Derivation of the plate bending equation using calculus of variations is a standard part of a course on engineering math and you can find the complete derivation in any responding graduate level textbook.
I would recommend you the following text, http://www.mm.bme.hu/~szeki/files/2012_vem-en.pdf
where the governing equation is derived and solved with the finite element method. Hope this helps.
Dear Iren,
Thank you for the nice reply. Actually, I know the governing equation and boundary conditions of kirchoff's plate bending problem. The problem was I could not define boundary conditions by calculus of variations but I found some clues. Additionally, Thank you for the book.
Best Regards.
Murat
Variational boundary conditions come up by asking that weak solutions once smooth turn out to be classical (pointwise meaning). In case of a bending plate, you have several boundary conditions which cast the problem in a variational framework among which: Dirichlet, Navier, Steklov...see the book by Gazzola-Grunau-Sweers for a dissertation on the subject.
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