The 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

The stability calculation of a П-shaped hinged frame is considered. The concept of r-like frames is introduced as frames with the same ratio r of the linear stiffnesses of the transom and the strut. It is shown that the parameter vcr , which determines the critical load on the frame, is the same for r-like frames. Approximate formulas allowing to determine the critical load parameter vcr and design lengths of compressed bars with an error not exceeding 2% are obtained.

Keywords: flat frame, stability, critical force, reduced length coefficient, r-like frames, approximation, least squares method

The shear calculation scheme of a multi-storey building, popular in design practice, is considered.The application of the method of displacements is shown, when the mutual displacements of the floors of the frame are taken as unknowns. The indicated technique leads to the solution of the system of equations of the displacement method with the upper two-diagonal matrix, which makes it possible to obtain an analytical solution of the problem.

Keywords: shear calculation scheme of the building, move method, tridiagonal matrix, sweep method, reciprocal movements, analytical solution of the equation system