## 4 Results and Discussion
### Kinematic Interaction and Incidence Parameter  
The kinematic interaction between the soil and foundation is generally represented by a transfer function, which is the ratio of the Fourier transform between foundation input motion (FIM) and FFM. To measure kinematic interaction and FIM at the foundation level without the effects of inertial interaction is a practically impossible requirement because both foundation and structure are massless. However, the effects of inertial interaction on foundation level motion could be reduced by evaluating transfer function at a frequency above the flexible-base frequency [5; 8]. In this study, the translational foundation motion (FMT) was used instead of FIM in estimating kinematic interaction to indicate results estimated from the centrifuge experiment clearly. Figure 8: (**a**) Relationship between overturning moment (M) and rocking angle (\(\theta\)), and (**b**) rocking stiffness for S1 structure on three soil profiles. To reduce unrealistic spikes at frequencies (\(f\)) for which the Fourier transform of FFM was close to zero, the smoothed power spectral density function of FFM (\(S_{xx}(f)\)) and the cross-power spectral density function of FFM and FM\({}_{\text{T}}\) (\(S_{yy}(f)\)) were used to estimate the transfer functions, as follows [8]:  
\[H(f)=\ \left|\frac{S_{xy}(f)}{S_{xx}(f)}\right| \tag{3}\]  
To assess the noise level and nonlinearities of the transfer function between FFM and FM\({}_{\text{T}}\), the cohesion function can be estimated as follows:  
\[\gamma^{2}(f)=\frac{\left|S_{xy}(f)\right|^{2}}{S_{xx}(f)S_{yy}(f)} \tag{4}\]  
where \(S_{yy}(f)\) is a smoothed power spectral density function of FM\({}_{\text{T}}\). The value of \(\gamma^{2}\) varies from zero to one, and the frequencies at which the value of \(\gamma^{2}\) is above 0.8 indicates that noise between the two motions is minor [8]. The translational transfer functions and its incoherence parameter (i.e., \(k_{t}\)), which were used to define transfer functions based on [PERSON]'s expression [6], were determined following the steps proposed by [PERSON] and [PERSON] [8] and [PERSON] [5] at the frequencies with coherence values above 0.8. Note that these parameters indicate stochastic incoherence between FM\({}_{\text{T}}\) and FFM caused by the averaging of incident waves across the foundation region. Figure 9 shows the translational transfer function for the horizontal motion of the S1 foundation at PGA\({}_{\text{BM}}\) of 0.08 \(g\) and 0.22 \(g\) with the corresponding, respective, estimated values of \(k_{t}\) at 0.142 and 0.305. The measured data surrounding the flexible frequency range (from 0.9 to 2.1 Hz, as shown in Figure 7) was excluded when estimating the k\({}_{\text{t}}\) parameter to reduce the effects of structural inertia on foundation motions. The translation transfer function was lower than unity at high frequency, which indicates the base-slab averaging effect of SSFI [8]. The translational transfer function was estimated from the results of numerical simulation for the S1 structure at PGA\({}_{\text{BM}}\) of 0.08 \(g\) and 0.22 \(g\) with a \(k_{t}\) parameter of 0.229 and 0.366, respectively, as shown in Figure 9. The base-slab averaging effect with a decrement of the transfer function at high frequency was obtained following centrifuge experiment results. Values for the k\({}_{\text{t}}\) parameter were estimated for S1, S2, and S3 structures located on the Uni ground and plotted against increases in input intensity in Figure 10a. The dashed line at the \(k_{t}\) value of 0.08 represents the value predicted for shallow foundations by [PERSON] and [PERSON] [8] at a shear wave velocity of about 95 m/s.