proportional to the most critical stress or displacement, it is easy to obtain the relationship
P, = Peo,( f)' (6-12)
where Ao=woto. This indicates that a one percent increase in area (corresponding to 0.5
percent increase in w and t) will improve the Pyfby about 1.5 percent. Since non-uniform
increases in the width and thickness may be more efficient than uniform scaling, we may
be able to do better than 1.5 percent. Thus, if we have Py-0.97, we can expect that we
can make the structure safe with a weight increase under two percent.
The probabilistic sufficiency factor gives a designer a measure of safety that can be used
more readily than the probability of failure or the safety index to estimate the required
weight increase to reach a target safety level. The Pyf of a beam design, presented in
section 4 in details, is 0.9733 for a target probability of failure of 0.00135, (6-12) indicate
that the deficiency in the Pyf can be corrected by scaling up the area by a factor of 1.0182.
Since the area A is equal to c2wt, the dimensions should be scaled by a factor c of 1.0091
(=1.01820.5) to w = 2.7123 and t = 3.5315. Thus the objective function of the scaled
design is 9.5785. The probability of failure of the scaled design is 0.001302 (safety index
of 3.0110 and probabilistic sufficiency factor of 1.0011) evaluated by MCS with
1,000,000 samples. Such estimation is readily available using the probability of failure
(0.003 14) and the safety index (2.7328) of the design.
Reliability Analysis Using Monte Carlo Simulation
Let g(x) denote the limit state function of a performance criterion (such as strength
allowable larger than stress), so that the failure event is defined as g(x) <0, where x is a
random variable vector. The probability of failure of a system can be calculated as