Deformation of a thin elastic plate with a fixed edge and attached rods. Part 1: The static problem

We construct asymptotics for the stress-strain state of a thin horizontal plate clamped along its edge, with vertical rods attached to it. Made of an isotropic and homogeneous elastic material, this structure is loaded by gravity. Using the dimension reduction procedure and analysis of the boundary layer, which exhibits exponential behavior near the plate edges and power-law behavior near the junction zones, we find the main and correction terms in the asymptotics for the deflection of the plate and rigid displacement of the rods, and their longitudinal deformation. We derive a weighted and anisotropic Korn inequality that is asymptotically sharp and provides justification for the asymptotic formulas.