Analysis of a Shear beam model with suspenders in thermoelasticity of type III

TL;DR

This paper analyzes a thermoelastic shear beam model with suspenders for a suspension bridge, establishing existence, stability, and providing numerical methods and simulations for the model.

Contribution

It introduces a comprehensive analysis of a thermoelastic shear beam with suspenders, including existence, stability, and numerical discretization methods.

Findings

Proved existence of a global solution.

Established exponential stability of the model.

Developed stable numerical schemes with error estimates.

Abstract

We conduct an analysis of a one-dimensional linear problem that describes the vibrations of a connected suspension bridge. In this model, the single-span roadbed is represented as a thermoelastic Shear beam without rotary inertia. We incorporate thermal dissipation into the transverse displacement equation, following Green and Naghdi's theory. Our work demonstrates the existence of a global solution by employing classical Faedo-Galerkin approximations and three a priori estimates. Furthermore, we establish exponential stability through the application of the energy method. For numerical study, we propose a spatial discretization using finite elements and a temporal discretization through an implicit Euler scheme. In doing so, we prove discrete stability properties and a priori error estimates for the discrete problem. To provide a practical dimension to our theoretical findings, we present a set of numerical simulations.