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

Lors d’une charge de torsion, les sections transversales des arbres circulaires restent planes et non déformées car elles sont symétriques sur le plan axial.
Cependant, en raison de l’absence d’axisymétrie dans une barre carrée, toute ligne de sa section transversale, à l’exception de ses diagonales et des lignes joignant les points médians de cette section transversale, se déformera lorsque la barre est tordue.
Considérons un petit élément cubique à un coin de la section transversale d'une barre carrée en torsion. La face de l'élément perpendiculaire à chaque axe fait partie de la surface libre de la barre, de sorte que toutes les contraintes sur ces faces et les coins de la section sont nulles. 
Par conséquent, on ne peut pas supposer ici que la contrainte de cisaillement varie linéairement avec la distance par rapport à l’axe.
La contrainte de cisaillement maximale se produit le long de la ligne médiane de la face la plus large de la barre et peut être exprimée en termes de largeur de ses faces plus larges et plus étroites. De même, l’angle de torsion est également défini en fonction de ces deux largeurs.
Ici, les coefficients, c1 et c2, dépendent uniquement du rapport des dimensions des deux faces.

torsion des membres non circulaires

Les arbres circulaires soumis à des contraintes de torsion conservent leur intégrité transversale en raison de leur nature axisymétrique. Cette symétrie assure une répartition uniforme des contraintes, permettant à l'arbre de résister à la torsion sans se déformer. En revanche, les barres carrées, dépourvues de cette symétrie axiale, subissent une distorsion significative sur toute leur section-transversale lorsqu'elles sont soumises à une torsion, à l'exception de leurs diagonales et des lignes reliant les milieux. Un examen détaillé d'un élément cubique au coin de la section transversale d'une barre carrée révèle que ses côtés tournés vers l'extérieur, qui font partie de l'extérieur de la barre, sont sans contrainte. Cela indique que la contrainte sur ces surfaces et aux coins de la section-transversale est nulle, ce qui conduit à la conclusion que la contrainte de cisaillement ne se répartit pas linéairement avec la distance à l'axe dans de telles barres.

Ceci peut être généralisé aux barres à sections rectangulaires. Dans ce cas, la contrainte de cisaillement atteint son maximum le long de la ligne médiane de la face la plus large de la barre. Cette contrainte maximale, ainsi que l'angle de torsion, dépendent des dimensions de la barre, notamment des largeurs de ses faces plus larges et plus étroites. La détermination de ces paramètres fait appel à des coefficients spécifiques, appelés c_1 et c_2, qui sont calculés en fonction du rapport des dimensions de la face de la barre.

<img alt="Equation 1" src="/files/ftp_upload/15622/15622_Equation_1.svg" style="text-align: center; height: 54px;">

Ce calcul met en évidence la relation entre les propriétés géométriques de la barre et sa réponse aux charges de torsion, soulignant l'importance de prendre en compte la forme et les dimensions des matériaux lors de l'évaluation de leur comportement en torsion.

Torsion des membres non circulaires

Lors d’une charge de torsion, les sections transversales des arbres circulaires restent planes et non déformées car elles sont symétriques sur le plan axial.
Cependant, en raison du manque ...

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Génie mécanique

Unit 1

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Torsion

KEY TERMS AND CONCEPTS

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Torsion of Noncircular Members

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

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...

deformation in a circular shaft

Shear Strain in a Circular Shaft

Shear strain in a circular shaft describes how the shaft’s surface changes shape under torsion. A circular shaft keeps each cross-section flat and undistorted as it twists. Each section behaves like a rigid slab that rotates in place. To study the strain inside the shaft, consider a cylindrical section with length L and radius R. One end of this section is fixed. The radius at a point inside the section is labeled r. Before the load is applied, a small square element lies on the surface of...

circular shaft - stresses in linear range

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...

angle of twist - elastic range

Elastic Twist in a Circular Shaft

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...

angle of twist: problem solving

Shaft Torsion Between Pulleys B and C

Shaft torsion between pulleys B and C is found by analyzing how torque twists an aluminum shaft. An electric motor applies 700 N·m of torque and produces stable rotation. The shaft carries pulleys B and C, which experience torques of 300 N·m and 400 N·m. The calculation starts with a section cut between pulleys B and C. A free-body diagram is used to study the cut cross-section. Because pulley B has an anticlockwise torque, the torque at the cut must balance it in the opposite direction. The...

design of transmission shafts

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...

stress concentrations in circular shafts

Why Shafts Develop Local Stress Peaks

Circular shafts can develop local stress peaks when torque is applied through real machine parts. The elastic torsion formula works best for a circular shaft with a uniform cross-section and rigid plates fixed to its ends. In practice, torque is often passed through flange couplings or gears, and these parts use keys in keyways to connect to the shaft. That loading method changes the stress pattern near the point where torque enters the shaft. The stress distribution no longer matches the...

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...

circular shafts - elastoplastic materials

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...

residual stresses in circular shafts

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...

torsion of noncircular members

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...

thin-walled hollow shafts

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

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

JoVE Core Mechanical Engineering offers concise, easy-to-understand animated videos covering key topics such as equilibrium, forces, truss analysis, and structural mechanics. Explore their real-world applications, and witness these concepts in action through scientist-led experiments.

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Torsion in Square and Rectangular Bars

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