Steel and Composite Structures

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Static and stress analyses of bi-directional FG porous plate using unified higher order kinematics theories

  • Mohamed, Salwa (Department of Engineering Mathematics, Faculty of Engineering, Zagazig University) ;
  • Assie, Amr E. (Mechanical Engineering Department, Faculty of Engineering, Jazan University) ;
  • Mohamed, Nazira (Department of Engineering Mathematics, Faculty of Engineering, Zagazig University) ;
  • Eltaher, Mohamed A. (Mechanical Engineering Department, Faculty of Engineering, King Abdulaziz University)
  • Received : 2022.02.01
  • Accepted : 2022.10.26
  • Published : 2022.11.10
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Abstract

This article aims to investigate the static deflection and stress analysis of bi-directional functionally graded porous plate (BDFGPP) modeled by unified higher order kinematic theories to include the shear stress effects, which not be considered before. Different shear functions are described according to higher order models that satisfy the zero-shear influence at the top and bottom surfaces, and hence refrain from the need of shear correction factor. The material properties are graded through two spatial directions (i.e., thickness and length directions) according to the power law distribution. The porosities and voids inside the material constituent are described by different cosine functions. Hamilton's principle is implemented to derive the governing equilibrium equation of bi-directional FG porous plate structures. An efficient numerical differential integral quadrature method (DIQM) is exploited to solve the coupled variable coefficients partial differential equations of equilibrium. Problem validation and verification have been proven with previous prestigious work. Numerical results are illustrated to present the significant impacts of kinematic shear relations, gradation indices through thickness and length, porosity type, and boundary conditions on the static deflection and stress distribution of BDFGP plate. The proposed model is efficient in design and analysis of many applications used in nuclear, mechanical, aerospace, naval, dental, and medical fields.

Keywords

References

  1. Abdollahi, M., Saidi, A.R. and Bahaadini, R. (2021), "Aeroelastic analysis of symmetric and non-symmetric trapezoidal honeycomb sandwich plates with FG porous face sheets", Aerosp. Sci. Technol., 119, 107211. https://doi.org/10.1016/j.ast.2021.107211.
  2. Abo-bakr, R.M., Shanab, R.A. and Attia, M.A. (2021), "Multiobjective optimization for lightweight design of bi-directional functionally graded beams for maximum frequency and buckling load", Compos. Struct., 278, 114691. https://doi.org/10.1016/j.compstruct.2021.114691.
  3. Akbas, S.D., Bashiri, A.H., Assie, A.E. and Eltaher, M.A. (2021), "Dynamic analysis of thick beams with functionally graded porous layers and viscoelastic support", J. Vib. Control, 27(13-14), 1644-1655. https://doi.org/10.1177/1077546320947302.
  4. Akbas, S.D., Fageehi, Y.A., Assie, A.E. and Eltaher, M.A. (2020), "Dynamic analysis of viscoelastic functionally graded porous thick beams under pulse load", Eng. Comput., 1-13. https://doi.org/10.1007/s00366-020-01070-3.
  5. Al-Furjan, M.S.H., Habibi, M., Ghabussi, A., Safarpour, H., Safarpour, M. and Tounsi, A. (2021a), "Non-polynomial framework for stress and strain response of the FG-GPLRC disk using three-dimensional refined higher-order theory", Eng. Struct., 228, 111496. https://doi.org/10.1016/j.engstruct.2020.111496.
  6. Al-Furjan, M.S.H., Habibi, M., Shan, L. and Tounsi, A. (2021b), "On the vibrations of the imperfect sandwich higher-order disk with a lactic core using generalize differential quadrature method", Compos. Struct., 257, 113150. https://doi.org/10.1016/j.compstruct.2020.113150.
  7. Al-Furjan, M.S.H., Safarpour, H., Habibi, M., Safarpour, M. and Tounsi, A. (2020), "A comprehensive computational approach for nonlinear thermal instability of the electrically FG-GPLRC disk based on GDQ method", Eng. Comput., 1-18. https://doi.org/10.1007/s00366-020-01088-7.
  8. Al-Osta, M.A., Saidi, H., Tounsi, A., Al-Dulaijan, S.U., AlZahrani, M.M., Sharif, A. and Tounsi, A. (2021), "Influence of porosity on the hygro-thermo-mechanical bending response of an AFG ceramic-metal plates using an integral plate model", Smart Struct. Syst., 28(4), 499-513. https://doi.org/10.12989/sss.2021.28.4.499.
  9. Alghanmi, R.A. and Zenkour, A.M. (2021), "An electromechanical model for functionally graded porous plates attached to piezoelectric layer based on hyperbolic shear and normal deformation theory", Compos. Struct., 274, 114352. https://doi.org/10.1016/j.compstruct.2021.114352.
  10. Alhaifi, K., Arshid, E. and Khorshidvand, A.R. (2021), "Large deflection analysis of functionally graded saturated porous rectangular plates on nonlinear elastic foundation via GDQM", Steel Compos. Struct., 39(6), 795-809. https://doi.org/10.12989/scs.2021.39.6.795.
  11. Alshorbagy, A.E., Eltaher, M.A. and Mahmoud, F. (2011), "Free vibration characteristics of a functionally graded beam by finite element method", Appl. Math. Model., 35(1), 412-425. https://doi.org/10.1016/j.apm.2010.07.006.
  12. Anamagh, M.R. and Bediz, B. (2020), "Free vibration and buckling behavior of functionally graded porous plates reinforced by graphene platelets using spectral Chebyshev approach", Compos. Struct., 253, 112765. https://doi.org/10.1016/j.compstruct.2020.112765.
  13. Arshid, E., Khorasani, M., Soleimani-Javid, Z., Amir, S. and Tounsi, A. (2021), "Porosity-dependent vibration analysis of FG microplates embedded by polymeric nanocomposite patches considering hygrothermal effect via an innovative plate theory", Eng. Comput., 1-22. https://doi.org/10.1007/s00366-021-01382-y.
  14. Assie, A.E., Eltaher, M.A. and Mahmoud, F.F. (2011), "Behavior of a viscoelastic composite plates under transient load", J. Mech. Sci. Technol., 25(5), 1129-1140. https://doi.org/10.1007/s12206-011-0302-6.
  15. Attia, M.A., El-Shafei, A.G. and Gad, S.I. (2021), "Investigation of indentation response of bi-directional functionally graded materials", Proceedings of the Institution of Mechanical Engineers, Part J: Journal of Engineering Tribology, 235(8), 1641-1658. https://doi.org/10.1177/1350650120971078.
  16. Barati, M.R. and Shahverdi, H. (2017), "Aero-hygro-thermal stability analysis of higher-order refined supersonic FGM panels with even and uneven porosity distributions", J. Fluids Struct, 73, 125-136. https://doi.org/10.1016/j.jfluidstructs.2017.06.007.
  17. Bekkaye, T.H.L., Fahsi, B., Bousahla, A.A., Bourada, F., Tounsi, A., Benrahou, K.H. and Al-Zahrani, M.M. (2020), "Porositydependent mechanical behaviors of FG plate using refined trigonometric shear deformation theory", Comput. Concrete, Int. J., 26(5), 439-450. https://doi.org/10.12989/cac.2020.26.5.439
  18. Bellman, R., Kashef, B.G. and Casti, J. (1972), "Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations", J. Comput. Phys., 10(1), 40-52. https://doi.org/10.1016/0021-9991(72)90089-7
  19. Carrera, E., Brischetto, S. and Robaldo, A. (2008), "Variable kinematic model for the analysis of functionally graded material plates", AIAA J., 46(1), 194-203. https://doi.org/10.2514/1.32490.
  20. Chanda, A. and Bora, S.N. (2020), "Effect of a porous sea-bed on water wave scattering by two thin vertical submerged porous plates", Europ. J. Mech. B/Fluids, 84, 250-261. https://doi.org/10.1016/j.euromechflu.2020.06.009.
  21. Chen, X., Chen, L., Huang, S., Li, M. and Li, X. (2021), "Nonlinear forced vibration of in-plane bi-directional functionally graded materials rectangular plate with global and localized geometrical imperfections", Appl. Math. Model., 93, 443-466. https://doi.org/10.1016/j.apm.2020.12.033.
  22. Cho, I.H. (2021), "Wave energy dissipation by a floating horizontal porous plate in oblique incident waves", Wave Motion, 102765. https://doi.org/10.1016/j.wavemoti.2021.102765.
  23. Coskun, S., Kim, J. and Toutanji, H. (2019), "Bending, free vibration, and buckling analysis of functionally graded porous micro-plates using a general third-order plate theory", J. Compos. Sci., 3(1), 15. https://doi.org/10.3390/jcs3010015.
  24. Daikh, A.A., Houari, M.S.A. and Eltaher, M.A. (2021c), "A novel nonlocal strain gradient Quasi-3D bending analysis of sigmoid functionally graded sandwich nanoplates", Compos. Struct., 262, 113347. https://doi.org/10.1016/j.compstruct.2020.113347.
  25. Daikh, A.A., Houari, M.S.A., Belarbi, M.O., Chakraverty, S. and Eltaher, M.A. (2021b), "Analysis of axially temperaturedependent functionally graded carbon nanotube reinforced composite plates", Eng. Comput., 1-22. https://doi.org/10.1007/s00366-021-01413-8.
  26. Daikh, A.A., Houari, M.S.A., Belarbi, M.O., Mohamed, S.A. and Eltaher, M.A. (2021a), "Static and dynamic stability responses of multilayer functionally graded carbon nanotubes reinforced composite nanoplates via quasi 3D nonlocal strain gradient theory", Defence Technol., https://doi.org/10.1016/j.dt.2021.09.011.
  27. Dhuria, M., Grover, N. and Goyal, K. (2021), "Influence of porosity distribution on static and buckling responses of porous functionally graded plates", Structures, 34, 1458-1474. https://doi.org/10.1016/j.istruc.2021.08.050.
  28. Do, D.T., Nguyen-Xuan, H. and Lee, J. (2020), "Material optimization of tri-directional functionally graded plates by using deep neural network and isogeometric multimesh design approach", Appl. Math. Model., 87, 501-533. https://doi.org/10.1016/j.apm.2020.06.002.
  29. El-Ashmawy, A.M., Xu, Y. and El-Mahdy, L.A. (2021), "Mechanical properties improvement of bi-directional functionally graded laminated MWCNT reinforced composite beams using an integrated tailoring-optimization approach", Microporous Mesoporous Mater., 314, 110875. https://doi.org/10.1016/j.micromeso.2021.110875.
  30. Eltaher, M.A., Fouda, N., El-midany, T. and Sadoun, A.M. (2018), "Modified porosity model in analysis of functionally graded porous nanobeams", J. Brazil. Soc. Mech. Sci. Eng., 40(3), 1-10. https://doi.org/10.1007/s40430-018-1065-0.
  31. Esen, I., Eltaher, M.A. and Abdelrahman, A.A. (2021), "Vibration response of symmetric and sigmoid functionally graded beam rested on elastic foundation under moving point mass", Mech. Based Des. Struct. Machines, 1-25. https://doi.org/10.1080/15397734.2021.1904255.
  32. Esmaeilzadeh, M. and Kadkhodayan, M. (2019), "Dynamic analysis of stiffened bi-directional functionally graded plates with porosities under a moving load by dynamic relaxation method with kinetic damping", Aerosp. Sci. Technol., 93, 105333. https://doi.org/10.1016/j.ast.2019.105333.
  33. Farsani, S.R., Jafari-Talookolaei, R.A., Valvo, P.S. and Goudarzi, A.M. (2021), "Free vibration analysis of functionally graded porous plates in contact with bounded fluid", Ocean Eng., 219, 108285. https://doi.org/10.1016/j.oceaneng.2020.108285.
  34. Gao, Z., Li, H., Zhao, J., Guan, J. and Wang, Q. (2021), "Analyses of dynamic characteristics of functionally graded porous (FGP) sandwich plates with viscoelastic materials-filled square-celled core", Eng. Struct., 248, 113242. https://doi.org/10.1016/j.engstruct.2021.113242.
  35. Ghandourah, E.E., Ahmed, H.M., Eltaher, M.A., Attia, M.A. and Abdraboh, A.M. (2021), "Free vibration of porous FG nonlocal modified couple nanobeams via a modified porosity model", Adv. Nano Res., 11(4), 405-422. https://doi.org/10.12989/anr.2021.11.4.405.
  36. Guellil, M., Saidi, H., Bourada, F., Bousahla, A.A., Tounsi, A., Al-Zahrani, M.M. and Mahmoud, S.R. (2021), "Influences of porosity distributions and boundary conditions on mechanical bending response of functionally graded plates resting on Pasternak foundation", Steel Compos. Struct., 38(1), 1-15. https://doi.org/10.12989/scs.2021.38.1.001.
  37. Hachemi, H., Bousahla, A.A., Kaci, A., Bourada, F., Tounsi, A., Benrahou, K.H. and Mahmoud, S.R. (2021), "Bending analysis of functionally graded plates using a new refined quasi-3D shear deformation theory and the concept of the neutral surface position", Steel Compos. Struct., 39(1), 51-64. https://doi.org/10.12989/scs.2021.39.1.051.
  38. Hamed, M.A., Abo-Bakr, R.M., Mohamed, S.A. and Eltaher, M.A. (2020), "Influence of axial load function and optimization on static stability of sandwich functionally graded beams with porous core", Eng. Comput., 36(4), 1929-1946. https://doi.org/10.1007/s00366-020-01023-w.
  39. Hamed, M.A., Sadoun, A.M. and Eltaher, M.A. (2019), "Effects of porosity models on static behavior of size dependent functionally graded beam", Struct. Eng. Mech., 71(1), 89-98. https://doi.org/10.12989/sem.2019.71.1.089.
  40. Javed, S. and Al Mukahal, F.H.H. (2021), "Free vibration of annular circular plates based on higher-order shear deformation theory: A spline approximation technique", Int. J. Aerosp. Eng., 2021. https://doi.org/10.1155/2021/5440376.
  41. Jha, D.K., Kant, T. and Singh, R.K. (2013), "Free vibration response of functionally graded thick plates with shear and normal deformations effects", Compos. Struct., 96, 799-823. http://dx.doi.org/10.1016/j.compstruct.2012.09.034.
  42. Joshi, K.K. and Kar, V.R. (2020), "Bending analysis of bidimensional functionally graded plate using FEA", Mater. Today: Proceedings, 26, 1766-1770. https://doi.org/10.1016/j.matpr.2020.02.37.
  43. Kaddari, M., Kaci, A., Bousahla, A.A., Tounsi, A., Bourada, F., Bedia, E.A. and Al-Osta, M.A. (2020), "A study on the structural behaviour of functionally graded porous plates on elastic foundation using a new quasi-3D model: Bending and free vibration analysis", Comput. Concrete, 25(1), 37-57. https://doi.org/10.12989/cac.2020.25.1.037.
  44. Karama, M., Afaq, K.S. and Mistou, S. (2003), "Mechanical behaviour of laminated composite beam by the new multilayered laminated composite structures model with transverse shear stress continuity", Int. J. Solids Struct., 40(6), 1525-1546. https://doi.org/10.1016/S0020-7683(02)00647-9.
  45. Karamanli, A. (2021), "Size-dependent behaviors of three directional functionally graded shear and normal deformable imperfect microplates", Compos. Struct., 257, 113076. https://doi.org/10.1016/j.compstruct.2020.113076.
  46. Karamanli, A. and Aydogdu, M. (2019), "Size dependent flapwise vibration analysis of rotating two-directional functionally graded sandwich porous microbeams based on a transverse shear and normal deformation theory", Int. J. Mech. Sci., 159, 165-181. https://doi.org/10.1016/j.ijmecsci.2019.05.047.
  47. Karamanli, A. and Aydogdu, M. (2020), "Vibration of functionally graded shear and normal deformable porous microplates via finite element method", Compos Struct., 237, 111934. https://doi.org/10.1016/j.compstruct.2020.111934.
  48. Karamanli, A. and Vo, T.P. (2021), "Bending, vibration, buckling analysis of bi-directional FG porous microbeams with a variable material length scale parameter", Appl. Math. Model., 91, 723-748. https://doi.org/10.1016/j.apm.2020.09.058.
  49. Karamanli, A., Aydogdu, M. and Vo, T.P. (2021), "A comprehensive study on the size-dependent analysis of strain gradient multi-directional functionally graded microplates via finite element model", Aerosp. Sci. Technol., 111, 106550. https://doi.org/10.1016/j.ast.2021.106550.
  50. Li, M., Soares, C.G. and Yan, R. (2020), "A novel shear deformation theory for static analysis of functionally graded plates", Compos. Struct., 250, 112559. https://doi.org/10.1016/j.compstruct.2020.112559.
  51. Li, M., Soares, C.G. and Yan, R. (2021a), "Free vibration analysis of FGM plates on Winkler/Pasternak/Kerr foundation by using a simple quasi-3D HSDT", Compos. Struct., 264, 113643. https://doi.org/10.1016/j.compstruct.2021.113643.
  52. Li, M., Yan, R., Xu, L. and Soares, C.G. (2021), "A general framework of higher-order shear deformation theories with a novel unified plate model for composite laminated and FGM plates", Compos. Struct., 261, 113560. https://doi.org/10.1016/j.compstruct.2021.113560.
  53. Li, M., Yan, R., Xu, L. and Soares, C.G. (2021b), "A general framework of higher-order shear deformation theories with a novel unified plate model for composite laminated and FGM plates", Compos. Struct., 261, 113560. https://doi.org/10.1016/j.compstruct.2021.113560.
  54. Lieu, Q.X., Lee, S., Kang, J. and Lee, J. (2018), "Bending and free vibration analyses of in-plane bi-directional functionally graded plates with variable thickness using isogeometric analysis", Compos. Struct., 192, 434-451. https://doi.org/10.1016/j.compstruct.2018.03.021.
  55. Liu, B., Ma, Z., Li, J., Xie, H., Wei, X., Wang, B. and Zhao, D. (2022), "Experimental study of a 3D printed permanent implantable porous Ta-coated bone plate for fracture fixation", Bioactive Mater., 10, 269-280. https://doi.org/10.1016/j.bioactmat.2021.09.009.
  56. Mahapatra, T.R., Kar, V.R., Panda, S.K. and Mehar, K. (2017), "Nonlinear thermoelastic deflection of temperature-dependent FGM curved shallow shell under nonlinear thermal loading", J. Thermal Stresses, 40(9), 1184-1199. https://doi.org/10.1080/01495739.2017.1302788.
  57. Mahapatra, T.R., Panda, S.K. and Kar, V.R. (2016), "Nonlinear flexural analysis of laminated composite panel under hygrothermo-mechanical loading-a micromechanical approach", Int. J. Comput. Meth., 13(03), 1650015. https://doi.org/10.1142/S0219876216500158.
  58. Mantari, J.L. and Soares, C.G. (2013), "A novel higher-order shear deformation theory with stretching effect for functionally graded plates", Compos. Part B: Eng., 45(1), 268-281. http://dx.doi.org/10.1016/j.compositesb.2012.05.036.
  59. Mantari, J.L., Oktem, A.S. and Soares, C.G. (2012), "Bending response of functionally graded plates by using a new higher order shear deformation theory", Compos. Struct., 94(2), 714-723. https://doi.org/10.1016/j.compstruct.2011.09.007.
  60. Mantari, J.L., Oktem, A.S. and Soares, C.G. (2012), "Bending response of functionally graded plates by using a new higher order shear deformation theory", Compos. Struct., 94(2), 714-723. https://doi.org/10.1016/j.compstruct.2011.09.007.
  61. Mehar, K., Panda, S.K. and Mahapatra, T.R. (2017a), "Thermoelastic nonlinear frequency analysis of CNT reinforced functionally graded sandwich structure", Europ. J. Mech. A/Solids, 65, 384-396. https://doi.org/10.1016/j.euromechsol.2017.05.005.
  62. Mehar, K., Panda, S.K., Bui, T.Q. and Mahapatra, T.R. (2017b), "Nonlinear thermoelastic frequency analysis of functionally graded CNT-reinforced single/doubly curved shallow shell panels by FEM", J. Thermal Stresses, 40(7), 899-916. https://doi.org/10.1080/01495739.2017.1318689.
  63. Melaibari, A., Daikh, A.A., Basha, M., Abdalla, A.W., Othman, R., Almitani, K.H. and Eltaher, M.A. (2022), "Free vibration of FG-CNTRCs nano-plates/shells with temperature-dependent properties", Mathematics, 10(4), 583. https://doi.org/10.3390/math10040583.
  64. Melaibari, A., Daikh, A.A., Basha, M., Wagih, A., Othman, R., Almitani, K.H. and Eltaher, M.A. (2022), "A dynamic analysis of randomly oriented functionally graded carbon nanotubes/fiber-reinforced composite laminated shells with different geometries", Mathematics, 10(3), 408. https://doi.org/10.3390/math10030408.
  65. Melaibari, A., Khoshaim, A.B., Mohamed, S.A. and Eltaher, M.A. (2020), "Static stability and of symmetric and sigmoid functionally graded beam under variable axial load", Steel Compos. Struct., 35(5), 671-685. https://doi.org/10.12989/scs.2020.35.5.671.
  66. Mohamed, S.A. (2020), "A fractional differential quadrature method for fractional differential equations and fractional eigenvalue problems", Math. Meth. Appl. Sci., https://doi.org/10.1002/mma.6753.
  67. Mohamed, S.A., Mohamed, N.A. and Abo-Hashem, S.I. (2021), "A novel differential-integral quadrature method for the solution of nonlinear integro-differential equations", Math. Meth. Appl. Sci., 44(18), 13945-13967. https://doi.org/10.1002/mma.7667.
  68. Neves, A.M.A., Ferreira, A.J., Carrera, E., Cinefra, M., Jorge, R. M.N. and Soares, C.M.M. (2012a), "Static analysis of functionally graded sandwich plates according to a hyperbolic theory considering Zig-Zag and warping effects", Adv. Eng. Softw., 52, 30-43. http://dx.doi.org/10.1016/j.advengsoft.2012.05.005.
  69. 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. (2012b), "A quasi3D hyperbolic shear deformation theory for the static and free vibration analysis of functionally graded plates", Compos. Struct., 94(5), 1814-1825. https://doi.org/10.1016/j.compstruct.2011.12.005
  70. Noor, A.K. (1973), "Free vibrations of multilayered composite plates", AIAA J., 11(7), 1038-1039. https://doi.org/10.2514/3.6868.
  71. Ohab-Yazdi, S.M.K. and Kadkhodayan, M. (2021), "Free vibration of bi-directional functionally graded imperfect nanobeams under rotational velocity", Aerosp. Sci. Technol., 119, 107210. https://doi.org/10.1016/j.ast.2021.107210.
  72. Pagano, N.J. (1970), "Exact solutions for rectangular bidirectional composites and sandwich plates", J. Compos. Mater., 4(1), 20-34. https://doi.org/10.1177/002199837000400102.
  73. Qiao, D., Yan, J., Feng, C., Liang, H., Ning, D., Li, B. and Johanning, L. (2021), "Numerical analysis on wave load reduction effect of a solid wall with porous plate by macroscopic CFD approach", Ocean Eng., 237, 109624. https://doi.org/10.1016/j.oceaneng.2021.109624.
  74. Quan, T.Q. and Duc, N.D. (2022), "Analytical solutions for nonlinear vibration of porous functionally graded sandwich plate subjected to blast loading", Thin-Wall. Struct., 170, 108606. https://doi.org/10.1016/j.tws.2021.108606.
  75. Rad, A.B. (2018), "Static analysis of non-uniform 2D functionally graded auxetic-porous circular plates interacting with the gradient elastic foundations involving friction force", Aerosp. Sci. Technol., 76, 315-339. https://doi.org/10.1016/j.ast.2018.01.036.
  76. Reddy, J. (1984), "A simple higher-order theory for laminated composite plates", J. Appl. Mech., 51(4), 745-752. https://doi.org/10.1115/1.3167719.
  77. 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.
  78. Shen, Z., Xia, J. and Cheng, P. (2019), "Geometrically nonlinear dynamic analysis of FG-CNTRC plates subjected to blast loads using the weak form quadrature element method", Compos. Struct., 209, 775-788. https://doi.org/10.1016/j.compstruct.2018.11.009.
  79. Shu, C. (2012), Differential Quadrature and Its Application in Engineering. Springer Science & Business Media.
  80. Sobhy, M. and Zenkour, A.M. (2019), "Porosity and inhomogeneity effects on the buckling and vibration of doubleFGM nanoplates via a quasi-3D refined theory", Compos. Struct., 220, 289-303. https://doi.org/10.1016/j.compstruct.2019.03.096.
  81. Tahir, S.I., Chikh, A., Tounsi, A., Al-Osta, M.A., Al-Dulaijan, S. U. and Al-Zahrani, M.M. (2021a), "Wave propagation analysis of a ceramic-metal functionally graded sandwich plate with different porosity distributions in a hygro-thermal environment", Compos. Struct., 269, 114030. https://doi.org/10.1016/j.compstruct.2021.114030.
  82. Tahir, S.I., Tounsi, A., Chikh, A., Al-Osta, M.A., Al-Dulaijan, S. U. and Al-Zahrani, M.M. (2021b), "An integral four-variable hyperbolic HSDT for the wave propagation investigation of a ceramic-metal FGM plate with various porosity distributions resting on a viscoelastic foundation", Waves Random Complex Media, 1-24. https://doi.org/10.1080/17455030.2021.1942310.
  83. Tahouneh, V. and Naei, M.H. (2014), "A novel 2-D six-parameter power-law distribution for three-dimensional dynamic analysis of thick multi-directional functionally graded rectangular plates resting on a two-parameter elastic foundation", Meccanica, 49(1), 91-109. https://doi.org/10.1007/s11012-013-9776-x.
  84. Taibi, F.Z., Benyoucef, S., Tounsi, A., Bachir Bouiadjra, R., Adda Bedia, E.A. and Mahmoud, S.R. (2015), "A simple shear deformation theory for thermo-mechanical behaviour of functionally graded sandwich plates on elastic foundations", J. Sandw. Struct. Mater., 17(2), 99-129. https://doi.org/10.1177/1099636214554904.
  85. Thai, H.T. and Choi, D.H. (2013), "Finite element formulation of various four unknown shear deformation theories for functionally graded plates", Finite Elements Anal. Des., 75, 50-61. http://dx.doi.org/10.1016/j.finel.2013.07.003.
  86. Thai, H.T. and Kim, S.E. (2012), "Analytical solution of a two variable refined plate theory for bending analysis of orthotropic Levy-type plates", Int. J. Mech. Sci., 54(1), 269-276. https://doi.org/10.1016/j.ijmecsci.2011.11.007.
  87. Touratier, M. (1991), "An efficient standard plate theory", Int.l J. Eng. Sci., 29(8), 901-916. https://doi.org/10.1016/0020-7225(91)90165-Y.
  88. Tran, L.V., Lee, J., Nguyen-Van, H., Nguyen-Xuan, H. and Wahab, M.A. (2015), "Geometrically nonlinear isogeometric analysis of laminated composite plates based on higher-order shear deformation theory", Int. J. Non-Linear Mech., 72, 42-52. http://dx.doi.org/10.1016/j.ijnonlinmec.2015.02.007.
  89. Wu, C.P. and Chiu, K.H. (2011), "RMVT-based meshless collocation and element-free Galerkin methods for the quasi-3D free vibration analysis of multilayered composite and FGM plates", Compos. Struct., 93(5), 1433-1448. https://doi.org/10.1016/j.compstruct.2010.11.015.
  90. Wu, C.P. and Yu, L.T. (2018), "Quasi-3D static analysis of twodirectional functionally graded circular plates", Steel Compos. Struct., 27(6), 789-801. https://doi.org/10.12989/scs.2018.27.6.789.
  91. Yu, T., Yin, S., Bui, T.Q., Xia, S., Tanaka, S. and Hirose, S. (2016), "NURBS-based isogeometric analysis of buckling and free vibration problems for laminated composites plates with complicated cutouts using a new simple FSDT theory and level set method", Thin-Wall. Struct., 101, 141-156. http://dx.doi.org/10.1016/j.tws.2015.12.008.
  92. Zarga, D., Tounsi, A., Bousahla, A.A., Bourada, F. and Mahmoud, S.R. (2019), "Thermomechanical bending study for functionally graded sandwich plates using a simple quasi-3D shear deformation theory", Steel Compos. Struct., 32(3), 389-410. https://doi.org/10.12989/scs.2019.32.3.389.
  93. Zenkour, A.M. (2006), "Generalized shear deformation theory for bending analysis of functionally graded plates", Appl. Math. Model., 30(1), 67-84. https://doi.org/10.1016/j.apm.2005.03.009.
  94. Zong, Z. (2009), Advanced Differential Quadrature Methods. Chapman and Hall/CRC.