3-6.  A shaft transmits torques applied to the gears. The three torques follow normal distributions, , and , respectively. Determine the distribution of the maximum shear stress on the shaft. Assume that  ,  and  are independent.

Solution:

The intrrnal torque is shown on the torque diagram.

From the torque diagram, we have . Then, applying torsion Formula

Thus,  also follows a normal distribution. We have

Thus,  follows a normal distribution .                                      Ans.