Cantilever Beam Equation for Almost Arbitrary Deflections: Derivation and Worked Examples

TL;DR

This paper derives a non-linear differential equation that accurately models the shape of cantilever beams under various deflections, depending on key parameters like boundary angle and material properties.

Contribution

It introduces a new exact equation for cantilever beam shapes and explores its parameter space for practical application.

Findings

Derived a non-linear 4th-order differential equation for beam shape

Identified key parameters influencing beam deflection

Mapped parameter space for practical use

Abstract

We derived a non-linear 4th-order ordinary differential equation the solutions of which lead to the exact shapes of the cantilever beam. The result of the equation in a non-dimensional form was found to depend on two parameters only: the angle of the beam at the fixed end, and the parameter encompassing the material characteristics and geometry of the beam. The parameter space was explored in detail and the results were used to suggest the areas in which they could be applied.