Mechanics of Composite Materials

, Volume 54, Issue 5, pp 605–620 | Cite as

Elasticity Theory Solution of the Problem on Bending of a Narrow Multilayer Cantilever with a Circular Axis by Loads at its End

  • S. B. Koval’chukEmail author
  • A. V. Goryk

An exact solution of the problem on plane bending of a narrow multilayer cantilever bar with a curved circular axis by tangential and normal loads distributed on its free end face is presented. The natural (for a bar structure) cylindrical circular coordinate system is used to describe the structure and geometry of the bar. The solution is obtained by directly integrating the equations of a plane elasticity theory problem using an analytical description of the mechanical characteristics of the discrete-inhomogeneous multilayer bar. Physical relations that take into account the cylindrical orthotropy of the material of bar layers and the conditions of absolutely rigid contact of layers are used in constructing the solution. The theoretical relations are realized for the test problem on determining the strain-stress state of a four-layer cantilever with a semiring axis. The solution obtained allows one to predict the strength and rigidity, to develop optimum design techniques, and to construct analytical solutions of different problems on bending of multilayer curved bars.


curved bar circular axis composite layer bending stress strain displacement 


  1. 1.
    H. Altenbach, “Theories for laminated ans sandwich plates. A review,” Mech. Compos. Mater., No. 3, 333-348 (1998).Google Scholar
  2. 2.
    S. A. Аmbartsumyan, General Theory of Anisotropic Shells [in Russian], M., Nauka (1974).Google Scholar
  3. 3.
    N. A. Alfutov, P. A. Zinovyev, and B. G. Popov, Calculation of Multilayered Plates and Shells Made of Composite Materials [in Russian], M., Mashinostroenie (1984).Google Scholar
  4. 4.
    V. V. Bolotin and Yu. N. Novickov, Mechanics of Multilayered Structures [in Russian], M., Mashinostroenie (1980).Google Scholar
  5. 5.
    V. V. Vasil’ev, Mechanics of Structures Made of Composite Materials[in Russian], M., Mashinostroenie (1988).Google Scholar
  6. 6.
    N. J. Pagano, “Exact solutions for rectangular bidirectional composites,” J. Compos. Mater., 4, 20-34 (1970).Google Scholar
  7. 7.
    A. N. Guz’, Ya. M. Grigorenko, G. A. Vanin, and I. Yu. Babich, Mechanics of Structural Elements, In 3 vol. Vol. 2 [in Russian], Mechanics of Composite Materials and Structural Elements [in Russian], Kiev. Nauk. Dumka (1983).Google Scholar
  8. 8.
    A. K. Malmeister, V. P. Tamuzh, and G. A. Teters, Strength of Polymer and Composite Materials [in Russian], Riga, Zinatne (1980).Google Scholar
  9. 9.
    A. V. Goryk, “Modeling the transverse compression of cylindrical bodies in bending,” Int. Appl. Mech., 37, Iss. 9, 1210-1221 (2001).Google Scholar
  10. 10.
    S. G. Lekhnitskii, Elasticity Theory of Anisotropic Bodies [in Russian], M., Nauka (1977).Google Scholar
  11. 11.
    N. I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity [in Russian], M., Nauka (1966).Google Scholar
  12. 12.
    A. V. Goryk and S. B. Koval’chuk, “Elasticity theory solution of the problem on plane bending of a narrow layered cantilever beam by loads at its free end,” Mech. Compos. Mater., 54, No. 2, 179-190 (2018).CrossRefGoogle Scholar
  13. 13.
    S. G. Lekhnitskii, “On the bending of a plane inhomogeneous curved beam,” J. Appl. Math. Mech., 43, No. 1, 182-183 (1979).CrossRefGoogle Scholar
  14. 14.
    G. Tolf, “Stresses in a cerved laminated beam,” Fiber Sci. and Technol., 19, No. 4, 243-267 (1983).CrossRefGoogle Scholar
  15. 15.
    W. L. Ko and R. H. Jackson, “Multilayer theory for delamination analysis of a composite curved bar subjected to end forces and end moments,” Composite Structures 5, Springer, Dordrecht, 173-198 (1989).Google Scholar
  16. 16.
    S. P. Timoshenko and J. N. Goodier, Elasticity Theory [Russian translation], ed. G. S. Shapiro, M., Nauka (1979).Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Poltava State Agrarian AcademyPoltavaUkraine

Personalised recommendations