5-4. A shear force  is applied to a square beam with the cross section as shown. If the allowable shear stress is , determine the probability of failure of the beam. Assume that  and  are independent.              

Solution:

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

Then, we have

Applying the shear formula,

Set , then , where

Thus, the probability of failure of the beam is

Ans.