The summary of the solution was presented in Table 4.5 Page 156 of Stresses in Beams, plates and shell according to the work of A. M. Wahl and G. Lobo, Jr (1930) titled “ Stresses and Deflections in Flat Circular Plates with Central Holes,” Trans. A.S.M .E., vol. 52, 1930.
You must propose a functional as the energy of deformation of the plate minus the work produced by the applied force.
Use coordinates functions for the deflection of the plate, and minimize the functional.
For example for a clamped square plate:
W(x,y)=A(x^2-1)^2*(y^2-1)^2+B*x*(x^2-1)^2*(y^2-1)^2+C*y*(x^2-1)^2*(y^2-1)^2
Try to look for vibration problems, replace the dynamic term by the work done by the applied force.
Here is an example http://dx.doi.org/10.1016/S0022-460X(81)80167-8
Thanks at all! I have been able to work out some exact solutions analytically. Though it isn't as complicated as it appears.
Look at Timoshenco' s book about plates and shells and use his approach.
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