## 4 Effect of Aftershocks on Residual Displacements  
It is worth noting that the residual-to-peak-inelastic displacement ratio as defined in Equation (4) has been extensively adopted for estimating residual displacements under single earthquake records [8; 10; 11; 12; 17; 18; 19; 21], and several empirical equations and response spectra have also been developed and incorporated into design specifications as guidelines for structural seismic design. However, it is not clear whether such equations or response spectra can be used to accurately estimate the residual displacement of structures when subjected to mainshock-aftershock sequences. To this end, it is necessary to investigate the effect of aftershocks on residual displacement demands.  
With or without considering the aftershock excitation, the residual displacement ratios can be computed from Equation (3) (i.e., \(C_{\mathrm{resq}}\)) and Equation (4) (i.e., \(C_{\mathrm{r,ms}}\)), respectively. Then the effect of aftershocks can be readily examined by comparing \(C_{\mathrm{r,ms}}\) with \(C_{\mathrm{r,seq}}\). As an illustration, \(144\)\(C_{\mathrm{r,ms}}\) and \(C_{\mathrm{r,seq}}\) samples are computed for a typical elastoplastic system  
Figure 7: Bilinear hysteresis model used in this study.  
(i.e., \(T=0.5\) s, \(\mu=4\)) subjected to mainshock records with or without aftershock ground motions shown in Appendix A. Then the cumulative frequency as well as the boxplot of 144 \(C_{\rm r,ms}\) and \(C_{\rm r,seq}\) samples is presented in Figure 8. A comparison between \(C_{\rm r,ms}\) (blue dot) and \(C_{\rm rseq}\) (red dot) cumulative frequency plots shows that the \(C_{\rm r,seq}\) values are larger than the \(C_{\rm r,ms}\) values across all quantiles. Particularly, the difference between the first quartile (\(Q1\)) of \(C_{\rm r,ms}\) and \(C_{\rm r,seq}\) samples is up to 58%. This implies that the aftershock can significantly increase the residual displacement ratios.  
To have a quantitative insight into the effect of aftershocks on residual displacement ratios, the ratios of the sample mean \(C_{\rm r,seq}\) to \(C_{\rm r,ms}\) (i.e., \(C_{\rm r,seq}/C_{\rm r,ms}\)) are shown in Figure 9, in which Figure 9 corresponds to different levels of displacement ductility, Figure 9 to different aftershock intensity, and Figure 9 to different levels of post-yield stiffness ratio.  
It is seen that the ratios \(C_{\rm r,seq}/C_{\rm r,ms}\) are always larger than the unit for all displacement ductility, relative intensity ratios, and post-yield stiffness ratios considered in this study, which thus implies that the aftershock can amplify the residual displacement ratios in the mean sense. Another observation, worthy to be noted, is that the effect of aftershocks largely depends on relative intensity ratios and post-yield stiffness ratios, especially for intense aftershock excitation and oscillators with high post-yield stiffness. For instance, for the structure with \(r=0.10\), the aftershock can amplify the residual displacement ratios by at most 72%.  
Figure 8: Cumulative frequency of residual displacement ratios with or without considering aftershock excitation.  
Figure 9: Effect of aftershocks on residual displacement ratios: (a) different levels of \(\mu\); (b) different levels of \(\kappa\); and (c) different levels of \(r\).  
In view of the significant effect of aftershocks on residual displacement ratios, it is obvious that conventional single-earthquake-record residual displacement spectra would underestimate demands for the situation of seismic sequence excitation. Therefore, a more reliable model for residual displacement estimation considering aftershocks needs to be developed. For this purpose, the residual displacement ratios of structures undergoing mainshock-aftershock sequences should be first statistically analyzed to investigate the influence of sequence characteristics and structural parameters.