67. A uniform load of  is applied to a beam with a round cross section. If the allowable bending stress is  and the maximum probability of failure is designed to be , estimate the minimum diameter of the beam and select a preferred one. Assume that  and  are independent.

Solution

Consider the free body diagram of the beam shown below.

Based on the moment equilibrium of the beam with respect to B,

The maximum bending moment occurs at point B and is given by

Thus the maximum bending stress is

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.