7. Two random forces and act on a beam in the figure. The allowable bending stress of the beam 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.