A bronze sleeve, 0.040 in. thick, was mounted on a 2.000-in. diameter steel shaft by heating it to 100?C, while the temperature of the shaft was maintained at 20?C Under these conditions, the sleeve just ?t on the shaft with zero clearance. Find the principal stresses in the sleeve after it cools to 20?C. Assume that friction between the shaft and the sleeve prevents any sliding at the interface during cooling, and assume that the shaft is so massive and stiff that strains in the shaft itself are negligible. For the bronze, E = 16 × 106 psi, υ = 0.30, and α = 18.4 × 10-6 (?C)-1.