9. A cylindrical pressure vessel made of tubing carries an internal pressure . It has an outside diameter of  in and a wall thickness  in. If the allowable tangential stress of the vessel is , determine the probability of failure using the First Order Second Moment Method. Use the theory of thin-walled vessels.

Solution

The inside diameter of the vessel is

Then

Applying the theory of thin-walled vessels, we have the maximum tangential stress as follows:

The limit-state function is the maximum tangential stress of the vessel subtracted from the allowable tangential stress. Failure occurs when.

where .

Using FOSM, we have

The probability of failure is then given by