## 5 Seismic Performance of the SSSI System
### Performance Maps of the SSSI System  
Figure 6 illustrates the performance maps of \(\alpha_{1}\) (left column) and \(\alpha_{2}\) (right column) for Frame 1 subjected to El Centro (top row), Kobe (middle row), and San Jose (bottom row) seismic excitations. It is noteworthy that the performance indexes across all parameter spaces are consistently below unity. Nevertheless, discernible regions within the maps exhibit enhanced performance of the SSSI system. Notably, these regions demonstrate consistent performance improvements across all three seismic events, suggesting the potential for identifying parameter values that yield robust performance regardless of the specific earthquake scenario. While this observation holds promise, its generalizability warrants further investigation through a comprehensive analysis encompassing a diverse range of seismic events. However, such an exhaustive study falls beyond the scope of the current paper. For the specified parameters (\(\alpha=0.2\), \(V_{s}=300\) m/s, and \(\chi=0.2\)), the SSSI system is shown to achieve a notable reduction of approximately 20% in both \(u_{1}\) and \(\Delta u\) compared to the performance of the standalone frame structure.  
Similar results are observed for Frame 2. Figure 7 is constructed similar to Figure 6 and presents the performance maps of \(\alpha_{1}\) (left column) and \(\alpha_{2}\) (right column) for Frame 2 under three selected earthquakes (El Centro in the first row, Kobe in the second, and San Jose in the third and last row). Consistently across the parameter space, the performance indexes remain below unity. Notably, distinct regions within these maps exhibit superior performance of the SSSI system. These regions demonstrate consistent performance enhancements across all three seismic scenarios, mirroring the observations made for Frame 2.  
Figure 6: Performance \(\alpha_{1}\) and \(\alpha_{2}\) maps of the 4-story frame structure under several earthquakes (\(\alpha=0.2\), \(V_{s}=300\) m/s, \(\chi=0.2\)).  
\(1\). However, it is worth highlighting that these performance-improving regions in Frame 2 are situated within parameter ranges different from those identified for Frame 1. This underscores the influence of the structural properties of the frame on the optimal parameter selection for ensuring the enhanced performance of the SSSI system. In essence, the identification of parameters that yield optimal SSSI system performance appears to be related to the mechanical characteristics of the frame structure being protected. In conclusion, the comparison between Frame 1 and Frame 2, considering that Frame 2 is larger than Frame 1 (consisting of 5 stories with an area of 350 m\({}^{2}\) compared to 4 stories with an area of 250 m\({}^{2}\) for Frame 1), reveals the following findings:  
* Achieving optimal performance of the SSSI system with Frame 2 necessitates higher values of \(\eta\) and \(\gamma\) compared to those required for Frame 1. This highlights the influence of the structural dimensions on the parameter settings that yield effective SSSI system performance.
* The extent of reduction in \(u_{1}\) and \(\Delta u\), as quantified by \(\alpha_{1}\) and \(\alpha_{2}\), is comparatively lower in Frame 2 than in Frame 1. This disparity indicates that, attributable to the greater mass of Frame 2 relative to Frame 1, the SSSI system exhibits enhanced performance for the smaller Frame 1.