## 5 Numerical Experimental Evaluation  
### Traditional Mainshock-Only and M-A Fragility Assessment  
The probabilities of the collapse of the pile foundation-supported bridge under mainshocks-only and mainshock-aftershock sequences are calculated, as shown in Figure 4 by using Equations (1)-(3). In this work, we mainly focused on the development of the mainshock-integrated aftershock fragility model, so only the collapse state is presented for simplification, for other damage states such as slight damage, moderate damage, and major damage can also be calculated. The probability of collapse induced by mainshock is calculated using Equation (1). As the results obtained by other researchers, the probability of collapse increases with the mainshock intensity. It is stated that in most cases, towards evaluating the vulnerability of a structure under a select intensity level or range of earthquakes, this traditional fragility analysis suffices the needs. Indeed, this practice does not provide empirical knowledge regarding the effects of aftershocks on the target structure.  
The collapse probability of the bridge under M-A sequences calculated using Equation (2) is also provided in Figure 4, in which the mainshock aftershock intensity is scaled with the uniform method. By comparing the fragility curve for mainshock only and mainshock-aftershock uniform scaling method, it can be known that the effect of aftershock is very small, which may attribute to that the intensity of scaled aftershock is still weak, but the advantage of this method is that the intensity relationship between mainshock and aftershock is not changed. The intensity of aftershock is scaled using the same scaling coefficient as mainshock. It is worth mentioning that by using the mainshock-aftershock uniform-scale, the horizontal coordinate PGA denotes mainshock intensity as in reference [7].  
The probability of collapse of the bridge system under mainshock aftershock sequences is also calculated using the method proposed by [PERSON] et al. [7]. Two specific ratios between aftershock and mainshock intensity are considered as presented in Figure 4, and the ratio is set as 0.5 and 0.9, respectively. For other ratios, it can also be easily calculated. In reference [7], the Park-Ang model is adopted to judge the failure of a structure under M-A sequences. In this work, the ductility demands are used, which does not affect the aim of this work for the development of the new fragility function model. From Figure 4, it can be known that by using the method proposed by [PERSON] et al. [7], the effect of aftershock can be known by comparing to the fragility curve of the bridge under mainshock only. Therefore, for this method, the probability of exceeding a damage state under mainshock only should be calculated first, or it is difficult to quantify the effect of aftershock. In addition, without calculating  
Figure 4: Collapse probability of the bridge under M-A sequences using different fragility models.  
the failure probability of a structure under only mainshocks, the effect of mainshocks in the probability of failure of the structure under mainshock-aftershock sequences is also unknown.