Shear Deformable Elastic Beam Models in Vibration and Sensitivity of Natural Frequencies to Warping Effects

A theory of shear deformable beams in vibration is formulated using a shear-warping theory whereby the cross section is allowed to warp according to a parametrically specified warping rule (parametric warping). A continuous family of beams is generated which is controlled by a warping parameter s ≥ 0 spanning from s = 0 (Timoshenko-Ehrenfest beam) to s = ∞ (Euler-Bernoulli beam) and intersecting the Levinson-Reddy model for s = 0:5. This enables one to express any response parameter as a function of s useful to describe the sensitivity of the beam’s behaviour to warping effects. The governing transverse displacement differential equation (DE) - of the fourth order in the case of no warping - is instead of the sixth order in the presence of warping effects, but remarkably the maximum order of time derivatives is still four. The vibration motion of the family’s general beam is characterized by two basic macroscopic space and time scales, which make it possible to ascertain that the terms of the governing DE with the fourth order time derivative are negligible with respect to the others. The simplified governing DE without fourth order time derivatives is applied to a beam case to derive the physically meaningful spectrum with warping effects and to assess the sensitivity of natural frequencies to the warping effects. Every frequency as a function of s exhibits a waved pattern featured by softening for 0 < s < sh (with smaller frequencies therein), by hardening for s > sh (with larger frequencies therein), sh varying with the vibration mode.