safety index, which replaces the probability by the distance, which is measured as the

number of standard deviations from the mean of a normal distribution that gives the same

probability. The safety index does not suffer from steep changes in magnitude, but it has

the same problems of accuracy as the probability of failure when based on Monte Carlo

simulations. However, the accuracy of probabilistic sufficiency factor is maintained in

the region of low probability. The probabilistic sufficiency factor also exhibits less

variation than probability of failure or safety index. Thus the probabilistic sufficiency

factor can be used to improve design response surface approximations for RBDO.

 The next section introduces the probabilistic sufficiency factor, followed by the

computation of the probabilistic sufficiency factor by Monte Carlo simulation. The

methodology is demonstrated by the reliability-based beam design problem.

 Probabilistic Sufficiency Factor

 The deterministic equivalent of reliability constraint in RBDO can be formulated as

 g, (, d < g (i d)(6-1)

where gr denotes a response quantity, go represent a capacity (e.g., strength allowable),

x is usually the mean value vector of random variables, d is the design vector. The

traditional safety factor is defined as

 g c(x, d)
 s(x, d) = (6-2)
 g, (x, d)

and the deterministic design problem requires

 s(i, d) > s, (6-3)

where s, is the required safety factor, which is usually 1.4 or 1.5 in aerospace

applications. The reliability constraint can be formulated as a requirement on the safety

factor