Name: why shafts develop local stress peaks
Uploaded: 2024-05-23T12:59:41.000Z
Duration: 1 min 18 s
Channel: JoVE

균일한 단면을 가진 원형 샤프트에 적용할 수 있는 탄성 비틀림 공식을 상기하십시오. 여기서, 샤프트는 단단히 부착 된 단단한 플레이트에 의해 끝에서 하중을 받는다고 가정합니다.
그러나 토크는 종종 플랜지 커플 링 또는 키 홈에 장착 된 키로 연결된 기어를 통해 샤프트에 적용됩니다.
이렇게 하면 토크 적용 점에 가까운 응력 분포가 변경되어 비틀림 공식에 의해 투영된 분포에서 벗어납니다.
직경의 급격한 변화는 특히 접합부 근처에서 응력 집중의 불균일한 분포로 이어질 수도 있습니다.
이러한 응력은 필렛을 구현하여 줄일 수 있습니다. 필렛에서 전단 응력의 최대값은 응력 집중 계수로 표현할 수 있습니다.
샤프트 직경 비율과 필렛 크기에 따라 달라지는 응력 집중 계수는 실제 사용을 위해 미리 계산하고 저장할 수 있습니다.
이 해석 방법은 최대 응력 값이 재료의 탄성 한계를 초과하지 않는 한 유효합니다. 소성 변형이 발생하면 최대 응력 값이 낮아집니다.

원형 샤프트의 응력 집중

단면이 일정한 원형 샤프트에 적용되는 탄성 비틀림 공식을 생각해 보세요. 이 공식은 샤프트의 끝 부분에 단단한 플레이트가 단단히 부착되어 있다고 가정합니다. 그러나 많은 경우 키 홈에 삽입된 키로 연결되는 플랜지 커플링이나 기어와 같은 메커니즘을 통해 샤프트에 토크가 적용됩니다. 이 적용 방법은 토크 적용 지점 근처의 응력 분포를 수정하여 비틀림 공식에 의해 예측된 분포에서 벗어나게 합니다. 또한 직경의 급격한 변화로 인해 특히 접합 부위 주변에 응력 집중이 불규칙하게 분포될 수도 있습니다.

이러한 응력은 필렛을 통합하여 완화할 수 있습니다. 응력 집중 계수는 모깎기에서 전단 응력의 가장 높은 값을 나타낼 수 있습니다. 샤프트 직경과 필렛 크기의 비율에 따라 달라지는 이 계수는 미리 계산하여 향후 참조 및 실제 적용을 위해 저장할 수 있습니다. 이 분석 방법은 최대 응력 값이 재료의 탄성 한계 내에 유지되는 경우 계속 효과적입니다. 소성 변형이 발생하면 최대 응력 값이 낮아집니다. 이러한 요인과 응력 분포에 미치는 영향을 이해하는 것이 탄성 비틀림 공식을 정확하고 실용적으로 적용하는 데 중요하다는 점을 강조합니다.

균일한 단면을 가진 원형 샤프트에 적용할 수 있는 탄성 비틀림 공식을 상기하십시오. 여기서, 샤프트가 그들로부터 하중을 받는다고 가정합니다 ...

Education

교육

JoVE Core

Mechanical Engineering

기계공학

Unit 1

제1단원

Torsion

비틀림

KEY TERMS AND CONCEPTS

주요 용어 및 개념

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 simple pattern predicted by the torsion formula. Sudden changes in shaft diameter can also create stress concentrations, especially near joints.

A fillet, which is a rounded transition, can reduce these concentrated stresses. The highest shearing stress at the fillet is described by a stress concentration factor. Engineers can calculate this factor in advance from the ratios of the shaft diameter and the fillet size, then save it for later use.

This approach is valid as long as the maximum stress stays within the material’s elastic limit. If plastic deformation occurs, the peak stress becomes lower than the elastic estimate. Understanding these effects helps make the elastic torsion formula more useful in practical designs.

Why Shafts Develop Local Stress Peaks

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

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 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 torque entry points, keyways, diameter changes, and fillets affect stress in circular shafts, and how stress concentration factors help predict peak shear stress.

Why Shafts Develop Local Stress Peaks | 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

Why Shafts Develop Local Stress Peaks

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