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

Meriem Chabekh, Nadhir Chougui, Delfim F. M. Torres

TL;DR

The paper addresses the dynamic response of a suspension-bridge roadbed modeled as a 1D extensible thermoelastic shear beam with suspenders under thermoelasticity of type III. It combines rigorous analysis with a practical numerical framework: global well-posedness and exponential stability are established via Faedo–Galerkin approximations and Lyapunov functionals, while a fully discrete finite element scheme with backward Euler time stepping is shown to be energy-stable and convergent with an error bound of $O(h^2+(\Delta t)^2)$ under high regularity. The results yield robust long-time behavior guarantees and provide concrete discretization guidance supported by numerical simulations that illustrate energy decay and convergence. This work provides a solid mathematical and computational foundation for stable simulations of large-span bridges under coupled thermoelastic dynamics and suspender interactions.

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.

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

TL;DR

The paper addresses the dynamic response of a suspension-bridge roadbed modeled as a 1D extensible thermoelastic shear beam with suspenders under thermoelasticity of type III. It combines rigorous analysis with a practical numerical framework: global well-posedness and exponential stability are established via Faedo–Galerkin approximations and Lyapunov functionals, while a fully discrete finite element scheme with backward Euler time stepping is shown to be energy-stable and convergent with an error bound of under high regularity. The results yield robust long-time behavior guarantees and provide concrete discretization guidance supported by numerical simulations that illustrate energy decay and convergence. This work provides a solid mathematical and computational foundation for stable simulations of large-span bridges under coupled thermoelastic dynamics and suspender interactions.

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.
Paper Structure (9 sections, 9 theorems, 170 equations, 11 figures, 1 table)

This paper contains 9 sections, 9 theorems, 170 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Global well‑posedness
  4. Exponential stability
  5. Numerical approximation
  6. Description of the discrete problem
  7. Study of the discrete energy
  8. Error estimate
  9. Simulations

Key Result

Lemma 2.1

Let $(u,\varphi,\psi,w)$ be the solution of (main:system). Then the energy functional $E$, defined by satisfies

Figures (11)

  • Figure 1: The evolution in time and space of $u$.
  • Figure 2: The evolution in time and space of $\varphi$.
  • Figure 3: The evolution in time and space of $\psi$.
  • Figure 4: The evolution in time and space of $w$.
  • Figure 5: The evolution in time of $u$ at $x=0.6$.
  • ...and 6 more figures

Theorems & Definitions (17)

  • Lemma 2.1
  • proof
  • Theorem 2.2: Aubin--Lions--Simon theorem, see p. 102 of Boyer
  • Theorem 3.1
  • proof
  • Theorem 4.1
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • ...and 7 more