Simulation Study on Spatial Form of the Suspended Roof Structure of Working Face in Shallow Coal Seam

Abstract

Longwall fully comprehensive mechanized mining is mainly used for the working faces of shallow coal seam with large mining height, which usually have a large suspended roof at the face end. The overhang at the face end leads to stress concentration, which affects the safe mining of the working face. In this paper, we use the 15210 working face with a suspended roof (overhanging area 50~70 m²) of the Zhangjiamao coal mine as study background, with physical simulation, numerical calculation and theoretical analysis, the spatial morphologies and changes in the roof structure at the face ends of the working face in shallow coal seam are obtained, in which the suspended roof increase from the bottom to top, forming step-laminated structures. The caving interval of the suspended roof at the face end is about two times the period weighting interval, and the suspended roof area at the tailgate is smaller than at the headgate. The distribution of the shear and the principal stress field at the face-end region is arc-shaped, and the distribution of the plastic zone shows that the collapse of a suspended ceiling has obvious hysteresis. According to the simplified analysis of the Marcus plate, when the layers of the stepped curved triangular plates increase, the length of the suspended roof on the solid coal side also increases, which is consistent with the results of the physical simulation and numerical calculations. The formation mechanism of the roof at the end of the working face provides a research foundation for the control of roofs found at face ends and further improves the theory of roof structure and the safety mining of suspended roof areas.

1. Introduction

Shallow coal seams are widely distributed in the north of the Shaanxi Province. The Shenfu-Dongsheng coalfield located at the edge of the Mu Us Desert is the largest proved coalfield in China. A shallow coal seam has many mineable coal seams, thick coal seams, simple structure, excellent coal quality, and good mining conditions [1,2]. The practice of large-height mining in shallow coal seams shows that the roadway of a working face is supported by bolts, cable mesh, and steel beams, and the roof control of the roadway is stable [3,4,5,6,7], so it is difficult for it to collapse at both ends of the working face after the working face is pushed over, resulting in an overhang at the end of the working face [8,9,10].

With the development of fully mechanized mining technology for large-height mining, the research on the ground control and roof structure theory of working faces with large mining height is deepening. Roof structure models for working faces with large mining height mainly include the stepped rock beam model [11,12,13], the main roof cantilever beam model [14], and the short cantilever beam-articulated rock beam model [15,16]. The immediate roof of a large mining height is divided into three categories [17], and the concept of an equivalent immediate roof [18] of a large mining height’s ground control is expounded. The above research enriches the theory of roof structures with large mining height in shallow coal seams.

The related research on the face-end hanging roof structure of shallow coal seam mainly includes research conducted using three-dimensional physical simulation, in which the roof slab periodically broke to form an arc shape, and both face ends of the strip-shaped rock slab were arc-shaped [19]. The shape and position of the hanging roof at the face end were studied in [20], an arc triangular hanging structure was observed [21,22]. The suspended roof fracture of the elastic foundation boundary in [23] and the elastic thin plate in [24] were purposed, and the influencing factors and weight relationship of the main roof fracture were calculated. The stability of suspended roof structures under different conditions has been studied: One study found that the cantilever beam structure instability of a roof under repeated mining is an important reason for accidents involving face-end roofs and supports [25]. In addition, [26] showed that the main roof forms a masonry beam structure after the roof collapses, and [27] found that hanging roofs are characterized by an increased confining pressure and reduced free surface, and the authors developed a new method based on shock wave collision and stress superposition to demolish hanging roofs. Using a similar material simulation experiment, the failure characteristics of the overlying strata under the influence of mining were analyzed to reveal the mechanical mechanism of the directional weakening of the end roof in [28], and, according to the numerical modeling of the stress–strain state, the nature of the distribution of vertical displacements and stresses in the vicinity of the face with a consistent increase in the length of the main hanging roof was established in [29,30]. Other studies focused on the suspension problem of hard roofs in deep mining, and the stress evolution of the rock, the best breaking position, and the shape of the main roof were obtained [31,32].

In mining practice, large overhangs occur at the end of working faces with large mining height in shallow coal seam. The hazards of an overhanging roof mainly involve the following aspects: ① The overhanging roof covering a large area will cause stress concentration, leading to an obvious strong pressure, and ② the suspended roof is likely to cause gas accumulation at the face end. When the suspended roof suddenly collapses, it may cause strong dynamic loads and gas to escape from the goaf. The increase in the size of the suspended roof will lead to an increase in the working resistance and an increase in the cost of the support, the statistical results of which are shown in

Table 1. Statistics of hanging roofs at the face end of working face in Shenfu mining area.

Coal Mine	Working Face	Depth/m	Mining Height/m	Suspended Size/m Length × Width	Lithology of Roof (Main/Immediate Roof)
Zhangjiamao	22210	90	6.0	4 × 3	Quartz sandstone/Quartz sandstone
Zhangjiamao	15210	210	5.8	18 × 10	Fine-grained sandstone/siltstone
Hongliulin	25210	175	5.8	10 × 6	Fine-grained sandstone/siltstone
Bulainta	22410	220	6.6	8 × 3	Medium-grain sandstone/siltstone
Halagou	22519-2	130	5.3	4 × 3	Medium-grain sandstone/siltstone
Daliuta	12201-1	80	3.6	10 × 3	Fine-grained sandstone/siltstone
Shigetai	31306	110	3.3	10 × 4	Medium grain sandstone/siltstone
Shangwan	12306	290	4.2	14 × 7	Coarse-grained sandstone/siltstone
Yujialiang	52209	150	3.9	10 × 4	Medium grain sandstone/siltstone

.

The currently used suspended roof control technologies mainly include hydraulic fracturing, withdrawal of anchor cables or bolts, carbon dioxide blasting, etc. Engineering practice shows that when the size of the suspended roof is large, the effect of hydraulic fracturing is better. A reasonable hydraulic fracturing position indicates that a shorter roof equates to a thicker roof, a larger elastic modulus, a smaller elastic buffer coefficient, and a fracturing position that is deeper in the coal wall [33,34,35]. Hydraulic fracturing on the face end can generate the main fractures and cracks inside the roof and weaken the roof structure, thus forming a weak fracture plane to roof caving [36,37].

However, in view of the unclear mechanism of the roof overhangs in working face with large mining height, field treatment is not effective. In this paper, we use a three-dimensional physical experiment device and a detachable equivalent load method, through which the spatial form of the suspended roof was simulated, and using three-dimensional numerical calculations, the stress field and the development law of the plastic zone in the suspended roof area were analyzed. The relationship between the structure at the face end and middle stope were also investigated in order to provide a theoretical basis for reasonable control technology, this research is very necessary for longwall mining.

2. Physical Simulation Experiment and Results

It is difficult to intuitively obtain a suspended roof’s shape during the actual measurement process; thus, we used a small three-dimensional physical simulation experimental device to analyze the law of roof caves, spatial shape, and the size of the suspended roof of the 15210 working face. The 15210 working face buried depth is 205 m, the mining height is 5.6 m, the length of the setup entry is 297 m, the mining distance is 1857 m, and the support model is ZY15000/29.5/63D. According to the comprehensive histogram of the 8-1 borehole strata, the main roof was mainly composed of fine-grained sandstone (FGS) and siltstone. The lithology of the 8-1 borehole is shown in

Table 2. Physical and mechanical parameters of No. 8-1 borehole.

Rock Stratum	Thickness h/m	Density ρ kg/m3	Elastic Modulus E/GPa	Poisson’s Ratio μ	Cohesion C/MPa	Internal Friction Angle φ/°	Tensile Strength Rc/MPa
Red soil	72	1580	0.06	0.25	0.09	30.0	0.12
Mudstone	3.0	2050	1.021	0.22	3.66	38.69	0.44
FGS	3.8	2450	0.803	0.21	2.54	38.13	1.87
Mudstone	12.6	2150	1.021	0.22	3.66	38.69	0.65
Siltstone	3.38	2470	0.826	0.19	3.13	39.29	1.75
FGS	28.97	2450	1.140	0.21	2.11	38.78	1.78
4−2 coal seam	3.95	1390	0.435	0.21	1.16	36.98	0.74
Mudstone	3.2	2150	1.021	0.22	3.66	38.69	0.65
FGS	17	2430	1.112	0.17	2.64	40.40	1.24
Mudstone	8.9	2490	1.021	0.22	3.66	38.69	0.65
Siltstone	9.7	2320	0.824	0.23	2.42	38.60	1.24
Mudstone	10.16	2150	1.021	0.22	3.66	38.69	0.65
FGS	7.44	2590	1.829	0.13	4.68	37.59	2.61
Siltstone	15.45	2440	0.880	0.18	2.10	39.37	1.65
5−2 coal seam	5.9	1390	0.266	0.14	1.45	37.60	0.82
Siltstone	10	2430	0.971	0.20	4.43	39.60	1.22

, which provides the basis for our physical simulation and numerical calculations.

In the physical simulation experiment, a rectangular parallelepiped model frame was designed based on the 15,210 working face, as shown in Figure 1. The size of the model is 100 cm (length) × 80 cm (width) × 50 cm (height); four sides of the model are made of 10 mm thick acrylic glass plates, and angle irons are used to fix the acrylic glass plates around. There is no cover plate on the top of the roof, and acrylic glass plates are also used in the floor structure. This setup allowed us to conveniently record and take photos during the mining experiment.

When the working face advanced to 60 m, the main roof incurred the first weighting, and the height of the caved roof became 26 m. It was observed that there were arc-shaped cracks in the roof, and the middle of the roof of the working face collapsed (Figure 2). Limited by the size of the model, this experiment was mainly simulated using the half-width of the working face to simulate the phenomenon of overhang of the face end.

When the working face advanced to 66 m, the roof completely collapsed behind the goaf, and the fracture was in the form of “O-X”, as shown in Figure 3a. The long side of the working face roof first broke, the short side lagged, and an arc transition zone emerged at the face end, leading to the “arc triangle” shape of the roof, as shown in Figure 3b. The arc-shaped roof size at the low level of the overhang area was small until the 26 m layer (the highest level of the model), at which point it reached the maximum overhang size, which is basically consistent with the on-site measurement and numerical calculation results.

When the working face advanced to 78 m, the roof’s periodic weighting interval was 12 m, and the cracks developed in advance to form new “O-rings”, with an interval of 24 m (two times the periodic weighting interval). The hanging end roof in the goaf of the working face caved in, and a new suspended roof was formed at the face end. The length of the suspended roof was about 24 m, and the inclined length was 22 m, as shown in Figure 4.

When the working face advanced to 96 m, the roof’s periodic weighting interval was 18 m. After the roof collapsed, the new suspended roof size of the working face end was 22 m × 16 m. The distance of the suspended roof was about 2 times the periodic weighting interval (18~24 m). When the working face advanced to 114 m, the roof periodically collapsed again, and the periodic weighting interval was 18 m; the suspended roof at the face end was unbroken, and the advanced cracks formed a new “O-ring” with a breaking distance of about 16 m. The advanced cracks on the working face were located 6 m in front of the coal wall, as shown in Figure 5.

During the mining process, the roof periodically collapsed, the average periodic weighting interval was 12 m, and the front fractures of the working face were obvious. The roof at the face end broke along the boundary, forming a stepped, arc-shaped stable fracture pattern around the goaf. To some extent, the existence of arc-shaped suspended roofs at face-end areas is a natural phenomenon. Even if the overload of the working face causes a full collapse, the suspended roof structure remains stable. In engineering practice, a reasonable control layer and the size of the suspended roof should be determined according to the breaking angle of the roof and the field-measured size of the suspended roof.

3. Numerical Simulation Calculations and Results

3.1. Establishment of the Model

Both the physical experiment and the numerical calculations were simulated by using the actual overlying rock and mining conditions. The physical experiment involved the spatial shape and breaking angle of the roof, while the numerical calculations allowed us to study the stress field and the plastic zone of the roof. Based on the physical simulation of the suspended roof, we used FLAC3D numerical simulation to reveal the formation and evolution mechanism of the suspended roof and provide a scientific basis for its control.

The size of the numerical calculation model was 518 m (length) × 300 m (width) × 144 m (height), and the thickness of the red soil layer was 72 m, which was equivalent to the load layer applied to the upper boundary of the model. The Mohr–Coulomb model was adopted, and the top of the overload was set as the free boundary, while the other sides were set with a normal horizontal displacement constraint on the surface and a normal vertical displacement constraint on the floor. The initial stress gradually reduced from the bottom to the top according to the lateral pressure coefficient on the front, back, left, and right sides. The coal pillar area was divided into finer sections, and the division unit was 2 m × 2 m × 0.5 m. The division unit for the direct roof and the main roof was 4 m × 2 m × 1 m. The numerical model first simulated the excavation of the 4⁻² coal seam’s 14208 working face and then simulated the excavation of the 15209 working face, followed by the excavation of the 15210 working face. The result after the excavation is shown in Figure 6. The inclined width of the simulated 15210 working face was 200 m, the mining height was 5.6 m, and the strike length was 200 m. The right side was the tailgate, and the left side was the headgate and auxiliary headgate. Border coal pillars were left on all sides.

Considering the boundary effect and the influence of the initial mining stage, we set the working face to advance to 96 m, 108 m, 120 m, and 132 m, in order to analyze the law of the shear stress (τ) and principal stress (σ₁) distribution at 4 m, 8 m, 12 m, 16 m, and 20 m layers above the roof. The calculation results for the direction and inclination of the overhang length are shown in Figure 7, Figure 8 and Figure 9. In these figures, the black line represents the coal wall of the 15210 working face, the left side is the headgate, and the right side is the tailgate. Due to the limited scope of this study, we did not repeat the simulation of the 15209 working face but only simulated the mining stress environment in the study area.

3.2. Distribution Law of Suspended Roof Stress Field

When the 15,210 working face advanced 96 m, the distribution law of the shear stress in different layers of the roof was analyzed, and its results are shown in Figure 7a–e.

①

In the mining process, the shear stress contours were distributed in an arc shape at the face end, which is consistent with the measured suspended roof shape;

②

The shear stress was larger on the side of the tailgate and smaller on the side of the headgate, and the largest shear stress occurred at the 8~12 m layer;

③

The shear stress kept increasing in the 4~8 m layer, and the 8~20 m layer kept decreasing, and the largest shear stress occurred at the 8 m layer;

④

As the roof layer increased, the shear stress contours moved to the back of the goaf. The 4~12 m layer was located 0~5 m in front of the coal wall, while the 12~20 m layer was located 0~5 m behind the coal wall in the 15210 working face.

The distribution law of the maximum principal stress in different layers of the roof was analyzed, and its results are shown in Figure 7f–j.

①

During the mining process, the maximum principal stress contour was arc-shaped at the face end, gradually decreasing from the coal wall to the goaf;

②

With the increase in the roof layer, the maximum principal stress decreased, and the maximum principal stress contour extended to the back of the goaf;

③

The maximum principal stress was more obvious on the headgate.

The stress field of the hanging roof when the working face advanced to 108 m~132 m was simulated, and the simulation results are shown in Figure 8.

①

When the working face advanced to 108 m, the shear stress distributions of different roof layers (4 m, 8 m, 12 m, 16 m, and 20 m) were plotted, which are shown in Figure 8a–e, and the basic laws were roughly the same. The arc-shaped transition area of the shear stress distribution at the 12 m layer had a small interval, which indicates that the hanging roof area collapsed at this time;

②

The maximum principal stress distributions of different layers of the roof (4 m, 8 m, 12 m, 16 m, and 20 m) are shown in Figure 8f–j, and the basic law of their distribution was roughly the same;

③

When the working face advanced to 120 m and 132 m, the law of the shear stress and maximum principal stress distribution at different layers (4 m, 8 m, 12 m, 16 m, and 20 m) of the roof were the same. Due to limitations in the scope of this paper, we will not list them one by one.

3.3. Evolution Law of the Shear Stress of Arc-Shaped Triangle Suspended Roof

A broken roof is caused by tensile–shear and compression–shear failures. For this reason, the evolution law of the shear stress distribution at the 8 m layer of the roof with difference advancements is shown in Figure 9. Figure 9a,b show that when the working face advanced to 96~108 m, the shear stress of the 8 m layer roof significantly reduced, indicating that the roof was damaged or broken. From Figure 9b–d, it can be seen that when the working face advanced to 108~132 m, there was no obvious change in the shear stress of the 8m layer roof at the headgate, indicating that there was a hanging roof at this stage. However, the shear stress at the end of the tailgate was small, indicating that there was no hanging roof in the tailgate.

As the advancing distance increased, the changes in the shear stress distribution of the 12 m layer roof were analyzed, as shown in Figure 10. Figure 10a,b show that when the working face advanced to 96~108 m, the shear stress of the 12 m layer roof in the headgate changed from small to large, but it had a certain hysteresis, compared with the 8 m layer. It can be seen from Figure 10b–d that when the working face advanced to 108~132 m, the shear stress at the 12 m layer roof of the headgate showed a decreasing trend, and the shear stress distribution was continuously in an arc shape, indicating that there was a hanging roof in this stage, but there was no obvious shear stress at the arc contour of the tailgate, indicating that there was no overhang phenomenon at the tailgate in this stage.

When the working face advanced to 96 m, 108 m, 120 m, and 132 m, the size changes of the suspended roof along the headgate, including the length of the suspended roof, the strike direction, and the inclined direction length of the suspended roof were analyzed, which are shown in Figure 11.

Figure 11 shows that ① when the working face advanced for 24 m, the hanging roof collapsed once, and ② when the hanging roof collapsed, the strike direction size of the 8 m layer was reduced by about 42~47%, and the inclined direction size was reduced by about 25%. The strike direction size of the 12 m layer roof was significantly reduced, by about 45~56%, while the inclined direction size was reduced by about 25~33%. However, the 16 m horizon of the strike direction size was reduced by about 30%, and the inclined direction size was reduced by about 20%. In addition, according to field measurement and engineering analogy statistics, the overhang area at the face end of the longwall face in the Shendong and Shennan mining areas is about 100 m², and the horizon of the hanging roof is 8~12 m. In actual practice, it is necessary to pay more attention to the rock formation of this layer, as the 8~12 m roof layers are the main control layers for hydraulic fracturing.

3.4. Distribution of the Plastic Zone at the Face End of Roof

According to the plastic zone of 4 m, 8 m, 12 m, 16 m and 20 m in the roof, the damage range and the size of the suspended roof of each layer can be further determined. When the working face advanced to 132 m, the plastic zone of the 4~20 m layer of the coal seam was analyzed, as shown in Figure 12. The analysis concluded as follows:

①

The roof of the 4 m layer underwent a tensile–shear failure. The roof on the side of the tailgate was more damaged than the auxiliary headgate, and there was an obvious shear failure on the front of the roof. At the 4 m layer, the suspended roof on the side of the tailgate was not obvious, and the size of the suspended roof was about 4 × 4 m; the size of the suspended roof on the headgate was about 8 × 8 m;

②

The roof at the 8 m layer incurred mainly a tensile–shear failure, and the roof in front of the coal wall had less shear failure. The hanging roof on the side of the tailgate was smaller, and the size of the tailgate was about 4 × 2 m. The stable roof was in the shape of an arc-shaped triangle behind the coal wall on the side of the headgate, and it revealed an obvious suspension phenomenon, the size of which was about 10 × 16 m;

③

The plastic zone at the 4~16 m layer developed upwards in the shape of a stepped arc. However, the size of the suspended roof at the 16~20 m layer was unchanged.

3.4.1. The Space Form of the Roof Overhang at Different Levels at the End of the Roof

According to the distribution of the plastic zone in each layer of the roof when the working face advanced to 132 m, the space morphology of the roof fracture was drawn, as shown in Figure 13. The 4~20 m layers are represented by black, red, green, pink, and light blue lines.

①

The roof damage developed toward the goaf and the inner side of the coal pillar according to a certain breaking angle, and the damaged area gradually decreased with the increase in the roof layer (Figure 13a);

②

On the side of the headgate, with the increase in the roof level, the damaged area decreased (Figure 13b), which indicates that the suspended roof area of the low-level roof was small, and the suspended roof area of the high-level roof was large;

③

On the side of the tailgate, affected by the concentrated stress of the coal pillars in the upper coal seam section and the lateral supporting pressure of the adjacent working face of the coal seam, the size of the suspended roof on the side of the tailgate was relatively small;

④

Along the slope of the working face, the upper and lower ends of the roof were broken to form a stepped, arc-shaped, three-hinged arch space structure, and the suspended roof periodically collapsed as the working face advanced.

3.4.2. Shape of the Plastic Zone along the Strike of the Overhanging Roof

When the working face advanced to 132 m, the plastic zone distribution of the roof along the upper part (the side of the headgate), the middle, and the lower part (the side of the tailgate) on the working face was simulated, which is shown in Figure 14.

The analysis concluded as follows:

①

The plastic zone of the roof on the side of the headgate developed into an arc-shaped arch, as shown in Figure 14a, which confirms the preceding analysis;

②

The roof fracture angle in the middle of the working face was larger, and the development height of the plastic zone was larger than that of the two ends, as shown in Figure 14b;

③

Affected by the concentrated stress of the coal pillars in the section, the plastic zone of the roof on the side of the tailgate was fully developed, indicating that the roof collapse on the side of the tailgate was relatively timely, as shown in Figure 14c.

When the working face advanced to 144 m (when the suspended roof had not collapsed) and 168 m (when the suspended roof collapsed), the positions of 3 m, 5 m, 10 m, 15 m, and 20 m behind the coal wall of the working face were selected for study, and the distribution of the roof plasticity along the slope of the working face area was simulated, as shown in Figure 15 and Figure 16. The analysis revealed the following observations:

①

When the working face advanced to 144 m, there were overhangs at both ends. The distribution of the roof plastic zone within 15 m from the goaf behind the coal wall was almost unchanged. The suspended roof area on the side of the transport chute was large, and the side of the return air chute was smaller;

②

When the working face advanced to 168 m, the heads at both ends were suspended and collapsed. At this time, the roof plastic zone gradually developed backward and upward within 20m behind the coal wall. This shows that after the overhang at both ends collapsed, the overlying rock on the upper part was further damaged;

③

Since the control roof distance of the support was 5.23~6.09 m, 3~5 m behind the coal wall was within the support range of the hydraulic support, and therefore the roof naturally did not break. Normally, there are overhangs 3 m and 5 m behind the coal wall in the inclined section.

The development height of the roof plastic zone (breaking position) at different positions in the goaf behind the coal wall is shown in Figure 17. The different positions from the coal wall are shown in red (1 m), orange (3 m), yellow (5 m), green (10 m), cyan (15 m), and blue (20 m). The following results can be intuitively drawn:

①

Before and after the roof of the end roof collapses, the size of the end hanging roof of the tailgate was small, and the side hanging roof always existed along the headgate;

②

The higher the roof level, the larger the arc-shaped suspended roof area, and at a certain time, the suspended roof of the goaf is basically broken as a whole;

③

The formation of the curved triangular suspended roof structure was affected by many factors, mainly including the direct roof and main roof thickness and strength, coal seam thickness and strength, coal pillar and roadway stability, etc.

Because the normal roof control distance of the 15,210 working face support is 5.23~6.09 m, the suspended roof in 1~5 m behind the coal wall was normal. The size of the suspended roof in the range of 10~20 m behind the coal wall decreased with the increase in the distance from the coal wall, and the size of the suspended roof after collapse decreased by about 45% in the strike direction.

4. Establishment of the Mechanical Structural Model at the Face End of the Roof

Through the numerical calculation of shear stress contour and physical similarity simulation analysis, the results showed that the real shape of the suspended roof was a combination of step-shaped arc plates with a certain thickness. Different rocks have different limits in terms of breaking positions, and their breaking angles are different. Therefore, the step-shaped arc triangle plate is more in line with reality.

After forming the arc triangular suspended roof, the area of the suspended roof increased with the continuous advancement of the working face. When the limit span was reached, the suspended roof suddenly collapsed, causing a strong dynamic load. After the formation of the end suspended roof, its shape was an arc plate structure, but the following conditions were observed: ① The plate structure had a certain thickness, which was not much smaller than the minimum edge length of the arc plate and, therefore, could not meet the theoretical boundary conditions of the elastic mechanics of the thin plate. ② It was difficult to solve the elastic mechanics of the arc thin plate, and an accurate solution for the elastic mechanics could not be obtained.

Therefore, according to the simplified solution of material mechanics, the arc triangular plate can be simplified into a triangular plate structure, and according to the simplified calculation method of the Marcus plate, the triangular cantilever plate can be regarded as a plate composed of several element strip beams. It is assumed that the length of rock beam AB is L, and it is subjected to uniform load q. Both ends of the rock beam were supported by a coal body as fixed beams. The mechanical model of the rock beam is shown in Figure 18, where the uniformly distributed load q can be decomposed into qsinα along the bedding plane and qcosα perpendicular to the bedding plane.

The simplified calculation of the Marcus slab considers the inclined rock beam [19,20,21]. The bending moment M_x at any point x in the inclined rock beam AB is

$$M_x=\frac{qcos\alpha }{12}(6x^2—6Lx+L^2)$$

(1)

where x is the distance from the section to end A, m; M_x is the bending moment, KN·m.

Under the action of the bending distance M_x, the bending stress σ₁ at any point on the inclined rock beam AB is

$${\sigma }_1=\frac{M_{x}y}{J}$$

(2)

where σ₁ the bending stress, MPa; y is a positive value above the neutral axis of the main roof; J is the moment of inertia, J = h³/12 m³; h is the thickness of the main roof, m.

In shallow coal seam mining, the inclination angle α of the coal seam and rock strata is α = 0, and only the vertical layer concentration load qcosα exists. The tensile strength is much lower than the compressive strength, so the rock beam AB is first broken at the lower surface of the neutral axis, and the stress σ₁ is

$${\sigma }_1=\frac{q\cos \alpha \left(6Lx-6x^2-L^2\right)}{2h^2}\cdot \left(0\le x\le L\right)$$

(3)

However, if the force of the concentrated load qsinα on the parallel plane does not exist, the stress σ₂ = 0. Generally, when the rock beam reaches the limited span L_max, its breaking point is the same as the maximum sinking point. Therefore, when the rock beam breaks, the maximum bending stress σ_max is

$${\sigma }_{\max }=\frac{q\cos \alpha }{2h^2}\left[6L_{\max }\left(\frac{1}{2}{L}_{\max }+\frac{h}{6}\tan \alpha \right)-6{\left(\frac{L_{\max }}{2}+\frac{h}{6}\tan \alpha \right)}^2-L_{\max }\right]$$

(4)

When the main roof’s rock tensile strength R_t = σ_max and the inclination angle α = 0, substituting into Equation (4), we can obtain

$$R_{\mathrm{t}}=\frac{q}{2h^2}\left[6L_{\max }\left(\frac{3}{2}{L}^2{}_{\max }-L_{\max }\right)\right]$$

(5)

In the process of mining shallow coal seams, the limit span of rock beams can be obtained through calculation. According to the results of the physical simulation and numerical calculation analysis, the fracture position of the rock beam above the roadway in the suspended roof area and the fracture position b of the solid coal roof are simplified to a ≈ 1.5b, which is derived from the Pythagorean theorem:

$$b=\frac{\sqrt{5}}{5}{L}_{\max }=\frac{\sqrt{5}}{5}\left(\sqrt{\frac{4R_{\mathrm{t}}{h}^2}{3q}+\frac{1}{9}}+\frac{1}{3}\right)$$

(6)

According to a comprehensive analysis, the suspended roof of the Zhangjiamao coal mine is mainly concentrated in the 8~18 m layer (1~3 times the mining height). Based on the above calculation, the strike direction size a of the 15,210 working face with the suspended roof can be obtained in the range of 15~30 m, and the range of the inclined direction size b was 8~18 m. When a = 1.5b, the suspended roof area was 45~270 m², which is basically consistent with the numerical calculation and physical simulation results.

According to the analysis of Equation (6), when the tensile strength of the suspended roof is constant, the smaller the suspended roof size, the greater the ability of the suspended roof to limit breakage. Conversely, the larger the suspended roof, the greater ability it has to break and, therefore, a lower limitation capacity, as shown in Figure 19.

It is concluded that the higher the level of the hanging roof, the smaller the uniform load q on the hanging roof, area, and the analysis results of the two were consistent. At the same time, according to the distribution curve of the ultimate load-bearing capacity of the hanging roof, it can be seen that the damage to the roof still developed in an arc-shaped arch, consistent with the numerical calculation results. This shows that it is feasible to use the Marcus plate to simplify a triangular plate structure to a curved triangular plate. Taking into account the mining practice in the Zhangjiamao coal mine, the tensile strength of the roof of 15210 working face was R_t = 2 MPa, the length of the suspended roof was 10~20 m, and the inclined length was 8~15 m, which reached the ultimate load-bearing capacity of the end roof. These values were basically consistent with the simulated results.

5. Conclusions

Using a three-dimensional physical simulation experiment, numerical calculations, and theoretical analysis, we systematically determined the mechanism of suspended roofs in the working faces of shallow coal seams. The following conclusions are mainly drawn:

(1)

Through the three-dimensional physical simulation experiment, it was concluded that the suspension roof collapse interval was twice the periodic weighting interval, the suspended roof had 5~15 m layers, and the fractures developed in a stepped arc shape. The size of the lower suspended roof was small, and the higher suspended roof was large, thus forming a “stepped arc-shaped laminated” suspended roof structure. The suspended roof fracture was located on the side of the coal pillar of the headgate.

(2)

FLAC3D numerical analysis results revealed that the shear stress contours in the suspended roof were distributed in an arc shape and continuously moved to the back of the goaf as the layer of the suspended roof increased. The maximum principal stress contour was distributed in an arc shape and gradually decreased toward the upper layer and the back of the goaf. Combined with the distribution characteristics of the plastic zone in the suspended roof area, it was found that when the suspended roof collapsed, the 8 m ~ 12 m layers significantly changed, and the size of the suspended roof collapses reduced by about 60%.

(3)

Using the theory of thin elastic slabs, we found that as the working face advanced, the suspended roof bending moment gradually increased and finally reached the breaking limit at the junction of the fixed side and the free side, and the roof slabs at different layers broke from the bottom to the top. The stability of the stepped arc triangular structure was only related to the roof and supporting conditions. When the overlying load of the end suspended roof structure was larger, the suspended roof area was smaller.