# Roof Fractures of Near-Vertical and Extremely Thick Coal Seams in Horizontally Grouped Top-Coal Drawing Method Based on the Theory of a Thin Plate

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{3D}finite element numerical simulation software, considering the characteristics of steeply inclined thick coal seam sub-level mining. It undertakes orthogonal numerical simulation experiment in three levels with different depths, coal seam angles, lateral pressure coefficient, and orientation of maximum horizontal principal stress, and translates roof stress of corresponding 9 simulation experiment into steeply inclined roof normal stress. We conclude that the distribution law of normal stress along dip and dip direction of a roof under the circumstance of different advancing distances and different sub-levels. The caving pace of first weight and periodical weight were counted under the effect of the roof uniform normal stress. It can better predict the weighting situation of the working face and ensure the safe, efficient, and sustainable mining of coal mines.

## 1. Introduction

## 2. Diagrams of the Logical Relation

^{3D}are substituted in the four thin plate models. By introducing the fracture criteria, the four models can be applied to simulate the fracture process of the immediate roof with HGTC. The diagram of the logical relation of the article analysis process is shown in Figure 1.

## 3. Engineering Background

## 4. Mechanical Model for the Roof of the HGTC Mining Face

#### 4.1. Pressure Characteristics of the Near-Vertical Coal Seam Mining Process

#### 4.2. The Thin Plate Mechanics Model in Different Mining Conditions

#### 4.3. Disturbance Equation of the Mechanics Models

#### 4.3.1. Galerkin Method

_{11}, σ

_{22}, σ

_{33}, σ

_{12}, σ

_{22}, σ

_{31}is stress, Pa; ε

_{11}, ε

_{22}, ε

_{33}is volume strain; γ

_{12}, γ

_{23}, γ

_{31}is torsional strain.

_{33}, ε

_{32}, ε

_{31}, so its strain energy can be expressed as [27]

^{2}.

^{3}/12(1 − υ

^{2}), N/m; h is elastic thin plates thickness, m.

_{ij}is the displacement function that meets all the displacement boundary conditions and stress boundary conditions. A

_{mn}indicates undetermined constants that satisfy the following:

#### 4.3.2. Disturbance Equation of Model A

_{1}is the undetermined constant of the model A, and m and n are positive integers.

#### 4.3.3. Disturbance Equation of Model B

_{2}is the undetermined constant of model B.

#### 4.3.4. Disturbance Equation of Model C

_{3}is the undetermined constant of model C.

#### 4.3.5. Disturbance Equation of Model D

_{4}is the undetermined constants of model D.

#### 4.4. The Roof Fracture Criterion

_{u}is the ultimate tensile strength of rock, Pa; σ

_{1}is the maximum tension stress of the rock, Pa.

_{x}. In the same manner, it can be shown that the displacement in the x-direction of adjacent point B on x-axis is ds

_{y}. According to the principle of calculating the arc length, the ds

_{x}and ds

_{y}can be expressed as:

_{z}, τ

_{xy}and τ

_{yz}are zero; thus, the stresses of the x-direction and the y-direction can be expressed as

_{x}and σ

_{y}, the immediate roof fracture criterion can be expressed as the following:

_{x}, σ

_{y}] < σ

_{u}, then the immediate roof is intact.

_{x}, σ

_{y}] = σ

_{u}, then the immediate roof is the critical level of the fracture and is intact.

_{x}, σ

_{y}] > σ

_{u}, then the immediate roof is fractured. The value of letter ‘a’ or letter ‘b’ is the fracture span of immediate roof. The high of sub-level is a constant, therefore the value of letter ‘a’ can be seen as the first or periodic fracture span of immediate roof.

## 5. The Stress Situation of the Roof through Numerical Simulation Analysis

#### 5.1. Numerical Simulation Model

^{3D}was used to calculate the normal stress in the immediate roof. To simulate the normal stress distribution of the four proposed models, the hybrid boundary conditions are applied at the boundary of the model. To study the normal stress distribution of the immediate roof in HGTC, the models are created to simulate the excavation process of HGTC based on the geological condition of the Adaohai Coal Mine.

_{H}= γH, where σ

_{H}is the pressure loading on the top boundary of the model, H is the buried depth, and γ is the bulk density of stratum (average value is 2.5 × 104 N/m

^{3}).

_{1}and σ

_{2}are the maximum principal stress and minimum principal stress in the plane, Pa; σ

_{x}and σ

_{y}are the stress loading along the x-axis and y-axis, Pa; τ

_{xy}is the shear stress in the plane, Pa; α

_{0}is the orientation of the maximum horizontal principal stress, degree.

_{x}and σ

_{y}are the stress loading on the left and right boundary of model, Pa.

_{1}” and “σ

_{2}”, the statistics of six mines, a total of 12 positions, and the in situ stress parameters were counted. According to the field in situ stress test, the ratio of “σ

_{1}” to “σ

_{2}” was 1.74 to 1.96, with an average of 1.89, as listed in Table 4; thus, the ratio of “σ

_{1}” and “σ

_{2}” in the orthogonal experiment design is 1.89.

#### 5.2. Coordinate Conversion

^{3D}could not be directly used in the proposed thin plate mechanics models because these two methods are based on different coordinate systems. Data conversion was required between these two coordinate systems. The stress information includes the vertical stress (σ

_{z}), the horizontal stress (σ

_{x}, σ

_{y}), and the shear stress (τ

_{xy}, τ

_{xz}, τ

_{yz}). According to the stress status at a point in different Cartesian coordinate system conversion relationships [36], the authors converted the stress data of FLAC

^{3D}to data in the new Cartesian coordinate system, for which the x-axis is the working face mining distance, the y-axis is the inclined direction of immediate roof, and the z-axis is the normal direction of immediate roof. The new Cartesian coordinate system was obtained by rotating the numerical simulation Cartesian coordinate system by θ degrees along the y-axis, as shown in Figure 7.

_{x’}, σ

_{z’}, τ

_{x’y’}, τ

_{y’z’}, and τ

_{x’z’}are stress and strain in the new Cartesian coordinate system.

#### 5.3. The Normal Stress Distribution of the Immediate Roof

#### 5.4. The Fractures Span of the Immediate Roof in HGTC

_{tx1}and σ

_{ty1}are the tensile stress of the x-axis and y-axis, Pa.

_{tx}

_{1max}) in the direction of the x-axis can be obtained:

_{ty}

_{max}) in the direction of the y-axis can be obtained by

_{txmax}or σ

_{tymax}reaches the ultimate tensile strength of immediate roof rock, the immediate roof is fractured. The symbol a is the first fracture span of the immediate roof in the first subsection.

_{tx}

_{2max}) in the direction of x-axis can be obtained by

_{ty}

_{2max}) in the direction of the y-axis can be obtained by

_{tx}

_{3max}) in the direction of the x-axis can be obtained by

_{ty}

_{3max}) in the direction of the y-axis can be obtained by

_{tx}

_{4max}and σ

_{ty}

_{4max}) in the directions of the x-axis and the y-axis can be obtained by

_{tx}

_{max}” and “σ

_{ty}

_{max}” pull in different directions on the immediate roof. At the same time, the tensile strength “σ

_{u}” and strength reduction coefficient “β” of the immediate roof can also be obtained through laboratory and field sonic tests. The maximum tensile stress of the immediate roof obtained by theoretical calculation and the maximum tensile stress of the rock obtained by the test are compared and analysed. According to the rock formation failure criterion established above, the stability of the immediate roof is judged. When the internal tensile stress of the immediate roof reaches the tensile strength of the rock mass, the tensile failure of the immediate roof will occur at the maximum tensile stress, and the internal cracks of the immediate roof will expand. The direction is perpendicular to the direction of tensile stress, and the corresponding advancing distance of the working face is the first or periodic failure span of the immediate roof rock stratum.

## 6. Discussion

^{3D}numerical simulation software was used to simulate the direct top stress distribution characteristics of the horizontal segmented top coal caving in the near-upright extra-thick coal seam. According to the simulation results, the normal stress distribution of the immediate roof is uniform. Thus, the deflection equations can be calculated under the uniform normal stress. The most dangerous position of the immediate roof and the first or periodic fracture span can be obtained by using the thin plate model and the new fracture criteria. It can better predict the weighting situation of the working face, and ensure the safe, efficient, and sustainable mining of coal mines.

## 7. Conclusions

- (1)
- Four mechanical models and the deflection equations for roof fractures in near-vertical and extremely thick coal seams have been established, which cover the possibility of stress on the roof when the roof is gradually damaged by a horizontally grouped top-coal drawing method in near-vertical and extremely thick coal seams.
- (2)
- The influence of the seam depth, the coal seam angle, the lateral pressure coefficient, and the maximum principal stress direction on the normal stress distribution of the immediate roof with HGTC were analysed by orthogonal experiment design and FLAC
^{3D}, and the normal stress could be simplified as uniform loading. The load could be set as a uniform load according to the numerical simulation results, when calculating the maximum tensile stress of the roof under the four states. - (3)
- A calculation method for roof stress distribution based on Hooke’s law and arc length theorem is proposed. Taking the maximum tensile stress strength criterion as the near-vertical immediate roof fracture criterion, the first fracture and periodic fractures span could be obtained, which can better predict the weighting situation of the working face, and ensure the safe, efficient, and sustainable mining of coal mines.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Tu, H.; Tu, S.; Yuan, Y.; Wang, F.; Bai, Q. Present situation of fully mechanized mining technology for steeply inclined coal seams in China. Arab. J. Geosci.
**2015**, 8, 4485–4494. [Google Scholar] [CrossRef] - Yang, S.; Zhao, B.; Li, L. Coal wall failure mechanism of longwall working face with false dip in steep coal seam. J. China Coal Soc.
**2019**, 44, 367–376. [Google Scholar] - Qin, D.; Wang, X.; Zhang, D.; Guan, W.; Zhang, L.; Xu, M. Occurrence characteristic and mining technology of ultra-thick coal seam in Xinjiang, China. Sustainability
**2019**, 11, 6470. [Google Scholar] [CrossRef] - Pan, Y.; Wang, Z.; Liu, A. Analytic solutions of deflection, bending moment and energy change of tight roof of advanced working surface during initial fracturing. Chin. J. Rock Mech. Eng.
**2012**, 31, 32–41. [Google Scholar] - Qian, M.; Miao, X.; He, F. Analysis of key block in the structure of voussoir beam in longwall mining. J. China Coal Soc.
**1994**, 19, 557–563. [Google Scholar] - Song, Y. Discussion on the law of old top pressure in steeply inclined horizontal section top coal mining. Coal Sci. Technol.
**1997**, 25, 35–38. [Google Scholar] - Shi, P.; Qi, T.; Zhang, J.; Wang, S.; Tian, Y. Scientific analysis on sublevel caving in thinner of the steep close and thick seam. Chin. J. Coal
**2004**, 29, 385–387. [Google Scholar] - Wang, N. Discussion on Reasonably Raising the Horizontal Section Height of Steeply Inclined Fully-Mechanized Caving Face. West. Prospect. Eng.
**2007**, 10, 149–153. [Google Scholar] - Cheng, W.; Xie, J.; Wang, G.; Xie, P. Comparison of strata movement between section and slicing of top coal mining in steep and thick seams. Chin. J. Coal
**2009**, 34, 478–481. [Google Scholar] - Lai, X.; Qi, T.; Jiang, D.; Cui, F.; Ma, J.; Shan, P. Comprehensive determination of dimension of segment pre-blasting of sub-level top coal caving in steep seams. Chin. J. Coal
**2011**, 36, 718–721. [Google Scholar] - Dai, H.; Guo, J.; Yi, S.; Wang, G.; Liu, A.; Kong, B.; Zou, B. The mechanism of strata and surface movements induced by extra-thick steeply inclined coal seam applied horizontal slice mining. Chin. J. Coal
**2013**, 39, 1109–1115. [Google Scholar] - Cui, F.; Lai, X.; Cao, J.; Shan, P. Mining disturbance of horizontal section full-mechanized caving face in steeply inclined coal seam. Chin. J. Min. Saf. Eng.
**2015**, 32, 610–616. [Google Scholar] - Shabanimashcool, M.; Li, C.C. Analytical approaches for studying the stability of laminated roof strata. Int. J. Rock Mech. Min. Sci.
**2015**, 79, 99–108. [Google Scholar] [CrossRef] - He, F.; Chen, D.; Xie, S. The kDL effect on the first fracture of main roof with elastic foundation boundary. Chin. J. Rock Mech. Eng.
**2017**, 36, 1384–1399. [Google Scholar] - Guo, W.B.; Wang, H.S.; Dong, G.W.; Li, L.; Huang, Y.G. A case study of effective support working resistance and roof support technology in thick seam fully-mechanized face mining with hard roof conditions. Sustainability
**2017**, 9, 935. [Google Scholar] [CrossRef] - Yang, W.H.; Lai, X.P.; Cao, J.T.; Xu, H.C.; Fang, X.W. Study on evolution characteristics of overburden caving and void during multi-horizontal sectional mining in steeply inclined coal seams. Therm. Sci.
**2020**, 24, 3915–3921. [Google Scholar] [CrossRef] - He, S.; Song, D.; Li, Z.; He, X.; Chen, J.; Zhong, T.; Lou, Q. Mechanism and Prevention of Rockburst in Steeply Inclined and Extremely Thick Coal Seams for Fully Mechanized Top-Coal Caving Mining and Under Gob Filling Conditions. Energies
**2020**, 13, 1362. [Google Scholar] [CrossRef] - Su, G.S.; Gan, W.; Zhai, S.B.; Zhao, G.F. Acoustic emission precursors of static and dynamic instability for coarse-grained hard rock. J. Cent. South Univ.
**2020**, 27, 2883–2898. [Google Scholar] [CrossRef] - Dong, L.; Zhang, Y.; Ma, J. Micro-crack mechanism in the fracture evolution of saturated granite and enlightenment to the precursors of instability. Sensors
**2020**, 20, 4595. [Google Scholar] [CrossRef] - Dong, L.; Chen, Y.; Sun, D.; Zhang, Y. Implications for rock instability precursors and principal stress direction from rock acoustic experiments. Int. J. Min. Sci. Technol.
**2021**, 31, 789–798. [Google Scholar] [CrossRef] - Chen, D.; Sun, C.; Wang, L. Collapse behavior and control of hard roofs in steeply inclined coal seams. Bull. Eng. Geol. Environ.
**2021**, 80, 1489–1505. [Google Scholar] [CrossRef] - Kong, D.; Xiong, Y.; Cheng, Z.; Wang, N.; Wu, G.; Liu, Y. Stability analysis of coal face based on coal face-support-roof system in steeply inclined coal seam. Geomech. Eng.
**2021**, 25, 233–243. [Google Scholar] - He, Z.L.; Lu, C.P.; Zhang, X.F.; Guo, Y.; Wang, C.; Zhang, H.; Wang, B.Q. Research on Mechanisms and Precursors of Slip and Fracture of Coal–Rock Parting–Coal Structure. Rock Mech. Rock Eng.
**2022**, 55, 1343–1359. [Google Scholar] [CrossRef] - Kirchhoff, G. Uber das Gleichewicht and die Bewegung einer elastischen Scheibe. J. Reine Angew. Math.
**1850**, 40, 51–88. [Google Scholar] - Xie, G.X.; Chang, J.C.; Yang, K. Investigations into stress shell characteristics of surrounding rock in fullymechanized top-coal caving face. Int. J. Rock Mech. Min. Sci.
**2009**, 46, 172–181. [Google Scholar] [CrossRef] - Shamshiri, R.; Ismail, W.W. Implementation of Galerkin’s method and modal analysis for unforced vibration response of a tractor suspension model. Res. J. Appl. Sci. Eng. Technol.
**2014**, 7, 49–55. [Google Scholar] [CrossRef] - Yang, G. An Introduction to Elastic-Plastic Mechanics; Tsinghua University Press: Beijing, China, 2004. [Google Scholar]
- Timoshenko, S.; Goodier, J.N. Theory of Elasticity; McGraw-Hill Book Company, Inc.: New York, NY, USA, 1951. [Google Scholar]
- Timoshenko, S.; Woinowsky-Krieger, S. Theory of Plates and Shells; McGraw-Hill Book Company, Inc.: New York, NY, USA, 1959; pp. 79–104. [Google Scholar]
- Huang, K.; Xia, Z. Theory of Plates and Shells; Tsinghua University Press: Beijing, China, 1987; pp. 1–5. [Google Scholar]
- Kharaghani, H. Arrays for orthogonal designs. J. Comb. Des.
**2000**, 8, 166–173. [Google Scholar] [CrossRef] - Wu, A.; Huang, M.; Han, B. Orthogonal design and numerical simulation of room and pillar configurations in fractured stopes. J. Cent. South Univ.
**2014**, 21, 3338–3344. [Google Scholar] [CrossRef] - Vieira, H., Jr.; Sanchez, S.; Kienitz, K.H.; Belderrain, M.C.N. Generating and improving orthogonal designs by using mixed integer programming. Eur. J. Oper. Res.
**2011**, 215, 629–638. [Google Scholar] [CrossRef] - Brady, B.H.G.; Brown, E.T. Rock Mechanics: For Underground Mining, 3rd ed.; Springer Science & Business Media: Beilin, Germany, 2006; pp. 142–161. [Google Scholar]
- Sun, X.; Fang, X.; Guan, L. Mechanics of Materials, 5th ed.; Higher Education Press: Beijing, China, 2009; pp. 216–217. [Google Scholar]
- Zhong, W. New Solution System for Elasticity; Dalian University of Technology Press: Dalian, China, 1995; pp. 7–9. [Google Scholar]
- Zhang, G.; Zhang, Y. Immediate roof first fracture characteristics of suberect and extremely thick coal. J. China Coal Soc.
**2018**, 43, 1220–1229. [Google Scholar]

**Figure 4.**The mechanical model of the roof under different mining conditions (h is the subsection height, a is mining length, and b is the width of immediate roof). (

**a**) model A; (

**b**) model B; (

**c**) model C; (

**d**) model D.

**Figure 5.**The deformation of the elastic thin plate, (

**a**) without deformation; (

**b**) small deformation.

**Figure 8.**The normal stress distribution of the immediate roof in the direction of dip (D is the seam depth, A is the coal seam angle, C is the lateral pressure coefficient, and M is the maximum principal stress direction).

Roof and Bottom | Lithology | Thickness |
---|---|---|

Main roof | Conglomerate | 25 m |

Immediate roof | Sandstone | 2.0 m |

Floor | Kaolin | 4.4 m |

Basic bottom | Pebbly sandstone, Conglomerate | 28 m |

NO. | Lithology | Thickness/m | Density Kg/m^{3} | Bulk Modulus/GPa | Shear Modulus/GPa | Friction Angle/Degree (°) | Cohesion/MPa | Tensile Strength/GPa |
---|---|---|---|---|---|---|---|---|

1 | Loose layer | 20 | 2100 | 7.0 | 3.5 | 25° | 5.5 | 1.6 |

2 | Sandy mudstone | Variable | 2600 | 8.1 | 6.0 | 36° | 18.8 | 3.5 |

3 | Mudstone | 48 | 2470 | 2.6 | 2.0 | 38° | 4.5 | 1.0 |

4 | Sandy mudstone | 24 | 2450 | 8.1 | 6.0 | 36° | 18.8 | 3.5 |

5 | Medium sandstone | 16 | 2430 | 10.9 | 6.9 | 31° | 39.5 | 5.1 |

6 | Sandstone | 8 | 2600 | 4.9 | 3.7 | 30° | 27.2 | 6.1 |

7 | Mudstone | 2 | 2430 | 2.6 | 2.0 | 38° | 4.5 | 1.0 |

8 | Coal seam | 28 | 1330 | 1.2 | 0.8 | 28° | 4.2 | 0.9 |

9 | Mudstone | 4 | 2400 | 2.6 | 2.0 | 38° | 4.5 | 1.0 |

10 | Sandstone | 12 | 2450 | 4.9 | 3.7 | 30° | 27.2 | 6.1 |

11 | Medium sandstone | 16 | 2650 | 10.9 | 6.9 | 31° | 39.5 | 5.1 |

12 | Sandy mudstone | 24 | 2500 | 8.1 | 6.0 | 36° | 18.8 | 3.5 |

13 | Coarse sandstone | 48 | 2500 | 12.5 | 9.4 | 35° | 35.6 | 3.5 |

**Table 3.**The simulations of nine representative combinations, based on the orthogonal array L9 (34).

NO | Seam Depth | Coal Seams Angle | Lateral Pressure Coefficient | Maximum Principal Stress Direction | Length | Width | Height | Blocks Number | Grid Points Number |
---|---|---|---|---|---|---|---|---|---|

1 | 100 m | 65° | 1 | 0° | 277 m | 200 m | 120 m | 553,750 | 586,921 |

2 | 100 m | 75° | 1.25 | 45° | 257 m | 200 m | 120 m | 516,250 | 550,685 |

3 | 100 m | 85° | 1.5 | 90° | 239 m | 200 m | 120 m | 515,000 | 549,155 |

4 | 300 m | 65° | 1.5 | 45° | 277 m | 200 m | 120 m | 553,750 | 586,921 |

5 | 300 m | 75° | 1 | 90° | 257 m | 200 m | 120 m | 516,250 | 550,685 |

6 | 300 m | 85° | 1.25 | 0° | 239 m | 200 m | 120 m | 515,000 | 549,155 |

7 | 500 m | 65° | 1.25 | 90° | 277 m | 200 m | 120 m | 553,750 | 586,921 |

8 | 500 m | 75° | 1.5 | 0° | 257 m | 200 m | 120 m | 516,250 | 550,685 |

9 | 500 m | 85° | 1 | 45° | 239 m | 200 m | 120 m | 515,000 | 549,155 |

In-Situ Stress Test Position | Depth/m | σ_{1}/MPa | σ_{2}/MPa | σ_{H}/MPa | σ_{1}/σ_{2} |
---|---|---|---|---|---|

The rock cross-cut at mining level + 1126 at Adaohai Mine | 367.5 | 18.03 | 10.9 | 9.25 | 1.95 |

The winch chamber at mining level + 1228 at Adaohai Mine | 206.9 | 12.26 | 6.84 | 6.27 | 1.96 |

The No. 2-1022 tail entry at Ganhe Mine | 461 | 16.18 | 8.38 | 11 | 1.93 |

The head entry of scope 2 at Ganhe Mine | 529 | 14.78 | 7.92 | 12.77 | 1.87 |

The No. 2151 head entry at Tuanbai Mine | 33 | 9.27 | 5.32 | 7.8 | 1.74 |

The No. 310 tail entry at Tuanbai Mine | 405 | 12.37 | 6.7 | 9.67 | 1.80 |

The No. 10-1021 head entry at Huipodi Mine | 367.9 | 9.32 | 4.99 | 8.77 | 1.87 |

The haulage roadway of east scope at Huipodi Mine | 355.1 | 10.32 | 5.57 | 8.33 | 1.85 |

The central of 1051 lane yard at Pangpangta Mine | 490 | 9.63 | 5.1 | 12.26 | 1.89 |

The No. 1092 tail entry at Pangpangta Mine | 592 | 11.68 | 6.16 | 14.8 | 1.90 |

The main haulage roadway at Changping Mine | 348.1 | 10.81 | 5.6 | 8.7 | 1.93 |

The first contact alley at Changping Mine | 343.9 | 9.64 | 4.99 | 8.6 | 1.93 |

NO. | The Bottom of Models | The Upper of Models (Mpa) | The Negative of x-Axis | The Direction of x-Axis (Mpa) | The Negative of y-Axis (Mpa) | The Direction of y-Axis (Mpa) |
---|---|---|---|---|---|---|

1 | Fixed | 0.25 | Fixed | 0.32 | Fixed | 0.18 |

2 | Fixed | 0.25 | Fixed | 0.31 | Fixed | 0.31 |

3 | Fixed | 0.25 | Fixed | 0.27 | Fixed | 0.48 |

4 | Fixed | 5.25 | Fixed | 7.88 | Fixed | 7.88 |

5 | Fixed | 5.25 | Fixed | 6.78 | Fixed | 3.72 |

6 | Fixed | 5.25 | Fixed | 8.47 | Fixed | 4.65 |

7 | Fixed | 10.25 | Fixed | 9.09 | Fixed | 16.54 |

8 | Fixed | 10.25 | Fixed | 19.85 | Fixed | 10.90 |

9 | Fixed | 10.25 | Fixed | 10.25 | Fixed | 10.25 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhang, G.; Li, Q.; Xu, Z.; Zhang, Y.
Roof Fractures of Near-Vertical and Extremely Thick Coal Seams in Horizontally Grouped Top-Coal Drawing Method Based on the Theory of a Thin Plate. *Sustainability* **2022**, *14*, 10285.
https://doi.org/10.3390/su141610285

**AMA Style**

Zhang G, Li Q, Xu Z, Zhang Y.
Roof Fractures of Near-Vertical and Extremely Thick Coal Seams in Horizontally Grouped Top-Coal Drawing Method Based on the Theory of a Thin Plate. *Sustainability*. 2022; 14(16):10285.
https://doi.org/10.3390/su141610285

**Chicago/Turabian Style**

Zhang, Guojun, Quansheng Li, Zhuhe Xu, and Yong Zhang.
2022. "Roof Fractures of Near-Vertical and Extremely Thick Coal Seams in Horizontally Grouped Top-Coal Drawing Method Based on the Theory of a Thin Plate" *Sustainability* 14, no. 16: 10285.
https://doi.org/10.3390/su141610285