Non‐linear response analyses of rectangular rigid bodies subjected to horizontal and vertical ground motion

This investigation deals with non‐linear seismic responses of free‐standing rectangular rigid bodies on horizontally and vertically accelerating rigid foundations. The responses are classified into two initial responses and four subsequent responses, accordingly the equations of motion governing the liftoff, slip and liftoff–slip interaction motions and boundary conditions corresponding to commencement and termination of the motions are defined. The time histories of responses presented herein show that the body is sensitive to small changes in the friction coefficient and slenderness, and to the wave properties and intensity of ground motions. Systematic trends are observed: the bodies on the low‐grip foundation avoid overturning while they are allowed to slip regardless of details of ground motions; the long period earthquakes tend to make the body overturn and slip largely. In contrast, the timing when liftoff and slip commences and terminates and their directions do not directly correspond with intensity of ground motions. Moreover, the vertical ground motion adds irregularities on the responses, since it excites or damps the responses. It is concluded that governing equations of motion and boundary conditions in view of discontinuous non‐linear systems are necessary to analyse actual motions of the rectangular rigid bodies subjected to horizontal and vertical ground motion. Copyright © 2002 John Wiley & Sons, Ltd.     

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.     

SLIDE ROTATION OF RIGID BODIES SUBJECTED TO A HORIZONTAL GROUND MOTION

Simultaneous sliding and rotation of the rigid block is studied, when it is subjected to a strong horizontal motion of the ground. The main purpose of the work is to develop an analytical approach, to predict qualitatively possible behaviour of the body. Two phases of the block rotation are investigated: (1) with a fixed direction of relative sliding, (2) after the friction force changes its direction. The conditions of the overturning for the first phase are formulated. In the second phase the main attention is paid to the analysis of conditions when the body returns back to the initial position and when the final overturning occurs. All the analytical results are compared with numerical calculations.     

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.     

Seismic response of slender rigid structures with foundation uplifting

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.     

Rocking instability of free-standing statues atop slender cantilevers under ground motion

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.     

Tsunami Excitation by Submarine Slides in Shallow-water Approximation

-- Landslide-induced tsunamis are receiving increased attention since there is evidence that recent large devastating events have been caused by underwater mass failures. Normally, numerical models are used to simulate tsunami excitation, most of which are based on shallow water, known also as long wave, approximation to the full equations of hydrodynamics. Analytical studies may handle only simplified problems, but help understand the basic features of physical processes. This paper is an analytical investigation of long-water waves excited by rigid bodies sliding on the sea bottom, based on the shallow-water approximation, which is here derived by properly scaling Euler equations for an inviscid, incompressible and irrotational ocean. In one-dimensional (1-D) cases (where motion depends only on one horizontal coordinate), under the further assumptions of small-height slide, which permits the recourse to linear theory, and of flat ocean floor, a solution for arbitrary body shape and velocity is deduced by applying the Duhamel theorem. It is also shown that this theorem can be advantageously used to obtain a general solution in case of a non-flat ocean floor, when the sea bottom follows a special power law, that can be adapted to study reasonable bottom profiles. The characteristics of the excited tsunamis are then evaluated by computing solutions in numerous examples, with special focus on wave pattern and wave evolution. The energy of the wave system is shown to depend on time: it grows expectedly in the initial phase of tsunami generation, when the moving body transfers energy to the water, but it may also diminish later, implying that a certain amount of energy may pass back from water waves to the slide.     

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.     

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.     

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.     

Rocking amplification and strong‐motion duration

This paper characterizes the ability of natural ground motions to induce rocking demands on rigid structures. In particular, focusing on rocking blocks of different size and slenderness subjected to a large number of historic earthquake records, the study unveils the predominant importance of the strong‐motion duration to rocking amplification (ie, peak rocking response without overturning). It proposes original dimensionless intensity measures (IMs), which capture the total duration (or total impulse accordingly) of the time intervals during which the ground motion is capable of triggering rocking motion. The results show that the proposed duration‐based IMs outperform all other examined (intensity, frequency, duration, and/or energy‐based) scalar IMs in terms of both “efficiency” and “sufficiency.” Further, the pertinent probabilistic seismic demand models offer a prediction of the peak rocking demand, which is adequately “universal” and of satisfactory accuracy. Lastly, the analysis shows that an IM that “efficiently” captures rocking amplification is not necessarily an “efficient” IM for predicting rocking overturning, which is dominated by the velocity characteristics (eg, peak velocity) of the ground motion.     

Modelling rocking response via equivalent viscous damping

The assessment of the out-of-plane response of masonry structures has been largely investigated in literature assuming that walls respond as rigid or semi-rigid bodies, and relevant equations of motion of single-degree-of-freedom and multi-degree of freedom systems have been proposed. Therein, energy dissipation has been usually modelled resorting to the classical hypotheses of impulsive dynamics, delivering a velocity-reduction coefficient of restitution applied at impact. In fewer works, a velocity-proportional damping force has been introduced, by means of a viscous coefficient being constant or variable. A review of such models is presented, a criterion for equivalence of dissipated energy is proposed, equations predicting equivalent viscous damping ratios are derived and compared with experimental responses. Finally, predictive equations are examined in terms of incremental dynamic analyses for large sets of natural ground motions.     

The relevance of energy damping in unreinforced masonry rocking mechanisms. Experimental and analytic investigations

Existing unreinforced masonry buildings frequently suffer out-of-plane local collapse mechanisms when undergoing earthquake ground motion. The energy damping that occurs during the motion, due to impacts of a wall against the foundation or against other walls, is a relevant parameter on the response. An experimental investigation has been carried out to estimate the dissipation of kinetic energy that takes place during free oscillations. Restraint conditions allow for two-sided rocking (wall resting on a foundation) and one-sided rocking (wall resting on a foundation adjacent to transverse walls). Five specimens have been tested, modelling walls acted out-of-plane (fa?ades). When one-sided rocking is under consideration, different depths of the contact surface between fa?ade and transverse walls are considered. In the case of two-sided rocking, the experimental coefficient of restitution is slightly lower than the analytic coefficient. In the case of one-sided rocking, an analytic formulation is proposed and this is compared against experimental data. Although the coefficient of restitution of one-sided rocking is less than half that of two-sided rocking, it is not equal to zero. Thus, it cannot induce a sudden stop of the motion. Hence, nonlinear time history analyses performed under this assumption may prove unsafe. Moreover, a comparison has been carried out between overturning maps, induced by twenty natural accelerograms, computed for the analytic coefficient of restitution and those computed for the experimental coefficient of restitution. The increased energy dissipation reduces the frequency of overturning and causes a more regular behaviour.     

Overturning of top profile of the Koyna Dam during severe ground motion

The Koyna Dam in India was subjected to a severe earthquake on 11 December 1967 with its epicentre very close to the dam site. During this earthquake, higher non-overflow monoliths of the dam suffered significant damage. In the highest non-overflow monolith, a horizontal crack occurred at the level where there was an abrupt change in the downstream slope. The dynamic behaviour of the top profile of this monolith of the dam above the crack has been investigated treating it as a rigid body. The study shows that the overturning of the cracked portion of the dam will not occur due to the severest anticipated ground motion at the site. However, to prevent the seepage of water and as a permanent remedial measure, strengthening of the dam is necessary but no emergency measures need be taken.     

Criteria for initiation of motion of rigid bodies to base excitation

It is known that when an unanchored rigid body is placed on a horizontal base which oscillates horizontally, it may undergo one of four modes of motion: rest, slide, slide–rock and rock. Initiation of a rigid body into these modes depends on the slenderness ratio of the body, the coefficient of friction between the body and the base and the acceleration of the base. In this study, the coefficient of friction and the base acceleration are considered random. For specific probability distributions of the coefficient of friction and the peak base acceleration, and for various values of the mean and standard deviation of the two random variables, the probabilities of occurrence of these modes of motion are obtained for a body of given slenderness ratio. It is shown that randomness of the coefficient of friction and base peak acceleration should not be ignored. Copyright © 2005 John Wiley & Sons, Ltd.     

Initiation of motion of a free‐standing body to base excitation

This study is concerned with the conditions for the initiation of various modes of response of a free‐standing rigid body, initially at rest, placed on a frictional horizontal base which undergoes earthquake‐like accelerations in both the horizontal and vertical directions. These conditions are derived using equations of motion appropriate for each mode of motion. The analysis shows that an equivalent horizontal base acceleration may be constructed by dividing the time history of horizontal base acceleration by the sum of gravitational acceleration and the time history of vertical base acceleration. The criteria governing each mode of response of a body of given aspect ratio are then presented graphically with the magnitude of the equivalent base acceleration as abscissa and the coefficient of friction between the body and the base as ordinate. The study shows that a body is more stable while the vertical base acceleration is upwards than when it is absent as expected. When the vertical base acceleration is downwards, although the body is very likely to be lifted off the base, it is nevertheless possible to rock, and slide and rock simultaneously provided the value of coefficient of friction is sufficiently high and the downward vertical base acceleration is not too different from gravitational acceleration. Copyright © 2007 John Wiley & Sons, Ltd.     

Evaluating simplified models in predicting global seismic responses of a shake table–test building isolated by triple friction pendulum bearings

This paper evaluates the ability of simplified superstructure models, including two shear frame models and a single-story model, in predicting global responses of a full-scale five-story steel moment-frame buildings isolated by triple friction pendulum bearings subjected to earthquake motions. The investigated responses include displacement of the isolation system, roof drift, story drift, and floor acceleration. Mechanical properties of the simplified superstructure models were derived from the modal information of a verified full 3-D model. The comparison between the analytical responses and experimental responses shows that the simplified models can well predict the displacement of the isolation system. Furthermore, the shear-frame models are adequate for predicting floor acceleration when the specimen is subjected to horizontal ground motions. However, when the specimen is subjected to 3-D motions, the shear-frame models un-conservatively predict floor acceleration. The full 3-D model improves the prediction of story drift compared with the simplified models for both horizontal and 3-D motions.     

Neural network analysis of overturning response under near-fault type excitation

Under strong seismic excitation, a rigid block will uplift from its support and undergo rocking oscillations which may lead to （complete） overturning. Numerical and analytical solutions to this highly nonlinear vibration problem are first highlighted in the paper and then utilized to demonstrate how sensitive the overturning behavior is not only to the intensity and frequency content of the base motion, but also to thc presence of strong pulses, to their detailed sequence, and even to their asymnletry. Five idealised pulses capable of representing ＂rupture-directivity＂ and ＂fling＂ affected ground motions near the fault, are utilized to this end ： the one-cycle sinus, the one-cycle cosinus, the Ricker wavelet, the truncated （T）-Ricker wavelet, and the rectangular pulse ＂Overturning-Acceleration Amplification＂ and ＂Rotation＂ spectra are introduced and presented. Artificial neural network modeling is then developed as an alternative numerical solution. The neural network analysis leads to closed-form expressions for predicting the overturning failure or survival of a rigid block, as a function of its geometric properties and the characteristics of the excitation time history. The capability of the developed neural network modeling is validated through comparisons with the numerical solution. The derived analytical expressions could also serve as a tool for assessing the destructiveness of near-fault ground motions, for structures sensitive to rocking with foundation uplift.     

Controlled overturning of unanchored rigid bodies

Typical small hospital and laboratory equipment and general supplies cannot be anchored to resist earthquake motions. In order to protect these non‐structural components, a common procedure is to provide barriers to restrain overturning of objects on shelves and other furniture. In many cases this option is not available, especially for hospital equipment, because of other functional requirements. This work presents an alternative approach. The method proposed here does not avoid overturning, but controls the direction of overturning by providing an inclination to the support base so that the overturning occurs in a preferential direction towards a safe area. For example, objects on shelves, could overturn towards the inside or a wall, and equipment on tables could overturn away from the edge. In both cases this would not only reduce the damage to the particular items, but reduce the amount of debris on the floor. In order to determine the proper inclination of the base, specific rigid bodies are analytically evaluated for bi‐directional excitation obtained from 314 earthquake records, in approximately 7500 cases. For each case, several inclination angles are evaluated. Finally, a parametric curve is adjusted to the data, given a relation between angle of inclination and percentage of controlled overturning cases. In all cases a 7° angle gives more than 98% confidence of controlled overturning. The design expressions were later compared with experimental results obtained on a six‐degree‐of‐freedom shake table; confirming the analytical expressions. Copyright © 2006 John Wiley & Sons, Ltd.