19.2
円形シャフトの変形
円形シャフトの際立った特性の 1 つは、ねじれていても断面の完全性を維持できることです。 言い換えれば、各断面は平らで変化のない実体として存在し続け、固体で硬いスラブのように単に回転するだけです。 このようなシャフト内のせん断応力の分布を理解するには、この円形シャフト内の円筒部分を考考えます。 この…
円形シャフトのユニークな特性は、ねじれの下で、すべての断面が平面で歪みなく、固体の剛性スラブとして回転することです。
せん断応力の分布を決定するには、長さ L と半径 R の円形シャフトの内側に円筒セクションがあり、一端に固定されているとします。円筒断面の半径はrです。
ここで、荷重が加えられる前に、円筒断面の表面に隣接する2つの円と直線によって形成された小さな正方形の要素について考えてみます。
ねじり荷重がシャフトに加えられると、正方形の要素がひし形に変形します。菱形の両側は固定されているため、せん断ひずみは線ABとA'Bの間の角度に等しくなります。
小さな角度近似と適切な形状を使用すると、ねじれのシャフトの任意の点でのせん断ひずみが、ねじれの角度とシャフトの軸からの距離rに比例することを示すことができます。シャフトの表面で最大になります。
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Q1: Why do circular shafts maintain their cross-sectional shape under torsion?
A unique property of circular shafts is that under torsion, every cross-section remains plane and undistorted, rotating as a solid rigid slab. This characteristic allows engineers to predict stress and strain distributions predictably. The cross-sectional integrity is maintained because the geometry of the circular shaft distributes torsional loads uniformly across the radius.
Q2: How does a small square element deform when torsional load is applied to a shaft?
When torsional load is applied to a circular shaft, a small square element on the cylindrical surface deforms into a rhombus shape. The shearing strain equals the angle between the original vertical line and the inclined line along the rhombus side. This deformation demonstrates how material particles shift relative to each other under torsional stress.
Q3: What is the relationship between shearing strain and distance from the shaft axis?
Shearing strain at any point in a shaft under torsion is directly proportional to both the angle of twist and the distance r from the shaft's axis. Using small angle approximation and appropriate geometry, this relationship can be mathematically demonstrated. The strain reaches its maximum at the shaft's surface, where the radial distance is greatest.
Q4: Where is shearing strain maximum in a circular shaft under torsion?
Shearing strain is maximum at the surface of a circular shaft under torsion. Since strain is proportional to the distance from the shaft's axis, the outermost fibers experience the greatest deformation. This distribution is critical for understanding stress concentrations and designing shafts to resist torsional failure.
Q5: How is shearing strain calculated from the geometry of a deformed element?
Shearing strain is determined by measuring the angle between the original vertical line AB and the inclined line A'B formed after the square element deforms into a rhombus. By applying small angle approximation and suitable geometry, this angular change quantifies the shearing strain. This method provides a geometric foundation for understanding stress distribution within the shaft.
Q6: What role does the angle of twist play in determining shearing strain distribution?
The angle of twist is a primary factor determining shearing strain at any point within a shaft. Shearing strain is directly proportional to the angle of twist multiplied by the radial distance from the shaft's axis. This proportional relationship allows engineers to predict strain magnitudes throughout the shaft's cross-section when the twist angle is known.
Q7: How does the cylindrical section model help explain stress distribution in torsion?
Analyzing a cylindrical section inside a circular shaft with fixed length L and radius R provides a simplified model for understanding torsional deformation. By examining how surface elements deform into rhombi, engineers can derive the relationship between angle of twist, radial distance, and shearing strain. This model forms the basis for circular shaft stresses in linear range analysis and design calculations.