8-13. A 22-ft column consists of four steel angles laced together. The cross-section is shown in the figure. The column is pin connected at both ends. Each of these angles has an area of ,   and moments of inertia of   . If an axial load  is applied to the column, and the modulus of elasticity follows , determine the distribution of the critical axial buckling load. Also, determine the probability of buckling. Assume that E and P are independent. 

Solution:

Section properties of the column

The critical axial buckling load is given by

,      where .

Since , we have

Thus, the critical axial buckling load follows .                      Ans.

Set , then , where

Thus, the probability of failure is

                     Ans.