3-3.  A coupling with 4 bolts connects the two shafts as shown. Assume that the shear stress in the bolts is uniform. The torque  applied on the shafts follows a normal distribution. Each bolt has a diameter , and the bolts are uniformly distributed at the radius . If the allowable shear stress of the bolt is, determine the probability of failure the bolt. Assume that  and  are independent.

Solution:

;         

in which  is the number of bolts, and  is the shear force in each bolt.

The shear stress developed in the bolt is

Set , then , where

Thus, the probability of failure of the bolt is

Ans.