## 1 Introduction
Furthermore, they pointed out that the aftershocks do not significantly increase the peak or the residual drift demands since the predominant period of the aftershocks is very different from the period of the frame models. [PERSON] et al. investigated the effects of duration and frequency contents of earthquakes on the collapse of buildings subjected to mainshocks. Their findings implied that both duration and frequency have a great impact on the structural collapse capacity [18]. The presence of aftershocks may increase the seismic demands of civil structures. The influence of aftershocks on the peak ductility demands caused by strong seismic sequences was investigated by [PERSON] and [PERSON]. They pointed out that using the real seismic sequences might underestimate the aftershock impact due to the limited number of the recorded dataset [19]. In reference [20], the findings showed that considering the damage from previous seismic events, the probability failure is affected significantly by the number of stories and the earthquake region. [PERSON] et al. studied the accumulative seismic damage of highway bridges under multiple earthquakes. They pointed out a significant increase in the exceedance probabilities of damage indices due to the aftershocks. Some researchers evaluated the effects of aftershocks on the maintenance cost [21]. In reference [22], the effects of aftershocks on the seismic loss of steel buildings are investigated. They found that aftershocks have a significant impact on seismic loss. [PERSON] and [PERSON] investigated the effects of aftershocks on the risk and resilience of bridges; the findings indicate that aftershocks have great effects on repair loss and residual functionality [23]. [PERSON] et al. investigated the performance of jackets for repairing the mainshock-induced damage on reinforced concrete columns of a bridge subjected to potential aftershock loads. Their findings showed that the jacket can improve the bridge collapse capacity due to aftershocks [24], and the basic characteristic of a structure may be changed when they are suffered from earthquake sequences or other hazards in their lifetime. [PERSON] and [PERSON] proposed a stochastic structural control method for structures under sequential excitations [25]. It is summarized with this literature review that traditional fragility modeling has been mostly adopted for estimating the vulnerability of structures due to either mainshocks or aftershocks. When using such traditional fragility modeling in these efforts, the cumulative distribution function of a normal or lognormal distribution is commonly used with two distribution parameters [26]; then the parameters of the likelihood functions are estimated using the method of moments or, more generally, the maximum likelihood estimation (MLE) method [27]. This traditional fragility modeling approach is reviewed for the sake of completeness and for understanding the effort in this paper (Section 2.1). In recent years, a few fragility-based efforts (Section 2.2) were proposed to assess the conjunct effects of main- and after-shock sequences. However, there are no or little efforts that provide a simplified and validated fragility modeling approach that can specifically fit the assessment of bridge structures subject to both mainshock and aftershock sequences. This paper aims to propose a new fragility model that generates fragility curves reflecting the contribution from both the mainshock and the aftershock events. This empirical model retains the basic form of a traditional fragility model while considering the effects of the mainshocks by introducing polynomials-based parameters (Section 3). To test the effectiveness, a bridge system is adopted to demonstrate the proposed fragility model, and real main- and after-shock sequences are selected (Section 4). Statistical analysis is conducted that confirms the accuracy and robustness of the model (Section 5). Remarks and conclusions are provided in Section 6.