Skip to main content
SpringerLink
Log in

Free vibration of functionally Euler-Bernoulli and Timoshenko graded porous beams using the transfer matrix method

  • Structural Engineering
  • Published:
KSCE Journal of Civil Engineering Aims and scope

Abstract

In this paper, an analytical method is developed to study the dynamic behavior of functionally imperfect Euler-Bernoulli and Timoshenko graded beams with differing boundary conditions, namely, hinged-hinged, clamped-clamped, clamped-hinged, and clamped-free. A transfer matrix method is used to obtain the natural frequency equations. The modified rule of mixture is used to describe the material properties of the functionally graded beams having porosities. The porosities are assumed to be evenly distributed over the beam cross-section. In this study, the effects of boundary conditions, material volume fraction index, slenderness ratio, beam theory, and porosity on natural frequency are determined. The present results are validated with results available in the literature.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Alshorbagy, Amal E., Eltaher, M. A., and Mahmoud, F. F. (2011). “Free vibration characteristics of a functionally graded beam by finite element method.” Applied Mathematical Modelling, Vol. 35, No. 1, pp. 412–425, DOI: 10.1016/j.apm.2010.07.006.

    Article  MathSciNet  MATH  Google Scholar 

  • Aydin, K. (2013). “Free vibration of functionally graded beams with arbitrary number of surface cracks.” European Journal of Mechanics-A/Solids, Vol. 42, pp. 112–124, DOI: 10.1016/j.euromechsol.2013.05.002.

    Article  MathSciNet  Google Scholar 

  • Aydogdu, M. and Taskin, V. (2007). “Free vibration analysis of functionally graded beams with simply supported edges.” Materials & Design, Vol. 28, No. 5, pp. 1651–1656, DOI: 10.1016/j.matdes.2006.02.007.

    Article  Google Scholar 

  • Ebrahimi, F. and Mokhtari, M. (2014). “Transverse vibration analysis of rotating porous beam with functionally graded microstructure using the differential transform method.” Journal of the Brazilian Society of Mechanical Sciences and Engineering, pp. 1–10, DOI: 10.1007/s40430-014-0255-7.

    Google Scholar 

  • Elishakoff, I. (2010). “An equation both more consistent and simpler than the Bresse-Timoshenko equation. Advances in Mathematical Modeling and Experimental Methods for Materials and Structures.” Solid Mechanics and Its Applications (R. Gilat and L. Banks-Sills, eds.), Springer, Berlin, pp. 249–254, DOI: 10.1007/978-481-3467-0_19.

    Google Scholar 

  • Falsone, G., Settineri, D., and Elishakoff, I. (2014). “A new locking-free-finite element method based on more consistent version of Mindlin plate equation.” Archive of Applied Mechanics, Vol. 84, pp. 967–983, DOI: 10.1007/s00419-014-0842-1.

    Article  MATH  Google Scholar 

  • Falsone, G., Settineri, D., and Elishakoff, I. (2015). “A new class of interdependent shape polynomials for the FE dynamic analysis of Mindlin plate and Timoshenko beam.” Meccanica, Vol. 50, pp. 767–780, ISSN: 0025-6455, DOI: 10.1007/s11012-014-0032-9.

    Article  MathSciNet  MATH  Google Scholar 

  • Huang, Y. and Li, X.-F. (2010). “A new approach for free vibration of axially functionally graded beams with non-uniform cross-section.” Journal of Sound and Vibration, Vol. 329, No. 11, pp. 2291–2303, DOI: 10.1016/j.jsv.2009.12.029.

    Article  Google Scholar 

  • Şimşek, M. (2010). “Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories.” Nuclear Engineering and Design, Vol. 240, No. 4, pp. 697–705, DOI: 10.1016/j.nucengdes.2009.12.013.

    Article  Google Scholar 

  • Ke, L., Wang, Y., Yang, J., Kitipornchai, S., and Alam, F. (2012). Nonlinear vibration of edged cracked FGM beams using differential quadrature method.” Science China Physics, Mechanics and Astronomy, Vol. 55, No. 11, pp. 2114–2121, DOI: 10.1007/s11433-012-4704-y.

    Article  Google Scholar 

  • Ke, L.-L., Yang, J., and Kitipornchai, S. (2010). “An analytical study on the nonlinear vibration of functionally graded beams.” Meccanica, Vol. 45, No. 6, pp. 743–752, DOI: 10.1007/s11012-009-9276-1.

    Article  MathSciNet  MATH  Google Scholar 

  • Khalili, S. M. R., Jafari, A. A., and Eftekhari, S. A. (2010). “A mixed Ritz-DQ method for forced vibration of functionally graded beams carrying moving loads.” Composite Structures, Vol. 92, No. 10, pp. 2497–2511, DOI: 10.1016/j.compstruct.2010.02.012.

    Article  Google Scholar 

  • Loy, C. T., Lam, K. Y., and Reddy, J. N. (1999). “Vibration of functionally graded cylindrical shells.” Int. J. Mech. Sci., Vol. 41, No. 3, pp. 309- 24, DOI: 10.1016/S0020-7403(98)00054-X.

    Article  MATH  Google Scholar 

  • Mashat, D. S., Carrera, E., Zenkour, A. M., Al Khateeb, S. A., and Filippi, M. (2014). “Free vibration of FGM layered beams by various theories and finite elements.” Composites Part B: Engineering, Vol. 59, pp. 269–278, DOI: 10.1016/j.compositesb.2013.12.008.

    Article  Google Scholar 

  • Nguyen, T.-K., Vo, T. P., and Thai, H.-T. (2013). “Static and free vibration of axially loaded functionally graded beams based on the first-order shear deformation theory.” Composites Part B: Engineering, Vol. 55, pp. 147–157, DOI: 10.1016/j.compositesb. 2013.06.011.

    Article  Google Scholar 

  • Pradhan, K. K. and Chakraverty, S. (2013). “Free vibration of Euler and Timoshenko functionally graded beams by Rayleigh–Ritz method.” Composites Part B: Engineering, Vol. 51, pp. 175–184, DOI: 10.1016/j.compositesb.2013.02.027.

    Article  Google Scholar 

  • Pradhan, K. K. and Chakraverty, S. (2014). “Effects of different shear deformation theories on free vibration of functionally graded beams.” International Journal of Mechanical Sciences, Vol. 82, pp. 149–160, DOI: 10.1016/j.ijmecsci.2014.03.014.

    Article  Google Scholar 

  • Sarkar, K. and Ganguli, R. (2014). “Closed-form solutions for axially functionally graded Timoshenko beams having uniform crosssection and fixed–fixed boundary condition.” Composites Part B: Engineering, Vol. 58, pp. 361–370, DOI: 10.1016/j.compositesb.2013.10.077.

    Article  Google Scholar 

  • Shahba, A., Attarnejad, R., and Hajilar, S. (2011). “Free vibration and stability of axially functionally graded tapered Euler-Bernoulli beams.” Shock and Vibration, Vol. 18, No. 5, pp. 683–696, DOI: 10.3233/SAV-2010-0589.

    Article  Google Scholar 

  • Shooshtari, A. and Rafiee, M. (2011). “Nonlinear forced vibration analysis of clamped functionally graded beams.” Acta Mechanica, Vol. 221, Nos. 1–2, pp. 23–38, DOI: 10.1007/s00707-011-0491-1

    Article  MATH  Google Scholar 

  • Sina, S. A., Navazi, H. M., and Haddadpour, H. (2009). “An analytical method for free vibration analysis of functionally graded beams.” Materials & Design, Vol. 30, No. 3, pp. 741–747, DOI: 10.1016/j.matdes.2008.05.015.

    Article  Google Scholar 

  • Stephen, N. G. (1982). “The second frequency spectrum of Timoshenko beams.” Journal of Sound and Vibration, Vol. 80, pp. 578–582.

    Article  Google Scholar 

  • Stephen, N. G. (2006). “The second spectrum of Timoshenko beam theory-Further assessment.” Journal of Sound Vibration, Vol. 292, pp. 372–389, DOI: 10.1016/j.jsv.2005.08.003.

    Article  MATH  Google Scholar 

  • Tang, A. Y., Wu, J. X., Li, X. F., and Lee, K. Y. (2014). “Exact frequency equations of free vibration of exponentially non-uniform functionally graded Timoshenko beams.” International Journal of Mechanical Sciences, Vol. 89, pp. 1–11, DOI: 10.1016/j.ijmecsci.2014.08.017.

    Article  Google Scholar 

  • Thai, Huu-Tai, and Thuc P. Vo (2012). “Bending and free vibration of functionally graded beams using various higher-order shear deformation beam theories.” International Journal of Mechanical Sciences, Vol. 62, No. 1, pp. 57–66, DOI: 10.1016/j.ijmecsci.2012.05.014.

    Article  Google Scholar 

  • Timoshenko, S. P. (1921). “On the correction for shear of the differential equation for transverse vibration of prismatic bars.” Phil. Mag. Ser., Vol. 6, No. 41, pp. 744–746, DOI: 10.1080/14786442108636264.

    Article  Google Scholar 

  • Wattanasakulpong, N. and Ungbhakorn, V. (2012). “Free vibration analysis of functionally graded beams with general elastically end constraints by DTM.” World Journal of Mechanics, Vol. 2, No. 6, pp. 297, DOI: 10.4236/wjm.2012.26036.

    Article  Google Scholar 

  • Wattanasakulpong, N. and Ungbhakorn, V. (2014). “Linear and nonlinear vibration analysis of elastically restrained ends FGM beams with porosities.” Aerospace Science and Technology, Vol. 32.1, No. 1, pp. 111–120, DOI: 10.1016/j.ast.2013.12.002.

    Article  Google Scholar 

  • Wattanasakulpong, N. and Chaikittiratana, A. (2015). “Flexural vibration of imperfect functionally graded beams based on Timoshenko beam theory: Chebyshev collocation method.” Meccanica, pp. 1–12, DOI: 10.1007/s11012-014-0094-8.

    Google Scholar 

  • Wattanasakulpong, N. and Mao, Q. (2015). “Dynamic response of Timoshenko functionally graded beams with classical and nonclassical boundary conditions using Chebyshev collocation method.” Composite Structures, Vol. 119, pp. 346–354, DOI: 10.1016/j.compstruct.2014.09.004.

    Article  Google Scholar 

  • Wei, D., Liu, Y., and Xiang, Z. (2012). “An analytical method for free vibration analysis of functionally graded beams with edge cracks.” Journal of Sound and Vibration, Vol. 331, No. 7, pp. 1686–1700, DOI: 10.1016/j.jsv.2011.11.020.

    Article  Google Scholar 

  • Yan, T., Yang, J., and Kitipornchai, S. (2012). “Nonlinear dynamic response of an edge-cracked functionally graded Timoshenko beam under parametric excitation.” Nonlinear Dynamics, Vol. 67, No. 1, pp. 527–540, DOI: 10.1007/s11071-011-0003-9.

    Article  MathSciNet  Google Scholar 

  • Zhu, J., Lai, Z., Yin, Z., Jeon, J., and Lee, S. (2001). “Fabrication of ZrO2–NiCr function-ally graded material by powder metal lurgy.” Mater. Chem. Phys., Vol. 68, Nos. 1–3, pp. 130–135, DOI: 10.1016/S0254-0584(00)00355-2.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Civil Engineering Department, Jordan University of Science and Technology, P.O. Box 3030, Irbid, 22110, Jordan

    Yousef S. Al Rjoub & Azhar G. Hamad

  2. University of Kufa, Najaf, Iraq

    Azhar G. Hamad

Authors
  1. Yousef S. Al Rjoub
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Azhar G. Hamad
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yousef S. Al Rjoub.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Al Rjoub, Y.S., Hamad, A.G. Free vibration of functionally Euler-Bernoulli and Timoshenko graded porous beams using the transfer matrix method. KSCE J Civ Eng 21, 792–806 (2017). https://doi.org/10.1007/s12205-016-0149-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12205-016-0149-6

Keywords

Search

Navigation