## 6 Constant-Ductility Residual Displacement Ratio Spectra  
### Establishment of Response Spectra  
To enable practical estimation of residual displacement demands under mainshock-aftershock sequences, this study proposes a residual displacement ratio response spectrum. As demonstrated by the parametric analysis, the sequence-induced residual displacement ratio, \(C_{\text{r,seq}}\), largely depends on the post-yield stiffness ratio \(r\), displacement ductility \(\mu\), relative intensity ratio \(\kappa\), and natural period \(T\). To mathematically characterize this  
Figure 14: Effect of displacement ductility on residual displacement ratios: (**a**) \(r=0.00\); (**b**) \(r=0.03\); and (**c**) \(r=0.10\). Figure 13: Effect of post-yield stiffness on residual displacement ratios. multivariate dependence while considering parameter coupling effects, a basic functional form, as presented in the following Equation (10), is adopted:  
\[C_{r,\text{seq}}=(\mu-1)(1-r)Q(r,\mu,\kappa,T) \tag{10}\]  
where the introduction of terms (\(\mu-1\)) and (\(1-r\)) is to satisfy the fundamental condition of \(C_{r,\text{seq}}(\mu=1)=0\) and \(C_{r,\text{seq}}(r=1)=0\), since the elastic oscillators do not sustain residual displacements after earthquake excitation. The term \(Q(r,\mu,\kappa,T)\) represents the adjustment coefficient related to \(r\), \(\mu\), \(\kappa\), and \(T\). To capture the dependence of \(Q(r,\mu,\kappa,T)\) on these parameters, the following functional form, as shown in Equation (11), is selected after testing many expressions. The test is conducted with the help of the nonlinear curve fit module in the 1 st Opt software (version 15.0), where the Levenberg-Marquardt algorithm is utilized to estimate the regression coefficients. Note that the adjusted determination coefficient (Adj. \(R^{2}\)) is adopted as the preliminary criterion to evaluate the goodness-of-fit of the different regression equations. The equation that maximizes the adjusted determination coefficient and possesses the simplified function form is considered a good candidate for the best regression model. \[Q(r,\mu,\kappa,T)=\lambda_{1}\mu^{\lambda_{2}}\Big{[}(1+2r)^{T}+\frac{\lambda_{ 3}}{\sqrt{T}}\Big{]} \tag{11}\]  
where \(\lambda_{1}\), \(  
between given points, then the parameters \(\lambda_{1}\), \(\lambda_{2}\), and \(\lambda_{3}\) can be determined approximately by the linear interpolation. As an alternative, the discrete data in tables can be used to develop the appropriate functional forms of \(\lambda_{1}\), \(\lambda_{2}\), and \(\lambda_{3}\) via regression analysis, which can facilitate the practical application. To this end, the discrete estimates \(\lambda_{1}\), \(\lambda_{2}\), and \(\lambda_{3}\) are plotted against \(r\) and \(\kappa\) to identify the potential trend. Figure 15 shows the tendency of \(\lambda_{1}\), \(\lambda_{2}\), and \(\lambda_{3}\) with respect to \(\kappa\) for different levels of \(r\). It can be observed that the dependence of \(\lambda_{1}\), \(\lambda_{2}\), and \(\lambda_{3}\) on \(r\) and \(\kappa\) may be characterized by the following equations:  
\[\lambda_{1}=0.5873+4.0321\;r-53.6939\;r^{2}+0.1609\;\kappa^{10r-1.2635} \tag{12}\]  
\[\lambda_{2}=-2.3448+1.2790\;\mathrm{exp}(-16.9347\;r)\frac{\kappa^{2}}{\kappa ^{2}+0.0534} \tag{13}\]  
\[\lambda_{3}=0.0656+0.5632\;r+16.3878\;r^{2}+\frac{0.0511+r}{1+\left(6.9219 \kappa-7.7095\right)^{2}} \tag{14}\]  
The fitted curves of \(\lambda_{1}\), \(\lambda_{2}\), and \(\lambda_{3}\) together with the corresponding adjusted determination coefficients are also presented in Figure 15. As can be seen, the proposed models can provide accurate estimates of \(\lambda_{1}\), \(\lambda_{2}\), and \(\lambda_{3}\).