## 3 Results
### Stress Distribution Characteristics
As \(\beta\) increases to \(15^{\circ}\) (Figure 4b), it can be observed that due to the inclination of the principal stress, the stress distribution along the opening boundary begins to exhibit slight distortion, with the positions of maximum and minimum stress shifting slightly. The stress distribution curve shows a certain degree of asymmetry. As \(\beta\) continues to increase to \(45^{\circ}\) (Figure 4d), this asymmetry in stress becomes more pronounced under low lateral pressure coefficients (\(\lambda\)). However, as \(\lambda\) increases, this asymmetry gradually decreases. In all cases, when \(\lambda=1\), the stress distribution along the opening boundary is symmetrical along the geometric symmetry axis of the opening. As \(\beta\) increases from \(45^{\circ}\) to \(90^{\circ}\) (Figure 4d-g), the stress distribution exhibits new characteristics, gradually returning to a symmetrical state. When \(\beta\) reaches \(90^{\circ}\) (Figure 4g), meaning the horizontal stress is the maximum principal stress, the stress distribution becomes symmetrical again but differs significantly from when \(\beta=0^{\circ}\). Although the two straight wall corners of the opening remain the areas of maximum compressive stress, the roof and floor have transitioned from regions of tensile stress concentration to regions of compressive stress concentration under low lateral pressure coefficients. Conversely, the original sides of the opening have shifted from compressive stress to tensile stress states. Figure 5 summarizes the variation in the maximum and minimum hoop stresses at the opening boundary with different \(\lambda\) as \(\beta\) changes. Overall, the maximum hoop stress exhibits a trend in first increasing and then decreasing with \(\beta\). For all \(\lambda\) values, the maximum hoop stress curve reaches a peak at \(\beta=45^{\circ}\), indicating that at this angle, the hoop stress on the tunnel boundary is the highest. This trend is most pronounced when \(\lambda=0\). As \(\lambda\) increases, while the overall trend remains consistent for \(\lambda<1\), the variation curve becomes flatter. This flattening indicates a reduced sensitivity in the maximum hoop stress to changes in \(\beta\) at higher lateral pressure coefficients, which may imply greater stability under varying stress orientations. When \(\lambda=1\), the principal stress orientation no longer has any impact on the stress at the opening boundary. The variation trend in the minimum hoop stress with \(\beta\) is relatively smooth. Near \(\beta=45^{\circ}\), the minimum hoop stress fluctuates slightly, but the changes are minor. When \(\lambda<0.4\), the minimum hoop stress is generally less than 0, indicating that tensile stress is likely to form at the opening boundary under low lateral pressure coefficients. However, as \(\lambda\) increases, the minimum hoop stress gradually rises, and the tensile stress regions \begin{table}
\begin{tabular}{c c c c c} \hline \multicolumn{2}{c}{**Principal Stress State**} & \multicolumn{3}{c}{**Non-Principal Stress State**} \\ \hline \(\lambda\) & \(\beta\) & \(\sigma_{x}\) & \(\sigma_{y}\) & \(\tau_{xy}\) \\ \hline \multirow{6}{*}{1.0} & 0 & 1.000 & 1.000 & 0.000 \\  & 15 & 1.000 & 1.000 & 0.000 \\  & 30 & 1.000 & 1.000 & 0.000 \\  & 45 & 1.000 & 1.000 & 0.000 \\  & 60 & 1.000 & 1.000 & 0.000 \\  & 75 & 1.000 & 1.000 & 0.000 \\  & 90 & 1.000 & 1.000 & 0.000 \\ \hline \end{tabular}
\end{table} 
Table 1: _Cont._transition to compressive stress regions. When \(\lambda=1.0\), the minimum hoop stress at the opening boundary stabilizes at around \(0.61p\). This analysis shows that the distribution of hoop stress around the three-center arch opening exhibits significant complexity with changes in the principal stress direction and lateral pressure coefficient. When the maximum principal stress is either vertical or horizontal, the stress distribution is relatively symmetrical. However, as the principal stress angle changes and \(\lambda<1\), the stress distribution gradually becomes asymmetrical.