Response Modification of Structures with Supplemental Rotational Inertia, PEER Report 2024-01


Tall, multistory, buildings are becoming increasingly popular in large cities as a result of growing
urbanization trends (United Nations Department of Economic and Social Affairs 2018). As cities
continue to grow, many of them along the coasts of continents which are prone to natural hazards,
the performance of tall, flexible buildings when subjected to natural hazards is a pressing issue
with engineering relevance. The performance of structures when subjected to dynamic loads can
be enhanced with various response modification strategies which have been traditionally achieved
with added stiffness, flexibility, damping and strength (Kelly et al. 1972; Skinner et al. 1973, 1974;
Clough and Penzien 1975; Zhang et al. 1989; Aiken 1990; Whittaker et al. 1991; Makris et al.
1993a,b; Skinner et al. 1993; Inaudi and Makris 1996; Kelly 1997; Soong and Dargush 1997;
Constantinou et al. 1998; Makris and Chang 2000a; Chang and Makris 2000; Black et al. 2002,
2003; Symans et al. 2008; Sarlis et al. 2013; Tena-Colunga 1997).

Together with the elastic spring that produces a force proportional to the relative displacement of
its end-nodes and the viscous dashpot that produces a force proportional to the relative velocity of
its end-nodes; the inerter produces a force proportional to the relative acceleration of its end-nodes
and emerges as the third elementary mechanical element (in addition to the spring and dashpot)
capable for modifying structural response. Accordingly, in this report we examine the seismic
performance of multistory and seismically isolated structures when equipped with inerters.

In view that the inerter emerges as the third elementary mechanical element for the synthesis of
mechanical networks, in Chapter 2 we derive the basic frequency- and time-response functions
of the inerter together with these of the two-parameter inertoelastic and inertoviscous mechanical

Chapter 3 examines the response of a two-degree-of-freedom (2DOF) structure where the first
story is equipped with inerters. Both cases of a stiff and a compliant support of the inerters are
examined. The case of two parallel clutching inerters is investigated and the study concludes that
as the compliance of the frame that supports the inerters increases, the use of a single inerter offers
more favorable response other than increasing the force transferred to the support frame.

Chapter 4 examines the seismic response analysis of the classical two-degree-of-freedom isolated
structure with supplemental rotational inertia (inerter) in its isolation system. The analysis shows
that for the “critical” amount of rotational inertia which eliminates the participation of the second
mode, the effect of this elimination is marginal on the structural response since the participation
of the second mode is invariably small even when isolation systems without inerters are used. Our
study, upon showing that the reaction force at the support of the inerter is appreciable, proceeds
with a non-linear response analysis that implements a state-space formulation which accounts for
the bilinear behavior of practical isolation system (single concave sliding bearings or lead-rubber
bearings) in association with the compliance of the support of the inerter. Our study concludes
that supplemental rotational inertia aggravates the displacement and acceleration response of the
elastic superstructure and as a result, for larger isolation periods (Tb > 2.5s) the use of inerters in
isolation systems is not recommended.

Chapter 5 first examines the response analysis of a SDOF elastoplastic and bilinear structure and
reveals that when the yielding structure is equipped with supplemental rotational inertia, the equal-
displacement rule is valid starting from lower values of the pre-yielding period given that the
presence of inerters lengthens the apparent pre-yielding period. The analysis concludes that sup-
plemental rotational inertia emerges as an attractive response modification strategy for elastoplastic
and bilinear SDOF structures with pre-yielding periods up to T1 = 1.5sec. For larger pre-yielding
periods (say T1 > 2.0sec), the effectiveness of inerters to suppress the inelastic response of 2DOF
yielding structures reduces; and for very flexible first stories; as in the case of isolated structures
examined in chapter 4, the use of inerter at the first level (isolation system) is not recommended.

Finally, chapter 6 shows that, in spite of the reduced role of inerters when placed at floor levels
other than the first level (they no-longer suppress the induced ground acceleration nor they can
eliminate the participation of higher modes), they still manifest a unique role since it is not possible
to replace a structure with solitary inerters at higher levels with an equivalent traditional structure
without inerters.

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Gholamreza Moghimi
Nicos Makris
Publication date: 
January 10, 2024
Publication type: 
Technical Report
Moghimi, G., Makris, N. (2024). Response Modification of Structures with Supplemental Rotational Inertia, PEER Report 2024/01. Pacific Earthquake Engineering Research Center, University of California, Berkeley, CA.