Derive the materials index for a torsion bar with a solid ci

Derive the materials index for a torsion bar with a solid circular cross section. The length, L, and stiffness, S*, are specified and the torsion bar is to be as light as possible. The torsional stiffness, S*=T/(/L).

Solution

Here the objective is to minimise the mass of the torsion bar which can be given as:

m = AL

where A is the cross-sectional area, is the density

Given, The torsional stiffness, S*=T/(/L),

Now, = (TL/GJ),

so, T/(/L) = GJ = G*(*d⁴)/32 = (G/2)*((/4)*d²)² = (G/2)*A²

S*=T/(/L) = (G/2)*A²

A = (2S*/G)

m = AL = L(2S*/G)

As L and S* are specified and 2 is constant,

so the material index would be M 1/m, M = G/