Equilibrium of circular von-K\'arm\'an plate bonded with Kirchhoff rod

TL;DR

This paper analyzes the buckling behavior of a circular von Kármán plate bonded to a Kirchhoff rod, formulating a bifurcation problem to identify critical mismatch values causing buckling, with implications for structural stability.

Contribution

It introduces a semi-analytic method to determine bifurcation points in a coupled rod-plate system considering geometric mismatch and explores how the rod influences buckling modes.

Findings

Critical mismatch values depend on the relative size of the rod and plate.

Rod buckles and influences plate deformation when the plate is smaller than the rod.

For larger plates, the system behaves as if the rod imposes a rigid boundary condition.

Abstract

A circular von Karman plate is considered bonded at its boundary to a circular Kirchhoff rod via a hinge like junction. There is a mismatch of dimension between the rod and the plate boundary in their respective stress free configurations. The process of gluing of the inextensible circular rod to the edge of the extensible plate causes the system to develop internal stress in the natural planar configuration. For some critical values of this mismatch the rod plate system, as expected, shows buckling behaviour in the interior or at the boundary of plate. In this paper, the corresponding bifurcation problem is formulated using the mismatch parameter as a bifurcation parameter. The non linear equations of equilibrium are linearized near the homogeneously deformed state and necessary conditions of bifurcation are solved for the critical geometric mismatch that is when non trivial solutions of the linearized equations exist. These critical points are candidates to be classified as bifurcation points. Additionally, the semi analytic nature of the analysis is exploited to perform a parametric study, relative to structure parameters in the problem formulation, of the critical points in the problem and the characteristic features are summarised. It is found that the rod plays an important role in determining the buckling behaviour when the plate's dimension is smaller than the looped rod, as in that case the rod buckles and the plate can bend or stretch depending on the planar or non planar nature of buckled mode of the rod. When the plate is larger, the rod effectively stimulates a rigid Dirichlet boundary condition due to its inextensibility; the critical points in this case are found to coincide with the case when buckling in the sole plate as a system is considered.