Journal Journal of Mechanical Engineering – Problemy Mashynobuduvannia
Publisher A. Podgorny Institute for Mechanical Engineering Problems
National Academy of Science of Ukraine
ISSN 0131-2928 (Print), 2411-0779 (Online)
Issue Vol. 21, no. 4, 2018 (December)
Pages 30-36
Cited by J. of Mech. Eng., 2018, vol. 21, no. 4, pp. 30-36



Stanislav B. Kovalchuk, Poltava State Agrarian Academy (1/3, Skovorody Str., Poltava, 36003, Ukraine), e-mail:, ORCID: 0000-0003-4550-431X

Aleksey V. Gorik, Poltava State Agrarian Academy (1/3, Skovorody Str., Poltava, 36003, Ukraine), ORCID: 0000-0002-2804-5580



The development of composite technologies contributes to their being widely introduced into the practice of designing modern different-purpose structures. Reliable prediction of the stress-strain state of composite elements is one of the conditions for creating reliable structures with optimal parameters. Analytical theories for determining the stress-strain state of multilayer rods (bars, beams) are significantly inferior in development to those for composite plates and shells, although their core structural elements are most common. The purpose of this paper is to design an analytical model for bending double support multilayer beams under concentrated load based on the previously obtained solution of the elasticity theory for a multi-layer cantilever. The first part of the article includes a statement of the problem, accepted prerequisites and main stages of constructing a model for bending a double-support multi-layer beam with a concentrated load (normal, tangential force and moment) and general-view supports in the extreme cross-sections. When building the model, the double support beam was divided across the loaded cross-section and presented in the form of two separate sections with equivalent loads on the ends. Using the general solution of the elasticity theory for a multilayer cantilever with a load on the ends, the main stress-strain state of the design sections was described without taking into account the local effects of changing the stress state near the concentrated load application points and supports. The obtained relations contain 12 unknown initial parameters. To determine them on the basis of the conditions of joint deformation (static and kinematic) of design sectors, a system of algebraic equations has been constructed. The constructed model allows one to determine the components of the main stress-strain state of double support beams each consisting of an arbitrary number of orthotropic layers, taking into account the amenability of their materials to lateral shear deformations and compression.


Keywords: multilayer beam, orthotropic layer, concentrated load, stresses, displacements.


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  1. Altenbakh, Kh. (1998). Theories for laminated and sandwich plates. A review. Mechanics of Composite Materials, vol. 34, iss. 3, pp. 243–252.
  2. Ambartsumyan, S. A. (1987). Teoriya anizotropnykh plastin [Theory of anisotropic plates]. Moscow: Nauka, 360 p. (in Russian).
  3. Bolotin, V. V. & Novichkov, Yu. N. (1980). Mekhanika mnogosloynykh konstruktsiy [Mechanics of multilayer structures]. Moscow: Mashinostroyeniye, 374 p. (in Russian).
  4. Vasilyev, V. V. (1988). Mekhanika konstruktsiy iz kompozitsionnykh materialov [Mechanics of structures made of composite materials]. Moscow: Mashinostroyeniye, 272 p. (in Russian).
  5. Grigolyuk, E. I. & Selezov, I. T. (1972). Neklassicheskaya teoriya kolebaniy sterzhney, plastin i obolochek. Itogi nauki i tekhniki [Non-classical theory of oscillations of rods, plates and shells. Results of science and technology]. Vol. 5. Moscow: Nauka, 271 p. (in Russian).
  6. Guz, A. N., Grigorenko, Ya. M., Vanin, G. A., & Babich, I. Yu. (1983). Mekhanika elementov konstruktsiy: V 3 t. T. 2: Mekhanika kompozitnykh materialov i elementov konstruktsiy [Mechanics of structural elements: In 3 vol. Vol. 2: Mechanics of composite materials and structural elements]. Kiyev: Naukova dumka, 484 p. (in Russian).
  7. Malmeyster, A. K., Tamuzh, V. P., & Teters, G. A. (1980). Soprotivleniye polimernykh i kompozitnykh materialov [Resistance of polymeric and composite materials]. Riga: Zinatne, 572 p. (in Russian).
  8. Rasskazov, A. O., Sokolovskaya, I. I., & Shulga, N. A. (1987). Teoriya i raschet sloistykh ortotropnykh plastin i obolochek [Theory and calculation of layered orthotropic plates and shells]. Kiyev: Vyshcha shkola, 200 p. (in Russian).
  9. Piskunov, V. G. (2003). An iterative analytical theory in the mechanics of layered composite systems. Mechanics of Composite Materials, vol. 39, iss. 1, pp. 1–16.
  10. Horyk, O. V., Piskunov, V. H., & Cherednikov, V. M. (2008). Mekhanika deformuvannia kompozytnykh brusiv [Mechanics of deformation of composite beams]. Poltava – Kyiv: ACMI, 402 p. (in Ukrainian).
  11. Goryk, A. V. (2001). Modeling transverse compression of cylindrical bodies in bending. International Applied Mechanics, vol. 37, iss. 9, pp. 1210–1221.
  12. Goryk, A. V. & Koval’chuk, S. B. (2018). Elasticity theory solution of the problem on plane bending of a narrow layered cantilever bar by loads at its end. Mechanics of Composite Materials, vol. 54, iss. 2, pp. 179–190.
  13. Goryk, A. V. & Koval’chuk, S. B. (2018). Solution of a transverse plane bending problem of a laminated cantilever beam under the action of a normal uniform load. Strength of Materials, vol. 50, iss. 3, pp. 406–418.
  14. Goldenveyzer, A. L. (1976). Teoriya uprugikh tonkikh obolochek [Theory of elastic thin shells]. Moscow: Nauka, 512 p. (in Russian).


Received 26 September 2018

Published 30 December 2018