Steel and Composite Structures

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Nonlinear primary resonance of functionally graded doubly curved shells under different boundary conditions

  • Jinpeng Song (College of Mechanical and Vehicle Engineering, Chongqing University) ;
  • Yujie He (College of Mechanical and Vehicle Engineering, Chongqing University) ;
  • Gui-Lin She (College of Mechanical and Vehicle Engineering, Chongqing University)
  • Received : 2023.03.28
  • Accepted : 2023.12.28
  • Published : 2024.01.25
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Abstract

Considering that different boundary conditions can have an important impact on structural vibration characteristics. In this paper, the nonlinear forced vibration behavior of functionally graded material (FGM) doubly curved shells with initial geometric imperfections under different boundary conditions is studied. Considering initial geometric imperfections and von Karman geometric nonlinearity, the nonlinear governing equations of FGM doubly curved shells are derived using Reissner's first order shear deformation (FOSD) theory. Three different boundary conditions of four edges simply supported (SSSS), four edges clamped (CCCC), clamped-clamped-simply-simply (CCSS) were studied, and a system of nonlinear ordinary differential equations was obtained with the help of Galerkin principle. The nonlinear forced vibration response of the FGM doubly curved shell is obtained by using the modified Lindstedt Poincare (MLP) method. The accuracy of this method was verified by comparing it with published literature. Finally, the effects of curvature ratio, power law index, void coefficient, prestress, and initial geometric imperfections on the resonance of FGM doubly curved shells under different boundary conditions are fully discussed. The relevant research results can provide certain guidance for the design and application of doubly curved shell.

Keywords

References

  1. Abo-bakr R.M., Mohamed N., Eltaher, M.A. and Emam S. (2023), "Multi-objective optimization for snap-through response of spherical shell panels", Appl. Math. Model., https://doi.org/10.1016/j.apm.2023.12.014.
  2. Ahmadi, H., Bayat, A. and Duc, N.D. (2021), "Nonlinear forced vibrations analysis of imperfect stiffened FG doubly curved shallow shell in thermal environment using multiple scales method", Compos. Struct., 256, 113090. https://doi.org/10.1016/j.compstruct.2020.113090.
  3. Ali, M.I. and Azam, M.S. (2021), "Dynamic stiffness formulation for out-of-plane natural vibration of elastically supported functionally graded plates", Int. J. Struct. Stab. Dyn., 21(05), 2150062. https://doi.org/10.1142/S0219455421500620.
  4. Alijani, F., Amabili, M., Karagiozis, K. and Bakhtiari-Nejad, F. (2011), "Nonlinear vibrations of functionally graded doubly curved shallow shells", J. Sound. Vib., 330(7), 1432-1454. https://doi.org/10.1016/j.jsv.2010.10.003.
  5. Amoozgar, M. and Gelman, L. (2022), "Vibration analysis of rotating porous functionally graded material beams using exact formulation", J. Vib. Control., 28(21-22), 3195-3206. https://doi.org/10.1177/10775463211027883.
  6. Arefi, M. (2020), "Smart analysis of doubly curved piezoelectric nano shells: electrical and mechanical buckling analysis", Smart. Struct. Syst., 25(4), 471-486. https://doi.org/10.12989/sss.2020.25.4.471.
  7. Asadijafari, M.H., Zarastvand, M.R. and Talebitooti, R. (2021), "The effect of considering Pasternak elastic foundation on acoustic insulation of the finite doubly curved composite structures", Compos. Struct., 256, 113064. https://doi.org/10.1016/j.compstruct.2020.113064.
  8. Babaei, H. (2022), "Thermomechanical analysis of snap-buckling phenomenon in long FG-CNTRC cylindrical panels resting on nonlinear elastic foundation", Compos. Struct., 286, 115199. https://doi.org/10.1016/j.compstruct.2022.115199.
  9. Babaei, H., Jabbari, M. and Eslami, M.R. (2021), "The effect of porosity on elastic stability of toroidal shell segments made of saturated porous functionally graded materials", J. Press. Vess-t. Asme., 143(3), 031501. https://doi.org/10.1115/1.4048418.
  10. Basturk, S. (2019), "The nonlinear dynamic response of functionally graded basalt/nickel composite plates", Mech. Adv. Mater. Struct., 26(20), 1719-1734. https://doi.org/10.1080/15376494.2018.1446109.
  11. Chen, S.H. and Cheung, Y.K. (1996), "A modified Lindstedt-Poincare method for a strongly nonlinear system with quadratic and cubic nonlinearities", Shock. Vib., 3(4), 279-285. https://doi.org/10.1155/1996/231241.
  12. Chen, X., Zhao, J.L., She, G.L., Jing, Y., Pu, H.Y. and Luo, J. (2022), "Nonlinear free vibration analysis of functionally graded carbon nanotube reinforced fluid-conveying pipe in thermal environment", Steel. Compos. Struct., 45(5), 641-652. https://doi.org/10.12989/scs.2022.45.5.641.
  13. Chorfi, S.M. and Houmat, A. (2010), "Non-linear free vibration of a functionally graded doubly-curved shallow shell of elliptical plan-form", Compos. Struct., 92(10), 2573-2581. https://doi.org/10.1016/j.compstruct.2010.02.001.
  14. Ding, H.X. and She, G.L. (2023a), "Nonlinear resonance of axially moving graphene platelet-reinforced metal foam cylindrical shells with geometric imperfection", Archives Civil Mech. Eng., 23, 97. https://doi.org/10.1007/s43452-023-00634-6
  15. Ding, H.X. and She, G.L. (2023b), "Nonlinear primary resonance behavior of graphene platelets reinforced metal foams conical shells under axial motion", Nonlinear Dyn., 111(15), 13723-13752. https://doi.org/10.1007/s11071-023-08564-x.
  16. Ding, H.X. and She, G.L. (2023c), "Nonlinear combined resonances of axially moving graphene platelets reinforced metal foams cylindrical shells under forced vibrations", Nonlinear Dyn. https://doi.org/10.1007/s11071-023-09059-5.
  17. Ding, H.X., Eltaher, M.A. and She, G.L. (2023a), "Nonlinear low-velocity impact of graphene platelets reinforced metal foams cylindrical shell: Effect of spinning motion and initial geometric imperfections", Aeros. Sci. Technol., 140, 108435. https://doi.org/10.1016/j.ast.2023.108435.
  18. Ding, H.X., Liu, H.B., She, G.L. and Wu, F. (2023c), "Wave propagation of FG-CNTRC plates in thermal environment using the high-order shear deformation plate theory", Comput. Concrete, 32(2), 207-215. https://doi.org/10.12989/cac.2023.32.2.207.
  19. Ding, H.X., She, G.L. and Zhang, Y.W. (2022a), "Nonlinear buckling and resonances of functionally graded fluid-conveying pipes with initial geometric imperfection", Eur. Phys. J. Plus, 137, 1329. https://doi.org/10.1140/epjp/s13360-022-03570-1.
  20. Ding, H.X., Zhang, Y.W. and She, G.L. (2022b), "On the resonance problems in FG-GPLRC beams with different boundary conditions resting on elastic foundations", Comput. Concrete, 30(6),433-443. https://doi.org/10.12989/cac.2022.30.6.433.
  21. Ding, H.X., Zhang, Y.W. and She, G.L. (2023b), "Propagation characteristics of guided waves in CNTRCs plates resting on elastic foundations in a thermal environment", Waves Random Complex Media. https://doi.org/10.1080/17455030.2023.2235611.
  22. Ding, H.X., Zhang, Y.W., Li, Y.P. and She, G.L. (2023d), "Nonlinear low-velocity impact response of graphene platelets reinforced metal foams doubly curved shells", Steel Compos. Struct., 49(3), 281-291. https://doi.org/10.12989/scs.2023.49.3.281.
  23. Dong, Y., Hu, H., Wang, L. and Mao, X. (2024), "Nonlinear coupled multi-mode vibrations of simply-supported cylindrical shells: Comparison studies", Commun. Nonlinear Sci. Numer. Simulat., 128, 107667. https://doi.org/10.1016/j.cnsns.2023.107667
  24. Eshraghi, I. and Dag, S. (2020), "Forced vibrations of functionally graded annular and circular plates by domain-boundary element method", Zamm-z. Angew. Math. Me., 100(8), e201900048. https://doi.org/10.1002/zamm.201900048.
  25. Ezzin, H., Mkaoir, M., Arefi, M., Qian, Z. and Das, R. (2021), "Analysis of guided wave propagation in functionally graded magneto-electro elastic composite", Wave Random Complex, 1-19. https://doi.org/10.1080/17455030.2021.1968541.
  26. Gan, L.L. and She, G.L. (2023), "Nonlinear snap-buckling and resonance of FG-GPLRC curved beams with different boundary conditions", Geomech. Eng., 32(5), 541-551. https://doi.org/10.12989/gae.2023.32.5.541.
  27. Gan, L.L. and She, G.L. (2024), "Nonlinear low-velocity impact of magneto-electro-elastic plates with initial geometric imperfection", Acta Astronautica, 214, 11-29. https://doi.org/10.1016/j.actaastro.2023.10.016.
  28. Gan, L.L., Xu, J.Q. and She, G.L. (2023), "Wave propagation of graphene platelets reinforced metal foams circular plates", Struct. Eng. Mech., 85(5), 645-654. https://doi.org/10.12989/sem.2023.85.5.645.
  29. Ghandourah, E.E., Daikh, A.A., Khatir, S., Alhawsawi, A.M., Banoqitah, E.M. and Eltaher, M.A. (2023), "A dynamic analysis of porous coated functionally graded nanoshells rested on viscoelastic medium", Mathematics-basel., 11(10), 2407. https://doi.org/10.3390/math11102407.
  30. Hao, Y.X., Cao, Z., Zhang, W., Chen, J. and Yao, M.H. (2019), "Stability analysis for geometric nonlinear functionally graded sandwich shallow shell using a new developed displacement field", Compos. Struct., 210, 202-216. https://doi.org/10.1016/j.compstruct.2018.11.027.
  31. Hong, C.C. (2021), "Vibration frequency of thick functionally graded material cylindrical shells with fully homogeneous equation and third-order shear deformation theory under thermal environment", J. Vib. Control., 27(17-18), 2004-2017. https://doi.org/10.1177/1077546320951663.
  32. Huang, X.L., Dong, L., Wei, G.Z. and Zhong, D.Y. (2019), "Nonlinear free and forced vibrations of porous sigmoid functionally graded plates on nonlinear elastic foundations", Compos. Struct., 228, 111326. https://doi.org/10.1016/j.compstruct.2019.111326.
  33. Karimiasl, M., Ebrahimi, F. and Akgoz, B. (2019), "Buckling and post-buckling responses of smart doubly curved composite shallow shells embedded in SMA fiber under hygro-thermal loading", Compos. Struct., 223, 110988. https://doi.org/10.1016/j.compstruct.2019.110988.
  34. Kim, S.E., Duc, N.D., Nam, V.H. and Van Sy, N. (2019), "Nonlinear vibration and dynamic buckling of eccentrically oblique stiffened FGM plates resting on elastic foundations in thermal environment", Thin. Wall. Struct., 142, 287-296. https://doi.org/10.1016/j.tws.2019.05.013.
  35. Kumar, S., Ranjan, V. and Jana, P. (2018), "Free vibration analysis of thin functionally graded rectangular plates using the dynamic stiffness method", Compos. Struct., 197, 39-53. https://doi.org/10.1016/j.compstruct.2018.04.085.
  36. Li, J. Q., Xue, Y. and Li, F. (2023b), "Active band gap control of magnetorheological meta-plate using frequency feedback control law", J. Sound Vib., 567, 118076. https://doi.org/10.1016/j.jsv.2023.118076.
  37. Li, Y.P., She, G.L., Gan, L.L. and Liu, H.B (2023a), "Nonlinear thermal post-buckling analysis of graphene platelets reinforced metal foams plates with initial geometrical imperfection", Steel Compos Struct., 46(5), 649-658. https://doi.org/10.12989/scs.2023.46.5.649.
  38. Li, Z.M., Liu, T. and Qiao, P. (2021), "Buckling and postbuckling of anisotropic laminated doubly curved panels under lateral pressure", Int. J. Mech. Sci., 206, 106615. https://doi.org/10.1016/j.ijmecsci.2021.106615.
  39. Loja, M.A.R. and Barbosa, J.I. (2020), "In-plane functionally graded plates: A study on the free vibration and dynamic instability behaviours", Compos. Struct., 237, 111905. https://doi.org/10.1016/j.compstruct.2020.111905.
  40. Mirjavadi, S.S., Forsat, M., Yahya, Y.Z., Barati, M.R., Jayasimha, A.N. and Hamouda, A.M.S. (2020), "Porosity effects on post-buckling behavior of geometrically imperfect metal foam doubly-curved shells with stiffeners", Struct. Eng. Mech., 75(6), 701-711. https://doi.org/10.12989/sem.2020.75.6.701.
  41. Mohamed, S.A., Assie, A.E. and Eltaher, M.A. (2023b), "Novel incremental procedure in solving nonlinear static response of 2D-FG porous plates", Thin. Wall. Struct., 189, 110779. https://doi.org/10.1016/j.tws.2023.110779.
  42. Mohamed, Y., Tharwan, A.A.D., Assie, A.E., Alnujaie, A. and Eltaher, M.A. (2023a), "A comprehensive study on static response of agglomerated microstructure-dependent coated functionally graded carbon nanotubes reinforced composite nanoshells rested on complex elastic foundation", Mech. Based Des. Struct., https://doi.org/10.1080/15397734.2023.2286484.
  43. Mohammadi, M., Mohseni, E. and Moeinfar, M. (2019), "Bending, buckling and free vibration analysis of incompressible functionally graded plates using higher order shear and normal deformable plate theory", Appl. Math. Model., 69, 47-62. https://doi.org/10.1016/j.apm.2018.11.047.
  44. Nazari, H., Babaei, M., Kiarasi, F. and Asemi, K. (2021), "Geometrically nonlinear dynamic analysis of functionally graded material plate excited by a moving load applying first-order shear deformation theory via generalized differential quadrature method", Sn. Appl. Sci., 3, 1-32. https://doi.org/10.1007/s42452-021-04825-9.
  45. Reddy, J.N. (1984), "Exact solutions of moderately thick laminated shells", J. Eng. Mech-asce., 110(5), 794-809. https://doi.org/10.1061/(ASCE)0733-9399(1984)110:5(794).
  46. Salehipour, H., Emadi, S., Tayebikhorami, S. and Shahmohammadi, M.A. (2021), "A semi-analytical solution for dynamic stability analysis of nanocomposite/fibre-reinforced doubly-curved panels resting on the elastic foundation in thermal environment", Eur. Phys. J. Plus., 137(1), 2. https://doi.org/10.1140/epjp/s13360-021-02190-5.
  47. Shakir, M. and Talha, M. (2019), "On the dynamic response of imperfection sensitive higher order functionally graded plates with random system parameters", Int. J. Appl. Mech., 11(03), 1950025. https://doi.org/10.1142/S175882511950025X.
  48. Shan, W.B. and She, G.L. (2023), "Nonlinear resonance of porous functionally graded nanoshells with geometrical imperfection", Struct. Eng. Mech., 88(4), 355-368. https://doi.org/10.12989/sem.2023.88.4.355.
  49. She, G.L. and Ding, H.X. (2023), "Nonlinear primary resonance analysis of initially stressed graphene platelet reinforced metal foams doubly curved shells with geometric imperfection", Acta Mech. Sin., 39, 522392. https://doi.org/10.1007/s10409-022-22392-x.
  50. She, G.L. and Li, Y.P. (2022), "Wave propagation in an FG circular plate in thermal environment", Geomech. Eng., 31(6), 615-622. https://doi.org/10.12989/gae.2022.31.6.615.
  51. She, G.L., Ding, H.X. and Zhang, Y.W. (2022), "Wave propagation in a FG circular plate via the physical neutral surface concept", Struct. Eng. Mech., 82(2), 225-232. https://doi.org/10.12989/sem.2022.82.2.225.
  52. Sobhy, M. and Zenkour, A.M. (2019), "Vibration analysis of functionally graded graphene platelet-reinforced composite doubly-curved shallow shells on elastic foundations", Steel Compos. Struct., 33(2), 195-208. https://doi.org/10.12989/scs.2019.33.2.195.
  53. Tharwan, M.Y., Daikh, A.A., Assie, A.E., Alnujaie, A. and Eltaher, M.A. (2023), "Refined quasi-3D shear deformation theory for buckling analysis of functionally graded curved nanobeam rested on Winkler/Pasternak/Kerr foundation", Mech. Based Des. Struct., https://doi.org/10.1080/15397734.2023.2270043.
  54. Van Long, N., Thinh, T.I., Bich, D.H. and Tu, T.M. (2022), "Nonlinear dynamic responses of sandwich-FGM doubly curved shallow shells subjected to underwater explosions using first-order shear deformation theory", Ocean. Eng., 260, 111886. https://doi.org/10.1016/j.oceaneng.2022.111886.
  55. Viet Hoang, V.N., Tien, N.D., Ninh, D.G., Thang, V.T. and Truong, D.V. (2021), "Nonlinear dynamics of functionally graded graphene nanoplatelet reinforced polymer doubly-curved shallow shells resting on elastic foundation using a micromechanical model", J. Sandw. Struct. Mater., 23(7), 3250-3279. https://doi.org/10.1177/1099636220926650.
  56. Wang, A., Chen, H. and Zhang, W. (2019), "Nonlinear transient response of doubly curved shallow shells reinforced with graphene nanoplatelets subjected to blast loads considering thermal effects", Compos. Struct., 225, 111063. https://doi.org/10.1016/j.compstruct.2019.111063.
  57. Wang, Y.Q. (2018), "Electro-mechanical vibration analysis of functionally graded piezoelectric porous plates in the translation state", Acta. Astronaut., 143, 263-271. https://doi.org/10.1016/j.actaastro.2017.12.004.
  58. Wu, F. and She, G.L. (2023), "Wave propagation in double nano-beams in thermal environments using the Reddy's high-order shear deformation theory", Adv. Nano Res., 14(6), 495-506. https://doi.org/10.12989/anr.2023.14.6.495.
  59. Xie, Z., Jiao, J. and Wrona, S. (2023), "The fluid-structure interaction lubrication performances of a novel bearing: experimental and numerical study", Tribology Int., 179, 108151. https://doi.org/10.1016/j.triboint.2022.108151.
  60. Xie, Z., Jiao, J., Zhao, B., Zhang, J. and Xu, F. (2024), "Theoretical and experimental research on the effect of bi-directional misalignment on the static and dynamic characteristics of a novel bearing", Mech. Syst. Signal Processing, 208, 111041. https://doi.org/10.1016/j.ymssp.2023.111041.
  61. Xu, J.Q. and She, G.L. (2022), "Thermal post-buckling analysis of porous functionally graded pipes with initial geometric imperfection", Geomech. Eng., 31(3), 329-337. https://doi.org/10.12989/gae.2022.31.3.329.
  62. Xu, J.Q. and She, G.L. (2023a), "Thermal post-buckling of graphene platelet reinforced metal foams doubly curved shells with geometric imperfection", Struct. Eng. Mech., 87(1), 85-94. https://doi.org/10.12989/sem.2023.87.1.085.
  63. Xu, J.Q. and She, G.L. (2023b), "The effects of temperature and porosity on resonance behavior of graphene platelet reinforced metal foams doubly-curved shells with geometric imperfection", Geomech. Eng., 35(1), 81-93. https://doi.org/10.12989/gae.2023.35.1.081.
  64. Xu, J.Q. and She, G.L. (2023c), "Resonance behavior of functionally graded carbon nanotube-reinforced composites shells with spinning motion and axial motion", Steel Compos. Struct., 49(3), 325-335. https://doi.org/10.12989/scs.2023.49.3.325.
  65. Xu, J.Q. and She, G.L. (2023d), "Thermal post-buckling and primary resonance of porous functionally graded beams: Effect of elastic foundations and geometric imperfection", Comput. Concrete, 32(6), 543-551. https://doi.org/10.12989/cac.2023.32.6.543.
  66. Xu, J.Q., She, G.L., Li. Y.P. and Gan, L.L. (2023), "Nonlinear resonances of nonlocal strain gradient nanoplates made of functionally graded materials considering geometric imperfection", Steel Compos. Struct., 47(6), 795-811. https://doi.org/10.12989/scs.2023.47.6.795.
  67. Xue, Y., Li, J., Li, F. and Song, Z. (2019), "Active control of plates made of functionally graded piezoelectric material subjected to thermo-electro-mechanical loads", Int. J. Struct. Stab. Dyn., 19(09), 1950107. https://doi.org/10.1142/S0219455419501074.
  68. Xue, Y., Li, J., Wang, Y., Song, Z. and Krushynska, A.O. (2023), "Widely tunable magnetorheological metamaterials with nonlinear amplification mechanism", Int. J. Mech. Sci., 264, 108830. https://doi.org/10.1016/j.ijmecsci.2023.108830.
  69. Yu, C., Meng, Z., Zhang, X., Li, S., Xu, W. and Chiu, C. (2022), "Orthogonal polynomials-ritz method for dynamic response of functionally graded porous plates using FSDT", Int. J. Struct. Stab. Dyn., 22(05), 2250057. https://doi.org/10.1142/S021945542250057.
  70. Zhai, Y., Ma, J. and Liang, S. (2021), "Dynamics properties of multi-layered composite sandwich doubly-curved shells", Compos. Struct., 256, 113142. https://doi.org/10.1016/j.compstruct.2020.113142.
  71. Zhang, J., Pan, S. and Chen, L. (2019), "Dynamic thermal buckling and postbuckling of clamped-clamped imperfect functionally graded annular plates", Nonlinear. Dyn., 95, 565-577. https://doi.org/10.1007/s11071-018-4583-5.
  72. Zhang, Y.W. and She, G.L. (2022), "Wave propagation and vibration of FG pipes conveying hot fluid", Steel. Compos, Struct., 42(3) 397-405. https://doi.org/10.12989/scs.2022.42.3.397.
  73. Zhang, Y.W. and She, G.L. (2023a), "Nonlinear low-velocity impact response of graphene platelet-reinforced metal foam cylindrical shells under axial motion with geometrical imperfection", Nonlinear Dyn., 111(7), 6317-6334. https://doi.org/10.1007/s11071-022-08186-9.
  74. Zhang, Y.W. and She, G.L. (2023b), "Nonlinear primary resonance of axially moving functionally graded cylindrical shells in thermal environment", Mech. Adv. Mater. Struct., https://doi.org/10.1080/15376494.2023.2180556
  75. Zhang, Y.W. and She, G.L. (2024), "Combined resonance of graphene platelets reinforced metal foams cylindrical shells with spinning motion under nonlinear forced vibration", Eng. Struct., 300, 117177. https://doi.org/10.1016/j.engstruct.2023.117177.
  76. Zhang, Y.W., Ding, H.X. and She, G.L. (2022), "Snap-buckling and resonance of functionally graded graphene reinforced composites curved beams resting on elastic foundations in thermal environment", J. Therm. Stresses, 45(12), 1029-1042. https://doi.org/10.1080/01495739.2022.2125137.
  77. Zhang, Y.W., Ding, H.X. and She, G.L. (2023a), "Wave propagation in spherical and cylindrical panels reinforced with carbon nanotubes", Steel Compos. Struct., 46(1), 133-141. https://doi.org/10.12989/scs.2023.46.1.133.
  78. Zhang, Y.W., Ding, H.X., She, G.L. and Tounsi, A. (2023d), "Wave propagation of CNTRC beams resting on elastic foundation based on various higher-order beam theories", Geomech. Eng., 33(4), 381-391. https://doi.org/10.12989/gae.2023.33.4.381.
  79. Zhang, Y.W., She, G.L. and Ding, H.X. (2023b), "Nonlinear resonance of graphene platelets reinforced metal foams plates under axial motion with geometric imperfections", Eur. J. Mech. A-Solid., 98, 104887. https://doi.org/10.1016/j.euromechsol.2022.104887.
  80. Zhang, Y.W., She, G.L., Gan, L.L. and Li, Y.P. (2023c), "Thermal post-buckling behavior of GPLRMF cylindrical shells with initial geometrical imperfection", Geomech. Eng., 32(6), 615-625. https://doi.org/10.12989/gae.2023.32.6.615.
  81. Zhang, Y.W., She, G.L.,and Eltaher, M.A. (2023e), "Nonlinear transient response of graphene platelets reinforced metal foams annular plate considering rotating motion and initial geometric imperfection", Aeros. Sci. Technol., 142, 108693. https://doi.org/10.1016/j.ast.2023.108693.
  82. Zhao, J.L., Chen, X., She, G.L., Jing, Y., Bai, R.Q., Yi, J., Pu, H.Y. and Luo, J. (2022a), "Vibration characteristics of functionally graded carbon nanotube-reinforced composite double-beams in thermal environments", Steel. Compos. Struct., 43(6), 797-808. https://doi.org/10.12989/scs.2022.43.6.797.
  83. Zhao, J.L., She, G.L., Wu, F., Yuan, S.J., Bai, R.Q., Pu, H.Y., Wang, S.L. and Luo, J. (2022b), "Guided waves of porous FG nanoplates with four edges clamped", Adv. Nano. Res., 13(5), 465-474. https://doi.org/10.12989/anr.2022.13.5.465.
  84. Zhao, Y.B. and Zheng, P.P. (2021), "Parameter analyses of suspended cables subjected to simultaneous combination, super and sub-harmonic excitations", Steel Compos. Struct., 40(2), 203-216. https://doi.org/10.12989/scs.2021.40.2.203.
  85. Zhao, Y.B., Peng, J., Zhao, Y.Y. and Chen, L.C. (2017), "Effects of temperature variations on nonlinear planar free and forced oscillations at primary resonance of suspended cables", Nonlinear Dyn., 89, 2815-2827. https://doi.org/10.1007/s11071-017-3627-6.