## 1 Introduction
### Train Axle Loads Identification
When a train travels a simply supported beam bridge, the strain time history curves at each measurement point typically exhibit a multi-peak shape, as shown in Figure 6c. This characteristic arises because each axle of the train induces strain responses as it moves along the bridge. The number and magnitude of these local peak strains in the strain time history are directly influenced by the train axle load vector \(\mathbf{P}\) and the strain influence line \(IL_{i}(x)\) of the measurement point \(i\). In practical engineering applications, there often exists a discrepancy between the theoretical strains and the actual strains measured on-site at specific measurement points on simply supported composite beams. Thus, taking the local peak strain in the strain time history curve into consideration, the axle loads identification of the given AC03 type train is carried out by minimizing the quadratic difference between the theoretical calculated and on-site measured local peak strains for the measurement point \(i\), which is defined by  
\[e_{i}=\sum_{j=1}^{N_{j}}\left(e_{j}^{pc}-\varepsilon_{j}^{pm}\right)^{2} \tag{6}\]  
where the \(\varepsilon_{j}^{pc}\) is the theoretically calculated \(j\)-th local peak strain, while the \(\varepsilon_{j}^{pm}\) is the on-site measured \(j\)-th local peak strain for the measurement point \(i\). The \(N_{\mathrm{j}}\) is the total number of the local peak strains in the strain time history curve of the measurement point \(i\). The minimization of Equation (6) utilizes a genetic algorithm known for its effectiveness in searching the global optimum across various optimization problems. This includes scenarios where the objective function is discontinuous, non-differentiable, stochastic, or highly nonlinear [30; 31]. The genetic algorithm initiates with multiple feasible solutions \(\mathbf{P}_{i}\) (\(i=1\) to \(N_{s}\), \(i\)\(\in\)\(N^{+}\)), each represented as an individual in the form of a binary string. These multiple feasible solutions \(\mathbf{P}_{i}\) constitute a population, with a size \(N_{s}\) of 60 tailored to address the optimization problem defined in Equation (6). The algorithm iteratively applies selection, crossover, and mutation operators to generate successive generations, guiding the population toward an optimal solution. A crossover ratio of 1.0 is maintained to preserve sufficient variability in the next population. Consequently, the global optimum solution \(\mathbf{P}_{b}=[p_{1}^{b}\;p_{2}^{b}\dots\;p_{N_{a}}^{b}]^{\mathrm{T}}\) for the axle loads of the AC03 type train is determined. The crossover ratio is set to 1.0 to keep adequate variability for the next population. The global optimum solution \(\mathbf{P}_{b}=[p_{1}^{b}\;p_{2}^{b}\dots\;p_{N_{a}}^{b}]^{\mathrm{T}}\) for the axle loads of the given AC03 type train is thus finally determined. Based on the AC03 type train passing over the simply supported composite beam bridge at speed \(v_{t}\), the strain responses at each measurement point incorporate dynamic impact effects. These effects are adjusted by the dynamic amplification factor \(\mu\) specific to each train (Equation (3)), as derived in Section 4.2. Therefore, the actual axle loads of the AC03-type train are ultimately represented by  
\[\mathbf{P}=\frac{1}{1+\mu}\mathbf{P}_{b}=\frac{1}{1+\mu}[p_{1}^{b}\;p_{2}^{b} \;\dots\;p_{N_{a}}^{b}]^{\mathrm{T}} \tag{7}\]  
According to the measured strain time history of each measurement point, the train axle load vector \(\mathbf{P}_{b}=[p_{1}^{b}\;p_{2}^{b}\dots\;p_{N_{a}}^{b}]^{\mathrm{T}}\) can be derived. The simply supported composite beam bridge features a total of \(N_{\mathrm{m}}\) strain measurement points located on the quarter and mid-span sections (Figure 3). Utilizing the redundant on-site measured strain time histories from these points allows for the computation of multiple estimations of the train axle load vector \(\mathbf{P}_{b}\) for the AC03 type train.