## 3 Description of the Case Study: An Existing RC Building
### Evaluation of the Seismic Demand and Non-Linear Dynamic Analyses  
The evaluation of the seismic demand in terms of elastic pseudo-acceleration \(S_{a}\) is performed considering the elastic design response spectrum related to 50 years reference period associated to nearly-collapse limit state according to [2] with reference to the seismic hazard site. The elastic response spectra have been defined adopting two different damping ratios as shown in Figure 7, in particular: \(\xi=5\%\) for the fixed-base (FB) NLN model and \(\xi\)= 2% as suggested by [17] for the base-isolated (BI) NLN model [17]. In general, the value of \(\xi=5\%\) is widely accepted for the dynamic analysis of FB RC structures [17]. With reference to the BI structure, the value of \(\xi\) is lower than the one related to the FB structure as the superstructure moves on the isolation level similarly to a rigid body. For instance, most of the energy in the BI structure is dissipated by hysteretic damping which is provided by the isolator devices. A value of \(\xi=2\%\) for BI RC structures is recognized according to common practice and literature references [3; 17]. The seismic inputs adopted to perform the non-linear dynamic analyses are represented by 21 recorded accelerograms [17] (each of them includes 3 accelerometric recordings in the respective directions) selected from European Strong Motion Database (ESM) [37] (Table 3) and then scaled in all directions. The selection has been carried out with the aim to define a set composed of ground motions with different characteristics to represent both a record-to-record variability and event-to-event variability, as commented in [3; 17]. The scaling factor has been defined in X direction with respect to the Intensity Measure (_IM_) related the elastic response spectrum in correspondence of the fundamental period of vibration \(T_{1}\) associated to 1 st eigenmode for both FB and BI structures. This is in order to ensure the compatibility of the selected ground motion inputs with elastic design response spectra  
Figure 6: Non-linear numerical models for FB (**a**) and BI (**b**) structure using SAP 2000 [25]. Figure 7: Elastic design response spectra according to nearly-collapse limit state [2] with two different damping ratios. defined by the design code [2] for nearly-collapse limit state. Note that the scaling factor has been defined along the direction (i.e., X direction) able to cause the worst effects. The scaled response spectra associated to the 21 recorded accelerograms are reported in Figure 8 with reference to both the FB and BI structures. The non-linear dynamic analyses have been performed adopting the direct integration of the motion equations [38; 39] which regulate the dynamic behavior of the structure with its non-linearities:  
\[[M]\left\{\frac{d^{2}u(t)}{dt^{2}}\right\}+[C]\left\{\frac{du(t)}{dt}\right\}+ [K][u(t)]=[F(t)] \tag{5}\]  
in which \(\{d^{2}u(t)/dt^{2}\}\), \(\{du(t)/dt\}\), \(\{u(t)\}\) are respectively the acceleration, velocity and displacement vector, having considered a multi-degree freedom (MDOF) system. While, \([M]\), \([C]\), \([K]\) are the mass, proportional Rayleigh damping and stiffness matrix evaluated according to [3; 17]. Finally, \(\{F(t)\}\) is the vector of external forces. In Figure 9 the hysteretic loops of one of the FP bearing subjected to the scaled motion of the earthquake EQ5 is depicted showing how different phenomena occur during the non-linear hysteretic response of the FPS device. In detail, it is possible to observe the influence of the velocity-dependent behavior of the FPS device because when the displacement is peak the sliding friction coefficient gradually decreases from \(\mu_{fast}\) to \(\mu_{slow}\). In addition, the floating shape of the curves is due to the vertical component of each ground motion.