2. A steel bar is used to be as a torsion spring. The diameter and length of the bar are  and , respectively. The shear modulus is  and the allowable torsional stress is . If the maximum probability of failure is desinged to be , determine the maximum angle of twist of the bar. Note that  and  are independent.

Solution

The angle of twist is given by

Thus

The maximum shear stress is

Thus the limit-state function is the maximum stress subtracted from allowable shear stress. Failure occurs when

where .

Using FOSM, we have

The probability of failure is then given by

Thus

Solving for  yields

Ans.