Rocking Response Of Rigid Blocks Research Articles

Seismic protection of rocking structures with inerters

SummaryThe seismic behaviour of a wide variety of structures can be characterized by the rocking response of rigid blocks. Nevertheless, suitable seismic control strategies are presently limited and consist mostly on preventing rocking motion all together, which may induce undesirable stress concentrations and lead to impractical interventions. In this paper, we investigate the potential advantages of using supplemental rotational inertia to mitigate the effects of earthquakes on rocking structures. The newly proposed strategy employs inerters, which are mechanical devices that develop resisting forces proportional to the relative acceleration between their terminals and can be combined with a clutch to ensure their rotational inertia is only employed to oppose the motion. We demonstrate that the inclusion of the inerter effectively reduces the frequency parameter of the block, resulting in lower rotation seismic demands and enhanced stability due to the well‐known size effects of the rocking behaviour. The effects of the inerter and inerter‐clutch devices on the response scaling and similarity are also studied. An examination of their overturning fragility functions reveals that inerter‐equipped structures experience reduced probabilities of overturning in comparison with uncontrolled bodies, while the addition of a clutch further improves their seismic stability. The concept advanced in this paper is particularly attractive for the protection of rocking bodies as it opens the possibility of nonlocally modifying the dynamic response of rocking structures without altering their geometry.

Out‐of‐plane behaviour of a full scale stone masonry façade. Part 1: specimen and ground motion selection

SUMMARYThe out‐of‐plane response of walls in existing stone masonry buildings is one of the major causes of vulnerability commonly observed in post‐earthquake damage surveys. In this context, a shaking table (ST) test campaign was carried out on a full‐scale masonry façade mainly focusing on the characterization of its out‐of‐plane overturning behaviour. The structure tested on the ST is a partial reproduction of an existing building from Azores, damaged during the 9 July 1998 Faial earthquake. The definition of the tested specimen as well as the selection of the input ground motion is reported in this paper. A specific emphasis is given to the definition of the time‐history to be applied during the tests because it was felt as an essential and crucial part of the work to obtain the desired overturning behaviour. The accelerogram to be imposed was selected from a large set of accelerograms (74) by means of a step‐by‐step procedure on the basis of several numerical analyses resorting to the rocking response of rigid blocks. A companion paper (Part 2) focuses on the ST test results and detailed data interpretation. Copyright © 2013 John Wiley & Sons, Ltd.

Analysis of the rocking response of rigid blocks standing free on a seismically isolated base

SUMMARYThis paper examines the rocking response and stability of rigid blocks standing free on an isolated base supported: (a) on linear viscoelastic bearings, (b) on single concave and (c) on double concave spherical sliding bearings. The investigation concludes that seismic isolation is beneficial to improve the stability only of small blocks. This happens because while seismic isolation increase the ‘static’ value of the minimum overturning acceleration, this value remains nearly constant as we move to larger blocks or higher frequency pulses; therefore, seismic isolation removes appreciably from the dynamics of rocking blocks the beneficial property of increasing stability as their size increases or as the excitation pulse period decreases. This remarkable result suggests that free‐ standing ancient classical columns exhibit superior stability as they are built (standing free on a rigid foundation) rather than if they were seismically isolated even with isolation system with long isolation periods. The study further confirms this finding by examining the seismic response of the columns from the peristyle of two ancient Greek temples when subjected to historic records. Copyright © 2011 John Wiley & Sons, Ltd.

Rocking of Rigid Block under Random Wave Excitations

This investigation deals with the rocking response of rigid blocks subjected to barrow band random excitations. The numerical procedure and computer program are developed to solve the nonlinear equations of motion governing the rocking motion of rigid blocks on rigid base subjected to horizontal excitations. The analytical results presented show that the response of the block is very sensitive to small changes in its slenderness ratio, the energy dissipation, the friction condition between block and base, and the details of ground motion. Systematic trends are not apparent : The stability of a block subjected to a particular ground motion does not necessarily increase monotonically with decreasing slenderness ratio or excitation intensity, occurrence of sliding motion, increasing bandwidth of random wave. Overturning of a block by a ground motion of particular intensity does not imply that the block will necessarily overturn under the action of more intense ground motion. In contrast, systematic trends are observed when the problem is studied from a probabilistic point of view with the ground motion modeled as a random process. The probability of a block exceeding any response level, as well as the probability that a block overturns, increases with increases with increase in ground motion intensity and slenderness ratio of the block, and occurrence of violent sliding motion.

Rocking response of rigid blocks under near-source ground motions

In this paper the transient rocking response of a rigid block subjected to trigonometric pulses and near-source ground motions is investigated in detail. First, the rocking response to a half-sine pulse motion is revisited. It is shown that the solution presented by Housner for the minimum acceleration amplitude of a half-sine pulse that is needed to overturn a rigid block is incorrect. In reality, under a half-sine pulse, a block overturns during its free vibration regime and not at the instant that the pulse expires, as was assumed by Housner. Within the limits of the linear approximation, the correct conditions for a block to overturn are established and the correct expression that yields the minimum acceleration required to overturn a block is derived. Subsequently, physically realizable cycloidal pulses are introduced and their resemblance to recorded near-source ground motions is illustrated. The study uncovers the coherent component of some near-source acceleration records, and the overturning potential of these motions is examined. It is found that the toppling of smaller blocks is more sensitive to the peak ground acceleration, whereas the toppling of larger blocks depends mostly on the incremental ground velocity. The kinematic characteristics of recorded near-source ground motions are examined in detail, and it is found that the high-frequency fluctuations that occasionally override the long-duration pulse will overturn a smaller block, whereas a larger block will overturn due to the long-duration pulse. A simple, yet dependable, method to determine the level of a recorded ground motion that is needed to overturn a given block is developed and illustrated through examples. The method examines the acceleration amplitude, and the duration and the type of distinct local pulses that are identifiable within the record, including the pulse that contains the peak ground acceleration. According to the type of each pulse identified, the method estimates the minimum acceleration amplitude needed to overturn a given block, and this value is compared with the recorded acceleration amplitude of each pulse identified within the record. In this light, the rocking response of rigid blocks subjected to strong near-source ground motions is shown to be quite ordered and predictable.

Sliding and rocking response of rigid blocks due to horizontal excitations

To study the dynamic response of a rigid block standing unrestrained on a rigid foundation which shakes horizontally, four modes of motion can be identified, i.e., rest, slide, rock, and slide and rock. The occurrence of each of these four modes and the transition between any two modes depend on the parametric values specified, the initial conditions, and the magnitude of ground acceleration. In this paper, a general two-dimensional theory is presented for dealing with the various modes of a free-standing rigid block, considering in particular the impact occurring during the rocking motion. Through selection of proper values for the system parameters, the occurrence of each of the four modes and the transition between different modes are demonstrated in the numerical examples.

Steady state rocking response of rigid blocks part 2: Experiment

AbstractThe results of an experimental investigation on the rocking response of rigid blocks subjected to sinusoidal base motion is presented. It is shown that two common types of steady state response are harmonic and 1/3 subharmonic response. The measured steady state response amplitudes correlate well with theoretical predictions for both harmonic and 1/3 subharmonic responses. Within each type of steady state response, theoretical studies show that the system may respond in either a symmetric or an unsymmetric mode. A symmetric mode denotes that the block oscillates about a mean zero position while an unsymmetric mode implies the mean position of oscillation is at an inclined position. One form of unsymmetric response was observed in the experiment. In addition, almost periodic responses were also observed in the experiment, although their existence has not been reported in theoretical studies.

Steady state rocking response of rigid blocks part 1: Analysis

AbstractThe result of a theoretical study on the rocking response of rigid blocks subjected to sinusoidal base motion is presented. The study indicates that, for a given excitation amplitude and frequency, a rigid block can respond in several different ways. Based on analysis, the regions of different classes of steady state symmetric response solutions are mapped on the excitation amplitude‐frequency parameter space. The steady state response solutions (both harmonic and subharmonic) are classified into two classes, out‐of‐phase and in‐phase with respect to the excitation. Only out‐of‐phase solutions are found to be stable. A parametric study shows that steady rocking response amplitude is highly sensitive to the size of the block and the excitation frequency in the low frequency range. It is relatively insensitive to the excitation amplitude and the system's coefficient of restitution of impact. For two blocks of the same aspect ratio and coefficient of restitution subjected to the same excitation, the larger block always responds in smaller amplitude than the smaller block. Computer simulation is carried out to study the stability of the symmetric steady state response solutions obtained from analysis. It is found that as the excitation frequency is decreased beyond the boundary of stable symmetric response, the response becomes unsymmetric where the mean amplitude of oscillation is non‐zero. Further decrease in excitation frequency beyond the stable unsymmetric response boundary causes instability in the form of overturning.

Rocking response of rigid blocks to earthquakes

AbstractThis investigation deals with the rocking response of rigid blocks subjected to earthquake ground motion. A numerical procedure and computer program are developed to solve the non‐linear equations of motion governing the rocking motion of rigid blocks on a rigid base subjected to horizontal and vertical ground motion.The response results presented show that the response of the block is very sensitive to small changes in its size and slenderness ratio and to the details of ground motion. Systematic trends are not apparent: The stability of a block subjected to a particular ground motion does not necessarily increase monotonically with increasing size or decreasing slenderness ratio. Overturning of a block by a ground motion of particular intensity does not imply that the block will necessarily overturn under the action of more intense ground motion.In contrast, systematic trends are observed when the problem is studied from a probabilistic point of view with the ground motion modelled as a random process. The probability of a block exceeding any response level, as well as the probability that a block overturns, increases with increase in ground motion intensity, increase in slenderness ratio of the block and decrease in its size.It is concluded that probabilistic estimates of the intensity of ground shaking may be obtained from its observed effects on monuments, minarets, tombstones and other similar objects provided suitable data in sufficient quantity is available, and the estimates are based on probabilistic analyses of the rocking response of rigid blocks, considering their non‐linear dynamic behaviour.