## 5 Comparison Studies  
Assuming the displacement condition at the contact surface between foundation and soil, the horizontal and rocking coefficients of dynamic flexibility for the normalized frequencies \(a_{0}\) (\(a_{0}\in[0,\ 8]\)) and soil Poisson ratio \(\
u=1/3\) are obtained by [PERSON] and [PERSON] [28] based on the elastic half space theory. The horizontal and rocking dynamic flexibility coefficients in Reference [28] are fitted by using the complex Chebyshev polynomials [32]. Figure 6 shows horizontal and rocking dynamic impedances for the circular surface foundation with regard to the normalized frequencies \(a_{0}\). Two cases with regard to Chebyshev polynomials of degrees \(N_{\rm s}=3\) and \(N_{\rm s}=5\) are illustrated for various degrees of precision compared with the elastic half space results [28]. It can be seen from Figure 6 that good agreement is achieved for \(N_{\rm s}=5\). In the following analysis, the Chebyshev polynomials of the degree \(N_{\rm s}=5\) are considered. Then, the selected dynamic flexibility can be expressed as the nested form depicted in Equation (38) according to the recursive function theory. In Table 1, the coefficients of mechanical elements of the nested LPM including the springs \(\chi_{j}\) and dashpots \(\delta_{j}\) in Equation (39) are presented with \(N_{\rm s}=5\). Figure 5: Soil–tank–liquid coupling model. [PERSON] and [PERSON] [30] utilized two groups of spring-dashpot systems to analyze soil-tank interaction under seismic excitation. The tank is rigid and without baffle. The ratio of maximum shear \(F_{\text{max}}\) under flexible foundation to that under rigid foundation is defined as \(\alpha_{1}\). The ratio of maximum moment \(M_{\text{max}}\) under flexible foundation to that under rigid foundation is defined as \(\alpha_{2}\). The maximum of the horizontal absolute acceleration for the base is normalized and associated with the maximum of the horizontal acceleration at the bedrock, that is to say, \(\alpha_{3}=|\left(\bar{u}_{0}+\bar{u}_{g}\right)|_{\text{max}}/|\,\bar{u}_{g} |_{\text{max}}\). The densities of the circular base and tank are fixed at 2500 kg/m\({}^{3}\) and 7800 kg/m\({}^{3}\), respectively. The density of soil is fixed at 2000 kg/m\({}^{3}\). The thickness of the circular base is equal to 0.05\(H\). The thicknesses of the tank and the baffles are obtained as 0.003\(R_{2}\). The base radius is the same as that of the cylindrical tank. The seismic record is El Centro earthquake (S 90 W component) in 1940. Table 2 depicts comparison results between the present model and the experimental results. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Coefficient**} & \multicolumn{2}{c}{**Stiffness**} & \multirow{2}{*}{**Coefficient**} & \multicolumn{2}{c}{**Damping**} \\ \cline{2-5}  & **Horizontal** & & **Rocking** & & **Horizontal** & **Rocking** \\ \hline \(\chi_{1}\) & \(-0.1400\) & \(-0.4977\) & \(\delta_{0}\) & \(0.6545\) & \(0.3927\) \\ \(\chi_{2}\) & \(0.6235\) & \(-2.7993\) & \(\delta_{1}\) & \(-0.1187\) & \(-0.4663\) \\ \(\chi_{3}\) & \(-0.1846\) & \(167.2036\) & \(\delta_{2}\) & \(0.0726\) & \(2.5569\) \\ \(\chi_{4}\) & \(0.1653\) & \(-0.0038\) & \(\delta_{3}\) & \(-0.0392\) & \(-2.4834\) \\ \(\chi_{5}\) & \(-0.4609\) & \(0.0038\) & \(\delta_{4}\) & \(0.0655\) & \(-0.0001\) \\ - & - & - & \(\delta_{5}\) & \(-0.0694\) & \(0.0188\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The coefficients for the springs and dashpots for the nested LPMs with the degree \(N_{\text{s}}=5\). Figure 6: Impedance of the circular surface foundation for varying dimensionless frequencies \(a_{0}\) in comparison with the elastic half space results [28]: (**a**) horizontal stiffness coefficient; (**b**) horizontal damping coefficient; (**c**) rocking stiffness coefficient; (**d**) rocking damping coefficient.