39. A shaft with a circular cross section is subjected to a torque . The allowable shear stress is . If the maximum probability of failure is designed to be , determine the minimum diameter using the First Order Second Moment Method.

Solution

For a round shaft in torsion, the maximum shear stress is given by

Thus the limit-state function is the maximum stress subtracted from allowable shear stress. Failure occurs when

where .

Using FOSM, we have

The probability of failure is

Thus

Solving for  yields

So  can be used.