Derivation of state-dependent fragility curves for reinforced concrete frame buildings using cloud analysis

Structures exposed to sequences of earthquakes—such as mainshock–aftershock events or seismic events occurring in short succession—often accumulate damage that significantly alters their seismic vulnerability. Traditional fragility models typically neglect this effect by assuming initially undamaged conditions, which can lead to an underestimation of damage. To address this limitation, state-dependent fragility models have been proposed in the literature. This study presents a comprehensive framework for assessing seismic fragility, with a particular focus on state-dependent fragility curves for reinforced concrete buildings representative of the Italian building stock. The proposed methodology incorporates the effect of prior damage by developing fragility curves conditioned on the initial damage state. Structural typologies are modelled as Equivalent Single Degree of Freedom (ESDoF) systems and analyzed through nonlinear dynamic analyses (NLDA) in OpenSees using a large suite of real ground motion records. Cloud Analysis is employed to establish probabilistic relationships between intensity measures (IMs) and engineering demand parameters (EDPs). Fragility curves are derived using Generalized Linear Models (GLMs) with probit link functions. A key contribution of this study is the development of modified versions of standard probit and ordinal probit regression models to ensure the correct logical ordering of fragility curves and eliminate overlapping—a common issue in both traditional and state-dependent fragility modelling. The results demonstrate that prior damage significantly increases structural vulnerability, and the proposed framework offers a more physically consistent and statistically robust approach to seismic risk assessment, particularly for post-earthquake decision-making and retrofit prioritization.