7-3. A box with a weight of  is placed in the center of a T-shape beam as shown in the figure. The ends support only vertical forces.  If the allowable bending stress of the beam is and the allowable shear stress is , determine the probabilities of failure of the beam caused by excessive bending stress and shear stress. Assume are independent.

Solution:

The neutral axis can be calculated by

Maximum moment develops at the center of the beam:

Thus, the maximum bending stress is

Set , then , where

The probability of failure of the beam caused by excessive bending stress is

Ans.

(2)  The maximum shear stress can be calculated by

Set , then , where

The probability of failure of the beam caused by excessive shear stress is

Ans.