On the Appropriate Rotary Inertia in Timoshenko Beam Theory

The rotary inertia defined by Timoshenko to account for the angular velocity effect in flexural vibration of beams has been questioned by some researchers in recent years, and it caused some confusions. This paper discusses the appropriate rotary inertia in Timoshenko beam theory (TBT) and evaluates the influence of the two forms of the rotary inertia on the prediction of the higher-mode frequencies of transversely vibrating beams. Based on the theory of elasticity and variational principle, this work shows that the rotary inertia in the original TBT, defined in terms of the rotation of beam cross-section induced by bending deformation, is variational consistent and is capable of yielding good results of the phase velocities of transversely vibrating beams even in the case where the wavelength of vibrating beams approaches the beam height. On the other hand, the so-called corrected TBT, in which the rotary inertia is defined in terms of the slope of beam deflection, is neither variational consistent nor accurate when the wavelength of vibrating beams approaches the beam height. Therefore, the rotary inertia in TBT defined by Timoshenko is correct and should be used in the dynamic analysis of beams.