determine maximum shearing stress in shaft a

Question

Determine maximum shearing stress in shaft: 

A propeller shaft 100 mm in the diameter, is 45 m long, transmits 10 MW at 80 rotation per minute. Determine maximum shearing stress in shaft. Also compute the stress at 20 mm, 40 mm, 60 mm and 80 mm diameters. Demonstrate the stress variation.

Sol.:  

d = 100mm

L = 45m

P = 10MW = 10 × 10⁶  W

N = 80RPM

Τ_max = ?

τ₂₀ = ?,  τ ₄₀ = ?,  τ ₆₀ = ?,  τ ₈₀ = ?

By using the relation

P = 2  .N.T_max/60 watts

10 × 10⁶ = 2  .80.T_max/60

T_max = 1193662.073 N                                                                                       ...(i)

For solid shaft    τ = 16T_max/  d³

τ = 16(1193662.073)/[  (100)³]

τ= 6.079 N/mm²

This is shear stress at outer surface that is;  τ₁₀₀  = 6.079 N/mm²                                             .......ANS

Now stress is varies linearly along diameter of shaft, the shear stress at inner diameter of

the shaft that is; τ₂₀ =  τ₂₀   × d₂₀/d₁₀₀  = 6.079 × 20/100 = 1.22 N/mm²

τ₁₀₀ = 1.22 N/mm²                                                                                                      ......ANS

τ₄₀ =  τ100 × d₄₀/d₁₀₀  = 6.079 × 40/100 = 2.44 N/mm²

τ₄₀ = 2.44 N/mm2                                                                             .......ANS

τ₄₀ =  τ100 × d₆₀/d₁₀₀  = 6.079 × 60/100 = 3.66 N/mm²

τ₄₀ = 3.66 N/mm2                                                                              .......ANS

τ₄₀=  τ100 × d₈₀/d₁₀₀  = 6.079 × 80/100 = 4.88 N/mm²

τ₄₀= 4.88 N/mm²                                                                                .......ANS