A trigonometric shear deformation theory for flexure of thick beams, taking into account transverse shear deformation effects, is developed. The number of variables in the present theory is same as that in the first order shear deformation theory. The sinusoidal function is used in displacement field in terms of thickness coordinate to represent the shear deformation effects. The noteworthy feature of this theory is that the transverse shear stresses can be obtained directly from the use of constitutive relations with excellent accuracy, satisfying the shear stress free conditions on the top and bottom surfaces of the beam. Hence, the theory obviates the need of shear correction factor. Governing differential equations and boundary conditions are obtained by using the principle of virtual work. The thick cantilever isotropic beams are considered for the numerical studies to demonstrate the efficiency of the. Results obtained are discussed critically with those of other theories.<\/p>\r\n","references":"[1]\tJ. A. C. Bresse, \"Cours de Mechanique Applique\u201d, Mallet-Bachelier, Paris, 1859.\r\n[2]\tJ. W. S. Lord Rayleigh, \"The Theory of Sound\u201d, Macmillan Publishers, London, 1877.\r\n[3]\tS. P. Timoshenko, \"On the correction for shear of the differential equation for transverse vibrations of prismatic bars\u201d, Philosophical Magazine, vol. 41, no. 6, pp. 742-746, 1921.\r\n[4]\tG. R. Cowper, \"The shear coefficients in Timoshenko beam theory\u201d, ASME J. of Applied Mechanic, vol. 33, no. 2, pp. 335-340, 1966.\r\n[5]\tG. R. Cowper, \"On the accuracy of Timoshenko beam theory\u201d, ASCE J. of Engineering Mechanics Division. Vol. 94, no. EM6, pp. 1447-1453, 1968.\r\n[6]\tM. Levinson, \"A new rectangular beam theory\u201d, J. of Sound and Vibration, vol. 74, no.1, pp. 81-87, 1981.\r\n[7]\tW. B. Bickford, \"A consistent higher order beam theory\u201d, Int. Proceeding of Dev. in Theoretical and Applied Mechanics (SECTAM), vol. 11, pp. 137-150, 1982.\r\n[8]\tL. W. Rehfield, P. L. N. Murthy, \"Toward a new engineering theory of bending: fundamentals\u201d, AIAA J. vol. 20, no. 5, pp. 693-699, 1982.\r\n[9]\tA. V. Krishna Murty, \"Towards a consistent beam theory\u201d, AIAA J., vol. 22, no. 6, pp. 811-816, 1984. \r\n[10]\tM. H. Baluch,, A. K. Azad, M. A. Khidir, \"Technical theory of beams with normal strain\u201d, ASCE J. of Engineering Mechanics, vol. 110, no. 8, pp. 1233-1237, 1984.\r\n[11]\tA. Bhimaraddi, K. Chandrashekhara, \"Observations on higher order beam Theory\u201d, ASCE J. of Aerospace Engineering, vol. 6, no.4, pp. 408-413, 1993.\r\n[12]\tH. Irretier, \"Refined effects in beam theories and their influence on natural frequencies of beam\u201d, Int. Proceeding of Euromech Colloquium, 219, on Refined Dynamical Theories of Beam, Plates and Shells and Their Applications, Edited by I. Elishak off and H. Irretier ,Springer-Verlag, Berlin, pp. 163-179, 1986.\r\n[13]\tT. Kant, A. Gupta, \"A finite element model for higher order shears deformable beam theory\u201d, J. of Sound and Vibration, vol. 125, no. 2, pp. 193-202, 1988.\r\n[14]\tP. R. Heyliger, J. N. Reddy, \"A higher order beam finite element for bending and vibration problems\u201d, J. of Sound and Vibration, vol. 126, no. 2, pp. 309-326, 1988. \r\n[15]\tR. C. Averill, J. N. Reddy, \"An assessment of four-noded plate finite elements based on a generalized third order theory\u201d, Int. J. of Numerical Methods in Engineering, vol. 33, pp. 1553-1572, 1992.\r\n[16]\tJ. N. Reddy, \"An Introduction to Finite Element Method\u201d. 2nd Ed., McGraw-Hill, Inc., New York, 1993.\r\n[17]\tV. Z. Vlasov, U. N. Leont\u2019ev, \"Beams, Plates and Shells on Elastic Foundations\u201d Moskva, Chapter 1, 1-8. Translated from the Russian by A. Barouch and T. Plez Israel Program for Scientific Translation Ltd., Jerusalem, 1966.\r\n[18]\tM. Stein, \"Vibration of beams and plate strips with three dimensional flexibility\u201d, ASME J. of Applied Mechanics, vol. 56, no. 1, pp. 228-231, 1989. \r\n[19]\tY. M. Ghugal. Y.M., R. Sharma, \"A hyperbolic shear deformation theory for flexure and vibration of thick isotropic beams\u201d, Int. J. of Computational Methods, vol. 6, no. 4, pp. 585-604, 2009. \r\n[20]\tY. M. Ghugal, R. P. Shmipi, \"A review of refined shear deformation theories for isotropic and anisotropic laminated beams\u201d, J. of Reinforced Plastics and Composites, vol. 20, no. 3, pp. 255-272, 2001. \r\n[21]\tF. B. Hildebrand, E. C. Reissner, \"Distribution of Stress in Built-In Beam of Narrow Rectangular Cross Section\u201d, J. of Applied Mechanics, vol. 64, pp. 109-116, 1942.\r\n","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 77, 2013"}