## 3 Analytical Results
### General Responses  
To investigate the influences of the soft-hard stratum junction on the tunnel responses, three cases are considered: (1) a uniform tunnel crossing soft-hard stratum junction; (2) a uniform tunnel buried in uniform hard layer with spring constants of foundation \(K_{1}\) = \(K_{2}\) = 0.63 GPa and wavelength \(\lambda_{1}\) = \(\lambda_{2}\) = 344 m; and (3) a uniform tunnel buried in uniform soft layer with spring constants of foundation \(K_{1}\) = \(K_{2}\) = 0.36 GPa and wavelength \(\lambda_{1}\) = \(\lambda_{2}\) = 215 m. The stiffness of the uniform tunnel is constant with 7.53 \(\times\) 10\({}^{12}\) N\(\cdot\)m\({}^{2}\), and no damping and no flexible joints are adopted for all three cases. Other parameters are same as those introduced in Section 4.1.  
Figure 4 shows the internal force responses of a uniform tunnel across a soft-hard stratum junction (Case 1). As it can be seen, two peak bending moments occur near sharp contact, which means the sudden change of the ground layers leads to significant bending moments in the contact area of non-homogeneous layers. This phenomenon is in agreement with the finding from [19]. It is interesting to see that the largest bending moment is located at the side of soft layer near the sharp contact area, which is accordance with shaking table test results [6]. To put it another way, the most unfavorable location of the bending moment is determined by the combination of the sharp contact and the weak surrounding rock,  
Figure 3: Input earthquake wave.  
which is the key position for aseismic design. Figure 4 shows that the maximum shear force appears at contact, and the farther from the contact, the smaller the shear force. Compared to the analytical solution and the existing research, the longitudinal responses have a lot in common, showing that the closed-form analytical solutions are accurate.  
In order to better describe the influence of sharp change of ground layers on the seismic response of the tunnel structure, the results of case 1 are normalized, where the normalized values are calculated with respect to those of case 2 in the hard layer and the normalized values are calculated with respect to those of case 3 in the soft layer. It can be clearly seen from Figure 5 that the internal forces are obviously amplified around the soft-hard interface. At the same time, the maximum internal force amplification factors occur in the range of about 1 \(D\) (\(D\) is the outer diameter of the tunnel). In terms of moment, the maximum amplified factor occurs in the hard layer near the interface along the tunnel axis, which indicates that the tunnel in the hard layer should also be adopted for anti-seismic measures when a tunnel passes through a soft-hard stratum. Therefore, the transition zone in relatively hard rock is needed as well. There are two peak amplified factors of shear force appearing near the interface where the largest one is located in the soft rock near the interface, as shown in Figure 5b. Above all, the influence of the soft-hard stratum junction on the tunnel responses is remarkable. The maximum amplified factor is in relatively hard rock for the bending moment, and is in soft rock for the shear force.  
Figure 4: Responses of a tunnel buried in non-homogeneous ground (case 1).  
Figure 5: Influences of the soft-hard stratum junction on normalized internal forces. (**a**) Normalized bending moment; (**b**) normalized shear force.