# Study on Structural Robustness of Isolated Structure Based on Seismic Response
## 2 Quantitative Evaluation Method of Structural Collapse Robustness
### Component Vulnerability Coefficient  
The existing studies are mainly aimed at studying the vulnerability of the entire structure under severe earthquakes. Structural vulnerability not only depends on the layout of the structure, the performance of building materials, but also depends on the strength of the key components of the structure.  
Compared with structural vulnerability, component vulnerability is used to reflect the probability of failure of components under external loadings. For a single member, if the load effect caused by a certain load is significantly less than the resistance of the component, the damage risk of the component under this loading is relatively low, the component vulnerability coefficient is also small; on the other side, this load increases the risk of damage to the component is large, and the corresponding vulnerability coefficient is greater.  
In the process of structural collapse, the axial force, shear force and bending moment load also appear together. Considering that the proposed method is too sensitive to the variation of the variables, the corresponding improvements are made. The damage coefficient \(\varphi_{i}\) is calculated as follows:  
\[\varphi_{i}=p\bigg{(}\frac{V}{V_{0}}\bigg{)}+p\bigg{(}\frac{M}{M_{0}}\bigg{)}+ q\bigg{(}\frac{N}{N_{0}}\bigg{)} \tag{6}\]  
where each coefficient shown in Equation (7) can be calculated as:  
\[\left\{\begin{array}{c}p=-\frac{4k}{(k-1)^{2}},q=\frac{4k^{2}}{(k-1)^{2}}\\ k=\frac{h+m-1}{n^{2}-n}\\ n=\frac{N_{b}}{N_{0}},m=\frac{M_{b}}{M_{0}}\end{array}\right. \tag{7}\]  
where \(V\), \(M\) and \(N\) are the shear force, bending moment and axial force respectively.  
\(M_{0}\) and \(N_{0}\) are the ultimate load-carrying capacity of the normal section and the ultimate load-carrying capacity of the axial compression under pure pressure, respectively, when the element is purely bent. The calculation method is shown in Equation (8):  
\[\left\{\begin{array}{c}M_{0}=f_{y}A_{s}(h_{0}-a_{s}^{\prime})\\ N_{0}=0.9\varphi\Big{(}f_{y}^{\prime}A_{s}^{\prime}+f_{c}A\Big{)}\end{array}\right. \tag{8}\]  
\(V_{0}\) is the shearing capacity of oblique section of pure shears, and the calculation method is based on the relationship between shear force and external load. The concrete calculation is shown in Equations (9) and (10). In general, the shear capacity is calculated according to Equation (9). When the members are subjected to concentrated loads, or subjected to complex loads but the shearing force caused by the concentrated load at the bearing portion is more than 75% of the total shear value, the shear bearing capacity of the member can be calculated according to Equation (10):  
\[V_{0}=0.7f_{b}bh_{0}+1.25f_{yv}\frac{A_{sv}}{s}h_{0} \tag{9}\]  
\[V_{0}=\frac{1.75}{\lambda+1.0}f_{b}bh_{0}+1.0f_{yv}\frac{A_{sv}}{s}h_{0} \tag{10}\]  
In the bottom frame, when a frame column loses its bearing capacity, the adjacent frame column is changed from the original axial compression state to the eccentric compression state. When the longitudinal reinforcement of the tension zone is produced, the calculation methods of ultimate load-carrying capacity \(M_{b}\) and ultimate load-carrying capacity \(M_{b}\) are shown in Equations (11) and (12), respectively:  
\[N_{b}=\alpha_{1}f_{c}bh_{0}\xi_{b} \tag{11}\]  
\[M_{b}=f_{y}A_{s}\big{(}h_{0}-a_{s}^{\prime}\big{)}-N_{b}\left(\frac{h}{2}-a_{ s}\right)+\alpha_{1}f_{c}bh_{0}^{2}\xi_{b}\Big{(}1-\frac{\xi_{b}}{2}\Big{)} \tag{12}\]  
where \(f_{c}\) and \(f_{t}\) are the axial compressive strength and the axial tensile strength, respectively; \(f_{y}\) and \(f_{yv}\) are the tensile strength of longitudinal reinforcement and stirrup; \(A_{yv}\) is the total cross-sectional area of stirrups in the same section; \(h_{0}\) is the cross-section effective height; \(b\) is the section width; \(\xi_{b}\) is the relative height of the compression zone of the section.