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http://dx.doi.org/10.12989/sem.2013.48.4.519

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)
Publication Information
Structural Engineering and Mechanics / v.48, no.4, 2013 , pp. 519-545 More about this Journal
Abstract
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.
Keywords
Timoshenko beam theory; flexural vibration; axial shear vibration; vibration parameter; analytical solution; double frequency phenomenon;
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Times Cited By KSCI : 1  (Citation Analysis)
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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.   DOI
3 De Rosa, M.A. (1995), "Free vibrations of Timoshenko beams on two-parametric elastic foundation", Comput. Struct., 57, 151-156.   DOI   ScienceOn
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.   DOI   ScienceOn
7 Levinson, M. (1981b), "Further results of a new beam theory", J. Sound Vib., 77(3), 440-444.   DOI   ScienceOn
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.   DOI   ScienceOn
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.   DOI   ScienceOn
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.   DOI   ScienceOn
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.   DOI   ScienceOn
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.   DOI   ScienceOn
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.   DOI   ScienceOn
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.   DOI   ScienceOn
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.   DOI   ScienceOn
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.   DOI   ScienceOn
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.   DOI   ScienceOn