8. A strip of steel is subjected to a torque . The steel is 12 in long,  in thick, and 1 in wide. The allowable shear stress of the steel is . Determine the probability of failure of the strip using the First Order Second Moment Method.

Solution

The length of the median line is 1 in. Based on the membrane analogy theory,

where  is the shear stress,  is the shear modulus,  is the angle of twist per unit length,  is torque,  is the length of the median line, and  is the wall thickness.

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

where .

Using FOSM, we have

The probability of failure is then given by