Energy-based procedures for seismic fragility analysis of mainshock-damaged buildings

In recent decades, significant research efforts have been devoted to developing fragility and vulnerability models for mainshock-damaged buildings, i.e., depending on the attained damage state after a mainshock ground motion (state-dependent fragility/vulnerability relationships). Displacement-based peak quantities, such as the maximum interstory drift ratio, are widely adopted in fragility analysis to define both engineering demands and structural capacities at the global and/or local levels. However, when considering ground-motion sequences, the use of peak quantities may lead to statistical inconsistencies (e.g., fragility curves' crossings) due to inadequate consideration of damage accumulation. In this context, energy-based engineering demand parameters (EDPs), explicitly accounting for cumulative damage, can help address this issue. This paper provides an overview of recent findings on the development of aftershock-fragility models of mainshock-damaged buildings. Particular focus is given to state-of-the-art frameworks for fragility analyses based on cumulative damage parameters. Moreover, a literature review on damage indices and energy-based concepts and approaches in earthquake engineering is reported to better understand the main advantages of the mostly adopted energy-based parameters, as well as their limitations. Different refinement levels of seismic response analyses to derive fragility relationships of mainshock-damaged buildings are also discussed. Finally, the benefits of adopting energy-based EDPs rather than, or in addition to, peak quantities in state-dependent fragility analyses are demonstrated on a reinforced concrete frame building. Specifically, a refined lumped plasticity modeling approach is adopted, and sequential cloud-based time-history analyses of a Multi-Degree-of-Freedom (MDoF) model are carried out. The results highlight that energy-based approaches for fragility analysis effectively capture damage accumulation during earthquake sequences without inconsistencies in the obtained statistical models. On the other hand, estimating global or local structural capacity in terms of cumulative EDPs is still challenging. Further experimental data are needed to better calibrate the quantification of energy-based damaged states.