Study of deep beams using finite element approach

Abstract

Beams are common structural elements in most structures and generally they are analysed using classical beam theories to evaluate the stress and strain characteristics of the beam. But in the case of deep beams, higher order shear deformation beam theories predicts more accurate results than classical beam theories due its more realistic assumption regarding the shear characteristics of the beam. In this study a hyperbolic shear deformation theory for thick isotropic beams is developed where the displacements are defined using a meaningful function which is more physical and directly comparable with other higher order theories. Governing variationally consistent equilibrium equations and boundary conditions are derived in terms of the stress resultants and displacements using the principle of virtual work. This theory satisfies shear stress free boundary condition at top and bottom of the beam and doesn’t need shear correction factor. .A displacement based finite element model of this theory is formulated using the variational principle. Displacements are approximated using the homogeneous solutions of the governing differential equations that describe the deformations of the cross-section according to the high order theory, which includes cubic variation of the axial displacements over the cross-section of the beam. Also, this model gives the exact stiffness coefficients for the high order isotropic beam element. The model has six degrees of freedom at the two ends, one transverse displacement and two rotations, and the end forces are a shear force and two components of end moments. Several numerical examples are discussed to validate the proposed shear deformation beam theory and finite element model of the beam theory. Results obtained for displacements using the present beam theory and the finite element model are compared with results obtained using other beam theories, 2D elastic theory and 2D and 3D finite element models. Solutions obtained using the proposed beam theory and finite element model are in close agreement with the solutions obtained using 2D elastic theory and 2D and 3D finite element models of ‘ABAQUS’.