## 5 Numerical Study  
For practical applications, the periodic MetaFoundation must be constructed by available materials. In this paper, it is assumed that the foundation is made of concrete, and the heavy cores are made of steel coated by rubber. The material properties of different parts of unit cells, including modulus of elasticity and density, are listed in Table 1.  
For the preliminary time history analysis, it is assumed that the MetaFoundation comprises unit cells with dimensions of 0.305 \(\times\) 0.305 \(\times\) 2.0 m corresponding to the length, width, and height, respectively. In addition, the cores of unit cells have a dimension of 0.10 m. The thickness of the rubber is assumed to be 0.05 m. The damping ratios corresponding to concrete and rubber are assumed to be 5% and 30%, respectively. A dispersion analysis has been conducted to calculate the dispersion relation and stop band corresponding to an infinite unit cell. The optical and acoustic branches obtained from the dispersion analysis are illustrated in Figure 6. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Part Name** & **Material** & **Modulus of Elasticity, \(E\) (GPa)** & **Density, \(\rho\) (kg/m3)** \\ \hline matrix & concrete & 24.85 & 2400 \\ soft material & rubber & 0.000137 & 1300 \\ core & steel & 210 & 7850 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material properties of a unit cell. The results show that the stop band forms in the frequency range of 18.733 to 18.984 Hz, where the elastic waves cannot propagate through the MF according to the Floquet-Bloch theory. However, the obtained stop band is valid for an infinite number of unit cells. For the case of the foundation comprising finite unit cells, additional modal analysis is required to calculate the stop band. Two different types of cylindrical tanks have been considered as case studies from [8]. The considered tanks have different aspect ratios (height to radius). The ratio of height to the radius of the assumed tanks (\(S=H/R\)) is 0.6 and 1.85, corresponding to squat and slender tanks, respectively. The filling fluid is assumed to be water, with a mass density of \(\rho_{w}=1000\) kg/m\({}^{3}\). Table 2 shows the geometrical properties of considered tanks. A foundation that has a square shape is considered for both squat and slender tanks. The dimension of the MF for the squat tank is assumed to be 50.0 \(\times\) 50.0 m, corresponding to its width and length, respectively; for the slender tank, a MetaFoundation with 15.0 m length and width is considered. As mentioned above, the height of each layer of MF is assumed to be 2.0 m.  
The natural period, relative masses, and relative heights of the simplified mass-spring model depend on the geometrical properties of the cylindrical tanks. The parameters of the equivalent mechanical model of the tank have been calculated and tabulated in Table 3. The damping ratios of the convective (\(\zeta_{c}\)) and impulsive masses (\(\zeta_{i}\)) are assumed to be 0.5% and 2%, respectively, as suggested by [8]. An adequate number of earthquakes should be considered for the required time history analysis to assess the efficacy of the proposed MF on the seismic responses of liquid storage tanks. Federal Emergency Management Agency (FEMA) [42] suggested three sets of ground motions for quantifying seismic performance factors of buildings. The three sets \begin{table}
\begin{tabular}{c c c c} \hline
**Tank Type** & \(H\) **(m)** & \(R\) **(m)** & \(t_{w}\) **(m)** \\ \hline Squat & 14.6 & 24.4 & 0.0203 \\ Slender & 11.3 & 6.1 & 0.0058 \\ \hline \end{tabular}
\end{table} 
Table 2: Geometrical properties of the selected tanks used as case studies. Figure 6: Dispersion analysis of considered unit cell. include far-field (FF), near-fault without pulse (NF-WO Pulse), and near-fault with pulse (NF-W Pulse) ground motions. In this paper, the FF set, which comprises twenty-two pairs of records with an average moment magnitude of \(M_{w}=7.0\), has been used.