Draft:Fragility function

In structural engineering, a fragility function expresses the occurrence or exceedance probability of an undesirable outcome (sometimes called "failure") to the degree of one or more measures of environmental excitation. It is usually conditioned on a clearly defined asset class. For example, a fragility function can express the probability that a window of a given size and with clearly defined materials and dimensions will at least crack when it is subjected to varying degrees of peak transient interstory drift (a measure of how much a story in a building has deformed). The undesirable outcome is "window cracks." Here, "occurrence or exceedance" means that the window could crack (occurrence) or a more severe outcome could happen (exceedance), such as glass falling out. In this example, the environmental excitation is peak transient interstory drift.

Structural engineers have used fragility functions as an essential element of probabilistic seismic risk assessment (a special case of probabilistic risk assessment) at least since the WASH-1400 study of 1975, which set out to estimate (among other things) the probability of a core meltdown in a nuclear reactor. Seismic fragility functions (a special case of fragility functions) use measures of earthquake excitation as their independent variable, their input. These can be measures of ground motion such as peak ground acceleration. They can be measures of structural response such as the peak transient interstory drift ratio. Or they can be other measures of damage such as structural collapse. A few fragility functions use vector inputs, i.e., with two or more measures of environmental excitation as the argument.

Seismic fragility functions represent the main analytical tool in the damage-analysis stage of second-generation performance-based earthquake engineering (PBEE-2). PBEE-2 was initially embodied in the so-called PEER methodology developed with National Science Foundation funding in the early 2000s by researchers affiliated with the Pacific Earthquake Engineering Research (PEER) Center. Later, PBEE-2 was encoded in a set of FEMA-funded guidelines initially called ATC-58 (by the Applied Technology Council of Redwood City, CA) and later published as FEMA P-58.

Fragility functions can take many mathematical forms, some parametric, some nonparametric. Common among parametric forms is the lognormal cumulative distribution function. In PBEE-2, when using the lognormal cumulative distribution function to approximate fragility, θ commonly denotes the median value of the environmental excitation at which failure occurs, which one can think of as the median capacity of an asset to resist that failure mode. That is, when an asset is subjected to environmental excitation x = θ, there is a 50% chance that the asset will fail. β commonly denotes the logarthmic standard deviation of capacity, i.e., the standard deviation of the natural logarithm of capacity. Some people call β the dispersion. Using this notation, in what one can call lognormal fragility function, one estimates failure probability P as follows.

$P=\Phi ((ln(x/\theta )/\beta )$

in which Φ denotes the standard normal cumulative distribution function of the term in parentheses and ln denotes the natural logarithm of the term in parentheses.