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BY-NC-ND 4.0 license Open Access Published by De Gruyter Open Access May 3, 2017

Nonlinear free vibration analysis of elastically supported carbon nanotube-reinforced composite beam with the thermal environment in non-deterministic framework

  • Virendra Kumar Chaudhari , Niranjan L. Shegokar and Achchhe Lal EMAIL logo


This paper deals with the investigation of nonlinear free vibration behavior of elastically supported carbon nanotube reinforced composite (CNTRC) beam subjected to thermal loading with random system properties. Material properties of each constituent’s material, volume fraction exponent and foundation parameters are considered as uncorrelated Gaussian random input variables. The beam is supported by a Pasternak foundation with Winkler cubic nonlinearity. The higher order shear deformation theory (HSDT) with von-Karman non-linearity is used to formulate the governing equation using Hamilton principle. Convergence and validation study is carried out through the comparison with the available results in the literature for authenticity and accuracy of the present approach used in the analysis. First order perturbation technique (FOPT),Second order perturbation technique (SOPT) and Monte Carlo simulation (MCS) methods are employed to investigate the effect of geometric configuration, volume fraction exponent, foundation parameters, distribution of reinforcement and thermal loading on nonlinear vibration characteristics CNTRC beam.The present work signifies the accurate analysis of vibrational behaviour influences by different random variables. Results are presented in terms of mean, variance (COV) and probability density function (PDF) for various aforementioned parameters.


[1] Iijima S., Helical microtubules of graphitic carbon, Nature (London), 1991, 354, 56-8.10.1038/354056a0Search in Google Scholar

[2] Thostenson E.T., Ren Z., Chou T.W., Advances in the science and technology of carbon nanotubes and their composites, a review, Compos Sci Techno, 2001,61, 1899-912.10.1016/S0266-3538(01)00094-XSearch in Google Scholar

[3] Lau K.T., Gu C., Gao G.H., Ling H.Y., Reid S.R., Stretching process of single- andmultiwalled carbon nanotubes for nanocomposite applications. Carbon, 2004, 42, 426-8.10.1016/j.carbon.2003.10.040Search in Google Scholar

[4] Wuite J., Adali S., Deflection and stress behavior of nanocomposite reinforced beams using a multiscale analysis, Composites Structures, 2005, 71, 388-96.10.1016/j.compstruct.2005.09.011Search in Google Scholar

[5] Vodenitcharova T., Zhang L.C., Bending and local buckling of a nanocomposite beam reinforced by a single-walled carbon nanotube, International Journal of Solids and Structures, 2006, 43, 3006-3024 .10.1016/j.ijsolstr.2005.05.014Search in Google Scholar

[6] Hu N., Fukunaga H., Lu C., Kameyama M., Yan B., Prediction of elastic properties of carbon nanotube reinforced composites, Proc Royal Soc A, 2005, 461,1685-710.10.1098/rspa.2004.1422Search in Google Scholar

[7] Han Y., Elliott J., Molecular dynamics simulations of the elastic properties of polymer/carbon nanotube composites, Comput Mater Sci, 2007, 39, 315-23.10.1016/j.commatsci.2006.06.011Search in Google Scholar

[8] Wan H., Delale F., Shen L., Effect of CNT length and CNT-matrix inter phase in carbon nanotube (CNT) reinforced composites, Mech Res Commun,2005, 32,481-9.10.1016/j.mechrescom.2004.10.011Search in Google Scholar

[9] Shooshtari A., Rafiee M., Nonlinear forced vibration analysis of clamped functionally graded beams, Acta Mech, 2011, Doi. 10.1007/s00707-011-0491-1.10.1007/s00707-011-0491-1Search in Google Scholar

[10] Yang J., Chen Y., Free vibration and buckling analysis of functionally graded beams with edge cracks, Compos Struct, 2008, 83 48-60.10.1016/j.compstruct.2007.03.006Search in Google Scholar

[11] Ke L. L, Yang J., Kitipornchai S., Nonlinear free vibration of functionally graded carbon nanotube-reinforced composite beams, Compos Struct,2010, 92, 676-83.10.1016/j.compstruct.2009.09.024Search in Google Scholar

[12] Kitipornchai S., Ke L-L., Yang J., Xiang Y., Nonlinear vibration of edge cracked functionally graded Timoshenko beams, J Sound Vib, 2009, 324, 962-982.10.1016/j.jsv.2009.02.023Search in Google Scholar

[13] Sina S.A., Navazi H.M., Haddadpour H., An analytical method for free vibration analysis of functionally graded beams.Mater Des, 2009, 30, 741-747.10.1016/j.matdes.2008.05.015Search in Google Scholar

[14] Aydogdu M., Vibration analysis of cross-ply laminated beams with general boundary conditions by Ritz method, Int J Mech Sci, 2005, 47, 1740-1755.10.1016/j.ijmecsci.2005.06.010Search in Google Scholar

[15] Shen H.S., Xiang Y., Nonlinear analysis of nanotube-reinforced composite beams resting on elastic foundations in thermal environments, Engineering Structures, 2013, 56, 698-708.10.1016/j.engstruct.2013.06.002Search in Google Scholar

[16] Boutaleb S., Zaïri F., Mesbah A., Abdelaziz M. N., Gloaguen J.M., Boukharouba T., Lefebvre J.M., Micromechanics-based modelling of stiffness and yield stress for silica/polymer Nanocomposite, International Journal of Solids and Structures, 2009,46, 1716-1726.10.1016/j.ijsolstr.2008.12.011Search in Google Scholar

[17] DeValve C., Pitchumani R., Analysis of vibration damping in a rotating composite beam with embedded carbon nanotubes, Composite Structures, 2014,110, 289-296.10.1016/j.compstruct.2013.12.007Search in Google Scholar

[18] Ke L.L., Jieyang, Kitipornchai S., Dynamic Stability of Functionally Graded Carbon Nanotube-Reinforced Composite Beams Mechanics of Advanced Materials and Structures, 2013, 20.10.1080/15376494.2011.581412Search in Google Scholar

[19] Yas M.H., Samadi N., Free vibrations and buckling analysis of carbon nanotube-reinforced composite Timoshenko beams on elastic foundation, International Journal of Pressure Vessels and Piping,2012,98, 119-128.10.1016/j.ijpvp.2012.07.012Search in Google Scholar

[20] Rao G.V., Varma R.R., Heuristic thermal post buckling and largeamplitude vibration formulations of beams. AIAA J, 2009,47, 1977-80.10.2514/1.43505Search in Google Scholar

[21] Mehdipour I., Barari A., Kimiaeifar A., Domairry G., Vibrational analysis of curved single-walled carbon nanotube on a Pasternak elastic foundation, Advances in Engineering Software,2012, 48,1-5.10.1016/j.advengsoft.2012.01.004Search in Google Scholar

[22] Lee H. L., Chang W. J., Vibration analysis of a viscous-fluidconveying single-walled carbon nanotube embedded in an elastic medium, Physic Low-dimensional Systems and Nanostructures, 2009, 41, 529-532.10.1016/j.physe.2008.10.002Search in Google Scholar

[23] Pradhan S. C., Murmu T., Differential Quadrature Method for Vibration Analysis of Beam on Winkler Foundation based on Nonlocal Elastic Theory. J. Inst. Engg, 2009, 89, 3-12.Search in Google Scholar

[24] Murmu T., Pradhan S. C., Thermo-mechanical Vibration of a Single-Walled Carbon Nanotube Embedded in an Elastic Medium based on Nonlocal Elasticity Theory, Comput. Mater. Sci, 2009, 46, 854-869.10.1016/j.commatsci.2009.04.019Search in Google Scholar

[25] Yoon J., Ru C.Q., Mioduchowski A., Vibration of an embedded multiwall carbon nanotube, Comput. Sci. Technol, 2003, 63, 1533-1542.10.1016/S0266-3538(03)00058-7Search in Google Scholar

[26] Timoshenko S.P., Young D.H., Weaver W., Vibration Problems in Engineering, New York, Wiley, 1974.Search in Google Scholar

[27] Chaudhari V. K., Lal A., Nonlinear free vibration analysis of elastically supported nanotube reinforced composite beam in thermal environment, Procedia Engineering, 2016,144, 928 - 935.10.1016/j.proeng.2016.05.119Search in Google Scholar

[28] Fantuzzi N., Tornabene F., Bacciocchi M., Dimitri R., Free vibration analysis of arbitrarily shaped functionally graded Carbon Nanotube-reinforced plates, Composites Part B, 2016, doi: 10.1016/j.compositesb.2016.09.021.Search in Google Scholar

[29] Tornabene F., Fantuzzi N., Bacciocchi M., Linear static response of nanocomposite plates and shells reinforced by agglomerated carbon nanotubes, Composites Part B 2016, 1-28.10.1016/j.compositesb.2016.07.011Search in Google Scholar

[30] Shegokar N. L., Lal A., Stochastic finite element nonlinear free vibration analysis of piezoelectric functionally gradedmaterials beam subjected to thermo-piezoelectric loadings with material uncertainties, Meccanica, 2013, DOI 10.1007/s1101 9852-2.10.1007/s11012-013-9852-2Search in Google Scholar

[31] Vanmarcke E., Grigoriu M., Stochastic finite element analysis of simple beams, J Eng Mech, 1983, 109 (5), 1203-1214.10.1061/(ASCE)0733-9399(1983)109:5(1203)Search in Google Scholar

[32] Kaminski M., Stochastic second-order perturbation approach to the stress-based finite element method, Int J Solids Struct, 2001, 38, 3831-3852.10.1016/S0020-7683(00)00234-1Search in Google Scholar

[33] Locke J.E., Finite element large deflection random response of thermally buckled plates. J Sound Vib, 1993,160, 301-312.10.1006/jsvi.1993.1025Search in Google Scholar

[34] Onkar A.K., Yadav D., Forced nonlinear vibration of laminated composite plates with random material properties, Compos Struct, 2005,70, 334-342.10.1016/j.compstruct.2004.08.037Search in Google Scholar

[35] Kitipornchai S., Yang J., Liew K. M., Random vibration of functionally graded laminates in thermal environments, Comput Methods Appl Mech Eng, 2006,195, 1075-1095.10.1016/j.cma.2005.01.016Search in Google Scholar

[36] Shaker A., Abdelrahman W., Tawfik M., Sadek E., Stochastic finite element analysis of the free vibration of functionally graded material plates, Comput Mech, 2008, 41, 707-714.10.1007/s00466-007-0226-2Search in Google Scholar

[37] Lal A., Singh B.N., Stochastic nonlinear free vibration response of laminated composite plates resting on elastic foundation in thermal environments, Comput Mech, 2009, 44, 15-29.10.1007/s00466-008-0352-5Search in Google Scholar

[38] Lal A., Singh B.N., Kumar R., Natural frequency of laminated composite plate resting on an elastic foundation with uncertain system properties, Struct Eng Mech, 2007, 27, 199-222.10.12989/sem.2007.27.2.199Search in Google Scholar

[39] Jagtap K. R., Lal A., Singh B.N., Stochastic nonlinear free vibration analysis of elastically supported functionally graded materials plate with system randomness in thermal environment, Compos Struct, 2011, 93, 3185-3199.10.1016/j.compstruct.2011.06.010Search in Google Scholar

Received: 2016-10-11
Accepted: 2016-11-24
Published Online: 2017-5-3
Published in Print: 2017-1-26

© 2017

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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