624.078
О.А. Кова ьчу
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O.A. Kovalchuk
THE SIMULATION
OF THREE-DIMENSIONAL
BAR SYSTEMS
BY THE FINITE ELEMENT
METHOD
The article considers the representation of framed building as a three-dimensional bar system and simulation of obtained bar system by the finite element method. Underway the bar state equations and the equations system solution in the Cauchy form were obtained. Also article deals with obtaining of stiffness matrix linking nodal forces and nodal dis-placements. The obtained results are applica-ble for both the statically determinate systems and the statically indeterminate ones.
Key words: three-dimensional bar sys-tem, finite element method, deflection meth-od, node displacement, node displacement vector, node force vector, stiffness matrix.
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REFERENCES
1. Aleksandrov A.V., Lashchenikov B.Ja., Shaposhnikov N.N. Stroitel'naya mekhanika. Tonkostennye prostranstvennye sistemy [Structural Mechanics. Thin-walled three-dimensional system]. Moscow, 1983, p. 488.
2. Aleksandrov A.V., Potapov V.D., Derzhavin B.P. Soprotivlenie materialov [Strength of the materials]. Moscow, 2003, p. 560.
3. Tolokonnikov L.A. Mehanika deformiruemogo tverdogo tela [Deformable solid mechan-ics]. Moscow, 1979. p. 318.
4. Korn G.A., Korn T.M. Spravochnik po matematike [References book of Mathematics]. Moscow, 1978. p. 720.
5. Shostak R.Ja. Operatsionnoe ischislenie [Operational calculus]. Moscow, 1972, p. 279. 6. Obraztsov I.F., Savel'ev L.M., Hazanov X.S. Metod konechnykh elementov v zadachakh stroitel'noy mekhaniki letatel'nykh apparatov [The finite element method for the problems of air-craft structural theory]. Moscow, 1985, p. 392.
7. Postnov V.A., Harhurim I.Ja. Metod konechnykh elementov v raschetahh sudovyhh kon-struktsiy [The finite element method for the ship structures calculation], Leningrad, 1974, p. 476.
8. Makarov E.G. Soprotivlenie materialov na baze Mathcad [Constructional materials based on Mathcad]. St. Petersburg, BHV, 2004, p. 512.
ю 2012 .
: а ьчу а ич,
, ,
,
« », 129337, . ,
-, . 26-, oko44@mail.ru
About the author: Kovalchuk Oleg Aleksandrovich, candidate of technical sciences, docent, director of the Fundamental Education Institute, Moscow State
University of Civil Engineering (MSUCE), 26
Yaro-slavskoye shosse, 129337, Moscow, Russian Federa-tion, oko44@mail.ru
:
. .
-// - - « . .
-». 2012. . 2. : http://www.nso-journal.ru. For citation:
Kovalchuk O.A. Modelirovanie prostranstvennykh sterzhnevykh system metodom konechnykh ele-mentov [The simulation of three-dimensional bar systems by the finite element method]
Nauchno-prakticheskiy Internet-гСurnКХ «NКuФК. StroТtОХ'stЯo. OЛrКгoЯКnТО» [Science, construction, education], 2012,
