Name: residual stress after shaft torsion
Uploaded: 2024-05-23T12:59:41.000Z
Duration: 1 min 10 s
Channel: JoVE

Nei materiali elastoplastici, la sollecitazione residua può svilupparsi quando il materiale subisce una deformazione plastica a causa di un'elevata sollecitazione di taglio o di grandi deformazioni.
Quando il carico esterno viene rimosso, le sollecitazioni interne sostenute all'interno del materiale dell'albero sono note come sollecitazioni residue.
Tracciando il diagramma coppia rispetto all'angolo di torsione, si può dimostrare che l'angolo di torsione non ritorna a zero quando la coppia viene rimossa. Una linea retta rappresenta lo scarico dell'albero.
Le tensioni residue nel materiale elastoplastico sono ottenute applicando il principio di sovrapposizione.
In primo luogo, nella fase di carico si considerano le sollecitazioni dovute alla coppia applicata, quindi si considerano le sollecitazioni dovute alla coppia uguale e contraria applicata per scaricare l'albero. La distribuzione delle tensioni residue nell'albero si ottiene sommando questi due gruppi di tensioni.
Tracciando la sollecitazione rispetto alla distanza radiale, si può osservare che alcune sollecitazioni residue condividono la stessa direzione delle sollecitazioni originali, mentre altre sono nella direzione opposta.

tensioni residue nell'albero circolare

Residual Stresses in Circular Shafts

Nei materiali che presentano un comportamento sia elastico che plastico, noti come materiali elastoplastici, quando questi materiali subiscono una deformazione plastica, le tensioni residue possono accumularsi. Questa deformazione deriva da elevati livelli di stress di taglio o da deformazioni significative. Le tensioni residue sono tensioni interne, che persistono all'interno di un materiale dopo aver rimosso la forza esterna che causa la deformazione. Questo fenomeno è dimostrato osservando il comportamento di un albero sottoposto a coppia; in particolare, l'angolo di torsione dell'albero non ritorna al suo stato originale dopo la rimozione della coppia, indicando la presenza di uno stress residuo. Questo comportamento è rappresentato graficamente su un diagramma coppia/angolo di torsione, dove il processo di scarico dell'albero è rappresentato come un percorso lineare.

La metodologia per calcolare queste tensioni residue incorpora il principio di sovrapposizione. Si compone di due fasi: inizialmente la valutazione delle sollecitazioni indotte dalla coppia applicata durante la fase di carico, seguita dalla valutazione delle sollecitazioni generate dall'applicazione di una coppia uguale e contraria per scaricare l'albero. La distribuzione delle tensioni residue all'interno dell'albero, viene determinata aggregando queste due risposte alle sollecitazioni. Un'ulteriore analisi, che traccia la sollecitazione rispetto alla distanza radiale, rivela che le sollecitazioni residue, possono allinearsi con la direzione della sollecitazione originale o contrastarla. Questa intuizione è fondamentale per valutare la resilienza e l'integrità strutturale dei materiali sotto stress.

Tensioni Residue nell'Albero Circolare

Nei materiali elastoplastici, la sollecitazione residua può svilupparsi quando il materiale subisce una deformazione plastica a causa di un'elevata sollecitazione di taglio o di grandi deformazioni.
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Ingegneria Meccanica

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Torsion

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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 present.

The behavior is shown on a torque versus angle of twist graph. During unloading, the path is a straight line. To calculate the residual stress, the principle of superposition is used. First, the stress from the applied torque is found during loading. Then an equal and opposite torque is applied to represent unloading.

The final residual stress distribution comes from adding these two stress results together. A stress versus radial distance plot shows how the residual stress changes from the center to the outside of the shaft. In some regions, the residual stress matches the original stress direction. In others, it acts in the opposite direction. This helps evaluate how well the material can resist damage and keep its structural integrity after loading.

Residual Stress After Shaft Torsion

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 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 can leave residual stress in circular shafts, how unloading is shown on a graph, and how superposition is used to find the final stress pattern.

Residual Stress After Shaft Torsion | JoVE K12

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Residual Stress After Shaft Torsion

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