## 2 Identification Methods
### Strain Influence Line Identification  
Next, a calibrated vehicle with a known axle weight and wheelbase information passes over the bridge deck, and the axle weight sensor collects the corresponding strain response; moreover, the strain influence line caused by the unit axle weight is identified by analyzing the strain response. Due to the local force characteristics of the orthotropic bridge deck, the strain influence line length of the measurement points on the U-rib is approximately the spacing between the front span and back span of the diaphragm [32]; thus, it can be taken as a three-span spacing of the diaphragm. The method is detailed as follows. 1. Assuming a calibrated vehicle, its axle weight is \(A_{n}\) (\(n\) = 1,2,...,N\({}_{\rm A}\), N\({}_{\rm A}\) is the number of axles), and the wheelbase is \(B_{n}\) (\(n\) = 1,2,...,N\({}_{\rm A}\), for uniformly descriptions, and \(B_{1}\) = 0 is defined). 2. Assuming that the strain influence line of measurement point \(i\) to be identified is \(\mathbf{I}_{i}\) and the length of influence line is N\({}_{\rm I}\), here, it denotes the sample point number. Take a moment when the first axle of the calibration vehicle passes the starting point of the strain influence line and define it as \(t_{0}=0\); then, denote the starting point position as \(d_{0}=0\). The sample frequency of signal is \(f\), and the vehicle spacing between the  
Figure 2: The strain response measured by FAD3 and FAD3\({}^{\prime}\). two consecutive sample points is calculated as \(\Delta d=\frac{v}{f}\). So at moment \(t_{k}\) (\(k\) = 0, 1, 2,... \(\times\)K, K is the number of points sampled when the last axle of the vehicle passes the end of the strain influence line), the longitudinal position of the vehicle on the bridge is \(d_{k}=k\cdot\Delta d\), and the strain response of measurement point \(i\) at moment \(t_{k}\) can be expressed as follows:  
\[\epsilon_{i}(t_{k})= \epsilon_{i}(d_{k})= \sum_{n=1}^{\text{N}_{A}}A_{n}\mathbf{I}_{i}(d_{k}-c_{n}) \tag{3}\]  
\[c_{n}=\sum_{n=1}^{\text{N}_{A}}B_{n}\cdot f/v \tag{4}\]  
where \(\mathbf{I}_{i}(d_{k}-c_{n})\) is the value of the influence line for the \(n\)th axle of the calibration vehicle at position \(d_{k}\), and \(c_{n}\) is the difference in sampling points between the \(n\)th axle and the first axle of the calibration vehicle. For all moments, the strain response of measurement point \(i\) can be expressed in matrix form as follows. \[\left[\mathbf{A}\right]_{(\text{N}_{\text{I}}+\text{C}_{\text{N}_{\text{A}}} -1)\times\text{N}_{\text{I}}}\left[\mathbf{I}_{i}\right]_{\text{N}_{\text{I} }\times 1}=\left[\epsilon_{i}\right]_{(\text{N}_{\text{I}}+\text{C}_{\text{N}_{ \text{A}}}-1)\times\text{N}_{\text{I}}} \tag{5}\]  
\[\left[\mathbf{A}\right]_{(\text{N}_{\text{I}}+\text{C}_{\text{N}_{\text{A}}} -1)\times\text{N}_{\text{I}}}=\left[\mathbf{A}_{1}\right]_{(\text{N}_{\text{I }}+\text{C}_{\text{N}_{\text{A}}}-1)\times\text{N}_{\text{I}}}+\left[\mathbf{A }_{2}\right]_{(\text{N}_{\text{I}}+\text{C}_{\text{N}_{\text{A}}}-1)\times \text{N}_{\text{I}}}+\ldots+\left[\mathbf{A}_{\text{N}_{A}}\right]_{(\text{N}_{ \text{I}}+\text{C}_{\text{N}_{\text{A}}}-1)\times\text{N}_{\text{I}}} \tag{6}\]  
\[\left[\mathbf{A}_{n}\right]_{(\text{N}_{\text{I}}+\text{C}_{n}-1)\times\text{ N}_{\text{I}}}=\left[ \begin{array}{ccccc}0&0&\ldots&0&0\\ \vdots&\vdots&[0]_{(\text{C}_{n}-1)\times\text{N}_{\text{I}}}&\vdots&\vdots\\ 0&0&\ldots&0&0\\ A_{n}&0&\ldots&0&0\\ 0&A_{n}&\ldots&0&0\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&\ldots&A_{n}&0\\ 0&0&\ldots&0&A_{n}\\ 0&0&\ldots&0&0\\ \vdots&\vdots&[0]_{(\text{C}_{\text{N}_{\text{A}}}-\text{C}_{n})\times\text{ N}_{\text{I}}}&\vdots&\vdots\\ 0&0&\ldots&0&0\\ \end{array} \right] \tag{7}\]  
Considering the effects of noise and vehicle vibration to the structural response, strain influence line \(\mathbf{I}_{i}\) can be estimated based upon the measured strain response of measurement point \(i\) using the least squares method.