He copper pipe has an outer diameter of 2.50 in. and an...

he copper pipe has an outer diameter of 2.50 in. and an inner diameter of 2.30 in. If it is tightly secured to the wall at $C$ and a uniformly distributed torque is applied to it as shown, determine the shear stress developed at points $A$ and $B$. These points lie on the pipe's outer surface. Sketch the shear stress on volume elements located at $A$ and $B$

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Key Concepts

Hollow Circular Cross-Section

A hollow circular cross-section, such as that of a pipe, consists of both an inner and outer diameter. This geometry affects the distribution of shear stresses under torsion, as the material distribution (or wall thickness) directly influences the polar moment of inertia and, consequently, the stress levels.

Polar Moment of Inertia

The polar moment of inertia is a geometrical property of a cross-sectional area that quantifies its ability to resist torsional deformation. For circular and hollow cross-sections, it is critical in calculating the shear stress generated by an applied torque.

Shear Stress

Shear stress is the force per unit area acting parallel to the surface within a material. Under torsion, the shear stress distribution is typically linear with respect to the distance from the center of the cross-section, reaching a maximum at the outer surface.

Torsion

Torsion refers to the twisting of an object when a torque is applied. It is a fundamental concept in mechanics that describes how applied torques introduce shear stresses throughout the material, particularly affecting circular or cylindrical structures.

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