Coal is a rock mineral that forms during the sedimentary evolution of surface plants from the ancient past. The deformation and ring breaking process of the coal body is manifested by the compaction, generation, and development of fissures. Figure 1 shows the stress–strain relation of coal under different confining pressures. Under different confining pressures, the failure process and residual strength of coal show remarkable differentiation, normally showing elastic–plastic softening characteristics.

The coal failure process with an elastic–plastic softening characteristic can be represented by five sections of the stress–strain curve, as shown in Figure 1.

The OA section is the compaction phase that occurs during the initial loading stage of the coal sample; the strain rate is larger than the stress rate, the in-situ cracks are compacted, and the curve trends upward.

The AB section is the stage of linear elastic deformation; the stress–strain relationship of the coal body is linear. B is the elastic limit point, and new cracks form after point B.

The BD section is the strain-hardening section. Due to the generation, expansion, and transfixion of the new cracks in coal, the strain rate increases, which is represented by remarkable dilatation. Before the stress achieves critical strength, cracks are expanding relatively slowly in the BC section. After point C, cracks develop very rapidly in the CD section, the coal volume is expanded, and many vertical cracks occur, and at point D, the critical strength is reached.

The DE section is the plastic softening section, in which stress decreases rapidly along with crack development. The deformation increases, cracks connect and cut through the coal, and the coal sample is crushed by compression. The deformation characteristics of the coal sample are closely related to the internal crack compaction, expansion, and evolution.

The EF section is the flow deformation section, in which coal still has certain residual strength after failure due to lateral stress and internal friction.

A simplified perfect linear elastic-plastic softening model shown in Figure 2A is proposed based on the coal stress-strain curve discussed in the above section. The deformation process is divided into elastic deformation before critical strength, plastic softening after critical strength, and flow deformation. The degree of plastic softening can be represented by α (α ≠ 0).

In the inelastic zone, based on the ground control theory and the practical conditions of floor deformation, the roof stratum in the inelastic zone deforms linearly. Figure 3A shows the mechanics model in the inelastic condition. The abutment pressure in the inelastic zone under limit equilibrium condition can be derived.

In Figure 3A, taking differential unit dx, the normal stress, lateral stress, and the coal-roof and coal-floor interface friction resistance acting on the differential unit satisfy the stress limit equilibrium condition. Neglecting gravity, the equilibrium equations of the differential unit are as follows. (5)∑FX=0 (6)∑FY=0 (7)T1=C1+f1σy (8)T2=C1+f1σyg The equilibrium equation in lateral direction is (see Appendix), (9)σx(h2+xtanθ)-(σx+dσx)[h2+(x+dx)tanθ]                      +T1cosθdxcosθ+σysinθdxcosθ+T2dx=0 where σ_y is the normal stress from the roof (MPa), C₁ is the cohesion between coal and roof, floor (MPa), f₁ is the friction coefficient between coal and roof, floor (°), H is the height of the coal seam (m), H = h₁ + h₂, h₁ is the coal deformation at the rib (m), and h₂ is the height of the coal seam at the rib (m).

The equilibrium equation in the vertical direction is, (10)σygdx+T1sinθdxcosθ-σycosθdxcosθ=0 where σ_yg is the normal stress from the floor (MPa).

Coal in the elastic zone has a single elastic stress distribution that shows high peak stress at the peak abutment pressure location. The ratio of lateral stress to abutment pressure is constant. In the whole elastic zone, coal is under an elastic compression condition. Therefore, as shown in Figure 3B, the stress equilibrium condition with ∑F_X = 0 and shearing stress T′=fσy of a differential unit in the elastic zone of coal is (33)(σx+dσx)H+2(fσy)dx-σxH=0 The lateral stress and vertical stress in the elastic zone have the following relationship (34)σy=βσx where β is the lateral stress coefficient.

For consideration of the friction resistance, T′|x=x0=KγH1 and T′|x=x0+x3=0, it is reasonable to assume that (35)f=(x0+x3-x)f1x3 where x₃ is the width of the elastic zone (m).

Substituting Equations (34) and (35) into Equation (33) and substituting the boundary condition σx|x=x0=KγH1β, the abutment pressure in the elastic zone can be derived by the following equation. (36)σy=KγH1eβf1Hx3(x0-x)(2x3+x0-x) When x = x₀ + x₃, substituting σ_y|_x=x0+x3 = γH₁ into Equation (36), the width of the elastic zone can be calculated by the following equation. (37)x3=Hβf1ln K

Substituting the parameters listed in Table 1 into Equations (20), (29), (31), (32), (36), and (37) with four mining depths of 200, 300, 400, and 500 m, the abutment pressure and the widths of different zones can be calculated. Figure 4 shows the abutment pressure curves vs. mining face location at various mining depths.

It can be discovered from Figure 4 that the widths of the crushed zone and inelastic zone increase by a logarithmic law with the increase in mining depth, and the mining depth has no impacts on the widths of the plastic zone and elastic zone. And at the same time, the abutment pressure changes at the crushed zone are identical. However, the peak abutment pressure increases with the increase in mining depth.

Four different deformation angles, i.e., 1°, 2°, 3°, and 4°, are substituted into the model to study the impacts on the abutment pressure. Figure 5 shows the abutment pressure curves vs. mining face location at various deformation angles.

It can be discovered from Figure 5 that the deformation angle has no impact on the widths of the elastic and inelastic zones. By substituting the deformation angle into the theoretical formula, it can be seen that when the deformation angle θ = 1°, the width of the crushed zone is 3.34 m, and the width of the plastic zone is 1.76 m; when the deformation angle θ = 2°, the width of the crushed zone is 4.28 m, and the width of the plastic zone is 0.88 m; when the deformation angle θ = 3°, the width of the crushed zone is 4.64 m, and the width of the plastic zone is 0.59 m; when the deformation angle θ = 4°, the width of the crushed zone is 4.85 m, and the width of the plastic zone is 0.44 m. The width of the crushed zone increases logarithmically with increase in the mining depth, and the width of the plastic zone decreases following a power law with increase in the mining depth. The abutment pressure curves at different deformation angles are similar, and the abutment pressure curves at the crushed zone and elastic zone almost overlap each other, which indicates that the deformation angle has no impact on abutment pressure.

To study the impacts of the peak abutment pressure coefficient on the abutment pressure, four different peak abutment pressure coefficients, i.e., K = 2, 3, 4, and 5, are discussed. Figure 6 shows the abutment pressure curves vs. mining face location at various peak abutment pressure coefficients.

It can be discovered from Figure 6 that the widths of the crushed zone and inelastic zone increase following a logarithmic law with the increase in peak abutment pressure coefficient, and the width of the elastic zone increases following a parabolic law with the increase in peak abutment pressure coefficient. The peak abutment pressure coefficient has no impact on the width of plastic zone. At the same time, the abutment pressure increases remarkably with the increase in the peak abutment pressure coefficient.

The impacts of the friction coefficient on the abutment pressure are discussed for four different friction coefficients, i.e., f₁ = 0.1, 0.2, 0.3, and 0.4. Figure 7 shows the abutment pressure curves vs. mining face location at various friction coefficients.

It can be discovered from Figure 7 that the widths of the crushed zone and inelastic zone decrease following a logarithmic law with the increase in friction coefficient. The friction coefficient has no significant impact on the width of the plastic zone, while it has a remarkable impact on the width of the elastic zone. At the same time, the abutment pressure curve shifts to the left as the friction angle increases. The friction coefficient is only related to the lithology of the coal, roof, and floor, and the roughness of the interface. The deformation of coal has a very small influence on friction angle.

Four different cohesions, i.e., 0.24, 0.48, 0.72, and 0.96 MPa, are substituted into the model to study the impacts on the abutment pressure. Figure 8 shows the abutment pressure curves vs. mining face location at various cohesions.

It can be discovered from Figure 8 that the widths of the crushed zone and the inelastic zone decrease following a logarithmic law with the increase in cohesion. The cohesion has no significant impact on the width of the plastic zone and elastic zone. At the same time, the abutment pressure curve shifts to the left as the cohesion increases. The cohesion relates to the bonding condition between coal and the roof/floor and the relative slippage at the interface.

To study the impacts of the UCS on the abutment pressure, four different UCS values, i.e., σ_c = 6, 9, 12, and 15 MPa, are discussed. Figure 9 shows the abutment pressure curves vs. mining face location at various UCS values.

It can be discovered from Figure 9 that the widths of the crushed zone and inelastic zone decrease almost linearly with the increase in UCS. The width of the plastic zone increases almost linearly with the increase in UCS. The UCS has no significant impact on the width of the elastic zone. At the same time, the abutment pressure curve shifts to the left as the UCS increases.

Four mining heights, i.e., 2, 3, 4, and 5 m, are substituted into the model to study the impacts on the abutment pressure. Figure 10 shows the abutment pressure curves vs. mining face location at various mining heights.

It can be discovered from Figure 10 that the widths of the crushed zone, plastic zone, inelastic zone, and elastic zone increase linearly with the increase in mining height. The width increase rate of the elastic zone is higher than those of the other three zones. The mining height has significant impacts on abutment pressure curves. The abutment pressure curve shifts to the right rapidly as the mining height increases.

Dongjiahe Coal Mine is one of the major producing coal mines in Chenghe Coalfield in Shaanxi province. It was constructed and started to produce in December of 1980 and has a 40-year mining history. The coal seam is in the Taiyuan Group of the Upper Carboniferous system and the Shanxi Group of the Lower Permian system. Currently, the minable seams are No. 5 Coal and No. 10 Coal. No. 5 Coal is buried at a depth of about 173.64–420.04 m, and it is the major minable coal seam. No. 10 Coal is buried at a depth of about 190.5–452.5 m, and it is an unstable coal seam in which most of the seam can be mined. The coal mine utilizes slope access, with multiple mining elevations and uphill/downhill developments. The major mining seam now is No. 5 Coal, and its average thickness is 3.71 m.

Longwall panel 07 is at No. 5 Coal at Dongjiahe Coal Mine, is separated by a 30-m barrier pillar from panel 06 in the south, and is surrounded by solid coal in the north. Coal in this panel is completely developed with an average thickness of about 3.0 m. The surface elevations are about +644.2~+680.7 m, while the panel elevations are about +255~+273 m. The longwall panel is 910 m in strike direction and 114 m in dip direction. The coal seam is high in the south and low in the north, with coal seam being inclined by about 0°-15°. No. 5 Coal is at the upper section of the Taiyuan Group. The immediate roof is hard gray K4 medium sandstone with a thickness of 10.9–18.15 m and major components of quartz and silicon sludge cementing and with fairly developed cracks. The immediate floor is hard dark gray coarse sandstone with a thickness of 0.2–3.21 m and contains many mica plates and localized sandstone stripes. The main floor is K3 fine quartz sandstone or siltstone with a thickness of about 3.4–7.8 m.

The sizes of the headgate and tailgate are 3.4 × 2.8 m, and they are supported by bolt, mesh, anchor, and beam. The open cut excavates along the roof with a length of 114 m. Its cross-section is 6.6 × 3.0 m, and it is supported by bolt, mesh, anchor, and beam. This panel utilizes longwall retreat mining on the No. 5 Coal floor. ZY4600/18/42 shields are adopted for roof support. The roof above the gob area caves as mining advances.

In order to obtain the stress distribution of the tailgate during retreat mining, an experiment was conducted at 80 m in front of the mining face in the tailgate of panel 07 at Dongjiahe Coal Mine. Figure 11 shows a layout diagram of the actual measurement of coal pillar advance bearing pressure.

Vibration wire borehole stress gauges (XL-YSYLJ) (as shown in Figure 11) and data loggers were installed in the tailgate perpendicular to the gate road direction for monitoring the abutment pressure distribution during retreat mining. Three boreholes were prepared, with one vibration wire borehole stress gauge (XL-YSYLJ) for each borehole, which were located at 60, 80, and 100 m from the open cut. The depths of the boreholes were 10 m to avoid the influence of stress concentration around the tailgate. The spacing between each group of survey lines was 20 m, and the borehole was about 1.6 m above the floor.

The data were recorded once per cut (0.6 m) when the longwall face was <15 m from the survey station, once per two cuts (1.2 m) when the distance from the survey station to the longwall face was about 15–30 m, and once per four cuts (2.4 m) when the longwall face was more than 30 m from the survey station.

Monitoring of the abutment pressure was conducted at the tailgate of the longwall panel. The data are processed to obtain the abutment pressure curves vs. distance of the working face and survey station. Figure 12 shows the abutment pressure distribution curves.

According to Figure 12A, the borehole stress gauge readings decrease after installation. This is because the coal surrounding the borehole deforms and becomes loose when subject to the initial pressure of the borehole stress gauge. As the deformation of the borehole continues, the stress will achieve a stable condition; this stress is the initial stress of the borehole. For borehole stress gauge 1, the peak abutment pressure is 22.73 MPa, occurring at about 5.8 m from the longwall face (inelastic zone width), the stress concentration coefficient is 3.12, the abutment pressure influence zone is 27.8 m (width of the inelastic zone and elastic zone), and the apparent influence distance is 4.2 m (crushed zone). For borehole stress gauge 2, the peak abutment pressure is 19.72 MPa, occurring at about 5.4 m from the longwall face (inelastic zone width), the stress concentration coefficient is 2.92, the abutment pressure influence zone is 30.2 m (width of the inelastic zone and elastic zone), and the apparent influence distance is 4.8 m (crushed zone). For borehole stress gauge 3, the peak abutment pressure is 23.84 MPa, occurring at about 6.2 m from the longwall face (inelastic zone width), the stress concentration coefficient is 2.90, the abutment pressure influence zone is 28.6 m (width of the inelastic zone and elastic zone), and the apparent influence distance is 4.2 m (crushed zone). Table 2 lists the data from the three stress gauges.

Table 3 shows a comparison of the theoretical calculations and the field measurements. The theoretical fitting curve of the bearing pressure distribution of the working face can be drawn using the figures in Table 3, as shown in Figure 12B.

Relative error analysis was applied to compare the theoretical calculations and field measurements. The lower the relative error, the higher the precision. It is shown in Table 3 that the abutment pressure distribution and the widths of the inelastic zone and the elastic zone coincide with the field measurements very well, which indicates that the theoretical model has very high accuracy.