11. A round column is 1.5 m long and is subjected to an axial force . The ends are pinned as shown in the figure, and the modulus of elasticity is  GPa. If the diameter of the column is , and  and  are independent, determine the probability of bending failure using the First Order Second Moment Method.

Solution

Based on the theory of Euler column formula, the critical load for unstable bending is

where  is a constant, depending on the end conditions shown in the figure,  is the moment of inertia, and   is the length of the column.

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