## 5 Conclusions  
A parameter study was conducted to investigate the effects of the ground-lining interaction in the tangential direction on lining response, using the beam-spring model and ground springs. Tangential ground spring stiffness \(k_{\rm t}\), load model type (the initial tangential earth pressure, \(\tau_{0}\)), normal ground spring stiffness \(k_{\rm n}\), and overburden depth \(h\) were used as parameters. As a result, the following was found:  
1. The distribution of normal effective earth pressure acting on lining \(\sigma_{\rm n}^{\prime}\) mainly obtains the influence of \(k_{\rm n}\) and \(h\), and the distribution of tangential earth pressure acting on lining \(\tau\) is influenced by \(k_{\rm t}\) and the load model type (i.e., \(\tau_{0}\)). Lining displacement is defined by \(\sigma_{\rm n}^{\prime}\), \(\tau\), and buoyancy and determines the distributions of the bending moment, \(M\). The distribution of \(N\) is determined by \(\sigma_{\rm n}^{\prime}\), \(\tau\), water pressure \(\sigma_{\rm w}\), and the lining self-weight. The influence of each parameter is propagated through the above mechanism. 2. The \(k_{\rm t}\) restricts the lining displacement in the tangential direction, and the axial force in the lining is generated in the opposite direction of the lining displacement. Therefore, as \(k_{\rm t}\)/\(k_{\rm n}\) changes from 0 to 1: (1) The lining shape and distribution shapes of \(M\) and \(N\) become more circular. (2) For \(k_{\rm n}\leq 100\)MN/m\({}^{3}\), the absolute \(M\) decreases at most to 23%. For \(k_{\rm n}\geq 500\) MN/m\({}^{3}\), the magnitude of \(M\) becomes close to zero. (3) The \(N_{\rm max}\) changes within 5%. (4) The change of the normalized eccentricity, \(e\)/\(t\), is less than 0.05. 3. The shear stress appears at the boundary toward the spring line for load model type 1 under \(K_{\rm h0}<1\). Therefore, as the load model changes from type 0 to type 1, (1) the lining shape and distribution shapes of \(M\) and \(N\) become flat in the horizontal direction; (2) for \(k_{\rm n}\leq 100\) MN/m\({}^{3}\), the absolute \(M\) increases to at most 58%; (3) the \(N_{\rm max}\) increases at most to 9%; and (4) under \(e\)/\(t>0.167\), \(e\)/\(t\) increases 1.4 times for the deep tunnel and 1.7 times for the shallow tunnel. The case of \(e\)/\(t>0.167\), which means tensile stress appears in the lining, corresponds to \(k_{\rm n}=10\) MN/m\({}^{3}\) for the deep tunnel and \(k_{\rm n}<50\) MN/m\({}^{3}\) for the shallow tunnel. This indicates that the load model has significant effects on sectional forces, especially in shallow tunnels in soft soils. 4. As \(k_{\rm n}\) increases, under the same displacement, the normal effective earth pressure \(\sigma_{\rm n}^{\prime}\) increases at the passive side and decreases at the active side. As a result, the lining shape and distribution shapes of \(M\) and \(N\) become more circular, and \(M\), \(N\), and \(e\)/\(t\) are drastically reduced at \({\rm mboxemphk_{\rm n}\geq 500\) MN/m\({}^{3}}\). 5. As the overburden depth, \(h\), decreases, the effective earth pressure \(\sigma_{\rm n}^{\prime}\) and water pressure \(\sigma_{\rm w}\) decrease, but the difference of the initial effective earth pressures at the crown and invert is constant. Then, (1) the lining shape and the distribution shape of \(M\) become more circular, (2) \(N\) decreases, and (3) \(e\)/\(t\) increases by about 1.6 times. 6. In the case of a shallow tunnel in soft soil where \(e\)/\(t>0.167\), tensile stress appears in the lining. Therefore, the tangential ground-lining interaction conditions, such as the initial tangential earth pressure due to load model and the tangential spring constant, should be considered carefully since these conditions significantly influence the bending moment in the segmental lining. 7.