline was smeared onto the probe, an ultrasonic wave with a frequency of 2 kHz was adopted, and the voltage was set to 1 kV. The initial arrival point of the acoustic wave was considered as the compressive wave velocity of the sample (Figure 3b). The sample was split up by the interface; in order to measure the whole sample’s wave velocity, the probes were placed on both sides of the interface. Because the horizontal crustal stress in a shallow tunnel is weak, there was no lateral confinement in this study.

3. Results

3.1. Uniaxial Compressive Strength

The UCS of the samples was tested after curing for 14 days under the same conditions (T = 25 °C, H = 35%), and the test results are shown in Figure 4. The load–displacement curve of a sample can be divided into three stages: the compaction stage, elastic deformation stage, and yield stage. The elastic modulus of the samples with different dip angles was similar. However, it was observed that the elastic modulus of the 45° samples was different compared with the other tested samples. The specimens with other angles mainly underwent compression, tension, and shear deformation, while the specimens with the angle of 45° mainly slipped along the interface (which can be seen in. Owing to the stress concentration at the tip of 45°, the specimens were constantly squeezed and broken, which resulted in large deformation. Additionally, another notable phenomenon was observed in the curve: multiple cracks existed in the samples, and each crack corresponded to a sudden change in the load–displacement curve. The step change in the curve is a manifestation of fracture development.

The UCS of the samples is shown in Figure 5. The strength of the no interface sample is 19.236 MPa, and is indicated by the dotted line in the figure. As the angle increases, the specimen strength first decreases and then increases, and the minimum strength was observed for the 60° sample. The strength of the 0°–75° sample was lower than that of the unbounded sample owing to the existence of the interface. When the angle was 90°, the sample was divided into two symmetrical parts by the interface, which is equivalent to the two parts bearing the load at the same time, resulting in the strength increase. This result is the same as most results reported in the literature: the peak strength shows a parabolic change with the increase in dip angle. Additionally, when α = 45° + φ/2, the UCS was at minimum [17,51]. In this paper, φ = 36.2° and α = 63.1°.

3.2. Deformation Characteristics

The deformation of samples with different inclination angles is significantly different. Figure 6 shows the horizontal deformed cloud map. For the no interface sample, the horizontal displacement field forms an hourglass shape, and the deformation range at the upper and lower ends is small, while the deformation range in the middle is large. The final failure also has the shape of an hourglass (Figure 6, Figure 7 and Figure 8), and the same deformation and failure characteristics as the intact concrete specimen. The main effect of the tunnel on the displacement field is embodied in the deformation of the rock surrounding the chamber. As can be seen in the figure, the deformation around the tunnel is different, particularly at the vault, foot, and bottom. The contact face with different dip angles has a significant effect on the tunnel. The main deformation of the tunnel occurs close to the interface and exhibits different deformation characteristics at different inclination angles.

When the angle is 0°, the specimen is mainly subjected to compressive stress, which results in lateral tensile stress and the specimen exhibits obvious deformation on both sides (Figure 7b). The deformed trace line is generally a straight longitudinal line. The interface divides the deformed area into two parts, and the maximum horizontal displacement reaches 19 mm.

When the angle was 15°, the sliding deformation and compression deformation of the specimen occurred along the interface (Figure 7c). On the one hand, this made the deformable track bend toward the slip direction; on the other hand, it changed the deformation characteristics of the tunnel, and the tunnel was subjected to deviatoric stress, which led to obvious deformation in the shoulder. Two distinct displacement boundaries appeared, with a maximum displacement of 28 mm.

Under the influence of the interface with an angle of 30°, three obvious curvilinear displacement boundaries appeared. The crack trace curve developed and bulged in the opposite direction of motion, which indicates that the material was pulled by a static friction force near the interface, causing deformation hysteresis. The lateral wall and arch foot of the chamber were obviously deformed, and the maximum deformation was approximately 30 mm.

When the inclination angle was 45°, sliding mainly occurred along the interface, and an obvious displacement trace did not appear on either side. The deformation on both sides was uniform, and large deformation existed at the bottom of the tunnel arch. The overall maximum deformation was approximately 35 mm. Notably, owing to the absence of lateral constraints in the test, the tip of the 45° sample was compressed and broken (Figure 7e). When a certain degree was reached, the entire sample was destroyed. When the inclination exceeded 45°, the specimen exhibited different deformation characteristics. When α was smaller than 45°, the main deformation was slippage and shear deformation; when α was greater than 45°, the main deformation was lateral tension.

When the angle was 60°, two obvious deformable lines existed and were distributed in a straight line, and the interface controlled the distribution of these deformable lines. Under vertical loading, compression deformation occurred, which led to the lateral tensile deformation of the specimen. The maximum deformation between the interface and the tunnel was approximately 15 mm.

When the angle was 75°, two displacement tracks appeared on both sides of the interface and a small degree of bending occurred. The convex direction of the curve pointed to the opposite direction of deformation. This shows that the specimen was influenced by the tunnel during the compression process, and bending and stretching deformation occurred.

The deformation characteristics of the specimen with an inclination of 90° were similar to those of the no interface specimen, and the deformation exhibited an hourglass shape. However, the deformation area was large, the displacement trace was close to the tunnel, and the curvature was small. Furthermore, the main crack was perpendicular to the minimum principal stress and the surface was rough, which was generated under tensile stress. These phenomena indicate strong tensile deformation characteristics.

3.3. Failure Mode

Figure 7 shows the morphology of the samples after failure from different angles. The dotted lines indicate the location of the main cracks. The relationship between the extension direction of the crack and the position of the interface can be considered as the failure mode of the sample. There are three main forms of deformation and failure: compression, tension, and shear. Shear deformation is the main deformation on the interface, and tensile fracture is the main fracture in the sample body. The different fracture development characteristics can be subdivided into three types according to the form of movement: compression fracture, slip fracture, and bending fracture. Compression cracking is distributed in all samples, mainly in the vertical direction, which is related to the sample material. The cracks generated by slippage and tension are evenly distributed near the interface and can be clearly observed in the 15°, 30°, and 45° samples (Figure 7c–e). The cracks are arranged in a pinnate pattern pointing to the direction of deformation. In the process of shear deformation, the sample tended to slide along the contact face. As the axial load continued to increase, the normal load on the interface also increased, the interface was tightly connected, and the friction resistance increased. The sliding force was not sufficient to resist friction but reached the shear strength of the sample material; therefore; many cracks were generated on both sides of the interface. The bending fracture was mainly caused by the influence of the tunnel, which changed the overall stress distribution, and the stress concentrated around the tunnel, resulting in bending cracking. It should be noted that crack development is more complex in samples at 45° and 60°. When the angle of the structural plane is 45°, it is easy to produce stress concentration and fracture at the tip of the sample under the action of load. This situation also occurs at the position where the tunnel contacts the structural plane. The shear slip deformation occurs along the structural plane. Under the influence of the principal stress and the friction force of the structural plane, a large number of tensile shear cracks are generated inside the rock, which extends outward from the structure plane. When the angle of structural plane is 60°, the influence of principal stress is more significant. According to the preliminary test, the included angle between the principal stress of the sample and the horizontal direction is 63.1°, which is very close to the dip angle of the structural plane. Therefore, in the process of compression deformation, the structural plane becomes the principal stress plane. A conjugate shear fracture is formed with the principal stress surface generated in the other direction. On the outside of this set of conjugate fracture planes, longitudinal tensile cracks through the whole rock mass appear, resulting in significant rock mass failure.

The three-dimensional distribution of fractures can effectively reflect the failure characteristics (Figure 8). The samples without an interface mainly developed vertical fractures, which constituted the fracture surfaces in the yz and xy directions, and the samples were cut into multiple blocks. Cracks appeared at the top, bottom, and arch of the tunnel.

The fracture development of the sample with an interface inclination of 0° was centered on the tunnel and distributed symmetrically, and fracture surfaces mainly developed in the yz and xy directions. Cracks developed at the top and bottom of the tunnel, and particularly near the interface. The rock surrounding the tunnel was severely fractured.

Out of the two fracture groups in the sample with an inclination angle of 15°, one group was parallel to the x to y direction and the other group was perpendicular to the x to y direction. In the fractures perpendicular to the x to y direction, the most obvious perforating fracture surface passed through the tunnel and was perpendicular to the interface. Additionally, owing to the interface sliding traction, as the distance to the interface decreased, the curvature increased and the fracture surface was thus formed.

The same fracture mode appeared in the 30°, 60°, and 75° samples. The curvature of the fracture surface increased with the dip angle.

A large number of incomplete cracks appeared in the sample with the 45° angle, and these cracks were distributed on both sides of the interface, which caused concentrated destruction of the rock surrounding the tunnel.

For the 90° sample, the number of fracture surfaces parallel to the y to z direction significantly increased, and many fractures appeared in the tunnel. The cracks in the tunnel gradually increased with the dip angle, and the damage degree gradually became more severe.

The test results revealed that, under the influence of an interface with different dip angles, the rock surrounding a tunnel without support mainly undergoes tensile failure (Figure 9a), shear failure (Figure 9b), slip along the interface (Figure 9c), collapse failure (Figure 9d), and bending failure (Figure 9e). A fracture surface parallel to the maximum principal stress is produced under axial pressure because the surrounding rock is brittle. The relationship between the ultimate strain and the loading speed is very small, and the brittle tensile fracture is controlled by tensile strain.

Shear failure contains brittle and plastic deformation. When the load provided by the maximum principal stress is higher than the maximum static friction force of the interface, the interface maintains stability, the surrounding rock near the interface is subjected to traction, and plastic shear deformation occurs. Consequently, a large number of parallel pinnate fissures appear along the interface. This phenomenon is obvious in the 15°–45° samples. Brittle deformation forms vertical cracks, and plastic deformation forms pinnate fractures. These two types of crack extend and connect, and form a curved fracture pattern. The degree of bending reflects the strength of the shear action on the surrounding rock, and as the curvature of the fracture surface increases, shear failure is the main failure of the surrounding rock.

Slide deformation occurs under axial stress, and is dominated by the internal friction angle of the interface. Slip deformation occurs when the dip angle of the interface is higher than its internal friction angle. The typical signs of slip deformation are the fracture of the slip plane and the occurrence of a broken rock mass at the inner wall of the tunnel.

Collapse failure often occurs in the rock surrounding the slab crack medium and the rock’s failure condition is closely related to the elastic curve shape of the slab crack deformation. When the inclination angle was 90°, the surrounding rock underwent elastic deformation toward the tunnel under compression stress, collapse failure occurred, and a curved fracture surface formed in the surrounding rock.

The failure mechanism of the bending failure was the same as that of the beam. The surrounding rock at the top of the chamber bent and deformed under the action of pressure. Failure occurred when the tensile strength of the rock was reached. The characteristic of bending failure is that the fracture surface converges to the bending middle point in the rock mass, and the most typical example is the sample with an inclination angle of 0°.

In summary, three kinds of cracks with different mechanical behaviors appeared in the course of the test, namely, tensile crack, shear crack, and tensile shear crack. Tensile cracks mainly developed in the samples of 0° and 90°, while shear cracks mostly developed near the structural plane, and were seen in the samples of 15° to 60°. Tensile shear cracks were widely distributed in all kinds of samples. In the process of shear slip, the rock mass on both sides of the structural plane is subjected to friction traction, and then tensile cracks are formed, which mainly develop near the structural plane.

Anchoring measures are typically adopted in engineering when perforating structural planes appear in the surrounding rock. The anchoring methods are different for different stress states. The anchorage method has been adopted in many projects [52,53,54]. For a large sliding belt passing through the tunnel, engineering disposal measures still focus on controlling the deformation, and the most common and effective method is to arrange the anti-slide piles. The position, size, depth, and other factors of the anti-slide pile determine the reinforcement effect and cost.

3.4. Failure Criteria

For a rock mass, the interface attitude greatly influences the strength. The relationship between the interface strength and the rock mass strength can be obtained based on the strength envelope of the Mohr–Coulomb strength criterion [17]. In this research, the formulas are derived based on the plane strain problem. Although the physical model is 3D, it has the same mechanic behavior along the z direction, which can be equal to the plane strain condition.

According to the geometrical relationship in Figure 10, the following expressions can be obtained:

$$\{\begin{array}{l}{\alpha }_1=\frac{{\phi }_i}{2}+\frac{1}{2}{\sin }^{-1}\left[\frac{2c_i\cos {\phi }_i+({\sigma }_1+{\sigma }_3)\sin {\phi }_i}{{\sigma }_1-{\sigma }_3}\right]\\ {\alpha }_2=\frac{{\phi }_i}{2}-\frac{1}{2}{\sin }^{-1}\left[\frac{2c_i\cos {\phi }_i+({\sigma }_1+{\sigma }_3)\sin {\phi }_i}{{\sigma }_1-{\sigma }_3}\right]+\frac{\pi }{2}\end{array}$$

(1)

where ${\phi }_i$ is the internal friction angle of the interface, and because the interface was flat in this study, ${\phi }_i$ = 36° was determined by a tilt test; $c_i$ is the cohesion of the interface, and $c_i$ = 780 kPa was obtained by a direct shear test. When ${\sigma }_3$ = 0 and ${\sigma }_1$ = ${\sigma }_c$, the UCS of the intact rock mass was 19.236 MPa. From this calculation, ${\alpha }_1$ = 38° and ${\alpha }_2$ = 88° can be obtained. When the dip angle α₁ < α < α₂, the structural planes slip. When α < α₁, and α > α₂, the structural planes do not slip. In this test, the structural plane of the samples of 0°, 15°, 30°, and 90° does not encounter slide deformation, while the structural plane of 45°, 60°, and 75° does encounter slide deformation.

This study investigated the effects of the different orientations of the contact faces on the chamber, and two models are proposed: a complete rock mass model with an interface and a chamber arch model. These two models are simplified as Problem 1 and Problem 2, respectively (Figure 11).

Problem 1:

In this study, two different interface forms appeared: the through–through interface (0°–45°) and the non-penetrating interface (60°–90°) (Figure 12).

For failure mode (a), the strength is expressed as follows:

$${\sigma }_{1m}=\{\begin{array}{l}{\sigma }_3·\frac{1+\sin \phi }{\left(1-\sin \phi \right)}\cos \alpha +c·\frac{2\cos \phi }{\left(1-\sin \phi \right)}\cos \alpha +c_i·\frac{1}{\tan {\phi }_i}\\ {\sigma }_3·\frac{\sin (2\alpha -{\phi }_i)+\sin {\phi }_i}{\sin (2\alpha -{\phi }_i)-\sin {\phi }_i}+c_i·\frac{2\cos {\phi }_i}{\sin (2\alpha -{\phi }_i)-\sin {\phi }_i}\\ {\sigma }_3·\frac{1-\mu }{\mu }+{\sigma }_c\end{array}\begin{array}{cc}& \end{array}\begin{array}{c}\begin{array}{c}\\ (\alpha <{\alpha }_1)\\ \end{array}\\ \begin{array}{c}({\alpha }_1<\alpha <{\alpha }_2)\\ \\ (\alpha <{\alpha }_2)\end{array}\end{array}$$

(2)

where $\phi$ is the internal friction angle of the rock (38°); c is the cohesion of rock (4 MPa); and $\mu$ is Poisson’s ratio of the rock (0.25).

For failure mode (b), the strength is expressed as follows:

$${\sigma }_{1m}=\frac{2a}{2a+b}{\sigma }_1+\frac{b}{2a+b}{\sigma }_{1i}$$

(3)

where ${\sigma }_{1i}$ is the strength of the structural plane.

Problem 2:

The ability of the tunnel model with an interface to resist the compression exerted by the arch effect, and its UCS, were lower than those of the complete model, and the UCS was variable (Figure 13). Three assumptions were made: (1) the surrounding rock material is a homogeneous elastomer; (2) the stress distribution of the surrounding rock is uniform; and (3) the stress is uniformly distributed in the plastic zone.

According to the Mohr–Coulomb criteria, the radius of the surrounding rock plastic zone can be calculated as follows [55,56,57]:

$$R_p=a{\left[\frac{\left({\sigma }_0+c\cot \phi \right)\left(1-\sin \phi \right)}{c\cot \phi }\right]}^{\frac{1-\sin \phi }{2\sin \phi }}$$

(4)

where ${\sigma }_0$ is the original stress field, and ${\sigma }_0$ = 0 in this study.

The surrounding rock pressure in a uniform stress field can be calculated using the Fenner formula [55,56,57], as follows:

$${\sigma }_r=\left[\left(1-\sin \phi \right)\left({\sigma }_0+c\cot \phi \right)\right]{\left(\frac{a}{R_p}\right)}^{\frac{2\sin \phi }{1-\sin \phi }}-c\cot \phi$$

(5)

By combining Problems 1 and 2, the anisotropy strength criteria of a different interface dip with the tunnel can be expressed as follows:

$${\sigma }_{1m}=\{\begin{array}{l}{\sigma }_3·\frac{1+\sin \phi }{\left(1-\sin \phi \right)}\cos \alpha +c·\frac{2\cos \phi }{\left(1-\sin \phi \right)}\cos \alpha +c_i·\frac{1}{\tan {\phi }_i}-{\sigma }_r\\ {\sigma }_3·\frac{\sin (2\alpha -{\phi }_i)+\sin {\phi }_i}{\sin (2\alpha -{\phi }_i)-\sin {\phi }_i}+c_i·\frac{2\cos {\phi }_i}{\sin (2\alpha -{\phi }_i)-\sin {\phi }_i}-{\sigma }_r\\ {\sigma }_3·\frac{1-\mu }{\mu }+{\sigma }_c-{\sigma }_r\end{array}\begin{array}{cc}& \end{array}\begin{array}{c}\begin{array}{c}\\ (\alpha <{\alpha }_1)\\ \end{array}\\ \begin{array}{c}({\alpha }_1<\alpha <{\alpha }_2)\\ \\ (\alpha <{\alpha }_2)\end{array}\end{array}$$

(6)

$${\sigma }_{1m}=\frac{2a}{2a+b}{\sigma }_1+\frac{b}{2a+b}{\sigma }_{1i}-{\sigma }_r$$

(7)

When the cell pressure is zero, the first principal stress is expressed as follows:

$${\sigma }_1={\sigma }_c+\frac{c_i}{\sin \beta \cos ({\phi }_i+\beta )}$$

(8)

The strength can be calculated using Equations (5)–(7):

In Figure 14, the fitting degree is lower than 45° and 90° (Figure 14a,b). In this study, when the inclination angle was 45°, under the uniaxial condition, the tip of the sample was first subjected to force, which resulted in a small contact area; therefore, the failure mode was fragmentation (Figure 7e). Shearing slip does not occur until the contact area of the rock mass is sufficient to bear the vertical load. There are two failure forms, which were explained in detail in the previous section. The calculated result for the strength of the interface reveals that the strength is lower than that obtained by testing.

Figure 14. Theory versus test. The calculation results were obtained by data fitting in Table 2.

Figure 14. Theory versus test. The calculation results were obtained by data fitting in Table 2.

Table 2. Calculation results.

$\mathit{\alpha }$	0	15	30	45	60	75	90
a/(cm)	\	\	\	\	2.12	3.66	5.00
b/(cm)	\	\	\	\	5.77	2.68	5.00
${\sigma }_{1m}$/(MPa)	17.47	17.24	15.60	5.70	3.10	16.26	19.236

a and b are the size of the rock mass after being cut by the structural plane, as shown in Figure 12b.

Table 2. Calculation results.

$\mathit{\alpha }$	0	15	30	45	60	75	90
a/(cm)	\	\	\	\	2.12	3.66	5.00
b/(cm)	\	\	\	\	5.77	2.68	5.00
${\sigma }_{1m}$/(MPa)	17.47	17.24	15.60	5.70	3.10	16.26	19.236

a and b are the size of the rock mass after being cut by the structural plane, as shown in Figure 12b.

In this study, because the interface with an inclination of 90° divided the sample into two parts, the test results reflect the strength of the individual parts; therefore, the UCS of the sample was larger than the calculated result.

4. Discussion

The influence of a single structure on the deformation and failure mode of the rock mass was studied in this paper. However, there are two or more structural planes in practical engineering. The mechanical behavior and deformation characteristics of the multiple structural planes in a rock mass are more complex. In this paper, the discontinuities with seven different dip angles were combined with each other to form 36 combinations of discontinuities.

The two sets of structural planes cut the rock mass into four blocks of different shapes. According to the deformation location, the overall deformation can be divided into two types: the compression deformation of the block itself and the shear slip deformation along the structural plane. The force of the rock mass in the two sets of structural planes is shown in Figure 15. There are four blocks: A, B, C, and D. Each block is subjected to four forces: gravity, external pressure, and the supporting forces of the structural planes.

Under the action of external load, the rock mass cannot maintain a stable state and will be deformed. In the calculation, it is assumed that the rock blocks are homogeneous and the structural plane is flat without filling. Under the action of load, the mechanical strength of the structural plane is the key criterion to determine the deformation and failure of the rock mass. However, the strength of the structural plane is far lower than that of the rock mass. Therefore, it is considered that the rock mass does not fail and the whole process is elastic deformation. In Figure 15, G is gravity, F is external load, F₁ is the supporting force from set 1 to the blocks, and F₂ is the supporting force from set 2 to the blocks. θ₁ is the angle between set 1 and the horizontal direction, and θ₂ is the angle between set 2 and the horizontal direction.

The supporting force of the structure planes to the block can be converted into the friction force between the block and the structural plane:

$$f=\{\begin{array}{l}{\mu }_1{F}_1\\ {\mu }_2{F}_2\end{array}$$

(9)

In Equation (9), μ is the dynamic friction coefficient of the structural plane. The deformation rate is small under the load, so it can be equivalent to the static friction coefficient. In this study, the two sets have the same properties, so μ₁ = μ₂.

The friction coefficient is taken as the tangent value of the friction angle of the structural plane:

$$\mu =\tan {\phi }_j$$

(10)

Based on this principle, a numerical simulation calculation was carried out. The material parameters of the numerical simulation calculation are listed in

Table 3. Materials parameters.

Type	Density ρ/(kg/m3)	Elasticity Modulus E/(GPa)	Poisson’s Ratio λ	Friction Angle φ/(°)	Cohesion c/(MPa)
Rock block	2500	15	0.25	32	10
Structural plane	1300	0.06	0.40	25	0.005

. In the numerical calculation, displacement loading is adopted, and deformation is applied to the top and bottom of the sample at the same time with a rate of 0.005 mm/s. The 36 structural plane combinations were regarded as the subjects for numerical simulation.

Due to the large amount of settlement result data, the horizontal displacement nephogram is shown here only, and the deformation trend chart is drawn based on the horizontal displacement. The deformations of different combinations are compared, and the final results are listed in Figure 16. In Figure 16a, the central axis of the deformable cloud image shows obvious deformation trace lines. Taking the dip = 0° as an example, the deformation trace coincides with the central axis and is perpendicular to the structural plane under one structural plane. The deformation trace deflects obviously after the appearance of the second structural plane. The deformation track in the area near the tunnel gradually deflects toward the direction of increasing the structural plane angle, forming an “S” shaped track line, which made the rock mass gradually evolve from lateral tensile deformation to shear deformation. The other dips show the same deformation characteristics. According to the maximum stress under different structural plane combinations, the stress nephogram was drawn, as shown in Figure 16b. In the Figure 16b, the rock mass strength is small under the combination conditions of the structural plane containing 45° and 60°, and the rock mass is in the weakest state under the combination conditions of 45°–45° and 45°–60°. This result has a certain correlation with the above test results.

Compared with the strength of a single structural plane, the strength under the influence of the combination of structural planes changes significantly, and according to the calculation, there is a certain mathematical relationship between them. Through the calculation and fitting results (Figure 17), the strength relationship between the single structural plane and the combination of structural planes can be expressed as:

$${\sigma }^{\prime }=a+b·{\sigma }_{\theta 1}+c·{\sigma }_{\theta 2}$$

(11)

In Equation (11), a, b, and c are the fitting parameters, and in this study, a = 12.36, b = −1.18, and c = 1.61. ${\sigma }_{\theta 1}$ is the strength of the rock mass with structural plane 1, and ${\sigma }_{\theta 2}$ is the strength of the rock mass with structural plane 2.

Complex tests and numerical simulations can be avoided by using this formula. As long as the rock mass strength with one single structural plane is known, the combined strength of a set of structural planes can be calculated. In this study, the strength of a single structural plane can be obtained by Formulas (6) and (7). By substituting Formulas (6) and (7) into Formula (11), the rock mass strength of the combination of two structural planes can be predicted. However, due to limited research data, the fitting degree of the formula is 60%, and the main error occurs in the small dip angle of the structural plane, which needs to be further improved and optimized in subsequent research.

5. Conclusions

In this study, the deformation and failure characteristics were captured through uniaxial compression tests on a tunnel with different structural plane dip angles, and the following conclusions were drawn:

(1) The dip angle of the interface has a significant effect on the strength of the rock surrounding the tunnel. As the angle of the interface increased, the UCS of the sample first decreased and then increased, and reached the minimum value at 60°. This result indicates that 60° is the most dangerous angle in the tunnel, and should be paid more attention in engineering.

(2) According to the deformation results, it was found that stress concentration can easily occur close to the interface, which is an area prone to damage. Under the influence of interfaces with different dip angles, the deformation modes in the surrounding rock of the tunnel are obviously different, and the deformation contour generally converges toward the tunnel cavity.

(3) There are five types of failure in the surrounding rock, namely, tensile failure, shear failure, contact slip failure, collapse failure, and bending failure. Tension failure was the main failure form and appeared in all samples. Shear failure occurred mainly in samples with a small dip angle. The slip failure of the interface mainly occurred in samples with α greater than 45°. The 90° dip angle was dominated by collapse failure, while the 0° dip angle was dominated by bending failure.

(4) In tunneling, rock masses generally incorporate two or three major joint sets instead of one discontinuity set. A wedge may form when two intersecting surfaces (such as bedding, faults, or joints) meet, and the block moves downward. The combined form of two structural planes is calculated by numerical simulation, and the strength prediction formula of the rock mass is obtained. When the combination of the structural plane is 45°+60°, the rock mass is at its weakest.