# Research on the Reasonable Width of Coal Pillar Driving along Goal under Thick Hard Roof
## 3 Method
### Theoretical Calculation of Coal Pillar Width under Thick Hard Roof  
The strength and stability of the coal pillar are the primary consideration when determining the width; there are many empirical formulas for calculating the strength of a coal pillar [35], such as the Obert-Kwvall/Wang formula (applicable to coal pillars with width-to-height ratios of 1-8), the Holland formula [36] (applicable to coal pillars with width-to-height ratios of 2-8), the Salamon-Munro formula [37] and Bieniawski formula [38]. It is obvious that the first two formulas are not suitable for calculating the strength of coal pillars in this paper. The Salamon-Munro formula is usually used to calculate the strength of coal pillars in the stable state, and the Bieniawski formula is used to calculate the strength of coal pillars considering in situ situation [39; 40; 41]. In this paper, \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Immediate Roof Lithology** & **Compressive Strength/MPa** & \(c_{1}\) & \(c_{2}\) \\ \hline hard & \textgreater{}40 & 2.1 & 16 \\ medium hard & 20–40 & 4.7 & 19 \\ weak & \textgreater{}20 & 6.2 & 32 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Table of values of height coefficients for collapse zones. adopting the Bieniawski formula to calculate the ultimate strength of coal pillars is more appropriate; the formula is shown as follows:  
\[\sigma_{p}=\sigma_{m}\bigg{(}0.64+0.36\frac{B}{h}\bigg{)} \tag{4}\]  
where \(\sigma_{p}\)--coal pillar strength, MPa; \(\sigma_{m}\)--uniaxial compressive strength of field sample, MPa; \(B\)--width of coal pillar, m; \(h\)--height of coal pillar, m.  
The compressive strength of the sample used in the laboratory and the compressive strength of the field sample will differ due to the different stress environments they are subjected to. Based on [PERSON]'s formula, [PERSON] gave a conversion formula for the uniaxial compressive strength of the laboratory sample and the uniaxial compressive strength of the field sample [42]:  
\[\sigma_{m}=0.2357\sigma_{c} \tag{5}\]  
where \(\sigma_{c}\)--uniaxial compressive strength of laboratory sample, MPa. The average compressive strength of coal samples from Jinniu Mine is 16.3 MPa, the height of the coal pillar is 5.1 m, the diameter of the laboratory specimen is 50 mm or 0.05 m, which can be calculated by substituting into the above formula, the uniaxial compressive strength of on-site critical cube specimen \(\sigma_{m}\) is 3.84 MPa. Calculation of load stress by the overlying rock strata of the coal pillar, the establishment of the coal pillar bearing model driving along goaf is shown in Figure 3 below, after the upper working face is mined back, according to the theory of masonry beams, the rock strata of the goaf is divided into caving zone, fracture zone, and bending subsidence zone. The rock strata above the coal pillar is an inverted step-like form; the angle between the edge of the inverted step and the plumb line is \(\theta\), which is called the lateral support angle. The load carried by the narrow coal pillar is the rock strata load in the red trapezoidal area shown in Figure 3, the width of the bottom of the trapezoid is the width of the coal pillar plus half of the width of the roadway, and the height of the trapezoid is the height of the fracture zone in the goaf minus the mining thickness of the working face [43]. The formula for calculating the stresses borne by the coal pillar is shown below:  
\[\sigma_{s}=\frac{\{d+2B+(H-h^{\prime})tan\theta\}\gamma(H-h^{\prime})}{2B} \tag{6}\]  
Figure 3: Coal pillar bearing structure in goaf excavation. where \(d\)--the roadway width, m; \(H\)--the height of the fracture zone (the calculated formula is shown in Table 2, the formula \(h\) is the height of mining), \(\theta\)--Lateral support angle, \({}^{\circ}\); \(h^{\prime}\)--height of roadway, m; \(\gamma\)--overlying rock mass, 0.025 N/mm\({}^{2}\).