57. A simply-supported beam is subjected to a force , as shown in the figure. The length of the beam is  and the modulus of elasticity is . If the allowable deflection is  and the maximum probability of failure is designed to be , determine the minimum diameter of the beam using . Assume that  and  are independent.

Solution

The maximum deflection of the beam is

Thus the limit-state function is the maximum deflection subtracted from the allowable deflection. 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.