8-6.  Rod AB is pin connected at it ends, and its diameter is . The force P follows . The modulus of elasticity follows . Find the probability of failure of the rod caused by buckling. Assume that Euler’s formula is valid. Also, assume that and  are independent.

Solution:

From the free body diagram of the bar CD

,  

Then, we can obtain .

The effective length factor . The moment of inertia of the rod is

Thus, the critical buckling load of the bar can be calculated by

Set , then , where

Thus, the probability of failure of the rod caused by buckling could be obtained by

Ans.