Derivation of conditions of complete contact for a beam on a tensionless Winkler elastic foundation with Mathematica

• Complete-contact conditions for beams on tensionless Winkler foundations are derived.
• Taylor–Maclaurin approximations are used for quantifier elimination in beam problems.
• Quantifier elimination algorithms in Mathematica are used in engineering mechanics.

The classical problem of a beam on a tensionless Winkler elastic foundation is reconsidered for the derivation of the conditions of complete contact between the beam and the foundation. This is achieved through the application of modern quantifier elimination software included in the computer algebra system Mathematica together with Taylor–Maclaurin series approximations to the deflection of the beam. Four particular beam problems have been considered in detail and the related QFFs (quantifier-free formulae) have been obtained for several values of the order in the series approximations. Additional approximation possibilities have also been investigated with an emphasis put on the use of the Galerkin method based on weighted residuals. The present results seem to constitute one more interesting application of modern quantifier elimination algorithms and the related software (here in Mathematica) to applied and engineering mechanics.