A composite shaft is fabricated by bonding an isotropic solid shaft having shear modulus \(G_{1}\), and outside radius \(r_{1}\), inside a hollow isotropic shaft having shear modulus \(G_{2}\) and outside radius \(r_{2}\). The composite shaft is to be loaded by a twisting moment, \(T\), that is distributed over the end of the shaft, as shown in Figure 3.11. Using an elementary mechanics of materials approach, derive the equations for the stresses and deformations at any radius and the equation for the effective torsional shear modulus of the composite shaft in terms of the material and geometrical properties of shafts 1 and 2.

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2r2 2 Material 1 Material 2 D FIGURE 3.11 Composite shaft under torsional load.