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

탄성 한계 내에서 솔리드 원형 샤프트의 최대 응력은 축으로부터의 반경 거리에 따라 선형으로 변합니다.
토크가 증가함에 따라 최대 응력은 항복 시작 시 포화 값에 도달합니다. 이 포화 값을 대체하면 변형이 완전히 탄성을 유지하는 해당 최대 탄성 토크를 계산할 수 있습니다.
J/c를 대체하면 반경 방향 거리와 최대 응력으로 토크를 표현합니다.
토크가 더 증가함에 따라 반지름 ρY의 탄성 코어 주위의 샤프트에서 플라스틱 영역이 발생합니다.
플라스틱 영역에서 응력은 τY와 균일하게 같고, 탄성 코어에서 응력은 ρ에 따라 선형으로 변합니다.
토크를 더 증가시키면 변형이 완전히 소성이 될 때까지 플라스틱 영역이 확장됩니다.
솔리드 원형 샤프트의 총 토크는 탄성 및 소성 영역에서의 토크의 중첩으로 표현될 수 있습니다.
방정식을 더 단순화하고 0에 가까워짐에 따라 ρY 를 대체하면 완전히 소성 변형에 해당하는 소성 토크의 한계 값을 결정할 수 있습니다.

원형 샤프트 - 탄소성 재료

Circular Shafts - Elastoplastic Materials

응력을 받는 솔리드 원형 샤프트에 대한 연구에 따르면 탄성 한계 내에서 응력은 샤프트 중심에서 멀어질수록 직접적으로 증가합니다. 이 관계는 샤프트가 임계 응력 지점에 도달할 때까지 유지되며, 그 지점을 넘어서면 항복이 시작되어 탄성 변형에서 소성 변형으로 전환됩니다. 이 중요한 시점에서 샤프트가 영구 변형 없이 견딜 수 있는 최대 토크가 결정되며 이는 샤프트의 탄성 동작의 한계를 나타냅니다.

샤프트의 토크가 증가함에 따라 내부 탄성 코어를 둘러싸는 소성 영역이 발달하며, 소성 영역의 일정한 응력 수준과 탄성 코어 내의 선형 응력 분포를 특징으로 합니다. 토크가 지속적으로 증가하면 소성 영역이 확장되어 샤프트 전체의 변형이 완전히 소성이 될 때까지 탄성 코어가 감소합니다.

샤프트에 가해지는 총 토크는 탄성 및 소성 변형 영역과 관련된 토크의 합입니다. 소성 영역의 확장에 초점을 맞춘 추가 분석 및 단순화를 통해 최종 소성 토크를 계산할 수 있습니다. 궁극적인 소성 토크는 샤프트가 완전히 소성 변형되어 원래 형태를 완전히 잃어버리기 전에 샤프트가 처리할 수 있는 최대 토크입니다.

탄성 한계 내에서 솔리드 원형 샤프트의 최대 응력은 축으로부터의 반경 거리에 따라 선형으로 변합니다.
토크가 증가함에 따라 ...

Education

교육

JoVE Core

Mechanical Engineering

기계공학

Unit 1

제1단원

Torsion

비틀림

KEY TERMS AND CONCEPTS

주요 용어 및 개념

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

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

Torsion Limits in Circular Shafts

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