A freely resting beam on a tensionless Winkler foundation loaded by forces at its ends
Abstract
A freely resting beam on a tensionless Winkler foundation loaded by forces at its ends
Incoming article date: 11.07.2025The article presents an analytical solution for a specific case of a system with unilateral constraints: a beam freely resting on a tensionless Winkler foundation, loaded with identical vertical forces F at both ends. It is shown that beam separation from the foundation occurs when the reduced beam length αl exceeds π (αl > π); the size of the detachment region does not depend on the magnitude of the applied force. An increase in foundation stiffness (for a constant beam length) leads to an expansion of the detachment zone. The bending moment at the midpoint of the beam with detachment is maximal at the «detachment moment» and equals 0.4F⁄α. The deflection at the midpoint also depends on the foundation stiffness and is proportional to the applied force. As the foundation stiffness increases, the bending moment and deflection decrease, approaching zero for an infinitely stiff foundation. The article provides a calculation algorithm for the described beam and an example demonstrating changes in the stress-strain state parameters of the beam with increasing foundation stiffness. The obtained results, in the author's opinion, are interesting in themselves and can also contribute to the set of verification problems for structurally nonlinear systems.
Keywords: system with unilateral constraints, tensionless Winkler foundation, beam, contact region, detachment region, Krylov functions