## 3 Statistical Approach Based on the Acceleration-Based BWIM System  
### Feasibility of the Concept  
The idea of using the bridge acceleration response for a BWIM system was first proposed by [PERSON] et al. [26]. They derived the bridge displacements by integrating the acceleration responses twice and used the derived signals in the conventional BWIM algorithm [26]. In this paper, the bridge acceleration signals are used directly in the BWIM algorithm. To simulate the acceleration response, the vehicle-bridge dynamic interaction model is programmed in MATLAB using a quarter-car vehicle and the bridge model. Displacement, velocity and acceleration at mid-span on the bridge are calculated. This location is chosen as it shows the maximum amplitude of response. The influence line response, by definition, is the bridge acceleration due to a unit single-axle load passing over the bridge and is required for the calibration of the BWIM system. The BWIM algorithm [19] uses a comparison of the calculated and measured bridge responses. The calculated response is found by summing the products of each axle weight and the influence line ordinate corresponding to its location; see Equation (4). This assumes the principles of scaling: the acceleration response to an axle with weight \(nW\) is \(n\) times the acceleration response to an axle with weight \(W\) and superposition: the acceleration response to a multi-axle vehicle is the sum of the responses to its individual axles. The calculated response for scan \(i\) is:  
\[R_{bi}^{T}=\sum\
olimits_{j=1}^{n_{a}}W_{j}I\left(x_{i}-d_{j}\right) \tag{4}\]  
where \(W_{j}\) is the weight of axle \(j\), \(I(x)\) is the influence line ordinate at distance \(x\) of unit-load from the start of bridge, \(x_{i}\) is the location of the first axle at scan \(i\), \(d_{j}\) is the distance of axle \(j\) from the first axle (\(d_{1}=0\)) and na is the number of axles. Acceleration responses can be used effectively for BWIM if the relationship between the signal amplitude and axle weight is linear or near to linear. This issue is explored by simulating the passage of a single quarter-car over the bridge with masses ranging from 5 to 30 \(t\). Accelerations on the bridge are calculated and plotted for a constant vehicle velocity of 80 km/h. Using Monte Carlo simulation, the vehicle mechanical properties are chosen randomly for each simulation run, assuming normal distribution with standard deviations of 20%. To begin with, the bridge initial conditions are kept at zero acceleration (static) in the model to represent ideal conditions, and a scanning frequency of 500 Hz is assumed. All the acceleration responses are plotted in the time domain. The amplitude of the first positive peak, which has the highest positive amplitude, is considered for the comparison. Figure 2 shows the bridge acceleration responses for different quarter-car masses (5 t to 30 t) using class 'A' road profile. The first positive peak amplitudes, normalised with respect to the 5 t axle value, are repeated for three different bridge profiles and plotted against axle mass in Figure 3. The best fits of the plots in Figure 3 show that the amplitudes follow a reasonably linear trend for each road profile which suggests that the bridge accelerations are scalable given that there is a constant road profile. The slopes of these fits are related to accuracy of the BWIM system. Ideally for an accurate BWIM system, the slope of the fit should be 1 [43]. As it can be seen from Figure 3, slope of the fits less than 1 suggests that the acceleration-based BWIM system will underestimate the inferred weights. However, because of the slope is constant for each given road profile, the underestimation will be constant.