## 5 Summary and Conclusions  
In this study, the seismic response of self-centering single-degree-of-freedom systems with flag-shaped hysteretic behavior considering soil-structure interaction is investigated under near-fault pulse-type ground motion records. The present study aims to develop new functional forms to estimate residual displacement demands and inelastic displacement ratios for self-centering SSI systems subjected to near-field pulse-like ground motions. To this purpose, extensive statistical analyses are carried out to examine the effects of ductility, strain hardening ratio, energy dissipation ratio, and aspect ratio. The following conclusions can be drawn from the results of this study.  
Pulse period and pulse period normalization are effective parameters for structural response. When the analyses are conducted based on structural period (T), there is an obvious difference between inelastic displacement ratios for almost every value of ductility. However, when C\({}_{\mu}\) spectra are analyzed with respect to the normalized structural period (T/T\({}_{\rm p}\)), the inelastic displacement values get close to each other and equal to unity, which is known as the equal displacement rule. Furthermore, C\({}_{\mu}\) spectra are strongly affected by ductility, especially for the short-medium-period region, and increase with an increase in ductility. However, for the long-period region, C\({}_{\mu}\) spectra are less sensitive to ductility variation. Residual displacement demand is significantly affected by pulse period normalization, since the values obtained by period normalization are much greater than the ones obtained by structural period. The post-yield stiffness ratio has a negligible effect on the considered structural response parameters. The energy dissipation ratio has a reducing effect on the inelastic displacement ratio for both fixed-base and interacting cases. Residual displacement demand decreases for increasing values of \(\beta\), but for \(\beta=1\), which corresponds to a bilinear elastoplastic system, residual displacement demand is extremely high because as the energy dissipation ratio increases, the self-centering capacity becomes weaker. For soil-structure interacting systems, as the aspect ratio increases, inelastic displacement spectra tend to decrease, especially for short-period regions. Two new functional forms to estimate the inelastic displacement ratio and the residual displacement of interacting systems are developed based on nonlinear regression analysis. The proposed equations are functions of the normalized period (T/T\({}_{\rm p}\)), ductility (\(\mu\)), energy dissipation ratio (\(\beta\)), post-yield stiffness ratio (\(\alpha\)), and aspect ratio (h/r). It is demonstrated that the proposed formulas can accurately predict the ductility-based inelastic displacement spectra and the residual displacement of self-centering systems with soil-structure interaction, both for the considered and verification databases.  
This research received no external funding.  
Not applicable.  
The data sets generated during and/or analyzed during the current study are available from the corresponding author upon reasonable request.  
The author declares no conflict of interest.