# Study on Structural Robustness of Isolated Structure Based on Seismic Response
## 2 Quantitative Evaluation Method of Structural Collapse Robustness
### Component Importance Coefficient  
The component importance coefficient is used to reflect the degree of influence of the loss of bearing capacity of individual members on the bearing capacity of the original structure under extreme conditions, and the determination of component importance coefficient is the basis for calculating the structural robustness coefficient.  
Without consideration of horizontal loads such as wind and earthquake, this study considers the coupling effects of various loads during structural collapse. Therefore, the degree of influence of the failure of the column under the coupling action and the failure of the rubber isolation bearing on the total energy distribution of the structure is taken as the evaluation index of the importance coefficient. The concrete calculation method is expressed as:  
\[\gamma_{i}=\frac{\alpha_{i}}{max\{\alpha_{i}\}} \tag{1}\]  
\[\alpha_{i}=\Delta E_{i}=\frac{1}{2}(r_{i}t^{\prime})^{T}K_{i}(r_{i}) \tag{2}\]  
where \(\gamma_{i}\) is the component importance coefficient; \(\alpha_{i}\) Which is used to show the effect of the failure of \(i\)-th frame column on the total strain energy of the structure; \(K_{i}\) is the stiffness matrix of the \(i\)-th frame column; \(r_{i}\) and \(r_{i}t^{\prime}\) are the displacement vectors of the \(i\)-th frame column before and after initial failure.  
The Element Stiffness Matrix of the \(i\)-th Bottom Column:  
\[K_{i}=\left[\begin{array}{ccccc}\frac{12 EI}{l^{3}}&0&-\frac{6 EI}{l^{2}}&- \frac{12 EI}{l^{3}}&0&-\frac{6 EI}{l^{2}}\\ 0&\frac{EA}{l}&0&0&-\frac{EA}{l}&0\\ -\frac{6 EI}{l^{2}}&0&\frac{4 EI}{l^{2}}&\frac{6 EI}{l^{2}}&0&\frac{2 EI}{l}\\ -\frac{12 EI}{l^{3}}&0&\frac{6 EI}{l^{2}}&\frac{12 EI}{l^{3}}&0&\frac{6 EI}{l^{2}} \\ 0&-\frac{EA}{l}&0&0&\frac{EA}{l}&0\\ -\frac{6 EI}{l^{2}}&0&\frac{2 EI}{l}&\frac{6 EI}{l^{2}}&0&\frac{4 EI}{l}\\ \end{array}\right] \tag{3}\]  
Since the bottom end of the column is connected with the foundation, the displacements generated at the bottom end of the column before and after the failure are too small to be compared with the displacement of the top end, and so the displacement vector of the bottom end node can be written as:  
\[r_{i}t^{\prime}=\left\{\begin{array}{c}\overline{u}_{i}t\\ \overline{v}_{i}t\\ \overline{\partial}_{i}t\\ 0\\ 0\\ 0\end{array}\right\}^{T}r_{i}=\left\{\begin{array}{c}\overline{u}_{i}\\ \overline{v}_{i}\\ \overline{\partial}_{i}\\ 0\\ 0\\ 0\end{array}\right\} \tag{4}\]  
Substituting Equation (4) into Equation (2), a formula to calculate the contribution of the base column to the total energy distribution of the structure is obtained:  
\[\alpha_{i}=\frac{1}{2}\left\{\begin{array}{c}\overline{u}_{i}t\\ \overline{v}_{i}t\\ \overline{\partial}_{i}t^{\prime}\end{array}\right\}^{T}\left[\begin{array}[ ]{ccc}\frac{12 EI}{l^{3}}&0&-\frac{6 EI}{l^{2}}\\ 0&\frac{EA}{l}&0\\ -\frac{6 EI}{l^{2}}&0&\frac{4 EI}{l}\\ \end{array}\right]\left\{\begin{array}{c}\overline{u}_{i}\\ \overline{v}_{i}\\ \overline{\partial}_{i}\\ \overline{\partial}_{i}\end{array}\right\} \tag{5}\]