## 3 Numerical Simulation
### Gross Weight Identification  
(1) Calibration of the strain response area  
Before identifying the total load weight, it was necessary to calibrate the strain response area by driving a vehicle with a known weight across the bridge from different transverse positions. Here, a two-axle calibration vehicle was simulated to pass over the bridge model from the left edge position of the lane at x = \(-68\) cm to the right edge position of the lane at x = \(68\) cm, every 4 cm (here, we defined it as calibration spacing \(d\)). The center position of the lane was defined as x = \(0\) cm. The calibration vehicle's front and rear axle weight was 14 kN and the wheelbase was 260 cm. By finite element calculating, the strain response \(\varepsilon_{ij}\) at the position of the measurement points shown in Figure 1 could be obtained, where \(i\) denotes the transverse position and \(j\) is the number of measurement points. Thus, all of the strain responses \(S_{ij}\) could be calculated, and the calibrated strain response area matrix \(\mathbf{B}^{C}=[\mathbf{B}^{C}_{1},\mathbf{B}^{C}_{2},\cdots,\mathbf{B}^{C}_{n }]\) could be established. To obtain a finer location of the transverse position, the strain response area vectors \(\mathbf{B}^{C}_{i}\) were refined at transverse positions every 1 cm using triple spline interpolation. (2) Locating the transverse position  
A two-axle test vehicle was simulated to pass over the model from the transverse position x = \(-42\) cm. The front and rear axle weights were 7 kN and 14 kN, respectively, and the wheelbase was 300 cm. The structural strain response could be calculated by FEA. Considering the influence of the noise, 10% and 20% (defined as noise variance/signal variance) Gaussian white noise signals were added to the response data. The strain response with 20% noise is shown in Figure 6.  
Figure 5: Finite element model. Figure 6: The strain response of the measurement point with 20% noise. According to the proposed method, the cosine similarity \(CI\) index could be calculated based on the strain response and the calibrated strain response area matrix. The relation between the evaluated \(CI\) index and the transverse position is shown in Figure 7.  
As seen from the figure, the cosine similarity \(CI\) index reached the maximum at the transverse position x = \(-\)42 cm under different noise cases, which is consistent with the practical position. Thus, the proposed index could effectively locate the transverse position with good anti-noise performance. Theoretically speaking, the strain response is nonlinear due to the dynamic effect. The proposed \(CI\) index is used to calculate the degree of linear correlation between the input vectors. However, here, the strain response was not directly used as an input. Instead, the strain response area was used as an input. In the example of a two-axle vehicle crossing the model at speed of 10 m/s, the front and rear axle weights were 10 kN and 15 kN, respectively, and the wheelbase was 320 cm. The static and dynamic responses of N2 are shown in Figure 8.  
As shown, the responses were different, albeit only slightly; the static and dynamic strain response areas were 18.2732 and 18.2752, respectively. From the above identification results of the transverse position, it also can be seen that the error was acceptable using cosine similarity. Figure 8: The static and dynamic response of N2. Figure 7: The variation curve of the \(CI\) index relative to the transverse position. Evaluating the gross vehicle weight  
To validate the gross weight identification effect, different types of vehicles were driven across the bridge model from different transverse positions, and the corresponding strain responses were obtained using finite element analysis. A total of five types of vehicles were simulated, and the wheelbase and axle weight parameters are listed in Tables 1 and 2, respectively.