Abstract
In this paper, a modified Fourier solution based on the first-order shear deformation theory is developed for the free vibration problem of moderately thick composite laminated annular sector plates with general boundary conditions, internal radial line and circumferential arc supports. In this solution approach, regardless of boundary conditions, the displacement and rotation components of the sector plate are written in the form of the trigonometric series expansion in which several auxiliary terms are added to ensure and accelerate the convergence of the series. Each of the unknown coefficients is taken as the generalized coordinate and determined using the Raleigh- Ritz method. The accuracy and reliability of the present solution are validated by the comparison with the results found in the literature, and numerous new results for composite laminated annular sector plates considering various kinds of boundary conditions are presented. Comprehensive studies on the effects of elastic restraint parameters, layout schemes and locations of line/arc supports are also made.New results are obtained for laminated annular sector plates subjected to elastic boundary restraints and arbitrary internal radial line and circumferential arc supports in both directions, and they may serve as benchmark solutions for future researches.
References
[1] Leissa, A.W., Vibration of plates. 1973, Washington DC: U. S. Government Printing Office.Search in Google Scholar
[2] Qatu, M.S., Vibration of laminated shells and plates. 2004: Elsevier.10.1016/B978-008044271-6/50006-5Search in Google Scholar
[3] Reddy, J.N., Mechanics of laminated composite plates and shells: theory and analysis. 2004: CRC press.10.1201/b12409Search in Google Scholar
[4] Carrera, E., S. Brischetto, and P. Nali, Plates and shells for smart structures: classical and advanced theories for modeling and analysis. Vol. 36. 2011: John Wiley & Sons.10.1002/9781119950004Search in Google Scholar
[5] Hosseini-Hashemi, S., et al., Differential quadrature analysis of functionally graded circular and annular sector plates on elastic foundation. Materials & Design, 2010. 31(4): p. 1871-1880.10.1016/j.matdes.2009.10.060Search in Google Scholar
[6] Hosseini-Hashemi, S., H.R.D. Taher, and H. Akhavan, Vibration analysis of radially FGM sectorial plates of variable thickness on elastic foundations. Composite Structures, 2010. 92(7): p. 1734-1743.10.1016/j.compstruct.2009.12.016Search in Google Scholar
[7] Aghelinejad, M., et al., Nonlinear Thermomechanical Post-Buckling Analysis of Thin Functionally Graded Annular Plates Based on Von-Karman’s Plate Theory. Mechanics of Advanced Materials and Structures, 2011. 18(5): p. 319-326.10.1080/15376494.2010.516880Search in Google Scholar
[8] Mirtalaie, S. and M. Hajabasi, 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, 2011. 225(3): p. 568-583.10.1243/09544062JMES2232Search in Google Scholar
[9] Zhou, Z., et al., Natural vibration of circular and annular thin plates by Hamiltonian approach. Journal of Sound and Vibration, 2011. 330(5): p. 1005-1017.10.1016/j.jsv.2010.09.015Search in Google Scholar
[10] Baferani, A.H., A. Saidi, and E. Jomehzadeh, Exact analytical solution for free vibration of functionally graded thin annular sector plates resting on elastic foundation. Journal of Vibration and Control, 2012. 18(2): p. 246-267.10.1177/1077546311402530Search in Google Scholar
[11] Jomehzadeh, E., A. Saidi, and S. Atashipour, An analytical approach for stress analysis of functionally graded annular sector plates. Materials & Design, 2009. 30(9): p. 3679-3685.10.1016/j.matdes.2009.02.011Search in Google Scholar
[12] Saidi, A. and A.H. Baferani, Thermal buckling analysis of moderately thick functionally graded annular sector plates. Composite Structures, 2010. 92(7): p. 1744-1752.10.1016/j.compstruct.2010.01.004Search in Google Scholar
[13] Malekzadeh, P., M.G. Haghighi, and M. Atashi, Free vibration analysis of elastically supported functionally graded annular plates subjected to thermal environment. Meccanica, 2011. 46(5): p. 893-913.10.1007/s11012-010-9345-5Search in Google Scholar
[14] Saidi, A., A.H. Baferani, and E. Jomehzadeh, Benchmark solution for free vibration of functionally graded moderately thick annular sector plates. Acta mechanica, 2011. 219(3-4): p. 309-335.10.1007/s00707-011-0459-1Search in Google Scholar
[15] Mousavi, S.M. and M. Tahani, Analytical solution for bending of moderately thick radially functionally graded sector plates with general boundary conditions using multi-term extended Kantorovich method. Composites Part B: Engineering, 2012. 43(3): p. 1405-1416.10.1016/j.compositesb.2011.11.068Search in Google Scholar
[16] Liew, K., S. Kitipornchai, and Y. Xiang, Vibration of annular sector Mindlin plates with internal radial line and circumferential arc supports. Journal of sound and vibration, 1995. 183(3): p. 401-419.10.1006/jsvi.1995.0262Search in Google Scholar
[17] Salehi, M. and A. Sobhani, Elastic linear and non-linear analysis of fiber-reinforced symmetrically laminated sector Mindlin plate. Composite structures, 2004. 65(1): p. 65-79.10.1016/j.compstruct.2003.10.006Search in Google Scholar
[18] Sharma, A., H. Sharda, and Y. Nath, Stability and vibration of thick laminated composite sector plates. Journal of sound and vibration, 2005. 287(1): p. 1-23.10.1016/j.jsv.2004.10.030Search in Google Scholar
[19] Houmat, A., Large amplitude free vibration of shear deformable laminated composite annular sector plates by a sector pelement. International Journal of Non-Linear Mechanics, 2008. 43(9): p. 834-843.10.1016/j.ijnonlinmec.2008.05.007Search in Google Scholar
[20] Andakhshideh, A., S.Maleki, and M. Aghdam, Non-linear bending analysis of laminated sector plates using generalized differential quadrature. Composite Structures, 2010. 92(9): p. 2258-2264.10.1016/j.compstruct.2009.08.007Search in Google Scholar
[21] Maleki, S. and M. Tahani, Bending analysis of laminated sector plates with polar and rectilinear orthotropy. European Journal of Mechanics-A/Solids, 2013. 40: p. 84-96.10.1016/j.euromechsol.2013.01.001Search in Google Scholar
[22] Golmakani, M. and M. Mehrabian, Nonlinear bending analysis of ring-stiffened circular and annular general angle-ply laminated plates with various boundary conditions. Mechanics Research Communications, 2014. 59: p. 42-50.10.1016/j.mechrescom.2014.04.007Search in Google Scholar
[23] Sharma, A., Free vibration of moderately thick antisymmetric laminated annular sector plates with elastic edge constraints. International Journal of Mechanical Sciences, 2014. 83: p. 124-132.10.1016/j.ijmecsci.2014.04.005Search in Google Scholar
[24] Hashemi, S.H., M. Es’haghi, and M. Karimi, Closed-form vibration analysis of thick annular functionally graded plates with integrated piezoelectric layers. International Journal of Mechanical Sciences, 2010. 52(3): p. 410-428.10.1016/j.ijmecsci.2009.10.016Search in Google Scholar
[25] Golmakani, M. and M. Kadkhodayan, Nonlinear bending analysis of annular FGM plates using higher-order shear deformation plate theories. Composite Structures, 2011. 93(2): p. 973-982.10.1016/j.compstruct.2010.06.024Search in Google Scholar
[26] Alipour, M. and M. Shariyat, Analytical stress analysis of annular FGM sandwich plates with non-uniform shear and normal tractions, employing a zigzag-elasticity plate theory. Aerospace Science and Technology, 2014. 32(1): p. 235-259.10.1016/j.ast.2013.10.007Search in Google Scholar
[27] Asemi, K., M. Salehi, and M. Akhlaghi, Post-buckling analysis of FGM annular sector plates based on three dimensional elasticity graded finite elements. International Journal of Non-Linear Mechanics, 2014. 67: p. 164-177.10.1016/j.ijnonlinmec.2014.08.014Search in Google Scholar
[28] Srinivasan, R. and V. Thiruvenkatachari, Free vibration analysis of laminated annular sector plates. Journal of sound and vibration, 1986. 109(1): p. 89-96.10.1016/S0022-460X(86)80024-4Search in Google Scholar
[29] Ding, H.-J. and R.-Q. Xu, Free axisymmetric vibration of laminated transversely isotropic annular plates. Journal of sound and vibration, 2000. 230(5): p. 1031-1044.10.1006/jsvi.1999.2666Search in Google Scholar
[30] Xu, R., Three-dimensional exact solutions for the free vibration of laminated transversely isotropic circular, annular and sectorial plates with unusual boundary conditions. Archive of Applied Mechanics, 2008. 78(7): p. 543-558.10.1007/s00419-007-0177-2Search in Google Scholar
[31] Malekzadeh, P., Three-dimensional free vibration analysis of thick laminated annular sector plates using a hybrid method. Composite Structures, 2009. 90(4): p. 428-437.10.1016/j.compstruct.2009.04.015Search in Google Scholar
[32] Malekzadeh, P., M.G. Haghighi, and M. Gholami, Dynamic response of thick laminated annular sector plates subjected to moving load. Composite Structures, 2010. 92(1): p. 155-163.10.1016/j.compstruct.2009.07.020Search in Google Scholar
[33] Fantuzzi, N., et al., Stability and accuracy of three Fourier expansion-based strong form finite elements for the free vibration analysis of laminated composite plates. International Journal for Numerical Methods in Engineering, 2017.10.1002/nme.5468Search in Google Scholar
[34] Li, W.L., Free vibrations of beams with general boundary conditions. Journal of Sound and Vibration, 2000. 237(4): p. 709-725.10.1006/jsvi.2000.3150Search in Google Scholar
[35] Li, W.L., Comparison of Fourier sine and cosine series expansions for beams with arbitrary boundary conditions. Journal of Sound and Vibration, 2002. 255(1): p. 185-194.10.1006/jsvi.2001.4108Search in Google Scholar
[36] Du, J., et al., An analytical method for the in-plane vibration analysis of rectangular plates with elastically restrained edges. Journal of Sound and Vibration, 2007. 306(3-5): p. 908-927.10.1016/j.jsv.2007.06.011Search in Google Scholar
[37] Du, J.T., et al., Free In-Plane Vibration Analysis of Rectangular Plates With Elastically Point-Supported Edges. Journal of Vibration and Acoustics-Transactions of the Asme, 2010. 132(3).10.1115/1.4000777Search in Google Scholar
[38] Shi, D., et al., A series solution for the in-plane vibration analysis of orthotropic rectangular plates with non-uniform elastic boundary constraints and internal line supports. Archive of Applied Mechanics, 2014: p. 1-23.10.1007/s00419-014-0899-xSearch in Google Scholar
[39] Shi, X., et al., A unified method for free vibration analysis of circular, annular and sector plates with arbitrary boundary conditions. Journal of Vibration and Control, 2014: p. 1077546314533580.10.1177/1077546314533580Search in Google Scholar
[40] Jin, G., et al., An exact solution for the free vibration analysis of laminated composite cylindrical shells with general elastic boundary conditions. Composite Structures, 2013. 106(0): p. 114-127.10.1016/j.compstruct.2013.06.002Search in Google Scholar
[41] Jin, G., et al., A unified approach for the vibration analysis of moderately thick composite laminated cylindrical shells with arbitrary boundary conditions. International Journal of Mechanical Sciences, 2013. 75(0): p. 357-376.10.1016/j.ijmecsci.2013.08.003Search in Google Scholar
[42] Chen, Y., G. Jin, and Z. Liu, Flexural and in-plane vibration analysis of elastically restrained thin rectangular plate with cutout using Chebyshev-Lagrangian method. International Journal of Mechanical Sciences, 2014. 89(0): p. 264-278.10.1016/j.ijmecsci.2014.09.006Search in Google Scholar
[43] Jin, G., et al., A modified Fourier series solution for vibration analysis of truncated conical shells with general boundary conditions. Applied Acoustics, 2014. 85(0): p. 82-96.10.1016/j.apacoust.2014.04.007Search in Google Scholar
[44] Jin, G., et al., A general Fourier solution for the vibration analysis of composite laminated structure elements of revolution with general elastic restraints. Composite Structures, 2014. 109(0): p. 150-168.10.1016/j.compstruct.2013.10.052Search in Google Scholar
[45] Wang, Q., et al., Free vibration of four-parameter functionally graded moderately thick doubly-curved panels and shells of revolution with general boundary conditions. Applied Mathematical Modelling.Search in Google Scholar
[46] Shi, D., et al., A series solution for the in-plane vibration analysis of orthotropic rectangular plates with non-uniform elastic boundary constraints and internal line supports. Archive of Applied Mechanics, 2015. 85(1): p. 51-73.10.1007/s00419-014-0899-xSearch in Google Scholar
[47] Shi, D., et al., An accurate solution method for the vibration analysis of Timoshenko beams with general elastic supports. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 2015. 229(13): p. 2327-2340.10.1177/0954406214558675Search in Google Scholar
[48] Lv, X., et al., 1953. A unified solution for the in-plane vibration analysis of multi-span curved Timoshenko beams with general elastic boundary and coupling conditions. Journal of Vibroengineering, 2016. 18(2).10.21595/jve.2015.16296Search in Google Scholar
[49] Shao, D., et al., Transient response analysis of cross-ply composite laminated rectangular plates with general boundary restraints by the method of reverberation ray matrix. Composite Structures, 2016. 152: p. 168-182.10.1016/j.compstruct.2016.05.035Search in Google Scholar
[50] Shao, D., et al., A unified analysis for the transient response of composite laminated curved beam with arbitrary lamination schemes and general boundary restraints. Composite Structures, 2016. 154: p. 507-526.10.1016/j.compstruct.2016.07.070Search in Google Scholar
[51] Shi, D., et al., 2111. A unified solution for free vibration of orthotropic circular, annular and sector plates with general boundary conditions. Journal of Vibroengineering, 2016. 18(5).10.21595/jve.2016.17004Search in Google Scholar
[52] Shi, D., et al., 1897. A unified solution for free vibration of orthotropic annular sector thin plates with general boundary conditions, internal radial line and circumferential arc supports. Journal of Vibroengineering, 2016. 18(1).Search in Google Scholar
[53] Shi, D., et al., A unified spectro-geometric-Ritz method for vibration analysis of open and closed shells with arbitrary boundary conditions. Shock and Vibration, 2016. 2016.10.1155/2016/4097123Search in Google Scholar
[54] Shi, X., et al., A unified method for free vibration analysis of circular, annular and sector plates with arbitrary boundary conditions. Journal of Vibration and Control, 2016. 22(2): p. 442-456.10.1177/1077546314533580Search in Google Scholar
[55] Wang, Q., 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. Journal of Composite Materials, 2016. 50(15): p. 2111-2135.10.1177/0021998315602138Search in Google Scholar
[56] Wang, Q., et al., An improved Fourier series solution for the dynamic analysis of laminated composite annular, circular, and sector plate with general boundary conditions. Journal of Composite Materials, 2016. 50(30): p. 4199-4233.10.1177/0021998316635240Search in Google Scholar
[57] Wang, Q., et al., A unified solution for free in-plane vibration of orthotropic circular, annular and sector plates with general boundary conditions. Applied Mathematical Modelling, 2016. 40(21): p. 9228-9253.10.1016/j.apm.2016.06.005Search in Google Scholar
[58] Wang, Q., et al., A unified solution for vibration analysis of moderately thick functionally graded rectangular plates with general boundary restraints and internal line supports. Mechanics of Advanced Materials and Structures, 2016(just-accepted): p. 00-00.10.1080/15376494.2016.1196797Search in Google Scholar
[59] Wang, Q., et al., A unified solution for vibration analysis of functionally graded circular, annular and sector plates with general boundary conditions. Composites Part B: Engineering, 2016. 88: p. 264-294.10.1016/j.compositesb.2015.10.043Search in Google Scholar
[60] Wang, Q., et al., Vibrations of Composite Laminated Circular Panels and Shells of Revolution with General Elastic Boundary Conditions via Fourier-Ritz Method. Curved and Layered Structures, 2016. 3(1): p. 105-136.10.1515/cls-2016-0010Search in Google Scholar
[61] Wang, Q., 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, 2016. 51(8): p. 1985-2017.10.1007/s11012-015-0345-3Search in Google Scholar
[62] Shao, D., et al., Free vibration of refined higher-order shear deformation composite laminated beams with general boundary conditions. Composites Part B: Engineering, 2017. 108: p. 75-90.10.1016/j.compositesb.2016.09.093Search in Google Scholar
[63] Wang, Q., et al., Benchmark solution for free vibration of thick open cylindrical shells on Pasternak foundation with general boundary conditions. Meccanica, 2017. 52(1): p. 457-482.10.1007/s11012-016-0406-2Search in Google Scholar
[64] Zhang, H., D. Shi, and Q.Wang, An improved Fourier series solution for free vibration analysis of the moderately thick laminated composite rectangular plate with non-uniform boundary conditions. International Journal of Mechanical Sciences, 2017. 121: p. 1-20.10.1016/j.ijmecsci.2016.12.007Search in Google Scholar
[65] Wang, Q., et al., Free vibration of four-parameter functionally graded moderately thick doubly-curved panels and shells of revolution with general boundary conditions. Applied Mathematical Modelling, 2017. 42: p. 705-734.10.1016/j.apm.2016.10.047Search in Google Scholar
[66] Wang, Q., et al., Free vibrations of composite laminated doubly-curved shells and panels of revolution with general elastic restraints. AppliedMathematical Modelling, 2017. 46: p. 227-262.10.1016/j.apm.2017.01.070Search in Google Scholar
[67] Wang, Q., et al., A unified formulation for free vibration of functionally graded carbon nanotube reinforced composite spherical panels and shells of revolution with general elastic restraints by means of the Rayleigh-Ritz method. Polymer Composites, 2017.10.1002/pc.24339Search in Google Scholar
[68] Shao, D., et al., An enhanced reverberation-raymatrix approach for transient response analysis of composite laminated shallow shells with general boundary conditions. Composite Structures, 2017. 162: p. 133-155.10.1016/j.compstruct.2016.11.085Search in Google Scholar
© 2017
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.