On the inadequacy of a stepped-beam approach in predicting shear stresses in tapered slender solids

The stress state in slender elastic cylinders with a straight axis and tapered cross-sections is investigated. Compared to the de Saint-Venant’s cylinder, the continuous variation in the dimensions of the cross-sections along the cylinder’s axis results in additional, non-trivial shear stress distributions within the cross-sections. This paper analytically investigates the dependence of these stresses on taper, a topic of significant practical interest for the design of tapered structural elements commonly employed in various engineering applications, such as components of wind turbines, aircraft, and bridges. The analytical study in this paper is based on a set of partial differential equations and boundary conditions, derived in a recent work, that govern the stress state in three-dimensional tapered cylinders undergoing cross-sectional warping, in- and out-of-plane. A new analytical solution is derived for rectangular cross-sectioned tapered cylinders, resembling the shear webs of large wind turbine blades, with external loads applied at the ends. By examining this paradigmatic case, the inadequacy of approaches based on stepped-beam models in predicting shear stresses in tapered slender solids is demonstrated analytically. Numerical examples, including comparisons to results from the literature and benchmark solutions based on finite element methods, are also provided and corroborate the analytical findings of this study.