Skip Navigation Links
Journal of Vibration Testing and System Dynamics

C. Steve Suh (editor), Pawel Olejnik (editor),

Xianguo Tuo (editor)

Pawel Olejnik (editor)

Lodz University of Technology, Poland


C. Steve Suh (editor)

Texas A&M University, USA


Xiangguo Tuo (editor)

Sichuan University of Science and Engineering, China


Comparing Different Theories for Dynamic Behavior of a Functionally Graded Beam

Journal of Vcibration Testing and System Dynamics 4(3) (2020) 287--296 | DOI:10.5890/JVTSD.2020.09.006

Peyman Beiranvand$^{1}$, Matin Abdollahifar$^{2}$, Farzad Akbarinia$^{3}$

$^{1}$ Department of Civil Engineering, Razi University, Kermanshah, Iran

$^{2}$ Department of Civil Engineering, Yasouj University, Yasouj, Iran

$^{3}$ Department of Civil Engineering, Imam Khomeini International University, Qazvin, Iran

Download Full Text PDF



Theoretical formulation, Navier’s solutions of rectangular plates based on a new higher order shear deformation model are presented for the static and dynamic analysis of functionally graded plates (FGPs). This theory enforces traction free boundary conditions at plate surfaces. Shear correction factors are not required because a correct representation of transverse shearing strain is given. Unlike any other theory, the number of unknown functions involved is only four, as against five in case of other shear deformation theories. The mechanical properties of the plate are assumed to vary continuously in the thickness direction by a simple power-law distribution in terms of the volume fractions of the constituents. Numerical illustrations concern flexural behavior of FG plates with Metal–Ceramic composition. Parametric studies are performed for varying ceramic volume fraction, volume fraction profiles, aspect ratios and length to thickness ratios. Results are verified with available results in the literature. It can be concluded that the proposed theory is accurate and simple in solving the static and dynamic behavior of functionally graded plates. This paper presents a theoretical investigation in free vibration of simply supported FG beam. Young’s modulus of beam varies in the thickness direction according to power law. Governing equations were found by applying Hamilton’s principle. Navier type solution method was used to obtain frequencies. Different higher order shear deformation theories and classical beam theories were used in the analysis. A free vibration frequency is given for different material properties.


  1. [1]  Aydogdu, M. and Timarci, T. (2003), Vibration analysis of cross-ply laminated square plates with general boundary conditions, Compos Sci Technol, 63, 1061-1070.
  2. [2]  Aydogdu, M. (2005), Vibration analysis of cross-ply laminated beams with general boundary conditions by Ritz method, Int J Mech Sci, 47, 1740-1755.
  3. [3]  Benatta, M.A., Mechab, I., Tounsi, A., and Adda bedia, E.A. (2008), Static analysis of functionally graded short beams including warping and shear deformation effects, Computational Materials Science, 44, 765-773.
  4. [4]  Karama, M., Afaq, K.S., and Mistou, S. (2003), Mechanical behaviour of laminated composite beam by the new multi-layered laminated composite structures model with transverse shear stress continuity, Int J Solids Struct, 40, 1525-1546.
  5. [5]  Kim, J. and Paulino, G.H. (2002), Finite element evaluation of mixed mode stress intensity factors in functionally graded materials, Int J Numerical Methods Eng, 53, 1903-1935.
  6. [6]  Messina, A. and Soldatos, K.P. (1999), Influence of edge boundary conditions on the free vibrations of cross-ply laminated circular panels, J Acoust Soc Am, 106, 2608-2620.
  7. [7]  Messina, A. and Soldatos, K.P. (1999), Vibration of completely free composite plates and cylindrical shell panels by a higher order theory, Int J Mech Sci, 41, 891-918.
  8. [8]  Reddy, J.N. (1984), A simple higher-order theory for laminated composite plates, J Appl Mech, 51, 745-752.
  9. [9]  Sallai, B-O., Tounsi, A., Mechab, I., Bachir, B. M., Meradjah, M., and Adda, B.E.A. (2009), A theoretical analysis of flexional bending of Al/Al2O3 S-FGM thick beams, Computational Materials Science, 44, 1344-1350.
  10. [10]  Sankarm, B.V. (2001), An elasticity solution for functionally graded beams, Compos Sci Technol, 61, 689-696.
  11. [11]  Soldatos, K.P. (1992), A transverse shear deformation theory for homogeneous monoclinic plates, Acta Mech, 94, 195-220.
  12. [12]  Soldatos, K.P. and Timarci, T. (1993), A unified formulation of laminated composite, shear deformable, five degrees of freedom cylindrical shell theories, Compos Struct, 25, 165-171.
  13. [13]  Soldatos, K.P. and Sophocles, C. (2001), On shear deformable beam theories: the frequency and normal mode equations of the homogeneous orthotropic Bickford beam, J Sound Vib, 242, 215-245.
  14. [14]  Timarci, T. and Soldatos, K.P. (1995), Comparative dynamic studies for symmetric cross-ply circular cylindrical shells on the basis of a unified shear deformable shell theory, J Sound Vib, 187, 609-624.
  15. [15]  Timoshenko, S.P. and Goodier, J.N. (1970), Theory of Elasticity, third ed. McGraw-Hill, New York.
  16. [16]  Vel, S.S. and Batra, R.C. (2002), Exact solution for the cylindrical bending vibration of functionally graded plates. In: Proceedings of the American Society of Composites, Seventh Technical Conference, October 21-23. West Lafayette, Indiana: Purdue University.
  17. [17]  Koizumi. (1997), FGM activities in Japan, Composites Part B: Engineering, 28(1-2), 1-4.
  18. [18]  Aydogdu, M. and Taskin, V. (2007), Free vibration analysis of functionally graded beams with simply supported edges, Mater Des, 28(5), 1651-1656.
  19. [19]  Calim, F.F. (2009), Free and forced vibrations of non-uniform composite beams, Composite Structures, 88, 413-423.
  20. [20]  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, 92, 2497-2511.
  21. [21]  Sina, S.A., Navazi, H.M., and Haddadpour, H. (2009), An analytical method for free vibration analysis of functionally graded beams, Materials and Design, 30, 741-747.
  22. [22]  Simsek, M. and Kocat¨urk, T. (2009), Free and forced vibration of a functionally graded beam subjected to a concentrated moving harmonic load, Composite Structures, 90(4), 465-473.
  23. [23]  Alshorbagy, A.E., Eltaher, M.A., and Mahmoud, F.F. (2011), Free vibration characteristics of a functionally graded beam by finite element method, Applied Mathematical Modelling, 35, 412-425.
  24. [24]  Simsek, M. (2010), Vibration analysis of a functionally graded beam under a moving mass by using different beam theories, Composite Structures, 92, 904-917.
  25. [25]  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, 329, 2291-2303.