## 1 Introduction
### Train Speed Estimation  
In this study, the signal cross-correlation function [28] was employed to estimate the speed of trains traveling across the simply supported composite beam bridge on Shanghai Metro Line 3. When a train passes through the quarter and mid-spans of the bridge, longitudinal strain time histories from corresponding measurement points on these sections are monitored (Figure 3). From a signal analysis perspective, there exists a correlation between the strain time history curves of corresponding measurement points at the quarter and mid-span sections under the same train. The correlation coefficient \(R\) quantifies the degree of correlation between strain values of the corresponding measurement points at the quarter-span \(\varepsilon_{i-j}^{Q}(t)\) and mid-span \(\varepsilon_{i-j}^{M}(t)\) sections at any given time \(t\) and \(t+\tau\). The cross-correlation function \(R(\tau)\) between these strain time series at a time difference \(\tau\) can be computed using the equation  
\[R(\tau)=\int_{-\infty}^{\infty}\varepsilon_{i-j}^{Q}(t)\varepsilon_{i-j}^{M}(t+\tau) \tag{2}\]  
According to the cross-correlation function, when a train sequentially passes through the quarter and mid-spans of the simply supported composite beam bridge, the cross-correlation function \(R(\tau)\) of the strain time history \(\varepsilon_{i-j}^{Q}(t)\) and \(\varepsilon_{i-j}^{M}(t)\) at the corresponding measurement points reaches its maximum at the time difference \(\tau=t_{d}\). This indicates that the strain time history \(\varepsilon_{i-j}^{M}(t)\) from the mid-span section, when shifted forward by time \(t_{d}\), most closely matches the strain history \(\varepsilon_{i-j}^{Q}(t)\) from the quarter-span section. The cross-correlation coefficient of strain time histories at corresponding measurement points is calculated for each train passing through the quarter and mid-spans of the bridge to determine the time difference \(t_{d}\) at which the cross-correlation coefficient reaches its maximum value. The average speed \(v_{t}\) of each train passing through the bridge is estimated using the ratio of the distance \(L/4\) between the quarter and mid-span sections to the time difference \(t_{d}\). When a train passes through the simply supported composite beam bridge at speed \(v_{t}\), the on-site measured strains at the quarter and mid-span sections both include dynamic amplification effects. For urban rail transit bridges, the dynamic amplification effect is generally calculated by multiplying the vertical static load effect of the train by a dynamic amplification coefficient \(\mu\). The dynamic amplification factor \(\mu\) is determined by the train speed \(v_{t}\) and the bridge span \(L\). For urban rail transit trains traveling at speeds generally below 80 km/h, the dynamic amplification factor \(\mu\) is 0.8 times that specified for railway bridges. According to the Design Specification for Railway Bridges and Culverts [29], the dynamic amplification factor \(\mu\) of the simply supported steel-concrete composite beam bridge for urban rail transit with a span of \(L\) can be obtained by  
\[1+\mu=1+0.8\times\frac{22}{40+L} \tag{3}\]