The solid cylinders A B and B C are bonded together at B and...

The solid cylinders $A B$ and $B C$ are bonded together at $B$ and are attached to fixed supports at $A$ and $C .$ Knowing that the modulus of rigidity is $3.7 \times 10^{6}$ psi for aluminum and $5.6 \times 10^{6}$ psi for brass, determine the maximum shearing stress $(a)$ in cylinder $A B,(b)$ in cylinder $B C$.

Key Concepts

Torsion in Circular Shafts

This concept involves understanding how a circular shaft behaves when subjected to a twisting moment (torque). Torsion causes an angular deformation along the shaft’s length and produces a shear stress distribution across the cross section. The analysis of torsion is essential for predicting deformations and stress concentrations in rotating or load-transmitting components.

Polar Moment of Inertia

The polar moment of inertia is a geometric property of a cross section that indicates its resistance to twisting. For circular shafts, this parameter is critical in determining how the applied torque is distributed over the cross-sectional area. A higher polar moment of inertia means a lower angle of twist and reduced shear stress for a given applied torque.

Modulus of Rigidity

Also known as the shear modulus, the modulus of rigidity measures a material’s resistance to shear deformation under torsional loading. It is a material property that, in combination with the polar moment of inertia, governs the relationship between applied torque and the resulting angle of twist, as well as the distribution of shear stress within the material.

Shear Stress under Torsion

When a shaft is subjected to torsion, the shearing stress at any point in its cross section can be computed using the formula that relates torque, radial distance from the center, and the polar moment of inertia. The maximum shear stress typically occurs at the outer surface and is a critical factor in the design to prevent material failure.

Compatibility in Composite Shafts

For shafts made of different materials bonded together, the compatibility conditions, such as equal angles of twist at the connection, must be satisfied. This ensures that the bonded sections deform together under torsion. The distribution of the applied torque between the different materials then depends on both their polar moments of inertia and their respective moduli of rigidity.

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