Structural Engineering and Mechanics

  • 제55권4호
  • /
  • Pages.871-884
  • /
  • 2015
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  • 1225-4568(pISSN)
  • /
  • 1598-6217(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

Effects of rotary inertia shear deformation and non-homogeneity on frequencies of beam

  • Avcar, Mehmet (Department of Civil Engineering, Faculty of Engineering, Suleyman Demirel University)
  • 투고 : 2015.05.09
  • 심사 : 2015.07.25
  • 발행 : 2015.08.25
⟨ 이전 논문 다음 논문 ⟩

초록

In the present study, separate and combined effects of rotary inertia, shear deformation and material non-homogeneity (MNH) on the values of natural frequencies of the simply supported beam are examined. MNH is characterized considering the parabolic variations of the Young's modulus and density along the thickness direction of the beam, while the value of Poisson's ratio is assumed to remain constant. At first, the equation of the motion including the effects of the rotary inertia, shear deformation and MNH is provided. Then the solutions including frequencies of the first three modes for various combinations of the parameters of the MNH, depth to length ratios, and shear corrections factors are reported. To show the accuracy of the present results, two comparisons are carried out and good agreements are found.

키워드

참고문헌

  1. Aghababaei, O., Nahvi, H. and Ziaei-Rad, S. (2009a), "Non-linear non-planar vibrations of geometrically imperfect inextensional beams, Part I: Equations of motion and experimental validation", Int. J. Nonlin. Mech., 44(2), 147-160. https://doi.org/10.1016/j.ijnonlinmec.2008.10.006
  2. Aghababaei, O., Nahvi, H. and Ziaei-Rad, S. (2009b), "Non-linear non-planar vibrations of geometrically imperfect inextensional beams, Part II: Equations of motion and experimental validation", Int. J. Nonlin. Mech., 44(2), 161-179. https://doi.org/10.1016/j.ijnonlinmec.2008.10.008
  3. Ait Amar Meziane, M., Abdelaziz, H.H. and Tounsi, A. (2014), "An efficient and simple refined theory for buckling and free vibration of exponentially graded sandwich plates under various boundary conditions" J. Sandw. Struct. Mater., 16(3), 293-318. https://doi.org/10.1177/1099636214526852
  4. Ait Yahia, S., Atmane, H.A., Houari, M.S.A. and Tounsi, A. (2015), "Wave propagation in functionally graded plates with porosities using various higher-order shear deformation plate theories", Struct. Eng. Mech., 53(6), 1143-1165. https://doi.org/10.12989/sem.2015.53.6.1143
  5. Akgoz, B. and Civalek, O. (2013), "Longitudinal vibration analysis of strain gradient bars made of functionally graded materials", Compos. B. Eng., 55, 263-268. https://doi.org/10.1016/j.compositesb.2013.06.035
  6. Al-Ansary, M.D. (1998), "Flexural vibrations of rotating beams considering rotary inertia", Comput. Struct., 69(3), 321-328. https://doi.org/10.1016/S0045-7949(98)00134-5
  7. Avcar, M. (2010), "Free vibration of randomly and continuously non- homogenous beams with clamped edges resting on elastic foundation", J. Eng. Sci. Des., 1, 33-38. (in Turkish)
  8. Avcar, M. (2014), "Free vibration analysis of beams considering different geometric characteristics and boundary conditions", Int. J. Appl. Mech., 4(3), 94-100.
  9. Avcar, M. and Saplioglu, K. (2015), "An artificial neural network application for estimation of natural frequencies of beams", Int. J. Adv. Comput. Sci. Appl., 6(6), 94-102.
  10. Bagdatli, S.M. and Uslu, B. (2015), "Free vibration analysis of axially moving beam under non-ideal conditions", Struct. Eng. Mech., 54(3), 597-605. https://doi.org/10.12989/sem.2015.54.3.597
  11. Belabed, Z., Houari, M.S.A., Tounsi, A., Mahmoud, S.R. and Beg, O.A. (2014), "An efficient and simple higher order shear and normal deformation theory for functionally graded material (FGM) plates", Compos. B Eng., 60, 274-283. https://doi.org/10.1016/j.compositesb.2013.12.057
  12. Becquet, R. and Elishakoff, I. (2001), "Class of analytical closed-form polynomial solutions for clamped- guided inhomogeneous beams", Chaos, Solit. Fract., 12(9), 1657-1678. https://doi.org/10.1016/S0960-0779(00)00125-9
  13. Berrabah, H.M., Tounsi, A., Semmah, A. and Adda Beida, E.A. (2013), "Comparison of various refined nonlocal beam theories for bending, vibration and buckling analysis of nanobeams", Struct. Eng. Mech., 48(3), 351-365. https://doi.org/10.12989/sem.2013.48.3.351
  14. Bourada, M., Kaci, A., Houari M.S.A. and Tounsi, A. (2015), "A new simple shear and normal deformations theory for functionally graded beams", Steel Compos. Struct., 18(2), 409-423. https://doi.org/10.12989/scs.2015.18.2.409
  15. Carrera, E., Giunta, G. and Petrolo, M. (2011), Beam Structures: Classical and Advanced Theories, John Wiley and Sons Ltd.
  16. Chakraverty, S. and Petyt, M. (1997), "Natural frequencies for free vibration of nonhomogeneous elliptic and circular plates using two-dimensional orthogonal polynomials", Appl. Math. Model., 21(7), 399-417. https://doi.org/10.1016/S0307-904X(97)00028-0
  17. Chakraverty, S., Jindal, R. and Agarwal, V.K. (2007), "Effect of non-homogeneity on natural frequencies of vibration of elliptic plates", Meccanica, 42(6), 585-599. https://doi.org/10.1007/s11012-007-9077-3
  18. Chang, C.H. and Yuan, Y.C. (1985), "Effect of rotatory inertia and shear deformation on vibration of an inclined bar with an end constraint", J. Sound Vib. 101(2), 171-180. https://doi.org/10.1016/S0022-460X(85)81213-X
  19. Chaudhuri, P. K. and Datta, S. (1989), "Note on the small vibration of beams with varying Young's modulus carrying a concentrated mass distribution", Indian J. Pure Appl. Math., 20(1), 75-88.
  20. Civalek, O. (2009), "Fundamental frequency of isotropic and orthotropic rectangular plates with linearly varying thickness by discrete singular convolution method", Appl. Math. Model., 33(10), 3825-3835. https://doi.org/10.1016/j.apm.2008.12.019
  21. Civalek, O. and Gurses, M. (2009), "Free vibration analysis of rotating cylindrical shells using discrete singular convolution technique" Int. J. Pres. Ves. Pip. 86, 677-683. https://doi.org/10.1016/j.ijpvp.2009.03.011
  22. Civalek, O. and Kiracioglu, O. (2009), "Free vibration analysis of Timoshenko beams by DSC method", Int. J. Numer. Meth. Biomed. Eng., 26(12), 1890-1898.
  23. Civalek, O. (2013), "Vibration analysis of laminated composite conical shells by the method of discrete singular convolution based on the shear deformation theory", Compos. B Eng., 45, 1001-1009. https://doi.org/10.1016/j.compositesb.2012.05.018
  24. De Silva, C.W. (2000), Vibration: Fundamentals and Practice, CRC Press LLC, Baco Raton.
  25. Demir, c., Civalek, O. and Akgoz, B. (2010), "Free vibration analysis of carbon nanotubes based on shear deformable beam theory by discrete singular convolution technique", Math. Comput. Appl., 15, 57-65.
  26. Ece, M.C., Aydogdu, M. and Taskin, V. (2007), "Vibration of a variable cross-section beam", Mech. Res. Commun., 34(1), 78-84. https://doi.org/10.1016/j.mechrescom.2006.06.005
  27. Elishakoff, I. and Becquet, I. (2000), "Closed-form solutions for natural frequency for inhomogeneous beams with one sliding support and the other clamped", J. Sound Vib., 238(3), 540-546. https://doi.org/10.1006/jsvi.2000.3010
  28. Elishakoff, I. and Candan, S. (2001), "Apparently first closed-form solution for vibrating: inhomogeneous beams", Int. J. Solid. Struct., 38, 3411-3441. https://doi.org/10.1016/S0020-7683(00)00266-3
  29. Elishakoff, I. and Guede, Z. (2001), "A remarkable nature of the effect of boundary conditions on closed-form solutions for vibrating inhomogeneous Bernoulli-Euler beams", Chaos, Solit. Fract., 12(4), 659-704. https://doi.org/10.1016/S0960-0779(00)00009-6
  30. Elishakoff, I. (2005), Eigenvalues of Inhomogeneous Structures: Unusual Closed-Form Solutions, Florida CRC Press, Boca Raton.
  31. Gan, B.S., Trinh, T.H., Le, T.H. and Nguyen, D.K. (2015), "Dynamic response of non-uniform Timoshenko beams made of axially FGM subjected to multiple moving point loads", Struct. Eng. Mech., 53(5), 981-995. https://doi.org/10.12989/sem.2015.53.5.981
  32. Grant, D.A. (1978), "The effect of rotary inertia and shear deformation on the frequency and normal mode equations of uniform beams carrying a concentrated mass", J. Sound Vib., 57(3), 357-365. https://doi.org/10.1016/0022-460X(78)90316-4
  33. Gupta, U.S., Lal, R. and Sharma, S. (2007), "Vibration of non-homogeneous circular Mindlin plates with variable thickness", J. Sound Vib., 302(1-2), 1-17. https://doi.org/10.1016/j.jsv.2006.07.005
  34. Gupta, A.K. and Kumar, L. (2010), "Vibration of non-homogeneous visco-elastic circular plate of linearly varying thickness in steady state temperature field", J. Theor. Appl. Mech., 48(1), 255-266.
  35. Gupta, A.K., Johri, T. and Vats, R.P. (2010), "Study of thermal gradient effect on vibrations of a non-homogeneous orthotropic rectangular plate having bi-direction linearly thickness variations", Meccanica, 45(3), 393-400. https://doi.org/10.1007/s11012-009-9258-3
  36. Han, M.S., Benaroya, H. and Wei, T. (1999), "Dynamics of transversely vibrating beams using four engineering theories", J. Sound Vib., 225(5), 935-988. https://doi.org/10.1006/jsvi.1999.2257
  37. Hebali, H., Tounsi, A., Houari, M.S.A, Bessaim, A. and Bedia, E.A.A. (2014), "A new quasi-3D hyperbolic shear deformation theory for the static and free vibration analysis of functionally graded plates", ASCE J. Eng. Mech., 140, 374-383. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000665
  38. Horr, A.M. and Schmidt, L.C. (1995), "Closed-form solution for the Timoshenko beam theory using a computer-based mathematical package", Comput. Struct., 55(3), 405-412. https://doi.org/10.1016/0045-7949(95)98867-P
  39. Hurty, W.C. and Rubinstein, M.F. (1964), "On the effect of rotatory inertia and shear in beam vibration", J. Franklin Inst., 278(2), 124-132. https://doi.org/10.1016/0016-0032(64)90113-9
  40. Irvine, T. (2010), "Transverse vibration of a beam simply supported at each end with bending, shear and rotary inertia", Vib. Data.
  41. Ji, Z.Y. and Yeh, K.Y. (1994), "The general solution for dynamic response of nonhomogeneous beam with variable cross section", Appl. Math. Mech., 15(5), 405-412. https://doi.org/10.1007/BF02451490
  42. Labuschagne, A., van Rensburg, N.F.J. and van der Merwe, A.J. (2009), "Comparison of linear beam theories", Math. Comput. Model., 49(1-2), 20-30. https://doi.org/10.1016/j.mcm.2008.06.006
  43. Lal, R. and Sharma, S. (2004), "Axisymmetric vibrations of non-homogeneous polar orthotropic annular plates of variable thickness", J. Sound Vib., 272(1-2), 245-265. https://doi.org/10.1016/S0022-460X(03)00329-8
  44. Lal, R. and Kumar, Y. (2013), "Transverse vibrations of nonhomogeneous rectangular plates with variable thickness", Mech. Adv. Mater. Struct., 20, 264-275. https://doi.org/10.1080/15376494.2011.584273
  45. Lin, H.Y. (2010), "An exact solution for free vibrations of a non-uniform beam carrying multiple elastic-supported rigid bars", Struct. Eng. Mech., 34(4), 399-416. https://doi.org/10.12989/sem.2010.34.4.399
  46. Liu, Z., Yin, Y., Wang, F., Zhao Y. and Cai L. (2013), "Study on modified differential transform method for free vibration analysis of uniform Euler-Bernoulli beam", Struct. Eng. Mech., 48(5), 697-709. https://doi.org/10.12989/sem.2013.48.5.697
  47. Leissa, A.W. and Qatu, M.S. (2011), Vibration of Continuous Systems, McGraw Hill.
  48. Mahi, A., Bedia, E.A.A. and Tounsi, A. (2015), "A new hyperbolic shear deformation theory for bending and free vibration analysis of isotropic, functionally graded, sandwich and laminated composite plates", Appl. Math. Model., 39, 2489-2508. https://doi.org/10.1016/j.apm.2014.10.045
  49. Mao, Q. (2015), "AMDM for free vibration analysis of rotating tapered beams", Struct. Eng. Mech., 54(3), 419-432. https://doi.org/10.12989/sem.2015.54.3.419
  50. Mazzei, A.J.J and Scott, R.A. (2012), "On the effects of non-homogeneous materials on the vibrations and static stability of tapered shafts", J. Vib. Control, 19(5), 771-786. https://doi.org/10.1177/1077546312438429
  51. Mohammadnejad, M., Saffari, H. and Bagheripour, M.H. (2014), "An analytical approach to vibration analysis of beams with variable properties", Arab. J. Sci. Eng., 39, 2561-2572. https://doi.org/10.1007/s13369-013-0898-1
  52. Nandi, P.K., Gorain, G.C. and Kar, S. (2012), "A note on stability of longitudinal vibrations of an inhomogeneous beam", Appl. Math., 3(1), 19-23. https://doi.org/10.4236/am.2012.31003
  53. Nayfeh, A.H. (1972), "Asymptotic behavior of eigenvalues for finite, inhomogeneous elastic rods", J. Appl. Mech., 39(2), 595-597. https://doi.org/10.1115/1.3422724
  54. Rao, S.S. (2007), Vibration of Continuous Systems, John Wiley and Sons Ltd.
  55. Saffari, H., Mohammadnejad, M. and Bagheripour, M.H. (2012), "Free vibration analysis of non-prismatic beams under variable axial forces", Struct. Eng. Mech., 43(5), 561-582. https://doi.org/10.12989/sem.2012.43.5.561
  56. Szyiko-Bigus, O. and Sniady P. (2015), "Dynamic response of a Timoshenko beam to a continuous distributed moving load", Struct. Eng. Mech., 54(4), 771-792. https://doi.org/10.12989/sem.2015.54.4.771
  57. Simsek, M. and Kocaturk, T. (2007), "Free vibration analysis of beams by using a third order shear deformation theory", Sadhana Acad. Proc. Eng. Sci., 32(3), 167-179.
  58. Simsek, M. (2010), "Vibration analysis of a functionally graded beam under a moving mass by using different beam theories", Compos. Struct., 92(4), 904-917. https://doi.org/10.1016/j.compstruct.2009.09.030
  59. Taha, M.H. and Abohadima, S. (2008), "Mathematical model for vibrations of non-uniform flexural beams", Eng. Mech., 15(1), 3-11.
  60. Tounsi, A., Houari, M.S.A, Benyoucef, S. and Bedia E.A.A. (2013), "A refined trigonometric shear deformation theory for thermoelastic bending of functionally graded sandwich plates", Aero. Sci. Tech., 24, 209-220. https://doi.org/10.1016/j.ast.2011.11.009
  61. Timoshenko, S.P. (1937), Vibration Problems in Engineering, Ed. Van Nostrand, D., Princeton, NJ.
  62. Tong, X., Tabarrok, B. and Yeh, K.Y. (1995), "Vibration analysis of Timoshenko beams with non-homogeneity and varying cross-section", J. Sound Vib., 186(5), 821-835. https://doi.org/10.1006/jsvi.1995.0490
  63. Wang, C.M., Reddy, J.N. and Lee, K.H. (2000), Shear Deformable Beams and Plates: Relationships with Classical Solutions, Elsevier Science Ltd.
  64. Wang, C.Y. and Wang, C.M. (2014), Structural Vibration: Exact Solutions for Strings, Membranes, Beams and Plates, CRC Press, Taylor and Francis Group, Boca Raton.
  65. Yesilce, Y. (2015), "Differential transform method and numerical assembly technique for free vibration analysis of the axial-loaded Timoshenko multiple-step beam carrying a number of intermediate lumped masses and rotary inertias", Struct. Eng. Mech., 53(3), 537-573. https://doi.org/10.12989/sem.2015.53.3.537
  66. Yildirim, V. and Kiral, E. (2000), "Investigation of the rotary inertia and shear deformation effects on the out-of-plane bending and torsional natural frequencies of laminated beams", Compos. Struct., 49(3), 313-320. https://doi.org/10.1016/S0263-8223(00)00063-5
  67. Zemri, A., Houari, M.S.A., Bousahla A.A. and Tounsi, A. (2015), "A mechanical response of functionally graded nanoscale beam: an assessment of a refined nonlocal shear deformation theory beam theory", Struct. Eng. Mech., 54(4), 693-710. https://doi.org/10.12989/sem.2015.54.4.693

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