Comparative analysis of finite element formulations at plane loading of an elastic body

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Abstract


The aim of the work - comparison of the results of determining the parameters of the stress-strain state of plane-loaded elastic bodies based on the finite element method in the formulation of the displacement method and in the mixed formulation. Methods. Algorithms of the finite element method in various formulations have been developed and applied. Results. In the Cartesian coordinate system, to determine the stress-strain state of an elastic body under plane loading, a finite element of a quadrangular shape is used in two formulations: in the formulation of the method of displacements with nodal unknowns in the form of displacements and their derivatives, and in a mixed formulation with nodal unknowns in the form of displacements and stresses. The approximation of displacements through the nodal unknowns when obtaining the stiffness matrix of the finite element was carried out using the form function, whose elements were adopted Hermite polynomials of the third degree. Upon receipt of the deformation matrix, the displacements and stresses of the internal points of the finite element were approximated through nodal unknowns using bilinear functions. The stiffness matrix of the quadrangular finite element in the formulation of the displacement method is obtained on the basis of a functional based on the difference between the actual workings of external and internal forces under loading of a solid. The matrix of deformation of the finite element was formed on the basis of a mixed functional obtained from the proposed functional by repla-cing the actual work of internal forces with the difference between the total and additional work of internal forces when loading the body. The calculation example shows a significant advantage of using a finite element in a mixed formulation.


About the authors

Natalia A. Gureeva

Financial University under the Government of the Russian Federation

Author for correspondence.
Email: aup-volgau@yandex.ru
49 Leningradsky Ave, GSP-3, Moscow, 125993, Russian Federation

Doctor of Physics and Mathematics, Associate Professor, Associate Professor of the Department of Data Analysis, Decision Making and Financial Technologies

Anatoly P. Nikolaev

Volgоgrad State Agrarian University

Email: aup-volgau@yandex.ru
26 Universitetskii Ave, Volgograd, 400002, Russian Federation

Doctor Of Technical Sciences, Professor, Professor of the Applied Geodesy, Environmental Engineering and Water Use Department

Vladislav N. Yushkin

Volgоgrad State Agrarian University

Email: aup-volgau@yandex.ru
26 Universitetskii Ave, Volgograd, 400002, Russian Federation

Candidate Of Technical Sciences, Associate Professor, Associate Professor of the Applied Geodesy, Environmental Engineering and Water Use Department

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