## 3 Experimental Results
### Dynamic SSSI Effects on S1 Structure  
#### 3.2.1 SSSI Effects on Responses of S1 Structure in Frequency Domain  
In frequency domain analysis of SSSI phenomena, a number of previous studies have reported changes in natural frequency (i.e., flexible-base frequency of the structure, \(f_{flex}\)) and seismic response of structure by plotting response spectra or transfer function [42; 43; 44]. In the present study, with an assumption of stability in ground amplification due to non-compacted ground and consistently applying input excitations, SSSI effects on the response of S1 structure were explored by comparing RM recorded for each SSSI test to that of the reference test which was response of single structure. Power Spectral Density (PSD) showing the distribution of power of a signal in frequency domain was calculated for S1 structure with a 1024-point hamming window for smoothing [33]. PSD values at a defined frequency (PSD\({}_{f}\)) could be obtained using [PERSON]'s method [45]:  
\[\text{PSD}_{f}=\frac{1}{K}\sum_{k=1}^{K}P_{m}(f) \tag{1}\]  
where \(K\) is the number of segments, \(k\) is the order of segment, and \(P_{m}(f)\) is modified periodogram value. \(P_{m}(f)\) can be defined from discrete Fourier transform of each segment as:  
\[P_{m}(f)=\frac{1}{\sum_{m=0}^{M}\omega^{2}(m)}\left|X_{k}(f)\right|^{2} \tag{2}\]  
where \(M\) is the number of points in each segment, \(X_{k}(f)\) is windowed discrete Fourier transform, and \(\omega(m)\) is window function. Figure 7 shows typical results of PSD of RM for S1 structure installed with S2 or S3 structures in various distances (i.e., B, 0.5B, and 0.03B) at medium earthquake input intensity. Solid lines indicate reference distribution of PSD for S1 structure (i.e., the single response for S1 structure). Compared to the reference response, SSSI effects caused a reduction in peak PSD of RM for S1 presented by solid points in Figure 7 when other structures such as S2 or S3 were located beside at all tested distances. Figure 7: Cont. Figure 7: Power Spectra Density (PSD) of S1 structure (a) S1-S2 tests with artificial EQ; (b) S1-S2 tests with Hachinobe EQ; (c) S1-S3 tests with artificial EQ; (d) S1-S3 tests with Hachinobe EQ at M\({}_{\ast}\)Input intensity. Solid circles indicate peak PSD at resonant frequency. _fdx_ indicates flexible-base frequency of S1 structure. For a more robust observation of change in response, the total of power spectral density (\(\Sigma\)PSD) indicating total energy of roof motion in the frequency domain was estimated for PSD of S1 structure. SSSI effects on S1 structure were evaluated using variation in \(\Sigma\)PSD of S1 structure in SSSI tests (\(\Sigma\)PSD(SSSI)) with that of reference test (\(\Sigma\)PSD(Ref)) as follows:  
\[\Delta\text{PSD}\;[\%]=\frac{\Sigma\text{PSD}(\text{SSSI})-\Sigma\text{PSD }(\text{Ref})}{\Sigma\text{PSD}(\text{Ref})}\times 100 \tag{3}\]  
Table 4 presents \(\Delta\)PSD of S1 with a combination of S2 and S3 structures at different distances. \(\Sigma\)PSD of S1 building reduced when S2 or S3 structure was placed at all tested distances regardless of the type and intensity of input motions. \(\Sigma\)PSD of S1 structure reduced more in tests with S2 structure compared to that in experiments involving S3 structure because S2 building had higher inertial characteristics (i.e., height and mass) than S3 [16; 17]. More decrement in \(\Sigma\)PSD was observed in tests at low input intensity compared to that at high input level. The most substantial decrement in \(\Sigma\)PSD of S1 was found to be approximately 30% in case of S2 structure located beside with distance B at L_Input. This observation could be attributed to decrement in inertial interaction of SFSI with increased input excitation level as observed in Figure 4. Experimental results with S1 and S3 structures at a distance of 0.03B, H_Input level of the artificial EQ, and all input levels of Hachinohe EQ could not be recorded because S3 structure collapsed during the test.