Much ado about shear correction factors in Timoshenko beam theory

Abstract

a b s t r a c tMany shear correction factors have appeared since the inception of Timoshenko beam theory in 1921. While rational bases for them have been offered, there continues to be some reluctance to their full acceptance because the explanations are not totally convincing and their efficacies have not been comprehensively evaluated over a range of application. Herein, three-dimensional static and dynamic information and results for a beam of general (both symmetric and non-symmetric) cross-section are brought to bear on these issues. Only homogeneous, isotropic beams are considered. Semi-analytical finite element (SAFE) computer codes provide static and dynamic response data for our purposes. Greater clarification of issues relating to the bases for shear correction factors can be seen. Also, comparisons of numerical results with Timoshenko beam data will show the effectiveness of these factors beyond the range of application of elementary (Bernoulli-Euler) theory.An issue concerning principal shear axes arose in the definition of shear correction factors for non-symmetric cross-sections. In this method, expressions for the shear energies of two transverse forces applied on the cross-section by beam and three-dimensional elasticity theories are equated to determine the shear correction factors. This led to the necessity for principal shear axes. We will argue against this concept and show that when two forces are applied simultaneously to a cross-section, it leads to an inconsistency. Only one force should be used at a time, and two sets of calculations are needed to establish the shear correction factors for a non-symmetrical cross-section.

Geometrically exact Cosserat rods with Kelvin–Voigt type viscous damping

“…However, the issue of correction factors for transverse shear in Timoshenko-type rod models is still subject of discussion and research activities (see e.g. Dong et al, 2010). density of the body in the reference volume, and v(X, t) = ∂ t x(X, t) is the velocity of the respective material point.…”

Force-based higher-order beam element with flexural–shear–torsional interaction in 3D frames. Part I: Theory

“…An account of the early history of the shear correction factor can be found in Kaneko [8]. on this field has continued throughout the following decades (e.g., Hutchinson and Zillmer [9]; Renton [10]; Hutchinson [11]) and up to the present day (Dong et al [12]). Within the framework of this paper, classical beam theories are considered to be of the firstorder, i.e., the cross-sectional displacement fields are linear functions on each of the cross-sectional coordinates.…”

Computational Methods in Earthquake Engineering

“…For instance, in the Timoshenko beam theory (TBT), the shear strain distribution is incorrectly assumed to be constant throughout the beam height; e.g., considering a simple rectangular cross-section, such hypothesis does not respect the zero shear strain and stress boundary conditions at its top and bottom. Therefore, a shear correction factor is required to accurately determine the strain energy of deformation, which has deserved the attention of researchers since the 1950s up to the present day [9,10,14,16]. Within the framework of this chapter, classical beam theories are considered to be of the first-order, i.e., those in which the displacement fields inside the cross-section are linear functions on each of the cross-sectional coordinates.…”