21. A
torsion 
is applied to a bar with a hollow
round cross-section. The bar has an outside diameter of 
and an inside diameter of 
. If the allowable stress of the bar
is 
, estimate the probability of
failure using the First Order Second Moment Method. Note that 
and
are independent.
Solution
The maximum shear stress is
where
is the polar second moment of area,
given by
The
limit-state function is the maximum shear stress subtracted from the allowable
stress. Failure occurs when
.
where
.
Using FOSM, we have
The probability of failure is then given by
