We present a methodology for developing fragilities for mainshock-damaged structures, "aftershock fragility", by performing incremental dynamic analysis (IDA) with a sequence of mainshock-aftershock ground motions. The aftershock fragility herein is distinguished from a conventional fragility for an intact structure. We estimate seismic response of a mainshock-damaged building by performing nonlinear time history analysis with a sequence of mainshock and aftershock ground motions (so-called "back-to-back" dynamic analysis). We perform the back-to-back dynamic analyses for a number of levels of mainshock response/damage, and a number of sequences of mainshock and aftershock ground motions. With estimated seismic responses from the back-to-back dynamic analyses, we compute various damage state transition probabilities, the probability of exceeding a higher damage state from an aftershock given a damage state due to a mainshock. For an illustration of the methodology, we develop an aftershock fragility for a typical New Zealand 5-storey reinforced concrete moment frame building. The building is modeled using a single-degree-of-freedom (SDOF) damped nonlinear oscillator with force-deformation behavior represented by a multi-linear capacity/pushover curve with moderate pinching hysteresis and medium cyclic deterioration.
