The conditions for plane sections remaining plane

Abstract

The Euler–Bernoulli hypothesis that plane sections before bending remain plane after bending is a cornerstone of the sciences, especially in mechanics and engineering, and an enabler of the Second Industrial Revolution. Although it is widely applied in science and engineering, especially in structural engineering, its general validity has never been established theoretically, except in some special cases such as linear elastic and small deformation conditions. Its applicability to a particular problem requires extensive experimental evaluation or practical observation. Through a thought experiment commonly used in the derivation of physics theory, this paper rigorously deduces the theorem that gives the conditions for the Euler–Bernoulli hypothesis. The new theorem is general and applies to flexural members composed of any number of different materials, regardless of the properties of the materials, magnitude of deformation, and elastic or plastic strain conditions. This new theorem paves the way for future secure applications of this hypothesis without the need for experimental verification before use.