# Development and Testing of a Railway Bridge Weigh-in-Motion System
## 2 RB-WIM Algorithm
### Influence Lines  
Influence lines (IL) are key properties of a bridge, defining how it responds to loading at a given measurement point [24]. It has been shown that influence lines should be calculated directly from measurements [34] since theoretical influence lines rarely provide an accurate description of bridge behaviour. Two methods of \"measuring\" influence lines for bridges are known, the SiWIM approach [33] and the Matrix Method [35]. The Matrix Method uses vehicles of known axle loads and spacings and an inverse Moses algorithm to derive the experimental influence lines. This method is more straightforward and less time-consuming but requires vehicles of known weight, which are not always available at the time of setting-up the B-WIM system. More recently, a variation on this approach addresses the issue of requiring a vehicle of known weight [36].  
The SiWIM system, used here, calculates the influence lines for the bridge using selected vehicles with unknown axle loads. Numerous evaluations of influence lines can be averaged to improve accuracy. Such ILs are normalised and require a scalar calibration factor to convert relative axle weights to actual weights (the \(C_{F}\) parameter in Equation (1)). A detailed explanation of this general procedure of IL calculation can be found in [33]. Briefly, the system models the IL with a cubic spline, chosen because its general characteristics match well with real influence lines--it is a curve of third order, continuous and smooth in first and second derivatives [32]. Some of the spline knots, representing supports and endpoints, are fixed, while some are allowed to vary. In order to determine the values and thus the shape of the IL, Equation (2) is used. Contrary to its use in weighing, where the only unknowns are axle loads, \(A_{i}\), the function \(I(x)\) is unknown when calculating the IL. Since the system is no longer a linear function of all unknowns, Powell's minimisation [32] is used to solve the problem.  
Figure 2: Raw axle detection measurement point (ADMP) signal and conditioned signal for a one-carriage passenger train.  
Figure 3 shows the IL calculated for the Nieporet Bridge, for sensors mounted at the base of the truss at mid-span. The white points represent the fixed knots, while the ordinates of the two grey points were varied in order to obtain the best fit. For this bridge, local bending of the stringer beams dominated over global truss bending, making the influence line similar to that for a continuous beam with supports at truss chord locations. This effect was accentuated by setting the fixed knots to zero at the chord locations (Figure 3).  
A theoretical influence line for an infinitely thin bridge (Euler Bernoulli beam) has a sharp peak at mid-span where the derivative is discontinuous. In contrast, the peak of the influence line for a real bridge has a rounded peak, which is approximated by a circular portion, whose radius corresponds roughly to the superstructure thickness. This peaked section is drawn in bold in Figure 3 and the radius was also varied to obtain the best fit.