49. A steel thin-wall tube is subjected to a torque  as shown in the figure. The tube is 400 mm long and has a square cross section with side length . The shear modulus of the tube is . Assume that the allowable shear stress is  and the maximum probabilty of failure is , estimate the minimum wall thickness of the tube using the First Order Second Moment Method. Note that  and  are independent.

Solution

Based on the theory of closed thin-walled tube, the shear stress is

where  is the area enclosed by section median line

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

Thus

Solving for  yields

Thus  can be used.                                                                                                          Ans.