## 1. Introduction

When the working face of the mining is a hard top, the initial pressure step of the old roof increases sharply, resulting in a large area above the working face. Once collapsed, accompanied by strong periodic pressure and obvious dynamic phenomena, it has a large slump area. Significant impacts, such as strong impact, serious equipment damage, and serious accidents, endanger the lives and property of production personnel [

1,

2,

3,

4,

5].

Solid filling, roadway support, and drilling pressure relief can reduce the effect of rock burst. Solid filling is to reduce the influence of overburden collapse by utilizing the compressibility of the filling material [

6,

7]; strengthening roadway support can effectively reduce the compressive strength [

8,

9]. The representative achievements of borehole pressure relief in the field of roadway control are: Li Shubin et al. [

10] determined the pressure-reducing parameters of the “three soft” coal seam mining roadway by establishing the functional relationship between the drilling parameters and the surrounding rock expansion deformation; Gao Mingshi et al. [

11] indexed the three-dimensional anchor into the coal roadway support project, supplemented with the roadway drilling and pressure relief technology, and solved the problem of supporting the full thick roadway in the soft thick coal seam and extra-thick coal seam along the bottom construction; using the numerical simulation method, the influence law of borehole diameter and length on the stability of high stress roadway was analyzed, and the pressure relief and support parameters of roadway were determined accordingly [

12,

13,

14]. Establishing broken beams over the key strata of the stope, and the mechanical model derives the formula for calculating the break angle of the key layer, and verifies the rationality and reliability of the formula through physical simulation experiments [

15,

16,

17]. Although the above work has achieved some useful conclusions in the application and parameter determination of the borehole pressure relief technology, it has not yet formed a reliable technical system, and the research process has been neglected. The interaction between the elevation angle of the borehole and the pressure relief effect leads to limited application and application of the research results. The critical layer breaking angle of on-site measurement and statistics is constrained by geological conditions and measuring instruments [

18,

19,

20]. The results measured at the site cannot be used as a theoretical basis for guiding the unloading of pressure relief holes in front of the hard top working face of the working face. Therefore, by establishing the mechanical structure model and the physical similarity simulation experiment to obtain the theory of the critical layer breaking angle, the field production practice can be better guided.

Based on the relevant theoretical methods of rock mechanics and material mechanics, this paper establishes the mechanical structure model of the key layer and deducts the expression of the breaking angle of the key layer. The applicability and rationality of the key layer break angle formula under different conditions of a single working face and double working face are verified by constructing the physical similarity simulation experiment of the double working face of Kuangou coal mine. The measured results are basically consistent with the theoretical calculation results. The 3DEC numerical simulation was used to establish the pressure relief gradient model, which verified the rationality of the theoretical calculation results. The breaking angle of the key layer calculated by the expression is of great significance for guiding the pressure relief of the hard roof working face.

## 3. Derivation of Breaking Angle Mechanics of Key Layers under Repeated Mining

The breaking of the key layer is regarded as the break of the beam “O-X”, and the upper surface is affected by the uniform load q. According to the two-dimensional stress analytical method of material mechanics [

23,

24], the normal stress

σ at any point in the beam is solved. The force of any unit in the key layer of the overlying rock is shown in

Figure 2. In

Figure 2, it is assumed that the stress components

σ_{x},

σ_{y},

τ_{xy},

τ_{yx} are all known. According to the shear stress reciprocal theorem and the stress balance equation, it can be concluded that:

Let

σ_{α} in Equation (1) be derived from

α:

If

α =

α_{0}, Equation (2) can be made zero, and the normal stress takes the extreme value on the cross section determined by

α_{0} (the beam is taken as the unit width). Substituting

α_{0} into Equation (2) and making it equal to zero yields:

According to the stress decomposition amount of the key layer rock beam derived by Xu Bin [

15], substituting into Equation (3), Equation (4) is obtained, where

L is the key layer breaking distance and h is the key layer average thickness.

The key layer is simplified to a simply supported beam, and the maximum normal stress

σ_{max} is used as the basis for rock fracture. When

σ_{max} = R

_{T}, the normal stress of the critical layer at this point reaches the tensile strength limit there, and the rock layer will be cracked there. The ultimate span when the beam is broken is:

Substituting Equation (5) into Equation (4) yields:

Equation (6) is the formula for calculating the angle between the principal stress plane and the horizontal plane of any unit body of the key layer. It is only necessary to substitute the tensile strength limit R

_{T} of the key layer and the uniform load q above the key layer, according to Mohr–Coulomb. The criterion is that the key layer is in a state of extreme equilibrium when it is broken. At this time, the angle

β′ between the fractured surface and the maximum normal stress

σ_{max} generated by the shear failure of the rock meets:

where

φ is the internal friction angle of the rock. The plane relationship between the broken section of the key layer and the horizontal plane is shown in

Figure 3.

It can be seen from

Figure 3 that the breaking angle of the key layer, that is, the angle between the broken section and the horizontal plane

θ (breaking angle of the key layer) is:

When the working face adopts the down-mining method for coal mining, the lower working face is affected by repeated mining when it is mining below the goaf of the upper working face, and the main instability mode of the working face roof is changed from the original turning instability to the sliding instability. When the roof of the lower working face is broken, and the friction is coupled with the rock layer of the goaf, the assumption of the theoretical calculation of the breaking angle is no longer valid. As shown in

Figure 4a, when the lower working face is pushed to the lower side of the upper goaf, the top plate of the working face is broken and affected by the broken rock in the goaf, and the original girder structure of the turning instability is transformed into the simply supported beam structure cut along the roof. As the upper goaf is not fully compacted, the hypothesis that the key layer is uniformly loaded is no longer valid. To simplify the analysis, the overlying strata are considered as simply-supported beam structures, and the red dotted line is regarded as the load reduction zone. The overlying rock stratum is regarded as a simple supported beam structure, and the formation law of the breaking line of the key layer is analyzed. Using the bending moment superposition of material mechanics, for the convenience of calculation, the mechanical model shown in

Figure 4a is regarded as a simply-supported beam model without repeated mining in

Figure 4b, and

Figure 4c is reduced by repeated mining stress unloading. The simple supported beams are superimposed.

Figure 4b simplifies the key layer as a fixed beam, where the normal stress of the critical layer reaches the tensile strength limit at that point, where the rock layer will be cracked, and the key layer that is not subjected to repeated mining is broken. The angle calculation formula is Equation (8). The actual meaning of the model in

Figure 4c is that the load on the working face is reduced by the effect of repeated mining when the working face is recovered to the goaf of the upper working face, partly because the top plate of the upper working face is crushed and relieved. The complete compaction of the goaf also results in a redistribution of the load. According to the equilibrium equation and the bending moment equation, the above simple bending beam bending moment formula is:

In Formula (9), q

_{1} is the maximum value of the pressure relief of the top plate due to the pressure relief.

M_{(x)} is determined by the derivative of

M_{(x)} and x is 0, and the maximum value and positive value of the bending moment

M are solved. The maximum value of the stress σ is (takes a unit width of 1):

The maximum normal stress

σ_{max} is used as the basis for rock fracture. When

σ_{max} = R

_{T}, that is, the normal stress of the critical layer at this point reaches the tensile strength limit there, and the rock layer will be cracked there. The ultimate span when the beam is broken is:

According to the reduction of the load-breaking cross-section and the horizontal plane relationship, the calculation formula of the breaking angle of the key layer breaking caused by the reduced load under the repeated mining is:

Calculate the breaking angle

θ_{1} of the key layer caused by the assumed unrepeated uniform load according to Equation (8), and then calculate the breaking angle

θ_{2} caused by the reduced load of the partial load indicated by the red dotted line according to Equation (12). The calculation of the breaking angle

θ of the key layer caused by the mining effect is:

Substituting Equations (8) and (12) into Equation (13) yields:

where R

_{T} is the ultimate tensile strength of the key layer, q

_{0} is the uniform load on the key layer, q

_{1} is the load reduced by repeated mining, and

λ is the repeated disturbance coefficient. When

λ = 1, Formula (14) represents the theoretical calculation formula for the breaking angle of the key layer of single coal seam; when

λ = 2, Formula (14) represents the theoretical calculation formula of the breaking angle of the key layer affected by repeated mining in the short-distance double-coal mining.

## 4. Experimental Verification of Similar Materials for Breaking Angle Calculation Formula

The W1145 working face and the W1123 working face of the west wing of the first mining area of the Kuangou coal mine are mainly used for the B4-1 coal seam and the B2 coal seam. First, the upper W1145 working face is recovered. After the W1145 working face is finished and the overburden is stable, the lower W1123 working face is recovered. When the W1145 working face is recovered, the overburden layer on the roof can be regarded as a uniform load (

λ = 1); when the W1123 working face is recovered, the working face is pushed to the lower side of the W1145 working face and is affected by repeated mining. The overlying load is redistributed, and the critical layer breaking can be regarded as the mechanical structure model shown in

Figure 4a (

λ = 2).

A similar simulation model was designed based on the W1145 and W1123 double working faces of the Kuangou Coal Mine. The experiment used a plane strain model frame with dimensions (length × width × height) = 5.0 × 0.3 × 1.5 m to determine the geometric similarity ratio (α

_{L} = L

_{H}/L

_{M}) of the simulation experiment, which was 1:200, and the model pavement size (long × width × height = 5.0 × 0.3 × 1.5 m), and a layer of iron brick was placed on the top instead of the unsimulated rock formation to apply a load to the model. According to the lithologic characteristics of the overburden strata of the B2 coal seam detected by the Z1201 borehole histogram of the W1123 working face in the main study of the Kuangou coal mine, the load of each layer of the rock layer on the old roof was calculated layer by layer. The position of the key layer was judged, as shown in

Table 1. The design and construction of the dynamic regulation of the overburden structure and the physical similarity simulation experiment provide support. The mining plan for the working face of the model is to move the W1145 and W1123 working faces to the model. According to the actual mining sequence of the mining face, firstly, the upper W1145 working face was recovered, and the cut hole was opened at 230 cm from the left boundary of the model B4-1 coal seam (8 cm), starting to recover the working face, mining to 30 cm away from the right boundary to stop mining, a total of 240 cm; in the model B4-1 coal seam W1145 working face mining end and overburden collapse stability, back to the lower W1123 working face in model B2. The coal seam was opened 38 cm from the left boundary and began to be harvested. When the mining was 30 cm away from the right boundary, the mining was stopped, and a total of 432 m was advanced.

In addition to its own weight, the load applied to any rock in the overburden is also affected by the interaction of the overlying strata. Assuming that the rock layer load q is evenly distributed, there are a total of m layers of rock layers directly above the top, each rock layer has a thickness of

h_{i} (

i = 1, 2, …, m), and the volume force is

γ_{i} (

i = 1, 2, …, m), and elasticity Modulus

Ei (

i = 1, 2, …, m). The rock layer controlled by the first layer has n layers. The first layer of rock and the n layer will be simultaneously deformed to form a composite beam. According to the theory of the overburden composite rock beam, the first layer can be obtained by the load that the n layer affects [

25]:

where (

q_{n})

_{1} refers to the load above the hard rock layer of the first layer;

γ_{i,} h_{i}, E_{i} refer to the bulk density, thickness, and elastic modulus of the i-th layer, respectively,

i = (

i = 1, 2, …,

n). The load q of each layer is calculated according to Formula (15). The results are shown in

Table 1.

#### 4.1. Verification of Breaking Angle of Key Layer in Single Coal Seam

The wide trench coal mine adopts the method of down mining to carry out mining. When the W1145 working face is recovered, it is not affected by repeated disturbances. Therefore,

λ = 1 can be used to calculate the breaking angle of the key layer of the W1145 working face. The mechanical parameters of the double key layer rock are substituted into Formula (14). The main key layer parameters are a compressive strength of 7.58 MPa, uniform load of 0.41 MPa, and internal friction angle of 22°. Subcritical parameters are a compressive strength of 5.31 MPa, the cloth load is 0.36 MPa, and the internal friction angle is 20°. The calculated primary bond layer breaking angle is 77.64°, and the subcritical layer breaking angle is 78.47°.

Figure 5 shows the collapse morphology of the overburden after the end of the W1145 working face. Because the overburden of the W1145 working face is only affected by the uniform load above the top surface of the working face, the measured breaking angle range is 76–78°. The results for the theoretical calculation are basically consistent, and the applicability of the theoretical formula of breaking angle in a single working face is verified.

#### 4.2. Key Layer Breaking Angle Verification under Repeated Mining

When mining the W1123 working face, take

λ = 2 when calculating the breaking angle of the key layer of the W1123 working face considering the influence of repeated disturbance. The compressive strength of the sub-critical layer rock mechanics parameters is 5.31 MPa, the uniform load is 0.36 MPa, the load is reduced by 0.30 MPa, and the internal friction angle is 20°; the compressive strength of the main key layer rock mechanics is 7.58 MPa, and the uniform load is 0.41 MPa. The breaking angle of the sub-critical layer calculated by substituting (14) is 69.88°, and the breaking angle of the main key layer is 69.82°. When the W1123 working surface is advanced to 73.2 cm, the sub-critical layer collapse characteristics are as shown in

Figure 6a. The left side of the model has a breaking angle of 78°, and the right side of the model has a breaking angle of 76°. When the W1123 working surface is advanced to 106.8 cm, the main key layer collapse is as shown in

Figure 6b; the left side of the model break angle is 78°, and the right side of the model angle is 76°. The difference between the measured results and the theoretical calculation results is large, which indicates that the

λ = 2 is not applicable when the W1123 working face is firstly collapsed, because when the W1123 working face is recovered to 106.8 cm, it is far away from the W1145 working face, and it is less affected by repeated mining. At this time, the initial collapse of the double key layer of the W1123 working face is still applicable to the case where the single coal seam is not subjected to repeated mining (

λ = 1), and when the

λ = 1, the physical parameters of the double key layer are substituted into Formula (14). The theoretical range of the breaking angle of the double key layer is 77–78.5°, which is basically consistent with the measured results. The rationality and applicability of the theoretical formula of the breaking angle of the key layer of the single coal seam are verified again.

As shown in

Figure 7a, when the W1123 working face is 190.8 cm, it is directly below the W1145 open-cut eye, and the right side of the model formed by the influence of repeated mining is 70°, and when

λ = 2 (14), the calculated 69.8° is basically consistent, which verifies the accuracy and applicability of the theoretical formula of the breaking angle of the key layer under repeated mining. As shown in

Figure 7b, when the W1123 working face is recovered to 308.4 cm, the fracture line on the right side of the W1123 working face and the breaking line above the W1145 goaf are connected, because the goaf is re-compacted by the overlying fractured rock. The formation of the key layer breaking line is no longer affected by the repeated mining effect, and the resulting breaking angle is increased to 78°.

The W1123 working face is self-opening and cutting to the breaking angle on both sides of the working face end model, and the resulting breaking angle is shown in

Figure 8. It can be seen from

Figure 8 that when the W1123 is recovered to the right side of the W1145 goaf model, the breaking angle suddenly drops to 70°, and as the working surface continues to advance to 308.4 cm, the right side breaking angle of the model rises to 78°. Because the broken line on the left side of the model tends to be stable and is less affected by the mining of the working face, and the breaking angle on the left side of the model is basically unchanged. When the broken line on the right side of the model and the overburden fracture are connected, the broken lines on both sides of the model are basically not affected by the W1145 goaf, and the breaking angle is 78°, which is consistent with the theoretical calculation results.

## 7. Conclusions

(1) Based on rock mechanics and material mechanics, the theoretical formula of the double critical layer breaking angle of the double working face was derived, and the repeated disturbance coefficient λ was proposed. When calculating the breaking angle of the key layer of a single coal seam, take λ = 1; calculate λ = 2 when calculating the breaking angle of the key layer affected by repeated mining in the short-distance double-coal mining.

(2) When the W1145 working face was recovered, the breaking angle of the key layer of the model was basically the same as that of the key layer of the single coal seam; when the W1123 working face was not advanced to the W1145 goaf, and the W1123 working face was fully harvested. In the future, the calculation result of the theoretical formula of the double-key breaking angle of the single working face and the measured result of the model should have less error. When the W1123 working face was plucked below the W1145 goaf, it was affected by the repeated mining, according to the breaking under the repeated mining. The results calculated by the angular theory formula were basically consistent with the experimental simulation results. The above shows the rationality and reliability of the formula for the double-key layer breaking angle of the double working face.

(3) According to the theoretical analysis and numerical simulation results, a pressure relief bore with an elevation angle of 78° was arranged on the working surface, and the pressure relief effect was tested by the borehole peep method. According to the analysis of the drilling peep results, the breaking angle of the key layer calculated according to the formula could be used as the theoretical basis for the elevation angle of the relief hole. It is indicated that the theoretical formula of the breaking angle of the key layer is well applied to the practice of pre-cracking and pressure relief in the roof of the working face of the wide trench coal mine.