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.