## 3 Numerical Simulation
### Influence of Calibration Spacing  
When we calibrated the stress response area in practical application, we needed to choose appropriate calibration spacing \(d\). The smaller the parameter \(d\) was, the higher the load identification accuracy that could be obtained; however, this may greatly increase the workload on site, and vice versa. In addition, it was difficult to accurately control the vehicle that travelled at a given spacing \(d\) on the deck. Here, we studied the influence of different calibration spacing \(d\) on identifying the GVW.  
(1) Constant calibration spacing  
In the numerical simulation above, the calibration spacing was set as \(d=4\) cm, and then the strain response area vectors were refined to 1 cm using triple spline interpolation. Then, we changed the calibration spacing \(d\) from 4 cm to 48 cm and established the corresponding calibrated strain response area matrix \(\mathbf{B}^{C}\). In the following, the same two-axle test vehicle was simulated to pass over the bridge from the transverse position x = \(-\)36 cm, and the structural strain response was obtained using FEA with different noise. Based on the calibrated strain response area matrix with different calibration spacing \(d\), we could locate the TP and identify the GVW of the test vehicle using the proposed method. The identified results are shown in Figures 9 and 10 below.  
It can be seen that when the calibration spacing \(d\) varied from 4 cm to 48 cm, the identification errors of the GVW and the TP showed an overall increasing trend. When  
Figure 10: Identification error of the GVW under different calibration spacing \(d\).  
Figure 9: Identification error of the TP under different calibration spacing \(d\).  
the calibration spacing \(d\) was less than 32 cm, the maximum identification error of the TP did not exceed 2 cm under all kinds of noise, and that of the GVW was below 5%. If the calibration spacing \(d\) exceeded 32 cm, both errors sharply increased. Thus, these results indicate that the calibration spacing should not exceed 32 cm.
2. Variable calibration spacing  
The influence of different calibration spacing \(d\) on the identification of the GVW was studied, and the maximum calibration spacing was determined as 32 cm in the above section. In practice, it is difficult to let the calibration vehicle accurately pass over the bridge from the specified transverse position, so we further studied the influence of random calibration spacing \(d\) on the identification of the GVW. The detailed method is listed below:  
1 The structural strain response was extracted under the two-axle calibration vehicle passing over the bridge model at the transverse position from x = \(-\)68 cm to x  
It can be seen that when the maximum calibration spacing was no more than 28 cm, most of the identified errors in the GVW were below 7.5% under 20% noise. With an increase in maximum calibration spacing, the identified errors became larger and more scattered. This was mainly because larger calibration spacing decreased the accuracy of the strain response area vectors when interpolated to 1 cm intervals. Thus, we could let the calibration vehicle randomly pass over the bridge, and select six groups of strain response where the maximum calibration spacing \(d\) was no more than 28 cm to build the strain response area matrix, which is convenient for practical engineering applications. Certainly, if the strain response of more positions with less calibration spacing was selected, better identified results could be obtained.