6. A circular shaft is subjected to an axial force . The tensile spring constant of the shaft is . If the allowable axial extension is , estimate the probability of failure using the First Order Second Moment Method. Assume  and  are independent.

Solution

The extension of the shaft is 

Thus the limit-state function is the actual extension subtracted from the allowable one. Failure occurs when.

where .

We can rewrite the limit-state function as

Because  and  are independently and normally distributed, their linear combination  is also normally distributed and

The probability of failure is then given by