A unified solution for vibration analysis of moderately thick, functionally graded rectangular plates with general boundary restraints and internal line supports

References

• A.W. Leissa, Vibration of plates, U.S. Government Printing Office, Washington DC, 1973.
   Google Scholar
• M.S. Qatu, Vibration of laminated shells and plates, Elsevier, Amsterdam, 2004.
   Google Scholar
• J.N. Reddy, Mechanics of laminated composite plates and shells: Theory and analysis, CRC Press, Boca Raton, FL, 2004.
   Google Scholar
• E. Carrera, S. Brischetto, and P. Nali, Plates and shells for smart structures: Classical and advanced theories for modeling and analysis, Vol. 36, John Wiley & Sons, Hoboken, NJ, 2011.
   Google Scholar
• J. Yang and H.-S. Shen, Dynamic response of initially stressed functionally graded rectangular thin plates, Compos. Struct., vol. 54, no. 4, pp. 497–508, 2001.
   Web of Science ®Google Scholar
• J. Yanga and H.-S. Shen, Non-linear analysis of functionally graded plates under transverse and in-plane loads, Int. J. Non Linear Mech., vol. 38, no. 4, pp. 467–482, 2003.
   Web of Science ®Google Scholar
• S. Abrate, Free vibration, buckling, and static deflections of functionally graded plates, Compos. Sci. Technol., vol. 66, no. 14, pp. 2383–2394, 2006.
   Web of Science ®Google Scholar
• K. Shukla, K. Ravi Kumar, R. Pandey, and Y. Nath, Postbuckling response of functionally graded rectangular plates subjected to thermo-mechanical loading, Int. J. Struct. Stab. Dyn., vol. 7, no. 3, pp. 519–541, 2007.
   Web of Science ®Google Scholar
• M. Amabili and S. Carra, Thermal effects on geometrically nonlinear vibrations of rectangular plates with fixed edges, J. Sound Vib., vol. 321, no. 3, pp. 936–954, 2009.
   Web of Science ®Google Scholar
• D. Liu, C. Wang, and W. Chen, Free vibration of FGM plates with in-plane material inhomogeneity, Compos. Struct., vol. 92, no. 5, pp. 1047–1051, 2010.
   Web of Science ®Google Scholar
• M. Mohammadi, A.R. Saidi, and E. Jomehzadeh, Levy solution for buckling analysis of functionally graded rectangular plates, Appl. Compos. Mater., vol. 17, no. 2, pp. 81–93, 2010.
   Web of Science ®Google Scholar
• A.H. Baferani, A. Saidi, and E. Jomehzadeh, An exact solution for free vibration of thin functionally graded rectangular plates, J. Mech. Eng. Sci., vol. 225, no. 3, pp. 526–536, 2011.
   Web of Science ®Google Scholar
• E. Carrera, S. Brischetto, M. Cinefra, and M. Soave, Effects of thickness stretching in functionally graded plates and shells, Composites Part B, vol. 42, no. 2, pp. 123–133, 2011.
   Web of Science ®Google Scholar
• Y. Kiani, E. Bagherizadeh, and M. Eslami, Thermal buckling of clamped thin rectangular FGM plates resting on Pasternak elastic foundation (three approximate analytical solutions), ZAMM, vol. 91, no. 7, pp. 581–593, 2011.
   Web of Science ®Google Scholar
• B. Yang, H. Ding, and W. Chen, Elasticity solutions for functionally graded rectangular plates with two opposite edges simply supported, Appl. Math. Modell., vol. 36, no. 1, pp. 488–503, 2012.
   Web of Science ®Google Scholar
• K. Dai, G. Liu, K. Lim, X. Han, and S. Du, A meshfree radial point interpolation method for analysis of functionally graded material (FGM) plates, Comput. Mech., vol. 34, no. 3, pp. 213–223, 2004.
   Web of Science ®Google Scholar
• W. Lanhe, Thermal buckling of a simply supported moderately thick rectangular FGM plate, Compos. Struct., vol. 64, no. 2, pp. 211–218, 2004.
   Web of Science ®Google Scholar
• K. Dai, G. Liu, X. Han, and K. Lim, Thermomechanical analysis of functionally graded material (FGM) plates using element-free Galerkin method, Comput. Struct., vol. 83, no. 17, pp. 1487–1502, 2005.
   Web of Science ®Google Scholar
• A.M. Zenkour, Generalized shear deformation theory for bending analysis of functionally graded plates, Appl. Math. Modell., vol. 30, no. 1, pp. 67–84, 2006.
   Web of Science ®Google Scholar
• W. Lanhe, W. Hongjun, and W. Daobin, Dynamic stability analysis of FGM plates by the moving least squares differential quadrature method, Compos. Struct., vol. 77, no. 3, pp. 383–394, 2007.
   Web of Science ®Google Scholar
• T.-L., Wu, K. Shukla, and J.H. Huang, Post-buckling analysis of functionally graded rectangular plates, Compos. Struct., vol. 81, no. 1, pp. 1–10, 2007.
   Web of Science ®Google Scholar
• M. Mahdavian, Buckling analysis of simply-supported functionally graded rectangular plates under non-uniform in-plane compressive loading, J. Solid Mech., vol. 1, no. 3, pp. 213–225, 2009.
   Google Scholar
• S. Hosseini-Hashemi, H. Taher, H. Akhavan, and M. Omidi, Free vibration of functionally graded rectangular plates using first-order shear deformation plate theory, Appl. Math. Modell., vol. 34, no. 5, pp. 1276–1291, 2010.
   Web of Science ®Google Scholar
• M. Mohammadi, A. Saidi, and E. Jomehzadeh, A novel analytical approach for the buckling analysis of moderately thick functionally graded rectangular plates with two simply-supported opposite edges, J. Mech. Eng. Sci., vol. 224, no. 9, pp. 1831–1841, 2010.
   Web of Science ®Google Scholar
• F. Alijani, F. Bakhtiari-Nejad, and M. Amabili, Nonlinear vibrations of FGM rectangular plates in thermal environments, Nonlinear Dyn., vol. 66, no. 3, pp. 251–270, 2011.
   Web of Science ®Google Scholar
• S. Hosseini-Hashemi, M. Fadaee, and S.R. Atashipour, A new exact analytical approach for free vibration of Reissner–Mindlin functionally graded rectangular plates, Int. J. Mech. Sci., vol. 53, no. 1, pp. 11–22, 2011.
   Web of Science ®Google Scholar
• S. Natarajan, P. Baiz, S. Bordas, T. Rabczuk, and P. Kerfriden, Natural frequencies of cracked functionally graded material plates by the extended finite element method, Compos. Struct., vol. 93, no. 11, pp. 3082–3092, 2011.
   Web of Science ®Google Scholar
• X. Zhao, Y. Lee, and K.M. Liew, Free vibration analysis of functionally graded plates using the element-free kp-Ritz method, J. Sound Vib., vol. 319, no. 3, pp. 918–939, 2009.
   Web of Science ®Google Scholar
• X. Zhao and K.M. Liew, Geometrically nonlinear analysis of functionally graded plates using the element-free kp-Ritz method, Comput. Methods Appl. Mech. Eng., vol. 198, no. 33, pp. 2796–2811, 2009.
   Web of Science ®Google Scholar
• S. Hosseini-Hashemi, H. Taher, H. Akhavan, and M. Omidi, Free vibration of functionally graded rectangular plates using first-order shear deformation plate theory, Appl. Math. Modell., vol. 34, no. 5, pp. 1276–1291, 2010.
   Web of Science ®Google Scholar
• R. Javaheri and M. Eslami, Thermal buckling of functionally graded plates based on higher order theory, J. Therm. Stresses, vol. 25, no. 7, pp. 603–625, 2002.
   Web of Science ®Google Scholar
• J. Yang and H.-S. Shen, Vibration characteristics and transient response of shear-deformable functionally graded plates in thermal environments, J. Sound Vib., vol. 255, no. 3, pp. 579–602, 2002.
   Web of Science ®Google Scholar
• S. Hosseini-Hashemi, M. Fadaee, and S. Atashipour, Study on the free vibration of thick functionally graded rectangular plates according to a new exact closed-form procedure, Compos. Struct., vol. 93, no. 2, pp. 722–735, 2011.
   Google Scholar
• H.-S. Shen, Thermal postbuckling behavior of shear deformable FGM plates with temperature-dependent properties, Int. J. Mech. Sci., vol. 49, no. 4, pp. 466–478, 2007.
   Web of Science ®Google Scholar
• R.S. Khabbaz, B.D. Manshadi, and A. Abedian, Nonlinear analysis of FGM plates under pressure loads using the higher-order shear deformation theories, Compos. Struct., vol. 89, no. 3, pp. 333–344, 2009.
   Web of Science ®Google Scholar
• M. Bodaghi and A. Saidi, Levy-type solution for buckling analysis of thick functionally graded rectangular plates based on the higher-order shear deformation plate theory, Appl. Math. Modell., vol. 34, no. 11, pp. 3659–3673, 2010.
   Web of Science ®Google Scholar
• H.-S. Shen and Z.-X. Wang, Nonlinear bending of FGM plates subjected to combined loading and resting on elastic foundations, Compos. Struct., vol. 92, no. 10, pp. 2517–2524, 2010.
   Web of Science ®Google Scholar
• M. Talha and B. Singh, Static response and free vibration analysis of FGM plates using higher order shear deformation theory, Appl. Math. Modell., vol. 34, no. 12, pp. 3991–4011, 2010.
   Web of Science ®Google Scholar
• C. Huang, O. McGee, and M. Chang, Vibrations of cracked rectangular FGM thick plates, Compos. Struct., vol. 93, no. 7, pp. 1747–1764, 2011.
   Web of Science ®Google Scholar
• B. Mirzavand and M. Eslami, A closed-form solution for thermal buckling of piezoelectric FGM rectangular plates with temperature-dependent properties, Acta Mech., vol. 218, no. 1–2, pp. 87–101, 2011.
   Web of Science ®Google Scholar
• D.-G. Zhang, Modeling and analysis of FGM rectangular plates based on physical neutral surface and high order shear deformation theory, Int. J. Mech. Sci., vol. 68, pp. 92–104, 2013.
   Web of Science ®Google Scholar
• S. Pradyumna and J.N. Bandyopadhyay, Free vibration analysis of functionally graded curved panels using a higher-order finite element formulation, J. Sound Vib., vol. 318, no. 1–2, pp. 176–192, 2008.
   Web of Science ®Google Scholar
• H. Matsunaga, Free vibration and stability of functionally graded plates according to a 2-D higher-order deformation theory, Compos. Struct., vol. 82, no. 4, pp. 499–512, 2008.
   Web of Science ®Google Scholar
• G. Nie and Z. Zhong, Semi-analytical solution for three-dimensional vibration of functionally graded circular plates, Comput. Methods Appl. Mech. Eng., vol. 196, no. 49, pp. 4901–4910, 2007.
   Web of Science ®Google Scholar
• C. Dong, Three-dimensional free vibration analysis of functionally graded annular plates using the Chebyshev–Ritz method, Mater. Des., vol. 29, no. 8, pp. 1518–1525, 2008.
   Web of Science ®Google Scholar
• P. Malekzadeh, Three-dimensional free vibration analysis of thick functionally graded plates on elastic foundations, Compos. Struct., vol. 89, no. 3, pp. 367–373, 2009.
   Web of Science ®Google Scholar
• S. Hosseini-Hashemi, M. Azimzadeh-Monfared, and H.R.D. Taher, A 3-D Ritz solution for free vibration of circular/annular functionally graded plates integrated with piezoelectric layers, Int. J. Eng. Sci., vol. 48, no. 12, pp. 1971–1984, 2010.
   Web of Science ®Google Scholar
• W. Yun, X. Rongqiao, and D. Haojiang, Three-dimensional solution of axisymmetric bending of functionally graded circular plates, Compos. Struct., vol. 92, no. 7, pp. 1683–1693, 2010.
   Web of Science ®Google Scholar
• V. Tajeddini, A. Ohadi, and M. Sadighi, Three-dimensional free vibration of variable thickness thick circular and annular isotropic and functionally graded plates on Pasternak foundation, Int. J. Mech. Sci., vol. 53, no. 4, pp. 300–308, 2011.
   Web of Science ®Google Scholar
• M. Yas and V. Tahouneh, 3-D free vibration analysis of thick functionally graded annular plates on Pasternak elastic foundation via differential quadrature method (DQM), Acta Mech., vol. 223, no. 1, pp. 43–62, 2012.
   Web of Science ®Google Scholar
• K. Asemi, M. Salehi, and M. Akhlaghi, Post-buckling analysis of FGM annular sector plates based on three dimensional elasticity graded finite elements, Int. J. Non Linear Mech., vol. 67, pp. 164–177, 2014.
   Web of Science ®Google Scholar
• J. Reddy and Z.-Q. Cheng, Three-dimensional thermomechanical deformations of functionally graded rectangular plates, Eur. J. Mech. A. Solids, vol. 20, no. 5, pp. 841–855, 2001.
   Web of Science ®Google Scholar
• M. Kashtalyan, Three-dimensional elasticity solution for bending of functionally graded rectangular plates, Eur. J. Mech. A. Solids, vol. 23, no. 5, pp. 853–864, 2004.
   Web of Science ®Google Scholar
• K.-S. Na and J.-H. Kim, Thermal postbuckling investigations of functionally graded plates using 3-D finite element method, Finite Elem. Anal. Des., vol. 42, no. 8, pp. 749–745, 2006.
   Web of Science ®Google Scholar
• Z. Huang, C. Lü, and W. Chen, Benchmark solutions for functionally graded thick plates resting on Winkler–Pasternak elastic foundations, Compos. Struct., vol. 85, no. 2, pp. 95–104, 2008.
   Web of Science ®Google Scholar
• Q. Li, V. Iu, and K. Kou, Three-dimensional vibration analysis of functionally graded material sandwich plates, J. Sound Vib., vol. 311, no. 1, pp. 498–515, 2008.
   Web of Science ®Google Scholar
• A. Alibeigloo, Exact solution for thermo-elastic response of functionally graded rectangular plates, Compos. Struct., vol. 92, no. 1, pp. 113–121, 2010.
   Web of Science ®Google Scholar
• S.M. Hasheminejad and B. Gheshlaghi, Three-dimensional elastodynamic solution for an arbitrary thick FGM rectangular plate resting on a two parameter viscoelastic foundation, Compos. Struct., vol. 94, no. 9, pp. 2746–2755, 2012.
   Web of Science ®Google Scholar
• K. Asemi, M. Shariyat, M. Salehi, and H. Ashrafi, A full compatible three-dimensional elasticity element for buckling analysis of FGM rectangular plates subjected to various combinations of biaxial normal and shear loads, Finite Elem. Anal. Des., vol. 74, pp. 9–21, 2013.
   Web of Science ®Google Scholar
• B. Bouderba, M.S.A. Houari, and A. Tounsi, Thermomechanical bending response of FGM thick plates resting on Winkler-Pasternak elastic foundations, Steel Compos. Struct., vol. 14, no. 1, pp. 85–104, 2013.
   Web of Science ®Google Scholar
• S.S. Vel and R.C. Batra, Three-dimensional exact solution for the vibration of functionally graded rectangular plates, J. Sound Vib., vol. 272, no. 3–5, pp. 703–730, 2004.
   Web of Science ®Google Scholar
• W.L. Li, Free vibrations of beams with general boundary conditions, J. Sound Vib., vol. 237, no. 4, pp. 709–725, 2000.
   Web of Science ®Google Scholar
• W.L. Li, Comparison of Fourier sine and cosine series expansions for beams with arbitrary boundary conditions, J. Sound Vib., vol. 255, no. 1, pp. 185–194, 2002.
   Web of Science ®Google Scholar
• T. Ye, G. Jin, Y. Chen, X. Ma, and Z. Su, Free vibration analysis of laminated composite shallow shells with general elastic boundaries, Compos. Struct., vol. 106, pp. 470–490, 2013.
   Web of Science ®Google Scholar
• G. Jin, X. Ma, S. Shi, T. Ye, and Z. Liu, A modified Fourier series solution for vibration analysis of truncated conical shells with general boundary conditions, Appl. Acoust., vol. 85, pp. 82–96, 2014.
   Web of Science ®Google Scholar
• G. Jin, Z. Su, S. Shi, T. Ye, and S. Gao, Three-dimensional exact solution for the free vibration of arbitrarily thick functionally graded rectangular plates with general boundary conditions, Compos. Struct., vol. 108, pp. 565–577, 2014.
   Web of Science ®Google Scholar
• G. Jin, X. Xie, and Z. Liu, The Haar wavelet method for free vibration analysis of functionally graded cylindrical shells based on the shear deformation theory, Compos. Struct., vol. 108, pp. 435–448, 2014.
   Web of Science ®Google Scholar
• Z. Su, G. Jin, and T. Ye, Three-dimensional vibration analysis of thick functionally graded conical, cylindrical shell and annular plate structures with arbitrary elastic restraints, Compos. Struct., vol. 118, pp. 432–447, 2014.
   Web of Science ®Google Scholar
• X. Xie, G. Jin, T. Ye, and Z. Liu, Free vibration analysis of functionally graded conical shells and annular plates using the Haar wavelet method, Appl. Acoust., vol. 85, pp. 130–142, 2014.
   Web of Science ®Google Scholar
• T. Ye, G. Jin, Y. Chen, and S. Shi, A unified formulation for vibration analysis of open shells with arbitrary boundary conditions, Int. J. Mech. Sci., vol. 81, pp. 42–59, 2014.
   Web of Science ®Google Scholar
• T. Ye, G. Jin, S. Shi, and X. Ma, Three-dimensional free vibration analysis of thick cylindrical shells with general end conditions and resting on elastic foundations, Int. J. Mech. Sci., vol. 84, pp. 120–137, 2014.
   Web of Science ®Google Scholar
• T. Ye, G. Jin, and Z. Su, Three-dimensional vibration analysis of laminated functionally graded spherical shells with general boundary conditions, Compos. Struct., vol. 116, pp. 571–588, 2014.
   Web of Science ®Google Scholar
• T. Ye, G. Jin, Z. Su, and Y. Chen, A modified Fourier solution for vibration analysis of moderately thick laminated plates with general boundary restraints and internal line supports, Int. J. Mech. Sci., vol. 80, pp. 29–46, 2014.
   Web of Science ®Google Scholar
• G. Jin, S. Shi, Z. Su, S. Li, and Z. Liu, A modified Fourier–Ritz approach for free vibration analysis of laminated functionally graded shallow shells with general boundary conditions, Int. J. Mech. Sci., vol. 93, pp. 256–269, 2015.
   Web of Science ®Google Scholar
• C. Yang, G. Jin, Z. Liu, X. Wang, and X. Miao, Vibration and damping analysis of thick sandwich cylindrical shells with a viscoelastic core under arbitrary boundary conditions, Int. J. Mech. Sci., vol. 92, pp. 162–177, 2015.
   Web of Science ®Google Scholar
• G. Jin, T. Ye, X. Wang, and X. Miao, A unified solution for the vibration analysis of FGM doubly-curved shells of revolution with arbitrary boundary conditions, Composites Part B, vol. 89, pp. 230–252, 2016.
   Web of Science ®Google Scholar
• Q. Wang, D. Shi, and X. Shi, A modified solution for the free vibration analysis of moderately thick orthotropic rectangular plates with general boundary conditions, internal line supports and resting on elastic foundation, Meccanica, vol. 51, no. 8, pp. 1985–2017, 2016.
   Web of Science ®Google Scholar
• D. Shi, Q. Wang, X. Shi, and F. Pang, Free vibration analysis of moderately thick rectangular plates with variable thickness and arbitrary boundary conditions, Shock Vib., vol. 2014, art. ID 572395, 2014.
   Web of Science ®Google Scholar
• D. Shi, Q. Wang, X. Shi, and F. Pang, An accurate solution method for the vibration analysis of Timoshenko beams with general elastic supports, J. Mech. Eng. Sci., vol. 229, no. 13, pp. 2327–2340, 2015.
   Google Scholar
• D. Shi, Q. Wang, X. Shi, and F. Pang, A series solution for the in-plane vibration analysis of orthotropic rectangular plates with non-uniform elastic boundary constraints and internal line supports, Arch. Appl. Mech., vol. 85, no. 1, pp. 51–73, 2015.
   Web of Science ®Google Scholar
• Q. Wang, D. Shi, and Q. Liang, Free vibration analysis of axially loaded laminated composite beams with general boundary conditions by using a modified Fourier–Ritz approach, J. Compos. Mater., vol. 50, no. 15, pp. 2111–2135, 2016.
   Web of Science ®Google Scholar
• Q. Wang, D. Shi, Q. Liang, and X. Shi, A unified solution for vibration analysis of functionally graded circular, annular and sector plates with general boundary conditions, Composites Part B, vol. 88, pp. 264–294, 2016.
   Google Scholar