Soil Dynamics and Earthquake Engineering xxx (xxxx) xxx Please cite this article as: Gholamreza Moghimi, Nicos Makris, Soil Dynamics and Earthquake Engineering, https://doi.org/10.1016/j.soildyn.2020.106474 0267-7261/© 2020 Elsevier Ltd. All rights reserved. Seismic response of yielding structures equipped with inerters Gholamreza Moghimi , Nicos Makris * Dept. of Civil and Environmental Engineering, Southern Methodist Univ., Dallas, 75275, TX, USA A R T I C L E INFO Keywords: Inerter Supplemental rotational inertia Response modifcation Seismic protection Nonlinear analysis Earthquake engineering ABSTRACT This paper investigates the seismic response of one- and two-degree-of-freedom yielding structures equipped with inerters at the frst story. Inerters are mechanical devices that their resisting force is proportional to the relative acceleration at their end-nodes. This class of response modifcation devices complements the traditional supplemental damping devices that their resisting force is proportional to the relative velocity at their end-nodes, also examined in this study for comparison. The paper develops a stable nonlinear response analysis method- ology that implements a state-space formulation. Given that the engagement with an inerter lengthens the apparent preyielding period of the inelastic structure, the paper shows that when a yielding structure is equipped with supplemental rotational inertia, the equal-displacement rule is valid starting from lower values of the preyielding period. The effectiveness of a single inerter versus the use of a pair of clutching inerters that can only resist the motion of the yielding structure is examined, and the paper concludes that a single inerter suppresses effectively the displacement response of inelastic structures by outperforming the response modifcation with supplemental damping in particular when the supporting frame of the response modifcation devices is compliant. 1. Introduction Response modifcation of civil structures has been traditionally achieved with added stiffness, fexibility, damping, and strength [1–11] and references reported therein. During the last two decades an alter- native response modifcation strategy has emerged-that of supplemental rotational inertia [12–15] and at present, it has enjoyed a handful of full-scale implementations [16,17]. Supplemental rotational inertia can be achieved either with a rack- and-pinion- fywheel arrangement schematically shown in Fig. 1, a ball-screw assembly [12,13,15,18–22] or fuid inerters [23–26]. The mechanical system shown in Fig. 1 is a mechanical analogue of the electrical capacitor in a force-current / velocity-voltage analogy be- tween mechanical and electrical networks. This missing analogy was frst recognized by Smith [14] who coined the term ’’inerter’’ for any mechanical arrangement where the output force is proportional only to the relative acceleration between its end-nodes. The constant of pro- portionality of the inerter is coined the ’’inertance’’ = M R [14] and has units of mass [M]. Accordingly, if F 1 , u 1 and F 2 , u 2 are the forces and displacements at the end-nodes of the inerter with inertance, M R , its constitutive relation is [27–29]: { F 1 (t) F 2 (t) } = [ M R M R M R M R ] { ¨ u 1 (t) ¨ u 2 (t) } (1) Following the pioneering work of Arakaki et al. [12,13] and sys- tematic theoretical and experimental studies in vehicle mechanics and dynamics [15,30–32], a growing number of publications have proposed the use of rotational inertia dampers for the wind [33,34] and seismic protection of civil structures [35–42] among others. More recently, Makris and Kampas [43] introduced the implementation of two parallel inerters together with the use of a clutch (pair of clutching inerters), so that the rotating fywheels only resist the motion of the structure without inducing any deformations. The benefts of using a pair of counter-rotating clutching inerters were subsequently examined on a 2DOF elastic structure [29]. About the same time, Domenico and Ric- cardi [44] examined the response of a multi-degree-of-freedom seismic isolated elastic structure where the isolation system was enhanced with a tuned mass damper inerter. In that study [44] the nonlinear behavior of the isolator is accounted for, and subsequently, the study proceeds with a stochastic linearization technique. All the aforementioned studies other than [44] examined invariably the response of elastic structures. In particular, the recent study by Makris and Moghimi [29] concluded that while the use of inerters may * Corresponding author. E-mail addresses: rmoghimi@smu.edu (G. Moghimi), nmakris@smu.edu (N. Makris). Contents lists available at ScienceDirect Soil Dynamics and Earthquake Engineering journal homepage: http://www.elsevier.com/locate/soildyn https://doi.org/10.1016/j.soildyn.2020.106474 Received 29 August 2020; Received in revised form 13 October 2020; Accepted 21 October 2020