## 2 Theoretical Analysis
### Transfer Matrix of a Damaged Bridge
It is also important to note that for structures exhibiting nonlinear characteristics, the accuracy of the reconstructed responses may not be guaranteed. Differentiating between reconstruction errors and divergences caused by damage becomes a challenging task in such scenarios. Moreover, the severity of the damage response can be evaluated through the derivation of Equation (7). Since we have assumed that \( \begin{bmatrix}\phi_{1}(x_{1})&\phi_{2}(x_{1})\\ \phi_{1}(x_{2})&\phi_{2}(x_{2})\end{bmatrix} ^{-1}\) would remain constant, the explicit form, for which and the values of elements are not necessary in this study, can be represented as  
\[ \begin{bmatrix}\phi_{1}(x_{1})&\phi_{2}(x_{1})\\ \phi_{1}(x_{2})&\phi_{2}(x_{2})\end{bmatrix} ^{-1}= \begin{bmatrix}a&b\\ c&d\end{bmatrix}  \tag{10}\]  
where \(a,b,c,d\) are corresponding elements with constant values. Further, the reconstructed responses at \(x_{3}\) should be  
\[R(x_{3})=[a\!\cdot\!\phi_{1}(x_{3})+c\!\cdot\!\phi_{2}(x_{3})]\!\cdot\!R(x_{1} )+[b\!\cdot\!\phi_{1}(x_{3})+d\!\cdot\!\phi_{2}(x_{3})]\!\cdot\!R(x_{2}) \tag{11}\]  
where \(R(\cdot)\) denotes the responses (displacement, inclination, etc.) at a specific location. For a damaged bridge, the modal coordinates at \(x_{3}\) for the first two modes were assumed to be \(\phi_{1}{}^{d}(x_{3})\) and \(\phi_{2}{}^{d}(x_{3})\). Then, the difference between the measured and reconstructed responses can be represented as  
\[\Delta R(x_{3})=[a\!\cdot\!\Delta\phi_{1}(x_{3})+c\!\cdot\!\Delta\phi_{2}(x_{ 3})]\!\cdot\!R(x_{1})+[b\!\cdot\!\Delta\phi_{1}(x_{3})+d\!\cdot\!\Delta\phi_{2 }(x_{3})]\!\cdot\!R(x_{2}) \tag{12}\]  
where  
\[\Delta\phi_{1}(x_{3})=\phi_{1}{}^{d}(x_{3})-\phi_{1}(x_{3}) \tag{13}\]  
\[\Delta\phi_{2}(x_{3})=\phi_{2}{}^{d}(x_{3})-\phi_{2}(x_{3}) \tag{14}\]  
It is important to note that \(R(x_{1})\) and \(R(x_{2})\) in Equation (13) represent the measured responses of the damaged bridge. The severity of the damage is directly proportional to the magnitudes of \(\Delta\phi_{1}(x_{3})\) and \(\Delta\phi_{2}(x_{3})\), implying that \(\Delta R(x_{3})\) would increase with more extensive damage. Specifically, one can assess the severity of damage by examining the maximum difference in the time history between the measured and reconstructed responses. A larger discrepancy is indicative of more severe damage. Furthermore, as previously established in studies, the modal coordinates' difference between the damaged and intact bridges at an observation point decreases as the distance from the observation point to the local damage increases. Consequently, it is anticipated that the observed difference between the reconstructed and measured responses will diminish as the observation location moves from the damaged region to the undamaged region. Additionally, the sensitivity of the proposed method should align with the measured physical index. As Equation (13) demonstrates, the target difference is a linear combination of \(\Delta\phi_{1}(x_{3})\) and \(\Delta\phi_{2}(x_{3})\). Consequently, the sensitivity of the method should mirror that of the measured physical index. For instance, if the measured response is displacement, the sensitivity should correspond to that of the displacement mode shape. Similarly, if inclination or strain is measured, the sensitivity should match that of the inclination and strain mode shapes, respectively.