# A Dynamic Analysis Method of Liquid-Filled Containers Considering the Fluid-Structure Interaction
## 4 Modal Analysis and Time-Historical Analysis of Cylindrical Liquid-Filled Containers
### Time-Historical Analysis of Cylindrical Liquid-Filled Containers
Taking the liquid level of 6 m as an example, the maximum displacements on the sidewall are separately 0.052, 0.049, and 0.046 mm under the El-Centro, Kobe, and Loma Prieta waves, and they all appear at a height of 3 m on the sidewall. The Rd values on the sidewall are, separately, 4.90, 4.95, and 4.62 under the El-Centro, Kobe, and Loma Prieta waves, and they all appear at a height of 1 m on the sidewall. The distribution of the maximum acceleration and Ra on the sidewall of containers with different liquid levels along the height are similar to that of the maximum displacement on the sidewall. Figures 27 and 28 show that as the liquid level rises, the maximum acceleration and Ra on the sidewall of cylindrical liquid-filled containers constantly grow. In the case that the liquid levels are 0, 1, and 2 m, the maximum acceleration on the sidewall appears on the top of the liquid-filled container; if the liquid levels are 3 and 4 m, the maximum acceleration on the sidewall is found at the height of 2.5 m; when the liquid levels are 5 and 6 m, the maximum acceleration on the sidewall occurs at a height of 3 m. The position of the maximum Rd remains almost unchanged. Under action of the three seismic waves, the maximum acceleration and Ra on the sidewall of containers with different liquid levels have a similar distribution along the height. The maximum acceleration and Ra on the sidewall of cylindrical liquid-filled containers always increase with rising liquid levels. Taking the liquid level of 6 m as an example, the maximum accelerations on the sidewall are 0.46, 0.20, and 0.058 m/s\({}^{2}\) under the El-Centro, Kobe, and Loma Prieta waves, respectively, and they all appear at a height of 3 m. The Ra values on the sidewall are, separately, 4.72, 8.58, and 6.35 under the El-Centro, Kobe, and Loma Prieta waves, and they all appear at a height of 1 m on the sidewall. Figure 25: The distribution of the maximum displacement on the sidewall of containers. (**a**) El-Centro wave. (**b**) Kobe wave. (**c**) Loma Prieta wave. Under the action of different seismic waves, the maximum acceleration on the sidewall of containers with different liquid levels differs greatly in the distribution height. On the one hand, this is because the frequencies of the input seismic waves differ greatly from the structural frequency of vibration. The main frequencies of the El-Centro, Kobe, and Loma Prieta waves are all concentrated within 5 Hz (Figure 24). In the case of different liquid levels, the minimum and maximum first frequencies of the cylindrical container are 13.17 Hz (liquid level of 6 m) and 19.94 Hz (liquid level of 0 m), respectively. This suggests an unobvious acceleration amplification effect. On the other hand, the El-Centro wave has more high-frequency components (above 15 Hz) compared with the Kobe and Loma Prieta waves, so the maximum acceleration on the sidewall under the El-Centro wave is greater than those under the Kobe and Loma Prieta waves. Figure 26: The distribution of Rd on the sidewall of containers. (**a**) El-Centro wave. (**b**) Kobe wave. (**c**) Loma Prieta wave. The distribution of the maximum von Mises stress and R\({}_{\sigma}\) on the sidewall of containers with different liquid levels along the height are shown in Figures 29 and 30. The maximum von Mises stress is always found at the bottom of the liquid-filled containers. The maximum R\({}_{\sigma}\) is always found at approximately 2 m of the liquid-filled containers. Apart from that at the bottom, the maximum von Mises stress on the sidewall of cylindrical liquid-filled containers constantly increases as the liquid rises. When the liquid levels are 0, 1, and 2 m, the maximum von Mises stress on the sidewall appears at a height of 1.0 m; if the liquid levels are 3 and 4 m, the maximum von Mises stress on the sidewall occurs at a height of 2.0 m; in the case that liquid levels are 5 and 6 m, the maximum von Mises stress on the sidewall occurs at a height of 2.5 m.  
Figure 27: Distribution of the maximum acceleration on the sidewall of containers.