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Combined influence of rotary inertia and shear coefficient on flexural frequencies of Timoshenko beam: numerical experiments

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Abstract

When dealing with the computation of bending frequencies for Timoshenko beams, it is common practice to assume a constant shear coefficient for a specific cross-sectional shape of the beam within the plane of bending vibration being considered. However, there have been various definitions, interpretations, and values assigned to it, depending on static and dynamic considerations, as well as experimental investigations, regardless of whether a wave approach or a mode approach is utilized. Currently, there is a lack of unanimity regarding the most suitable value for the shear coefficient. This study examines the impact of varying both the shear coefficient \(\kappa\) and the rotary inertia parameter \(\overline{r}_{{\text{g}}}\) on the first four bending modes of a uniform Timoshenko beam, considering numerous boundary conditions. The present numerical experiments reveal that increasing the shear coefficient tends to increase the frequency parameter values, while increasing the rotary inertia parameter is seen to decrease them. Both effects are very small at lower modes but become more pronounced at higher modes. The combined influence is seen to depress the frequency parameter values for all kinds of end conditions considered in this work.

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Abbreviations

\(A\) :

Cross-sectional area of the beam

\(\left[ B \right]\) :

Strain displacement matrix

\(b\) :

Beam width

\(E\) :

Modulus of elasticity

\(G\) :

Modulus of rigidity

\(h\) :

Beam height

\(I\) :

Second moment of area

\(i\) :

Mode number

\(j\) :

Complex number \(= \sqrt { - 1}\)

\(\left[ K \right]\) :

Stiffness matrix \(= \left[ {K_{{\text{e}}} } \right] + \left[ {K_{{\text{s}}} } \right]\)

\(\left[ {K_{{\text{e}}} } \right]\) :

Elastic stiffness matrix

\(\left[ {K_{{\text{s}}} } \right]\) :

Shear stiffness matrix

\(\kappa\) :

Shear coefficient

\(L\) :

Length of the beam

\({\mathfrak{L}}\) :

Lagrangian

\(\left[ M \right]\) :

Mass matrix \(= \left[ {M_{{\text{t}}} } \right] + \left[ {M_{{\text{r}}} } \right]\)

\(\left[ {M_{{\text{t}}} } \right]\) :

Translational mass matrix

\(\left[ {M_{{\text{r}}} } \right]\) :

Rotary inertia mass matrix

\(\left[ {N_{w} } \right]\) :

Transverse displacement shape functions matrix

\(\left[ {N_{\theta } } \right]\) :

Rotation shape functions matrix

\(r_{{\text{g}}}\) :

Radius of gyration = \(\sqrt {{I \mathord{\left/ {\vphantom {I A}} \right. \kern-0pt} A}}\)

\(\overline{r}_{{\text{g}}}\) :

Rotary inertia parameter (ratio of radius of gyration of section to beam length) \(= {{r_{{\text{g}}} } \mathord{\left/ {\vphantom {{r_{{\text{g}}} } L}} \right. \kern-0pt} L}\)

\(Q\) :

Shear force

\(\left\{ q \right\}\) :

Vector of nodal coordinates

\(\left\{ {\overline{q}} \right\}\) :

Vector of displacement amplitudes of vibration

\(s\) :

Slenderness ratio \(= {L \mathord{\left/ {\vphantom {L {r_{{\text{g}}} }}} \right. \kern-0pt} {r_{{\text{g}}} }}\)

\(T\) :

Kinetic energy

\(t\) :

Time

\(U\) :

Strain energy

\(\gamma\) :

Angle of distortion due to shear

\(\varepsilon\) :

Normal strain

\(\tau\) :

Shear strain

\(\theta\) :

Rotation angle due to bending

\(\nu\) :

Poisson’s ratio

\(\rho\) :

Mass density

\(\tau\) :

Shear stress

\(\omega\) :

Natural frequency

\(\overline{\omega }\) :

Frequency parameter \(= \omega L^{2} \sqrt {{{\rho A} \mathord{\left/ {\vphantom {{\rho A} {EI}}} \right. \kern-0pt} {EI}}}\)

\(\overline{\omega }_{{\text{E}}}\) :

Euler–Bernoulli frequency parameter

\(\overline{\omega }_{{\text{T}}}\) :

Timoshenko frequency parameter

\(\eta\) :

Aspect ratio, (\(= {h \mathord{\left/ {\vphantom {h b}} \right. \kern-0pt} b}\), height/width)

\(\zeta\) :

Non-dimensional length \(= {x \mathord{\left/ {\vphantom {x L}} \right. \kern-0pt} L}\)

[]T :

Transpose of []

References

  1. Rayleigh, J.W.S.B.: The Theory of Sound. Macmillan, London (1877)

    MATH  Google Scholar 

  2. Timoshenko, S.P.: On the correction for shear of the differential equation for transverse vibration of prismatic bars. Philos. Mag. 41, 744–746 (1921). https://doi.org/10.1080/14786442108636264

    Article  Google Scholar 

  3. Timoshenko, S.P.: On the transverse vibrations of bars of uniform cross-section. Philos. Mag. 43, 379–384 (1922). https://doi.org/10.1080/14786442208633855

    Article  Google Scholar 

  4. Elishakoff, I.: Who developed the so—called Timoshenko beam theory? Math. Mech. Solids 25(1), 97–116 (2020). https://doi.org/10.1177/1081286519856931

    Article  MathSciNet  MATH  Google Scholar 

  5. Elishakoff, I.: Handbook on Timoshenko–Ehrenfest Beam and Uflyand–Mindlin Plate Theories. World Scientific, Singapore (2020). https://doi.org/10.1142/10890

    Book  Google Scholar 

  6. Dym, C.L., Shames, I.H.: Solid Mechanics: A Variational Approach Augmented Edition. Springer, New York. https://doi.org/10.1007/978-1-4614-6034-3. ISBN: 978–1–4614–6034–3

  7. Mindlin, R.D., Dersiewicz, H.: Timoshenko's shear coefficient for flexural vibrations of beams. In: Proceedings of 2nd U.S. National Congress of Applied Mechanics, pp. 175–178. ASME, New York (1954).

  8. Pai, P.F., Schulz, M.J.: Shear correction factors and an energy-consistent beam theory. Int. J. Solids Struct. 36, 1523–1540 (1999). https://doi.org/10.1016/S0020-7683(98)00050-X

    Article  MATH  Google Scholar 

  9. Sutherland, J.G., Goodman, L.E.: Vibrations of prismatic bars including rotatory inertia and shear corrections. Technical Report of a Cooperative Investigation with the Office of Naval Research, University of Illinois, Urbana, USA (1951). https://www.ideals.illinois.edu/items/13525

  10. Goodman, L.E.: Discussion: Flexural vibrations in uniform beams according to the Timoshenko theory, by R. A. Anderson. Trans. ASME J. Appl. Mech. 21(2), 203–204 (1954). https://doi.org/10.1115/1.4010878

    Article  Google Scholar 

  11. Stephen, N.G.: Timoshenko’s shear coefficient from a beam subjected to gravity loading. J. Appl. Mech. 47(1), 121–127 (1980). https://doi.org/10.1115/1.3153589

    Article  MATH  Google Scholar 

  12. Dong, S.B., Alpdogan, C., Taciroglu, E.: Much ado about shear correction factors in Timoshenko beam theory. Int. J. Solids Struct. 47(13), 1651–1665 (2010). https://doi.org/10.1016/j.ijsolstr.2010.02.018

    Article  MATH  Google Scholar 

  13. Rosinger, H.E., Ritchie, I.G.: On Timoshenko’s correction for shear in vibrating isotropic beams. J. Phys. D: Appl. Phys. 10(11), 1461–1466 (1977). https://doi.org/10.1088/0022-3727/10/11/009

    Article  Google Scholar 

  14. Filon, L.N.G.: On an approximate solution for the bending of a beam of rectangular cross-section under any system of load, with special reference to points of concentrated or discontinuous loading. Proc. R. Soc. Lond 70, 459–466 (1902). https://doi.org/10.1098/rspl.1902.0050

    Article  Google Scholar 

  15. Goens, V.E.: Uber die bestimmung des elastizitatsmodule von staben mit hilfe von biegungsschwingungen. Ann. Phys. Ser. 5(11), 649–678 (1931). https://doi.org/10.1002/andp.19314030602

    Article  MATH  Google Scholar 

  16. Mindlin, R.D.: Influence of rotatory inertia and shear on flexural motions of isotropic elastic plates. J. Appl. Mech. 18(1), 31–38 (1951). https://doi.org/10.1115/1.4010217

    Article  MATH  Google Scholar 

  17. Olsson, R.G.: Zur Berechnung der frequenz der transversalschwingungen des prismatischen stabes. Z. Angew. Math. und Mech. 15, 245–245 (1935). https://doi.org/10.1002/zamm.19350150410

    Article  MATH  Google Scholar 

  18. Cowper, G.R.: The shear coefficient in Timoshenko’s beam theory. J. Appl. Mech. 33(2), 335–340 (1966). https://doi.org/10.1115/1.3625046

    Article  MATH  Google Scholar 

  19. Stephen, N.G.: On the variation of Timoshenko’s shear coefficient with frequency. J. Appl. Mech. 45(3), 695–697 (1978). https://doi.org/10.1115/1.3424391

    Article  Google Scholar 

  20. Stephen, N.G., Levinson, M.: A second order beam theory. J. Sound Vib. 67(3), 293–305 (1979). https://doi.org/10.1016/0022-460X(79)90537-6

    Article  MATH  Google Scholar 

  21. Hutchinson, R.: Shear coefficients for Timoshenko beam theory. J. Appl. Mech. 68(1), 87–92 (2001). https://doi.org/10.1115/1.1349417

    Article  MATH  Google Scholar 

  22. Faghidian, S.A.: Unified formulations of the shear coefficients in Timoshenko beam theory. J. Eng. Mech. 143(9), 06017013 (2017). https://doi.org/10.1061/(ASCE)EM.1943-7889.0001297

    Article  Google Scholar 

  23. Chan, K.T., Lai, K.F., Stephen, N.G., Young, K.: A new method to determine the shear coefficient of Timoshenko beam theory. J. Sound Vib. 330, 3488–3497 (2011). https://doi.org/10.1016/j.jsv.2011.02.012

    Article  Google Scholar 

  24. Kim, J.G., Lee, J.K., Yoon, H.J.: On the effect of shear coefficients in free vibration analysis of curved beams. J. Mech. Sci. Technol. 28(8), 3181–3187 (2014). https://doi.org/10.1007/s12206-014-0727-9

    Article  Google Scholar 

  25. Kaneko, T.: On Timoshenko’s correction for shear in vibrating beams. J. Phys. D Appl. Phys. 8, 1927–1936 (1975). https://doi.org/10.1088/0022-3727/8/16/003

    Article  Google Scholar 

  26. Freund, J., Karakoç, A.: Shear and torsion correction factors of Timoshenko beam model for generic cross sections. Res. Eng. Struct. Mat. 2(1), 19–27 (2016). https://doi.org/10.17515/resm2015.19me0827

    Article  Google Scholar 

  27. Kaneko, T.: An experimental study of the Timoshenko’s shear coefficient for flexurally vibrating beams. J. Phys. D Appl. Phys. 11(14), 1979–1988 (1978). https://doi.org/10.1088/0022-3727/11/14/010

    Article  Google Scholar 

  28. Puchegger, S., Bauer, S., Loidl, D., Kromp, K., Peterlik, H.: Experimental validation of the shear correction factor. J. Sound Vib. 261(13), 177–184 (2003). https://doi.org/10.1016/S0022-460X(02)01181-1

    Article  MATH  Google Scholar 

  29. Méndez-Sánchez, R.A., Morales, A., Flores, J.: Experimental check on the accuracy of Timoshenko’s beam theory. J. Sound Vib. 279(1–2), 508–512 (2005). https://doi.org/10.1016/j.jsv.2004.01.050

    Article  Google Scholar 

  30. Franco-Villafañe, J.A., Méndez-Sánchez, R.A.: On the accuracy of the Timoshenko beam theory above the critical frequency: best shear coefficient. J. Mech. 32(5), 515–518 (2016). https://doi.org/10.1017/jmech.2015.104

    Article  Google Scholar 

  31. Kawashima, H.: The shear coefficient for quartz crystal of rectangular cross section in Timoshenko’s beam theory. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 43(3), 434–440 (1996). https://doi.org/10.1109/58.489402

    Article  Google Scholar 

  32. Carnegie, W., Thomas, J.: The effects of shear deformation and rotary inertia on the lateral frequencies of cantilever beams in bending. ASME. J. Eng. Ind. 94(1), 267–278 (1972). https://doi.org/10.1115/1.3428121

    Article  Google Scholar 

  33. Ghosh, N.G.: Research note: the effect of variation of shear stress factor on vibrating beams. J. Mech. Eng. Sci. 16(5), 346–348 (1974). https://doi.org/10.1243/JMES_JOUR_1974_016_062_02

    Article  Google Scholar 

  34. Rossi, R.E., Laura, P.A.A., Maurizi, M.J.: Numerical experiments on the effect of the value of the shear coefficient upon the natural frequencies of a Timoshenko beam. J. Sound Vib. 154(2), 374–379 (1992). https://doi.org/10.1016/0022-460X(92)90590-T

    Article  Google Scholar 

  35. Filipich, C.P., Rosales, M.B., Cortinez, V.H.: Natural frequencies of Timoshenko beam: exact values by means of a generalized solution. Mec. Comput. 14(1), 134–143 (1994)

    Google Scholar 

  36. Khan, M.U.: Vibration analysis of rotating tapered Timoshenko beam using Fourier-\(p\) superelement. Master Thesis, King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia (2021)

  37. Thomas, D.L., Wilson, J.M., Wilson, R.R.: Timoshenko beam finite elements. J. Sound Vib. 31(3), 315–330 (1973). https://doi.org/10.1016/S0022-460X(73)80276-7

    Article  MATH  Google Scholar 

  38. Petyt, M.: Introduction to Finite Element Vibration Analysis, 2nd ed. Cambridge University Press, Cambridge (2010). ISBN 978-0-521-19160–9

    MATH  Google Scholar 

  39. Ginsberg, J.H.: Mechanical and Structural Vibrations: Theory and Applications. Wiley, New York (2001). ISBN: 978-0-471-37084-0

    Google Scholar 

  40. Wilkinson, J.H.: The Algebraic Eigenvalue Problem. Clarendon Press, Oxford (1965)

    MATH  Google Scholar 

  41. Wittrick, W.H., Williams, F.W.: A general algorithm for computing natural frequencies of elastic structures. Q. J. Mech. Appl. Math. 24(3), 263–284 (1971). https://doi.org/10.1093/qjmam/24.3.263

    Article  MathSciNet  MATH  Google Scholar 

  42. LAPACK 3.11.0 Linear Algebra PACKkage. https://www.netlib.org/lapack/explore-html/d8/d70/group__lapack.html. Accessed 14 June 2023

  43. Murty, A.V.: Vibrations of short beams. AIAA J. 8(1), 34–38 (1970). https://doi.org/10.2514/3.5602

    Article  MATH  Google Scholar 

  44. El Ouatouati, A., Johnson, D.A.: A new approach for numerical modal analysis using the element-free method. Int. J. Numer. Methods Eng. 46(1), 1–27 (1999). https://doi.org/10.1002/(SICI)1097-0207(19990910)46:1%3C1::AID-NME659%3E3.0.CO;2-G

    Article  MATH  Google Scholar 

  45. Downs, B.: Reference frequencies for the validation of numerical solutions of transverse vibrations of non-uniform Beams. J. Sound Vib. 61(1), 71–78 (1978). https://doi.org/10.1016/0022-460X(78)90042-1

    Article  MATH  Google Scholar 

  46. Davis, R., Henshell, R.D., Warburton, G.B.: A Timoshenko Beam Element. J. Sound Vib. 22(4), 475–487 (1972). https://doi.org/10.1016/0022-460X(72)90457-9

    Article  Google Scholar 

  47. Dawe, D.J.: A finite element for the vibration analysis of Timoshenko beams. J. Sound Vib. 60(1), 11–20 (1978). https://doi.org/10.1016/0022-460X(78)90397-8

    Article  MATH  Google Scholar 

  48. Downs, B.: Transverse vibration of a uniform, simply supported Timoshenko beam without transverse deflection. J. Appl. Mech. 43(4), 671–674 (1976). https://doi.org/10.1115/1.3423953

    Article  MATH  Google Scholar 

  49. Bhashyam, G.R., Prathap, G.: The second frequency spectrum of Timoshenko beams. J. Sound Vib. 76(3), 407–420 (1981). https://doi.org/10.1016/0022-460X(81)90520-4

    Article  Google Scholar 

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Acknowledgements

The support of King Fahd University of Petroleum & Minerals is greatly appreciated.

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  1. Mechanical Engineering Department, King Fahd University of Petroleum and Minerals, KFUPM BOX 579, 31261, Dhahran, Saudi Arabia

    Abdelaziz Bazoune

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Bazoune, A. Combined influence of rotary inertia and shear coefficient on flexural frequencies of Timoshenko beam: numerical experiments. Acta Mech 234, 4997–5013 (2023). https://doi.org/10.1007/s00707-023-03648-6

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