66. A beam with a round cross section is subjected to a uniform load of . The diameter of the beam 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

Consider the force equilibrium of beam shown below.

According to the moment equilibrium of beam with respect to B,

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

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