stringent weight requirements, it is also important to provide guidelines for controlling

the magnitudes of uncertainties for the purpose of reducing structural weight.

 Deterministic structural optimization is computationally expensive due to the

need to perform multiple structural analyses. However, reliability-based optimization

adds an order of magnitude to the computational expense, because a single reliability

analysis requires many structural analyses. Commonly used reliability analysis methods

are based on either simulation techniques such as Monte Carlo simulation, or moment-

based methods such as the first-order-reliability-method (e.g., Melchers, 1999). Monte

Carlo simulation is easy to implement, robust, and accurate with sufficiently large

samples, but it requires a large number of analyses to obtain a good estimate of low

failure probability. Monte Carlo simulation also produces a noisy estimate of probability

and hence is difficult to use with gradient-based optimization. Moment-based methods do

not have these problems, but they are not well suited for problems with many competing

critical failure modes.

 Response surface approximations solve the two problems of Monte Carlo

simulation, namely simulation cost and noise from random sampling. Response surface

approximations (Khuri and Cornell 1996) typically fit low order polynomials to a number

of response simulations to approximate response. Response surface approximations

usually fit the structural response such as stresses in terms of random variables for

reliability analyses. The probability of failure can then be calculated inexpensively by

Monte Carlo simulation using the fitted response surfaces. Response surface

approximations can also be fitted to probability of failure in terms of design variables,

which replace the reliability constraints in RBDO to filter out numerical noise in the