
Andreev Vladimir Igorevich 
Moscow State University of Civil Engineering (National Research University) (MGSU)
Doctor of Technical Sciences, Professor, Head of the Resistance of Materials Department, Moscow State University of Civil Engineering (National Research University) (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation.

Kapliy Daniil Aleksandrovich 
Moscow State University of Civil Engineering (National Research University) (MGSU)
Postgraduate student, Resistance of Materials Department, Moscow State University of Civil Engineering (National Research University) (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation.
Subject: one of the promising trends in the development of structural mechanics is the development of methods for solving problems in the theory of elasticity for bodies with continuous inhomogeneity of any deformation characteristics: these methods make it possible to use the strength of the material most fully. In this paper, we consider the twodimensional problem for the case when a vertical, locally distributed load acts on the hemisphere and the inhomogeneity is caused by the influence of the temperature field. Research objectives: derive governing system of equations in spherical coordinates for determination of the stress state of the radially inhomogeneous hemispherical shell under locally distributed vertical load. Materials and methods: as a mechanical model, we chose a thickwalled reinforced concrete shell (hemisphere) with inner and outer radii a and b, respectively, b > a. The shell’s parameters are a = 3.3 m, b = 4.5 m, Poisson’s ratio ν = 0.16; the load parameters are f = 10MPa  vertical localized load distributed over the outer face, θ0 = 30°, temperature on the internal surface of the shell Ta = 500 °C, temperature on the external surface of the shell Tb = 0 °C. The resulting boundaryvalue problem (a system of differential equations with variable coefficients) is solved using the Maple software package. Results: maximal compressive stresses σr with allowance for material inhomogeneity are reduced by 10 % compared with the case when the inhomogeneity is ignored. But it is not so important compared with a 3fold decrease in the tensile stress σθ on the inner surface and a 2fold reduction in the tensile stress σθ on the outer surface of the hemisphere as concretes generally have a tensile strength substantially smaller than the compressive strength. Conclusions: the method presented in this article makes it possible to reduce the deformation characteristics of the material, i.e. it leads to a reduction in stresses, which allows us to reduce the thickness of the reinforced concrete shell, and also more rationally distribute the reinforcement across the crosssection, increase the maximum values of the mechanical loads.
DOI: 10.22227/19970935.2017.12. 13261332

Andreev Vladimir Igorevich 
Moscow State University of Civil Engineering (National Research University) (MGSU)
Doctor of Technical Sciences, Professor, corresponding member of Russian Academy of Architecture and Construction Sciences, chair, Department of Strength of Materials, Moscow State University of Civil Engineering (National Research University) (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation;
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.

Polyakova Lyudmila Sergeevna 
Moscow State University of Civil Engineering (National Research University) (MGSU)
Master student, Department of Strength of Materials, Moscow State University of Civil Engineering (National Research University) (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation;
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.
Among the classical works devoted to Solid Mechanics a significant place is occupied by the studies taking into account the physical and geometric nonlinearity. Also there is enough of works, which concern linear problems taking into account the inhomogeneity of the material. At the same time there are very few publications, which take into account both effects (nonlinearity and inhomogeneity). This is due to the lack of experimental data on the influence of various factors on the parameters defining the nonlinear behavior of the materials. Thus it is of great importance to study the influence of inhomogeneity when solving the problems of structures made of physically nonlinear materials. This article provides a solution to one of the problems of the nonlinear theory of elasticity taking into account the inhomogeneity. The problem is solved in an axisymmetric formulation, i.e. all the parameters of the nonlinear relationship between the intensities of stresses and strains are functions of the radius. The article considers an example  the stress distribution in the inhomogeneous soil massif with a cylindrical cavity.
DOI: 10.22227/19970935.2015.11.3845
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