## 2 Methodology  
This paper considers the 3 DOF methodology proposed in [21] that may assess the two main effects (period elongation and damping increase) by assuming three equivalent-linear models (structure, isolation system and soil) and by setting only four parameters that synthetize the soil properties and the isolation characteristics (see Figure 1). The entire system implements two formulations to reproduce the effects of the soil on the structure by considering the applying the theoretical model proposed by [PERSON] [22] and applied in [23] to assess the effects of SSI on structural configurations. The second formulation is based on the analytical approach proposed by [24; 25] that models the isolated system (structure + isolator) with a 2 DOF linear system. It is worth noting here that the assumption that the structure is in both formulations represented as a single-DOF is strictly valid only when the fundamental period is representative of the behavior of the entire structure (i.e., vertical and horizontal regularity). Another important limitation consists of neglecting the torsional responses of isolated structures that depends on the structural flexibility [26]. It is worth noting that the 3 DOF system is not an alternative to the most advanced methodologies, such as numerical simulations. The aim of this method is to make the design procedure faster by considering two steps. Step 1  
The 3 DOF system is used for defining the period elongation by avoiding time- and cost-consuming numerical simulations at a preliminary stage. Step 2  
Advanced numerical simulations are needed to be performed to assess the performance of the system (structure, BI and soil). Therefore, the number of these simulations will be significantly reduced because of the application of step 1. Considering the soil as a homogenous damped material, [23], the period elongation may be calculated as:  
\[T_{1}=\frac{T_{\mathrm{S}}\,\sqrt{2}}{\sqrt{2a+\beta}} \tag{1}\]  
where:  
\(\mathrm{T_{\mathrm{S}}}\) is the fundamental period of the fixed base structure (no BI and no SSI);  
\(\mathrm{T_{1}}\) is the fundamental period (elongated) of the system (soil, isolation, structure);  
\(\beta\) is the ratio between the structural mass (ms) and the mass of the foundation soil (mg):  
\[\beta=\frac{m_{\mathrm{s}}}{m_{g}} \tag{2}\]  
\(\alpha\) is the squared ratio between the frequency of the soil (\(\omega_{g}\)) and that of the structure (\(\omega_{\mathrm{s}}\)):  
\[\alpha=\left(\frac{T_{\mathrm{s}}}{T_{\mathrm{g}}}\right)^{2}=\left(\frac{ \omega_{g}}{\omega_{\mathrm{s}}}\right)^{2} \tag{3}\]  
The second formulation assumes an equivalent linear isolation that is a simplification of more refined applications presented in literature [27; 28; 29; 30]. However, such an assumption  
Figure 1: The considered 3 DOF system. \(\mathrm{T_{\mathrm{s}}}\): the fundamental period of the fixed base structure (no BI and no SSI); n: isolation ratio; \(\mathrm{T_{\mathrm{S}}}\): shear wave velocity of the foundation soil. may be considered valid for the scope of investigating the key parameters for preliminary design. Therefore, two parameters are defined:  
\[\gamma=\frac{M_{s}}{M_{s}+M_{b}} \tag{4}\]  
\[\varepsilon=\left( \begin{array}{c}\omega_{b}\\ \omega_{s}\end{array} \right)^{2} \tag{5}\]  
where:  
\(\mathrm{M_{b}}\) is the mass of the base floor above the isolation system,  
\(\omega_{\mathrm{b}}\) is the isolation system frequency. The equivalent 3-DOF model (soil + isolation + structure) may be used to represent the entire system and the characteristic equation for the frequencies results (more details in [21]):  
\[ \begin{array}{c}\omega^{6}+(\alpha+1+\varepsilon+\frac{\gamma}{1-\gamma}+ \varepsilon\cdot\beta\cdot\frac{1-\gamma}{\gamma})\cdot{\omega_{s}}^{2}\cdot \omega^{4}\\ -[\alpha\cdot\left(\varepsilon+1+\frac{\gamma}{1-\gamma}\right)+\varepsilon \cdot(\beta+1+\beta\cdot\frac{1-\gamma}{\gamma})]\cdot{\omega_{s}}^{4}\cdot \omega^{2}+\alpha\cdot\varepsilon\cdot{\omega_{s}}^{6}.\end{array}