# Nonlinear dynamic response of base isolated rigid blocks under horizontal motions

The first term in Eq. Model Description The adopted model consists of a rectangular-prismatic rigid block of mass m and centroid mass-moment of inertia ICstanding free on a seismically-isolated rigid base of mass mb Fig. It is hoped that the two-blocks model used herein can facilitate the development of more sophisticated multi-block structural models.

The potential energy of the system is obtained as: A rigorous formulation of the impact problem is presented in this paper based on the study originally published by Roussis et al.

Based on the developed model, a computer program was developed in Matlab to calculate the system response to simple acceleration pulses and pulse-type earthquake motions of various amplitudes and frequency content.

The governing equations for each regime of motion are derived next through a large-displacement formulation not restricted to slender blocks by means of the Lagrange method, for two seismic-isolation models. This feature is demonstrated by using accelerograms from the Northridge, CA,earthquake.

Penzien, "Rocking response of rigid blocks to earthquakes", Earthquake Eng. Considering a slender rigid block resting upon a rigid foundation, based on the assumption of perfectly-inelastic impact and sufficient friction to prevent sliding, Housner investigated the free- and forced-vibration rocking response to a rectangular pulse, a half-sine pulse, and an earthquake-type excitation.

The study examines in depth the system response based on a large-displacement formulation that combines the nonlinear equations of motion together with a rigorous model governing impact.

Stella, "Rocking behaviour of multi-block columns subjected to pulse-type ground motion accelerations", Open Const. An extensive numerical investigation was carried out for different geometric characteristics of the block and isolation-system parameters, aiming to identify potential trends in the rocking response and stability of the system.

The impact problem is addressed separately in Section 2. The rigid block will be set into rocking on top of the moving base when the overturning moment due to external loads exceeds the resisting moment due to gravity, yielding the following condition Fig.

Assuming no sliding of the block against the rigid base, when subjected to ground excitation the system can exhibit two possible oscillation patterns: Previous article in issue. The equations governing the rocking response of the system to horizontal and vertical ground accelerations are derived for each pattern, and an impact model is developed by conservation of angular momentum considerations.

A measure of the dynamic characteristics of the block, albeit in an approximate sense since the natural frequency is amplitude-dependent, is given by the size parameter [ 1 G.

Makris, "Analysis of the rocking response of rigid blocks standing free on a seismically isolated base", Earthquake Eng. Housner, "The behavior of inverted pendulum structures during earthquakes", Bulletin of the Seismological Society of America, vol.

Derived from first principles, the model assumes point-impact, perfectly-inelastic impact i. The mathematical treatment of the problem is broad in scope in that it is neither restricted to small rotations nor slender blocks.

Assuming no sliding, the rocking response of the system standing free on a rigid foundation is investigated. The derivation of the equations of motion accounts for the consecutive transition from one pattern of motion to another, each being governed by a set of highly nonlinear differential equations.

Nonlinear Model for Isolation System Consider the bilinear hysteretic model to represent the mechanical behavior of friction-pendulum-type isolation system, described by: This paper presents a comprehensive mathematical formulation for the nonlinear rocking response of seismically-isolated free-standing rigid blocks to base excitation.

Di Cintio, "Base isolation for seismic protection of statues", In: Evidently, the mutually-coupled equations governing the rocking regime are highly nonlinear and not amenable to closed-form solution even for the simplest form of ground excitation.

Two isolation-system models are utilized in the analysis: Linear Model for Isolation System Consider first the block isolated with a linear isolation system composed of a linear spring with stiffness kb and a linear viscous damper with coefficient cb, by interposing a rigid base of mass mb.

The behavior of such a linear-viscoelastic model in terms of the lateral force developed in the isolation system is described by: The system behavior is described in terms of four possible patterns of response and impact between either the two blocks or the base block and the ground.

Numerical results are obtained by developing an ad hoc computational scheme that is capable of determining the response of the system under an arbitrary base excitation. Equations of Motion When subjected to horizontal ground accelerationthe supporting base will oscillate in the horizontal direction with a displacement u t relative to the foundation.

More recently, motivated primarily by the need to mitigate the seismic risk of objects of cultural heritage, a rather limited number of studies on the rocking response of a rigid block with its base seismically isolated have been pursued e.

The 14th World Conference on Earthquake Engineering. The model elucidates the inherent base-block dynamic interaction, a fundamental response feature that distinguishes the problem at hand from the classic Housner-type problem of a rocking block impacting a rigid foundation with infinite mass.

Despite the apparent geometric simplicity of the problem, the mathematical description of the system dynamics is profoundly complex, mainly due to the inherent nonlinear nature of the impact phenomenon and the potential transition from one oscillation pattern to another, each one governed by different equations of motion.

Pisiara, "Base-isolation technology for earthquake protection of art objects", In:A study on the rocking response of a rigid block resting on a nonlinear flexible foundation has been made. To account for the additional features of the rocking behavior on very flexible foundations, a novel nonlinear model has been proposed for the base-foundation interaction, based on the Hunt and Crossleyâ€™s nonlinear impact force model.

Nonlinear Dynamic Response of RC Buildings with Different Base Isolation Systems Subjected to Horizontal and Vertical Components of Near-Fault Ground Motions.

A three-dimensional rigid body on the shape of a parallelepiped is modelled in order to rock on a side or a vertex of the base, in order to evaluate the seismic response of rigid blocks lying on a horizontal support.

The center of mass of the body is considered as eccentric with respect to its.

Rocking Response of Seismically-Isolated Rigid Blocks Under Simple Acceleration Pulses and Earthquake Excitations can be set into rocking on top of the moving base under dynamic excitation. The study examines in depth the motion of the block/base system with a large-displacement formulation that combines the nonlinear equations of.

Seismic response characteristics of a base isolated cable-stayed bridge under moderate and strong ground motions Nonlinear dynamic analysis Isolation system Lead-rubber bearing horizontal deck displacement under earthquake excitation and.

The dynamic behavior of structures of two stacked rigid blocks subjected to ground excitation has been examined. Assuming no sliding, the rocking response of the system standing free on a rigid foundation has been investigated.

The analytical formulation of this nonlinear problem has proved challenging.

Nonlinear dynamic response of base isolated rigid blocks under horizontal motions
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