A direct analytical derivation of the multi-dimensional fragility spaces of structures under nonstationary mainshock-multi-aftershock sequences

Performance-based earthquake engineering (PBEE) is a popular direction in the earthquake community, and at this stage, risk-based PBEE has become mainstream. In the risk-based probabilistic framework, seismic fragility analysis constitutes the most important link, and corresponding research on the mainshock–aftershock sequence has received widespread attention in recent years. Since a mainshock is often accompanied by multiple aftershocks and there is great uncertainty in the vibration characteristics of aftershocks, a seismic fragility analysis of structures under a stochastic mainshock-multi-aftershock sequence is meaningful and necessary. The corresponding questions, such as how to derive the multi-dimensional analytical fragility form under a stochastic mainshock-multi-aftershock sequence and how to correlate multiple intensity measures with multiple demand parameters, still require further investigation. In this paper, a direct analytical derivation of the multi-dimensional seismic fragility spaces of structures under nonstationary stochastic mainshock-multi-aftershock sequences is introduced. The methodology framework, implementation steps, and application examples are also provided in detail. Moreover, two scenarios, the one-mainshock-one-aftershock and one-mainshock-two-aftershocks, are considered, and the obtained multi-dimensional analytical fragility spaces for both scenarios are validated. In general, the matching accuracy of the fragility results for both scenarios is proven to be high, and the direct analytical derivation of the multi-dimensional fragility spaces is validated to be ideally consistent, which further provides a reference for multi-dimensional risk analysis under nonstationary stochastic mainshock-multi-aftershock sequences in future work.