Beams with an intermediate pier: Spectral properties, asymmetry and stability

Maurizio Garrione; Maurizio Garrione

doi:10.3934/mine.2021016

Mathematics in Engineering

2021, Volume 3, Issue 2: 1-21. doi: 10.3934/mine.2021016

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Research article Special Issues

Beams with an intermediate pier: Spectral properties, asymmetry and stability

Maurizio Garrione ^,

Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci, 32 - 20133 Milano, Italy
^†This contribution is part of the Special Issue: Qualitative Analysis and Spectral Theory for Partial Differential Equations
Guest Editor: Veronica Felli
Link: www.aimspress.com/mine/article/5511/special-articles

Received: 16 March 2020 Accepted: 28 May 2020 Published: 29 June 2020

We deal with beams with an intermediate pier, motivated by the investigation of the stability of suspension bridges with two spans. First, we provide a complete spectral theorem for the associated linear stationary fourth-order problem with hinged boundary conditions, determining the eigenvalues and discussing their optimality (in a suitable sense) in terms of the position of the pier. Then, we consider a related nonlinear model with a restoring force of superquadratic displacement type, discussing its stability both from a linear and from a suitable nonlinear point of view. We determine the position of the pier maximizing the stability of the structure and we compare the energy thresholds of instability under hinged, clamped or mixed (left-clamped and right-hinged) boundary conditions. In any case, we highlight that an asymmetric structure is in general more stable.
- multi-point ODEs,
- beams,
- intermediate pier,
- asymmetry,
- stability,
- optimal position
Citation: Maurizio Garrione. Beams with an intermediate pier: Spectral properties, asymmetry and stability[J]. Mathematics in Engineering, 2021, 3(2): 1-21. doi: 10.3934/mine.2021016

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Abstract

We deal with beams with an intermediate pier, motivated by the investigation of the stability of suspension bridges with two spans. First, we provide a complete spectral theorem for the associated linear stationary fourth-order problem with hinged boundary conditions, determining the eigenvalues and discussing their optimality (in a suitable sense) in terms of the position of the pier. Then, we consider a related nonlinear model with a restoring force of superquadratic displacement type, discussing its stability both from a linear and from a suitable nonlinear point of view. We determine the position of the pier maximizing the stability of the structure and we compare the energy thresholds of instability under hinged, clamped or mixed (left-clamped and right-hinged) boundary conditions. In any case, we highlight that an asymmetric structure is in general more stable.

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