Name: torsion in square and rectangular bars
Uploaded: 2024-05-23T12:59:41.000Z
Duration: 1 min 16 s
Channel: JoVE

비틀림 하중 동안 원형 샤프트의 단면은 축 대칭이기 때문에 평면으로 유지되고 왜곡되지 않습니다.
그러나 정사각형 막대에 축 대칭이 없기 때문에 대각선과 해당 단면의 중간점을 연결하는 선을 제외한 단면의 모든 선은 막대가 비틀릴 때 왜곡됩니다.
비틀림에서 정사각형 막대의 단면 모서리에 있는 작은 입방 요소를 고려하십시오. 각 축에 수직인 요소의 면은 바의 자유 표면의 일부이므로 이러한 면과 단면의 모서리에 대한 모든 응력은 0입니다. 
결과적으로, 여기에서는 전단 응력이 축으로부터의 거리에 따라 선형적으로 변한다고 가정할 수 없습니다.
최대 전단 응력은 바의 넓은 면의 중심선을 따라 발생하며 더 넓은 면과 더 좁은 면의 너비로 표현할 수 있습니다. 마찬가지로, 비틀림 각도도 이 두 너비로 정의됩니다.
여기서 계수 c1 및 c2는 두 면의 치수 비율에만 의존합니다.

비원형 부재의 비틀림

비틀림 응력을 받는 원형 줄기는 축대칭 특성으로 인해 단면적 무결성을 유지합니다. 이러한 대칭은 응력의 균일한 분포를 보장하여 줄기가 비틀림 없이 비틀림을 견딜 수 있게 해줍니다. 대조적으로, 이러한 축 대칭이 결여된 정사각형 막대는 비틀림을 받을 때 대각선과 중간점을 연결하는 선을 제외하고 단면에 걸쳐 상당한 왜곡을 경험합니다. 정사각형 막대 단면의 모서리에 있는 입방체 요소를 자세히 조사해 보면 막대 외부의 일부인 바깥쪽을 향한 측면에 응력이 없음을 알 수 있습니다. 이는 이러한 표면과 단면 모서리의 응력이 없음을 나타내며, 전단 응력이 이러한 막대의 축으로부터의 거리에 따라 선형적으로 분포되지 않는다는 결론으로 ​​이어집니다.

이는 직사각형 단면을 가진 막대로 일반화될 수 있습니다. 이 경우 전단 응력은 바의 더 넓은 면의 중심선을 따라 최고조에 이릅니다. 비틀림 각도와 함께 이 최대 응력은 바의 치수, 특히 더 넓은 면과 더 좁은 면의 너비에 따라 달라집니다. 이러한 매개변수를 결정하는 데에는 막대의 면 치수 비율을 기준으로 계산되는 c_1 및 c_2라는 특정 계수가 포함됩니다.

<img alt="Equation 1" src="/files/ftp_upload/15622/15622_Equation_1.svg" style=" height: 54px;">

이 계산은 막대의 기하학적 특성과 비틀림 하중에 대한 반응 사이의 관계를 강조하며, 비틀림 시 재료의 거동을 평가할 때 재료의 모양과 치수를 고려하는 것의 중요성을 강조합니다.

비틀림 하중 동안 원형 샤프트의 단면은 축 대칭이기 때문에 평면으로 유지되고 왜곡되지 않습니다.
그러나 부족으로 인해 ...

Education

교육

JoVE Core

Mechanical Engineering

기계공학

Unit 1

제1단원

Torsion

비틀림

KEY TERMS AND CONCEPTS

주요 용어 및 개념

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 the twisting pattern is not uniform across the whole bar.

A close look at a small cubic element at the corner of a square bar shows another key detail. The outer sides of that corner element are part of the bar’s surface, and they are stress-free. So the stress at the surface and at the corners is zero. That also shows shearing stress does not increase in a simple linear way with distance from the axis in these bars.

The same idea applies to bars with rectangular cross-sections. In a rectangular bar, the shearing stress is greatest along the centerline of the wider face. The maximum stress and the angle of twist both depend on the bar’s dimensions, especially the widths of the wider and narrower faces. Engineers use coefficients called c1 and c2 to find these values from the ratio of the face dimensions.

Torsion in Square and Rectangular Bars

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 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 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

See how square and rectangular bars twist under torsion, why corners can be stress-free, and how bar dimensions affect maximum shear stress and angle of twist.

Torsion in Square and Rectangular Bars | 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 in Square and Rectangular Bars

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