Flexure of Thick Cantilever Beams using Trigonometric Shear Deformation Theory

Abstract

A trigonometric shear deformation theory for flexural analysis 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 isotropic beams are considered for the numerical studies to demonstrate the efficiency of the theory. It has been shown that the theory is capable of predicting the local effect of stress concentration due to fixity of support. The cantilever isotropic beams subjected to linearly varying loads are examined using the present theory. Results obtained are discussed critically with those of other theories.

Keywords: Thick beam, Trigonometric shear deformation, Principle of virtual work, Equilibrium equations, Displacements, Stresses