10. A toruqe  is applied to a ductile shaft. The yield strength in tension of the shaft is  and the one in compression is . If the maximum probability of failure is designed to be , determine the minimum diameter of the shaft using the First Order Second Moment Method. Assume that ,  and  are independent.

Solution

According to the Coulomb-Mohr theory, the shear yield strength of ductile shaft is

The maximum shear stress is given by

So the limit-state function is the maximum shear stress subtracted from the shear yield strength. 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.