55. A grinding wheel has a weight of , a diameter of , and a bore with a diameter of . The thickness of the wheel is , and the Poisson’s ration is . If the allowable tangential stress is  and the maximum probability of failure is designed to be , determine the maximum working speed of the grinding wheel using the theory of rotating rings.

Solution

The outside radius and inside radius of the wheel are

Thus the mass density of the wheel is

And the angular velocity of the wheel is

According to the theory of rotating rings, the maximum tangential stress is

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

where .

Using FOSM, we have

The probability of failure is then given by

Thus

Solving for  yields

Ans.