This form uses the theory of plates with large deflections, where the forces acting in the plane of the plate are not negligible.
This theory is represented, among others, by equations (234) page 402 of Timoshenko-Krieger 'Theory of Plates and Shells' McGraw Hill 2nd ed. The improved approximation of the radial strain is used here, as given in note 2 page 396 op.cit.
This theory is more realistic in representing plate deflections with respect to the pure bending theory when the center deflection is larger than a fraction of plate thickness.
Please note that plate behaviour according to this theory is far from being linear: hence the principle of superposition is not applicable.
The solution is obtained with a finite difference scheme, using 140 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.