## 3 Vehicle-Bridge Coupling Analysis for Corrugated Steel Web Box
### Calculate the Impact Factor  
The impact effect of vehicle load on the bridge is generally represented by multiplying the standard value of the vehicle load by the impact factor. In the Chinese standard (JTG D60-2015) [46], the impact factor is defined as:  
\[\mu=\frac{Y_{d\max}}{Y_{j\max}}-1 \tag{8}\]  
Here, \(Y_{d\max}\) represents the maximum peak of the sectional dynamic response when the vehicle load crosses the bridge, and \(Y_{j\max}\) represents the maximum static response  
Figure 10: Mid-span vertical displacement at different speeds. (**a**) 20 km/h; (**b**) 40 km/h; (**c**) 60 km/h; (**d**) 80 km/h; (**e**) 100 km/h; (**f**) 120 km/h. at the same section when the same vehicle load is applied statically. However, vehicle-bridge coupling vibration is a dynamic problem. If the bridge happens to be in an upward vibration phase when the vehicle passes the measurement point, \(Y_{\text{dmax}}\) will decrease. In this case, directly calculating the impact factor using the definition method may not accurately reflect the impact effect of the vehicle load on the bridge. In the Chinese standard (JTG/T [J21-01-2015]) [47], the impact factor is calculated based on the time history curve of the bridge's dynamic deflection or dynamic strain, using the following formula:  
\[\mu=\frac{f_{d\text{max}}}{f_{j\text{max}}}-1=\frac{f_{d\text{max}}}{\frac{f_{ d\text{max}}+f_{d\text{min}}}{2}}-1 \tag{9}\]  
where \(f_{d\text{max}}\) represents the maximum dynamic deflection amplitude, \(f_{j\text{max}}\) is the peak value of the waveform amplitude center trajectory, or it can be obtained through low-pass filtering, and \(f_{d\text{min}}\) is the dynamic deflection valley value corresponding to \(f_{d\text{max}}\). Drawing on the idea of separating dynamic and static responses in the methods above, the dynamic load effect of the vehicle on the bridge is obtained by processing the dynamic deflection time history curve through filtering. This approach is used to calculate the impact factor. Deriving Formula (9) yields the following:  
\[\mu=\frac{f_{d\text{max}}}{f_{j\text{max}}}-1=\frac{f_{d\text{max}}-f_{j\text {max}}}{f_{j\text{max}}}=\frac{f_{i\text{max}}}{f_{j\text{max}}} \tag{10}\]  
where \(f_{i\text{max}}\) is the maximum value of the bridge vibration amplitude obtained through high-pass filtering, and \(f_{j\text{max}}\) is the maximum value of the static deflection time history curve at the corresponding position from static analysis. The purpose of high-pass filtering is to remove low-frequency static deflection from the bridge vibration response. After this operation, the remaining high-frequency response is the dynamic deflection caused by the impact effect, which will be used to calculate the impact factor. The determination of the cutoff frequency is crucial before performing high-pass filtering. According to the abovementioned principle, the ideal filtering condition is achieved when the removed low-frequency response is exactly equal to the static response obtained from the bridge static analysis. The specific method can be divided into two steps. First, a spectral analysis is performed on the dynamic deflection time history curve to estimate the approximate frequency range. Then, the high-pass filter cutoff frequency is determined based on the principle that the maximum static deflection value obtained by low-pass filtering should be equal to the maximum static deflection value calculated by the static analysis module. Based on trial calculations, a cutoff frequency of 2 Hz is selected to process the mid-span node deflection time history curve with high-pass filtering in this paper. As shown in Figure 11, the high-pass filtering effect of the mid-span deflection under class A pavement is presented. Due to space limitations, only a partial set of response results is presented. After high-pass filtering, the time history curve effectively reflects the vibration situation in the bridge's dynamic response.