Project ForgeA shaft of length L is made of two materials, one in the inner core and the other in the outer rim, and the two are perfectly joined together (no slip at the interface) along the entire length of the shaft. The diameter of the inner core is di and the external diameter of the rim is do. The modulus of rigidity of the core and rim materials are Gi and Go, respectively. It is given that do=2di and Gi=3Go. When the shaft is twisted by application of a torque along the shaft axis, the maximum shear stress developed in the outer rim and the inner core turn out to be τo and τi, respectively. All the deformations are in the elastic range and stress-strain relations are linear. Then the ratio τi/τo is ______ (round off to 2 decimal places).
Explanation
Since the core and rim are bonded with no slip, both share the same angle of twist per unit length, dθ/dx. Maximum shear stress in each material occurs at its own outer radius: τ=G·r·(dθ/dx). So τi/τo = (Gi·ri)/(Go·ro) = (Gi/Go)×(di/do) = 3×(1/2) = 1.5.
Source: GATE ME 2022 Session 2 (ME-2) (2022)
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