## Definition

The *most likely failure point* is also known as *design point* in structural reliability analysis. It is the point in the failure domain that has the smallest distance to the origin in the underlying standard Normal space; i.e.:

$$

\mathbf{u}^* = \underset{\mathbf{u}\in\mathbf{U}_\mathcal{F}}{\operatorname{arg\,min}}\left(\|\mathbf{u}\|\right)\;,

$$ where $\mathbf{u}^*$ is the design point in standard Normal space and $\mathbf{U}_\mathcal{F}=\left\{\mathbf{u}\in\mathrm{R}^N|G(\mathbf{u})\le0\right\}$ the failure domain in standard Normal space.

The coordinates of the design point $\mathbf{x}^*$ in the original space are then obtained by

$$

\mathbf{x}^* = T^{-1}(\mathbf{u}^*)\;,

$$ where $T^{-1}(\cdot)$ is the transformation from the standard Normal space to the original space.

## Interpretation

Of all the points in the failure domain, the design point corresponds to the most likely explanation of failure — therefore the name "most likely failure point". Generally, the area around the design point contributes most to the integral of the probability of failure. The design point is thus a logical choice for the reference point of the system of interest. The popular first-order reliability method (FORM) linearizes the limit state function around this point.

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