6-6.  A random force  is acting on a bar, which has a radius of 16 mm as shown in the figure. Determine the probabilities of failure at point A located at the left end due to excessive normal stress and shear stress. The allowable normal stress follows , and the allowable shear stress follows . Assume that ,  and  are independent.

Solution:

Section Properties

Normal Stress at Point A

Since , we have

Set , then , where

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

Ans.

Shear Stress at Point A

Thus, we have

Set , then , where

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

Ans.