## 4 Numerical Simulation and Analysis of Result
### Results of Different Seismic Excitations  
In this section, different seismic excitations were used for simulation. In addition to the seismic acceleration data of the 2008 Wenchuan earthquake (east-west component) as shown in Figure 10, the 1940 El Centro earthquake data as shown in Figure 12 were also used for simulation. Thus, there are six cases using different seismic excitations in this part: Wenchuan EW, El Centro EW, El Centro NS, El Centro vertical, El Centro EW + vertical, and El Centro NS + vertical. All simulations were executed at \(t=40\) s using 2,000,000 time steps, which consumed approximately 160 CPU hours in the series run by using a Xeon E5-2697 (2.6 GHz) Workstation (14 cores). In the cases of Wenchuan EW, El Centro EW, and El Centro NS, the excitation consists of only a horizontal component, while in the three other cases (El Centro vertical, El Centro EW + vertical, El Centro NS + vertical), the excitation consists of a vertical component. Figure 12: Time series of the seismic acceleration of the 1940 El Centro earthquake. (Upper: east-west component, middle: north-south component, lower: vertical component). For the horizontal vibration cases, as shown in Figure 13, the pressure oscillations at the symmetrical points on the left and right sides are opposite in phase. We compared all the symmetrical points from B1&B10 to B9&B18, and the pressure oscillations are all opposite in phase for all of these cases. In Figure 13, we only show some of the symmetrical points at different depths and on both the upper and lower surfaces of the arch rib. This type of pressure oscillation will cause the arch rib to undergo vertical antisymmetric bending, which corresponds to the second-order vibration shape shown in Table 2. However, only in the vertical vibration cases (El Centro vertical), as shown in Figure 14, the pressure evolution at the symmetrical points is in the same phase. This type of pressure oscillation will cause the arch rib to undergo vertical symmetric bending, which corresponds to the fourth-order vibration shape shown in Table 2. For the cases including both horizontal and vertical excitations (El Centro EW + vertical, El Centro NS + vertical), both second-and fourth-order vibration shapes should be considered. Figure 13: Comparison of pressure evolution at the symmetrical boundary points. (Wenchuan EW excitation). In order to analyze whether the inner water pressure oscillation can cause resonance in the second-or fourth-order vibration shape, a fast Fourier transform (FFT) was carried out for the pressure oscillation data for all boundary monitoring points B1 to B18 for all six cases. Figure 15 shows the fast Fourier transform results of all six cases. The zero-frequency fraction indicates static pressure on the points, which has no influence on the arch rib. The frequencies of the first four spikes are in Figure 15, and cases are listed in Table 3. The frequency of the first spike is 1.125 for all cases. The subsequent spike frequencies are about 3, 5, and 7 times the first spike frequency, respectively, which means that all spikes will contribute to the possible resonance if the first spike frequency causes resonance in the second-or fourth-order vibration shape. Therefore, we only compared the first spike frequency with the natural frequency of the arch bridge in the second-or fourth-order vibration shapes. The first spike frequency (1.125 Hz) is different from the frequency of the second (0.941722 Hz) and fourth (1.661838 Hz) order vibration shapes. It seems there is no risk of resonance at the designed water level for all excitations. Figure 14: Comparison of pressure evolution at the symmetrical boundary points. (El Centro vertical excitation). Figure 15: The fast Fourier transform of pressure time\(-\)series at the monitoring boundary points for different seismic excitations at the designed water level.