Strategies for finding the design point under bounded random variables

• Inverse of standard Gaussian cumulative probability distribution Φ-1(u)Φ-1(u), collapses when u→1u→1.
• Mapping to standard Gaussian space may collapse as bounded random variable is pushed beyond its support.
• Issues to be addressed during iterative design point search.
• Benchmark problem is introduced to address above issues.
• Formal mapping to standard Gaussian space smoothens the above problems.

The First Order Reliability Method is well accepted as an efficient way to solve structural reliability problems with linear or moderately non-linear limit state functions. Bounded random variables introduce strong non-linearities in the mapping to standard Gaussian space, making search for design points more demanding. Since standard Gaussian space is unbounded, two particular problems have to be addressed: a. the limiting behavior of the probability mapping as a random variable approaches its upper or lower limits; b. how to impose the bounds if design point search leaves the problems supporting domain. Both problems have been overlooked elsewhere, and are addressed in the present article. Based on the Principle of Normal Tail Approximation, two alternatives for the mapping to standard Gaussian space are studied. Three different schemes are proposed to impose the limits of bounded random variables, in the reverse mapping to original design space. Several algorithms are investigated with respect to their ability to find the design point in highly non-linear problems involving bounded random variables. A challenging benchmark reliability problem is also presented herein, and used as a test bed to explore the proposed strategies and the performance of the optimization algorithms.