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

均一な断面を持つ円形シャフトに適用可能な弾性ねじり式を思い出してください。ここでは、シャフトの端部には、シャフトにしっかりと取り付けられた剛性プレートによって負荷がかかっていると仮定します。
ただし、トルクは、キー溝に取り付けられたキーで接続されたフランジカップリングまたはギアを介してシャフトに加えられることがよくあります。
これにより、トルク印加点付近の応力分布が、ねじり式によって予測される分布から逸脱して変化します。
直径の急激な変化は、特にジョイントの近くで応力集中が不均一に分布する可能性もあります。
これらのストレスは、フィレットを実装することで軽減できます。フィレットでのせん断応力の最大値は、応力集中係数で表すことができます。
応力集中係数は、シャフトの直径比とフィレットサイズに依存し、事前に計算して保存して実用化することができます。
この解析方法は、最大応力値が材料の弾性限界を超えない限り有効です。塑性変形が発生すると、最大応力値が低くなります。

円形シャフトの応力集中

一貫した断面を持つ円形シャフトに適用される弾性ねじり公式を考えてみましょう。 この式は、シャフトの端にしっかりと取り付けられた剛性のプレートによって荷重がかかることを前提としています。 しかし、多くの場合、トルクは、キー溝に挿入されたキーによって接続されたフランジカップリングやギアなどの機構を介してシャフトに加えられます。 この適用方法では、トルクの適用点付近の応力分布が変更され、ねじり公式によって予測される分布から逸脱します。 さらに、直径の突然の変化により、特に接合領域の周囲で応力集中が不規則に分布する可能性があります。

これらの応力はフィレットを組み込むことで軽減できます。 応力集中係数は、フィレットにおけるせん断応力の最高値を表すことができます。 この係数は、シャフトの直径とフィレットのサイズの比率に依存するため、事前に計算し、将来の参照や実用化のために保存できます。 この解析方法は、最大応力値が材料の弾性限界内に収まる場合に有効です。 塑性変形が発生すると、ピーク応力値の低下につながります。これらの要因と応力分布への影響を理解することが、弾性ねじり公式を正確かつ実際に適用するために重要であることを強調します。

均一な断面を持つ円形シャフトに適用可能な弾性ねじり式を思い出してください。ここでは、シャフトが彼らの...

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

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

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Why Shafts Develop Local Stress Peaks

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