## 3 Numerical Simulation
### Calibration of Influence Line  
Taking the center position of the bridge deck as x = 0, a two-axle vehicle with a gross weight of 25 kN, a front and rear axle weight of 10 kN and 15 kN, respectively, and a wheelbase of 320 cm is driven across the bridge every 5 cm from x = \(-0.8\) m to x = 0.8 m along the transverse position. The strain response at the position of the axle weight sensors shown in Figure 5 is calculated by FEA. Thus, the strain influence lines of each sensor under vehicle load at different transverse positions can be identified. The identified influence lines of Z1 and Z2 is shown in Figure 7 below.  
As seen from the above figure, the influence line will be affected by the transverse position of the vehicle's load. For precisely identifying the vehicle's load, it is necessary to accurately identify the transverse position of the vehicle's load, which is not convenient for practical engineering applications. Therefore, the proposed GA optimization method is adopted to find an optimal combination of measurement point responses that are insensitive to the vehicle load's transverse position. The relevant parameter of GA is listed in Table 1.  
The result of the optimization iteration is shown in Figure 8.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Maximum number of genetic generations & 1000 \\ Population size & 50 \\ Encoding method & Binary encoding \\ Selection method & Roulette \\ Crossover approach & Uniform crossover \\ Mutation method & Multipoint mutation \\ Crossover probability & 0.8 \\ Mutation probability & 0.2 \\ Range of independent variables & \((-20,20)\) \\ Deviation of fitness function value & \(1\times 10^{-6}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: The relevant parameter of GA.  
Figure 7: The identified strain influence lines of part of sensors under vehicle loads at different transverse positions:(**a**) Z1 and (**b**) Z2.  
After GA global optimization, we can obtain the optimal combination coefficients of six measurement points as \(k_{i}=[-2.599,4.758,4.233,4.965,\,-2.302,\,-1.865]\).  
The optimal combined strain response is used to identify the corresponding combined strain influence line under the vehicle load at different transverse positions. The identified result is shown in Figure 9.  
As observed from the figure, the optimal combined strain influence line does not change much when the vehicle passes the bridge at different transverse positions, so it is not sensitive to the lateral position of the load. It can be used for vehicle load identification without identifying the transverse position of the load. The optimal combined strain influence line identified under load at different transverse positions is averaged, which will be used for the following identification of axle weights.  
Figure 8: Iterative optimization curve.  
Figure 9: The optimal combination strain influence line under vehicle loads at different transverse positions.  
It can be seen from the results that the maximal identified error of the axle's weight is below 2% and that of the gross weight is below 1%. The main cause is that the identified errors of some axles are positive while other axles are negative. This indicated that the optimized combined influence line is insensitive to the transverse position of the vehicle. The proposed algorithm can be used to identify vehicle loads without estimating the transverse position of the vehicle in advance.