Name: shear flow in thin-walled shafts
Uploaded: 2024-05-23T12:59:41.000Z
Duration: 1 min 15 s
Channel: JoVE

ねじり荷重を受ける厚さ「t」の薄肉中空シャフトからの幅「dx」の小さな部分を考えてみましょう。
部分が平衡状態にあるため、部分に作用する力は、部分の端にかかるせん断力だけです。
せん断力を小面の縦方向のせん断応力とその面の面積の積として表すと、部材全体で一定のままのせん断流の表現を導き出すことができます。
ここで、長さ 'ds' の中空シャフト壁セクションの小さな要素を考えてみます。要素に加えられるせん断力の大きさは、要素のせん断流と長さの積として表されます。
中空シャフトのキャビティ内の任意の点Oに関するこの力のモーメントは、力に点Oから力の作用線までの垂直距離を力に掛けることによって得られます。
モーメント方程式を積分すると、薄肉の中空円筒全体に適用されるトルクの式が得られます。

薄肉中空軸（薄肉中空シャフト）

Thin-Walled Hollow Shafts

ねじり荷重を受ける薄肉中空軸を解析する場合、幅 dx のセグメントが分離されて検査されます。平衡状態にもかかわらず、このセグメントはその端部でねじりせん断力にさらされます。これらの力は、セグメントの小さい表面に沿った長手方向のせん断応力とその面積の積によって定量的に記述され、せん断流の概念につながります。このせん断流れは構造全体で一貫しており、ねじり応力が均一に分布していることを示しています。 


さらなる分析には、dsと識別される軸壁の小さな部分を調べます。せん断流によってこの部分に加えられる力は、せん断流に部分の長さを乗じることによって計算されます。軸に対するねじり衝撃は、この力が軸の中空内の特定の点 O の周りに生成するモーメントを計算することによって決定されます。この計算には、力に、O から力の方向に垂直な力の作用線までの距離を掛けることが含まれます。


このモーメントを軸全体で積分すると、中空円筒に影響を与える合計トルクの式が得られます。この積分計算により、ねじり力が薄肉中空軸の構造的挙動と完全性にどのような影響を与えるかについてのより深い理解が得られ、ねじり荷重下での設計と解析に対する重要な洞察が得られます。


<img alt="Equation 1" src="/files/ftp_upload/15623/15623_Equation_1.svg" style="text-align: center; height: 18px;">

ねじり荷重を受ける厚さ「t」の薄肉中空シャフトからの幅「dx」の小さな部分を考えてみましょう。
その部分から...

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

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 on that section is found by multiplying the shear flow by the section length.

The torque on the shaft is then found by taking the moment of that force about a point O inside the hollow. This means using the force and its perpendicular distance from O to the line of action. Integrating this moment around the full shaft gives the total torque on the hollow cylinder and helps show how thin-walled shafts respond to twisting load.

Shear Flow in Thin-Walled 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

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 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 shear flow and torsional shearing forces act in a thin-walled hollow shaft, and how integrating the moment gives the total torque.

Shear Flow in Thin-Walled Shafts | JoVE K12

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mechanical engineering

Shear Flow in Thin-Walled Shafts

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