## 2 Theoretical Development
### Improvement of Static Residual Force  
According to Equation (5), the static residual force vector \(\xi\) depends on the applied static load vector \(l\). Thus, the loading position and level of the static load both need to be accurately measured in advance for structural damage assessment. This operation increases the cost and time of damage evaluation. In view of this, an improved residual force algorithm independent of static load vector is proposed in this section. As stated previously, the static residual force vector \(\xi\) has nonzero coefficients associated with the damaged DOFs, allowing the following:  
\[\eta=Ku_{d} \tag{19}\]  
Then, Equation (5) can be rewritten as the following:  
\[\eta=\xi+l \tag{20}\]  
In engineering practice, static loading often adopts single-point or multiple-point loading modes. This means that the static load vector \(l\) only has a few nonzero coefficients in the DOFs corresponding to the loading positions. According to the above analysis, the vector \(\eta\) will only have nonzero coefficients corresponding to the damaged DOFs and loading positions. If the nonzero coefficients of \(\eta\) corresponding to the loading positions are set to zero, the resulting vector \(\overline{\eta}\) after this modification can also be used for damage localization and quantification. Compared with \(\xi\), this new vector \(\overline{\eta}\) can be computed by and \(u_{d}\) without \(l\), as long as the loading positions are known approximately in advance. The above operation can be realized by the following formula:  
\[\overline{\eta}=\Omega\cdot\eta \tag{21}\]  
\[\Omega=\left[\begin{array}{ccccc}1&&&&\\ &1&&&&\\ &&\ddots&&&\\ &&&0&&\\ &&&0&&\\ &&&\ddots&\\ &&&&1\end{array}\right] \tag{22}\]  
where \(\Omega\) is a diagonal matrix. Most of the diagonal elements in \(\Omega\) are one, except the diagonal elements corresponding to the static loading region, which are zero. However, the modification from \(\eta\) to \(\overline{\eta}\) may lead to leakage judgment of damage when the loading positions are exactly the locations of damaged DOFs. This problem can be solved by considering together two static loading modes with different loading positions. Without loss of generality, assuming \(\overline{\eta}_{1}\) and \(\overline{\eta}_{2}\) denote two modified vectors obtained by two different loading modes, the new damage indicator vector \(\gamma\) can be defined as the following:  
\[\gamma=\left|\overline{\eta}_{1}\right|+\left|\overline{\eta}_{2}\right| \tag{23}\]  
where \(\left|\right|\) denotes the absolute value. Using Equation (23), structural damage areas can be assessed initially by the nonzero coefficients of \(\gamma\). Subsequently, damage extents can also be calculated by a similar process to that in Equations (12) to (18).  
For single-damage cases, the damage quantification formula is modified as the following:  
\[\alpha_{q}=\frac{\delta_{i}}{k_{i}^{q}\cdot u_{d}} \tag{24}\]  
where \(\delta_{i}\) denotes a nonzero coefficient in \(\overline{\eta}\). For multiple-damage cases, the damage quantification equation is modified as the following:  
\[\left[\begin{array}{ccccc}k_{1}^{1}u_{d}&k_{1}^{2}u_{d}&\cdots&k_{1}^{r}u_{ d}\\ k_{2}^{1}u_{d}&k_{2}^{2}u_{d}&\cdots&k_{2}^{r}u_{d}\\ \vdots&\vdots&\vdots&\vdots\\ k_{r}^{r}u_{d}&k_{r}^{r}u_{d}&\cdots&k_{r}^{r}u_{d}\end{array}\right]\left\{ \begin{array}{c}\alpha_{1}\\ \alpha_{2}\\ \vdots\\ \alpha_{r}\end{array}\right\}=\left\{\begin{array}{c}\delta_{1}\\ \delta_{2}\\ \vdots\\ \delta_{r}\end{array}\right\} \tag{25}\]  
According to the above derivation, the improved residual force method realizes the operation of damage identification without a known size and location of static load. This innovation is beneficial to realizing rapid damage detection in bridge structure since any load vehicle can be used in testing. In summary, a flowchart as shown in Figure 1 is given to illustrate the whole technique more clearly.