5. If  and , what is the distribution of the stress acting on rod BC? The diameter of rod BC is 0.05 m. Assume  and  are independent.

Solution

Consider the force equilibrium of rod AB as shown in the figure below

According to the moment equilibrium of rod AB with respect to point A,

                                                             (1)

where  is the moment resulted from ,  is the moment resulted from ,  is the moment resulted from , and  is the moment resulted from .

Then

                                                    (2)

Solving for F_B yields

                                                                       (3)

Thus the stress acting on rod BC is

                                                                 (4)

where A is the cross-sectional area of rod BC, and D is the diameter of rod BC.

Combining Eqs. (3) and (4) yields

                                                                 (5)

Because  and  are independently and normally distributed, their linear sum,, is also normally distributed. The mean and standard deviation of  are given by

So the distribution of the stress acting on rod BC is Ans.