PSI - Issue 78

Lorenzo Hofer et al. / Procedia Structural Integrity 78 (2026) 1927–1934 L. Hofer, K.Toska, M.A. Zanini, F. Faleschini, C. Pellegrino / Structural Integrity Procedia 00 (2025) 000–000

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commonly calibrated based on the outcomes of a set of numerical non-linear structural analyses (NLTHA). For both seismic hazard and structural fragility, several input parameters must be a priori fixed and different models (e. g., empirical, numerical, etc.) have to be assumed. In general uncertainties can be classified into two types: epistemic uncertainty, associated with the choice of the models for describing the hazard or the structural behaviour (e.g., choice of ground motion prediction equations, or the structural model itself), and aleatory uncertainty due to the intrinsic randomness of natural events such as earthquakes and structural deterioration (McGuire and Shedlock, 1981). Currently, the so-called logic tree approach is one of the most adopted methods for handling all the different uncertainty sources. In this method, every branch takes into account possible parameters’ values and/or possible alternative models. This methodology underpins several national and regional hazard models, including the Italian seismic hazard map (Stucchi et al. 2004) and the European Seismic Hazard Map (Giardini et al. 2014). More recently, Zanini et al. (2019) introduced a semi-analytical formulation to explicitly incorporate multiple sources of uncertainty in hazard quantification. On the structural side, numerous studies have investigated the influence of uncertainty on fragility assessment (e.g., Borgonovo et al. 2013; Pang et al. 2021; Chen et al. 2023), demonstrating that both aleatory and epistemic uncertainties can significantly affect fragility curve estimates (Dolšek 2008). Despite this, the literature remains limited in addressing the combined impact of uncertainties from both the hazard and fragility components on the overall seismic reliability assessment. The present study aims to advance this framework by systematically analyzing the principal sources of both aleatory and epistemic uncertainty that structural engineers and risk analysts must confront in seismic reliability assessments. Further details are provided in Hofer et al. (2023). 2. Mathematical formulation In scientific literature, the mean failure rate � is widely adopted for the seismic risk assessment of a structural system because it simply summarizes the site seismicity and the structural fragility and can be easily computed as: � = ∫ [ | ] ∙ | �� | �� (1) where �� represents the hazard curve of a specific site and [ | ] is the fragility curves representing the probabilistic structural behaviour of the analysed structure, i.e. the probability of reach and exceed a specific damage state for a given value of intensity measure . In Eq. (1) | �� | represents the mean number of seismic events per year producing a shaking of exactly , and can be computed as | �� | = − �� ⁄ ( ) ( ) . Then, the failure probability �,�,� in a specific time window T can be computed from � as �,�,� =1− �� � ∙� and finally, the corresponding reliability index �,� can be computed as the inverse of the standard normal cumulative density function evaluated in �,�,� . 3. Uncertainty sources 3.1. Seismic hazard In general, each quantity involved in the PSHA integral depends on several parameters that can be treated as random variables � (RVs) thus making �� itself a RV. �� ( � )=∑ � ���,� ∫ ∫ [ > | , , ���� ] � � ( , � ) � ���,� � ���,� � � ( , � ) � ���,� � ���,� � �� ��� (2) In the magnitude distribution of the i th seismogenic zone (SZ) � � ( , � ) common parameters that can be treated as random are the maximum ���,� and minimum ���,� earthquake magnitude and the slope � of the Gutenberg– Richter (G-R) occurrence law. Furthermore, also � ���,� can be considered as a RV ( � ���,� ) related to the G-R law and thus included the parameters vector � = � ���,� , ���,� , � , � ���,� � . Then also the source-to-site distance distribution � � ( , � ) can depends on some parameters (e.g., those describing the SZ geometry) that can be treated as RVs as well as [ > | , ] that represents the exceedance probability of a given value conditioned on a seismic event with a specific magnitude and occurring at a distance , and it is computed by a Ground Motion