Dynamic response of FG-CNTRC beams subjected to a moving mass

Ismail Esen, Thi Thom Tran, Dinh Kien Nguyen
Author affiliations


  • Ismail Esen Department of Mechanical Engineering, Karabuk University, Makina Mühendisliği Bölümü, Karabük, Turkey
  • Thi Thom Tran Institute of Mechanics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet Street, Cau Giay District, Ha Noi, Viet Nam
  • Dinh Kien Nguyen Graduate University of Science and Technology, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet Street, Cau Giay District, Ha Noi, Viet Nam




FG-CNTRC beams, third-order shear deformation theory, finite element method, moving mass


This article presents the forced vibration of composite beams reinforced by single-walled carbon nanotubes (SWCNTs) and subjected to a moving mass. Considering the distribution of carbon nanotubes such as uniform (UD-CNT), functionally graded Λ (FGΛ-CNT) and X (FGX-CNT), three different beams are studied. Based on a third-order shear deformation theory (TSDT), the motion equations of the beams are derived using Hamilton's principle. Including mass interaction forces, the motion equations are transformed into a finite element equation in which a two-node beam element with eight degrees of freedom is utilized. To improve the efficiency of the beam element, the transverse shear rotation is employed as an independent variable in the derivation of the beam element. The vibration characteristics, including the dynamic magnification factors and the time histories for mid-span deflections are computed by using the Newmark method.  Numerical result reveal that the vibration of the beams is clearly influenced of the CNT reinforcement, and the dynamic magnification is significantly decreased by increasing the CNT volume fraction. It is also shown that the FGX-CNT beam is the best in dynamic resistance in terms of the lowest dynamic deflection and dynamic magnification factors. The effects of the total volume fraction and the moving load velocity on the dynamic behaviour of the functionally graded carbon nanotube reinforced composites (FG-CNTRC) beams are examined in detail and highlighted. 


Download data is not yet available.


Bohlén M. and Bolton K. - Molecular dynamics studies of the influence of single wall carbon nanotubes on the mechanical properties of Poly(vinylidene fluoride), Comput. Mater. Sci. 68 (2013) 73-80. DOI: https://doi.org/10.1016/j.commatsci.2012.10.010

Han Y. and Elliott J. - Molecular dynamics simulations of the elastic properties of polymer/carbon nanotube composites, Comput. Mater. Sci. 39 (2007) 315-323. DOI: https://doi.org/10.1016/j.commatsci.2006.06.011

Griebel M. and Hamaekers J. - Molecular dynamics simulations of the elastic moduli of polymer–carbon nanotube composites, Comput. Methods Appl. Mech. Engrg. 193 (2004) 1773-1788. DOI: https://doi.org/10.1016/j.cma.2003.12.025

Lu X. and Hu Z. - Mechanical property evaluation of single-walled carbon nanotubes by finite element modeling, Compos. Part B: Eng. 43 (2012) 1902-1913. DOI: https://doi.org/10.1016/j.compositesb.2012.02.002

Giannopoulos G. I., Kakavas P. A., and Anifantis N. K. - Evaluation of the effective mechanical properties of single walled carbon nanotubes using a spring based finite element approach, Comput.Mater. Sci. 41 (2008) 561-569. DOI: https://doi.org/10.1016/j.commatsci.2007.05.016

Shen H. S. - Nonlinear bending of functionally graded carbon nanotube-reinforced composite plates in thermal environments, Compos. Struct. 91 (2009) 9-19. DOI: https://doi.org/10.1016/j.compstruct.2009.04.026

Ke L. L., Yang J., and Kitipornchai S. - Nonlinear free vibration of functionally graded carbon nanotubereinforced composite beams, Compos. Struct. 92 (2010) 676-683. DOI: https://doi.org/10.1016/j.compstruct.2009.09.024

Ke L. L., Yang J., and Kitipornchai S. - Dynamic stability of functionally graded carbon nanotube- reinforced composite beams, Mech. Adv. Mater. Struct. 20 (2013) 28-37. DOI: https://doi.org/10.1080/15376494.2011.581412

Yas M. H. and Heshmati M. - Dynamic analysis of functionally graded nanocomposite beams reinforced by randomly oriented carbon nanotube under the action of moving load, Appl. Math. Model. 36 (2012) 1371-94. DOI: https://doi.org/10.1016/j.apm.2011.08.037

Lin F. and Xiang Y. - Vibration of carbon nanotube reinforced composite beams based on the first and third order beam theories, Appl. Math. Model. 38 (2014) 3741-3754. DOI: https://doi.org/10.1016/j.apm.2014.02.008

Ansari R., Faghih Shojaei M., Mohammadi V., Gholami R., and Sadeghi F. - Nonlinear forced vibration analysis of functionally graded carbon nanotube-reinforced composite Timoshenko beams, Compos. Struct. 13 (2014) 316-327. DOI: https://doi.org/10.1016/j.compstruct.2014.03.015

Aydogdu M. - On the vibration of aligned carbon nanotube reinforced composite beams, Adv. Nano Res. 2 (2014) 199-210. DOI: https://doi.org/10.12989/anr.2014.2.4.199

Wu. H. L., Yang J., and Kitipornchai S. - Nonlinear vibration of functionally graded carbon nanotube-reinforced composite beams with geometric imperfections, Compos. Part B: Eng. 90 (2016) 86-96. DOI: https://doi.org/10.1016/j.compositesb.2015.12.007

Chaudhari V. K. and Lal A. - Nonlinear Free Vibration Analysis of Elastically Supported Nanotube-reinforced Composite Beam in Thermal Environment, Proc. Eng. 144 (2016) 928-935. DOI: https://doi.org/10.1016/j.proeng.2016.05.119

Wu H., Kitipornchai S. and Yang J. - Imperfection sensitivity of thermal post-buckling behaviour of functionally graded carbon nanotube-reinforced composite beams, Appl. Math. Model. 42 (2017) 735-752. DOI: https://doi.org/10.1016/j.apm.2016.10.045

Gholami R., Ansari R., and Gholami Y. - Nonlinear resonant dynamics of geometrically imperfect higher-order shear deformable functionally graded carbon-nanotube reinforced composite beams, Compos. Struct. 174 (2017) 45-58. DOI: https://doi.org/10.1016/j.compstruct.2017.04.042

Shafiei H. and Setoodeh A. R. - Nonlinear free vibration and post-buckling of FG-CNTRC beams on nonlinear foundation, Steel. Compos. Struct. 24 (2017) 65-77. DOI: https://doi.org/10.12989/scs.2017.24.1.065

Vo-Duy T., Ho-Huu V. and Nguyen-Thoi T. - Free vibration analysis of laminated FG-CNT reinforced composite beams using finite element method, Front. Struct. Civil. Eng. 13 (2019) 324-336. DOI: https://doi.org/10.1007/s11709-018-0466-6

Ranjbar M. and Feli S. - Temperature-dependent analysis of axially functionally graded CNT reinforced micro-cantilever beams subjected to low velocity impact, Mech. Adv. Mater. Struct. 26 (2019) 1154-1168. DOI: https://doi.org/10.1080/15376494.2018.1432788

Fallah A., Dehkordi M. B., Nourbakhsh H., and Beni Y. T. - Semi-exact solution for nonlinear dynamic analysis of graded carbon nanotube-reinforced beam with graded shape memory wires, Mech. Adv. Mater. Struct. 28 (2019) 1-15. DOI: https://doi.org/10.1080/15376494.2019.1578012

Palacios J. A. and Ganesan R. - Dynamic response of Carbon-Nanotube-Reinforced-Polymer materials based on multiscale finite element analysis, Compos. Part B: Eng. 166 (2019) 497-508. DOI: https://doi.org/10.1016/j.compositesb.2019.02.039

Shi G. - A new simple third-order shear deformation theory of plates, Int. J. Solids Struct. 44 (2007) 4399-417. DOI: https://doi.org/10.1016/j.ijsolstr.2006.11.031

Shi G., Lam K. Y. and Tay T. E. - On efficient finite element modeling of composite beams and plates using higher-order theories and an accurate composite beam element, Compos. Struct. 41 (1998) 159-165. DOI: https://doi.org/10.1016/S0263-8223(98)00050-6

Esen I. -Dynamic response of a functionally graded Timoshenko beam on two-parameter elastic foundations due to a variable velocity moving mass, Int. J. Mech. Sci. 153–154 (2019) 21-35. DOI: https://doi.org/10.1016/j.ijmecsci.2019.01.033

Esen I. - A new finite element for transverse vibration of rectangular thin plates under a moving mass, Finite Elem. Anal. Des. 66 (2013) 26-35. DOI: https://doi.org/10.1016/j.finel.2012.11.005




How to Cite

Ismail Esen, T. T. Tran, and D. K. Nguyen, “Dynamic response of FG-CNTRC beams subjected to a moving mass”, Vietnam J. Sci. Technol., vol. 60, no. 5, pp. 853–868, Nov. 2022.



Mechanical Engineering - Mechatronics