## 1 Introduction
Collapse margin ratio is selected as the normalization parameter because it is the basis for the code-defined maximum considered earthquake at most sites [28]. Critical demand-to-capacity ratio can take the uncertainty induced by the ground motion as well as the uncertainty associated with the definition of capacity into consideration [29]. In order to explore the overall deformation and collapse resistance of the structure under the action of main aftershocks, previous research studies considered the maximum inter-story drift ratio as the main index to define the limit state of the structure [30; 31; 32; 33], and analyzed the vulnerability of wooden structures, concrete structures, and steel frame structures with masonry filling under main aftershocks. [PERSON] et al. [34] considered the roof drift of the structure as the structural response index, and used three different statistical methods (IDA method, maximum likelihood fit for truncated IDA, and maximum likelihood estimation (MLE) method) to process IDA data, completed the vulnerability analysis under the action of main shock and main aftershock, and compared the difference in structural response. In addition, [PERSON] et al. [35] considered the Park-Ang damage index as the structural response parameter to study the impact of aftershocks on the damage accumulation of frame step-terrace structures. [PERSON] et al. [36] considered the maximum compressive strain of concrete as the measuring standard, and the incremental dynamic analysis of the reinforced concrete nuclear island plant under the aftershock was carried out to discuss the impact of aftershock on the vulnerability of the plant. Several scholars considered two indices to analyze the structures based on the IDA method. [PERSON] and [PERSON] [37] selected the maximum inter-story drift ratio as the index to carry out the IDA for the main and aftershocks of the moment-resisting frame, before and after the addition of the super elastic viscous damper. In order to explore the recovery performance of the two types of structures, the residual inter-story drift ratio was selected as the index to carry out the IDA of the frame damaged by the aftershock. [PERSON] et al. [38] selected the maximum inter-story drift ratio and Park-Ang damage index to analyze the vulnerability of a concrete frame structure under main aftershock based on the IDA method. The authors found that the probability of reaching the same performance level of the structure is different when the two indices are selected to analyze the structure. [PERSON] et al. [39] selected the residual inter-story drift ratio and the maximum inter-story drift ratio as the structural response indices. Based on the IDA, the vulnerability analysis of the moment-resisting frame before and after the addition of the shape memory alloy frame, under the action of the main aftershock, was carried out. Among them, the excess value of the residual inter-story drift ratio was directly calculated by the maximum transient story drift and story drift at yield. In the IDA quantile calculation results, although the median of the ground motion intensity corresponding to the residual inter-story drift ratio and the maximum inter-story drift ratio was not much different, the traditional vulnerability analysis method cannot be used to calculate the subsequent excess probability without assuming the correlation between the two sets of data. In conclusion, existing studies on IDA using two indices mainly focused on the impact of aftershocks on the structure or the change of structural performance corresponding to different indices after the change of building system, and did not analyze the correlation of the two indices. Therefore, based on the limitations of previous research studies, a double-index IDA method is presented considering the correlation between two indices. The maximum inter-story drift ratio and the residual inter-story drift ratio are selected as two indices, which can measure the seismic performance and the repairability of structures, respectively.