5-1.  A shear force  is applied to a member with the cross section as shown. If the allowable shear stress is , determine the probability of failure of the member. Assume that  and  are independent.              

Solution:

The cordinate system is shown in the figure, then

Using the Parallel Axis Theorem, the moment of inertia about the centroidal axis is

Thus, the first moment of the cross section area at the neutral axis is

The maximum shear stress is

Set , then , where

Thus, the probability of failure of the member is

Ans.