A New Beam Theory Considering Horizontal Shear Strain

Abstract

Methods of setting up and solving problems of flexural members, considering the horizontal shear strain, have been studied since the 1970s but there has not been any complete theory. When considering the influence of horizontal shear strain, with the horizontal shear strain approaching zero (when shear elastic modulus $G \to \infty$ or the ratio h/l is very small), the presented solutions do not converge to the case of zero horizontal shear strain, due to the shear locking phenomenon. Many authors have conducted studies to overcome this problem. Although they have achieved acceptable solutions, theoretical mistakes are unavoidable. In this article, the author will present a new method, in which the displacement and shear force functions are considered as functions that need to be determined to set up a new Beam Theory Considering Horizontal Shear Strain. To develop beam problems based on the Method of Gauss’s Principle of Least Constraint, the author uses the calculus of variations and partial integral to establish two differential equations to determine two unknown functions and beams’ boundary conditions. The beam theory (not considering the horizontal shear strain) is a separated condition of this theory. Using this theory in calculating beams and frames does not encounter shear locking phenomenon. The author will present equations of elastic line; analytic formulas determining deflection, angle of rotation, moment and shear force of beams, with different supports and static loads. When considering horizontal shear strain, changes occur in both the displacement and internal forces of beams and frames. However, while the displacement increases considerably, the redistribution of internal forces is quite insignificant.