## 3 Example and Discussion
### Comparison of Results with and without Considering External Water Pressure \(p_{1}\)  
As discussed earlier, numerous calculation methods employed in the design of water conveyance tunnel lining structures tend to err on the side of caution, disregarding the reinforcing effect of external water pressure (\(p_{1}\)) on the lining's bearing capacity and merely  
Figure 2: Stresses on boundary \(L_{1}\), \(L_{2}\).  
teresting it as a safety margin. This section aims to quantify the influence of external water pressure (\(p_{1}\)) on the lining's bearing capacity by comparing stress results (\(\sigma_{1}^{r}\), \(\sigma_{2}^{r}\), \(\sigma_{1}^{\theta}\), \(\sigma_{2}^{\theta}\), \(\sigma_{3}^{\theta}\)) at the boundaries of \(L_{1}\) and \(L_{2}\) with and without internal water pressure \(p_{1}\)). The parameters utilized are consistent with those outlined in Section 3.1, and the findings are depicted in Figures 3 and 4.  
Figure 4: Comparison of tangential normal stress results with and without considering external hydraulic pressure \(p_{1}\).  
Figure 3: Comparison of radial normal stress results at \(L_{1}\) boundary with and without considering external water pressure \(p_{1}\).  
From Figure 3, we can observe the radial stress results at the L\({}_{1}\) interface between the surrounding rock and the lining. Without considering external water pressure, the radial stress results of the surrounding rock and the lining coincide, indicating solely the interaction force stemming from tunnel excavation. However, when external water pressure is taken into account, several changes occur. The radial contact compressive stress on the surrounding rock decreases, signifying a reduction in the interaction force between the lining and the surrounding rock. Conversely, the radial compressive stress on the lining increases, satisfying \(\sigma_{1}^{r}-\sigma_{2}^{r}=-0.2\) MPa (being equal to the external water pressure). It is worth noting that the radial stress of the lining's outer boundary reflects the superposition of geotechnical pressure and external water pressure induced by tunnel excavation.  
Figure 4 illustrates the comparison of the influence of external water pressure (\(p_{1}\)) on the tangential normal stress distribution of the surrounding rock and lining:  
(1) The tangential normal stress \(\sigma_{1}^{\theta}\) of the surrounding rock at \(L_{1}\) exhibits close alignment in both cases. This suggests that under the parameter conditions of this example, the influence of external water pressure (\(p_{1}\)) on the tangential normal stress of the surrounding rock is very small;  
(2) With consideration of external water pressure (\(p_{1}\)), the tangential normal stress \(\sigma_{2}^{\theta}\) at the outer boundary (\(L_{1}\)) of the lining remains in a compressive stress state across the entire domain. Conversely, when external water pressure (\(p_{1}\)) is not considered, a tension-compression stress state coexists. The maximum difference between the two scenarios can reach approximately 2 MPa. Similarly, when external water pressure (\(p_{1}\)) is considered, the tangential normal stress \(\sigma_{3}^{\theta}\) at the inner boundary (\(L_{2}\)) of the lining remains in a compressive stress state throughout the domain. Conversely, without consideration of external water pressure (\(p_{1}\)), a tension-compression stress state is evident. The difference between the two scenarios can also reach about 2 MPa.  
In summary, the presence of external water pressure can mitigate the phenomenon of lining expansion induced by high internal water pressure. Furthermore, the greater the external water pressure, the more pronounced its effect, thereby enhancing the performance of the lining structure operating under high internal water pressure conditions.