## 2 Seismic Fragility Analysis  
Seismic fragility analysis is a powerful tool for evaluating the failure probability of bridges or buildings subjected to earthquake loads, which is conditional on a particular seismic intensity [50; 51; 52; 53; 54; 55; 56; 57]. The fragility function is expressed as Equation (1).  
\[P(C<D)|PGA=\Phi\bigg{(}\frac{\ln x-\ln\sigma}{\xi}\bigg{)} \tag{1}\]  
where \(P(C<D)|PGA\) is the probability of exceeding a damage state, \(C\) denotes the structure's capacity, \(D\) denotes the seismic demand, \(\Phi\) is the cumulative normal distribution function, \(x\) is the intensity of ground motions, \(\sigma\), and \(\xi\) is the mean and standard deviation of seismic intensity resulting in a damaged state, respectively.  
Over the past ten years, researchers have mainly focused on the seismic fragility of structures under multiple hazards [58; 59; 60; 61; 11]. In this work, we focus on the fragility of strengthened corroded bridges under mainshock and aftershock sequences. Equation (2) is the fragility function of a corroded structure,  
\[P(C<D)|PGA=\Phi\bigg{(}\frac{\ln x-\ln\sigma_{c}}{\xi_{c}}\bigg{)} \tag{2}\]  
where \(\sigma_{c}\) and \(\xi_{c}\) are functions of the corrosion ratio \(\eta_{m}\).  
The fragility function for a corroded structure reinforced by steel plates is a function of the ground-motion intensity, corrosion level, and thickness (_th_) of the bonded steel plates as in Equation (3).  
\[P(C<D)|PGA=\Phi\bigg{(}\frac{\ln x-\ln\sigma_{c,th}}{\xi_{c,th}}\bigg{)} \tag{3}\]  
where \(\sigma_{c,th}\) and \(\xi_{c,th}\) are functions of the corrosion ratio \(\eta_{m}\) and the thickness of the steel plates, respectively. For a specific corrosion level, \(\sigma_{c,th}\) and \(\xi_{c,th}\) can be calculated using the maximum likelihood method as in [62]. In [4], a time-variant quadratic model is used to estimate the mean and standard deviation at different years, and the concept is borrowed in this work for a thickness-variant quadratic model. By calculating \(\sigma_{c,th}\) and \(\xi_{c,th}\) at different _th_ values, fitting equations for \(\sigma_{c,th}\) and \(\xi_{c,th}\) can be obtained using the least-squares method. The fragility curves for any expected thickness of the strengthened steel plates at a specific corrosion level can be calculated as shown in Equation (6):  
\[\sigma_{c,th} =p_{1}th^{2}+p_{2}th+p_{3} \tag{4}\] \[\xi_{c,th} =p_{4}th^{2}+p_{5}th+p_{6}\] (5) \[P(C<D)|PGA =\Phi\Bigg{(}\frac{\ln x-\ln\big{(}p_{1}th^{2}+p_{2}th+p_{3}\big{)} }{p_{4}th^{2}+p_{5}th+p_{6}}\Bigg{)} \tag{6}\]  
where \(p_{1}\), \(p_{2}\), and \(p_{3}\) are the coefficients for \(\sigma_{c,th}\) at a specific corrosion ratio; \(p_{4}\), \(p_{5}\), and \(p_{6}\) are the coefficients for \(\xi_{c,th}\) at a specific corrosion ratio. For a specific corrosion level, the \(\sigma_{c,th}\) and \(\xi_{c,th}\) of the structure reinforced by steel plates of different thicknesses were calculated. A least-squares fitting method was used to calculate \(\sigma_{c,th}\) and \(\xi_{c,th}\) at any expected thickness of the steel plates. Thereafter, using Equation (6), the fragility curves and surface at any expected thickness of the steel plate can be calculated efficiently without performing nonlinear dynamics a second time.