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

Bir şaft PQ, her iki tarafta eşit ve zıt torklara maruz kaldığında bükülür.
Rastgele bir R noktasından şaftın eksenine dik bir kesit düşünün. Kısım QR'nin serbest cisim diyagramı, şaft bükülürken kısım PR'nin QR üzerine uyguladığı kesme kuvvetlerini gösterir.
Denge denklemlerini QR kısmına uygulayarak, kesit içindeki kesme kuvvetlerinin iç tork ile ilişkili olduğu gösterilebilir. Burada r, milin ekseninden kesme kuvvetine olan dik mesafeyi temsil eder.
Şimdi, kesme kuvvetinin kesme gerilimi ve alan elemanının ürünü olarak ifade edilebileceği milin küçük bir alan elemanını düşünün.
Bu ilişkinin yerine konarak, kesme gerilimi cinsinden tork ifadesi elde edilir.
Bu ilişki, şaftın herhangi bir enine kesitindeki kesme gerilmeleri tarafından karşılanmalıdır. Ancak bu gerilmelerin kesitteki dağılımı hakkında bilgi vermemektedir.
Elastik bir şafttaki kesme gerilmelerinin dağılımı yalnızca statik ile belirsizdir ve deformasyon analizi gerektirir.

Şafttaki gerilimler

Stresses in a Shaft

PQ şaftı, her iki tarafa da eşit ve zıt torklar uygulandığında bir burkulma kuvvetine maruz kalır. Bunu anlamak için herhangi bir R noktasında şaft eksenine dik olarak kesen bir kesit incelenir. QR segmentinin serbest cisim diyagramı analiz edildiğinde, şaft burulma yaşadıkça PR kısmının QR segmentine uyguladığı kesme kuvvetlerini ortaya çıkarır.

Denge koşullarının QR segmentine uygulanması, kesit içindeki iç kesme kuvvetlerinin iç tork ile doğrudan ilişkili olduğunu tespit eder. Burada 'r' şaft ekseninden kesme kuvvetine olan dik mesafeyi ifade eder. Daha sonra şaftın küçük bir alan elemanı dikkate alınır. Kesme kuvveti, kesme gerilimi ile alan elemanının çarpımı olarak ifade edilebilir. Bu ilişkinin değiştirilmesi üzerine, kesme gerilimi cinsinden tork için bir ifade türetilir.

<img alt="Equation 1" src="/files/ftp_upload/15612/15612_Equation_1.svg" style="height: 28px;">

Bu türetilmiş ilişki herhangi bir şaft kesitindeki kesme gerilmeleri için geçerli olmak zorundadır. Ancak bu gerilimlerin kesit boyunca dağılımına dair fikir vermez. Son olarak, elastik bir şafttaki kayma gerilmelerinin dağılımının yalnızca statik ile belirlenemeyeceğine dikkat etmek önemlidir. Doğru tespit için deformasyon analizi gerektirir.

Şafttaki Gerilimler

Bir şaft PQ, her iki tarafta eşit ve zıt torklara maruz kaldığında bükülür.
Bir şaft boyunca şaftın eksenine dik bir bölüm düşünün ...

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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 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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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

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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...

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

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 Makine Mühendisliği, denge, kuvvetler, kafes kiriş analizi ve yapısal mekanik gibi temel konuları kapsayan kısa, anlaşılması kolay animasyonlu videolar sunar. Gerçek dünyadaki uygulamalarını keşfedin ve bilim insanlarının önderlik ettiği deneylerle bu kavramların uygulanmasına tanık olun.

mechanical engineering

Torque and Shear in a Twisted Shaft

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