## 4 Results and Discussion
### Results Comparison and Method Validation  
A series of studies on the earthquake-induced dynamic responses of the horizontal-cylindrical liquid vessels have been carried out by [PERSON] (15,16). Numerical solutions about the sloshing frequencies, the impulsive masses and the convective masses of the horizontal-cylindrical liquid storage tanks with respect to different liquid depths under longitudinal and transverse seismic excitation are obtained. Besides, an \"equivalent rectangular container\" method is proposed, which means that the externally induced dynamic responses of a horizontal-cylindrical container with arbitrary liquid depth can be computed quite accurately by replacing the cylindrical container with an equivalent rectangular container which has the same free-surface dimensions and owns the same liquid volume (15). The effectiveness and correctness of this equivalence have been numerically (38) and experimentally (39) demonstrated. In this study this equivalence technique is adopted to demonstrate the practicability of the proposed ALE based FSI approach for the interaction between tunnel linings and inner water. Assuming that the inner diameter of the tunnel is \(r\) and the free-surface height of the liquid is \(h\), based on the equivalence theorem, the height of the liquid (\(h_{\text{eq}}\)) in the equivalent rectangular container can be calculated using Equation (18). Figure 11 illustrates the layout of the equivalent relationship. \[h_{eq}=\frac{h-r}{2}+\frac{r^{2}}{2}\left\{\frac{\arcsin\left[\left(h-r \right)/r\right]+\pi/2}{\sqrt{h\left(2r-h\right)}}\right\} \tag{18}\]  
Once the cylindrical container is treated as its equivalent rectangular container, the equivalent masses can be calculated using the analytical solution proposed by [PERSON] (12). For liquid storage containers with rectangular cross section, the hydrodynamic pressure (\(p_{i}\)) along the height direction can be expressed as Equation (19), where \(c\left(y\right)\) can be expressed as Equation (20) and \(y\) means the vertical coordinate of the location where the pressure is calculated. The equivalent mass attached to each node can be obtained by using the steps as follows. Firstly, through the mapping of equivalent rectangular surfaces and the original cylinder surface, the transformation of the pressure on the equivalent rectangular face to that on the original cylindrical face is completed. Then, the total equivalent mass on an element can be calculated by dividing the product which can be obtained through multiplying the area of the element's face bearing pressure by the pressure value, by the  
Figure 11: Schematic layout of the equivalent relationship. \(r\), \(h\) and \(h_{eq}\) mean the inner diameter of the tunnel, the free-surface height and the equivalent height, respectively. current acceleration \(a_{n}=\ddot{u}\left(t\right)\). At last, the obtained total equivalent mass is distributed to the four nodes owned by the element using Equation (21), where \(S_{i}\) is the area of the face with pressure owned by the element; \(m_{i}\), \(i=1,2,3,4\) means the equivalent mass at each node. Once the procedure is programmed, the whole finite element model can be processed automatically. \[p_{i}=\sqrt{3}p\ddot{u}\left(t\right)c\left(y\right) \tag{19}\]  
where  
\[c\left(y\right)=h_{eq}\left(\frac{y}{h_{eq}}-\frac{1}{2}\left(\frac{y}{h_{eq}} \right)^{2}\right)\tanh\frac{\sqrt{3h_{eq}\left(2r-h_{eq}\right)}}{h_{eq}} \tag{20}\]  
\[m_{i}=\frac{1}{4}\frac{p_{i}S_{i}}{a_{n}}=\frac{\sqrt{3}}{4}pc\left(y\right)S_ {i} \tag{21}\]  
Without loss of generality, the free-surface height of the liquid is assumed to be \(1.5r\), where \(r\) is the diameter of the tunnel lining which is \(2.75\) m for island and land part and \(2.92\) m for cross river part. The equivalent masses are calculated and applied by using the above-mentioned procedure.