Unilateral buckling of thin plates by complementarity eigenvalue analyses

In this work, the analysis by the finite element method of thin plates subjected to buckling in the presence of unilateral punctual obstacles, that is, supports that allow the plate to move in one direction but prevent the motion in the opposite direction, is addressed. An appropriate algorithm based on the solution of a semi-smooth system of equations, that results from the formulation of the unilateral buckling problem as a complementarity eigenvalue problem, is used. Rectangular and square plates are analysed under various membrane loadings, including compression and shear. The conformal Bogner-Fox-Schmit (BFS) finite element is employed to compute the bifurcation loads and the corresponding instability modes in scenarios with and without unilateral obstacles. For each plate and type of loading, the six lowest bifurcation loads and corresponding modes are computed for different levels of mesh refinement. The results confirm that the convergence of bifurcation loads obtained using the BFS element is monotonically decreasing as the mesh is refined. It is also confirmed that, when unilateral obstacles are present, the lowest bifurcation load, known as the critical load, can never be lower than the one of the homologous problem without unilateral obstacles.