CALCULATION OF ELASTIC DEFLECTIONS OF THIN STIFF SHELLS BASED ON THE FINITE ELEMENT METHOD OUT OF THE KIRCHHOFF’S THEORY

GEVORGYAN Hrant Ararat, Ph. D. in Eng., Researcher, Institute of Mechanics of the National Academy of Sciences of the Republic of Armenia, Yerevan, Republic of Armenia, This email address is being protected from spambots. You need JavaScript enabled to view it.">This email address is being protected from spambots. You need JavaScript enabled to view it.

The computational method of determining thin stiff shells deflections formulated on the basis of the plane-spatial problem of the FEM without using Kirchhoff’s hypothesis is developed; in virtue of the geometric properties of the finite element stiffness matrix, a tensor of flexion stiffness is introduced. A linear and a nonlinear modification of the plane-spatial problem of the FEM for calculation of small elastic deflections of thin shells are formulated. An example of calculation of fragment of sloping conical shell is given in accordance with the common principles of two-dimensional domain discretization and some elements of fractal geometry.

finite element method, antiplane shear, plane-spatial problem, Kirchhoff’s hypothesis, flexion stiffness, tensor of flexion stiffness, fractal geometry, fragment of shell

• Gevorgyan H. Plosko-prostranstvennaya zadacha metoda konechnykh elementov [A plane-spatial problem in the finite element method]. Mekhanika mashin, mekhanizmov i materialov [Mechanics of Machines, Mechanisms and Materials], 2014, no. 1(26), pp. 49–52.
• Gevorgyan H. Traktovka geometricheskogo smysla konechnykh raznostey i proizvodnoy funktsii na osnove ispolzovaniya apparata MKE [An interpretation of the geometric meaning of the finite difference and the function derivative through
  the use of the finite element method tools]. Mekhanika mashin, mekhanizmov i materialov [Mechanics of Machines, Mechanisms and Materials], 2016, no. 2(35), pp. 95–98.
• Gevorgyan H. Raschet uprugikh progibov tonkikh zhestkikh plastin na osnove MKE bez ucheta gipotezy Kirkhgofa [Calculation of elastic deflections of thin stiff plates based on the finite element method out of the Kirchhoff’s theory]. Mekhanika mashin, mekhanizmov i materialov [Mechanics of Machines, Mechanisms and Materials], 2017, no. 1(38), pp. 39–44.
• Zienkiewicz O. Metod konechnykh elementov v tekhnike [The Finite Element Method in Engineering Science]. Moscow, Mir Publ., 1975. 543 p.
• Bathe K.J., Wilson E.L. Numerical Methods in Finite Element Analysis. Englewood Cliffs, Prentice-Hall, 1976. 528 p.
• Reddy J.N. An Introduction to the Finite Element Method. New York, McGraw-Hill, 2006. 761 p.
• Daryl L. Logan. A First Course in the Finite Element Method. Florence, USA, Nelson Engineering, 2011. 752 p.
• Conley R., Delaney T.J., Jiao X. Overcoming element quality dependence of finite elements with adaptive extended stencil FEM. International Journal for Numerical Methods in Engineering, 2016, vol. 108, no. 9, pp. 1054–1085.
• Natarajan S., Bordas S., Ooi E.T. Virtual and smoothed finite elements: A connection and its application to polygonal/polyhedral finite element methods. International Journal for Numerical Methods in Engineering, 2015, vol. 104, no. 13, pp. 1173–1199.
• Alvares Dias L., Vampa V., Martin M.T. The construction of plate finite elements using wavelet basis functions. Revista investigacion operacional, 2009, vol. 30, no. 3, pp. 193–204.
• Morozov N.F., Tovstik P.Е., Tovstik T.P. Obobshchennaya model Timoshenko–Reysnera dlya mnogosloynoy plastiny [Generalized model of Timoshenko–Reissner for multiplayer plate]. Izvestiya RAN, Mekhanika tverdogo tela [News of RAS, Mechanics of solids], 2016, no. 5, pp. 22–35.
• Zveryaev Е.М. Neprotivorechivaya teoriya tonkikh uprugikh obolochek [Noncontradictory theory of thin elastic shells]. Prikladnaya matematika i mekhanika [Applied mathematics and mechanics], 2016, no. 5, pp. 580–596.
• Gevorgyan H. Trivialnyy metod konechnykh elementov [Trivial finite element method]. Saarbrucken, LAP LAMBERT Academic Publishing, 2016. 208 p.
• Mandelbrot B. The fractal geometry of nature. New York, W.H. Freeman & Co, 1982. 498 p.
• Timoshenko S., Woinowsky-Krieger S. Plastinki i obolochki [Theory of plates and shells]. Moscow, Nauka Publ., 1966. 636 p.