## 5 Numerical Experimental Evaluation
### Mainshock-Integrated Fragility Function Model  
In the nonlinear dynamic analysis, the intensity of the mainshock is scaled from 0.4 g to 1.6 g with incremental 0.2 g. To account for the aftershock effects, the intensity of the aftershock is scaled from a smaller value to the mainshock intensity. In this work, we assume that the maximum aftershock intensity is up to the mainshock intensity. For example, the aftershock intensity is scaled from 0.1 g to 0.4 g when the mainshock intensity is 0.4 g. Furthermore, when the mainshock intensity is 1.6 g, aftershock intensity is scaled from 0.2 g to 1.6 g. In this section, we focus on the conditional probability of collapse of bridge systems under M-A sequences, and hence, the probability of collapse is both the function of mainshock and aftershock intensity. Through numerical simulation, the fragility curves for the investigated bridge system are obtained by using mainshock-dependent aftershock fragility and mainshock integrated aftershock fragility function model, respectively. In this section, the mainshock-dependent aftershock fragility model (Equation (4)) was used first for fitting the fragility data, as shown in Figure 5. Further, the probability of failure using the proposed new model is shown in Figure 6.  
Figure 5: Seismic fragility curves of bridge system under M-A sequences traditional fragility function model for different mainshock intensities. Figure 5 shows the fragility curves for mainshock intensity at 0.6 g, 0.8 g, 1.0 g, and 1.2 g, respectively. From Figure 5, it can be known that the numerical simulation data has an initial value; for example, the initial value is about 1.7% when the mainshock intensity is 0.6 g and aftershock intensity is smaller than 0.5 g. However, the fitting curve shows that the initial value is 0 and then increases sharply with aftershock intensity. From Figure 5a to Figure 5d, it is concluded that the cumulative normal distribution function cannot fit the numerical simulation data well. Figure 5 also implies that the initial value is 0.6 g and 0.8 g.  
Figure 6: Seismic fragility curves for bridge systems under M-A sequences using the proposed new fragility function model with different mainshock intensity. with increased mainshock intensity, the probability of collapse of the bridge system increases as well. The numerical simulation data also indicates that aftershock increases the probability of the structure if its intensity is higher than a threshold value. The probability of collapse is 0 when the mainshock intensity is 0.4 g and aftershock intensity from 0.1 g to 0.4 g, so the result is not presented in this section. Therefore, for small mainshock and aftershock intensity, they have fewer effects on the probability of collapse, but it may affect other damage states. When the mainshock intensity is 1.4 g and 1.6 g, the cumulative distribution function cannot fit the numerical simulation data for numerical errors. The error analysis between numerical simulation data and the traditional fragility function model is conducted; the results show that the error is very large between the numerical simulation data and the fragility function model. Based on Equation (5) and numerical simulation data, the fragility curves are generated and presented in Figure 6. From Figure 6, it is concluded that the proposed new fragility function model fits the numerical simulation data well for all the example cases regardless of the magnitude of the intensity of the mainshock being small or large. Figure 6a-f shows that there is an initial value that is larger than 0, and the value increases with the mainshock intensity. This initial value is a function of the mainshock intensity. From Figure 6a-f, it is concluded that the developed mainshock-dependent fragility function model almost exactly matches the numerical simulation data. It is worth to mention that for a mainshock intensity, to reduce the calculation cost, the aftershock just scaled to several intensities.