# 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  
Time-historical analysis was conducted by taking cylindrical liquid-filled containers with different liquid levels. The El-Centro (1940), Kobe (1995), and Loma Prieta (1989) waves were selected and applied in the \(x\) direction, with the peak acceleration of 0.1 g. The acceleration time-history curves and Fourier spectra are shown in Figure 24. Structural  
Figure 23: The first five \(R_{f}\) of cylindrical containers. \begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{**Order**} & \multicolumn{6}{c}{**Liquid Level**} \\ \cline{2-7}  & **0 m** & **1 m** & **2.0 m** & **3.0 m** & **4.0 m** & **5.0 m** & **6.0 m** \\ \hline
1 & 19.94 & 19.94 & 19.88 & 19.45 & 18.05 & 15.93 & 13.71 \\
2 & 20.11 & 20.11 & 20.02 & 19.48 & 18.23 & 16.30 & 14.20 \\
3 & 26.81 & 26.81 & 26.72 & 26.13 & 24.39 & 21.46 & 18.37 \\
4 & 29.29 & 29.28 & 29.09 & 27.85 & 24.93 & 21.88 & 19.41 \\
5 & 38.30 & 38.30 & 38.16 & 37.10 & 34.10 & 30.44 & 26.34 \\ \hline \end{tabular}
\end{table} 
Table 5: The first five frequencies of cylindrical containers (unit: Hz). analysis generally focuses on the displacement, acceleration, and stress of structures. Hence, the maximum displacement, maximum acceleration, and maximum von [PERSON] stress on the sidewall at different heights of liquid-filled containers at \(x=D/2\) and \(y=0\) were monitored. Figure 24: Acceleration time-history curves and Fourier spectra. (**a**) El-Centro wave. (**b**) Fourier spectrum under the El-Centro wave. (**c**) Kobe wave. (**d**) Fourier spectrum under the Kobe wave. (**e**) Loma Prieta wave. (**f**) Fourier spectrum under the Loma Prieta wave. In order to measure the influence of different water depths on the seismic response of the container, the dimensionless coefficient is used to represent the influence coefficient of the dynamic response of the container caused by water depth, and the coefficient of displacement along the sidewall height is as follows:  
\[R_{d}=\frac{d_{i}-d_{0}}{d_{0}} \tag{13}\]  
where \(d_{i}\) is the maximum displacement of the sidewall at different water depths, and \(d_{0}\) is the maximum displacement of the sidewall without water. The coefficient of acceleration along the sidewall height is  
\[R_{a}=\frac{a_{i}-a_{0}}{a_{0}} \tag{14}\]  
where \(a_{i}\) is the maximum acceleration of the sidewall at different water depths, and \(a_{0}\) is the maximum acceleration of the sidewall without water. The coefficient of von Mises stress along the sidewall height is  
\[R_{\sigma}=\frac{\sigma_{i}-\sigma_{0}}{\sigma_{0}} \tag{15}\]  
where \(\sigma_{i}\) is the maximum von Mises stress of the sidewall at different water depths, and \(\sigma_{0}\) is the maximum von Mises stress of the sidewall without water. The distribution of the maximum displacement and Rd on the sidewall of containers with different liquid levels along the height are displayed in Figures 25 and 26. As the liquid level rises, the maximum displacement and Rd on the sidewall of cylindrical liquid-filled containers increase. When the liquid levels are 0 (no liquid contained), 1, and 2 m, the maximum displacement on the sidewall appears on the top of the container; under conditions with liquid levels of 3 and 4 m, the maximum displacement on the sidewall appears at the height of 2.5 m; if the liquid levels are 5 and 6 m, the maximum displacement on the sidewall occurs at the height of 3 m. Unlike the distribution of the maximum displacement on the sidewall, the position of the maximum Rd remains largely unchanged. The maximum displacement and Rd on the sidewall have a similar distribution along the height at different liquid levels under the three input seismic waves, and they always increase on the sidewall of the cylindrical liquid-filled container with rising liquid levels, whereas the values of the maximum displacement and Rd are different.