Max Shear Stress in a Hollow Shaft

Equation: for a shaft in torsion, τmax = T·c / J, where c is the outer radius and J is the polar moment of inertia. For a hollow circular section, J = π(do⁴ − di⁴)/32. Values: T = 1.5 kN·m = 1500 N·m is the applied torque. The largest shear stress is at the outer surface, so c = do/2 = 60/2 = 30 mm = 0.030 m. do = 0.060 m and di = 0.040 m come from the section. Step 1: J = π(0.060⁴ − 0.040⁴)/32 = π(1.296 × 10⁻⁵ − 2.56 × 10⁻⁶)/32 = 1.021 × 10⁻⁶ m⁴. Step 2: τmax = 1500 × 0.030 / 1.021 × 10⁻⁶ = 44.1 × 10⁶ Pa = 44 MPa. Using the outside diameter as c instead of the radius doubles the answer to 88 MPa.