Sliding Impact Mechanism of Square Roadway Based on Complex Function Theory

To clarify the process of stress change and plastic zone evolution of square roadways under high-stress conditions, the rotational square expansion plastic zone evolution model of square roadway was established by theoretical analysis, numerical simulation, and engineering verification. The shear slip impact stress criterion of square roadway based on complex variable function theory was studied, and the law of surrounding rock stress distribution, plastic zone expansion, elastic energy density, local energy release rate (LERR), and total energy release of square roadway were analyzed. The results show that the compressive stress is concentrated in the four corners of the roadway after the roadway excavated and transfers with the change of plastic zone. Main shear failures start from the four corners and develop in a rotating square shape, forming square failure zones I and II. The square failure zone I is connected with the roadway contour and rotated 45°. The square failure zone II is connected with the square failure zone I and rotated 45°. When the original rock stress is low, the surrounding rock tends to be stable after the square shear slip line field formed. When the original rock stress is high, the shear failure of the surrounding rock continues to occur after the square failure zone II formed, showing a spiral slip line. Corners of the square roadway and square failure zones I and II are the main energy accumulation and release areas. The maximum elastic energy density and LERR increase exponentially with the ratio of vertical stress to uniaxial compressive strength (Ic). When square corners of the roof are changed to round corners, the plastic zone of the roof expands to form an arch structure. The maximum elastic energy density decreases by 22%, which reduces the energy level and possibility of rock burst. This study enriches the failure mechanism of roadway sliding impact. It can provide a basic theoretical reference for the design of the new roadway section and support form based on the prevention of rock burst.


Introduction
e causes of rock burst are various, and their manifestations are also different [1][2][3][4]. e cross section of many premining roadways in a coal mine is square or rectangular, and a lot of rock burst happened in these roadways. For example, on February 22, 2020, a rock burst occurred in Xinjulong Coal Mine, resulting in four deaths and extensive roadway damage. Many scholars have studied the failure of rectangular and square roadways and achieved fruitful results.
Shi et al. [5] studied the stress distribution law of rectangular roadways under different span height ratios and lateral pressure coefficients. Guo et al. [6] studied the evolution law of plastic zone and large-scale failure criterion of roadway surrounding rock by FLAC numerical simulation method. Wu et al. [7,8] found that V-shaped belt failure occurred on both sides of the tunnel through true triaxial experiment and numerical simulation. Wang et al. [9] established the elastic energy spatial zoning evolution model of surrounding rock under different driving speeds. Yang et al. [10] divided the rock burst roadway into a head-on area, dynamic evolution area of a plastic circle, and stable area of a plastic circle. Li et al. [11] divided the rectangular roadway roof into fracture through the area, fracture development area, microfracture area, and nonfracture area. Yu et al. [12] studied the loose range of rectangular roadway surrounding rock under different lithologic conditions by combining the method of multipoint displacement of deep base point and borehole peeping. Yu et al. [13] studied the distribution and evolution law of stress, displacement, and plastic zone of a roadway with a composite roof. Pan et al. [14] studied the stress distribution, surrounding rock displacement, and plastic zone distribution characteristics of the bottom gas drainage roadway through numerical simulation. Meng et al. [15] obtained the evolution process of roadway surrounding rock impact by studying the stress, displacement, and plastic zone distribution of soft rock roadway. Wang et al. [16] studied the analytical expressions of stress distribution, plastic zone, and disturbed zone width on both sides of the roadway in a gas coal seam. Hou et al. [17] studied the deformation characteristics and acoustic emission characteristics of rectangular roadway surrounding rock under different initial in situ stresses through a laboratory test. Yin et al. [18] studied the critical plastic softening zone depth and critical load of the rectangular roadway under rock burst. Rock burst occurs when the softening zone of tunnel rock mass with rock burst tendency reaches the critical depth [19]. Li et al. [20] studied the deformation and failure process of rectangular roadways through numerical simulation and field observation. Zheng et al. [21] analyzed the calculation formula of the Hoek Brown constitutive model based on the stress of rectangular roadways and studied the effect of anchorage system on roadway roof under different preloads. Chen et al. [22] studied the working face stability of shallow buried square tunnels in heterogeneous soil. Zhao et al. [23] applied the theory of complex variable function to solve the stress of square tunnel surrounding rock in a homogeneous isotropic elastic rock mass. Zhang et al. [24] established the friction work calculation model of roadway plastic zone and studied the influence of the relationship between total energy and friction work in roadway impact area on rock burst. Yi et al. [25] studied the transmission and dissipation of strain energy in the surrounding rock of deep roadway by numerical simulation. Wen et al. [26] studied the rock burst evaluation method based on the ratio of released energy to the absorbed energy. However, the research on the law of plastic zone evolution and energy dissipation around square roadway needs to be further studied.
In this paper, the rotational square expansion plastic zone evolution model of square roadway was established, and the shear slip stress conditions of square roadway based on the theory of complex variable function are studied. e stress distribution, plastic zone expansion, elastic energy density, local energy release rate, and total energy released by surrounding rock are analyzed, and the optimization scheme of the roadway section is proposed.

Roadway Shear Slip Impact Model and
Surrounding Rock Stress Distribution 2.1. Shear Slip Impact Model of Roadway. After the excavation of the square roadway, the high-stress concentration is formed at the corner, and the shear slip occurs and expands continuously from the corner, as shown in Figure 1.
When the shear slip zone passes through, a square failure zone is formed, which is called square failure zone I. After the formation of square failure zone I, the stress concentration zones shift from the corners of the roadway to the corners of square failure zone I. And the shear slip continued to expand in a square form from the corners of square failure zone I, forming square failure zone II. Either square failure zone is circumscribed with the tunnel contour (or the previous square failure zone) and rotated 45°, and the expansion of plastic failure area is called rotational square expansion, and the shear slip line field is rotational square slip line field. After the formation of square failure zone II, the stress distribution at the corner of square failure zone II changes due to the reaction of the destroyed coal and rock mass in the roadway. e shear failure does not propagate according to the rotational square but tends to the logarithmic spiral [27,28] shear slip mode.
When the roadway failure does not form square failure zone I, the plastic zone is small and will not form a largescale rock burst. When the roadway failure forms square failure zone I, the failure area is connected for the first time, forming a large range of overall weak blocks. If there are high elastic energy accumulations around the roadway, it is easy to form a large-scale roadway rock burst. e center of two sides, roof, and floor are the main impact areas, while the impact at the corner is weak. When the roadway failure forms square failure zone II, the overall weak block of the surrounding rock further increases, which is easy to form large-scale roadway rock burst. e two sides, roof, floor, and corners are the main impact areas.

Stress Distribution and Impact Determination of Surrounding Rock Based on Complex Function eory.
In [23], the stress distribution of square roadway surrounding rock is studied by complex variable function and integral transformation theory. e circular stress distribution function of surrounding rock in polar coordinates of square roadway is given as follows: where e length of the roadway is much larger than the cross section of the roadway. e deformation and failure of the roadway can be approximated as a plane strain problem. According to the plane strain characteristics, equation (2) can be obtained by universal Hooke's law.
where σ z is the stress along the length of the roadway and σ r is the radial stress of the roadway. σ r is 0 on the surface of the roadway, and σ m , which is the average stress of roadway, is as follows: According to the characteristics along the slip line, Δσ m is directly proportional to Δω. where where Δω′ is the angle from the starting point of a spiral line to any point of the spiral line. According to the Mohr-Coulomb criterion, under the limit equilibrium state, the stress state at any point can be expressed as where σ is the normal stress, τ is the shear stress, and φ is the internal friction angle.

Shock and Vibration 3
Substituting equation (5) into equation (6), the stress state at any point in the limit equilibrium state is equation (7). When the stress state of the surrounding rock breaks through the limit equilibrium state, the shear slip impact occurs in the corresponding region.

Modeling.
e strain-softening model was established by FLAC 3D software. e model length × width × height is 100 m × 1 m × 50 m. e physical and mechanical parameters of rock strata are mainly taken from Xinjulong Coal Mine, and the conventional parameters are used to supplement the missing coal and rock parameters. Table 1 lists the physical and mechanical parameters of each rock stratum.
Fix the bottom and lateral boundaries, and apply different forces on the top of the model, respectively. 21 groups of different initial stress states are formed, and the vertical stress at the boundary between the coal seam and roof is 1.0-3.0 [σ c ], respectively (the ratio of vertical stress to uniaxial compressive strength Ic � 1.0-3.0). e coefficient of lateral pressure is 1.0.

Law of Stress Change.
After the model is balanced, the roadway is excavated along the coal seam floor with a section size of 4 m × 4 m. Four vertical stress measuring points are arranged in the far field of the roadway sidewall to monitor the change of regional stress. e distance between the stress measuring point and the roadway sidewall is 28-37 m, and the vertical direction is located on the horizontal plane of the middle point of the roadway. When Ic � 1.5, the vertical stress change of each measuring point is shown in Figure 2. Each measuring point has experienced two obvious stress reduction and recovery stages, which indicates that the surrounding rock of the roadway has a twice obvious failure, causing regional stress adjustment. e evolution of compressive stress concentration area of roadway surrounding rock is shown in Figure 3. When the roadway is just excavated, the compressive stress is concentrated near the four corners of the roadway. After the first obvious failure of the roadway, the compressive stress concentration area rotates 45°and expands outward. After the second obvious failure of the roadway, the compressive stress concentration area rotated 45°again and expanded outward.

Evolution Law of Failure Area.
When Ic � 1.5, the plastic zone expansion of the surrounding rock is shown in Figure 4. After the first obvious failure of the roadway, square failure zone I is formed, and square failure zone I is mainly shear and tensile failure. After the second obvious failure of the roadway, square failure zone II is formed, and square failure zone II is mainly shear failure. e main shear failure areas of the surrounding rock are shown in Figure 5, and the main shear failure zones extend in a rotating square shape.
When Ic � 1.5, the shear strain nephogram of the roadway surrounding rock is shown in Figure 6. e shear strain is the largest at the four corners of the roadway, and a square shear slip line field connected with the roadway and rotated 45°is formed, which is the boundary of square failure zone I in the theoretical analysis, and square failure zone II with a rotation of 45°is formed outside square failure zone I.
When the original rock stress is low, the surrounding rock tends to be stable after the formation of square failure zone II. When the original rock stress is high, after the formation of square failure zone II, the failure area continues to expand outward, as shown in Figure 7, which shows the shear strain nephogram of surrounding rock with Ic � 3.0. After the formation of square failure zone II, the stress distribution has changed, and the propagation direction of the failure zone has changed. e slip line field is similar to the spiral expansion form.

Energy Monitoring Methods.
When the elastic moduli of the three principal stress directions are the same, the elastic strain energy released by the element can be expressed as [29,30] where U e is the elastic strain energy released by the element, E 0 is the elastic modulus, σ 1 , σ 2 , and σ 3 are the three principal stresses of the element, and v is the Poisson ratio. e elastic strain energy released by the element is expressed by the local energy release rate (LERR), and the sum of elastic strain energy released by all elements is the total energy released by the surrounding rock, which is recorded as ERE. e calculation method is as follows [31]: where LERR i is the local energy release rate of the ith element and U e i max is the peak value of elastic strain energy 4 Shock and Vibration density of the ith element before failure. U e i min is the valley value of elastic strain energy density after the failure of the ith element and V i is the volume of the ith unit.
According to the above method, FLAC 3D software is used to program Fish language to monitor the energy parameters of the model.

Energy Evolution Law.
When Ic � 1.5, the evolution law of elastic energy density of surrounding rock is shown in Figure 8. After the formation of square failure zone I, the elastic energy accumulates at its four corners. After the formation of square failure zone II, the elastic energy accumulated at the four corners of square failure zone II.
When Ic � 1.5, the local energy release rate of the surrounding rock is shown in Figure 9. Four corners of the roadway and the four corners of square failure zone I are the areas where a lot of energy is released. e maximum local energy release rate is 30.6 kJ/m 3 .
When Ic � 1.5, the total energy release curve of the surrounding rock is shown in Figure 10. ere are two peaks in the total energy released by the surrounding rock. When the total energy released by the surrounding rock reaches the peak value I, the square failure zone I is basically formed. e surrounding rock tends to be stable and the stress is readjusted, resulting in the accumulation of new strain energy, which leads to the decrease of the total energy released in the region. When the total energy released by the surrounding rock reaches peak II, the square failure zone II is basically formed. e stress is adjusted again, and the surrounding rock tends to the final stable state and the total energy released tends to be stable. e statistical results of maximum elastic energy density (PE) and local energy release rate (LERR) of each model are shown in Figure 11. rough data fitting, it can be seen that PE and LERR increase exponentially with Ic.

Case 1. Liangbaosi Coal Mine is located in Juye coalfield.
e average buried depth of 3507 working face is 1067 m, and the average thickness of the coal seam is 6.2 m. e coal seam, roof, and floor have a weak impact tendency. A coal blasting occurred during the tunneling of the 3507 track gateway, which resulted in roof subsidence of the right shoulder and bolt fracture, as shown in Figure 12. e roof subsidence of the right shoulder of the roadway is 200-300 mm, while the middle part of the roof has no deformation, which is consistent with the theoretical analysis that the corner of the roadway is damaged first.  Step   Figure 13. e middle part of the sides and the middle part of the roof protrude. e deformation of roadway corners is relatively small. It is consistent with the theoretical analysis that the main impact areas are the roadway side and middle roof.

Optimization of Roadway Section
e right angle in the structure is easy to produce stress concentration, while the fillet stress concentration is low. e filleted corner should be kept as far as possible in the roadway excavation. e roadway driving along the bottom shall avoid forming a right angle at the top angle, while the roadway driving along the top shall avoid forming a right angle at the bottom angle. e two roof corners of the roadway in the model, where Ic � 1.5, are changed into rounded corners, and the fillet radius is 0.5 m. When the model is recalculated, the law of plastic zone expansion has changed. e surrounding rock forms an arch failure area. e newly formed arched surrounding rock structure can bear more force and reduce deformation. As shown in Figures 14 and  15, the maximum depth of the plastic zone on the roof of the fillet roadway is reduced by 20%, the maximum depth of plastic zone on both sides is reduced by 10%, and the maximum horizontal displacement is moved from the middle point of the two sides to the bottom angle of the two sides. Figure 16 shows the cloud chart of the maximum elastic energy density after roadway excavation. e maximum elastic energy density of the square roadway is 61.7 kJ/m 3 , which is located on the roadway floor. e maximum elastic energy density of the round corner roadway is 48.2 kJ/m 3 , which is located at the top coal of the roadway. e maximum elastic energy density of the filleted corner roadway is 22% less than that of the square roadway, and the location of the maximum elastic energy density is transferred from the floor to the top coal. Figure 17 shows the local energy release rate nephogram of the round corner roadway. e energy release rate and range of the floor are larger than those of top coal. e corners of the roadway floor are still right angle, and the plastic zone still expanded in a rotating square shape in the early stage. A large amount of energy is released at the corner of square failure zone I. When the corners of the roadway roof are changed to round corners, the plastic zone of the Step Figure 10: Curve of the total energy released from surrounding rock. 8 Shock and Vibration roof expands to form an arch structure, which increases the bearing capacity and releases less energy. erefore, the accumulated elastic energy of the floor is less than that of the roof, but a greater energy release and damage occur. At the same time, it can be determined that the ability of top coal with arch structure is higher than that of floor sandstone with right angle corner.

Discussion
is paper studies the impact failure mechanism of the square roadway under high stress. e roadway mainly  presents tensile shear failure. e shear failure starts from the four corners of the roadway, develops at 45°with the roadway contour, and finally integrates into a square shear failure area, which is connected with the roadway contour and rotated 45°to the roadway contour. en, the four corners of the square shear failure zone begin to expand, forming a new square shear failure zone, which is connected with the original square shear failure zone and rotated 45°to form a rotating square slip line field. After two times of rotation square shear failure, the stress distribution at the corner changed due to the reaction of the damaged coal and rock mass in the roadway. e shear   failure does not expand in the form of rotation square, and the expansion form of shear failure tends to logarithmic spiral shear slip, which is similar to the spiral shear slip line studied by scholars before.
When the corners of the square roadway are changed into rounded corners, the evolution law of the plastic zone changes, and the square plastic zone is transformed into an arch shape, which is more conducive to bearing capacity and can reduce the accumulation of elastic energy of surrounding rock. In addition, the construction of shear bolts in the main shear area of the roadway can strengthen the ability of the roadway to withstand shear failure. More prevention and control technologies need more in-depth research. e research results enrich the failure mechanism of roadway sliding and impact and can provide a basic theoretical reference for the design of new roadway cross section and support forms based on rock burst prevention and control.

Conclusions
In this paper, the rotational square expansion plastic zone evolution model of square roadway was established, and the stress criterion of shear slip impact of square roadway based on complex function theory is given.
When the roadway is just excavated, the compressive stress is concentrated near the four corners of the roadway, and the main shear failure starts from the four corners and gradually forms the square failure zones I and II. Square failure zone I is connected with the roadway contour and rotated by 45°, and square failure zone II is connected with square failure zone I and rotated 45°. When the original rock stress is low, after the formation of square failure zone II, the surrounding rock tends to be stable, forming a rotating square shear slip line field. When the original rock stress is high, the shear failure of the surrounding rock continues to occur after the formation of square failure zone II and tends to expand in the form of a logarithmic spiral.
When square failure zone I is not formed, the plastic zone is small and large-scale rock burst will not occur. When square failure zone I is formed, the failure area is connected for the first time, forming a large-scale overall weak block. If there is a high elastic energy accumulation around the roadway, it is easy to form a large-scale roadway rock burst. e center of two sides and the center of roof and floor are the main impact areas, and the impact at each corner is weak. When square failure zone II is formed, the weak block of surrounding rock of roadway will further increase, which is easy to form large-scale roadway rock burst. e two sides, roof, and floor and corners of the roadway are the main impact areas. e corners of the square roadway and the corners of square failure zones I and II are the main energy accumulation and release areas. e maximum elastic energy density and local energy release rate increase exponentially with the ratio of vertical stress to uniaxial compressive strength. When the corners of the roof are changed to round corners, the plastic zone of the roof expands to form an arch structure, the bearing capacity increases, and the plastic zone becomes smaller when it reaches stability. e maximum elastic energy density of the round corner roadway is 22% less than that of the square roadway, which reduces the energy level and possibility of rock burst.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.