Abstract
An invariant vector approximation of unknown quantities is proposed and implemented to construct the stiffness matrix of a quadrilateral curved finite element in the form of a fragment of the mid-surface of an elliptic cylinder with 18 degrees of freedom per node. Numerical examples show that the vector approximation has significant advantages over the scalar one as applied to arbitrary shells with considerable mid-surface curvature gradients.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. V. Novozhilov, K. F. Chernykh, and E. I. Mikhailovskii, Linear Theory of Thin Shells (Politekhnika, Leningrad, 1991) [in Russian].
K. Z. Galimov and V. N. Paimushin, Theory of Shells of Complex Geometry (Kazan. Gos. Univ., Kazan, 1985) [in Russian].
E. I. Grigolyuk and V. V. Kabanov, Stability of Shells (Nauka, Moscow, 1978) [in Russian].
E. N. Grigolyuk and E. A. Lopanitsyn, Finite Deflections, Stability, and Supercritical Behavior of Thin Hollow Shells (Mosk. Gos. Tekh. Univ. “MAMI,” Moscow, 2004) [in Russian].
V. L. Yakushev, Nonlinear Deformations and Stability of Thin Shells (Nauka, Moscow, 2004) [in Russian].
V. G. Bazhenov, V. K. Lomunov, S. L. Osetrov, and E. V. Pavlenkova, “Experimental and computational method of studying large elastoplastic deformations of cylindrical shells in tension to rupture and construction of strain diagrams for an inhomogeneous stress-strain state,” Prikl. Mekh. Tekh. Fiz., No. 1, 116–124 (2013).
A. I. Golovanov, O. N. Tyuleneva, and A. F. Shigabutdinov, Finite Element Method in Statics and Dynamics of Thin-Walled Structures (Fizmatlit, Moscow, 2006) [in Russian].
N. M. Yakupov and M. N. Serazutdinov, Computation of Elastic Thin-Walled Structures of Complex Geometry (Inst. Mekh. Mashinostr. Kazan. Nauchn. Tsentr Ross. Akad. Nauk, Kazan, 1993) [in Russian].
V. A. Postnov and I. Ya. Kharkhurim, Finite Element Method in Computation of Ship Structures (Sudostroenie, Leningrad, 1974) [in Russian].
K.-J. Bathe, Finite Element Procedures (Prentice Hall, Upper Saddle River, N.J., 2007; Fizmatlit, Moscow, 2010).
A. P. Nikolaev, Yu. V. Klochkov, A. P. Kiselev, and N. A. Gureeva, Vector Interpolation of Displacement Fields in Computing Shells (Volgograd. Gos. Agrar. Univ., Volgograd, 2012) [in Russian].
L. I. Sedov, A Course in Continuum Mechanics (Wolters-Noordhoff, Groningen, 1971; Nauka, Moscow, 1976), Vol. 1.
Yu. V. Klochkov, A. P. Nikolaev, and T. A. Kiseleva, “Comparison of stresses produced by scalar and vector FEM interpolations in jointed shells consisting of different materials,” Stroit. Mekh. Raschet Sooruzh., No. 5(250), 7–75 (2013).
N. M. Belyaev, Mechanics of Materials (Nauka, Moscow, 1976) [in Russian].
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © T.A. Kiseleva, Yu.V. Klochkov, A.P. Nikolaev, 2015, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2015, Vol. 55, No. 3, pp. 418–428.
Rights and permissions
About this article
Cite this article
Kiseleva, T.A., Klochkov, Y.V. & Nikolaev, A.P. Comparison of scalar and vector FEM forms in the case of an elliptic cylinder. Comput. Math. and Math. Phys. 55, 422–431 (2015). https://doi.org/10.1134/S0965542515030094
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0965542515030094