## 2 Materials and Methods
### Procedure for Analysis  
This study focuses on the seismic response of single-degree-of-freedom systems exhibiting flag-shaped hysteretic behavior, considering soil-structure interaction in relation to the inelastic displacement ratio and residual displacement. Equation (1) provides the dynamic equation of motion for a system with an SDOF.  
\[\mathrm{m\ddot{u}}+\mathrm{c\dot{u}}+\mathrm{f_{s}}=-\mathrm{m\ddot{u}_{g}} \tag{1}\]  
In Equation (1), the SDOF system's mass is denoted by m, its relative velocity and acceleration responses are represented by \(\mathrm{\dot{u}}\) and \(\mathrm{\ddot{u}}\), the viscous damping coefficient is denoted by c, the self-centering system's restoring force is denoted by \(\mathrm{f_{s}}\), and the ground motion acceleration is represented by \(\mathrm{\ddot{u}_{g}}\). [PERSON]'s time-integration technique is implemented into a computer program for inelastic time history analyses. As emphasized by [PERSON] and [PERSON] [53], the time-integration step size is a critical and effective parameter to predict the residual displacements. The time step chosen for time history analyses is the minimum of the following: the original earthquake ground motion time step, 1/25 of the structural period, and 0.01 s. In the case of interaction, the foundation is portrayed as a rigid circular disk with a radius of r. The soil beneath the foundation is assumed to be a uniform half-space defined by shear wave velocity \(\mathrm{V_{s}}\), dilatational wave velocity  
Figure 2: Force–displacement hysteretic model for varying values of energy dissipation ratio.  
\(\mathrm{V_{p}}\), mass density \(\rho\), and Poisson's ratio \(\upsilon\). The supportive soil is substituted with springs and dampers for horizontal and rocking movements. A spring-dashpot-mass model with frequency-independent coefficients is employed to represent the foundation for all motions. The modeling of the deformable soil foundation is performed in a similar manner to that of the structure and is coupled for dynamic SSI analysis [54]. The stiffness and damping coefficients for the horizontal (\(\mathrm{K_{x}}\), \(\mathrm{C_{x}}\)) and rocking modes (\(\mathrm{K_{\theta}}\), \(\mathrm{C_{\theta}}\)) of soil medium are defined as follows [55]:  
\[\mathrm{K_{x}}=\frac{8\rho\mathrm{V_{s}^{2}r}}{2-\upsilon} \tag{2}\]  
\[\mathrm{K_{\theta}}=\frac{8\rho\mathrm{V_{s}^{2}r^{3}}}{3(1-\upsilon)} \tag{3}\]  
\[\mathrm{C_{x}}=\rho\mathrm{V_{s}}\pi\pi^{2} \tag{4}\]  
\[\mathrm{C_{\theta}}=\rho\mathrm{V_{p}}\pi\frac{r^{4}}{4} \tag{5}\]  
The prevailing method for addressing elastic soil-structure interaction remains consistent, entailing the use of a replacement oscillator characterized by the system's effective period and damping. This equivalent oscillator's mass is equivalent to that of the actual structure, and under harmonic base excitation, it is mandated that the resonant period and peak response of the interacting system align with those of the oscillator. Eurocode 8 stipulates the inclusion of dynamic soil-structure interaction effects for structures impacted significantly by P-8 (2 nd order) effects, those with substantial or deep-seated foundations like bridge piers, offshore caissons, and silos, slender tall structures such as towers and chimneys, and structures supported by very soft soils with an average shear wave velocity less than 100 m/s [56]. Effective periods and damping of the interacting system are given below:  
\[\widetilde{\mathrm{T}}=\mathrm{T}\sqrt{1+\frac{k}{K_{x}}\left(1+\frac{K_{x}h^{ 2}}{K_{\theta}}\right)} \tag{6}\]  
\[\widetilde{\beta}=\beta_{0}+\frac{0.05}{\left(\frac{\widetilde{\mathrm{T}}}{ \widetilde{\mathrm{T}}}\right)^{3}} \tag{7}\]  
In current codes, one should refer to the figure for the appropriate values of the foundation damping factor denoted by \(\beta_{0}\)[57; 58].