Beam Design Example
The details of the beam design problem mentioned in section 2 are presented here.
Since the limit state of the problem is available in closed form as shown by (6-6) and (6-
7), the direct Monte Carlo simulation with a sufficient large number of samples is used
here (without analysis response surface) in order to in order to better demonstrate the
advantage of probabilistic sufficiency factors over probability of failure or safety index
better. By using the exact limit state function, the errors in the results of Monte Carlo
simulation are purely due to the convergence errors, which can be easily controlled by
changing the sample size. In applications where analysis response surface approximation
must be used, the errors introduced by approximation can be reduced by sequentially
improving the approximation as the optimization progresses.
The reliability constraints, shown by (6-8) to (6-10), are approximated by design
response surface approximates that fit to probability of failure, safety index, and
probabilistic sufficiency factor. The accuracy of the design response surface
approximations is then compared. The design response surface approximations are in two
design variables w and t. A quadratic polynomial in two variables has six coefficients to
be estimated. Since Face Center Central Composite Design (FCCCD, Khuri and Cornell
1996) is often used to construct quadratic response surface approximation, a FCCCD
with 9 points was employed here first with poor results. Based on our previous
experience, higher-order design response surface approximations are needed to fit the
probability of failure or the safety index, and the number of points of a typical design of
experiments should be about twice the number of coefficients. A cubic polynomial in two
variables has 10 coefficients that require about 20 design points. Latin Hypercube
sampling can be used to construct higher order response surface (Qu et al. 2000). We