47. An axial force  is applied to a round Euler column with a length of . The ends of column are pined as shown in the figure. The modulus of elasticity is  kpsi. If the maximum probability of failure is designed to be , determine the minimum diameter of the column using the First Order Second Moment Method. Note that  and  are independent.

Solution

According to the theory of Euler column, the critical load for unstable bending is

where , depending on the end conditions shown in the figure.

The limit-state function is the actual load of the column subtracted from the critical load. Failure occurs when.

where .

Using FOSM, we have

The probability of failure is then given by

Thus

Solving for  yields

Thus  can be used.                                                                                                                    Ans.