6-9. A random force  is acting on the center of a shaft that has a radius of 0.5 in. Determine the probability of failure at point B due to excessive shear stress. The allowable shear stress follows . Assume that and  are independent.

Solution:

Section Properties

Normal Stress at Point B

Shear Stress at Point B

where

, 

Thus

Since , we have

Set , then , where

Thus, the probability of failure due to excessive normal stress is

Ans.