19.2
Déformation dans un arbre circulaire
L'une des caractéristiques distinctives des arbres circulaires est leur capacité à maintenir leur intégrité de section transversale sous torsion. En d…
Une propriété unique des arbres circulaires est qu’en torsion, chaque section transversale reste plane et non déformée, tournant comme une dalle rigide solide.
Pour déterminer la répartition de la contrainte de cisaillement, considérons une section cylindrique à l’intérieur d’un arbre circulaire de longueur L et de rayon R, fixé à une extrémité. Le rayon de la section cylindrique est r.
Considérons maintenant le petit élément carré formé par deux cercles adjacents et des lignes droites à la surface de la section cylindrique avant d’appliquer une charge.
Lorsque la charge de torsion est appliquée à l’arbre, l’élément carré se déforme en un losange. Comme les deux côtés du losange sont fixes, la déformation de cisaillement est égale à l'angle entre les lignes AB et A'B.
En utilisant une approximation de petit angle et une géométrie appropriée, on peut montrer que la déformation par cisaillement en un point donné d’un arbre en torsion est proportionnelle à l’angle de torsion et à la distance r par rapport à l’axe de l’arbre. Il est maximal à la surface de l’arbre.
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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.