Steel and Composite Structures

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Mechanical and thermal buckling analysis of laminated composite plates

  • Kettaf, Fatima Zohra (Departement de Genie Mecanique, Faculte de Genie Mecanique, Universite des Sciences et de la Technologie d'Oran Mohamed-Boudiaf) ;
  • Beguediab, Mohamed (Laboratoire des Materiaux et Systemes Reactifs, Universite de Sidi Bel Abbes, Faculte de Technologie) ;
  • Benguediab, Soumia (Material and Hydrology Laboratory, Faculty of Technology, Civil Engineering Department, University of SidiBel Abbes) ;
  • Selim, Mahmoud M. (Department of Mathematics, Al-Aflaj College of Science and Humanities, Prince Sattam bin Abdulaziz University) ;
  • Tounsi, Abdelouahed (Material and Hydrology Laboratory, Faculty of Technology, Civil Engineering Department, University of SidiBel Abbes) ;
  • Hussain, Muzamal (Department of Mathematics, Govt. College University Faisalabad)
  • Received : 2020.03.30
  • Accepted : 2021.07.22
  • Published : 2021.09.10
⟨ Previous Next ⟩

Abstract

The mechanical and thermal buckling analysis of laminated composite plates is presented in this document. Different theories of thick plates taking into account the parabolic distribution of transverse shear stresses and satisfying the condition of zero shear stresses on the top and bottom surfaces without using shear correction factor are presented and a comparison between the results obtained by these theories is also illustrated. The high order nonlinear stress-displacement relation of the plates was taken into consideration. The principle of potential energy is used to obtain the equations of equilibrium. The closed-form solutions of symmetric and antisymmetric cross-ply are obtained using Navier solution. Using math software Maple, the temperatures and the critical loads of buckling are determined. Finally, a parametric study of the influence of the various parameters such as: mode of buckling, the geometrical ratios a / b and a / h, Young's modulus, Coefficient of thermal expansion, loading type, the orientation of the fibers and the number of layers on the critical buckling temperature and the critical buckling charge is shown and discussed. Numerical results indicate that deformation due to transverse shear has a significant effect on both mechanical and thermal behavior of buckling of laminated simply supported plates.

Keywords

References

  1. 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.
  2. Akbas, S.D., et al. (2020), "Dynamic analysis of viscoelastic functionally graded porous thick beams under pulse load", Eng. with Comput., https://doi.org/10.1007/s00366-020-01070-3.
  3. 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.
  4. Ambartsumyan, S.A. (1970), Theory of Anisotropic Plate, (1st edition), Technomic Publishing Co.
  5. Arefi, M. and Zur, K.K. (2020), "Free vibration analysis of functionally graded cylindrical nanoshells resting on Pasternak foundation based on two-dimensional analysis", Steel Compos. Struct., 34(4), 615-623. http://dx.doi.org/10.12989/scs.2020.34.4.615.
  6. Asghar, S., Naeem, M.N., Hussain, M. and Tounsi, A. (2020), "Nonlocal vibration of DWCNTs based on Flugge shell model using wave propagation approach", Steel Compos. Struct., 34(4), 599-613. https://doi.org/10.12989/scs.2020.34.4.599.
  7. Ashraf, M.A., et al. (2020), "Effects of elastic foundation on the large-amplitude vibration analysis of functionally graded GPLRC annular sector plates", Eng. with Comput., https://doi.org/10.1007/s00366-020-01068-x.
  8. 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.
  9. Bamdad, M., Mohammadimehr, M. and Alambeigi, K. (2020), "Bending and buckling analysis of sandwich Reddy beam considering shape memory alloy wires and porosity resting on Vlasov's foundation", Steel Compos. Struct., 36(6), 671-687. http://dx.doi.org/10.12989/scs.2020.36.6.671.
  10. Batou, B., Nebab, M., Bennai, R., AitAtmane, H., Tounsi, A. and Bouremana, M. (2019), "Wave dispersion properties in imperfect sigmoid plates using various HSDTs", Steel Compos. Struct., 33(5), 699-716. https://doi.org/10.12989/scs.2019.33.5.699.
  11. Birman, V. (1995), "Stability of functionally graded hybrid composite plates", Compos. Eng., 5(7), 913-921. https://doi.org/10.1016/0961-9526(95)00036-M.
  12. Birman, V. and Bert, C.W. (1993), "Buckling and Post-buckling of Composite Plates and Shells Subjected to Elevated Temperature", J. Appl. Mech., 60(2), 514-519. https://doi.org/10.1115/1.2900823.
  13. Boulal, A., Bensattalah, T., Karas, A., Zidour, M., Heireche, H., and AddaBedia, E.A. (2020), 'Buckling of carbon nanotube reinforced composite plates supported by Kerr foundation using Hamilton's energy principle", Struct. Eng. Mech., 73(2), 209-223. https://doi.org/10.12989/sem.2020.73.2.209.
  14. Bouazza, M., Lairedj, A., Benseddiq, N. and Khalki, S. (2016), "A refined hyperbolic shear deformation theory for thermal buckling analysis of cross-ply laminated plates", Mech. Res. Commun., 73, 117-126. http://dx.doi.org/10.1016/j.mechrescom.2016.02.015.
  15. Dahsin, L. and Xiaoyu, L. (1996), "An overall view of laminate theories based on displacement hypothesis", J. Compos. Mater., 30(14), 1539-1561. https://doi.org/10.1177/002199839603001402.
  16. Duc, N.D., Seung-Eock, K. and Chan, D.Q. (2018), "Thermal buckling analysis of FGM sandwich truncated conical shells reinforced by FGM stiffeners resting on elastic foundations using FSDT", J. Therm. Stresses, 41(3), 331-365. https://doi.org/10.1080/01495739.2017.1398623.
  17. Eltaher, M.A. and Mohamed, S.A. (2020), "Buckling and stability analysis of sandwich beams subjected to varying axial loads", Steel Compos. Struct., 34(2), 241-260, http://dx.doi.org/10.12989/scs.2020.34.2.241.
  18. Eslami, M.R. and Javaheri, R. (1999), "Buckling of composite cylindrical shells under mechanical and thermal loads", J. Therm. Stresses, 22(6), 527-545. https://doi.org/10.1080/014957399280733.
  19. Gafour, Y., Hamidi, A., Benahmed, A., Zidour, M. and Bensattalah, T. (2020), "Porosity-dependent free vibration analysis of FG nanobeam using non-local shear deformation and energy principle", Adv. Nano Res., 8(1), 49-58. https://doi.org/10.12989/anr.2020.8.1.049.
  20. Ghandourah, E. and Abdraboh, A.M. (2020), "Dynamic analysis of functionally graded nonlocal nanobeam with different porosity models", Steel Compos. Struct., 36(3), 293-305. http://dx.doi.org/10.12989/scs.2020.36.3.293.
  21. Hamed, M.A., Mohamed, S.A. and Eltaher, M.A. (2020), "Buckling analysis of sandwich beam rested on elastic foundation and subjected to varying axial in-plane loads", Steel Compos. Struct., 34(1), 75-89, http://dx.doi.org/10.12989/scs.2020.34.1.075.
  22. Hirwani, C.K. and Panda, S.K. (2019), "Nonlinear thermal free vibration frequency analysis of delaminated shell panel using FEM", Compos. Struct., 224, 111011. https://doi.org/10.1016/j.compstruct.2019.111011.
  23. Jafari, A.A. and Eftekhari, S.A. (2011), "An efficient mixed methodology for free vibration and buckling analysis of orthotropic rectangular plates", Appl. Math. Comput., 218(6), 2670-2692. https://doi.org/10.1016/j.amc.2011.08.008.
  24. Kar, V.R. and Panda, S.K. (2016a), "Postbuckling analysis of shear deformable FG shallow spherical shell panel under nonuniform thermal environment", J. Therm. Stresses, 40(1), 25-39. https://doi.org/10.1080/01495739.2016.1207118.
  25. Kar, V.R. Mahapatra, T.R. and Panda, S.K. (2016b), "Effect of Different Temperature Load on Thermal Postbuckling Behaviour of Functionally Graded Shallow Curved Shell Panels", Compos. Struct., 160, 1236-1247. http://dx.doi.org/10.1016/j.compstruct.2016.10.125.
  26. Kar, V.R., Panda, S.K. and Mahapatra, T.R. (2016c), "Thermal Buckling Behaviour of Shear Deformable Functionally Graded Single/Doubly Curved Shell Panel with TD and TID Properties", Adv. Mater. Res., 5(4), 205-221. http://dx.doi.org/10.12989/amr.2016.5.4.205.
  27. Kar, V.R. and Panda, S.K. (2016d), "Post-buckling behaviour of shear deformable functionally graded curved shell panel under edge compression", Int. J. Mech. Sci., 115-116, 318-324. https://doi.org/10.1016/j.ijmecsci.2016.07.014.
  28. Kar, V.R. and Panda, S.K. (2020), "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.
  29. 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. Solid. Struct., 40(6), 1525-1546. https://doi.org/10.1016/S0020-7683(02)00647-9.
  30. Katariya, P.V. and Panda, S.K. (2016), "Thermal buckling and vibration analysis of laminated composite curved shell panel", Aircraft Eng. Aerosp. Technol., 88(1), 97-107. https://doi.org/10.1108/AEAT-11-2013-0202.
  31. Katariya, P.V., Panda, S.K., Hirwani, C.K., Mehar, K. and Thakare, O. (2017a), "Enhancement of thermal buckling strength of laminated sandwich composite panel structure embedded with shape memory alloy fibre", Smart Struct. Syst., 20(5), 595-605. http://dx.doi.org/10.12989/sss.2017.20.5.595.
  32. Katariya, P.V., Panda, S.K. and Mahapatra, T.R. (2017b), "Nonlinear thermal buckling behaviour of laminated composite panel structure including the stretching effect and higher-order finite element", Adv. Mater. Res., 6(4), 349-361. http://dx.doi.org/10.12989/amr.2017.6.4.349.
  33. Katariya, P.V. and Panda, S.K. (2019), "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.
  34. Katariya, P.V. and Panda, S.K. (2020), "Numerical analysis of thermal post-buckling strength of laminated skew sandwich composite shell panel structure including stretching effect", Steel Compos. Struct., 34(2), 279-288. http://dx.doi.org/10.12989/scs.2020.34.2.279.
  35. Kim, S.E., Thai, H.T. and Lee, J. (2009), "A two variable refined plate theory for laminated composite plates", Compos. Struct., 89, 197-205. https://doi.org/10.1016/j.compstruct.2008.07.017.
  36. Levinson, M. (1980), "An accurate simple theory of the statics and dynamics of elastic plates", Mech. Res. Commun., 7(6), 343-350. https://doi.org/10.1016/0093-6413(80)90049-X.
  37. Liew, K.M., Wang, J., Ng, T.Y. andTan, M.J. (2004), "Free vibration and buckling analyses of shear-deformable plates based on FSDT meshfree method", J. Sound Vib., 276, 997-1017. https://doi.org/10.1016/j.jsv.2003.08.026.
  38. Mallikarjuna, M. and Kant, T.A. (1993), "Critical review and some results of recently developed refined theories of fiber-reinforced laminated composites and sandwiches", Compos. Struct., 23(4), 293-312. https://doi.org/10.1016/0263-8223(93)90230-N.
  39. Mantari, J.L., Oktem, A.S. and GuedesSoares, C. (2011), "A new higher order shear deformation theory for sandwich and composite laminated plates", Compos. part B-Eng., 43(3), 1489-1499. https://doi.org/10.1016/j.compositesb.2011.07.017.
  40. Matsunaga, H. (2005), "Thermal buckling of cross-ply laminated composite and sandwich plates according to a global higher-order deformation theory", Compos. Struct., 68(4), 439-454. https://doi.org/10.1016/j.compstruct.2004.04.010.
  41. Mehar, K. and Panda, S.K. (2019), "Multiscale modeling approach for thermal buckling analysis of nanocomposite curved structure", Adv. Nano Res., 7(3), 181-190. http://dx.doi.org/10.12989/anr.2019.7.3.181.
  42. Mehar, K., Panda, S.K., Devarajan, Y. and Choubey, G. (2019), "Numerical buckling analysis of graded CNT-reinforced composite sandwich shell structure under thermal loading", Compos. Struct., 216(15), 406-414. https://doi.org/10.1016/j.compstruct.2019.03.002.
  43. Mehar, K., Panda, S.K. and Sharma, N. (2020), "Numerical investigation and experimental verification of thermal frequency of carbon nanotube-reinforced sandwich structure", Eng. Struct., 211, 110444. https://doi.org/10.1016/j.engstruct.2020.110444.
  44. Mehar, K. and Panda, S.K. (2020), "Numerical investigation of thermal frequency responses of graded hybrid smart nanocomposite (CNT-SMA-Epoxy) structure", Mech. Adv. Mater. Struct., https://doi.org/10.1080/15376494.2020.1725193.
  45. Meyers,C.A. and Hyer, M.W. (1991), "Thermal buckling and postbuckling of symmetrically laminated composite plates", J. Therm. Stresses, 14(4), 519-540. https://doi.org/10.1080/01495739108927083.
  46. Mindlin, R.D. (1951), "Influence of rotary inertia and shear on flexural motions of isotropic, elastic plates", J. Appl. Mech., 18(1), 31-38. https://doi.org/10.1115/1.4010217
  47. Na, K.S. and Kim, J.H. (2006), "Three-dimensional thermomechanical buckling analysis for functionally graded composite plates", Compos. Struct., 73(4), 413-422. https://doi.org/10.1016/j.compstruct.2005.02.012.
  48. Narita, Y. and Leissa, A.W. (1990), "Buckling studies for simply supported symmetrically laminated rectangular plates", Int. J. Mech. Sci., 32(11), 909-924. https://doi.org/10.1016/0020-7403(90)90063-O.
  49. Noor, A.K. and Burton, W.S. (1992), "Three-dimensional solutions for the thermal buckling and sensitivity derivatives of temperature-sensitive multilayered angle-ply plates", J. Appl. Mech., 59(4), 848-856. https://doi.org/10.1115/1.2894052.
  50. 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. https://doi.org/10.1108/MMMS-08-2017-0082.
  51. Ounis, H., Tati, A. and Benchabane, A. (2014), "Thermal buckling behavior of laminated composite plates: a finite-element study", Front. Mech. Eng., 9(1), 41-49. https://doi.org/10.1007/s11465-014-0284-z.
  52. Panc, V. (1975), Theories of Elastic Plates (Mechanics of Surface Structures, 2), Noordhof International Publishing, Leyden, Netherlands.
  53. Panda, S.K. and Singh, B.N. (2009a), "Thermal post-buckling behaviour of laminated composite cylindrical/hyperboloid shallow shell panel using nonlinear finite element method", Compos. Struct., 91(3), 366-374. https://doi.org/10.1016/j.compstruct.2009.06.004.
  54. Panda, S.K and Singh, B.N. (2009b), "Nonlinear free vibration of spherical shell panel using higher order shear deformation theory - A finite element approach", Int. J. Pressure Vessels Piping, 86(6), 373-383. https://doi.org/10.1016/j.ijpvp.2008.11.023.
  55. Panda, S.K. and Singh, B.N. (2010a), "Nonlinear free vibration analysis of thermally postbuckled composite spherical shell panel", Int. J. Mech. Mater. Des., 6, 175-188. https://doi.org/10.1007/s10999-010-9127-1
  56. Panda, S.K and Singh, B.N. (2010b), "Thermal post-buckling analysis of a laminated composite spherical shell panel embedded with shape memory alloy fibres using non-linear finite element method", Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 224(4), 757-769. https://doi.org/10.1243/09544062JMES1809.
  57. Panda, S.K and Singh, B.N. (2011), "Large amplitude free vibration analysis of thermally post-buckled composite doubly curved panel using nonlinear FEM", Finite Elem. Anal. Des., 47(4), 378-386. 0168-874X. https://doi.org/10.1016/j.finel.2010.12.008..
  58. Panda, S.K. and Singh, B.N. (2013a), "Nonlinear finite element analysis of thermal post-buckling vibration of laminated composite shell panel embedded with SMA fibre", Aerosp. Sci. Technol., 29(1), 47-57. https://doi.org/10.1016/j.ast.2013.01.007.
  59. Panda, S.K. and Singh, B.N. (2013b), "Thermal Postbuckling Behavior of Laminated Composite Spherical Shell Panel Using NFEM#", Mech. Based Des. Struc., 41(4), 468-488. https://doi.org/10.1080/15397734.2013.797330
  60. Panda, S.K. and Singh, B.N. (2013c), "Post-Buckling Analysis of Laminated Composite Doubly Curved Panel Embedded with SMA Fibers Subjected to Thermal Environment", Mech. Adv. Mater. Struct., 20(10), 842-853. https://doi.org/10.1080/15376494.2012.677097.
  61. Panda, S.K. and Singh, B.N. (2013d), "Large amplitude free vibration analysis of thermally post-buckled composite doubly curved panel embedded with SMA fibers", Nonlinear Dyn, 74, 395-418. https://doi.org/10.1007/s11071-013-0978-5.
  62. Panda, S.K. and Katariya, P.V. (2015), "Stability and Free Vibration Behaviour of Laminated Composite Panels Under Thermo-mechanical Loading", Int. J. Appl. Comput. Math., 1, 475-490. https://doi.org/10.1007/s40819-015-0035-9.
  63. Pandya, B.N. and Kant, T. (1988), "Higher-order shear deformable theories for flexure of sandwich plates-finite element evaluations", Int. J. Solid. Struct., 24(12), 419-451. https://www.civil.iitb.ac.in/~tkant/papers/TKant-IITB-JP016.pdf
  64. Panjehpour, M., Loh, E.W.K. and Deepak, T.J. (2018), "Structural Insulated Panels: State-of-the-Art", Trends Civil Eng. Architect., 3(1), 336-340. https://doi.org/10.32474/TCEIA.2018.03.000151.
  65. Peng, L.X., Liew, K.M. and Kitipornchai, S. (2006), "Buckling and free vibration analyses of stiffened plates using the FSDT mesh-free method", J. Sound Vib., 289(3), 421-449. https://doi.org/10.1016/j.jsv.2005.02.023.
  66. Pham, Q.H., et al. (2020), "Geometrically nonlinear analysis of functionally graded shells using an edge-based smoothed MITC3 (ES-MITC3) finite elements", Eng. with Comput., 36, 1069-1082. https://doi.org/10.1007/s00366-019-00750-z.
  67. Qin, Y., Luo, K.R. and Yan, X. (2020), "Buckling analysis of steel plates in composite structures with novel shape function", Steel Compos. Struct., 35(3), 405-413. https://doi.org/10.12989/scs.2020.35.3.405.
  68. Rad, M.H.G., Shahabian, F. and Hosseini, S.M. (2020), "Geometrically nonlinear dynamic analysis of FG graphene platelets-reinforced nanocomposite cylinder: MLPG method based on a modified nonlinear micromechanical model", Steel Compos. Struct., 35(1), 77-92. http://dx.doi.org/10.12989/scs.2020.35.1.077.
  69. 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.
  70. Reissner, E. (1945), "The Effect of Transverse Shear Deformation on the Bending of Elastic Plates", ASME J. Appl. Mech., 12, 68-77. https://doi.org/10.1016/j.compstruct.2008.07.017.
  71. Salah, F., Boucham, B., Bourada, F., Benzair, A., Bousahla, A.A. and Tounsi, A. (2019), "Investigation of thermal buckling properties of ceramic-metal FGM sandwich plates using 2D integral plate model", Steel Compos. Struct., 33(6), 805-822. http://dx.doi.org/10.12989/scs.2019.33.6.805.
  72. Selmi, A. (2019), "Effectiveness of SWNT in reducing the crack effect on the dynamic behavior of aluminium alloy", Adv. Nano Res., 7(5), 365-377. https://doi.org/10.12989/anr.2019.7.5.365.
  73. Shariyat, M. (2007), "Thermal buckling analysis of rectangular composite plates with temperature-dependent properties based on a layerwise theory", Thin-Wall. Struct., 45(4), 439-452. https://doi.org/10.1016/j.tws.2007.03.004.
  74. She, G.L. (2020), "Wave propagation of FG polymer composite nanoplates reinforced with GNPs", Steel Compos. Struct., 37(1), 27-35. https://doi.org/10.12989/scs.2020.37.1.027.
  75. Shi, Y., Lee, R.Y.Y. and Mei, C. (1999), "Thermal postbuckling of composite plates usingthe finite element modal coordinates method", J. Therm. Stresses, 22(6), 595-614. https://doi.org/10.1080/014957399280779.
  76. Shiau, L.C., Kuo, S.Y. and Chen, C.Y. (2010), "Thermal buckling behavior of composite laminated plates", Compos. Struct., 92(2), 508-514. https://doi.org/10.1016/j.compstruct.2009.08.035.
  77. Shojaee, S., Valizadeh, N., Izadpanah, E., Bui, T. and Vu, T. (2012), "Free vibration and buckling analysis of laminated composite plates using the NURBS-based isogeometric finite element method", Compos. Struct., 94(5), 1677-1693. https://doi.org/10.1016/j.compstruct.2012.01.012.
  78. Tayeb, T.S., Zidour, M., Bensattalah, T., Heireche, H., Benahmed, A. and Bedia, E.A. (2020), "Mechanical buckling of FG-CNTs reinforced composite plate with parabolic distribution using Hamilton's energy principle", Adv. Nano Res., 8(2), 135-148. https://doi.org/10.12989/anr.2020.8.2.135.
  79. Thanh, C.L., et al. (2020), "A geometrically nonlinear size-dependent hypothesis for porous functionally graded micro-plate", Eng. with Comput., https://doi.org/10.1007/s00366-020-01154-0.
  80. Touratier, M. (1991), "An efficient standard plate theory", Int. J. Eng. Sci., 29(8), 901-916. https://doi.org/10.1016/0020-7225(91)90165-Y.
  81. Tran, V.K., Pham, Q.H. and Nguyen-Thoi, T. (2020), "A finite element formulation using four-unknown incorporating nonlocal theory for bending and free vibration analysis of functionally graded nanoplates resting on elastic medium foundations", Eng. with Comput., https://doi.org/10.1007/s00366-020-01107-7.
  82. Wang, J., Liew, K.M., Tan, M.J. and Rajendran, S. (2002), "Analysis of rectangular laminated composite plates via FSDT meshless method", Int. J. Mech. Sci., 44(7), 1275-1293. https://doi.org/10.1016/S0020-7403(02)00057-7.
  83. Yaghoobi, H. and Fereidoon, A. (2014), "Mechanical and thermal buckling analysis of functionally graded plates resting on elastic foundations: An assessment of a simple refined nth-order shear deformation theory", Compos.Part B: Eng., 62, 54-64. https://doi.org/10.1016/j.compositesb.2014.02.014.
  84. 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, https://doi.org/10.1016/j.tws.2015.12.008.
  85. Zhang, L.W., Zhu, P. and Liew, K.M. (2014), "Thermal buckling of functionally graded plates using a local Krigingmeshless method", Compos. Struct., 108, 472-492 https://doi.org/10.1016/j.compstruct.2013.09.043.
  86. Zhao, X., Lee, Y.Y. and Liew, K.M. (2009), "Mechanical and thermal buckling analysis of functionally graded plates", Compos. Struct., 90(2), 161-171. https://doi.org/10.1016/j.compstruct.2009.03.005.