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Unsteady elastic diffusion bending model for a Timoshenko beam on a Winkler foundation: unsteady elastic diffusion bending model for a Timoshenko beam on a Winkler foundation

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Abstract

We consider an unsteady elastic diffusion vibration problem of an orthotropic Timoshenko beam on an elastic foundation under a distributed transverse load. The Winkler model is used as an elastic foundation model. We use the system of Timoshenko beam bending equations taking into account diffusion for the mathematical problem formulation. These equations are obtained with the d’Alembert variational principle applied to the elastic diffusion continuum model. The resulting model considers the diffusion fluxes relaxation. The problem solution is sought as convolutions of Green’s functions with functions defining unsteady distributed disturbances. The integral Laplace transform in time and the expansion in the Fourier series in the longitudinal coordinate are used to find the Green’s functions. A calculation example for the beam with a rectangular section is considered. The beam deflections and the concentration increments under the action of an impulsively applied distributed transverse load are found. Finally, the main conclusions about the coupling effect of the stress–strain state and mass transfer in the beam are represented.

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Funding

This work was funded by the subsidy from RFBR (Project No. 20-08-00589 A).

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Authors and Affiliations

  1. Moscow Aviation Institute, Research Institute of Mechanics Lomonosov Moscow State University, Moscow, Russia

    A. V. Zemskov

  2. Research Institute of Mechanics Lomonosov Moscow State University, Moscow Aviation Institute, Moscow, Russia

    D. V. Tarlakovskii

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  1. A. V. Zemskov
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  2. D. V. Tarlakovskii
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Zemskov, A.V., Tarlakovskii, D.V. Unsteady elastic diffusion bending model for a Timoshenko beam on a Winkler foundation: unsteady elastic diffusion bending model for a Timoshenko beam on a Winkler foundation. Arch Appl Mech 92, 1355–1366 (2022). https://doi.org/10.1007/s00419-022-02112-6

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