Three alternative versions of Bresse–Timoshenko theory for beam on pure Pasternak foundation

Elishakoff, Isaac; Tonzani, Giulio Maria; Marzani, Alessandro

doi:10.1016/j.ijmecsci.2017.10.043

Cited by 7 publications

(2 citation statements)

References 21 publications

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“…The static and dynamic properties of a Cosserat-type lattice interface studied by Vasiliev et al [14]. Calculation of eigenfrequencies for a Bresse-Timoshenko beam with any boundary conditions and elastic interaction with a rigid medium is obtained by Wang and Stephens [15], Manevich [16] or Elishakoff et al [17] (see more recently Elishakoff [18] and Challamel and Elishakoff [19]). Bresse-Timoshenko beam theory is merely a one-dimensional Cosserat continuum medium by considering two independent translational and rotational degrees of freedom (Rubin [20] and Exadaktylos [21]).…”

Section: Introductionmentioning

confidence: 99%

Exact solutions for the vibration of finite granular beam using discrete and gradient elasticity cosserat models

Massoumi

Challamel

Lerbet

2021

Journal of Sound and Vibration

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The present study theoretically investigates the free vibration problem of a discrete granular system. This problem can be considered as a simple model to rigorously study the effects of the microstructure on the dynamic behavior of the equivalent continuum structural model. The model consists of uniform grains confined by discrete elastic interactions, to take into account the lateral granular contributions. This repetitive discrete system can be referred to discrete Cosserat chain or a lattice elastic model with shear interaction. First for the simply supported granular beam resting on Winkler foundations, due to the critical frequencies which concern the nature of the dynamic results, the natural frequencies are exactly calculated, starting from the resolution of the linear difference eigenvalue problem. The natural frequencies of such a granular model are analytically calculated for whatever modes. It is shown that the difference equations governed to the discrete system converge to the differential equations of the Bresse-Timoshenko beam resting on Winkler foundation (also classified as a continuous Cosserat beam model) for an infinite number of grains. A gradient Bresse-Timoshenko model is constructed from continualization of the difference equations. This continuous gradient elasticity Cosserat model is obtained from a polynomial or a rational expansion of the pseudo-differential operators, stemming from the continualization process. Scale effects of the granular chain are captured by the continuous gradient elasticity model. The natural frequencies of the continuous gradient Cosserat models are compared with those of the discrete Cosserat model associated with the granular chain. The results clarify the dependency of the beam dynamic responses to the beam length ratio.

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Section: Introductionmentioning

confidence: 99%

Exact solutions for the vibration of finite granular beam using discrete and gradient elasticity cosserat models

Massoumi

Challamel

Lerbet

2021

Journal of Sound and Vibration

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“…More recently, Elishakoff et al [15][16][17] proposed a Timoshenko beam based on slope inertia (SIBT) by using deflection slope instead of bending rotation in the kinetic energy expression. The fourth time derivative term of the Timoshenko beam is eliminated while a new sixth order term is introduced.…”

Section: Introductionmentioning

confidence: 99%

Effect of appropriate rotational inertia on high frequency vibrations in Timoshenko beam theory

Huang,

Chen,

Zhong

2024

Proceedings of the 10th Convention of the European Acoustics Association Forum Acusticum 2023

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In this paper, a simple Timoshenko beam theory (STBT) is proposed and a radiative energy transfer model (RETM) based on STBT theory is established. The appropriate rotational inertia of Timoshenko beam theory (TBT) is discussed and the effects of two forms of rotational inertia on the high frequency vibration of a transverse vibrating beam are evaluated. Based on elastic theory and wave propagation theory, it is found that when the frequency is higher than the critical frequency, there is a second spectrum in TBT theory, which has no obvious physical significance. On the other hand, the so-called slope inertia Bresse Timoshenko (SIBT) is neither consistent nor accurate when the wavelength of the vibrating beam is close to the height of the beam. The theory cannot be used for vibration analysis above the critical frequency. By numerical analysis of the surface, the simple Timoshenko theory established in this paper and the energy model of high frequency vibration are very direct and effective. The effectiveness and simplicity of this model are verified by comparing it with several beam theory wave propagation methods.

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Generalized foundation Timoshenko beam and its calculating methods

Xia

2022

Arch Appl Mech

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Three alternative versions of Bresse–Timoshenko theory for beam on pure Pasternak foundation

Cited by 7 publications

References 21 publications

Exact solutions for the vibration of finite granular beam using discrete and gradient elasticity cosserat models

Exact solutions for the vibration of finite granular beam using discrete and gradient elasticity cosserat models

Effect of appropriate rotational inertia on high frequency vibrations in Timoshenko beam theory

Generalized foundation Timoshenko beam and its calculating methods

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