Nonlinear Dynamic Analysis of Rectangular FGM Plates of Variable Thickness Subjected to Mechanical Load
Main Article Content
Abstract
This paper establishes the governing equations of rectangular plates of variable thickness subjected to mechanical load by using the classical plate theory, the geometrical nonlinearity in von Karman-Donnell sense. Solutions to the problem are derived according to Galerkin method. Nonlinear dynamic responses, critical dynamic loads are obtained by using Runge-Kutta method and the Budiansky–Roth criterion. The effect of volume-fraction index k and some geometric factors are considered and numerical results are presented.
Keywords:
Dynamic responses, nonlinear vibration, rectangular FGM plate, variable thickness
References
[1] V. Ungbhakorn, N. Wattanasakulpong, Thermo-elastic Vibration Analysis of Third-order Shear Deformable Functionally Graded Plates with Distributed Patch Mass Under Thermal Environment, J. Appl. Acoust. 74 (2013) 1045–1059. http://doi.org/10.1016/j.apacoust.2013.03.010.
[2] M. Talha, B.N. Singh, Static Response and Free Vibration Analysis of FGM Plates Using Higher Order Shear Deformation Theory, Appl. Math. Model. 34 (2010) 3991–4011. http://doi.org/10.1016/j.apm.2010.03.034.
[3] N.D. Duc, P.H. Cong, Nonlinear Postbuckling of Symmetric S-FGM Plates Resting on Elastic Foundations Using Higher Order Shear Deformation Plate Theory in Thermal Environments, Compos. Struct. 100 (2013) 566–574. https://doi.org/10.1016/j.compstruct.2013.01.006.
[4] N.D. Duc, H.V. Tung, Mechanical and Thermal Postbuckling of Higher Order Shear Deformable Functionally Graded Plates on Elastic Foundations, Compos. Struct. 93 (2011) 2874–2881.
https://doi.org/10.1016/j.compstruct.2011.05.017.
[5] Dao Huy Bich, Vu Hoai Nam, Nguyen Thi Phuong, Nonlinear postbuckling of eccentrically stiffened functionally graded plates and shallow shells. Vietnam J Mech. 33 (3) (2011) 132–47.
https://doi.org/10.15625/0866-7136/33/3/207
[6] H. Hebali, A. Tounsi, M.S. A. Houari, A. Bessaim, E.A.A. Bedia, New Quasi-3D Hyperbolic Shear Deformation Theory for the Static and Free Vibration Analysis of Functionally Graded Plates, Journal of Engineering Mechanics 140 (2014) 374-383. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000665.
[7] A. Mahi, E.A.A. Bedia, A. Tounsi, A new hyperbolic shear deformation theory for bending and free vibration analysis of isotropic, functionally graded, sandwich and laminated composite plates, Applied Mathematical Modelling 39(9) (2015) 2489-2508. https://doi.org/10.1016/j.apm.2014.10.045.
[8] R. Benferhat, T.H. Daouadji, M.S. Mansour, L. Hadji, Free vibration analysis of FG plates resting on an elastic foundation and based on the neutral surface concept using higher-order shear deformation theory, Comptes Rendus Mécanique 344(9) (2016) 631-641. https://doi.org/10.12989/eas.2016.10.5.1033.
[9] R. Kandasamy, R. Dimitri, F. Tornabene, Numerical study on the free vibration and thermal buckling behavior of moderately thick functionally graded structures in thermal environments, Composite Structures, 157 (2016) 207-221. https://doi.org/10.1016/j.compstruct.2016.08.037.
[10] E. Efraim, M. Eisenberger, Exact vibration analysis of variable thickness thick annular isotropic and FGM plates. Journal of Sound and Vibration 299 (2007) 720–738. https://doi.org/10.1016/j.jsv.2006.06.068.
[11] S.H. Hosseini-Hashemi, H.R.D. Taher, H. Akhavan, Vibration analysis of radially FGM sectorial plates of variable thickness on elastic foundations, Composite Structures 92 (2010) 1734–1743.
https://doi.org/10.1016/j.compstruct.2009.12.016.
[12] M. Shariyat, M.M. Alipou, A power series solution for vibration and complex modal stress analyses of variable thickness viscoelastic two-directional FGM circular plates on elastic foundations, Applied Mathematical Modelling 37 (2013) 3063–3076. https://doi.org/10.1016/j.apm.2012.07.037.
[13] V. Tajeddini, A. Ohadi, M. Sadighi, Three-dimensional free vibration of variable thickness thick circular and annular isotropic and functionally graded plates on Pasternak foundation, Int J of Mech Sci. 53 (2011) 300–308. https://doi.org/10.1016/j.ijmecsci.2011.01.011.
[14] F. Tornabene, N. Fantuzzi, M. Bacciocchi, E. Viola, J.N. Reddy, A Numerical Investigation on the Natural Frequencies of FGM Sandwich Shells with Variable Thickness by the Local Generalized Differential Quadrature Method, Applied Sciences 7 (2) (2017) 131. https://doi.org/10.3390/app7020131.
[15] A.H. Sofiyev, The buckling of an orthotropic composite truncated conical shell with continuously varying thickness subject to a time dependent external pressure, Composites: Part B. 34 (2003) 227–233.
https://doi.org/10.1016/S1359-8368(02)00105-1.
[16] R.A. Akbari, S.A. Ahmadi, Buckling Analysis of Functionally Graded Thick Cylindrical Shells with Variable Thickness Using DQM, Arabian Journal for Science and Engineering 39 (11) (2014) 8121-8133.
https://doi.org/10.1007/s13369-014-1356-4.
[17] P.T. Thang, N.D. Duc, N.T. Trung, Effects of variable thickness and imperfection on nonlinear buckling of sigmoid-functionally graded cylindrical panels, Composite Structures 155 (2016) 99-106.
https://doi.org/10.1016/j.compstruct.2016.08.007.
[18] P.T. Thang, N.T. Trung, J. Lee, Closed-form expression for nonlinear analysis of imperfect sigmoid-FGM plates with variable thickness resting on elastic medium, Composite Structures 143 (2016)143-150.
https://doi.org/10.1016/j.compstruct.2016.02.002.
[19] D.O. Brush, Almroth, Buckling of Bars, Plates and Shells, New York, Mc Graw-Hill, Inc., 1975.
[20] A.S. Volmir, Nonlinear Dynamics of Plates and Shells, Science Edition, Moscow, 1972.
[21] B. Budiansky, R.S. Roth, Axisymmetric dynamic buckling of clamped shallow spherical shells, NASA Technical Note D. 510 (1962) 597–609.
[22] B. Uymaz, M. Aydogdu, Three-Dimensional Vibration Analyses of Functionally Graded Plates under Various Boundary Conditions. Journal of Reinforced Plastics and Composites 26(18) (2007) 1847–1863.
https://doi.org/10.1177/0731684407081351 22.
[2] M. Talha, B.N. Singh, Static Response and Free Vibration Analysis of FGM Plates Using Higher Order Shear Deformation Theory, Appl. Math. Model. 34 (2010) 3991–4011. http://doi.org/10.1016/j.apm.2010.03.034.
[3] N.D. Duc, P.H. Cong, Nonlinear Postbuckling of Symmetric S-FGM Plates Resting on Elastic Foundations Using Higher Order Shear Deformation Plate Theory in Thermal Environments, Compos. Struct. 100 (2013) 566–574. https://doi.org/10.1016/j.compstruct.2013.01.006.
[4] N.D. Duc, H.V. Tung, Mechanical and Thermal Postbuckling of Higher Order Shear Deformable Functionally Graded Plates on Elastic Foundations, Compos. Struct. 93 (2011) 2874–2881.
https://doi.org/10.1016/j.compstruct.2011.05.017.
[5] Dao Huy Bich, Vu Hoai Nam, Nguyen Thi Phuong, Nonlinear postbuckling of eccentrically stiffened functionally graded plates and shallow shells. Vietnam J Mech. 33 (3) (2011) 132–47.
https://doi.org/10.15625/0866-7136/33/3/207
[6] H. Hebali, A. Tounsi, M.S. A. Houari, A. Bessaim, E.A.A. Bedia, New Quasi-3D Hyperbolic Shear Deformation Theory for the Static and Free Vibration Analysis of Functionally Graded Plates, Journal of Engineering Mechanics 140 (2014) 374-383. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000665.
[7] A. Mahi, E.A.A. Bedia, A. Tounsi, A new hyperbolic shear deformation theory for bending and free vibration analysis of isotropic, functionally graded, sandwich and laminated composite plates, Applied Mathematical Modelling 39(9) (2015) 2489-2508. https://doi.org/10.1016/j.apm.2014.10.045.
[8] R. Benferhat, T.H. Daouadji, M.S. Mansour, L. Hadji, Free vibration analysis of FG plates resting on an elastic foundation and based on the neutral surface concept using higher-order shear deformation theory, Comptes Rendus Mécanique 344(9) (2016) 631-641. https://doi.org/10.12989/eas.2016.10.5.1033.
[9] R. Kandasamy, R. Dimitri, F. Tornabene, Numerical study on the free vibration and thermal buckling behavior of moderately thick functionally graded structures in thermal environments, Composite Structures, 157 (2016) 207-221. https://doi.org/10.1016/j.compstruct.2016.08.037.
[10] E. Efraim, M. Eisenberger, Exact vibration analysis of variable thickness thick annular isotropic and FGM plates. Journal of Sound and Vibration 299 (2007) 720–738. https://doi.org/10.1016/j.jsv.2006.06.068.
[11] S.H. Hosseini-Hashemi, H.R.D. Taher, H. Akhavan, Vibration analysis of radially FGM sectorial plates of variable thickness on elastic foundations, Composite Structures 92 (2010) 1734–1743.
https://doi.org/10.1016/j.compstruct.2009.12.016.
[12] M. Shariyat, M.M. Alipou, A power series solution for vibration and complex modal stress analyses of variable thickness viscoelastic two-directional FGM circular plates on elastic foundations, Applied Mathematical Modelling 37 (2013) 3063–3076. https://doi.org/10.1016/j.apm.2012.07.037.
[13] V. Tajeddini, A. Ohadi, M. Sadighi, Three-dimensional free vibration of variable thickness thick circular and annular isotropic and functionally graded plates on Pasternak foundation, Int J of Mech Sci. 53 (2011) 300–308. https://doi.org/10.1016/j.ijmecsci.2011.01.011.
[14] F. Tornabene, N. Fantuzzi, M. Bacciocchi, E. Viola, J.N. Reddy, A Numerical Investigation on the Natural Frequencies of FGM Sandwich Shells with Variable Thickness by the Local Generalized Differential Quadrature Method, Applied Sciences 7 (2) (2017) 131. https://doi.org/10.3390/app7020131.
[15] A.H. Sofiyev, The buckling of an orthotropic composite truncated conical shell with continuously varying thickness subject to a time dependent external pressure, Composites: Part B. 34 (2003) 227–233.
https://doi.org/10.1016/S1359-8368(02)00105-1.
[16] R.A. Akbari, S.A. Ahmadi, Buckling Analysis of Functionally Graded Thick Cylindrical Shells with Variable Thickness Using DQM, Arabian Journal for Science and Engineering 39 (11) (2014) 8121-8133.
https://doi.org/10.1007/s13369-014-1356-4.
[17] P.T. Thang, N.D. Duc, N.T. Trung, Effects of variable thickness and imperfection on nonlinear buckling of sigmoid-functionally graded cylindrical panels, Composite Structures 155 (2016) 99-106.
https://doi.org/10.1016/j.compstruct.2016.08.007.
[18] P.T. Thang, N.T. Trung, J. Lee, Closed-form expression for nonlinear analysis of imperfect sigmoid-FGM plates with variable thickness resting on elastic medium, Composite Structures 143 (2016)143-150.
https://doi.org/10.1016/j.compstruct.2016.02.002.
[19] D.O. Brush, Almroth, Buckling of Bars, Plates and Shells, New York, Mc Graw-Hill, Inc., 1975.
[20] A.S. Volmir, Nonlinear Dynamics of Plates and Shells, Science Edition, Moscow, 1972.
[21] B. Budiansky, R.S. Roth, Axisymmetric dynamic buckling of clamped shallow spherical shells, NASA Technical Note D. 510 (1962) 597–609.
[22] B. Uymaz, M. Aydogdu, Three-Dimensional Vibration Analyses of Functionally Graded Plates under Various Boundary Conditions. Journal of Reinforced Plastics and Composites 26(18) (2007) 1847–1863.
https://doi.org/10.1177/0731684407081351 22.