8-12. A 5.5-m long
steel hollow circular tube has an outer diameter of 
and
inner diameter of 
. The tube is pinned at
both ends, and an axial force 
is acting on
it. 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
The critical axial buckling load is
, where 
.
Since , we have
Thus, the critical
axial buckling load follows 
.
Ans.
Set
, then 
,
where
Thus, the probability of failure is
Ans.