Shear and Bending Stresses in Prismatic, Non-Circular-Profile Shafts with Epitrochoidal Contours Under Shear Force Loading

This article deals with an analytical approach to determining the stress state in trochoidal profiles under shear bending. The transverse shear stresses cannot be neglected, especially in the case of (space-saving) shafts with a low aspect ratio (l/d). In addition to bending stresses, the transverse shear stresses are a significant component of the principal stresses. The approach is based on a formulation of the elastic behavior of a non-circular-profiled shaft using complex stress functions, according to Muskhelishvili. In order to determine the shear stresses, similar to the torsion load case, a conformal mapping must be found that can completely transform the unit circle in the model plane to the profile area in the real plane. For simple and higher epiotrochoidal profiles, an exact and complete mapping was determined from the contour equation. The method was applied to such profiles as an example. Equations were derived to determine the maximum shear and bending stresses and the bending deformations. For practical applications, additional equations for the main stresses that are decisive in dimensioning were determined. Accompanying numerical investigations were carried out and compared with the analytical results, where a very good agreement was observed.