Abstract
Based on Reissner’s mixed variational theorem, the authors develop a finite annular prism method (FAPM) for the three-dimensional (3D) free vibration analysis of bi-directional functionally graded (FG) annular plates with assorted boundary conditions. In this formulation, the FG annular plate is divided into a number of finite annular prisms with triangular cross-sections, in which Fourier functions and Lagrange polynomials are used to interpolate the circumferential direction and radial-thickness surface variations of primary field variables in each individual prism, respectively. The material properties of the FG annular plate are assumed to obey an exponential function distribution varying doubly exponentially through the radial-thickness surface. These FAPM solutions for the frequency parameters and their corresponding mode shapes of the FG annular plate closely agree with the solutions obtained using other 3D approaches available in the literature.
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References
M. Koizumi, Concept of FGM, Ceramic Trans., 34 (1993) 3–10.
M. Koizumi, FGM activities in Japan, Compos. Part B-Eng., 28 (1–2) (1997) 1–4.
J. N. Reddy, C. M. Wang and S. Kitipornchai, Axisymmetric bending of functionally graded circular and annular plates, Eur. J. Mech. A/Solids, 18 (2) (1999) 185–199.
K. Swaminathan, D. T. Naveenkumar, A. M. Zenkour and E. Carrera, Stress, vibration and buckling analyses of FGM plates-A state-of-the art review, Compos. Struct., 120 (2015) 10–31.
E. Carrera, S. Brischetto and A. Robaldo, Variable kinematic model for the analysis of functionally graded material plates, AIAA J., 46 (1) (2008) 194–203.
M. Cinefra, S. Belouettar, M. Soave and E. Carrera, Variable kinematic models applied to free-vibration analysis of functionally graded material shells, Eur. J. Mech. A/Solids, 29 (6) (2010) 1078–1087.
D. K. Jha, T. Kant and R. K. Singh, A critical review of recent research on functionally graded plates, Compos. Struct., 96 (2013) 833–849.
F. Tornabene and S. Brischetto, 3D capability of refined GDQ models for the bending analysis of composite and sandwich plates, spherical and doubly-curved shells, Thin-Walled Struct., 129 (2018) 94–124.
C. P. Wu, K. H. Chiu and Y. M. Wang, A review on the three-dimensional analytical approaches of multilayered and functionally graded piezoelectric plates and shells, CMC-Comput. Mater. Continua, 8 (2) (2008) 93–132.
C. P. Wu and Y. C. Liu, A review of semi-analytical numerical methods for laminated composite and multilayered functionally graded elastic/piezoelectric plates and shells, Compos. Struct., 147 (2016) 1–15.
Y. Ootao, Y Tanigawa and O. Ishimaru, Optimization of material composition of functionally graded plate for thermal stress relaxation using a genetic algorithm, J. Therm. Stresses, 23 (3) (2000) 257–271.
S. Ding and C. P. Wu, Optimization of material composition to minimize the thermal stresses induced in FGM plates with temperature-dependent material properties, Int. J. Mech. Mater. Des., 14 (4) (2018) 527–549.
O. S. Hussein and S. B. Mulani, Optimization of in-plane functionally graded panels for buckling strength: Unstiffened, stiffened panels, and panels with cutouts, Thin-Walled Struct., 122 (2018) 173–181.
L. F. Qian and R. C. Batra, Design of bidirectional functionally graded plate for optimal natural frequencies, J. Sound Vib., 280 (2005) 415–424.
Y. Kumar and R. Lal, Prediction of frequencies of free axisymmetric vibration of two-directional functionally graded annular plates on Winkler foundation, Eur. J. Mech. A/Solids, 42 (2013) 219–228.
R. Lal and N. Ahlawat, Buckling and vibrations of two-directional functionally graded circular plates subjected to hydrostatic in-plane force, J. Vib. Control, 23 (13) (2017) 2111–2127.
F. Tornabene, E. Viola and D. J. Inman, 2-D differential quadrature solution for vibration analysis of functionally graded conical, cylindrical shell and annular plate structures, J. Sound Vib., 328 (3) (2009) 259–290.
Z. Su, G. Jin, S. Shi, T. Ye and X. Jia, A unified solution for vibration analysis of functionally graded cylindrical, conical shells and annular plates with general boundary conditions, Int. J. Mech. Sci., 80 (2014) 62–80.
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, Compos. Part B-Eng., 88 (2016) 264–294.
F. Tornabene, Free vibration analysis of functionally graded conical, cylindrical shell and annular plate structures with a four-parameter power-law distribution, Comput. Methods Appl. Mech. Engrg., 198 (37–40) (2009) 2911–2935.
M. H. Amini, M. Soleimani, A. Altafi and A. Rastgoo, Effects of geometric nonlinearity on free and forced vibration analysis of moderately thick annular functionally graded plate, Mech. Adv. Mater. Struct., 20 (9) (2013) 709–720.
A. R. Saidi, A. Rasouli and S. Sahraee, Axisymmetric bending and buckling analysis of thick functionally graded circular plates using unconstrained third-order shear deformation plate theory, Compos. Struct., 89 (2009) 110–119.
M. Talha and B. N. Singh, Static response and free vibration analysis of FGM plates using higher order shear deformation theory, Appl. Math. Modell., 34 (12) (2010) 3991–4011.
S. Hosseini-Hashemi, M. Es’haghi, H. R. D. Taher and M. Fadaie, Exact closed-form frequency equations for thick circular plates using a third-order shear deformation theory, J. Sound Vib., 329 (16) (2010) 3382–3396.
S. Sahraee and A. R. Saidi, Axisymmetric bending analysis of thick functionally graded circular plates using fourth-order shear deformation theory, Eur. J. Mech. A/Solids, 28 (5) (2009) 974–984.
R. C. Batra, Higher-order shear and normal deformable theory for functionally graded incompressible linear elastic plates, Thin-Walled Struct., 45 (12) (2007) 974–982.
A. J. M. Ferreira, R. C. Batra, C. M. C. Roque, L. F. Qian and R. M. N. Jorge, Natural frequencies of functionally graded plates by a meshless method, Compos. Struct., 75 (2006) 593–600.
R. Lal and R. Rani, On radially symmetric vibrations of non-uniform annular sandwich plates, Thin-Walled Struct., 94 (2015) 562–576.
P. Malekzadeh and N. S. Hamzehkolaei, A 3D discrete layer-differential quadrature free vibration of multilayered FG annular plates in thermal environment, Mech. Adv. Mater. Struct., 20 (4) (2013) 316–330.
E. Carrera, Theories and finite elements for multilayered, anisotropic, composite plates and shells, Arch. Comput. Meth Engng., 9 (2) (2002) 87–140.
K. Mercan, A. K. Baltacioglu and Ö. Civalek, Free vibration of laminated and FGM/CNT composites annular thick plates with shear deformation by discrete singular convolution method, Compos. Struct., 186 (2018) 139–153.
X. Shi, C. Li, F. Wang and F. Wei, A unified formulation for free transverse vibration analysis of orthotropic plates of revolution with general boundary conditions, Mech. Adv. Mater. Struct., 25 (2) (2018) 87–99.
T. J. R. Hughes, J. A. Cottrell and Y. Bazilevs, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput. Methods Appl. Mech. Eng., 194 (39–41) (2005) 4135–4195.
J. A. Cottrell, T. J. R. Hughes and Y. Bazilevs, Isogeometric Analysis: Toward Integration of CAD and FEA, Chichesrer, United Kindom: John Wiley & Sons (2009).
X. Qin, G. Jin, M. Chen and S. Yin, Free in-plane vibration analysis of circular, annular, and sector plates using isogeometric approach, Shock Vib., 4314761 (2018).
Q. X. Lieu, S. Lee, J. Kang and J. Lee, Bending and free vibration analyses of in-plane bi-directional functionally graded plates with variable thickness using isogeometric analysis, Compos. Struct., 192 (2018) 434–451.
J. So and A. W. Leissa, Three-dimensional vibrations of thick circular and annular plates, J. Sound Vib., 209 (1) (1998) 15–41.
J. H. Kang and A. W. Leissa, Three-dimensional vibrations of thick, linearly tapered, annular plates, J. Sound Vib., 217 (5) (1998) 927–944.
K. M. Liew and B. Yang, Three-dimensional elasticity solutions for free vibrations of circular plates: A polynomials-Ritz analysis, Comput. Methods Appl. Mech. Eng., 175 (1–2) (1999) 189–201.
K. M. Liew and B. Yang, Elasticity solutions for free vibrations of annular plates from three-dimensional analysis, Int. J. Solids Struct., 37 (52) (2000) 7689–7702.
D. Zhou, F. T. K. Au, Y. K. Cheung and S. H. Lo, Three-dimensional vibration analysis of circular and annular plates via the Chebyshev-Ritz method, Int. J. Solids Struct., 40 (12) (2003) 3089–3105.
D. Zhou, S. H. Lo, F. T. K. Au and Y. K. Cheung, Three-dimensional free vibration of thick circular plates on Pasternak foundation, J. Sound Vib., 292 (3–5) (2006) 726–741.
P. Shi and C. Y. Dong, Vibration analysis of functionally graded annular plates with mixed boundary conditions in thermal environment, J. Sound Vib., 331 (15) (2012) 3649–3662.
C. Y. Dong, Three-dimensional free vibration analysis of functionally graded annular plates using the Chebyshev-Ritz method, Mater. Des., 29 (8) (2008) 1518–1525.
G. Nie and Z. Zhong, Dynamic analysis of multi-directional functionally graded annular plates, Appl. Math. Modell., 34 (3) (2010) 608–616.
G. J. Nie and Z. Zhong, Vibration analysis of functionally graded annular sectorial plates with simply supported radial edges, Compos. Struct., 84 (2) (2008) 167–176.
I. D. Kermani, M. Ghayour and H. R. Mirdamadi, Free vibration analysis of multi-directional functionally graded circular and annular plates, J. Mech. Sci. Technol., 26 (11) (2012) 3399–3410.
P. Malekzadeh, S. A. Shahpari and H. R. Ziaee, Three-dimensional free vibration of thick functionally graded annular plates in thermal environment, J. Sound Vib., 329 (4) (2010) 425–442.
S. S. Vel and R. C. Batra, Three-dimensional exact solution for the vibration of functionally graded rectangular plates, J. Sound Vib., 272 (3–5) (2004) 703–730.
J. Zhao, Y. Zhang, K. Choe, X. Qu, A. Wang and Q. Wang, Three-dimensional exact solution for the free vibration of thick functionally graded annular sector plates with arbitrary boundary conditions, Compos. Part B, 159 (2019) 418–436.
E. Carrera, Historical review of zig-zag theories for multi-layered plates and shells, Appl. Mech. Rev., 56 (3) (2003) 287–308.
E. Carrera, Assessment of theories for free vibration analysis of homogeneous and multilayered plates, Shock Vib., 11 (3–4) (2004) 261–270.
C. P. Wu and H. Y. Li, An RMVT-based finite rectangular prism method for the 3D analysis of sandwich FGM plates with various boundary conditions, CMC-Comput. Mater. Continua., 34 (1) (2013) 27–62.
C. P. Wu and H. Y. Li, RMVT-based finite cylindrical prism methods for multilayered functionally graded circular hollow cylinders with various boundary conditions, Compos. Struct., 100 (2013) 592–608.
C. P. Wu and L. T. Yu, Quasi-3D static analysis of two-directional functionally graded circular plates, Steel Compos. Struct., 27 (2018) 789–801.
C. F. Liu and Y. T. Lee, Finite element analysis of three-dimensional vibrations of thick circular and annular plates, J. Sound Vib., 233 (1) (2000) 63–80.
D. Zhou, F. T. K. Au, Y. K. Cheung and S. H. Lo, Three-dimensional vibration analysis of circular and annular plates via the Chebyshev-Ritz method, Int. J. Solids Struct., 40 (12) (2003) 3089–3105.
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Recommended by Associate Editor Heung Soo Kim
Chih-Ping Wu obtained his Ph.D. degree from the Engineering Mechanics, Ohio State University, USA, in 1988. Since then, he worked as an Associate Professor at the Department of Civil Engineering (CE), National Cheng Kung University (NCKU), Taiwan. In 1993 and 2003, he was promoted to be the Full Professor and Distinguished Professor at NCKU, respectively. From 2003 to 2006, he served as the Head of the CE Department, NCKU. He majored in the research areas including FGM plates/shells, nonlocal continuum mechanics, computational mechanics, finite element methods, meshless methods, perturbation methods, and three-dimensional elasticity/ piezoelectricity.
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Wu, CP., Yu, LT. Free vibration analysis of bi-directional functionally graded annular plates using finite annular prism methods. J Mech Sci Technol 33, 2267–2279 (2019). https://doi.org/10.1007/s12206-019-0428-5
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DOI: https://doi.org/10.1007/s12206-019-0428-5