Control of the rock surrounding a deep roadway is one of the theoretical bottlenecks and key problems in deep mining. Roadway support is a core issue in coal mining, and it is also the key issue in rock burst prevention (Kang et al., 2015; Li and Wu, 2019). The load corresponding to the unstable equilibrium point of the roadway system is called the critical rock burst load on the tunnel. The critical rock burst load of the tunnel is one of the most important indexes to characterize the problem of a rock burst when the roadway is disturbed in a specific stress environment. It is also the theoretical basis of stress control theory in rock burst prevention theory.

Scholars have studied and advanced the understanding of the critical load of rock bursts in roadways (Pan, 1999; Pan et al., 2014). Li et al. examined the rock burst instability theory and concluded that the modulus ratio was an important parameter affecting the critical load of a rock burst (Li et al., 2018). Ma et al. used the roof shear beam model to deduce that the critical load decreased with an increase in the width-height ratio of a rectangular section roadway but increased with the increase of roof thickness. The critical load decreased with an increase in the stiffness ratio of the coal seam to the roof but increased with the increase in the modulus ratio (Ma et al., 2011). Several studies found that the internal friction angle had a relatively strong influence on the damage radius, stress, and displacement of the surrounding rock, while the influence of cohesion is small (Pan and Wang, 2004a; Pan and Wang, 2004b; Pan et al., 2006; Pan et al., 2007). Guo et al. proposed that the rock burst occurrence conditions were related to the ratio of the ascending and descending modulus of rock mass and the development degree of rock fracture (Guo et al., 2011). Chen et al. conducted a theoretical analysis on the critical rock burst of a circular section of a roadway and concluded that the critical rock burst increases with the increase of unified strength theoretical parameters, supporting force, modulus ratio, and internal friction angle and decreases with the increase of dilatancy parameters (Chen et al., 2019; Chen et al., 2020). Meng et al. concluded that part of the input energy in the surrounding rock of a deep circular roadway is released in the form of work conducted by surrounding rock pressure and is used to resist support pressure. When there is no support, this part of the energy is converted into surrounding rock kinetic energy (Meng et al., 2014). Several studies proposed the shape method of the airfoil arch section roadway to improve the bearing capacity and optimize the support parameters (Xiao et al., 2016; Xiao et al., 2017; Ding et al., 2020). Wang et al. used the theory of elastic-plastic mechanics, defined the ratio of pre-peak plastic strain to elastic strain as the pre-peak damage coefficient (K), and established the constitutive relationship of coal and rock based on the Scott model. The results show that the smaller the K value is, the greater the impact tendency is (Wang et al., 2019). Yin et al. found that the lateral pressure coefficient, roadway radius, and mechanical parameters of surrounding rock (cohesion and internal friction angle) were the main factors affecting the shape and size of the plastic zone in a circular roadway (Yin et al., 2018a; Yin et al., 2018b). Huang and Gao proposed that the critical condition of a roadway rock burst was closely related to changes in the surrounding rock support mode, initial strength parameters, and strength parameters of the plastic zone. When the critical plastic zone depth and critical load are small, the rock burst occurs easily, the occurrence frequency is high, and the damage is small. When the critical plastic zone depth and critical load are large, a rock burst does not occur easily, and its occurrence frequency is low. Once it occurs, its strength is destructive (Huang and Gao, 2001). Liu et al. analyzed the airfoil tension crack at the pre-existing crack tip in the roadway coal wall, determined the critical stress of rock burst, and analyzed its influencing factors (Liu et al., 2018; Liu et al., 2019; Liu et al., 2020; Liu et al., 2021). Several studies used acoustic emission and infrared monitoring to conduct experimental research on rock failures, which was of great significance in detecting rock mass failure on the engineering scale (Dong et al., 2019; Dong et al., 2020a; Dong et al., 2020b; Dong et al., 2021; Dong and Luo, 2022).

Due to the initial stress state of the surrounding rock before roadway excavation, the stress redistribution after excavation reaches the secondary stress state. Therefore, the surrounding rock is not only the main load source but also an important part of the bearing structure. An analytical solution is the most powerful means to analyze such roadway engineering problems. Li et al. believed that once the stress in a certain area of the surrounding rock exceeds the strength of the rock mass, the surrounding rock in this part will enter the plastic or failure state. The elastic solutions of a circular tunnel under a nonaxisymmetric external load and radial and shear internal load were derived. Using these solutions, the new elastic–plastic solution and plastic zone radius equation of a circular tunnel under non-axisymmetric external load and radial internal load were derived (Li et al., 2022). Guo et al. thought that the form of a plastic zone of roadway surrounding rock determines its failure form and degree. Based on the implicit equation of a plastic zone boundary, the variation law of the general form of the plastic zone of rock surrounding a circular roadway was comprehensively studied (Guo et al., 2016; Guo et al., 2018).

In conclusion, the critical load of roadway rock burst is one of the main research topics of tunnel anti-shock support design, but the influence trend of the properties of the surrounding rock on the critical load of a rock burst is not clear. This article considers the small disturbance and three-dimensional equivalent strain caused by the external load of the surrounding rock system in the equilibrium state and adopts the bilinear constitutive relationship. In other words, the overall stress is characterized by a linear climb before the peak and a rapid decline after the peak, which is mainly reflected in the elastic stage and the softening stage after the peak. Based on the Coulomb yield criterion and the rock burst initiation criterion, the critical load formulas with and without support are obtained. According to the analytical solution of the critical rock burst load of a circular roadway, the influence of surrounding rock properties such as uniaxial compressive strength, softening modulus, elastic modulus, and internal friction angle on the critical rock burst load of a roadway was studied by controlling single and double variables. FLAC3D software was used for numerical simulation, and the theoretical and numerical solutions were compared and analyzed to provide a theoretical basis for analyzing roadway anti-impact engineering problems.

Assuming that the coal–rock mass system is subjected to external force P (Pa), the characteristic depth of the plastic softening deformation zone is ρ (m). If the coal–rock mass system in the equilibrium state produces a small disturbance Δ P due to the external load, the corresponding plastic softening zone of the coal–rock mass is ρ + Δ ρ . If the system has a stable equilibrium state, the disturbance Δ P must be bounded. If the equilibrium state of the coal–rock mass system is unstable, even in the case of a small disturbance, it will lead to the infinite expansion of the plastic softening zone. The discriminant relationship of the extreme point of the disturbance response when the rock burst occurs is derived: Δ ρ Δ P = d ρ d P = ∞ . (1)

Assuming that the roadway is a circular cavity with radius a (m), under the action of hydrostatic pressure P (Pa), the softening zone is generated; that is, the depth of the damage zone is ρ (m), and the inner surface of the roadway is subjected to a constant support stress P 0 (Pa). The roadway is simplified as an axisymmetric plane strain problem, as shown in Figure 1. We define a polar coordinate system r , θ through the roadway center. The Z-axis perpendicular to r , θ plane, takes unit thickness in this study. The radial stress and circumferential stress at the interface between the elastic zone and softening zone are σ r p (Pa) and σ θ p (Pa), and the stress at the interface can be obtained according to the stress solution in the elastic zone. σ r = σ r p , (2) σ θ = 2 P − σ r p . (3)

In Eq. 2 (Pan, 1999), σ r (Pa) is radial stress; σ θ (Pa) is circumferential stress.

The coal rock at the junction of softening zone and the elastic zone has just entered the yield state, so the damage variable D = 0 is obtained. According to the Coulomb yield criterion, σ θ = q σ r + σ c . (4)

In Eq. 4, σ c (Pa) is the uniaxial compressive strength of coal and rock; in the limit equilibrium of rock failure, the angle between the resultant force formed by the normal stress and the internal friction on the shear plane and the normal stress is called the internal friction angle (Pan and Dai, 2021). It is expressed by φ (°), which reflects the size of the internal friction of the rock, and the internal friction angle variable q = 1 + sin ⁡ φ 1 − sin ⁡ φ is defined.

According to the deformation geometry equation, u = B r , ε r = − B r 2 , ε θ = B r 2 . (5)

In Eq. 5, B is the integral constant, u(m) is radial displacement, ε r is the radial strain component, and ε θ is the circumferential strain component. According to the geometric relationship and the incompressible volume property, the equivalent effect in the softening region becomes ε ¯ = 2 3 B r 2 . (6)

The von Mises yield criterion indicates that under certain deformation conditions when the equivalent effect force at a point within a stressed object reaches a certain value, that point begins to enter a plastic state. The equivalent effect variation is a physical quantity used to determine the location of the yield surface of a material after strengthening and is calculated using the same formula as the fourth strength theory formula for calculating the equivalent effect force, except that the stress is changed to a strain. For coal and rock materials, it is considered that the strain ε is equal to that under the uniaxial condition, and the equivalent strain ε ¯ under the three-dimensional condition is ε ¯ = 2 9 ε r − ε θ 2 + ε θ − ε z 2 + ε z − ε r 2 + 6 γ r θ 2 + γ θ Z 2 + γ Z r 2 . (7)

In Eq. 7, ε r , ε θ , and ε Z are the positive strain component and γ r θ , γ θ Z , and γ Z r are the shear strain component.

In the area where the elastic zone and the failure damage zone intersect, the coal rock reaches peak strength and peak strain. Based on the plastic mechanics theory, the uniaxial strain is transformed into the triaxial strain, and the equivalent strain ε ¯ = ε c is obtained. Let r = ρ , and B = 3 2 ρ 2 ε c . (8)

In Eq. 8, ε c is the peak strain.

Without considering the volume force, the equilibrium equation in the softening zone is d σ r d r + σ r − σ θ r = 0 . (9)

Based on the equilibrium equation and the yield condition, the stress partial differential equation of the damage zone is d σ r d r − q − 1 σ r r = σ c + λ ε c r − λ ε c r 3 ρ 2 . (10)

In Eq. 10, λ (Pa) is the softening modulus.

By solving this equation, the stress is σ r = Q r q − 1 + λ ε c q + 1 ρ 2 r 2 − σ c + λ ε c q − 1 . (11)

In Eq. 11, Q is the integral constant.

Boundary conditions on the roadway surface are as follows: σ r = P 0 . (12)

In the area where the elastic zone and the failure damage zone intersect, the continuous condition of radial stress is σ r = 2 P − σ c 1 + q . (13)

According to Eq. 11, combined with boundary conditions (Chen et al., 2019; Chen et al., 2020), ρ q − 1 a q − 1 ⋅ σ c + λ ε c q − 1 − λ ε c q + 1 ⋅ ρ q + 1 a q + 1 + P 0 ρ q − 1 a q − 1 = 2 P − σ c q + 1 + σ c + λ ε c q − 1 − λ ε c q + 1 . (14)

By solving the aforementioned formula d ρ d p , it is concluded that the initiation criterion of a rock burst is ρ * 2 a 2 = 1 + E λ + P 0 λ q − 1 . (15)

In Eq. 15, ρ * (m) is the critical softening zone depth of the roadway when a rock burst occurs; E (Pa) is the elastic modulus of coal and rock materials. Before the triaxial compression stress–strain curve of coal and rock reaches the peak, the coal and rock are in the elastic deformation stage, and the elastic modulus is E. When the curve reaches the peak, it is simplified to a linear deformation. The softening modulus is defined as λ (Pa), and its expression is λ = 񥔳 σ f − σ r ′ ) . ε f is the strain corresponding to the peak strength σ f (Pa) of the stress–strain curve under triaxial compression; σ r ′ (Pa) and ε r ′ are the termination strength and termination strain at the end of the triaxial compression test, respectively.

When support stress is not considered: P 0 = 0 ρ * 2 a 2 = 1 + E λ . (16)

Corresponding to Eq. (15), it is deduced that the critical load P * (Pa) of a rock burst when the section shape of the roadway is circular is equal to P * σ c = − 1 q − 1 1 + λ E + 0.5 q + 1 q − 1 1 + λ E 1 + E λ + P 0 q − 1 q − 1 2 − 0.5 λ E 1 + E λ + P 0 λ q − 1 q + 1 2 + P 0 σ c 1 + E λ + P 0 λ q − 1 q − 1 2 . (17)

When support stress is not considered: P 0 = 0 P * σ c = − 1 q − 1 1 + λ E + 0.5 q + 1 q − 1 1 + λ E 1 + E λ q − 1 2 − 0.5 λ E 1 + E λ q + 1 2 . (18)

According to Eq. (18), the critical rock burst load of a circular roadway is related to the mechanical properties of the surrounding rock, including uniaxial compressive strength σ _c , softening modulus λ, elastic modulus E, internal friction angle φ, and support stress P₀. According to the Mining Engineering Design Manual (Zhang, 2003), the rock grades of the surrounding rock are soft, medium-hard, and hard. The rock mass test parameters are as follows: the uniaxial compressive strength is 95.6 MPa, the softening modulus is 25 GPa, the elastic modulus is 24 GPa, and the internal friction angle is 30°. The mechanical properties of surrounding rock, such as uniaxial compressive strength, softening modulus, elastic modulus, internal friction angle, and the influence of support stress on the critical load of roadway rock burst, were studied by the control variable method. The control variable method means that when a physical quantity is affected and restricted by multiple factors at the same time, the multi-factor problem is transformed into multiple single-factor problems. When studying the relationship between this physical quantity and one of the factors, only one factor is allowed to change, and the other factors are controlled to remain unchanged to determine the relationship between the relevant physical quantities.

According to the results of the analysis of the factors affecting the critical load of the rock burst in Section 3, the uniaxial compressive strength, elastic modulus, and internal friction angle are selected for numerical simulation. Using Flac3D software, a circular roadway model is established that is 400 m underground, 12 m long, and has a radius of 3 m, as shown in Figure 9.

The constitutive model adopts the Mohr–Coulomb model, which requires six parameters: bulk modulus, shear modulus, internal friction angle, dilatancy angle, cohesion, and tensile strength. The bulk modulus and shear modulus are obtained from the elastic modulus and Poisson’s ratio, as shown in Eq. 19. K * = E 3 1 − 2 υ , G = E 2 1 + υ . (19)

In the formula, K ^* is the bulk modulus, G is the shear modulus, and v is Poisson’s ratio.

According to the Griffith strength criterion, the tensile strength of rock is generally one-tenth of the compressive strength. For soil or rock, the dilatancy angle ranges from 0° to 20°. In the Flac3D model, the default value is 0°. The outside of the circular roadway is homogeneous surrounding rock. The rock mass test parameters are as follows: volume weight = 25 KN/M³, cohesion = 0.8 Mpa, internal friction angle = 30°, Poisson’s ratio = 0.3, and elastic modulus = 24.9 GPa.

(1) A mechanical model of a circular “roadway rock-support” system considering damage is established. The critical load formula of the rock burst in the roadway with and without supporting conditions is derived according to the instability theory of rock burst disturbance response.