# Design point

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*physical space*) to the standard normal space . This can be accomplished by using a mapping such that:

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 () 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* (). A schematic representation of the design point location in a reliability problem is shown below.

The task of finding the design point () involves the solution of the following constrained optimization problem:

Closed form solutions for finding the design point exist for a few cases only , e.g. if the performance function is linear with respect to the uncertain parameters and if 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

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