Browse > Article
http://dx.doi.org/10.12989/sem.2015.56.5.835

Exact mathematical solution for free vibration of thick laminated plates  

Dalir, Mohammad Asadi (Mechanical Engineering Department, Bu-Ali Sina University)
Shooshtari, Alireza (Mechanical Engineering Department, Bu-Ali Sina University)
Publication Information
Structural Engineering and Mechanics / v.56, no.5, 2015 , pp. 835-854 More about this Journal
Abstract
In this paper, the modified form of shear deformation plate theories is proposed. First, the displacement field geometry of classical and the first order shear deformation theories are compared with each other. Using this comparison shows that there is a kinematic relation among independent variables of the first order shear deformation theory. So, the modified forms of rotation functions in shear deformation theories are proposed. Governing equations for rectangular and circular thick laminated plates, having been analyzed numerically so far, are solved by method of separation of variables. Natural frequencies and mode shapes of the plate are determined. The results of the present method are compared with those of previously published papers with good agreement obtained. Efficiency, simplicity and excellent results of this method are extensible to a wide range of similar problems. Accurate solution for governing equations of thick composite plates has been made possible for the first time.
Keywords
classical theory; first order shear deformation; modified form; laminated plate; vibration; accurate solution;
Citations & Related Records
연도 인용수 순위
  • Reference
1 Navvier, L.M.H. (1819), Resume des Lecons de M'echanique, Ecole Polytechnique, Paris, France.
2 Nyfeh, A.H. and Frank Pai, P. (2004), Linear and Nonlinear Structural Mechanics, John wiley & Sons Inc, New Jersy, US.
3 Omer, C. (2008), "Free vibration analysis of symmetrically laminated composite plates with first-order shear deformation theory (FSDT) by discrete singular convolution method", Finite Elem. Anal. Des., 44, 725-731.   DOI
4 Omer, C. (2009), "Fundamental frequency of isotropic and orthotropic rectangular plates with linearly varying thickness by discrete singular convolution method", Appl. Math. Model., 33(26), 3825-3835.   DOI
5 Pagano, N.J. (1970), "Exact solution for rectangular bidirectional composite and sandwich plates", J. Compos. Mater., 4, 20-34.   DOI
6 Pagano, N.J. and Hatfield, S.J. (1972), "Elastic behavior of multilayer bidirectional composites", AIAA J., 10, 931-933.   DOI
7 Qatu, M.S. (2004), Vibration of Laminated Shells and Plates, Elsevier. Oxford. UK.
8 Rao, S.S. and Prasad, A.S. (1980), "Natural frequencies of Mindlin circular plates", J. Appl. Mech., 47, 652-655.   DOI
9 Reddy, J.N. (1984), "A simple higher order theory for laminated composite plates", J. Appl. Mech., 51, 745-752.   DOI
10 Reddy, J.N. (2004), Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, 2nd Edition, CRC Press, Washington D.C, US.
11 Reissner, E. (1944), "On the theory of bending of elastic plates", J. Math. Phys., 23, 184-191.   DOI
12 Seide, P. (1975), Small Elastic Deformation of Thin Shells, Noordhoff, Leyden, The Netherlands.
13 Shimpi, R.P. (2002), "Refined plate theory and its variants", AIAA J., 40, 137-146.   DOI
14 Shooshtari, A. and Razavi, S. (2010), "A closed form solution for linear and nonlinear free vibrations of composite and fiber metal laminated rectangular plates", Compos. Struct., 92, 2663-2675.   DOI
15 Srinivas, S., Rao, C.J. and Rao, A.K. (1970), "An exact analysis fpr vibration of simply-supported and laminated thick rectangular plates", J. Sound. Vib., 12, 187-199.   DOI
16 Srinivas, S. and Rao, A.K. (1970), "Bending, vibration and buckling of simply-supported thick orthotropic rectangular plates and laminates", Int. J. Solid. Struct., 6, 1463-1481.   DOI
17 Szilard, R. (2004), Theories and Applications of Plate Analysis: Classical, Numerical and Engineering Methods, John Wiley & Sons, Inc, New Jersy, US.
18 Thai, H.T. and Choi, D.H. (2014), "Finite element formulation of a refined plate theory for laminated composite plates", J. Compos Mater., 48, 3521-3538.   DOI
19 Vinson, J.R. and Sierakowski, R.L. (1986), The Behavior of Structures Composed of Composite Materials, Nijhoff, Boston, Massachusetts.
20 Viswanathan, K.K., Kyung, K.S. and Jang, L.H. (2009), "Asymmetric free vibrations of laminated annular cross-ply circular plates including the effects of shear deformation and rotary inertia: spline method", Forschung im Ingenieurwesen, 73, 205-217.   DOI
21 Viswanathan, K.K. and Lee, S.K. (2007), "Free vibration of laminated cross-ply plates including shear deformation by spline method", Int. J. Mech. Sci., 49, 352-363.   DOI
22 Washizu, K. (1975), Variational Method in Elasticity and Plasticity, 2nd Edition, PERGAMON Press.
23 Zenkour, A.M. (2009), "The refined sinusoidal theory for FGM plates on elastic foundations", Int. J. Mech. Sci., 51, 869-880.   DOI
24 Whitney, J.M. (1987), Structural Analysis of Laminated Anisotripic Plates, Technomic, Lancaster, Pennsylvania.
25 Yang, P.C., Norris, C.H. and Stavsky, Y. (1966), "Elastic wave propagation in heterogeneous plates", Int. J. Solid. Struct., 2, 665-684.   DOI
26 Zenkour, A.M. (2004), "Analytical solution for bending cross-ply laminated plates under thermo-mechanical loading", Compos Struct., 65, 367-379.   DOI
27 Chen, W.Q. and Lue, C.F. (2005), "3D vibration analysis of cross-ply laminated plates with one pair of opposite edges simply supported", Compos. Struct., 69, 77-87.   DOI
28 Basset, A.B. (1890), "On the extension and flexure of cylindrical and spherical thin elastic shells", Philos. Trans. R. Soc. London, B, Biol. Sci., 181, 433-480.   DOI
29 Bernoulli, J. (1789), "Essai theorique sur les vibrations de plaques elastiques rectangulaires et Libres", Nova Acta Acad, Petropolit, 5, 197-219.
30 Chakraverty, S. (2009), Vibration of Plates, CRC Press, New York, US.
31 Chladni, E.F.F. (1802), Die Akustik. Breitkopf & Hartel, Leipzig.
32 Euler, L. (1766), "De motu vibratorio tympanorum", Novi Commentari Acad. Petropolit, 10, 243-260.
33 Green, A.E. and Naghdi, P.M. (1981), "A theory of laminated composite plate and road", Report UCB/AM-81-3. University of California, Berkeley. California.
34 Kirchhoff, G. (1850), "Uber das gleichgwich und die bewegung einer elastischen scheibe", Journal Fur Die Reine und Angewandte Mathematik, 40, 51-88.
35 Hencky, H. (1947), "Uber die berucksichtigung der schubverzerrung in ebenen platen", Ingeieur Archiv, 16 72-76.   DOI
36 Hildebrand, F.B. and Reissner, E. (1949), "Thomas GB. Notes on The Foundations of The Theory of Small Displacements of Orthotropic Shells", NASA Technical Note, No: 1833.
37 Hosseini-Hashemi, S., Es'haghi, M., Rokni Damavandi Taher, H. and Fadaie, M. (2010), "Exact closed-form frequency equations for thick circular plates using a third-order shear deformation theory", J. Sound Vib., 329, 3382-3396.   DOI
38 Kirchhoff, G. (1876), Vorlesungen Uber Mathematische Physik, BG. Teubner. Leipzig.
39 Lekhnitski, S.T. (1968), Anisotropic Plates, Gordon and Breach, New York, US.
40 Liew, K.M., Han, J.B. and Xiao, Z.M. (1997), "Vibration analysis of circular Mindlin plates using the differential quadrature method", J. Sound Vib., 205(5), 617-630.   DOI
41 Lo, K.H., Christensen, R.M. and Wu, E.M. (1977), "A high-order theory of plate deformation-Part 1: Homogeneous plates", J. Appl. Mech., 44, 663-668.   DOI
42 Lo, K.H., Christensen, R.M. and Wu, E.M. (1977), "A high-order theory of plate deformation-Part 2: Laminated plates", J. Appl. Mech., 44, 669-676.   DOI
43 Mbakogu, F.C. and Pavlovic, M.N. (1998), "Closed-form fundamental-frequency estimates for polar orthotropic circular plates", Appl. Acoust., 54, 207-228.   DOI
44 Mindlin, R.D. (1951), "Influence of rotary inertia and shear on flexural motion of isotropic elastic plates", J. Appl. Mech., ASME, 18, 31-38.