## 4 Strength Model of Inclined Pillar
### Verification of Inclined Pillar Shear Strength Model  
In total, 18 cases of inclined pillars [32; 33] (5 failed pillars and 13 stable pillars) were used to verify the shear strength model of inclined pillars. The relevant parameters of the 18 inclined pillars are shown in Tables 1 and 2, in which the internal friction angle of the ore-body is 31\({}^{\circ}\) and the bulk density of overburden rock mass is calculated according to 27 kN/m\({}^{3}\). To verify the shear strength model of the inclined pillar, the performance of the pillar expressed by the shear strength is expressed by the factor of safety (_FOS_). The factor of safety is expressed as the ratio of the shear strength to the shear load,  
\[FOS=\frac{T_{p}}{\tau_{\theta}}=\frac{\frac{(1-\sin\theta_{m})\sin\beta}{\frac {1-\sin(\theta_{m}-\beta)}{(1-k)\gamma H\sin\theta\cos\theta}}}{1-\eta} \tag{18}\]  
In Equation (18), when the _FOS_ is greater than 1.0, the pillar is regarded as stable, and when the _FOS_ is less than 1.0, the pillar fails. The shear strength model of the inclined pillar was derived based on the strength of the vertical pillar and Mohr-Coulomb criterion; thus, it is particularly important to choose the strength model of the vertical pillar. So far, many empirical formulas have been put forward for vertical pillars (including coal pillars and hard rock pillars). Some of the commonly used strength formulas are shown in Table 3. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Pillar Case** & **Depth/m** & **Pillar Height/m** & **Pillar Width/m** & **Dip/Degree** & **In-Situ Stress Ratio (k)** & **Roadway Width/m** & **UCS of Intact Ore-Body/MPa** \\ \hline
1 & 95 & 2.0 & 3.0 \(\times\) 7.15 & 22.0 & 2.10 & 4.2 & 27.6 \\
2 & 103 & 1.8 & 2.0 \(\times\) 7.15 & 18.3 & 1.33 & 4.2 & 8.3 \\
3 & 73 & 1.8 & 2.0 \(\times\) 9.20 & 14.0 & 1.90 & 3.6 & 18.3 \\
5 & 235 & 1.5 & 2.5 \(\times\) 13.15 & 12.4 & 0.81 & 4.5 & 8.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Failed inclined pillar cases [32; 33]. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Pillar Case** & **Depth/m** & **Pillar Height/m** & **Pillar Width/m** & **Dip/Degree** & **In-Situ Stress Ratio (k)** & **Roadway Width/m** & **UCS of Intact Ore-Body/MPa** \\ \hline
1 & 95 & 2.0 & 18.3 \(\times\) 18.3 & 22.0 & 2.10 & 4.2 & 27.6 \\
2 & 103 & 1.8 & 20.8 \(\times\) 18.3 & 18.3 & 1.33 & 4.2 & 8.30 \\
3 & 73 & 1.8 & 20.8 \(\times\) 18.3 & 18.3 & 1.71 & 4.2 & 44.70 \\
4 & 65 & 1.8 & 22.4 \(\times\) 22.4 & 14.0 & 1.90 Among many vertical strength formulas, it is still a challenge to choose a suitable formula to calculate vertical pillar strength. The research shows that the vertical coal pillar strength calculated by Equation (21) is in good agreement with the actual strength in India [7]. Equation (21) was used to calculate the strength of the vertical pillar; according to Equation (15), the shear strength was calculated. Then, Equation (18) was used to calculate the factor of safety of each inclined pillar, and the results are shown in Figure 9. For stable inclined pillars, all the factors of safety are greater than 1.0, while for failed inclined pillars, the factors of safety are much less than 1.0. From the results, it is concluded that the proposed shear strength model is well validated with the field cases, which shows that the proposed shear strength model is reasonable and reliable. It should be pointed out that the proposed shear model was only verified by some coal pillar cases. Next, this model will be verified using different rock types of inclined pillars or on-site engineering verification. Moreover, the key to using this model to calculate the strength of inclined pillars is to select a suitable formula to calculate the strength of vertical pillars. Therefore, the selection of appropriate formula for the vertical pillar may be a challenge for this research.