## 5 Statistical Analysis of Residual Displacement Ratios  
### Effect of Ground Motion Duration  
The duration of earthquake ground motion is a key parameter for characterizing seismic records, with its impact on the inelastic response of structures having been systematically quantified through prior studies [53; 54]. However, the specific influence of ground motion duration--particularly in sequential earthquakes--on the residual displacement response of structures remains insufficiently explored, despite its critical implications for post-earthquake functionality and repair strategies.  
In this section, the concept of significant duration, as defined by [PERSON] and [PERSON] [55], is adopted to quantify the time interval of strong shaking during an earthquake. Mathematically, it is determined by first normalizing the cumulative squared acceleration of the ground motion _a(t)_:  
\[F(t)=\frac{\int_{0}^{t}a(\tau)^{2}d\tau}{\int_{0}^{t_{\text{end}}}a(\tau)^{2}d\tau} \tag{8}\]  
where \(F(t)\) represents the fraction of total energy released up to time \(t\).  
The 5-95% significant duration \(D_{\text{s}}\) is the time interval between the first arrival at 5% (\(t_{5}\)) and 95% (\(t_{95}\)) of the normalized cumulative squared acceleration \(F(t)\), that is written as follows:  
\[D_{\text{s}}=t_{95}-t_{5} \tag{9}\]  
According to the significant duration of mainshock (\(D_{\text{s,ms}}\)) and aftershock (\(D_{\text{s,as}}\)) determined in terms of Equations (8) and (9), all the selected mainshock-aftershock sequences are divided into four subsets, including: (a) long mainshock and long aftershock group (\(D_{\text{s,ms}}\geq 8\) s & \(D_{\text{s,as}}\geq 8\) s), abbreviated by Long-Long, containing 50 sequences; (b) long mainshock and short aftershock group (\(D_{\text{s,ms}}\geq 8\) s & \(D_{\text{s,as}}<8\) s), abbreviated by Long-Short, containing 47 sequences; (c) short mainshock and long aftershock group (\(D_{\text{s,ms}}<8\) s & \(D_{\text{s,as}}\geq 8\) s), abbreviated by Short-Long, containing six sequences; (d) short mainshock and short aftershock group (\(D_{\text{s,ms}}<8\) s & \(D_{\text{s,as}}<8\) s), abbreviated by Short-Short, containing 41 sequences. To ensure the validity of statistical results, the Short-Short group with a sample size that is too small is excluded from the analysis.  
Figure 10 shows the residual displacement ratios \(C_{\text{r,seq}}\) for three groups of mainshock-aftershock sequences. It is seen that the spectral shape as well as the ordinate of \(C_{\text{r,seq}}\) is not significantly influenced by the sequence duration for post-yield stiffness ratios larger than zero. Significant duration seems to affect the amplitude of \(C_{\text{r,seq}}\) for zero post-yield stiffness ratio, particularly for natural periods between 0.5 and 2.5 s. As can be seen, the longer the duration of mainshocks and aftershocks, the larger the residual displacement ratios. This observation, however, does not necessarily apply to other period ranges. For instance, for \(r=0.00\) and \(T>2.5\) s, the amplitude of \(C_{\text{r,seq}}\) computed from the Short-Short group is larger than that from the Long-Long group. In general, it is concluded that the significant duration has a slighter effect on hardening structures than on elastic-perfectly plastic structures.