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Nonlinear free and forced vibrations of oblique stiffened porous FG shallow shells embedded in a nonlinear elastic foundation

  • Kamran Foroutan (Industrial Systems Engineering, University of Regina) ;
  • Liming Dai (Industrial Systems Engineering, University of Regina)
  • Received : 2023.09.20
  • Accepted : 2023.12.05
  • Published : 2024.01.10

Abstract

The present research delves into the analysis of nonlinear free and forced vibrations of porous functionally graded (FG) shallow shells reinforced with oblique stiffeners, which are embedded in a nonlinear elastic foundation (NEF) subjected to external excitation. Two distinct types of PFG shallow shells, characterized by even and uneven porosity distribution along the thickness direction, are considered in the research. In order to model the stiffeners, Lekhnitskii's smeared stiffeners technique is implemented. With the stress function and first-order shear deformation theory (FSDT), the nonlinear model of the oblique stiffened shallow shells is established. The strain-displacement relationships for the system are derived via the FSDT and utilization of the von-Kármán's geometric assumptions. To discretize the nonlinear governing equations, the Galerkin method is employed. The model such developed allows analysis of the effects of the stiffeners with various angles as desired, in addition to the quantitative investigation on the influence of the surrounding nonlinear elastic foundations. To numerically solve the problem of vibrations, the 4th-order P-T method is used, as this method, known for its enhanced accuracy and reliability, proves to be an effective choice. The validation of the present research findings includes a comprehensive comparison with outcomes documented in existing literature. Additionally, a comparative analysis of the numerical results against those obtained using the 4th Runge-Kutta method is performed. The impact of stiffeners with varying angles and material parameters on the vibration characteristics of the present system is also explored. The researchers and engineers working in this field may use the results of this study as benchmarks in their design and research for the considered shell systems.

Keywords

Acknowledgement

The authors greatly appreciate the supports of the Natural Sciences and Engineering Research Council of Canada (NSERC), and the University of Regina to the present research.

References

  1. Ahmadi, H. and Foroutan, K. (2019), "Nonlinear vibration of stiffened multilayer FG cylindrical shells with spiral stiffeners rested on damping and elastic foundation in thermal environment", Thin Wall. Struct., 145, 106388. https://doi.org/10.1016/j.tws.2019.106388.
  2. 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.
  3. Bahranifard, F., Malekzadeh, P. and Haghighi, M.G. (2022), "Moving load response of ring-stiffened sandwich truncated conical shells with GPLRC face sheets and porous core", Thin Wall. Struct., 180, 109984. https://doi.org/10.1016/j.tws.2022.109984.
  4. Bich, D.H. and Ninh, D.G. (2017), "An analytical approach: Nonlinear vibration of imperfect stiffened FGM sandwich toroidal shell segments containing fluid under external thermo-mechanical loads", Compos. Struct., 162, 164-181. https://doi.org/10.1016/j.compstruct.2016.11.065.
  5. Bich, D.H., Duc, N.D. and Quan, T.Q. (2014), "Nonlinear vibration of imperfect eccentrically stiffened functionally graded double curved shallow shells resting on elastic foundation using the first order shear deformation theory", Int. J. Mech. Sci., 80, 16-28. https://doi.org/10.1016/j.ijmecsci.2013.12.009.
  6. Bich, D.H., Van Dung, D., Nam, V.H. and Phuong, N.T. (2013), "Nonlinear static and dynamic buckling analysis of imperfect eccentrically stiffened functionally graded circular cylindrical thin shells under axial compression", Int. J. Mech. Sci., 74, 190-200. https://doi.org/10.1016/j.ijmecsci.2013.06.002.
  7. Brush D.D and Almroth B.O. (1975), Buckling of Bars, Plates and Shells, Mc. Graw-Hill, New York.
  8. 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.
  9. Civalek, O. (2006), "Free vibration analysis of composite conical shells using the discrete singular convolution algorithm", Steel Compos. Struct., 6(4), 353. https://doi.org/10.12989/scs.2006.6.4.353.
  10. Cong, P.H. and Duc, N.D. (2021), "Nonlinear dynamic analysis of porous eccentrically stiffened double curved shallow auxetic shells in thermal environments", Thin Wall. Struct., 163, 107748. https://doi.org/10.1016/j.tws.2021.107748.
  11. Cong, P.H., Trung, V.D., Khoa, N.D. and Duc, N.D. (2022), "Vibration and nonlinear dynamic response of temperature-dependent FG-CNTRC laminated double curved shallow shell with positive and negative Poisson's ratio", Thin Wall. Struct., 171, 108713. https://doi.org/10.1016/j.tws.2021.108713.
  12. Cuong-Le, T., Nguyen, K.D., Nguyen-Trong, N., Khatir, S., Nguyen-Xuan, H. and Abdel-Wahab, M. (2021), "A three-dimensional solution for free vibration and buckling of annular plate, conical, cylinder and cylindrical shell of FG porous-cellular materials using IGA", Compos. Struct., 259, 113216. https://doi.org/10.1016/j.compstruct.2020.113216.
  13. Daemi, H. and Eipakchi, H. (2020), "Effect of different viscoelastic models on free vibrations of thick cylindrical shells through FSDT under various boundary conditions", Struct. Eng. Mech., 73(3), 319-330. https://doi.org/10.12989/sem.2020.73.3.319.
  14. Dai, L. (2008), Nonlinear Dynamics of Piecewise Constant Systems and Implementation of Piecewise Constant Arguments, World Scientific Publishing Co, New Jersey, USA.
  15. Dai, L. and Foroutan, K. (2023), "Nonlinear stability and vibration analyses of functionally graded variable thickness toroidal shell segments reinforced with spiral stiffeners", Int. J. Appl. Mech., 15(8), 2350061. https://doi.org/10.1142/S1758825123500618.
  16. Duc, N.D., Nguyen, P.D. and Khoa, N.D. (2017), "Nonlinear dynamic analysis and vibration of eccentrically stiffened S-FGM elliptical cylindrical shells surrounded on elastic foundations in thermal environments", Thin Wall. Struct., 117, 178-189. https://doi.org/10.1016/j.tws.2017.04.013.
  17. Farahani, H. and Barati, F. (2015), "Vibration of sumberged functionally graded cylindrical shell based on first order shear deformation theory using wave propagation method", Struct. Eng. Mech., 53(3), 575-587. https://doi.org/10.12989/sem.2015.53.3.575.
  18. Farid, M., Zahedinejad, P. and Malekzadeh, P. (2010), "Three-dimensional temperature dependent free vibration analysis of functionally graded material curved panels resting on two-parameter elastic foundation using a hybrid semi-analytic, differential quadrature method", Mater. Des., 31(1), 2-13. https://doi.org/10.1016/j.matdes.2009.07.025.
  19. Foroutan, K. and Ahmadi, H. (2022), "Nonlinear vibration of SSMFG cylindrical shells with internal resonances resting on the nonlinear viscoelastic foundation", Struct. Eng. Mech., 84(6), 767-782. https://doi.org/10.12989/sem.2022.84.6.767.
  20. Foroutan, K. and Dai, L. (2022), "Subharmonic and superharmonic resonances of five-layered porous functionally graded sandwich cylindrical shells with two-layered viscoelastic cores", J. Vib. Control, 29(19-20), 4643-4658. https://doi.org/10.1177/10775463221122091.
  21. Foroutan, K. and Dai, L. (2023a), "Nonlinear dynamic responses of porous FG sandwich cylindrical shells with a viscoelastic core resting on a nonlinear viscoelastic foundation", Mech. Adv. Mater. Struct., 30(15), 3184-3203. https://doi.org/10.1080/15376494.2022.2070803.
  22. Foroutan, K. and Dai, L. (2023b), "Free vibration of porous FG shallow shells reinforced with oblique stiffeners", The 2023 World Congress on Advances in Structural Engineering and Mechanics (ASEM23) GECE, Seoul, Korea, August.
  23. Foroutan, K., Shaterzadeh, A. and Ahmadi, H. (2019), "Nonlinear dynamic analysis of spiral stiffened cylindrical shells rested on elastic foundation", Steel Compos. Struct., 32(4), 509-519. https://doi.org/10.12989/scs.2019.32.4.509.
  24. Gao, K., Gao, W., Wu, B., Wu, D. and Song, C. (2018), "Nonlinear primary resonance of functionally graded porous cylindrical shells using the method of multiple scales", Thin Wall. Struct., 125, 281-293. https://doi.org/10.1016/j.tws.2017.12.039.
  25. Heydarpour, Y., Malekzadeh, P., Golbahar Haghighi, M.R. and Vaghefi, M. (2012), "Thermoelastic analysis of rotating laminated functionally graded cylindrical shells using layerwise differential quadrature method", Acta Mechanica, 223(1), 81-93. https://doi.org/10.1007/s00707-011-0551-6.
  26. Hoang, V.N.V., Ninh, D.G., Van Bao, H. and Le Huy, V. (2021), "Behaviors of dynamics and stability standard of graphene nanoplatelet reinforced polymer corrugated plates resting on the nonlinear elastic foundations", Compos. Struct., 260, 113253. https://doi.org/10.1016/j.compstruct.2020.113253.
  27. Jooybar, N., Malekzadeh, P., Fiouz, A. and Vaghefi, M. (2016), "Thermal effect on free vibration of functionally graded truncated conical shell panels", Thin Wall. Struct., 103, 45-61. https://doi.org/10.1016/j.tws.2016.01.032.
  28. 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.
  29. Malekzadeh, P. and Daraie, M. (2014), "Dynamic analysis of functionally graded truncated conical shells subjected to asymmetric moving loads", Thin Wall. Struct., 84, 1-13. https://doi.org/10.1016/j.tws.2014.05.007.
  30. Malekzadeh, P. and Heydarpour, Y. (2012a), "Response of functionally graded cylindrical shells under moving thermo-mechanical loads", Thin Wall. Struct., 58, 51-66. https://doi.org/10.1016/j.tws.2012.04.010.
  31. Malekzadeh, P. and Heydarpour, Y. (2012b), "Free vibration analysis of rotating functionally graded cylindrical shells in thermal environment", Compos. Struct., 94(9), 2971-2981. https://doi.org/10.1016/j.compstruct.2012.04.011.
  32. Malekzadeh, P., Heydarpour, Y., Haghighi, M.G. and Vaghefi, M. (2012), "Transient response of rotating laminated functionally graded cylindrical shells in thermal environment", Int. J. Press. Vessel. Pip., 98, 43-56. https://doi.org/10.1016/j.ijpvp.2012.07.003.
  33. Matsunaga, H. (2008), "Free vibration and stability of functionally graded shallow shells according to a 2D higher-order deformation theory", Compos. Struct., 84(2), 132-146. https://doi.org/10.1016/j.compstruct.2007.07.006.
  34. Ninh, D.G. and Hoang, V.N.V. (2022), "A new shell study for dynamical characteristics of nanocomposite shells with various complex profiles-Sinusoidal and cosine shells", Eng. Struct., 251, 113354. https://doi.org/10.1016/j.engstruct.2021.113354.
  35. Ninh, D.G., Ha, N.H., Long, N.T., Tan, N.C., Tien, N.D. and Dao, D.V. (2023), "Thermal vibrations of complex-generatrix shells made of sandwich CNTRC sheets on both sides and open/closed cellular functionally graded porous core", Thin Wall. Struct., 182, 110161. https://doi.org/10.1016/j.tws.2022.110161.
  36. Ninh, D.G., Quan, N.M. and Hoang, V.N.V. (2022b), "Thermally vibrational analyses of functionally graded graphene nanoplatelets reinforced funnel shells with different complex shapes surrounded by elastic foundation", Mech. Adv. Mater. Struct., 29(26), 4654-4676. https://doi.org/10.1080/15376494.2021.1934763.
  37. Ninh, D.G., Van Vang, T., Ha, N.H., Long, N.T., Nguyen, C.T. and Dao, D.V. (2022a), "Effect of cracks on dynamical responses of double-variable-edge plates made of graphene nanoplatelets-reinforced porous matrix and sur-bonded by piezoelectric layers subjected to thermo-mechanical loads", Eur. J. Mech. A/Solid., 96, 104742. https://doi.org/10.1016/j.euromechsol.2022.104742.
  38. Qu, Y., Wu, S., Chen, Y. and Hua, H. (2013), "Vibration analysis of ring-stiffened conical-cylindrical-spherical shells based on a modified variational approach", Int. J. Mech. Sci., 69, 72-84. https://doi.org/10.1016/j.ijmecsci.2013.01.026.
  39. Rahmani, M. and Mohammadi, Y. (2021), "Vibration of two types of porous FG sandwich conical shell with different boundary conditions", Struct. Eng. Mech., 79(4), 401. https://doi.org/10.12989/sem.2021.79.4.401.
  40. Reddy, J.N. (2004), Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, Boca Raton, CRC Press.
  41. Setoodeh, A.R., Shojaee, M. and Malekzadeh, P. (2018), "Application of transformed differential quadrature to free vibration analysis of FG-CNTRC quadrilateral spherical panel with piezoelectric layers", Comput. Meth. Appl. Mech. Eng., 335, 510-537. https://doi.org/10.1016/j.cma.2018.02.022.
  42. Shakeri, M., Eslami, M.R. and Yas, M.H. (1999), "Elasticity solution and free vibrations analysis of laminated anisotropic cylindrical shells", Struct. Eng. Mech., 7(2), 181-202. https://doi.org/10.12989/sem.1999.7.2.181.
  43. Shen, H.S. and Xiang, Y. (2012), "Nonlinear vibration of nanotube-reinforced composite cylindrical shells in thermal environments", Comput. Meth. Appl. Mech. Eng., 213, 196-205. https://doi.org/10.1016/j.cma.2011.11.025.
  44. Sofiyev, A.H. (2007), "Vibration and stability of composite cylindrical shells containing a FG layer subjected to various loads", Struct. Eng. Mech., 27(3), 365-391. https://doi.org/10.12989/sem.2007.27.3.365.
  45. Sofiyev, A.H., Keskin, S.N. and Sofiyev, A.H. (2004), "Effects of elastic foundation on the vibration of laminated non-homogeneous orthotropic circular cylindrical shells", Shock Vib., 11(2), 89-101. https://doi.org/10.1155/2004/424926.
  46. Soong, T.C. (1969), "Buckling of cylindrical shells with eccentric spiral-type stiffeners", AIAA J., 7(1), 65-72. https://doi.org/10.2514/3.5036.
  47. 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.
  48. Wang, X. and Guo, W. (2016), "Dynamic modeling and vibration characteristics analysis of submerged stiffened combined shells", Ocean Eng., 127, 226-235. https://doi.org/10.1016/j.oceaneng.2016.10.008.
  49. Wang, X.H. and Redekop, D. (2011), "Free vibration analysis of moderately-thick and thick toroidal shells", Struct. Eng. Mech., 39(4), 449-463. https://doi.org/10.12989/sem.2011.39.4.449.
  50. Wang, Y. and Wu, D. (2017). "Free vibration of functionally graded porous cylindrical shell using a sinusoidal shear deformation theory", Aerosp. Sci. Technol., 66, 83-91. https://doi.org/10.1016/j.ast.2017.03.003.
  51. Wang, Y.Q., Guo, X.H., Chang, H.H. and Li, H.Y. (2010), "Nonlinear dynamic response of rotating circular cylindrical shells with precession of vibrating shape-Part I: Numerical solution", Int. J. Mech. Sci., 52(9), 1217-1224. https://doi.org/10.1016/j.ijmecsci.2010.05.008.
  52. Wattanasakulpong, N. and Ungbhakorn, V. (2014), "Linear and nonlinear vibration analysis of elastically restrained ends FGM beams with porosities", Aerosp. Sci. Technol., 32(1), 111-120. https://doi.org/10.1016/j.ast.2013.12.002.
  53. Wu, M.Q., Zhang, W. and Niu, Y. (2021), "Experimental and numerical studies on nonlinear vibrations and dynamic snap-through phenomena of bistable asymmetric composite laminated shallow shell under center foundation excitation", Eur. J. Mech. A-Solid., 89, 104303. https://doi.org/10.1016/j.euromechsol.2021.104303.
  54. Yazdani, R., Mohammadimehr, M. and Navi, B.R. (2019), "Free vibration of Cooper-Naghdi micro saturated porous sandwich cylindrical shells with reinforced CNT face sheets under magneto-hydro-thermo-mechanical loadings", Struct. Eng. Mech., 70(3), 351-365. https://doi.org/10.12989/sem.2019.70.3.351.
  55. Zahedinejad, P., Malekzadeh, P., Farid, M. and Karami, G. (2010), "A semi-analytical three-dimensional free vibration analysis of functionally graded curved panels", Int. J. Press. Vessel. Pip., 87(8), 470-480. https://doi.org/10.1016/j.ijpvp.2010.06.001.
  56. Zhang, W., Liu, T., Xi, A. and Wang, Y.N. (2018), "Resonant responses and chaotic dynamics of composite laminated circular cylindrical shell with membranes", J. Sound Vib., 423, 65-99. https://doi.org/10.1016/j.jsv.2018.02.049.
  57. Zhu, J., Lai, Z., Yin, Z., Jeon, J. and Lee, S. (2001), "Fabrication of ZrO2-NiCr functionally graded material by powder metallurgy", Mater. Chem. Phys., 68(1-3), 130-135. https://doi.org/10.1016/S0254-0584(00)00355-2.