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 d…
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.
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Q1: Why do circular shafts remain undistorted during torsional loading while square bars distort?
Circular shafts maintain plane, undistorted cross-sections because they are axis-symmetric, ensuring even stress distribution. Square bars lack this symmetry, causing significant cross-sectional distortion when twisted. Only the diagonals and lines connecting midpoints remain undistorted. This fundamental difference in geometry determines how each shape responds to torsional stress.
Q2: What happens to stress at the corners of a square bar's cross-section under torsion?
At the corners of a square bar's cross-section, all stresses are zero because these faces are part of the bar's free surface. This stress-free condition at the corners means shearing stress cannot be assumed to vary linearly with distance from the axis, unlike in circular shafts where linear variation is valid.
Q3: Where does maximum shearing stress occur in a rectangular bar under torsion?
Maximum shearing stress in a rectangular bar occurs along the centerline of the wider face, not at the corners or edges. This stress concentration depends on the dimensions of both the wider and narrower faces. The location and magnitude are expressed using coefficients c1 and c2, which depend solely on the ratio of the bar's face dimensions.
Q4: How do the coefficients c1 and c2 relate to a noncircular bar's torsional behavior?
Coefficients c1 and c2 are dimensionless factors that depend exclusively on the ratio of the wider face width to the narrower face width. These coefficients determine both the maximum shearing stress and the angle of twist in rectangular bars. They allow engineers to calculate torsional response without assuming linear stress distribution.
Q5: What lines in a square bar's cross-section remain undistorted during torsion?
During torsion, only the diagonals and lines connecting the midpoints of opposite faces remain undistorted in a square bar's cross-section. All other lines in the cross-section experience distortion due to the bar's lack of axial symmetry. This selective distortion pattern distinguishes noncircular members from symmetric circular shafts.
Q6: Why can't shearing stress be assumed to vary linearly with distance from the axis in square bars?
Shearing stress cannot be assumed linear in square bars because the corners and free surfaces experience zero stress. In circular shafts, axisymmetry ensures uniform stress distribution allowing linear variation. The asymmetric geometry of square bars creates a nonlinear stress gradient that requires coefficients c1 and c2 for accurate calculation.
Q7: How does the angle of twist in a rectangular bar depend on its dimensions?
The angle of twist in a rectangular bar is defined in terms of both the wider and narrower face widths, similar to how maximum shearing stress is determined. The relationship involves coefficients c1 and c2, which are calculated from the ratio of the bar's face dimensions. This dimensional dependence reflects how geometry directly influences torsional deformation.