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Mechanics of Thin-Walled Rods of Open Profile

  • Yury Vetyukov
Chapter
Part of the Foundations of Engineering Mechanics book series (FOUNDATIONS)

Abstract

We consider rods, whose cross section is a thin open-ended strip. The effect of warping, when the points of the rod move axially at torsion, needs to be included in the analysis. Beginning with the discussion of traditional approaches, based on certain hypotheses and approximations, we proceed with the asymptotic study of the corresponding model of a shell. The procedure of asymptotic splitting justifies known equations with an additional force factor (bi-moment) and warping, related to the rate of twist. Moreover, it allows for accurate recovery of the stressed state in a cross section first in terms of the shell theory, and then in the three-dimensional body. Further, we extend the results to the geometrically nonlinear range with the direct approach to a material line with additional degrees of freedom of particles due to the warping. The incremental analysis results in linear equations in the vicinity of a pre-stressed state. Solutions of particular problems show the importance of cubic terms in the strain energy of the rod, which is deduced from the equations of the nonlinear shell theory. Considered examples include linear problems, several types of buckling behavior as well as the analysis of higher order effects in comparison to numerical solutions with the shell model. The presented form of the direct approach was originally suggested by Vetyukov and Eliseev (Sci. Tech. Bull. St. Petersbg. State Polytechn. Univ., 1:49–53, 2007). This chapter quotes extensively from Vetyukov (Acta Mech., 200(3–4):167–176, 2008; J. Elast., 98(2):141–158, 2010) with kind permission from Springer Science and Business Media.

Supplementary material

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References

  1. 22.
    Bîrsan M (2004) The solution of Saint-Venant’s problem in the theory of Cosserat shells. J Elast 74:185–214 zbMATHGoogle Scholar
  2. 43.
    Davini C, Paroni R, Puntel E (2008) An asymptotic approach to the torsion problem in thin walled beams. J Elast 93:149–176 zbMATHMathSciNetGoogle Scholar
  3. 44.
    Di Egidio A, Vestroni F (2011) Static behavior and bifurcation of a monosymmetric open cross-section thin-walled beam: numerical and experimental analysis. Int J Solids Struct 48:1894–1905 CrossRefGoogle Scholar
  4. 50.
    Eliseev V (2003) Mechanics of elastic bodies. St Petersburg State Polytechnical University Publishing House, St Petersburg (in Russian) Google Scholar
  5. 59.
    Freddi L, Morassi A, Paroni R (2007) Thin-walled beams: a derivation of Vlassov theory via γ-convergence. J Elast 86:263–296 zbMATHMathSciNetGoogle Scholar
  6. 67.
    Golubev O (1963) Generalization of the theory of thin rods. Tr LPI 226:83–92 (in Russian) MathSciNetGoogle Scholar
  7. 70.
    Hamdouni A, Millet O (2006) An asymptotic non-linear model for thin-walled rods with strongly curved open cross-section. Int J Non-Linear Mech 41:396–416 CrossRefzbMATHMathSciNetGoogle Scholar
  8. 74.
    Hodges D (2006) Nonlinear composite beam theory. Progress in astronautics and aeronautics. American Institute of Aeronautics and Astronautics Google Scholar
  9. 119.
    Pietraszkiewicz W (1989) Geometrically nonlinear theories of thin elastic shells. Adv Mech 12(1):51–130 MathSciNetGoogle Scholar
  10. 127.
    Reismann H (1988) Elastic plates. Theory and application. Wiley, New York zbMATHGoogle Scholar
  11. 141.
    Simitses GJ, Hodges DH (2006) Fundamentals of structural stability. Elsevier, New York zbMATHGoogle Scholar
  12. 144.
    Simo J, Vu-Quoc L (1991) A geometrically exact rod model incorporating shear and torsion-warping deformation. Int J Solids Struct 27(3):371–393 CrossRefzbMATHMathSciNetGoogle Scholar
  13. 150.
    Timoshenko S (1945) Theory of bending, torsion and buckling of thin-walled members of open cross-section. J Franklin Inst 239(3–5):201–219, 249–268, 343–361 CrossRefzbMATHMathSciNetGoogle Scholar
  14. 151.
    Timoshenko S, Gere J (1961) Theory of elastic stability, 2nd edn. McGraw-Hill, New York Google Scholar
  15. 152.
    Timoshenko S, Woinowsky-Krieger S (1959) Theory of plates and shells, 2nd edn. McGraw-Hill, New York Google Scholar
  16. 158.
    Vetyukov Y (2008) Direct approach to elastic deformations and stability of thin-walled rods of open profile. Acta Mech 200(3–4):167–176 CrossRefzbMATHGoogle Scholar
  17. 159.
    Vetyukov Y (2010) The theory of thin-walled rods of open profile as a result of asymptotic splitting in the problem of deformation of a noncircular cylindrical shell. J Elast 98(2):141–158 zbMATHMathSciNetGoogle Scholar
  18. 163.
    Vetyukov Y, Eliseev V (2007) Elastic deformations and stability of equilibrium of thin-walled rods of open profile. Sci Tech Bull St Petersbg State Polytechn Univ 1:49–53 (in Russian) Google Scholar
  19. 168.
    Vlasov V (1961) Thin-walled elastic beams, 2nd edn. Israel Program for Scientific Translations, Jerusalem Google Scholar
  20. 170.
    Washizu K (1974) Variational methods in elasticity and plasticity. Pergamon, Elmsford Google Scholar
  21. 176.
    Yu W, Hodges D, Volovoi V, Fuchs E (2005) A generalized Vlasov theory for composite beams. Thin-Walled Struct 43(9):1493–1511 CrossRefGoogle Scholar
  22. 177.
    Yu W, Liao L, Hodges D, Volovoi V (2005) Theory of initially twisted, composite, thin-walled beams. Thin-Walled Struct 43(8):1296–1311 CrossRefGoogle Scholar
  23. 179.
    Ziegler F (1995) Mechanics of solids and fluids, 2nd edn. Mechanical engineering series. Springer, Vienna CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Wien 2014

Authors and Affiliations

  • Yury Vetyukov
    • 1
  1. 1.Institute of Technical MechanicsJohannes Kepler UniversityLinzAustria

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