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 the stress field or deflections of the elastic beam based on this theory. In contrast, Timoshenko theory is not so much used by engineers. However, in some cases, Euler–Bernoulli theory, which neglects the effect of transversal shear deformation, yields unacceptable results. For instance, when dealing with visco-elastic behavior, shear deformations play a fundamental role. Recent studies on the response evaluation of a visco-elastic Euler–Bernoulli beam under quasi-static and dynamic loads have been stressed that for better capturing of the visco-elastic behavior, a fractional constitutive law has to be considered. In this context, it has been provided that if the 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 mentioned before, when dealing with visco-elasticity the Timoshenko beam model is more appropriate than Euler–Bernoulli’s one, and 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. Ready-to-use tables and a straightforward formulation have been provided leading to significant analysis cost savings and improved quality.