## 7 Conclusions  
A comprehensive study was implemented to investigate the influence of aftershocks on residual displacement ratios by comparing the seismic response of bilinear oscillators under mainshock-only and mainshock-aftershock sequences. The effect of ground motion sequence characteristics and structural parameters is examined based on the statistical analysis of nonlinear time-history analysis results. Finally, the constant-ductility residual displacement ratio response spectra are developed for simple SDOF oscillators subjected to mainshock-aftershock sequences. The main conclusions are summarized as follows:  
1. The aftershock can significantly amplify the residual displacement ratios in a statistical mean sense. Conventional single-record-based models exhibit non-conservative predictions, underestimating residual displacement ratios by 3-42% across natural periods, thereby validating the necessity of sequence-dependent analysis. 2. Given the identical mainshock, seismic sequences with stronger aftershocks tend to result in larger residual displacement ratios. Such observation is particularly true for systems with low levels of post-yield stiffness ratio. 3. Sequences combining high-frequency mainshock with low-frequency aftershock (mean period \(T_{\rm m,ms}<T_{\rm m,as}\)) generate 5-40% higher residual displacement ratios than reverse configurations (i.e., \(T_{\rm m,ms}>T_{\rm m,as}\)) for elastic-perfectly plastic systems, highlighting spectral incompatibility risks. 4. For elastic-perfectly plastic systems, the larger the displacement ductility, the higher the residual displacement ratios. However, for hardening oscillators (\(r>0\)), the larger the displacement ductility, the lower the residual displacement ratios. 5. The proposed response spectra, which can reasonably account for the parametric dependencies of residual displacement ratios on post-yield stiffness ratios, displacement ductility, natural periods, and relative intensity ratios, provide a reliable and rapid tool for repairability assessment of structures against mainshock-aftershock sequences in terms of residual displacements. Based on the findings, the following practical recommendations are proposed to guide when aftershock effects should be prioritized in seismic design:  
1. Oscillators with softening behavior (\(r\leq 3\%\)) require explicit consideration of aftershocks, as absolute residual displacements escalate disproportionately. 2. Structures located in regions prone to sequences with strong aftershocks (\(\kappa\geq 1/2\)) require explicit sequence-based evaluation to avoid underestimation. 3. Regions prone to spectrally incompatible sequences (e.g., high-frequency mainshock followed by low-frequency aftershock) warrant sequence-specific assessments to mitigate residual displacement amplifications. Conceptualization, [PERSON] and J.G. ; methodology, all authors; data curation, [PERSON]; software, [PERSON] and H.H. ; formal analysis, [PERSON] and H.H. ; investigation, [PERSON] and J.W. ; validation, [PERSON] and H.H. ; writing--original draft, [PERSON]; writing--review and editing, all authors; visualization, [PERSON] All authors have read and agreed to the published version of the manuscript. This research was funded by the Natural Science Basic Research Program of Shaanxi Province (No. 2024 JC-YBQN-0434), the Natural Science Foundation of Shaanxi Provincial Department of Education (No. 23 JK0474), the Shaanxi Provincial Science Foundation for Outstanding Young Scholars (No. 2023 JC-QJ-47), the Shaanxi Province International Science and Technology Cooperation Program Project (No. 2024 GH-ZDXM-18), Shaanxi University Youth Innovation Team Construction Project (2023), the Scientific research project of young innovation team of the Shaanxi Provincial Department of Education (24 JP076), the Xi'an Science and Technology Plan Project (No. 23 GXFW0035), and the National Natural Science Foundation of China (Nos. 51978125 and 51978060). Not appliable. **Informed Consent Statement:** Not applicable.