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1.
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.  相似文献   

2.
Numerous structures uplift under the influence of strong ground motion. Although many researchers have investigated the effects of base uplift on very stiff (ideally rigid) structures, the rocking response of flexible structures has received less attention. Related practical analysis methods treat these structures with simplified ‘equivalent’ oscillators without directly addressing the interaction between elasticity and rocking. This paper addresses the fundamental dynamics of flexible rocking structures. The nonlinear equations of motion, derived using a Lagrangian formulation for large rotations, are presented for an idealized structural model. Particular attention is devoted to the transition between successive phases; a physically consistent classical impact framework is utilized alongside an energy approach. The fundamental dynamic properties of the flexible rocking system are compared with those of similar linear elastic oscillators and rigid rocking structures, revealing the distinct characteristics of flexible rocking structures. In particular, parametric analysis is performed to quantify the effect of elasticity on uplift, overturning instability, and harmonic response, from which an uplifted resonance emerges. The contribution of stability and strength to the collapse of flexible rocking structures is discussed. Copyright © 2012 John Wiley & Sons, Ltd.  相似文献   

3.
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.  相似文献   

4.
Seismic response of slender rigid structures with foundation uplifting   总被引：2，自引：0，他引：2  
The rocking of rigid structures uplifting from their support under strong earthquake shaking is investigated. The structure is resting on the surface of either a rigid base or a linearly elastic continuum. A large-displacement approach is adopted to extract the governing equations of motion allowing for a rigorous calculation of the nonlinear response even under near-overturning conditions. Directivity-affected near-fault ground motions, idealized as Ricker wavelets or trigonometric pulses, are used as excitation. The conditions under which uplifting leads to large angles of rotation and eventually to overturning are investigated. A profoundly nonlinear rocking behavior is revealed for both rigid and elastic soil conditions. This geometrically nonlinear response is further amplified by unfavorable sequences of long-duration pulses in the excitation. Moreover, through the overturning response of a toppled tombstone, it is concluded that the practice of estimating ground accelerations from overturning observations is rather misleading and meaningless.  相似文献   

5.
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.  相似文献   

6.
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).  相似文献   

7.
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.  相似文献   

8.
In this paper the rocking response of slender/rigid structures stepping on a viscoelastic foundation is revisited. The study examines in depth the motion of the system with a non‐linear analysis that complements the linear analysis presented in the past by other investigators. The non‐linear formulation combines the fully non‐linear equations of motion together with the impulse‐momentum equations during impacts. The study shows that the response of the rocking block depends on the size, shape and slenderness of the block, the stiffness and damping of the foundation and the energy loss during impact. The effect of the stiffness and damping of the foundation system along with the influence of the coefficient of restitution during impact is presented in rocking spectra in which the peak values of the response are compared with those of the rigid block rocking on a monolithic base. Various trends of the response are identified. For instance, less slender and smaller blocks have a tendency to separate easier, whereas the smaller the angle of slenderness, the less sensitive the response to the flexibility, damping and coefficient of restitution of the foundation. Copyright © 2008 John Wiley & Sons, Ltd.  相似文献   

9.
The investigation of structural single rocking walls (SRWs) continues to gain interest as they produce self-centering lateral load responses with reduced structural damage. The simple rocking model with modifications has been shown to capture these responses accurately if the SRW and its underlying base are infinitely rigid. This paper advances previous rocking models by accounting for (1) the inelastic actions at or near the base of the SRW and (2) the flexural responses within the wall. Included in the proposed advancements are hysteretic and inherent viscous damping associated with these two deformation components so that the total dynamic responses of SRWs can be captured with good accuracy. A system of nonlinear equations of motion is developed, in which the rocking base is discretized into fibers using a zero-length element to locate the associated compressive deformations and damage. The flexural deformations of the rocking body are captured using an elastic term, while the impact events are modeled using impulse-momentum equations. Comparisons with experiments of structural precast concrete and masonry SRWs show that the proposed approach accurately estimates the dynamic responses of different SRWs with and without unbonded posttensioning, for various dynamic excitations and degrees of hysteretic action. Using the proposed approach, a numerical investigation employs different configurations of structural SRWs to quantify the various sources of energy loss, including hysteretic action and impact damping, during various horizontal ground motions.  相似文献   

10.
Shaking table tests were conducted to investigate the response of rectangular wooden blocks and block assemblies of various sizes and slenderness to harmonic and earthquake base excitation. The shaking tests were followed by an analytical and a numerical study of response of single blocks and block assemblies. The analytical study was aimed at establishing criteria for the initiation of rocking and of overturning in response to harmonic base motion and consisted of solving numerically the differential equations of motion of a rigid block on a rigid foundation. The numerical study, in the course of which the response of both single blocks and block assemblies was examined, was implemented by means of the Distinct Element Method (DEM). Prior to the DE simulation of actual shaking tests, preliminary analyses were conducted to confirm numerical stability and to evaluate material and damping parameters. Comparing the recorded time histories with those given by the analytical study and the DE simulation, good agreement was found. The distinct element model in use appeared to follow the highly non-linear motion of rigid body assemblies faithfully to reality. On the basis of the results, provided that the necessary parameters are carefully estimated, the employed DE model can be regarded as an appropriate tool to simulate response of rigid body assemblies to dynamic base excitation.  相似文献   

11.
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.  相似文献   

12.
The rocking response of a rigid, freestanding block in two dimensions typically assumes perfect contact at the base of the block with instantaneous impacts at two distinct, symmetric rocking points. This paper extends the classical two‐dimensional rocking model to account for an arbitrary number of rocking points at the base representing geometric interface defects. The equations of motion of this modified rocking system are derived and presented in general terms. Energy dissipation is modeled assuming instantaneous point impacts, yielding a discrete angular velocity adjustment. Whereas this factor is always less than unity in the classical model, it is possible for this factor to exceed unity in the presented model, yielding a finite increase in the angular velocity at impact and a markedly different rotational response than the classical model predicts. The derived model and the classical model are numerically integrated and compared to the results of recent shake table tests. These comparisons show that the new model significantly enhances agreement in both peak angular displacement and motion decay. The equations of motion and the energy dissipation of the presented model are further investigated parametrically considering the size of the defect, the number of rocking points, and the aspect ratio and size of the block.  相似文献   

13.
The highly complex rocking response of free-standing statues atop multi-drum columns underground excitation resulting in insuperable difficulties for obtaining reliable solution is reexamined analytically. This is achieved after simulating the columns by monolithic viscoelastic cantilevers having structural damping, based on experiments, equivalent to the energy dissipation due to impact and sliding of multi-drum columns. Subsequently, the conditions of rocking (overturning) instability of free-standing rigid blocks (representing the statues) after their uplift from the top surface of the laterally vibrating cantilevers, are established, including overturning with or without impact. Attention focuses on the minimum amplitude ground acceleration which leads to an escaped motion through the vanishing of the angular velocity and acceleration. Maximization of such a minimum amplitude (implying stabilization) of the rigid block is obtained by seeking the optimum combination of values of the slenderness ratio of the column and its height. Analytically derived results based on linearised analyses are in excellent agreement with those obtained via nonlinear numerical analyses.  相似文献   

14.
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.  相似文献   

15.
In order to use rocking as a seismic response modification strategy along both directions of seismic excitation, a three‐dimensional (3D) rocking model should be developed. Since stepping or rolling rocking structural members out of their initial position is not a desirable performance, a rocking design should not involve these modes of motion. To this end, a model that takes the aforementioned constraint into account needs to be developed. This paper examines the 3D motion of a bounded rigid cylinder that is allowed to uplift and sustain rocking and wobbling (unsteady rolling) motion without sliding or rolling out of its initial position (i.e., a 3D inverted pendulum). Thus, the cylinder is constrained to zero residual displacement at the end of its 3D motion. This 3D dynamic model of the rocking rigid cylinder has two DOFs (three when damping is included), making it the simplest 3D extension of Housner's classical two‐dimensional (2D) rocking model. The development of models with and without damping is presented first. They are simple enough to perform extensive parametric analyses. Modes of motion of the cylinder are identified and presented. Then, 3D rocking and wobbling earthquake response spectra are constructed and compared with the classical 2D rocking earthquake response spectra. The 3D bounded rocking earthquake response spectra for the ground motions considered seem to have a very simple linear form. Finally, it is shown that the use of a 2D rocking model may lead to unacceptably unconservative estimates of the 3D rocking and wobbling seismic response. Copyright © 2017 John Wiley & Sons, Ltd.  相似文献   

16.
This paper presents the results of an experimental investigation on the rocking behavior of rigid blocks. Two types of test specimens have been tested, namely M and C types. Nine blocks of the M type and two blocks of the C type with different aspect ratios were tested with varying initial rotational amplitudes and with different materials at the contact interface, namely concrete, timber, steel, and rubber. The results showed that the interface material has significant influence on the free rocking performance of the blocks. Blocks tested on rubber had the fastest energy dissipation followed by concrete and timber bases, respectively. Analysis of the test results has shown that the energy dissipation in the case of tests on a rubber base is a continuous mechanism whereas in the case of tests on rigid bases, i.e. timber and concrete, energy dissipation is a discrete function. Finally, the rocking characteristics of the blocks were calculated using piecewise equations of motion and numerical analysis. It was possible to predict the correct free rocking amplitude response when a reliable value for the coefficient of restitution was used. Copyright © 2010 John Wiley & Sons, Ltd.  相似文献   

17.
Performance-based earthquake engineering requires accurate estimation of structural response associated with different damage states because of strong ground motion. In recent work (Meza-Fajardo and Papageorgiou, 2018, EESD), we demonstrated that a significant contribution to the response of elastic soil-structure systems for high-rise buildings is attributed to base rocking associated with Rayleigh waves. The present paper presents results of a study investigating the effects of Rayleigh waves on the response of soil-structure systems with nonlinear behavior at the level of the superstructure. By introducing a rigid-elastic rotational spring at the base of the building, we take into account the stiffness reduction due to damage to the lateral load-resisting system at its root, and with it, increased displacement demands. Considering different levels of ductility and post-yield stiffness, we investigate the impact of rocking because of Rayleigh waves on maximum and residual interstory drift ratios. Our results indicate that rocking due to surface waves should be an important consideration for design and evaluation of tall buildings, as inelastic action elongates their effective natural period, and consequently, they are more prone to be damaged by resonance and excitation of extended duration because of Rayleigh waves.  相似文献   

18.
Response of pendulums to complex input ground motion   总被引：1，自引：0，他引：1  
Dynamic response of most seismological instruments and many engineering structures to ground shaking can be represented via response of a pendulum (single-degree-of-freedom oscillator). In most studies, pendulum response is simplified by considering the input from uni-axial translational motion alone. Complete ground motion however, includes not only translational components but also rotations (tilt and torsion). In this paper, complete equations of motion for three following types of pendulum are described: (i) conventional (mass-on-rod), (ii) mass-on-spring type, and (iii) inverted (astatic), then their response sensitivities to each component of complex ground motion are examined. The results of this study show that a horizontal pendulum similar to an accelerometer used in strong motion measurements is practically sensitive to translational motion and tilt only, while inverted pendulum commonly utilized to idealize multi-degree-of-freedom systems is sensitive not only to translational components, but also to angular accelerations and tilt. For better understanding of the inverted pendulum's dynamic behavior under complex ground excitation, relative contribution of each component of motion on response variants is carefully isolated. The systematically applied loading protocols indicate that vertical component of motion may create time-dependent variations on pendulum's oscillation period; yet most dramatic impact on response is produced by the tilting (rocking) component.  相似文献   

19.
This paper deals with the dynamic response of free-standing statues on the top surface of slender elastically supported cantilevers subjected to horizontal ground motion. Given that there is no link between the base of the statue and the top surface of the monolithic cantilever the statue is in equilibrium in the vertical direction under its own weight. Attention is focused on the determination of the minimum amplitude ground acceleration which leads to the rocking (overturning) instability of the statue whose mass and rotatory inertia are a priory known. It is assumed that the friction between the base of the statue and the top surface of the cantilever is sufficiently large to prevent sliding so that rocking prevails. After simulating the statue by a rigid block freely supported on the top surface of the elastically restrained monolithic cantilever, a theoretical dynamic analysis of the cantilever–rigid block system under horizontal ground motion is comprehensively presented. Two modes of overturning instability of the free standing rigid block are discussed: instability without or with impact. Criteria for overturning instability of the rigid block associated with the minimum amplitude ground acceleration which leads through the vanishing of the angular velocity to an escaped motion in the phase-plane portrait, are properly assessed.  相似文献   

20.
The classical problem of rocking of a rigid, free-standing block to earthquake ground shaking containing distinct pulses, as is the case of near-fault earthquake motions, is revisited. A rectangular block resting on a rigid base is considered, subjected to a range of idealized single-lobe ground acceleration pulses expressed by a generalized function controlled by a single shape parameter. The problem is treated analytically in the realm of the linearized equations of motion under the assumption of slender block geometry and rocking without slipping. Peak rocking response and overturning criteria for different waveforms are presented in terms of dimensionless closed-form expressions and graphs. Two parameters are employed to this end: dimensionless pulse duration f (i.e., actual pulse duration times characteristic block frequency) and dimensionless uplift strength η (i.e., ratio of minimum required acceleration for initiation of uplift over peak pulse acceleration). The linearized response is compared analytically with the fully non-linear one using an ad hoc energy formulation leading to an approximate closed-form solution. It is shown that the non-linear equations of motion yield more stable response than their linearized counterparts. A brief discussion on scaling laws is provided.  相似文献   

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