Technical Note Are Some Top-Heavy Structures More Stable? Nicos Makris, M.ASCE 1 ; and Michalis F. Vassiliou 2 Abstract: This technical note investigates the dynamic response and stability of a rocking frame that consists of two identical free-standing slender columns capped with a freely supported rigid beam. Part of the motivation for this study is the emerging seismic design concept of allowing framing systems to uplift and rock along their plane in order to limit bending moments and shear forces— together with the need to stress that the rocking frame is more stable the more heavy is its cap-beam, a finding that may have significant implications in the pre- fabricated bridge technology. In this technical note, a direct approach is followed after taking dynamic force and moment equilibrium of the components of the rocking frame, and the remarkable results obtained in the past with a variational formulation (by the same authors) is confirmed—that the dynamics response of the rocking frame is identical to the rocking response of a solitary, free-standing column with the same slenderness, yet with larger size, which produces a more stable configuration. The motivation for reworking this problem by following a direct approach is to show, in the simplest possible way, that the heavier the freely supported cap beam, the more stable is the rocking frame, regardless of the rise of the center of gravity of the cap beam. The conclusion is that top-heavy rocking frames are more stable that when they are top-light. DOI: 10.1061/(ASCE)ST.1943-541X.0000933. © 2014 American Society of Civil Engineers. Author keywords: Rocking frame; Seismic isolation; Articulated structures; Prefabricated bridges; Seismic design; Seismic effects. Introduction It is common experience that a small, slender, free-standing, top- heavy object (such as a vase) may easily overturn due to a horizon- tal shaking, while a racing car with a low center of gravity remains stable even under the large horizontal forces that develop during a sharp turn. Whenever the stability of a free-standing object is an issue, the obvious, intuitive measure is to lower its center of gravity. At the same time, ancient free-standing columns with aspect ratio as high as 6=1 supporting heavy free-standing epistyles to- gether with the even heavier frieze atop have survived the test of time and remain stable for several hundred years in areas with appreciable seismic hazard (Konstantinidis and Makris 2005). The remarkable seismic stability of tall, free-standing solitary columns was understood some 50 years ago by Housner (1963), who uncovered a size-frequency scale effect that explained why (1) the larger of two geometrically similar blocks survives an ex- citation that will topple the smaller block, and (2) of two acceler- ation pulses with the same amplitude, the one with the longer duration is more capable of overturning. Following Housner’ s pioneering work, several studies from other investigators (Aslam et al. 1980; Yim et al. 1980; Spanos and Koh 1984; Tso and Wong 1989; Shenton 1996; Makris and Rousos 2000; Zhang and Makris 2001; Dimitrakopoulos and DeJong 2012) showed that the uplifting and rocking of solitary, tall, free-standing columns has beneficial effect to their seismic resistance, similar to the way that sliding reduces the base shears of heavy low-rise structures (Skinner and Robinson 1993; Kelly 1997; Konstantinidis and Makris 2009, 2010). Results on the dynamic response of two free-standing columns capped with a freely supported beam have been presented by Allen et al. (1986), who adopted a Lagrangian formulation. In the Allen et al. (1986) paper, it was assumed that the mass of each column, m c , is much less than the mass of the freely supported beam, m b , and therefore, the equation of motion derived was for m b =m c → ∞. Furthermore, the results presented were obtained by solving the linearized equation of motion. In this technical note, it is shown that the exact nonlinear equation of motion can be derived and solved without making any approximations and results are offered for any finite value of γ ¼ m b =2m c . Furthermore, while the governing equation for the rocking frame appearing in the Allen et al. (1986) paper shows clearly that the response involves the slenderness, α, and size, R, of the columns of the rocking frame, the Allen et al. (1986) paper does not make any attempt to associate the dynamic response/stability of the rocking frame with that of the solitary rocking column. Within the context of a planar rocking motion, this technical note shows that the dynamic response of the four-hinge free- standing rocking frame shown in Fig. 1 is more stable than the dynamic response of one of its columns when standing alone. Most importantly, this technical note shows that the heavier the freely supported beam, the more stable is the rocking frame, regardless of the rise of the center of gravity of the system. The conclusion is that rocking frames are more stable than when they are top-heavy than when they are top-light. Numerical studies with the discrete element method by Papaloizou and Komodromos (2009) are in agreement with our analytical result—that the planar response of free-standing columns supporting epistyles is more stable than the response of the solitary, free-standing column. Dynamics of the Rocking Frame With reference to Fig. 1, and assuming that the coefficient of friction is large enough that there is no sliding, the equations of motion of a free-standing column with size R ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b 2 þ h 2 p and slenderness α ¼ tan 1 ðb=hÞ, and subjected to a horizontal ground acceleration ¨ u g ðtÞ when rocking around pivot points O and O 0 , 1 Professor, Dept. of Civil Engineering, Univ. of Patras, 265 04 Rio Patras, Greece (corresponding author). E-mail: nmakris@upatras.gr 2 Postdoctoral Researcher, IBK, Wolfgang-Pauli-Strasse 15, ETH Zürich, CH-8093 Zürich, Switzerland. E-mail: vassiliou@ibk.baug.ethz.ch Note. This manuscript was submitted on December 20, 2012; approved on August 29, 2013; published online on February 3, 2014. Discussion period open until July 3, 2014; separate discussions must be submitted for individual papers. This technical note is part of the Journal of Struc- tural Engineering, © ASCE, ISSN 0733-9445/06014001(5)/$25.00. © ASCE 06014001-1 J. Struct. Eng. J. Struct. Eng. Downloaded from ascelibrary.org by University of Patras on 02/20/14. Copyright ASCE. For personal use only; all rights reserved.