# Intermediate Isolation System with Nonlinear Lower Structure and Isolation System
## 3 Examination of the Existing Building
### Seismic Assessment of the LS  
The first step of the proposed procedure (Section 2) aims to characterize the dynamic behaviour and lateral strength capacity of the LS.  
The three-dimensional Finite Element (3D FE) model of the LS, built by means of the software SAP2000 [43], is shown in Figure 1b. As observed in Figure 1a, the masonry buildings in aggregate show vertical setbacks that are removed to provide the same rooftop level, corresponding to the isolation layer. For this aim, a \"filling\" steel structure has been added on the roof of the first building on the left side; furthermore, the masonry walls of the second and third buildings have been extended with the same masonry material so that they all reach the same height (Figure 1b).  
The structural system of the unreinforced masonry (URM) building is made of vertical walls connected by horizontal spandrels. The modelling approach adopted in this paper is the equivalent frame model (EFM) with weak spandrels [44], a simplified approach that is very common in engineering practice [44; 45; 46]. Neglecting the flexural contribution of the  
Figure 1: (**a**) Front view of the existing building with the volume to be reconstructed (depicted in red); (**b**) 3D FE model of the lower structure; (**c**) plastic hinge for flexural and shear behaviour of masonry piers; (**d**) capacity curves for the weakest direction of the existing building.  
spandrels, the model is composed of vertical cantilever beams (the piers) connected to each other at floor levels through a rigid diaphragm.  
A concentrated plasticity model is used for describing the flexural and shear nonlinear behaviour of masonry piers [44; 47; 48; 49]. Two flexural hinges are assigned at both ends of each pier, while one shear hinge is assigned at the centre of each pier. Only the diagonal cracking (Equation (2)) has been considered as a possible shear failure mechanism [50] since it is the most likely failure mode for URM with irregular blocks, as also suggested by the Italian code [51]. The sliding shear failure, which is related to the un-cracked section and thus, in turn, to the flexural behaviour, has not been considered in the model. For this reason, the shear-flexure interaction in the plastic hinges is neglected. The flexural and diagonal shear resistances are calculated using the formulas reported in the Italian code [51; 52] and recalled below:  
\[\mathrm{M_{u}}=\left(\mathrm{I^{2}\!\cdot\!t}\frac{\sigma_{0}}{2}\right)\left( 1\!-\!\frac{\sigma_{0}}{0,85\mathrm{f_{d}}}\right) \tag{1}\]  
\[\mathrm{V_{u}}\!=\!\mathrm{I\!\cdot\!t}\frac{\mathrm{f_{td}}}{\mathrm{b}} \sqrt{1\!+\!\frac{\sigma_{0}}{\mathrm{f_{td}}}} \tag{2}\]  
where: \(\mathrm{I}\), \(\mathrm{t}\), and \(\sigma_{0}\) are the length, the thickness and the stress compression due to gravity load of the masonry piers, respectively; \(\mathrm{b}\) is a factor that takes into account the shear stress distribution in the cross section and is related to the aspect ratio of the wall [53].  
The plastic hinge properties at yielding and ultimate limit states are given in Figure 1c.  
Push over analyses were carried out on the 3D FE model of the LS (Figure 1b) by means of the software SAP2000 [43]. The push over curve for the weakest Y direction (Figure 1b) and uniform loading pattern, which corresponds to the lowest capacity of LS, is provided in Figure 1d, along with the bilinear force-displacement relationship obtained according to the N2 method [54]. The dynamic properties of the equivalent single degree-of-freedom model (SDOF LS, Figure 2a) are: mass \(\widetilde{m}_{LS}\) = 2472 kNs\({}^{2}\)/m, lateral stiffness \(\widetilde{k}_{LS}\) = 639,883 kN/m, fundamental period \(T_{LS}=2\pi\left(\widetilde{m}_{LS}/\widetilde{k}_{LS}\right)^{0.5}\) = 0.39 s, damping constant \(\widetilde{c}_{LS}=2\xi_{LS}\omega_{LS}\widetilde{m}_{LS}\) = 3983 kNs/m, equivalent viscous damping ratio \(\xi_{LS}\) = 0.05, and fundamental circular frequency \(\omega_{LS}=2\pi/T_{LS}=16.11\) rad/s, where the \(\sim\) symbol refers to the reduced order model.