Effect of Winkler Foundation on Radially Symmetric Vibrations of Bi-Directional FGM Non-Uniform Mindlin’s Circular Plate Subjected to In-Plane Peripheral Loading

Document Type: Research Paper

Authors

1 Department of Mathematics, Jaypee Institute of Information Technology, Noida, India

2 Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, India

10.22034/jsm.2019.1873720.1466

Abstract

An analysis has been presented of the effect of elastic foundation and uniform in-plane peripheral loading on the natural frequencies and mode shapes of circular plates of varying thickness exhibiting bi-directional functionally graded characteristics, on the basis of first order shear deformation theory. The material properties of the plate are varying following a power-law in both the radial and transverse directions. The numerical solutions of the coupled differential equations leading the motion of simply supported and clamped plates acquired by using Hamilton’s principle, is attained by harmonic differential quadrature method. The effect of different plate parameters namely gradient index, heterogeneity parameter, density parameter, taper parameter and thickness parameter is illustrated on the vibration characteristics for the first three modes of vibration for various values of in-plane peripheral loading parameter together with foundation parameter. Critical buckling loads in compression are calculated for both the boundary conditions by putting the frequencies to zero. The reliability of the present technique is confirmed by comparing the results with exact values and results of published work.                   

Keywords

[1] Amini M.H., Soleimani M., Rastgoo A., 2009, Three-dimensional free vibration analysis of functionally graded material plates resting on an elastic foundation, Smart Materials and Structures 18(8): 085015.
[2] Ansari R., Gholami R., Shojaei M.F., Mohammadi V., Darabi M.A., 2013, Thermal buckling analysis of a Mindlin rectangular FGM microplate based on the strain gradient theory, Journal of Thermal Stresses 36(5): 446-465.
[3] Baferani A.H., Saidi A.R., Jomehzadeh E., 2011, An exact solution for free vibration of thin functionally graded rectangular plates, Proceedings of the Institution of Mechanical Engineers Part C, Journal of Mechanical Engineering Science 225(3): 526-536.
[4] Batra R.C., Aimmanee S., 2005, Vibrations of thick isotropic plates with higher order shear and normal deformable plate theories, Computers and Structures 83: 934-955.
[5] Bisadi H., Es' haghi M., Rokni H., Ilkhani M., 2012, Benchmark solution for transverse vibration of annular Reddy plates, International Journal of Mechanical Sciences 56(1): 35-49.
[6] Bowles J.E., 1982, Foundation Analysis and Design, McGraw-Hill, Inc.
[7] Chen S.S., Xu C.J., Tong G.S., Wei X., 2015, Free vibration of moderately thick functionally graded plates by a meshless local natural neighbor interpolation method, Engineering Analysis with Boundary Elements 61: 114-126.
[8] Civalek Ö., 2008, Free vibration analysis of symmetrically laminated composite plates with first-order shear deformation theory (FSDT) by discrete singular convolution method, Finite Elements in Analysis and Design 44(12): 725-731.
[9] Deresiewicz H., Mindlin R.D., 1955, Axially symmetric flexural vibrations of a circular disk, Journal of Applied Mechanics 22: 86-88.
[10] Efraim E., 2011, Accurate formula for determination of natural frequencies of FGM plates basing on frequencies of isotropic plates, Procedia Engineering 10: 242-247.
[11] Efraim E., Eisenberger M., 2007, Exact vibration analysis of variable thickness thick annular isotropic and FGM plates, Journal of Sound and Vibration 299(4): 720-738.
[12] Eftekhari S.A., Jafari A.A., 2013, Modified mixed Ritz-DQ formulation for free vibration of thick rectangular and skew plates with general boundary conditions, Applied Mathematical Modelling 37(12): 7398-7426.
[13] Fallah A., Aghdam M.M., Kargarnovin M.H., 2013, Free vibration analysis of moderately thick functionally graded plates on elastic foundation using the extended Kantorovich method, Archive of Applied Mechanics 83(2): 177-191.
[14] Ferreira A.J.M., Roque C.M.C., Jorge R.M.N., 2005, Free vibration analysis of symmetric laminated composite plates by FSDT and radial basis functions, Computer Methods in Applied Mechanics and Engineering 194(39): 4265-4278.
[15] Gorman D.G., 1983, Vibration of thermally stressed polar orthotropic annular plates, Earthquake Engineering and Structural Dynamics 11(6): 843-855.
[16] Gupta U.S., Lal R., 1979, Axisymmetric vibrations of linearly tapered annular plates under an in-plane force, Journal of Sound and Vibration 64(2): 269-276.
[17] Gupta U.S., Ansari A.H., Sharma S., 2006, Buckling and vibration of polar orthotropic circular plate resting on Winkler foundation, Journal of Sound and Vibration 297(3): 457-476.
[18] Gupta U.S., Lal R., Sharma S., 2007, Vibration of non-homogeneous circular Mindlin plates with variable thickness, Journal of Sound and Vibration 302(1): 1-17.
[19] Han J.B., Liew K.M., 1999, Axisymmetric free vibration of thick annular plates, International Journal of Mechanical Sciences 41(9): 1089-1109.
[20] Hong G.M., Wang C.M., Tan M.T., 1993, Analytical buckling solutions for circular Mindlin plates: inclusion of inplane prebuckling deformation, Archive of Applied Mechanics 63(8): 534-542.
[21] Hosseini-Hashemi S., Derakhshani M., Fadaee M., 2013, An accurate mathematical study on the free vibration of stepped thickness circular/annular Mindlin functionally graded plates, Applied Mathematical Modelling 37(6): 4147-4164.
[22] Hosseini-Hashemi S., 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(8): 1025-1035.
[23] Hosseini-Hashemi S., Taher H.R.D., Akhavan H., 2010, Vibration analysis of radially FGM sectorial plates of variable thickness on elastic foundations, Composite Structures 92(7): 1734-1743.
[24] Irie T., Yamada G., Aomura S., 1979, Free vibration of a Mindlin annular plate of varying thickness, Journal of Sound and Vibration 66(2): 187-197.
[25] Irie T., Yamada G., Aomura S., 1980, Natural frequencies of Mindlin circular plates, Journal of Applied Mechanics 47(3): 652-655.
[26] Lal R., Gupta U.S., 1982, Influence of transverse shear and rotatory inertia on axisymmetric vibrations of polar orthotropic annular plates of parabolically varying thickness, Indian Journal of Pure and Applied Mathematics 13(02): 205-220.
[27] Leissa A.W., 1982, Advances and Trends in Plate Buckling Research , Ohio State University Research Foundation Columbus.
[28] Li S.Q., Yuan H., 2011, Quasi-Green’s function method for free vibration of clamped thin plates on Winkler foundation, Applied Mathematics and Mechanics 32: 265-276.
[29] Liew K.M., Hung K.C., Lim M.K., 1995, Vibration of Mindlin plates using boundary characteristic orthogonal polynomials, Journal of Sound and Vibration 182(1): 77-90.
[30] Mindlin R.D., 1951, Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates, Journal of Applied Mechanics 18: 31-38.
[31] Miyamoto Y., Kaysser W.A., Rabin B.H., Kawasaki A., Ford R.G., 2013, Functionally Graded Materials: Design, Processing and Applications, Springer Science & Business Media.
[32] Naderi A., Saidi A.R., 2011, An analytical solution for buckling of moderately thick functionally graded sector and annular sector plates, Archive of Applied Mechanics 81(6): 809-828.
[33] Najafizadeh M.M., Heydari H.R., 2008, An exact solution for buckling of functionally graded circular plates based on higher order shear deformation plate theory under uniform radial compression, International Journal of Mechanical Sciences 50: 603-612.
[34] Rao G.V., Raju K.K., 1986, A study of various effects on the stability of circular plates, Computers and Structures 24(1): 39-45.
[35] Rao L.B., Rao C.K., 2014, Frequency analysis of annular plates with inner and outer edges elastically restrained and resting on Winkler foundation, International Journal of Mechanical Sciences 81: 184-194.
[36] Saidi A.R., Baferani A.H., Jomehzadeh E., 2011, Benchmark solution for free vibration of functionally graded moderately thick annular sector plates, Acta Mechanica 219(3-4): 309-335.
[37] Satouri S., Asanjarani A., Satouri A., 2015, Natural frequency analysis of 2D-FGM sectorial plate with variable thickness resting on elastic foundation using 2D-DQM, International Journal of Applied Mechanics 7(02): 1550030.
[38] Striz A.G., Wang X., Bert C.W., 1995, Harmonic differential quadrature method and applications to analysis of structural components, Acta Mechanica 111(1-2): 85-94.
[39] Su Z., Jin G., Wang X., 2015, Free vibration analysis of laminated composite and functionally graded sector plates with general boundary conditions, Composite Structures 132: 720-736.
[40] Tajeddini V., Ohadi A., 2011, Three-dimensional vibration analysis of functionally graded thick, annular plates with variable thickness via polynomial-Ritz method, Journal of Vibration and Control 18(11): 1698-1707.
[41] Thai H.T., Choi D.H., 2013, A simple first-order shear deformation theory for the bending and free vibration analysis of functionally graded plates, Composite Structures 101: 332-340.
[42] Thai H.T., Kim S.E., 2015, A review of theories for the modeling and analysis of functionally graded plates and shells, Composite Structures 128: 70-86.
[43] Wang C.M., Xiang Y., Kitipornchai S., Liew K.M., 1993, Axisymmetric buckling of circular Mindlin plates with ring supports, Journal of Structural Engineering 119(3): 782-793.
[44] Wang Q., Shi D., Liang Q., Shi X., 2016, A unified solution for vibration analysis of functionally graded circular, annular and sector plates with general boundary conditions, Composites Part B-Engineering 88: 264-294.
[45] Xiang Y., 2003, Vibration of rectangular Mindlin plates resting on non-homogenous elastic foundations, International Journal of Mechanical Sciences 45(6): 1229-1244.
[46] Xiang Y., Wei G.W., 2004, Exact solutions for buckling and vibration of stepped rectangular Mindlin plates, International Journal of Solids and Structures 41(1): 279-294.
[47] Xue K., Wang J.F., Li Q.H., Wang W.Y., Wang P., 2014, An exact series solution for the vibration of Mindlin rectangular plates with elastically restrained edges, Key Engineering Materials 572: 489-493.
[48] Zamani M., Fallah A., Aghdam M.M., 2012, Free vibration analysis of moderately thick trapezoidal symmetrically laminated plates with various combinations of boundary conditions, European Journal of Mechanics-A/Solids 36: 204-212.
[49] Zenkour A.M., 2005, A comprehensive analysis of functionally graded sandwich plates: Part 2—Buckling and free vibration, International Journal of Solids and Structures 42(18): 5243-5258.
[50] Zhang L.W., Lei Z.X., Liew K.M., 2015, Buckling analysis of FG-CNT reinforced composite thick skew plates using an element-free approach, Composites Part B: Engineering 75: 36-46.
[51] Zhang L.W., Lei Z.X., Liew K.M., 2015, Computation of vibration solution for functionally graded carbon nanotube-reinforced composite thick plates resting on elastic foundations using the element-free IMLS-Ritz method, Applied Mathematics and Computation 256: 488-504.
[52] Zhao X., Lee Y.Y., Liew K.M., 2009, Mechanical and thermal buckling analysis of functionally graded plates, Composite Structures 90: 161-171.