## 4 Results and Discussion
### Cross-Sectional Forces  
#### 4.2.1 Bending Moment  
Figures 7d and 8d show the distribution of bending moment \(M\) for the deep tunnel and the shallow tunnel. To make each parameter's effect clear, Figure 9 shows the maximum and minimum moment (\(M_{\mathrm{max}}\), \(M_{\mathrm{min}}\)), respectively. From these figures, the following was found:  
1. The distribution shape of \(M\) is flat in the horizontal direction.
2. The distribution shape of \(M\) becomes more uniform as \(k_{\mathrm{t}}\) and \(k_{\mathrm{n}}\) increase, load model type 0 is compared with load model type 1, and the overburden decreases.
3. For \(k_{\mathrm{n}}\leq 100\) MN/m\({}^{3}\), the maximum value of the absolute \(M\) shows a maximum of a 23% reduction as \(k_{\mathrm{t}}/k_{\mathrm{n}}\) increases from 0 to 1, and they show approximately a 58% increment as the load model changes from type 0 to type 1. For \(k_{\mathrm{n}}\geq 500\) MN/m\({}^{3}\), the magnitude of \(M\) becomes close to zero. This can be explained as the same as in Section 4.1.1 (Lining Displacement) since the distribution of \(M\) basically reflects the lining deformation in Figures 7a and 8a.
4. The \(M\) at the longitudinal joints of the ring is close to 0; on the other hand, around these longitudinal joints, the \(M\) in the next ring has a maximum/minimum value, especially \(k_{\mathrm{n}}\leq 100\) MN/m\({}^{3}\). This is because (1) the bending stiffness of the longitudinal joints is smaller than that of the segment itself, and (2) a certain moment in one ring is transferred through the next ring by the circumferential joints in the case of a staggered arrangement.
5. In the case of the deep tunnel, a very stiff ground (\(k_{\mathrm{n}}=1000\) MN/m\({}^{3}\)), \(k_{\mathrm{t}}/k_{\mathrm{n}}=0\), and load model type 1, \(M\) fluctuates significantly at the lower part while in other cases, the \(M\) distributions are more stable. This is because (1) when \(k_{\mathrm{t}}/k_{\mathrm{n}}=0\) and the load model is type 1 (with \(\tau_{0}\)), the lining deforms most flat in the horizontal direction; (2) when \(k_{\mathrm{n}}\) is larger, ground self-stabilization isexpected with a small active displacement; (3) in the deep tunnel, high water pressure causes lining shrinkage; that is, the displacement in the active side appears; (4) because of buoyancy, the tunnel moves upward; (5) the segments are installed in staggered arrangements; and (6) therefore, while the upper part of the lining is supported by the ground, the lower part of the lining can move freely in the normal direction because of the gap between the lining and ground, as shown in Figure 10 which presents the lining displacements in the normal and tangential directions, \(u_{\rm n}\) and \(u_{\rm t}\), in the case of \(k_{\rm n}=1000\) MN/m\({}^{3}\) in the deep tunnel. Thus, \(M\) has peaks at the longitudinal joints of the next rings at the lower part.  
Figure 10: Lining displacement for ground at \(k_{\rm n}=1000\) MN/m\({}^{3}\) (deep tunnel). (**a**) Normal displacement (+: outward); (**b**) Tangential displacement (+: counterclockwise).  
Figure 9: Maximum and minimum bending moment (kN-m/m).