# Design point

The first step for identifying the design point is the transformation of the reliability problem at hand from the original random variable space $\mathbf{y} \in \Re^d$ (also known as the physical space) to the standard normal space $\mathbf{x} \in \Re^d$. This can be accomplished by using a mapping $\mathbf{y} \leftrightarrow \mathbf{x}$ such that:

$\mathbf{x}=T_{y x}(\mathbf{y})$

$\mathbf{y}=T_{x y}(\mathbf{x})$

A frequently used procedure to perform such a mapping is the Nataf's model [1]. Once the random variables involved have been mapped into the standard normal space, it is possible to define the design point ($\mathbf{x}^{\ast}$) using a geometrical or probabilistic interpretation [2]

• Geometrical Interpretation: The design point is the realization in the standard normal space which lies on the limit state surface with the minimum Euclidean norm with respect to the origin.
• Probabilistic Interpretation: The design point is the failure point with highest probability density.

The Euclidean norm of the design point with respect to the origin is generally denoted as the reliability index ($\beta$). A schematic representation of the design point location in a reliability problem is shown below.

Schematic representation of design point

The task of finding the design point ($\mathbf{x}^*$) involves the solution of the following constrained optimization problem:

$\mbox{min}~\beta=\sqrt{x_1^2+x_2^2+\ldots+x_d^2}$
$\mbox{subject to}~g(T_{x y}(\mathbf{x}))\leq0 ,~~\mathbf{x} \in \Re^d$

Closed form solutions for finding the design point exist for a few cases only , e.g. if the performance function $g(\mathbf{y})$ is linear with respect to the uncertain parameters and if $\mathbf{y}$ follows a Gaussian distribution. However, when dealing with more general structural reliability problems, it is necessary to apply an optimization algorithm to estimate the design point.

## References

1. P.L. Liu and A. Der Kiureghian. Multivariate distribution models with prescribed marginals and covariances. Probabilistic Engineering Mechanics, 1(2):105–112, 1986. [1]
2. A.M. Freudenthal. Safety and the probability of structural failure. ASCE Transactions, 121:1337–1397, 1956, ISBN 0-87262-263-0.