# Evaluating Collapse Fragility Curves for Existing Buildings Retrofitted Using Seismic Isolation
## 1 Introduction  
Collapse fragility functions (CFFs) constitute one of the main components of performance-based earthquake engineering assessment procedures [1; 2]. A rigorous approach for the evaluation of CFF is based on non-linear response-time history analysis (NTHA), typically incremental dynamic analysis (IDA) or multiple-stripe dynamic analysis (MSA), performed adopting a specific set of seismic ground motion records, and accurate structural modeling approaches.  
The CFF is a cumulative probability distribution (typically in the log-normal form), derived by fitting NTHA (IDA or MSA) results. Basically, CFFs define the structural collapse probability as a function of a reference seismic intensity measure (e.g., the spectral acceleration associated to the fundamental period of the structure, \(\mathrm{S_{a}}\)(T)). As a matter of fact, CFFs can be identified through a set of two main parameters, i.e., the median \"collapse capacity\" (i.e., ground motion intensity with 50% probability of exceedance) and the dispersion depending on several uncertainties, including aleatory (record-to-record variability) and epistemic uncertainties (lack of knowledge, design and modeling uncertainties).  
In the last decade, several studies aimed at the evaluation of the CFFs for new and existing fixed-base reinforced-concrete buildings [3; 4] have been proposed. Conversely, a few studies [5; 6] investigated the collapse performances and capacity of existing buildings retrofitted with different isolation systems.  
Among those, [PERSON] et al. [6] proposed CFFs for existing reinforced concrete (RC) framed buildings, located in medium or high-seismicity areas, designed for vertical loads only and equipped with different base-isolation systems.  
Recently, the RINTC (Implicit Risk of code-conforming Italian buildings) research project [7], funded within the ReLuis 2015-2018 program (promoted by the Italian Department of Civil Protection), developed a systematic methodology for the estimation of collapse probabilities of different building typologies, included existing buildings retrofitted using seismic isolation. Pragmatic choices regarding the definition of the collapse conditions and the numerical non-linear modeling of the isolation system have been adopted within [7]. The preliminary results of the aforementioned project pointed out that base-isolated systems play a key role in the collapse performances of the whole structure. As a matter of fact, a limited \"residual resistance\" of the isolation system has been observed after the attainment of the design intensity level, especially for high seismicity areas, in particular for rubber-based systems. Therefore, it can be expected that the definition of the collapse conditions and the numerical non-linear modeling of the rubber-based isolation system can potentially affect the definition of the CFF.  
In this paper, the main limitations of the aforementioned work are questioned and further developments of the described methodology, in terms of non-linear modeling and collapse conditions, are proposed for the upcoming version (2019-2021) of the RINTC research project. Such developments are then applied for the estimation of the collapse fragility functions of two archetypes of existing buildings retrofitted with seismic isolation. The CFFs derived from the RINTC research project [7] are finally compared with those obtained adopting the proposed developments. The outcomes of the present study could represent an addendum to the scientific literature providing enhanced tools, potentially suitable for refined performance-based assessment and design procedures.