DOI QR코드

DOI QR Code

Physical insight into Timoshenko beam theory and its modification with extension

  • Senjanovic, Ivo (Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb) ;
  • Vladimir, Nikola (Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb)
  • 투고 : 2013.07.04
  • 심사 : 2013.11.01
  • 발행 : 2013.11.25

초록

An outline of the Timoshenko beam theory is presented. Two differential equations of motion in terms of deflection and rotation are comprised into single equation with deflection and analytical solutions of natural vibrations for different boundary conditions are given. Double frequency phenomenon for simply supported beam is investigated. The Timoshenko beam theory is modified by decomposition of total deflection into pure bending deflection and shear deflection, and total rotation into bending rotation and axial shear angle. The governing equations are condensed into two independent equations of motion, one for flexural and another for axial shear vibrations. Flexural vibrations of a simply supported, clamped and free beam are analysed by both theories and the same natural frequencies are obtained. That fact is proved in an analytical way. Axial shear vibrations are analogous to stretching vibrations on an axial elastic support, resulting in an additional response spectrum, as a novelty. Relationship between parameters in beam response functions of all type of vibrations is analysed.

키워드

참고문헌

  1. Carrera, E., Giunto, G. and Petrolo M. (2011), Beam Structures, Classical and advanced Theories, John Wiley & Sons Inc., New York, NY, USA.
  2. Cowper, G.R. (1966), "The shear coefficient in Timoshenko's beam theory", J. Appl. Mech., 33, 335-340. https://doi.org/10.1115/1.3625046
  3. De Rosa, M.A. (1995), "Free vibrations of Timoshenko beams on two-parametric elastic foundation", Comput. Struct., 57, 151-156. https://doi.org/10.1016/0045-7949(94)00594-S
  4. Geist, B. and McLaughlin, J.R. (1997), "Double eigenvalues for the uniform Timoshenko beam", Appl. Math. Letters, 10(3), 129-134.
  5. Inman, D.J. (1994), Engineering Vibration, Prentice Hall, Inc., Englewood Cliffs, New Jersey.
  6. Levinson, M. (1981a), "A new rectangular beam theory", J. Sound Vib., 74(1), 81-87. https://doi.org/10.1016/0022-460X(81)90493-4
  7. Levinson, M. (1981b), "Further results of a new beam theory", J. Sound Vib., 77(3), 440-444. https://doi.org/10.1016/S0022-460X(81)80180-0
  8. Li, X.F. (2008), "A unified approach for analysing static and dynamic behaviours of functionally graded Timoshenko and Euler-Bernoulli beams", J. Sound Vib., 32(5), 1210-1229.
  9. Matsunaga, H. (1999), "Vibration and buckling of deep beam-columns on two-parameter elastic foundations", J. Sound Vib., 228, 359-376. https://doi.org/10.1006/jsvi.1999.2415
  10. Mindlin, R.D. (1951), "Influence of rotary inertia and shear on flexural motions of isotropic elastic plates", J. Appl. Mech., 18(1), 31-28.
  11. Pavazza, R. (2005), "Torsion of thin-walled beams of open cross-sections with influence of shear", Int. J. Mech. Sci., 47, 1099-1122. https://doi.org/10.1016/j.ijmecsci.2005.02.007
  12. Pavazza, R. (2007), Introduction to the Analysis of Thin-Walled Beams, Kigen, Zagreb, Croatia. (in Croatian)
  13. Pilkey, W.D. (2002), Analysis and Design of Elastic Beams, John Wiley & Sons Inc., New York, NY, USA.
  14. Reddy, J.N. (1997), "On locking free shear deformable beam elements", Comput. Meth. Appl. Mech. Eng., 149, 113-132. https://doi.org/10.1016/S0045-7825(97)00075-3
  15. Senjanovic, I. and Fan, Y. (1989), "A higher-order flexural beam theory", Comput. Struct., 10, 973-986.
  16. Senjanovic, I. and Fan, Y. (1990), "The bending and shear coefficients of thin-walled girders", Thin-Wall. Struct., 10, 31-57. https://doi.org/10.1016/0263-8231(90)90004-I
  17. Senjanovic, I. and Fan, Y. (1993), "A finite element formulation of ship cross-sectional stiffness parameters", Brodogradnja, 41(1), 27-36.
  18. Senjanovic, I. and Tomasevic, S. (1999), "Longitudinal strength analysis of a Cruise Vessel in early design stage", Brodogradnja, 47(4), 350-355.
  19. Senjanovic, I., Toma?evic, S. and Vladimir, N. (2009), "An advanced theory of thin-walled structures with application to ship vibrations", Mar. Struct., 22(3), 387-437. https://doi.org/10.1016/j.marstruc.2009.03.004
  20. Simsek, M. (2011), "Forced vibration of an embedded single-walled carbon nanotube traversed by a moving load using nonlocal Timoshenko beam theory", Steel. Comp. Struct., 11(1), 59-76. https://doi.org/10.12989/scs.2011.11.1.059
  21. Sniady, P. (2008), "Dynamic response of a Timoshenko beam to a moving force", J. Appl. Mech., 75(2), 0245031-0245034.
  22. Stojanovic, V. and Kozic, P. (2012), "Forced transverse vibration of Rayleigh and Timoshenko double-beam system with effect of compressive axial load", Int. J. Mech. Sci., 60, 59-71. https://doi.org/10.1016/j.ijmecsci.2012.04.009
  23. Stojanovic, V., Kozic, P. and Janevski, G. (2013), "Exact closed-form solutions for the natural frequencies and stability of elastically connected multiple beam system using Timoshenko and higher-order shear deformation theory", J. Sound Vib., 332, 563-576. https://doi.org/10.1016/j.jsv.2012.09.005
  24. Timoshenko, S.P. (1921), "On the correction for shear of the differential equation for transverse vibration of prismatic bars", Phylosoph. Magazine, 41(6), 744-746.
  25. Timoshenko, S.P. (1922), "On the transverse vibrations of bars of uniform cross section", Phylosoph. Magazine, 43, 125-131.
  26. Timoshenko, S.P. (1937), Vibration Problems in Engineering, 2nd Edition, D. van Nostrand Company, Inc. New York, NY, USA.
  27. van Rensburg, N.F.J. and van der Merve, A.J. (2006), "Natural frequencies and modes of a Timoshenko beam", Wave Motion, 44, 58-69. https://doi.org/10.1016/j.wavemoti.2006.06.008
  28. Zhou, D. (2001), "Vibrations of Mindlin rectangular plates with elastically restrained edges using static Timoshenko beam functions with Rayleigh-Ritz method", Intl. J. Solids Struct., 38, 5565-5580. https://doi.org/10.1016/S0020-7683(00)00384-X

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