19.2
圆轴的变形
圆轴的显著特征之一是：它们能够在扭转的状态下保持其横截面的完整性。换句话说，每个横截面都能够作为一个平坦的、不变的实体存在着，同时还要像一个坚固的、刚性的板一样进行旋转。如果要了解此类轴内的剪切应力分布，则需要对该圆形轴内的圆柱形截面来进行分析。该截面的长度为 L，半径为 R，其中的一端是固定的。并…
圆轴的一个独特性质是，在受到扭转时，每个横截面均保持平面且不发生畸变，如同一个实心刚性板一样旋转。
为了确定剪切应力的分布情况，考虑在长度为 L、半径为 R 的圆轴内部存在一个圆柱形截面，该圆轴一端固定，圆柱形截面的半径为 r。
现在，考虑在载荷施加之前，圆柱段表面由两个相邻圆和直线所形成的微小正方形元素。
当扭转载荷施加到轴上时，方形单元会变形为菱形。由于菱形的两条边是固定的，因此剪切应变等于 AB 与 A'B 两条线之间的夹角。
利用小角度近似和适当的几何关系可知，圆轴在扭转变形中任意一点的切应变与扭转角以及该点到轴线的距离 r 成正比，在轴的表面处达到最大值。
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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.