numbers, control variates, and rotation sampling. Conditional expectation methods utilize
the independence of random variables to reduce the order of probabilistic integration to
achieve higher efficiency. Some common techniques are conditional expectation,
generalized conditional expectation, and adaptive conditional expectation. Specific
methods include response surface method and internal control variables techniques. The
VRTs can be combined further to increase the efficiency of simulation. A comparison of
the accuracy and efficiency of several common VRT methods can be found in Kamal and
Ayyub (2000). Latin hypercube sampling and response surface methods are studied in
this dissertation.
The VRT requires fewer limit state function evaluations to achieve the desired level
of accuracy, but the simplicity of simulation is lost, and the computational complexity of
each simulation cycle is increased.
Moment-Based Methods
Besides VRT, moment-based methods also reduce the computational cost
drastically compared to MCS. The first-order-reliability method (FORM) and second
order reliability-method (SORM) are well-established methods that can solve many
practical applications (Rackwitz 2000). FORM and SORM methods first transform the
random variables from the original space (X-space) to the uncorrelated standard normal
space (U-space). An optimization problem is then solved to find the minimum distance
point (most probable point, MPP) on the limit state surface (Z=0) to the origin of the U-
space. The minimum distance, P, is called the safety index. The probability of failure is
then calculated by using the normal cumulative distribution function Pf, = c(-P) in