Name: torsion in square and rectangular bars
Uploaded: 2024-05-23T12:59:41.000Z
Duration: 1 min 16 s
Channel: JoVE

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&#39;s cross-section in torsion. The element&#39;s face perpendicular to each axis is part of the bar&#39;s free surface, so all stresses on these faces and the cross-section&#39;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.

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.
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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.

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 ...

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Mechanical Engineering

Unit 1

Torsion

KEY TERMS AND CONCEPTS

torsion in square and rectangular bars

Torsion in square and rectangular bars shows how shape changes the way a material twists. Circular shafts keep their cross-sections nearly unchanged because they are axis-symmetric. That symmetry helps the stress spread evenly, so the shaft can resist torsion without distorting.

Square bars behave differently because they do not have the same axial symmetry. When twisted, their cross-sections distort, although the diagonals and the lines connecting the midpoints stay as exceptions. This means the twisting pattern is not uniform across the whole bar.

A close look at a small cubic element at the corner of a square bar shows another key detail. The outer sides of that corner element are part of the bar’s surface, and they are stress-free. So the stress at the surface and at the corners is zero. That also shows shearing stress does not increase in a simple linear way with distance from the axis in these bars.

The same idea applies to bars with rectangular cross-sections. In a rectangular bar, the shearing stress is greatest along the centerline of the wider face. The maximum stress and the angle of twist both depend on the bar’s dimensions, especially the widths of the wider and narrower faces. Engineers use coefficients called c1 and c2 to find these values from the ratio of the face dimensions.

Torsion in Square and Rectangular Bars

stresses in a shaft

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plastic deformation in circular shafts

Torsional Failure and Plastic Strain in Shafts

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Torsion Limits in Circular Shafts

Circular shafts under torsion show a clear change from elastic behavior to plastic deformation as torque increases. In the elastic limit, stress rises in direct proportion to the distance from the shaft’s center. The outer material carries the highest stress first, and the shaft can still return to its original shape. At a critical stress level, the shaft begins to yield. This marks the maximum torque the shaft can carry without permanent deformation. Beyond this point, a plastic region forms...

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Residual Stress After Shaft Torsion

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Torsion in square and rectangular bars shows how shape changes the way a material twists. Circular shafts keep their cross-sections nearly unchanged because they are axis-symmetric. That symmetry helps the stress spread evenly, so the shaft can resist torsion without distorting. Square bars behave differently because they do not have the same axial symmetry. When twisted, their cross-sections distort, although the diagonals and the lines connecting the midpoints stay as exceptions. This means...

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Shear Flow in Thin-Walled Shafts

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key terms and concepts

See how square and rectangular bars twist under torsion, why corners can be stress-free, and how bar dimensions affect maximum shear stress and angle of twist.

Torsion in Square and Rectangular Bars | JoVE K12

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Torsion in Square and Rectangular Bars

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