## 2 Predictions of the Stability of Headings in the LGCB Mines
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \multirow{2}{*}{**Location**} & \multirow{2}{*}{**Rock Type**} & \(h\) & \(\rho\) & \(R_{c}\) & \(R_{r}\) & \(E_{i}\) & \(\
u\) \\  & & **[m]** & **[MPa]** & **[MPa]** & **[MPa]** & **[MPa]** & **[MPa]** & **[-]** \\ \hline \multirow{8}{*}{Roof} & Anhydrite I & 0.70 & 2.94 & 132.49 & 6.43 & 42,420.00 & 0.24 \\  & Anhydrite II & 1.80 & 2.94 & 117.61 & 5.25 & 38,240.00 & 0.24 \\  & Anhydrite III & 3.00 & 2.94 & 97.88 & 5.58 & 40,600.00 & 0.24 \\  & Anhydrite IV & 9.30 & 2.94 & 86.93 & 5.94 & 38,300.00 & 0.24 \\  & Dolomite I & 1.20 & 2.81 & 175.86 & 7.20 & 68,400.00 & 0.26 \\  & Dolomite II & 1.70 & 2.79 & 156.21 & 8.03 & 47,290.00 & 0.25 \\  & Dolomite III & 0.60 & 2.63 & 194.22 & 10.05 & 89,800.00 & 0.27 \\  & Dolomite IV & 0.70 & 2.64 & 186.60 & 8.98 & 82,050.00 & 0.28 \\  & Dolomite V & 1.00 & 2.70 & 165.69 & 4.55 & 65,050.00 & 0.24 \\  & Dolomite VI & 0.70 & 2.73 & 126.89 & 9.59 & 23,400.00 & 0.22 \\  & Dolomite VII & 0.60 & 2.65 & 188.90 & 10.87 & 84,400.00 & 0.27 \\  & Dolomite VIII & 0.70 & 2.80 & 118.44 & 5.45 & 39,150.00 & 0.23 \\ \hline \multirow{2}{*}{Walls} & Dolomite IX & 1.60 & 2.71 & 95.00 & 10.16 & 27,170.00 & 0.22 \\  & Dolomite shale & 0.60 & 2.69 & 111.49 & 9.06 & 28,890.00 & 0.23 \\  & Sandstone I & 1.20 & 2.40 & 47.85 & 2.93 & 16,940.00 & 0.17 \\ \hline \multirow{2}{*}{Floor} & Sandstone II & 9.50 & 2.33 & 36.19 & 2.78 & 13,600.00 & 0.14 \\ \hline \end{tabular} The symbols used in the above table are as follows: \(h\)—thickness of rock layers, \(\rho\)—volume density, \(R_{c}\)—rock sample strength to uniaxial compression, \(R_{r}\)—tensile strength of the rock sample, \(E_{i}\)—longitudinal modulus of elasticity, \(\rho\)—Poisson’s ratio.  
\end{table} 
Table 2: Mean strength and strain rock parameters for the Jm-15/H-173 borehole and the Jm-15-460 borehole. and strain parameters) for each of the rock layers present in the Jm-15/H-173 and Jm-15-460 boreholes (Table 3). The numerical model was constructed in the RS2 application, in a triaxial stress state and in a plane strain state. Numerical calculations were performed for an isotropic and for a uniform medium. The rock medium was described with the elastic-plastic model with softening (layers in the roof and walls) and with the elastic-plastic model (layers in the floor). Table 4 shows the strength and strain parameters of the rocks in the model. The numerical model was developed on the basis of the Mohr-Coulomb failure criterion, which states that rock may reach threshold effort under the following condition:  
\[\sigma_{1}=\sigma_{3}.\frac{1+\sin\varphi}{1-\sin\varphi}+\frac{2c\cdot\cos \varphi}{1-\sin\varphi} \tag{3}\]  
or  
\[\sigma_{3}=-\sigma_{t} \tag{4}\]  
where:  
\(\sigma_{1}\)--effective maximum stress at failure,  
\(\sigma_{3}\)--effective minimum stress at failure,  
\(\varphi\)--internal friction angle,  
\(c\)--cohesion,  
\(\sigma_{t}\)--uniaxial tensile strength of the rock mass. Numerical analyses were performed for a group of four headings. The excavations have a trapezoidal shape. In the headings, the side walls were inclined at an angle of 10\({}^{\circ}\). The adjacent excavations are separated by pillars 20 m in width. Table 5 lists the dimensions of headings in their assumed cross-sections. The headings were protected with full-length-grounted rockbolts in a 1.5 m \(\times\) 1.5 m bolting grid (Figure 1). The parameters of the employed RM-18 1.8 m long bolts are shown in Table 6.