19.11:
Torsion of Noncircular Members
During torsional loading, the cross-sections of the circular shafts remain plane and undistorted since they are axis-symmetric.
However, due to the lack of axisymmetry in a square bar, any line in its cross-section, apart from its diagonals and the lines joining the midpoints of that cross-section, will distort when the bar is twisted.
Consider a small cubic element at a corner of a square bar's cross-section in torsion. The element's face perpendicular to each axis is part of the bar's free surface, so all stresses on these faces and the cross-section's corners are zero.
As a result, here, it cannot be assumed that shearing stress varies linearly with distance from the axis.
The maximum shearing stress occurs along the center line of the wider face of the bar and can be expressed in terms of the width of its wider and narrower faces. Similarly, the angle of twist is also defined in terms of these two widths.
Here, the coefficients, c1 and c2, depend solely on the ratio of dimensions of the two faces.
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.