## 2 Evaluation Index and Analysis Method of Broken Cable  
### Evaluation Index  
The research shows that most scholars study the cable fracture of a cable-stayed bridge by analyzing the internal force response of the structure. However, the actual internal force of a cable-stayed bridge in service is not easy to obtain, and thus, the structural internal force is not suitable for analyzing the stress state of a broken cable of a cable-stayed bridge in service. The component stress of a cable-stayed bridge is more suitable. Because the two ends of the cable are anchored on the main beam and the cable tower and the initial stress is applied, the main beam will be subjected to a strong axial force and become a compression-bending member. Therefore, Formula (1) was adopted to calculate the stress of the main girder and pylon:  
\[\sigma=\pm\frac{\mathrm{N}}{\mathrm{A}}+\frac{\mathrm{My}}{\mathrm{I}} \tag{1}\]  
where N--axial force of the main girder and cable tower, M--bending moment of the main girder and cable tower, A--cross-sectional area, I--moment of inertia of the section, y--the distance from the calculated position to the neutral axis.  
In a cable-stayed bridge structure, the sudden breaking of the cable means that the cable directly exits the structure. At the same time, it also causes structural vibration, and the stress and deformation of the structure reach the maximum response value in an instant. Then, the response value fluctuates within a certain range, finally tends to be in equilibrium under the damping effect and the remaining cables are forced to share the stress transferred by the broken cables. In order to study the influence of cable breakage on structural stress and deformation, the following three indexes (3D indexes) were proposed to analyze the dynamic response of an asymmetric single-tower cable-stayed bridge after a cable breakage. In the formulas, \(S_{0}\) is the initial response value of structural vibration, \(S_{\mathrm{dyn}}\) is the maximum response value in the vibration process and \(S_{\mathrm{s}}\) is the stress value when the vibration tends to be static.  
In order to quantify the maximum amplitude of structural stress and deformation during structural vibration, the dynamic influence coefficient (DIC) of a cable-stayed bridge was defined, and its value was greater than 1, which indicates that cable breakage is unfavorable to structural stress.  
\[\mathrm{DIC}=\left|\frac{S_{\mathrm{dyn}}}{S_{0}}\right| \tag{2}\]  
To study the influence of structural vibration caused by a cable breaking on cable stress, the definition of dynamic amplification factor by PTI [18] was introduced:  
\[\mathrm{DAF}_{1}=\left|\frac{S_{\mathrm{dyn}}-S_{0}}{S_{\mathrm{s}}-S_{0}}\right| \tag{3}\]  
To quantify the influence of structural vibration caused by the sudden breaking of cables on the stress and deformation of components other than cables and consider the static effect, the second dynamic amplification factor of cable-stayed bridges was introduced [19], and the formula is as follows:  
\[\mathrm{DAF}_{2}=\left|\frac{S_{\mathrm{dyn}}}{S_{\mathrm{s}}}\right| \tag{4}\]  
In order to quantify the influence of sudden cable breakage on the bearing capacity of cable-stayed bridge members, the definition of demand-capacity ratio is introduced, and it is considered that \(\mathrm{DCR}>1\) members were destroyed due to material yield, and the calculation formula is as follows:  
\[\mathrm{DCR}=\left|\frac{\sigma_{\mathrm{max}}}{\sigma_{\mathrm{y}}}\right| \tag{5}\]  
where \(\sigma_{\mathrm{max}}=S_{\mathrm{dyn}}\), which is the maximum response value of structural stress. \(\sigma_{\mathrm{y}}\) is the yield strength, which is the strength standard value of the material. In this paper, the tensile strength standard value of cable steel wire and the compressive strength standard value of concrete were selected [20].