The polygon-circle paradox and convergence in thin plate theory

The solution for a simply supported many-sided polygonal plate does not agree with that for the corresponding circular plate. This paper describes the earlier work of Rao and Rajaiah on polygonal plates and then explains why best convergence of series solutions occurs when the boundary conditions are denned as w = Vw> = 0. Notation D — operator in original problems, D = operator adjoint to D, Fx, F2 = boundary terms, K = parameter which can be given arbitrary values, s = number of sides of the regular polygon, t — number of terms used in a truncated series, u = eigenfunction (see equation (7)), v = solution of adjoint problem, w = lateral deflection of plate, x, y = Cartesian coordinates, A = eigenvalue (see equation (7)), v = Poisson's ratio. The following suffices are also used: m indicates with eigenfunction or eigenvalue, n indicates directional normal to plate boundary, nn indicates double differentiation in direction normal to plate boundary, t indicates direction along plate boundary, tt indicates double differentiation in direction along plate boundary.