PEER Report 2024/01: "Response Modification of Structures with Supplemental Rotational Inertia"

January 31, 2024

PEER has published Report No. 2024/01: "Response Modification of Structures with Supplemental Rotational Inertia." It was authored by Gholamreza Moghimi and Nicos Makris, Department of Civil and Environmental Engineering, Southern Methodist University, Dallas, Texas.

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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 networks.

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 iii reveals that when the yielding structure is equipped with supplemental rotational inertia, the equaldisplacement 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 supplemental 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.

Keywords: Supplemental rotational inertia, Response modification, Inerter, Nonlinear analysis, Seismic protection, Earthquake Engineering.