# Model Updating of Bridges Using Measured Influence Lines
## 2 Materials and Methods
### Measurements
#### 2.2.2 Extraction of the Strain Influence Line (IL) from the Strain Measurements  
B-WIM refers to a method that uses an instrumented bridge or culvert to weigh-in-motion the passing vehicles. The B-WIM analysis is an inverse-type problem where the structural response (mid-span strains) is measured and the traffic load causing this response is calculated [19]. Most operational B-WIM algorithms, including the one in this study, work on static assumptions and follow the principles proposed by [PERSON] [20]. The key parameter in the B-WIM analysis is the IL, a function that describes the response of a structure at a given point under a moving unit load. IL illustrates how various load effects, like displacements and strains, change under the moving traffic loads. Several methods of calculating ILs exist [21]; the method used in this paper has been described in detail in [16]. The following is a short recap of the method and the adjustments required for its use in FEMU. The goal of B-WIM is to solve a system of equations, defined in (1) as:  
\[s(t_{j})=\sum_{i=0}^{N}A_{i}I\big{(}v_{i}(t_{j}-t_{i})\big{)};\ j=1\ldots M \tag{1}\]  
where \(s(t_{j})\) is the average value of the strain signals from sensors at \(M\) different times, \(A_{i}\) is the unknown axle weight, \(I(x)\) is the known strain IL, \(v_{i}\) is the measured axle velocity, \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**1 st Axle**} & \multicolumn{3}{c}{**2 nd Axle**} & \multicolumn{3}{c}{**3 rd Axle**} & \multicolumn{3}{c}{**4 th Axle**} & \multicolumn{3}{c}{**5 th Axle**} \\ \hline \multirow{2}{*}{**Vehicle 1**} & **Load** & **Spacing** & **Load** & **Spacing** & **Load** & **Spacing** & **Load** & **Spacing** & **Load** & **GVW** \\  & **(kN)** & **(m)** & **(kN)** & **(m)** & **(kN)** & **(m)** & **(kN)** & **(m)** & **(kN)** & **(kN)** \\ \hline V1\_F & 68.67 & 3.42 & 73.58 & 1.37 & 71.61 & / & / & / & / & 213.86 \\ V1\_E & 57.88 & 3.42 & 37.28 & 1.37 & 36.30 & / & / & / & / & 131.45 \\ V2\_F & 69.65 & 3.60 & 90.25 & 2.82 & 64.75 & 1.31 & 61.80 & 1.31 & 59.84 & 346.29 \\ V2\_E & 56.90 & 3.60 & 29.43 & 2.82 & 14.72 & 1.31 & 15.70 & 1.31 & 14.72 & 131.45 \\ \hline \hline \end{tabular}  
* F represents a fully loaded truck, and E an empty truck.  
\end{table} 
Table 1: Axle loads, axle spacing, and gross vehicle weights (GVW) of the calibration vehicles. Figure 7: Calibration vehicles: V1 (**a**) and V2 (**b**). \(t_{i}\) is the arrival time of an individual axle, \(N\) is the number of axles, and \(M\) is the number of measurements, which must be greater than or equal to \(N\). The strain IL is specified in advance. Thus, the equation is linear in the unknown terms, meaning that relatively simple methods suffice to obtain the axle weights. The same equation can be used to calculate the strain IL by treating the function \(I(x)\) as an unknown, in addition to axle loads. Cubic splines are used to model the strain ILs. Splines are curves of the third order, smooth in derivatives and continuous in second derivatives, thereby matching the characteristics of observed strain ILs. The solved system is not linear in the unknown terms, necessitating iterative methods. Figure 8 demonstrates the principle of strain IL modelling for a hypothetical three-span bridge, for the strain sensor located at the mid-span of the inner span. The strain IL is defined by the three types of points. The white points represent supports, the abscissas are defined by geometry, and the ordinates are 0. The exception is point M, located at the measurement point and whose ordinate is also optimised. The grey points define the radius d/2 of the rounded cap that improves the fit between the measured and modelled responses. Finally, the optional black points A through F can fine-tune the match. Their x-coordinates are selected, and y-coordinates are optimised. The ordinates of the curve between the control points are obtained with cubic spline interpolation. This study required an adaptation to the original method for calculating the strain IL.