Two shafts are assembled according to the scheme presented below. They are made of steel $(E={2\mathrm{.}10}^{11}Pa,v=0\mathrm{.}3)$ with an allowable shear stress of $80\frac{N}{m}{m}^2.$ One shaft if fully solid $($diameter $d_1)$ and the other is hollow $($internal diameter $d_i,$ external de$)$

$1-d_1$ and $d_e$ are given. Calculate $d_i$ in order to get a hollow shaft with the same resistance as the solid one.

$2-$ Compare the weight of both shafts.

$3-$ The length of the shafts is $1$ m and a torque of $1963$ Nm is applied to the assembly. Calculate the torsional deformation induced on both shafts $($degrees$)$

$4-$ Each shaft is linked to a solid disc and those discs are assembled using $12$ bolts uniformly distributed around a perimeter $($diameter D$).$ Characteristics of the bolts are: $E={2\mathrm{.}10}^{11}Pa,v=0\mathrm{.}3,$ allowable shear stress of $200\frac{N}{m}{m}^2$

Calculate the diameter needed for each bolt to transmit the $1963$ Nm torque considering the following hypotheses: Bolts are assimilated to simple uniform pins and all sources of stress concentrations are neglected. Friction acting at the interface of the discs is also neglected. Discs are considered to be perfectly rigid.