Timoshenko vs Euler-Bernoulli beam: fractional visco-elastic behaviour

Abstract

The Euler-Bernoulli beam theory is well established in such a way that engineers are very confident with the determination of stress field or def lections of the elastic beam based on this theory. Conversely Timoshenko theory is not so much used by engineers. However in such cases Euler-Bernoulli theory that n eglects the effect of transversal shear deformation leads to unacceptable results. For inst ance when dealing with the visco-elastic behaviour the shear deformations play a fundamental role. Recent studies [1]-[2] on the response evaluation of visco-elastic Euler-Bernoulli beam under quasi-static and dynamic loads, have been stressed that for better capturing the visco-elastic behavior a fractional constitutive law has to be considered. In this context it has been provided that, for homogeneous beam both correspondence principles also hold, then the study of a fractional visco-elastic Euler-Bernoulli beam may be derived from the elastic one. As aforementioned in dealing with visco-elasticity the Timoshenko beam model is more proper than Euler-Bernoulli’s one, then this paper provides important informations to engineering designers, introducing exact linking relationships between visco-elastic Euler-Bernoulli beam solutions and visco-elastic Timoshenko beam solutions. Thus the effect of transverse shear deformations may be taken into account for, without the need of a studying a more complicated model.