Equilibria of a clamped Euler beam (Elastica) with distributed load: large deformations

TL;DR

This paper investigates the equilibrium shapes of a clamped Euler beam under uniform load, characterizing minimizers, deriving conditions, and analyzing stability for large deformations.

Contribution

It introduces novel equilibrium shapes, characterizes energy minimizers, and provides stability criteria for curled configurations of the elastica.

Findings

Existence of curled local minimizers proven.

Sufficient conditions for stability and instability established.

Numerical applicability of stability conditions demonstrated.

Abstract

We present some novel equilibrium shapes of a clamped Euler beam (Elastica from now on) under uniformly distributed dead load orthogonal to the straight reference configuration. We characterize the properties of the minimizers of total energy, determine the corresponding Euler-Lagrange conditions and prove, by means of direct methods of calculus of variations, the existence of curled local minimizers. Moreover, we prove some sufficient conditions for stability and instability of particular solutions of the Euler-Lagrange conditions that can be applied to numerically found curled shapes.