## 3 Mainshock-Integrated Aftershock Fragility  
### Rationale and Formulation  
The fragility expression in Equation (3) defines the probability of exceedance conditional on two hazard variables, which, if defined generically, can be written as P(\(C<D|H_{1}\), \(H_{2}\)), where \(H_{1}\) and \(H_{2}\) are the representative hazard-intensity variables. If this expression is treated as a fragility model in terms of two hazard variables, it leads to a fragility surface model that lacks straightforwardness towards readily interpreting the vulnerability of a structure as a fragility curve does so. There are a number of efforts that extended the traditional single-hazard fragility-modeling to a multi-hazard case. In reference [29], the authors considered the effects of seismic hazards and aging. To integrate the two hazards, they proposed a time-dependent fragility model for bridge structures under aging and seismic loads, in which the time variable \(t\) for aging is incorporated into the model parameters as: \(\sigma_{t}=p_{1\_m}t^{2}+p_{2\_m}t+p_{3\_m}\), \(\xi_{t}=p_{1\_\xi}t^{2}+p_{2\_\xi}t+p_{3\_\xi}\).  
The proposed fragility model in this paper is inspired by the mathematical treatment of time-dependency in reference [29]. To integrate the mainshock effects and emphasize the aftershocks, and more importantly, to retain a one-dimensional fragility-curve model, the authors treat the intensity of aftershocks as the independent function variable, then correlate the model parameters with the mainshock variations. Two differences are highlighted. First, it is noted that the physical relations between main- and aftershocks are much different from the time-dependent aging and earthquakes. Second, to account for the mainshock effects when treating the weak aftershock intensity, an intercept term is introduced into the formulation. Equation (5) defines the proposed probabilistic fragility model:  
\[P[C<D]|(PGAa,PGAm)=\frac{\Phi\Big{(}\frac{\ln x-\ln\mu_{m}}{\sigma_{m}}\Big{)}+ \gamma_{m}}{1+\gamma_{m}} \tag{5}\]  
where \(P[C<D]\big{|}(PGAa,PGAm)\) is the probability of collapse under an aftershock intensity (\(PGAa\)) and a mainshock intensity (\(PGAm\)); \(C\) is the capacity of a structure; \(D\) is seismic demand under the given seismic intensity; \(\Phi\) is the cumulative normal distribution function; \(\mu_{m}\) and \(\sigma_{m}\) are the mean and standard deviation of seismic intensity of the mainshock; \(\gamma_{m}\) is a non-negative parameter that is the function of mainshock intensity. This parametric structure in Equation (5) ensures that first, the function is bounded by zero and unit; and second, when \(\Phi(\cdot)\) approaches to zero (namely, when the aftershock intensity \(x\) is small), the fragility approaches to an intercept value of \(\gamma_{m}/(1+\gamma_{m})\). This implies that when the aftershock intensity is weak, this intercept accounts for the non-zero probability of failure that the mainshock triggers.  
Given the fragility formulation in Equation (5), the model parameters \(\mu_{m}\), \(\sigma_{m}\), and \(\gamma_{m}\), depend on the mainshock intensities. When a strong mainshock intensity is selected, it leads to a certain degree of damage with uncertainties due to their intra-record uncertainties. In the meantime, with the aftershock variations and uncertainties, the model parameters vary dependent on the mainshock intensity. To consider the potential nonlinearity of this dependency, quadratic polynomials, which feature the least degree of nonlinearity in terms of the mainshock intensity, are empirically prescribed to model the relation between the model parameters and the mainshock intensity:  
\[\mu_{m} =p_{1}PGAm^{2}+p_{2}PGAm+p_{3} \tag{6}\] \[\sigma_{m} =p_{4}PGAm^{2}+p_{5}PGAm+p_{6}\] (7) \[\gamma_{m} =p_{7}PGAm^{2}+p_{8}PGAm+p_{9} \tag{8}\]where \(p_{1}\), \(p_{2}\),..., and \(p_{9}\) are the polynomial constants estimated by the least-squares method for the computation of \(\mu_{m}\), \(\sigma_{m}\), and \(\gamma_{m}\) at different mainshock intensities.