# Seismic Response of a Large LNG Storage Tank Based on a Shaking Table Test
## 3 Simplified Mechanical Model of the Storage Tank
### Equations of Motion for Non-Isolated Storage Tank  
Considering the mass and stiffness distribution characteristics of the outer tank, the non-isolated storage tank is simplified as a 3-DOF model. Among them, the mass and height of the dome are \(M^{*}\) and \(h^{*}\). The equivalent mass and equivalent height of the outer tank wall are \(M_{1}\) and \(h_{1}\), respectively. The mass and height of the platform are \(M_{2}\) and \(h_{2}\), respectively. The simplified model of the non-isolated storage tank is shown in Figure 7.  
Assuming that the uniform mass of the outer tank wall along the height direction is \(m\), lateral displacement is \(x(z,t)\), which is:  
\[x(z,t)=\sin(\frac{\pi z}{2h})\sin(\omega t) \tag{2}\]  
Under the excitation of horizontal load, according to the principle of equivalent base shear force and overturning moment, Equations (3) and (4) can be obtained. The equivalent mass and equivalent height of the outer tank wall particle can be obtained by combining the two equations:  
\[\int_{0}^{h}m\sin(\frac{\pi z}{2h})dz\times\omega^{2}\sin(\omega t)=M_{1}\sin (\frac{\pi h_{1}}{2h})\times\omega^{2}\sin(\omega t) \tag{3}\]  
\[\int_{0}^{h}m\sin(\frac{\pi z}{2h})zdz\times\omega^{2}\sin(\omega t)=M_{1} \sin(\frac{\pi h_{1}}{2h})h_{1}\times\omega^{2}\sin(\omega t) \tag{4}\]  
The outer tank wall is regarded as a shear wall with an annular section, and the stiffness of the outer tank wall is calculated according to the calculation method of the equivalent stiffness of the shear wall with small openings Equation (5) is the calculation formula for the lateral displacement of the shear wall under the action of the concentrated \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{-} & **Experimental** & \multirow{2}{*}{**1-DOF**} & \multirow{2}{*}{**2-DOF**} & \multirow{2}{*}{**3-DOF**} & \multirow{2}{*}{**4-DOF**} \\  & **Data** & & & & \\ \hline Acceleration & & & & & \\ amplitude (m/s\({}^{2}\)) & 20.71 & 16.83 & 9.68 & 17.29 & 25.93 \\ Fitting degree (\%) & - & 81.24\% & 46.72\% & 83.46\% & 79.87\% \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Acceleration amplitude fitting of different simplified models. Figure 7: Simplified mechanical model of the storage tank. force \(P\), and the stiffness \(K_{1}\) of the outer tank wall is the reciprocal of the lateral displacement under the action of the unit concentration force:  
\[u=1.2\times\frac{Ph^{3}}{3 EI}(1+\frac{3\mu EI}{Gah^{2}}) \tag{5}\]  
In the formula, \(EI\) is the bending stiffness, \(GA\) is the shear stiffness, \(\mu\) is the shear non-uniformity coefficient, and \(h\) is the height of the wall. The engineering frequency of the concrete outer tank can be obtained by the approximate method [8], as shown in Formula (6):  
\[\frac{1}{f_{ot}^{2}}=\frac{1}{f_{F}^{2}}+\frac{1}{f_{S}^{2}}+\frac{1}{f_{R}^{2 }}+(\frac{1}{f_{cf}^{2}}+\frac{1}{f_{cs}^{2}}) \tag{6}\]  
In the formula, \(f_{F}\) is the engineering frequency for the curved cantilever beam; \(f_{S}\) is the engineering frequency of the shear cantilever beam; \(f_{R}\) is the engineering frequency of the cantilever beam assumed based on a series of independent rings; \(f_{cf}\) is the engineering frequency of a bent cantilever beam with lumped mass at the rod end; and \(f_{cs}\) is the engineering frequency for a sheared cantilever beam with lumped mass at the rod ends. The above frequencies are calculated by Equations (7)-(13). \[f_{F}=\frac{0.20D_{c}}{L^{2}}\sqrt{\frac{E_{c}}{\rho_{c}}} \tag{7}\]  
\[f_{s}=\frac{1}{8L\sqrt{1+v_{c}}}\sqrt{\frac{E_{c}}{\rho_{c}}} \tag{8}\]  
\[f_{R}=\frac{1}{D_{c}\pi\sqrt{1-v_{c}^{2}}}\sqrt{\frac{E_{c}}{\rho_{c}}} \tag{9}\]  
\[f_{cf}=\frac{1}{2\pi}\sqrt{\frac{k_{cf}}{m_{d}}} \tag{10}\]  
\[k_{cf}=3\pi E_{c}(\frac{D_{c}}{2L})^{3}t_{c} \tag{11}\]  
\[f_{cs}=\frac{1}{2\pi}\sqrt{\frac{k_{cS}}{m_{d}}} \tag{12}\]  
Figure 7:0  
In Equations (7)-(13), \(D_{c}\) is the diameter of the outer tank. \(L\) is the height of the outer tank wall.