## 4 Parametric Analysis
### The Role of the Parameters Characterizing the SSSI System  
In order to assess the sensitivity of the structural dynamics to the various parameters defining the SSSI (structure-soil-structure interaction) system, the response of the frame structure is analyzed with \(\eta\) held constant at a value of \(\eta=50\). This specific value corresponds to the sections delineated by dash-dot lines in Figure 4. Figure 5 juxtaposes these sections for \(u_{1}\) (on the left) and \(\Delta u\) (on the right) with analogous curves derived by individually varying the remaining parameters. All the graphs of Figure 5 are represented in the same scale, facilitating a direct comparison of the impacts stemming from modifications in the different parameters. This common scale enables a concise assessment of the influence of parameter variations on the structural response, aiding in the identification of the key factors affecting the system dynamics. Figure 5 illustrates the sensitivity analysis with respect to \(\alpha\) (refer to Equation (14)) while maintaining fixed values for \(V_{s}=500\) m/s and \(\chi=1.0\). The bold curve corresponds to the section extracted from the surfaces depicted in Figure 4, representing the case where \(\alpha=1.0\). The results indicate that variations in the parameter \(\alpha\) (which corresponds to the separation distance \(d\) between the two foundation platforms) have a negligible impact on the structural response of the frame, as evidenced by the close proximity of the different curves. Figure 4: Displacement surfaces and maps of the 4-story frame structure under the Kobe earthquake. (**a**) Displacement \(u_{1}\). (**b**) Drift \(\Delta u\) (\(V_{s}=500\) m/s, \(\alpha=1.0\), \(\chi=1.0\)). Figure 5b demonstrates the impact of the shear wave velocity \(V_{s}\) of the soil on the structural response. It is important to note that this analysis is not intended to alter the soil characteristics, but rather to assess the performance of the proposed interaction as a function of \(V_{s}\). This approach allows for determining in advance whether the method is suitable for a specific case and soil type. The various curves in the graphs correspond to different values of \(V_{s}\) while maintaining fixed values for \(\alpha=0.2\) and \(\chi=1.0\). The bold curves coincide with those presented in Figure 5a, representing the case where \(\alpha=0.2\) and \(V_{s}=500\) m/s. Notably, the curve associated with the highest value of \(V_{s}=800\) m/s exhibits a nearly constant profile compared to the other curves. A decrease in the shear wave velocity \(V_{s}\) significantly alters the system behavior, leading to notable changes in the minimum values of both \(u_{1}\) and \(\Delta u\). This trend suggests improved performance of the SSSI system with lower values of \(V_{s}\). However, for smaller virtual mass parameter \(\gamma\), the structural response shows an increase in both displacement and drift. Nevertheless, such increments are confined to specific ranges of \(\gamma\), highlighting the complex interplay between system parameters and structural behaviour. Finally, Figure 5c investigates the influence of \(\chi\) (refer to Equation (11)), which represents the reduction in the soil's stiffness beneath the foundation of the external device resulting from the implementation of the GSI (Ground Stiffness Improvement) technique. The curves in the graphs correspond to varying values of \(\chi\) while maintaining fixed values for \(\alpha=0.2\) and \(V_{s}=300\) m/s. The bold curves align with those depicted in Figure 5b, representing the scenario with \(\chi=1\) and \(V_{s}=300\) m/s. Both graphs include the response of the frame structure in the absence of the external device (standalone structure); such a response, identified by the dashed straight line, accounts solely for the interaction of the structure with the soil beneath its foundation platform. The findings indicate that the minimum values of both \(u_{1}\) and \(\Delta u\) decrease as \(\chi\) is reduced.