This flat zero probability of failure or infinite safety index cannot provide useful gradient
information to the optimization.
x2 g(x)<0
Unsafe Region
g(x)= 0
*. **Limit State
,' ** g (x) >
** *Safe Region
X1
Figure 6-3. Monte Carlo simulation of problem with two random variables
Calculation of Probabilistic Sufficiency Factor by Monte Carlo Simulation
Here we propose the use of probabilistic sufficiency factor to solve the problems
associated with probability calculation by MCS. Pyf can be estimated by MCS as follows.
Define the nth safety factor of MCS as
s~n =nt m ein sx (6-14)
where M~ is the sample size of MCS, and the nth min means the nth Smallest safety factor
among M safety factors from MCS. Thus so, is the nt'h-order statistics ofM safety factors
from MCS, which corresponds to a probability of nM \ of s(x) < s ,. That is, we seek to
find the safety factor that is violated with the required probability Pr. The probabilistic
sufficiency factor is then given as
P~= st, for n =PSIM (6-15)