## 5 Application of Train Axle Loads Identification Method
### Train Axle Load Analysis for Urban Rail Transit  
The axle loads of a total of 68 trains traveling on the simply supported composite beam bridge of Shanghai Metro Line 3 were identified and analyzed. Histograms depicting the axle loads of locomotives (Tc) and middle train carriages (Mp) for forward trains are shown in Figure 11. There is a certain degree of dispersion in the axle loads of locomotives and middle train carriages for forward trains. The axle loads of locomotives for forward trains follow a logarithmic normal distribution characterized by an \(R^{2}\) value of 0.89. Specifically, the axle load range for locomotives of forward trains is 8.0 to 12.5 t (Figure 11a), with a statistical mean of 9.64 t and a variance of 0.8. Similarly, the axle loads of middle train carriages for forward trains exhibit a normal distribution, with an \(R^{2}\) value of 0.95. The axle load range for middle train carriages of forward trains spans 9.0 to 14.5 t (Figure 11b), with a statistical mean of 10.77 t and a variance of 0.8. Notably, the mean axle load of middle train carriages is higher than that of locomotives for forward trains, which is consistent with passenger traveling habits, as well as the functional settings of different train carriages. The equal variance values of axle loads between locomotives and middle train carriages for forward trains suggest comparable variability in passenger loads across these compartments during operations. Overall, considering the train design configuration (Figure 2) and relevant literature [32], the mean values and distributions of axle loads for locomotives (Tc) and middle train carriages (Mp) for forward trains on Shanghai Metro Line 3 are reasonable.  
Figure 10: Histogram of train speed distribution on Shanghai Metro Line 3.  
The histograms displaying the axle loads of locomotives (Tc) and middle train carriages (Mp) for reverse trains traveling on the simply supported composite beam bridge of Shanghai Metro Line 3 are presented in Figure 12. There is a noticeable degree of dispersion in the axle loads of both locomotives and middle train carriages for reverse trains. The axle loads of locomotives for reverse trains follow a normal distribution, characterized by an \(R^{2}\) value of 0.89. Specifically, the axle load range for locomotives of reverse trains spans from 8.0 to 12.5 t (Figure 12a), with a statistical mean of 10.28 t and a variance of 1.36. Similarly, the axle loads of middle train carriages for reverse trains also exhibit a normal distribution, with an \(R^{2}\) value of 0.90. The axle load range for middle train carriages of reverse trains falls between 9.0 and 14.5 t (Figure 12b), with a statistical mean of 10.77 t and a variance of 0.8. Interestingly, unlike forward trains, the mean values of axle loads for locomotives and middle train carriages in reverse trains are nearly consistent. Furthermore, the variance of axle loads for locomotives is larger than that of middle train carriages for reverse trains, indicating significant variability in passenger loads in locomotives during operations. In light of justifications from the train design configuration (Figure 2) and relevant literature [32], the mean values and distributions of axle loads for locomotives (Tc) and middle train carriages (Mp) in reverse trains on Shanghai Metro Line 3 are also reasonable.  
Figure 11: Histogram of axle load distribution for forward trains on Shanghai Metro Line 3: (**a**) locomotives (Tc); (**b**) middle train carriages (Mp).  
Figure 12: Histogram of axle load distribution for reverse trains on Shanghai Metro Line 3: (**a**) locomotives (Tc); (**b**) middle train carriages (Mp).