Variational Asymptotic Based Shear Correction Factor for Isotropic Circular Tubes

An analytical closed-form expression is derived for the shear correction factor of beams with annular cross sections. The beam theory is formulated using the variational asymptotic method, which uses inherently occurring small parameters to convert a variational statement from three-dimensional elasticity to an engineering beam theory without any ad hoc assumptions on the beam kinematics. Comparisons are made with existing solutions in the literature; then, the critical speeds of a rotor are shown to depend not only on the inclusion of transverse shear but also on the choice of the shear correction factor, establishing its importance. The motivation of this work is twofold: first, rotors that can be modeled as beams with annular cross sections are often encountered, and this expression for the shear correction factor can be readily incorporated into any rotor dynamics solver; and second, closed-form expressions, when available, always provide more insight into structural mechanics as opposed to numerical solutions.