Steel and Composite Structures

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Finite element based free vibration analysis of sandwich FGM plates under hygro-thermal conditions using zigzag theory

  • Aman Garg (State Key Laboratory of Intelligent Manufacturing Equipment and Technology, School of Mechanical Science and Engineering, Huazhong University of Science and Technology) ;
  • Neeraj Kumar Shukla (Department of Electrical Engineering, College of Engineering, King Khalid University) ;
  • M.Ramkumar Raja (Department of Electrical Engineering, College of Engineering, King Khalid University) ;
  • Hanuman D. Chalak (Department of Civil Engineering, National Institute of Technology Kurukshetra) ;
  • Mohamed-Ouejdi Belarbi (Laboratoire de recherche en Genie Civil, LRGC, Universite de Biskra) ;
  • Abdelouahed Tounsi (YFL (Yonsei Frontier Lab), Yonsei University) ;
  • Li Li (State Key Laboratory of Intelligent Manufacturing Equipment and Technology, School of Mechanical Science and Engineering, Huazhong University of Science and Technology) ;
  • A.M. Zenkour (Department of Mathematics, Faculty of Science, King Abdulaziz University)
  • Received : 2022.04.17
  • Accepted : 2023.11.21
  • Published : 2023.12.10
⟨ Previous Next ⟩

Abstract

In the present work, a comparative study has been carried out between power, exponential, and sigmoidal sandwich FGM plates for free vibration conditions under hygro-thermal conditions. Rules of mixture is used to determine effective material properties across the thickness for power-law and sigmoid sandwich FGM plates. Exponential law is used to plot effective material properties for exponentially graded sandwich FGM plates. Temperature and moisture dependent material properties were used during the analysis. Free vibration analysis is carried out using recently proposed finite element based HOZT. Present formulation satisfies interlayer transverse stress continuity conditions at interfaces and transverse shear stress-free conditions at the plate's top and bottom surfaces. The present model is free from any penalty or post-processing requirements. Several new results are reported in the present work, especially for unsymmetric sandwich FGM plates and exponential and sigmoidal sandwich FGM plates.

Keywords

Acknowledgement

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through Small Group Research Project under grant number (RGP.1 / 311 / 44).

References

  1. Abazid, M.A., Alotebi, M.S. and Sobhy, M. (2018), "A novel shear and normal deformation theory for hygrothermal bending response of FGM sandwich plates on Pasternak elastic foundation", Struct. Eng. Mech., 67(3), 219-232. https://doi.org/10.12989/sem.2018.67.3.219.
  2. Aria, A.I. and Friswell, M.I. (2019), "Computational hygro-thermal vibration and buckling analysis of functionally graded sandwich microbeams", Compos. Part B Eng., 165, 785797. https://doi.org/10.1016/j.compositesb.2019.02.028.
  3. Attia, A., Berrabah, A.T., Bousahla, A.A., Bourada, F., Tounsi, A. and Mahmoud S.R. (2021), "Free vibration analysis of FG plates under thermal environment via a simple 4-unknown HSDT", Steel Compos. Struct., 41(6), 899-910. https://doi.org/10.12989/scs.2021.41.6.899.
  4. Belarbi, M.-O., Tati, A., Ounis, H. and Khechai, A. (2017), "On the free vibration analysis of laminated composite and sandwich plates: a layerwise finite element formulation", Lat. Am. J. Solids Struct., 14, 2265-2290. https://doi.org/10.1590/1679-78253222.
  5. Beldjelili, Y., Tounsi, A. and Mahmoud, S.R. (2016), "Hygro-thermo-mechanical bending of S-FGM plates resting on variable elastic foundations using a four-variable trigonometric plate theory", Smart Struct. Syst., 18(4), 755-786. https://doi.org/10.12989/sss.2016.18.4.755.
  6. Biswas, D. and Ray, C. (2019), "An improved isoparametric quadratic element based on refined zigzag theory to compute interlaminar stresses of multilayered anisotropic plates", Int. J. Numer. Methods Eng., 119(12), 1245-1278. https://doi.org/10.1002/nme.6090.
  7. Brundaban, S., Kulmani, M, Bamadev, S., Nitin, S. and Panda, S.K. (2021), "Thermal frequency analysis of FG sandwich structure under variable temperature loading", Struct. Eng. Mech., 77(1), 57-74. https://doi.org/10.12989/sem.2021.77.1.057.
  8. Carrera, E. (2003), "Historical review of Zig-Zag theories for multilayered plates and shells", Appl. Mech. Rev., 56, 287-308. https://doi.org/10.1115/1.1557614.
  9. Chalak, H.D., Chakrabarti, A., Iqbal, M.A. and Sheikh, A.H. (2013), "Free vibration analysis of laminated soft core sandwich plates", J. Vib. Acoust., 135, 011013. https://doi.org/10.1115/1.4007262.
  10. Chen, SX., Sahmani, S. and Safaei, B. (2021), "Size-dependent nonlinear bending behavior of porous FGM quasi-3D microplates with a central cutout based on nonlocal strain gradient isogeometric finite element modelling", Eng. Comput. 37, 1657-1678. https://doi.org/10.1007/s00366-021-01303-z.
  11. Dinh-Cong, D., Truong, T.T. and Nguyen-Thoi, T. (2022), "A comparative study of different dynamic condensation techniques applied to multi-damage identification of FGM and FG-CNTRC plates", Eng. Comput., 38(Suppl 5), 3951-3975. https://doi.org/10.1007/s00366-021-01312-y.
  12. D'Angela, D. and Ercolino, M. (2021), "Fatigue crack growth in metallic components: Numerical modelling and analytical solution", Struct. Eng. Mech., 79(5), 541-556. https://doi.org/10.12989/sem.2021.79.5.541.
  13. Erfani, S., Vakili, A. and Akrami V. (2021), "Numerical investigation on the flexural links of eccentrically braced frames with web openings", Steel Compos. Struct., 39(2), 171-188. https://doi.org/10.12989/scs.2021.39.2.171.
  14. Garg, A. and Chalak, H.D. (2019), "A review on analysis of laminated composite and sandwich structures under hygrothermal conditions", Thin-Walled Struct., 142, 205-226. https://doi.org/10.1016/j.tws.2019.05.005.
  15. Garg, A. and Chalak, H.D. (2020), "Free vibration analysis of laminated sandwich plates under thermal loading", IOP Conf. Ser. Mater. Sci. Eng., 872, 012055. https://doi.org/10.1088/1757-899X/872/1/012055.
  16. Garg, A. and Chalak, H.D. (2021a), "Analysis of non-skew and skew laminated composite and sandwich plates under hygro-thermo-mechanical conditions including transverse stress variations", J. Sandw. Struct. Mater., 23(8), 3471-3494. https://doi.org/10.1177/1099636220932782.
  17. Garg, A. and Chalak, H.D. (2021b), "Novel higher-order zigzag theory for analysis of laminated sandwich beams", Proc. Inst. Mech. Eng. Part L J. Mater. Des. Appl., 235, 176-194. https://doi.org/10.1177/1464420720957045.
  18. Garg, A., Chalak, H.D. and Chakrabarti, A. (2020), "Comparative study on the bending of sandwich FGM beams made up of different material variation laws using refined layerwise theory", Mech. Mater., 151, 103634. https://doi.org/10.1016/j.mechmat.2020.103634.
  19. 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.
  20. 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. Inst. Eng. India Ser. C, 102, 1167-1201. https://doi.org/10.1007/s40032-021-00740-5.
  21. Garg, A., Chalak, H.D. and Chakrabarti, A. (2022a), "Bending analysis of functionally graded sandwich plates using HOZT including transverse displacement effects", Mech. Based Des. Struct. Mach., 50(10), 3563-3577. https://doi.org/10.1080/15397734.2020.1814157.
  22. Garg, A., Belarbi, M.O., Li, L. an Tounsi, A. (2022b), "Bending analysis of power-law sandwich FGM beams under thermal conditions", Adv. Aircr. Spacecr. Sci., 9(3), 243-261. https://doi.org/10.12989/aas.2022.9.3.243.
  23. Garg, A., Chalak, H.D., Zenkour, A.M., Belarbi, M.O. and Houari, M.S.A. (2022c), "A review of available theories and methodologies for the analysis of nano isotropic, nano functionally graded, and CNT reinforced nanocomposite structures", Arch. Comput. Methods Eng., 29, 2237-2270. https://doi.org/10.1007/s11831-021-09652-0.
  24. Garg, A., Gupta, S., Chalak, H.D., Belarbi, M.O., Tounsi, A. and Zenkour, A.M. (2023), "Free vibration analysis of power-law and sigmoidal sandwich FG plates using refined zigzag theory", Adv. Mater. Res., 12(1), 43-65. https://doi.org/10.12989/amr.2023.12.1.043.
  25. Ghatage, P.S., Kar, V.R. and Sudhagar, P.E. (2020), "On the numerical modelling and analysis of multi-directional functionally graded composite structures: A review", Compos. Struct., 236, 111837. https://doi.org/10.1016/j.compstruct.2019.111837.
  26. Hadji, L. and Avcar, M. (2021), "Nonlocal free vibration analysis of porous FG nanobeams using hyperbolic shear deformation beam theory", Adv. Nano Res., 10(3), 281-293. https://doi.org/10.12989/anr.2021.10.3.281.
  27. Hadji, L., Amoozgar, M. and Tounsi, A. (2022), "Non-linear thermal buckling of FG plates with porosity", Steel Compos. Struct., 42(5), 711-722. https://doi.org/10.12989/scs.2022.42.5.711.
  28. Hajlaoui, A., Triki, E., Frikha, A., Wali, M. and Dammak, F. (2017), "Nonlinear dynamics analysis of FGM shell structures with a higher order shear strain enhanced solid-shell element", Lat. Am. J. Solids Struct., 14(1), 72-91. https://doi.org/10.1590/1679-78253323.
  29. Hajlaoui, A., Chebbi, E., Wali, M. and Dammak, F. (2020a), "Static analysis of carbon nanotube-reinforced FG shells using an efficient solid-shell element with parabolic transverse shear strain", Eng. Comput., 37(3), 823-849. https://doi.org/10.1108/EC-02-2019-0075.
  30. Hajlaoui, A., Chebbi, E., Wali, M. and Dammak, F. (2020b), Geometrically nonlinear analysis of FGM shells using solid-shell element with parabolic shear strain distribution. Int. J. Mech. Mater. Des., 16, 351-366. https://doi.org/10.1007/s10999-019-09465-x.
  31. Hajlaoui, A., Chebbi, E. and Dammak, F. (2021), "Three-dimensional thermal buckling analysis of functionally graded material structures using a modified FSDT-based solid-shell element", Int. J. Press. Vessels Pip., 194, 104547. https://doi.org/10.1016/j.ijpvp.2021.104547.
  32. Hajlaoui, A. and Dammak, F. (2022), "A modified first shear deformation theory for three-dimensional thermal post-buckling analysis of FGM plates", Meccanica, 57, 337-353. https://doi.org/10.1007/s11012-021-01427-y.
  33. Iurlaro, L., Gherlone, M. and Di Sciuva, M. (2014), "Bending and free vibration analysis of functionally graded sandwich plates using the Refined Zigzag Theory", J. Sandw. Struct. Mater., 16, 669-699. https://doi.org/10.1177/1099636214548618.
  34. Jha, D.K., Kant, T. and Singh, R.K. (2013), "A critical review of recent research on functionally graded plates", Compos. Struct., 96, 833-849. https://doi.org/10.1016/j.compstruct.2012.09.001.
  35. Kandasamy, R., Dimitri, R. and Tornabene, F. (2016), "Numerical study on the free vibration and thermal buckling behavior of moderately thick functionally graded structures in thermal environments", Compos. Struct., 157, 207-221. https://doi.org/10.1016/j.compstruct.2016.08.037.
  36. Khechai, A., Tati, A., Belarbi, M.O. and Guettala, A. (2019), "Numerical analysis of stress concentration in isotropic and laminated plates with inclined elliptical holes", J. Inst. Eng. Ser. C, 100, 511-522. https://doi.org/10.1007/s40032-018-0448-4.
  37. Kiarasi, F., Babaei, M., Dimitri, R. and Tornabene, F. (2020), "Hygrothermal modeling of the buckling behavior of sandwich plates with nanocomposite face sheets resting on a Pasternak foundation", Contin. Mech. Thermodyn., 33, 911-932. https://doi.org/10.1007/s00161-020-00929-6.
  38. Kiarasi, F., Babaei, M., Mollaei, S., Mohammadi, M. and Asemi, K. (2021), "Free vibration analysis of FG porous joined truncated conical-cylindrical shell reinforced by graphene platelets", Adv. Nano Res., 11(4), 361-380. https://doi.org/10.12989/anr.2021.11.4.361.
  39. Koizumi, M. and Niino, M. (1995), "Overview of FGM Research in Japan", MRS Bull., 20(1), 19-21. https://doi.org/10.1557/S0883769400048867.
  40. Laoufi, I., Ameur, M., Zidi, M., Bedia, E.A.A. and Bousahla, A.A. (2016), "Mechanical and hygrothermal behaviour of functionally graded plates using a hyperbolic shear deformation theory", Steel Compos. Struct., 20(4), 889-912. https://doi.org/10.12989/scs.2016.20.4.889.
  41. Li, D. (2020), "Layerwise theories of laminated composite structures and their applications: a review", Arch. Comput. Methods Eng., 28, 577-600. https://doi.org/10.1007/s11831-019-09392-2.
  42. Luong, V.D., Abbes, F., Hoang, M.P., Duong, P.T.M. and Abbes, B. (2021), "Finite element elastoplastic homogenization model of a corrugated-core sandwich structure", Steel Compos. Struct., 41(3), 437-445. https://doi.org/10.12989/scs.2021.41.3.437.
  43. Ma, X., Sahmani, S. and Safaei, B. (2022), "Quasi-3D large deflection nonlinear analysis of isogeometric FGM microplates with variable thickness via nonlocal stress-strain gradient elasticity", Eng. Comput. 38, 3691-3704. https://doi.org/10.1007/s00366-021-01390-y.
  44. Madenci, E. (2021a), "Free vibration analysis of carbon nanotube RC nanobeams with variational approaches", Adv. Nano Res., 11(2), 157-171. https://doi.org/10.12989/anr.2021.11.2.157.
  45. Madenci, E. (2021b), "Free vibration and static analyses of metal-ceramic FG beams via high-order variational MFEM" Steel Compos. Struct., 39(5), 493-509. https://doi.org/10.12989/scs.2021.39.5.493.
  46. Madenci, E. and Gulcu, S. (2020), "Optimization of flexure stiffness of FGM beams via artificial neural networks by mixed FEM", Struct. Eng. Mech., 75(5), 633-642. https://doi.org/10.12989/sem.2020.75.5.633.
  47. Madenci, E. and Ozkili, Y.O. (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.
  48. Madenci, E. and Ozutok, A. (2020), "Variational approximate for high order bending analysis of laminated composite plates", Struct. Eng. Mech., 73(1), 97-108. https://doi.org/10.12989/sem.2020.73.1.097.
  49. Mantari, J.L. and Ore, M. (2015), "Free vibration of single and sandwich laminated composite plates by using a simplified FSDT", Compos. Struct., 132, 952-959. https://doi.org/10.1016/j.compstruct.2015.06.035.
  50. Merdaci, S., Tounsi, A., Houari, M.S.A., Mechab, I., Hebali, H. and Benyoucef, S. (2011), "Two new refined shear displacement models for functionally graded sandwich plates", Arch. Appl. Mech., 81, 1507-1522. https://doi.org/10.1007/s00419-010-0497-5.
  51. Mota, M.T., Fairbairn, E.M.R., Ribeiro, F.L.B., Rossi, P., Tailhan, J.L., Andrade, H.C.C. and Rita, M.R. (2021), "A 3D probabilistic model for explicit cracking of concrete", Comput. Concr., 27(6), 549-562. https://doi.org/10.12989/cac.2021.27.6.549.
  52. Natarajan, S., Deogekar, P.S., Manickam, G. and Belouettar, S. (2014), "Hygrothermal effects on the free vibration and buckling of laminated composites with cutouts", Compos. Struct., 108, 848-855. https://doi.org/10.1016/j.compstruct.2013.10.009.
  53. Natarajan, S. and Manickam, G. (2012), "Bending and vibration of functionally graded material sandwich plates using an accurate theory", Finite Elem. Anal. Des., 57, 32-42. https://doi.org/10.1016/j.finel.2012.03.006.
  54. Neves, A.M.A., Ferreira, A.J.M., Carrera, E., Cinefra, M., Jorge, R.M.N., Soares, C.M.M and Araujo, A.L. (2017), "Influence of zig-zag and warping effects on buckling of functionally graded sandwich plates according to sinusoidal shear deformation theories", Mech. Adv. Mater. Struct., 24, 360-376. https://doi.org/10.1080/15376494.2016.1191095.
  55. Neves, A.M.A., Ferreira, A.J.M., Carrera, E., Cinefra, M., Jorge, R.M.N. and Soares, C.M.M. (2012), "Static analysis of functionally graded sandwich plates according to a hyperbolic theory considering Zig-Zag and warping effects", Adv. Eng. Softw., 52, 30-43. https://doi.org/10.1016/j.advengsoft.2012.05.005.
  56. Neves, A.M.A., Ferreira, A.J.M., Carrera, E., Cinefra, M., Roque, C.M.C., Jorge, R.M.N. and Soares, C.M.M. (2013a), "Free vibration analysis of functionally graded shells by a higher-order shear deformation theory and radial basis functions collocation, accounting for through-the-thickness deformations", Eur. J. Mech. A/Solids., 37, 24-34. https://doi.org/10.1016/j.euromechsol.2012.05.005.
  57. Neves, A.M.A., Ferreira, A.J.M., Carrera, E., Cinefra, M., Roque, C.M.C., Jorge, R.M.N. and Soares, C.M.M. (2013b.), "Static, free vibration and buckling analysis of isotropic and sandwich functionally graded plates using a quasi-3D higher-order shear deformation theory and a meshless technique", Compos. Part B Eng., 44, 657-674. https://doi.org/10.1016/j.compositesb.2012.01.089.
  58. Nguyen, D.H., Bui-Tien, T., Roeck, G.D. and Wahab, M.A. (2021), "Damage detection in structures using modal curvatures gapped smoothing method and deep learning", Struct. Eng. Mech., 77(1), 47-56. https://doi.org/10.12989/sem.2021.77.1.047.
  59. Nguyen, T.K., Nguyen, B.D., Vo, T.P. and Thai, H.T. (2017), "Hygro-thermal effects on vibration and thermal buckling behaviours of functionally graded beams", Compos. Struct., 176, 1050-1060. https://doi.org/10.1016/j.compstruct.2017.06.036.
  60. Nguyen, T.K., Sab, K. and Bonnet, G. (2008), "First-order shear deformation plate models for functionally graded materials", Compos. Struct., 83(1), 25-36. https://doi.org/10.1016/j.compstruct.2007.03.004.
  61. Pandey, S. and Pradyumna, S. (2015), "Free vibration of functionally graded sandwich plates in thermal environment using a layerwise theory", Eur. J. Mech. A/Solids, 51, 55-66. https://doi.org/10.1016/j.euromechsol.2014.12.001.
  62. Pandey, S. and Pradyumna, S. (2018), "Analysis of functionally graded sandwich plates using a higher-order layerwise theory", Compos. Part B Eng., 153, 325-336. https://doi.org/10.1016/j.compositesb.2018.08.121.
  63. Pham, Q.-H., Nguyen, P.-C., Tran, V.-K. and Nguyen-Thoi, T. (2021), "Finite element analysis for functionally graded porous nano-plates resting on elastic foundation", Steel Compos. Struct., 41(2), 149-166. https://doi.org/10.12989/scs.2021.41.2.149.
  64. Plagianakos, T.S. and Papadopoulos, E.G. (2015), "Higher-order 2-D/3-D layerwise mechanics and finite elements for composite and sandwich composite plates with piezoelectric layers", Aerosp. Sci. Technol., 40, 150-163. https://doi.org/10.1016/j.ast.2014.10.015.
  65. Priyanka, R., Twinkle, C.M. and Pitchaimani, J. (2022), "Stability and dynamic behavior of porous FGM beam: influence of graded porosity, graphene platelets, and axially varying loads", Eng. Comput., 38(Suppl 5), 4347-4366. https://doi.org/10.1007/s00366-021-01478-5.
  66. Razieh, H., Mostafa, M. and Adlparvar, M.R. (2021), "On thermally induced instability of FG-CNTRC cylindrical panels", Adv. Nano Res., 10(1), 43-57. https://doi.org/10.12989/anr.2021.10.1.043.
  67. Reddy, J.N. (2000), "Analysis of functionally graded plates", Int. J. Numer. Methods Eng., 47(1-3), 663-684. https://doi.org/10.1002/(SICI)1097-0207(20000110/30)47:1/3<663::AID-NME787>3.0.CO;2-8.
  68. Refrafi, S., Bousahla, A.A., Bouhadra, A., Menasria, A., Bourada, F., Tounsi, Abdeldjebbar, Bedia, E.A.A., Mahmoud, S.R., Benrahou, K.H. and Tounsi, A. (2020), "Effects of hygro-thermo-mechanical conditions on the buckling of FG sandwich plates resting on elastic foundations", Comput. Concr., 25(4), 311-325. https://doi.org/10.12989/cac.2020.25.4.311.
  69. Sharma, R., Jadon, V.K. and Singh, B. (2015), "A review on the finite element methods for heat conduction in functionally graded materials", J. Inst. Eng. Ser. C, 96, 73-81. https://doi.org/10.1007/s40032-014-0125-1.
  70. Sobhy, M. (2017), "Hygro-thermo-mechanical vibration and buckling of exponentially graded nanoplates resting on elastic foundations via nonlocal elasticity theory", Struct. Eng. Mech., 63(3), 401-415. https://doi.org/10.12989/sem.2017.63.3.401.
  71. Swaminathan, K., Naveenkumar, D.T., Zenkour, A.M. and Carrera, E. (2015), "Stress, vibration and buckling analyses of FGM plates-A state-of-the-art review", Compos. Struct., 120, 10-31. https://doi.org/10.1016/j.compstruct.2014.09.070.
  72. Tahir, S.I., Tounsi, A., Chikh, A., Al-Osta, M.A., Al-Dulaijan, S.U. and Al-Zahrani, M.M. (2022), "The effect of three-variable viscoelastic foundation on the wave propagation in functionally graded sandwich plates via a simple quasi-3D HSDT", Steel Compos. Struct., 42(4), 501-511. https://doi.org/10.12989/scs.2022.42.4.501.
  73. Taleb, O., Houari, M.S.A., Bessaim, A., Tounsi, A. and Mahmoud, S.R. (2018), "A new plate model for vibration response of advanced composite plates in thermal environment", Struct. Eng. Mech., 67(4), 369-383. https://doi.org/10.12989/sem.2018.67.4.369.
  74. Thai, C.H., Zenkour, A.M., Wahab, M.A. and Nguyen-Xuan, H. (2016), "A simple four-unknown shear and normal deformations theory for functionally graded isotropic and sandwich plates based on isogeometric analysis", Compos. Struct., 139, 77-95. https://doi.org/10.1016/j.compstruct.2015.11.066.
  75. Thai, H.-T., Nguyen, T.-K., Vo, T.P. and Lee, J. (2014), "Analysis of functionally graded sandwich plates using a new first-order shear deformation theory", Eur. J. Mech. - A/Solids, 45, 211-225. https://doi.org/10.1016/j.euromechsol.2013.12.008.
  76. Tran, T.T., Nguyen, N.H., Do, T.V., Minh, P.V. and Duc, N.D. (2021), "Bending and thermal buckling of unsymmetric functionally graded sandwich beams in high-temperature environment based on a new third-order shear deformation theory", J. Sandw. Struct. Mater., 13(3), 906-930. https://doi.org/10.1177/1099636219849268.
  77. Wali, M., Hajlaoui, A. and Dammak, F. (2014), "Discrete double directors shell element for the functionally graded material shell structures analysis", Comput. Methods Appl. Mech. Eng., 278, 388-403. https://doi.org/10.1016/j.cma.2014.05.011.
  78. Wu, C.L., Kan, J.C., Wang, Q.H., Liu, J.M. and Li, Z.Q. (2021), "FEM analysis of the modular prefabricated steel-concrete composite beam-column internal joint under reciprocating action", Steel Compos. Struct., 41(1), 45-64. https://doi.org/10.12989/scs.2021.41.1.045.
  79. Yang, X., Sahmani, S. and Safaei, B. (2021), "Postbuckling analysis of hydrostatic pressurized FGM microsized shells including strain gradient and stress-driven nonlocal effects", Eng. Comput., 37, 1549-1564. https://doi.org/10.1007/s00366-019-00901-2.
  80. Zenkour, A.M. (2005a), "A comprehensive analysis of functionally graded sandwich plates: Part 1-Deflection and stresses", Int. J. Solids Struct., 42, 5224-5242. https://doi.org/10.1016/j.ijsolstr.2005.02.015.
  81. Zenkour, A.M. (2005b), "A comprehensive analysis of functionally graded sandwich plates: Part 2-Buckling and free vibration", Int. J. Solids Struct., 42, 5243-5258. https://doi.org/10.1016/j.ijsolstr.2005.02.016.
  82. Zenkour, A.M. (2012), "Hygrothermal analysis of exponentially graded rectangular plates", J. Mech. Mater. Struct., 7, 687-700. https://doi.org/10.2140/jomms.2012.7.687.
  83. Zenkour, A.M. (2013), "Bending analysis of functionally graded sandwich plates using a simple four-unknown shear and normal deformations theory", J. Sandw. Struct. Mater., 15(6), 629-656. https://doi.org/10.1177/1099636213498886.
  84. Zenkour, A.M. and Radwan, A.F. (2020), "Bending and buckling analysis of FGM plates resting on elastic foundations in hygrothermal environment", Arch. Civ. Mech. Eng., 20(112), 1-23. https://doi.org/10.1007/s43452-020-00116-z.
  85. Zhang, N., Khan, T., Guo, H., Shi, S., Zhong, W. and Zhang, W. (2019), "Functionally graded materials: an overview of stability, buckling, and free vibration analysis", Adv. Mater. Sci. Eng., 2019, 1-18. https://doi.org/10.1155/2019/1354150.