Confirmation of Lagrange Hypothesis for Twisted Elastic Rod

TL;DR

This paper proves that the Lagrange hypothesis, stating the optimal elastic rod has a constant cross-section, holds true for Greenhill's torque buckling problem, confirming a longstanding assumption in structural optimization.

Contribution

It provides a rigorous proof that the Lagrange hypothesis applies to Greenhill's problem, affirming the isoperimetric inequality in this context.

Findings

Lagrange hypothesis is valid for Greenhill's torque buckling.

The isoperimetric inequality is affirmed for this problem.

Supports the assumption of constant cross-section in optimal elastic rods.

Abstract

The history of structural optimization as an exact science begins possibly with the celebrated Lagrange problem: to find a curve which by its revolution about an axis in its plane determines the rod of greatest efficiency. The Lagrange hypothesis, that the optimal rod possesses the constant cross-section was abandoned for Euler buckling problem. In this Article the Lagrange hypothesis is proved to be valid for Greenhill's problem of torque buckling. The corresponding isoperimetric inequality is affirmed.