Beams on elastic foundation using modified Betti׳s theorem

• We introduce the modified Betti׳s theorem to the solution of beams on an elastic foundation.
• We introduce the reference beam on an elastic foundation.
• We derived fundamental formulae relating to the reference beam on an elastic foundation.
• The derived formulae might be used for the analytical solution of any other beams on an elastic foundation.
• We provide three illustrative examples of the introduced method.

In this paper, the restricted form of the principle of quasi work – the so-called modified Betti׳s theorem – is used for developing an alternative analytical solution of beams on an elastic foundation. A brief review of this principle is provided along with the reminder of the classical Winkler׳s model. A methodology is based on the calculation of the deflection of beam on an elastic foundation from the deflection of a reference beam which is topologically equivalent. Fundamental formulae for the reference beam on an elastic foundation are derived and thoroughly discussed. Applying the modified Betti׳s theorem, these formulae can be used for analysis of any arbitrary topologically equivalent beam on an elastic foundation. The applicability of this methodology is proved and a detailed guidance for its use is provided as well. The methodology is illustrated by three representative examples. The beam on an elastic foundation is a structure which is often used in many areas of technical practice including civil engineering, aerospace engineering etc; thus, the presented alternative way of its analytical solution might be found helpful, mainly for quick calculation and for its simple algorithmization.