Bending of plates

For large-deflection plate theory, we consider the inclusion of membrane strains

This equation was first derived by Lagrange in December 1811 in correcting the work of Germain who provided the basis of the theory.

For rectangular plates, Navier in 1820 introduced a simple method for finding the displacement and stress when a plate is simply supported. The idea was to express the applied load in terms of Fourier components, find the solution for a sinusoidal load (a single Fourier component), and then superimpose the Fourier components to get the solution for an arbitrary load.

If we apply these boundary conditions and solve the plate equation, we get the solution

The classical rectangular plate equation for small deflections thus becomes:

The deflection of a simply-supported plate (of corner-origin) with general load is given by

The deflection of a simply-supported plate (of corner-origin) with uniformly-distributed load is given by

This is an ordinary differential equation which has the general solution

We can superpose the symmetric and antisymmetric solutions to get more general solutions.

The solutions to the governing equations can be found if one knows the corresponding Kirchhoff-Love solutions by using the relations