Abstract
A large number of references dealing with the geometry, static, vibration and buckling analysis of elliptic paraboloid shells exist in the literature. This review work attempts to organize and summarize the extensive published literature on the basic achievements in investigations of thin-walled structures in the form of elliptic paraboloids. Possibilities of elliptic paraboloids with reference to machine-building and construction designs and to the apparatuses used in theoretical physics are briefly considered. The geometric part of the review is extended due to consideration of optimization of surface’s sizes, researches of representation of a surface on the plane and introducing bibliographic material on fractal geometry. Several existent analytical and numeral methods of calculation of the examined shells on durability give a possibility to choose one of the methods for the solution of new two-dimensional or three-dimensional tasks. Geometrical researches, approximation and bending of elliptic paraboloid surfaces, research of the stress-strain state of shells by analytical and numerical methods, natural and forced vibration of a shell, forming and setting of surfaces, application of shells in the form of elliptic paraboloids are the main problems which are considered in this review.
Highlights
Banerjee [64] proved that using of a stress function for analysis of elliptic paraboloid shells gives results not satisfying the fundamental equations for the equilibrium conditions
Apraksina [81] carried out a comparative research of a stress-strain state of an elliptic paraboloid shell with the help of a theory of shallow and non-shallow shells given in arbitrary curvilinear coordinates
The hyperbolic paraboloid is the most popular type of shells, due to its attractive form and simple structure [2], many researchers carry out investigations in the field of strength, vibration, stability, and application of elliptic paraboloid shells that are very attractive from an architectural point of view
Summary
The application of new shapes in the architecture of spatial shell structures is the second motive of regaining of their former popularity. The reinforced concrete elliptic paraboloid shells, examined in this review, are used mainly in civil engineering because concrete is the most widely used material in large-span shell construction [12 - 15]. As an example of the application of elliptic paraboloids in mechanical engineering, one can present a modernized centrifugal mixer of powder materials declared in a patent No 51-167-99, 27.11.2003, RF. This mixer has inside a structure in the form of an elliptic paraboloid. In preparing of this review, the authors used materials published in scientific and technical journals and proceedings, and studied monographs, reports of scientific conferences, and other scientific and technical literatures mainly for the period 1970-2016
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