Circular plates behaving as pure membranes: notes
This form uses the theory of plates behaving as (pure) flexible membranes, where the resistance of the plate to bending is negligible.
This theory is represented, among others, by equations (234) page 402 of Timoshenko-Krieger 'Theory of Plates and Shells' McGraw Hill 2nd ed.
by putting zero in place of the left hand side of the second of the equations. Moreover the improved approximation of the radial strain is used here,
as given in note 2 page 396 op.cit.
This theory is realistic in representing plate deflections only for very thin plates when the center deflection is much larger than plate thickness
(though still substantially smaller than plate outer radius for the results to be realistic).
Please note that plate behaviour according to membrane theory is far from being linear: hence the principle of superposition is not applicable.
The solution is obtained with a finite difference scheme, using 500 equal subdivisions of the plate radius.
Please note also that a solution will not necessarily be found for any load value, given the geometry and material, due to the limited precision of computer calculations.
If you get an all zero result, try with a lower load. If you then get a result, this means that your load is too high for the calculation to come to the end.