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The authors deal with the vibration of the pipeline that occurs during the transportation of a gas-containing liquid due to the uneven distribution of gas accumulation along the length of the pipeline, which produces pulsating movements of the center of gravity of the moving liquid, when the conditions for supporting the middle part change. The solution is constructed using the calculation scheme of a beam lying on an elastic foundation described by the Winkler model. To assess the impact on the stress-strain state of the beam of the sudden exclusion of the elastic foundation under the middle part of the beam, the circular frequencies and forms of natural transverse vibrations of the beam that occur in the absence of disturbing forces are determined. The given solution of the problem of vibrations of a pipe, through which a gas-containing liquid is transported, with a change in the conditions of support of the middle part of the beam, can be used to predict the durability of the pipe.

An efficient method for constructing an approximate analytical solution to the contact problem of the penetration of a spherical indenter into the soft elastic layer lying on a rigid elastic foundation is proposed. The method permits to obtain the problem solution effective for the full range of the problem characteristic geometrical and physical parameter values. Some cases when Young’s moduli of the layer and the substrate differ more than a hundredfold are analyzed. The obtained contact stresses for different values of the foundation stiffness are compared with the formerly known results for the nondeformable foundation case. It is shown that when the elastic properties of the layer and the substrate differ a hundredfold, maximum difference between these results is less than 3%. The relationship between the pressing force and the radius of the sphere is determined. The question is put owing to the control problem of the thin coatings elastic properties.

To evaluate the force to damage the ice covers is necessary for estimation of icebreaking capability of vessels, as well as of hull strength of icebreakers, and navigation of ships in ice conditions. On the other hand, the use of ice cover support to arrange construction works from the ice is also of practical interest.By the present moment a great deal of investigations of ice cover deformation have been carried out to result, usually, in approximate calculations formula which was obtained after making a variety of assumptions. Nevertheless, we believe that it is possible to make further improvement in calculations. Application numerical methods, and, for example, FEM, makes possible to avoid numerous drawbacks of analytical methods dealing with both complex boundaries and load application areas and other problem peculiarities.The article considers an application of mixed models of FEM for investigating ice cover deformation. A simple flexible triangle element of mixed type was taken to solve this problem. Vector of generalized coordinates of the element contains apices flexures and normal bending moments in the middle of its sides. Compared to other elements mixed models easily satisfy compatibility requirements on the boundary of adjacent elements and do not require numerical displacement differentiation to define bending moments, because bending moments are included in vector of element generalized coordinates.The method of account of rigid support plate is proposed. The resulting ratio, taking into account the "stiffening", reduces the number of resolving systems of equations by the number of elements on the plate contour.To evaluate further the results the numerical realization of ice cover stress-strained problem it becomes necessary and correct to check whether calculation results correspond to accurate solution. Using an example of circular plate the convergence of numerical solutions to analytical solutions is showed.The article presents just a small part of the obtained results. However, that is enough to confirm an acceptability of FEM mixed models in this class of problems.

A rod-chain model was used to study the dynamics of a tree stem. A numerical method is developed for simulating the mechanical behavior of a flexible tree stem. The motion of the flexible tree stem is determined under the resistance forces exerting on the tree crown. Flexible tree stem interaction with an elastic foundation and a forest machine is investigated as well. The numerical model is applied for calculating the impulse of force on the elastic foundation and the forest machine.

The bending of three-layer elastic beam partial support pattern on the elastic basis with stepped variable thickness of one of two stiff shells is considered. To describe kinematics of asymmetrical over thickness core pack the hypotheses of the broken normal are accepted. A system of equilibrium equations in terms of displacements has been derived. Analytical deciding a problem are received and their numeric analysis are conducted.

The axisymmetrical vibrations of elastic circular three-layer metal-polymeric plate on en elastic foundation ander heat stroke are explored. For the kinematics description asymmetrical on the thickness of package accepted hypothesises broken normal. Filler is light. Analytical deciding a problem are received and their numeric analysis are conducted.

The problem of buckling of a thin corrugated shell lying on elastic foundation under action of the non-uniform hydrostatic pressure is considered. The influence of the elastic foundation is taken into account as the additional pressure within the framework of Winkler`s model. The semi-moment equations of the thin elastic shell theory are assumed as the governing ones. The rigid clamped conditions are considered at the edges of the shell. Assuming that the hydrostatic pressure depends upon the circumferential co-ordinate, solutions of the governing equations are constructed in the form of functions decaying far from some generator on the shell surface. Using the complex WKB-method, the boundary-value problem is reduced to the sequence of the one-dimensional boundary-value problems. Its numeric solution has been obtained. The influence of parameters of corrugation of the shell and modulus of subgrade reaction on the critical pressure has been analyzed.

The axisymmetrical forced vibrations of elastic circular three-layer metal-polymeric plate on en elastic foundation are explored under action of divided local loadings: rectangular, sinusoidal, parabolic. Reaction of foundation was described on the base of Winkler's model. For the kinematics description asymmetrical on the thickness of package accepted hypothesises broken normal. Filler is light. Analytical deciding a problem are received and their numeric analysis are conducted.

The problem on the parametric vibrations of a thin, long cylindrical shell lying on an elastic foundation under action of compressive static axial forces and pulsing pressure is considered. Under an assumption that a thickness and physical characteristics of the shell as well as a foundation factor of an elastic basis are functions of a longitudinal coordinate, solutions of the Flugge type equations are constructed in the form of functions decaying far from some parallel on the shell surface.

The problem of the analysis of the stress-strain state of hinged layered plates on elastic foundation is considered. Bending of the layered plate on the Pasternak foundation at its mechanical loading is investigated. Original twodimensional discrete-structural generalized theory of multilayered plates is used to study the orthotropic plate bending. This theory is based on developing the displacement vector components of each layer in the power series along transverse coordinate. It allows to calculate displacement and stress tensor components of each layer with the required accuracy. The possibilities of the proposed approach are demonstrated on the examples of studying the response of layered plates on elastic foundation. The reliability of the results, obtained on its basis is determined through comparison with the data, presented in known scientific publications. The effect of the elastic foundation parameters on the stress-strain state of multi-layer structures is investigated. It was found that with an increase in the number of power series terms, displacement and stress demonstrate good convergence. It is shown that the elastic foundation shear stiffness significantly affects the stress-strain state of the hinged layered plates. [ABSTRACT FROM AUTHOR]