Abstract
The stress induced in a loaded beam will not exceed some threshold, but also its maximum deflection, as for all the elastic systems, will be controlled. Nevertheless, the linear beam theory fails to describe the large deflections; highly flexible linear elements, namely, rods, typically found in aerospace or oil applications, may experience large displacements—but small strains, for not leaving the field of linear elasticity—so that geometric nonlinearities become significant. In this article, we provide analytical solutions to large deflections problem of a straight, cantilevered rod under different coplanar loadings. Our researches are led by means of the elliptic integrals, but the main achievement concerns the Lauricella hypergeometric functions use for solving elasticity problems. Each of our analytic solutions has been individually validated by comparison with other tools, so that it can be used in turn as a benchmark, that is, for testing other methods based on the finite elements approximation.
Highlights
As early as 1691, Jakob Bernoulli proposed to find out the deformed centerline planar “elastica” of a thin, homogeneous, straight, and flexible rod under a force applied at its end, and ignoring its weight
Almost half a century ago, the analytic treatment of flexible rods inspired a nice book, 6, which collects many problems ordered according to the constraint type, and all solved by the elliptic integrals of I and II kind, while the III kind and Theta functions appear there marginally in the 6th Chapter concerning the three dimensional deformations
We are looking for the shape of a L, EJ slender and thin cantilever clamped at the bottom of a pool of calm water
Summary
As early as 1691, Jakob Bernoulli proposed to find out the deformed centerline planar “elastica” of a thin, homogeneous, straight, and flexible rod under a force applied at its end, and ignoring its weight. Bend it, stretch it and shake it, but we cannot break it, it is completely elastic It is a mathematical object, and does not exist in the physical world; yet it finds wide application in structural and biological mechanics: columns, struts, cables, thread, and DNA have all been modeled with rod theory. A rod is modelled as a curve with effective mechanical properties such as bending and torsional stiffnesses, see 20. Such problems belong to statics, they have a strong relation with dynamics; arclength along the rod plays a role similar to time in a dynamical system. The closed form solutions for these problems are still important to the practical point that the accuracy of the approximate methods can be precisely evaluated by these solutions, to say nothing of the mathematical importance
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