Variational formulation of the Melan equation

TL;DR

This paper derives a variational formulation of the nonlinear, nonlocal Melan equation for suspension bridges, proving existence of solutions and highlighting differences from simplified models.

Contribution

It introduces a variational approach to the Melan equation, establishing solution existence and analyzing qualitative differences from simplified models.

Findings

Existence of at least one solution is proven.

The problem is nonlinear and nonlocal with hinged endpoints.

Numerical results suggest potential uniqueness of solutions.

Abstract

The Melan beam equation modeling suspension bridges is considered. A slightly modified equation is derived by applying variational principles and by minimising the total energy of the bridge. The equation is nonlinear and nonlocal, while the beam is hinged at the endpoints. We show that the problem always admits at least one solution whereas the uniqueness remains open although some numerical results suggest that it should hold. We also emphasize the qualitative difference with some simplified models.