Rocking Response and Stability Analysis of an Array of Free-Standing Columns Capped With a Free-Standing Rigid Beam

Abstract

Abstract. This paper investigates the planar rocking response of an array of free-standing columns capped with a freely supported rigid beam in an effort INTRODUCTIONUnder base shaking slender objects and tall rigid structures may enter into rocking motion that occasionally results in overturning. Early studies on the seismic response of a slender rigid block were presented by Milne [1]; however, it was Housner [2] who uncovered a sizefrequency scale effect which explained why: (a) the larger of two geometrically similar blocks can survive the excitation that will topple the smaller block; and (b) out of two same acceleration amplitude pulses, the one with the longer duration is more capable to induce overturning. Following Housner's seminal paper a number of studies have been presented to address the complex dynamics of one of the simplest man-made structures-the free standing rigid column.Yim et al. [3] conducted numerical studies by adopting a probabilistic approach, Aslam et al. [4] confirmed with experimental studies that the rocking response of rigid blocks is sensitive to system parameters; while Psycharis and Jennings [5] examined the uplift of rigid bodies supported on viscoelastic foundation. Subsequent studies by Spanos and Koh [6] investigated the rocking response due to harmonic steady-state loading and identified "safe" and "unsafe" regions together with the fundamental and suharmonic modes of the system. Their study was extended by Hogan [7], [8] who further elucidated the mathematical structure of the problem by introducing the concepts of orbital stability and Poincare sections. The transient rocking response of free-standing rigid blocks was examined in depth by Zhang and Makris [9] who showed that there exist two modes of overturning: (a) by exhibiting one or more impacts; and (b) without exhibiting any impact. The existence of the second mode of overturning results in a safe region that is located on the acceleration-frequency plane above the minimum overturning acceleration spectrum. The fundamental differences between the response of a rocking rigid column (inverted pendulum) and the response of the linear elastic oscillator (regular pendulum) led to the development of the rocking spectrum (Makris and Konstantinidis In this paper we investigate the planar rocking response of an array of free-standing columns capped with a freely supported rigid beam as shown schematically in Figure 1. Herein we use the term "rocking frame" for the one degree of freedom structure shown in Figure 1. Sliding does not occur either at the pivot points at the base or at the pivot points at the capbeam. Our interest to this problem was partly motivated from the need to explain the remarkable seismic stability of ancient free-standing columns which support heavy free standing epistyles together with the even heavier frieze atop. As an example, Figure 2 shows the entrance view of the late archaic temple of Aphaia in the island of Aegina nearby Athens, Greece. Dates ranging from 510BC to 470BC have been proposed fo...

Dynamics of rocking podium structures

“…Therefore, at the moment of impact, the pivot point instantaneously moves to the other edge of the column. This assumption has been previously used to calculate the coefficient of restitution in several rocking systems [46,50,55]. The aforementioned assumption results in Conservation of Angular Momentum (CoAM) for each column of the rocking podium frame during impact.…”

Finite element analysis of the seismic response of ancient columns

“…[18][19][20][21][22][23] Analytical approaches published in the recent years should also be mentioned. [24][25][26][27] The basic conclusions drawn from these studies are (a) the seismic response of multidrum columns is very sensitive to geometric parameters, as well as to the characteristics of the seismic action. (b) The probability of a column to collapse increases with its slenderness; a similar observation is made when the size of the column decreases, keeping the slenderness unaltered.…”

“…In several cases, the behavior of ancient columns was studied as part of wider approaches related, usually, with ongoing restoration projects of ancient monuments . Analytical approaches published in the recent years should also be mentioned . The basic conclusions drawn from these studies are (a) the seismic response of multidrum columns is very sensitive to geometric parameters, as well as to the characteristics of the seismic action.…”