## 1 Introduction
### Train Axle Loads Identification  
The identification of train axle loads for the AC03 type train operating on Shanghai Metro Line 3 involves solving an inverse structural response problem. In this method, the known structural responses (strain measurements) are used to determine the axle loads that cause these responses. This approach is rooted in strain influence line theory, where the effect of a specific axle load is proportional to the product of the influence line ordinate and the magnitude of the axle load at the corresponding position. The AC03 type train configuration (Figure 2) and its operational speed on Shanghai Metro Line 3 have been established. The unknowns in this context are solely the axle loads of the AC03 train, denoted as \(\textbf{P}=[p_{1}\,p_{2}\,\dots\,p_{\rm Na}]^{\mathsf{T}}\), where \(p_{1}\) represents the load of the train axle \(i\), and \(N_{\rm a}\) is the total number of train axles. Therefore, theoretically, the axle loads \(p_{1}\) (\(i=1\) to \(N_{\rm a}\)) can be determined by \(N_{a}\) time's measurement of the structural responses (strain measurement) at distinct \(N_{a}\) longitudinal positions. As depicted in Figure 6a, the given AC03 type train on Shanghai Metro Line 3 travels across the simply supported composite beam bridge at a velocity of \(v_{t}\). Time is designated as 0 when the first train axle of the AC03 type train enters the bridge. At time \(t\), the train will arrive at the longitudinal position \(x=v_{t}t\). At any given longitudinal position \(x\), the coefficient matrix \(\mathbf{I}(x)\) representing the strain influence line for all the strain measurement points is  
\[\mathbf{I}(x)= \begin{bmatrix}IL_{1}(x)&IL_{1}(x-l_{1})&\cdots&IL_{1}(x-\sum_{ k=1}^{N_{a}}l_{k})\\ IL_{2}(x)&IL_{2}(x-l_{1})&\cdots&IL_{2}(x-\sum_{k=1}^{N_{a}}l_{k})\\ \vdots&\vdots&\ddots&\vdots\\ IL_{N_{m}}(x)&IL_{N_{m}}(x-l_{1})&\cdots&IL_{N_{m}}(x-\sum_{k=1}^{N_{a}}l_{k}) \end{bmatrix} _{N_{m}\times N_{a}} \tag{4}\]  
where \(IL_{i}(x)\) represents the strain influence line function for measurement point \(i\), as detailed in Section 4.1. The coefficient matrix \(\mathbf{I}(x)\) of the strain influence line is an \(N_{m}\times N_{a}\) matrix, where each element \(IL_{i}(x)\) is determined by the longitudinal position of each axle of the AC03 type train along with the strain influence line for measurement point \(i\). The \(N_{m}\) denotes the total number of strain measurement points on the simply supported composite beam bridge (Figure 3). The \(l_{k}\) signifies the distance between the first and the \(k\)-th axle of the AC03-type train. Based on the calculated strain influence lines (Figure 2), the theoretical strains \(\varepsilon^{c}(t)\) at all strain measurement points at time \(t\) can be derived relative to the longitudinal position \(x\) of the first train axle using  
\[\varepsilon^{c}(t)=[\varepsilon_{1}^{t}\;\varepsilon_{2}^{t}\;\cdots\; \varepsilon_{N_{m}}^{t}]^{T}=\mathbf{I}(t)\times\mathbf{P} \tag{5}\]  
The theoretical strains \(\varepsilon^{c}(t)\) form a column vector that includes the calculated strain values at all measurement points at time \(t\). Therefore, based on Equation (5), it becomes feasible to establish a system comprising \(N_{a}\) equations with \(N_{a}\) unknowns by selecting \(N_{a}\) positions for the first train axle. This approach facilitates the determination of the axle loads \(p_{i}\) for the given AC03 type train. In practical scenarios involving on-site strain measurements on actual bridges, the number of data samples collected is typically much greater than \(N_{a}\), owing to continuous strain data acquisition by the measurement system. This data abundance ensures a robust basis for accurate train axle load identification. Figure 6: Structural effect of train axle loads on a simply supported composite beam bridge: (**a**) actual AC03 type train running on the bridge; (**b**) strain influence line theory; (**c**) theoretical strain time history curve.