A generalized differential quadrature-based computational model for describing free vibrations behavior of functionally graded circular plates around buckled configuration

Document Type : Research Note

Authors

1 Department of Mechanical Engineering, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran

2 Laboratory Unit, Stam Sanat Company, Karaj, Iran

3 Department of Mechanical Engineering, University of Kurdistan, Sanandaj, Iran

4 Department of Mechanical Engineering, Ahrar Institute of Technology and Higher Education, Rasht, Iran

Abstract

In the present article, the vibrational behavior of buckled functionally graded (FG) circular plates with clamped and simply-supported edge conditions is described. Considering von Kármán’s assumptions, the geometric nonlinearity is incorporated into the Kirchhoff plate theory and the nonlinear governing equations of motion are then derived using Hamilton’s principle. Critical buckling load and linear natural frequencies are first calculated using the generalized differential quadrature (GDQ) method. Afterward, the postbuckling characteristics of the circular plate are obtained via solving the nonlinear governing equations, directly. By several comparative studies, the reliability of the presented model is revealed. Finally, the fundamental natural frequency of the plate is evaluated for prebuckled and postbuckled configurations. The effects of material property and boundary conditions on the static bifurcation diagram and the natural frequency of the initial undeflected and bucked plate are studied. It is found that the trend of the fundamental natural frequency changes with the applied radial load around the prebuckled configuration is unlike the one around the buckled configuration.

Keywords


  1. References:

    1. Sofiyev, A.H. "The vibration and stability behavior of freely supported FGM conical shells subjected to external pressure", Composite Structures, 89(3), pp. 356-366 (2009).
    2. Najafizadeh, M.M., and Hedayati, B. "Refined theory for thermoelastic stability of functionally graded circular plates", Journal of Thermal Stresses, 27(9), pp. 857-880 (2004).
    3. Allahverdizadeh, A., Naei, M.H., and Nikkhah Bahrami, M. "Nonlinear free and forced vibration analysis of thin circular functionally graded plates", Journal of Sound and Vibration, 310(4-5), pp. 966-984 (2008).
    4. Sepahi, O., Forouzan, M.R., and Malekzadeh, P. "Thermal buckling and postbuckling analysis of functionally graded annular plates with temperature-dependent material properties", Materials Design, 32(7), pp. 4030-4041 (2011).
    5. Fallah, F., Vahidipoor, M.K., and Nosier, A. "Post-buckling behavior of functionally graded circular plates under asymmetric transverse and in-plane loadings", Composite Structures, 125, pp. 477-488 (2015).
    6. Ansari, R., Shahabodini, A., and Shojaei, M.F. "Vibrational analysis of carbon nanotube-reinforced composite quadrilateral plates subjected to thermal environments using a weak formulation of elasticity", Composite Structures, 139, pp. 167-187 (2016).
    7. Żur, K.K. "Green’s function in frequency analysis of circular thin plates of variable thickness", Journal of Theoretical and Applied Mechanics, 53(4), pp. 873-884 (2015).
    8. Żur, K.K. "Green's function for frequency analysis of thin annular plates with nonlinear variable thickness", Applied Mathematical Modelling, 40 (5-6), pp. 3601-3619 (2016).
    9. Żur, K.K. "Green’s function approach to frequency analysis of thin circular plates", Bulletin of the Polish Academy of Sciences. Technical Sciences, 64(1), (2016).
    10. Żur, K.K. "Free vibration analysis of elastically supported functionally graded annular plates via quasi-Green's function method", Composites Part B: Engineering, 144, pp. 37-55 (2018).
    11. Żur, K.K. "Multiparametric analytical solution for the eigenvalue problem of FGM porous circular plates", Symmetry, 11(3), p. 429 (2019).
    12. Shahabodini, A., Gholami, Y., Ansari, R., et al. "Vibration analysis of graphene sheets resting on Winkler/Pasternak foundation: A multiscale approach", The European Physical Journal Plus, 134(10), pp. 1-15 (2019).
    13. Gholami, Y., Shahabodini, A., Ansari, R., et al. "Nonlinear vibration analysis of graphene sheets resting on Winkler–Pasternak elastic foundation using an atomistic-continuum multiscale model", Acta Mechanica, 230(12), pp. 4157-4174 (2019).
    14. Nikbakht, S., Kamarian, S., and Shakeri, M. "A review on optimization of composite structures Part II: Functionally graded materials", Composite Structures, 214, pp. 83-102 (2019).
    15. Smita, P., and Mohanty, S.C. "Nonlinear free vibration analysis of functionally graded plate resting on elastic foundation in thermal environment using higher-order shear deformation theory", Scientia Iranica, 26(2), pp. 815-833 (2019).
    16. Gholami, R., and Ansari, R. "On the vibration of postbuckled functionally graded-carbon nanotube reinforced composite annular plates", Scientia Iranica, 26(2), pp. 3857-3874 (2019).
    17. Atri, H.R., and Shojaee, S. "Analysis of laminated composite plates based on THB-RKPM method using the higher order shear deformation plate theory", Scientia Iranica, 26(4), pp. 2057-2078 (2019).
    18. Li, M., Soares, C.G., and Yan, R. "A novel shear deformation theory for static analysis of functionally graded plates", Composite Structures, 250, p. 112559 (2020).
    19. Sharma, T.K. "Free vibration analysis of functionally graded circular piezoelectric plate using COMSOL multiphysics", In AIP Conference Proceedings, AIP Publishing LLC, 2220(1), p. 080017 (2020).
    20. Radaković, A., Čukanović, D., Bogdanović, G., et al. "Thermal buckling and free vibration analysis of functionally graded plate resting on an elastic foundation according to high order shear deformation theory based on new shape function", Applied Sciences, 10(12), p. 4190 (2020).
    21. Lal, R., and Saini, R. "Vibration analysis of functionally graded circular plates of variable thickness under thermal environment by generalized differential quadrature method", Journal of Vibration and Control, 26(1-2), pp. 73-87 (2020).
    22. Xu, P., and Wellens, P. "Effects of static loads on the nonlinear vibration of circular plates", Journal of Sound and Vibration, 504, p. 116111 (2021).
    23. Imran, M., Khan, R., and Badshah, S. "Experimental, analytical, and finite element vibration analyses of delaminated composite plates", Scientia Iranica, 28(1), pp. 231-240 (2021).
    24. Qin, X., Shen, Y., Chen, W., et al. "Bending and free vibration analyses of circular stiffened plates using the FSDT mesh-free method", International Journal of Mechanical Sciences, 202, p. 106498 (2021).
    25. Javani, M., Kiani, Y., and Eslami, M.R. "Geometrically nonlinear free vibration of FG-GPLRC circular plate on the nonlinear elastic foundation", Composite Structures, 261, p. 113515 (2021).
    26. Sobhy, M. "3-D elasticity numerical solution for magneto-hygrothermal bending of FG graphene/metal circular and annular plates on an elastic medium", European Journal of Mechanics-A/Solids, 88, p. 104265 (2021).
    27. Hilali, Y., and Bourihane, O. "A mixed MLS and Hermite-type MLS method for buckling and postbuckling analysis of thin plates", Structures, 33, pp. 2349-2360 (2021).
    28. Singh, S.D., and Sahoo, R. "Analytical solution for static and free vibration analysis of functionally graded CNT-reinforced sandwich plates", Archive of Applied Mechanics, pp. 1-16 (2021).
    29. Singh, V., Kumar, R., and Patel, S.N. "Parametric instability analysis of functionally graded CNT-reinforced composite (FG-CNTRC) plate subjected to different types of non-uniform in-plane loading", Emerging Trends of Advanced Composite Materials in Structural Applications, pp. 291-312 (2021).
    30. Sladek, J., Sladek, V., Krahulec, S., et al. "Analyses of circular magnetoelectroelastic plates with functionally graded material properties", Mechanics of Advanced Materials and Structures, 22(6), pp. 479-489 (2015).
    31. Saadatmand, M., and Kook, J. "Multi-objective optimization of a circular dual back-plate MEMS microphone: tradeoff between pull-in voltage, sensitivity and resonance frequency", Microsystem Technologies, 25(8), pp. 2937-2947 (2019).
    32. Ansari, R., Faghih Shojaei, M., Mohammadi, V., et al. "Size-dependent vibrations of post-buckled functionally graded Mindlin rectangular microplates", Latin American Journal of Solids and Structures, 11, pp. 2351-2378 (2014).
    33. Leissa, A.W. "Conditions for laminated plates to remain flat under inplane loading", Composite Structures, 6(4), pp. 261-270 (1986).
    34. Hui-Shen, S. "Thermal postbuckling behavior of shear deformable FGM plates with temperature-dependent properties", International Journal of Mechanical Sciences, 49(4), pp. 466-478 (2007).
    35. Hui-Shen S. "Nonlinear bending response of functionally graded plates subjected to transverse loads and in thermal environments", International Journal of Mechanical Sciences, 44(3), pp. 561-584 (2002).
    36. Piotr, J., Żur, K.K., Kim, J., et al. "On the bifurcation buckling and vibration of porous nanobeams", Composite Structures, 250, p. 112632 (2020).
    37. Shu, C. "Differential quadrature and its application in engineering", Springer Science & Business Media (2000).
    38. Nayfeh, A.H, and Emam, S.A. "Exact solution and stability of postbuckling configurations of beams", Nonlinear Dynamics, 54(4), pp. 395-408 (2008).