At the northernmost point of the A’ai mining region, Qiuci Mining Co., Ltd. is situated on the west bank of the Kuche River, 101 km north of Kuche County, Xinjiang. The direct roof is made with a typical thickness of 5.06 m of fine sandstone, and the basic roof is made of medium-sized sandstone, often about 7.92 m. The A503 fully mechanised caving face is the third coal mining face in the A5 mining area. The northern part of the face is the original A5 goaf, the southern part is the A502 working face, the western part is the boundary protection coal pillar of the mine field, the eastern part is the A5 mining area protection coal pillar, and the upper part is the A602 working face goaf. The burial depth of the working face is about 150 m. The direct bottom is made of grey siltstone, with a thickness of approximately 4.40 m, and the basic bottom is made of light medium grey-fine sandstone, with a thickness of around 6.89 m. The bottom plate is complete, and there are no cracks.

Only the irregularly shaped area depicted in Figure 1 remains in the A5 coal seam in an unmined state following the mining of the Qiuci Mine’s A502 working face. The chamber and pillar portion of this location is in the west, and the coal seam fire region created by mining is in the south.

Based on the characteristics of the working face, four preliminary layout schemes have been designed: regular layout, asymmetric handle (one-time expansion of the working face) layout scheme, symmetric handle (expansion-contraction) layout scheme, and extremely irregular (multiple expansion-contraction) layout scheme. As shown in Figure 2.

Through economic and technical comparison, although the extremely irregular working face has some complicated processes and is difficult to mine, it can mine 438,400 tons more coal than the conventional regular layout, and the economic benefits are significant. Choose an extremely irregular layout as the mining plan for the A503 working face of Qiuzi Mine. The variation of working face length can be divided into areas with unchanged face length, areas with increased face length, and areas with decreased face length. Hydraulic support technology will be increased while hydraulic support technology will be decreased to complete the mining of quite erratic coal seams.

As is clear from this technique, an appropriate segment pillar width is crucial for the secure mining of relatively erratic working faces. The load on the pillar of coal is affected by mining the working face, and as variations in the working face’s forward motion occur, so does the mining width (Zhu et al., 2014; Gao et al., 2019). There will be a significant loss of coal resources in a relatively limited working face width if the coal pillars are left in place in compliance with the limit load. To ensure safe mining at the working face without losing significant resources available for coal, the coal’s dimensions and pillars should be modified together with the working face’s width.

From the Qiuci Coal Mine in Kuche, Xinjiang Province, China, coal samples were taken from whole coal. The samples were constructed in accordance with the experimental specification advised by ISRM and were cylindrical, as depicted in Figure 3, with a diameter of 50 mm and heights of 15, 20, 30, 35, 50, 75, and 100 mm. The tests were carried out. using electro-hydraulic servo rock testing equipment created by Changchun Corporation that has a maximum axial pressure of 1,000 kN.

Using an ultrasonic detector RSM-SY5, the P-wave velocities of each coal sample were determined using an instrument with a timing precision of 0.05 μs, a test range of 0–629,000 μs, an amplifier power of 82 dB, and a bandwidth of 10 kHz–250 kHz. For testing, three coal samples for each ratio with P-wave velocities between 1,550 and 1,750 m/s were chosen. All measurements were carried out until the sample was destroyed at a 0.002 mm/s constant loading rate.

Coal samples’ stress-strain curves at various ratios under UCT are shown in Figures 4, 5.

It is depicted in Figure 3 based on the coal samples’ stress-strain curves, which show various proportions. It is clear that the compression stage, the linear elastic stage, the fracture propagation stage, the yield stage, and the post-peak stage can be distinguished in the stress-strain curves of coal samples with various proportions. The compression stage of the coal sample is shortened, and the elastic stage is prolonged with the increase in length-diameter ratio. The wider the coal sample, the greater the peak strength and the greater the residual strength.

Figure 4 shows that as the aspect ratio rises, the average durability of the coal samples decreases. The coal’s durability sample has the following relationship to D: σ s = 0.78 1 D 3 + 11.35 (1)

When D < 0.7, the strength of the coal sample increases greatly with the decrease in the height-diameter ratio.

The relationship between the height-diameter ratio and the compressive strength of coal samples was obtained by laboratory experiments. However, because the laboratory experiment is carried out on small-sized coal samples, it can only reflect the law of the influence of width on strength and cannot represent the strength of large-scale coal pillars. It is still necessary to construct the strength formula for the on-site coal pillar scales because the aspect ratio plays a significant role in the coal sample strength.

The pillar width between the roadway and room goaf is altered with the working face’s length once the very irregular working face is prepared. The layout of an appropriate coal pillar width is based on the strength of coal pillars with various aspect ratios (Jiang et al., 2011; Wang et al., 2011). Therefore, the strength of coal pillars with various aspect ratios was investigated using FLAC^3D numerical simulation software. The model has a floor, pillar, and roof. Based on the dimensions of panel A503 from the Qiuci Coal Mine, four coal pillar numerical models were created, with a total width along the x, y, and z axes of 9 m, 14 m, 19 m, and 24 m, and a height in the z direction of 10 m. According to Figure 6, one zone’s coal pillar length is 0.1 m × 0.1 m × 0.1 m. To examine the strength of coal pillars with various aspect ratios, software was employed.

The x, y, and z directions, as well as the model’s top, were set in the vertical direction to maintain a constant vertical velocity of displacement, which is confined to the roof and floor boundaries, respectively. The pillar is modelled using the Mohr Coulomb yield criterion with train softening (DuncanTruemanCraig, 1995).

Through experimental testing, the physical and mechanical characteristics of samples of ordinary coal were determined. The charcoal samples’ elastic moduli were 0.5 GPa, 10.7 MPa for uniaxial compressive strength, 0.27 for Poisson ratio, 5.3 MPa for cohesive strength, and 25.2° for friction, respectively. The carbon pillar’s plastic stress softening characteristics are represented in Table 1.

Figure 7 illustrates the curves of stress and strain, and failure traits of coal samples with various aspect ratios when using numerical simulation tools.

Figure 7 shows how the stress-strain curves of coal samples with different aspect ratios are similar to the shape of laboratory tests. It includes the post-peak stage, the yield stage, and the elastic stage. In the early stage of coal pillar deformation, the upper and lower boundaries first undergo compaction deformation and then extend to the middle of the coal sample. With the loading of pressure, the plastic zone expands, so that the column is less damaged in the elastic stage. The failure of the coal pillar is mainly a shear failure. Stress concentration occurs at the diagonal, so the diagonal is first destroyed. Both sides occur symmetrically. The coal pillar experienced shear failure of the X-shaped conjugate slope. The higher the D, the more obvious the shear failure characteristics of the double-inclined plane of the coal pillar.

The strength design formula suggested by Bieniawski (1992), Holland (Pati, 2011), Bunting, and Bunschinger is employed as the calculation formula for pillar strength in the mine. Table 2 displays the strength of coal pillars with various aspect ratios.

Figure 8 illustrates how the coal pillar’s strength decreases as W rises. The Bieniawski formula fits the results of the numerical model better than the other four strength formulas, which can roughly reflect the relationship’s trend. The right coal pillar strength formula in the Qiuci mine can be obtained by using the Bieniawski formula to correct the coal pillar strength. σ s = 0.968 × 12.15 0.64 + 0.36 W (2)

Figure 9 shows that the curve of the coal pillar’s adjusted strength and W agrees exactly with the outcomes of the numerical model, with a correlation degree of 1.

The roof can be reduced to a beam with a very erratic working face. The support of the coal pillar received below is simplified as the elastic foundation, and the weight of the rock formation above the beam is simplified as the load qc. The roof mechanical model is constructed, and the two sides of the beam are fixed, as shown in Figure 10.

The differential equation for the beam’s curved surface is: E I ∂ 4 w x ∂ x 4 = q x − p x (3)

The cross-sectional moment of inertia of the beam is I, m⁴, and the elastic modulus of the beam is E, GPa. The load received below the beam is represented by P(x), and the load received above the beam is represented by q(x).

The foundation’s reaction force for an elastic foundation can be expressed as follows: p x = k 0 w x (4)

When L ₃<x≤ L ₃ + L ₄, taking the characteristic coefficient α = k 0 4 E I 4 , the general answer to Eq. 3 is as follows Huang et al. (2020): w x = e α x d 1 ⁡ cos α x + d 2 ⁡ sin α x + e − α x d 3 ⁡ cos α x + d 4 ⁡ sin α x + q c k 0 (5)

Similarly, the general solution of other area elastic foundation equation can also be obtained.

When 0<x≤ L ₃, the general answer to Eq. 3 is as follows Huang et al. (2020): w x = q c 24 x 4 + d 5 6 x 3 + d 6 2 x 2 + d 7 x + d 8 (6)

The corner’s relationship to the bending moment, shear force, and deflection of a particular segment of the beam is as follows Huang et al. (2020): θ x = d w x d x M x = − E I d 2 w x d x 2 Q x = − E I d 3 w x d x 3 (7)

Both ends of the beam are fixed ends, so the boundary condition is: w x = − L 1 − L 2 = 0 , w x = L 3 + L 4 + L 5 + L 6 + L 7 + L 8 = 0 θ x = − L 1 − L 2 = 0 , θ x = L 3 + L 4 + L 5 + L 6 + L 7 + L 8 = 0 (8)

Additionally, the bending, shear, corner, and deflection moments are equivalent at x = –L ₂, x = 0, x = L ₃, x = L ₃ + L ₄, x = L ₃ + L ₄+ L ₅, x = L ₃ + L ₄+ L ₅+ L ₆, x = L ₃ + L ₄+ L ₅+ L ₆ + L ₇ in the model. From the above boundary conditions, unknown coefficients can be determined.

The tension of the coal pillars in the regional domain can be determined using Eq. 4 in place of Eq. 8 to model the region of the coal pillars as a continuously distributed Winkler elastic foundation Zhang et al. (2017): p 0 x = q c + k 0 d 1 e α x ⁡ cos α x + d 2 e α x ⁡ sin α x + d 3 e − α x ⁡ cos α x + d 4 e − α x ⁡ sin α x (9)

The stress in a single coal pillar will be the integration for stress in this range of area, such as: Assuming that the length and width of the coal room make up the range of coordinates in the load-bearing area of a single coal pillar, [x ₁, x ₂], the stress in the single coal pillar will be Zhang et al. (2017): Q = ∫ x 1 x 2 p 0 x d x (10)

Equation 10, when integrated, yields the single coal pillar’s stress as: Q = q c x 2 − x 1 + k 0 2 α ( − e α x 1 d 1 + e α x 1 d 2 + e − α x 1 d 3 + e − α x 1 d 4 cos α x 1 + − e α x 1 d 1 − e α x 1 d 2 − e − α x 1 d 3 + e − α x 1 d 4 sin α x 1 + e α x 1 d 1 − e α x 1 d 2 − e − α x 1 d 3 − e − α x 1 d 4 cos α x 2 − e α x 1 d 1 + e α x 1 d 2 + e − α x 1 d 3 − e − α x 1 d 4 sin α x 2 ) (11)

The foundation coefficient k₀ is 8×10^9, the overlaying uniform load qc is 3.2 × 10⁶ N/m², the basic top elastic modulus E is 20.9 GPa, and the fundamental thickness of the A5 coal seam is 11 m, according to the Qiuci Mine’s geological information. Roadway widths in the A503 working face L ₅ are 4 m, coal pillar widths in the A503 working face L ₂ are 6 m, room-pillar goaf widths in L ₆ and L ₈, mining width in L ₇ is 5 m, and the length of the suspended roof in the A502 working face L ₁ is 10 m. The concentration factor (k) is 2. The link between the working face length and coal pillar strength by bringing these parameters into Eq. 11 is shown in Figure 11.

As seen in Figure 10, the coal pillar’s peak tension rises as the face length of the working face lengthens. The following equation determines the face length and peak stress of the pillar created by fitting: σ = 0.0618 L + 7.0763 (12)

The proportion of the length of the working face and the aspect ratio can be determined as follows by combining formulas (2) and (12): w / h ≥ 0.0146 L − 0.102 (13)

The breadth of coal pillars varies as a result of the A503 working face of Qiuci Mine adopting a very atypical arrangement to extract as many coal resources as possible. The surface length of each area is entered into formula (13), and the theoretical working face’s length and the minimal width of the pillar of coal are calculated as shown in Figure 12 in the instance where the height of the pillar of coal is h = 6 m.

The alteration in face length is primarily divided into four sections in the actual manufacturing process: the beginning working face, whose length is constant; the length rising working face; the length constant working face in the intermediate area; and the length dropping working face. The length increase working face has a minimum coal pillar in width 13.4 m, the length constant working face has a minimum coal pillar in width 13.4 m, and the length decrease working face has a minimum coal pillar in width from 3.68 m to 13.4 m, according to formula (12) and Figure 12. The coal pillar’s smallest width is 4.91 m in the initial face. As a result, Figure 13 depicts the theoretical border of the actual coal pillar and the minimum coal pillar width boundary line.

Figure 13 indicates that the theoretical limit of the smallest width of the coal pillar is within the bounds of the actual coal pillar boundary line.