Junction of elastic plates and beams (Preliminary version)
Antonio Gaudiello, Regis Monneau, Jacqueline Mossino, Francois Murat, Ali Sili
TL;DR
This paper analyzes the linearized elasticity problem in a thin multidomain composed of a horizontal plate and a vertical beam, with thickness $ε$ and beam radius $r^ε$, under clamped boundaries. By applying a dual-scale rescaling and introducing $q^ε = k^ε ε^3 /(r^ε)^2$, it identifies a rigorous limit problem as $ε→0$, $r^ε→0$ with $r^ε ≫ ε^2$, yielding a coupled 2D plate–1D beam system connected by six junction conditions. The limit behavior depends on the limit $q ∈ [0,∞]$: for $0<q<∞$ the plate and beam remain coupled; for $q=∞$ the beam decouples and the plate dominates; for $q=0$ the plate decouples and the beam dominates. The work provides detailed a priori estimates, compactness arguments, and density results, culminating in energy convergence statements that justify reduced-dimension models with precise junction transmission conditions.
Abstract
We consider the linearized elasticity system in a multidomain of the three dimensional space. This multidomain is the union of a horizontal plate, with fixed cross section and small thickness "h", and of a vertical beam with fixed height and small cross section of radius "r". The lateral boundary of the plate and the top of the beam are assumed to be clamped. When "h" and "r" tend to zero simultaneously, with "r" much greater than the square of "h", we identify the limit problem. This limit problem involves six junction conditions.