19.11 : Torsion of Noncircular Members

Circular shafts undergoing torsional stress maintain their cross-sectional integrity due to their axisymmetric nature. This symmetry ensures an even distribution of stress, allowing the shaft to withstand torsion without distorting. In contrast, square bars, lacking this axial symmetry, experience significant distortion across their cross-sections when subjected to torsion, with the exception of along their diagonals and at lines connecting midpoints. A detailed examination of a cubic element at the corner of a square bar's cross-section reveals that its outward-facing sides, which are part of the bar's exterior, are stress-free. This indicates that the stress on these surfaces and at the corners of the cross-section is null, leading to the conclusion that shearing stress does not distribute linearly with the distance from the axis in such bars.

This can be generalized to bars with rectangular cross-sections. In this case, the shearing stress reaches its peak along the centerline of the bar's broader face. This maximum stress, along with the angle of twist, depends on the dimensions of the bar, notably the widths of its wider and narrower faces. Determining these parameters involves specific coefficients, referred to as c₁ and c₂, which are calculated based on the ratio of the bar's face dimensions.

This calculation highlights the relationship between the bar's geometric properties and its response to torsional loading, underscoring the importance of considering the shape and dimensions of materials when evaluating their behavior under torsion.