This paper examines in depth the transient rocking response of free-standing rigid blocks subjected to physically realizable trigonometric pulses. First, the expressions for the dynamic horizontal and vertical reactions at the pivot point of a rocking block are derived and it is shown that the coefficient of friction needed to sustain pure rocking motion is, in general, an increasing function of the acceleration level of the pulse. Subsequently, this paper shows that under cycloidal pulses a free-standing block can overturn with two distinct modes: (1) by exhibiting one or more impacts; and (2) without exhibiting any impact. The existence of the second mode results in a safe region that is located on the acceleration-frequency plane above the minimum overturning acceleration spectrum. The shape of this region depends on the coefficient of restitution and is sensitive to the nonlinear nature of the problem. This paper concludes that the sensitive nonlinear nature of the problem, in association with the presence of the safe region that embraces the minimum overturning acceleration spectrum, complicates further the task of estimating peak ground acceleration by only examining the geometry of free-standing objects that either overturned or survived a ground shaking.

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Rocking Response of Free-Standing Blocks under Cycloidal Pulses

The effect of material and ground motion uncertainty on the seismic vulnerability curves of RC structure

The dynamic behavior of rigid-block structures resting on a rigid foundation subjected to horizontal harmonic excitation is examined. For slender structures, the nonlinear equation of motion is approximated by a piecewise linear equation. Using this approximation for an initially quiescent structure, safe or no-toppling and unsafe regions are identified in an excitation amplitude versus excitation frequency plane. Furthermore, several possible modes of steady-state response are detected, and analytical procedures are developed for determining the amplitudes of the predominant modes and for performing stability analyses. It is shown that the produced stability diagrams can be beneficial to assessing the toppling potential of a rigid-block structure under a given amplitude-frequency combination of harmonic excitation; in this manner the integration of the equation of motion is circumvented.

Closure of "Rocking of Rigid Blocks Due to Harmonic Shaking"

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 pote...

Rocking response of rigid blocks under near-source ground motions

The paper presents an overview of the progress in research regarding seismic response of plan and vertically irregular building structures. Three areas of research are surveyed. The first is the study of the effects of plan-irregularity by means of single-storey and multi-storey building models. The second area encompasses passive control as a strategy to mitigate torsional effects, by means of base isolation and other types of devices. Lastly, the third area concerns vertically irregular structures and setback buildings. Although fewer papers have been published in this last area with respect to the former ones, this state-of-the-art reports extensively on research efforts and progress into the seismic behaviour of irregular buildings in elevation to show the growing interest among specialists in the field.

/pdf/a-review-of-research-on-seismic-behaviour-of-irregular-3wulwfe6am.pdf

A review of research on seismic behaviour of irregular building structures since 2002

The 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.

Steady state rocking response of rigid blocks part 1: Analysis

The 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 2: Experiment

Using a three element single mass model, this paper presents the ductility demands on the elements of torsionally unbalanced systems when subjected to strong earthquake shaking. Torsionally unbalanced systems based on nine structural configurations are considered, ranging from torsionally stiff systems with the centre of rigidity (CR) centrally located to torsionally flexible systems with CR eccentrically located. The strength of the elements is designed based on the Canadian and New Zealand codes, and the Uniform Building Code (UBC) of the United States. It is shown that all three codes can limit the ductility demands on the elements to that of a similar but torsionally balanced system when the system is torsionally stiff. However, substantial additional ductility demands on the element at the stiff edge of the system exist for torsionally flexible systems when the New Zealand code or UBC is used. The large ductility demand is caused by the low strength of the stiff-edge element permitted by these codes.

Design of torsionally unbalanced structural systems based on code provisions I: Ductility demand

In order to mitigate the effect of torsion during earthquakes, most seismic codes of the world provide design guidelines for strength distribution based on the traditional perception that element stiffness and strength are independent parameters. Recent studies have pointed out that for an important class of widely used structural elements such as reinforced concrete flexural walls, stiffness is a strength-dependent parameter. This implies that the lateral stiffness distribution in a wall-type system cannot be defined prior to the assignment of elements' strength. Consequently, stiffness eccentricity cannot be computed readily and the current codified torsional provisions cannot be implemented in a straightforward manner. In this study, an alternate guideline for strength distribution among lateral force resisting elements is presented. To develop such a guideline, certain issues related to the dynamic behaviour of asymmetric wall-type systems during a damaging earthquake were examined. It is shown that both stiffness and strength eccentricity are important parameters affecting the seismic response of asymmetric wall-type systems. In particular, results indicate that torsional effects can be minimized by using a strength distribution that results in the location of the centre of strength CV and the centre of rigidity CR on the opposite sides of the centre of mass CM. Copyright © 2001 John Wiley & Sons, Ltd.

A strength distribution criterion for minimizing torsional response of asymmetric wall‐type systems

Using a single mass monosymmetric model, this paper examines the additional seismic inelastic deformations and displacement caused by structural asymmetry of the model. Stiffness eccentricity and resistance eccentricity are used as measures of asymmetry in the elastic and inelastic range respectively. Seven ways of specifying strength distribution among resisting elements are considered, including code provisions from Canada, Mexico, New Zealand and the United States. These specifications are related t o the model resistance eccentricity.

It is shown that when torsional shears are included in the strength design of the elements, the structure in general will have small resistance eccentricity, even if it has large stiffness eccentricity in the elastic range. For structures which are designed with allowance for torsional shears, the ductility demands on the elements are similar to those when the structure is symmetrical. However, the edge displacements can be up to three times that if the system is symmetrical. This finding has significant implications in evaluating adequate separation between buildings to avoid the pounding problem during earthquakes.

W. K. Tso

Papers

Steady state rocking response of rigid blocks part 1: Analysis

Steady state rocking response of rigid blocks part 2: Experiment

Design of torsionally unbalanced structural systems based on code provisions I: Ductility demand

A strength distribution criterion for minimizing torsional response of asymmetric wall‐type systems

Additional seismic inelastic deformation caused by structural asymmetry