Table 8-7. Probabilities of failure calculated by Monte Carlo simulation with 1x106
samples
Probability of failure of Probability of failure of System probability of
load case 1 load case 2 failure
0 693 x10-6 693 x10-6
Design Response Surfaces
In order to filter the noise in the results of MCS, design response surface
approximation (DRS) are constructed to approximate the reliability constraints. Using the
ARS constructed in previous section, the probability of failure at each design point of the
design of experiment of DRS can be evaluated inexpensively by MCS. Since the
probability of failure is highly nonlinear, the reliability constraint of Equation (8-1) is
replaced by an equivalent form of constraint in terms of probabilistic sufficiency factor
introduced in chapter six.
The range of the cubic DRS is shown in Table 8-8, which is a subset of the range of
ARS shown in Table 8-5 to avoid extrapolation in probability calculation. The error
statistics are summarized in Table 8-9. It is seen that the accuracy of the DRS to
probability of failure is poor, while DRS to PSF is accurate enough for RBDO.
Table 8-8. Range of design response surface approximations (inch)
b h tl t2
9.6-10.7 1.9854 -2.1084 0.0974-0.0986 0.1246-0.1271
Table 8-9. Cubic design response surface approximation to the probability of failure and
probabilistic sufficiency factor (calculated by Monte Carlo sampling of lx106
samples)
Probabilistic sufficiency
Probability of failure
factor
Rsquare adj. 0.6452 0.9990
RMSE predictor 0.000211 0.00176
Mean of response 0.0000580 1.079