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

弾性限界内では、ソリッド円形シャフトの最大応力は、その軸からの半径方向の距離に比例して変化します。
トルクが増加すると、最大応力は降伏の開始時に飽和値に達します。この飽和値を代入すると、変形が完全に弾性のままである対応する最大弾性トルクを計算できます。
J/cを代入すると、トルクを半径方向の距離と最大応力で表します。
トルクがさらに増加すると、半径ρYの弾性コアの周囲にシャフトに塑性領域が発達します。
塑性領域では、応力は一様にτYに等しくなりますが、弾性コアでは、応力はρとともに直線的に変化します。
トルクをさらに増加させると、変形が完全にプラスチックになるまでプラスチック領域が拡大します。
中実の円形シャフトの合計トルクは、弾性領域と塑性領域のトルクの重ね合わせとして表すことができます。
方程式をさらに単純化し、ゼロに近づくにつれてρYを代入すると、完全に塑性変形に対応する塑性トルクの限界値を決定できます。

円形シャフト - 弾塑性材料

Circular Shafts - Elastoplastic Materials

シャフトのトルクが増加すると、内側の弾性コアを取り囲む塑性領域が発達します。これは、塑性領域内の一定の応力レベルと弾性コア内の線形応力分布を特徴とします。 トルクが継続的に増加すると、塑性ゾーンが広がり弾性コアが減少し、シャフト全体の変形が完全に塑性となるまで続きます。 

応力がかかった中実シャフトの研究では、弾性限界内では、シャフトの中心からの距離に応じて応力が直接増加することが示されています。 この関係は、シャフトが応力の臨界点に達するまで維持され、それを超えると降伏が始まり、弾性変形から塑性変形への移行が始まります。 この重要な時点で、シャフトが永久変形することなく耐えることができる最大トルクが決定され、シャフトの弾性挙動の限界が決まります。



シャフトにかかる合計トルクは、弾性変形領域と塑性変形領域に関連するトルクの合計です。 塑性領域の拡大に焦点を当ててさらに分析および単純化することにより、最終的な塑性トルクを計算できます。 極限塑性トルクとは、シャフトが完全に塑性変形して元の形状を完全に失う前にシャフトが処理できる最大トルクです。</p

弾性限界内では、ソリッド円形シャフトの最大応力は、その軸からの半径方向の距離に比例して変化します。
トルクが増加すると、...

Education

JoVE Core

Mechanical Engineering

機械工学

Unit 1

ユニット1

Torsion

ねじれ

KEY TERMS AND CONCEPTS

主要用語と概念

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

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mechanical engineering

Torsion Limits in Circular Shafts

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