###### Abstract  
This study aims to examine the seismic response of self-centering single-degree-of-freedom (SDOF) systems exhibiting flag-shaped hysteretic behavior, while considering soil-structure interaction, in contrast to prior research that predominantly addressed conventional hysteretic behavior and overlooked soil flexibility. The inelastic displacement ratio, residual displacement, and residual displacement ratio are used to analyze the seismic response of interacting structures. These structural response parameters are calculated based on the nonlinear dynamic analyses of SDOF systems subjected to 56 near-fault pulse-type ground motions. Analyses are conducted for varying values of ductility, energy dissipation coefficient, strain hardening ratio, aspect ratio, structural period, and normalized vibration period by pulse period of the record. New formulas to estimate the inelastic displacement ratio and residual displacement of self-centering SDOF systems with soil-structure interaction are developed based on a statistical analysis of the findings. p 2025 13 June 20  
Soil-structure interaction effects and variations of the inelastic displacement ratio with soil flexibility have been the topics of numerous research studies as well [10; 11; 12; 13; 14; 15; 16; 17]. After previous destructive earthquakes, it was demonstrated that not only peak displacements but also the structural system's residual displacement occurring at the end of the ground motion play a crucial role in determining whether the system needs strengthening, repairs, or rebuilding. In many design guidelines, including Vision 2000, FEMA356, and FEMA P-58 [18; 19; 20], the importance of taking into account residual displacements and residual drifts in seismic performance assessment is emphasized. As a result, estimating residual structural displacement demands is crucial for the assessment and repair of existing structures. Several studies have focused on inelastic displacement ratios and residual displacement ratios up to now. The first well-recognized study on residual displacements was conducted by [PERSON] and [PERSON] [21; 22]. [PERSON] and [PERSON] [23], [PERSON] and [PERSON] [24], and [PERSON] et al. [25] studied the residual displacement demands of SDOF systems, whereas [PERSON] and [PERSON] [26; 27; 28] carried out the most thorough studies on residual displacement ratios for both single-story and multistory structures. Since then, various parametric studies have been conducted by many researchers focused on residual displacements and normalized residual displacement ratios [29; 30; 31; 32; 33; 34]. Different normalization alternatives have been developed for residual displacement demands by various researchers. For example, yield displacement was used by [PERSON] and [PERSON] [23] and [PERSON] and [PERSON] [35] to normalize residual displacement, which was referred to as the residual displacement ductility. [PERSON] and [PERSON] [24] utilized the maximum possible residual displacement demand for the purpose of normalization. The ratio of residual displacement to maximum inelastic displacement was adapted by several researchers [5; 25; 26; 30]. In a similar way, [PERSON] and [PERSON] [27; 28] normalized residual displacement with elastic spectral displacement. Most of these above-mentioned studies generally considered conventional elastoplastic and/or degrading hysteretic behavior for structural response. The self-centering structure, which has been developed and proposed recently and has an excellent self-centering capability, is gaining more attention as a way to ensure the seismic resilience of a structure by limiting or reducing residual deformation in a system. Self-centering systems can improve the seismic resilience of a structure and decrease repair costs due to their capacity for energy dissipation and smaller residual deformation. The self-centering system can be thought of as consisting of two parts put in parallel: the self-centering part combined with the energy dissipation part [36].