4-6.  The forces acting on a shaft are shown in the figure. P is a random force with , and  is a distributed load with . The allowable bending stress of the shaft is . , , and  are independent. Determine the diameter of the shaft to make sure that the probability of failure is less than .

Solution:

The shear and moment diagrams are shown below.

As indicated in the moment diagram, .

The moment of inertia of the cross section about the neutral axis is .

Applying the flexure formula, we have

Set , then , where

The probability of failure of the shaft is then given by

Then, . We have

Thus, .                                                                                                                        Ans.