## 4 Numerical Calculation Results and Analysis
### Ground Motion Records Used for Dynamic Calculation  
Three earthquake ground motions are selected for the present study. Table 2 shows the characteristics of the three earthquake ground motions along with their corresponding peak ground acceleration (PGA) in terms of g. According to the Chinese code for seismic design of buildings, the horizontal X-direction component amplitude is adjusted to 0.4 g, the horizontal Y-direction component amplitude is adjusted to 0.34 g, and the vertical Z direction component amplitude is adjusted to 0.26 g. The three components of the El Centro ground motion are present in Figure 2 in terms of m/s\({}^{2}\). The corresponding Fourier spectrum curves of the three seismic waves are shown in Figure 3.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Natural Frequency**} & \multirow{2}{*}{**Numerical Results/Hz**} & \multicolumn{2}{c}{**Theoretical Results**} & \multicolumn{2}{c}{**Code Results**} \\ \cline{3-6}  & & **Value/Hz** & **Error/\%** & **Value/Hz** & **Error/\%** \\ \hline Fluid Convective & & 0.228 & 0.216 & 5.56 & 0.216 & 5.56 \\ Frequency,\(f_{\rm w}\) & & & & & \\ Fluid– tank coupling & 7.046 & 7.084 & 0.54 & 7.188 & 1.98 \\ impulsive frequency,\(f_{\rm c}\) & & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Comparison of the natural frequency by a numerical solution and other literature.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Earthquake**} & \multirow{2}{*}{**Date**} & \multirow{2}{*}{**Recording Station**} & \multicolumn{2}{c}{**X-Direction**} & \multicolumn{1}{c}{**Y-Direction**} & \multicolumn{1}{c}{**Z-Direction**} \\  & & & **Component** & **Component** & **Component** \\  & & & **PGA/g** & **PGA/g** & **PGA/g** \\ \hline Imperial Valley, & & & & & \\ California & 19 May 1940 & El Centro & 0.281 & 0.211 & 0.178 \\ Tangshan, & 9 August 1976 & Qian’an Lanhe Bridge & 0.162 & 0.153 & 0.081 \\ Hebi & 17 January 1994 & CDMG Station 23595 & 0.072 & 0.060 & 0.036 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Characteristics of the earthquake ground motions.  
For comparison purposes, the base excitation was applied along the unidirectional (Referred to as \"X\"), horizontal bi-directional (Referred to as \"X + Y\"), horizontal and vertical bi-directional (Referred to as \"X + Z\"), and tri-directional (Referred to as \"X + Y + Z\") components of the earthquake, as shown as Figure 4.  
For each case, the location with the largest seismic response can be found as the observation point on a certain height by cycling along the tank circumference. For the X excitation and X + Z excitation, the peak response quantities are measured at the nodes on the plane where the XOZ plane intersects the tank wall; that is, along the X-direction (\(\theta=0^{\circ}\)). For the X + Y excitation and X + Y + Z excitation, the nodes of the peak response quantities on the tank wall will change due to the Y-directional component, as shown in Figure 5.  
Figure 4: A schematic diagram of the four calculation conditions (the dotted line on the axis indicates that there is no excitation in this direction).  
Figure 3: Fourier spectra of the El Centro ground motion: (**a**) X-direction; (**b**) Y-direction; (**c**) Z-direction.  
Figure 2: Acceleration time-history curves of the El Centro ground motion: (**a**) X-direction; (**b**) Y-direction; (**c**) Z-direction.