PSI - Issue 78

Joud Habib et al. / Procedia Structural Integrity 78 (2026) 799–806

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1. Introduction In the context of seismic risk assessment, the conditional probability of exceeding a certain damage state under a single seismic event given a certain value of ground motion Intensity Measure (IM) is called state-independent fragility. This definition is commonly used in Performance Based Earthquake Engineering (PBEE) (Cornell et al., 2002; Baker, 2007). It assumes that structures are initially undamaged, and it is suited to analyze the effects of mainshocks. However, real-world observations from recent earthquakes have shown that structures can accumulate damage through sequences of events, such as mainshock – aftershock pairs or multiple closely spaced main shocks (Monteferrante et al., 2025; Wu et al., 2024; Cattari et al., 2014; Dolce et al., 2018). To incorporate prior damage in seismic fragility, state-dependent fragility models have been introduced (Jeon et al., 2015). This approach accounts for the pre-existing damage state in a building when estimating the probability of reaching higher damage levels in subsequent shocks. A recent study by Iervolino et al. (2020), has emphasized the importance of damage accumulation in fragility assessments, particularly in mainshock – aftershock sequences, and have proposed frameworks that evaluate how sequential seismic events influence structural performance. For example, preliminary results on state-dependent seismic fragility functions for Italian reinforced concrete (RC) buildings were presented by Orlacchio et al. (2021), highlighting the significance of considering prior damage. This work was later extended by Orlacchio et al. (2024), who developed state-dependent fragility functions for various Italian building classes, enhancing the accuracy of seismic risk evaluations across different structural typologies. The findings revealed that aftershocks following a significant mainshock considerably increase the probability of failure, especially when the structure has already been weakened. Similar conclusions were reached by Lee and Ju (2020), and Amini and Vamvatsikos (2021), who developed methodologies that refine fragility estimates by incorporating cumulative damage effects and the interactions of sequential seismic events. Despite these advancements, there remains a need for a systematic and physically consistent methodology that integrates state-dependent fragility with robust statistical treatment to overcome issues such as curve overlap and loss of logical ordering among damage states. The present work proposes a methodology based on Cloud Analysis (Jalayer et al. 2009); to ensure statistical robustness and physical consistency, this study introduces modified forms of probit and ordinal probit regression, which address the challenge of overlapping curves and preserve the logical progression across damage states. The proposed methodology is used to develop state-dependent fragility curves for archetypical reinforced concrete frame building typologies in Italy, as classified by the SERA project (Seismology and Earthquake Engineering Research Infrastructure Alliance for Europe) by Romão et al. (2019). These typologies are idealized as Equivalent Single Degree of Freedom (ESDoF) systems, enabling efficient simulation of nonlinear structural responses under multiple seismic scenarios. The ESDoF models considered in this study are representative of Italian reinforced concrete (RC) residential buildings with infilled frames, classified within the broader taxonomy of the SERA. They are grouped into three main design categories based on seismic code provisions: Code Design Null (CDN-0) with no seismic design, Code Design Low with a 5% seismic coefficient (CDL-5), and Code Design Medium with a 10% seismic coefficient (CDL-10). Within each category, six building typologies are defined, varying in the number of stories from one to six, resulting in a total of eighteen structural typologies. These ESDoF models follow the modelling approach proposed by Suzuki et al. (2021). In which both the dynamic and static capacities of each structure are represented by an equivalent capacity curve. To ensure consistency and applicability across different structural configurations, each ESDoF is characterized by a normalized force – displacement curve, expressed in terms of displacement and force per unit mass (F/m). This curve is defined by four key points: Yield, Peak, Degradation, and Ultimate, as shown in Fig. 1. (a). The values of these characteristic points for each typology are taken from Orlacchio et al. (2024). Fig. 1. (b) represents the normalized capacity curves for the six typologies of the category of Code Design Null (CDN-0). 2. Description of the ESDoF models 2.1. Capacity curves and damage states