10. A grinding wheel has a diameter of , a thickness of , and a bore with a diameter of . The speed of the wheel is  rev/min. The material is isotropic, and the Poisson’s ration is . If the weight of the wheel is , the allowable tangential stress is , and  and  are independent, determine the probability of failure using the First Order Second Moment Method.

Solution

The angular velocity of the wheel is

The outside radius and inside radius of the wheel are

The mass density of the wheel is

Based on 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