Elliptic integral solution of the extensible elastica with a variable length under a concentrated force

In problems of beams that may move relative to one of their spatially fixed supports, the length of the reference configuration of the beam that is currently located in between the supports depends on the deformation and therefore on the intensity of the external loads. In the present paper, the large deformation of a slender beam under a concentrated force is considered, for which one end is clamped while the other may slide through another clamping device in horizontal direction. The geometric symmetry properties allow the complete structure to be partitioned into four similar parts, each of which represents a cantilever beam under a tip force. In the analysis, a beam theory based on Reissner’s geometrically exact relations for the plane deformation of beams is utilized, from which the equation of a cantilever beam with an extensible axis is obtained by adopting a linear dependence of the stress resultants on the generalized strain measures. For this non-linear equation, a closed-form solution in terms of elliptic integrals is presented, from which the equilibrium shape of the entire structure is constructed afterward. The mid-span deflection and the additional length, by which the reference configuration exceeds the distance between the supports, are discussed and compared with the relations of the inextensible elastica, which exhibits bending deformation only. Moreover, a critical load is found for which no equilibrium configuration exists.

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