Boundary Layer Solutions Induced by Displacement Boundary Conditions of Shear Deformable Beams and Accuracy Study of Several Higher-Order Beam Theories

Abstract

Although a number of the higher-order/refined beam theories had been proposed for the analysis of shear deformable beams, these beam theories cannot properly take account of the displacement boundary conditions of shear deformable beams, and they are not capable of characterizing the boundary layer effects resulting from the displacement boundary conditions. A new beam theory with the sixth-order differential equilibrium equations is employed in this paper to analytically solve the boundary layer solutions of three typical beams with clamped ends. The influences of the boundary layer solutions given by the new sixth-order beam theory on the deflections, shear forces and stresses are studied, and the resulting solutions are evaluated by the finite-element results. The accuracy assessment of the deflections given by various higher-order/refined beam theories is also carried out by comparing these analytical solutions of deflections with the numerical results. The numerical results show that the new sixth-order beam theory can properly deal with the displacement boundary conditions of shear deformable beams and yield accurate solutions of both displacements and stresses. Furthermore, the new sixth-order beam theory is also capable of properly characterizing the boundary layer effect induced by the displacement boundary conditions. The current study also shows that the boundary layer solutions at the zone of clamped ends do not considerably affect the deflections; however, they contribute significantly to the localized distributions of the shear force and the normal stresses at the zone of the clamped ends. In particular, the resulting normal stresses at the beam surfaces of a clamped end are much larger than those predicted by Timoshenko beam theory.