Structural Engineering and Mechanics

  • 제75권6호
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  • Pages.737-746
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  • 2020
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  • 1225-4568(pISSN)
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  • 1598-6217(eISSN)
테크노프레스 (Techno-Press)
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DOI QR코드

DOI QR Code

Free vibration analysis of functionally graded beams with variable cross-section by the differential quadrature method based on the nonlocal theory

  • Elmeiche, Noureddine (Civil Engineering and Environmental Laboratory, Sidi Bel Abbes University) ;
  • Abbad, Hichem (Civil Engineering and Environmental Laboratory, Sidi Bel Abbes University) ;
  • Mechab, Ismail (LMPM, Department of Mechanical Engineering, University of Sidi Bel Abbes) ;
  • Bernard, Fabrice (Universite de Rennes, INSA de Rennes - LGCGM)
  • 투고 : 2019.04.23
  • 심사 : 2020.04.19
  • 발행 : 2020.09.25
⟨ 이전 논문 다음 논문 ⟩

초록

This paper attempts to investigate the free vibration of functionally graded material beams with nonuniform width based on the nonlocal elasticity theory. The theoretical formulations are established following the Euler-Bernoulli beam theory, and the governing equations of motion of the system are derived from the minimum total potential energy principle using the nonlocal elasticity theory. In addition, the Differential Quadrature Method (DQM) is applied, along with the Chebyshev-Gauss-Lobatto polynomials, in order to determine the weighting coefficient matrices. Furthermore, the effects of the nonlocal parameter, cross-section area of the functionally graded material (FGM) beam and various boundary conditions on the natural frequencies are examined. It is observed that the nonlocal parameter and boundary conditions significantly influence the natural frequencies of the functionally graded material beam cross-section. The results obtained, using the Differential Quadrature Method (DQM) under various boundary conditions, are found in good agreement with analytical and numerical results available in the literature.

키워드

참고문헌

  1. Aydogdu, M. (2009), "A general nonlocal beam theory: Its application to nanobeam bending, buckling and vibration", Physica E, 41(9), 1651-1655. https://doi.org/10.1016/j.physe.2009.05.014
  2. Bambill, D.V., Felix, D.H. and Rossi, R.E. (2010), "Vibration analysis of rotating Timoshenko beams by means of the differential quadrature method", Struct. Eng. Mech., 34(2), 231-245. https://doi.org/10.12989/sem.2010.34.2.231
  3. Bellman, R.E., Kashef, B.G., and Casti, J. (1972), "Differential quadrature: A technique for the rapid solution of nonlinear partial differential equations", J. Comput. Phys, 10(1), 40-52. https://doi.org/10.1016/0021-9991(72)90089-7.
  4. Bozdogan, K.B. (2012), "Differential quadrature method for free vibration analysis of coupled shear walls". Struct. Eng. Mech., 41(1), 67-81. https://doi.org/10.12989/sem.2012.41.1.067
  5. Cem Ece, M., Aydogdu, M., and Taskin, V. (2007), "Vibration of a variable cross section beam", Mech. Res. Comm., 34(1), 78-84. https://doi.org/10.1016/j.mechrescom.2006.06.005.
  6. Eringen, A.C (1983), "On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves", J. Appl. Phys., 54(9), 4703-4710. https://doi.org/10.1063/1.332803
  7. Eringen, A.C. (2002), Nonlocal Continuum Field Theories, Springer-Verlag, New York, USA.
  8. Garijo, D. (2015), "Free vibration analysis of non-uniform Euler-Bernoulli beams by means of Bernstein pseudo spectral collocation", Eng. Comput., 31(4), 813-823. https://doi.org/10.1007/s00366-015-0401-6
  9. Ghazaryan, D., Burlayenko, D., Avetisyan, V.N. and Bhaskar, A. (2017), "Free vibration analysis of functionally graded beams with non-uniform cross-section using the differential transform method", J. Eng. Math., 110(1), 97-121. https://doi.org/10.1007/s10665-017-9937-3
  10. Koizumi, M. (1997), "FGM activities in Japan", Compos. Part B, 28(1-2), 1-4. https://doi.org/10.1016/S1359-8368(96)00016-9.
  11. Mechab, I., El Meiche, N., and Bernard, F., (2016), "Free vibration analysis of higher-order shear elasticity nanocomposite beams with consideration of nonlocal elasticity and Poisson effect", J. Nanomech. Micromech., 6(3). https://doi.org/10.1061/(ASCE)NM.2153-5477.0000110
  12. Murmu, T. and Pradhan, S.C. (2008), "Buckling analysis of beam on winkler foundation by using MDQM and nonlocal theory", J. Aerosp. Sci. Technol., 60(3), 206-215.
  13. Nedri, K., El Meiche, N., and Tounsi, A. (2014), "Free vibration analysis of laminated composite plates resting on elastic foundations by using a refined hyperbolic shear deformation theory", J. Mech. Sci., 49(6), 629-640. https://doi.org/10.1007/s11029-013-9379-6
  14. Pradhan, S.C. and Murmu, T. (2009), "Differential quadrature method for vibration analysis of beam on Winkler foundation based on nonlocal elastic theory", J. Inst. Eng. (India) Metallurgy Mater. Eng. Div., 89, 3-12.
  15. Rajasekaran, S., Gimena, L., Gonzaga, P. and Gimena, F.N. (2009), "Solution method for the classical beam theory using differential quadrature", Struct. Eng. Mech., 33(6), 675-696. https://doi.org/10.12989/sem.2009.33.6.675.
  16. Rajasekaran, S., and Khaniki, H.B. (2018), "Bending, buckling and vibration analysis of functionally gradednon-uniform nanobeams via finite element method", J. Brazilian Soc. Mech. Sci. Eng., 40, 549. https://doi.org/10.1007/s40430-018-1460-6.
  17. Reddy, J.N. (2007), "Nonlocal theories for bending, buckling and vibration of beams", J. Eng. Sci., 45(2-8), 288-307. https://doi.org/10.1016/j.ijengsci.2007.04.004.
  18. Tahouneh, V. (2014), "Free vibration analysis of bidirectional functionally graded annular plates resting on elastic foundations using differential quadrature method", Struct. Eng. Mech., 52(4), 663-686. https://doi.org/10.12989/sem.2014.52.4.663.
  19. Tahouneh, V. (2018). "Vibrational analysis of sandwich sectorial plates with functionally graded sheets reinforced by aggregated carbon nanotube", J. Sandwich Struct. Mater., 22(5), 1-46. https://doi.org/10.1177/1099636218785972.
  20. Tahouneh, V. (2019), "Using IGA and trimming approaches for vibrational analysis of L-shape graphene sheets via nonlocal elasticity theory", Steel Compos. Struct., 33(5), 717-727. https://doi.org/10.12989/scs.2019.33.5.717.
  21. Tahouneh, V.(2020), "Influence of vacancy defects on vibration analysis of graphene sheets applying isogeometric method: Molecular and continuum approaches", Steel Compos. Struct., 34(2), 261-277. https://doi.org/10.12989/scs.2020.34.2.261.
  22. Yang, J. and Chen, Y (2008),"Free vibration and buckling analyses of functionally graded beams with edge cracks", Compos. Struct., 83(1), 48-60. https://doi.org/10.1016/j.compstruct.2007.03.006.
  23. Yas, M.H., Sobhani Aragh, B. and Heshmati, M. (2011), "Three-dimensional free vibration analysis of functionally graded fiber reinforced cylindrical panels using differential quadrature method", Struct. Eng. Mech., 37(5), 529-542. https://doi.org/10.12989/sem.2011.37.5.529.