Beam on a two-parameter elastic foundation: simplified finite element model

When calculating beams resting on a solid elastic foundation, the simplest foundation models proposed by Winkler-Zimmerman and Vlasov-Leontyev are often used. These hypotheses have been repeatedly subjected to well-founded criticism, because they do not take into account the inclusion in the work of some areas of the base and do not allow determining reactive pressures at the ends of the foundation beam and beyond it. In order to clarify these hypotheses, many authors have proposed some other models that allow smoothing out the shortcomings of these models to varying degrees. This article proposes a new numerical approach to solving the problem of a beam on a two-parameter elastic foundation. To calculate the beam, the finite element method has been used. A separate rod has been proposed as a finite element for solving the bending state of the beam on a two-parameter model of an elastic foundation. There has been presented the construction of the stiffness matrix of this finite element. The elastic foundation is assumed to be linear, homogeneous and isotropic and is taken into account using the parameters r, s. The reactions of the elastic base, deflections and angles of rotation, the formulas for calculating bending moments and transverse forces have been determined. There have been given examples of static calculation of a beam on an elastic two-parameter foundation for the action of various loads. These examples demonstrate the effectiveness of the developed method. Reliability of the method proposed by the authors has been verified on test examples, and good agreement has been obtained with the well-known models of Winkler and Vlasov.