Name: torsion formula for circular shafts
Uploaded: 2024-05-23T12:59:40.000Z
Duration: 1 min 13 s
Channel: JoVE

Consider a case when the torque applied to the circular shaft is within Hooke&#39;s law limit, so there is no permanent deformation.
Now, recall the expression for shearing strain. Multiplying it by the modulus of rigidity and using Hooke&#39;s Law for shearing stress and strain, an expression for shearing stress in a shaft can be determined.
Recall that the sum of the moments of the elementary forces exerted on any cross-section of the shaft must be equal to the magnitude of the torque exerted on the shaft.
Substituting for shearing stress and rearranging the terms gives an expression with an integral term, which represents the polar moment of inertia of the cross-section with respect to its center.
Further rearrangements and substitutions for maximum shearing stress give the elastic torsion formula for the shearing stress in a rigid uniform circular shaft.
However, for a hollow shaft with r1 and r2 as the inner and outer radii, the polar moment of inertia is expressed as a difference in the fourth power of two radii.

circular shaft - stresses in linear range

Consider a scenario where a circular shaft is subject to torque that remains within the boundaries of Hooke's Law, avoiding any permanent deformation. So, the formula for shearing strain is revisited. This formula is multiplied by the modulus of rigidity, and then Hooke's Law for the shearing stress and strain is applied. As a result, the equation for shearing stress in a shaft can be derived.
<img alt="Shear stress formula, &#964;=Gr/R&#964;max, in material mechanics equations." src="/files/ftp_upload/15614/15614_Equation_1.svg" style="text-align: center; height: 28px;" data-altupdatedat="1762173717000" data-alt="Equation 1" />
Furthermore, it is crucial to remember that the sum of the moments of the elementary forces acting on any cross-section of the shaft must be identical to the torque applied on that shaft. An integral term emerges when the equation is adjusted to substitute for shearing stress. This term signifies the polar moment of inertia of the cross-section concerning its center. After more adjustments and substitutions for the maximum shearing stress, the elastic torsion formula can be derived for the shearing stress in a uniformly rigid circular shaft.
<img alt="Torque formula &#964;=Tr/J; equation diagram for mechanical engineering calculations." src="/files/ftp_upload/15614/15614_Equation_2.svg" style="text-align: center; height:40px;" data-altupdatedat="1762173717000" data-alt="Equation 2" />
However, the scenario differs for a hollow shaft where r1 and r2 are represented as the inner and outer radii. In this case, the polar moment of inertia is expressed as the difference in the fourth power of both radii.

Circular Shaft - Stresses in Linear Range

Consider a case when the torque applied to the circular shaft is within Hooke&#39;s law limit, so there is no permanent deformation.
Now, recall the ...

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Torsion

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torsion formula for circular shafts

Torsion in a circular shaft is analyzed when the applied torque stays within Hooke's law. In this linear range, the shaft does not undergo permanent deformation. The discussion uses shearing strain, shearing stress, and the modulus of rigidity to build the stress equation.

The derivation also requires the moments of the small forces on a cross-section to equal the applied torque. When the shearing stress expression is rewritten, an integral term appears. This term represents the polar moment of inertia, which measures how the cross-section resists twisting about its center.

After the maximum shearing stress is substituted, the elastic torsion formula is obtained for a uniformly rigid circular shaft. The same idea is then applied to a hollow shaft. For that case, the polar moment of inertia depends on the inner radius r1 and outer radius r2, and it is given by the difference of their fourth powers.

Torsion Formula for Circular Shafts

stresses in a shaft

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Torsion in a circular shaft is analyzed when the applied torque stays within Hooke's law. In this linear range, the shaft does not undergo permanent deformation. The discussion uses shearing strain, shearing stress, and the modulus of rigidity to build the stress equation. The derivation also requires the moments of the small forces on a cross-section to equal the applied torque. When the shearing stress expression is rewritten, an integral term appears. This term represents the polar moment...

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plastic deformation in circular shafts

Torsional Failure and Plastic Strain in Shafts

Plastic deformation in circular shafts happens when torsional force pushes the material past its yield strength. At that point, the shaft does not return to its original shape. Instead, it keeps a permanent strain. To measure this change, the stress pattern inside the shaft must be understood. The maximum shearing stress is calculated first. Then a shearing-stress-strain diagram can be drawn to find the maximum shearing strain. Shearing strain changes linearly with the distance from the shaft...

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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

Residual stress can remain in a circular shaft after torque causes plastic deformation. This happens in elastoplastic materials, which show both elastic and plastic behavior. The plastic change can come from high shearing stress or large strain. Residual stress is the internal stress that stays in the material after the external force is removed. In a shaft under torque, the angle of twist does not return fully to its original value. That lasting change shows that residual stress is still...

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Shear Flow in Thin-Walled Shafts

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key terms and concepts

Learn how torque, shearing stress, and polar moment of inertia lead to the elastic torsion formula for solid and hollow circular shafts in the linear range.

Torsion Formula for Circular Shafts | JoVE K12

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Torsion Formula for Circular Shafts

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