Non-linear solution for beams in terms of the Weierstrass [formula omitted]-function using new methodology

Abstract

As structures become slender their non-linear aspects become more apparent and needing of assessment. In that spirit, the authors proposed a theory for addressing the effects of these non-linearities in a highly flexible beam akin to an wing in aeroservoelastic analyses regarding piezoelectric control for flutter suppression. This framework was proven quite efficient for it allowed large displacements to be naturally incorporated by means of a set of generalized variables that encoded the beam mechanics (membrane and bending) and in which space some mechanical features could be linearized. Therefore, the authors investigated the consequences of solving analytically a cantilever beam problem subjected to a material load at its free tip by means of that theory and demonstrated the connection between that problem (in particular when it comes to the buckling problem) and the Weierstrass elliptic ℘-function, a relationship not yet demonstrated to the best of the authors’ knowledge. That demonstration is the subject of this article, as well as a comprehensive study of the solutions for some loading conditions in a reference slender beam and the suggestion of further applications that could be developed from the solution found, in particular in FE analysis.