## 4 Inferred Weight as an Indicator of Structural Health
### Bridge Damage Detection Using Inferred Vehicle GVW Statistics  
Figure 7(a) shows the histogram of inferred weights for the healthy bridge. Figure 7(b) presents the histogram of inferred GVW for a population of 500 vehicles crossing the bridge for various levels of damage. Figure 7 represents ideal conditions with none of the sources of error outlined in Section 3.3 included. While the force histories and the inferred GVW for individual vehicles are inaccurately inferred, there is an underlying trend for inferred weight to decrease with increasing bridge damage (Figure 7(b)). This may be explained by a poor match between measured data from the damaged bridge and the theoretical equivalent healthy bridge. While there are significant inaccuracies for individual vehicles, a Normal distribution fit to the inferred vehicles is an excellent match with the Normal distribution fit to the true weights. Figure 7(b) shows that this distribution is visibly and consistently affected by bridge damage, even for very low damage levels. In general, bridge damage causes the peak to reduce, the standard deviation to increase, and, in most cases, the mean to shift slightly to the left. Table 4 confirms the increasing trend in standard deviation (STD) with damage and the slight falling trend in mean inferred GVW. The trend in the mean is not fully consistent--there is a slight increase between healthy and 2.5% damage, followed by falls between 2.5%, 5%, 7.5%, and 10%. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Accurate Measurements**} & \multicolumn{2}{c}{**Measurements with Various Sources of Inaccuracy in Table 5**} \\  & & & **(\% Error in Brackets *)** \\ \hline Damage level & STD (t) & Mean (t) & STD (t) & Mean (t) \\ \hline Actual & 1.86 & 8.77 & 1.94 & 9.06 \\
0\% & 2.07 & 8.68 & 2.27 (17\%) & 8.12 (\(-\)10\%) \\
2.5\% & 2.18 & 8.71 & 2.55 (31\%) & 8.04 (\(-\)11\%) \\
5\% & 2.71 & 8.49 & 2.89 (49\%) & 7.71 (\(-\)15\%) \\
7.5\% & 3.45 & 8.17 & 3.34 (72\%) & 7.42 (\(-\)18\%) \\
10\% & 4.14 & 7.92 & 3.88 (100\%) & 7.28 (\(-\)20\%) \\ \hline \hline \end{tabular}  
* Note: numbers in the brackets are the percentage difference from true value.  
\end{table} 
Table 4: Inferred GVW statistics for different damage levels. Figure 7: MFI results with 5% and 10% global damage levels for; (**a**) first axle; and (**b**) second axle. The trends are less consistent when the results are repeated, considering various sources of potential inaccuracy. In particular, in this case, a different batch of 500 vehicles is used for each damage level, thus allowing for the natural GVW variability in the vehicle population. Table 5 summarizes the various sources of simulated inaccuracy, and Figure 9 shows the resulting statistical distributions of inferred weight. The increasing standard deviation with damage is again evident, and the slight downward trend in the mean--see Table 4.  
Figure 8: Inferred GVWs for 500 vehicles: (**a**) Histogram for healthy and fitted Gaussian distributions of inferred and true vehicle weights; (**b**) fitted Gaussian distributions for healthy and damaged bridges. \begin{table}
\begin{tabular}{c c} \hline \hline
**Number of Vehicles Per Batch** & **500** \\ \hline Batches of vehicles & Randomly chosen from WIM database \\ Velocity error (mean, std) (\%) & (0\%, 3\%) \\ Pre-existing deflection (time intervals) & After first vehicle leaves bridge 0.5–2 s \\ Measurement noise & 3\% \\ Profile & Class A, carpet profile, the same for each batch of vehicles \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Variables in simulation. This entire set of simulations, including errors, has been carried out three times to establish the repeatability of the trends. The variations in the means and standard deviations with damage are presented in Figure 10. There is considerable variation in the inferred mean weights--the coefficient of determination (R\({}^{2}\)) of the trendline (linear regression) is only 0.63.