A Unified Variational Framework for Planar Elastica with General Distributed Loads

TL;DR

The paper addresses integrating general distributed loads into planar elastica by formulating a variational framework that places all load effects directly in the energy functional. Using a Fubini-based integral-reduction, nested load integrals are converted to single-integral expressions, yielding compact Euler–Lagrange equations with clearly separated load contributions. It demonstrates the method on hard magnetic rods and the 2D heavy elastica, reproducing classical results and enabling straightforward inclusion of magnetic and gravitational effects; the approach is extended to the X–Z plane with gravity. This creates a modular, extensible tool for designing planar elastic-rod structures under multiple body forces, while remaining restricted to planar deformations and requiring known a priori field distributions.

Abstract

We present a simple variational framework for planar elastica that enables distributed energies, such as gravitational loading or magnetic body torques, to be incorporated in a modular and unified manner. The formulation is based on expressing all load induced contributions directly at the level of the energy functional, which avoids the force balance constructions used in classical treatments such as Wang (1986) and makes the inclusion of additional physical effects straightforward. The resulting planar energy functional yields compact governing equations in which the contributions of individual load types remain clearly separated. We demonstrate that the framework reproduces the classical heavy elastica equations exactly and naturally accommodates magnetic energy terms commonly used in hard magnetic rod models. Although mathematically elementary, the formulation provides a clean and extensible structure for describing planar rod deformations under general distributed loads.

Figures (1)

• Figure 1: Comparison of the corrected equation with the classical Euler elastica