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Nowadays there are various calculation methods for solving a wide range of problems in construction, hydrodynamics, thermal conductivity, aerospace research and many other areas of industry. Analytical methods that make up one class for solving problems, and numerical calculation methods that make up another class, including those implemented in computing complexes, are used for the design and construction of various thin-walled structures such as shells. Due to the fact that thin-walled spatial structures in the form of various shells are widely used in many areas of human activity it is useful to understand and know the capabilities of different calculation methods. Research works on the study of the stress-strain state of the torse shell of equal slope with an ellipse at the base are not widely available at the moment. For the first time the derivation of the differential equations of equilibrium of momentless theory of shells to determine the normal force Nu from the action of uniformly distributed load tangentially directed along rectilinear generatrixes to the middle surface of the torse of equal slope with a directrix ellipse is presented in this article. The parameters of the stress state of the studied torse are also obtained by the finite element method and the variational-difference method. The SCAD software based on the finite element method and the program SHELLVRM written on the basis of the variational-difference method are used. The numerical results of the parameters of the stress state of the studied torse are analyzed, and the advantages and disadvantages of the analytical method and two numerical calculation methods are determined.

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.

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.

At the classical literature of the design of the shell there is used the coordinated system of the middle surfaces of principle curvatures or common non-orthogonal coordinate system. The system of equations of the shells of complex forms is complicated and it’s difficult to receive the analytical decision. For analyses of the shells with non-orthogonal coordinate system and the most shells of complex form in the coordinate system of principle curvatures there is used usually finite element method which usually uses only the equation of the surface but don’t take into account the geometrical characteristics of the middle surfaces of the shell. At the chair of strength of materials and constructions of Engineering Academy RUDN there is worked up the complex program VRMSHELL for analyses of the shells of complex forms by variation-difference method. The complex includes the library of curves on the base of which there are calculated the geometrical characteristics of the classes of the surfaces included at the program complex. The complex includes the classes of surfaces as cylindrical, surfaces of rotation, Joachimsthal’s surfaces, Monge’s surfaces. The coordinate system of surfaces of these classes is the coordinate system of principle curvatures.

The variation-difference method is a convenient numerical method for shells of complex forms. It is enough when only cinematic boundary conditions are satisfied because the method is based on the principle of Lagrange. Another advantage of the variation-difference method is the better opportunity to create computer programs based on it. For shell analysis in orthogonal coordinate system as well as for shell analysis in principal curvatures the system of equations describing stress-strain state can be simplified. In this paper the difference between analysis in orthogonal coordinate system and analysis in principal curvatures of the surface is considered. The main distinction of the analysis of shells in orthogonal curvilinear coordinate system is the necessity of determination of components which include curvature of torsion of coordinate lines. The addition of these components in the equations of the theory of shells for the coordinate system in principal curvatures gives possibility to analyze shells in common orthogonal coordinate system. In this article shell analysis in orthogonal coordinate system is applied to shells based on normal cyclic surfaces.

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 authors suggest a simple analytical method to analyze right helicoid stress-strain state based on a corrected and simplified Rekach’s method of two Reissner’s equations reduced into one eight-order equation with solutions found by the trigonometric Fourier series. In the study of this technique, some inaccuracies were found in the formulas that have a significant impact on the results. A direct approach to the definition of the tangential displacements is also introduced. Improvement of the method allows to simplify the equations and get compact enough formulas which are suitable for programming. It is proven that the numerical results obtained by this modified analytical method are similar to those obtained by finite element method, while the original version of this technique leads to incorrect results.

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.