19.2
Vervorming in een cirkelvormige as
Een van de onderscheidende kenmerken van ronde assen is hun vermogen om hun dwarsdoorsnede-integriteit onder torsie te behouden. Met andere woorden, e…
A unique property of circular shafts is that under torsion, every cross-section remains plane and undistorted, rotating as a solid rigid slab.
To determine the distribution of shearing stress, consider a cylindrical section inside a circular shaft of length L and radius R, fixed at one end. The radius of the cylindrical section is r.
Now, consider the small square element formed by two adjacent circles and straight lines on the surface of the cylindrical section before any load is applied.
As the torsional load is applied to the shaft, the square element deforms into a rhombus. Since the two sides of the rhombus are fixed, the shearing strain is equal to the angle between lines AB and A'B.
Using a small angle approximation and suitable geometry, it can be shown that shearing strain at any given point of a shaft in torsion is proportional to the angle of twist and the distance r from the axis of the shaft. It is maximum at the surface of the shaft.
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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.