40. A round shaft is subjected to a bending moment . The yield strength of the shaft is . If the maximum probability of failure is designed to be , determine the minimum diameter using the First Order Second Moment Method. Note that  and  are independent.

Solution

For a round shaft in bending, the bending stress is given by

Thus the limit-state function is the bending stress subtracted from the yield strength. Failure occurs when

where .

Using FOSM, we have

The probability of failure is

Thus

Solving for  yields

Thus  can be used.