## 4 Conclusions  
This study investigates the effects of aftershocks on seismic responses of unidirectional eccentric structures and bidirectional eccentric structures with different eccentricities subjected to unidirectional and bidirectional loadings. In the investigation, three-dimensional models of unidirectional and bidirectional eccentric structures with eccentricities of 0%, 5%, 10%, 15%, 20%, 25%, and 30% are established by using the OpenSEES software. A representative mainshock-aftershock sequence is selected and scaled according to the performance level indicated by the IDR. Seismic capacities of the symmetric structural models and eccentric structural models are estimated by a static pushover analysis and a nonlinear time history analysis. Seismic response results under a total of six types of cases (called UUM, UUS UBM, UBS, BBM, and BBS) are discussed. The main conclusions are as follows:  
1. For unidirectional eccentric structures, the displacement response during the unidirectional mainshock increases obviously with an improvement in eccentricity, and the displacement growth rate increases first and then decreases. The structural response increases further under the unidirectional MSAS. The peak displacement and maximum inter-story drift ratio during the aftershock can reach up to 1.4 times and 1.5 times those of the mainshock, respectively, when the structure experiences a mainshock of 0.15 g. When the amplitude of the mainshock is low, the aftershock shows a more significant influence on the structural response. 2. For unidirectional eccentric structures, bidirectional horizontal loadings are more likely to cause damage to the structures and have a more adverse effect on the structural displacement responses in the main direction. The symmetric structure is most affected by the aftershock under the bidirectional MSAS. The peak displacement and maximum inter-story drift ratio during the aftershock can reach up to 1.35 times and 1.3 times those of the mainshock, respectively, when the structure experiences a mainshock of 0.15 g. Compared with the structural response under bidirectional loading, the structural response under unidirectional loading is more sensitive to the intensity of aftershock ground motions. 3. For bidirectional eccentric structures, the peak displacements and maximum inter-story drift ratios of the structures tend to ascend with an increase in eccentricity. Compared with unidirectional eccentric structures, the responses of bidirectional eccentric structures are more complex under the same bidirectional horizontal earth quakes, and the aftershocks have a more significant influence on the responses of bidirectional eccentric structures. When the structure experienced a mainshock of 0.15 g, the peak displacement during the aftershock can reach 1.5 times that of the mainshock, and the maximum inter-story drift ratio during the aftershock can reach 1.4 times that of the mainshock. The non-synchronous nonlinear behaviors of different columns of the perimeter frames may have an influence on the change in eccentricities, which need to have further investigations. Different nonlinear behaviors of columns could produce different increased trends of structural demands (i.e., torsional angle and inter-story drift ratio). Investigation, P.S. ; methodology, P.S. and W.W.; project administration, W.W.; supervision, W.W.; writing--original draft preparation, P.S. and S.Z. ; writing--review and editing, P.S., S.Z. and W.W. All authors have read and agreed to the published version of the manuscript. This investigation is supported by the National Natural Science Foundation of China (Nos. 52222811, U1939210, and 52178469), Natural Science Foundation of Heilongjiang Province (No. LH2021E075), the Fundamental Research Funds for the Central Universities (No. HIT.BRET.2022010). These supports are greatly appreciated. No new data were created or analyzed in this study. Data sharing is not applicable to this article. The authors declare no conflict of interest.