8-1.  A vertical force P acts on a bar as shown in the figure. If the length of the bar L follows a normal distribution , determine the distribution of its critical buckling load of the bar. Assume that the spring has a stiffness of .

Solution:

The free body diagram is shown below. The disturbing force F can be determined by summing moments about A.

,   

Then, . The restoring spring force can be determined by

For the mechanism to be on the verge of buckling, the disturbing force F must be equal to the spring force . Then, we have

Since ,  also follows normal distribution , in which

Thus, the critical buckling load of the bar  follows .                            Ans.