Flexure and vibration of thick beams using trigonometric shear deformation theory

Abstract

A trigonometric shear deformation theory is used for flexural analysis of thick beams, taking into account transverse shear deformation effect. The theory accounts for cosine distribution of transverse shear strain through the thickness of the beam. The governing differential equations and boundary conditions are obtained by using the principle of virtual work. The general solutions for the transverse displacement and rotation are presented for transversely loaded beams. The solutions for static flexure are given for beams with fixed end and subjected to uniformly distributed load and concentrated load. The solutions obtained address the accurate flexural behavior of shear deformable beams.  It has been shown that the theory is capable of predicting the local effects due to concentrated load and fixity of support. The results of displacements and stresses obtained from static flexure of cantilever and fixed beams and results of free vibration frequencies for simply supported beam are presented and discussed critically with those of other theories. The results are found to agree well with the exact elasticity results wherever applicable. It is observed that the present theory yields the exact shear coefficient (p²/12) from the free vibration analysis of the simply supported beam. 

Keywords: Thick beam, trigonometric shear deformation, principle of virtual work, equilibrium equations, transverse deflection, axial stress, transverse shear stress, free vibration.