## 4 Numerical Simulation and Analysis of Result  
The SPH model was employed to study the influence of inner water on the arch bridge under seismic excitation. A sketch of the arch is shown in Figure 5. The length of the arch rib is 140 m, and the vector height of the arch ring is 23.333 m. The arch rib has a catenary curve shape. In order to reduce self-gravity and to balance the water pressure, the arch rib is designed to be hollow, and water can enter the rib. The designed water level is 15.56 m. Most of the time, the inner water exists in the arch rib. Here, we consider the arch rib as a 2D hollow rib as shown in Figure 5 because the width of the bridge is much larger than the depth of the arch rib box. As preliminary research on the inner water in the arch rib, the 2D simulation will reduce the computational cost drastically, and the 2D simulation results can help us obtain a conceptual understanding of the inner water effect on the arch rib.  
Figure 4: Comparisons of SPH\(-\)computed pressure time series at sensor points with experimental data for cases A and B.  
The initial particle spacing is \(\Delta x\). The initial fluid particle pressure is set up following the static pressure distribution. The initial pressure on boundary particles is interpolated from the fluid particles as per the boundary treatment proposed by [PERSON] et al. [17]. The density of the water is taken as 1000 kg/m\({}^{3}\). The coefficient in the equation of state is \(\mathrm{B}=2.8\times 10^{6}\). The artificial viscosity coefficient is \(\alpha=0.01\). The XSPH coefficient is \(\varepsilon=0.1\). In order to study the pressure evolution on the boundary, we defined some monitoring points on the solid boundary as shown in Figure 6. B1 to B9 were set on the right side of the arch rib, B10 to B18 on the left side of the arch rib, and were symmetrical about the y-axis.  
The objective of this investigation was to find out whether there would be any possibility of damage caused to the arch bridge. Several studies [24,25] on the dynamic response analysis of arch bridges have shown that results with high accuracy can be obtained by using a smaller number of beam cells. As the arch bridge's vibration shape and frequency were analyzed using the FEM method, and the finite element model of the arch bridge was established by software (Figure 7), it is composed of 949 beam elements with 1089 nodes. One main arch circle is composed of 6 pieces of the arch box, the cross-section of the arch ring is shown in Figure 8, and the material of the arch circle is C40 concrete; its density is 2420-2440 kg/m\({}^{3}\), modulus of elasticity is \(2.0\times 10^{10}\) Pa, Poisson's ratio 0.2, and damping ratio 0.05. Because the arch bridge adopts an equal cross-sectional suspension line box-shaped hingeless arch, the two ends of the arch circle are cemented to the abutment, so the boundary condition is cemented at both ends. The first-to fourth-order vibration shapes are shown in Figure 9. The frequency and the illustration of each vibration shape are listed in Table 2. In this paper, the dynamic pressure of the inner water is simulated in 2D. Therefore, the inner water's influence could only cause the vibration shape in the vertical plane, which means only second-and fourth-order vibration shapes will be considered in our study.  
Figure 6: Position of boundary particles for monitoring.  
Figure 7: The finite element model of the arch bridge.