## 4 Results and Discussion
### Inertial Interaction, Flexible-Base Frequency, and Rocking Effects
In this study, the overturning moment at the surface of the foundation and the rocking angle of the structure in the time domain were identified based on measured values from two vertical accelerations attached to the foundation and three pairs of strain gauges attached to the structure's walls [23; 24]. The rocking stiffness (\(k_{\theta}\)) can be estimated from the hysteretic backbone curve of the overturning moment and rocking angle. An example is illustrated in Figure 8a for the S1 structure at a PGABM of 0.1 g and 0.3 g. The overturning moment was normalized to the ultimate moment capacity (Mult = Wp-L(1 - 1/FSv)/2), which is defined based on structure weight (Wp), foundation length (L), and soil strength through the vertical safety factor (FSv) [24; 39; 40]. The estimated overturning moment increased with an increase in rocking angle; however, it was limited by the Mult of the SFS system that reduced the structure response during high-intensity earthquakes [38]. The rocking stiffness (\(k_{\theta}\)) was calculated from these backbone curves and then normalized to the maximum rocking stiffness (\(k_{\theta,Max}\)), as shown in Figure 8b [24; 39]. The \(k_{\theta,Max}\) can be calculated from the initial shear modulus of soil beneath structure (\(G_{max}\)), moment of  
Figure 7: Comparison of period lengthening ratio (PLR) for the S1 structure on L1 and L2 profiles to those on Uni profile. inertia at the centroid of the foundation in the direction of rocking (_I_), and foundation dimensions (i.e., length of \(L\) and width of _B_) as [41]:  
\[k_{\theta,Max}=\frac{G_{max}}{1-\theta}\textsuperscript{0.75}\left(\frac{L}{B} \right)\textsuperscript{0.25}\left[2.4+0.5\frac{B}{L}\right]\left(\text{KN} \cdot\text{m}\right) \tag{2}\]  
The rocking stiffness values were obtained in the earthquake with a PGABM above 0.1 \(g\) since the rocking behavior dominated during strong motion. The lines in Figure 8b indicate the power fitting functions for normalized rocking stiffness (\(k_{\theta}\)/\(k_{\theta,Max}\)) with the rocking angle at a high regression number (i.e., values of R2 were above 0.93). Due to the reduction in the contact area between the soil and foundation surface, the rocking stiffness significantly decreased with increasing rocking angle [36]. At low earthquake intensity with small values of rocking angle, the rocking stiffness for the S1 structure on the layered profiles was higher than those obtained with the Uni profile because of the higher stiffness of the lower layer and the ultimate moment capacity that implied an increase in ultimate bearing capacity of the foundation when it was located on layered ground [41; 42]. Moreover, as the thickness of the soft upper layer decreased to approximately 0.5 B, the rocking stiffness sharply increased. At high rocking angles, the rocking stiffness of the S1 structure on the three soil profiles converged. This can be partially attributed to the reduction in soil stiffness observed in Figure 6 with period lengthening effects. The rocking stiffness for the structure located on the Uni profile was slightly higher than for those on layered profiles at a rocking angle above 0.008 rad. This can be attributed to the large decrease in the soil's stiffness underneath the structure, as indicated in Figures 5b and 7.