32. A torsion bar consists of a circular cross secion with a diameter . The bar has a lenghth of  and is subjected to a toque  applied at the central portion of the span. The shear modulus of the bar is . If ,  and  are independent, what is the mean and standard deviation of the spring rate using the First Order Second Moment Method?

Solution

For a torsion bar, the spring rate is

For this problem with two springs in parallel, the total spring rate is given by

Let

where .

Using FOSM, we have