In structural reliability analysis, the *limit state function* (also referred to as *performance function*) states whether for a particular sample point $\mathbf{x}$, the expected system behavior is *acceptable* or *undesired*. More formally, let $g(\mathbf{x})$ be a function such that $g(\mathbf{x})\le 0$ if and only if the system is in an undesired state, and $g(\mathbf{x})>0$ otherwise.

For the definition of the undesired system response, one typically distinguishes between the ultimate, damage and the serviceability limit-state [Melchers, 1999], where *ultimate* refers to at least partial collapse of the structure, and *serviceability* means disruption of normal use.

For some problems, the limit state function can be expressed in terms of demand $S$ and capacity $R$ of the system of interest. Failure occurs if the demand exceeds the capacity. In this case, the limit state function can be expressed as:

$$

g(\mathbf{x}) = R(\mathbf{x}) - S(\mathbf{x})\,.

$$ An equivalent alternative representation of the limit state function for the above example case is:

$$

g(\mathbf{x}) = 1 - \frac{S(\mathbf{x})}{R(\mathbf{x})}\,.

$$

Note that the performance of numerical methods for structural reliability analysis can depend on the particular formulation of the limit state function.

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