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

비틀림 하중을 받는 두께 't'의 얇은 벽의 중공축에서 너비 'dx'의 작은 부분을 고려하십시오.
부분이 평형 상태에 있기 때문에 부분에 작용하는 유일한 힘은 부분의 끝에 가해지는 전단력입니다.
전단력을 작은 면과 그 면의 영역에 대한 종방향 전단 응력의 곱으로 표현하면 전단 흐름에 대한 표현이 도출될 수 있으며, 이는 부재 전체에서 일정하게 유지됩니다.
이제 길이가 'ds'인 중공축 벽 단면의 작은 요소를 고려하십시오. 요소에 가해지는 전단력의 크기는 전단 흐름과 요소의 길이의 곱으로 표현됩니다.
중공축의 캐비티 내에서 임의의 점 O에 대한 이 힘의 모멘트는 점 O에서 힘의 작용선까지의 수직 거리를 곱하여 얻습니다.
모멘트 방정식을 적분하면 얇은 벽의 중공 실린더 전체에 적용된 토크에 대한 표현이 제공됩니다.

벽이 얇은 중공축

Thin-Walled Hollow Shafts

비틀림 하중을 받는 벽이 얇은 중공축을 해석할 때 너비 dx를 갖는 마디를 분리하여 검사합니다. 평형 상태에도 불구하고 이 마디는 끝 부분에서 비틀림 전단력에 직면합니다. 이러한 힘은 마디의 작은 표면에 대한 세로 방향 전단 응력과 이 표면의 면적을 곱하여 정량적으로 설명되며 전단 흐름의 개념을 이끌어냅니다. 이 전단 흐름은 구조 전체에 걸쳐 일관되며 이는 비틀림 응력의 균일한 분포를 나타냅니다.

추가 분석에는 ds로 식별되는 줄기 벽의 작은 부분을 검사하는 작업이 포함됩니다. 전단 흐름에 의해 이 단면에 가해지는 힘은 전단 흐름에 단면 길이를 곱하여 계산됩니다. 줄기에 대한 비틀림 충격은 줄기의 빈 공간 내 특정 지점 O에 대해 이 힘이 생성되는 순간을 계산하여 결정됩니다. 이 계산에는 힘의 방향에 수직인 O에서 힘의 작용선까지의 거리를 힘에 곱하는 작업이 포함됩니다.

전체 줄기에 걸쳐 이 모멘트를 통합하면 중공 실린더에 영향을 미치는 총 토크에 대한 표현식이 생성됩니다. 이 통합 계산은 비틀림 힘이 벽이 얇은 중공 줄기의 구조적 거동과 무결성에 어떻게 영향을 미치는지에 대한 더 깊은 이해를 제공하여 비틀림 하중 하에서 설계 및 분석에 대한 주요 통찰력을 제공합니다.

<img alt="Equation 1" src="/files/ftp_upload/15623/15623_Equation_1.svg" style="height: 18px;">

비틀림 하중을 받는 두께 't'의 얇은 벽의 중공축에서 너비 'dx'의 작은 부분을 고려하십시오.
그 부분이 ...

Education

JoVE Core

Mechanical Engineering

Unit 1

Torsion

KEY TERMS AND CONCEPTS

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

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.

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Shear Flow in Thin-Walled Shafts

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