extrapolation problem was solved, and the side constraints are set as the range of the ARS
shown in Table 5-3. The error of 0.000012 is much lower than the allowable failure
probability of 0.0001.
Table 5-6 compares the reliability-based optimum with the three deterministic
optima from chapter 4 and their failure probabilities. The optimal thickness increased
from 0.100 to 0.120, while the failure probability decreased by about one order of
magnitude.
Table 5-6. Comparison of reliability-based optimum with deterministic optima
Optimal Design Laminate Failure probability Allowable
[81, 62, tl, t2] thickness from MCS of ARS probability of
(degree and inch) (inch) 1,000,000 samples failure
[24.89, 25.16, 0.015, 0.015] 0.120 (0.120) 0.000055 0.0001
[0.00, 28.16, 0.005, 0.020] 0.100 (0.103) 0.019338a
Deterministic
[27.04, 27.04, 0.010, 0.015] 0.100 (0.095) 0.000479
optima
[25.16, 27.31, 0.005, 0.020] 0.100 (0.094) 0.000592
a This deterministic optimum is out of the range of the analysis response surfaces; the
probability of failure was calculated by Monte Carlo simulation based on another set of
analysis response surfaces.
Refining the Reliability-Based Design
The reliability-based designs in Table 5-6 show that the ply angles close to 250
offer designs with low failure probability. Furthermore, good designs require only a
single ply-angle allowing simplification of the configuration of the laminate from
[f 6,/ B O]s to [f B ]s. Table 5-7 shows the failure probabilities of some chosen designs
calculated with Monte Carlo simulation using ARS. The laminates with ply-angles of
240, 250, and 260 offer lower probabilities of failure than the rest. These three laminates
will be further studied.