Effect of boundary conditions in three alternative models of Timoshenko–Ehrenfest beams on Winkler elastic foundation

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

doi:10.1007/s00707-017-2034-x

Cited by 12 publications

(6 citation statements)

References 20 publications

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“…An important initial motivation for dynamic analysis originates from the needs of railway engineering in the evaluation of the effect of moving loads [7]. In this case, apart from the aforementioned Euler-Bernoulli beam theory, refined Timoshenko-type formulations have also been adapted, see, e.g., [8][9][10][11]; and also more recent publications concerned with harmonic vibrations of a Timoshenko-type beam supported by a Winkler foundation, see [12,13].…”

Section: Introductionmentioning

confidence: 99%

Asymptotic derivation of a refined equation for an elastic beam resting on a Winkler foundation

Erbaş

Kaplunov

Elishakoff

2021

Mathematics and Mechanics of Solids

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A two-dimensional mixed problem for a thin elastic strip resting on a Winkler foundation is considered within the framework of plane stress setup. The relative stiffness of the foundation is supposed to be small to ensure low-frequency vibrations. Asymptotic analysis at a higher order results in a one-dimensional equation of bending motion refining numerous ad hoc developments starting from Timoshenko-type beam equations. Two-term expansions through the foundation stiffness are presented for phase and group velocities, as well as for the critical velocity of a moving load. In addition, the formula for the longitudinal displacements of the beam due to its transverse compression is derived.

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Section: Introductionmentioning

confidence: 99%

Asymptotic derivation of a refined equation for an elastic beam resting on a Winkler foundation

Erbaş

Kaplunov

Elishakoff

2021

Mathematics and Mechanics of Solids

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“…(Here we reproduce the results contained in the work of Manevich and Kolakowski [31] according Figure 1.) Secondly, based on Elishakoff's papers and collaborators and their studies on truncated versions for classical Timoshenko equations [2,[17][18][19] (see also recent contributions of Elishakoff et al [20,[22][23][24][25] ), Almeida Júnior and Ramos [3] showed that the total energy for viscous damping acting on angle rotation of the simplified Timoshenko system given by is exponentially stable regardless of any values of coefficients 1 , 2 , and . It is noteworthy that both set of Equations (1.1)-(1.2) and (1.5)-(1.6) without damping ( = 0) have all main features considered by Timoshenko [43] from modelling point of 1 The second spectrum constitutes an important issue in literature and it has been noticed and studied since the classical work due to Timoshenko.…”

Section: F I G U R Ementioning

confidence: 99%

A new scenario for stability of nonlinear Bresse‐Timoshenko type systems with time dependent delay

et al. 2019

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In this paper, we study the asymptotic behavior for a Bresse-Timoshenko type system with time-dependent delay terms. Our results follow a recent approach given by Almeida Júnior and Ramos (Zeitschrift f̈r angewandte Mathematik und Physik. 68, 1-31. 2017) where they showed that the viscous damping acting on angle rotation of the classical Bresse-Timoshenko model eliminates the damage consequences of the so called second spectrum of frequency. Moreover, for Bresse-Timoshenko type systems which are free of these consequences the assumption of equal wave speeds is not necessary anymore to get the exponential stability. Here, by introducing an appropriate Lyapunov functional we prove the exponential stability for time dependent delay cases regardless of any relationship between wave propagation velocities. K E Y W O R D SBresse-Timoshenko-type systems, exponential decay, phase velocities, second spectrum, time dependent delay 2 0 1 0 M S C 35B35,

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“…However, this frequency does not divide the branch of natural frequencies in two parts. Indeed, it is only related to the case in which relation (20) does not hold.…”

Section: Unified Formulation Of the Problemmentioning

confidence: 99%

“…We underline here that, in order to write a unified formulation for the slope inertia version of Timoshenko-Ehrenfest equation, we assume that inequality given by (20) holds. If not, the obtained slope inertia formulation reduces to the following:…”

Section: Unified Formulation Of the Problemmentioning

confidence: 99%

“…Current paper is devoted to extension of the work by Cazzani, Stochino and Turco to include the Winkler elastic foundation. Specifically, in this paper we deal with the simply supported beam on Winkler foundation, whereas its companion (Elishakoff, Tonzani and Marzani) deals with the remaining boundary conditions.This reference also demonstrates that the current theory of beams that takes into account rotary inertia and shear deformation was developed jointly by Stephen P. Timoshenko (1878‐1972) and Paul Ehrenfest (1880‐1933).…”

Section: Introductionmentioning

confidence: 99%

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Contrasting three alternative versions of Timoshenko‐Ehrenfest theory for beam on Winkler elastic foundation – simply supported beam

et al. 2018

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In this paper the free vibration frequencies of beam on one‐parameter (Winkler) elastic foundation are analyzed via original Timoshenko‐Ehrenfest theory as well as two truncated versions of the Timoshenko‐Ehrenfest beam theory. The traditional version of Timoshenko‐Ehrenfest beam theory is contrasted with the truncated version that lacks the fourth‐order time derivative, as well as with the recently developed version that incorporates slope inertia effect. Advantages and disadvantages of each approach are indicated in the context of free vibrations of the beam on Winkler foundation. In addition an intriguing intermingling phenomenon is discussed in detail, and its implications on three theories.

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Effect of boundary conditions in three alternative models of Timoshenko–Ehrenfest beams on Winkler elastic foundation

Cited by 12 publications

References 20 publications

Asymptotic derivation of a refined equation for an elastic beam resting on a Winkler foundation

Asymptotic derivation of a refined equation for an elastic beam resting on a Winkler foundation

A new scenario for stability of nonlinear Bresse‐Timoshenko type systems with time dependent delay

Contrasting three alternative versions of Timoshenko‐Ehrenfest theory for beam on Winkler elastic foundation – simply supported beam

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