Abstract
In this paper, the dynamics and the buckling loads for an Euler–Bernoulli beam resting on an inhomogeneous elastic, Winkler foundation are studied. An analytical, asymptotic method is proposed to determine the stability of the Euler–Bernoulli beam for various types of inhomogeneities in the elastic foundation taking into account different types of damping models. Based on the Rayleigh variation principle, beam buckling loads are computed for cases of harmonically perturbed types of inhomogeneities in the elastic foundation, for cases of point inhomogeneities in the form of concentrated springs in the elastic foundation, and for cases with rectangular inclusions in the elastic foundation. The investigation of the beam dynamics shows the possibility of internal resonances for particular values of the beam rigidity and longitudinal force. Such types of resonances, which are usually typical for nonlinear systems, are only possible for the beam due to its inhomogeneous foundation. The occurrence of so-called added mass effects near buckling instabilities under the influence of damping have been found. The analytical expressions for this “added mass” effect have been obtained for different damping models including space hysteresis types. This effect arises as a result of an interaction between the main mode, which is close to instability, and all the other stable modes of vibration.
Highlights
Buckling of an Euler–Bernoulli beam (E–B) resting on an elastic Winkler foundation has been thoroughly studied from various points of view in many engineering fields for more than 80 years
We consider the problem on how to determine the buckling load of an Euler–Bernoulli beam resting on an inhomogeneous elastic Winkler foundation taking into account different damping models
The dynamics of and the buckling load for an Euler–Bernoulli beam resting on an inhomogeneous elastic Winkler foundation are studied
Summary
Buckling of an Euler–Bernoulli beam (E–B) resting on an elastic Winkler foundation has been thoroughly studied from various points of view in many engineering fields for more than 80 years. The dynamics and buckling loads for an E–B beam resting on an inhomogeneous elastic Winkler foundation plays an importing role in the study of problems of soil–solid interaction [2]. In [12] free vibrations of an Euler–Bernoulli beam resting on a variable Winkler foundation is considered. In [21] the finite difference and the finite element methods are applied to determine the natural frequencies of non-prismatic and non-homogeneous beams, subjected different boundary conditions and resting on a variable Winkler foundation. We consider the problem on how to determine the buckling load of an Euler–Bernoulli beam resting on an inhomogeneous elastic Winkler foundation taking into account different damping models (including space hysteresis type of models). This interaction is induced by damping, and we will discuss how it depends on the type of damping model
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