Steel and Composite Structures

  • 제44권1호
  • /
  • Pages.81-89
  • /
  • 2022
  • /
  • 1229-9367(pISSN)
  • /
  • 1598-6233(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

The effect of transverse shear deformation on the post-buckling behavior of functionally graded beams

  • Meksi, Ali (Laboratoire d'Etude des Structures et de Mecanique des Materiaux, Departement de Genie Civil, Faculte des Sciences et de la Technologie, Universite Mustapha Stambouli) ;
  • Youzera, Hadj (Laboratoire d'Etude des Structures et de Mecanique des Materiaux, Departement de Genie Civil, Faculte des Sciences et de la Technologie, Universite Mustapha Stambouli) ;
  • Sadoun, Mohamed (Laboratoire d'Etude des Structures et de Mecanique des Materiaux, Departement de Genie Civil, Faculte des Sciences et de la Technologie, Universite Mustapha Stambouli) ;
  • Abbache, Ali (Laboratoire d'Etude des Structures et de Mecanique des Materiaux, Departement de Genie Civil, Faculte des Sciences et de la Technologie, Universite Mustapha Stambouli) ;
  • Meftah, Sid Ahmed (Laboratoire de Modelisation et Simulation Multi-echelle, Universite de Sidi Bel Abbes) ;
  • Tounsi, Abdelouahed (YFL (Yonsei Frontier Lab), Yonsei University) ;
  • Hussain, Muzamal (Department of Mathematics, Govt. College University Faisalabad)
  • 투고 : 2021.11.03
  • 심사 : 2022.05.17
  • 발행 : 2022.07.10
⟨ 이전 논문 다음 논문 ⟩

초록

The purposes of the present work it to study the effect of shear deformation on the static post-buckling response of simply supported functionally graded (FGM) axisymmetric beams based on classical, first-order, and higher-order shear deformation theories. The behavior of postbuckling is introduced based on geometric nonlinearity. The material properties of functionally graded materials (FGM) are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. The equations of motion and the boundary conditions derived using Hamilton's principle. This article compares and addresses the efficiency, the applicability, and the limits of classical models, higher order models (CLT, FSDT, and HSDT) for the static post-buckling response of an asymmetrically simply supported FGM beam. The amplitude of the static post-buckling obtained a solving the nonlinear governing equations. The results showing the variation of the maximum post-buckling amplitude with the applied axial load presented, for different theory and different parameters of material and geometry. In conclusion: The shear effect found to have a significant contribution to the post-buckling behaviors of axisymmetric beams. As well as the classical beam theory CBT, underestimate the shear effect compared to higher order shear deformation theories HSDT.

키워드

참고문헌

  1. Abbache, A., Youzera, H., Abualnour, M., Houari, M.S., Meftah, S. and Tounsi, A. (2021), "Superharmonic vibrations of sandwich beams with fibre composite core layer based on the multiple scale method", Struct. Eng. Mech., 80(2), 201-210. https://doi.org/10.12989/sem.2021.80.2.201.
  2. Abouelregal, A.E., Mohammed, W.W. and Mohammad-Sedighi, H. (2021), "Vibration analysis of functionally graded microbeam under initial stress via a generalized thermoelastic model with dual-phase lags", Archive Appl. Mech., 91(5), 2127-2142. https://doi.org/10.1007/s00419-020-01873-2.
  3. Ahmed, R.A., Khalaf, B.S., Raheef, K.M., Fenjan, R.M. and Faleh, N.M. (2021), "Investigating dynamic response of nonlocal functionally graded porous piezoelectric plates in thermal environment", Steel Compos. Struct., 40(2), 243-254. https://doi.org/10.12989/scs.2021.40.2.243.
  4. Akbas, S. D. (2021), "Dynamic analysis of axially functionally graded porous beams under a moving load", Steel Compos. Struct., 39(6), 811-821. https://doi.org/10.12989/scs.2021.39.6.811.
  5. Akbas, S.D. (2017), "Post-buckling responses of functionally graded beams with porosities", Steel Compos Struct., 24(5), 579-589. https://doi.org/10.12989/scs.2017.24.5.579.
  6. Al-Basyouni, K.S. and Mahmoud, S.R. (2021), "Mathematical approach for the effect of the rotation, the magnetic field and the initial stress in the non-homogeneous an elastic hollow cylinder", Struct. Eng. Mech., 79(5), 593-599. https://doi.org/10.12989/sem.2021.79.5.593.
  7. Avcar, M. (2019), "Free vibration of imperfect sigmoid and power law functionally graded beams", Steel and Compos. Struct., 30(6), 603-615. https://doi.org/10.12989/scs.2019.30.6.603
  8. Barati, M.R. (2017), "Dynamic response of porous functionally graded material nanobeams subjected to moving nanoparticle based on nonlocal strain gradient theory", Mater. Res. Express, 4(11), 115017. https://doi.org/10.1088/2053-1591/aa9765.
  9. Barati, M.R., Zenkour, A.M. (2017), "Post-buckling analysis of refined shear deformable graphene platelet reinforced beams with porosities and geometrical imperfection", Compos. Struct., 181, 194-202. https://doi.org/10.1016/j.compstruct.2017.08.082.
  10. Belarbi, M.O., Daikh, A.A., Garg, A., Merzouki, T., Chalak, H.D. and Hirane, H. (2021a), "Nonlocal finite element model for the bending and buckling analysis of functionally graded nanobeams using a novel shear deformation theory", Compos. Struct., 264, 113712. https://doi.org/10.1016/j.compstruct.2021.113712.
  11. Belarbi, M.O., Garg, A., Houari, M.S.A., Hirane, H., Tounsi, A. and Chalak, H.D. (2021b), "A three-unknown refined shear beam element model for buckling analysis of functionally graded curved sandwich beams", Eng. Comput., 1-28. https://doi.org/10.1007/s00366-021-01452-1.
  12. Ben-Oumrane S., Tounsi, A., Ismail, M., Mohamed, B.B., Mustapha, M. and Bedia, E.A. (2009), "A theoretical analysis of flexional bending of Al/Al2O3 S-FGM thick beams", Comp. Mater. Sci., 44(4), 1344-1350. https://doi.org/10.1016/j.commatsci.2008.09.001.
  13. Benaoum, A., Youzera, H., Abualnour, M., Houari, M.S.A., Meftah, S.A. and Tounsi, A. (2021), "Superharmonic vibrations of sandwich beams with viscoelastic core layer with the multiple scale method", Struct. Eng. Mech., 80(6), 727. https://doi.org/10.12989/sem.2021.80.6.727.
  14. Benatta M., Mechab I., Tounsi A. And Bedia, E.A. (2008), "Static analysis of functionally graded short beams including warping and shear deformation effects", Comput. Mater. Sci., 44(2), 765-773. https://doi.org/10.1016/j.commatsci.2008.05.020.
  15. Bui, T.Q., Khosravifard, .A., Zhang, .C., Hematiyan, M.R. and Golub, M.V. (2013), "Dynamic analysis of sandwich beams with functionally graded core using a truly mesh free radial point interpolation method", Eng. Struct., 47(0), 90-104. https://doi.org/10.1016/j.engstruct.2012.03.041.
  16. David James Lloyd. (2005), "Functionally graded aluminum alloy sheet", U.S. Patent Application No. 10/696,877. https://patents.google.com/patent/US20050092403A1/en.
  17. Dehshahri, K., Nejad, M.Z., Ziaee, S., Niknejad, A. And Hadi, A. (2020), "Free vibrations analysis of arbitrary threedimensionally FGM nanoplates", Adv. Nano Res., 8(2), 115-134. https://doi.org/10.12989/anr.2020.8.2.115.
  18. Emam, S.A. and Nayfeh, A.H. (2009), "Postbuckling and free vibrations of composite beams", Compos Struct., 88(4), 636- 642. https://doi.org/10.1016/j.compstruct.2008.06.006.
  19. Eslami, M.R., Eslami, J. and Jacobs, M. (2018). Buckling and Postbuckling of Beams, Plates, and Shells, Switzerland: Springer International Publishing.
  20. Faleh, N.M., Ahmed, R.A. and Fenjan, R.M. (2018), "On vibrations of porous FG nanoshells", Int. J. Eng. Sci., 133, 1-14. https://doi.org/10.1016/j.ijengsci.2018.08.007.
  21. Fallah, A. and Aghdam, M.M. (2011), "Nonlinear free vibration and post-buckling analysis of functionally graded beams on nonlinear elastic foundation", Eur. J. Mech. A/Solids, 30(4), 571-583. https://doi.org/10.1016/j.euromechsol.2011.01.005.
  22. Fang, W., Yu, T, Lich, L.V. and Bui, T.Q. (2019), "Analysis of thick porous beams by a Quasi-3D theory and isogeometric analysis", Compos. Struct., 221, 110890. https://doi.org/10.1016/j.compstruct.2019.04.062
  23. Garg, A., Belarbi, M.O., Chalak, H.D. and Chakrabarti, A. (2021a), "A review of the analysis of sandwich FGM structures", Compos. Struct., 258, 113427. https://doi.org/10.1016/j.compstruct.2020.113427.
  24. Garg, A., Chalak, H.D., Belarbi, M.O., Chakrabarti, A. and Houari, M.S.A. (2021b), "Finite element-based free vibration analysis of power-law, exponential and sigmoidal functionally graded sandwich beams", J. Institution Eng. (India) Series C., 102(5), 1167-1201. https://doi.org/10.1007/s40032-021-00740-5.
  25. Garg, A., Chalak, H.D., Li, L., Belarbi, M.O., Sahoo, R. and Mukhopadhyay, T. (2022), "Vibration and buckling analyses of sandwich plates containing functionally graded metal foam core", Acta Mechanica Solida Sinica., 1-16. https://doi.org/10.1007/s10338-021-00295-z.
  26. Garg, A., Chalak, H.D., Zenkour, A.M., Belarbi, M.O. and Houari, M.S.A. (2021c), "A review of available theories and methodologies for the analysis of nano isotropic, nano functionally graded, and CNT reinforced nanocomposite structures", Archives Comput. Methods Eng., 1-34. https://doi.org/10.1007/s11831-021-09652-0.
  27. Ghayesh, M.H. (2018), "Nonlinear vibration analysis of axially functionally graded shear-deformable tapered beams", Appl. Math Model, 59, 583-596. https://doi.org/10.1016/j.apm.2018.02.017.
  28. Giunta, G., Belouettar, S. and Carrera, E. (2010), "Analysis of FGM beams by means ofclassical and advanced theories", Mech Adv. Mater. Struct., 17(8), 622-35. https://doi.org/10.1080/15376494.2010.518930.
  29. Gupta, R.K.., Babu, G.J., Janardhan, G.R. and Rao, G.V. (2009), "Relatively simple finite element formulation for the large amplitude free vibrations of uniform beams", Finite Elem Anal Des., 45(10), 624-631. https://doi.org/10.1016/j.finel.2009.04.001.
  30. Huang, Y.Q. and Li, Q.S. (2004), "Bending and buckling analysis of antisymmetric laminates using the moving least square differential quadrature method", Comput. Methods Appl. Mech. Eng., 193(33-35), 3471-392. https://doi.org/10.1016/j.cma.2003.12.039.
  31. Jena, S.K., Chakraverty, S. and Malikan, M. (2021), "Application of shifted Chebyshev polynomial-based Rayleigh-Ritz method and Navier's technique for vibration analysis of a functionally graded porous beam embedded in Kerr foundation", Eng. Comput., 37(4)., 3569-3589. https://doi.org/10.1007/s00366-020-01018-7.
  32. Jena, S.K., Chakraverty, S., Malikan, M. and Sedighi, H. (2020), "Implementation of Hermite-Ritz method and Navier's technique for vibration of functionally graded porous nanobeam embedded in Winkler-Pasternak elastic foundation using bi-Helmholtz nonlocal elasticity", J. Mech. Mater. Struct., 15(3), 405-434. https://doi.org/10.2140/jomms.2020.15.405 16.
  33. Ke, L.L., Yang, J., Kitipornchai, S. and Xiang, Y. (2009b), "Flexural vibration and elastic buckling ofa cracked Timoshenko beam made of functionally graded materials", Mech. Adv. Mater. Struct., 16(6), 488-502. https://doi.org/10.1080/15376490902781175.
  34. Ke, LL., Yang, J. and Kitipornchai, S. (2009a), "Postbuckling analysis of edge cracked functionally graded Timoshenko beams under end shortening", Compos. Struct., 90(2), 152-160. https://doi.org/10.1016/j.compstruct.2009.03.003.
  35. Khorshidi, M.A., Shariati, M. and Emam, S.A. (2016), "Postbuckling of functionally graded nanobeams based on modified couple stress theory under general beam theory", Int. J. Mech. Sci., 110, 1601-1669. https://doi.org/10.1016/j.ijmecsci.2016.03.006.
  36. Kiani, Y. and Eslami, M.R. (2013), "Thermomechanical buckling oftemperature-dependent FGM beams", Lat. Am. J. Solids Struct., 10(2), 223-246. http://dx.doi.org/10.1590/S1679-78252013000200001.
  37. Kitipornchai, S., Yang, J. and Liew, K.M. (2006), "Random Vibration of the Functionally Graded Laminates in Thermal Environments", Comput. Method Appl. M., 195, 1075-1095. https://doi.org/10.1016/j.cma.2005.01.016.
  38. Koochi, A. and Goharimanesh, M. (2021), "Nonlinear oscillations of CNT nano-resonator based on nonlocal elasticity: The energy balance method", Report. Mech. Eng., 2(1), 41-50. https://doi.org/10.31181/rme200102041g.
  39. Kou, M., Bi, J., Yuan, B. and Wang, Y. (2020), "Peridynamic analysis of dynamic fracture behaviors in FGMs with different gradient directions", Struct. Eng. Mech., 75(3), 339-356. https://doi.org/10.12989/sem.2020.75.3.339.
  40. Laib, S., Meftah, S.A., Youzera, H., Ziane, N. and Tounsi, A. (2021), "Vibration and damping characteristics of the masonry wall strengthened with bonded fibre composite patch with viscoelastic adhesive layer", Comput. Concrete., 27(3), 253-268. https://doi.org/10.12989/cac.2021.27.3.253.
  41. Lee, J.W. and Lee, J.Y. (2017), "Free vibration analysis of functionally graded Bernoulli-Euler beams using an exact transfer matrix expression", Int. J. Mech. Sci., 122, 1-17. https://doi.org/10.1016/j.ijmecsci.2017.01.011.
  42. Lei, J., He, Y., Li, Z., Guo, S. and Liu, D. (2019), "Postbuckling analysis of bi-directional functionally graded imperfect beams based on a novel third-order shear deformation theory", Compos. Struct., 209, 811-829. https://doi.org/10.1016/j.compstruct.2018.10.106.
  43. Li, C., Shen, H.S. and Wang, H. (2019), "Thermal post-buckling of sandwich beams with functionally graded negative Poisson's ratio honeycomb core", Int. J. Mech. Sci., 152, 289-297. https://doi.org/10.1016/j.ijmecsci.2019.01.002.
  44. Li, S.R. and Batra, R.C. (2012), "Relations between buckling loads of functionally graded Timoshenko and homogeneous Euler-Bernoulli beams", Compos. Struct., 95, 5-9. https://doi.org/10.1016/j.compstruct.2012.07.027.
  45. 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.
  46. Liu, W.Q., Liu, S.J., Fan, M.Y., Tian, W., Wang, J.P. and Tahouneh, V. (2020), "Influence of internal pores and graphene platelets on vibration of non-uniform functionally graded columns", Steel Compos. Struct., 35(2), 295-306. https://doi.org/10.12989/scs.2020.35.2.295.
  47. Ma, L.S. and Lee, D.W. (2011), "Exact solutions for nonlinear static responses of a shear deformable FGM beam under an inplane thermal loading", Eur. J. Mech. A-Solid, 31(1), 13-20. https://doi.org/10.1016/j.euromechsol.2011.06.016.
  48. Madenci, E. and Ozkili, Y.P. (2021), "Free vibration analysis of open-cell FG porous beams: analytical, numerical and ANN approaches", Steel Compos. Struct., 40(2), 157-173. https://doi.org/10.12989/scs.2021.40.2.157.
  49. 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.
  50. Meksi, A., Belakhdar, K., Bouguenina, O. and Tounsi, A. (2018), "Effect of parabolic-concave thickness variation on the mechanical buckling resistance of simply supported FGM lates", Jordan J. Civil Eng., 12(2). https://jjce.just.edu.jo/issues/paper.php?p=4245.pdf.
  51. Mirjavadi, S.S., Forsat, M., Nia, A.F., Badnava, S. and Hamouda, A.M.S. (2020), "Nonlocal strain gradient effects on forced vibrations of porous FG cylindrical nanoshells", Adv. Nano Res., 8(2), 149-156. https://doi.org/10.12989/anr.2020.8.2.149.
  52. Nejati, M., Eslampanah, A. and Najafizadeh, M. (2016), "Buckling and vibration analysis of functionally graded carbon nanotubereinforced beam under axial load", J. Appl. Mech., 8(1), 1650008. https://doi.org/10.1142/S1758825116500083.
  53. Nguyen, TK., Vo, T.P. and Thai, H.T. (2013), "Static and free vibration of axially loaded functionally graded beams based on the first-order shear deformation theory", Compos. Part B: Eng., 55, 147-157. https://doi.org/10.1016/j.compositesb.2013.06.011.
  54. Noroozi, R., Barati, A., Kazemi, A., Norouzi, S. and Hadi, A. (2020), "Torsional vibration analysis of bi-directional FG nanocone with arbitrary cross-section based on nonlocal strain gradient elasticity", Adv. Nano Res., 8(1), 13-24. https://doi.org/10.12989/anr.2020.8.1.013.
  55. Pagano, N.J. (1969), "Exact solutions for composite laminates in cylindrical bending", J. Compos. Mats., 3(3) 398-411. https://doi.org/10.1177/002199836900300304.
  56. Parvin, N. and Yusefi, A. (2017), U.S. Patent Application, No. 15/456,501. https://patents.google.com/patent/US20170368607A1/en.
  57. Pei, Y.L., Geng, P.S. and Li, L.X. (2018), "A modified higherorder theory for FG beams", Eur. J. Mech. A/Solids, 72, 186-197. https://doi.org/10.1016/j.euromechsol.2018.05.008.
  58. Pham, Q.H., Pham, T.D., Trinh, Q.V. and Phan, D.H. (2020), "Geometrically nonlinear analysis of functionally graded shells using an edge-based smoothed MITC3 (ES-MITC3) finite elements", Eng. Comput., 36(3), 1069-1082. https://doi.org/10.1007/s00366-019-00750-z.
  59. Pradhan, K.K. and Chakraverty, S. (2013), "Free vibration of euler and Timoshenko functionally graded beams by Rayleigh-Ritz method", Compos. Part B: Eng., 51, 175-184. https://doi.org/10.1016/j.compositesb.2013.02.027.
  60. Rahimi, G.H., Gazor, M.S., Hemmatnezhad, M. and Toorani, H. (2013), "On the post buckling and free vibrations of FG Timoshenko beams", Compos. Struct., 95, 247-253. https://doi.org/10.1016/j.compstruct.2012.07.034.
  61. Reddy, J.N. (1984), "A simple higher-order theory for laminated composite plates", J. Appl. Mech., 51(4), 745-752. https://doi.org/10.1115/1.3167719.
  62. Reissner, E. (1945), "The effect of transverse shear deformation on the bending of elastic plates", J. Appl. Mech., A69-A77.
  63. Rostami, R. and Mohammadimehr, M. (2020), "Vibration control of rotating sandwich cylindrical shell-reinforced nanocomposite face sheet and porous core integrated with functionally graded magneto-electro-elastic layers", Eng. Comput., 1-14. https://doi.org/10.1007/s00366-020-01052-5.
  64. Sarkar, K. and Ganguli, R. (2014), "Closed-form solutions for axially functionally graded Timoshenko beams having uniform cross-section and fixed-fixed boundary condition", Compos. Part B. Eng., 58, 361-370. https://doi.org/10.1016/j.compositesb.2013.10.077.
  65. Sedighi, H.M., Shirazi, K.H., Noghrehabadi, A.R. and Yildirim, A. H.M.E.T. (2012), "Asymptotic investigation of buckled beam nonlinear vibration", Iran. J. Sci. Technol., Transactions Mech. Eng., 36(M2), 107-116.
  66. Shahmohammadi, M.A., Azhari, M. and Saadatpour, M.M. (2020), "Free vibration analysis of sandwich FGM shells using isogeometric B-spline finite strip method", Steel Compos. Struct., 34(3), 361-376. https://doi.org/10.12989/scs.2020.34.3.361.
  67. Shanab, R.A. and Attia, M.A. (2020), "Semi-analytical solutions for static and dynamic responses of bi-directional functionally graded nonuniform nanobeams with surface energy effect", Eng. Comput., 1-44. https://doi.org/10.1007/s00366-020-01205-6.
  68. Shariati, A., Jung, D.W., Mohammad-Sedighi, H., Zur, K.K., Habibi, M. and Safa, M. (2020), "On the vibrations and stability of moving viscoelastic axially functionally graded nanobeams", Materials., 13(7), 1707. https://doi.org/10.3390/ma13071707
  69. Shariati, A., Jung, D.W., Mohammad-Sedighi, H., Zur, K.K., Habibi, M. and Safa, M. (2020), "Stability and dynamics of viscoelastic moving rayleigh beams with an asymmetrical distribution of material parameters", Symmetry, 12(4), 586. https://doi.org/10.3390/sym12040586.
  70. She, G.L., Yuan, F.G. and Ren, Y.R. (2017), "Thermal buckling and post-buckling analysis of functionally graded beams based on a general higher-order shear deformation theory", Appl. Math. Model, 47, 340-357. https://doi.org/10.1016/j.apm.2017.03.014.
  71. Shen, H.S., Wang, Z.X. (2014), "Nonlinear analysis of shear deformable FGM beams resting on elastic foundations in thermal environments", Int. J. Mech. Sci., 81, 195-206. https://doi.org/10.1016/j.ijmecsci.2014.02.020.
  72. Shokouhifard, V., Mohebpour, S., Malekzadeh, P. and Alighanbari, H. (2020), "An inclined FGM beam under a moving mass considering Coriolis and centrifugal accelerations", Steel Compos. Struct., 35(1), 61-76. https://doi.org/10.12989/scs.2020.35.1.061.
  73. Simsek, M. (2010), "Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories", Nuclear Eng. Des., 240(4), 697-705. https://doi.org/10.1016/j.nucengdes.2009.12.013.
  74. Timoshenko, S.P.X. (1922), "On the transverse vibrations of bars of uniform cross-section", Lond. Edinb. Dubl. Phil. Mag., 43(253), 125-131. https://doi.org/10.1080/14786442208633855.
  75. Ton-That, H.L. (2021b), "A new C0 third-order shear deformation theory for the nonlinear free vibration analysis of stiffened functionally graded plates", Facta Universitatis, Series: Mech. Eng., 19(2), 285-305. https://doi.org/10.22190/FUME200629040T.
  76. Ton, L.H.T. (2021a), "A modified shear deformation theory associated with the four-node quadrilateral element for bending and free vibration analyses of plates", Int. J. Eng. Appl. Phys., 1(3), 235-241. https://orcid.org/0000-0002-5195-1856.
  77. Trinh, L.C., Vo, T.P., Thai, H.T. and Nguyen, T.K. (2016), "An analytical method for the vibration and buckling of functionally graded beams under mechanical and thermal loads", Compos. Part B. Eng., 100, 152-163. https://doi.org/10.1016/j.compositesb.2016.06.067.
  78. Vo, T.P., Thai, H.T., Nguyen, T.K. and Inam, F. (2014), "Static and vibration analysis of functionally graded beams using refined shear deformation theory", Meccanica., 49, 155-168. https://doi.org/10.1007/s11012-013-9780-1.
  79. Yaghoobi, H. and Torabi, M. (2013), "Post-buckling and nonlinear free vibration analysis of geometrically imperfect functionally graded beams resting on nonlinear elastic foundation", Appl. Math. Model, 37(18-19), 8324-8340. https://doi.org/10.1016/j.apm.2013.03.037.
  80. Yang, J. and Chen, Y. (2008), "Free vibration and buckling analyses of functionally graded beams with edge cracks", Compos. Struct., 83(1), 48-60. https://doi.org/10.1016/j.compstruct.2007.03.006.
  81. Youzera, H. and Meftah, S.A. (2017b), "Nonlinear damping and forced vibration behaviour of sandwich beams with transverse normal stress", Compos. Struct., 179, 258-268. https://doi.org/10.1016/j.compstruct.2017.07.038.
  82. Youzera, H., Meftah, S.A. and Daya, E.M. (2017a), "Superharmonic resonance of cross-ply laminates by the method of multiple scales", J. Comput. Nonlin. Dyn., 12(5). 054503. https://doi.org/10.1115/1.4036914.
  83. Youzera, H., Meftah, S.A., Challamel, N. and Tounsi A. (2012), "Nonlinear damping and forced vibration analysis of laminated composite beams", Compos. Part B. Eng., 43(3), 1147-1154. https://doi.org/10.1016/j.compositesb.2012.01.008.
  84. Youzera, H., Meftah, S.A., Selim, M.M. and Tounsi, A. (2021), "Finite element method for axial and bending coupling effect on free vibration response of functionally graded beams under thermal environment", Mech. Adv. Mater. Struct., 1-15. https://doi.org/10.1080/15376494.2021.1979140.
  85. Yu, T., Hu, H., Zhang, J.H. and Bui, T.Q. (2019b), "Isogeometric analysis of size-dependent effects for functionally graded microbeams by a non-classical quasi-3D theory", Thin-Wall. Struct., 138, 1-14. https://doi.org/10.1016/j.tws.2018.12.006.
  86. Yu, T., Zhang, J., Hu, H. and Bui, T.Q. (2019a), "A novel size dependent quasi-3D isogeometric beam model for two directional FG microbeams analysis", Compos. Struct., 211, 76-88. https://doi.org/10.1016/j.compstruct.2018.12.014.
  87. Yuan, Y., Zhao, K., Zhao, Y. And Kiani, K. (2020), "Nonlocalintegro-vibro analysis of vertically aligned monolayered nonuniform FGM nanorods", Steel Compos. Struct., 37(5), 551-569. https://doi.org/10.12989/scs.2020.37.5.551.