Revisiting the rocking block: closed-form solutions and similarity laws

Dimitrakopoulos, Elias G.; DeJong, Matthew J.

doi:10.1098/rspa.2012.0026

Cited by 159 publications

(173 citation statements)

References 24 publications

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“…The introduction of two-step transformation of the magnitude of the equivalent sinusoidal waves (introduction of the one-cycle sinusoidal wave equivalent to the double impulse and magnification of the equivalent one-cycle sinusoidal wave) is due to the viewpoint of the equivalence of the maximum Fourier amplitude and from the viewpoint of connection to the previously related works by the present authors. (6) The proposed magnified overturning limit exhibits a fairly good correspondence to the other available data for resonance (Makris and Kampas, 2016;Dimitrakopoulos and DeJong, 2012b).…”

Section: Numerical Examples and Discussionsupporting

confidence: 60%

“…Other data from Makris and Kampas (2016) and Dimitrakopoulos and DeJong (2012b) (coefficient of restitution = 0.8) and the West's formula (Milne, 1885) have also been plotted for comparison. The plots from Makris and Kampas (2016) and Dimitrakopoulos and DeJong (2012b) are limited because only the resonant one is picked up. It can be observed that the proposed magnified expression correspond fairly well to other results for resonance.…”

Section: Numerical Examples and Discussionmentioning

confidence: 99%

“…This scenario seems valid because the input limit on the overturning corresponding to this scenario provides a lower limit in general. As for more detailed scenarios, see Ishiyama (1982) and Dimitrakopoulos and DeJong (2012b).…”

Section: Maximum Rotation Of Rigid Block Subjected To Critical Doublementioning

confidence: 99%

See 2 more Smart Citations

Closed-Form Overturning Limit of Rigid Block under Critical Near-Fault Ground Motions

Nabeshima

Taniguchi

Kojima

et al. 2016

Front. Built Environ.

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A closed-form limit on the input level of the double impulse as a substitute of a near-fault ground motion is derived for the overturning of a rigid block. The rocking vibration of the rigid block is formulated by using the conservation law of angular momentum and the conservation law of mechanical energy. The initial rotational velocity after the first impulse and the rotational velocity after the impact are determined by the conservation law of angular momentum. The velocity change after the second impulse is also characterized by the conservation law of angular momentum. The maximum angles of rotation of the rigid block in both the clockwise and anti-clockwise directions, which are needed for the computation of the overturning limit, are derived by the conservation law of mechanical energy. This enables us to avoid the computation of complicated non-linear time-history responses. The critical timing of the second impulse to the first impulse is characterized by the time of impact after the first impulse. It is clarified that the action of the second impulse just after the impact corresponds to the critical timing. It is derived from the closed-form expression of the critical velocity amplitude limit of the double impulse that its limit is proportional to the square root of size, i.e., the scale effect.

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Section: Numerical Examples and Discussionsupporting

confidence: 60%

Section: Numerical Examples and Discussionmentioning

confidence: 99%

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Closed-Form Overturning Limit of Rigid Block under Critical Near-Fault Ground Motions

Nabeshima

Taniguchi

Kojima

et al. 2016

Front. Built Environ.

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“…[5], [6], [7], [8]), as well as approximate formulas for the design of controlled rocking systems. However, few computational models exist for modeling the response of deformable rocking bodies (e.g.…”

Section: Introductionmentioning

confidence: 99%

A New Computational Approach for the Rocking Response of Deformable Bodies Considering Material Nonlinearity

Avgenakis¹,

Psycharis²

2017

Proceedings of the 6th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COM

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Abstract. An increasing interest in the use of rocking members in earthquake resistant structural systems has been observed in recent years. The benefits of such members include the limitation of the damage, re-centering capabilities and reduction of seismic forces transmitted to the rest of the structure due to a yield-like response.Despite the importance of such members, few models able to describe the response of flexible rocking members in structural systems realistically can be found in literature, since most of the research focuses on the response of rigid bodies or ignores the stress nonlinearity near the contact area, which is considered crucial to the accurate determination of the rocking response of deformable rocking bodies. A new macro-element formulation for elastic rocking members has been proposed by the authors in previous papers which is able to take into account this stress nonlinearity, producing results which are very close to the ones produced by conventional finite element programs.Rocking members in structural systems are expected to behave inelastically for large seismic excitations. In this paper, the macroelement presented previously by the authors, which considered the element material to be elastic, is extended to also take material nonlinearity into account. The effect of the nonlinear stress distribution across the rocking interface on the member displacements is determined and approximate formulas for their determination are included in the macro-element formulation. The results produced by the macro-element with the inelastic material are finally compared to the ones of conventional finite element codes, showing very good agreement between the results of the two methods.

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“…If the restoring moment is M rst = I o✓ (t) + mgR sin [↵sgn✓ (t) ✓ (t)] , and the overturning moment is given from the expression M ovtn = mü g (t) R cos [↵sgn✓ (t) ✓ (t)], the equation of motion will be [1,10,11,12]:…”

Section: Seismic Response Of Undamped Rigid Rocking Blocksmentioning

confidence: 99%

Simple Models for the Seismic Response of Rigid or Flexible Blocks

Diamantopoulos¹,

Fragiadakis²

2017

Proceedings of the 6th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COM

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Abstract. The paper examines different approaches for modeling the seismic response of blocks that are either rigid or flexible. Simple models that are based on beam-column elements and can be implemented with common finite element (FE) software are developed. We first study rigid blocks and we compare our findings with results obtained solving directly the block's equation of motion as derived by Housner's theory. We compare different single-degree-of-freedom (SDOF) models which consist of a rigid beam connected to a zero-length, non-linear rotational spring with negative-stiffness, while different options for modelling energy dissipation are considered. The proposed models are compared using simple wavelets and natural ground motion records producing accurate results in all cases. The proposed FE approach allows to also study flexible blocks offering a modeling that is simple and able to provide very accurate solutions using tools that are widely available in existing software, or easy to implement in a FE code.

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Revisiting the rocking block: closed-form solutions and similarity laws

Cited by 159 publications

References 24 publications

Closed-Form Overturning Limit of Rigid Block under Critical Near-Fault Ground Motions

Closed-Form Overturning Limit of Rigid Block under Critical Near-Fault Ground Motions

A New Computational Approach for the Rocking Response of Deformable Bodies Considering Material Nonlinearity

Simple Models for the Seismic Response of Rigid or Flexible Blocks

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