Fractional visco-elastic Timoshenko beam from elastic Euler-Bernoulli beam

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 deflections of the elastic beam based on this theory. Conversely Timoshenko theory is not so much used by engineer freelancers. However in such cases Euler-Bernoulli theory, that neglects the effect of transversal shear deformation leads to unacceptable results. For instance when dealing with the visco-elastic behavior, shear deformations play a fundamental role. Recent studies 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 correspondence principle also holds, 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 information to engineering designers, introducing exact linking relationships between elastic Euler-Bernoulli beam response and fractional visco-elastic Timoshenko beam response. The ready-to-use tables and straightforward formulation have been provided leading to tremendous cost analysis savings and improved quality.