For a solid circular shaft of radius R under torque T, what is the maximum shear stress formula (in terms of T and R)?

For a solid circular shaft of radius R under torque T, what is the maximum shear stress formula (in terms of T and R)?

• A. tau_max = 2T/(π R^3)
• B. tau_max = T/(π R^2)
• C. tau_max = T R^2 / J
• D. tau_max = 4T/(π R^3)

Answer: (A)

In torsion of a solid circular shaft, the shear stress at a radius r is tau(r) = T r / J, where J is the polar moment of inertia. For a solid circle, J = π R^4 / 2. The maximum shear stress occurs at the outer surface (r = R), so tau_max = T R / J = T R / (π R^4 / 2) = 2T / (π R^3). This matches the given formula for the maximum shear stress in terms of T and R. Other forms miss either substituting J for a solid shaft or use an incorrect coefficient.

In torsion of a solid circular shaft, the shear stress at a radius r is tau(r) = T r / J, where J is the polar moment of inertia. For a solid circle, J = π R^4 / 2. The maximum shear stress occurs at the outer surface (r = R), so tau_max = T R / J = T R / (π R^4 / 2) = 2T / (π R^3). This matches the given formula for the maximum shear stress in terms of T and R. Other forms miss either substituting J for a solid shaft or use an incorrect coefficient.