## 4 Results and Discussion
### Cross-Sectional Forces
#### 4.2.2 Axial Force  
The axial force in the segmental lining, \(N\), results from the normal force on the lining, such as normal effective earth pressure and water pressure; the tangential force on the lining, such as the tangential force due to the tangential springs and the tangential earth pressure due to the load model; and the vertical force due to the self-weight of the lining. Therefore, the axial force distribution is determined by the normal effective earth pressure described in Section 4.1.2, the tangential earth pressure described in Section 4.1.3, the water pressure, and the self-weight of the lining. Figures 7e and 8e show the distribution of axial force for the deep tunnel and the shallow tunnel, respectively. To make each parameter's effect clear, Figure 11 shows the maximum axial force \(N_{\text{max}}\), and Figure 12 shows the \(F_{\text{h}}\)/\(F_{\text{v}}\) ratio defined as follows:  
\[\frac{F_{\text{h}}}{F_{\text{v}}} =\frac{N_{\text{C}}+N_{\text{I}}}{N_{\text{L}}+N_{\text{R}}} \tag{7}\]  
where \(N_{\text{L}}\), \(N_{\text{R}}\), \(N_{\text{C}}\), and \(N_{\text{I}}\) are axial forces in the lining at the left and right spring line, crown, and invert, respectively. \(F_{\text{h}}\)/\(F_{\text{v}}\) shows the shape of the axial force distribution, which becomes horizontally long for \(F_{\text{h}}\)/\(F_{\text{v}}\)\(<\) 1 and vertically long for \(F_{\text{h}}\)/\(F_{\text{v}}\)\(>\) 1. From these figures, the following was found:  
1. The distribution shape of \(N\) is flat in the horizontal direction, while that of \(\sigma_{\text{n}}^{\prime}\) is flat in the vertical direction. This is because the \(N\) comes from the \(\sigma_{\text{n}}^{\prime}\) initial tangential stress \(\tau_{0}\), tangential stress \(\tau\) due to \(k_{\text{t}}\), water pressure \(\sigma_{\text{w}}\), and the self-weight of the lining. 2. As the tangential spring stiffness, \(k_{\text{t}}\), increases, \(F_{\text{h}}\)/\(F_{\text{v}}\) increases; that is, the distribution shape of \(N\) deforms to the vertical outward direction and the horizontal inward direction. This is because the tangential spring restricts the lining deformation to be flat in the horizontal direction under \(K_{\text{h}0}\)\(=\) 0.5, and it causes compression force to the lining in the horizontal direction and tensile force to the lining in the vertical direction. 3. As the load model changes from type 0 to type 1, \(F_{\text{h}}\)/\(F_{\text{v}}\) decreases; that is, the distribution shape of \(N\) changes from being flat in the vertical direction to being flat in the horizontal direction. This is because the shear earth pressure generates shear force on the lining to the horizontal outward direction, and this causes an increase in \(N\) in the vertical direction and a decrease in \(N\) in the horizontal direction. This tendency is the reverse of that caused by \(k_{\text{t}}\). 4. The coefficient of subgrade reaction \(k_{\text{n}}\) and the overburden depth influence the magnitude of \(N_{\text{max}}\) in Figure 11 but do not affect the distribution shape of \(N\) as shown in Figures 7 and 8. This can be explained the same way as \(\sigma_{\text{n}}^{\prime}\) in Section 4.1.2 (Normal Effective Earth Pressure). 5. As for \(N_{\text{max}}\) in Figure 11, as \(k_{\text{t}}\) increases, \(N_{\text{max}}\) increases for load model type 0, and \(N_{\text{max}}\) decreases for load model type 1. This is because \(N_{\text{max}}\) appears at the invert for load model type 0 and at the spring line for load model 1, as shown in Figures 7, 8 and 12, and the influences of \(k_{\text{t}}\) on \(F_{\text{h}}\) and \(F_{\text{v}}\) are reversed. 6. As \(k_{\text{n}}\) increases and overburden depth decreases, \(N_{\text{max}}\) in Figure 11 decreases. This can be explained the same way as \(\sigma_{\text{n}}^{\prime}\) in Section 4.1.2 (Normal Effective Earth Pressure). 7.