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

Bei der Torsionsbelastung bleiben die Querschnitte der Kreiswellen flach und unverzerrt, da sie achssymmetrisch sind.
Aufgrund des Mangels an Achsensymmetrie in einem quadratischen Stab wird jedoch jede Linie in seinem Querschnitt, abgesehen von seinen Diagonalen und den Linien, die die Mittelpunkte dieses Querschnitts verbinden, verzerrt, wenn der Stab verdreht wird.
Stellen Sie sich ein kleines kubisches Element an einer Ecke des Querschnitts eines quadratischen Stabes in Torsion vor. Die Fläche des Elements senkrecht zu jeder Achse ist Teil der freien Fläche des Stabes, sodass alle Spannungen auf diese Flächen und die Ecken des Querschnitts Null sind. 
Daher kann hier nicht davon ausgegangen werden, dass die Schubspannung linear mit dem Abstand von der Achse variiert.
Die maximale Schubspannung tritt entlang der Mittellinie der breiteren Fläche des Stabes auf und kann in der Breite der breiteren und schmaleren Flächen ausgedrückt werden. In ähnlicher Weise wird auch der Verdrehungswinkel in Bezug auf diese beiden Breiten definiert.
Hier hängen die Koeffizienten c1 und c2 ausschließlich vom Verhältnis der Dimensionen der beiden Flächen ab.

torsion nicht kreisförmiger elemente

Kreisförmige Wellen, die einer Torsionsbeanspruchung ausgesetzt sind, behalten aufgrund ihrer achsensymmetrischen Beschaffenheit ihre Querschnittsintegrität. Diese Symmetrie sorgt für eine gleichmäßige Spannungsverteilung, sodass der Schaft Torsion standhalten kann, ohne sich zu verziehen. Im Gegensatz dazu kommt es bei quadratischen Stäben, denen diese Achsensymmetrie fehlt, zu erheblichen Verformungen über ihren Querschnitt, wenn sie einer Torsion ausgesetzt werden, mit Ausnahme entlang ihrer Diagonalen und an Linien, die die Mittelpunkte verbinden. Eine detaillierte Untersuchung eines kubischen Elements an der Ecke des Querschnitts eines quadratischen Stabs zeigt, dass seine nach außen gerichteten Seiten, die Teil der Außenseite des Stabs sind, spannungsfrei sind. Dies weist darauf hin, dass die Spannung auf diesen Oberflächen und an den Ecken des Querschnitts Null ist, was zu dem Schluss führt, dass sich die Scherspannung in solchen Stäben nicht linear mit dem Abstand von der Achse verteilt.

Dies lässt sich auf Stäbe mit rechteckigem Querschnitt verallgemeinern. In diesem Fall erreicht die Scherspannung ihren Höhepunkt entlang der Mittellinie der breiteren Fläche des Stabes. Diese maximale Spannung hängt zusammen mit dem Verdrehungswinkel von den Abmessungen der Stange ab, insbesondere von der Breite ihrer breiteren und schmaleren Flächen. Zur Bestimmung dieser Parameter sind spezifische Koeffizienten erforderlich, die als c_1 und c_2 bezeichnet werden und auf der Grundlage des Verhältnisses der Flächenabmessungen der Stange berechnet werden.

<img alt="Equation 1" src="/files/ftp_upload/15622/15622_Equation_1.svg" style="text-align: center; height: 54px;">

Diese Berechnung verdeutlicht den Zusammenhang zwischen den geometrischen Eigenschaften des Stabes und seiner Reaktion auf Torsionsbelastung und unterstreicht die Bedeutung der Berücksichtigung der Form und Abmessungen von Materialien bei der Bewertung ihres Verhaltens unter Torsion.

Torsion nicht kreisförmiger Elemente

Bei der Torsionsbelastung bleiben die Querschnitte der Kreiswellen flach und unverzerrt, da sie achssymmetrisch sind.
Aufgrund des Mangels an ...

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Einheit 1

Torsion

KEY TERMS AND CONCEPTS

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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

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Torsion in Square and Rectangular Bars

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