7-1. Two independent random
forces 
and 
act
on
the beam shown in the figure. If the allowable normal 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.
The free-body diagram of this shaft is shown in Fig. 1. The shear and moment diagrams are shown in Fig. 2.
 |
, 
,
, 
,
As
indicated in the figure 
, 
The
moment of inertia of the cross section about the neutral axis is 
.
(1) The maximum bending stress can be calculated by
.
Set
, then 
,
where
The probability of failure of the bean caused by excessive bending stress is
Ans.
(2) The maximum shear stress can be calculated by
in
which 
.
Set
, then 
,
where
The probability of failure of the beam caused by excessive shear stress is
Ans.