Name: torque and shear in a twisted shaft
Uploaded: 2024-05-23T12:59:40.000Z
Duration: 1 min 18 s
Channel: JoVE

Wał PQ jest skręcany, gdy jest poddawany równym i przeciwstawnym momentom obrotowym po obu stronach.
Rozważmy przekrój prostopadły do osi wału przechodzący przez dowolny punkt R. Wykres swobodnego korpusu części QR pokazuje siły ścinające wywierane przez część PR na QR podczas skręcania wału.
Stosując równania równowagi do porcji QR, można wykazać, że siły ścinające w przekroju są związane z wewnętrznym momentem obrotowym. Tutaj r oznacza prostopadłą odległość od osi wału do siły ścinającej.
Rozważmy teraz element wału o małej powierzchni, w którym siła ścinająca może być wyrażona jako iloczyn naprężenia ścinającego i elementu powierzchniowego.
Podstawiając tę zależność, otrzymuje się wyrażenie na moment obrotowy w postaci naprężenia ścinającego.
Zależność ta musi być spełniona przez naprężenia ścinające w dowolnym przekroju wału. Nie dostarcza jednak informacji o rozkładzie tych naprężeń w przekroju.
Rozkład naprężeń ścinających w wale sprężystym jest nieokreślony wyłącznie przez statykę i wymaga analizy odkształceń.

naprężenia w wale

Stresses in a Shaft

Wał PQ jest poddawany działaniu siły skręcającej, gdy po obu stronach przyłożone są równe i przeciwne skierowane momenty obrotowe. Aby to zrozumieć, bada się przekrój przecinający prostopadle do osi wału w dowolnym punkcie R. Kiedy analizuje się diagram swobodnego ciała segmentu QR, ujawnia on siły ścinające wywierane przez część PR na segment QR, gdy wał ulega skręceniu.

Zastosowanie warunków równowagi do segmentu QR pozwala ustalić, że wewnętrzne siły ścinające w przekroju są bezpośrednio skorelowane z wewnętrznym momentem obrotowym. Tutaj „r” oznacza prostopadłą odległość od osi wału do siły ścinającej. Następnie uwzględniany jest element o małej powierzchni wału. Siłę ścinającą można wyrazić jako iloczyn naprężenia ścinającego i elementu powierzchniowego. Podstawiając tę zależność, wyprowadza się wyrażenie na moment obrotowy w postaci naprężenia ścinającego.

<img alt="Equation 1" src="/files/ftp_upload/15612/15612_Equation_1.svg" style="height: 28px;">

Ta wyprowadzona zależność musi być prawdziwa dla naprężeń ścinających w dowolnym przekroju wału. Nie zapewnia jednak wglądu w rozkład tych naprężeń w przekroju. Na koniec należy zauważyć, że rozkładu naprężeń ścinających w elastycznym wale nie można określić wyłącznie na podstawie statyki. Dokładne określenie wymaga analizy odkształceń.

Naprężenia w wale

Wał PQ jest skręcany, gdy jest poddawany równym i przeciwstawnym momentom obrotowym po obu stronach.
Rozważmy przekrój prostopadły do osi wału przez ...

Education

JoVE Core

Mechanical Engineering

Inżynieria mechaniczna

Unit 1

Jednostka 1

Torsion

Skręcenie

KEY TERMS AND CONCEPTS

KLUCZOWE TERMINY I POJĘCIA

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 perpendicular distance from the shaft axis to the shearing force.

A small area element on the cross-section is then considered. The shearing force on that element can be written as shearing stress times area. Substituting this relation gives an expression for torque in terms of shearing stress. This relation must hold for the shearing stresses in any shaft cross-section.

Even so, the relation does not show how the stress is spread across the cross-section. The distribution of shearing stress in an elastic shaft cannot be found from statics alone. It requires deformation analysis for an accurate result.

Torque and Shear in a Twisted Shaft

stresses in a 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...

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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...

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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...

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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...

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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

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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...

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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...

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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...

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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

Explore how equal and opposite torques create internal shear forces in a shaft, and how torque relates to shearing stress in a cut section.

Torque and Shear in a Twisted Shaft | 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

Torque and Shear in a Twisted Shaft

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