31. A torsion-bar with a round cross section is subjected to a torque . The diameter of the cross section is  and the allowable shear stress is , determine the probability of failure using the First Order Second Moment Method. Note that ,   and  are independent.

Solution

The maximum shear stress is

where  is the polar second moment of area.

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

where .

Using FOSM, we have

The probability of failure is then given by