### Free Vibration of Thick Isotropic Plates Using Trigonometric Shear Deformation Theory

Document Type: Research Paper

Authors

Department of Applied Mechanics, Government Engineering College, Aurangabad-431005 (Maharashtra State)

Abstract

In this paper a variationally consistent trigonometric shear deformation theory is presented for the free vibration of thick isotropic square and rectangular plate. In this displacement based theory, the in-plane displacement field uses sinusoidal function in terms of thickness coordinate to include the shear deformation effect. The cosine function in terms of thickness coordinate is used in transverse displacement to include the effect of transverse normal strain. Governing equations and boundary conditions of the theory are obtained using the principle of virtual work. Results of frequency of bending mode, thickness-shear mode and thickness-stretch mode are obtained from free vibration of simply supported isotropic square and rectangular plates and compared with those of other refined theories and frequencies from exact theory. Present theory yields exact dynamic shear correction factor π2/12 from thickness shear motion of the plate.

Keywords

[1] Kirchhoff G.R., 1850, Uber das gleichgewicht und die bewegung einer elastischen scheibe, Journal of Reine Angew. Math.(Crelle) 40: 51-88.

[2] Kirchhoff G.R., 1850, Uber die schwingungen einer kriesformigen elastischen scheibe, Poggendorffs Annalen 81: 258-264.

[3] Reissner E., 1944, On the theory of bending of elastic plates, Journal of Mathematics and Physics 23: 184-191.

[4] Reissner E., 1945, The effect of transverse shear deformation on the bending of elastic plates, ASME Journal of Applied Mechanics 12: 69-77.

[5] Mindlin R.D., 1951, Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates, ASME Journal of Applied Mechanics 18: 31-38.

[6] Wang C.M., Lim G.T., Reddy J.N., Lee K.H., 2001, Relationships between bending solutions of Reissner and Mindlin plate theories, Engineering Structures 23 (7): 838-849.

[7] Librescu L., 1975, Elastostatic and Kinetics of Anisotropic and Heterogeneous Shell-Type Structures, Nooedhoff International, Leyden, The Netherlands.

[8] Donnell L.H., 1976, Beams, Plates and Shells, McGraw-Hill, New York.

[9] Lo K.H., Christensen R.M., Wu E.M., 1977, A high-order theory of plate deformation, Part-1: Homogeneous plates, ASME Journal of Applied Mechanics 44: 663-668.

[10] Lo K.H., Christensen R.M., Wu E.M., 1977, A high-order theory of plate deformation, Part-2: Laminated plates, ASME Journal of Applied Mechanics 44: 669-676.

[11] Levinson M., 1980, An accurate, simple theory of the statics and dynamics of elastic plates, Mechanics Research Communications 7: 343-350.

[12] Murty A.V.K., Vellaichamy S., 1988, Higher-order theory of homogeneous plate flexure, AIAA Journal 26: 719-725.

[13] Reddy J.N., 1984, A refined nonlinear theory of plates with transverse shear deformation, International Journal of Solids and Structures 20(9/10): 881-896.

[14] Reddy J.N., 1984, A simple higher order theory for laminated composite plates, ASME Journal of Applied Mechanics 51: 745-752.

[15] Reddy J.N., Phan N.D., 1985, Stability and vibration of isotropic, orthotropic and laminated plates according to higher order deformation theory, Journal of Sound and Vibration 98: 157-170.

[16] Srinivas S., Rao A.K., Joga Rao C.V., 1969, Flexure of simply supported thick homogenous and laminated rectangular plates, ZAMM: Zeitschrift fur Angewandte Mathematic und Mchanik 49(8): 449-458.

[17] Srinivas S., Joga Rao C.V., Rao A. K., 1970, An exact analysis for vibration of simply supported homogeneous and laminated thick rectangular plates, Journal of sound and vibration 12(2): 187-199.

[18] Noor A.K., Burton W.S., 1989, Assessment of shear deformation theories for multilayered composite plates, Applied Mechanics Reviews 42: 1-13.

[19] Reddy J.N., 1989, On the generalization of displacement-based laminate theories, Applied Mechanics Reviews 42:S213-S222.

[20] Reddy J.N., Robbins D.H.Jr., 1994, Theories and computational models for composite laminates, Applied Mechanics Reviews 47: 147-169.

[21] Liu D., Li X., 1996, An overall view of laminate theories based on displacement hypothesis, Journal of Composite Materials 30:1539-561.

[22] Liew K.M., Xiang Y., Kitipornchai S., 1995, Research on thick plate vibration, Journal of Sound and Vibration 180:163-176.

[23] Ghugal Y.M., Shimpi R.P., 2002, A review of refined shear deformation theories for isotropic and anisotropic laminated plates, Journal of Reinforced Plastics and Composites 21: 775-813.

[24] Kreja I., 2011, A literature review on computational models for laminated composite and sandwich panels, Central European Journal of Engineering 1(1): 59-80.

[25] Levy M., 1877, Memoire sur la theorie des plaques elastique planes, Journal des Mathematiques Pures et Appliquees 30: 219-306.

[26] Stein M., Jegly D.C., 1987, Effect of transverse shearing on cylindrical bending, vibration and buckling of laminated plates, AIAA Journal 25: 123-129.

[27] Shimpi R.P., Arya H., Naik N.K., 2003, A higher order displacement model for the plate analysis, Journal of Reinforced Plastics and Composites 22: 1667-1688.

[28] Shimpi R.P., Patel H.G., 2006, Free vibration of plate using two variable refined plate theory, Journal of Sound and Vibration 296:979-999.

[29] Ghugal Y.M., Pawar M.D., 2011, Buckling and vibration of plates by hyperbolic shear deformation theory, Journal of Aerospace Engineering and Technology 1(1):1-12.

[30] Ghugal Y.M., Pawar M.D., 2011, Flexural analysis of thick plates by hyperbolic shear deformation theory, Journal of Experimental and Applied Mechanics 2(1):1-21.

[31] Ghugal Y.M., Sayyad A.S., 2010, Free vibration of thick orthotropic plates using trigonometric shear deformation theory, Latin American Journal of Solids and Structures 8: 229-243.

[32] Ghugal Y.M., Sayyad A.S., 2010, A flexure of thick isotropic plate using trigonometric shear deformation theory, Journal of Solid Mechanics 2(1): 79-90.

[33] Ghugal Y.M., Kulkarni S.K., 2011, Thermal stress analysis of cross-ply laminated plates using refined shear deformation theory,Journal of Experimental and Applied Mechanics: An International Journal 2(1): 47-66.

[34] Timoshenko S.P., Goodier J.N., 1970, Theory of Elasticity, Third edition, McGraw Hill, New York.

[35] Manjunatha B.S., Kant T., 1993, Different numerical techniques for the estimation of multiaxial stresses in symmetric/unsymmetric composite and sandwich beams with refined theories, Journal of Reinforced Plastics and Composites 12: 2-37.

[36] Vinayak R.U., Prathap G., Naganarayana B.P., 1996, Beam elements based on a higher order theory — I: Formulation and analysis of performance, Computers and Structures 58: 775-789.

[37] Lamb Horace, 1917, On waves in an elastic plate, Proceedings of Royal Society London, Series A 93: 114-128.