On the Out-of-Plane Deviation of Bending Deformation States in Moderately Thick Bars with Asymmetric Cross-Sections

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Abstract

A characteristic feature of the six-parameter theories of bars is the coupled form of the constitutive equations; in particular the equations linking transverse forces with transverse shear deformations cannot be, in general, decoupled while keeping a separate form of the remaining constitutive equations. The mentioned feature of the constitutive equations implies that, within the six-parameter theories of straight elastic prismatic bars, there do not exist, in general, plane states of bending/shearing deformations. Thus, any vertical load causes lateral deflections, the only exception being the pure bending problem. The present paper delivers analytical solutions: closed-form formulae for shape functions, i.e., deformation states associated with kinematic loads at the ends, and solutions to selected static problems corresponding to transverse span load. Although elementary, the presented solutions seem to be derived for the first time. In particular, the hitherto published shape functions concerned the theories of moderately thick bars in which all the constitutive equations are decoupled.

Keywords:

six-parameter theory of bars, moderately thick bars, shape functions

References


  1. Vlasov V.Z., Thin-walled bars [in Russian], [in:] Collected Works of V.Z. Vlasov, Sokolovskii V.V. et al. [Eds], Vol. II, The Publisher House of the Academy of Sciences of SSSR, Moscow, 1963.

  2. Bazoune A., Khulief Y.A., Stephen N.G., Shape functions of three-dimensional Timoshenko beam element, Journal of Sound and Vibration, 259(2): 473–480, 2003, https://doi.org/10.1006/jsvi.2002.5122

  3. Petrolo A.S., Casciaro R., 3D beam element based on Saint Venant’s rod theory, Computers & Structures, 82(29): 2471–2481, 2004, https://doi.org/10.1016/j.compstruc.2004.07.004

  4. Lewiński T., Czarnecki S., On incorporating warping effects due to transverse shear and torsion into the theories of straight elastic bars, Acta Mechanica, 232(1): 247–282, 2021, https://doi.org/10.1007/s00707-020-02849-7

  5. Love A.E.H., Treatise on the Mathematical Theory of Elasticity, Cambridge University Press, Cambridge, 1927.

  6. Iesan D., On Saint–Venant’s problem, Archive for Rational Mechanics and Analysis, 91(4): 363–373, 1986, https://doi.org/10.1007/BF00282340

  7. Librescu L., Song O., Thin-Walled Composite Beams: Theory and Application, Springer, Dordrecht, 2006, https://doi.org/10.1007/1-4020-4203-5

  8. Dikaros I.C., Sapountzakis E.J., Argyridi A.K., Generalized warping effect in the dynamic analysis of beams of arbitrary cross section, Journal of Sound and Vibration, 369: 119–146, 2016, https://doi.org/10.1016/j.jsv.2016.01.022

  9. El Fatmi R., Non-uniform warping including the effects of torsion and shear forces. Part I: A general beam theory, International Journal of Solids and Structures, 44(18–19): 5912–5929, 2007, https://doi.org/10.1016/j.ijsolstr.2007.02.006

  10. El Fatmi R., Non-uniform warping including the effects of torsion and shear forces. Part II: Analytical and numerical applications, International Journal of Solids and Structures, 44(18–19): 5930–5952, 2007, https://doi.org/10.1016/j.ijsolstr.2007.02.005

  11. Schramm U., Kitis L., Kang W., Pilkey W.D., On the shear deformation coefficient in beam theory, Finite Elements in Analysis and Design, 16(2): 141–162, 1994, https://doi.org/10.1016/0168-874X(94)00008-5

  12. Kączkowski Z., Statics of bars and systems of bars, [in:] Technical Mechanics Strength of Structural Elements [in Polish: Mechanika Techniczna. Wytrzymałość Elementow Konstrukcyjnych], Życzkowski M. [Ed.], pp. 19–179, Warszawa, 1988.

  13. Pełczyński J., Gilewski W., Algebraic formulation for moderately thick elastic frames, beams, trusses, and grillages within Timoshenko theory, Mathematical Problems in Engineering, 2019(1): 7545473, 2019, https://doi.org/10.1155/2019/7545473

  14. Luo Y., An efficient 3D Timoshenko beam element with consistent shape functions, Advances in Applied Mechanics, 1(3): 95–106, 2008.