Find the inner diameter of the tubular steel shaft:

A solid aluminium shaft having 1 m long & of 50 mm diameter is to be replaced by a tubular steel shaft of the similar length and similar outside diameter so that either shaft might carry the same torque and contain the similar angle of twist, over the overall length. What should be the inner diameter of the tubular steel shaft?

Modulus of rigidity of steel might be taken as 0.85 × 10⁵ N/mm² and that of aluminium as 0.28 × 105 N/mm².

Solution

Solid Aluminium Shaft l_A = 1000 mm

d_A = 50 mm

G_A = 0.28 × 10⁵ N/mm²

Tabular Steel Shaft

Ls = 100 cm

D_s0 = 50 mm,  d_si = ?

Gs = 0.85 × 105 N/mm²

Here,   T_A = Ts and θ_A = θs

J _A=  π (d _A)4  = (π/32) × 50⁴  = 6.25 π × 10⁶ / 32 mm⁴          

We know,

T/J  = G θ/l or θ = T l/JG

Therefore,

θ_A  = (T_A l_A   /  ((625 π/32) × 0.28 × 10) ) radian

Since θ_A = θ_s and also T_A  = Ts , l_A  = ls

    d_si  = 45. 243 mm