found that Latin Hypercube sampling might fail to sample points near some corners of
the design space, leading to poor accuracy around these corners. To deal with this
extrapolation problem, all four vertices of the design space were added to 16 Latin
Hypercube sampling points for a total of 20 points. Mixed stepwise regression (Myers
and Montgomery 1995) was employed to eliminate poorly characterized terms in the
response surface models.
Design with Strength Constraint
The range for the design response surface, shown in Table 6-2, was selected based
on the mean-based deterministic design, w = 1.9574" and t = 3.9149". The probability of
failure was calculated by direct Monte Carlo simulation with 100,000 samples based on
the exact stress in (6-6).
Table 6-2. Range of design variables for design response surface
System variables w t
Range 1.5" to 3.0" 3.5" to 5.0"
Cubic design response surfaces with 10 coefficients were constructed and their
statistics are shown in Table 6-3. An R a4, close to one and an average percentage error
(defined as the ratio of root mean square error (RMSE) predictor and mean of response)
close to zero indicate good accuracy of the response surfaces. It is seen that the design
response surfaces for the probabilistic sufficiency factor has the highest R a4, and the
smallest average percentage error. The standard error in probability calculated by Monte
Carlo simulation can be estimated as
p(1- p)
cr (6-19)
IM
where p is the probability of failure, and M~ is the sample size of the Monte Carlo