44. A force  is applied to a beam shown in the figure. The beam is cantileverd at point O and has a circular cross section. The allowable bending stress of the beam is . If the maximum probability of failure is designed to be , determine the minimum diameter of the beam and select a preferred diameter using the First Order Second Moment Method. Note that  and  are independent.

Solution

Based on the moment equilibrium of beam OC, the bending moment acting at O is

Thus the maximum bending stress is given by

The limit-state function is the maximum bending stress subtracted from the allowable stress. 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.