The condition of minimum energy requires that , where is the potential energy of the external forces and . After some algebraic manipulation, the details of which will not be discussed here, following equations are obtained for finding the displacements , and :

	(17.8)
	(17.9)

We can introduce the stress function

,    ,

and simplify equation (10.13a) as,

(17.10)

Let us write the strain components in terms of stresses:

In which we can substitute the expressions for the strains from equation 17.6

	(17.11)

Deriving the three equations of 10.26 by , and respectively, we obtain,

	(17.12)

By adding these three, we obtain,

(17.13)

Equations 17.10 and 17.13 represent the complete system of equations for solving problems of large deflections of thin plate. These nonlinear set of equations are called Föppl-von Karman equations. These are not amenable for solving analytically except in some one dimensional cases, so that one needs to use numerical or semi-analytical techniques to handle them.