In mining engineering, the thick and hard roof threatens the safe production. Based on Reissner plate theory and combined with weighted residual method, with four edges clamped as the boundary conditions, this paper deduces the theoretical formula of the first breaking span of thick and hard roof. Based on Vlasov plate theory, with four edges simply supported as the boundary conditions, this paper deduces the theoretical formula of the periodic breaking span of thick and hard roof. The two formulas are used to verify the breaking span of thick and hard roof of Tashan Coal Mine, proving that its accuracy is higher than that of traditional beam theory. This paper studies the distribution characteristics of strain energy density in front of the coal seam during the mining process by numerical simulation, which is compared with the results of field microseismic experiments. It is found that the strain energy density of the coal seam has a good correlation with the probability of microseismic events. This paper provides theoretical support for more precise calculation of breaking span of the thick and hard roof and technical support for the practical stability analysis of the surrounding rock under the thick and hard roof.
The characteristics of thick and hard roof are large thickness, high strength, poor joint fissure development, strong integrity, and strong selfbearing ability [
Usually, the beam theory is used to calculate the breaking span of roof, using the mechanical mode of fixed beam with line loaded to get the first and periodic breaking span, and the stress distribution in rock beam is analyzed by using the material mechanics theory [
The thick and hard roof breaking has an important effect on the stability of surrounding rocks and coal seam [
Given the above, according to the mechanical model suitable for thick and hard roof, this paper is intended to get the theoretical formula of the first breaking span of thick and hard roof by using Reissner plate theory and combining with weighted residual method, with four edges clamped as the boundary conditions and get the theoretical formula of the periodic breaking span of thick and hard roof by using the the modified Vlasov plate theory, with four edges simply supported as the boundary conditions. Furthermore, this paper is intended to study the distribution law of strain energy of coal seam under the action of thick and hard roof through the 3DEC numerical simulation software and the corresponding theoretical formula. At last, taking one of the most representative thick and hard roof in the world, Tashan Coal Mine roof, as example, the breaking span formulas are going to be verified, and the correlation between the strain energy of coal seam and field measured microseismic activity is also going to be discussed as well. So the purpose of this paper is to provide theoretical support for more precise calculation of breaking span of the thick and hard roof and provide technical support for the practical stability analysis of the surrounding rock through the distribution law of the strain energy in the coal seam.
The classification criteria for the plate is as follows [
In the study, the ratio of thickness and span of thick and hard roof is between 1/5 and 1/8; therefore, thick plate theory should be used to calculate the breaking span of roof.
The stope roof is simplified to a rectangular plate. With the excavation of the coal seam, the boundary condition of rectangular roof is also changing. At the beginning during the coal seam mining, no cracks in the boundary of roof rock mass appear, where boundary condition is fouredge fixed. As the excavation continues, the size of the goaf roof increases, and the midpoint of the long side of the roof will firstly enter the plastic state and then extend to form a plastic hinge along the long side. And then, the midpoint of the short side of the roof plate will also enter the plastic state and form a plastic hinge along the short side, where boundary condition is fouredge simply supported. When the boundary condition changes from fouredge fixed to fouredge simply supported, which means weakening the boundary constraint, the roof still remains whole without breaking at this time. As the coal mining face continues to advance, a plastic hinge along the
Three stages of roof breaking.
Due to the different boundary conditions, it is necessary to derive the theoretical formulas of the first breaking span and the periodic breaking span of the thick and hard roof, respectively. Reissner theory is a relatively mature thick plate theory when the boundary condition is fouredge fixed, while Vlasov theory is a relatively simple plate theory with high precision when the boundary condition is fouredge simply supported. Therefore, Reissner thick plate theory and Vlasov plate theory are used to derive the first breaking span and the theoretical breaking span, respectively.
Reissner made a straight line hypothesis instead of straight normal, that is, the line which is perpendicular to the middle surface before deformation is still a line after deformation. Considering the effect of shear strain
The short side of the rectangular plate is
And
In order to meet the above boundary conditions, the following trial function is selected:
According to its formulation, the trial function is confirmed if the form of
Expand the load on the thick plate into a double triangular series; that is,
With the increase of
We can get the deflection function of the roof when substituting (
The bending moments
The relationship between the tensile strength
The thick plate breaking discriminant under fouredge fixed condition can be obtained when substituting (
After the first breaking of the roof, as the coal face continues to move forward, periodic breaking will occur. The breaking of thick and hard roof often results in a large dynamic load, leading to cracks in the boundary of the front unbroken roof, which helps the boundary enter into the plastic state quickly. Meanwhile, it is easy to form a hinge relationship between the unbroken roof and broken roof. Therefore, the boundary condition of the roof can be simplified as fouredge simply supported condition.
Under this boundary condition, the Vlasov plate theory which considers horizontal shear deformation has the advantages of high precision and simple form [
Considering the horizontal shear deformation, it is assumed that the true displacement of each point on the middle surface normal is nonlinear, and the relationship between the shear stress and the displacement is as follows:
The rotation angle of straight normal on the middle surface can be obtained from formula (
For rectangular plate with fouredge simply supported condition, the boundary conditions are as follows:
Based on formula (
And the differential equations of plate bending are rewritten as follows:
The deflection and rotation angle displacement functions are transformed into the following double trigonometric series:
At this time, all the board boundary conditions have been met. Furtherly, the horizontal load
Substituting (
As with the roof first breaking, the load on the roof is uniform load too. With the increase of
The value of
In order to verify the accuracy of the derived formulas when calculating the first and the periodic breaking span of thick and hard roof and also compared with the accuracy of traditional beam theory, the 20meterthick hard lamprophyre of 8212 working face of Tashan Coal Mine in Datong, China, is selected as the calculation object. Datong mining area is dual system coal seam; as the upper Jurassic resource exhausted, the main coal seam is shifting to the lower Carboniferous coal seam. The distance between the double coal seams is 150~350 m. The depth of Carboniferous coal seam is 400~600 m. In Carboniferous coal seam, it is 3–5#. The distribution of the double coal seam diagram in Datong mining area is shown in Figure
The distribution of the double coal seam diagram in Datong mining area.
The working face is mining 3–5# coal seam, with the average buried depth 474 meters, dip angle 3° which is flat seam. The average thickness of 3–5# coal seam is 12 meters, which is typical extremely thick coal seam in China. Conventional longwall topcoal caving (CLTCC) is the mining method used in Tashan Coal Mine [
The histogram of 8212 fully mechanized caving face.
In field measurement, before forced caving measures such as hydraulic fracturing and presplitting blasting, the first breaking span of the 20meterthick lamprophyre can reach about 110 m, and the periodic breaking span of the 20meterthick lamprophyre can reach between 65 m and 75 m. So, the two ratios, one being thickness and the first breaking span of roof and the other being thickness and the periodic breaking span of roof, are all between 1/5 and 1/8, which is applicable to the thick plate theory. Every time the thick and hard roof breaks, the pressure of the working face increases obviously, and it lasts a long time. The safety valves of hydraulic supports open frequently and the maximum descent speed of the support column is 320 mm/h. The resistance of the hydraulic support can reach 14000 kN, which can easily cause the breakage of the support. At this time, the collapse phenomenon is serious, and the pressure in the middle of the working face is extremely obvious. Figure
The strong pressure behavior when 20 m thick roof is broken.
The mechanical parameters of thick and hard roof are
In addition to the dead weight, the effective load of any rock stratum in the overlying strata is usually subjected to the interaction of the overlying adjacent rock layers. Generally speaking, the load of stratum is not uniformly distributed, but in order to analyze the problem conveniently, it is assumed that rock load is uniformly distributed [
The effective load calculation diagram of stratum.
Suppose that the first layer is a key stratum, the number of upper strata is
The key strata theory shows that [
According to the strata histogram and the key strata theory, the upper lamprophyre controlled height is 54.1 m, getting to the sandy mudstone layer, while the lower lamprophyre controlled height is 4 m, that is, the thickness of 2# coal seam. Uniform load on the upper lamprophyre can be obtained through formula (
At this stage, all the parameter values of the first and the periodic breaking span of the thick and hard roof have been determined, as shown in Table
The value of parameters involved in calculation.
Parameters involved in calculation  Value 


0.21 

20 m 

12 MPa 

230 m 

1.85 MPa 
Through formula (
When using the beam theory to calculate the first breaking span of the roof, the formula is
When using the beam theory to calculate the periodic breaking span of the roof, the formula is
Through formula (
The error rate
According to formula (
In order to verify the correctness of the deduced theoretical formula again, the 8206 working face of Tashan Coal Mine is selected as the calculation object too. The histogram of 8206 working faces is basically the same as that of the 8212 working faces; there is only a slight difference in rock thickness; the mechanical parameters of the corresponding rock strata are consistent. Because of the limited space, the histogram of 8206 working faces is no longer given here. The width of the working face is 205 m, the advance speed of the working face is 5.6 m/d, and the thickness of the roof is 21 m; according to the previous calculation method, the uniform load on the upper lamprophyre can be obtained through formula (
In conclusion, the accuracy of derived calculation formulas based on the thick plate theory in this paper is higher than the beam theory to get the breaking span of the thick and hard roof.
Because of good stability and large breaking span, the thick and hard roof can accumulate a large amount of strain energy in the coal seam. Once the thick and hard roof collapses, it will cause huge energy release, forming strong pressure. The energy release is sudden, transient, and destructive, as shown in Figure
Substituting (
Therefore, in order to know the distribution of strain energy density of coal seam under thick and hard roof deformation and failure, it is necessary to know the three principal stress values of the coal seam. Although it is difficult to obtain the three principal components of stress of each point over time, approximate values are obtained through numerical simulation methods. Specifically, we apply here the large scale threedimensional discrete element software 3DEC. This software can help to study the evolution law of roof strata migration and the stress of the coal seam during the working face advancing [
Similarly, the thick and hard roof of 8212 working face of Tashan Coal Mine is selected as the simulation object. According to the stratigraphic histogram, the design model has height of 200 m. and length of 300 m. and has a length boundary coal pillar of length of 30 m. Because the coal seam is flat seam, it is assumed to be a horizontal coal seam in the numerical model. The model uses brick units to simulate coal seams and surrounding rocks. And it is assumed that the boundary condition does not change during the excavation; that is, the four sides and the bottom of the model are fixed constraint; the upper part of the model is free boundary; at the top of the model, the uniform load applied by the overlying strata is
Mechanic parameters of coal seam and strata.
Strata  Density/(kg/m^{3})  Bulk modulus/GPa  Shear modulus/GPa  Internal friction angle/°  Cohesion/MPa 

Sandstone  2810  60  54.6  42  1.97 
Sandy mudstone  2678  36.3  31.2  38  1.55 
Mudstone  2504  2.5  1.21  31  1.38 
Siltstone  2740  45.4  39.2  42  1.98 
Mudstone  2500  2.78  1.2  30  1.378 
4# coal seam  1456  2.71  1.2  31  1.349 
Siltstone  2736  44.4  41.2  45  1.88 
Lamprophyre  2825  46.1  42.8  49  4.87 
2# coal seam  1452  2.7  1.19  30  1.347 
Lamprophyre  2812  44.1  41.6  47  4.63 
3–5# coal seam  1450  2.68  1.18  30  1.346 
Sandstone  2800  58  54.4  41  1.92 
Mechanic parameters of interface of coal seam and strata.
Strata  Normal stiffness/GPa  Shear stiffness/GPa  Cohesion/MPa  Internal friction angle/° 

Sandstone  4.2  2.2  0.49  15 
Sandy mudstone  2  1.1  0.26  12 
Mudstone  1.52  0.83  0.2  12 
Siltstone  4.2  4  0.39  9 
Mudstone  1.5  0.8  0.19  10 
4# coal seam  0.33  0.12  0.14  10 
Siltstone  3.9  3.9  0.37  7 
Lamprophyre  4.2  4.2  0.67  17 
2# coal seam  0.31  0.11  0.135  9 
Lamprophyre  4  4  0.63  15 
3–5# coal seam  0.3  0.1  0.11  9 
Sandstone  4  2  0.47  14 
As the field microseismic experiments were carried out when the working face advancing distance was 30 m, 60 m, 90 m, 120 m, and 150 m, respectively, therefore, the numerical simulation experiments also focus on the energy distribution characteristics of coal seam in front of the mining coal wall when the working face advancing distance is 30 m, 60 m, 90 m, 120 m, and 150 m, respectively. In order to obtain the distribution of strain energy density in front of the coal seam during the mining process, the dense principal stress measuring points are arranged in the coal seam, the schematic diagram of the measuring points arrangement is shown in Figure
The schematic diagram of the measuring points arrangement.
The numerical model is shown in Figure
The numerical model.
The simulation results.
The face’s advancing distance is 30 m
The face’s advancing distance is 60 m
The face’s advancing distance is 90 m
The face’s advancing distance is 120 m
The face’s advancing distance is 150 m
The face’s advancing distance is 180 m
As can be seen from Figure
Distribution of strain energy density of coal seam ahead of working face when working face is advancing different distances.
The following conclusions can be obtained from Figure
The distribution of the strain energy density in front of the working face is similar to that of the stress distribution, which shows the obvious decreasing, increasing, and stable areas. The decreasing area is close to the working face, whose stress value is lower than original rock stress, with little strain energy accumulation. As the distance from the coal wall increases, the strain energy density increases sharply. And the strain energy density is at a higher level when the distance from the working face is between 20 m and 60 m, where the peak values are obtained. A large amount of strain energy is accumulated in the coal seam in the increasing area. When the distance is far from the working face, the coal seam is almost unaffected by the working face mining, and the strain energy accumulated in the coal seam is generally stable.
When the advance distance increases from 30 m to 90 m, the strain energy density of the front coal seam at the same distance from the working face increases as the advance distance increases; however, when the advance distance increases from 90 m to 150 m, the strain energy density of the front coal seam at the same distance from the working face decreases firstly and then increases. The reason is that when the advance distance is lower than 120 m, the roof does not break, and energy has always been in the condition of accumulation; when the working face advancing distance is up to 120 m, the roof breaks, leading to the energy accumulated in the coal seam release.
In the field microseismic experiment, the microseismic events of the surrounding rock were recorded when the distance of the face advance was 30 m, 60 m, 90 m, 120 m, and 150 m, respectively. The microseismic system is made up of ground monitoring mainframe, ground photoelectric converter, underground photoelectric converter, underground monitoring mainframe, signal cable, and detector. The microseismic system composition and experimental results are shown in Figures
Microseismic system.
Microseismic experiment results.
The experiment result shows that microseismic events are few when measuring points are located in 0–20 m in front of the working face, and microseismic events increase with the increase of the distance from the working face; microseismic events’ most intensive area is 20–60 m ahead of the working face. When the distance from the working face is larger than 60 m, the microseismic events decrease with the increase of the distance. Comparing Figure
By making reasonable assumptions on the basis of the plate theory and combining with powerful mathematical software, this paper proves that the accuracy of the derived formula is higher than that of the traditional beam theory when calculating the breaking span of the thick and hard roof. However, some deviations still exist between the calculated results and the field data, and reasons can be summarized as follows. Firstly, the traditional thick plate theory is based on reasonable assumptions, and the paper makes further assumptions on the basis of thick plate theory. Though these assumptions are as much as possible close to the real situation, assumptions can never be completely consistent with the real situation. Secondly, the mechanical parameters involved in the calculation of coal and rock are obtained in the laboratory, which may be a little different from the real ones in the field.
The study of the energy distribution of the coal seam contributes to a better understanding and prevention of the dynamic disasters of the coal and rock mass, especially under the action of thick and hard roof, which can accumulate a lot of energy, easily leading to mine disasters because of the big breaking span of roof. Strain energy has an important position at all kinds of energy in the surrounding rock, because a series of dynamic disasters such as coal bumps, rock burst, and mine earthquake are closely related to the gathering and release of strain energy. Therefore, this paper focuses on the distribution of strain energy in coal seam under thick and hard roof action. In the future, we can use the same method combining numerical simulation with theory to get the distribution of strain energy in any region by changing the positions of measuring points. Future studies may involve other energy forms, such as dissipation energy, kinetic energy, and surface energy. And the same research method may help to study other energy forms above, that is, obtaining the parameter values required in the energy expression through numerical simulation and substituting these parameter values into the energy expression to get the specific energy value. In this way, we can avoid the defects that the parameters required in the expression are difficult or even unable to obtain completely in theory.
Based on Reissner plate theory and combined with weighted residual method, with fouredge fixed condition as the boundary condition, this paper derives the theoretical formula of the first breaking span of thick and hard roof. Based on Vlasov plate theory, with fouredge simply supported condition as the boundary condition, this paper derives theoretical formula of the periodic breaking span of thick and hard roof. The two formulas above are used to calculate the first breaking span and the periodic breaking span of thick and hard roof, respectively, in Tashan Coal Mine, whose results are compared with those of the traditional beam theory. In detail, for 8212 working faces, the error rate is reduced from 34.5% to 9.69% when calculating the first breaking span, and the biggest error rate is reduced from 21.6% to 7.47% when calculating the periodic breaking span; for 8206 working faces, the error rate is reduced from 38.77% to 10.2% when calculating the first breaking span, and the biggest error rate is reduced from 21.3% to 8.43% when calculating the periodic breaking span. Therefore, the accuracy of the formula deduced in this paper is higher than that of the traditional beam theory when calculating the breaking span of the thick and hard roof.
According to the definition of strain energy density and combined with 3DEC numerical simulation software, this paper studies the distribution of strain energy density of the coal seam in front of the working face with different advance distance. The distribution of the strain energy density in front of the working face is similar to that of the stress distribution, which shows obvious decreasing, increasing, and stable areas. When the thick and hard roof does not break, the strain energy density of the front coal seam at the same distance from the working face increases as the advance distance increases. When the thick and hard roof breaks, energy release leading the strain energy density of the front coal seam at the same distance from the working face decreases firstly and then increases again.
The strain energy accumulated in the coal seam has a good correlation with its microseismic events. That is, when the strain energy of the surrounding rock is large, the probability of occurrence of microseismic events is greater; when the strain energy of the surrounding rock is small, the probability of occurrence of microseismic events is lower. The numerical simulation method helps to study the distribution of surrounding rocks energy in the future.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This study was financially supported by the National Natural Science Foundation of China (51604006) and the National Program on Key Basic Research Project (973 Program) (2010CB226802).