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

Вал PQ поворачивается под действием одинаковых и противоположных крутящих моментов с обеих сторон.
Рассмотрим сечение, перпендикулярное оси вала через произвольную точку R. На диаграмме свободного тела части QR показаны поперечные силы, действующие частью PR на QR при вращении вала.
Применяя уравнения равновесия к порции QR, можно показать, что поперечные силы в сечении связаны с внутренним крутящим моментом. Здесь r представляет собой перпендикулярное расстояние от оси вала до силы сдвига.
Теперь рассмотрим небольшой элемент вала, где сила сдвига может быть выражена как произведение напряжения сдвига на элемент площади.
Подставляя это соотношение, выражение крутящего момента получается в терминах напряжения сдвига.
Это соотношение должно удовлетворяться напряжениями сдвига в любом поперечном сечении вала. Однако он не дает информации о распределении этих напряжений в сечении.
Распределение касательных напряжений в упругом валу не определяется только статикой и требует анализа деформаций.

Напряжения в валу

Stresses in a Shaft

Вал PQ подвергается скручивающей силе, когда с обеих сторон приложены равные и противоположные крутящие моменты. Чтобы понять это, исследуют сечение, которое расположено перпендикулярно оси вала в любой произвольной точке R. При анализе диаграммы свободного тела сегмента QR, рассматриваются силы сдвига, действующие со стороны части PR на сегмент QR, когда стержень испытывает скручивание.

Применяя условия равновесия к сегменту QR, можно установить, что внутренние силы сдвига внутри сечения напрямую коррелируют с внутренним крутящим моментом. Здесь «r» означает перпендикулярное расстояние от оси вала до силы сдвига. Далее учитывается элемент малой площади вала. Силу сдвига можно выразить как произведение напряжения сдвига на элемент площади. Подставив это соотношение, получим выражение для крутящего момента через напряжение сдвига.

<img alt="Equation 1" src="/files/ftp_upload/15612/15612_Equation_1.svg" style="text-align: center; height: 28px;">

Это производное соотношение должно сохраняться для сдвиговых напряжений в любом поперечном сечении вала. Однако, это не дает представления о распределении этих напряжений по поперечному сечению. Наконец, важно отметить, что распределение напряжений сдвига в упругом валу не может определяться исключительно статикой. Для точного определения требуется анализ деформации.

Вал PQ поворачивается под действием одинаковых и противоположных крутящих моментов с обеих сторон.
Рассмотрим сечение, перпендикулярное оси вала через ...

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Unit 1

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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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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...

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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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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

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Torque and Shear in a Twisted Shaft

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