## 4 Implementation and Case Study
### Bridge Modeling  
A reinforced-concrete two-span bridge model is considered in this work. The total length of the pile-foundation supported bridge is 60.0 m, with the same length for each span. The height of the column is 9.5 m, and its diameter is 1.6 m. The length of the pile foundation is 20.0 m, and its diameter is 1.6 m. The soil around the pile foundations is sand of Class D. The geometric model of the bridge is presented in Figure 2a,b. The finite-element (FE) model of the soil-pile-b  
Figure 1: Flowchart for assessing the vulnerability of a structure under earthquake sequences.  
OpenSees (version 2.4.5, Pacific Earthquake Engineering (PEER) Center, Berkeley, CA, USA) [30]. In the FE model, the columns are simulated by using a nonlinear beam-column element. Elastic beam elements are assigned to the bridge deck. The concrete material model developed by [PERSON] is utilized to describe the stress-strain relationship of confined concrete [31]. In the OpenSees program, _steel02_ material is used for longitudinal steel, and _concrete02_ material is applied for the cover and confining layer of concrete [32]. Idealized roller-like abutments are applied to simulate the abutments.  
To consider the soil-structure interaction (SSI), the pile foundation is simulated as a Nonlinear Winkler Foundation that is available in OpenSees [32]. With this Winkler approach, two vertical friction-resistant springs along with the pile (\(t\)-\(z\)), one vertical-bearing spring at the bottom of the pile (\(q\)-\(z\)), and three lateral-resistance (\(p\)-\(y\)) springs are used to simulate the integrated nonlinear soil-pile interaction effects. The stiffness of those springs around the pile foundation is calculated according to the empirical equations reported in [33]. Figure 2b provides a schematic illustration of these springs.  
After the FE model for the bridge is created, incremental dynamic analysis (IDA) is conducted, and the maximum displacement at the top of the column is captured and denoted as the ductility  
Figure 2: Pile-supported bridge model and the finite-element distributed springs for modeling the piles.  
demand (\(\mu_{\Delta}\)) as a damage index to determine whether the bridge systems is collapse or not. For the collapse state, \(\mu_{\Delta}\geq 7\) is calculated in this work. In the IDA, the mainshock is scaled into different intensity levels. For each mainshock intensity, the aftershock intensity is increased from a smaller value to the intensity of mainshock for nonlinear dynamic analysis (by doing this, the intensity of aftershock is smaller than that of mainshock in the numerical simulation, which is more realistic), in which \(P\)-\(\Delta\) effect is considered. In this work, the uncertainty of materials such as concrete, steel, and soil are considered by using the Latin hypercube sampling method. It is worth to mention that in the IDA, the intensity of aftershock is scaled to different target levels. It is noted that some values may be as high as nearly equals to that of the mainshocks. However, in a real earthquake event, the recorded intensity of aftershock may be higher than that of the mainshock, even though this is not a common phenomenon. Since probabilistic hazard analysis for the mainshocks and aftershocks are not of concern in this study, and this paper aims to propose a fragility function model, the intensity relation between the mainshocks and aftershocks do not affect the results in this study.