65. A rectangular-cross-section beam is simply-supported and is subjected to a concentrated load of  as shown in the figure. The width and height of the cross section is  and, respectively. If the allowable bending stress is  and the maximum probability of failure is designed to be , determine the maximum length of the beam. Assume that  and  are independent.

Solution

The reaction force is given by

And the maximum bending moment is

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

Thus

Solving for  yields

Ans.