63. A force  is applied to a tapered bar shown in the figure. The bar has a diameter of  at one end and a diameter of  at the other end. The length is  and the modulus of elasticity is . If the allowable axial elongation is , determine the probability of failure using the First Order Second Moment Method. Note that the elongation of tapered portion is . And assume that  and   are independent.

Solution

For the tapered portion, the elongation is given by

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

where .

Using FOSM, we have

The probability of failure is then given by

Ans.