## 3 Results and Discussion  
### Modal Analysis of LNG Tanks  
Before carrying out dynamic analyses of structure, it is imperative to initiate a modal analysis to comprehend the inherent vibrational characteristics of the structure, laying the groundwork for subsequent dynamic analyses. Typically, the modalities of a structure can be categorized into dry modes and wet modes, depending on whether the influence of surrounding fluids is taken into consideration. Given the presence of liquid within the tank, the modal analysis for tanks differs from that of general structures, necessitating the inclusion of the impact of liquid on the tank, thus constituting wet modal analysis.  
Within the Ansys Workbench, the application of the acoustic-structural coupling algorithm in the Modal Acoustics module facilitates the completion of wet modal analysis.  
Figure 6: Vertical acceleration time histories of the Taft earthquake (\(Z\)-axis).  
Figure 7: (**1**) LNG surface wave height measuring point; (**2**) tank structure measuring point.  
Disregarding the influence of the soil and focusing solely on the modal analysis of the fluid-structure coupling model, the tank bottom is subjected to fixed constraints, with the internal liquid being water. Moreover, our analysis did not account for the influence of air within the tank. The depth of a full LNG tank is 41.6 m. The natural frequencies corresponding to the dry and wet modes of the storage tank, as computed, are presented in Table 3. Figures 8 and 9 show the first four modes of empty and full tanks, respectively. The modal frequency of a tank filled with LNG is notably lower than that of an empty tank.  
According to the method proposed by [PERSON] [24] and [PERSON] et al. [25], the frequencies of the tank structure were calculated separately for the empty-tank and full-tank states.  
Empty tank modal [24]:  
\[\begin{split} f_{r}&=f_{1}\cdot\xi_{1}\cdot\alpha\\ f_{1}&=\frac{1}{4H}\sqrt{\frac{G}{2\rho_{s}}}\\ k_{1}&=1-\frac{\pi^{2}}{6H}\left(\frac{R}{H}\right) ^{2}(1-\mu)\\ \alpha&=\left[1+\frac{0.1}{1+\mu}\left(\frac{H}{R} \right)^{2}\right]^{-\frac{1}{2}}\end{split} \tag{5}\]  
Figure 8: The first four modal shapes of a storage tank structure in the empty state.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Modal** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** \\ \hline \hline Frequency of & & & & & & & & & & \\ empty tanks (Hz) & 3.714 & 6.935 & 7.019 & 7.026 & 7.028 & 7.045 & 7.070 & 7.080 & 7.106 & 7.113 \\ \hline Frequency of full & & & & & & & & & & \\ tanks (Hz) & 3.193 & 4.976 & 4.978 & 5.031 & 5.035 & 5.192 & 5.205 & 5.972 & 5.974 & 6.283 \\ \hline \hline \end{tabular}
\end{table}
Table 3: The first 10 modal frequencies of empty and full tanks.  
Figure 9: The first four modal shapes of a storage tank structure in the full state.  
where \(f_{1}\) is the fundamental period of shear vibration for the beam; \(\alpha\) is the influence coefficient for bending deformation; \(\upvarepsilon_{1}\) represents the influence coefficient for cross-sectional deformation; \(\uprho_{\text{s}}\) is the density of the material used for the outer tank wall; \(H\) is the height of the outer tank wall; \(\mu\) represents the Poisson's ratio of the tank wall material; and \(G\) represents the shear modulus of the tank wall material.  
Full tank modal [25]:  
\[T_{i}=C_{i}\frac{h\sqrt{\rho}}{\sqrt{tE/D}} \tag{6}\]  
where \(C_{i}\) is a coefficient related to \(h/D\), set at 4.4; \(D\) is the inner diameter of the storage tank; \(t\) is the thickness of the tank wall; \(h\) represents the height of the liquid surface (m); \(\rho\) is the liquid density; and \(E\) is the elastic modulus of the tank wall material.  
The calculated results are shown in Table 4. The results indicate that the error between the simulation results obtained using ANSYS and the results calculated using the theoretical method is less than 10%.