Rectangular plates on (bi)elastic foundation

This form uses 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 w of the plate. This model is
also exactly applicable to plates floating on a liquid.

The elastic foundation is here referred to as '(bi)elastic', to highlight the distinction with respect to a (mono)elastic foundation, that is treated elsewhere, where the subgrade reacts only when compressed, and not if the plate goes upwards.

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 V minimization ('Options'->'Show formulae' in the menu to see).

When present, the button 'Data' will download the matrix of the coefficients (without dimensions) aij as defined in sheet's formulae. Note that it is up to the user to link the downloaded matrix to the input data: when you download it, the matrix corresponds to the data in the sheet as you view it.

The elastic foundation is here referred to as '(bi)elastic', to highlight the distinction with respect to a (mono)elastic foundation, that is treated elsewhere, where the subgrade reacts only when compressed, and not if the plate goes upwards.

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 V minimization ('Options'->'Show formulae' in the menu to see).

When present, the button 'Data' will download the matrix of the coefficients (without dimensions) aij as defined in sheet's formulae. Note that it is up to the user to link the downloaded matrix to the input data: when you download it, the matrix corresponds to the data in the sheet as you view it.