Name: elastic twist in a circular shaft
Uploaded: 2024-05-23T12:59:40.000Z
Duration: 1 min 13 s
Channel: JoVE

Considere un eje circular de longitud L y una sección transversal uniforme de radio r sometido a un par en su extremo libre.
La deformación máxima de cizallamiento en el eje es proporcional tanto al ángulo de torsión como a la distancia radial desde el eje del eje.
En el rango elástico, esta deformación cortante se puede expresar en términos del par aplicado, la distancia radial, el momento polar de inercia y el módulo de rigidez.
Al igualar estas dos ecuaciones, derivamos una expresión para el ángulo de torsión dentro del rango elástico. Esta ecuación se aplica a un eje homogéneo con una sección transversal uniforme cuando el par solo se aplica en uno de sus extremos.
Si el eje está sujeto a pares en diferentes ubicaciones, o si consta de varias partes con diferentes secciones transversales o materiales, el ángulo de torsión debe considerarse por separado para cada parte.
El ángulo total de torsión se calcula sumando todos los valores individuales de cada parte del eje o integrando a lo largo de la longitud para ejes con secciones transversales no uniformes.

Ángulo de torsión - rango elástico

Angle of Twist - Elastic Range

<div>
 Considere un eje cilíndrico con una longitud indicada por L y un radio de sección transversal consistente denominado r. Este eje sufre un torque en el extremo libre. La tensión de corte más alta dentro del eje es directamente proporcional al ángulo de torsión y la distancia radial desde el eje del eje. Cuando el eje se comporta elásticamente, esta deformación cortante se puede articular utilizando variables como el torque aplicado, la distancia radial, el momento polar de inercia y el módulo de rigidez. Al igualar estas dos ecuaciones, es posible formular una expresión para el ángulo de torsión dentro del rango elástico.
 
 <img alt="Equation 1" src="/files/ftp_upload/15615/15615_Equation_1.svg" style="text-align: center; height: 30px;">
 Esta ecuación se aplica a un eje uniforme y homogéneo, donde el torque solo se ejerce en sus extremos. Sin embargo, si el eje está expuesto a momentos de torsión en diferentes puntos o está compuesto por diferentes partes con diferentes secciones transversales o materiales, el ángulo de torsión debe evaluarse de forma distinta para cada sección. La suma de todos los valores individuales de cada segmento del eje determina el ángulo de torsión total. Alternativamente, se puede calcular integrando a lo largo de la longitud de ejes con secciones transversales no uniformes. Este enfoque presenta una comprensión integral del comportamiento de los ejes en diferentes condiciones.
 
 <img alt="Equation 2" src="/files/ftp_upload/15615/15615_Equation_2.svg" style="text-align: center; height: 46px;">
</div>

Ángulo de torsión - Rango elástico

Considere un eje circular de longitud L y una sección transversal uniforme de radio r sometido a un par en su extremo libre.
La deformación máxima de cizallamiento ...

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Unit 1

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Torsion

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KEY TERMS AND CONCEPTS

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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 shaft resists twisting. The modulus of rigidity describes how strongly the material resists shear deformation.

Setting the strain relation from twist equal to the elastic strain relation gives an expression for the angle of twist in the elastic range. This formula applies to a uniform, homogeneous shaft when torque is applied only at its ends. If torque acts at different points, or if the shaft has sections with different shapes or materials, the twist must be found for each section separately.

The total angle of twist is then the sum of the twist in each shaft segment. For a shaft with a non-uniform cross-section, the same result can be found by integrating along its length. This gives a complete way to analyze torsion in shafts under changing conditions.

Elastic Twist in a Circular Shaft

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 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

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

Learn how torque creates shear strain and angle of twist in a circular shaft. See how elastic range formulas work for uniform shafts and segmented cases.

Elastic Twist in a Circular Shaft | JoVE K12

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Elastic Twist in a Circular Shaft

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