Inequality constraints in one-dimensional finite elements for an elastic beam on a tensionless Winkler foundation

The problem of an isotropic elastic beam under bending conditions on a tensionless Winkler elastic foundation is revisited. The beam is assumed partitioned into several finite elements and the deflection of the beam is required to be a positive quantity along the whole beam so that the related fundamental fourth-order ordinary differential equation can continuously hold true. Assuming four related arbitrary conditions at the element tips (e.g. the deflections and the rotations) and approximating to the transcendental (exponential-trigonometric) terms in the solution of the differential equation by simple polynomials (by using a Chebyshev approximation), we reach a quantifier elimination problem in elementary algebra and geometry concerning the continuous positivity of the deflection of the beam. By using classical Sturm-Habicht sequences, the related theorem and elementary Boolean/logical minimization techniques inside the computer algebra system Maple V, we show how the quantified variable (the length variable along the finite beam element) can be eliminated and related quantifier-free formulae, including only the four arbitrary boundary conditions at the element tips, can be constructed. The case of the continuous positivity of the general quintic polynomial is studied in detail by this approach. Further generalizations are also suggested in brief.