# Study on the Method of Moving Load Identification Based on Strain Influence Line
## 2 Theoretical Background
### Moving Load Identification Method Considering the Load Transverse Distribution  
When the vehicle load acts on the bridge, the load is not only transmitted in the longitudinal direction, but also in the horizontal direction. Therefore, the force analysis of the bridge under the vehicle load is a space calculation problem. Then, the internal force analysis of the bridge section can be carried out through the influence surface. The influence surface of the bridge internal force can be expressed by a two-valued function \(\eta(x,y)\), then the internal force value of section \(a\) can be expressed as \(S=P\cdot\eta(x,y)\), in which \(S\) is the internal force value of the section, and \(P\) is the vehicle load. In addition, \(\eta(x,y)\) can be separated into the product of two single-valued functions by the separation variable method. That is \(\eta(x,y)=\eta_{1}(x)\cdot\eta_{2}(y)\), in which \(\eta_{1}(x)\) is the internal force influence line of the beam section, and \(\eta_{2}(y)\) is the change curve of the load ratio when the unit load acts in different positions along the horizontal direction. Then, the internal force value \(P^{\prime}\) of the beam section can be expressed as \(P^{\prime}=P\cdot\eta_{2}(y)\), equivalent to assigned the load to the beam along the horizontal direction when the load \(P\) acts on point \(a(x,y)\). For the simply supported T-beam bridge, as shown in Figure 3, it is approximately assumed that \(S=P\cdot\eta(x,y)\approx P\cdot\eta_{1}(x)\cdot\eta_{2}(y)\), which neglects the spatial effect of the bridge and turns it into a plane problem. When a moving load acts on the bridge deck and its position changes with the \(x\) coordinate but \(y\) coordinate is constant, then the \(P\cdot\eta_{2}(y)\) is constant too. That is the direction of the load transverse distribution coefficient along the beam span does not change. Therefore, each beam can be analyzed individually when analyzing the internal force influence line of the beam section, and the equivalent load of each beam can be obtained according to the load transverse distribution coefficient. Take the two-axle vehicle as an example to analyze the internal force (as shown in Figure 4). The axle weight is \(P_{11}\), \(P_{12}\), \(P_{21}\), and \(P_{22}\), respectively, and its action position is \((x_{1},y_{1})\), \((x_{1},y_{2})\), \((x_{2},y_{1})\), \((x_{2},y_{2})\), respectively. The \(y\) coordinate values of the four wheel loads are constant when the vehicle travels parallel to the \(x\) coordinate on the bridge, that is, the transverse distribution coefficient of each wheel load is constant. When analyzing the internal force of a single beam, the equivalent load \(\left(P_{1}^{n},P_{2}^{n}\right)\) acting on it can be obtained by the following equation:  
\[\left\{ \begin{array}{l}P_{1}^{n}=P_{11}\cdot\eta_{2}(y_{1})+P_{12}\cdot\eta _{2}(y_{2})\\ P_{2}^{n}=P_{21}\cdot\eta_{2}(y_{1})+P_{22}\cdot\eta_{2}(y_{2})\end{array}  \right.\left(n=1,\;2,\;3,\;4,\;5\right) \tag{10}\]  
The total weight of the vehicle can be expressed as:  
\[P=P_{11}+P_{12}+P_{21}+P_{22}=\sum_{n=1}^{5}(P_{1}^{n}+P_{2}^{n}) \tag{11}\]  
Figure 4: Wheel load transverse distribution on the bridge. Figure 3: The internal force calculation under vehicle load. According to the influence line theory, when the strain integral coefficient of one beam section is known, the total weight of the moving load can be calculated through the monitored strain integral value on the section. Assuming that the above vehicle loads drive parallel to the \(x\) coordinate from one end to the other end of the beam, the measured strain integral values of the mid-span section of each beam bottom are \(A_{2}\), \(A_{3}\), \(A_{4}\) and \(A_{5}\), respectively. In addition, it is assumed that the mid-span strain integral coefficients of each beam bottom are \(\alpha_{1}\), \(\alpha_{2}\), \(\alpha_{3}\), \(\alpha_{4}\), and \(\alpha_{5}\), respectively.