Nonlinear Equations for Beams and Degenerate Plates with Piers

This book develops a full theory for hinged beams and degenerate plates with multiple intermediate piers with the final purpose of understanding the stability of suspension bridges. New models are proposed and new tools are provided for the stability analysis. The book opens by deriving the PDE's based on the physical models and by introducing the basic framework for the linear stationary problem. The linear analysis, in particular the behavior of the eigenvalues as the position of the piers varies, enables the authors to tackle the stability issue for some nonlinear evolution beam equations, with the aim of determining the "best position" of the piers within the beam in order to maximize its stability. The study continues with the analysis of a class of degenerate plate models. The torsional instability of the structure is investigated, and again, the optimal position of the piers in terms of stability is discussed. The stability analysis is carried out by means of both analytical tools and numerical experiments. Several open problems and possible future developments are presented. The qualitative analysis provided in the book should be seen as the starting point for a precise quantitative study of more complete models, taking into account the action of aerodynamic forces. This book is intended for a two-fold audience. It is addressed both to mathematicians working in the field of Differential Equations, Nonlinear Analysis and Mathematical Physics, due to the rich number of challenging mathematical questions which are discussed and left as open problems, and to Engineers interested in mechanical structures, since it provides the theoretical basis to deal with models for the dynamics of suspension bridges with intermediate piers. More generally, it may be enjoyable for readers who are interested in the application of Mathematics to real life problems.

Autorentext

Maurizio Garrione is Assistant Professor of Mathematical Analysis in the Department of Mathematics of the Politecnico di Milano, Italy. His research focus is in ordinary and partial differential equations, differential models and applications.

Filippo Gazzola is Professor of Mathematical Analysis in the Department of Mathematics of the Politecnico di Milano, Italy. His research focus is in partial differential equations in a broad sense, in calculus of variations and, in particular, in models for suspension bridges.

Inhalt

1 The physical models.- 2 Functional setting and vibrating modes for symmetric beams.- 3 Nonlinear evolution equations for symmetric beams.- 4 Nonlinear evolution equations for degenerate plates.- 5 Final comments and perspectives.