68. A torque  is applied to a welded thin-wall steel tube as shown in the figure. The tube is of round section with a diameter of . It has a a thickness of   and a length of . The shear modulus is . If the allowable anlge of twist is , estimate probability of failure using the First Order Second Moment Method. Note that  and  are independent.

Solution

Based on the theory of closed thin-walled tube, the angular twist per unit of length is

where  is the section median line and  is the area enclosed by section median line.

Thus the actual angle of twist is

The limit-state function is the actual angle of twist subtracted from the allowable one. Failure occurs when.

where .

Using FOSM, we have

The probability of failure is then given by

Ans.