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

考虑一种情况，当施加在圆轴上的扭矩处于胡克定律极限范围内时，不会发生永久变形。
现在，回顾一下切应变的表达式。将其乘以剪切模量，并结合剪切应力与应变的胡克定律，即可确定轴中剪切应力的表达式。
请记住，作用在轴的任意横截面上的初等力的力矩之和必须等于作用在该轴上的扭矩的大小。
代入剪切应力并重新整理各项后，可得到一个包含积分项的表达式，该积分项表示横截面相对于其中心的极惯性矩。
经过进一步的重新排列和代入，得到刚性均匀圆轴切应力的弹性扭转公式，用于计算最大切应力。
然而，对于内径和外径分别为 r1 和 r2 的空心轴，其极惯性矩表示为两个半径的四次方之差。

圆轴 - 线性范围内的应力

Circular Shaft - Stresses in Linear Range

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 假设有这样一种情况：圆轴所承受的扭矩能够保持在胡克定律的范围之内，以此来避免圆轴出现任何永久性的变形。因此，需要对剪切应变的公式进行重新整理。将该公式与刚性模量进行相乘，然后将其应用到剪切应力和应变的胡克定律中。最终便可以推导出圆轴中剪切应力的方程。
 
 <img alt="Equation 1" src="/files/ftp_upload/15614/15614_Equation_1.svg" style="text-align: center; height: 28px;">
 此外，需要记住作用在圆轴中任何横截面上基本力的力矩总和必须与施加在该轴上的扭矩是相同的。当调整方程来代替剪应力时，便会出现一个积分项。该术语表示了横截面相对于轴中心的极惯性矩。对最大剪应力进行多次调整和代入后，便能够推导出均匀刚性圆轴中剪应力的弹性扭转公式。
 
 <img alt="Equation 2" src="/files/ftp_upload/15614/15614_Equation_2.svg" style="text-align: center; height:40px;">
 然而对于空心轴来说，情况便会有所不同，其中的 r_1 和 r_2 能够表示为内半径和外半径。在这种情况下，极惯性矩也可以表示为两个半径的四次方之差。
 
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考虑一种情况：当施加在圆轴上的扭矩处于胡克定律极限范围内时，不会产生永久变形。  
现在，回顾……

Education

教育

JoVE Core

Mechanical Engineering

机械工程

Unit 1

第一单元

Torsion

扭转

KEY TERMS AND CONCEPTS

关键术语与概念

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

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 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

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, 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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mechanical engineering

Torsion Formula for Circular Shafts

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