# Pc + + 55 4 50 Pi+ Pc Compound cylinder Compound cylinder with misfit loaded by without misfit loaded internal pressure by internal pressure (a) (b)

Consider two cylinders made of steel (E = 200 GPa, ν = 0.3 and σY = 500 MPa) shrink fit on each other, as shown in Fig. 2.11 page 74 Lecture Notes. Let a = 400 mm and b = 200 mm. Before they were shrunk, the outer radius of the inner cylinder was bigger than the inner radius of the outer cylinder by δ (radial interference). The compound cylinder is then subjected to an internal pressure p

Our objective is to design an optimum compound cylinder i.e. find such δ and c which will provide the highest strength (such cylinder can sustain a higher internal pressure than, for example, a single cylinder with the same dimensions and made of the same material). (50 Marks)

You also need to calculate the critical pressure ratio of the compound to the one of the single cylinder having the same dimensions (a and b) and made of the same material. These two will obviously have the same weight but different strength. Assume plane stress conditions in all calculations. (10 marks)

Hints:

(i) Derive an expression for the maximum shear stress in the inner cylinder in terms of p and δb⁄.

(ii) Similarly, derive an expression for the maximum shear stress in the outer cylinder. In order to obtain an optimal design, we choose δ such that these two maximum shear stresses are equal. Demonstrate that this would be the optimal design.

(iii) If σY = 300 MPa is the yield stress of the material; calculate the critical pressure ratio of the compound to the one of the single cylinder having the same dimensions and made of the same material.

Pc++55 4 50Pi+PcCompound cylinderCompound cylinderwith misfit loaded bywithout misfit loadedinternal pressureby internal pressure(a)(b)cErsaCompound cylinderwith misfit only(c)