## 4 Results and Discussion
### Inertial Interaction, Flexible-Base Frequency, and Rocking Effects  
Figure 6 shows the spectral acceleration of ratio roof motion (RM) to translational foundation motion (FM\({}_{\text{T}}\)) for the S1 structure, referred to as RM/FM\({}_{\text{T}}\), at the PGA\({}_{\text{BM}}\) of 0.1 \(g\) and 0.22 \(g\). Results from numerical simulation showed a good agreement with the centrifuge experiments. The frequency at the peak of the RM/ FM\({}_{\text{T}}\) ratio indicated the flexible-base frequency (\(f_{flex}\)) of the soil-foundation-structure (SFS) system [2, 33]. \(f_{flex}\) could be related to the foundation's sliding and rocking stiffness (i.e., \(k_{s}\) and \(k_{\theta}\)) and unchanged structure properties, such as fixed-base frequency (\(f_{n}\)), height (\(h\)) and stiffness (\(k\)), as follows [1]:  
\[f_{flex}=\frac{f_{n}}{\sqrt{1+\frac{k}{k_{s}}+\frac{hh^{2}}{k_{\theta}}}} \tag{1}\]  
A reduction in \(f_{flex}\) implies that \(k_{s}\) and \(k_{\theta}\) of the SFS system reduced during the seismic load. At a relatively low input motion with a PGA\({}_{\text{BM}}\) of 0.1 \(g\), \(f_{flex}\) of the S1 structure located on the Uni profile was 1.4 Hz, which was smaller than the fixed-base frequency (\(f_{n}\) of 2.19 Hz) because of period lengthening effects. Moreover, \(f_{flex}\) significantly increased in the layered profiles: It was 1.4 Hz for Uni profile and about 1.9 Hz for L1 and L2 soil profiles. A similar trend was observed in numerical analyses. At low input intensity (Figure 6a), the increase in global stiffness of the soil in the layered profiles compared to the Uni soil contributed to this increment in translation and rocking stiffness of the foundation, and as a result, the \(f_{flex}\) of the S1 structure increased. However, at higher earthquake intensity (Figure 6b), the S1 structure experienced similar \(f_{flex}\) in all soil profiles because of the reduction in the stiffness of soil beneath the structure in layered cases. The  
Figure 6: Ratio of response spectra (RRS) of roof motion (RM) to foundation motion (FMT) for three soil profiles at (**a**) PGA = 0.1 \(g\) and (**b**) PGA = 0.22 \(g\). Solid markers indicate peak RRS corresponding to \(f_{\text{flex}}\) for each case. stiffness of soil in the thin layer underneath the structure decreased, which resulted in the \(f_{flex}\) of the structure above the layered ground decreasing until it reached that of uniform soil. The variations in period lengthening ratio (PLR = \(f_{n}\)/\(f_{flex}\)), are plotted in Figure 7. The dashed line indicates a PLR value of 1.31 for the S1 structure, which was calculated based on the equation suggested by [PERSON], [PERSON] [1]. In general, the value of \(f_{flex}\) for the S1 structure on the three soil profiles increased with an increase in PGABM. The PLR values for layered profiles during low-intensity earthquakes were significantly higher than those for the Uni profile because of the increase in soil stiffness. However, with increasing earthquake intensity, the stiffness of the soil beneath the structure in layered profiles decreased sharply due to stress concentration, and the response of the structures. This caused PLR for the S1 structure on the layered profiles to become similar to that for the Uni profile. At the highest input excitation with a PGABM of 0.3 g, the PLR of the S1 structure on the Uni profile was slightly higher than for the layered profiles because of the reduction in stiffness of SFS system in layered profiles at high earthquake intensity. Recently, rocking behavior and its effects on SFSI have been defined and recognized [34; 35; 36; 37; 38]. During a strong earthquake, the overturning moment (M) at the foundation level, which is generated by the inertial response of the above structure, is restricted by the ultimate moment capacity of the foundation, and, therefore, the seismic response is limited.