For a point load applied in the middle of the plate, a trial function would be "w(x,y) = B*(y^2+x^2-R^2) in cartesian coordinates (centered on the origo of the plate), where B is the parameter to be solved (Ritz, Galerkin) and R is the radius of the plate.
Since that provides constant moment (Moment = function of second derivatives of deflection) in the plate, it is not suitable for this problem. Perhaps something like
w(x,y) = B*(y^4+x^4-R^4) + C*(y^3+x^3-R^3) + D*(y^2+x^2-R^2)
would be more appropriate. B, C and D are the constants to be solved, then input into the approximation again and used to calculate moments and shears (functions of displacement partial derivatives), normal and shear stresses, and finally the von Mises stress. This approximation is kinematically admissible (at edges, R=y or R=x and thus deflection is zero), and provides quadratic moment and linear shear.
Please note that these are my spontaneous thoughts and the result of some googling "circular plate trial function". Some book might have a better approximation than what I have presented, but the idea is this:
1. The trial function should satisfy essential boundary conditions.
2. The trial function is a sum of several linearly independent deflection interpolants multiplied by unknown parameters (w.i) to be solved: w(x,y)= SUM(i=1, n) w.i*polynomial (or parameter*trigonometric function)
3. The solution should be complete in the energy. This criterion is mathematical (exact solution minus approximation, measured in energy, is smaller than a constant), and not important for practical problems.
