24. A torque  is applied to a steel thin-wall tube. The tube is 40 in long and has a square cross section with side length  and wall thickness . The shear modulus of the tube is . If the allowable shear stress is , determine the probability of failure using the First Order Second Moment Method. Note that  and  are independent.

Solution

According to the theory of closed thin-walled tube, the shear stress is

where  is the area enclosed by section median line, given by

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

where .

Using FOSM, we have

The probability of failure is then given by