8-4. The steel bar BC has a rectangular cross section as shown in the figure. It is pin connected at its ends. The force P acting on the bar AB follows a normal distribution  and the modulus of elasticity follows . Determine the smallest moment of inertia of the bar BC to make sure that the probability of failure of bar BC caused by x-x axis buckling is no more than . Assume that E and P are independent, and .

Solution:

From the free body diagram of the bar AB

,  

Then, we can obtain .

The effective length factor of x-x axis buckling is. Thus, the x-x axis critical buckling load can be calculated by

The effective length factor of y-y axis buckling is. Since , the y-y axis critical buckling load can be calculated by

Thus, the probability of failure of the bar BC caused by buckling is given by

Set , then , where

Thus,  could be rewritten as

                                                          Ans.