## 4 Distribution of the Stresses around the Tunnel  
In engineering studies, it is often necessary to determine the magnitude and orientation of the maximum, intermediate, and minimum stresses acting on the rock mass. Through the aforementioned inversion calculations, the normal and shear stresses in the rock mass were obtained. According to [PERSON] [22] and [PERSON] [23], if we can obtain the six stress components (\(\sigma_{x},\sigma_{y},\varepsilon_{z},\tau_{xy},\tau_{yz},\tau_{xz}\)) of the in situ stress field in the \(xyz\) coordinate system, the principal stresses (both magnitude and orientation) of the stress field could be indirectly calculated. In this study, based on the principles of elasticity and previous research [24; 25], the formulae for calculating the principal stresses (\(\sigma_{1}\), \(\sigma_{2}\), \(\sigma_{3}\)) and the direction cosines (\(l_{x}\), \(m_{y}\), \(n_{z}\)) were derived as described below. The formulae for calculating the principal stresses are as follows:  
\[ \begin{array}{l}\sigma_{1}=\frac{l_{1}}{3}+R\cos\frac{\sigma}{3}\\ \sigma_{2}=\frac{l_{1}}{3}+R\cos\frac{\sigma+2\pi}{3}\\ \sigma_{3}=\frac{l_{1}}{3}+R\cos\frac{\sigma+4\pi}{3}\end{array}  \tag{7}\]  
where  
\[ \begin{array}{l}I_{1}=\sigma_{x}+\sigma_{y}+\sigma_{z}\\ I_{2}=\sigma_{x}\sigma_{y}+\sigma_{y}\sigma_{z}+\sigma_{z}\sigma_{x}-(\sigma_ {xy}^{2}+\sigma_{yz}^{2}+\sigma_{zx}^{2})\\ I_{3}=\sigma_{x}\sigma_{y}\sigma_{z}+2\tau_{xy}\tau_{yz}\tau_{xz}-(\sigma_{x} \sigma_{yz}^{2}+\sigma_{y}\sigma_{zx}^{2}+\sigma_{z}\sigma_{xy}^{2})\\ R=\frac{2}{3}\sqrt{I_{1}^{2}-3I_{2}}\\ \cos\varphi=\frac{2l_{1}l_{2}+2I_{3}}{2(l_{1}^{2}-3I_{2})^{\frac{1}{2}}}\end{array}  \tag{8}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Measurement Point** & & \(\varepsilon_{x}\) & \(\varepsilon_{y}\) & \(\varepsilon_{z}\) & \(\gamma_{xy}\) & \(\gamma_{yz}\) \\ \hline \multirow{2}{*}{A} & Inversion value (\(\mu\)e) & - & 78.95 & \(-\)16.74 & - & \(-\)80.62 \\  & Error (\%) & - & 1.31 & 4.59 & - & 2.87 \\ \hline \multirow{2}{*}{B} & Inversion value (\(\mu\)e) & - & \(-\)58.41 & 18.60 & - & \(-\)75.56 \\  & Error (\%) & - & 2.66 & 3.34 & - & 0.74 \\ \hline \multirow{2}{*}{C} & Inversion value (\(\mu\)e) & 0.76 & 216.69 & - & \(-\)125.87 & - \\  & Error (\%) & - & 1.05 & - & 0.7 & - \\ \hline \multirow{2}{*}{D} & Inversion value (\(\mu\)e) & 7.91 & \(-\)372.88 & - & \(-\)38.75 & - \\  & Error (\%) & - & 0.78 & - & 7.74 & - \\ \hline \multirow{2}{*}{E} & Inversion value (\(\mu\)e) & 11.63 & \(-\)166.29 & - & \(-\)24.81 & - \\  & Error (\%) & - & 1.02 & - & 0.76 & - \\ \hline \multirow{2}{*}{F} & Inversion value (\(\mu\)e) & 14 & \(-\)129.91 & - & \(-\)107.93 & - \\  & Error (\%) & - & 1.05 & - & 0.06 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Analysis of strain errors. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \(\sigma_{x}\) & \(\sigma_{y}\) & \(\sigma_{z}\) & \(\tau_{xy}\) & \(\tau_{xz}\) & \(\tau_{yz}\) \\ \hline Inverted Values (MPa) & 34.720 & 25.380 & 15.982 & \(-\)3.400 & \(-\)6.954 & 6.065 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Inferred stress values from field inversion. The direction cosines of the principal stress vector relative to the \(x\), \(y\), and \(z\)-axes are as follows:  
\[l_{xi} =\frac{A_{i}}{\sqrt{A_{i}^{2}+B_{i}^{2}+C_{i}^{2}}}\] \[m_{yi} =\frac{B_{i}}{\sqrt{A_{i}^{2}+B_{i}^{2}+C_{i}^{2}}} \tag{9}\] \[n_{zi} =\frac{C_{i}}{\sqrt{A_{i}^{2}+B_{i}^{2}+C_{i}^{2}}}\]  
where  
\[A_{i} =\tau_{xy}\tau_{yz}-(\sigma_{y}-\sigma_{i})\times\tau_{zx}\] \[B_{i} =\tau_{xy}\tau_{zx}-(\sigma_{x}-\sigma_{i})\times\tau_{xz}\qquad i =1,2,3 \tag{10}\] \[C_{i} =(\sigma_{x}-\sigma_{i})*(\sigma_{y}-\sigma_{i})-\tau_{xy}{}^{2}\]  
By substituting the six stress components \((\sigma_{x},\sigma_{y},\sigma_{z},\tau_{xy},\tau_{yz},\tau_{xz})\) obtained from the inversion into these equations, the principal stresses and their direction cosines with respect to the coordinate axes for the \(-930\) level of Macheng Iron Mine were calculated using Equations (7)-(10).