Design point

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Remove reference to the Nataf Transformation
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.

See Also