Thermal vibration analysis of FGM beams using an efficient shear deformation beam theory

  • Safa, Abdelkader (Department of Civil Engineering, Ahmed Zabana University Centre) ;
  • Hadji, Lazreg (Department of Mechanical Engineering, Ibn Khaldoun University) ;
  • Bourada, Mohamed (Material and Hydrology Laboratory, University of Sidi Bel Abbes, Faculty of Technology, Civil Engineering Department) ;
  • Zouatnia, Nafissa (Department of Civil Engineering, Ibn Khaldoun University)
  • Received : 2019.04.04
  • Accepted : 2019.08.03
  • Published : 2019.09.25


An efficient shear deformation beam theory is developed for thermo-elastic vibration of FGM beams. The theory accounts for parabolic distribution of the transverse shear strains and satisfies the zero traction boundary conditions on the on the surfaces of the beam without using shear correction factors. The material properties of the FGM beam are assumed to be temperature dependent, and change gradually in the thickness direction. Three cases of temperature distribution in the form of uniformity, linearity, and nonlinearity are considered through the beam thickness. Based on the present refined beam theory, the equations of motion are derived from Hamilton's principle. The closed-form solutions of functionally graded beams are obtained using Navier solution. Numerical results are presented to investigate the effects of temperature distributions, material parameters, thermal moments and slenderness ratios on the natural frequencies. The accuracy of the present solutions is verified by comparing the obtained results with the existing solutions.


  1. Attia, A., Bousahla, A.A., Tounsi, A., Mahmoud, S.R. and Alwabli, A.S. (2018), "A refined four variable plate theory for thermoelastic analysis of FGM plates resting on variable elastic foundations", Struct. Eng. Mech., 65(4), 453-464.
  2. Bennai, R., Fourn, H., Ait Atmane, H., Tounsi, A. and Bessaim, A. (2019), "Dynamic and wave propagation investigation of FGM plates with porosities using a four variable plate theory", Wind Struct., 28(1), 49-62.
  3. Chen, Y., Jin, G., Zhang, C., Ye, T. and Xue, Y. (2018), "Thermal vibration of FGM beams with general boundary conditions using a higher-order shear deformation theory", Compos. Part B, 153, 376-386.
  4. Ebrahimi, F. and Safarpour, H. (2018), "Vibration analysis of inhomogeneous nonlocal beams via a modified couple stress theory incorporating surface effects", Wind Struct., 27(6), 431-438.
  5. Ebrahimi, F. and Salari, E. (2015), "Nonlocal thermo-mechanical vibration analysis of functionally graded nanobeams in thermal environment", Acta Astronaut, 113, 29-50.
  6. El-Haina, F., Bakora, A., Bousahla, A.A., Tounsi, A. and Mahmoud, S.R. (2017), "A simple analytical approach for thermal buckling of thick functionally graded sandwich plates", Struct. Eng. Mech., 63(5), 585-595.
  7. Giunta, G., Crisafulli, D., Belouettar, S. and Carrera, E. (2013), "A thermo-mechanical analysis of functionally graded beams via hierarchical modeling", Compos. Struct., 95, 676-690.
  8. Hamdi, H. and Farah, K. (2018), "Beam finite element model of a vibrate wind blade in large elastic deformation", Wind Struct., 26(1), 25-34.
  9. Karami, B., Janghorban, M., Shahsavari, D. and Tounsi, A. (2018), "A size-dependent quasi-3D model for wave dispersion analysis of FG nanoplates", Steel Compos. Struct., 28(1), 99-110.
  10. Menasria, A., Bouhadra, A., Tounsi, A., Bousahla, A.A. and Mahmoud, S.R. (2017), "A new and simple HSDT for thermal stability analysis of FG sandwich plates", Steel Compos. Struct., 25(2), 157-175.
  11. Mouli, C.B., K. Ramji, Kar, V.R., Panda, S.K., Lalepalli, A.K. and Pandey, H.K. (2018), "Numerical study of temperature dependent eigenfrequency responses of tilted functionally graded shallow shell structures", Struct. Eng. Mech., 68(5), 527-536.
  12. Rong, X.N., Xu, R.Q., Wang, H.Y. and Feng, S.Y. (2018), "Analytical solution for natural frequency of monopile supported wind turbine towers", Wind Struct., 25(5), 459-474.
  13. Sankar, B.V. and Tzeng, J.T. (2002), "Thermal stresses in functionally graded beams", AIAA J., 40(6), 1228-1232.
  14. Semmah, A., Heireche, H., Bousahla, A.A. and Tounsi, A. (2019), "Thermal buckling analysis of SWBNNT on Winkler foundation by non local FSDT", Adv. Nano Res., 7(2), 89-98.
  15. Simsek, M. (2010), "Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories", Nucl. Eng. Des., 240, 697-705.
  16. Sina, S.A., Navazi, H.M. and Haddadpour, H. (2009), "An analytical method for free vibration analysis of functionally graded beams", Mater. Des., 30, 741-747.
  17. Trinh, L.C., Vo, T.P., Thai, H.T., T.K. and Nguyen, T.K. (2016), "An analytical method for the vibration and buckling of functionally graded beams under mechanical and thermal loads", Compos. Part B, 100, 152-163.
  18. Wattanasakulpong, N., Prusty, B.G. and Kelly, D.W. (2011), "Thermal buckling and elastic vibration of third-order shear deformable functionally graded beams", Int. J. Mech. Sci., 53, 734-743.
  19. Wu, H., Kitipornchai, S. and Yang, J. (2018), "Free vibration of thermo-electro-mechanically postbuckled FG-CNTRC beams with geometric imperfections", Steel Compos. Stuct., 39(3), 319-332.