4-3.  Two independent vertical random forces  and  act on a shaft as shown in the following figure. The sleeve bearings at A and B support only vertical forces. The allowable bending stress of the shaft is . Determine the diameter of the shaft so that the probability of failure is less than .

Solution

The free-body diagram of this shaft is shown in Fig. 1. The shear and moment diagrams are shown in Fig. 2.

,     ,       

,     ,   

As indicated in the moment diagram, .

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

Given , we have .

Set , then , where

The probability of failure of the shaft is computed by

Then,

Thus,   in.                                                                                                                 Ans.