Beams on elastic foundation: notes

This form uses the classical theory of beams on elastic soil, where the beam, over its width, 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 beam.

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 body floating on a liquid the theory is exact, and the 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 (with exponential terms for infinite beams) obtained from strain energy V minimization ('Options'->'Show formulae' in the menu to see, relationships enforcing boundary conditions not shown).

For long finite beams a solution might not be found due to numerical precision: choose a semi-infinite beam instead.

When present, the button 'Data' will download the coefficients (without dimensions) ai,bi,ci,A,B 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 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 body floating on a liquid the theory is exact, and the 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 (with exponential terms for infinite beams) obtained from strain energy V minimization ('Options'->'Show formulae' in the menu to see, relationships enforcing boundary conditions not shown).

For long finite beams a solution might not be found due to numerical precision: choose a semi-infinite beam instead.

When present, the button 'Data' will download the coefficients (without dimensions) ai,bi,ci,A,B 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.