# Evaluating Collapse Fragility Curves for Existing Buildings Retrofitted Using Seismic Isolation
## 4 Methodology  
The seismic performances of the case-study buildings have been evaluated by means of MSA carried out considering 10 intensity levels and 20 ground motions per stripe [7]. A preliminary probabilistic seismic hazard analysis (PSHA) has been carried out using OPENQUAKE [19] to derive the hazard curves at the building sites as a function of the spectral acceleration, S\({}_{\text{a}}\)(T), associated to the reference period (corresponding to the fundamental period of vibration of the case-study, approximately equal to 3 sec in both cases).  
Hazard curves are further discretized at the 10 Intensity Measure (IM) values corresponding to the following return periods: 10, 50, 100, 250, 500, 1000, 2500, 5000, 10,000 and 100,000 years.  
Then, 20 Seismic records pairs have been selected for each IM based on the proper conditional mean spectra (CMS), assuming an adequate magnitude-distance-deviation disaggregation and a proper attenuation relationship for the selected sites [20].  
Assumed that the superstructure and the isolation system can be considered as two components of an in-series system, the collapse of either component induces the collapse of the entire structure.  
In line with the mentioned RINTC research project, the collapse of the superstructure has been conventionally associated to the attainment of a top displacement corresponding to a 50% decrease of the lateral strength, computed via pushover analysis in the fixed base configuration [7].  
The failure modes considered for HDRBs are cavitation, shear failure and buckling. The same failure criterion adopted in [7] are considered herein for cavitation and shear failure. In particular, cavitation occurs in the post-cavitation branch for an axial tensile strain of the order of 25% while a limit shear strain equal to 350% has been assumed as shear failure deformation threshold.  
By contrast, a different failure criterion has been adopted for the buckling mode. In the first edition of RINTC research project [7] (using the HDR element), buckling is deemed to occur when  
Figure 3: Cyclic response of HDRB n. 20 (SLD case-study) subjected to a certain ground motion record (n. 3), HDR element vs. Kikuchi bearing element.  
the P/P\({}_{\rm cr}\) ratio between the current axial load (P) and the critical buckling load (P\({}_{\rm cr}\)) is equal to 1, (P\({}_{\rm cr}\) was computed step by-step as a function of the effective shear rigidity and effective flexural rigidity of the device). The latter criterion underestimates the effective capacity of HDRBs neglecting their post-buckling behavior, thus leading to an excessively conservative collapse condition. As a consequence, in this paper, where HDRBs are modeled with the Kikuchi bearing element (see Section 3), a more \"rational\" criterion has been assumed considering that the buckling failure is deemed to occur for an axial compressive strain of the order of 50%.  
Finally, the failure of FSBs has been associated to the attainment of a certain limit displacement, assumed to be equal to the displacement capacity of the device increased by an extra-displacement equal to a fraction (50%) of the bearing pot radius. In line with the European/Italian seismic Code [9], the connection resistance should be (minimum) two times larger than the maximum shear force value transmitted by the seismic device, as a consequence connections failure has been neglected.  
All that considered, the collapse of the isolation system is deemed to occur when at least 50% of the devices reaches one of the failure conditions mentioned above. Failure modes and collapse conditions for the selected case-studies are summarized in Table 3.