4-5.  The forces acting on a beam are shown in the figure. P is a random force with , and  is a distributed load with . The beam has a square cross section of  on each side, and its allowable bending stress is . , , and  are independent. Determine the probability of failure of the beam.

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 beam is then given by

Ans.