## 2 Simplified Analytical Solution
### Closed-Form Solution for Responses of Tunnel Lining under Earthquake Wave  
Using the Hamilton principle and employing the shear beam theory, the governing equation of the tunnel is given by [22,23]:  
\[\left\{ \begin{array}{l}EI\varphi^{\prime\prime}+\kappa GA(w^{\prime}-\varphi)=0 \\ \kappa GA(w^{\prime\prime}-\varphi^{\prime})-\mu\ddot{w}-K(w-w_{g})-c(\dot{w}- \dot{w}_{g})=0\end{array} \right. \tag{2}\]  
where \(\varphi\) is the bending angle; \(\kappa\) is the shear correction coefficient; \(G\) is the shear modulus; \(\mu=\rho A\) and \(\gamma=\rho I\) represent the mass and the rotation inertia of the beam per unit length, respectively; \(K\) and \(c\) are the foundation modulus and damping coefficient, respectively; the prime indicates a derivative with respect to the spatial coordinate \(x\); and the dot stands for a derivative with respect to time. In the case where the load is a harmonic load \(w_{g}(x,t)=W_{g}(x)\cdot e^{i\Omega t}\), the displacement and the rotation angle can be also expressed as \(w(x,t)=W(x)\cdot e^{i\Omega t}\) and \(\varphi(x,t)=\Phi(x)\cdot e^{i\Omega t}\), respectively. Inserting those expressions in Equation (1) will eliminate the time variable, and the equation will be recast to:  
\[\left\{ \begin{array}{l}EI\Phi^{\prime\prime}+\kappa GA(W^{\prime}-\Phi)=0\\ \kappa GA(W^{\prime\prime}-\Phi^{\prime})+\mu\Omega^{2}W-KW-ic\Omega W=-KW_{g} -ic\Omega W_{g}\end{array} \right. \tag{3}\]  
where \(i\) is the imaginary unit, \(\Omega\) is loading frequency, and \(t\) is time. From Equation (3), it can be simplified to:  
\[W^{\prime\prime\prime}+\frac{\mu\Omega^{2}-K-ic\Omega}{\kappa GA}W^{\prime \prime}+(\frac{-\mu\Omega^{2}+K+ic\Omega}{EI})W=-\frac{K+ic\Omega}{\kappa GA}W ^{\prime\prime}_{g}+\frac{K+ic\Omega}{EI}W_{g} \tag{4}\]  
Then it can be symbolically presented as:  
\[W^{\prime\prime\prime}+a_{1}W^{\prime\prime}+a_{2}W=b_{1}W^{\prime\prime}_{g} (x)+b_{2}W_{g}(x) \tag{5}\]  
where  
\[ \begin{array}{l}a_{1}=\frac{\mu\Omega^{2}-K-ic\Omega}{\kappa GA},\ a_{2}=- \frac{\mu\Omega^{2}}{EI}+\frac{K}{EI}+\frac{ic\Omega}{EI}\\ b_{1}=-(\frac{K}{\kappa GA}+\frac{ic\Omega}{\kappa GA}),\ b_{2}=\frac{K+ic \Omega}{EI}\end{array}  \tag{6}\]  
By means of the superposition principle, the solution of Equation (5) can be input in the following form [24]:  
\[W(x)=\int\limits_{0}^{L}W_{g}(x_{0})G(x,x_{0})dx_{0} \tag{7}\]  
Figure 2: Analytical model. where \(G(x,x_{0})\) is the Green's function to be determined. \(G(x,x_{0})\) is the deflection of the tunnel at point \(x\) due to a unit concentrated load acting at an arbitrary position \(x=x_{0}\). Mathematically, \(G(x,x_{0})\) is the solution of the following equation:  
\[W^{m\,n}-a_{1}W^{m}+a_{2}W=b_{1}\delta^{n}(x-x_{0})+b_{2}\delta(x-x_{0}) \tag{8}\]  
Though Laplace transform and Laplace inverse transform, Equation (8) will lead to the explicit expression of the Green's function \(G(x,x_{0})\):  
\[ \begin{array}{ll}W(x;x_{0})=G(x;x_{0})&=H(x-x_{0})\phi_{1}(x-x_{0})+W(0)\phi _{2}(x)+W^{\prime}(0)\phi_{3}(x)\\ &+W^{\prime\prime}(0)\phi_{4}(x)+W^{\prime\prime}(0)\phi_{5}(x)\end{array}  \tag{9}\]  
where \(H(\cdot)\) is the Heaviside function, and \(W(0)\), \(W^{\prime}(0)\), \(W^{\prime\prime}(0)\), \(W^{\prime\prime\prime}(0)\) are the constants, which are determined by the boundary conditions of the tunnel. The expressions of \(\phi_{1},\phi_{2},\phi_{3},\phi_{4}\) and \(\phi_{5}\) are:  
\[ \begin{array}{l}\phi_{1}(x)=\sum\limits_{l=1}^{4}\psi_{l}(x)(b_{1}s^{2}+b_{ l})\text{, }\phi_{2}(x)=\sum\limits_{l=1}^{4}\psi_{l}(x)(s_{l}{}^{3}+a_{1}s_{l})\text{, }\phi_{3}(x)=\sum\limits_{l=1}^{4}\psi_{l}(x)(s_{l}{}^{2}+a_{1})\text{,}\\ \phi_{4}(x)=\sum\limits_{l=1}^{4}\psi_{l}(x)s_{l}\text{, }\phi_{5}(x)=\sum\limits_{l=1}^{4}\psi_{l}(x)\end{array}  \tag{10}\]  
where  
\[