Large deformation and stability of an extensible elastica with an unknown length

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Abstract

A beam resting on spatially fixed supports may slide relatively to these as soon as external forces are applied. Consequently, the length of the portion of the reference configuration, which is currently located between the supports, depends on the loading and therefore is not known in advance. In the present paper, the problem of a slender beam under a uniformly distributed force is investigated, which is clamped at one side but may slide through another clamping device in axial direction at the opposite side. In combination with a suitable coordinate transformation by which the numerical treatment is simplified, a finite element approach is utilized to determine the equilibrium shape for the maximum critical load that can be imposed on the beam. In the course of this, the influence of the extensibility of the beam axis is studied. A theory based on Reissner's geometrically exact relations for the plane deformation of beams is adapted such that it allows constitutive relations on stress-strain level to be integrated consistently. In addition to the classical equations of the extensible elastica, a constitutive model derived from the St. Venant-Kirchhoff material of non-linear continuum mechanics is studied. The results obtained in this survey are finally compared to those from the linear beam theory, which turns out to be incapable of describing the problem under consideration in a satisfactory manner. © 2011 Elsevier Ltd. All rights reserved.

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Humer, A., & Irschik, H. (2011). Large deformation and stability of an extensible elastica with an unknown length. International Journal of Solids and Structures, 48(9), 1301–1310. https://doi.org/10.1016/j.ijsolstr.2011.01.015

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