## 3 Numerical Analysis
### Incremental Dynamic Analysis
#### 3.3.3 Using Two Inter-Story Drift Ratios as DMs  
To further explore the relationship between the two inter-story drift ratios and earthquake intensity, and to analyze the correlation between the two indices, this section uses the above two types of inter-story drift ratios as the independent variables and the \(S_{a}\)(\(T_{1}\),5%) as the dependent variable, according to the data distribution under different earthquake intensity, to explore the influence of double-index model for vulnerability analysis. Considering the randomness of the inter-story drift ratio of the steel frame structure under the seismic action, it is assumed that the distribution of the maximum inter-story drift ratio and the residual inter-story drift ratio is a 2-D lognormal distribution under the condition that the intensity of ground motion is determined. Therefore, the probability density function can be expressed as follows:  
\[f(x,y)=\left(2\pi\sigma_{1}\sigma_{2}\sqrt{1-\rho^{2}}\right)^{-1}\exp\left[- \frac{1}{2(1-\rho^{2})}\left(\frac{(x-\mu_{1})^{2}}{\sigma_{1}^{2}}-\frac{2 \rho(x-\mu_{1})(y-\mu_{2})}{\sigma_{1}\sigma_{2}}+\frac{(y-\mu_{2})^{2}}{ \sigma_{2}^{2}}\right)\right] \tag{8}\]  
where \(\sigma_{1}\) and \(\sigma_{2}\) represent the standard deviations of data \(x\) and \(y\), respectively; \(\mu_{1}\) and \(\mu_{2}\) represent the mean values of data \(x\) and \(y\), respectively; and \(\rho\) stands for the correlation coefficient with a value range of [-1, 1]. The calculation method is as follows:  
\[\rho=\frac{Cov(x,y)}{\sigma_{1}\sigma_{2}} \tag{9}\]  
where \(Cov(x,y)\) represents the covariance of data \(x\) and \(y\). The data when \(S_{a}\)(\(T_{1}\),5%) = 2.0 g were analyzed as an example, and the joint probability density diagram was obtained, as shown in Figures 13 and 14. Figure 13: Joint probability density function of structural response under the main shock when \(S_{a}\)(\(T_{1}\),5%) = 2.0 g: (**a**) joint probability density function diagram; and (**b**) top view of the joint probability density function. It can be seen from Figures 13 and 14 that the peak value of the joint probability distribution function under the action of the main shock is significantly greater than that under the action of the main aftershock, indicating that the aftershock increases the discrete degree of multiple structural responses. It is found that the maximum inter-story drift ratio did not significantly change, but the residual inter-story drift ratio increased slightly under the aftershock. Comparing the top view, it is found that under the action of the main shock, the probability density function is relatively uniform. However, under the action of the main aftershock, the graphics significantly elongated, and the variation range of the residual inter-story drift ratio is more significant. It indicates that the structure's control over the maximum inter-story drift ratio and residual inter-story drift ratio will be weakened, and its control over the residual inter-story drift ratio will be weaker under the action of the main aftershocks. The results show that under the action of the main shock and the main aftershock, there is an obvious positive linear correlation between the maximum inter-story drift ratio and the residual inter-story drift ratio. In other words, the residual inter-story drift ratio is largely affected by the maximum inter-story drift ratio during an earthquake. Therefore, using the two-dimensional lognormal distribution to analyze the structural vulnerability can improve the calculation accuracy, while using the one-dimensional lognormal distribution of a single index to analyze separately will increase the first type error probability. In addition, in the normal joint distribution, the transcendence probability calculation results for the maximum inter-story drift ratio and the residual inter-story drift ratio will be accurate, rather than the calculation of probability multiplication using the two- and one-dimensional normal distributions alone, which improves the accuracy of the calculation.