## 2 Damage Identification Method of Hangers  
The redistribution of cable force caused by the hanger's damage will lead to the deflection change of the tie-beam before and after the hanger's damage. If the loads that cause the deflection change of the tie-beam can be identified, then the change of the cable force can be inferred, and the damage identification of the hanger can be carried out. According to the theory of elastic foundation beam, the main girder of the bridge of load-bearing cable system can be regarded as a continuous beam with elastic supports, and the through-tied arch bridge can be simplified to a continuous beam with elastic supports [27; 28]. In the theoretical derivation of this paper, the elastically supported continuous beam is used as the mechanical model of the through-arch bridge. As shown in Figure 1, it is a through-tied arch bridge with \(n\) hangers that bears uniformly distributed loads. In Figure 1a,b, the damaged hangers are hanger _Ni_ and hanger _Nj_, respectively, as they are supposed to be completely damaged, so the corresponding mechanical model removes the damaged hanger. In Figure 1,\(w^{u}(x)\) and \(w^{d}(x)\) are the deflection curve before and after the hanger's damage. When the hanger is wholly damaged (the change of cable force of the damaged hanger is 100%), the difference of the deflection obtained from the healthy state and the wholly damaged conditions are expressed using Equation (1). \[\Delta f(j)=w^{d}(j)-w^{u}(j)\ \ \ \ \ (j=1\cdots n) \tag{1}\]  
where \(\Delta f(i)\) is the deflection change at the anchorage point of the hanger and the tie-beam. When the damaged state of the hanger is unknown (see Figure 1c), the deflection difference at the anchorage point of hanger _Ni_ and the tie-beam can be expressed as Equation (2). \[\Delta w(i)=f_{i1}\eta_{1}+f_{Z}\eta_{2}+\cdots+f_{ii}\eta_{i}+\cdots+f_{ij} \eta_{j}+\cdots+f_{in}\eta_{n}+\chi\ \ \ \ \ (i=1\cdots n) \tag{2}\]  
where \(\Delta w(i)\) is the deflection change at the anchorage point of the hanger _Ni_ and the tie-beam, \(f_{ii}\) and \(f_{jj}\) are the deflection difference at the anchorage point of the tie-beam and the completely damaged hanger _Ni_ and _Nj_ (see Figure 1a,b), respectively, \(f_{ij}\) is the deflection difference at the anchorage point of the tie-beam and the hanger _Ni_ when the hanger _Nj_ is completely damaged (see Figure 1b), and \(\{\eta_{i}\}\) is a column vector composed of the reduction ratio of cable force of each hanger. When a hanger is damaged alone, it is  
Figure 1: Mechanical model: (**a**) the hanger _Ni_ is completely damaged; (**b**) the hanger _Nj_ is completely damaged; (**c**) unknown damaged state of the hanger. a column vector with only one non-zero element, and when the number of the damaged hangers is \(m\), it is a column vector with \(m\) non-zero elements. \(\chi_{i}\) is caused by the deflection difference of multiple hangers damaged at the same time and is not equal to the sum of the deflection differences corresponding to the separate damage of each damaged hanger. To illustrate the existence of \(\chi_{i}\), we take the simultaneous damage of two hangers as an example, to prove that the deflection difference of simultaneous damage is not equal to the sum of the deflection difference of two hangers damaged separately. The cable loss occurs separately at hanger \(Ni\) and \(Nj\) in Figure 2a,b. The damage degree is 10% and 20%, respectively, while the hangers \(Ni\) and \(Nj\) are simultaneously damaged in Figure 2c, and the damage degree is 10% and 20%, respectively. It can be seen that the corresponding structures of the three cases are different after the hanger is damaged. Therefore, these damage conditions do not conform to the superposition principle, as the premise of the superposition principle requires that the structure does not change. Therefore, the sum of the deflection difference corresponding to Figure 2a,b is not equal to the deflection difference corresponding to Figure 2c, then Equation (3) can be obtained.