## 4 Application to a Plate Girder Bridge under Vehicle Load
### Optimization Results
Using strains at the optimized locations, the MSE (\(\times 10^{-3}\)) for four, three, and two strains was 0.7987, 0.8830, and 1.1249, respectively, for 40 km/h. The MSE increased slightly as the number of strains decreased but was still close to zero. From the results, as with the training accuracy, it could be confirmed that the prediction accuracy of the test data was very high. For an intuitive comparison, the predicted displacements are presented in Figures 13-18. Figure 13 shows the change in displacements over time at the midpoint of the center girder (CG) for strains at arbitrary and optimized locations for 40 km/h, which is expected to produce the maximum displacements. In addition, the differences between the numerical analysis results and predicted displacements are plotted. In both arbitrary and optimized locations, as the number of strains decreased, the difference from the numerical analysis results increased. In particular, in the case of arbitrary locations, the difference according to the number of strains was large, and it could be seen that there was an overall fluctuation. In the case of the optimized locations, the difference in the number of strains was not large and predicted displacements showed good agreements with the overall numerical analysis. Figure 14 shows the overall displacement shape of the center girder (CG) at t = 6.52 s, at which time the maximum displacement occurred. As in Figure 14, for arbitrary locations, the difference between predicted displacements and numerical results was very large. For the optimized locations, predicted displacements showed good agreements with numerical analysis regardless of the number of strains. The same results are shown for the test datasets of 60 km/h and 80 km/h (Figures 15-18). From the results, as in the simple beam, the effectiveness of the GA was confirmed in optimizing the locations of the strain-measurement points required for displacement prediction of a bridge under vehicle load. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & & \multicolumn{3}{c}{**Locations of Strain-Measurement Points (m)**} & \multicolumn{3}{c}{**Training**} \\ \cline{3-8}
**No. of Strains(= n\({}_{\text{s}}\))** & & & & & & \\ \cline{3-8}  & **Point 1** & **Point 2** & **Point 3** & **Point 4** &
\begin{tabular}{c} **MSE** \\ **(\(\times 10^{-3}\))** \\ \end{tabular} & **R** \\ \hline \multicolumn{2}{c}{11} & & & 0\(\sim\)36 (interval 3.6 m) & & 1.2007 & 0.9988 \\ \hline \multirow{3}{*}{**Arbitrary**} & 4 & 7.2 & 14.4 & 25.2 & 28.8 & 5.4946 & 0.9947 \\  & 3 & 7.2 & 25.2 & 32.4 & - & 31.1106 & 0.9696 \\  & 2 & 14.4 & 21.6 & - & - & 44.0195 & 0.9575 \\ \hline \multirow{3}{*}{**Optimized**} & 4 & 7.2 & 14.4 & 21.6 & 32.4 & 1.4268 & 0.9986 \\  & 3 & 7.2 & 18 & 25.2 & - & 1.7511 & 0.9983 \\ \cline{1-1}  & 2 & 10.8 & 21.6 & - & - & 2.5370 & 0.9975 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The results of ANN training with optimized strain-measurement points. Figure 12: The optimized locations of strain-measurement points on the girders (Unit: mm). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**No. of Strains Per Grider**} & \multicolumn{2}{c}{**ANN Structure**} & \multicolumn{2}{c}{**Mean Squared Error (MSE, \(\times 10^{-3}\))**} \\ \cline{3-6}  & **(Input-Hidden-Output)** & **40 km/h** & **60 km/h (Test 2)** & **80 km/h (Test 3)** \\ \hline \(11\) & & 55–55–55 & 0.6874 & 0.3712 & 4.4094 \\ \hline \multirow{3}{*}{Arbitrary} & 4 & 20–20–55 & 1.7815 & 1.7924 & 7.6117 \\  & 3 & 15–15–55 & 15.0731 & 11.8733 & 31.2476 \\  & 2 & 10–10–55 & 15.0141 & 10.7792 & 43.6243 \\ \hline \multirow{3}{*}{Optimized} & 4 & 20–20–55 & 0.7987 & 0.4805 & 4.7009 \\  & 3 & 15–15–55 & 0.8830 & 0.5789 & 5.0831 \\  & 2 & 10–10–55 & 1.1249 & 0.8025 & 5.7813 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The results of the ANN test for three test datasets. Figure 13: Displacements over time at center girder for 40 km/h test dataset. Figure 16: Overall displacements shape at t = 1.75 s (max. displacement) for the 60 km/h test dataset.