Relevance. Thin-walled structures of shells constitute a large class in architecture, in civil and industrial construction, mechanical engineering and instrument making, aircraft, rocket and shipbuilding, etc. Each surface has certain ad-vantages over the other. So the torso surface can be deployed on the plane of all its points without folds and breaks, with the length of the curves and the angles between any curves belonging to the surface, do not change. The investigation of the stressstrain state of the equal slope shell with a director ellipse at the base is presented to date in a small volume. The aim of the work. Obtaining data for comparative analysis of the results of the stress-strain state of equal slope shells by the finite element method and the variational-difference method. Methods. To assess the stressstrain state of the equal slope shell, the SCAD Office computer complex based on the finite element method and the “PLATEVRM” program, written on the basis of the variational-difference method, are used. Results. The numerical results of the stress-strain state of the equal slope shell are obtained and analyzed, the pros and cons of the results of calculations by the finite element method and the variational-difference method are revealed.
The aim of this work is to receive the geometrical equations of strains of shells at the common orthogonal not conjugated coordinate system. At the most articles, textbooks and monographs on the theory and analysis of the thin shell there are considered the shells the coordinate system of which is given at the lines of main curvatures. Derivation of the geometric equations of the deformed state of the thin shells in the lines of main curvatures is given, specifically, at monographs of the theory of the thin shells of V.V. Novozhilov, K.F. Chernih, A.P. Filin and other Russian and foreign scientists. The standard methods of mathematic analyses, vector analysis and differential geometry are used to receive them. The method of tensor analysis is used for receiving the common equations of deformation of non orthogonal coordinate system of the middle shell surface of thin shell. The equations of deformation of the shells in common orthogonal coordinate system (not in the lines of main curvatures) are received on the base of this equation. Derivation of the geometric equations of deformations of thin shells in orthogonal not conjugated coordinate system on the base of differential geometry and vector analysis (without using of tensor analysis) is given at the article. This access may be used at textbooks as far as at most technical institutes the base of tensor analysis is not given.
Relevance. Architects and engineers, designing shells of revolution, use in their projects, as a rule, spherical shells, paraboloids, hyperboloids, and ellipsoids of revolution well proved themselves. But near hundreds of other surfaces of revolution, which can be applied with success in building and in machine-building, are known. Methods. Optimization problem of design of axisymmetric shell subjected to given external load is under consideration. As usual, the solution of this problem consists in the finding of shape of the meridian and in the distribution of the shell thickness along the meridian. In the paper, the narrower problem is considered. That is a selection of the shell shape from several known types, the middle surfaces of which can be given by parametrical equations. The results of static strength analyses of the domes of different Gaussian curvature with the same overall dimensions subjected to the uniformly distributed surface load are presented. Variational-difference energy method of analysis is used. Results. Comparison of results of strength analyses of six selected domes showed that a paraboloid of revolution and a dome with a middle surface in the form of the surface of rotation of the z = -acosh(x/b) curve around the Oz axis have the better indices of stress-strain state. These domes work almost in the momentless state and it is very well for thin-walled shell structures. New criterion of optimality can be called "minimum normal stresses in shells of revolution with the same overall dimensions, boundary conditions, and external load".Keywords: dome; shell of revolution; paraboloid of revolution; the forth order paraboloid of revolution; catenary line; variational-difference energy method of analysis; linear shell theory; geometrical modeling; optimal design Conflict of interestThe authors confirm that this article content has no conflict of interest.For citation Krivoshapko S.N., Ivanov V.N. (2019). Simplified selection of optimal shell of revolution. Structural Mechanics of Engineering Constructions and Buildings, 15(6), 438-448. http://dx.
ЭПИ-ГИПОЦИКЛОИДЫ И ЭПИ-ГИПОЦИКЛОИДАЛЬНЫЕ КАНАЛОВЫЕ ПОВЕРХНОСТИ В.Н. ИВАНОВРоссийский университет дружбы народов ул. Миклухо-Маклая, д. 6, Москва, Российская Федерация, 117198 (поступила в редакцию: 12 января 2018 г.; принята к публикации: 5 марта 2018 г.) В статье рассматриваются кривые -эпи-и гипоциклоиды, образующиеся движением точек, связанных с окруж-ностями одинакового радиуса, катящимися одновременно по внешней и внутренней сторонам неподвижной окружно-сти. Показывается взаимосвязь этих кривых. Рассматривается качение окружностей с постоянным углом наклона к плоскости неподвижной окружности. При полном вращении подвижной окружности вокруг касательной к неподвиж-ной окружности точка, связанная с подвижной окружностью, описывает окружность вокруг касательной к неподвиж-ной окружности. При этом начальная точка в горизонтальной плоскости, принадлежащая эпициклоиде, при повороте на 180° переходит в точку гипоциклоиды. При качении подвижной окружности и полном вращении вокруг касатель-ной в каждой точке касания к подвижной окружности образуются эпи-гипоциклоидальные циклические поверхности. В статье доказывается, что окружности эпи-гипоциклоидальных циклических поверхностей являются линиями глав-ных кривизн, и, следовательно, поверхности относятся к классу каналовых поверхностей. Приводятся рисунки эпи-гипоциклоид и эпи-гипоциклоидальных циклических поверхностей с различными параметрами -отношением радиу-сов подвижной и неподвижной окружностей, положением точки, описывающей эпи-гипоциклоиды.Ключевые слова: эпи-гипоциклоиды, циклические поверхности, каналовые поверхности, эпи-гипоциклоидальные каналовые поверхности EPI-HYPOCYCLOIDS AND EPI-HYPOCYCLOIDAL CANAL SURFACES V.N. IVANOV Peoples' Friendship University of Russia (RUDN University)6 Miklukho-Maklaya St., Moscow, 117198, Russia (received: January 12, 2018; accepted: March 05, 2018) In the article are regarded the curves -epi-and hypocycloids, which are formed by the moving of the generating points, linked with the circles of the same radius and which are at the same time outside and inside of the unmoving circle. There is shown the relation of those curves. The moving of the circles with constant angle to the plane of the unmoving circle is also regarded. At full rotation of the moving circle the generating point linked with moving circle described a circle around the tangent of the unmoving circle. And the initial point laying in horizontal plane on epicycloid moving to the point on hypocycloid when the moving circle rotates on around the tangent of the unmoving circle. When the circle made a full rotation around the unmoving circle with full rotation around the tangent to the unmoving circle the epi-hypocycloidal cyclic surfaces are formed. In the article is proofed that the circles of the epi-hypocycloidal cyclic surfaces are the coordinate lines of the main curvatures of the surface and so the surfaces belongs to the class of canal surfaces. The drawings of the epi-hypocycloidal canal surfaces with different parameters -relation of the radius of ...
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