Determine the maximum shear stress and angle of twist:

A hollow circular shaft of 6 m length and inner & outer diameter ois equal to 80 mm and

100 mm is subjected to a torque of 20 kN-m. If G = 80 GPa, determine the maximum shear stress and the angle of twist.

Solution

l = 6 m;            do = 100 mm;  di = 80 mm

T = 20 kN-m = 20 × 10³ N-m

G = 80 GPa = 0.8 × 10¹¹ N/m²

R = 50 mm

Z P       = J / R = 5.8 × 10^-6 /0.05   = 1.16 × 10^{- 4} m³

T/ J = τ_max / R     = G θ/ l

τ_max = T /Z_P =    20 × 10³ / 1.16 × 10^{- 4}

 = 17.2 × 10⁷ N/m2

 = 172 N/mm²

θ= τ_max /R ×  (l/G) = 172 × 10 ⁶ /0.05 × (6/ (0.8 × 10¹¹))

= 0.258 radian = 14.78o.