## 4 Influence of Lining Thinning on Rock Pressure in Four-Track HSR Tunnels
### Stress Analysis Results  
The stress distribution in the tunnel lining under the influence of thinning defects was analyzed for a shallow four-track HSR tunnel. The representative results are shown in Figure 14, based on engineering conditions of 34 m burial depth and Grade IV rock. Whether viewed from the outer or inner surface, the presence of thinning defects leads to a significant stress redistribution in both the defect-affected zone and the adjacent active zone of the shallow four-track HSR tunnel. Pronounced stress concentration is observed in the lining structure. Moreover, based on the area of the stress concentration zone, it is evident that the thinning defect has a greater impact on the inner side of the lining than on the outer side. This is attributed to the modeling assumption that the thinning defect occurs on the inner surface of the lining in the numerical simulation. Under the thinning defect conditions illustrated in Figure 14a,c,d, the maximum principal stress on the outer side of the lining reaches its peak within the thinning-affected zone, while on the inner side, the maximum stress appears in the active thinning zone. The peak values of maximum principal stress under these three scenarios are 2.066 MPa, 5.015 MPa, and 4.496 MPa, respectively. In contrast, under the scenario shown in Figure 14b, the highest maximum principal stress occurs on the inner side of the vault lining, reaching a peak value of 6.674 MPa. Based on the stress contour maps of the tunnel lining, the distribution curves of the principal stress along the axial direction of the thinning defect were extracted. The influence of different thinning defect parameters--including position (\(P_{\text{j}}\)), longitudinal extent (\(L_{\text{j}}\)), arc length (\(L^{\text{arc}}\)), and thinning-thickness ratio (\(R_{\text{j}}\))--on the maximum principal stress is illustrated in Figure 15. The analysis is based on the engineering conditions of 34 m burial depth and Grade IV rock. As shown in Figure 15a, the presence of thinning defects causes the distribution of the maximum principal stress along the defect axis to exhibit a W-shaped pattern at positions \(P_{\text{j}}=1\) to 6 and an M-shaped pattern at \(P_{\text{j}}=7\), 8, and 9. The peak maximum principal stress reaches 3.89 MPa at \(P_{\text{j}}=6\), while the lowest value of 0.54 MPa occurs at \(P_{\text{j}}=5\). When the defect is located at \(P_{\text{j}}=7\) or beyond, the variation in maximum principal stress is minimal, and the curve approaches a nearly horizontal line. Figure 15b shows that the peak maximum principal stress of the lining affected by thinning defects typically appears near the boundary of the active thinning zone, whereas the minimum stress is observed within the thinning-affected zone. As the longitudinal extent \(L_{\text{j}}\) increases, the W-shaped stress distribution widens. However, the maximum value of principal stress does not increase correspondingly; for instance, when \(L_{\text{j}}=8\) m, the peak stress reaches 5.53 MPa, while increasing \(L_{\text{j}}\) to 24 m causes a reduction in the peak stress to 3.73 MPa. According to Figure 15c, when the thinning arc length \(L^{\text{arc}}_{\text{j}}\) is 4 m, the maximum principal stress is 1.13 MPa. When \(L^{\text{arc}}_{\text{j}}\) increases to 8 m and 16 m, the maximum stress rises sharply to 5.01 MPa. However, further increases to 32 m and 40 m result in decreased \(\sigma_{\text{max}}\) values of 4.31 MPa and 4.04 MPa, respectively. As illustrated in Figure 15d, when the thinning-thickness ratio \(R_{\text{j}}<\) is 0.75, the W-shaped stress distribution is nearly flat, and the peak principal stress is 2.18 MPa. When \(R_{\text{j}}\) is reduced to 0.50, \(\sigma_{\text{max}}\) increases to 3.81 MPa. A further reduction to \(R_{\text{j}}=\) 0.25 causes a more pronounced W-shape with a greater vertical variation, although \(\sigma_{\text{max}}\) decreases to 1.84 MPa. According to the railway tunnel design code [16], the flexural tensile strength of C30 concrete is 0.55 MPa.