37. A torque  is applied to a bar with a round cross section. The diameter of the bar is . If the allowable shear stress is , determine the probability of failrue using the First Order Second Moment Method. Note that ,  and  are independent.

Solution

The maximum shear stress is given by

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

where .

Using FOSM, we have

The probability of failure is then given by