42. A
beam with a round cross section is cantilevered at O shown in the figure.
It is subjected to an uniform load of 
. If the diameter of the cross
section and the allowable bending stress are 
and 
, respectively, estimate the
probability of failure using the First Order Second Moment Method. Note that 
,
and 
are independent.
Solution
According to the moment equilibrium of beam OC, the bending moment acting at O is
Thus the maximum bending stress is given by
The
limit-state function is the maximum bending stress subtracted from the
allowable stress. Failure occurs when
.
where
.
Using FOSM, we have
The probability of failure is then given by
