Free Vibration Analysis of Bidirectional Functionally Graded Conical/Cylindrical Shells and Annular Plates on Nonlinear Elastic Foundations, Based on a Unified Differential Transform Analytical Formulation
الموضوعات :M Molla-Alipour 1 , M Shariyat 2 , M Shaban 3
1 - Department of Mechanical Engineering, University of Mazandaran, Babolsar , Iran
2 - Faculty of Mechanical Engineering, K.N. Toosi University of Technology, Tehran, Iran
3 - Mechanical Engineering Department, Bu-Ali Sina University, Hamadan, Iran
الکلمات المفتاحية: Bidirectional functionally graded, Annular plates, Free vibration, Non-uniform elastic foundation, Conical and cylindrical shells, Non-uniform elastic foundationAnnular plates,
ملخص المقالة :
In the present research, a unified formulation for free vibration analysis of the bidirectional functionally graded conical and cylindrical shells and annular plates on elastic foundations is developed. To cover more individual cases and optimally tailored material properties, the material properties are assumed to vary in both the meridian/radial and transverse directions. The shell/plate is assumed to be supported by a non-uniform Winkler-type elastic foundation in addition to the edge constraints. Therefore, the considered problem contains some complexities that have not been considered together in the available researches. The proposed unified formulation is derived based on the principle of minimum total potential energy and solved using a differential transform analytical method whose center is located at the outer edge of the shell or plate; so that the resulting semi-analytical solution can be employed not only for truncated conical shells and annular plates, but also for complete conical shells and circular plates. Accuracy of results of the proposed unified formulation is verified by comparing the results with those of the three-dimensional theory of elasticity extracted from the ABAQUS finite element analysis code. A variety of the edge condition combinations are considered in the results section. A comprehensive parametric study including assessment of influences of the material properties indices, thickness to radius ratio, stiffness distribution of the elastic foundation, and various boundary conditions, is accomplished. Results reveal that influence of the meridian variations of the material properties on the natural frequencies is more remarkable than that of the transverse gradation.
[1] Qatu M.S., 2004, Vibration of Laminated Shells and Plates, San Diego, Elsevier.
[2] Reddy J.N., 2003, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, Florida, CRC Press.
[3] Shen H-S., 2009, Functionally Graded Materials: Nonlinear Analysis of Plates and Shells, Taylor & Francis Group, LLC.
[4] Carrera E., Brischetto S., Nali P., 2011, Plates and Shells for Smart Structures: Classical and Advanced Theories for Modeling and Analysis, John Wiley & Sons.
[5] Tornabene F., Fantuzzi N., 2014, Mechanics of Laminated Composite Doubly-Curved Shell Structures, Società Editrice Esculapio, Milan.
[6] Efraim E., Eisenberger M., 2007, Exact vibration analysis of variable thickness thick annular isotropic and FGM plates, Journal of Sound and Vibration 29: 720-738.
[7] Hosseini-Hashemi Sh., Fadaee M., Es’haghi M., 2010, A novel approach for in-plane/out-of-plane frequency analysis of functionally graded circular/annular plates, International Journal of Mechanical Sciences 52: 1025-1035.
[8] Nie G.J., Zhong Z., 2010, Dynamic analysis of multi-directional functionally graded annular plates, Applied Mathematical Modelling 34: 608-616.
[9] Malekzadeh P., Golbahar Haghighi M.R., Atashi M.M., 2011, Free vibration analysis of elastically supported functionally graded annular plates subjected to thermal environment, Meccanica 46: 893-913.
[10] Jodaei A., Jalal M., Yas M.H., 2012, Free vibration analysis of functionally graded annular plates by state-space based differential quadrature method and comparative modeling by ANN, Composites Part B: Engineering 43: 340-353.
[11] Dong C.Y., 2008, Three-dimensional free vibration of functionally graded annular plates using the Chebyshev-Ritz method, Materials & Design 29: 1518-1525.
[12] Shariyat M., Alipour M.M., 2011, Differential transform vibration and modal stress analyses of circular plates made of two-directional functionally graded materials resting on elastic foundations, Archive of Applied Mechanics 81(9): 1289-1306.
[13] Alipour M.M., Shariyat M., 2011, A power series solution for free vibration of variable thickness Mindlin circular plates with two-directional material heterogeneity and elastic foundations, Journal of Solid Mechanics 3(2): 183-197.
[14] Alipour M.M., Shariyat M., 2010, Stress analysis of two-directional FGM moderately thick constrained circular plates with non-uniform load and substrate stiffness distributions, Journal of Solid Mechanics 2(4): 316-331.
[15] Shariyat M., Alipour M.M., 2013, 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(5): 3063-3076.
[16] Shariyat M., Alipour M.M., 2014, A novel shear correction factor for stress and modal analyses of annular FGM plates with non-uniform inclined tractions and non-uniform elastic foundations, International Journal of Mechanical Sciences 87: 60-71.
[17] Lezgy-Nazargah M., 2015, Fully coupled thermo-mechanical analysis of bi-directional FGM beams using NURBS isogeometric finite element approach, Aerospace Science and Technology 45: 154-164.
[18] Shariyat M., Jafari A.A., Alipour M.M., 2013, Investigation of the thickness variability and material heterogeneity effects on free vibration of the viscoelastic circular plates, Acta Mechanica Solida Sinica 26(1): 83-98.
[19] Alipour M.M., Shariyat M., Shaban M., 2010, A semi-analytical solution for free vibration and modal stress analyses of circular plates resting on two-parameter elastic foundations, Journal of Solid Mechanics 2(1): 63-78.
[20] Lezgy-Nazargah M., Cheraghi N., 2017, An exact Peano Series solution for bending analysis of imperfect layered FG neutral magneto-electro-elastic plates resting on elastic foundations, Mechanics of Advanced Materials and Structures 24(3): 183-199.
[21] Lezgy-Nazargah M., Meshkani Z., 2018, An efficient partial mixed finite element model for static and free vibration analyses of FGM plates rested on two-parameter elastic foundations, Structural Engineering and Mechanics 66(5): 665-676.
[22] Najafizadeh M.M., Isvandzibaei M.R., 2007, Vibration of functionally graded cylindrical shell based on higher order shear deformation plate theory with ring support, Acta Mechanica 191: 75-91.
[23] Matsunaga H., 2008, Free vibration and stability of functionally graded circular cylindrical shells according to a 2D higher-order deformation theory, Composite Structures 88: 519-531.
[24] Santos H., MotaSoares C.M., MotaSoares C.A., Reddy J.N., 2009, A semi-analytical finite element model for the analysis of cylindrical shells made of functionally graded materials. Composite Structures 91: 427-432.
[25] Zhao X., Lee Y.Y., Liew K.M., 2009, Thermoelastic and vibration analysis of functionally graded cylindrical shells, International Journal of Mechanical Sciences 51: 694-707.
[26] Vel S.S., 2010, Exact elasticity solution for the vibration of functionally graded anisotropic cylindrical shells, Composite Structures 92: 2712-2727.
[27] Shen H.S., 2012, Nonlinear vibration of shear deformable FGM cylindrical shells surrounded by an elastic medium, Composite Structures 94: 1144-1154.
[28] Ebrahimi M.J., Najafizadeh M.M., 2014, Free vibration analysis of two-dimensional functionally graded cylindrical shells, Applied Mathematical Modelling 38: 308-324.
[29] Ng T.Y., He X.Q., Liew K.M., 2002, Finite element modeling of active control of functionally graded shells in frequency domain via piezoelectric sensors and actuators, Computational Mechanics 28: 1-9.
[30] Pradhan S.C., Loy C.T., Lam K.Y., Reddy J.N., 2000, Vibration characteristics of functionally graded cylindrical shells under various boundary conditions, Applied Acoustics 61:111-129.
[31] Viola E., Rossetti L., Fantuzzi N., 2012, Numerical investigation of functionally graded cylindrical shells panels using the generalized unconstrained third order theory coupled with the stress recovery, Composite Structures 94: 3736-3758.
[32] Bhangale R.K., Ganesan N., Chandramouli P., 2006, Linear thermoelastic buckling and free vibration behavior of functionally graded truncated conical shells, Journal of Sound and Vibration 292: 341-371.
[33] Tornabene F., 2009, Free vibration analysis of functionally graded conical, cylindrical shell and annular plate structures with a four-parameter power-law distribution, Computer Methods in Applied Mechanics and Engineering 198: 2911-2935.
[34] Tornabene F., Viola E., Inman D.J., 2009, 2-D differential quadrature solution for vibration analysis of functionally graded conical, cylindrical shell and annular plate structures, Journal of Sound and Vibration 328: 259-290.
[35] Qu Y.G., Long X.H., Yuan G.Q., Meng G., 2013, A unified formulation for vibration analysis of functionally graded shells of revolution with arbitrary boundary conditions, Composites Part B: Engineering 50: 381-402.
[36] Sofiyev A.H., 2009, The vibration and stability behavior of freely supported FGM conical shells subjected to external pressure, Composite Structures 89: 356-366.
[37] Sofiyev A.H., 2011, On the vibration and stability of clamped FGM conical shells under external loads, Journal of Composite Materials 45: 771-788.
[38] Malekzadeh P., Fiouz A.R., Sobhrouyan M., 2012, Three-dimensional free vibration of functionally graded truncated conical shells subjected to thermal environment, International Journal of Pressure Vessels and Piping 89: 210-221.
[39] Sofiyev A.H., 2012, The non-linear vibration of FGM truncated conical shells, Composite Structures 94: 2237-2245.
[40] Najafov A.M., Sofiyev A.H., 2013, The non-linear dynamics of FGM truncated conical shells surrounded by an elastic medium, International Journal of Mechanical Sciences 66: 33-44.
[41] Malekzadeh P., Heydarpour Y., 2013, Free vibration analysis of rotating functionally graded truncated conical shells, Composite Structures 97: 176-188.
[42] Sofiyev A.H., 2014, On the dynamic buckling of truncated conical shells with functionally graded coatings subject to a time dependent axial load in the large deformation, Composites: Part B 58: 524-533.
[43] Sofiyev A.H., 2014, The combined influences of heterogeneity and elastic foundations on the nonlinear vibration of orthotropic truncated conical shells, Composites: Part B 61: 324-339.
[44] Xiang X., Guoyong J., Tiangui Y., Zhigang L., 2014, Free vibration analysis of functionally graded conical shells and annular plates using the Haar wavelet method, Applied Acoustics 85: 130-142.
[45] Zhu S., Guoyong J., Shuangxia Sh., Tiangui Y., Xingzhao J., 2014, A unified solution for vibration analysis of functionally graded cylindrical, conical shells and annular plates with general boundary conditions, International Journal of Mechanical Sciences 80: 62-80.
[46] Zhu S., Guoyong J., Tiangui Y., 2014, Three-dimensional vibration analysis of thick functionally graded conical, cylindrical shell and annular plate structures with arbitrary elastic restraints, Composite Structures 118: 432-447.
[47] Tornabene F., Fantuzzi N., Bacciocchi M., 2014, Free vibrations of free-form doubly-curved shells made of functionally graded materials using higher-order equivalent single layer theories, Composites: Part B 67: 490-509.
[48] Sofiyev A.H., Kuruoglu N., 2015, On a problem of the vibration of functionally graded conical shells with mixed boundary conditions, Composites: Part B 70: 122-130.
