## 2 Identification Methods
### Optimal Combined Strain Influence Line  
The strain influence line can be identified separately for different measurement points. Theoretically, when the vehicle passes through the bridge in a determined transverse position, only one measurement point datum is needed during vehicle load identification using the influence line. However, when the transverse position of the vehicle changes, it is necessary to identify the influence lines of multiple transverse positions in advance. Then, the influence line corresponding to the transverse position of the vehicle load is used to identify the unknown load. Moreover, when the wheels are transversely far from the measurement point, the measured strain response is smaller, and the signal-to-noise ratio is lower. In order to facilitate the actual vehicle load's identification, here, an optimizing analysis is carried out to find a combination of measurement point responses that is insensitive to the vehicle's transverse position, and the optimal combined strain response is used to identify the corresponding combined strain influence line. Finally, the unknown load is identified based on the combined strain influence line. The detailed method is as follows. Let the calibration vehicle pass over the bridge at transverse position \(k\), we can extract the maximum strain response measured by the axle's weight sensor. Calculate the maximum of the combined strain response at the measurement points:  
\[y_{j}=\sum_{i=1}^{\text{N}_{\text{s}}}x_{i}\cdot s_{i} \tag{9}\]  
where N\({}_{\text{s}}\) is the number of measurement points, \(x_{i}\) is the combined coefficient of measurement points, and \(s_{i}\) is the maximum value of strain response of measurement point \(i\). Let the vehicle pass through the bridge deck several times from different transverse positions \(k\), we can obtain the corresponding maximum combined strain response, and then we calculate its variation coefficient according to the following Formula [14]:  
\[c_{\text{p}}=\frac{\sigma(y_{k})}{\mu(y_{k})} \tag{10}\]  
where \(\sigma\) and \(\mu\) is the mean and standard deviation of the maximum value of combined responses, respectively. The smaller the variation coefficient, the less sensitive the corresponding combination strain to the transverse position of the vehicle load; thus, the optimal combination can be determined by searching for the combination with the smallest variation coefficient. The objective function to be optimized is as follows:  
\[f_{o}(\textbf{X})=c_{\text{p}}=\frac{\sigma(\sum\limits_{i=1}^{\text{N}_{ \text{s}}}x_{i}\cdot s_{i})}{\mu(\sum\limits_{i=1}^{\text{N}_{\text{s}}}x_{i} \cdot s_{i})} \tag{11}\]  
where X= \([x_{1},x_{2},\dots,x_{\text{N}_{\text{s}}}]\) denotes independent variable parameters consisting of N\({}_{\text{s}}\) measurement points. Here, the genetic algorithm is used to perform the optimal global search. A genetic algorithm (GA) is proposed by [PERSON] [33] based on the natural selection and genetic mechanism of Darwinian biological evolution to obtain the optimal solution, which has been well applied in many fields. It first encodes the solution (called chromosome) of the problem, usually using binary encoding or real number encoding to map the solution space to the chromosome space as follows:  
\[x_{i}=\text{U}_{1}+(\sum\limits_{j=1}^{m}b_{j}\cdot 2^{j-1})\cdot\frac{\text{U}_{ 2}-\text{U}_{1}}{2^{m}-1} \tag{12}\]  
where U\({}_{1}\) and U\({}_{2}\) denote the lower and upper limits of the value of parameter \(x_{i}\), respectively, \(m\) is the length of the chromosome of encoded variable, and \(b_{j}\) is the genetic coding at position \(j\) of the variable. Subsequently, reasonable initial populations are generated in these solution spaces, and the initial populations are processed with crossover and mutation. Moreover, the \(k\)th individual fitness function is calculated as follows:  
\[f_{c}(\textbf{X}_{k})=\frac{1}{1+f_{o}(\textbf{X}_{k})},k=1,2,\dots,\text{M} \tag{13}\]  
where M is the number of populations.