Name: torque and shear in a twisted shaft
Uploaded: 2024-05-23T12:59:40.000Z
Duration: 1 min 18 s
Channel: JoVE

Un arbre PQ est tordu lorsqu’il est soumis à des couples égaux et opposés de chaque côté.
Considérons une section perpendiculaire à l'axe de l'arbre par un point R arbitraire. Le schéma de corps libre de la portion QR montre les forces de cisaillement exercées par la portion PR sur la partie QR lorsque l’arbre se tord.
En appliquant les équations d’équilibre à la portion QR, on peut montrer que les forces de cisaillement à l’intérieur de la section sont liées au couple interne. Ici, r représente la distance perpendiculaire entre l'axe de l'arbre et la force de cisaillement.
Considérons maintenant un élément de petite surface de l’arbre où la force de cisaillement peut être exprimée comme le produit de la contrainte de cisaillement et de l’élément de surface.
En substituant cette relation, l’expression du couple est obtenue en termes de contrainte de cisaillement.
Cette relation doit être satisfaite par les contraintes de cisaillement dans n’importe quelle section transversale de l’arbre. Cependant, il ne fournit pas d’informations sur la répartition de ces contraintes dans la section transversale.
La répartition des contraintes de cisaillement dans un arbre élastique est indéterminée par la seule statique et nécessite une analyse de déformation.

contraintes dans un puits

Stresses in a Shaft

L'arbre PQ est soumis à une force de torsion lorsque des couples égaux et opposés sont appliqués de part et d'autre. Une section qui coupe perpendiculairement à l'axe de l'arbre en tout point arbitraire R est examinée pour comprendre cela. Lorsque le diagramme du corps libre du segment QR est analysé, il révèle les forces de cisaillement exercées par la partie PR sur le segment QR lorsque la tige subit une torsion.

L'application de conditions d'équilibre au segment QR établit que les forces de cisaillement internes à l'intérieur de la section sont directement corrélées au couple interne. Ici, « r » signifie la distance perpendiculaire entre l’axe de l’arbre et la force de cisaillement. Ensuite, un élément de petite surface de l'arbre est pris en compte. La force de cisaillement peut être exprimée comme la multiplication de la contrainte de cisaillement et de l'élément de surface. En remplaçant cette relation, une expression du couple en termes de contrainte de cisaillement est dérivée.

<img alt="Equation 1" src="/files/ftp_upload/15612/15612_Equation_1.svg" style="text-align: center; height: 28px;">

Cette relation dérivée doit être vraie pour les contraintes de cisaillement dans n'importe quelle section transversale de l'arbre. Cependant, cela ne donne pas d’informations sur la répartition de ces contraintes sur la section transversale. Enfin, il est important de noter que la répartition des contraintes de cisaillement dans un arbre élastique ne peut être déterminée uniquement par la statique. Cela nécessite une analyse de déformation pour une détermination précise.

Contraintes dans un puits

Un arbre PQ est tordu lorsqu’il est soumis à des couples égaux et opposés de chaque côté.
Considérons une section perpendiculaire à l'axe de l'arbre par un ...

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torque and shear in a twisted shaft

A shaft under torque develops internal shear forces as it twists. In this setup, equal and opposite torques act on the shaft PQ. To study the effect, a section is cut perpendicular to the shaft axis at an arbitrary point R.

The free-body diagram of the QR segment shows the shearing forces applied by the PR portion. These forces act as the shaft resists twisting. Using equilibrium on the QR segment links the internal shear forces in the cut section to the internal torque. Here, r is the perpendicular distance from the shaft axis to the shearing force.

A small area element on the cross-section is then considered. The shearing force on that element can be written as shearing stress times area. Substituting this relation gives an expression for torque in terms of shearing stress. This relation must hold for the shearing stresses in any shaft cross-section.

Even so, the relation does not show how the stress is spread across the cross-section. The distribution of shearing stress in an elastic shaft cannot be found from statics alone. It requires deformation analysis for an accurate result.

Torque and Shear in a Twisted Shaft

stresses in a shaft

A shaft under torque develops internal shear forces as it twists. In this setup, equal and opposite torques act on the shaft PQ. To study the effect, a section is cut perpendicular to the shaft axis at an arbitrary point R. The free-body diagram of the QR segment shows the shearing forces applied by the PR portion. These forces act as the shaft resists twisting. Using equilibrium on the QR segment links the internal shear forces in the cut section to the internal torque. Here, r is the...

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Elastic twist in a circular shaft describes how a cylindrical shaft rotates when torque is applied. The shaft has a length L and a uniform radius r. A torque at the free end creates shearing strain, which is greatest at points farther from the shaft axis. When the shaft stays within the elastic range, the shearing strain can be written in terms of the applied torque, radial distance, polar moment of inertia, and modulus of rigidity. The polar moment of inertia is a geometric measure of how the...

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Sizing a Transmission Shaft by Torque

A transmission shaft is sized from the power it must carry and the speed it rotates at. These two design limits guide the choice of shaft material and cross-sectional size. The goal is to keep the maximum shearing stress within the elastic limit while still transmitting the needed power. Power is directly linked to torque, which is the twisting force applied to the shaft. By rearranging the power relation, the torque can be found from the required power and the shaft speed. That torque value...

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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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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...

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Shear Flow in Thin-Walled Shafts

Shear flow in a thin-walled hollow shaft describes how torsional force is carried through the wall. When a small segment of width dx is isolated, it is still in equilibrium, but it experiences torsional shearing forces at its ends. These forces come from the longitudinal shearing stress on the segment's minor surface multiplied by that surface area. The result is a uniform shear flow through the structure. To study the wall more closely, a small section with length ds is considered. The force...

key terms and concepts

Explore how equal and opposite torques create internal shear forces in a shaft, and how torque relates to shearing stress in a cut section.

Torque and Shear in a Twisted Shaft | JoVE K12

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Torque and Shear in a Twisted Shaft

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