## 4 Results and Discussion
### Cross-Sectional Forces
#### 4.2.4 Support Rate of the Initial Earth Pressure by the Lining  
To make the influence of each parameter on the arching effect clear, Figure 14 shows the support rate of the initial effective earth pressure by the lining in the vertical and horizontal directions, \(r_{\rm v}\) and \(r_{\rm h}\), defined as:  
\[F_{\rm i0}=\sigma^{\prime}_{\rm i0}\times D\ ({\rm i}\ =\ {\rm h},\ {\rm v}) \tag{9}\]  
\[\left\{\begin{array}{c}F_{\rm ev}=N_{\rm L}+N_{\rm R}-\sigma_{\rm w}\times D -W/2\\ F_{\rm eh}=N_{\rm C}+N_{\rm I}-\sigma_{\rm w}\times D\end{array}\right. \tag{10}\]  
\[r_{\rm i}=\frac{F_{\rm ei}}{F_{\rm i0}}\ ({\rm i}\ =\ {\rm h},\ {\rm v}) \tag{11}\]  
where \(F_{0}\) is the initial force on the tunnel section due to the initial effective earth pressure \(\sigma^{\prime}_{0}\); \(F_{\rm e}\) is the force due to the effective earth pressure supported by the lining; \(N_{\rm L}\), \(N_{\rm R}\), \(N_{\rm C}\), and \(N_{\rm I}\) are axial forces in  
Figure 13: Normalized eccentricity.  
the lining at the left and right spring lines, crown, and invert, respectively; \(\alpha_{\rm w}\) is the water pressure at the tunnel center; \(D\) is the tunnel diameter; \(W\) is the self-weight of the lining; and the suffixes v and h show the vertical and horizontal directions, respectively. Here, in the case where the support rate of the initial effective earth pressure \(\sigma^{\prime}_{i0}\) by the lining \(r_{\rm i}\) (i = h and v) is less than 1, the rest of \(\sigma^{\prime}_{i0}\) is expected to be supported by the surrounding ground. On the other hand, in the case where \(r_{\rm i}\) (i = h and v) is larger than 1, the ground reaction force acting on the lining is larger than \(\sigma^{\prime}_{i0}\). For the deep tunnel, \(F_{\rm v0}\) = 2864 kN/m and \(F_{\rm h0}\) = 1432 kN/m, and for the shallow tunnel, \(F_{\rm v0}\) = 1176 kN/m and \(F_{\rm h0}\) = 588 kN/m.  
From Figure 14, the following was found:  
1. \(r_{\rm v}<1<r_{\rm h}\) is for \(k_{\rm n}\leq 100\) MN/m\({}^{3}\) while \(r_{\rm v}\) is close to 0, and \(r_{\rm h}<1\) for \(k_{\rm n}\) = 1000 MN/m\({}^{3}\). This is because (1) for \(k_{\rm n}\leq 100\) MN/m\({}^{3}\), the effective earth pressure acting on the lining, \(\sigma^{\prime}_{\rm n}\) at the crown and invert are at the active side, and the \(\sigma^{\prime}_{\rm n}\) at the spring line is at the passive side due to the ground reaction, and (2) for \(k_{\rm n}\) = 1000 MN/m\({}^{3}\), \(\sigma^{\prime}_{\rm n}\) is always at the active side and \(\sigma^{\prime}_{\rm n}\) is close to zero for the deep tunnel due to ground self-stabilization, as shown in Figures 7b and 8b.
2. As the tangential spring constant, \(k_{\rm t}\), decreases and the load model type changes from type 0 to type 1, \(r_{\rm v}\) increases and \(r_{\rm h}\) decreases. This tendency can be explained the same as in Section 4.1.3 (Tangential Earth Pressure).
3. As the coefficient of the subgrade reaction, \(k_{\rm n}\), increases, \(r_{\rm v}\) and \(r_{\rm h}\) decrease. This is because as \(k_{\rm n}\) increases, \(\sigma^{\prime}_{\rm n}\) drops more in the active side as described in Section 4.1.2 (Normal Effective Earth Pressure). Furthermore, \(r_{\rm v}\) and \(r_{\rm h}\) have almost the same value for the deep tunnel and shallow tunnel. This means that the influence rate of the overburden depth on the initial effective earth pressure \(\sigma^{\prime}_{0}\) and the effective earth pressure \(\sigma^{\prime}_{\rm n}\) is almost the same in this analysis condition.  
Figure 14: Support rate of initial effective earth pressure by lining in the vertical and horizontal direction \(r_{\rm v}\), \(r_{\rm h}\).