The shear correction factor (k) and the rotary inertia parameter (r_g/L) are both known to reduce the bending frequencies of Timoshenko beams. However, the effect of increasing each parameter on the bending frequencies is different. Increasing r_g/L tends to decrease the bending frequencies, while increasing k tends to increase the frequency values. The combined effect of these two parameters is to further decrease the bending frequencies. What is the physical reasoning behind this? For further details, consult the following article:

A. Bazoune, Combined influence of rotary inertia and shear coefficient on flexural frequencies of Timoshenko beam: Numerical experiments, Acta Mech., (2023) 234 4997–5013, https://doi.org/10.1007/s00707-023-03648-6