## 2 Theoretical Analysis
### Transfer Matrix of a Damaged Bridge  
In the context of concrete bridge structures, a common form of damage is cracking within the concrete layers. For steel bridges, fatigue failure is a typical issue. Both cracking in concrete and fatigue in steel lead to a reduction in the stiffness of the affected cross-sections. Therefore, in this study, we consider bridge damage to be represented by a decrease in stiffness at certain locations, as referenced in [30]. Moreover, the model of local stiffness reduction has been extensively employed in previous research to simulate bridge damage scenarios, as documented in studies [10; 12]. As existing research [31] has shown, the curvature of the bridge's mode shapes exhibits local peaks around the damaged area. Considering that the curvature of bridge displacement is linearly proportional to bridge strain, and that the inclination mode shape can be derived by integrating the strain mode shape, it follows that there would be a significant divergence in the inclination mode shape between an intact beam and a damaged one. Furthermore, existing research [32] also points out that for regions of the bridge that are far from the damaged area and remain intact, the divergence in mode shape functions between the intact and damaged beams is minimal. Consequently, it can be inferred that in undamaged regions, the inclination mode shapes of both damaged and intact bridges would be nearly identical. The basic damage localization method in this study is to detect the difference between the reconstructed bridge responses and the measured responses. For an intact bridge, the responses at three observation points, specifically, \(x_{1}\), \(x_{2}\), and \(x_{3}\), would be measured. Then, the transfer matrix from \(x_{1}\), and \(x_{2}\) to \(x_{3}\) would be estimated. It is assumed that, during the service period, no damage occurs at \(x_{1}\), and \(x_{2}\). Therefore, the mode shape coordinates at \(x_{1}\)and \(x_{2}\) would remain constant, i.e., the \( \begin{bmatrix}\phi_{1,incl}(x_{1})&\phi_{2,incl}(x_{1})\\ \phi_{1,incl}(x_{2})&\phi_{2,incl}(x_{2})\end{bmatrix} ^{-1}\) in Equation (8) would be a constant matrix. If no damage occurs at \(x_{3}\) and existing damage is far away from \(x_{3}\), then the mode coordinates at \(x_{3}\), specifically, \(\phi_{1,incl}(x_{3})\) and \(\phi_{2,incl}(x_{3})\) would also be invariable. Consequently, the transfer matrix \(T_{low,incl}\), as indicated in Equation (8), would remain constant. Specifically, for the transfer matrix, it can be inferred from Equation (7) that  
\[T_{low,incl,damage}= \begin{bmatrix}\phi_{1,damage}(x_{2})&\phi_{2,damage}(x_{ 2})\\ \phi_{1,damage}(x_{3})&\phi_{2,damage}(x_{3})\end{bmatrix}\begin{bmatrix}\phi_ {1,damage}(x_{1})&\phi_{2,damage}(x_{1})\\ \phi_{1,damage}(x_{2})&\phi_{2,damage}(x_{2})\end{bmatrix} ^{-1} \tag{8}\]  
where the subscript _damage_ indicates the corresponding mode shape functions of the damaged beam. When the observation points \(x_{1}\), \(x_{2}\), and \(x_{3}\) are far away from the damaged region, it can be inferred that  
\[T_{low,incl,intact}\approx T_{low,incl,damage} \tag{9}\]  
Therefore, even though potential damage might have occurred at the other locations on the bridge, the reconstructed responses should almost be identical with the measured responses, and the corresponding time history curves should totally overlap. However, if damage occurs near \(x_{3}\), \(\phi_{1,incl}(x_{3})\) and \(\phi_{2,incl}(x_{3})\), as well as the transfer matrix from \(x_{1}\) and \(x_{2}\) to \(x_{3}\), of the damaged bridge, would be different for those of the intact bridge. Consequently, the reconstructed responses at \(x_{3}\) would be different from the measured responses. Assume that there are multiple observation locations on the bridge; by checking whether or not difference can be found in the comparison between measured and reconstructed response at all these locations, one can determine the location of the potential damage.