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Abstract Introduction. This article offers a methodology for the analysis of a two-layer composite beam on an elastic foundation that represents a one-parameter Winkler model. The behaviour of two-layer beams is described by A.R. Rzhanitsyn in his theory of composite rods that have rigid transverse and elastic-yielding longitudinal connections between layers. The theory of composite rods allows to study the stress-strain state (SSS) of multilayer foundation beams, having a layer featuring low thermal conductivity, and perforated beams. Materials and methods. However, analytical solutions to these problems involve certain difficulties; therefore, they are often inapplicable. We propose to apply a numerical method, a method of successive approximations (MSA), developed by professor R.F. Gabbasov at the Department of structural and theoretical mechanics of the Moscow State University of Civil Engineering. MSA has proven to be an effective and highly accurate method designated for analyzing static/dynamic loads, applied to beams, slabs and shells, and for making stability calculations. The difference-based variation of the MSA method has a number of advantages over difference equations of the “classical” finite difference method (FDM). The proposed methodology allows to take into account various types of boundary conditions without involving contour points. Сoncentrated forces, concentrated moments, and piecewise distributed loads can be taken into account as loading types. The article describes a problem solving algorithm. The system of initial differential equations is solved using difference analogs. Typical difference equations for regular and boundary points are provided. Results. The analysis of a composite free-lying beam on an elastic foundation illustrates the proposed approach. The qua-lity of the analysis results is controlled by performing a numerical study of the solution convergence using several nested meshes. Conclusions. The proposed method can be used in the engineering practice of design organizations and the educational process of higher educational institutions training civil engineering specialists.

In structural engineering, the analysis of plate elements is one of the important problems that based on theory of plates. The analytical solutions using traditional methods and manual calculations are determined in the cases with simple calculation schemes. It will be difficult to solve problems with the complicatedness in loading and boundary conditions. Thereof, the application of numerical methods is necessary to deal with that. The finite difference method and successive approximation method are popular numerical methods to solve relatively thoroughly the plate problems. In addition, with the development of construction technology, the dimensions of the current building projects become greater and lead to increase of building displacements that is required to take into account in analysis. The analysis of plates with non-linear geometrical is relatively complicated and iterative algorithms can help to solve this problem. In this paper, the authors used the difference equations of successive approximation method (MSA) and generalized equations of finite difference method (MFD) to solve the problems of non-linear plates with different boundary conditions and loading. From the comparison of obtained results with Volmir’s analytical results the conclusions and recommendations are proposed.

Beams are considered the most popular bending elements used in building structures. With the load-bearing capacity and production characteristics, it is possible to use many different materials to combine them into beams to receive high economic-technical efficiency. Besides, because of the development of structural material industry, the manufacturing and construction technology, the form of beams is as diverse as composite wood beams, foundation beams, force-sensitive composite beams prior to installation grafting, steel-concrete combination beams. So there are many researches about solving the problems of multi-layer structures in general and multi-layer beams in particular by different calculation methods. In this paper, based on the theory of multi-layer composite rods and plates of A.R Rzhanitsyn the solutions of calculation analysis of three-layer beams that is subjected to discontinuous loads using difference equations of successive approximation method (MSA) are presented. The obtained results with good convergence show high accuracy of the numerical method with the use of difference equations of successive approximation method.

Currently, plates with characteristics of high thermal insulation, sound insulation and strength are sufficiently and widely used in structural engineering. To describing the connect of the plates with other elements, it is possible to give many calculation schemes on the different views. In this paper, it is considered the problem of rectangular plates, each side of which are fixed with hinged support or fixed support. This paper deals with the discontinuous points of boundary conditions with the use generalized equations of finite difference method. The obtained results of the analysis of stress state in discontinuous area of boundary conditions are compared with the results of work of Smirnov V.A.