14. A cantilever with a circular
cross-section is subjected to an axial tension 
and a torsion 
acting at point B. The
length of the cantilever is 
and the diameter is 
. If the yield strength of the cantilever
is 
, and 
, 
and 
are independent, determine the
probability of failure using the First Order Second Moment Method. Use the
distortion-energy theory.
Solution
The critical stress element appears on the top surface at point A, shown in the following figure.
The normal stress resulted from axial force is given by
where 
is the area of
the cross-section.
The torsion stress at the critical stress element is
where 
is the radius of the
circular cross-section, and 
is the polar second
moment of inertial.
According to the distortion-energy theory, the von Mises stress is found to be
So
the limit-state function is the von Mises stress subtracted from the yield
strength. Failure occurs when
.
where
.
Using FOSM, we have
The probability of failure is then given by
