Unsteady elastic diffusion bending model for a bernully-euler beam on a winkler foundation | Mekhanika | kompozitsionnykh | materialov i konstruktsii

Unsteady elastic diffusion bending model for a bernully-euler beam on a winkler foundation

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We consider the unsteady elastic diffusion vibrations problem of an orthotropic Bernoulli-Euler beam on an elastic foundation under the action of a distributed transverse load. The Winkler model is used as an elastic foundation model. For the mathematical formulation, we use the system of Bernoulli-Euler beam bending equations taking into account diffusion. These equations are obtained with use the d’Alembert variational principle, which is applied to the elastic diffusion model for a continuum. The resulting model consider the diffusion fluxes relaxation. The problem formulation is closed by homogeneous boundary conditions, which expressing the simple support conditions and zero initial conditions (internal disturbances absence at the initial time). The problem solution is sought using the Green’s functions method and is represented as convolutions of Green’s functions with functions defining unsteady volumetric disturbances. To find the Green’s functions, the integral Laplace transform in time and the expansion in Fourier series in the longitudinal coordinate are used. As a result, the original system of equations for elastic diffusion beam vibrations is reduced to the linear system of algebraic equations with respect to the sought functions Fourier coefficients in the Laplace transformation. The Laplace transform inversion is done analytically due to residues and operational calculus tables. Calculation examples for a beam with rectangular section are considered. The beam deflections and the diffusants concentration increments under the action of a impulsively applied distributed transverse load are found. Numerical study of the mechanical and diffusion fields interaction in a beam is performed. We used three-component continuum as an example. The solution is presented in analytical form and in the graphs form of the displacement fields and concentration increments as functions of time and coordinate. At the end of the article, the main conclusions about the coupling effect of the stress-strain state and mass transfer in the beam are represented.