Nonlinear Analysis 106 (2014) 18–34 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na Bending and stretching energies in a rectangular plate modeling suspension bridges Mohammed Al-Gwaiz a , Vieri Benci a,b , Filippo Gazzola c,∗ a Department of Mathematics, King Saud University, Riyadh, Saudi Arabia b Dipartimento di Matematica, Università di Pisa, Italy c Dipartimento di Matematica, Politecnico di Milano, Italy article info Article history: Received 25 January 2014 Accepted 16 April 2014 Communicated by Enzo Mitidieri MSC: 35G30 74B20 35J35 Keywords: Nonlinear elasticity Prestressed plates Bending and stretching energies Variational methods abstract 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. © 2014 Published by Elsevier Ltd. 1. Introduction Consider a rectangular plate hinged at two opposite edges and free on the remaining two edges. We have in mind a suspension bridge and our purpose is to study several different models to describe its behavior. We view the roadway of the bridge as a long narrow rectangular thin plate, hinged on its short edges where the bridge is supported by the ground, and free on its long edges. Let L denote its length and 2ℓ denote its width; a realistic assumption is that 2ℓ ∼ = L 100 . For simplicity, we take L = π so that, in the sequel, Ω = (0,π) × (−ℓ, ℓ) ⊂ R 2 . (1) This model is considered in [1] where the analysis of the bending energy of the plate leads to a fourth order elliptic equation. However, motivated by the presence of free parts of the boundary, the stretching energy was neglected. A more accurate analysis would have to take account of the stretching energy. From a mathematical point of view, one may notice that H 2 ⊂ H 1 and that the H 2 -norm bounds the H 1 -norm; whence, the stretching energy may be considered as a ‘‘lower order term’’ when compared with the bending energy. But, as we shall see, the former plays an important role in the model and a deep motivation to introduce the stretching energy comes from structural engineering and physics. Concrete is weakly elastic and heavy loads can produce cracks. On the other hand, metals are more elastic and react to loads by bending. ∗ Corresponding author. Tel.: +39 0223994637. E-mail addresses: malgwaiz@ksu.edu.sa (M. Al-Gwaiz), benci@dma.unipi.it (V. Benci), filippo.gazzola@polimi.it (F. Gazzola). http://dx.doi.org/10.1016/j.na.2014.04.011 0362-546X/© 2014 Published by Elsevier Ltd.