A Critical Study of Howell et al.'s Nonlinear Beam Theory

TL;DR

This paper critically examines Howell et al.'s nonlinear beam theory, demonstrating its inconsistency with fundamental beam equations and deriving a corrected model based on nonlinear plate theory.

Contribution

It identifies flaws in Howell et al.'s nonlinear beam theory and proposes a valid alternative derived from nonlinear plate equations.

Findings

Howell et al.'s theory is invalid for constant curvature deformations.

The linear Euler-Bernoulli beam equation remains valid under such deformations.

A new nonlinear beam equation is derived from Ciarlet's nonlinear plate equations.

Abstract

In our analysis, we show that Howell et al.'s nonlinear beam theory does not depict a representation of the Euler-Bernoulli beam equation, nonlinear or otherwise. The authors' nonlinear beam theory implies that one can bend a beam in to a constant radius of deformation and maintain that constant radius of deformation with zero force. Thus, the model is disproven by showing that it is invalid when the curvature of deformation is constant, while even the linear Euler-Bernoulli beam equation stays perfectly valid under such deformations. To conclude, we derive a nonlinear beam equation by using Ciarlet's nonlinear plate equations and show that our model is valid for constant radius of deformations.