Geomechanics and Engineering

  • 제28권1호
  • /
  • Pages.49-64
  • /
  • 2022
  • /
  • 2005-307X(pISSN)
  • /
  • 2092-6219(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

Effect of the variable visco-Pasternak foundations on the bending and dynamic behaviors of FG plates using integral HSDT model

  • Hebali, Habib (Material and Hydrology Laboratory, University of Sidi Bel Abbes, Faculty of Tech nology, Civil Engineering Department) ;
  • Chikh, Abdelbaki (Material and Hydrology Laboratory, University of Sidi Bel Abbes, Faculty of Tech nology, Civil Engineering Department) ;
  • Bousahla, Abdelmoumen Anis (Laboratoire de Modelisation et Simulation Multi-echelle, Universite de Sidi Bel Abbes) ;
  • Bourada, Fouad (Material and Hydrology Laboratory, University of Sidi Bel Abbes, Faculty of Tech nology, Civil Engineering Department) ;
  • Tounsi, Abdeldjebbar (Material and Hydrology Laboratory, University of Sidi Bel Abbes, Faculty of Tech nology, Civil Engineering Department) ;
  • Benrahou, Kouider Halim (Material and Hydrology Laboratory, University of Sidi Bel Abbes, Faculty of Tech nology, Civil Engineering Department) ;
  • Hussain, Muzamal (Department of Mathematics, Govt. College University Faisalabad) ;
  • Tounsi, Abdelouahed (Material and Hydrology Laboratory, University of Sidi Bel Abbes, Faculty of Tech nology, Civil Engineering Department)
  • 투고 : 2020.06.26
  • 심사 : 2021.12.01
  • 발행 : 2022.01.10
⟨ 이전 논문 다음 논문 ⟩

초록

In this work, the bending and dynamic behaviors of advanced composite plates resting on variable visco-Pasternak foundations are studied using a simple shear deformation integral plate model. The research is carried out with a view to a three-parameter foundation including the influences of the variable Winkler coefficient, the constant Pasternak coefficient and the damping coefficient of the elastic medium. The present theory uses a displacement field with integral terms instead of derivative terms by including also the shear deformation effect without introducing the shear correction factors. The equations of motion for advanced composite plates are obtained using the Hamilton principle. Analytical solutions for the bending and dynamic analysis are deduced for simply supported plates resting on variable visco-Pasternak foundations. Some numerical results are presented to demonstrate the impact of material index, elastic foundation type, and damping coefficient of the foundation, on the bending and dynamic responses of advanced composite plates.

키워드

참고문헌

  1. Abdelrahman, W.G. (2020), "Effect of material transverse distribution profile on buckling of thick functionally graded material plates according to TSDT", Struct. Eng. Mech., 74(1), 83-90. https://doi.org/10.12989/sem.2020.74.1.08.
  2. Abdulrazzaq, M.A. Fenjan, R.M Ahmed, R.A. and Faleh, N.M. (2020), "Thermal buckling of nonlocal clamped exponentially graded plate according to a secant function based refined theory", Steel Compos. Struct., 35(1), 147-157. http://doi.org/10.12989/scs.2020.35.1.147.
  3. Abed, Z.A.K. and Majeed, W.I. (2020), "Effect of boundary conditions on harmonic response of laminated plates", Compos. Mater. Eng., 2(2). 125-140. https://doi.org/10.12989/cme.2020.2.2.125.
  4. Ahmed, R.A., Fenjan, R.M. and Faleh, N.M. (2019), "Analyzing post-buckling behavior of continuously graded FG nanobeams with geometrical imperfections", Geomech. Eng., 17(2), 175-180. https://doi.org/10.12989/gae.2019.17.2.175.
  5. Ahmed, R.A., Moustafa, N.M., Faleh, N.M. and Fenjan, R.M. (2020), "Nonlocal nonlinear stability of higher-order porous beams via Chebyshev-Ritz method", Struct. Eng. Mech., 76(3), 413-420. http://doi.org/10.12989/sem.2020.76.3.413.
  6. Akavci, S.S. and Tanrikulu, A.H. (2015), "Static and free vibration analysis of functionally graded plates based on a new quasi-3D and 2D shear deformation theories", Compos. Part B, 83, 203-215. https://doi.org/10.1016/j.compositesb.2015.08.043.
  7. Akbas, S.D. (2020a), "Dynamic responses of laminated beams under a moving load in thermal environment", Steel Compos. Struct., 35(6), 729-737. https://doi.org/10.12989/scs.2020.35.6.729.
  8. Akbas, S.D. (2015), "Wave propagation of a functionally graded beam in thermal environments", Steel Compos. Struct., 19(6), 1421-1447. https://doi.org/10.12989/scs.2015.19.6.1421.
  9. Akbas, S.D. (2020b), "Modal analysis of viscoelastic nanorods under an axially harmonic load", Adv. Nano Res., 8(4), 277-282. http://doi.org/10.12989/anr.2020.8.4.277.
  10. Akgun, G. and Kurtaran, H. (2019), "Large displacement transient analysis of FGM super-elliptic shells using GDQ method", Thin-Wall. Struct., 141, 133-152. https://doi.org/10.1016/j.tws.2019.03.049.
  11. Al-Basyouni, K.S., Ghandourah, E., Mostafa, H.M. and Algarni, A. (2020), "Effect of the rotation on the thermal stress wave propagation in non-homogeneous viscoelastic body", Geomech. Eng., 21(1), 1-9. https://doi.org/10.12989/gae.2020.21.1.001.
  12. Avcar, M. (2019), "Free vibration of imperfect sigmoid and power law functionally graded beams", Steel Compos. Struct., 30(6), 603-615. https://doi.org/10.12989/scs.2019.30.6.603.
  13. Baferani, H.A., Saidi, A.R. and Ehteshami, H. (2011), "Accurate solution for free vibration analysis of functionally graded thick rectangular plates resting on elastic foundation", Compos. Struct., 93(7), 1842-1853. http://doi.org/10.1016/j.compstruct.2011.01.020.
  14. Carrera, E., Brischetto, S. andRobaldo, A. (2008), "Variable Kinematic Model for the Analysis of Functionally Graded Material plates", AIAA J., 46(1), 194-203. http://doi.org/10.2514/1.32490.
  15. Carrera, E., Brischetto, S., Cinefra, M. and Soave, M. (2011), "Effects of thickness stretching in functionally graded plates and shells", Compos. Part B: Eng., 42(2), 123-133. http://doi.org/10..1016/j.compositesb.2010.10.005.
  16. Chen, C.S., Hsu, C.Y. and Tzou, GJ. (2009), "Vibration and stability of functionally graded plates based on a higher-order deformation theory", J. Reinf. Plast. Compos., 28(10), 1215-1234. https://doi.org/10.1177/0731684408088884.
  17. Dehshahri, K., Nejad, M. Z., Ziaee, S., Niknejad, A. and Hadi, A. (2020), "Free vibrations analysis of arbitrary three-dimensionally FGM nanoplates", Adv. Nano Res., 8(2), 115-134. https://doi.org/10.12989/anr.2020.8.2.115.
  18. Ebrahimi, T., Nejad, M.Z., Jahankohan, H. and Hadi, A. (2021), "Thermoelastoplastic response of FGM linearly hardening rotating thick cylindrical pressure vessels", Steel Compos. Struct., 38(2), 189-211. https://doi.org/10.12989/scs.2021.38.2.189.
  19. Fallah, A., Aghdam, M.M. and Kargarnovin, M.H. (2013), "Free vibration analysis of moderately thick functionally graded plates on elastic foundation using the extended Kantorovich method", Arch. Appl. Mech., 83(2), 177-191. https://doi.org/10.1007/s00419-012-0645-1.
  20. Fares, M.E., Elmarghany, M.K. and Atta, D. (2009), "An efficient and simple refined theory for bending and vibration of functionally graded plates", Compos. Struct., 91(3), 296-305. https://doi.org/10.1016/j.compstruct.2009.05.008.
  21. Fenjan, R.M., Faleh, N.M. and Ahmed, R.A. (2020b), "Geometrical imperfection and thermal effects on nonlinear stability of microbeams made of graphene-reinforced nanocomposites", Adv. Nano Res., 9(3). 147-156. http://doi.org/10.12989/anr.2020.9.3.147.
  22. Fenjan, R.M., Faleh, N.M. and Ahmed R.A. (2020c), "Strain gradient based static stability analysis of composite crystalline shell structures having porosities", Steel Compos. Struct., 36(6), 631-642. http://doi.org/10.12989/scs.2020.36.6.631.
  23. Fenjan, R.M., Moustafa, N.M. and Faleh, N.M. (2020a), "Scale-dependent thermal vibration analysis of FG beams having porosities based on DQM", Adv. Nano Res., 8(4), 283-292. http://doi.org/10.12989/anr.2020.8.4.283.
  24. Ferreira, A.J.M., Carrera, E., Cinefra, M., Roque, C.M.C. and Polit, O. (2011), "Analysis of laminated shells by a sinusoidal shear deformation theory and radial basis functions collocation, accounting for through-the-thickness deformations", Compos. Part B: Eng., 42(5), 1276-1284. https://doi.org/10.1016/j.compositesb.2011.01.031.
  25. Ferreira, A.J.M., Roque, C.M.C. and Jorge, R.M.N. (2005), "Analysis of composite plates by trigonometric shear deformation theory and multiquadrics", Comput. Struct., 83(27), 2225-2237. https://doi.org/10.1016/j.compstruc.2005.04.002.
  26. Hirwani, C.K., Panda, S.K. and Mahapatra, T.R. (2018a). "Nonlinear finite element analysis of transient behaviour of delaminated composite plate", J. Vib. Acoust., 140(2). https://doi.org/10.1115/1.4037848.
  27. Hirwani, C.K., Panda, S.K., Mahapatra, T.R. and Mahapatra, S.S. (2017), "Nonlinear transient finite-element analysis of delaminated composite shallow shell panels", AIAA J., 55(5), 1734-1748. https://doi.org/10.2514/1.J055624.
  28. Hirwani, C.K., Panda, S.K. and Mahapatra, T.R. (2018b). "Thermomechanical deflection and stress responses of delaminated shallow shell structure using higher-order theories", Compos. Struct., 184, 135-145. S0263822317325023 https://doi.org/10.1016/j.compstruct.2017.09.071.
  29. Hirwani, C.K. and Panda, S.K. (2018), "Numerical and experimental validation of nonlinear deflection and stress responses of pre-damaged glass-fibre reinforced composite structure", Ocean Eng., 159, 237-252. https://doi.org/10.1016/j.oceaneng.2018.04.035.
  30. Hirwani, C.K. and Panda, S.K. (2019a), "Nonlinear finite element solutions of thermoelastic deflection and stress responses of internally damaged curved panel structure", Appl. Math. Model., 65, 303-317. S0307904X18304037. https://doi.org/10.1016/j.apm.2018.08.014.
  31. Hirwani, C.K. and Panda, S.K. (2019b), "Nonlinear transient analysis of delaminated curved composite structure under blast/pulse load", Eng. with Comput., 1-14. https://doi.org/10.1007/s00366-019-00757-6.
  32. Hirwani, C.K., Panda, S.K., Mahapatra, S.S., Mandal, S.K., Srivastava, L. and Buragohain, M.K. (2018c), "Flexural strength of delaminated composite plate - An experimental validation", Int. J. Damage Mech., 27(2), 296-329. 1056789516676515. https://doi.org/10.1177/1056789516676515.
  33. Hirwani, C.K., Panda, S.K., Mahapatra, T.R., Mandal, S.K., Mahapatra, S.S. and De, A.K. (2018e), "Delamination effect on flexural responses of layered curved shallow shell panel-experimental and numerical analysis", Int. J. Comput. Method., 15(4), 1850027. https://doi.org/10.1142/S0219876218500275.
  34. Hirwani, C.K., Panda, S.K., Mahapatra, T.R. and Mahapatra, S.S. (2017b). "Numerical study and experimental validation of dynamic characteristics of delaminated composite flat and curved shallow shell structure", J. Aerosp. Eng., 30(5), 04017045. https://doi.org/10.1061/(ASCE)AS.1943-5525.0000756.
  35. Hirwani, C.K., Panda, S.K. and Patle, B.K. (2018d), "Theoretical and experimental validation of nonlinear deflection and stress responses of an internally debonded layer structure using different higher-order theories", Acta Mechanica, 229(8), 3453-3473. https://doi.org/10.1007/s00707-018-2173-8.
  36. Hosseini-Hashemi, S., Rokni Damavandi Taher, H., Akhavan, H. and Omidi, M. (2010), "Free vibration of functionally graded rectangular plates using first-order shear deformation plate theory", Appl. Math. Model., 34(5), 1276-1291. https://doi.org/10.1016/j.apm.2009.08.008.
  37. 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. https://doi.org/10.1016/j.compstruct.2012.09.034.
  38. Kar, V.R. and Panda, S.K. (2015), "Nonlinear flexural vibration of shear deformable functionally graded spherical shell panel", Steel Compos. Struct., 18(3), 693-709. https://doi.org/10.12989/scs.2015.18.3.693.
  39. Karakoti, A., Pandey, S. and Kar, V.R. (2021), "Dynamic responses analysis of P and S-FGM sandwich cylindrical shell panels using a new layerwise method", Struct. Eng. Mech., 80(4), 417-432. https://doi.org/10.12989/sem.2021.80.4.417.
  40. Karami, B. and Janghorban, M. (2019), "A new size-dependent shear deformation theory for wave propagation analysis of triclinic nanobeams", Steel Compos. Struct., 32(2), 213-223. https://doi.org/10.12989/scs.2019.32.2.213.
  41. Karami, B. and Karami, S. (2019), "Buckling analysis of nanoplate-type temperature-dependent heterogeneous materials", Adv. Nano Res., 7(1), 51-61. https://doi.org/10.12989/anr.2019.7.1.051.
  42. Katariya, P.V., Hirwani, C.K. and Panda, S.K. (2019), "Geometrically nonlinear deflection and stress analysis of skew sandwich shell panel using higher-order theory", Eng. with Compu., 35 (2), 467-485. https://doi.org/10.1007/s00366-018-0609-3.
  43. Katariya, P.V., Mehar, K. and Panda, S.K. (2020), "Nonlinear dynamic responses of layered skew sandwich composite structure and experimental validation", Int. J. Nonlinear Mech., 125, 103527. https://doi.org/10.1016/j.ijnonlinmec.2020.103527.
  44. Katariya, P.V. and Panda, S.K. (2019a), "Numerical evaluation of transient deflection and frequency responses of sandwich shell structure using higher order theory and different mechanical loadings", Eng. with Comput., 35(3), 1009-1026. https://doi.org/10.1007/s00366-018-0646-y.
  45. Katariya, P.V. and Panda, S.K. (2019b), "Numerical frequency analysis of skew sandwich layered composite shell structures under thermal environment including shear deformation effects", Struct. Eng. Mech., 71(6), 657-668. https://doi.org/10.12989/sem.2019.71.6.657.
  46. Koizumi, M. (1997), "FGM activities in Japan", Compos. Part B: Eng, 28, 1-4. https://doi.org/10.1016/S1359-8368(96)00016-9.
  47. Kolahchi, R., Safari, M. and Esmailpour, M. (2016), "Dynamic stability analysis of temperature-dependent functionally graded CNT-reinforced visco-plates resting on orthotropic elastomeric medium", Compos. Struct., 150, 255-265. https://doi.org/10.1016/j.compstruct.2016.05.023.
  48. Liu, W., Liu, S., Fan, M., Tian, W., Wang, J. and Tahouneh, V. (2020), "Influence of internal pores and graphene platelets on vibration of non-uniform functionally graded columns", Steel Compos. Struct., 35(2), 295-303. https://doi.org/10.12989/scs.2020.35.2.295.
  49. Madenci, E. (2019), "A refined functional and mixed formulation to static analyses of fgm beams", Struct. Eng. Mech., 69(4), 427-437. https://doi.org/10.12989/sem.2019.69.4.427
  50. Mansour, W and Fayed, S. (2021), "Flexural rigidity and ductility of RC beams reinforced with steel and recycled plastic fibers", Steel Compos. Struct., 41(3), 317-334. https://doi.org/10.12989/scs.2021.41.3.317.
  51. Mantari, J.L. and Guedes Soares, C. (2012), "Generalized hybrid quasi-3D shear deformation theory for the static analysis of advanced composite plates", Compos. Struct., 94(8), 2561-2575. https://doi.org/10.1016/j.compstruct.2012.02.019.
  52. Mantari, J.L. and Guedes Soares, C. (2013), "A novel higher-order shear deformation theory with stretching effect for functionally graded plates", Compos. Part B: Eng., 45(1), 268-281. https://doi.org/10.1016/j.compositesb.2012.05.036.
  53. Mantari, J.L., Oktem, A.S. and Guedes Soares, C. (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.
  54. Mercan, K., Ebrahimi, F. and Civalek, O. (2020), "Vibration of angle-ply laminated composite circular and annular plates", Steel Compos. Struct., 34(1), 141-154. https://doi.org/10.12989/scs.2020.34.1.141.
  55. Mindlin, R.D. (1951), "Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates", J. Appl. Mech., 18(1), 31-38. https://doi.org/10.1115/1.4010217
  56. 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.
  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. (2013), "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(1), 657-674. ttps://doi.org/10.1016/j.compositesb.2012.01.089.
  58. 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.
  59. Neves, A.M.A., Ferreira, A.J.M., Carrera, E., Roque, C.M.C., Cinefra, M., Jorge, R.M.N. and Soares, C.M.M. (2012a), "A quasi-3D sinusoidal shear deformation theory for the static and free vibration analysis of functionally graded plates", Compos. Part B: Eng., 43(2), 711-725. ttps://doi.org/10.1016/j.compositesb.2011.08.009.
  60. 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.
  61. Othman, M. and Fekry, M. (2018), "Effect of rotation and gravity on generalized thermo-viscoelastic medium with voids", Multidiscip. Model. Mater. Struct., 14(2), 322-338. ttps://doi.org/10.1108/MMMS-08-2017-0082.
  62. Pandey, H.K., Agrawal, H., Panda, S.K., Hirwani, C.K., Katariya, P.V. and Dewangan, H.C. (2020), "Experimental and numerical bending deflection of cenosphere filled hybrid (Glass/Cenosphere/Epoxy) composite", Struct. Eng. Mech., 73 (6), 715-724. https://doi.org/10.12989/sem.2020.73.6.715.
  63. Panjehpour, M., Loh, E.W.K. and Deepak, T.J. (2018), "Structural insulated panels: state-of-the-art", Trends Civil Eng. its Architect., 3(1) 336-340. ttps://doi.org/10.32474/TCEIA.2018.03.000151.
  64. Pasternak, P.L. (1954), "On a new method of analysis of an elastic foundation by means of two foundation constants", Gosuedarstvennoe Izadatelstvo Literatim po Stroitelstvu i Arkhitekture, 1, 1-56.
  65. Patle, B.K., Hirwani, C.K., Panda, S.K., Katariya, P.V., Dewangan, H.C. and Sharma, N. (2020), "Nonlinear deflection responses of layered composite structure using uncertain fuzzified elastic properties", Steel Compos. Struct., 35(6), 753-763. https://doi.org/10.12989/scs.2020.35.6.753.
  66. Patnaik, S.S., Swain, A. and Roy, T. (2020), "Creep compliance and micromechanics of multi-walled carbon nanotubes based hybrid composites", Compos. Mater. Eng., 2(2), 141-152. ttps://doi.org/10.12989/cme.2020.2.2.141.
  67. Qian, L.F. and Batra, R.C. (2005), "Three-dimensional transient heat conduction in a functionally graded thick plate with a higher-order plate theory and a meshless local Petrov-Galerkin Method", Comput. Mech., 35(3), 214-226. https://doi.org/10.1007/s00466-004-0617-6.
  68. Rachedi, M.A., Benyoucef, S., Bouhadra, A., Bachir Bouiadjra, R., Sekkal, M. and Benachour, A. (2020), "Impact of the homogenization models on the thermoelastic response of FG plates on variable elastic foundation", Geomech. Eng., 22(1), 65-80. https://doi.org/10.12989/gae.2020.22.1.065.
  69. Rashidpour, P., Ghadiri, M. and Zajkani, A. (2021), "The response of viscoelastic composite laminated microplate under low-velocity impact based on nonlocal strain gradient theory for different boundary conditions", Steel Compos. Struct., 41(3), 335-351. https://doi.org/10.12989/scs.2021.41.3.335.
  70. Reddy, J.N. (2000), "Analysis of functionally graded plates", Int. J. Numer. Method. 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.
  71. Reddy, J.N. (2011), "A general nonlinear third-order theory of functionally graded plates", Int. J. Aerosp. Lightweight Struct., 1(1), 1-21. https://doi.org/10.3850/S201042861100002X.
  72. Reissner, E. (1945), "The effect of transverse shear deformation on the bending of elastic plates", J. Appl. Mech., 12(2), 69-72.
  73. Sadoughifar, A., Farhatnia, F., Izadinia, M. and Talaeetaba, S.B. (2020), "Size-dependent buckling behaviour of FG annular/circular thick nanoplates with porosities resting on Kerr foundation based on new hyperbolic shear deformation theory", Struct. Eng. Mech., 73(3), 225-238. https://doi.org/10.12989/sem.2020.73.3.225.
  74. Sahoo, S.S., Panda, S.K., Mahapatra, T.R. and Hirwani, C.K. (2019), "Numerical analysis of transient responses of delaminated layered structure using different mid-plane theories and experimental validation", Iran J. Sci. Technol. T. Mech. Eng., 43,41-56. https://doi.org/10.1007/s40997-017-0111-3.
  75. Selmi, A. (2020), "Dynamic behavior of axially functionally graded simply supported beams", Smart Struct. Syst., 25(6), 669-678. https://doi.org/10.12989/sss.2020.25.6.669.
  76. 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.
  77. Shakouri, M. (2021), "Analytical solution for stability analysis of joined cross-ply thin laminated conical shells under axial compression", Compos. Mater. Eng., 3(2), 117-134. https://doi.org/10.12989/cme.2021.3.2.117.
  78. Singh, V.K., Hirwani, C.K., Panda, S.K., Mahapatra, T.R. and Mehar, K. (2019), "Numerical and experimental nonlinear dynamic response reduction of smart composite curved structure using collocation and non-collocation configuration", Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 233(5), 095440621877436-. doi:10.1177/0954406218774362.
  79. Sobhy, M. (2013), "Buckling and free vibration of exponentially graded sandwich plates resting on elastic foundations under various boundary conditions", Compos. Struct., 99, 76-87. https://doi.org/10.1016/j.compstruct.2012.11.018.
  80. Talha, M. and Singh, B.N. (2010), "Static response and free vibration analysis of FGM plates using higher order shear deformation theory", Appl. Math. Model., 34(12), 3991-4011. https://doi.org/10.1016/j.apm.2010.03.034.
  81. Thai, H.T. and Choi, D.H. (2012), "A refined shear deformation theory for free vibration of functionally graded plates on elastic foundation", Compos. Part B: Eng., 43(5), 2335-2347. https://doi.org/10.1016/j.compositesb.2011.11.062.
  82. Thai, H.T. and Choi, D.H. (2014), "Zeroth-order shear deformation theory for functionally graded plates resting on elastic foundation", Int. J. Mech. Sci., 78, 35-43. ttps://doi.org/10.1016/j.ijmecsci.2013.09.020.
  83. Thai, H.T. and Choi, D.H. (2011), "A refined plate theory for functionally graded plates resting on elastic foundation", Compos. Sci. Technol., 71(16), 1850-1858. https://doi.org/10.1016/j.compscitech.2011.08.016.
  84. Thai, H.T. and Kim, S.E. (2013), "A simple quasi-3D sinusoidal shear deformation theory for functionally graded plates", Compos. Struct., 99, 172-180. https://doi.org/10.1016/j.compstruct.2012.11.030.
  85. Timesli, A. (2020), "An efficient approach for prediction of the nonlocal critical buckling load of double-walled carbon nanotubes using the nonlocal Donnell shell theory", SN Appl. Sci., 2, 407. https://doi.org/10.1007/s42452-020-2182-9
  86. Wang, H., Yan, W. and Li, C. (2020), "Response of angle-ply laminated cylindrical shells with surface-bonded piezoelectric layers", Struct. Eng. Mech., 76(5), 599-611. https://doi.org/10.12989/SEM.2020.76.5.599.
  87. Winkler, E. (1867), "Die lehre von der elasticitaet und festigkeit", Prague: Prag Dominicus; 1867.
  88. Xiang, S., Jin, Y.X., Bi, Z.Y., Jiang, S.X. and Yang, M.S. (2011), "A n-order shear deformation theory for free vibration of functionally graded and composite sandwich plates", Compos. Struct., 93(11), 2826-2832. https://doi.org/10.1016/j.compstruct.2011.05.022.
  89. Xiang, S. and Kang, G.W. (2013), "A nth-order shear deformation theory for the bending analysis on the functionally graded plates", Eur. J. Mech.-A/Solids, 37, 336-343. https://doi.org/10.1016/j.euromechsol.2012.08.005.
  90. Yuan, Y., Zhao, K., Zhao, Y. and Kiani, K. (2020), "Nonlocal-integro-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.
  91. Zenkour, A.M. (2009), "The refined sinusoidal theory for FGM plates on elastic foundations", Int. J. Mech. Sci., 51(11-12), 869-880. https://doi.org/10.1016/j.ijmecsci.2009.09.026.