Steady state rocking response of rigid blocks part 1: Analysis

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.     

Linearization of rigid body dynamics on frictional interfaces under harmonic loading

The non-linear dynamic response of the rigid block is linearized by means of a friction model that implicitly considers block response through an experimental parameter obtained from shaking table experiments. In view of the great difficulty in carrying out shaking table experiments, in this technical note some recommendations to estimate friction model parameters are given. The selection of parameters considers the sliding response mode of the block–plane-excitation system: stick–slip or continuous sliding. Once all the friction parameters and block response mode were estimated, a methodology was proposed to compute rigid block dynamic response. The numerical results were then compared to actual experimental data for a rigid block sliding on a geotextile–wood interface, along an inclined plane subjected to base harmonic acceleration. Experiments were carried out for both the stick–slip and the continuous sliding modes. Computed and measured responses for both cases showed good agreement, thus indicating that the methodology developed in this research is adequate to capture the physics (of non-linear nature) of rigid blocks sliding on frictional interfaces subjected to complex harmonic loading. The findings encourage the extension of the linearization technique to the more general seismic loading case.     

Rocking response of rigid blocks to earthquakes

This 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.     

Sliding-uplifting response of rigid blocks to base excitation

In the present paper a two-degree-of-freedom system is considered which allows the simulation of rigid blocks uplifting and sliding on frictional foundations; the monolateral constraint between block and base is schematized by means of a joint model, which allows the contact problem to be discretized. The joint model is governed by normal and shear constitutive laws, which have been derived by the phenomenological behaviour of stone blocks and rock joints, as given by rock mechanics. Furthermore, a numerical procedure has been developed in order to solve the non-linear equations governing the motion of the block-base system, and to analyse the dynamic response of this system under seismic excitation; particular attention has been paid to the influence of the vertical displacement on the slip response.     

Seismic performance of an existing bridge with scoured caisson foundation

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 geometric center.As seismic input,three Italian recorded accelerograms,with different spectral content,are used.The study is mainly conducted to highlight the differences between the seismic response of 2D and 3D models of rigid blocks,with the aim to understand if,in some cases,the use of the 3D model of rigid block is required to obtain safer results.In fact,the outcomes show that in some ranges of the geometrical and mechanical parameters that characterize the excitation and the body,a two-dimensional model,which is not able to consider the 3D rocking on a vertex,can provide unsafe results.In particular,it is found that the overturning process of the three-dimensional block can occur under excitations which are lower than those which overturn a corresponding two-dimensional block.     

Investigation of nonlinear sloshing effects in seismically excited tanks

Nonlinear behavior of liquid sloshing inside a partially filled rectangular tank is investigated. The nonlinearity in the numerical modeling of the liquid sloshing originates from the nonlinear terms of the governing equations of the fluid flow and the liquid free surface motion as a not known boundary condition. The numerical simulations are performed for both linear and nonlinear conditions. The computed results using linear conditions are compared with readily available exact solution. In order to verify the results of the nonlinear numerical solution, a series of the shaking table tests on rectangular tank were conducted. Having verified linear and nonlinear numerical models, they are used for computation of near wall sloshing height at a series of real scale tanks (with various dimensions) under the both harmonic and earthquake base excitation. Finally, the nonlinear effects on liquid sloshing modeling are discussed and the practical limitations of the linear solution in evaluating the response of seismically excited liquids are also addressed.     

Dynamic response analysis of solitary flexible rocking bodies: modeling and behavior under pulse‐like ground excitation

Results obtained for rigid structures suggest that rocking can be used as seismic response modification strategy. However, actual structures are not rigid: structural elements where rocking is expected to occur are often slender and flexible. Modeling of the rocking motion and impact of flexible bodies is a challenging task. A non‐linear elastic viscously damped zero‐length spring rocking model, directly usable in conventional finite element software, is presented in this paper. The flexible rocking body is modeled using a conventional beam‐column element with distributed masses. This model is verified by comparing its pulse excitation response to the corresponding analytical solution and validated by overturning analysis of rocking blocks subjected to a recorded ground motion excitation. The rigid rocking block model provides a good approximation of the seismic response of solitary flexible columns designed to uplift when excited by pulse‐like ground motions. Guidance for development of rocking column models in ordinary finite element software is provided. Copyright © 2014 John Wiley & Sons, Ltd.     

Analysis of block random rocking

The stochastic rocking of a rigid block due to shaking of its foundation is examined. The foundation is flexible (Winkler) and its independent springs and dashpots separate from the block base when tension is incipient. Modulated white noise is used as a model of horizontal acceleration of the foundation. The statistics of the rocking response are found by an analytical procedure which involves a combination of static condensation and stochastic linearization. The analytical solution is computationally efficient and compares well with pertinent data obtained by numerical simulations.     

Dynamic analysis of stacked rigid blocks

The dynamic behavior of a structural model of two stacked rigid blocks subjected to ground excitation is examined. 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. 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. 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. 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. This feature is demonstrated by using accelerograms from the Northridge, CA, 1994, earthquake. It is hoped that the two-blocks model used herein can facilitate the development of more sophisticated multi-block structural models.     

Periodic response of a rigid block resting on a footing subjected to harmonic excitation

This study presents the dynamic behaviour of a rigid block which rests on a footing supported by a spring and a dashpot on a rigid base. The response of the rigid body is examined carefully when the base is excited by a harmonic force. It is found that a periodic motion appears in three different modes: stick-stick, stick-slip and slip-slip. The conditions that initiate the stick-stick and slip-slip modes are derived in explicit forms and the maximum sliding displacement is also obtained analytically. Useful dimensionless parameters are proposed for the presentation of the dynamic behaviour. The accuracy of results is confirmed by the response history computed by the Nigam-Jennings method.     

Dynamics of a sliding-rocking block considering impact with an adjacent wall

A freestanding rigid block subjected to base excitation can exhibit complicated motion described by five response modes: rest, pure rocking, pure sliding, combined sliding-rocking, and free flight. Previous studies on the dynamics of a rocking block have assumed that the block does not interact with neighboring objects. However, there are many applications in which the block may start or come in contact with an adjacent boundary during its motion, for example, a bookcase or cabinet colliding with a partition wall in an earthquake. This paper investigates the dynamics of a sliding-rocking block considering impact with an adjacent wall. A model is developed in which the base and wall are assumed rigid, and impact is treated using the classical impulse and momentum principle. The model is verified by comparing its predictions in numerical simulations against those of an existing general-purpose rigid-body model in which impact is treated using a viscoelastic impact model. The developed model is used to investigate the effects of different parameters on the stability of a block subjected to analytical pulse excitations. It is found that wall placement (left or right) has a dominant effect on the shape of the overturning acceleration spectra for pulse excitations. In general, decreasing the gap distance, base friction coefficient, and wall coefficient of restitution enhance the stability of the block. Similar observations are made when evaluating the overturning probability of a block using earthquake floor motions.     

In-plane and anti-plane strong shaking of soil systems and structures

The concept of in-plane and anti-plane shaking is introduced with a rigid block on a plane surface with Coulomb friction. Using a hypoplastic constitutive relation to model the mechanical behaviour of the soil, numerical solutions for a rigid block on a thin dry or saturated soil layer are obtained. The coupled nature of dynamic problems involving granular materials is shown, i.e. the motion of the block changes the soil state—skeleton stresses and density—which in turn affects the block motion. Motions of the block as well as soil response can be more realistically calculated by the new model. The same constitutive equation is applied to the numerical simulation of the propagation of plane waves in homogeneous and layered level soil deposits induced by a wave coming from below. Experiments with a novel laminar shake box as well as real seismic records from well-documented sites during strong earthquakes are used to verify the adequacy of the hypoplasticity-based numerical model for the prediction of soil response during strong earthquakes. The response of a homogeneous earth dam subjected to in-plane and anti-plane shaking is investigated numerically. In-plane and anti-plane shaking is shown to cause nearly the same spreading of a sand dam under drained conditions, whereas under undrained conditions anti-plane shaking causes stronger spreading of the dam. The dynamic behaviour of a breakwater founded on rockfill and soft clay during the 1995 Kobe earthquake is back-calculated to show the good performance of the proposed numerical model also with a structure. Section 9 deals with buildings on mattresses of densified cohesionless soils or fine-grained soils with granular columns, slopes with ‘hidden’ dams and structures on piles traversing clayey slopes to show the suitability of hypoplasticity-based models for the earthquake-resistant design and safety assessment of geotechnical systems.     

Experimental study of the dynamic interaction between the foundation of the NEES/UCSD Shake Table and the surrounding soil: Reaction block response

An extensive experimental study of the dynamic interaction between the foundation block for the NEES/UCSD Large High Performance Outdoor Shake Table and the surrounding soil was conducted in 2003. The vibrations induced by the two NEES@UCLA large eccentric mass shakers were recorded at multiple stations within the reinforced concrete foundation block and on the surface of the surrounding soil up to distances of 270 m from the block. The present paper focuses on analysis of the data recorded within the reaction block including the average rigid body motion of the foundation and its dependence on frequency, and the deformation of the block for longitudinal (EW), transverse (NS), and torsional excitation. Comparison of the reaction block response during shaker induced vibrations with that for the much stronger actuator forces shows that linearity holds for the range of forces involved. Comparisons with analytical results for a simplified model of the foundation show good agreement between experimental and theoretical results.     

Steady state rocking response of rigid blocks part 2: Experiment

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.     

On the seismic performances of rigid block‐like structures coupled with an oscillating mass working as a TMD

In this paper, the effects of a mass damper on the rocking motion of a non‐symmetric rigid block‐like structure, subject to different seismic excitation, are investigated. The damper is modelled as a single degree of freedom oscillating mass, running at the top of the block and connected to it by a linear visco‐elastic device. The equations of rocking motion, the uplift and the impact conditions are derived. A nondimensionalisation of the governing equations is performed with the aim to obtain an extensive parametric analysis. The results are achieved by numerical integration of these equations. The slenderness and the base of the rigid block, and the eccentricity of the centre of mass are taken as variable parameters in the analyses. The main objective of the study is to check the performance of the damper versus the spectral characteristics of the seismic input. Three earthquake registrations with different frequency contents are used in the analyses. The results show that the presence of the mass damper leads to different levels of improvement of the response of the system, depending on the spectral characteristics of the seismic input. Curves providing the overturning slenderness of blocks of specific sizes versus the characteristics of the TMD are obtained. Copyright © 2017 John Wiley & Sons, Ltd.     

A finite element model for seismic response analysis of deformable rocking frames

A new finite element model to analyze the seismic response of deformable rocking bodies and rocking structures is presented. The model comprises a set of beam elements to represent the rocking body and zero‐length fiber cross‐section elements at the ends of the rocking body to represent the rocking surfaces. The energy dissipation during rocking motion is modeled using a Hilber–Hughes–Taylor numerically dissipative time step integration scheme. The model is verified through correct prediction of the horizontal and vertical displacements of a rigid rocking block and validated against the analytical Housner model solution for the rocking response of rigid bodies subjected to ground motion excitation. The proposed model is augmented by a dissipative model of the ground under the rocking surface to facilitate modeling of the rocking response of deformable bodies and structures. The augmented model is used to compute the overturning and uplift rocking response spectra for a deformable rocking frame structure to symmetric and anti‐symmetric Ricker pulse ground motion excitation. It is found that the deformability of the columns of a rocking frame does not jeopardize its stability under Ricker pulse ground motion excitation. In fact, there are cases where a deformable rocking frame is more stable than its rigid counterpart. Copyright © 2016 John Wiley & Sons, Ltd.     

Shaking table test and dynamic response prediction on an earthquake-damaged RC building

This paper presents the results from shaking table tests of a one-tenth-scale reinforced concrete (RC) building model. The test model is a protype of a building that was seriously damaged during the 1985 Mexico earthquake. The input ground excitation used during the test was from the records obtained near the site of the prototype building during the 1985 and 1995 Mexico earthquakes. The tests showed that the damage pattern of the test model agreed well with that of the prototype building. Analytical prediction of earthquake response has been conducted for the prototype building using a sophisticated 3-D frame model. The input motion used for the dynamic analysis was the shaking table test measurements with similarity transformation. The comparison of the analytical results and the shaking table test results indicates that the response of the RC building to minor and the moderate earthquakes can be predicated well. However, there is difference between the predication and the actual response to the major earthquake.     

The masonry arch as a four-link mechanism under base motion

The dynamic response of an unreinforced masonry arch is examined, modelling the rigid body motions of arch segments under the influence of gravitational and inertial forces. This extends earlier studies of single rocking blocks, stacked blocks, and portal mechanisms of blocks; the masonry arch is analysed as another kinematic form of such a system. In this first effort a part-circular planar arch ring is studied and excitation is restricted to horizontal ground acceleration of the base. The mechanism kinematics are presented and the governing equation of motion is derived in non-linear form. The instantaneous form is determined for small rotations about the initial geometry and is used to study the conditions for the onset of mechanism motion. Possible failure conditions are posed and bounding principles are stated. One possible failure condition, direct overturning as a four-link mechanism, is studied for one simplified base motion. The results show that an arch geometry establishes good resistance to earthquake excitation in that ground acceleration must exceed a rather high threshold before any mechanism motion would develop; however, once that threshold has been passed the arch has relatively modest resistance before failure. Other possible failure conditions are discussed; one emerges from pounding effects between segments at impact, and another develops from sliding of blocks over one another as the internal forces (normal and tangential to the masonry joint) vary with the inertial forces.     

On the rocking–sliding instability of rigid blocks under ground excitation: Some new findings

Rocking (overturning) instability analyses of rigid blocks based on the assumption that the friction between the block and the ground is sufficiently large to exclude the effect of sliding, are reconsidered by including the effect in question. Both modes of overturning instability – without impact and after one impact – are thoroughly discussed in connection with small sliding, whose value depends on the values of kinetic (dry) friction coefficient and the external frequency excitation. Using an energy approach the analytical derivation of the nonlinear differential equations of motion of free-standing rigid blocks under one-sine ground pulse including the effect of sliding, are comprehensively established. The serious difficulties in solving this problem on one hand the change of the kinetic friction coefficient during the motion and on the other hand the reliable evaluation of the actual friction effect when rocking is included, are effectively confronted. This is achieved through a reliable approximation of an equivalent (reduced) coefficient assuming that the major part of friction takes place from the initiation of motion and terminates shortly after the onset of rocking. In cases of slender blocks closed form solutions for overturning due to simultaneous rocking–sliding without or after one impact, are conveniently derived. Among other findings, it was explored that the single block in question for small values of the external frequency (long periods of excitation) the sliding effect is beneficial (stabilizing the block), while for large values of external frequency this effect is detrimental (destabilizing the block).     

Comparison between experimental and analytical results for seesaw energy dissipation systems using fluid viscous dampers

This paper presents results of experimental and numerical investigations of a seesaw energy dissipation system (SEDS) using fluid viscous dampers (FVDs). To confirm the characteristics of the FVDs used in the tests, harmonic dynamic loading tests were conducted in advance of the free vibration tests and the shaking table tests. Shaking table tests were conducted to demonstrate the damping capacity of the SEDS under random excitations such as seismic waves, and the results showed SEDSs have sufficient damping capacity for reducing the seismic response of frames. Free vibration tests were conducted to confirm the reliability of simplified analysis. Time history response analyses were also conducted and the results are in close agreement with shaking table test results.