## 2 Damage Identification Method of Hangers
\[f_{(ii)}^{a}+f_{(ij)}^{b}\
eq f_{(ij)}^{c} \tag{3}\]  
If superposing the three deflections, using the deflection difference caused by the simultaneous damage of two cables to subtract the deflection difference corresponding to the individual damage of each cable, the result will not be zero, shown in Figure 2d. This process shows that the deflection difference caused by two hangers damaged simultaneously is not equal to the sum of the deflection difference caused by the two hangers damaged separately. Furthermore, the difference of deflection caused by simultaneous damage of multiple hangers is not equal to the sum of the deflection difference of multiple hangers damaged separately. Therefore, the difference between them at the anchorage point of hanger \(Ni\) and the tie-beam is defined as \(\chi_{i}\). Thus, Equation (3) can be rewritten  
Figure 2: The difference of deflection changes between the simultaneous damage of two hangers and the two hangers damaged separately: (**a**) the damaged hanger is \(Ni\); (**b**) the damaged hanger is \(Nj\); (**c**) the damaged hangers are \(Ni\) and \(Nj\);(**d**) the result of superposing the three deflections of (**a–c**). into Equation (4). It is easy to find that when a single hanger is damaged, \(\chi_{i}\) is equal to zero. Otherwise, it is not equal to zero. \[f_{(ii)}^{a}+f_{(ij)}^{b}=f_{(ij)}^{c}+\chi_{i} \tag{4}\]  
Based on this, \(n\) displacement equations can be established as Equation (5). \[\left\{ \begin{array}{l}\Delta w(1)=f_{11}\eta_{1}+f_{12}\eta_{2}+\cdots+f_{1 1}\eta_{i}+\cdots+f_{1j}\eta_{j}+\cdots+f_{1n}\eta_{n}+\chi_{1}\\ \Delta w(2)=f_{21}\eta_{1}+f_{22}\eta_{2}+\cdots+f_{2i}\eta_{i}+\cdots+f_{2j} \eta_{j}+\cdots+f_{2n}\eta_{n}+\chi_{2}\\ \cdots\quad\quad\cdots\quad\cdots\quad\cdots\\ \Delta w(n)=f_{n1}\eta_{1}+f_{n2}\eta_{2}+\cdots+f_{ni}\eta_{i}+\cdots+f_{nj} \eta_{j}+\cdots+f_{nn}\eta_{n}+\chi_{n}\end{array} \right. \tag{5}\]  
Write Equation (5) in the form of a matrix as Equation (6). \[\left( \begin{array}{c}\Delta w(1)\\ \Delta w(2)\\ \vdots\\ \Delta w(n)\end{array} \right)=\left[ \begin{array}{cccc}f_{11}&f_{12}&\cdots &f_{1n}\\ f_{21}&f_{22}&\cdots&f_{2n}\\ \vdots&\vdots&&\vdots\\ f_{n1}&f_{n2}&\cdots&f_{nn}\end{array} \right]\left( \begin{array}{c}\eta_{1 }\\ \eta_{2}\\ \vdots\\ \eta_{n}\end{array} \right)+\left( \begin{array}{c}\chi_{1}\\ \chi_{2}\\ \vdots\\ \chi_{n}\end{array} \right) \tag{6}\]  
Or rewrite it to Equation (7). \[\Delta W=F\eta+\chi \tag{7}\]  
where \(F\) is the deflection difference influence matrix for hanger damage identification, \(\Delta W\) is the deflection difference series vector at the anchorage point of each hanger and tie-beam under arbitrary damage state, and \(\chi\) is the difference vector between the deflection change caused by simultaneous damage of multiple hangers and multiple hangers damaged separately. Equation (8) can be obtained from Equation (7). \[\Delta W=F\eta+F\chi^{\prime}=F(\eta+\chi^{\prime}) \tag{8}\]  
Solve Equation (8) to get Equation (9). \[\eta+\chi^{\prime}=F^{-1}\Delta W \tag{9}\]  
When a single hanger is damaged, \(\chi^{\prime}\) is a zero vector \(\eta=F^{-1}\Delta W\). Otherwise, \(\eta\
eq F^{-1}\Delta W\).