Bending Analysis of Laminated Composite Plates with Arbitrary Boundary Conditions

Subject Areas : Engineering

A.M Naserian Nik ¹ , M Tahani ²

1 - Department of Mechanical Engineering, Faculty of Engineering, Ferdowsi University of Mashhad
2 - Department of Mechanical Engineering, Faculty of Engineering, Ferdowsi University of Mashhad

Received: 2008-12-11 Accepted : 2009-02-10 Published : 2009-04-30

Keywords: Laminated plates, Analytical solution, Arbitrary boundary conditions, First-order shear deformation theory,

Abstract :

It is well known that for laminated composite plates a Levy-type solution exists only for cross-ply and antisymmetric angle-ply laminates. Numerous investigators have used the Levy method to solve the governing equations of various equivalent single-layer plate theories. It is the intension of the present study to introduce a method for analytical solutions of laminated composite plates with arbitrary lamination and boundary conditions subjected to transverse loads. The method is based on separation of spatial variables of displacement field components. Within the displacement field of a first-order shear deformation theory (FSDT), a laminated plate theory is developed. Two systems of coupled ordinary differential equations with constant coefficients are obtained by using the principle of minimum total potential energy. Since the procedure used is simple and straightforward it can, therefore, be adopted in developing higher-order shear deformation and layerwise laminated plate theories. The obtained equations are solved analytically using the state-space approach. The results obtained from the present method are compared with the Levy-type solutions of cross-ply and antisymmetric angle-ply laminates with various admissible boundary conditions to verify the validity and accuracy of the present theory. Also for other laminations and boundary conditions that there exist no Levy-type solutions the present results may be compared with those obtained from finite element method. It is seen that the present results have excellent agreements with those obtained by Levy-type method.

References:

[1] Smith C.S., 1990, Design of Marine Structures in Composite Materials, Elsevier Applied Science, London.

[2] Haresceugh R.I. 1989, Aircraft and aerospace applications of composites, in: Concise Encyclopedia of Composite Materials, edited by A. Kelly,Pergamon Press, Oxford.

[3] Beardman P., 1989, Automotive components: fabrication, in: Concise Encyclopedia of Composite Materials, edited by A. Kelly, Pergamon Press, Oxford.

[4] Bowen D.H., 1989, Applications of composites: an overview, in: Concise Encyclopedia of Composite Materials, edited by A. Kelly,Pergamon Press, Oxford.

[5] Fujihara K., Teo K., Gopal R., Loh P.L., Ganesh V.K., Ramakrishna S., Foong K.W.C., Chew C.L., 2004, Fibrous composite materials in dentistry and orthopedics: review and applications, Composites Science and Technology 64: 775-788.

[6] Bhaskar K., Kaushik B., 2004, Simple and exact series solutions for flexure of orthotropic rectangular plates with any combination of clamped and simply supported edges, Composite Structures 63: 63–68.

[7] Bose P., Reddy J.N., 1998, Analysis of composite plates using various plate theories, Part 1: Formulation and analytical solutions, Structural Engineering and Mechanics, An International Journal 6(6): 583-612.

[8] Bose P., Reddy J.N., 1998, Analysis of composite plates using various plate theories, Part 2: Finite element model and numerical results, Structural Engineering and Mechanics, An International Journal 6(6): 727-746.

[9] Kant T., Swaminathan K., 2002, Analytical solutions for static analysis of laminated composite and sandwich plates based on a higher order refined theory, Composite Structures 56(4): 329-344.

[10] Nelson R.B., Lorch D.R., 1974, A refined theory for laminated orthotropic plates, ASME Journal of Applied Mechanics 41: 177-183.

[11] Pagano N.J., 1970, Exact solutions for rectangular bidirectional composites and sandwich plates, Journal of Composite Materials 4: 20-34.

[12] Qianl L.F., Batra R.C., Chen L.M., 2003, Elastostatic deformations of a thick plate by using a higher-order shear and normal deformable plate theory and two meshless local Petrov-Galerkin (MLPG) methods, CMES 4(l): 161-175.

[13] Ren J.G., 1986, Bending theory of laminated plate, Composites Science and Technology 27: 225-248.

[14] Savithri S., Varadan T.K., 1992, Laminated plates under uniformly distributed and concentrated loads, ASME Journal of Applied Mechanics 59: 211-214.

[15] Soldatos K.P., Watson P., 1997, Accurate stress analysis of laminated plates combining a two-dimensional theory with the exact three-dimensional solution for simply supported edges, Mathematics and Mechanics of Solids 2: 459-489.

[16] Soni S.R., Pagano N.J., 1982, Elastic response of composite laminates, in: Mechanics of Composite Materials-Recent Advances, edited by Z. Hashin, C.T. Herakovich, Pergamon Press, New York.

[17] Shu X.P. Soldatos K.P., 2000, Cylindrical bending of angle-ply laminates subjected to different sets of edge boundary conditions, International Journal of Solids and Structures 37: 4289-4307.

[18] Swaminathan K., Ragounadin D., 2004, Analytical solutions using a higher-order refined theory for the static analysis of antisymmetric angle-ply composite and sandwich plates, Composite Structures 64: 405-417.

[19] Wenbin Yu., 2005, Mathematical construction of a Reissner–Mindlin plate theory for composite laminates, International Journal of Solids and Structures 42: 6680-6699.

[20] Timoshenko S.P., Krieger S.W., 1959, Theory of Plates and Shells, McGraw-Hill, New York, Second Edition.

[21] Bhaskar K., Kaushik B., 2004, Simple and exact series solutions for flexure of orthotropic rectangular plates with any combination of clamped and simply supported edges, Composite Structure 63: 63-68.

[22] Khalili M.R., Malekzadeh K., Mittal R.K., 2005, A new approach to static and dynamic analysis of composite plates with different boundary conditions, Composite Structure 69: 149-155.

[23] Green A.E., 1944, Double fourier series and boundary value problems, in: Proceedings of Cambridge Philosophical Society 40: 222-228.

[24] Kabir H.R.H., Chaudhuri R.A., 1992, Boundary-continuous fourier solution for clamped mindlin plates, ASCE Journal of Engineering Mechanics 118: 1457-1467.

[25] Green A.E., Hearmon R.F.S., 1945, The buckling of flat rectangular plywood plates, Philosophical Magazine 36: 659-687.

[26] Chaudhuri R.A., Kabir H.R.H., 1992, Fourier analysis of clamped moderately thick arbitrarily laminated plates, AIAA Journal 30(11): 2796-2798.

[27] Vel S.S., Batra R.C., 1999, Analytical solutions for rectangular thick laminated plates subjected to arbitrary boundary conditions, AIAA Journal 37: 1464-1473.

[28] Eshelby J.D., Read W.T., Shockley W., 1953, Anisotropic elasticity with applications to dislocation theory, Acta Metallurgica 1: 251-259.

[29] Reddy J.N., 2004, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, CRC Press, Boca Raton, FL, Second Edition.

[30] Fung Y.C., 1965, Foundation of Solid Mechanics, Prentice-Hall, Englewood Cliffs, New Jersey, First Edition.

[31] Goldberg J.L., Schwartz A.J., 1972, Systems of Ordinary Differential Equations, An Introduction, Harper and Row, New York.