## 2 Case studies
### Design of Isolation Systems  
The design displacement \(\mathrm{d_{Ed}}\) of the isolation systems has been estimated referring to the simplified linear analysis method considering the horizontal dynamic translation of DCCSS bearings (Figure 4a), without superimposing static torsional effect, and assuming the superstructure as a rigid body. Referring to Figure 4b), the main parameters of the linear equivalent model of the isolation system based on DCCSS bearings are effective stiffness \(\mathrm{K_{e}}\), effective period \(\mathrm{T_{e}}\) and effective damping \(\mathrm{\xi_{e}}\) as reported in Equation (1), where \(\mathrm{R_{e}}\) is the effective radius, \(\mathrm{\mu}\) is the friction coefficient, \(\mathrm{g}\) is the gravity acceleration. The friction pendulum forces are proportional to the device axial load and the sliding velocity.  
\[\mathrm{K_{e}}=\mathrm{N_{Sd}}\bigg{(}\frac{1}{\mathrm{R_{e}}}+\frac{\mathrm{ \mu}}{\mathrm{d_{Ed}}}\bigg{)},\ \mathrm{T_{e}}=\pi\sqrt{\frac{1}{\mathrm{g}\bigg{(}\frac{1}{\mathrm{R_{e}}}+ \frac{\mathrm{\mu}}{\mathrm{d_{Ed}}}\bigg{)}}},\ \mathrm{\xi_{e}}=\frac{2}{\pi}\cdot\frac{1}{\frac{ \mathrm{d_{Ed}}}{\mathrm{i}\cdot\mathrm{R_{e}}}+1}, \tag{1}\]  
The design procedure for the new construction case studies for L'Aquila (AQ_new) and Naples (NA_new) sites is shown in Figure 5, in terms of Acceleration-Displacement Response Spectra (ADRS). The design displacement \(\mathrm{d_{Ed}}\) at the CLS and the characteristics of designed DCCSS bearings are summed up Figure 5.  
The design procedure of the isolation system for existing building case studies starts from the spectral acceleration associated with the occurrence of the first plastic hinge (\(\mathrm{S_{e,min}}=\mathrm{V_{e}}/\mathrm{M}\)). The elastic limit of the base shear \(\mathrm{V_{e}}\) of the superstructure in the fixed-base configuration has been identified from pushover curves (as described later). To take into account the structural uncertainties, the aforesaid spectral acceleration has been divided by a safety factor equal to 1.1, the design acceleration at the LLS is \(\mathrm{S_{e}^{*}}=\mathrm{S_{e,min}}/1.1\) as shown in Figure 6 (red dashed lines). The minimum value of the fundamental period \(\mathrm{T_{iso,min}}\) of the base-isolated system is then found. The design procedure for the existing case studies (AQ_LSD; NA_GLD) is shown in Figure 6.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline
**IM** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** \\ \hline \(\mathrm{T_{R}}\) (**Years**) & **10** & **50** & **100** & **250** & **500** & **1000** & **2500** & **5000** & **10,000** & **100,000** \\ \hline L’Aquila, \(\mathrm{S_{a}}\) (\(\mathrm{T=3\,s}\)) & 0.0002 & 0.011 & 0.031 & 0.062 & 0.11 & 0.177 & 0.271 & 0.384 & 0.576 & 1.053 \\ Naples, \(\mathrm{S_{a}}\) (\(\mathrm{T=3\,s}\)) & 0.001 & 0.009 & 0.026 & 0.044 & 0.067 & 0.093 & 0.126 & 0.162 & 0.216 & 0.348 \\ \hline \end{tabular}
\end{table}
Table 2: Spectral acceleration Sa(T = 3 s) expressed in [g] for the IMs.  
Figure 3: Response spectrum of ground motion for IM 6 (\(\mathrm{CLS}-\mathrm{T_{R}}\) = 1000 years)—site of L’Aquila (**a**) and site of Napoli (**b**).  
The characteristics of designed DCCSS bearings for all case studies are reported in Table 3. The maximum displacement d\({}_{\text{max}}\) has been assumed by catalogue, taking into account torsional effects.  
Figure 4: (**a**) DCCSS bearing and (**b**) linear equivalent model.  
Figure 5: Isolation system design procedure on CLS and LLS spectra for the cases of New constructions building in the site of L’Aquila (**a**) and Naples (**b**).  
Figure 6: Elastic limits from pushover curves of existing buildings and isolation system design procedure for the cases of L’Aquila (**a**) and Naples (**b**).