Modern rocking and stepping cores have been known as the efficient self-centering earthquake-resisting systems (SC-ERSs). The current article proposes an approximate equivalent linear (EL) model for rapid estimation of the SC-ERS displacement. An equivalent damping ratio and effective stiffness are formulated for flag-shaped hysteresis of a fully SC-ERS. The approximate EL model is first established using secant stiffness and Jacobsen's damping model. Nonlinear response history analyses are carried out to compare exact and approximated peak displacements. Findings reveal that EL analysis of the SC-ERS based on Jacobsen's damping leads to underestimation of the maximum inelastic displacement. Accordingly, a new optimal damping formula is proposed using a genetic algorithm and nonlinear regression analyses. The improved EL model is validated by practical examples, and the results show acceptable accuracy in design-level displacement estimation. KEYWORDS equivalent linear model, flag-shaped hysteresis, optimization analysis, secant stiffness, selfcentering systems, viscous damping model 1 | INTRODUCTIONOn the basis of the traditional ductility-based design concept, structural members allow for sustaining of uniform damage. Nevertheless, buildings experience severe damage following a large earthquake, which lead to socioeconomic losses due to business downtime and repair costs. In recent years, self-centering earthquake-resisting systems (SC-ERSs) have been developed to provide a sustainable system through directing seismic damage to replaceable devices. Figure 1a shows examples of rocking and stepping self-centering systems including single or coupled stepping braced frames, [1][2][3][4][5][6][7][8] pin-supported rocking frame or wall, [9][10][11] rocking precast concrete wall, [12][13][14][15] and self-centering timber frames. [16,17] These lateral resisting systems are generally composed of bounded or unbounded posttensioning (PT), replaceable energy dissipation (ED) devices, rigid strut beam, bumper, and special diaphragm connections. Regardless of the SC-ERS components, a flag-shaped hysteresis is representative of their total seismic responses. Figure 1b depicts a sample of push-pull curves of the system characterized by uplift ( F up − Δ up ) and yield ( F y − Δ y ) parameters along with postyield stiffness (α) and ED index (β). Prior to unloading the system ( F d − Δ d ), the response is generally composed of three main branches. In the first branch that terminates when the system is uplifted, the SC-ERS has a large initial elastic stiffness and behaves as conventional buildings. Nevertheless, in the second branch, the system stiffness is reduced and sways on its foundation, which leads to the system yielding point. The yielding point is the beginning of the third branch with the hardening stiffness of α. Note that geometric nonlinearity is controlled using PT and ED devices. Accordingly, the PT cable provides adjustable restoring forces, and the ED device produces required ED capacity during the unloading ph...