6. A torsion bar AB is supported at B and is fixed at A. The bar is connected to a cantilever BC. The spring rate of bar AB is , and the spring rate of the cantilever BC is . If  and  are independent, what is the overall spring rate with respect to the vertical deflection  at C? Find its mean and standard deviation using the First Order Second Moment Method.

Solution

For the torsion bar AB, the acting torque is

                                                               (1)

Thus the angle of twisting is

                                                           (2)

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

                                                           (3)

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

                                                                       (4)

Then the overall deflecting at C is

                                                         (5)

Solving for the overall spring rate

                              (6)

Ans.

Now using FOSM, we have

Ans.