# Stress Analysis and Spalling Failure Simulation on Surrounding Rock of Deep Arch Tunnel
## 4 Discussion
### Stress Distribution Analysis of Surrounding Rock under Different Working Conditions
The lateral pressure coefficient (\(\lambda\)) was calculated according to the initial failure stresses (see Table 3), which is 0.395, 0.386, and 0.494 for specimens A12-42.2-27.1, A13-22.2-22.2, and A14-17.3-28.9, respectively. Substituting the analytical functions in Equations (23), (25) and (27) into Equation (4), the stress at different hole boundary locations of the three specimens was obtained, as shown in Figure 17. The visual stress distribution in the hole boundary is shown in Figure 18. It can be seen from Figure 17 that the stress at the arch foot is the highest, followed by that at the arch shoulder. The highest stress concentration coefficients at the foot of sidewalls of reached 6.85 \(p\), 8.24 \(p\), and 8.53 \(p\). At this moment, the tangential stresses at the foot sidewalls are 471.9 MPa, 473.4 MPa, and 499.5 MPa, which are 3.26, 3.27, and 3.45 times the UCS respectively. A slight buckling occurred at the arch foot, as shown in Figures 7-9. When the lateral pressure coefficient is in the range of 0.38-0.50, tensile stress acted on the floor, and the tensile stress was transferred to compressive stress on the roof. However, the floor always showed a higher stress than the roof. Although tensile stress was present on the roof, there was less room for tensile deformation to occur on the roof, and thus tensile failure did not occur. The comparative tangential stress occurred at both corners, which acts as the compressive stresses and decreases to the minimum at the sidewalls. The experimental results showed that the spalling of rock flakes occurred on both sidewalls, and the particle ejection occurred at the four corners. The particle ejection is caused by the stress concentration, while the spalling of rock flakes is caused by the axial micro-cracks generated at the four corners, which is a spalling course towards the free face under tangential stress. Comparing the initial failure stress \(\sigma_{\rm Zi}\) and the maximum tangential stress \(\sigma_{\rm omax}\) of specimens A12-42.2-27.1 and A13-22.2-22.2, with similar lateral pressure coefficient, it was found that the excavation in the direction of the maximum horizontal principal stress can improve the stability of the tunnel, i.e., \(\sigma_{\rm X}>\sigma_{\rm Y}\). The \(\sigma_{\rm omax}\) of holes can be reduced in this manner. Comparing the stress states of specimens A12-42.2-27.1 and A14-17.3-28.9, it can be seen that the initial failure stress \(\sigma_{\rm Zi}\) of the semi-circular arch with \(f/B=1/2\) is greater than that of the three-centered arch with \(f/B=1/4\) under constant \(\sigma_{\rm Y}\). This indicates that the pressure-bearing capacity of the semi-circular arch tunnel is greater than that of the three-centered arch tunnel. Figure 17 shows that when the polar angle is in the range of 0-90\({}^{\circ}\) (corresponding to the arc segment of the arch hole), the growth rate of the tangential stress follows the semi-circular arch hole \(<\) three-centered arch hole combined with the tangential stress distribution of the arc segment in Figure 17. The growth rate of the tangential stress accelerates with the increase in arch height. Comparing the \(\sigma_{\rm Zi}\) of \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Specimen** & \(\sigma_{\rm X}\)**/(MPa)** & \(\sigma_{\rm Y}\)**/(MPa)** & \(\sigma_{\rm Z}\)**/(MPa)** & \(\sigma_{\rm Z}\)**/(MPa)** & \(\sigma_{\rm Z}\)**/(MPa)** \\ \hline A12-42.2-27.1 & 42.2 & 27.1 & 21.6 & 68.6 & 68.6 \\ A13-22.2-22.2 & 22.2 & 22.2 & 17.6 & 57.5 & 65.6 \\ A14-17.3-28.9 & 17.3 & 28.9 & 13.5 & 58.5 & 60.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The initial failure stresses of tunnel walls. specimens A13-22.2-22.2 and A14-17.3-28.9 shows that, although the arch height decreases by 4.17 mm, the initial failure stress of the three-centered arch holes is similar.