7. A round shaft is subjected to an axial force . The length of the shaft is  and the modulus of elasticity is . If the allowable axial extension is  and the maximum probability of failure is designed to be , determine the minimum diameter of the shaft using the First Order Second Moment Method and then select a preferred one. Assume that  and  are independent.

Solution

The total extension of the shaft is 

Thus the limit-state function is the actual extension 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

Then  can be used.

Ans.