Impulsive Dirac‐delta forces in the rocking motion

Prieto, Francisco Ureña; Lourenço, Paulo B.; Oliveira, Carlos Sousa

doi:10.1002/eqe.381

Cited by 58 publications

(47 citation statements)

References 18 publications

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“…This force behaves as a Dirac-delta force, which is in agreement with the results reported elsewhere [13].…”

Section: Discussionsupporting

confidence: 92%

“…The implementation of such an impulsive force has reported excellent results in a previous work [13].…”

Section: Impact Model Phase Dynamicsmentioning

confidence: 87%

“…This average results in adding a factor of 1/2 (for a more detailed discussion, please, see [13]). Therefore, a factor of 2 should be included in the impulsive term in order to make theoretical and numerical results compatible.…”

Section: Implementation Of Harmonic Forcingmentioning

confidence: 99%

“…being g(r) an arbitrary function and r min given by equation (13). Through a change of variable y = r/r min , and, after some manipulation, the following expression holds …”

Section: Appendix: Dirac-delta Forces and Phase Dynamicsmentioning

confidence: 99%

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On the Rocking Behavior of Rigid Objects

Prieto

Lourenço

2005

Meccanica

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Abstract.A novel formulation for the rocking motion of a rigid block on a rigid foundation is presented in this work. The traditional piecewise equations are replaced by a single ordinary differential equation. In addition, damping effects are no longer introduced by means of a coefficient of restitution but understood as the presence of impulsive forces. The agreement with the classical formalism is very good for both free rocking regime and harmonic forcing excitation.

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“…This force behaves as a Dirac-delta force, which is in agreement with the results reported elsewhere [13].…”

Section: Discussionsupporting

confidence: 92%

“…The implementation of such an impulsive force has reported excellent results in a previous work [13].…”

Section: Impact Model Phase Dynamicsmentioning

confidence: 87%

Section: Implementation Of Harmonic Forcingmentioning

confidence: 99%

“…being g(r) an arbitrary function and r min given by equation (13). Through a change of variable y = r/r min , and, after some manipulation, the following expression holds …”

Section: Appendix: Dirac-delta Forces and Phase Dynamicsmentioning

confidence: 99%

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On the Rocking Behavior of Rigid Objects

Prieto

Lourenço

2005

Meccanica

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“…The fundamental differences between the response of a rocking rigid column (inverted pendulum) and the response of the linear elastic oscillator (regular pendulum) led to the development of the rocking spectrum (Makris and Konstantinidis [10]). More recent studies pertinent to the rocking response of rigid columns have focused on more practical issues such as representation of the impact (Prieto et al [11]), the effect of the flexibility-yielding of the supporting base (Apostolou et al [12], Palmeri and Makris [13]) or the effect of seismic isolation (Vassiliou and Makris [14]). …”

Section: Introductionmentioning

confidence: 99%

Rocking Response and Stability Analysis of an Array of Free-Standing Columns Capped With a Free-Standing Rigid Beam

Makris¹,

Vassiliou²

2014

Proceedings of the 4th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COM

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Abstract. This paper investigates the planar rocking response of an array of free-standing columns capped with a freely supported rigid beam in an effort INTRODUCTIONUnder base shaking slender objects and tall rigid structures may enter into rocking motion that occasionally results in overturning. Early studies on the seismic response of a slender rigid block were presented by Milne [1]; however, it was Housner [2] who uncovered a sizefrequency scale effect which explained why: (a) the larger of two geometrically similar blocks can survive the excitation that will topple the smaller block; and (b) out of two same acceleration amplitude pulses, the one with the longer duration is more capable to induce overturning. Following Housner's seminal paper a number of studies have been presented to address the complex dynamics of one of the simplest man-made structures-the free standing rigid column.Yim et al. [3] conducted numerical studies by adopting a probabilistic approach, Aslam et al. [4] confirmed with experimental studies that the rocking response of rigid blocks is sensitive to system parameters; while Psycharis and Jennings [5] examined the uplift of rigid bodies supported on viscoelastic foundation. Subsequent studies by Spanos and Koh [6] investigated the rocking response due to harmonic steady-state loading and identified "safe" and "unsafe" regions together with the fundamental and suharmonic modes of the system. Their study was extended by Hogan [7], [8] who further elucidated the mathematical structure of the problem by introducing the concepts of orbital stability and Poincare sections. The transient rocking response of free-standing rigid blocks was examined in depth by Zhang and Makris [9] who showed that there exist two modes of overturning: (a) by exhibiting one or more impacts; and (b) without exhibiting any impact. The existence of the second mode of overturning results in a safe region that is located on the acceleration-frequency plane above the minimum overturning acceleration spectrum. The fundamental differences between the response of a rocking rigid column (inverted pendulum) and the response of the linear elastic oscillator (regular pendulum) led to the development of the rocking spectrum (Makris and Konstantinidis In this paper we investigate the planar rocking response of an array of free-standing columns capped with a freely supported rigid beam as shown schematically in Figure 1. Herein we use the term "rocking frame" for the one degree of freedom structure shown in Figure 1. Sliding does not occur either at the pivot points at the base or at the pivot points at the capbeam. Our interest to this problem was partly motivated from the need to explain the remarkable seismic stability of ancient free-standing columns which support heavy free standing epistyles together with the even heavier frieze atop. As an example, Figure 2 shows the entrance view of the late archaic temple of Aphaia in the island of Aegina nearby Athens, Greece. Dates ranging from 510BC to 470BC have been proposed fo...

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Effects of interface material on the performance of free rocking blocks

ElGawady

Quincy

Butterworth

et al. 2011

Earthq Engng Struct Dyn

112

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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.the piecewise equations of motions [3]. Pena et al. [6] used complex coupled rocking rotations and discrete element methods to predict the rocking response of four specimens. Both methods are extremely sensitive to the rocking parameters. Finally, it was found that repeatability of the rocking tests under random vibration does not exist.Shenton and Jones [7] showed that under horizontal ground excitation a rigid block has five modes of response namely rest, rocking, sliding, rocking-sliding, and free-flight. Shenton [8] analytically derived the criteria that govern the initiation of these modes in the static frictionpeak ground acceleration parameter space. Oliveto et al. [9] derived analytical expressions for the minimum acceleration impulses for uplift and overturning. They showed that the minimum overturning impulse of a flexible system is always smaller than the corresponding impulse for the rigid system. Taniguchi [10] showed that the effect of the vertical ground acceleration component on the response criteria is extremely important and went on to derive the criteria for initiation of rocking, sliding, and rocking-sliding.The rocking problem is stiff and highly nonlinear in nature, resulting in a variety of rocking responses even for relatively simple harmonic excitation. For an undamped rocking system, Jeong et al. [11] found that quasi periodic and chaotic motions dominated the response. For a damped rocking system, Wong and Tso [12] showed that out-of-phase harmonic and sub-harmonic responses can be stable, and that all in-phase steady-state rocking responses were unstable. When in-phase periodic vertical acceleration was added to a horizontal periodic excitation, the response changed to quasi periodic and chaotic [11]. The stability regions of harmonic/sub-harmonic responses as well as possible chaotic responses were determined using a discrete mapping technique by Hogan ...

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Impulsive Dirac‐delta forces in the rocking motion

Cited by 58 publications

References 18 publications

On the Rocking Behavior of Rigid Objects

On the Rocking Behavior of Rigid Objects

Rocking Response and Stability Analysis of an Array of Free-Standing Columns Capped With a Free-Standing Rigid Beam

Effects of interface material on the performance of free rocking blocks

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