Equivalent linear model for fully self‐centering earthquake‐resisting systems

Abstract

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...

An equivalent damping ratio that accounts for ground-motion spectral shape for displacement-based design of flag-shaped hysteresis systems

“…Previous researchers have found that the assumption of harmonic excitation at the maximum level of deformation (Jacobsen’s approach) can lead to overestimates of damping, because the hysteretic response under actual earthquake motions has many cycles with ductility demands that are smaller than the target peak ductility (Blandon and Priestley, 2005; Gulkan and Sozen, 1974). However, due to its simplicity, Jacobsen’s approach is still being used in designs of self-centering structures (Fan et al, 2019; Hu and Wang, 2021; Liu et al, 2016; Rahgozar et al, 2019; Xu and Yang, 2019; Zhu et al, 2020), with modifications.…”

“…The EDR was selected so that the energy dissipated for the nonlinear system and viscously damped system would be the same for complete loops of the hysteretic models under steady-state harmonic excitations. Due to its simplicity, this method has been implemented to develop EDR equations for a wide variety of systems (Fan et al, 2019;Hu and Wang, 2021;Kowalsky, 1994;Rahgozar et al, 2019;Zhu et al, 2020). Several studies have been carried out to improve the accuracy of area-based strategies.…”

Extension of direct displacement‐based design for quantifying higher mode effects on controlled rocking steel cores

“…This damping equation is modified by researchers to consider the influence of the period. [ 46 ] The authors have proposed the modification factor of to account for the effect of equivalent periods. Figure 5 shows the correlation between the proposed modification factors and T e , n for different flag‐shaped responses.…”

Evaluating the Accuracy of an Equivalent Linear Model in Predicting Peak Displacement of Seismic Isolation Systems using Single Friction Pendulum Bearings

“…The study proposed equations for predicting the accuracy of the predicted peak displacement as functions of isolation system's property and ground motion type. Beside these researches on isolation systems, there have been also many studies on the accuracy, limitation and improvement of the secant stiffness approach on other nonlinear systems [14][15][16][17][18][19][20].…”