9. A shaft has a circular cross section with a diameter of  and is designed to transmit a power . If the allowable shear stress of  the shaft is  and the maximum probability of failure is designed to be , determine the minimum shaft working speed to transmit the required power.

Solution

The transmitted power is given by

Thus the required torque is

And the maximum shear stress resulting from the torque is

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

where .

Using FOSM, we have

The probability of failure is then given by

Thus

Solving for  yields

Ans.