The primary breaking pattern of main roof above the first working face based on main bending moment

Scientific mining is based on breaking regularity of roof above underground working face in coal mine. In order to explore the primary breaking pattern of main roof above the first working face, on account of theory of thin elastic plate, development of the breaking in each region of main roof is analyzed and the breaking sequence of each region is explored in virtue of main bending moment taken as the breaking parameter. The results indicate that the first broken point of main roof is midpoint of the long side. The breaking, which occurs on the top surface of main roof, is caused by the second main bending moment. The fracture in long side region starts from midpoint of the long side and develops along the length direction of the working face. The fracture in middle region starts from the center of main roof and develops along the length direction of the working face. The fracture in short side region starts from midpoint of the short side and develops along advance direction of the working face. There always is an extreme value order of control moment in each region, M c > M z > M d , when a single parameter is within a reasonable range. Due to this, the breaking sequence is the long side region, the middle region and the short side region although they end up with the same breaking pattern O-X. M c , M z and M d depend on the advance distance of working face and increase linearly with transverse loading. Besides, the short side of main roof becomes stable with the increase of the length of working face. Revealing the primary breaking pattern of main roof above the first working face contributes to learning breaking behavior of main roof and providing theoretical support for design of the working face and roof management.


Introduction
Although exploitation of coal resources can bring valuable coal products for people, it may also raise some thorny issues in the mining operations. For example, strata movement not only pose a serious threat to the safety of mining production of the working face and adjacent stopes, but also can result in ground surface subsidence or even collapse. Revealing the laws of strata movement can contribute to addressing these problems and promote the development of mining technology. Because the movement of strata is closely related to the failure rule of the roof, it is crucial important to learn the failure pattern of the roof (Qian and Xu 2019;Zuo et al. 2019).
Several researchers have used the plate model to study the failure rule of the roof. For example, Zhao et al. (2018) have given analytic solution of their plate model, and presented the evolution of boundary condition of main roof on the base of calculating results of the bending moment. The model developed by Yu (2018) has shown that the midpoint of the long side in the thin plate with four simply clamped edges, and the midpoint of the clamped side in the thin plate with three simply clamped edges are more likely to break. Based on the stress and the bending moment in X direction and deflection of the roof, Xue et al. (2016) have investigated initial and periodic collapse behavior of the roof burdened with underground pressure. Yang et al. (2015)' results, which were based on the stress in Y direction of the thin plate have indicated that the centre and the midpoint of the long side in the roof with four simply clamped edges are prone to break when initial collapse happens. Wang and Gao (2015) have presented inner failure mechanism of the roof according to calculated values of the bending moment and the stress in both X and Y direction, and shear stress of variable length of the working face. Liu and Li (2014)' theoretical analysis and numerical simulation results have given stress distribution of the island coal face and the evolution of the breaking roof with time. Pu et al. (2011) have analyzed the bending moment of the thin plate with four simply clamped edges, and they have obtained the collapsing, developing and transferring pattern of the roof. Zhang et al. (2010) have presented stress distribution of initial and periodic collapse obtained by using ANSYS, and revealed failure mechanism of the roof with highly inclined up-dip working face. Similarly, taking advantage of Mathcad software and then getting the contour line of the stress in both X and Y direction of the roof, Li et al. (2008) have showed the initial and periodic collapse pattern. Wang et al. (2008) have built the elastic plate model based on the roof supported by pillars, and showed the failure pattern in different stages in the light of the bending moment value. Wang et al. (2005) have calculated stress in the thin plates with four simply clamped edges and three simply clamped edges. They have had a good explanation of segmental, stage and migration pressure, which happens in hard strata. On the base of analyzing stress distribution of the roof in Bai Laping, a coal mine, Tang and Ye (2003) have drawn the conclusion that the first broken point of main roof is midpoint of the long clamped side. Some studies (Zhao et al. 2018;Yu 2018;Xue et al. 2016;Yang et al. 2015;Wang and Gao 2015;Liu and Li 2014;Pu et al. 2011;Zhang et al. 2010;Li et al. 2008;Wang et al. 2008;Wang et al. 2005;Tang and Ye 2003) mentioned above were carried out under the assumption that normal stress reaching the critical values causes failure of the roof. It is not reasonable since the maximum value of normal stress is first principal stress, and sometimes stress in both X and Y direction are not the maximum value of normal stress. Similarly, the main moment is the true cause of failure of the roof when the bending moment is taken as the failure criteria. Besides, some previous works (Zhao et al. 2018;Xue et al. 2016;Wang and Gao 2015;Zhang et al. 2010;Wang et al. 2008;Wang et al. 2005) are limited to broken points of the roof instead of the whole development process of the roof failure. In consideration of theses limits, this study will be conducted. Firstly, the paper obtains distribution of the main bending moment and failure direction of broken points, and subsequently discusses the development of failure regions of main roof. Finally, parameter effect analysis is used to explore the breaking sequence of each region, and to get the primary breaking pattern of main roof above the first working face.

Development and analysis of the model
Beginning at open-off cut, the working face is advancing. As a result of it, stress applied on the immediate roof is increasing until stress reaches the maximum value, and then the immediate roof collapses. Similarly, after that, main roof also experiences the first collapse with the increasing exposure roof surface, and fitful collapse of the immediate roof. Several studies (Zhao et al. 2018;Yu 2018;Xue et al. 2016;Yang et al. 2015;Wang and Gao 2015;Liu and Li 2014;Pu et al. 2011;Zhang et al. 2010;Li et al. 2008;Wang et al. 2008;Wang et al. 2005;Tang and Ye 2003;He et al. 2017) show that mechanical behavior of the roof above underground working face can be studied by using the thin elastic plate theory.

Deflection surface equation
Before the first collapse, main roof can be simplified to a thin elastic plate, and its four sides are restricted by coal, the immediate roof, and overlying strata, which are regarded as fixed boundaries in the paper, and have no movement. As showed in Fig.1, a thin elastic plate model with four fixed boundaries is built, and its x, y and z axis coincide with length, width and height direction of the working face, respectively. S denotes advanced distance of the working face, and t is the length. It is noted that s < t. F1, F2, G1and G2 are midpoints of its corresponding side, and O is the central point of main roof. According to the thin elastic plate theory (Xu 2006;Timshenko and Goodier 1970;Timshenko and Woinowsky-krieger 1959), the fixed boundary condition is presented as follows: Where ω is deflection of the thin plate. The trial function of the above equation can be written as: Where C is a parameter. Since the equation of (3) satisfies displacement boundary conditions and inner force boundary conditions from (1) and (2), according to Galerkin method, there is: Where D is bending stiffness of the thin elastic plate. Consequently, the deflection surface equation of the thin elastic fixed plate can be written as:

Failure criteria
According to rock mechanical theory (Qian et al. 2010;Sun et al. 2009), when Mu is used as ultimate moment of main roof, the failure criteria is presented as follows； The equation of (6) shows that main roof collapses when main bending moment of a point in main roof reaches the maximum absolute value. According to elastic thin plate theory (Xu 2006), plate stress can be expressed as function of ω and bending moment Mx and My, and torque Mxy can be calculated as: In addition to main bending moment M1 and M2, they can be expressed as: Since bending moment and torque are internal force applied on unit width, their dimension and unit are LMT -2 and MN, respectively, instead of L 2 MT -2 and MN•m (Xu 2006).  Fig.2 (b), which is caused by the second principal bending moment. By comparing the results from Fig.2 (a) and Fig.2

The development of failure of each region
From the discussion made above, five extreme points (F1, F2, O, G1 and G2) and F1 (or F2, -43.43 MN), which is more likely to become failure, are obtained. In this section, failure patterns will be analyzed. To simplify the problem, main roof is divided into three parts, which are the short side region, the middle region and the long side region, as showed in Fig.3. Since main roof collapses when main bending moment of a point in main roof reaches the maximum absolute value (Mk, the controlled bending moment) according to Eq. (6), and the sign of the bending moment represents just bending direction, the focus of the paper is absolute value of main bending moment. Fig.4 (b) shows variation of Mk (the second principal moment)of some points. Their locations can be got through orient angle β (0, 30, 60 and 90°) and distance d (1, 2, 3, 4, 5 and 6 m) between the targeted point and F2, as presented in Fig.4 (a).  Fig.4 ( which means that normal direction of action surface of M1 is vertical to that of M2 and x direction , which are parallel to each other. Failure of the points on the long side is caused by the second principal moment. In this way, there is tension damage along x direction in these points of main roof. In fact, since Mxy of the points on the long side is 0 MN, and Mx and My are main bending moment according to the definition of main bending moment, normal direction of action surface is x direction and y direction. Mx is the second principal bending moment if the bigger one among Mx and My is My. Similarly, the failure situation along x direction of the points on the long side can also be obtained. Distribution of torque is shown in Fig.5. The obtained failure pattern above is that after failure of F2, points along length direction of the working face will successively break. Meanwhile, the same failure pattern will happen in the upper half of the main owing to the symmetry axis of x. Fig.6 (b) gives variation of Mk (the first principal moment) of some points. Their locations can be got through orient angle β (0, 30, 60 and 90°) and distance d (1, 2, 3, 4, 5 and 6 m) between the targeted point and reference point O, as presented in Fig.6 (a). long symmetry axis of main roof is null, and Mx is the first principal bending moment since the bigger one among Mx and My is Mx. The first principal bending moment causes failure of points on the long symmetry axis of main roof. So, tensile failure will happen in these points along x direction. The obtained result above is that after failure of O, points along length direction of the working face will successively break. Fig.7 (b) presents variation of Mk (the second principal moment) of some points. Their locations can be got through orient angle β (0, 30, 60 and 90°) and distance d (1, 2, 3, 4, 5 and 6 m) between the targeted point and reference point G1, as presented in Fig.7 (a). G1 (or G2) is more likely to break in the short side region.

Failure of the short side region
advanced direction of the working face are more likely to break while points along length direction keeps stable. Owing to the symmetry distribution of Mk, main bending moment in the left side of main roof has the same pattern. So, after failure of G1 (or G2), points on the short side of main roof will break. From Fig 5, it can be found that Mxy of points on the long symmetry axis of main roof is null, and My is the second principal bending moment since the bigger one among Mx and My is Mx. The second principal bending moment causes failure of points on the short symmetry axis of main roof. So, tensile failure will happen in these points along y direction. The obtained result above is that after failure of G1, points along advanced direction of the working face will successively break. Meanwhile, the same failure pattern will happen in the other side of the main owing to the symmetry axis.

Analysis of breaking sequence of each region
After getting the failure evolution of each region, breaking sequence will be discussed below. Since the points in which Mk reaches extremum is the breaking points, and their breaking sequence represents corresponding region', extremum of Mk of these points will be analyzed. Substituting Eq. (5) into Eq. (12) and (13)

The effect of advanced distance of the working face
Before the first collapse, advanced distance of the working face is equal to width of main roof. Affected by geological conditions and mining technology, main roof initially collapses when the advanced distance of the working face reaches 30 to 60 m. The effect of advanced distance of the working face on extremum of Mk of each region is presented in Fig.8.

The effect of length of the working face
Length of the working face is equal to length of main roof. Due to satisfactory performances in both economic and efficient terms, the longwall face has achieved breakthroughs in length for recent years. For example, Length of the working face in some mines with good geological conditions even reached 350 m, which is far greater than length that previous ones can have, such as 80 m. The evolution of Mk of each region with length of the working face is presented in Fig.9.

The effect of lateral load
Before the first collapse of main roof, overlying strata is in a higher position owing to support from main roof. When the working face is advancing, load from overlying strata is small given that main roof bends, and abscission layer happens. Fig.10 gives the effect of lateral load (q) on Mc, Md and Mz.

The effect of Poisson's ratio
Poisson's ratio has positive correlation with material strength. Poisson's ratio of main roof, overlying strata and coal successively increases, respectively. Fig.11 shows the effect of Poisson's ratio (ν) on Mc, Md and Mz.

The breaking pattern of main roof
The working face is dynamic, and progressively approaches the main transportation road. The primary breaking pattern of main roof above the first working face shown in Fig.12 is obtained according to the development of failure of each region and breaking sequence of each region. Fig.12 The primary breaking pattern of main roof above the first working face Fig.12 shows that after failure of F1 (or F2), points along length direction of the working face will successively break. With the advancing working face, Mk of the point of O reaches ultimate moment, and then it collapses with the points along the long symmetry axis. After the similar failure happens in the point of G1 (or G2), there is formation of breaking pattern O-X (Zuo et al. 2019;Pu et al. 2011;Qian et al. 2010). Due to relative difference of Mk of each region of main roof, there is no probability that the long side firstly collapse, then the long symmetry axis, and finally the long symmetry axis.

Conclusions
(1) The first broken point of main roof is midpoint of the long side. The breaking, which occurs on the top surface of main roof, is caused by the second main bending moment.
(2) The failure in long side region starts from midpoint of the long side and develops along the length direction of the working face. The failure in middle region starts from the center of main roof and develops along the length direction of the working face. The failure in short side region starts from midpoint of the short side and develops along advance direction of the working face. The long side region firstly collapse, then the middle region, and finally the short side region. Together, there is formation of breaking pattern O-X.