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Structural instability of nonlinear plates modelling suspension bridges: Mathematical answers to some long-standing questions |
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| Affiliation: | 1. Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy;2. Dipartimento di Scienze e Innovazione Tecnologica, Università del Piemonte Orientale “Amedeo Avogadro”, Viale Teresa Michel 11, 15121 Alessandria, Italy;3. Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy;1. Institute of Mathematics, University of Bordeaux, UMR CNRS 5251, 3 ter place de la Victoire, 33076 Bordeaux Cedex, France;2. Laboratoire: Systèmes Dynamiques et Applications, Faculty of Sciences, Department of Mathematics, University of Tlemcen, Algeria;3. Faculty of Sciences, Departement of Mathématics, University of Tlemcen, Algeria;1. Laboratoire de Mathématiques et Applications Université Sultan Moulay slimane, Faculté des Sciences et Techniques, B.P.523, Béni-Mellal, Morocco;2. Laboratoire de Mathématiques Jean Leray UMR6629 CNRS / Université de Nantes 2 rue de la Houssinière, BP92208 44322 Nantes, France |
| Abstract: | We model the roadway of a suspension bridge as a thin rectangular plate and we study in detail its oscillating modes. The plate is assumed to be hinged on its short edges and free on its long edges. Two different kinds of oscillating modes are found: longitudinal modes and torsional modes. Then we analyze a fourth order hyperbolic-like equation describing the dynamics of the bridge. In order to emphasize the structural behavior we consider an isolated equation with no forcing and damping. Due to the nonlinear behavior of the cables and hangers, a structural instability appears. With a finite dimensional approximation we prove that the system remains stable at low energies while numerical results show that for larger energies the system becomes unstable. We analyze the energy thresholds of instability and we show that the model allows to give answers to several questions left open by the Tacoma collapse in 1940. |
| Keywords: | Higher order equations Boundary value problems Nonlinear evolution equations |
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