Vibrations And Stability Of Bernoulli-Euler And Timoshenko Beams On Two-Parameter Elastic Foundation

  • 1 Kielce University of Technology, Faculty of Civil and Environmental Engineering, Al. 1000-lecia PP 7, 25-314 Kielce, Poland

Abstract

The vibration and stability analysis of uniform beams supported on two-parameter elastic foundation are performed. The second foundation parameter is a function of the total rotation of the beam. The effects of axial force, foundation stiffness parameters, transverse shear deformation and rotatory inertia are incorporated into the accurate vibration analysis. The work shows very important question of relationships between the parameters describing the beam vibration, the compressive force and the foundation parameters. For the free supported beam, the exact formulas for the natural vibration frequencies, the critical forces and the formula defining the relationship between the vibration frequency and the compressive forces are derived. For other conditions of the beam support conditional equations were received. These equations determine the dependence of the frequency of vibration of the compressive force for the assumed parameters of elastic foundation and the slenderness of the beam.

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  • 1. Bołotin, W.W.: Dynamic Stability of Elastic Systems, Moskwa, 1956. (in Russian).

  • 2. Timoshenko, S.P., Gere, J.M.: Theory of elastic stability, McGraw–Hill, New York, 1961.

  • 3. Wolmir, A.C.: Stability of Elastic Systems, Moskwa, 1963. (in Russian).

  • 4. Timoshenko, S.P.: On the correction for shear of the differential equation for transverse vibrations of prismatic bars, Philosophical Magazine, Vol. 41, 744–746, 1921.

  • 5. Timoshenko, S.P.: On the transverse vibrations of bars of uniform cross– section, Philosophical Magazine, Vol. 43, 125–131, 1922.

  • 6. Gryczmański, M., Jurczyk, P.: The Subsoil Models and their Evaluation, Inżynieria i Budownictwo, Vol. 2, No. 95, 98–104, 1995. (in Polish).

  • 7. Jemielita, G., Szcześniak, W.: Methods for Foundation Modeling, Prace Naukowe Politechniki Warszawskiej, Vol. 120, 1–33, 1993. (in Polish).

  • 8. Winkler, E.: Die Lehre von der Elastizität und Festigkeit, Dominicus, Prague, (1867).

  • 9. Thambiratnam, D., Zbuge, Y.: Free vibration analysis of beam on elastic foundation, Computers and Structures, Vol. 60, No. 6, 971–980, 1996.

  • 10. Chen, C.N.: DQEM vibration analyses of non–prismatic shear deformable beams resting on elastic foundations, Journal of Sound and Vibration, Vol. 255, No. 5, 989–999, 2002.

  • 11. De Rosa, M.A.: Stability and dynamics of beams on Winkler elastic foundation, Earthquake Engineering and Structural Dynamics, Vol. 18, 377–388, 1989.

  • 12. Fargitaly, S.H., Zeid, K.M.: An exact frequency equation for an axially loaded beam–mass–spring system resting on a Winkler elastic foundation, Journal of Sound and Vibration, Vol. 185, No. 2, 357–363, 1995.

  • 13. Nageswara, Rao B., Venkateswara, Rao G.: Post–critical behaviour of Euler and Beck columns resting on an elastic foundation, Journal of Sound and Vibration, Vol. 2764, 1150–1158, 200.

  • 14. Song Xi, Soi– Rong Li.: Thermal buckling and post–buckling of pinned–fixed Euler–Bernoulli beams on an elastic foundation, Mechanics Research Communications, Vol. 34, 164–171, 2006.

  • 15. Kim, S.M.: Vibration and stability of axial loaded beams on elastic foundation under moving harmonic loads, Engineering Structures, Vol. 26, 95–10, 20045.

  • 16. Kim, S.M., Cho, Y. H.: Vibration and dynamic buckling of shear beam–columns on elastic foundation under moving harmonic loads, International Journal of Solids and Structures, Vol. 43, 393–412, 2006.

  • 17. Sato, M., Kanie, S., Mikami, T.: Mathematical analogy of a beam on elastic supports as a beam on elastic foundation, Applied Mathematical Modelling, Vol. 32, 688–699, 2008.

  • 18. Pasternak, P.Ł.: On a new method of analysis of an elastic foundation by means of two foundation constants, Gosstrojizdat, Moscow, 1954. (in Russian).

  • 19. Filonienko – Borodich, M.M.: Some approximate theories of elastic foundation, Uchenyie Zapiski Moskovskogo Gosudarstvennogo Universtiteta, Mechanika, Vol. 46, 3–18, 1940. (in Russian).

  • 20. Vlasow, V.Z., Loentiev, U.N.: Beams, plates and shells on elastic foundation, Gosfizmat, Moskow, 1966. (in Russian).

  • 21. Gomuliński, A.: Determination of eigenvalues for circular plates resting on elastic foundation with two moduli, Archives of Civil Engineering, Vol. XIII, No. 2, 183–203, 1967. (in Polish).

  • 22. Ayvaz, Y., Daloglu, A.: Earthquake analysis of beams resting on elastic foundations by using a modified Vlasov model, Journal of Sound and Vibration, Vol. 200, No. 3, 315–325, 1997.

  • 23. Ayvaz, Y.: Application of modified Vlasov model to free vibration analysis of beams resting on elastic foundations, Journal of Sound and Vibration, Vol. 255, No. 1, 111–127, 2002.

  • 24. Naidu, N.R., Rao, G.V.: Stability behaviour of uniform beams on a class of two–parameter elastic foundation, Computers and Structures, Vol. 57, No. 3, 551–553, 1995.

  • 25. Naidu, N.R., Rao, G.V.: Vibrations of initially stressed uniform beams on two–parameter elastic foundation, Computers and Structures, Vol. 57, No. 2, 941–943, 1995.

  • 26. Yokoyama, T.: Vibration analysis of Timoshenko beam–columns on two–parameter elastic foundations, Computers and Structures, Vol. 61, No. 6, 995–1007, 1995.

  • 27. El – Mously, M.: Foundamental frequencies of Timoshenko beams mounted on Pasternak foundation, Journal of Sound and Vibration, Vol. 228, No. 2, 452–157, 1999.

  • 28. Arboleda– Monsalve, L.G., Apata– Medina, D.G., Aristizabal – Ochoa, J.D.: Timoshenko beam–column with generalized and conditions on elastic foundation: Dynamic–stiffness matrix and load vector, Journal of Sound and Vibration, Vol. 310, 1057–1079, 2008.

  • 29. De Rosa, M.A.: Free vibrations of Timoshenko beams on two–parameter elastic foundation, Computers and Structures, Vol. 57, No. 1, 151–156, 1995.

  • 30. Filipich, C.P., Rosales, M.B.: A further study about the behaviour of foundation piles and beams in a Winkler–Pasternak soil, International Journal of Mechanical Sciences, Vol. 44, 21–36, 2002.

  • 31. Chen, W.Q., Lü, C.F., Bian, Z.G.: A mixed method for bending and free vibration of beams resting on a Pasternak elastic foundation, Applied Mathematical Modelling, Vol. 28, 877–890, 2004.

  • 32. Matsunaga, H.: Vibration and buckling of deep beam–columns on two–parameter elastic foundations, Journal of Sound and Vibration, Vol. 228, No. 2, 359–376, 1999.

  • 33. Ying, C.F. Lü, Chen, W.Q.: Two–dimensional elasticity solutions for functionally graded beams resting on elastic foundations, Composite Structures, Vol. 84, 209–219, 2008.

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