Approximate solution to the Föppl-Kármán-Marguerre equations (Q1111067)

From the engineering point of view it is of interest to know a simple and yet sufficiently precise approximate solution to the thin plate equation when initial deflection has the form of buckling mode and load ranges from zero up to few times the bucklng load. We review: (i) the formula based on classical small solution expansion near the critical load (Koiter's initial postbuckling analysis), (ii) solution of one Galerkin equation assuming initial as well as the overall deflection in the form of the buckling mode, (iii) the formal correction to (ii) suggested by \textit{A. C. Walker} [The postbuckling behaviour of simply supported square plates. Aeron. Q. 20, 203-222 (1969)]. Our aim is to combine the power of small solution analysis with the practical usefulness of approximate solution (ii) by a consistent approach. We realize it taking the Galerkin equation (ii) as an equality constraint into a constructive small solution analysis. Although derived in a neighbourhood of the buckling load the resulting formula yields an exact deflection function at zero load value. Illustrative numerical results are given.