Bending and stretching energies in a rectangular plate modeling suspension bridges

A rectangular plate modeling the roadway of a suspension bridge is considered. Both the contributions of the bending and stretching energies are analyzed. The latter plays an important role due to the presence of the free edges. A linear model is first considered; in this case, separation of variables is used to determine explicitly the deformation of the plate in terms of the vertical load. Moreover, the same method allows us to study the spectrum of the linear operator and the least eigenvalue. Then the stretching energy is introduced without linearization and the equation becomes quasilinear; the nonlinear term also affects the boundary conditions. We consider two quasilinear models; the surface increment model (SIM) in which the stretching energy is proportional to the increment of the surface and a nonlocal model (NLM) introduced by Berger in the 50s (see Berger (1955)). The SIM and the NLM are studied in detail. According to the strength of prestressing we prove the existence of multiple equilibrium positions.