Problem $3$

Consider a solid circular shaft A$-$D with a varied diameter between the gears: d$\_($AB$),$d$\_($BC$),$ and d$\_($CD$).$

The shaft is fixed to a rigid wall at D and three external torques are applied to each gear at A$,$B$,$

and C $.$ The torques are applied in the senses shown below:

ad$\_($hole $)$A to D d$\_($AB$),$d$\_($BC$),$ and

d$\_($CD$)$ remain the same, as do the external torques T$\_($A$),$T$\_($B$),$ and T$\_($C$).$

For this hollow shaft, find the maximum and minimum shear stress in each segment of

the shaft. Then, determine which segment has the overall maximum shear stress in the

shaft as well as which segment has the overall minimum shear stress in the shaft. Finally,

plot a cross$-$sectional shear stress distribution at the section where the maximum shear

stress in the shaft occurs.

Hint: For internal torques, we obey the same sign convention as for axially

loaded members.

However, when calculating the torsional shear stress, we instead consider the

absolute value of the shear stress. Therefore, you should calculate positive

quantities for all torsional shear stresses.

Take T$\_($A$)=200$lb$**$in$,$T$\_($B$)=1400$lb$*$in$,$T$\_($C$)=300$lb$**$in$.$

Take d$=0\mathrm{.}1$in and d$\_($hole $)=0\mathrm{.}25$in$.$

Find the internal torque in segments AB$,$ BC$,$ and CD in lb$*$in$.$