Acta Mechanica

, Volume 229, Issue 4, pp 1649–1686 | Cite as

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

  • Isaac Elishakoff
  • Giulio Maria Tonzani
  • Alessandro Marzani
Original Paper


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.


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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.


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Copyright information

© Springer-Verlag GmbH Austria 2017

Authors and Affiliations

  • Isaac Elishakoff
    • 1
  • Giulio Maria Tonzani
    • 2
  • Alessandro Marzani
    • 2
  1. 1.Department of Ocean and Mechanical EngineeringFlorida Atlantic UniversityBoca RatonUSA
  2. 2.Dipartimento di Ingegneria Civile, Chimica, Ambientale e dei Materiali, Scuola di Ingegneria e ArchittetturaUniversitá di BolognaBolognaItaly

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