Exact curvature elastica of a thin cantilever under terminal loads. (English) Zbl 1157.74024

Scarpello, Giovanni Mingari; Ritelli, Daniele

J. Geom. Symmetry Phys. 12, 75-92 (2008).

Summary: We study a thin flexible one side-built-in rod under a concentrated terminal force in its elastic equilibrium configuration. In order to make the problem more tractable, a proper set of state variables is chosen, facing with a second-order nonlinear but autonomous boundary value problem, in the rotation \(\varphi(s)\) pertaining to each \(s\)-section. The search of the free end rotation \(\varphi_0\), following the isoperimetric assumption, leads to a numerical sub-problem inside the main boundary value problem. Furthermore, if \(x(s)\) and \(y(s)\) mean the elastica coordinates parametrized with the arclength \(s\), one obtains \(x_0(s)\) and \(y_0(s)\) as elliptic functions of \(s\). Finally, some minor changes have been shown in order to pass from a loading force to a more general free-end load combination, consisting of a force and a couple.

MSC:

74K10

Rods (beams, columns, shafts, arches, rings, etc.)

Keywords:

autonomous boundary value problem; isoperimetric assumption; elliptic functions