## 2 Simplified Analytical Solution  
In this section, a simplified analytical solution is developed to evaluate the responses of a tunnel with flexible joints through a soft-hard stratum junction under earthquake motion, as shown in Figure 1. A tunnel with three regions is designed where the two regions on the right side are thickened, which is often adopted for the sudden change in rock type and soft rock. In addition, the flexible connection is set between the regions. The transition Region 2 consists of two segments and the flexible connection is also set between them, which is intended to mitigate the influence of the sharp change in properties of the surrounding rock on the tunnel responses. The other two regions, which consist of one segment each, are embedded in hard layer (Rock 1) and soft layer (Rock 2), respectively. Note that Rock 1 is harder than Rock 2. To simplify the problem, the tunnel is assumed to behave as a shear beam on viscoelastic foundation. The following assumptions are made:  
1. Each lining segment has homogeneous isotropic linear elasticity and its sounding rock is viscoelastic.
2. The axial deformation of the tunnel is not considered.
3. The seismic input motion is assumed as an incident sinusoidal wave.  
Considering an idealized model of a tunnel with flexible joints, a mechanical model shown in Figure 2 is built. In the model, a mechanical model of shear spring and rotation spring is assumed to model the flexible joint between lining segments, which considers the differences of both displacement and rotation angle, as suggested by Ref [9,20]. This can accommodate the movement caused by non-homogeneous ground. \(C_{\text{w}k}\) and \(C_{\text{w}k}\) are the stiffness coefficients of extensional spring and rotational spring, respectively. Three joints were designed and denoted as Joint 1, Joint 2, and Joint 3. The lining segments are supported by foundations with different spring coefficients \(K_{1}\) and \(K_{2}\) and different damping coefficients \(c_{1}\) and \(c_{2}\), respectively. The stiffness and mass per unit length of the lining segment are \(EI_{j}\) and \(\rho A_{j}\) (j = 1,2,3,4), respectively, where \(E\) is the Young's modulus of elasticity, \(I_{j}\) and \(A_{j}\) are the moment of inertia and area of the cross section, respectively, and \(\rho\) is the density of the lining. For the sake of simplicity, the transverse shear wave is considered here. The transverse displacements of the free field under sinusoidal shear motions, which is propagating parallel to the tunnel axis, is defined as \(w_{g}\) (\(x\),\(t\)). \(w_{g}\) is expressed as:  
\[\begin{array}{l}w_{g1}=w_{\text{max1}}\cos\theta\sin(\frac{2\pi}{\lambda_{1}/ \cos\theta}x+\alpha_{0})\\ w_{g2}=w_{\text{max2}}\cos\theta\sin(\frac{2\pi}{\lambda_{2}/\cos\theta}x+ \alpha_{0})\end{array} \tag{1}\]  
where \(w_{\text{max1}}\)(\(w_{\text{max2}}\)) is the peak free field transverse displacement in Rock 1 (Rock 2), which can be obtained by a design response spectrum or site response analysis techniques [19], e.g., DEEPSOIL [21]; \(\lambda_{1}\) and \(\lambda_{2}\) are the wavelength of the transverse motions in hard and soft layers, respectively; \(\theta\) is the incident angle earthquake wave; and \(\alpha_{0}\) is the displacement phase angle.  
Figure 1: Longitudinal profile of the tunnel.