Steel and Composite Structures

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

A computational investigation on flexural response of laminated composite plates using a simple quasi-3D HSDT

  • Draiche, Kada (Material and Hydrology Laboratory, University of Sidi Bel Abbes, Faculty of Technology, Civil Engineering Department) ;
  • Selim, Mahmoud M. (Department of Mathematics, Al-Aflaj College of Science and Humanities, Prince Sattam bin Abdulaziz University) ;
  • Bousahla, Abdelmoumen Anis (Laboratoire de Modelisation et Simulation Multi-echelle, Universite de Sidi Bel Abbes) ;
  • Tounsi, Abdelouahed (Material and Hydrology Laboratory, University of Sidi Bel Abbes, Faculty of Technology, Civil Engineering Department) ;
  • Bourada, Fouad (Material and Hydrology Laboratory, University of Sidi Bel Abbes, Faculty of Technology, Civil Engineering Department) ;
  • Tounsi, Abdeldjebbar (Material and Hydrology Laboratory, University of Sidi Bel Abbes, Faculty of Technology, Civil Engineering Department) ;
  • Mahmoud, S.R. (GRC Department, Jeddah Community College, King Abdulaziz University)
  • Received : 2021.04.03
  • Accepted : 2021.10.24
  • Published : 2021.12.10
⟨ Previous Next ⟩

Abstract

In this work, a simple quasi 3-D parabolic shear deformation theory is developed to examine the bending response of antisymmetric cross-ply laminated composite plates under different types of mechanical loading. The main feature of this theory is that, in addition to including the transverse shear deformation and thickness stretching effects, it has only five-unknown variables in the displacement field modeling like Mindlin's theory (FSDT), yet satisfies the zero shear stress conditions on the top and bottom surfaces of the plate without requiring a shear correction factor. The static version of principle of virtual work was employed to derive the governing equations, while the bending problem for simply supported antisymmetric cross-ply laminated plates was solved by a Navier-type closed-form solution procedure. The adequacy of the proposed model is handled by considering the impact of side-to-thickness ratio on bending response of plate through several illustrative examples. Comparison of the obtained numerical results with the other shear deformation theories leads to the conclusion that the present model is more accurate and efficient in predicting the displacements and stresses of laminated composite plates.

Keywords

References

  1. 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. https://doi.org/10.12989/SCS.2020.35.1.147.
  2. Akbas, S.D. (2020), "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.
  3. Akgun, G. and Kurtaran, H. (2018), "Geometrically nonlinear transient analysis of laminated composite super-elliptic shell structures with generalized differential quadrature method", Int. J. Nonlinear Mech., 105, 221-241. https://doi.org/10.1016/j.ijnonlinmec.2018.05.016.
  4. Alasadi, A.A., Ahmed, R.A. and Faleh, N.M. (2019), "Analyzing nonlinear vibrations of metal foam nanobeams with symmetric and non-symmetric porosities", Adv. Aircraft Spacecraft Sci., 6(4), 273-282. https://doi.org/10.12989/aas.2019.6.4.273.
  5. 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.
  6. Ashton, J.E. and Whitney, J.M. (1970), "Theory of Laminated Plates", Progress in materials science series, Vol. 4, Lancaster, PA: Technomic Publishing Company, Inc.
  7. 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.
  8. Ayat, H., Kellouche, Y., Ghrici, M. and Boukhatem, B. (2018), "Compressive strength prediction of limestone filler concrete using artificial neural networks", Adv. Comput. Des., 3(3), 289-302. https://doi.org/10.12989/acd.2018.3.3.289.
  9. Barbero, E.J. (2011), "Introduction to composite materials design", Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business, 552 pages.
  10. Behera, S. and Kumari, P. (2018), "Free vibration of Levy-type rectangular laminated plates using efficient zig-zag theory", Adv. Comput. Des., 3(3), 213-232. https://doi.org/10.12989/acd.2018.3.3.213.
  11. Bensaid, I., Bekhadda, A. and Kerboua, B. (2018), "Dynamic analysis of higher order shear-deformable nanobeams resting on elastic foundation based on nonlocal strain gradient theory", Adv. Nano Res., 6(3), 279-298. https://doi.org/10.12989/anr.2018.6.3.279.
  12. Broutman, L.J. and Krock, R.H. (Eds.), (1974), "Composite Materials", 1-7, Academic Press, New York.
  13. Chandrashekhar, M. and Ganguli, R. (2010), "Nonlinear vibration analysis of composite laminated and sandwich plates with random material properties", Int. J. Mech. Sci., 52(7), 874-891. https://doi.org/10.1016/j.ijmecsci.2010.03.002.
  14. Chen, H., Song, H., Li, Y. and Safarpour, M. (2020), "Hygro-thermal buckling analysis of polymer-CNT-fiber-laminated nanocomposite disk under uniform lateral pressure with the aid of GDQM", Eng. with Comput., https://doi.org/10.1007/s00366-020-01102-y.
  15. Cheng, X., Zhang, J., Cheng, Y., Guo, X. and Huang, W. (2020), "Effect of curing condition on mechanical properties of scarf-repaired composite laminates", Steel Compos. Struct., 37(4), 419-429. https://doi.org/10.12989/scs.2020.37.4.419.
  16. Daniel, G., Suong, V.H. and Stephen W.T. (2003), "Composite materials: design and applications", Boca Raton, FL: CRC Press, Taylor & Francis Group, 531 pages.
  17. Daouadji, T.H. (2017), "Analytical and numerical modeling of interfacial stresses in beams bonded with a thin plate", Adv. Comput. Des., 2(1), 57-69. https://doi.org/10.12989/acd.2017.2.1.057.
  18. Dash, S., Mehar, K., Sharma, N., Mahapatra, T.R. and Panda, S.K. (2019), "Finite element solution of stress and flexural strength of functionally graded doubly curved sandwich shell panel", Earthq. Struct., 16(1), 55-67. https://doi.org/10.12989/eas.2019.16.1.055.
  19. Do, V.N.V. and Lee, C.H. (2018), "Quasi-3D higher-order shear deformation theory for thermal buckling analysis of FGM plates based on a meshless method", Aerosp. Sci. Technol., 82-83, 450-465. https://doi.org/10.1016/j.ast.2018.09.017
  20. 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. https://doi.org/10.12989/scs.2020.34.2.241.
  21. Fenjan, R.M., Moustafa, N.M. and Faleh, N.M. (2020), "Scale-dependent thermal vibration analysis of FG beams having porosities based on DQM", Adv. Nano Res., 8(4), 283-292. https://doi.org/10.12989/anr.2020.8.4.283.
  22. Ghugal, Y.M. and Sayyad, A.S. (2013), "Stress analysis of thick laminated plates using trigonometric shear deformation theory", Int. J. Appl. Mech., 5(1), 23 pages. https://doi.org/10.1142/S1758825113500038.
  23. Ghugal, Y.M. and Shimpi, R.P. (2002), "A review of refined shear deformation theories of isotropic and anisotropic laminated plates", J. Reinf. Plast. Compos., 21(9), 775-813. https://doi.org/10.1177/073168402128988481.
  24. Ghumare, S.M. and Sayyad, A.S. (2019), "A new quasi-3D model for functionally graded plates", J. Appl. Comput. Mech., 5(2), 367-380. https://doi.org/10.22055/jacm.2018.26739.1353.
  25. Ghumare, S.M. and Sayyad, A.S. (2017), "A new fifth-order shear and normal deformation theory for static bending and elastic buckling of P-FGM beams", Lat. Am. J. Solids Struct., 14(11), 1-19. https://doi.org/10.1590/1679-78253972.
  26. Han, B., Hui, W.W., Zhang, Q.C., Zhao, Z.Y., Jin, F., Zhang, Q., Lu, T.J. and Lu, B.H. (2018), "A refined quasi-3D zigzag beam theory for free vibration and stability analysis of multilayered composite beams subjected to thermo-mechanical loading", Compos. Struct., 204, 620-633. https://doi.org/10.1016/j.compstruct.2018.08.005.
  27. HassaineDaouadji, T. and Adim, B. (2017), "Mechanical behaviour of FGM sandwich plates using a quasi-3D higher order shear and normal deformation theory", Struct. Eng. Mech., 61(1), 49-63. https://doi.org/10.12989/sem.2017.61.1.049.
  28. Hirwani, C.K., Panda, S.K. and Mahapatra, S.S. (2018b), "Flexural strength of delaminated composite plate-An experimental validation", Int. J. Damage Mech., 27(2), 296-329. https://doi.org/10.1177/10567895 16676515.
  29. Hirwani, C.K., Panda, S.K. and Patle, B.K. (2018a), "Theoretical and experimental validation of nonlinear deflection and stress responses of an internally debonded layer structure using different higher-order theories", Acta Mech., 229(8), 3453-3473. https://doi.org/10.1007/s00707-018-2173-8.
  30. Hussain, M. and Naeem, M.N. (2019), "Rotating response on the vibrations of functionally graded zigzag and chiral single walled carbon nanotubes", Appl. Math. Model., 75, 506-520. https://doi.org/10.1016/j.apm. 2019.05.039.
  31. Kalita, K., Dey, P., Haldar, S. and Gao, X.Z. (2020), "Optimizing frequencies of skew composite laminates with metaheuristic algorithms", Eng. with Comput., 36, 741-761. https://doi.org/10.1007/s00366-019-00728-x.
  32. Karama, M., Afaq, K.S. and Mistou, S. (2009), "A new theory for laminated composite plates, Proceedings of the institution of mechanical engineers", Part L: J. Mater.: Design Appl., 223(2), 53-62. https://doi.org/10.1243/14644207JMDA189.
  33. Karamanli, A. and Aydogdu, M. (2019), "Free vibration and buckling analysis of laminated composites and sandwich micro beams using a transverse shear-normal deformable beam theory", J. Vib. Control, 26(3-4), 214-228. https://doi.org/10.1177%2F1077546319878538. https://doi.org/10.1177%2F1077546319878538
  34. Karamanli, A. and Vo, T. (2018), "Size dependent bending analysis of two directional functionally graded micro beams via a quasi-3D theory and finite element method", Compos. Part B: Eng., 144, 171-183. https://doi.org/10.1016/j.compositesb.2018.02.030.
  35. 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.
  36. Katariya, P.V. and Panda, S.K. (2019b), "Frequency and deflection responses of shear deformable skew sandwich curved shell panel: A finite element approach", Arabian J. Sci. Eng., 44 (2), 1631-1648. https://doi.org/10.1007/s13369-018-3633-0.
  37. 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 Comput., 35(2), 467-485. https://doi.org/10.1007/s00366-018-0609-3.
  38. Katariya, P.V., Panda, S.K. and Mahapatra, T.R. (2018), "Bending and vibration analysis of skew sandwich plate", Aircr. Eng. Aerosp. Tech., 90(6), 885-895. https://doi.org/10.1108/AEAT05-2016-0087.
  39. Kaw, A.K. (2005), "Mechanics of composite materials", 2nd Edition, Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, 490. https://doi.org/10.1201/9781420058291.
  40. Khelifa, Z., Hadji, L., Hassaine Daouadji, T. and Bourada, M. (2018), "Buckling response with stretching effect of carbon nanotube-reinforced composite beams resting on elastic foundation", Struct. Eng. Mech., 67(2), 125-130. https://doi.org/10.12989/sem.2018.67.2.125.
  41. Kiani, Y. (2019), "NURBS-based thermal buckling analysis of graphene platelet reinforced composite laminated skew plates", J. Therm. Stresses, 1-19. https://doi.org/10.1080/01495739.2019.1673687.
  42. Kirchhoff, G.R. (1850), "Uber das gleichgewicht und die bewegung einer elastischen scheibe", J. Pure App. Math., 40, 51-88.
  43. 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.
  44. Kollar, L.P. and Springer, G.S. (2003), "Mechanics of Composite Structures", Published in the United States of America, Cambridge university press, New York, 480. . https://doi.org/10.1017/CBO9780511547140
  45. Lal, A., Jagtap, K.R. and Singh, B.N. (2017), "Thermo-mechanically induced finite element based nonlinear static response of elastically supported functionally graded plate with random system properties", Adv. Comput. Des., 2(3), 165-194. https://doi.org/10.12989/acd.2017.2.3.165.
  46. Li, D.H., Guo, Q.R., Xu, D. and Yang, X. (2017), "Three-dimensional micromechanical analysis models of fiber reinforced composite plates with damage", Comput. Struct., 191, 100-114. https://doi.org/10.1016/j.compstruc. 2017.06.005.
  47. 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.
  48. Mahapatra, T.R., Kar, V.R. and Panda, S.K. (2016), "Large amplitude bending behaviour of laminated composite curved panels", Eng. Comput., 33(1), 116-138. https://doi.org/10.1108/EC-05-2014-0119.
  49. Mantari, J.L. and Soares, C.G. (2013), "Generalized layerwise HSDT and finite element formulation for symmetric laminated and sandwich composite plates", Compos. Struct., 105(8), 319-331. https://doi.org/10.1016%2Fj.compstruct.2013.04.042. https://doi.org/10.1016%2Fj.compstruct.2013.04.042
  50. Matsunaga, H. (2007) "Free vibration and stability of angle-ply laminated composite and sandwich plates under thermal loading", Compos. Struct., 77(2), 249-262. https://doi.org/10.1016/j.compstruct.2005.07.002.
  51. Mehar, K. and Panda, S.K. (2016), "Free vibration and bending behaviour of CNT reinforced composite plate using different shear deformation theory", IOP conference series: Mat. Sci. Eng., 115 (1), 012014. https://doi.org/10.1088/1757-899X/115/1/012014.
  52. Mehar, K. and Panda, S.K. (2017a), "Thermoelastic analysis of FG-CNT reinforced shear deformable composite plate under various loadings", Int. J. Comput.Meth., 14(2), 1750019. https://doi.org/10.1142/S0219876217500190.
  53. Mehar, K. and Panda, S.K. (2017b), "Numerical investigation of nonlinear thermo-mechanical deflection of functionally graded CNT reinforced doubly curved composite shell panel under different mechanical loads", Compos. Struct., 161, 287-298. https://doi.org/10.1016/j.compstruct.2016.10.135.
  54. Mehar, K. and Panda, S.K. (2018), "Thermoelastic flexural analysis of FG-CNT doubly curved shell panel", Aircraft Eng. Aerosp. Tech., 90 (1), 11-23. https://doi.org/10.1108/AEAT-11-2015-0237.
  55. Mehar, K. and Panda, S.K. (2019), "Theoretical deflection analysis of multi-walled carbon nanotube reinforced sandwich panel and experimental verification", Compos. Part B: Eng., 167, 317-328. https://doi.org/10.1016/j.compositesb.2018.12.058.
  56. Mehar, K., Panda, S.K. and Patle, B.K. (2018), "Stress, deflection, and frequency analysis of CNT reinforced graded sandwich plate under uniform and linear thermal environment: A finite element approach", Polym. Compos., 39(10), 3792-3809. https://doi.org/10.1002/pc.24409.
  57. 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.
  58. Mindlin, R.D. (1951), "Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates", J. Appl. Mech., 18, 31-38. https://doi.org/10.1115/1.4010217
  59. Narwariya, M., Choudhury, A. and Sharma, A.K. (2018), "Harmonic analysis of moderately thick symmetric cross-ply laminated composite plate using FEM", Adv. Comput. Des., 3(2), 113-132. https://doi.org/10.12989/acd.2018.3.2.113.
  60. Nguyen, N.D., Nguyen, T.K., Vo, T.P. and Thai, H.T. (2018), "Ritz-based analytical solutions for bending, buckling and vibration behavior of laminated composite beams", Int. J. Struct. Stab. Dyn. 18(11), 1850130. https://doi.org/10.1142/S0219455418501304.
  61. 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. https://doi.org/10.32474/TCEIA.2018.03.000151.
  62. Patle, B.K., Hirwani, C.K., Singh, R.P. and Panda, S.K. (2018), "Eigenfrequency and deflection analysis of layered structure using uncertain elastic properties-a fuzzy finite element approach", Int. J. Approximate Reasoning, 98, 163-176. https://doi.org/10.1016/j.ijar.2018.04.013.
  63. Reddy, J.N. (1984), "A simple high-order theory of laminated composite plate", J. Appl. Mech., 51, 745-752. https://doi.org/10.1115/1.3167719.
  64. Reddy, J.N. (2004), "Mechanics of laminated composite plates and shells: theory and analysis", 2nd Ed., CRC Press LLC, 858 Pages.
  65. Remil, A., Benrahou, K.H., Draiche, K., Bousahla, A.A. and Tounsi, A. (2019), "A simple HSDT for bending, buckling and dynamic behavior of laminated composite plates", Struct. Eng. Mech., 70(3), 325-337. https://doi.org/10.12989/sem.2019.70.3.325.
  66. Sadoun, M., Houari, M.S.A., Bakora, A., Tounsi, A., Mahmoud, S. R., and Alwabli, A. S. (2018), "Vibration analysis of thick orthotropic plates using quasi 3D sinusoidal shear deformation theory", Geomech. Eng., 16(2), 141-150. https://doi.org/10.12989/gae.2018.16.2.141.
  67. Safaei, B. (2020), "The effect of embedding a porous core on the free vibration behavior of laminated composite plates", Steel Compos. Struct., 35(5), 659-670. https://doi.org/10.12989/scs.2020.35.5.659.
  68. 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. https://doi.org/10.12989/scs.2019.33.6.805.
  69. Sarangan, S. and Singh, B.N. (2016), "Higher-order closed-form solution for the analysis of laminated composite and sandwich plates based on new shear deformation theories", Comp. Struct., 138, 391-403. https://doi.org/10.1016/j.compstruct.2015.11.049.
  70. Sayyad, A.S. and Ghugal, Y.M. (2014), "A new shear and normal deformation theory for isotropic, transversely isotropic, laminated composite and sandwich plates", Int. J. Mech. Mater. Des., 10, 247-267. https://doi.org/1010.1007/s10999-014-9244-3.
  71. Sayyad, A.S. and Ghugal, Y.M. (2015), "On the free vibration analysis of laminated composite and sandwich plates: A review of recent literature with some numerical results", Compos. Struct., 129, 177-201. https://doi.org/10.1016/j.compstruct.2015.04.007.
  72. Sayyad, A.S., Ghugal, Y.M. and Mhaske, B.A. (2015) "A four-variable plate theory for thermoelastic bending analysis of laminated composite plates", J. Therm. Stresses, 38(8), 904-925. https://doi.org/10.1080/01495739.2015.1040310.
  73. Selmi, A. (2020), "Exact solution for nonlinear vibration of clamped-clamped functionally graded buckled beam". Smart Struct. Syst., 26(3), 361-371. https://doi.org/10.12989/sss.2020.26.3.361.
  74. Shimpi, R.P. and Patel, H.G. (2006), "A two variable refined plate theory for orthotropic plate analysis", Int. J. Solids Struct., 43, 6783-6799. https://doi.org/10.1016/j.ijsolstr.2006.02.007.
  75. Shinde, B.M. and Sayyad, A.S. (2017), "A quasi-3D polynomial shear and normal deformation theory for laminated composite, sandwich, and functionally graded beams", Mech. Adv. Compos. Struct., 4, 139-152. https://doi.org/10.22075/macs.2017.10806.1105.
  76. Sobhy, M. and Radwan, A.F. (2017), "A new quasi 3D nonlocal plate theory for vibration and buckling of FGM nanoplates", Int. J. Appl. Mech., 9(1), 1-29. https://doi.org/10.1142/S1758825117500089.
  77. Swaminathan, K. and Fernandes, R. (2013), "Higher order computational model for the thermoelastic analysis of cross-ply laminated composite plates", Int. J. Sci. Eng. Res., 4(5), 119-122.
  78. Tanzadeh, H. and Amoushahi, H. (2021), "Analysis of laminated composite plates based on different shear deformation plate theories", Struct. Eng. Mech., 75(2), 247-269. https://doi.org/10.12989/sem.2020.75.2.247.
  79. 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(2020). https://doi.org/10.1007/s42452-020-2182-9.
  80. Vinyas, M. (2020), "On frequency response of porous functionally graded magneto-electro-elastic circular and annular plates with different electro-magnetic conditions using HSDT", Compos. Struct., 240, 112044. https://doi.org/10.1016/j.compstruct.2020.112044.
  81. Vo, T.P., Thai, H.T., Nguyen, T.K., Lanc, D. and Karamanli, A. (2017), "Flexural analysis of laminated composite and sandwich beams using a four unknown shear and normal deformation theory", Compos. Struct., 176, 388-397. https://doi.org/10.1016/j.compstruct.2017.05.041.
  82. Yaylaci, M. and Avcar, M. (2020), "Finite element modeling of contact between an elastic layer and two elastic quarter planes", Comput. Concrete, 26(2), 107-114. https://doi.org/10.12989/cac.2020.26.2.107.
  83. Zamani, H.A., Aghdam, M.M. and Sadighi, M. (2017), "Free vibration analysis of thick viscoelastic composite plates on visco-Pasternak foundation using higher-order theory", Compos. Struct., 182, 25-35. https://doi.org/10.1016/j.compstruct.2017.08.101.
  84. Zenkour, A.M. (2007), "Three-dimensional elasticity solution for uniformly loaded cross-ply laminates and sandwich plates", J. Sandw. Struct. Mater.,9(3), 213-238. https://doi.org/10.1177%2F1099636 207065675. https://doi.org/10.1177%2F1099636207065675
  85. Zeverdejani, M.K. and Beni, Y.T. (2020), "Effect of laminate configuration on the free vibration/buckling of FG Graphene/PMMA composites", Adv. Nano Res., 8(2), 103-114. https://doi.org/10.12989/anr.2020.8.2.103.
  86. Zhang, P., Qi, C., Fang, H. and Sun, X. (2020), "Bending and free vibration analysis of laminated piezoelectric composite plates", Struct. Eng. Mech., 75(6), 747-769. https://doi.org/10.12989/sem.2020.75.6.747.