The aluminum rod A B(G=27 GPa) is bonded to the brass rod B D (G=39 GPa) . Knowing that portion

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The aluminum rod A B(G=27 GPa) is bonded to the brass rod B...

The aluminum $\operatorname{rod} A B(G=27 \mathrm{GPa})$ is bonded to the brass rod $B D$ $(G=39 \mathrm{GPa}) .$ Knowing that portion $C D$ of the brass rod is hollow and has an inner diameter of $40 \mathrm{mm}$, determine the angle of twist at $A$.

Key Concepts

Torsion in Circular Shafts

This concept deals with the deformation of a shaft subjected to a twisting moment (torque). The fundamental relationship is given by the torsion formula, where the angle of twist is proportional to the applied torque, the length of the shaft, and inversely proportional to both the shear modulus and the polar moment of inertia. It provides the basis for calculating angular displacements in response to twisting loads.

Shear Modulus

Shear modulus is a material property that measures resistance to shear deformation. In the context of torsion, it determines how much a material will twist under a given torque. Different materials, such as aluminum and brass, have different shear moduli, affecting how they deform when used in composite shafts.

Polar Moment of Inertia

The polar moment of inertia is a geometric property of a cross-section that quantifies its ability to resist torsional deformation. It takes into account the distribution of material around the center of the shaft, and its calculation differs for solid and hollow sections. For hollow sections, the calculation involves the difference between the fourth powers of the outer and inner radii.

Composite Shafts

Composite shafts are structural elements composed of two or more different materials bonded together to act as a single unit. In torsion problems involving composite shafts, the condition of compatibility (equal angle of twist) is enforced across the bonded materials, and each material's contribution to the overall torsional rigidity is considered individually.

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