48. A bar has a hollow round cross section and an inner diameter of . The bar is subjected to a torsion . And the allowable stress of the bar is . If the maximum probability of failure is designed to be , determine the minimum outer diameter of the bar using the First Order Second Moment Method. Note that  and  are independent.

Solution

The maximum shear stress is

where  is the polar second moment of area, given by

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

where .

Using FOSM, we have

The probability of failure is then given by

Thus

Solving for  yields

Thus  can be used.                                                                                                          Ans.