60. As shown in the figure, a torsion bar AB fixed at A is simply supported at B and connected to a cantilever BC. A force  is applied at C. Bar AB has a spring rate of  and a length of . Cantilever AB has a spring rate of . If the allowable deflection at C is  and the maximum probability of failure is designed to be , determine the maximum length of BC using First Order Second Moment Method. Assume that ,  and  are independent.

Solution

The torque of torsion bar AB is

Thus the angle of twisting is

Then the deflection  at C resulted from the twisting of torsion bar AB is

For the cantilever BC, the deflection  at C resulted from force F is

Then the overall deflecting at C is

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

where .

Using FOSM, we have

The probability of failure is then given by

Thus

Solving for  yields

Ans.