surface approximation is very close to the exact optimum. Note that the values of Pyffor
the probability based optimum and safety index based optimum provide a good estimate
to the required weight increments. For example, with a Pyf-0.9663 the safety index based
design has a safety factor shortfall of 3.37 percent, indicating that it should not require
more than 2.25 percent weight increment to remedy the problem. Indeed the optimum
design is 2.08 percent heavier. This would have been difficult to infer from a probability
of failure of 0.00408, which is three times larger than the target probability of failure.
Table 6-5. Comparisons of optimum designs based on cubic design response surface
approximations of probabilistic sufficiency factor, safety index and
probability of failure
Desgn espnseMinimize obj ective function F while P 2 3 or 0.0013 5 > pof
surface of Obj ective Pof/Safety index/Safety factor
Optima
function F=wt from MCS of 100,000 samples
w=2.6350,
Probability '9.2225 0.00690/2.4624/0. 9481
t-3.5000
w=2.6645,
Safety index '9.3258 0.00408/2.6454/0.9663
t-3.5000
Probabilistic w=2.4526,
'9.5367 0.00128/3.0162/1.0021
sufficiency factor t-3.8884
Exact optimum w=2.4484,
'9.5204 0.00135/3.00/1.00
(Wu et al. 2001) t-3.8884
Design with Strength and Displacement Constraints
For system reliability problem with strength and displacement constraints, the
probability of failure is calculated by direct Monte Carlo simulation with 100,000
samples based on the exact stress and exact displacement in (6-6) and (6-7). The
allowable tip displacement Do is chosen to be 2.25" in order to have two competing
constraints (Wu et al. 2001). The three cubic design response surface approximations in
the range of design variables shown in Table 6-2 were constructed and their statistics are
shown in Table 6-6.