Abstract
In this paper, analytical solution of a refined plate theory is developed for free vibration analysis of functionally graded plates under various boundary conditions. The theory accounts for a quadratic variation of the transverse shear strains across the thickness, and satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate without using shear correction factors. The mechanical properties of functionally graded material are assumed to vary according to power law distribution of the volume fraction of the constituents. The Levy-type solution in conjunction with the state space concept is used to solve the equations of motion of rectangular plates with two opposite edges simply supported and the other two edges having arbitrary boundary conditions. The accuracy of the present solutions is verified by comparing the present results with those obtained using the classical theory, first-order theory, and higher-order theory.
Similar content being viewed by others
References
Baferani, A. H., Saidi, A. R., and Ehteshami, H. (2011). “Accurate solution for free vibration analysis of functionally graded thick rectangular plates resting on elastic foundation.” Composite Structures, Vol. 93, No. 7, pp. 1842–1853.
Brischetto, S. and Carrera, E. (2010). “Advanced mixed theories for bending analysis of functionally graded plates.” Computers & Structures, Vol. 88, Nos. 23–24, pp. 1474–1483.
Ferreira, A. J. M., Batra, R. C., Roque, C. M. C., Qian, L. F., and Jorge, R. M. N. (2006). “Natural frequencies of functionally graded plates by a meshless method.” Composite Structures, Vol. 75, Nos. 1–4, pp. 593–600.
He, X. Q., Ng, T. Y., Sivashanker, S., and Liew, K. M. (2001). “Active control of FGM plates with integrated piezoelectric sensors and actuators.” International Journal of Solids and Structures, Vol. 38, No. 9, pp. 1641–1655.
Hosseini-Hashemi, S., Fadaee, M., and Atashipour, S. R. (2011a). “A new exact analytical approach for free vibration of Reissner-Mindlin functionally graded rectangular plates.” International Journal of Mechanical Sciences, Vol. 53, No. 1, pp. 11–22.
Hosseini-Hashemi, S., Fadaee, M., and Atashipour, S. R. (2011b). “Study on the free vibration of thick functionally graded rectangular plates according to a new exact closed-form procedure.” Composite Structures, Vol. 93, No. 2, pp. 722–735.
Hosseini-Hashemi, S., Rokni Damavandi Taher, H., Akhavan, H., and Omidi, M. (2010). “Free vibration of functionally graded rectangular plates using first-order shear deformation plate theory.” Applied Mathematical Modelling, Vol. 34, No. 5, pp. 1276–1291.
Jha, D. K., Kant, T., and Singh, R. K. (2013). “A critical review of recent research on functionally graded plates.” Composite Structures, Vol. 96, pp. 833–849.
Kim, Y. W. (2005). “Temperature dependent vibration analysis of functionally graded rectangular plates.” Journal of Sound and Vibration, Vol. 284, Nos. 3–5, pp. 531–549.
Liu, D. Y., Wang, C. Y., and Chen, W. Q. (2010). “Free vibration of FGM plates with in-plane material inhomogeneity.” Composite Structures, Vol. 92, No. 5, pp. 1047–1051.
Mantari, J. L. and Guedes Soares, C. (2013). “Finite element formulation of a generalized higher order shear deformation theory for advanced composite plates.” Composite Structures, Vol. 96, pp. 545–553.
Matsunaga, H. (2008). “Free vibration and stability of functionally graded plates according to a 2-D higher-order deformation theory.” Composite Structures, Vol. 82, No. 4, pp. 499–512.
Mirtalaie, S. H. and Hajabasi, M. A. (2011). “Free vibration analysis of functionally graded thin annular sector plates using the differential quadrature method.” Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, Vol. 225, No. 3, pp. 568–583.
Mori, T. and Tanaka, K. (1973). “Average stress in matrix and average elastic energy of materials with misfitting inclusions.” Acta Metallurgica, Vol. 21, No. 5, pp. 571–574.
Natarajan, S. and Manickam, G. (2012). “Bending and vibration of functionally graded material sandwich plates using an accurate theory.” Finite Elements in Analysis and Design, Vol. 57,pp. 32–42.
Nguyen-Xuan, H., Tran, L. V., Nguyen-Thoi, T., and Vu-Do, H. C. (2011). “Analysis of functionally graded plates using an edge-based smoothed finite element method.” Composite Structures, Vol. 93, No. 11, pp. 3019–3039.
Nguyen-Xuan, H., Tran, L. V., Thai, C. H., and Nguyen-Thoi, T. (2012). “Analysis of functionally graded plates by an efficient finite element method with node-based strain smoothing.” Thin-Walled Structures, Vol. 54, pp. 1–18.
Oyekoya, O. O., Mba, D. U., and El-Zafrany, A. M. (2009). “Buckling and vibration analysis of functionally graded composite structures using the finite element method.” Composite Structures, Vol. 89, No. 1, pp. 134–142.
Pradyumna, S. and Bandyopadhyay, J. N. (2008). “Free vibration analysis of functionally graded curved panels using a higher-order finite element formulation.” Journal of Sound and Vibration, Vol. 318, Nos. 1–2, pp. 176–192.
Qian, L. F., Batra, R. C., and Chen, L. M. (2003). “Free and forced vibrations of thick rectangular plates using higher-order shear and normal deformable plate theory and meshless Petrov-Galerkin (MLPG) method.” Computer Modeling in Engineering and Sciences, Vol. 4, No. 5, pp. 519–534.
Qian, L. F., Batra, R. C., and Chen, L. M. (2004). “Static and dynamic deformations of thick functionally graded elastic plates by using higher-order shear and normal deformable plate theory and meshless local Petrov-Galerkin method.” Composites Part B: Engineering, Vol. 35, Nos. 6–8, pp. 685–697.
Reddy, J. N. (1984). “A simple higher-order theory for laminated composite plates.” Journal of Applied Mechanics, Vol. 51, No. 4, pp. 745–752.
Reddy, J. N. (2000). “Analysis of functionally graded plates.” International Journal for Numerical Methods in Engineering, Vol. 47, Nos. 1–3, pp. 663–684.
Roque, C. M. C., Ferreira, A. J. M., and Jorge, R. M. N. (2007). “A radial basis function approach for the free vibration analysis of functionally graded plates using a refined theory.” Journal of Sound and Vibration, Vol. 300, Nos. 3–5, pp. 1048–1070.
Shimpi, R. P. (2002). “Refined plate theory and its variants.” AIAA Journal, Vol. 40, No. 1, pp. 137–146.
Sundararajan, N., Prakash, T., and Ganapathi, M. (2005). “Nonlinear free flexural vibrations of functionally graded rectangular and skew plates under thermal environments.” Finite Elements in Analysis and Design, Vol. 42, No. 2, pp. 152–168.
Talha, M. and Singh, B. N. (2010). “Static response and free vibration analysis of FGM plates using higher order shear deformation theory.” Applied Mathematical Modelling, Vol. 34, No. 12, pp. 3991–4011.
Thai, H. T. and Choi, D. H. (2011). “A refined plate theory for functionally graded plates resting on elastic foundation.” Composites Science and Technology, Vol. 71, No. 16, pp. 1850–1858.
Thai, H. T. and Choi, D. H. (2012). “An efficient and simple refined theory for buckling analysis of functionally graded plates.” Applied Mathematical Modelling, Vol. 36, No. 3, pp. 1008–1022.
Thai, H. T. and Choi, D. H. (2013a). “Efficient higher-order shear deformation theories for bending and free vibration analyses of functionally graded plates.” Archive of Applied Mechanics, Vol. 83, No. 12, pp. 1755–1771.
Thai, H. T. and Choi, D. H. (2013b). “Finite element formulation of various four unknown shear deformation theories for functionally graded plates.” Finite Elements in Analysis and Design, Vol. 75, pp. 50–61.
Thai, H. T. and Choi, D. H. (2013c). “A simple first-order shear deformation theory for the bending and free vibration analysis of functionally graded plates.” Composite Structures, Vol. 101, pp. 332–340.
Thai, H. T. and Choi, D. H. (2013d). “Zeroth-order shear deformation theory for functionally graded plates resting on elastic foundation.” International Journal of Mechanical Sciences, Vol. 73, pp. 40–52.
Thai, H. T. and Choi, D. H. (2014). “Improved refined plate theory accounting for effect of thickness stretching in functionally graded plates.” Composites Part B: Engineering, Vol. 56, pp. 705–716.
Thai, H. T. and Kim, S. E. (2010). “Free vibration of laminated composite plates using two variable refined plate theory.” International Journal of Mechanical Sciences, Vol. 52, No. 4, pp. 626–633.
Thai, H. T. and Kim, S. E. (2011). “Levy-type solution for buckling analysis of orthotropic plates based on two variable refined plate theory.” Composite Structures, Vol. 93, No. 7, pp. 1738–1746.
Thai, H. T. and Kim, S. E. (2013a). “A simple higher-order shear deformation theory for bending and free vibration analysis of functionally graded plates.” Composite Structures, Vol. 96, pp. 165–173.
Thai, H. T. and Kim, S. E. (2013b). “A simple quasi-3D sinusoidal shear deformation theory for functionally graded plates.” Composite Structures, Vol. 99, pp. 172–180.
Thai, H. T., Taehyo, P., and Choi, D. H. (2013). “An efficient shear deformation theory for vibration of functionally graded plates.” Archive of Applied Mechanics, Vol. 83, No. 1, pp. 137–149.
Thai, H. T. and Vo, T. P. (2013). “A new sinusoidal shear deformation theory for bending, buckling, and vibration of functionally graded plates.” Applied Mathematical Modelling, Vol. 37, No. 5, pp. 3269–3281.
Tran, L. V., Ferreira, A. J. M. and Nguyen-Xuan, H. (2013). “Isogeometric analysis of functionally graded plates using higher-order shear deformation theory.” Composites Part B: Engineering, Vol. 51, pp. 368–383.
Valizadeh, N., Natarajan, S., Gonzalez-Estrada, O. A., Rabczuk, T., Bui, T. Q., and Bordas, S. P. A. (2013). “NURBS-based finite element analysis of functionally graded plates: Static bending, vibration, buckling and flutter.” Composite Structures, Vol. 99, pp. 309–326.
Vel, S. S. and Batra, R. C. (2004). “Three-dimensional exact solution for the vibration of functionally graded rectangular plates.” Journal of Sound and Vibration, Vol. 272, Nos. 3–5, pp. 703–730.
Woo, J., Meguid, S. A., and Ong, L. S. (2006). “Nonlinear free vibration behavior of functionally graded plates.” Journal of Sound and Vibration, Vol. 289, No. 3, pp. 595–611.
Zhao, X., Lee, Y. Y., and Liew, K. M. (2009). “Free vibration analysis of functionally graded plates using the element-free kp-Ritz method.” Journal of Sound and Vibration, Vol. 319, Nos. 3–5, pp. 918–939.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Thai, HT., Choi, DH. Levy solution for free vibration analysis of functionally graded plates based on a refined plate theory. KSCE J Civ Eng 18, 1813–1824 (2014). https://doi.org/10.1007/s12205-014-0409-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12205-014-0409-2