64. A square-cross-section beam is simply-supported and is subjected to an uniform load of  as depicted in the figure. The length of the beam is  and the side length of cross section is . If the allowable bending stress is , estimate the probability of failure using the First Order Second Moment Method. Assume that  and  are independent.

Solution

The reaction force is

The maximum bending moment occurs at  and is given by

Thus the bending stress is

The limit-state function is the actual bending stress subtracted from the allowable one. Failure occurs when.

where .

Using FOSM, we have

The probability of failure is then given by

Ans.