Hi Ehab magdy yehia,

Your question concerns not only the plate but any structure having free surfaces. If we use a FEM package then we shouldn't prescribe zero forces/moments at the nodes belonging to the free surfaces (package will do this automatically). But when we program the system of FE equations manually, we must write this zeros at the corresponding parts of r.h.s.

Best regards.

May I ask about your cantilever plate under what load ?

Ehab magdy yehia

In FEA We have what you call natural boundary conditions and enforced boundary conditions. An example of an enforced boundary condition is the setting of displacements to zero at plate edges. A natural boundary condition is one that can be realized without explicitly implementing it in the FE code. In the thin plate theory M and Q are dependent parameters that can be computed from derivatives of the displacement function. Hence, boundary conditions such M = Q = 0 are natural and need not be implemented explicitly. When the analysis results are output by the program it will be found that M and Q will approach zero at free edges as the number of elements is increased.

On the other hand if the plate is analyzed using Mindlin thick plate theory in which shear rotations are independent parameters then M=Q = 0 becomes an essential boundary condition and these need to be enforced in the code.