Name: torque and shear in a twisted shaft
Uploaded: 2024-05-23T12:59:40.000Z
Duration: 1 min 18 s
Channel: JoVE

シャフトPQは、両側で等しく反対のトルクを受けるとねじれます。
シャフトの軸に垂直なセクションを任意の点Rで考える。部分QRのフリーボディ図は、シャフトがねじれるときに部分PRがQRに及ぼすせん断力を示しています。
平衡方程式を部分QRに適用すると、セクション内のせん断力が内部トルクに関連していることを示すことができます。ここで、r はシャフトの軸からせん断力までの垂直距離を表します。
次に、せん断力をせん断応力と面積要素の積として表すことができるシャフトの小さな面積要素について考えます。
この関係を代入することにより、トルクの表現はせん断応力の観点から得られます。
この関係は、シャフトの任意の断面のせん断応力によって満たされる必要があります。ただし、断面におけるこれらの応力の分布に関する情報は提供されません。
弾性シャフトのせん断応力の分布は、静力学だけでは不確定であり、変形解析が必要です。

シャフト内の応力

Stresses in a Shaft

シャフト PQ は、どちらの側にも等しい逆向きのトルクがかかると、ねじりの力を受けます。 これを理解するために、任意の点 R でシャフトの軸に垂直に切断した断面を調べます。 QR セグメントの自由体図を分析すると、シャフトがねじれるときに PR 部分によって QR セグメントに加えられるせん断力が明らかになります。

QR セグメントに平衡条件を適用すると、セクション内の内部せん断力が内部トルクと直接相関することが確立されます。 ここで、「r」はシャフトの軸からせん断力までの垂直距離を意味します。 次にシャフトの小面積要素を考慮します。 せん断力は、せん断応力と面積要素の積で表すことができます。 この関係を代入すると、せん断応力によるトルクの式が導出されます。

<img alt="Equation 1" src="/files/ftp_upload/15612/15612_Equation_1.svg" style="height: 28px;">

この導出された関係は、あらゆるシャフト断面のせん断応力に当てはまります。 ただし、断面全体にわたるこれらの応力の分布についての洞察は得られません。 最後に、弾性シャフトのせん断応力の分布は静力学だけでは決定できないことに注意することが重要です。 正確に判断するには変形解析が必要です。

シャフトPQは、両側で等しく反対のトルクを受けるとねじれます。
シャフトの軸に垂直なセクションを考えてみましょう...

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

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 perpendicular distance from the shaft axis to the shearing force.

A small area element on the cross-section is then considered. The shearing force on that element can be written as shearing stress times area. Substituting this relation gives an expression for torque in terms of shearing stress. This relation must hold for the shearing stresses in any shaft cross-section.

Even so, the relation does not show how the stress is spread across the cross-section. The distribution of shearing stress in an elastic shaft cannot be found from statics alone. It requires deformation analysis for an accurate result.

Torque and Shear in a Twisted Shaft

stresses in a 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...

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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...

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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...

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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...

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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...

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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...

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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

Explore how equal and opposite torques create internal shear forces in a shaft, and how torque relates to shearing stress in a cut section.

Torque and Shear in a Twisted Shaft | JoVE K12

JoVE Core 機械工学は、平衡、力、トラス解析、構造力学などの重要なトピックをカバーする簡潔でわかりやすいアニメーション ビデオを提供します。実際のアプリケーションを探索し、科学者主導の実験を通じてこれらのコンセプトが実際に動作しているのを目撃してください。

mechanical engineering

Torque and Shear in a Twisted Shaft

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