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Stress Concentration

Example: The stepped shaft shown below is subjected to a torque, T, of 2.5 kip in. Find

the maximum shear stress in the shaft if the radius, r, of the fillet joining the two portions of

the shaft is 0.125 in. The shaft diameters are: d₁ = 2 in. , d₂ = 1.5 in.

Strategy: Calculate the nominal shear stress in the shaft using Tc/J for the smaller diameter

shaft. Then use the chart for torsion of circular shafts with fillets to find the stress concentration

factor, K. The stress concentration factor, K, depends on the ratio of the larger diameter to the

smaller diameter, d₁/d₂, as well as the ratio of the radius of the fillet weld to the smaller diameter,

r/d₂. So these values need to be determined. Finally apply K = τmax / τave.

So for the given data:

d₁/d₂ = 2/1.5 = 4/3 = 1.33, r/d₂ = 0.125/1.5 = 0.083

For the smaller shaft J = π (d₂/ 2)⁴ / 2, c = d₂/2 , T = 2500 lb in

τave = T c / J = (2500 lbin)(0.75)/[π(0.75)⁴ / 2] = 3772.56 psi

From the chart for torsion of circular shafts connected by a fillet radius weld

For d₁/d₂ = 1.33, r/d₂ = 0.083 K = 1.42

τmax = K τave (1.42) (3772.56) = 5360 psi = 5.36 ksi (result)

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