Nguyen Dinh Kien


Free vibration analysis of two-directional functionally graded material (2-D FGM) beams in thermal environment based on a new third-order shear deformation theory is presented. The material properties are assumed to be graded in both the thickness and longitudinal directions by a power law distribution, and they are considered to be temperature-dependent. Equations of motion, in which the shear rotation rather than the cross-sectional rotation is considered to be an independent variable, are constructed from Hamilton's principle. A finite element formulation is derived and employed to compute the vibration characteristics of the beams. The numerical results reveal that the developed formulation is accurate, and it is capable to give accurate natural frequencies by using a small number of elements. A parametric study is carried out to highlight the effects of material composition, temperature rise on the vibration characteristics of the beams.


2-D FGM beam, temperature-dependent properties, third-order shear deformation theory, shear rotation, free vibration analysis.


A. Chakraborty, S. Gopalakrishnan, and J. N. Reddy. A new beam finite element for the analysis of functionally graded materials, International Journal of Mechanical Sciences, 45, (3),

(2003), pp. 519–539.

H. K. Ching and S. C. Yen. Transient thermoelastic deformations of 2-D functionally graded beams under nonuniformly convective heat supply, Composite Structures, 73, (2006), pp. 381-393.

H. J. Xiang and J. Yang. Free and forced vibration of a laminated FGM Timoshenko beam of variable thickness under heat conduction, Composites Part B: Engineering, 39, (2008), pp. 292-303.

R. Kadoli, K. Akhtar, and N. Ganesan. Static analysis of functionally graded beams using higher order shear deformation theory, Applied Mathematical Modelling, 32, (12), (2008), pp. 2509–2525.

S. C. Pradhan and T. Murmu. Thermo-mechanical vibration of FGM sandwich beam under variable elastic foundations using differential quadrature method, Journal of Sound and Vibration, 321, (2009), pp. 342-362.

P. Malekzadeh. Two-dimensional in-plane free vibrations of functionally graded circular arches with temperature-dependent properties, Composite Structures, 91, (2009), pp.38-47.

A. Mahi, E. A. Adda Bedia, A. Tounsi, and I. Mechab. An analytical method for temperature-dependent free vibration analysis of functionally graded beams with general boundary conditions, Composite Structures, 92, (2010), pp. 1877-1887.

G. Shi. A new simple third-order shear deformation theory of plates, International Journal of Solids and Structures, 44, (2007), pp. 4399-4417.

N. Wattanasakulpong, B. G. Prusty, and D. W. Kelly. Thermal buckling and elastic vibration of third-order shear deformable functionally graded beams, International Journal of Mechanical Science, 53, (2011), pp. 734-743.

A. E. Alshorbagy, M. A. Eltaher, and F. F. Mahmoud. Free vibration chatacteristics of a functionally graded beam by finite element method, Applied Mathematical Modelling, 35, (1), (2011), pp. 412–425.

A. Shahba, R. Attarnejad, M. T. Marvi, and S. Hajilar. Free vibration and stability analysis of axially functionally graded tapered Timoshenko beams with classical and non-classical boundary conditions, Composites: part B, 42, (4), (2011), pp. 801–808.

H. T. Thai and T. P. Vo. Bending and free vibration of functionally graded beams using various higher-order shear deformation beam theories, International Journal of Mechanical Science, 62, (1), (2012), pp. 57–66.

H. T. Thai, P. T. Vo, T. K. Nguyen. Static and vibration analysis of functionally graded beams

using refined shear deformation theory, Meccanica, 49, (2014), pp.155–168. DOI 10.1007/s11012-013-9780-1.

T. -K. Nguyen, T. P. Vo, and H. -T. Thai. Static and free vibration of axially loaded functionally graded beams based on the first-order shear deformation theory, Composites Part B: Engineering, 55, (2013), pp. 147-121.

S. E. Esfahani, Y. Kiani, and M. R. Eslam. Non-linear thermal stability analysis of temperature dependent FGM beams supported on non-linear hardening elastic foundations, International Journal of Mechanical Science, 69, (2013), pp.10-20.

A. Fallah and M. M. Aghdam. Thermo-mechanical buckling and nonlinear free vibration analysis of functionally graded beams on nonlinear elastic foundation, Composites Part B: Engineering, 43, (2012), pp. 1523-1530.

H. S. Shen and Z. X. Wang. Nonlinear analysis of shear deformable FGM beams resting on elastic foundations in thermal environments, International Journal of Mechanical Science, 81, (2014), pp. 195-206.

Y. Kiani, M. Sadighi, S. Jedari Salami, and M. R. Eslami. Low velocity impact response of thick FGM beams with general boundary conditions in thermal field, Composite Structures, 104, (2013), pp. 293-303.

M. Komijani, S. E. Esfahani, J. N. Reddy, Y. P. Liu, and M. R. Eslami. Nonlinear thermal stability and vibration of pre/postbuckled temperature- and microstructure-dependent functionally graded beams resting on elastic foundation, Composite Structures, 112, (2014), pp. 292-307.

L. C. Trinh, P. T. Vo, H. T. Thai, and T. K. Nguyen. An analytical method for the vibration and buckling of functionally graded beams under mechanical and thermal loads, Composites Part B: Engineering, 100, (2016), pp.152-163.

C. F. Lü, W. Q. Chen, R. Q. Xu, and C. W. Lim. Semi-analytical elasticity solutions for bi-directional functionally graded beams, International Journal of Solids and Structures, 45, (2008), pp. 258–275.

Z. Wang, X. Wang, G. Xu, S. Cheng, and T. Zeng. Free vibration of two-directional functionally graded beams, Composite Structures, 135, (2016), pp. 191–198.

M. Şimşek. Bi-directional functionally graded materials (BDFGMs) for free and forced vibration of Timoshenko beams with various boundary conditions, Composite Structures, 133, (2015), pp. 968–978.

D. K. Nguyen, Q. H. Nguyen, T. T. Tran, and V. T. Bui. Vibration of bi-dimensional functionally graded Timoshenko beams excited by a moving load, Acta Mechanica, 228, (2017), pp. 141-155.

A. Karamanli. Bending behaviour of two directional functionally graded sandwich beams by using a quasi-3d shear deformation theory, Composite Structures. 174, (2017), pp. 70–86.

T. K. Nguyen, K. Sab, and G. Bonnet. First-order shear deformation plate models for functionally graded materials, Composite Structures, 83, (2008), pp. 25-36.

G. Shi, K. Y. Lam, and T. E. Tay. On efficient finite element modeling of composite beams and plates using higher-order theories and an accurate composite beam, Composite Structures, 41, (1998), pp. 159-165.

G. Shi and K. Y. Lam. Finite element vibration analysis of composite beams based on higher-oder beam theory, Journal of Sound and Vibration, 219, (1999), pp. 707-721.

Y. S. Touloukian. Thermophysical properties of high temperature solids materials. New York: MacMillan, (1967).

Y.-W. Kim. Temperature dependent vibration analysis of functionally graded rectangular plates, Journal of Sound and Vibration, 284 (2005), pp. 531-549.

F. Ebrahimi, F. Ghasemi, and E.Salari. Investigating thermal effects on vibration behavior of temperature-dependent compositionally graded Euler beams with porosities, Meccanica, (2015), DOI 10.1007/s11012-015-0208-y.



  • There are currently no refbacks.



Editorial Office of Vietnam Journal of Mechanics

3rd Floor, A16 Building, 18B Hoang Quoc Viet Street, Cau Giay District, Hanoi, Vietnam

Tel: (+84) 24 3791 7103