Steel and Composite Structures

  • 제30권6호
  • /
  • Pages.493-516
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  • 2019
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  • 1229-9367(pISSN)
  • /
  • 1598-6233(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

Nonlinear vibration analysis of carbon nanotube reinforced composite plane structures

  • Rezaiee-Pajand, Mohammad (Department of Civil Engineering, Ferdowsi University of Mashhad) ;
  • Masoodi, Amir R. (Department of Civil Engineering, Ferdowsi University of Mashhad) ;
  • Rajabzadeh-Safaei, Niloofar (Department of Civil Engineering, Ferdowsi University of Mashhad)
  • 투고 : 2018.07.12
  • 심사 : 2019.03.08
  • 발행 : 2019.03.25
⟨ 이전 논문 다음 논문 ⟩

초록

This paper is dedicated to nonlinear static and free vibration analysis of Uniform Distributed Carbon Nanotube Reinforced Composite (UD-CNTRC) structures under in-plane loading. The authors have suggested an efficient six-node triangular element. Mixed Interpolation of Tensorial Components (MITC) approach is employed to alleviate the membrane locking phenomena. Moreover, the behavior of the well-known LST element is considerably improved by applying an additional linear interpolation on the strain fields. Based on the rule of mixture, the properties of CNTRC are obtained. In this study, only the uniform distributed CNTs are employed through the thickness direction of element. To achieve the natural frequencies and shape modes, the eigenvalue problem is also solved. Using Total Lagrangian Principles, large amplitude free vibration is considered based on the first normalized mode shape of structure. Different well-known plane problem benchmarks and some proposed ones are studied to validate the accuracy and capability of authors' formulations. In addition, the effects of length to the height ratio of beam, CNT's characteristics, support conditions and normalized amplitude parameter on the linear and nonlinear vibration parameters are investigated.

키워드

참고문헌

  1. Alijani, F. and Amabili, M. (2014), "Non-linear vibrations of shells: A literature review from 2003 to 2013", Int. J. Non-Linear Mech., 58, 233-257. https://doi.org/10.1016/j.ijnonlinmec.2013.09.012
  2. Allahkarami, F., Nikkhah-Bahrami, M. and Saryazdi, M.G. (2018), "Nonlinear forced vibration of FG-CNTs-reinforced curved microbeam based on strain gradient theory considering out-ofplane motion", Steel Compos. Struct., Int. J., 26(6), 673-691.
  3. Allaoui, A., Bai, S., Cheng, H.-M. and Bai, J. (2002), "Mechanical and electrical properties of a MWNT/epoxy composite", Compos. Sci. Technol., 62(15), 1993-1998. https://doi.org/10.1016/S0266-3538(02)00129-X
  4. Allman, D. (1984), "A compatible triangular element including vertex rotations for plane elasticity analysis", Comput. Struct., 19(1-2), 1-8. https://doi.org/10.1016/0045-7949(84)90197-4
  5. Amabili, M. (2017), "Nonlinear damping in large-amplitude vibrations: modelling and experiments", Nonlinear Dyn., 1-14.
  6. Ansari, R. and Hemmatnezhad, M. (2012), "Nonlinear finite element analysis for vibrations of double-walled carbon nanotubes", Nonlinear Dyn., 67(1), 373-383. https://doi.org/10.1007/s11071-011-9985-6
  7. Bergan, P. and Felippa, C.A. (1985), "A triangular membrane element with rotational degrees of freedom", Comput. Methods Appl. Mech. Eng., 50(1), 25-69. https://doi.org/10.1016/0045-7825(85)90113-6
  8. Bhashyam, G. and Prathap, G. (1980), "Galerkin finite element method for non-linear beam vibrations", J. Sound Vib., 72(2), 191-203. https://doi.org/10.1016/0022-460X(80)90652-5
  9. Billings, S.A. (2013), Nonlinear system identification: NARMAX methods in the time, frequency, and spatio-temporal domains, John Wiley & Sons.
  10. Brebbia, C.A. and Walker, S. (2016), Boundary Element Techniques in Engineering, Elsevier.
  11. Calio, I. and Greco, A. (2014), "Free vibrations of spatial Timoshenko arches", J. Sound Vibr., 333(19), 4543-4561. https://doi.org/10.1016/j.jsv.2014.04.019
  12. Chavan, S.G. and Lal, A. (2017), "Bending behavior of SWCNT reinforced composite plates", Steel Compos. Struct., Int. J., 24(5), 537-548.
  13. Chen, S., Cheung, Y. and Xing, H. (2001), "Nonlinear vibration of plane structures by finite element and incremental harmonic balance method", Nonlinear Dyn., 26(1), 87-104. https://doi.org/10.1023/A:1012982009727
  14. Ding, H., Zhu, M.-H. and Chen, L.-Q. (2018), "Nonlinear vibration isolation of a viscoelastic beam", Nonlinear Dyn., 92(2), 325-349. https://doi.org/10.1007/s11071-018-4058-8
  15. Dumir, P. and Bhaskar, A. (1988), "Some erroneous finite element formulations of non-linear vibrations of beams and plates", J. Sound Vib., 123(3), 517-527. https://doi.org/10.1016/S0022-460X(88)80167-6
  16. Erik, T.T. and Chou, T.W. (2002), "Aligned multi-walled carbon nanotube-reinforced composites: processing and mechanical characterization", J. Phys. D: Appl. Phys., 35(16), L77. https://doi.org/10.1088/0022-3727/35/16/103
  17. Fajman, P. (2002), "New triangular plane element with drilling degrees of freedom", J. Eng. Mech., 128(4), 413-418. https://doi.org/10.1061/(ASCE)0733-9399(2002)128:4(413)
  18. Felippa, C.A. (2003), "A study of optimal membrane triangles with drilling freedoms", Comput. Methods Appl. Mech. Eng., 192(16-18), 2125-2168. https://doi.org/10.1016/S0045-7825(03)00253-6
  19. Feng, Y. and Bert, C.W. (1992), "Application of the quadrature method to flexural vibration analysis of a geometrically nonlinear beam", Nonlinear Dyn., 3(1), 13-18. https://doi.org/10.1007/BF00045468
  20. Gupta, K. (1978), "Development of a finite dynamic element for free vibration analysis of two-dimensional structures", Int. J. Numer. Methods Eng., 12(8), 1311-1327. https://doi.org/10.1002/nme.1620120808
  21. Gupta, K.K. (1979), "Finite dynamic element formulation for a plane triangular element", Int. J. Numer. Methods Eng., 14(10), 1431-1448. https://doi.org/10.1002/nme.1620141002
  22. Heydari, M.M., Hafizi Bidgoli, A., Golshani, H.R., Beygipoor, G. and Davoodi, A. (2015), "Nonlinear bending analysis of functionally graded CNT-reinforced composite Mindlin polymeric temperature-dependent plate resting on orthotropic elastomeric medium using GDQM", Nonlinear Dyn., 79(2), 1425-1441. https://doi.org/10.1007/s11071-014-1751-0
  23. Hirwani, C.K. and Panda, S.K. (2018), "Numerical nonlinear frequency analysis of pre-damaged curved layered composite structure using higher-order finite element method", Int. J. Non-Linear Mech., 102, 14-24. https://doi.org/10.1016/j.ijnonlinmec.2018.03.005
  24. Hirwani, C., Mahapatra, T., Panda, S., Sahoo, S., Singh, V. and Patle, B. (2017), "Nonlinear free vibration analysis of laminated carbon/epoxy curved panels", Defence Sci. J., 67(2), 207. https://doi.org/10.14429/dsj.67.10072
  25. Iu, V., Cheung, Y. and Lau, S. (1985), "Non-linear vibration analysis of multilayer beams by incremental finite elements, Part I: Theory and numerical formulation", J. Sound Vib, 100(3), 359-372. https://doi.org/10.1016/0022-460X(85)90292-5
  26. Kolahdouzan, F., Arani, A.G. and Abdollahian, M. (2018), "Buckling and free vibration analysis of FG-CNTRC-micro sandwich plate", Steel Compos. Struct., Int. J., 26(3), 273-287.
  27. Kumar, P. and Srinivas, J. (2017a), "Free vibration, bending and buckling of a FG-CNT reinforced composite beam: Comparative analysis with hybrid laminated composite beam", Multidiscipl. Model. Mater. Struct., 13(4), 590-611. https://doi.org/10.1108/MMMS-05-2017-0032
  28. Kumar, P. and Srinivas, J. (2017b), "Vibration, buckling and bending behavior of functionally graded multi-walled carbon nanotube reinforced polymer composite plates using the layerwise formulation", Compos. Struct., 177, 158-170. https://doi.org/10.1016/j.compstruct.2017.06.055
  29. Leung, A. and Fung, T. (1989), "Non-linear steady state vibration of frames by finite element method", Int. J. Numer. Methods Eng., 28(7), 1599-1618. https://doi.org/10.1002/nme.1620280710
  30. Leung, A., Zhu, B., Zheng, J. and Yang, H. (2004), "Analytic trapezoidal Fourier p-element for vibrating plane problems", J. Sound Vib., 271(1-2), 67-81. https://doi.org/10.1016/S0022-460X(03)00263-3
  31. Lewandowski, R. (1994), "Non-linear free vibrations of beams by the finite element and continuation methods", J. Sound Vib., 170(5), 577-593. https://doi.org/10.1006/jsvi.1994.1088
  32. Liew, K., Rajendran, S. and Wang, J. (2006), "A quadratic plane triangular element immune to quadratic mesh distortions under quadratic displacement fields", Comput. Methods Appl. Mech. Eng., 195(9-12), 1207-1223. https://doi.org/10.1016/j.cma.2005.04.012
  33. Liew, K., Lei, Z. and Zhang, L. (2015), "Mechanical analysis of functionally graded carbon nanotube reinforced composites: a review", Compos. Struct., 120, 90-97. https://doi.org/10.1016/j.compstruct.2014.09.041
  34. Lin, F. and Xiang, Y. (2014a), "Numerical analysis on nonlinear free vibration of carbon nanotube reinforced composite beams", Int. J. Struct. Stabil. Dyn., 14(1), 1350056. https://doi.org/10.1142/S0219455413500569
  35. Lin, F. and Xiang, Y. (2014b), "Vibration of carbon nanotube reinforced composite beams based on the first and third order beam theories", Appl. Math. Model., 38(15-16), 3741-3754. https://doi.org/10.1016/j.apm.2014.02.008
  36. Liu, G.-R. and Gu, Y. (2001a), "A point interpolation method for two-dimensional solids", Int. J. Numer. Methods Eng., 50(4), 937-951. https://doi.org/10.1002/1097-0207(20010210)50:4<937::AID-NME62>3.0.CO;2-X
  37. Liu, G. and Gu, Y. (2001b), "A local radial point interpolation method (LRPIM) for free vibration analyses of 2-D solids", Journal of Sound and Vibration, 246(1), 29-46. https://doi.org/10.1006/jsvi.2000.3626
  38. Liu, G., Nguyen-Thoi, T. and Lam, K. (2009), "An edge-based smoothed finite element method (ES-FEM) for static, free and forced vibration analyses of solids", J. Sound Vib., 320(4-5), 1100-1130. https://doi.org/10.1016/j.jsv.2008.08.027
  39. Mahapatra, T.R., Mehar, K., Panda, S.K., Dewangan, S. and Dash, S. (2017), "Flexural strength of functionally graded nanotube reinforced sandwich spherical panel", IOP Conference Series: Materials Science and Engineering, IOP Publishing.
  40. Marur, S. and Prathap, G. (2005), "Non-linear beam vibration problems and simplifications in finite element models", Computat. Mech., 35(5), 352-360. https://doi.org/10.1007/s00466-004-0622-9
  41. Masoodi, A.R. and Arabi, E. (2018), "Geometrically nonlinear thermomechanical analysis of shell-like structures", J. Thermal Stress., 41(1), 37-53. https://doi.org/10.1080/01495739.2017.1360166
  42. Mayandi, K. and Jeyaraj, P. (2015), "Bending, buckling and free vibration characteristics of FG-CNT-reinforced polymer composite beam under non-uniform thermal load", Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials: Design and Applications, 229(1), 13-28. https://doi.org/10.1177/1464420713493720
  43. Mehar, K. and Panda, S.K. (2016a), "Free vibration and bending behaviour of CNT reinforced composite plate using different shear deformation theory", IOP Conference Series: Materials Science and Engineering, IOP Publishing.
  44. Mehar, K. and Panda, S.K. (2016b), "Geometrical nonlinear free vibration analysis of FG-CNT reinforced composite flat panel under uniform thermal field", Compos. Struct., 143, 336-346. https://doi.org/10.1016/j.compstruct.2016.02.038
  45. Mehar, K. and Panda, S.K. (2016c), "Nonlinear static behavior of FG-CNT reinforced composite flat panel under thermomechanical load", J. Aerosp. Eng., 30(3), 04016100. https://doi.org/10.1061/(ASCE)AS.1943-5525.0000706
  46. Mehar, K. and Panda, S.K. (2017), "Thermoelastic analysis of FGCNT reinforced shear deformable composite plate under various loadings", Int. J. Computat. Methods, 14(2), 1750019. https://doi.org/10.1142/S0219876217500190
  47. Mehar, K. and Panda, S.K. (2018a), "Elastic bending and stress analysis of carbon nanotube-reinforced composite plate: Experimental, numerical, and simulation", Adv. Polym. Technol., 37(6), 1643-1657. https://doi.org/10.1002/adv.21821
  48. Mehar, K. and Panda, S.K. (2018b), "Thermal free vibration behavior of FG-CNT reinforced sandwich curved panel using finite element method", Polym. Compos., 39(8), 2751-2764. https://doi.org/10.1002/pc.24266
  49. Mehar, K. and Panda, S.K. (2018c), "Nonlinear finite element solutions of thermoelastic flexural strength and stress values of temperature dependent graded CNT-reinforced sandwich shallow shell structure", Struct. Eng. Mech., Ing. J., 67(6), 565-578.
  50. Mehar, K., Panda, S.K., Dehengia, A. and Kar, V.R. (2016), "Vibration analysis of functionally graded carbon nanotube reinforced composite plate in thermal environment", J. Sandw. Struct. Mater., 18(2), 151-173. https://doi.org/10.1177/1099636215613324
  51. Mehar, K., Panda, S.K. and Mahapatra, T.R. (2017a), "Theoretical and experimental investigation of vibration characteristic of carbon nanotube reinforced polymer composite structure", Int. J. Mech. Sci., 133, 319-329. https://doi.org/10.1016/j.ijmecsci.2017.08.057
  52. Mehar, K., Panda, S.K. and Mahapatra, T.R. (2017b), "Thermoelastic nonlinear frequency analysis of CNT reinforced functionally graded sandwich structure", Eur. J. Mech.-A/Solids, 65, 384-396. https://doi.org/10.1016/j.euromechsol.2017.05.005
  53. Mehar, K., Panda, S.K. and Mahapatra, T.R. (2018a), "Nonlinear Frequency Responses of Functionally Graded Carbon Nanotube-Reinforced Sandwich Curved Panel Under Uniform Temperature Field", Int. J. Appl. Mech., 10(3), 1850028. https://doi.org/10.1142/S175882511850028X
  54. Mehar, K., Panda, S.K. and Mahapatra, T.R. (2018b), "Thermoelastic deflection responses of CNT reinforced sandwich shell structure using finite element method", Scientia Iranica, 25(5), 2722-2737.
  55. Mehar, K., Panda, S.K. and Patle, B.K. (2018c), "Stress, deflection, and frequency analysis of CNT reinforced graded sandwich plate under uniform and linear thermal environment: A finite element approach", Polym. Compos., 39(10), 3792-3809. https://doi.org/10.1002/pc.24409
  56. Mei, C. (1972), "Nonlinear vibration of beams by matrix displacement method", AIAA Journal, 10(3), 355-357. https://doi.org/10.2514/3.6595
  57. Mei, C. (1973), "Finite element displacement method for large amplitude free flexural vibrations of beams and plates", Comput. Struct., 3(1), 163-174. https://doi.org/10.1016/0045-7949(73)90081-3
  58. Mei, C. (1986), "Discussion of finite element formulations of nonlinear beam vibrations", Comput. Struct., 22(1), 83-85. https://doi.org/10.1016/0045-7949(86)90087-8
  59. Mir, M., Tahani, M. and Hassani, B. (2017), "Analytical prediction of Young's modulus of carbon nanotubes using a variational method", Appl. Math. Model., 45, 1031-1043. https://doi.org/10.1016/j.apm.2017.01.038
  60. Mohamed, N., Eltaher, M.A., Mohamed, S.A. and Seddek, L.F. (2018), "Numerical analysis of nonlinear free and forced vibrations of buckled curved beams resting on nonlinear elastic foundations", Int. J. Non-Linear Mech., 101, 157-173. https://doi.org/10.1016/j.ijnonlinmec.2018.02.014
  61. Qaisi, M. (1997), "A power series solution for the non-linear vibration of beams", J. Sound Vib., 199(4), 587-594. https://doi.org/10.1006/jsvi.1996.0696
  62. Rajendran, S. and Zhang, B. (2007), "A "FE-meshfree" QUAD4 element based on partition of unity", Comput. Methods Appl. Mech. Eng., 197(1-4), 128-147. https://doi.org/10.1016/j.cma.2007.07.010
  63. Rao, S.S. (2007), Vibration of Continuous Systems, John Wiley & Sons.
  64. Rao, G.V. and Raju, K.K. (2003), "Large amplitude free vibrations of beams-an energy approach", ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift fur Angewandte Mathematik und Mechanik, 83(7), 493-498. https://doi.org/10.1002/zamm.200310024
  65. Rehfield, L.W. (1973), "Nonlinear free vibrations of elastic structures", Int. J. Solids Struct., 9(5), 581-590. https://doi.org/10.1016/0020-7683(73)90071-1
  66. Rezaiee-Pajand, M. and Masoodi, A.R. (2016), "Exact natural frequencies and buckling load of functionally graded material tapered beam-columns considering semi-rigid connections", J. Vib. Control, 24(9), 1787-1808. https://doi.org/10.1177/1077546316668932
  67. Rezaiee-Pajand, M. and Rajabzadeh-Safaei, N. (2016a), "An explicit stiffness matrix for parabolic beam element", Latin Am. J. Solids Struct., 13, 1782-1801. https://doi.org/10.1590/1679-78252820
  68. Rezaiee Pajand, M. and Rajabzadeh Safaei, N. (2016b), "Static and dynamic analysis of circular beams using explicit stiffness matrix", Struct. Eng. Mech., Ing. J., 60(1), 111-130. https://doi.org/10.12989/sem.2016.60.1.111
  69. Rezaiee-Pajand, M. and Yaghoobi, M. (2014), "An efficient formulation for linear and geometric non-linear membrane elements", Latin Am. J. Solids Struct., 11(6), 1012-1035. https://doi.org/10.1590/S1679-78252014000600007
  70. Rezaiee-Pajand, M., Arabi, E. and Masoodi, A.R. (2018a), "A triangular shell element for geometrically nonlinear analysis", Acta Mechanica, 229(1), 323-342. https://doi.org/10.1007/s00707-017-1971-8
  71. Rezaiee-Pajand, M., Mokhtari, M. and Masoodi, A.R. (2018b), "Stability and free vibration analysis of tapered sandwich columns with functionally graded core and flexible connections", CEAS Aeronaut. J., 9(4), 629-648. https://doi.org/10.1007/s13272-018-0311-6
  72. Rezaiee-Pajand, M., Rajabzadeh-Safaei, N. and Hozhabrossadati, S.M. (2018d), "Three-dimensional deformations of a curved circular beam subjected to thermo-mechanical loading using green's function method", Int. J. Mech. Sci., 142-143, 163-175. https://doi.org/10.1016/j.ijmecsci.2018.04.045
  73. Rezaiee Pajand, M., Masoodi, A. and Arabi, E. (2018e), "Geometrically nonlinear analysis of FG doubly-curved and hyperbolical shells via laminated by new element", Steel Compos. Struct., Int. J., 28(3), 389-401.
  74. Rezaiee-Pajand, M., Arabi, E. and Masoodi, A.R. (2019), "Nonlinear analysis of FG-sandwich plates and shells", Aerosp. Sci. Technol. [In Press]
  75. Sadri, M., Younesian, D. and Esmailzadeh, E. (2016), "Nonlinear harmonic vibration and stability analysis of a cantilever beam carrying an intermediate lumped mass", Nonlinear Dyn., 84(3), 1667-1682. https://doi.org/10.1007/s11071-016-2596-5
  76. Salvetat, J.-P., Bonard, J.-M., Thomson, N.H., Kulik, A.J., Forro, L., Benoit, W. and Zuppiroli, L. (1999), "Mechanical properties of carbon nanotubes", Appl. Phys. A, 69(3), 255-260. https://doi.org/10.1007/s003390050999
  77. Sarma, B. and Varadan, T. (1982), "Certain discussions in the finite element formulation of nonlinear vibration analysis", Comput. Struct., 15(6), 643-646. https://doi.org/10.1016/S0045-7949(82)80004-7
  78. Sarma, B., Varadan, T. and Prathap, G. (1988), "On various formulations of large amplitude free vibrations of beams", Comput. Struct., 29(6), 959-966. https://doi.org/10.1016/0045-7949(88)90321-5
  79. Shafiei, H. and Setoodeh, A.R. (2017), "Nonlinear free vibration and post-buckling of FG-CNTRC beams on nonlinear foundation", Steel Compos. Struct., Int. J., 24, 65-77. https://doi.org/10.12989/scs.2017.24.1.065
  80. Shang, H.Y., Machado, R.D., Abdalla Filho, J.E. and Arndt, M. (2017), "Numerical analysis of plane stress free vibration in severely distorted mesh by Generalized Finite Element Method", Eur. J. Mech.-A/Solids, 62, 50-66. https://doi.org/10.1016/j.euromechsol.2016.11.006
  81. Shen, H.-S. (2009), "Nonlinear bending of functionally graded carbon nanotube-reinforced composite plates in thermal environments", Compos. Struct., 91(1), 9-19. https://doi.org/10.1016/j.compstruct.2009.04.026
  82. Shen, H.-S., Lin, F. and Xiang, Y. (2017), "Nonlinear vibration of functionally graded graphene-reinforced composite laminated beams resting on elastic foundations in thermal environments", Nonlinear Dyn., 90(2), 899-914. https://doi.org/10.1007/s11071-017-3701-0
  83. Singh, G., Rao, G.V. and Iyengar, N. (1990a), "Re-investigation of large-amplitude free vibrations of beams using finite elements", J. Sound Vib., 143(2), 351-355. https://doi.org/10.1016/0022-460X(90)90958-3
  84. Singh, G., Sharma, A. and Rao, G.V. (1990b), "Large-amplitude free vibrations of beams-a discussion on various formulations and assumptions", J. Sound Vib., 142(1), 77-85. https://doi.org/10.1016/0022-460X(90)90583-L
  85. Tahouneh, V. (2017), "Vibration and mode shape analysis of sandwich panel with MWCNTs FG-reinforcement core", Steel Compos. Struct., Int. J., 25(3), 347-360.
  86. Tahouneh, V. (2018), "3-D Vibration analysis of FGMWCNTs/Phenolic sandwich sectorial plates", Steel Compos. Struct., Int. J., 26(5), 649-662.
  87. Tian, R. and Yagawa, G. (2007), "Allman's triangle, rotational DOF and partition of unity", Int. J. Numer. Methods Eng., 69(4), 837-858. https://doi.org/10.1002/nme.1790
  88. Tu, Z.-c. and Ou-Yang, Z.-c. (2002), "Single-walled and multiwalled carbon nanotubes viewed as elastic tubes with the effective Young's moduli dependent on layer number", Phys. Review B, 65(23), 233407. https://doi.org/10.1103/PhysRevB.65.239901
  89. Valentini, L., Biagiotti, J., Kenny, J.M. and Lopez Manchado, M.A. (2003), "Physical and mechanical behavior of singlewalled carbon nanotube/polypropylene/ethylene-propylenediene rubber nanocomposites", J. Appl. Polym. Sci., 89(10), 2657-2663. https://doi.org/10.1002/app.12319
  90. Vodenitcharova, T. and Zhang, L. (2006), "Bending and local buckling of a nanocomposite beam reinforced by a singlewalled carbon nanotube", Int. J. Solids Struct., 43(10), 3006-3024. https://doi.org/10.1016/j.ijsolstr.2005.05.014
  91. Wan, H., Delale, F. and Shen, L. (2005), "Effect of CNT length and CNT-matrix interphase in carbon nanotube (CNT) reinforced composites", Mech. Res. Commun., 32(5), 481-489. https://doi.org/10.1016/j.mechrescom.2004.10.011
  92. Wattanasakulpong, N. and Ungbhakorn, V. (2013), "Analytical solutions for bending, buckling and vibration responses of carbon nanotube-reinforced composite beams resting on elastic foundation", Computat. Mater. Sci., 71, 201-208. https://doi.org/10.1016/j.commatsci.2013.01.028
  93. Weaver, Jr. W., Timoshenko, S.P. and Young, D.H. (1990). Vibration Problems in Engineering, John Wiley & Sons.
  94. Wielentejczyk, P. and Lewandowski, R. (2017), "Geometrically nonlinear, steady state vibration of viscoelastic beams", Int. J. Non-Linear Mech., 89, 177-186. https://doi.org/10.1016/j.ijnonlinmec.2016.12.012
  95. Yang, F., Sedaghati, R. and Esmailzadeh, E. (2008), "Free in-plane vibration of general curved beams using finite element method", J. Sound Vib., 318(4-5), 850-867. https://doi.org/10.1016/j.jsv.2008.04.041
  96. Yas, M. and Samadi, N. (2012), "Free vibrations and buckling analysis of carbon nanotube-reinforced composite Timoshenko beams on elastic foundation", Int. J. Pressure Vessels Piping, 98, 119-128. https://doi.org/10.1016/j.ijpvp.2012.07.012
  97. Yazdani Sarvestani, H. and Ghayoor, H. (2016), "Free vibration analysis of curved nanotube structures", Int. J. Non-Linear Mech., 86, 167-173. https://doi.org/10.1016/j.ijnonlinmec.2016.09.001
  98. Yu, Z., Guo, X. and Chu, F. (2010), "A multivariable hierarchical finite element for static and vibration analysis of beams", Finite Elem. Anal. Des., 46(8),625-631. https://doi.org/10.1016/j.finel.2010.03.002
  99. Zhang, L. (2017), "On the study of the effect of in-plane forces on the frequency parameters of CNT-reinforced composite skew plates", Compos. Struct., 160, 824-837. https://doi.org/10.1016/j.compstruct.2016.10.116
  100. Zhang, B. and Rajendran, S. (2008), "'FE-Meshfree'QUAD4 element for free-vibration analysis", Comput. Methods Appl. Mech. Eng., 197(45-48), 3595-3604. https://doi.org/10.1016/j.cma.2008.02.012
  101. Zhang, L., Song, Z. and Liew, K. (2015), "State-space Levy method for vibration analysis of FG-CNT composite plates subjected to in-plane loads based on higher-order shear deformation theory", Compos. Struct., 134, 989-1003. https://doi.org/10.1016/j.compstruct.2015.08.138
  102. Zhong, H. and Guo, Q. (2003), "Nonlinear vibration analysis of Timoshenko beams using the Differential Quadrature Method", Nonlinear Dyn., 32(3), 223-234. https://doi.org/10.1023/A:1024463711325
  103. Zhou, X., Huang, K. and Li, Z. (2018), "Geometrically nonlinear beam analysis of composite wind turbine blades based on quadrature element method", Int. J. Non-Linear Mech., 104, 87-99. https://doi.org/10.1016/j.ijnonlinmec.2018.05.007

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