Let us now assume that bending occurs only in one dimension, so that is a function of only and the above expression for the free energy simplifies to,

(16.6)

If the bending of the plate is increased slightly so that increases by , then the differential amount of increase in free energy of plate is

(16.7)

Integration by parts of the above equation yields:

(16.8)

If be the external force acting on the plate per unit area normal to the surface, then the work done by due to displacement of the point of application of the force is .

Then it can be shown after some algebraic manipulation that the minimization of the total free energy of the system which includes the free energy of the plate and the work done by the external load results in the following equilibrium equation of a bent plate:

(16.9)

where is the flexural rigidity of the plate. The boundary condition for solving this equation is different depending on whether the plate is clamped or supported.

(a) For plate whose edge is clamped, the boundary conditions are (no vertical displacement) and (the edge remains horizontal). The reaction force and the reaction moment are written as

,

(16.10)

(b) For a supported plate the moment of force becomes zero where it is supported, so that the following boundary conditions prevail: and .