Non-linear free vibrations of buckled plates with deformable loaded edges

In the great majority of papers dealing with non-linear dynamic buckling of plates it is assumed that the edge at which the in-plane load is applied is not deformable (rigid). In those few references in which buckling with deformable edges is examined normally a variational approach is used, based on “trial” functions for the in-plane displacements, which approach often leads to incorrect results. In this paper an exact form of the solution is found for the in-plane displacements, it being assumed that the membrane force is, at all times, exactly equal to the load applied at the corresponding loaded edge. As is well known, when the plate is buckled through a rigid beam these two forces are only on average equal to each other. It is shown here that the deformability of the loaded edges for plates undergoing large deformations should be taken into account for square-shaped plates, especially for those close to the so-called “golden-cut” shape (the aspect ratio between the edges equal to √2), while for long rectangular plates this effect may be disregarded.