69. A welded steel tube is in torsion . The tube is thin-wall and 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 shear stress is , estimate the mean and standard deviation of the angle of twist 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 and is given by

The allowable torque is given by

Thus the angular twist per unit of length is

where  is the section median line.

So the angle of twist can be obtained by

Let

where .

Using FOSM, we have

Ans.