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Fractal-like overturning maps for stacked rocking blocks with numerical and experimental validation

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Fractal-like overturning maps for stacked rocking blocks with numerical and experimental validation. / Anagnostopoulos, Sokratis; Norman, James; Mylonakis, George.

In: Soil Dynamics and Earthquake Engineering, Vol. 125, 105659, 01.10.2019.

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Anagnostopoulos, Sokratis ; Norman, James ; Mylonakis, George. / Fractal-like overturning maps for stacked rocking blocks with numerical and experimental validation. In: Soil Dynamics and Earthquake Engineering. 2019 ; Vol. 125.

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@article{b83e95c0315b4da3b45ecd393ed8944f,
title = "Fractal-like overturning maps for stacked rocking blocks with numerical and experimental validation",
abstract = "A novel, compact mathematical formulation is presented to describe the dynamic rocking response of single and double block systems subjected to gravity and/or ground excitation. The derivation of the closed-form solutions for impact and motion is based on the Euler-Lagrange equation and the conservation of angular momentum, and combines all the different cases of possible block relative rotating and impact modes (16 in total) into a single set of equations without the need of transient expressions. The derived equations that describe the impact modes are the equivalent to the expression derived by Housner and depend on the angular velocity of the blocks before impact. The analytical model is integrated numerically via an ad hoc algorithm and its reliability & accuracy are verified after various self-consistency tests and comparisons with the literature. In addition, several shaking table experiments were conducted in EQUALS laboratory in Bristol, set-up constructed to test free and forced rocking motion of single and double block configurations. The error margins of the measurements are determined, and the extracted data are in good agreement with the numerical results for most examined cases. The ideal Housner restitution coefficient of single block impact to a rigid base is adjusted to match experimental conditions, and it is found to be correlated with the block aspect ratio. The forced rocking of a two-block system is shown to exhibit numerous different response patterns depending on the excitation conditions. The integrated model is finally applied to produce normalised overturning maps for double block systems, subjected to single-pulse sine inputs, which uncover the existence of a fractal-type behaviour. This previously unsuspected trait of multi-block systems is reminiscent of the chaotic behaviour exhibited by a classical double pendulum and suggests that the risk of overturning can only be evaluated on a probabilistic sense.",
keywords = "Analytical model of motion and impacts, Compact formulation, Dynamic response, Euler-Lagrange equation, Fractal-like overturning maps, Laboratory measurements of free and excited blocks, Restitution coefficient, Rocking rigid blocks, Stacked blocks",
author = "Sokratis Anagnostopoulos and James Norman and George Mylonakis",
year = "2019",
month = "10",
day = "1",
doi = "10.1016/j.soildyn.2019.04.033",
language = "English",
volume = "125",
journal = "Soil Dynamics and Earthquake Engineering",
issn = "0267-7261",
publisher = "Elsevier",

}

RIS - suitable for import to EndNote

TY - JOUR

T1 - Fractal-like overturning maps for stacked rocking blocks with numerical and experimental validation

AU - Anagnostopoulos, Sokratis

AU - Norman, James

AU - Mylonakis, George

PY - 2019/10/1

Y1 - 2019/10/1

N2 - A novel, compact mathematical formulation is presented to describe the dynamic rocking response of single and double block systems subjected to gravity and/or ground excitation. The derivation of the closed-form solutions for impact and motion is based on the Euler-Lagrange equation and the conservation of angular momentum, and combines all the different cases of possible block relative rotating and impact modes (16 in total) into a single set of equations without the need of transient expressions. The derived equations that describe the impact modes are the equivalent to the expression derived by Housner and depend on the angular velocity of the blocks before impact. The analytical model is integrated numerically via an ad hoc algorithm and its reliability & accuracy are verified after various self-consistency tests and comparisons with the literature. In addition, several shaking table experiments were conducted in EQUALS laboratory in Bristol, set-up constructed to test free and forced rocking motion of single and double block configurations. The error margins of the measurements are determined, and the extracted data are in good agreement with the numerical results for most examined cases. The ideal Housner restitution coefficient of single block impact to a rigid base is adjusted to match experimental conditions, and it is found to be correlated with the block aspect ratio. The forced rocking of a two-block system is shown to exhibit numerous different response patterns depending on the excitation conditions. The integrated model is finally applied to produce normalised overturning maps for double block systems, subjected to single-pulse sine inputs, which uncover the existence of a fractal-type behaviour. This previously unsuspected trait of multi-block systems is reminiscent of the chaotic behaviour exhibited by a classical double pendulum and suggests that the risk of overturning can only be evaluated on a probabilistic sense.

AB - A novel, compact mathematical formulation is presented to describe the dynamic rocking response of single and double block systems subjected to gravity and/or ground excitation. The derivation of the closed-form solutions for impact and motion is based on the Euler-Lagrange equation and the conservation of angular momentum, and combines all the different cases of possible block relative rotating and impact modes (16 in total) into a single set of equations without the need of transient expressions. The derived equations that describe the impact modes are the equivalent to the expression derived by Housner and depend on the angular velocity of the blocks before impact. The analytical model is integrated numerically via an ad hoc algorithm and its reliability & accuracy are verified after various self-consistency tests and comparisons with the literature. In addition, several shaking table experiments were conducted in EQUALS laboratory in Bristol, set-up constructed to test free and forced rocking motion of single and double block configurations. The error margins of the measurements are determined, and the extracted data are in good agreement with the numerical results for most examined cases. The ideal Housner restitution coefficient of single block impact to a rigid base is adjusted to match experimental conditions, and it is found to be correlated with the block aspect ratio. The forced rocking of a two-block system is shown to exhibit numerous different response patterns depending on the excitation conditions. The integrated model is finally applied to produce normalised overturning maps for double block systems, subjected to single-pulse sine inputs, which uncover the existence of a fractal-type behaviour. This previously unsuspected trait of multi-block systems is reminiscent of the chaotic behaviour exhibited by a classical double pendulum and suggests that the risk of overturning can only be evaluated on a probabilistic sense.

KW - Analytical model of motion and impacts

KW - Compact formulation

KW - Dynamic response

KW - Euler-Lagrange equation

KW - Fractal-like overturning maps

KW - Laboratory measurements of free and excited blocks

KW - Restitution coefficient

KW - Rocking rigid blocks

KW - Stacked blocks

UR - http://www.scopus.com/inward/record.url?scp=85067898352&partnerID=8YFLogxK

U2 - 10.1016/j.soildyn.2019.04.033

DO - 10.1016/j.soildyn.2019.04.033

M3 - Article

VL - 125

JO - Soil Dynamics and Earthquake Engineering

JF - Soil Dynamics and Earthquake Engineering

SN - 0267-7261

M1 - 105659

ER -