## 2 Traditional Fragility Analysis and Adaption
### Simply Adapted Fragility Analysis for M-A Effects  
One direct adaption to Equation (1) for assessing the effects of M-A sequences can be obtained by treating a pair of main- and after-shock ground motions as one record to simulate the demands and define the damage state (\(C<D\)). With such treatment in constructing fragility curves, a uniform scale is used to scale up or down any input sequence when performing the seismic simulation. With this, considering that nearly in all events, the PGA of the total M-A sequence is the mainshock PGA, \(PGA_{m}\) is used as the independent variable of the fragility model as shown in Equation (2):  
\[P(C<D)|PGA_{m}=\Phi\bigg{(}\frac{\ln x-\ln\mu_{m}}{\sigma_{m}}\bigg{)} \tag{2}\]where \(\mu_{m}\) and \(\sigma_{m}\) are model parameters by fitting the data as a result of uniformly scaled main- and after-shock sequences, and the function variable \(x\) is for the mainshock intensity \(PGA_{m}\). It is obvious that by adopting Equation (2) with the mainshock's PGA, the effects of aftershocks are implicitly hidden or mixed into the mainshock effects in the resulting fragility curve, and an end-user cannot discern the effects of aftershocks on the probable damage. In this paper, this fragility approach, as expressed in Equation (2), is termed mainshock-aftershock uniform-scale (MA-US) fragility modeling. [PERSON] et al. [7] representatively proposed a methodology for quantifying the M-A effects by fragility analysis (mainshock-aftershock specific-ratio fragility model). In this work, the formulation starts with a bivariate formulation with \(IM_{ms}\) and \(IM_{as}\) denoting the intensity measures (IMs) of the mainshock and the aftershock, respectively. This leaves potentially a research gap that expects a relationship between the IMs of the mainshocks and the aftershocks. However, very limited knowledge exists that can bridge such a gap. The widely used relation that weakly connects a mainshock and its aftershock was proposed in [28]. This relation describes the rate of occurrence of an aftershock with a magnitude greater than or equal to \(M_{a}\) at a time immediately after a mainshock of magnitude \(M_{m}\). Essentially, this empirical relation can be used to calculate how many aftershocks greater than a designated magnitude given a mainshock in a prescribed duration. On the other hand, there is no acceptable relation that describes the mainshock intensity (\(IM_{ms}\)) and the intensity (\(IM_{as}\)) of an arbitrary aftershock of the mainshock. To explicitly evaluate the seismic vulnerability of structures due to the main-/after-shock sequences, [PERSON] et al. [7] introduced a relative intensity measure, termed \(\
abla IM\) as a ratio between the mainshock and aftershock intensity, in the modeling. This parameter \(\
abla IM\) is treated as a user-defined parameter, which specifies a constant ratio between the mainshocks and the aftershocks for one scenario of vulnerability assessment by selecting the main-/after-shock motions. With a given \(\
abla IM\), the M-A sequences fragility model is:  
\[F_{sqq}(x,\
abla IM)=P\Big{[}EDP_{seq}\geq EDP_{LSi}\big{|}IM_{ms}=x,\
abla IM \Big{]} \tag{3}\]  
It is noted that in this fragility model, the mainshock's PGA is treated as the primary variable as in the traditional fragility model in Equation (1). The difference is that one may flexibly select a different \(\
abla IM\). In reference [7], by setting \(\
abla IM\) as 0.1, 0.5, and a few other values smaller than 1.0, the intensity of aftershocks is determined by \(\
abla IM\times IM_{ms}\) before scaling and being applied to the simulation, further providing data for the fragility formulation. With different \(\
abla IM\), the fragility curves as a function of the mainshock PGA can be created for different specific levels of \(\
abla IM\). In this study, the fragility model in reference [7] is called the mainshock-aftershock ratio-specific (MA-RS) fragility model. Another adaption is to consider the variability of aftershocks after selecting a certain level of mainshock intensity.