Skip to main content
Log in

Effect of boundary conditions in three alternative models of Timoshenko–Ehrenfest beams on Winkler elastic foundation

  • Original Paper
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

In this paper we discuss about the free vibrations of a beam on Winkler foundation via original Timoshenko–Ehrenfest beam theory, as well as one of its truncated versions, and a model based on slope inertia. Differences between the three models are indicated. We analyze five different sets of boundary conditions, which are derived from the most typical end constraints: simply supported end, clamped end and free end. A detailed proof about the non-existence of zero frequencies for the free–free beam and for the simply supported–free beam is given. Differences between the models are indicated in the context of free vibrations of the beam on Winkler foundations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abohadima, S., Taha, M., Abdeen, M.A.M.: General analysis of Timoshenko beam on elastic foundation, Hindawi Publishing Corporation. Math. Probl. Eng. 2015(182523), 1–11 (2015)

    Article  Google Scholar 

  2. Auciello, N.M.: Vibrations of Timoshenko beams on two-parameter elastic soil. Eng. Trans. 56(3), 187–200 (2008)

    MathSciNet  Google Scholar 

  3. Bresse, J.A.A.: Cours de Mecanique Appliquee. Mallet-Bachelier, Paris (1859). (in French)

    MATH  Google Scholar 

  4. Cazzani, A., Stochino, F., Turco, E.: On the whole spectrum of Timoshenko beams. Part I: a theoretical revisitation. Z. Angew. Math. Phys. 67(24), 1–30 (2016)

    MathSciNet  MATH  Google Scholar 

  5. Cazzani, A., Stochino, F., Turco, E.: On the whole spectrum of Timoshenko beams. Part II: further application. Z. Angew. Math. Phys. 67(25), 1–21 (2016)

    MathSciNet  MATH  Google Scholar 

  6. Cazzani, A., Stochino, F., Turco, E.: An analytical assessment of finite element and isogeometric analyses of the whole spectrum of Timoshenko beams. Z. Angew. Math. Mech. (ZAMM)/J. Appl. Math. Mech. 96, 1220–1224 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. Crandall, S.H.: The Timoshenko beam on elastic foundation. In: Proceedings of the Third Midwestern Conference on Solid Mechanics, pp. 146–159. The University of Michigan Press, Ann Arbor, MI (1957)

  8. De Rosa, M.A.: Free vibrations of Timoshenko beams on two-parameter elastic foundation. Comput. Struct. 57(1), 151–156 (1995)

    Article  MATH  Google Scholar 

  9. Downs, B.: Transverse vibration of a uniform simply supported Timoshenko beam without transverse deflection. J. Appl. Mech. 43(4), 671–674 (1976)

    Article  MATH  Google Scholar 

  10. Elishakoff, I., Livshits, D.: Some closed-form solutions in random vibration of Timoshenko–Ehrenfest beams. Probab. Eng. Mech. 4(1), 49–54 (1989)

    Article  Google Scholar 

  11. Elishakoff, I.: An equation both more consistent and simpler than the Timoshenko–Ehrenfest equation. In: Advances in Mathematical Modeling and Experimental Methods for Materials and Structures, pp. 249–254. Springer, Dordrecht (2010)

  12. Elishakoff, I., Kaplunov, J., Nolde, E.: Celebrating the centenary of Timoshenko’s study of effects of shear deformation and rotary inertia. Appl. Mech. Rev. 67(6), 060802 (2015)

    Article  Google Scholar 

  13. Elishakoff, I., Hache, F., Challamel, N.: Critical contrasting of three versions of vibrating Bresse–Timoshenko beam with a crack. Int. J. Solids Struct. 109, 143–151 (2017)

    Article  Google Scholar 

  14. Ghannadiasl, A., Mofid, M.: An analytical solution for free vibration of elastically restrained Timoshenko beam on an arbitrary variable Winkler foundation and under axial load. Lat. Am. J. Solids Struct. 12(13), 2417–2438 (2015)

    Article  Google Scholar 

  15. Goens, E.: Über die Bestimmung des Elastizitätsmodulus von Stäben mit Hifle von Biegungsschwingungen. Ann. Phys. 403(6), 649–678 (1931)

    Article  MATH  Google Scholar 

  16. Hassan, M.T., Nassar, M.: Analysis of stressed Timoshenko beams on two parameters foundation. KSCE J. Civ. Eng. 19(1), 173–179 (2015)

    Article  Google Scholar 

  17. Huang, T.C.: The effect of rotary inertia and of shear deformation on the frequency and normal mode equations of uniform beams with simple end conditions. J. Appl. Mech. 28(4), 579–584 (1961)

    Article  MATH  Google Scholar 

  18. Kang, J.-H.: Exact characteristic equations in closed-form for vibration of completely free Timoshenko beams. Int. J. Struct. Stab. Dyn. 16(10), 1550078 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Malekzadeh, P., Karami, G., Farid, M.: DQEM for free vibration analysis of Timoshenko beams on elastic foundation. Comput. Mech. 31, 219–228 (2003)

    MATH  Google Scholar 

  20. Manevich, A.I.: Transverse waves in a Timoshenko beam of visco-elastic material. Theor. Found. Civ. Eng. (Warsaw) 17, 217–228 (2009). (in Russian)

    Google Scholar 

  21. Manevich, A.I.: Dynamics of Timoshenko beam on linear and nonlinear foundation: phase relations, significance of the second spectrum, stability. J. Sound Vib. 334, 209–220 (2015)

    Article  Google Scholar 

  22. Muravsky, G.B.: Oscillation of the Timoshenko type beam lying on the elastic-hereditary foundation. Mekhanika Tverdogo Tela (Mechanics of Solids), 13(5):167–169 (1981) (in Russian)

  23. Noga, S., Bogacz, R.: Free vibration of the Timoshenko beam interacting with the Winkler foundation. Symulacja w Badaniach I Rozw. 2(4), 209–223 (2011)

    Google Scholar 

  24. Obara, P.: Vibrations and stability of Bernoulli–Euler and Timoshenko beams on two-parameter elastic foundation. Arch. Civ. Eng. 60(4), 421–439 (2014)

    Google Scholar 

  25. Pielorz, A.: Discrete-continuous models in the analysis of low structures subject to kinematic excitations caused by transversal waves. J. Theor. Appl. Mech. 34(3), 547–566 (1996)

    Google Scholar 

  26. Rayleigh L. (J.W.S. Strut): The Theory of Sound. Macmillan, London (see also Dover, New York, 1945) (1878)

  27. Sadeghian, M., Ekhterai, Toussi H.: Frequency analysis for a Timoshenko beam located on elastic foundation. Int. J. Eng. Trans. A Basics 24(1), 87–105 (2010)

    Google Scholar 

  28. Timoshenko, S.P.: A Course in Theory of Elasticity, Part II: Rods and Plates. “Kollins” Publishers, Petrograd (1916). (in Russian)

    Google Scholar 

  29. Timoshenko, S.P.: On the differential equation for the flexural vibrations of prismatical rods. Glas. Hrvat. Prir. Drus. 32(2), 55–57 (1920)

    Google Scholar 

  30. Timoshenko, S.P.: On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Philos. Mag. 41, 744–746 (1921)

    Article  Google Scholar 

  31. Timoshenko, S.P.: History of Strength of Materials. With a Brief Account of the History of Theory of Elasticity and Theory of Structures. Dover Publications, New York (1953)

    MATH  Google Scholar 

  32. Timoshenko, S.P.: A Course in Theory of Elasticity, 2nd edn. “Naukova Dumka” Publishers, Kiev (1972). (in Russian)

    Google Scholar 

  33. Vlasov, V.Z., Leontiev, U.N.: Beams, plates, and shells on elastic foundations, (translated from Russian). Israel Program for Scientific Translation, Jerusalem (1966)

    Google Scholar 

  34. Wang, C.M., Lam, K.Y., He, X.O.: Exact solution for Timoshenko beams on elastic foundations using Green’s functions. J. Struct. Mech. 26(1), 101–113 (1998)

    Google Scholar 

  35. Wang, Y.H., Tham, L.G., Cheung, Y.K.: Beams and plates on elastic foundations: a review. Prog. Struct. Eng. Mater. 7(4), 174–182 (2005)

    Article  Google Scholar 

  36. Wu, J.-S.: Analytical and Numerical Methods for Vibration Analyses, Section 2.12: Vibration of a Timoshenko Beam on the Elastic Foundation. Wiley, Singapore (2013)

    Google Scholar 

  37. Yerofeev, V.I., Kazhaev, V.V., Lisenkova, Y.Y., Semerikova, N.P.: Comparative analysis of dynamical behavior of beams in Bernoulli–Euler, Rayleigh and Timoshenko models lying on elastic foundation. Vestn. Nauchno-Tekhnicheskogo Razvit. Nizhnyi Novgorod 8(24), 18–26 (2009). (in Russian)

    Google Scholar 

  38. Yin, J.H.: Closed-form solution for reinforced Timoshenko beam on elastic foundation. J. Eng. Mech. 126(8), 868–874 (2000)

    Article  Google Scholar 

  39. Yokoyama, T.: Vibration analysis of Timoshenko beam-columns on two-parameter elastic foundations. Comput. Struct. 61(6), 995–1007 (1996)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

First author acknowledges helpful discussion with Prof. Arkady Manevich of the Ukrainian Chemical Technology University, Dnepropetrovsk, Ukraine. We are thankful to two anonymous reviewers for providing insightful and constructive comments.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Isaac Elishakoff.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Elishakoff, I., Tonzani, G.M. & Marzani, A. Effect of boundary conditions in three alternative models of Timoshenko–Ehrenfest beams on Winkler elastic foundation. Acta Mech 229, 1649–1686 (2018). https://doi.org/10.1007/s00707-017-2034-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-017-2034-x

Navigation