
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

Osilenker Boris Petrovich 
Moscow State University of Civil Engineering (MGSU)
Doctor of Physical and Mathematical Sciences, Professor, Professor, Department of Higher Mathematics, Moscow State University of Civil Engineering (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation;
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.
The article has findings on convergence and additivity (uniform and almost universal) of Fourier series in respect of loaded orthonormalized polynomials. The findings are applied to the Fourier series in respect of loaded Jacobi polynomials. The objective of research into loaded systems of mathematical physics was formulated in the classical book by R. Courant and D. Hilbert “Methods of Mathematical Physics”. Many researchers drive attention to polynomial systems, as they are used in the study of the Sturm–Liouville problem with a parameter in the boundary conditions, loaded integral equations and Schrodinger point potentials.As for applied problems, they are immediately related to important and frequent types of problems concerning concentrated loads, including oscillations of a heterogeneous loaded rod, torsional oscillations of a rod having pulleys at the ends, propagation of heat inside the rod having concentrated heat sources at the ends, etc.
DOI: 10.22227/19970935.2013.8.3514
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