卷 89, 编号 2 (2025)

The translational–rotational motions of a symmetrical dumb-bell are considered in a generalized elliptical Sitnikov problem. We describe the equations of the dumb-bell motion and their integral manifold “gravitational propeller” which contains the motion such that the dumb-bell barycenter moves along the normal to the motion plane of two primaries, whilst the dumb-bell itself rotates continuously around the normal keeping a constant angle π/2 with normal. We obtained the equation of plane dumb-bell oscillations when its barycenter coincides with the barycenter of primaries. It is shown that this equation coincides with the Beletskii equation if the dumb-bell has an infinitesimal length. Small plane oscillations of dumb-bell are investigated by introducing two small parameters: e (the eccentricity of primaries orbit) and ɛ (a measure of the deviation of the phase point from the origin). The regions of singular and regular small oscillations and different types of equations for the regular domain are described. We have an increase in the frequency of oscillations to infinity with an increase in the length of the dumb-bell and the tendency of its point masses to pass near the primaries.

An axisymmetric problem is formulated for determining the upper (kinematic) limit of the bearing capacity of spherical shallow shells of annular shape in plan, the internal openings of which are closed by rigid inserts. Such compound structures are in contact with an incompressible fluid. The shells are reinforced with fibers along spiral trajectories symmetrical with respect to the meridian, as well as along meridional and/or circumferential directions. The materials of the composition components are assumed to be rigid-plastic and have different yield strengths under tension and compression. Plastic flow in the phases of the composition is determined by piecewise linear flow conditions. A two-layer model of a thin-walled structure is used, the kinematics of which in the limit state is described by the relations of the classical theory of shallow shells. The extreme problem of determining the ultimate load is formulated on the basis of the application of the principle of virtual power. An unconventional discretization of this problem was carried out, the solution of which was obtained using methods of linear programming theory. The convergence of the numerical solution is tested and compared with exact solutions of similar problems for homogeneous isotropic plates. Good accuracy of the numerical solution is demonstrated. The influence of the reinforcement structure parameters, the magnitude of the shallow shell lift and boundary conditions on the value of the ultimate load is investigated. It is shown that for annular plates the best arrangement of fibers is in the radial direction, and for shallow shells the rational one is a meridional-circumferential structure with specially selected reinforcement densities. It has been demonstrated that with an increase in the lifting height of a shallow shell, its load-bearing capacity more than doubles compared to a plate of the same geometry in plan and the same thickness.

We construct asymptotics of eigenfrequencies and eigenmodes of a composite anisotropic body with a group of small inclusions while mass of each is larger or equal in order the mass of the surrounding material. If a part of the body surface is rigidly clamped, modes of the natural oscillations are localized in main near the inclusions while the principal asymptotic terms of eigenfrequencies are described by the spectrum of problems about inclusions of unit size in the weightless space. In the case when the body surface is traction free, an interaction of small heavy inclusions is observed, namely the limiting problem consists of system of equations for inclusions in the space which are combined into a single spectral problem with integral terms at the spectral parameter. The structure of the integro-differential equations depends on the mass concentration coefficient as well as disposition of the inclusions. Justification of the derived asymptotic expansions is performed in a representable and most complicated case of superheavy concentrated masses distributed along a line and other situations are considered in the same way.

The study is devoted to the problem of applying the method of variable elasticity parameters, which uses the provisions of the deformation theory of plasticity, to solving problems of autofreting cylindrical shells loaded with internal pressure. The paper considers two cases of autofreting thick-walled cylindrical shells: with longitudinal stretching and without longitudinal stretching. When determining the stress-strain state, the shell material was considered incompressible and dependencies in the form of a power function and linear power functions were used to describe the deformation diagram of the material. The analysis of the loading process was carried out by studying the loading trajectories of various points of the shell wall in the Ilyushin stress space and the Nadai–Lode parameter for stresses. As studies have shown, in the case of autofreting with longitudinal tension, as well as when loading the shell with internal pressure up to destruction, loading is simple for all functions describing the deformation diagram, which proves the validity of solving such problems by the method of variable elasticity parameters. When autofreting the shell without longitudinal stretching, using a power approximation of the deformation diagram, the loading process up to destruction can be considered simple, which corresponds to Ilyushin’s theorem on simple loading. With the linear-power approximation of the deformation diagram, the process of loading the shell is not simple, but a comparative analysis of the stress state obtained with the power-law and linear-power approximation of the deformation diagram showed a slight difference at all stages of loading. Moreover, these differences decrease with increasing pressure, which allows us to conclude that the method of variable elasticity parameters can be applied to solving problems of autofreting cylindrical shells without longitudinal stretching, as well as loading such shells with internal pressure up to destruction.

An exact solution to an elastic boundary value problem is constructed for a half-plane with a one-dimensional, semi-infinite stiffener perpendicular to its straight boundary. A point force is applied at the top of the stiffener. This solution is compared with a numerical simulation of a finite-length pile using three-dimensional finite element (FE) analysis. By applying correction factors, the transition from a two-dimensional problem to a three-dimensional one is achieved in the analytical solution.