## 4 Conclusions  
In Section 2, a methodology for assessing the seismic vulnerability of mainshock-aftershock sequences was proposed, with a particular focus on the influence of the mainshock-aftershock time interval. In Section 3, a case study of seismic vulnerability  
Figure 12: Functional function curves of the mainshock-aftershock structure at different recovery times.  
assessment for a reinforced concrete frame using the proposed mainshock-aftershock framework was presented. Finally, Section 4 summarized the conclusions of this study.  
Taking into account the incremental structural damage, this study uses the finite element model to calculate the structural vulnerability curve. The calculation method for normalizing the economic loss of the building structure and the analysis approach for the seismic resilience index in the building seismic design code are enhanced. Given that aftershocks can induce secondary damage to the structure, this research analyzes the seismic resilience of the structure under the mainshock-aftershock and compares it with the seismic resilience during the main earthquake alone. The innovation points and conclusions of this study are summarized as follows:  
1. The calculation method for the seismic resilience index of the structure under the mainshock-aftershock was proposed. This method takes into account the loss of structural function and incorporates boundary conditions. In this study, seismic resilience was analyzed based on three different functional recovery models: linear, triangular, and exponential functions. Moreover, three random factors were taken into account to analyze the structure under the mainshock-aftershock: the strength of the mainshock-aftershock, the time interval between the main earthquake and the aftershock, and the incremental damage. The results reveal that the seismic resilience index during the mainshock-aftershock is lower compared to that of a single earthquake event;
2. The seismic resilience of reinforced concrete frames was analyzed using the Monte Carlo method. Time interval samples were obtained through random sampling based on the model for mainshock-aftershock time intervals. The seismic resilience of structures under the main earthquake alone and the mainshock-aftershocks was calculated considering multiple, fortified, and rare earthquake intensities. The study findings revealed that if aftershocks occur before the complete repair of structural damage caused by the mainshock, a shorter mainshock-aftershock time interval results in increased damage and decreased seismic resilience of the reinforced concrete frame.  
Future research developments entail extending the proposed method to other types of structures such as bridges and enhancing the accuracy of parameter values utilized in the analysis.  
Conceptualization, P.L. and D.W.; Data curation, X.L.; Formal analysis, P.L. and X.W.; Writing-original draft, P.L., X.W. and D.W.; Writing-review &editing, P.L. and D.W. All authors have read and agreed to the published version of the manuscript.  
This research was funded by [Natural Science Foundation of Hebei Province] grant number [E20203103] and [Science and Technology Project of the Hebei Education Department] grant number [BJ2021043].  
Not applicable.  
All the data utilized during the study are included in the submitted article.  
Author [PERSON] was employed by the company Heilongjiang Transportation Investment Group Co., Ltd.. Author [PERSON] was employed by the company Beijing Shunjie Xinghong Engineering Design Consulting Co., Ltd.. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.