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

当轴PQ两侧受到大小相等、方向相反的扭矩作用时，该轴会发生扭转变形。
考虑通过任意点 R 且垂直于轴线的横截面。部分 QR 的受力图显示了当轴发生扭转变形时，部分 PR 作用在 QR 上的剪切力。
将平衡方程应用于 QR 段，可以证明该截面内的剪切力与内扭矩相关。其中，r 表示从轴线到剪切力的垂直距离。
现在考虑轴的一个微小面积单元，该处的剪切力可表示为剪切应力与面积单元的乘积。
通过代入该关系式，可得到以剪切应力表示的扭矩表达式。
该关系式必须满足轴在任意横截面上的剪应力要求，但它并未提供关于这些应力在横截面上分布情况的信息。
弹性轴中剪应力的分布仅通过静力学无法确定，需要结合变形分析。

轴中的应力

Stresses in a Shaft

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 当在两侧施加大小相等且方向相反的扭矩时，轴 PQ 便会受到扭转力的作用。通过检查位于任意点 R 处垂直于轴的轴线切割的截面来了解这一点。当分析位于 QR 段的受力分析图时，可以发现当轴发生扭转时位于 PR 中的一部分剪切力会施加到 QR 段。
 
 将平衡条件应用到 QR 段便可以确定位于该段内的内部剪切力与内部扭矩是直接相关的。在这里，“r”则表示了从轴的轴线到剪切力之间的垂直距离。接下来，对轴上的小面积元件进行分析。剪切力可以表示为剪切应力与面积元素之间的乘积。将该关系进行代入后，便能够推导出用剪切应力来表示的扭矩表达式。
 
 <img alt="Equation 1" src="/files/ftp_upload/15612/15612_Equation_1.svg" style="text-align: center; height: 28px;">
 这个推导出的表达式对于任何轴所在横截面的剪切应力都必须是成立的。但是，它无法深入了解这些应力在横截面上的分布。最后，值得注意的是，弹性轴中剪切应力的分布不能仅由静力学来确定。因此需要进行变形分析才能够准确的对其进行确定。
 
</div>

当轴 PQ 两侧受到大小相等、方向相反的扭矩作用时，该轴将发生扭转变形。  
考虑通过轴轴线的垂直截面……

Education

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JoVE Core

Mechanical Engineering

机械工程

Unit 1

第一单元

Torsion

扭转

KEY TERMS AND CONCEPTS

关键术语与概念

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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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 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

Torque and Shear in a Twisted Shaft

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