Elliptic integral solutions of spatial elastica of a thin straight rod bent under concentrated terminal forces

Mingari Scarpello, Giovanni ; Ritelli, Daniele (2006) Elliptic integral solutions of spatial elastica of a thin straight rod bent under concentrated terminal forces. Meccanica, 41 (5). pp. 519-527.
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In this article we solve in closed form a system of nonlinear differential equations modelling the elastica in space of a thin, flexible, straight rod, loaded by a constant thrust at its free end. Common linearizations of strength of materials are of course not applicable any way, because we analyze great deformations, even if not so large to go off the linear elasticity range. By passing to cylindrical coordinates ρ, θ, z, we earn a more tractable differential system evaluating ρ as elliptic function of polar anomaly θ and also providing z through elliptic integrals of I and III kind. Deformed rod’s centerline is then completely described under both tensile or compressive load. Finally, the planar case comes out as a degeneracy, where the Bernoulli lemniscatic integral appears.

Document type
Mingari Scarpello, Giovanni
Ritelli, Daniele
Elliptic integrals, Spatial elastica, Nonlinear differential equations, Linear elasticity
Deposit date
16 Sep 2011 07:24
Last modified
16 Sep 2011 07:24

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