## 1 Introduction  
Historical data show that aftershocks often occur after an earthquake [1], and prominent examples include the Tangshan Earthquake in 1976 [2], the Northridge Earthquake in 1994 [3], the Wenchuan Earthquake in 2008 [4], the Tohoku Earthquake [5] in 2011, and the Christchurch Earthquake in 2011 [6]. Among them, the Tohoku earthquake caused a direct economic loss of 200 billion dollars [7]. After the Christchurch earthquake, more than 100,000 houses were damaged, and 10,000 houses needed to be demolished. The total cost of reconstruction estimated by the New Zealand government reached 40 billion dollars [8]. For buildings, when structural members undergo the plastic deformation stage under the action of the main shock and later experience the effect of aftershocks again, the damage accumulates, and the possibility of building collapse increases significantly. Therefore, in the study of the seismic performance of structures, it is of great practical significance to consider the collapse performance of a structure under the effect of aftershocks. In order to explore the influence of main aftershocks on seismic performance and seismic resilience of different types of structures, incremental dynamic analysis (IDA), vulnerability analysis, response surface method, Monte Carlo simulations, random vibration approaches, and adaptive support vector regression models are used [9, 10, 11, 12]. The methods of IDA and vulnerability analysis are used in this paper. The IDA was first proposed by [PERSON] [13] in 1977, and it is based on the dynamic elastoplastic time history analysis. Later, and after a research study by [PERSON] et al. [14], the method has been widely used in seismic capacity analysis and overall collapse performance evaluation of frames. In recent years, scholars have carried out various research on IDA. In terms of improving computational efficiency, a simplified modal incremental dynamic analysis (MIDA) method based on modal pushover analysis (MPA) [15; 16], a simplified IDA method based on equivalent dual degree of freedom (EDDOF) [17], a simplified IDA method based on fast nonlinear analysis (FNA) [18] and an IDA method based on force analogy method (FAM) [19] were proposed. In terms of evaluation and application of IDA, [PERSON] et al. [20] applied the IDA method to the performance evaluation of high-rise hybrid structures, and considered that it is reasonable to use the inter-story drift ratios as the response parameter. [PERSON] et al. [21] used the IDA method to evaluate the seismic performance of a steel frame structure, and found that the IDA curve obtained by Sa as the IM parameter had better convergence. [PERSON] [22] demonstrated the advantages of using IDA in the sensitivity analysis of structures to a variety of parameters. As far as research objects are concerned, IDA has been widely used to analyze various structures such as buildings, stations, docks, and railways [23; 24; 25]. In the IDA under the action of main aftershocks, structural deformation, structural energy consumption, and local structural damage are generally used as a basis for evaluation. Generally, one of the above indices is used to evaluate and judge the seismic performance of a structure. The maximum inter-story drift ratio can accurately reflect the damage degree and the performance stage of the structure, which is directly related to the collapse resistance of the structure and the damage degree of the component [26]. The Park-Ang damage index takes into account the energy dissipation capacity of the structure, which can reflect the cumulative damage of buildings under aftershocks, and is mostly used in concrete structures. The maximum compressive strain is an important criterion for judging the local damage of structures. [PERSON] et al. [27] proposed median column drift ratio (CDR) as the criterion for judging whether the column in the structure can have shear failure and vertical bearing capacity loss, and ultimate roof drift ratio (RDR\({}_{\text{tilt}}\)) as the basis for measuring structural ductility.