## 3 Statistical Approach Based on the Acceleration-Based BWIM System
### The BWIM Analysis Using the Half-Car Vehicle Model  
The acceleration-based BWIM system analysis is carried out using the half-car vehicle model. The bridge midspan acceleration responses for a half-car at 4 m axle spacing at a constant velocity of 80 km/h and using 1 st and 2 nd axle weights of 9 and 11 t, respectively, are simulated using first a smooth profile, then a class 'A' road profile. A 'calculated' bridge acceleration response is found for each simulated'measured' acceleration response using the BWIM equations and the bridge influence line response. The simulated'measured' and the 'calculated' bridge accelerations are shown in Figure 5a,b, using smooth profile and class 'A' road [27], respectively. To simulate realistic bridge conditions, pre-existing bridge vibration is assumed on arrival of the vehicle on the bridge. During many repetitions of this exercise, simulated'measured' and 'calculated' responses were seen to be consistently similar, although some bias is present when using a road profile, as mentioned in the section above. The BWIM algorithm is tested using the half-car vehicle model with smooth and class 'A' road profile. [PERSON] algorithm, illustrated in Equations (5) and (6) for a 2-axle vehicle, is based on the minimization of the sum of squared differences between the'measured' and the 'calculated' responses [19]. The axle weights are inferred by setting the partial derivatives of the sum of squared differences to zero, as shown in Equation (6). This process results in a system of linear equations in terms of the unknown axle weights, \(W_{j}\). \[E=\sum\
olimits_{i=1}^{n_{s}}\left(R_{i}^{m}-(W_{1}I(x_{i})+W_{1}I(x_{i}-d_{2}) )\right)^{2} \tag{5}\]  
\[\frac{\partial E}{\partial W_{j}}=0;\ \ j=1,2 \tag{6}\]  
Figure 4: Bridge acceleration response to a two quarter-car vehicle and sum of the individual axle responses using the class ’A’ road profile. Figure 5: Mid-span ‘simulated measured’ and corresponding ‘calculated’ bridge acceleration responses using: (**a**) smooth profile; (**b**) class ’A’ road profile. where \(\rm R_{i}^{m}\) is the measured bridge acceleration response for scan i, and \(\rm n_{s}\) is the total number of scans. The accuracy of the acceleration-based BWIM system is found by comparing the actual gross weights used to generate the accelerations with the corresponding inferred values. Because of the consistent biasness between the'measured' and the 'calculated' responses, the accuracy of individual vehicle weights is poor. Therefore, a statistical approach to the acceleration-based BWIM system is considered in this paper. The acceleration responses to three sets of 500 half-car vehicles are used to test the acceleration-based BWIM system under healthy bridge conditions and the results are plotted in the form of histograms in Figure 6a,b using a smooth profile and a class 'A' road, respectively. The axle weights, velocities and axle spacings of the half-car vehicles are chosen randomly, using Monte Carlo simulation, assuming normal distributions with means and standard deviations of typical two-axle traffic data. In this section, initial free-vibration of the bridge is generated using the acceleration response to an earlier passing two quarter-car vehicle with randomly chosen vehicle properties. A single random profile of class 'A' is used for the simulations. It can be seen in Figure 6a that the inferred weights are fairly accurate and consistent when a smooth road profile is used in the analysis and the average of the inferred GVW accuracies for each set are \(-5.9\%\), \(-5.8\%\) and \(-6\%\), respectively. In Figure 6b, it can be seen that the acceleration-based BWIM system underestimates the axle weights quite significantly but this inaccuracy is very consistent. The averages of the 500 inferred GVW accuracies of the BWIM system for each set in Figure 6b with the half-car vehicle model are \(-61.5\%\), \(-61.8\%\) and \(-61.7\%\), respectively. The road profile has a significant influence on the accuracy of the acceleration-based BWIM.