### Free vibration analysis of 2-D FGM beams in thermal environment based on a new third-order shear deformation theory

#### Abstract

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.

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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. doi:10.1016/s0020-7403(03)00058-4.

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. doi:10.1016/j.apm.2007.09.015.

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, (1), (2009), pp. 342–362. doi:10.1016/j.jsv.2008.09.018.

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, (8), (2010), pp. 1877–1887. doi:10.1016/j.compstruct.2010.01.010.

G. Shi. A new simple third-order shear deformation theory of plates. International Journal of Solids and Structures, 44, (13), (2007), pp. 4399–4417. doi:10.1016/j.ijsolstr.2006.11.031.

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 Sciences, 53, (9), (2011), pp. 734–743. doi:10.1016/j.ijmecsci.2011.06.005.

A. E. Alshorbagy, M. A. Eltaher, and F. F. Mahmoud. Free vibration characteristics of a functionally graded beam by finite element method. Applied Mathematical Modelling, 35, (1), (2011), pp. 412–425. doi:10.1016/j.apm.2010.07.006.

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 Sciences, 62, (1), (2012), pp. 57–66. doi:10.1016/j.ijmecsci.2012.05.014.

T. P. Vo, H. T. Thai, T. K. Nguyen, and F. Inam. Static and vibration analysis of functionally graded beams using refined shear deformation theory. Meccanica, 49, (1), (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. 147–157. doi:10.1016/j.compositesb.2013.06.011.

L. C. Trinh, T. P. 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. doi:10.1016/j.compositesb.2016.06.067.

Z. H. Wang, X. H. Wang, G. D. Xu, S. Cheng, and T. Zeng. Free vibration of twodirectional functionally graded beams. Composite Structures, 135, (2016), pp. 191–198. doi:10.1016/j.compstruct.2015.09.013.

M. S¸ims¸ek. 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. doi:10.1016/j.compstruct.2015.08.021.

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, (1), (2017), pp. 141–155. doi:10.1007/s00707-016-1705-3.

A. Karamanlı. Bending behaviour of two directional functionally graded sandwich beams by using a quasi-3d shear deformation theory. Composite Structures, 174, (2017), pp. 70–86. doi:10.1016/j.compstruct.2017.04.046.

T. V. Do, D. K. Nguyen, N. D. Duc, D. H. Doan, and T. Q. Bui. Analysis of bi-directional functionally graded plates by FEM and a new third-order shear deformation plate theory. Thin-Walled Structures, 119, (2017), pp. 687–699. doi:10.1016/j.tws.2017.07.022.

T. K. Nguyen, K. Sab, and G. Bonnet. First-order shear deformation plate models for functionally graded materials. Composite Structures, 83, (1), (2008), pp. 25–36. doi:10.1016/j.compstruct.2007.03.004.

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 element. Composite Structures, 41, (2), (1998), pp. 159–165. doi:10.1016/s0263-8223(98)00050-6.

G. Shi and K. Y. Lam. Finite element vibration analysis of composite beams based on higher-order beam theory. Journal of Sound and Vibration, 219, (4), (1999), pp. 707–721. doi:10.1006/jsvi.1998.1903.

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

Y. W. Kim. Temperature dependent vibration analysis of functionally graded rectangular plates. Journal of Sound and Vibration, 284, (3), (2005), pp. 531–549. doi:10.1016/j.jsv.2004.06.043.

F. Ebrahimi, F. Ghasemi, and E. Salari. Investigating thermal effects on vibration behavior of temperature-dependent compositionally graded euler beams with porosities. Meccanica, 51, (1), (2016), pp. 223–249. doi:10.1007/s11012-015-0208-y.

DOI: https://doi.org/10.15625/0866-7136/10503 Display counter: Abstract : 168 views. PDF : 3 views.

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