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

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 on 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 then given by

Ans.