A Static Flexure of Thick Isotropic Plates Using Trigonometric Shear Deformation Theory

Document Type : Research Paper


Department of Applied Mechanics, Government Engineering College


A Trigonometric Shear Deformation Theory (TSDT) for the analysis of isotropic plate, taking into account transverse shear deformation effect as well as transverse normal strain effect, is presented. The theory presented herein is built upon the classical plate theory. In this displacement-based, trigonometric shear deformation 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. It accounts for realistic variation of the transverse shear stress through the thickness and satisfies the shear stress free surface conditions at the top and bottom surfaces of the plate. The theory obviates the need of shear correction factor like other higher order or equivalent shear deformation theories. Governing equations and boundary conditions of the theory are obtained using the principle of virtual work. Results obtained for static flexural analysis of simply supported thick isotropic plates for various loading cases are compared with those of other refined theories and exact solution from theory of elasticity.


[1] Kirchhoff G.R., 1850, Uber das gleichgewicht und die bewegung einer elastischen Scheibe, Journal für die reine und angewandte Mathematik (Crelle's Journal) 40: 51-88.
[2] Reissner E., 1944, On the theory of bending of elastic plates, Journal of Mathematics and Physics 23: 184-191.
[3] Reissner E. 1945, The effect of transverse shear deformation on the bending of elastic plates, ASME Journal of Applied Mechanics 12: 69-77.
[4] 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.
[5] Naghdi P.M., 1957, On the theory of thin elastic shells, Quarterly of Applied Mathematics 14: 369-380.
[6] Pister K.S., Westmann R.A., 1962, Bending of plates on an elastic foundation, ASME Journal of Applied Mechanics 29: 369-374.
[7] Whitney J.M., Sun, C.T., 1973, A higher order theory for extensional motion of laminated composites, Journal of Sound and Vibration 30: 85-97.
[8] Nelson R.B., Lorch, D.R., 1974, A refined theory for laminated orthotropic plates, ASME Journal of Applied Mechanics 41: 177-183.
[9] Reissner E., 1963, On the derivation of boundary conditions for plate theory, in: Proceedings of Royal Society of London, Series A 276: 178-186.
[10] Provan J.W., Koeller R.C., 1970, On the theory of elastic plates, International Journal of Solids and Structures 6: 933-950.
[11] 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.
[12] Lo K.H., Christensen R.M., Wu E.M., 1978, Stress solution determination for higher order plate theory, International Journal of Solids and Structures 14: 655-662.
[13] Levinson M., 1980, An accurate, simple theory of the statics and dynamics of elastic plates, Mechanics Research Communications 7: 343-350.
[14] Reddy J.N., 1984, A simple higher order theory for laminated composite plates, ASME Journal of Applied Mechanics 51: 745-752.
[15] Krishna Murty A.V., 1977, Higher order theory for vibrations of thick plates, AIAA Journal 15: 1823-1824.
[16] Krishna Murty A.V., 1986, Toward a consistent plate theory, AIAA Journal 24: 1047-1048.
[17] Savithri S., Varadan, T.K., 1992, A simple higher order theory for homogeneous plates, Mechanics Research Communications 19: 65- 71.
[18] Soldatos K.P., 1988, On certain refined theories for plate bending, ASME Journal of Applied Mechanics 55: 994-995.
[19] Reddy J.N., 1990, A general non-linear third-order theory of plates with moderate thickness, International Journal of Nonlinear Mechanics 25:677-686.
[20] Noor A.K., Burton W.S., 1989, Assessment of shear deformation theories for multilayered composite plates, Applied Mechanics Reviews 42: 1-13.
[21] 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.
[22] Levy M., 1877, Memoire sur la theorie des plaques elastique planes, Journal des Mathematiques Pures et Appliquees, 30: 219-306.
[23] Stein M., 1986, Nonlinear theory for plates and shells including effect of shearing, AIAA Journal 24: 1537-1544.
[24] Shimpi R.P., 2002, Refined plate theory and its variants, AIAA Journal 40 (1): 137-146.
[25] Shimpi R.P., Patel H.G., 2006, A two variable refined plate theory for orthotropic plate analysis, International Journal of Solids and Structures 43: 6783-6799.
[26] Shimpi R.P., Patel, H.G. Arya, H., 2007, New first order shear deformation plate theories, Journal of Applied Mechanics 74: 523-533.
[27] Srinivas S., Joga Rao C.V., Rao A.K., 1970, Bending, vibration and buckling of simply supported thick orthotropic rectangular plate and laminates, International Journal of Solids and Structures 6: 1463-1481.
[28] Timoshenko S.P., Goodier J.M., 1970, Theory of Elasticity, McGraw-Hill, 3rd International Edition, Singapore.