# 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.

### 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

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

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

# References

1. Galimov K.Z., Paimushin V.N. Teoriya obolochek slozhnoj geometrii [The theory of shells of complex geometry]. Kazan, Kazan University Publ.; 1985. (In Russ.)
2. Petrov V.V. Nelinejnaya inkremetal'naya stroitel'naya mekhanika [Nonlinear incremental structural mechanics]. Vologda, Infra-Inzheneriya Publ.; 2014. (In Russ.)
3. Bate K.-U. Metody konechnyh elementov [Finite Element Methods]. Moscow, Fizmatlit Publ.; 2010. (In Russ.)
4. Golovanov A.I., Tyuleneva O.N., Shigabutdinov A.F. Metod konechnyh elementov v statike i dinamike tonkostennyh konstrukcij [Finite element method in the statics and dynamics of thin-walled structures]. Moscow, Fizmatlit Publ.; 2006. (In Russ.)
5. Kiselev A.P., Gureeva N.A., Kiseleva R.Z. Raschet mnogoslojnoj obolochki s ispol'zovaniem ob"emnogo konechnogo elementa [Calculation of a multilayer shell using a volumetric finite element]. Izvestia VSTU [Bulletin of the Volgograd State Technical University]. 2010;4(4):125–128. (In Russ.)
6. Kayumov R.A. K resheniyu zadach neodnorodnoi teorii uprugosti metodom konechnykh elementov [To the solution of problems of the heterogeneous theory of elasticity by the finite element method]. Trudy Vtoroi Vserossiiskoi nauchnoi konferentsii (1–3 iyunya 2005 g.). Ch. 1. Matematicheskie modeli mekhaniki, prochnost' i nadezhnost' konstruktsii [Proceedings of the Second All-Russian Scientific Conference (1–3 June 2005). Part 1. Mathematical models of mechanics, strength and reliability of structures]. Samara, SamGTU Publ.; 2005. p. 143–145. (In Russ.)
7. Kiselev A.P., Kiseleva R.Z., Nikolaev A.P. Account of the shift as rigid body of shell of revolution axially symmetric loaded on the base of FEM. Structural Mechanics of Engineering Constructions and Buildings. 2014;(6):59–64. (In Russ.)
8. Klochkov Yu.V., Nikolaev A.P., Ischanov T.R. Finite element analysis of stress-strain state of shells of revolution with taking into account the strain of transversal shearing. Structural Mechanics of Engineering Constructions and Buildings. 2016;(5):48–56. (In Russ.)
9. Klochkov Yu.V., Nikolaev A.P., Sobolevskaya T.A., Klochkov M.Yu. Comparative analysis of efficiency of use of finite elements of different dimensionality in the analysis of the stress-strain state of thin shells. Structural Mechanics of Engineering Constructions and Buildings. 2018;14(6): 459–466. (In Russ.)
10. Gureeva N.A., Arkov D.P. Flat problem of theory of jump in base method of final elements in mixed understanding in account physical nonlinearity. Structural Mechanics of Engineering Constructions and Buildings. 2010;(4): 32–36. (In Russ.)
11. Beirão da Veiga L., Lovadina C., Mora D. A virtual element method for elastic and inelastic problems on polytope meshes. Computer methods in applied mechanics and engineering. 2015;(295):327–346.
12. Klochkov Y.V., Nikolaev A.P., Vakhnina O.V., Kiseleva T.A. Stress-strain analysis of a thin-shell part of fuselage using a triangular finite element with Lagrange multipliers. Russian Aeronautics. 2016;59(3):316–323.
13. Klochkov Y.V., Nikolaev A.P., Vakhnina O.V. Calculation of rotation shells using finite triangular elements with Lagrange multipliers in variative approximation of displacements. Journal of Machinery Manufacture and Reliability. 2016;45(1):51–58.
14. Magisano D., Liabg K., Garcea G., Leonetti L., Ruess M. An efficient mixed variational reduced order model formulation for nonlinear analyses of elastic shells. International Journal for Numerical Methods in Engineering, 2018;113(4):634–655.
15. Gureeva N.A., Klochkov Yu.V., Nikolaev A.P. Analysis of a shell of revolution subjected to axisymmetric loading taking into account geometric nonlinearity on the basis of the mixed finite element method. Russian Aeronautics. 2014;57(3):232–239.
16. Bandurin N.G., Gureeva N.A. Determination of plain stress condition of shells applying mixed formulation of finite-element method in terms of geometrical nonlinearity. Cosmonautics and rocket engineering. 2013;(1):69–75. (In Russ.)
17. Ignatiev V.A., Ignatiev A.V. Plane problem solution of elasticity theory by the finite element method in the form of classical mixed method. Bulletin of the Volgograd State University of Architecture and Civil Engineering. Series: Construction and Architecture. 2013;31–2(50):337–343. (In Russ.)

# Statistics

#### Views

Abstract - 130

PDF (Russian) - 140

#### Dimensions

Copyright (c) 2020 Gureeva N.A., Nikolaev A.P., Yushkin V.N.

You consent to our cookies if you continue to use our website.