## 4 Lab Experiment
### Validation of the Damage Localization Method  
The BP-ANNs, which had been trained on the monitoring data from the intact bridge, were subsequently applied to reconstruct the responses of the damaged bridge at the \(-2.4\) m and \(2.4\) m cross-sections. For this reconstruction, the measured inclination data from the damaged bridge at \(-4.4\) m and \(4.4\) m were utilized as inputs. It is important to note that no new BP-ANNs were trained using the measured responses from the damaged bridge. Additionally, other details, such as the bridge's physical parameters (span, frequency, mode shape, etc.) and the vehicle's physical parameters (vehicle mass, speed, and direction), were not factored into the reconstruction process. The comparisons between the reconstructed and measured inclination time histories at \(-2.4\) m and \(2.4\) m are presented in Figure 15a, b, respectively. Figure 14: Comparison between reconstructed and measured inclination time histories at \(-2.4\) m and \(2.4\) m of the intact bridge (ellipses denote where the peak and valley amplitudes were selected). (**a**) Reconstruction result at \(-2.4\) m. (**b**) Reconstruction result at \(2.4\) m.  
As outlined in the theoretical derivations in Section 2.2, the presence of potential damage can be pinpointed by comparing the reconstructed and measured bridge responses across various locations. When damage is proximate to an observation point, a significant divergence between the measured and reconstructed responses, particularly in terms of the maximum values, is expected at that location. In contrast, if the damage is distant from the observation point, the reconstructed and measured responses should align closely, with minimal divergence in their maximum values. Figure 15a illustrates a pronounced divergence between the reconstructed and actual measurement data at the \(-2.4\) m location, especially concerning the amplitude of the peak value around \(230\) s. This discrepancy is likely due to changes in the mode shape function. Since the damaged area is near the \(-2.4\) m point, it is anticipated that the mode shape coordinates would be significantly altered in this region, leading to a corresponding shift in the transfer matrix and the observed divergence. Specifically, in the damaged region, the amplitudes of the mode shape functions at \(-2.4\) m are expected to be considerably higher than those of the intact bridge, resulting in measured responses of the damaged bridge being larger than the reconstructed responses. Complementing the comparison shown in Figure 15a, further validation of the proposed method is provided by the comparison between reconstructed and measured responses at the \(2.4\) m location, where no damage is present. Figure 15b demonstrates that the reconstructed results at the \(2.4\) m location closely match and nearly overlap with the measured data for the damaged bridge. This congruence can be attributed to the fact that the damaged region, which is only \(0.1\) m in length, is approximately \(5\) m away from the  
Figure 15: Comparison of inclination time histories between reconstructed and measured results of the damaged bridge (ellipses denote where the peak and valley amplitudes were selected). (**a**) Reconstruction result at \(-2.4\) m (the arrow marks the time point around \(230\) s, which corresponds to the highest amplitude). (**b**) Reconstruction result at \(2.4\) m.  
2.4 m point. Thus, the local stiffness change at the damaged region has a negligible effect on the modal coordinates at the 2.4 m location. Additionally, as deduced from Section 2.2, the divergence would gradually decrease from the damaged region to the non-damaged region.