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2015 | Buch

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Mathematical Models for Suspension Bridges

Nonlinear Structural Instability

verfasst von: Filippo Gazzola

Verlag: Springer International Publishing

Buchreihe : MS&A

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This work provides a detailed and up-to-the-minute survey of the various stability problems that can affect suspension bridges. In order to deduce some experimental data and rules on the behavior of suspension bridges, a number of historical events are first described, in the course of which several questions concerning their stability naturally arise. The book then surveys conventional mathematical models for suspension bridges and suggests new nonlinear alternatives, which can potentially supply answers to some stability questions. New explanations are also provided, based on the nonlinear structural behavior of bridges. All the models and responses presented in the book employ the theory of differential equations and dynamical systems in the broader sense, demonstrating that methods from nonlinear analysis can allow us to determine the thresholds of instability.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Brief History of Suspension Bridges

Abstract

The sound modeling of any mechanical system requires careful experimental observations. For complex structures such as suspension bridges, the observations have to be taken from history and not just from lab experiments. In this chapter we survey several historical events and we attempt to classify the observed phenomena in suitable categories. The most instructive event is certainly the Tacoma Narrows Bridge collapse which is analysed in great detail, together with many different attempts of explanations. None of them seems to answer to all the questions raised by the collapse.

Filippo Gazzola

Chapter 2. One Dimensional Models

Abstract

The first attempts to model suspension bridges were to view the roadway as a beam. Although this point of view rules out an important degree of freedom, the torsion, it appears to be a reasonable approximation since the width of the roadway is much smaller than its length. In this chapter we review classical modeling of beams and cables and of their interaction.

Filippo Gazzola

Chapter 3. A Fish-Bone Beam Model

Abstract

In the previous chapter we saw beam models for the main span of a bridge, within different equations and different coupling with the sustaining cable. However, modeling the roadway of a suspension bridge as a beam prevents to highlight the most dangerous oscillations in bridges, the torsional oscillations which are considered responsible for the TNB collapse. If one wishes to give an answer to question (Q1) raised in Sect. 1.6, the bridge cannot be seen as a simple one dimensional beam.

Filippo Gazzola

Chapter 4. Models with Interacting Oscillators

Abstract

In this chapter we model a suspension bridge through a number of coupled oscillators. This generates second order Hamiltonian systems which can be tackled with ODE methods. We first analyse a single cross section of the bridge and we model it as a nonlinear double oscillator able to describe both vertical and torsional oscillations. By means of a suitable Poincaré map we show that its conserved internal energy may transfer from the vertical oscillation of the barycenter to the torsional oscillation of the cross section. This happens when enough energy is present in the system, as for the fish-bone model considered in Chap. 3 We name again flutter energy the critical energy threshold where this transfer may occur.

Filippo Gazzola

Chapter 5. Plate Models

Abstract

The most natural way to describe the bridge roadway is to view it as a rectangular elastic plate. Rocard (Dynamic instability: automobiles, aircraft, suspension bridges. Crosby Lockwood, London, 1957, p. 150) writes:

The plate as a model is perfectly correct and corresponds mechanically to a vibrating suspension bridge.

In this chapter we make some attempts to model suspension bridges with nonlinear plate equations. We discuss both material nonlinearities, such as the behavior of the restoring force due to the hangers and the sustaining cables, and geometric nonlinearities due to possible wide oscillations which bring the plate (roadway) far away from its equilibrium position. The results in the present chapter should be seen as a prelude of more detailed studies aiming to increase the knowledge of the qualitative behavior of suspension bridges through plate models. Still, these results are sufficient to highlight a torsional instability and the existence of a flutter energy similar to those described in the previous chapters for different models.

Filippo Gazzola

Chapter 6. Conclusions

Abstract

In the previous chapters we have analysed several different models for suspension bridges. Each model has highlighted a form of instability due to the presence of some nonlinearity. The purpose of this final chapter is to put all together the observed phenomena and to afford possible explanations as well as to give answers to the questions raised in Chap. 1

Filippo Gazzola

Backmatter

Titel: Mathematical Models for Suspension Bridges
verfasst von: Filippo Gazzola
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-15434-3
Print ISBN: 978-3-319-15433-6
DOI: https://doi.org/10.1007/978-3-319-15434-3

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