On the Derivation of the Equation for the Deflection of Thin Beams

Abstract

It is shown that the usual derivation of the equation for the deflection of thin beams that employs the well-known equation for the radius of curvature of a plane curve in known Cartesian coordinates is inconsistent with the assumption of an unstrained neutral axis. A derivation is presented that is consistent with that assumption and it is shown that with the inconsistent derivation, the lowest-order nonlinear description obtained softens the beam while the equivalent description obtained with the consistent derivation stiffens the beam. This latter nonlinear description gives the lowest order nonlinear correction to the Euler buckling load that results from the general Kirchhoff solution.