An Analytical Approach to the Influence of Rotating Shear Stress on the Stress State in Combined-Load H-Profile Shafts: A Contribution to DIN 3689—Part 2

This article presents analytical approaches for determining the stress state in hypotrochoidal profiles (H-profiles) under torsional, bending, and shear loading. The focus lies on shear loading. Using conformal mapping, a formulation of elasticity theory is first adapted to H-profile cross-sections. Closed-form solutions for the respective stress components are then derived by solving a reformulated surface integral. Suitable conformal mappings for hypotrochoidal contours are obtained through a successive method applied to their parametric description. These mappings are essential for the elasticity-theoretical formulation used to determine the stress state in the profile bar. Building on this, the transverse shear stresses for H-profile cross-sections are determined for the first time. The influence of shear stresses on the overall stress state is discussed in detail. These stresses generally act in a rotational manner and superimpose on the torsional stresses. This effect proves more pronounced here than in circular cross-sections. Accompanying finite element analyses (FEA), carried out for several examples, showed very good agreement with the analytical solutions. From the resulting maximum stress values and stress gradients, form and notch factors are calculated. The proposed procedure also applies to combined loading cases. It is intended for use in the new standard covering hypotrochoidal profile contours.