An Analytical Solution to Rectangular Kirchhoff Plates Supported at All Corner Points Using Two Single Series and the Boundary Collocation Method

The aim of this paper is the analysis of arbitrarily loaded isotropic rectangular Kirchhoff plates supported at all corner points, whereby the edges are arbitrarily supported (simply supported, clamped, or free). Analytical methods commonly used include the single trigonometric series of Lévy, the double trigonometric series of Navier, the power series expansions, etc. In this paper the deflection surface was approximated with the sum of a particular solution to the governing differential equation (GDE) and two single series. The terms of the single series were the product of an unknown function of an independent variable and a trigonometric function of the other independent variable, whereby the trigonometric functions were chosen to vanish at the corner points. On the one hand the terms of the series were required to satisfy the homogeneous GDE, leading to two uncoupled differential equations, one for each unknown function, and so the approximate solution satisfied exactly the GDE. On the other hand the boundary conditions were satisfied only at selected collocation points along the boundary, the number of collocation points in each direction corresponding to the number of terms of the associated series. Results for several plates supported at all corner points with arbitrary edge support conditions were presented and showed good agreement with the exact results, the accuracy increasing the more terms of series were considered. Cantilevered plates and plates having two adjacent edges free will be analyzed using this approach in future research.