Fractional calculus applied to evaluate stress concentration and shear effects in simply supported beams

Abstract

Simply supported prismatic beams submitted to uniformly distributed static load are modeled in the ANSYS environment under linear elasticity limits. However, the beam deflection profile may behave in disagreement with the predictions from either Euler-Bernoulli (EB) or even Timoshenko-Ehrenfest (TE) linear theories, depending on the load magnitude and geometrical properties of the cross section. Careful examination on the ANSYS stress output data reveals, for those cases, local stress concentrations on the support contacts areas due to the intensive force reactions, which end up affecting the resulting transversal deflection along the entire beam, although the linear regime is ensured for all finite elements. Another way to verify these effects is to employ fractional calculus modeling on the EB or TE equations so that to confront the general fractional solutions with the ANSYS deflection outputs, which leads to a fractional order model whenever the stress concentration is present. The deviation of the fractional order with respect to the integer order of reference allows to measure the degree of relevance of the supporting reactions on the structure behavior. Both EB and TE theories lead to similar results, whose only difference relies on the shear effects already predicted analytically in the TE modeling, which may also be estimated on its magnitude by comparison using fractional calculus. Several case studies have been conducted and led to the development of a new estimate for structural analysis.