Name: torsion limits in circular shafts
Uploaded: 2024-05-23T12:59:41.000Z
Duration: 1 min 24 s
Channel: JoVE

在弹性限度内，实心圆轴中的最大应力与其轴线的径向距离成线性关系。
随着扭矩的增加，最大应力在屈服开始时达到饱和值。将该饱和值代入，可计算出相应最大弹性扭矩，此时变形仍完全处于弹性范围内。
用径向距离和最大应力来表示扭矩，可替代 J/c。
随着扭矩进一步增大，轴内会在半径为 ρY 的弹性核心周围形成塑性区域。
在塑性区域，应力均匀地等于 τY，而在弹性核心区域，应力随 ρ 线性变化。
进一步增加扭矩，塑性区域不断扩大，直至变形完全为塑性变形。
实心圆轴中的总扭矩可以表示为弹性区域和塑性区域扭矩的叠加。
进一步简化方程，并将 ρY &#160;趋近于零时进行代入，可确定塑性扭矩的极限值，该值对应于完全塑性变形。

圆轴 - 弹塑性材料

Circular Shafts - Elastoplastic Materials

<div>
 对实心圆轴在应力作用下的研究表明了：在弹性极限内，应力会随着与轴心之间距离的增加而直接增大。这种关系一直持续到实心圆轴达到一个应力的临界点处，当超过该临界点时便会开始屈服，这标志着从弹性形变到塑性形变之间所发生的转变。在这个关键时刻，能够确定轴在不发生永久形变的情况下所能承受的最大扭矩，这标志着其弹性行为到达了极限。
 
 随着轴上扭矩的增加，围绕内部弹性芯的塑性区域便逐渐形成，其特征是塑性区域中的应力水平是恒定的，弹性芯内的应力呈线性分布。随着扭矩的持续增加，塑性区域会不断扩大，弹性芯体会逐渐减小，直到整个轴上所发生的形变完全变为塑性。
 
 施加在轴上的总扭矩是与弹性形变和塑性形变区域相关的扭矩之和。通过进行进一步的分析和化简，并以塑性区域的扩展作为其中的重点，便能够计算出极限塑性扭矩。极限塑性扭矩是指轴在屈服于完全塑性变形、完全失去其原始形状之前所能够承受的最大扭矩。
 
</div>

在弹性限度内，实心圆轴中的最大应力与其轴线的径向距离成线性关系。  
随着扭矩的增加，……

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 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 near the outside of the shaft, while an inner elastic core remains. In the plastic area, stress stays constant, but the elastic core still follows a linear stress pattern.

As torque keeps increasing, the plastic zone grows and the elastic core shrinks. The shaft eventually becomes fully plastic across its cross section. The total torque at any stage comes from both the elastic and plastic parts together. By examining how the plastic region expands, it is possible to calculate the ultimate plastic torque, which is the maximum torque the shaft can withstand before it loses its original form completely.

Torsion Limits in 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 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

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 circular shafts react to torsion, when yielding begins, and how elastic and plastic regions combine to determine ultimate plastic torque.

Torsion Limits in Circular Shafts | 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.

mechanical engineering

Torsion Limits in Circular Shafts

Description

Transcript