13. A
ductile shaft is subjected to a toruqe 
. The diameter of the shaft is 
. If the yield strength in tension
of the shaft is 
and the yield strength in
compression is 
, and 
,
and 
are independent, determine the
probability of failure using the First Order Second Moment Method.
Solution
Based on the Coulomb-Mohr theory for ductile materials, the shear yield strength is given by
The maximum shear stress is
So the
limit-state function is the maximum shear stress subtracted from the shear yield
strength. Failure occurs when
.
where
.
Using FOSM, we have
The probability of failure is then given by
