This form uses a variation of the classical theory of plates on elastic foundation, where the plate rests on a subgrade, also called a Winkler foundation, behaving as a bed of independent springs, the soil reaction being proportional to the deflection y of the plate. This model is also exactly applicable to plates floating on a liquid.
The elastic foundation is here referred to as '(mono)elastic', as the subgrade reacts only when compressed, and not if the plate goes upwards.
The behavior is no more linear, hence the principle of superposition is not valid here.
The constant k relating the deflection to the reacting pressure is called the modulus of the foundation, and has units of a force over a volume.
In the case of a plate floating on a liquid, this modulus equals the unit weight of the liquid.
Typical values of the modulus in daN/cm³ are as follows (this is only a suggestion, actual values will depend also on the size of the loaded area and other parameters):
-dry or wet sand: 1.3 (loose) to 4.1 (medium) to 16 (compact)
-submerged sand : 0.8 (loose) to 2.5 (medium) to 9.6 (compact)
-compact clay : 2.4 (compr.strength 1-2 daN/cm²) to 4.8 (2-4 daN/cm²) to over 9.6 (>4 daN/cm²)

The solution is based onto a polynomial approximation formula obtained from strain energy minimization (hit the 'Formulas' button to see), solved by means of an iterative procedure.