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Geometric characteristics of the deformation state of the shells with orthogonal coordinate system of the middle surfaces
- Authors: Ivanov V.N.1, Shmeleva A.A.1
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Affiliations:
- Peoples’ Friendship University of Russia (RUDN University)
- Issue: Vol 16, No 1 (2020)
- Pages: 38-44
- Section: Theory of thin elastic shells
- URL: https://journals.rudn.ru/structural-mechanics/article/view/23008
- DOI: https://doi.org/10.22363/1815-5235-2020-16-1-38-44
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Abstract
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.
About the authors
Vyacheslav N. Ivanov
Peoples’ Friendship University of Russia (RUDN University)
Author for correspondence.
Email: i.v.ivn@mail.ru
SPIN-code: 3110-9909
Doctor of Technical Sciences, Professor, Department of Civil Engineering, Engineering Academy
6 Miklukho-Maklaya St., Moscow, 117198, Russian FederationAlisa A. Shmeleva
Peoples’ Friendship University of Russia (RUDN University)
Email: i.v.ivn@mail.ru
graduate student of Department of Civil Engineering, Engineering Academy
6 Miklukho-Maklaya St., Moscow, 117198, Russian FederationReferences
- Aron H. Das Gleichgewlcht und die Bewegund einer unendlich dunnen beliebig gekrummten elastischen Schale. J. fur reine und angew, Math. 1874;78:136–174.
- Love A. The small free vibrations and deformation of thin elastic shell. Pfill. Transs Roy. Soc. 1888;179(A): 491–546.
- Vlasov V.Z. Obshchaya teoriya obolochek i ee prilozheniya v tekhnike [General theory of shells and its application at technology]. Moscow; Leningrad: Gostehizdat Publ.; 1949. (In Russ.)
- Novozhilov V.V. Teoriya tonkih obolochek [Theory of thin shells]. Leningrad: GSIPS Publ.; 1962. (In Russ.)
- Chernih C.F. Linejnaya teoriya obolochek. Chast' 1. Obshchaya teoriya obolochek [Linear theory of shells. Part 1. Total theory of shells]. Leningrad: Leningrad University; 1962. (In Russ.)
- Chernih C.F. Linejnaya teoriya obolochek. Chast' 2. Nekotorye voprosy teorii [Linear theory of shells. Part 2. Some question of theory]. Leningrad: Leningrad University; 1964. (In Russ.)
- Goldeveizer А.L. Teoriya uprugih tonkih obolochek [Theory of elastic thin shells]. Moscow: Nauka Publ.; 1976. (In Russ.)
- Novozhilov V.V., Chernih К.F., Michailovskiy Е.I. Linejnaya teoriya tonkih obolochek [Linear theory of thin shells]: monograph. Leningrad: Politechnika Publ.; 1991. (In Russ.)
- Klochkov Yu.V., Nikolaev A.H., Ishchanov T.R. Comparative analysis of scalar and vector forms of approximations in a FEM after the V.V. Novozhilov's relations for elliptic cylinders. Structural Mechanics of Engineering Constructions and Buildings. 2015;(2):51–57. (In Russ.)
- Klochkov Yu.V., Vakhnina O.V., Kiseleva T.A. Calculation of thin shells on the basis of the triangular final element with the correcting Lagrange's coefficients. Structural Mechanics of Engineering Constructions and Buildings. 2015;(5):55–59. (In Russ.)
- Ivanov V.N. The base of finite element method and variation-difference method: textbook. Moscow: RUDN University Publ.; 2008. (In Russ.)
- Ivanov V.N., Krivoshapko S.N. Analiticheskie metody rascheta obolochek nekanonicheskoj formy [Analytical methods for calculating shells of non-canonical form]: textbook. Moscow: RUDN Publ.; 2010. (In Russ.)
- Ivanov V.N., Abbushy N.U. Raschet obolochek slozhnoj geometrii variacionno-raznostnym metodom [Analysis of the complex geometry using the variationaldifference method]. Structural Mechanics of Engineering Constructions and Buildings: Intercollegiate collection of scientific papers. 2000;(9):25–34. (In Russ.)
- Abovskiy А.P., Andreev N.P., Deruga А.P. Variacionnye principy teorii uprugosti i teorii obolochek [Variation principle of the theory of elasticity and theory of the shells]. Moscow: Nauka Publ.; 1978. (In Russ.)
- Washizu K. Variational methods in elasticity and plasticity. Oxford: Pergamon Press; 1968.
- Krivoshapko S.N., Ivanov V.N. Encyclopedia of Analytical Surfaces. Switzerland: Springer International Publishing; 2015.
- Ivanov V.N. Geometry and forming of the normal surfaces with system of plane coordinate lines. Structural Mechanics of Engineering Constructions and Buildings. 2011;(4):6–14. (In Russ.)
- Ivanov V.N., Shmeleva A.A. Geometry and formation of the thin-walled space shell structures on the base of normal cyclic surfaces. Structural Mechanics of Engineering Constructions and Buildings. 2016;(6):3–8. (In Russ.)
- Ivanov V.N. Raschet obolochek nekanonicheskoj formy [Analyses of the shells of noncanonic forms]: textbook complex. Moscow: RUDN Publ.; 2013. (In Russ.)
