Series Solutions for Beams on Elastic Foundations

In this paper series solutions are derived for beams on elastic foundation, subjected to a variety of end and loading conditions. These series solutions have the following advantages over the “formal” solutions of the differential equations of the corresponding problems: 1. The parameters of the beams (EI and l) and the modulus of the foundation (k) appear in these formulas in an explicit form and are not incorporated in the arguments of trigonometric and hyperbolic functions, as it is in the case of the formal solutions of the corresponding differential equations. For this reason, the series formulas are very suitable for design purposes and can also be used to obtain any desired degree of accuracy. 2. These formulas are valid for the entire lengths of the beams, irrespective of the discontinuities in the derivatives of the elastic line caused by the loadings. Solutions, for hinged-hinged beams, which reduce to the form of simple sine series, are not discussed here because they can be found in the related literature in a large variety of loading conditions.