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Introduction. The definition of surfaces of congruent sections was first formulated in the work written by I.I. Kotov. These and several other types of surfaces, generated by the motion of a curve, belonged to the class of kinematic surfaces. Such kinematic surfaces as those of plane parallel displacement, surfaces of rotation, Monge surfaces, cyclic surfaces with ge-nerating circles having constant radius, rotative and spiroidal surfaces, helical some helix-shaped surfaces can be included into the class of surfaces that have congruent sections. Materials and methods. Using I.I. Kotov’s methodology, the authors first derived parametrical and vector equations for eight surfaces of congruent pendulum type cross sections of circular, elliptic, and parabolic cylinders and several helix-shaped surfaces. Circles, ellipses, and parabolas, located in the plane of the generating curve of a guiding cylinder or in the planes of a bundle that passes through the longitudinal axis of a cylinder, generate plane curves. Ellipses, analyzed in the article, can be easily converted into circles and this procedure can increase the number of shapes analyzed here. Results. Formulas are provided in the generalized form, so the shape of a plane generating curve can be arbitrary. Some surfaces of congruent sections are determined by two varieties of parametric equations. In one case, the central angle of the guiding cylindrical surface was used as an independent parameter, but in the other case, one of rectangular coordinates of the cylinder’s guiding curve served as an independent parameter. Two types of surfaces are analyzed: 1) when local axes of generating curves remain parallel in motion; 2) when these axes rotate. Conclusions. The analysis of the sources and the results, recommendations and proposals for application of surfaces, having congruent sections, is made with a view to their use in architecture and technology. The list of references has 27 positions, and it shows that the surfaces considered in this paper are being analyzed by architects, engineers, and geometricians both in Russia and abroad.

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 = - a cosh( 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”.

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.

In the article are regarded the curves - epiand 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 the moving and unmoving circles λ, the position of the generating point μ - are shown.

In 1960, the University of friendship of peoples was created by the Soviet government’s decision. In 1962 at the University’s faculty of engineering started the "Resistance of Materials, and Structural Analysis" chair. It was headed by doctor of engineering, professor Vladimir Germanovich Rekach. Simultaneously with the educational process at the chair was opened the postgraduate education in the field of study "Structural Mechanics". One of the main tasks set by V.G. Rekach before graduate students was developing of methods for calculating of shells for new complex forms. It was necessary to begin this work with the study of available in the scientific literature information on the considered class of surfaces, derivation of equations for the surfaces of considered forms, derivation of formulas for the quadratic forms’ coefficients, drawings for this class of surfaces and the concrete calculated construction. Currently, the work in this field of study at the “Resistance of Materials” chair (since 2006 the “Strength of Materials and Structures” chair) is carried out under the supervision of doctors of engineering, professors V.N. Ivanov and S.N. Krivoshapko – the disciples of V.G. Rekach. In 1990–2016 have been prepared and defended 12 dissertations on analysis of shells for surfaces with non-canonical forms. A large amount of material on surfaces geometry that has been accumulated at the RUDN University’s “Resistance of Materials” chair, and works variety in world literature has necessitated to systematize available materials, to create a reference manual on analytical surfaces. In 2006 has been published the 1st version of such reference manual. In 2011 has been published “Encyclopedia of Analytical Surfaces” [10]. This paper gives a short history of the encyclopedia creation, as well as its main tasks and principles of formation, the character of treatment for materials on surfaces classes and the concrete surfaces, and examples for classes and surfaces from encyclopedia. The possibilities for encyclopedia materials usage at architecture, building constructions, machine manufacturing and other areas of science and techniques are considered too.

The visualization problem of the formation process of surfaces with the same slope (generator angle) is considered. Among them, there are surfaces with different guides, such as a parabola, a circular cosine wave, a spherical evolvent. The possibility of helicoidal surface formation is also considered, since their generators have the same angle of inclination in all positions to the horizontal plane. The principles and techniques of surface formation in the AutoCAD environment are described using the functional language AutoLisp. The tasks of visualizing the formation of surfaces with the same slope with the guide parabola and the general cosine, as well as Monge surfaces and different types of helicoids are solved by means of authors’ programs coded in AutoLisp language, tools and the color palette of the AutoCAD program. The generator in all presented cases is a straight line with the same inclination angle α to the guide line (surfaces with parabola, circle cosine and evolvent) or to the horizontal plane xy (helicoids).

The paper is about the stress-strain state of the thin shell in the form of Monge surface with parabola generatrix and sinusoid guide. Coordinate system of the Monge surface is a system of coordinate lines of the principal curvatures of the surface. The variational-difference method is used for analysis. Variational-difference method allows using the geometric characteristics of the middle surface of the shell, which is important in the calculation of shells of complex shape. In the finite element method, which is often used in the shells of complex shape analysis, the equation of the middle surface of the shell is used only for finite element mesh.