*3.1. Theoretical Analysis*

#### 3.1.1. Stress Distribution Pattern below Remaining Coal Pillars

After the 2-2 coal seam was extracted, the rock formation above the gob collapsed, causing the abutment pressure on the remaining coal pillar to rise sharply. These stresses transferred to the underlying coal seam, forming a larger area under the effects of the coal pillar abutment stress, i.e., the rock mass below the coal pillar entered the high abutment pressure state earlier. The abutment pressure distribution on the T-shaped remaining coal pillar [24] is shown in Figure 4. When the coal pillar was narrow (20 m), the roof fracture position moved toward the inside of the coal pillar, i.e., the range of the limit equilibrium area of the coal pillar increased significantly. As a result, the vertical stress on the coal pillar increased significantly, peaking at 70 MPa. As the coal pillar became wider (70 m), the pressure-bearing range increased. The peak vertical stress decreased significantly compared with that of the narrow coal pillar, but the range of effects on the underlying coal seam was increased relatively.

**Figure 3.** Flowchart of the research procedure.

**Figure 4.** Abutment stress distribution on the T-shaped remaining coal pillar in the 2-2 coal seam.

To investigate the stress transfer pattern of the 2-2 coal seam to the underlying coal seam, the coal pillar abutment pressure can be simplified to the mechanical model shown in Figure 5. Since the abutment pressure is symmetrically distributed on the wide coal pillar about its center axis, the center of the wide coal pillar is selected as the coordinate origin. For the narrow coal pillar, the minimum stress point in the internal elastic core is selected as the coordinate origin. The horizontal cross-sectional direction of the coal seam floor is selected as the x-axis direction, and the vertically downward direction is selected as the y-axis direction. γ is the volumetric weight of the rock, H is the depth, and *k*1, *k*2, *k*3, *k*4, and *k*<sup>5</sup> are the maximum stress concentration factors.

**Figure 5.** Coal pillar force models. (**a**) Wide coal pillar force model; (**b**) Narrow coal pillar force model.

In the figure, <sup>ᬅ</sup>–<sup>ᬎ</sup> are stress-concentrated areas of the coal pillar, and the vertical stress components of the floor rock mass are:

$$\sigma\_{\mathcal{Y}} = -\frac{2}{\pi} \int\_{-\infty}^{+\infty} \frac{q\chi(y-\xi)^2 d\xi}{\left[\chi^2 + (y-\xi)^2\right]^2} \tag{1}$$

where *q* is the micro-element load collection degree, and *σ<sup>y</sup>* is the vertical stress (MPa); *x* and *y* are the vertical and horizontal distances of the concentrated force q to any point M on the floor (m).

As shown in Figure 5, the additional stress generated by the coal pillar at point M on the underlying coal seam is calculated based on the superposition principle, and the calculation results are as follows.

The additional vertical stress generated by the wide coal pillar is:

$$\sigma\_{y1} = -\frac{2}{\pi} \left\{ \begin{array}{ll} \int\_{-c}^{-b} \frac{q\_1 x (y-\frac{\pi}{\xi})^2 d\xi}{\left[x^2 + (y-\frac{\pi}{\xi})^2\right]} + \int\_{-b}^{-a} \frac{q\_2 x (y-\frac{\pi}{\xi})^2 d\xi}{\left[x^2 + (y-\frac{\pi}{\xi})^2\right]} + \int\_{-a}^{0} \frac{q\_3 x (y-\frac{\pi}{\xi})^2 d\xi}{\left[x^2 + (y-\frac{\pi}{\xi})^2\right]} \\ + \int\_{0}^{a} \frac{q\_4 x (y-\frac{\pi}{\xi})^2 d\xi}{\left[x^2 + (y-\frac{\pi}{\xi})^2\right]} + \int\_{a}^{b} \frac{q\_5 x (y-\frac{\pi}{\xi})^2 d\xi}{\left[x^2 + (y-\frac{\pi}{\xi})^2\right]} + \int\_{b}^{a} \frac{q\_6 x (y-\frac{\pi}{\xi})^2 d\frac{\pi}{\xi}}{\left[x^2 + (y-\frac{\pi}{\xi})^2\right]} \end{array} \right\} + \gamma y \tag{2}$$

where *<sup>q</sup>*<sup>1</sup> = *<sup>k</sup>*1*γ<sup>H</sup> <sup>c</sup>* [*<sup>x</sup>* <sup>−</sup> (*<sup>a</sup>* <sup>+</sup> *<sup>b</sup>* <sup>+</sup> *<sup>c</sup>*)], *<sup>q</sup>*<sup>2</sup> <sup>=</sup> (*k*2−*k*1)*γ<sup>H</sup> <sup>b</sup>* [*x* − (*a* + *b*)], *q*<sup>3</sup> = *k*1*γH*, *q*<sup>4</sup> = *k*1*γH*, *<sup>q</sup>*<sup>5</sup> <sup>=</sup> (*k*2−*k*1)*γ<sup>H</sup> <sup>b</sup>* [*<sup>x</sup>* <sup>−</sup> (*<sup>a</sup>* <sup>+</sup> *<sup>b</sup>*)], *<sup>q</sup>*<sup>6</sup> <sup>=</sup> *<sup>k</sup>*1*γ<sup>H</sup> <sup>c</sup>* [*x* − (*a* + *b* + *c*)], and *y* is the coal seam spacing. The additional vertical stress generated by the narrow coal pillar is:

$$\sigma\_{\mathfrak{F}} = -\frac{2}{\pi} \left\{ \int\_{-\mathfrak{r}}^{-d} \frac{q\_{\mathfrak{F}} \mathbf{x} (\mathbf{y} - \boldsymbol{\xi})^{2} d\mathbf{f}}{\left[ \mathbf{x}^{2} + (\mathbf{y} - \boldsymbol{\xi})^{2} \right]} + \int\_{-d}^{0} \frac{q\_{\mathfrak{F}} \mathbf{x} (\mathbf{y} - \boldsymbol{\xi})^{2} d\mathbf{f}}{\left[ \mathbf{x}^{2} + (\mathbf{y} - \boldsymbol{\xi})^{2} \right]} + \int\_{f}^{\mathfrak{F}} \frac{q\_{\mathfrak{F}} \mathbf{x} (\mathbf{y} - \boldsymbol{\xi})^{2} d\mathbf{f}}{\left[ \mathbf{x}^{2} + (\mathbf{y} - \boldsymbol{\xi})^{2} \right]} \right\} + \gamma y \tag{3}$$
 
$$\text{where } q\_{\mathfrak{F}} = \frac{k\_{\mathfrak{F}} \mathcal{H}}{\mathfrak{r}} [\mathbf{x} - (\mathbf{d} + \mathbf{e})], \ q\_{\mathfrak{F}} = \frac{(k\_{\mathfrak{k}} - k\_{\mathfrak{k}}) \gamma H}{d} (\mathbf{x} - \mathbf{e}), \ q\_{\mathfrak{F}} = \frac{(k\_{\mathfrak{k}} - k\_{\mathfrak{k}}) \gamma H}{f} (\mathbf{x} - \mathbf{f}),$$
  $q\_{10} = \frac{k\_{\mathfrak{k}} \gamma H}{\mathcal{F}} [\mathbf{x} - (\mathbf{f} + \mathbf{g})]$ .
