Advances in nano research

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

An efficient C1 beam element via multi-scale material adaptable shape function

  • El-Ashmawy, A.M. (Department of Aircraft Mechanics, Military Technical College) ;
  • Xu, Yuanming (School of Aeronautic Science and Engineering, Beihang University)
  • Received : 2020.12.16
  • Accepted : 2022.07.03
  • Published : 2022.10.25
⟨ Previous Next ⟩

Abstract

Recently, promising structural technologies like multi-function, ultra-load bearing capacity and tailored structures have been put up for discussions. Finite Element (FE) modelling is probably the best-known option capable of treating these superior properties and multi-domain behavior structures. However, advanced materials such as Functionally Graded Material (FGM) and nanocomposites suffer from problems resulting from variable material properties, reinforcement aggregation and mesh generation. Motivated by these factors, this research proposes a unified shape function for FGM, nanocomposites, graded nanocomposites, in addition to traditional isotropic and orthotropic structural materials. It depends not only on element length but also on the beam's material properties and geometric characteristics. The systematic mathematical theory and FE formulations are based on the Timoshenko beam theory for beam structure. Furthermore, the introduced element achieves C1 degree of continuity. The model is proved to be convergent and free-off shear locking. Moreover, numerical results for static and free vibration analysis support the model accuracy and capabilities by validation with different references. The proposed technique overcomes the issue of continuous properties modelling of these promising materials without discarding older ones. Therefore, introduced benchmark improvements on the FE old concept could be extended to help the development of new software features to confront the rapid progress of structural materials.

Keywords

References

  1. Anandrao, K.S., Gupta, R.K., Ramachandran, P. and Rao, G.V. (2012), "Free vibration analysis of functionally graded beams", Defence Sci. J., 62(3), 139-146, https://doi.org/10.14429/dsj.62.1326.
  2. Ansari, R., Shojaei, M.F., Mohammadi, V., Gholami, R. and Sadeghi, F. (2014), "Nonlinear forced vibration analysis of functionally graded carbon nanotube-reinforced composite timoshenko beams", Compos. Struct., 113, 316-327, https://doi.org/10.1016/j.compstruct.2014.03.015.
  3. Aydogdu, M. (2014), "On the vibration of aligned carbon nanotube reinforced composite beams", Adv. Nano Res., 2(4), 199-210. http://doi.org/10.12989/anr.2014.2.4.199.
  4. Baiges, J., Codina, R., Castanar, I. and Castillo, E. (2020), "A finite element reduced-order model based on adaptive mesh refinement and artificial neural networks", Int. J. Numer. Meth. Eng., 121(4), 588-601, https://doi.org/10.1002/nme.6235.
  5. Barati, M.R. and Shahverdi, H. (2020), "Finite element forced vibration analysis of refined shear deformable nanocomposite graphene platelet-reinforced beams", J. Brazil. Soc. Mech. Sci. Eng., 42(1), 33, https://doi.org/10.1007/s40430-019-2118-8.
  6. Bazoune, A., Khulief, Y. and Stephen, N. (2003), "Shape functions of three-dimensional Timoshenko beam element", J. Sound Vib., 259(2), 473-480, https://doi.org/10.1006/jsvi.2002.5122.
  7. Berghouti, H., Adda Bedia, E., Benkhedda, A. and Tounsi, A. (2019), "Vibration analysis of nonlocal porous nanobeams made of functionally graded material", Adv. Nano Res., 7(5), 351-364, https://doi.org/10.12989/ANR.2019.7.5.351.
  8. Bogue, R. (2014), "Smart materials: A review of capabilities and applications", Assembly Auto., 34(3). https://doi.org/10.1108/AA-10-2013-094.
  9. Caraballo, S. (2011), "Thermo-mechanical beam element for analyzing stresses in functionally graded materials", Ph.D. Thesis, University of South Florida, Florida, U.S.A.
  10. Chakraborty, A., Gopalakrishnan, S. and Reddy, J. (2003), "A new beam finite element for the analysis of functionally graded materials", Int. J. Mech. Sci., 45(3), 519-539. https://doi.org/10.1016/S0020-7403(03)00058-4.
  11. Chakraborty, A., Mahapatra, D.R. and Gopalakrishnan, S. (2002), "Finite element analysis of free vibration and wave propagation in asymmetric composite beams with structural discontinuities", Compos. Struct., 55(1), 23-36. https://doi.org/10.1016/S0263-8223(01)00130-1.
  12. Chandrashekhara, K. and Bangera, K.M. (1992), "Free vibration of composite beams using a refined shear flexible beam element", Comput. Struct., 43(4), 719-727. https://doi.org/10.1016/0045-7949(92)90514-Z.
  13. Chandrashekhara, K., Krishnamurthy, K. and Roy, S. (1990), "Free vibration of composite beams including rotary inertia and shear deformation", Compos. Struct., 14(4), 269-279. https://doi.org/10.1016/0263-8223(90)90010-C.
  14. Cook, R.D. (2007), Concepts and Applications of Finite Element Analysis, John wiley & sons.
  15. Dabbagh, A., Rastgoo, A. and Ebrahimi, F. (2020), "Post-buckling analysis of imperfect multi-scale hybrid nanocomposite beams rested on a nonlinear stiff substrate", Eng. Comput., 1-14. ttps://doi.org/10.1007/s00366-020-01064-1.
  16. Dabbagh, A., Rastgoo, A. and Ebrahimi, F. (2021), "Thermal buckling analysis of agglomerated multiscale hybrid nanocomposites via a refined beam theory", Mech. Based Des. Struct., 49(3), 403-429. https://doi.org/10.1080/15397734.2019.1692666.
  17. Ebrahimi, F. and Dabbagh, A. (2018a), "Effect of humid-thermal environment on wave dispersion characteristics of singlelayered graphene sheets", Appl. Phys. A, 124(4), 1-11. https://doi.org/10.1007/s00339-018-1734-y.
  18. Ebrahimi, F. and Dabbagh, A. (2018b), "On wave dispersion characteristics of double-layered graphene sheets in thermal environments", J. Electromagnet. Waves., 32(15), 1869-1888. https://doi.org/10.1080/09205071.2017.1417918.
  19. Ebrahimi, F. and Dabbagh, A. (2018c), "Wave dispersion characteristics of embedded graphene platelets-reinforced composite microplates", Eur. Phys. J. Plus, 133(4), 1-13. https://doi.org/10.1140/epjp/i2018-11956-5.
  20. Ebrahimi, F. and Dabbagh, A. (2019a), "Application of the nonlocal strain gradient elasticity on the wave dispersion behaviors of inhomogeneous nanosize beams", Eur. Phys. J. Plus, 134(3), 112. https://doi.org/10.1140/epjp/i2019-12464-x.
  21. Ebrahimi, F. and Dabbagh, A. (2019b), "Vibration analysis of graphene oxide powder-/carbon fiber-reinforced multi-scale porous nanocomposite beams: A finite-element study", Eur. Phys. J. Plus, 134(5), 1-15. https://doi.org/10.1140/epjp/i2019-12594-1.
  22. Ebrahimi, F. and Dabbagh, A. (2019c), "Wave dispersion characteristics of heterogeneous nanoscale beams via a novel porosity-based homogenization scheme", Eur. Phys. J. Plus, 134(4), 1-8. https://doi.org/10.1140/epjp/i2019-12510-9.
  23. Ebrahimi, F. and Dabbagh, A. (2019d), Wave Propagation Analysis of Smart Nanostructures, CRC Press. https://doi.org/10.1201/9780429279225.
  24. Ebrahimi, F. and Dabbagh, A. (2020a), "A brief review on the influences of nanotubes' entanglement and waviness on the mechanical behaviors of cntr polymer nanocomposites", J. Comput. Appl. Mech., 51(1), 247-252. https://doi.org/10.22059/jcamech.2020.304476.517.
  25. Ebrahimi, F. and Dabbagh, A. (2020b), Mechanics of Nanocomposites: Homogenization and Analysis, CRC Press. https://doi.org/10.1201/9780429316791.
  26. Ebrahimi, F. and Dabbagh, A. (2020c), "Vibration analysis of multi-scale hybrid nanocomposite shells by considering nanofillers' aggregation", Waves Random Complex Med., 1-19. https://doi.org/10.1080/17455030.2020.1810363.
  27. Ebrahimi, F. and Dabbagh, A. (2020d), "Viscoelastic wave propagation analysis of axially motivated double layered graphene sheets via nonlocal strain gradient theory", Waves Random Complex Med., 30(1), 157-176. https://doi.org/10.1080/17455030.2018.1490505.
  28. Ebrahimi, F. and Dabbagh, A. (2021), "An analytical solution for static stability of multiscale hybrid nanocomposite plates", Eng. Comput., 37(1), 545-559. https://doi.org/10.1007/s00366-019-00840-y.
  29. Ebrahimi, F., Dabbagh, A., Rabczuk, T. and Tornabene, F. (2019a), "Analysis of propagation characteristics of elastic waves in heterogeneous nanobeams employing a new two-step porositydependent homogenization scheme", Adv. Nano Res., 7(2), 135-143. https://doi.org/10.12989/anr.2019.7.2.135.
  30. Ebrahimi, F., Dabbagh, A. and Rastgoo, A. (2019b), "Vibration analysis of porous metal foam shells rested on an elastic substrate", J. Strain Anal. Eng., 54(3), 199-208. https://doi.org/10.1177/0309324719852555.
  31. Ebrahimi, F., Dabbagh, A. and Rastgoo, A. (2020), "Static stability analysis of multi-scale hybrid agglomerated nanocomposite shells", Mech. Based Des. Struct., 1-17. https://doi.org/10.1080/15397734.2020.1848585.
  32. Ebrahimi, F., Dabbagh, A. and Rastgoo, A. (2021a), "Free vibration analysis of multi-scale hybrid nanocomposite plates with agglomerated nanoparticles", Mech. Based Des. Struct., 49(4), 487-510. https://doi.org/10.1080/15397734.2019.1692665.
  33. Ebrahimi, F., Dabbagh, A. and Taheri, M. (2021b), "Vibration analysis of porous metal foam plates rested on viscoelastic substrate", Eng. Comput., 37(4), 3727-3739. https://doi.org/10.1007/s00366-020-01031-w.
  34. Ebrahimi, F. and Habibi, S. (2017), "Low-velocity impact response of laminated fg-cnt reinforced composite plates in thermal environment", Adv. Nano Res., 5(2), 69-97. http://doi.org/10.12989/anr.2017.5.2.069.
  35. Ebrahimi, F., Haghi, P. and Dabbagh, A. (2018), "Analytical wave dispersion modeling in advanced piezoelectric double-layered nanobeam systems", Struct. Eng. Mech., Int. J, 67(2), 175-183. https://doi.org/10.12989/sem.2018.67.2.175.
  36. Ebrahimi, F., Khosravi, K. and Dabbagh, A. (2021c), "Wave dispersion in viscoelastic fg nanobeams via a novel spatialtemporal nonlocal strain gradient framework", Waves Random Complex Med., pages 1-23. https://doi.org/10.1080/17455030.2021.1970282.
  37. Ebrahimi, F., Nopour, R. and Dabbagh, A. (2021d), "Effect of viscoelastic properties of polymer and wavy shape of the cnts on the vibrational behaviors of cnt/glass fiber/polymer plates", Eng. Comput., 1-14. https://doi.org/10.1007/s00366-021-01387-7.
  38. Ebrahimi, F., Seyfi, A. and Dabbagh, A. (2019c), "Dispersion of waves in fg porous nanoscale plates based on nsgt in thermal environment", Adv. Nano Res., 7(5), 325-335. https://doi.org/10.12989/anr.2019.7.5.325.
  39. Ebrahimi, F., Seyfi, A. and Dabbagh, A. (2019d), "A novel porosity-dependent homogenization procedure for wave dispersion in nonlocal strain gradient inhomogeneous nanobeams", Eur. Phys. J. Plus, 134(5), 1-11. https://doi.org/10.1140/epjp/i2019-12547-8.
  40. Ebrahimi, F., Seyfi, A. and Dabbagh, A. (2021e), "The effects of thermal loadings on wave propagation analysis of multi-scale hybrid composite beams", Waves Random Complex Med., 1-24. https://doi.org/10.1080/17455030.2021.1956015.
  41. Eisenberger, M. (1994), "Derivation of shape functions for an exact 4-dof timoshenko beam element", Commun. Numer. Meth. Eng., 10(9), 673-681. https://doi.org/10.1002/cnm.1640100902.
  42. Eisenberger, M. (2003), "An exact high order beam element", Comput. Struct., 81(3), 147-152. https://doi.org/10.1016/S0045-7949(02)00438-8.
  43. El-Ashmawy, A., Kamel, M. and Elshafei, M. (2016a), "A generalized non-conventional finite element model for analysis of isotropic, orthotropic and function graded beams", ERJFaculty Eng. Shoubra, 28(5), 63-83.
  44. El-Ashmawy, A., Kamel, M. and Elshafei, M.A. (2016b), "Thermo-mechanical analysis of axially and transversally function graded beam", Compos. Part B Eng., 102, 134-149. https://doi.org/10.1016/j.compositesb.2016.07.015.
  45. El-Ashmawy, A. and Xu, Y. (2020), "Longitudinal modeling and properties tailoring of functionally graded carbon nanotube reinforced composite beams: A novel approach", Appl. Math. Modell., 88, 161-174. https://doi.org/10.1016/j.apm.2020.06.043.
  46. El-Ashmawy, A., Xu, Y. and Aziz, L. (2021), "Mechanical properties improvement of bidirectional functionally graded laminated mwcnt reinforced composite beams using an integrated tailoring-optimization approach", Micropor. Mesopor. Mater., 314, 110875. https://doi.org/10.1016/j.micromeso.2021.110875.
  47. El-Ashmawy, A.M. and Xu, Y. (2021), "Combined effect of carbon nanotubes distribution and orientation on functionally graded nanocomposite beams using finite element analysis", Mater. Res. Exp., 8(1), 015012. https://doi.org/10.1088/2053-1591/abc773.
  48. Elshafei, M.A. (2013), "Fe modeling and analysis of isotropic and orthotropic beams using first order shear deformation theory", Mater. Sci. Appl., 4(1), 26. https://doi.org/10.4236/msa.2013.41010.
  49. Emdadi, M., Mohammadimehr, M. and Navi, B.R. (2019), "Free vibration of an annular sandwich plate with cntrc facesheets and fg porous cores using ritz method", Adv. Nano Res., 7(2), 109-123. http://doi.org/10.12989/anr.2019.7.2.109.
  50. Farzad Ebrahimi, Ali Dabbagh, A.R.T.R. (2020), "Agglomeration effects on static stability analysis of multiscale hybrid nanocomposite plates", Comput. Mater. Continua, 63(1), 41-64. https://doi.org/10.32604/cmc.2020.07947.
  51. Feng, D.C. and Wu, J.Y. (2020), "Improved displacement-based timoshenko beam element with enhanced strains", J. Struct. Eng., 146(3), 04019221. https://doi.org/10.1061/(ASCE)ST.1943-541X.0002549.
  52. Filippi, M., Carrera, E. and Zenkour, A. (2015), "Static analyses of fgm beams by various theories and finite elements", Compos. Part B Eng., 72, 1-9. https://doi.org/10.1016/j.compositesb.2014.12.004.
  53. Friedman, Z. and Kosmatka, J. B. (1993), "An improved two-node timoshenko beam finite element", Comput. Struct., 47(3), 473-481. https://doi.org/10.1016/0045- 949(93)90243-7.
  54. Frikha, A., Hajlaoui, A., Wali, M. and Dammak, F. (2016), "A new higher order c0 mixed beam element for fgm beams analysis", Compos. Part B Eng., 106, 181-189. https://doi.org/10.1016/j.compositesb.2016.09.024.
  55. Gibson, R.F. (2016), Principles of Composite Material Mechanics, CRC press.
  56. Han, Y. and Elliott, J. (2007), "Molecular dynamics simulations of the elastic properties of polymer/carbon nanotube composites", Comput. Mater. Sci., 39(2), 315-323. https://doi.org/10.1016/j.commatsci.2006.06.011.
  57. Haskul, M. (2020), "Elastic state of functionally graded curved beam on the plane stress state subject to thermal load", Mech. Based Des. Struct., 48(6), 739-754. https://doi.org/10.1080/15397734.2019.1660890.
  58. Heshmati, M. and Yas, M. (2013), "Free vibration analysis of functionally graded cnt-reinforced nanocomposite beam using eshelby-mori-tanaka approach", J. Mech. Sci. Technol., 27(11), 3403-3408. https://doi.org/10.1007/s12206-013-0862-8.
  59. Hocaoglu, M. and Karagulle, H. (2020), "Effect of carbon nanotube reinforcement on the natural frequencies and damping ratios of nanocomposite beams", Mater. Res. Express, 7(2), 25021. https://doi.org/10.1088/2053-1591/ab721a.
  60. Hou, H. and He, G. (2018), "Static and dynamic analysis of twolayer timoshenko composite beams by weak-form quadrature element method", Appl. Math. Modell., 55, 466-483. https://doi.org/10.1016/j.apm.2017.11.007.
  61. Huang, Y. and Ouyang, Z.Y. (2020), "Exact solution for bending analysis of two-directional functionally graded timoshenko beams", Arch. Appl. Mech., 1-19. https://doi.org/10.1007/s00419-019-01655-5.
  62. Iwai, R. and Kobayashi, N. (2003), "A new flexible multibody beam element based on the absolute nodal coordinate formulation using the global shape function and the analytical mode shape function", Nonlinear Dyn., 34(1-2), 207-232. https://doi.org/10.1023/B:NODY.0000014560.78333.76.
  63. Katili, I., Syahril, T. and Katili, A.M. (2020), "Static and free vibration analysis of fgm beam based on unified and integrated of timoshenko's theory", Compos. Struct., 112130. https://doi.org/10.1016/j.compstruct.2020.112130.
  64. Ke, L.L., Yang, J. and Kitipornchai, S. (2013), "Dynamic stability of functionally graded carbon nanotube reinforced composite beams", Mech. Adv. Mater. Struct., 20(1), 28-37. https://doi.org/10.1080/15376494.2011.581412.
  65. Kennedy, G.J., Hansen, J.S. and Martins, J.R. (2011), "A timoshenko beam theory with pressure corrections for layered orthotropic beams", Int. J. Solid Struct., 48(16-17), 2373-2382. https://doi.org/10.1016/j.ijsolstr.2011.04.009.
  66. Khan, M.A., Yasin, M., Beg, M.S. and Khan, A. (2020), "Free and forced vibration analysis of functionally graded beams using finite element model based on refined third-order theory", Emerging Trends Mech. Engi., 603-612. https://doi.org/10.1007/978-981-32-9931-3_58.
  67. Khdeir, A. and Reddy, J. (1997), "An exact solution for the bending of thin and thick cross-ply laminated beams", Compos. Struct., 37(2), 195-203. https://doi.org/10.1016/S0263-8223(97)80012-8.
  68. Kocaturk, T. and Simsek, M. (2005), "Free vibration analysis of timoshenko beams under various boundary conditions", Sigma, 1, 30-44.
  69. Kumar, P. and Srinivas, J. (2017), "Free vibration, bending and buckling of a fg-cnt reinforced composite beam", Multidiscip. Model. Mater. Struct., 13(4), 590-611. https://doi.org/10.1108/MMMS-05-2017-0032.
  70. Lee, J. and Schultz, W. (2004), "Eigenvalue analysis of timoshenko beams and axisymmetric mindlin plates by the pseudospectral method", J. Sound Vib., 269(3-5), 609-621. https://doi.org/10.1016/S0022- 0X(03)00047-6.
  71. Lees, A. and Thomas, D. (1982), "Unified timoshenko beam finite element", J. Sound Vib., 80(3), 355-366. https://doi.org/10.1016/0022-460X(82)90276-0.
  72. Lezgy-Nazargah, M. (2020), "A four-variable global-local shear deformation theory for the analysis of deep curved laminated composite beams", Acta Mechanica, pages 1-32. https://doi.org/10.1007/s00707-019-02593-7.
  73. Li, N., Huang, S., Zhang, G., Qin, R., Liu, W., Xiong, H., Shi, G. and Blackburn, J. (2019), "Progress in additive manufacturing on new materials: A review", J. Mater. Sci. Technol., 35(2), 242-269. https://doi.org/10.1016/j.jmst.2018.09.002.
  74. Li, N., Li, Z. and Xie, L. (2013), "A fiber-section model based timoshenko beam element using shear-bending interdependent shape function", Earthq. Eng. Eng. Vib., 12(3), 421-432. https://doi.org/10.1007/s11803-013-0183-z.
  75. Li, W. and Han, B. (2018), "Research and application of functionally gradient materials", IOP Conference Series Mater. Sci. Eng., 22065. https://doi.org/10.1088/1757-899X/394/2/022065.
  76. Li, X.F. (2008), "A unified approach for analyzing static and dynamic behaviors of functionally graded timoshenko and euler-bernoulli beams", J. Sound Vib., 318(4-5), 1210-1229. https://doi.org/10.1016/j.jsv.2008.04.056.
  77. Liu, H., Wu, H. and Lyu, Z. (2020), "Nonlinear resonance of fg multilayer beam-type nanocomposites: Effects of graphene nanoplatelet-reinforcement and geometric imperfection", Aerosp. Sci. Technol., 98, 105702. https://doi.org/10.1016/j.ast.2020.105702.
  78. Mansouri, L., Arezki, D., Khatir, S., Behtani, A., Tiachacht, S., Slimani, M. and Wahab, M.A. (2020), "A comparative study of the behavior of glass fiber-reinforced polyester composite laminates under static loading", Proceedings of the 13th International Conference on Damage Assessment of Structures, 875-886. https://doi.org/10.1007/978-981-13-8331-1_70.
  79. Minghini, F., Tullini, N. and Laudiero, F. (2007), "Locking-free finite elements for shear deformable orthotropic thin-walled beams", Int. J. Numer. Meth. Eng., 72(7), 808-834. https://doi.org/10.1002/nme.2034.
  80. Mojiri, H. and Salami, S. J. (2020), "Free vibration and dynamic transient response of functionally graded composite beams reinforced with graphene nanoplatelets (gpls) resting on elastic foundation in thermal environment", Mech. Based Des. Struct., 1-21. https://doi.org/10.1080/15397734.2020.1766492.
  81. Nabi, S. M. and Ganesan, N. (1994), "A generalized element for the free vibration analysis of composite beams", Comput. Struct., 51(5), 607-610. https://doi.org/10.1016/0045-7949(94)90068-X.
  82. Panchore, V., Ganguli, R. and Omkar, S. (2015), "Meshless local petrov-galerkin method for rotating timoshenko beam: A locking-free shape function formulation", Comput. Model. Eng. Sci., 108(4), 215-237.
  83. Ping, L. (2005), "Generation of hermitian shape functions for straight beam element using constructing function method", J. Struct. Eng., 31(4), 243-248.
  84. Rajasekaran, S. and Khaniki, H.B. (2018), "Free vibration analysis of bi-directional functionally graded single/multi-cracked beams", Int. J. Mech. Sci., 144, 341-356. https://doi.org/10.1016/j.ijmecsci.2018.06.004.
  85. Ranjbar, M. and Feli, S. (2018), "Low velocity impact analysis of an axially functionally graded carbon nanotube reinforced cantilever beam", Polym. Compos., 39(S2), E969-E983. https://doi.org/10.1002/pc.24386.
  86. Reddy, J. (1997), "On locking-free shear deformable beam finite elements", Comput. Meth. Appl. Mech. Eng., 149(1-4), 113-132. https://doi.org/10.1016/S0045-7825(97)00075-3.
  87. Reddy, J. and Chin, C. (1998), "Thermomechanical analysis of functionally graded cylinders and plates", J. Therm. Stress., 21(6), 593-626. https://doi.org/10.1080/01495739808956165.
  88. Reddy, J.N. (2003), Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, CRC press.
  89. Sahin, S., Karahan, E., Kilic, B. and Ozdemir, O. (2019), "Finite element method for vibration analysis of timoshenko beams", Proceedings in 2019 9th International Conference on Recent Advances in Space Technologies (RAST), 673-679. https://doi.org/10.1109/RAST.2019.8767827.
  90. Selvaraj, R. and Ramamoorthy, M. (2020), "Experimental and finite element vibration analysis of cnt reinforced mr elastomer sandwich beam", Mech. Based Des. Struct., 1-13. https://doi.org/10.1080/15397734.2020.1778487.
  91. Sevilla, R., Fernandez-Mendez, S. and Huerta, A. (2011), "Nurbsenhanced finite element method (nefem)", Arch. Comput. Meth. Eng., 18(4), 441. https://doi.org/10.1007/s11831-011-9066-5.
  92. 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.
  93. Shen, H.S. and Xiang, Y. (2013), "Nonlinear analysis of nanotubereinforced composite beams resting on elastic foundations in thermal environments", Eng. Struct., 56, 698-708. https://doi.org/10.1016/j.engstruct.2013.06.002.
  94. Shen, H.S. and Zhang, C.L. (2010), "Thermal buckling and postbuckling behavior of functionally graded carbon nanotubereinforced composite plates", Mater. Des., 31(7), 3403-3411. https://doi.org/10.1016/j.matdes.2010.01.048.
  95. Simsek, M. (2009), "Static analysis of a functionally graded beam under a uniformly distributed load by ritz method", Int. J. Eng. Appl. Sci., 1(3), 1-11.
  96. Simsek, M. and Kocaturk, T. (2007), "Free vibration analysis of beams by using a third-order shear deformation theory", Sadhana, 32(3), 167-179. https://doi.org/10.1007/s12046-007-0015-9.
  97. Sinha, G.P. and Kumar, B. (2020), "Review on vibration analysis of functionally graded material structural components with cracks", J. Vib. Eng. Technol., 1-27. https://doi.org/10.1007/s42417-020-00208-3.
  98. Soni, S.K., Thomas, B. and Kar, V.R. (2020), "A comprehensive review on cnts and cnt-reinforced composites: Syntheses, characteristics and applications", Mater. Today Commun., 101546. https://doi.org/10.1016/j.mtcomm.2020.101546.
  99. Suresh, S. and Mortensen, A. (1998), Fundamentals of Functionally Graded Materials, The Institute of Materials.
  100. Takahashi, Y. (2006), "Study on the shape function of the 2-dimensional beam element formulated by absolute nodal coordinates", Proceedings of Dynamics and Design Conference 2006 of the Japan Society of Mechanical Engineers, 70.
  101. Talo, M., Carboni, B., Formica, G., Lanzara, G., Snyder, M. and Lacarbonara, W. (2020), Nonlinear Dynamic Response of Nanocomposite Cantilever Beams, in New Trends in Nonlinear Dynamics, 49-57. https://doi.org/10.1007/978-3-030-34724-6_6.
  102. Tayeb, T.S., Zidour, M., Bensattalah, T., Heireche, H., Benahmed, A. and Bedia, E. (2020), "Mechanical buckling of fg-cnts reinforced composite plate with parabolic distribution using hamilton's energy principle", Adv. Nano Res., 8(2), 135-148. https://doi.org/10.12989/anr.2020.8.2.135.
  103. Thostenson, E.T., Ren, Z. and Chou, T.W. (2001), "Advances in the science and technology of carbon nanotubes and their composites: A review", Compos. Sci. Technol., 61(13), 1899-1912. https://doi.org/10.1016/S0266-3538(01)00094-X.
  104. Tudjono, S., Han, A., Nguyen, D.K., Kiryu, S. and Gan, B.S. (2017), "Exact shape functions for timoshenko beam element", J. Comput. Eng., 19(3), 12-20. https://doi.org/10.9790/0661-1903041220.
  105. Vo-Duy, T., Ho-Huu, V. and Nguyen-Thoi, T. (2019), "Free vibration analysis of laminated fg-cnt reinforced composite beams using finite element method", Front. Struct. Civil Eng., 13(2), 324- 336. https://doi.org/10.1007/s11709-018-0466-6.
  106. Wang, Y., Xie, K. and Fu, T. (2020), "Vibration analysis of functionally graded graphene oxide-reinforced composite beams using a new ritz-solution shape function", J. Brazil. Soc. Mech. Sci. Eng., 42(4), 1-14. https://doi.org/10.1007/s40430-020-2258-x.
  107. Wattanasakulpong, N. and Ungbhakorn, V. (2013), "Analytical solutions for bending, buckling and vibration responses of carbon nanotube-reinforced composite beams resting on elastic foundation", Comput. Mater. Sci., 71, 201-208. https://doi.org/10.1016/j.commatsci.2013.01.028.
  108. Wu, H., Yang, J. and Kitipornchai, S. (2016), "Nonlinear vibration of functionally graded carbon nanotubereinforced composite beams with geometric imperfections", Compos. Part B Eng., 90, 86-96. https://doi.org/10.1016/j.compositesb.2015.12.007.
  109. Wu, Z., Zhang, Y., Yao, G. and Yang, Z. (2019), "Nonlinear primary and super-harmonic resonances of functionally graded carbon nanotube reinforced composite beams", Int. J. Mech. Sci., 153, 321-340. https://doi.org/10.1016/j.ijmecsci.2019.02.015.
  110. Yarali, E., Farajzadeh, M.A., Noroozi, R., Dabbagh, A., Khoshgoftar, M.J. and Mirzaali, M.J. (2020), "Magnetorheological elastomer composites: Modeling and dynamic finite element analysis", Compos. Struct., 254, 112881. https://doi.org/10.1016/j.compstruct.2020.112881.
  111. Yas, M. and Heshmati, M. (2012), "Dynamic analysis of functionally graded nanocomposite beams reinforced by randomly oriented carbon nanotube under the action of moving load", Appl. Math. Modell., 36(4), 1371-1394. https://doi.org/10.1016/j.apm.2011.08.037.
  112. Yas, M. and Samadi, N. (2012), "Free vibrations and buckling analysis of carbon nanotube-reinforced composite timoshenko beams on elastic foundation", Int. J. Press. Vessels Piping, 98, 119-128. https://doi.org/10.1016/j.ijpvp.2012.07.012.
  113. Zerrouki, R., Karas, A. and Zidour, M. (2020), "Critical buckling analyses of nonlinear fgcnt reinforced nano-composite beam", Adv. Nano Res., 9(3), 211-220. https://doi.org/10.12989/anr.2020.9.3.211.
  114. Zhou, T., Chazot, J.-D., Perrey-Debain, E. and Cheng, L. (2019), "Performance of the partition of unity finite element method for the modeling of timoshenko beams", Comput. Struct., 222, 148-154. https://doi.org/10.1016/j.compstruc.2019.07.004.
  115. Zhou, Z., Chen, M. and Xie, K. (2020), "Nurbs-based free vibration analysis of axially functionally graded tapered timoshenko curved beams", Appl. Math. Mech., 1-20. https://doi.org/10.1007/s11012-013-9847-z.